Properties

Label 21.30.g.a.5.1
Level $21$
Weight $30$
Character 21.5
Analytic conductor $111.884$
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] = [21,30,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, 30, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 30);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 30 \)
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: \(111.883889004\)
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 21.30.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)
 
\(f(q)\) \(=\) \(q+(7.17445e6 - 4.14217e6i) q^{3} +(-2.68435e8 - 4.64944e8i) q^{4} +(-1.24856e12 - 1.28880e12i) q^{7} +(3.43152e13 - 5.94357e13i) q^{9} +O(q^{10})\) \(q+(7.17445e6 - 4.14217e6i) q^{3} +(-2.68435e8 - 4.64944e8i) q^{4} +(-1.24856e12 - 1.28880e12i) q^{7} +(3.43152e13 - 5.94357e13i) q^{9} +(-3.85176e15 - 2.22381e15i) q^{12} +1.93779e16i q^{13} +(-1.44115e17 + 2.49615e17i) q^{16} +(-1.96231e18 - 1.13294e18i) q^{19} +(-1.42962e19 - 4.07465e18i) q^{21} +(9.31323e19 + 1.61310e20i) q^{25} -5.68558e20i q^{27} +(-2.64060e20 + 9.26470e20i) q^{28} +(-4.74078e21 + 2.73709e21i) q^{31} -3.68457e22 q^{36} +(-3.75709e22 + 6.50747e22i) q^{37} +(8.02665e22 + 1.39026e23i) q^{39} +9.11042e23 q^{43} +2.38780e24i q^{48} +(-1.02088e23 + 3.21829e24i) q^{49} +(9.00963e24 - 5.20171e24i) q^{52} -1.87713e25 q^{57} +(-9.20660e25 - 5.31543e25i) q^{61} +(-1.19445e26 + 2.99838e25i) q^{63} +1.54743e26 q^{64} +(-1.75561e26 - 3.04081e26i) q^{67} +(1.71067e27 - 9.87658e26i) q^{73} +(1.33635e27 + 7.71540e26i) q^{75} +1.21648e27i q^{76} +(-3.19532e27 + 5.53446e27i) q^{79} +(-2.35506e27 - 4.07909e27i) q^{81} +(1.94312e27 + 7.74070e27i) q^{84} +(2.49742e28 - 2.41945e28i) q^{91} +(-2.26750e28 + 3.92742e28i) q^{93} -7.63732e28i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14348907 q^{3} - 536870912 q^{4} - 2497125299549 q^{7} + 68630377364883 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14348907 q^{3} - 536870912 q^{4} - 2497125299549 q^{7} + 68630377364883 q^{9} - 77\!\cdots\!84 q^{12}+ \cdots - 45\!\cdots\!15 q^{93}+O(q^{100}) \) 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 \(0.666667\pi\)
\(3\) 7.17445e6 4.14217e6i 0.866025 0.500000i
\(4\) −2.68435e8 4.64944e8i −0.500000 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −1.24856e12 1.28880e12i −0.695807 0.718229i
\(8\) 0 0
\(9\) 3.43152e13 5.94357e13i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −3.85176e15 2.22381e15i −0.866025 0.500000i
\(13\) 1.93779e16i 1.36498i 0.730893 + 0.682492i \(0.239104\pi\)
−0.730893 + 0.682492i \(0.760896\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.44115e17 + 2.49615e17i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −1.96231e18 1.13294e18i −0.563429 0.325296i 0.191092 0.981572i \(-0.438797\pi\)
−0.754521 + 0.656276i \(0.772131\pi\)
\(20\) 0 0
\(21\) −1.42962e19 4.07465e18i −0.961701 0.274101i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 9.31323e19 + 1.61310e20i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 5.68558e20i 1.00000i
\(28\) −2.64060e20 + 9.26470e20i −0.274101 + 0.961701i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.74078e21 + 2.73709e21i −1.12488 + 0.649448i −0.942642 0.333807i \(-0.891667\pi\)
−0.182236 + 0.983255i \(0.558333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −3.68457e22 −1.00000
\(37\) −3.75709e22 + 6.50747e22i −0.685372 + 1.18710i 0.287947 + 0.957646i \(0.407027\pi\)
−0.973320 + 0.229453i \(0.926306\pi\)
\(38\) 0 0
\(39\) 8.02665e22 + 1.39026e23i 0.682492 + 1.18211i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 9.11042e23 1.88038 0.940191 0.340649i \(-0.110647\pi\)
