Properties

Label 21.23.h.a.2.1
Level $21$
Weight $23$
Character 21.2
Analytic conductor $64.409$
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] = [21,23,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, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.2");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 23 \)
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: \(64.4085613168\)
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 21.23.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)
 
\(f(q)\) \(=\) \(q+(88573.5 + 153414. i) q^{3} +(-2.09715e6 - 3.63237e6i) q^{4} +(9.98966e8 - 1.70643e9i) q^{7} +(-1.56905e10 + 2.71768e10i) q^{9} +O(q^{10})\) \(q+(88573.5 + 153414. i) q^{3} +(-2.09715e6 - 3.63237e6i) q^{4} +(9.98966e8 - 1.70643e9i) q^{7} +(-1.56905e10 + 2.71768e10i) q^{9} +(3.71504e11 - 6.43464e11i) q^{12} +2.09432e12 q^{13} +(-8.79609e12 + 1.52353e13i) q^{16} +(-1.03370e13 + 1.79042e13i) q^{19} +(3.50271e14 + 2.11109e12i) q^{21} +(-1.19209e15 - 2.06477e15i) q^{25} -5.55906e15 q^{27} +(-8.29336e15 - 4.99841e13i) q^{28} +(-1.85408e16 - 3.21136e16i) q^{31} +1.31622e17 q^{36} +(-4.22284e16 + 7.31418e16i) q^{37} +(1.85501e17 + 3.21297e17i) q^{39} +1.26850e18 q^{43} -3.11640e18 q^{48} +(-1.91396e18 - 3.40932e18i) q^{49} +(-4.39210e18 - 7.60734e18i) q^{52} -3.66233e18 q^{57} +(3.36973e19 - 5.83654e19i) q^{61} +(3.07009e19 + 5.39234e19i) q^{63} +7.37870e19 q^{64} +(-1.07065e20 - 1.85442e20i) q^{67} +(-3.10685e20 - 5.38121e20i) q^{73} +(2.11176e20 - 3.65767e20i) q^{75} +8.67128e19 q^{76} +(5.18791e20 - 8.98573e20i) q^{79} +(-4.92385e20 - 8.52837e20i) q^{81} +(-7.26904e20 - 1.27674e21i) q^{84} +(2.09215e21 - 3.57379e21i) q^{91} +(3.28445e21 - 5.68883e21i) q^{93} -1.61889e21 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 177147 q^{3} - 4194304 q^{4} + 1997931923 q^{7} - 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 177147 q^{3} - 4194304 q^{4} + 1997931923 q^{7} - 31381059609 q^{9} + 743008370688 q^{12} + 4188631572094 q^{13} - 17592186044416 q^{16} - 20673935603651 q^{19} + 700542277678314 q^{21} - 23\!\cdots\!25 q^{25}+ \cdots - 32\!\cdots\!84 q^{97}+O(q^{100}) \) 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 \(0.666667\pi\)
\(3\) 88573.5 + 153414.i 0.500000 + 0.866025i
\(4\) −2.09715e6 3.63237e6i −0.500000 0.866025i
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) 9.98966e8 1.70643e9i 0.505210 0.862996i
\(8\) 0 0
\(9\) −1.56905e10 + 2.71768e10i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 3.71504e11 6.43464e11i 0.500000 0.866025i
\(13\) 2.09432e12 1.16860 0.584299 0.811538i \(-0.301369\pi\)
0.584299 + 0.811538i \(0.301369\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.79609e12 + 1.52353e13i −0.500000 + 0.866025i
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) −1.03370e13 + 1.79042e13i −0.0887368 + 0.153697i −0.906977 0.421179i \(-0.861616\pi\)
0.818241 + 0.574876i \(0.194950\pi\)
\(20\) 0 0
\(21\) 3.50271e14 + 2.11109e12i 0.999982 + 0.00602689i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) −1.19209e15 2.06477e15i −0.500000 0.866025i
\(26\) 0 0
\(27\) −5.55906e15 −1.00000
\(28\) −8.29336e15 4.99841e13i −0.999982 0.00602689i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.85408e16 3.21136e16i −0.729710 1.26389i −0.957006 0.290069i \(-0.906322\pi\)
0.227296 0.973826i \(-0.427011\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.31622e17 1.00000
\(37\) −4.22284e16 + 7.31418e16i −0.237348 + 0.411099i −0.959953 0.280163i \(-0.909612\pi\)
0.722604 + 0.691262i \(0.242945\pi\)