0.940191 + 0.340649i \(0.110647\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 2.38780e24i 1.00000i
\(49\) −1.02088e23 + 3.21829e24i −0.0317054 + 0.999497i
\(50\) 0 0
\(51\) 0 0
\(52\) 9.00963e24 5.20171e24i 1.18211 0.682492i
\(53\) 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
\(57\) −1.87713e25 −0.650592
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −9.20660e25 5.31543e25i −1.19348 0.689058i −0.234389 0.972143i \(-0.575309\pi\)
−0.959095 + 0.283085i \(0.908642\pi\)
\(62\) 0 0
\(63\) −1.19445e26 + 2.99838e25i −0.969908 + 0.243472i
\(64\) 1.54743e26 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.75561e26 3.04081e26i −0.583908 1.01136i −0.995011 0.0997693i \(-0.968190\pi\)
0.411103 0.911589i \(-0.365144\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.71067e27 9.87658e26i 1.64054 0.947164i 0.659894 0.751359i \(-0.270601\pi\)
0.980643 0.195805i \(-0.0627320\pi\)
\(74\) 0 0
\(75\) 1.33635e27 + 7.71540e26i 0.866025 + 0.500000i
\(76\) 1.21648e27i 0.650592i
\(77\) 0 0
\(78\) 0 0
\(79\) −3.19532e27 + 5.53446e27i −0.974814 + 1.68843i −0.294265 + 0.955724i \(0.595075\pi\)
−0.680549 + 0.732703i \(0.738259\pi\)
\(80\) 0 0
\(81\) −2.35506e27 4.07909e27i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.94312e27 + 7.74070e27i 0.243472 + 0.969908i
\(85\) 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
\(91\) 2.49742e28 2.41945e28i 0.980371 0.949766i
\(92\) 0 0
\(93\) −2.26750e28 + 3.92742e28i −0.649448 + 1.12488i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.63732e28i 1.18782i −0.804531 0.593911i \(-0.797583\pi\)
0.804531 0.593911i \(-0.202417\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000e28 8.66025e28i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 1.87902e29 + 1.08485e29i 1.22403 + 0.706693i 0.965774 0.259384i \(-0.0835196\pi\)
0.258254 + 0.966077i \(0.416853\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −2.64347e29 + 1.52621e29i −0.866025 + 0.500000i
\(109\) 2.21424e29 + 3.83517e29i 0.634658 + 1.09926i 0.986587 + 0.163233i \(0.0521924\pi\)
−0.351929 + 0.936027i \(0.614474\pi\)
\(110\) 0 0
\(111\) 6.22500e29i 1.37074i
\(112\) 5.01640e29 1.25925e29i 0.969908 0.243472i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.15174e30 + 6.64956e29i 1.18211 + 0.682492i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.93155e29 + 1.37378e30i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 2.54519e30 + 1.46946e30i 1.12488 + 0.649448i
\(125\) 0 0
\(126\) 0 0
\(127\) 4.62357e30 1.44485 0.722427 0.691447i \(-0.243027\pi\)
0.722427 + 0.691447i \(0.243027\pi\)
\(128\) 0 0
\(129\) 6.53623e30 3.77369e30i 1.62846 0.940191i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 9.89935e29 + 3.94356e30i 0.158401 + 0.631014i
\(134\) 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
\(139\) 1.83699e31i 1.55021i 0.631831 + 0.775107i \(0.282304\pi\)
−0.631831 + 0.775107i \(0.717696\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 9.89068e30 + 1.71312e31i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 1.25983e31 + 2.35123e31i 0.472291 + 0.881443i
\(148\) 4.03414e31 1.37074
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 3.93538e31 + 6.81629e31i 0.999589 + 1.73134i 0.524625 + 0.851334i \(0.324206\pi\)
0.474964 + 0.880005i \(0.342461\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 4.30928e31 7.46389e31i 0.682492 1.18211i
\(157\) 1.05949e31 6.11698e30i 0.152951 0.0883065i −0.421571 0.906795i \(-0.638521\pi\)
0.574522 + 0.818489i \(0.305188\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.17537e31 + 3.76785e31i −0.182311 + 0.315772i −0.942667 0.333734i \(-0.891691\pi\)
0.760356 + 0.649506i \(0.225024\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.73964e32 −0.863182
\(170\) 0 0
\(171\) −1.34674e32 + 7.77539e31i −0.563429 + 0.325296i
\(172\) −2.44556e32 4.23583e32i −0.940191 1.62846i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 9.16141e31 3.21434e32i 0.274101 0.961701i
\(176\) 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
\(181\) 2.39694e32i 0.439863i 0.975515 + 0.219931i \(0.0705833\pi\)
−0.975515 + 0.219931i \(0.929417\pi\)
\(182\) 0 0
\(183\) −8.80697e32 −1.37812
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −7.32755e32 + 7.09880e32i −0.718229 + 0.695807i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 1.11019e33 6.40970e32i 0.866025 0.500000i
\(193\) −1.19349e33 2.06719e33i −0.863453 1.49554i −0.868575 0.495557i \(-0.834964\pi\)
0.00512254 0.999987i \(-0.498369\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.52373e33 8.16437e32i 0.881443 0.472291i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.54923e33 + 8.94450e32i −0.719030 + 0.415132i −0.814395 0.580310i \(-0.802931\pi\)
0.0953657 + 0.995442i \(0.469598\pi\)
\(200\) 0 0
\(201\) −2.51911e33 1.45441e33i −1.01136 0.583908i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −4.83701e33 2.79265e33i −1.18211 0.682492i
\(209\) 0 0
\(210\) 0 0
\(211\) −9.41087e33 −1.86868 −0.934338 0.356388i \(-0.884008\pi\)
−0.934338 + 0.356388i \(0.884008\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.44671e33 + 2.69247e33i 1.24915 + 0.356029i
\(218\) 0 0
\(219\) 8.18210e33 1.41718e34i 0.947164 1.64054i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.73361e34i 1.54359i 0.635872 + 0.771794i \(0.280641\pi\)
−0.635872 + 0.771794i \(0.719359\pi\)
\(224\) 0 0
\(225\) 1.27834e34 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 5.03888e33 + 8.72760e33i 0.325296 + 0.563429i
\(229\) 2.52851e34 + 1.45984e34i 1.53197 + 0.884484i 0.999271 + 0.0381794i \(0.0121558\pi\)
0.532700 + 0.846304i \(0.321177\pi\)
\(230\) 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
\(237\) 5.29423e34i 1.94963i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 3.24242e34 1.87201e34i 0.936747 0.540831i 0.0478081 0.998857i \(-0.484776\pi\)
0.888939 + 0.458025i \(0.151443\pi\)
\(242\) 0 0
\(243\) −3.37926e34 1.95102e34i −0.866025 0.500000i
\(244\) 5.70740e34i 1.37812i
\(245\) 0 0
\(246\) 0 0
\(247\) 2.19539e34 3.80253e34i 0.444024 0.769072i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.60041e34 + 4.74866e34i 0.695807 + 0.718229i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −4.15384e34 7.19466e34i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 1.30778e35 3.28286e34i 1.32950 0.333738i
\(260\) 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
\(268\) −9.42538e34 + 1.63252e35i −0.583908 + 1.01136i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) −1.14928e35 6.63536e34i −0.605857 0.349792i 0.165485 0.986212i \(-0.447081\pi\)
−0.771342 + 0.636420i \(0.780414\pi\)
\(272\) 0 0
\(273\) 7.89581e34 2.77030e35i 0.374144 1.31271i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.76431e34 8.25203e34i −0.182828 0.316668i 0.760014 0.649906i \(-0.225192\pi\)
−0.942843 + 0.333238i \(0.891859\pi\)
\(278\) 0 0
\(279\) 3.75695e35i 1.29890i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −6.02910e35 + 3.48090e35i −1.69570 + 0.979013i −0.745950 + 0.666002i \(0.768004\pi\)
−0.949750 + 0.313010i \(0.898663\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.40984e35 4.17397e35i 0.500000 0.866025i
\(290\) 0 0
\(291\) −3.16351e35 5.47936e35i −0.593911 1.02868i
\(292\) −9.18411e35 5.30245e35i −1.64054 0.947164i
\(293\) 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
\(300\) 8.28435e35i 1.00000i
\(301\) −1.13749e36 1.17415e36i −1.30838 1.35054i