\(38\) 0 0
\(39\) 1.85501e17 + 3.21297e17i 0.584299 + 1.01204i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 1.26850e18 1.36502 0.682509 0.730877i \(-0.260889\pi\)
0.682509 + 0.730877i \(0.260889\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −3.11640e18 −1.00000
\(49\) −1.91396e18 3.40932e18i −0.489525 0.871989i
\(50\) 0 0
\(51\) 0 0
\(52\) −4.39210e18 7.60734e18i −0.584299 1.01204i
\(53\) 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
\(57\) −3.66233e18 −0.177474
\(58\) 0 0
\(59\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(60\) 0 0
\(61\) 3.36973e19 5.83654e19i 0.774403 1.34131i −0.160727 0.986999i \(-0.551384\pi\)
0.935130 0.354306i \(-0.115283\pi\)
\(62\) 0 0
\(63\) 3.07009e19 + 5.39234e19i 0.494771 + 0.869023i
\(64\) 7.37870e19 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.07065e20 1.85442e20i −0.876646 1.51839i −0.854999 0.518629i \(-0.826443\pi\)
−0.0216461 0.999766i \(-0.506891\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −3.10685e20 5.38121e20i −0.990303 1.71526i −0.615463 0.788166i \(-0.711031\pi\)
−0.374840 0.927089i \(-0.622302\pi\)
\(74\) 0 0
\(75\) 2.11176e20 3.65767e20i 0.500000 0.866025i
\(76\) 8.67128e19 0.177474
\(77\) 0 0
\(78\) 0 0
\(79\) 5.18791e20 8.98573e20i 0.693577 1.20131i −0.277081 0.960847i \(-0.589367\pi\)
0.970658 0.240464i \(-0.0772997\pi\)
\(80\) 0 0
\(81\) −4.92385e20 8.52837e20i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −7.26904e20 1.27674e21i −0.494771 0.869023i
\(85\) 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
\(91\) 2.09215e21 3.57379e21i 0.590388 1.00850i
\(92\) 0 0
\(93\) 3.28445e21 5.68883e21i 0.729710 1.26389i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.61889e21 −0.226322 −0.113161 0.993577i \(-0.536098\pi\)
−0.113161 + 0.993577i \(0.536098\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.00000e21 + 8.66025e21i −0.500000 + 0.866025i
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 1.49051e21 2.58164e21i 0.107678 0.186503i −0.807151 0.590344i \(-0.798992\pi\)
0.914829 + 0.403841i \(0.132325\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 1.16582e22 + 2.01926e22i 0.500000 + 0.866025i
\(109\) −1.25416e22 2.17227e22i −0.486029 0.841828i 0.513842 0.857885i \(-0.328222\pi\)
−0.999871 + 0.0160575i \(0.994889\pi\)
\(110\) 0 0
\(111\) −1.49613e22 −0.474697
\(112\) 1.72109e22 + 3.02294e22i 0.494771 + 0.869023i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.28609e22 + 5.69168e22i −0.584299 + 1.01204i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.07014e22 + 7.04968e22i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) −7.77658e22 + 1.34694e23i −0.729710 + 1.26389i
\(125\) 0 0
\(126\) 0 0
\(127\) 2.76555e23 1.99499 0.997496 0.0707258i \(-0.0225315\pi\)
0.997496 + 0.0707258i \(0.0225315\pi\)
\(128\) 0 0
\(129\) 1.12356e23 + 1.94606e23i 0.682509 + 1.18214i
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 0 0
\(133\) 2.02258e22 + 3.55249e22i 0.0878088 + 0.154229i
\(134\) 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
\(139\) −7.03681e23 −1.88024 −0.940119 0.340845i \(-0.889287\pi\)
−0.940119 + 0.340845i \(0.889287\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −2.76031e23 4.78099e23i −0.500000 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) 3.53511e23 5.95603e23i 0.510402 0.859936i
\(148\) 3.54238e23 0.474697
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −8.35858e23 1.44775e24i −0.898227 1.55578i −0.829759 0.558122i \(-0.811522\pi\)