\(302\) 0 0
\(303\) 0 0
\(304\) 5.65596e35 3.26547e35i 0.563429 0.325296i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.29328e36i 1.98130i 0.136412 + 0.990652i \(0.456443\pi\)
−0.136412 + 0.990652i \(0.543557\pi\)
\(308\) 0 0
\(309\) 1.79746e36 1.41339
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −2.40350e36 1.38766e36i −1.56838 0.905506i −0.996358 0.0852700i \(-0.972825\pi\)
−0.572025 0.820236i \(-0.693842\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.43095e36 1.94963
\(317\) 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
\(324\) −1.26437e36 + 2.18995e36i −0.500000 + 0.866025i
\(325\) −3.12584e36 + 1.80471e36i −1.18211 + 0.682492i
\(326\) 0 0
\(327\) 3.17719e36 + 1.83435e36i 1.09926 + 0.634658i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.46610e36 + 4.27141e36i −0.715327 + 1.23898i 0.247506 + 0.968886i \(0.420389\pi\)
−0.962833 + 0.270097i \(0.912944\pi\)
\(332\) 0 0
\(333\) 2.57850e36 + 4.46610e36i 0.685372 + 1.18710i
\(334\) 0 0
\(335\) 0 0
\(336\) 3.07739e36 2.98132e36i 0.718229 0.695807i
\(337\) 6.54162e36 1.46235 0.731176 0.682189i \(-0.238972\pi\)
0.731176 + 0.682189i \(0.238972\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.27518e36 3.88666e36i 0.739929 0.672685i
\(344\) 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
\(349\) 2.11165e36i 0.284219i −0.989851 0.142109i \(-0.954612\pi\)
0.989851 0.142109i \(-0.0453884\pi\)
\(350\) 0 0
\(351\) 1.10174e37 1.36498
\(352\) 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
\(361\) −3.49782e36 6.05840e36i −0.288365 0.499463i
\(362\) 0 0
\(363\) 1.31415e37i 1.00000i
\(364\) −1.79530e37 5.11692e36i −1.31271 0.374144i
\(365\) 0 0
\(366\) 0 0
\(367\) −1.07606e37 + 6.21261e36i −0.698517 + 0.403289i −0.806795 0.590832i \(-0.798800\pi\)
0.108278 + 0.994121i \(0.465466\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 2.43471e37 1.29890
\(373\) −9.84732e36 + 1.70561e37i −0.505289 + 0.875187i 0.494692 + 0.869068i \(0.335281\pi\)
−0.999981 + 0.00611850i \(0.998052\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.76969e37 −1.94188 −0.970939 0.239329i \(-0.923073\pi\)
−0.970939 + 0.239329i \(0.923073\pi\)
\(380\) 0 0
\(381\) 3.31716e37 1.91516e37i 1.25128 0.722427i
\(382\) 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
\(387\) 3.12626e37 5.41484e37i 0.940191 1.62846i
\(388\) −3.55093e37 + 2.05013e37i −1.02868 + 0.593911i
\(389\) 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
\(397\) −7.91562e37 4.57009e37i −1.64446 0.949428i −0.979221 0.202795i \(-0.934997\pi\)
−0.665237 0.746633i \(-0.731669\pi\)
\(398\) 0 0
\(399\) 2.34371e37 + 2.41924e37i 0.452686 + 0.467274i
\(400\) −5.36871e37 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −5.30390e37 9.18662e37i −0.886487 1.53544i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.18314e37 2.41514e37i 0.564306 0.325802i −0.190566 0.981674i \(-0.561032\pi\)
0.754872 + 0.655872i \(0.227699\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.16485e38i 1.41339i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 7.60912e37 + 1.31794e38i 0.775107 + 1.34252i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −2.21956e38 −1.96869 −0.984344 0.176258i \(-0.943601\pi\)
−0.984344 + 0.176258i \(0.943601\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.64450e37 + 1.85021e38i 0.335533 + 1.33665i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 1.41920e38 + 8.19378e37i 0.866025 + 0.500000i
\(433\) 2.59179e38i 1.52942i 0.644376 + 0.764709i \(0.277117\pi\)
−0.644376 + 0.764709i \(0.722883\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 1.18876e38 2.05899e38i 0.634658 1.09926i
\(437\) 0 0
\(438\) 0 0
\(439\) 2.57263e38 + 1.48531e38i 1.24349 + 0.717928i 0.969803 0.243891i \(-0.0784239\pi\)
0.273686 + 0.961819i \(0.411757\pi\)
\(440\) 0 0
\(441\) 1.87778e38 + 1.16504e38i 0.849737 + 0.527206i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 2.89428e38 1.67101e38i 1.18710 0.685372i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.93206e38 1.99432e38i −0.695807 0.718229i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 5.64685e38 + 3.26021e38i 1.73134 + 0.999589i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.72451e38 + 2.98694e38i −0.465459 + 0.806199i −0.999222 0.0394353i \(-0.987444\pi\)
0.533763 + 0.845634i \(0.320777\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −5.37446e38 −1.20063 −0.600317 0.799762i \(-0.704959\pi\)
−0.600317 + 0.799762i \(0.704959\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 7.13991e38i 1.36498i
\(469\) −1.72700e38 + 6.05928e38i −0.320100 + 1.12309i
\(470\) 0 0
\(471\) 5.06752e37 8.77720e37i 0.0883065 0.152951i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.22052e38i 0.650592i
\(476\) 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
\(481\) −1.26101e39 7.28044e38i −1.62037 0.935522i
\(482\) 0 0
\(483\) 0 0
\(484\) 8.51643e38 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 9.15738e38 + 1.58610e39i 0.983108 + 1.70279i 0.650059 + 0.759884i \(0.274744\pi\)
0.333049 + 0.942909i \(0.391922\pi\)
\(488\) 0 0
\(489\) 3.60430e38i 0.364622i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.57782e39i 1.29890i
\(497\) 0 0
\(498\) 0 0
\(499\) 1.53440e38 2.65765e38i 0.115739 0.200466i −0.802336 0.596873i \(-0.796410\pi\)
0.918075 + 0.396407i \(0.129743\pi\)
\(500\) 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
\(507\) −1.24810e39 + 7.20590e38i −0.747538 + 0.431591i
\(508\) −1.24113e39 2.14970e39i −0.722427 1.25128i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) −3.40877e39 9.71558e38i −1.82178 0.519237i
\(512\) 0 0
\(513\) −6.44140e38 + 1.11568e39i −0.325296 + 0.563429i
\(514\) 0 0
\(515\) 0 0
\(516\) −3.50911e39 2.02599e39i −1.62846 0.940191i
\(517\) 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
\(523\) 4.49490e39 + 2.59513e39i 1.71572 + 0.990570i 0.926361 + 0.376636i \(0.122919\pi\)
0.789357 + 0.613934i \(0.210414\pi\)
\(524\) 0 0
\(525\) −6.74154e38 2.68559e39i −0.243472 0.969908i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.54553e39 2.67694e39i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 1.56780e39 1.51885e39i 0.467274 0.452686i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.65732e39 + 4.60262e39i −0.620990 + 1.07559i 0.368312 + 0.929702i \(0.379936\pi\)
−0.989302 + 0.145883i \(0.953398\pi\)
\(542\) 0 0
\(543\) 9.92852e38 + 1.71967e39i 0.219931 + 0.380932i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.99595e39 1.99071 0.995355 0.0962743i \(-0.0306926\pi\)
0.995355 + 0.0962743i \(0.0306926\pi\)
\(548\) 0 0
\(549\) −6.31852e39 + 3.64800e39i −1.19348 + 0.689058i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.11223e40 2.79200e39i 1.89096 0.474680i
\(554\) 0 0
\(555\) 0 0
\(556\) 8.54096e39 4.93112e39i 1.34252 0.775107i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 1.76541e40i 2.56669i
\(560\) 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
\(567\) −2.31667e39 + 8.12820e39i −0.274101 + 0.961701i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) 9.34796e39 + 1.61911e40i 0.998834 + 1.73003i 0.541231 + 0.840874i \(0.317959\pi\)
0.457603 + 0.889157i \(0.348708\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 5.31002e39 9.19722e39i 0.500000 0.866025i
\(577\) −2.01974e39 + 1.16610e39i −0.185458 + 0.107074i −0.589855 0.807509i \(-0.700815\pi\)
0.404396 + 0.914584i \(0.367482\pi\)
\(578\) 0 0
\(579\) −1.71253e40 9.88730e39i −1.49554 0.863453i
\(580\) 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 \(0.5\pi\)
\(588\) 7.55008e39 1.21690e40i 0.527206 0.849737i
\(589\) 1.24038e40 0.845052
\(590\) 0 0
\(591\) 0 0
\(592\) −1.08291e40 1.87565e40i −0.685372 1.18710i
\(593\) 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
\(597\) −7.40993e39 + 1.28344e40i −0.415132 + 0.719030i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 3.36080e40i 1.70908i 0.519385 + 0.854540i \(0.326161\pi\)
−0.519385 + 0.854540i \(0.673839\pi\)
\(602\) 0 0
\(603\) −2.40977e40 −1.16782
\(604\) 2.11279e40 3.65947e40i 0.999589 1.73134i
\(605\) 0 0
\(606\) 0 0
\(607\) 3.05609e39 + 1.76443e39i 0.134564 + 0.0776907i 0.565771 0.824562i \(-0.308579\pi\)
−0.431207 + 0.902253i \(0.641912\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −9.86435e39 1.70856e40i −0.376610 0.652308i 0.613957 0.789340i \(-0.289577\pi\)
−0.990567 + 0.137032i \(0.956244\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 4.88869e40 2.82248e40i 1.62061 0.935658i 0.633849 0.773457i \(-0.281474\pi\)
0.986758 0.162201i \(-0.0518593\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −4.62705e40 −1.36498
\(625\) −1.73472e40 + 3.00463e40i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) −5.68811e39 3.28403e39i −0.152951 0.0883065i
\(629\) 0 0
\(630\) 0 0
\(631\) −3.60894e40 −0.905636 −0.452818 0.891603i \(-0.649581\pi\)
−0.452818 + 0.891603i \(0.649581\pi\)
\(632\) 0 0
\(633\) −6.75178e40 + 3.89814e40i −1.61832 + 0.934338i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.23636e40 1.97826e39i −1.36430 0.0432774i
\(638\) 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
\(643\) 6.72150e40i 1.28353i −0.766900 0.641767i \(-0.778202\pi\)
0.766900 0.641767i \(-0.221798\pi\)
\(644\) 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
\(651\) 7.89277e40 1.98129e40i 1.25981 0.316245i
\(652\) 2.33579e40 0.364622
\(653\) 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
\(657\) 1.35567e41i 1.89433i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 6.40088e40 3.69555e40i 0.819065 0.472888i −0.0310288 0.999518i \(-0.509878\pi\)
0.850094 + 0.526631i \(0.176545\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 7.18092e40 + 1.24377e41i 0.771794 + 1.33679i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.74264e41 −1.71786 −0.858932 0.512089i \(-0.828872\pi\)
−0.858932 + 0.512089i \(0.828872\pi\)
\(674\) 0 0
\(675\) 9.17139e40 5.29511e40i 0.866025 0.500000i
\(676\) 4.66982e40 + 8.08836e40i 0.431591 + 0.747538i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) −9.84296e40 + 9.53568e40i −0.853128 + 0.826494i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 7.23024e40 + 4.17438e40i 0.563429 + 0.325296i
\(685\) 0 0
\(686\) 0 0
\(687\) 2.41876e41 1.76897
\(688\) −1.31295e41 + 2.27410e41i −0.940191 + 1.62846i
\(689\) 0 0
\(690\) 0 0
\(691\) 2.50166e41 + 1.44433e41i 1.68188 + 0.971036i 0.960409 + 0.278595i \(0.0898688\pi\)
0.721475 + 0.692441i \(0.243464\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −1.74041e41 + 4.36889e40i −0.969908 + 0.243472i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 1.47451e41 8.51309e40i 0.772317 0.445898i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.63119e40 8.02145e40i 0.214448 0.371435i −0.738654 0.674085i \(-0.764538\pi\)
0.953102 + 0.302650i \(0.0978715\pi\)
\(710\) 0 0
\(711\) 2.19296e41 + 3.79832e41i 0.974814 + 1.68843i
\(712\) 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
\(721\) −9.47919e40 3.77618e41i −0.344120 1.37085i
\(722\) 0 0
\(723\) 1.55084e41 2.68613e41i 0.540831 0.936747i
\(724\) 1.11444e41 6.43423e40i 0.380932 0.219931i
\(725\) 0 0
\(726\) 0 0
\(727\) 4.63965e40i 0.149361i 0.997208 + 0.0746804i \(0.0237937\pi\)
−0.997208 + 0.0746804i \(0.976206\pi\)
\(728\) 0 0
\(729\) −3.23258e41 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 2.36410e41 + 4.09475e41i 0.689058 + 1.19348i
\(733\) −4.94052e41 2.85241e41i −1.41177 0.815087i −0.416217 0.909265i \(-0.636644\pi\)
−0.995555 + 0.0941784i \(0.969978\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −3.82697e41 6.62851e41i −0.971652 1.68295i −0.690568 0.723268i \(-0.742639\pi\)
−0.281085 0.959683i \(-0.590694\pi\)
\(740\) 0 0
\(741\) 3.63748e41i 0.888048i
\(742\) 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
\(751\) −2.90692e41 + 5.03494e41i −0.584325 + 1.01208i 0.410634 + 0.911800i \(0.365307\pi\)
−0.994959 + 0.100281i \(0.968026\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 5.26752e41 + 1.50133e41i 0.961701 + 0.274101i
\(757\) −1.10661e42 −1.98201 −0.991004 0.133833i \(-0.957271\pi\)
−0.991004 + 0.133833i \(0.957271\pi\)
\(758\) 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
\(763\) 2.17814e41 7.64215e41i 0.347921 1.22070i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −5.96030e41 3.44118e41i −0.866025 0.500000i
\(769\) 1.40269e42i 2.00000i 0.00120402 + 0.999999i \(0.499617\pi\)
−0.00120402 + 0.999999i \(0.500383\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.40751e41 + 1.10981e42i −0.863453 + 1.49554i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −8.83039e41 5.09823e41i −1.12488 0.649448i
\(776\) 0 0
\(777\) 8.02277e41 7.77231e41i 0.984508 0.953773i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −7.88620e41 4.89287e41i −0.849737 0.527206i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.65774e42 9.57099e41i 1.68999 0.975716i 0.735475 0.677552i \(-0.236959\pi\)
0.954515 0.298164i \(-0.0963741\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.03002e42 1.78404e42i 0.940554 1.62909i
\(794\) 0 0
\(795\) 0 0
\(796\) 8.31738e41 + 4.80204e41i 0.719030 + 0.415132i
\(797\) 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
\(804\) 1.56166e42i 1.16782i
\(805\) 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
\(811\) 2.48784e42i 1.64066i 0.571889 + 0.820331i \(0.306211\pi\)
−0.571889 + 0.820331i \(0.693789\pi\)
\(812\) 0 0
\(813\) −1.09939e42 −0.699584
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.78774e42 1.03215e42i −1.05946 0.611680i
\(818\) 0 0
\(819\) −5.81023e41 2.31459e42i −0.332336 1.32391i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 1.86909e42 + 3.23735e42i 0.996166 + 1.72541i 0.573844 + 0.818965i \(0.305452\pi\)
0.422322 + 0.906446i \(0.361215\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 3.33636e42 1.92625e42i 1.60041 0.923998i 0.609008 0.793164i \(-0.291568\pi\)
0.991404 0.130834i \(-0.0417656\pi\)
\(830\) 0 0
\(831\) −6.83626e41 3.94692e41i −0.316668 0.182828i
\(832\) 2.99858e42i 1.36498i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.55619e42 + 2.69541e42i 0.649448 + 1.12488i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 2.56769e42 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.52621e42 + 4.37552e42i 0.934338 + 1.61832i
\(845\) 0 0
\(846\) 0 0
\(847\) 2.76083e42 6.93040e41i 0.969908 0.243472i
\(848\) 0 0
\(849\) −2.88370e42 + 4.99471e42i −0.979013 + 1.69570i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 6.15806e42i 1.95291i −0.215717 0.976456i \(-0.569209\pi\)
0.215717 0.976456i \(-0.430791\pi\)
\(854\) 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
\(859\) 4.01322e42 + 2.31703e42i 1.14972 + 0.663790i 0.948819 0.315821i \(-0.102280\pi\)
0.200900 + 0.979612i \(0.435613\pi\)
\(860\) 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
\(867\) 3.99279e42i 1.00000i
\(868\) −1.28398e42 5.11495e42i −0.316245 1.25981i
\(869\) 0 0
\(870\) 0 0
\(871\) 5.89245e42 3.40201e42i 1.38049 0.797025i
\(872\) 0 0
\(873\) −4.53929e42 2.62076e42i −1.02868 0.593911i
\(874\) 0 0
\(875\) 0 0
\(876\) −8.78547e42 −1.89433
\(877\) −4.23941e42 + 7.34287e42i −0.899106 + 1.55730i −0.0704672 + 0.997514i \(0.522449\pi\)
−0.828639 + 0.559783i \(0.810884\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −1.00856e43 −1.93762 −0.968812 0.247797i \(-0.920293\pi\)
−0.968812 + 0.247797i \(0.920293\pi\)
\(884\) 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
\(889\) −5.77282e42 5.95885e42i −1.00534 1.03774i
\(890\) 0 0
\(891\) 0 0
\(892\) 8.06033e42 4.65363e42i 1.33679 0.771794i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −3.43152e42 5.94357e42i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) −1.30244e43 3.71218e42i −1.80836 0.515414i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7.05605e42 + 1.22214e43i 0.918874 + 1.59154i 0.801129 + 0.598492i \(0.204233\pi\)
0.117745 + 0.993044i \(0.462434\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 2.70523e42 4.68559e42i 0.325296 0.563429i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.56749e43i 1.76897i
\(917\) 0 0
\(918\) 0 0
\(919\) −4.27677e42 + 7.40758e42i −0.460300 + 0.797262i −0.998976 0.0452508i \(-0.985591\pi\)
0.538676 + 0.842513i \(0.318925\pi\)
\(920\) 0 0
\(921\) 9.49917e42 + 1.64531e43i 0.990652 + 1.71586i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.39962e43 −1.37074
\(926\) 0 0
\(927\) 1.28958e43 7.44538e42i 1.22403 0.706693i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) 3.84645e42 6.19960e42i 0.342996 0.552832i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 8.90131e42i 0.723152i 0.932343 + 0.361576i \(0.117761\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(938\) 0 0
\(939\) −2.29918e43 −1.81101
\(940\) 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 \(0.666667\pi\)
\(948\) 2.46152e43 1.42116e43i 1.68843 0.974814i
\(949\) 1.91387e43 + 3.31492e43i 1.29286 + 2.23931i
\(950\) 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
\(961\) 6.10238e42 1.05696e43i 0.343566 0.595073i
\(962\) 0 0
\(963\) 0 0
\(964\) −1.74076e43 1.00503e43i −0.936747 0.540831i
\(965\) 0 0
\(966\) 0 0
\(967\) −3.22345e43 −1.65820 −0.829099 0.559101i \(-0.811146\pi\)
−0.829099 + 0.559101i \(0.811146\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 2.09489e43i 1.00000i
\(973\) 2.36750e43 2.29359e43i 1.11341 1.07865i
\(974\) 0 0
\(975\) −1.49508e43 + 2.58956e43i −0.682492 + 1.18211i
\(976\) 2.65362e43 1.53207e43i 1.19348 0.689058i
\(977\) 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
\(981\) 3.03928e43 1.26932
\(982\) 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
\(988\) −2.35729e43 −0.888048
\(989\) 0 0
\(990\) 0 0
\(991\) 2.76601e43 + 4.79086e43i 0.997207 + 1.72721i 0.563287 + 0.826261i \(0.309537\pi\)
0.433920 + 0.900951i \(0.357130\pi\)
\(992\) 0 0
\(993\) 4.08601e43i 1.43065i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 4.25437e43 2.45626e43i 1.40526 0.811326i 0.410331 0.911936i \(-0.365413\pi\)
0.994926 + 0.100611i \(0.0320797\pi\)
\(998\) 0 0
\(999\) 3.69987e43 + 2.13612e43i 1.18710 + 0.685372i
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 21.30.g.a.5.1 2
3.2 odd 2 CM 21.30.g.a.5.1 2
7.3 odd 6 inner 21.30.g.a.17.1 yes 2
21.17 even 6 inner 21.30.g.a.17.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.30.g.a.5.1 2 1.1 even 1 trivial
21.30.g.a.5.1 2 3.2 odd 2 CM
21.30.g.a.17.1 yes 2 7.3 odd 6 inner
21.30.g.a.17.1 yes 2 21.17 even 6 inner