−0.0684684 0.997653i \(-0.521811\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 7.78047e23 1.34762e24i 0.584299 1.01204i
\(157\) 1.27068e24 + 2.20088e24i 0.889487 + 1.54064i 0.840483 + 0.541838i \(0.182271\pi\)
0.0490037 + 0.998799i \(0.484395\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.61163e24 2.79143e24i 0.746797 1.29349i −0.202553 0.979271i \(-0.564924\pi\)
0.949350 0.314219i \(-0.101743\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.17432e24 0.365622
\(170\) 0 0
\(171\) −3.24385e23 5.61851e23i −0.0887368 0.153697i
\(172\) −2.66024e24 4.60768e24i −0.682509 1.18214i
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) −4.71423e24 2.84127e22i −0.999982 0.00602689i
\(176\) 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
\(181\) −1.20332e25 −1.76163 −0.880817 0.473458i \(-0.843006\pi\)
−0.880817 + 0.473458i \(0.843006\pi\)
\(182\) 0 0
\(183\) 1.19388e25 1.54881
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.55331e24 + 9.48612e24i −0.505210 + 0.862996i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 6.53557e24 + 1.13199e25i 0.500000 + 0.866025i
\(193\) 1.20825e25 + 2.09275e25i 0.873023 + 1.51212i 0.858854 + 0.512220i \(0.171177\pi\)
0.0141686 + 0.999900i \(0.495490\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −8.37008e24 + 1.41021e25i −0.510402 + 0.859936i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.75881e25 + 3.04635e25i 0.907476 + 1.57180i 0.817558 + 0.575846i \(0.195327\pi\)
0.0899181 + 0.995949i \(0.471339\pi\)
\(200\) 0 0
\(201\) 1.89662e25 3.28504e25i 0.876646 1.51839i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.84218e25 + 3.19075e25i −0.584299 + 1.01204i
\(209\) 0 0
\(210\) 0 0
\(211\) −7.12329e25 −1.93007 −0.965037 0.262115i \(-0.915580\pi\)
−0.965037 + 0.262115i \(0.915580\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.33212e25 4.41907e23i −1.45939 0.00879577i
\(218\) 0 0
\(219\) 5.50368e25 9.53266e25i 0.990303 1.71526i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.78784e24 0.0705970 0.0352985 0.999377i \(-0.488762\pi\)
0.0352985 + 0.999377i \(0.488762\pi\)
\(224\) 0 0
\(225\) 7.48183e25 1.00000
\(226\) 0 0
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 7.68045e24 + 1.33029e25i 0.0887368 + 0.153697i
\(229\) 3.72331e25 6.44896e25i 0.409958 0.710067i −0.584927 0.811086i \(-0.698877\pi\)
0.994885 + 0.101019i \(0.0322102\pi\)
\(230\) 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
\(237\) 1.83805e26 1.38715
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.03143e26 1.78649e26i −0.647517 1.12153i −0.983714 0.179741i \(-0.942474\pi\)
0.336197 0.941792i \(-0.390859\pi\)
\(242\) 0 0
\(243\) 8.72246e25 1.51077e26i 0.500000 0.866025i
\(244\) −2.82673e26 −1.54881
\(245\) 0 0
\(246\) 0 0
\(247\) −2.16489e25 + 3.74970e25i −0.103698 + 0.179610i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.31486e26 2.24603e26i 0.505210 0.862996i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −1.54743e26 2.68022e26i −0.500000 0.866025i
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 8.26263e25 + 1.45126e26i 0.234866 + 0.412522i
\(260\) 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
\(268\) −4.49062e26 + 7.77799e26i −0.876646 + 1.51839i
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −5.37914e26 + 9.31695e26i −0.929075 + 1.60920i −0.144202 + 0.989548i \(0.546061\pi\)
−0.784873 + 0.619657i \(0.787272\pi\)
\(272\) 0 0
\(273\) 7.33578e26 + 4.42128e24i 1.16858 + 0.00704302i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.51225e26 + 4.35135e26i 0.341025 + 0.590672i 0.984623 0.174691i \(-0.0558927\pi\)
−0.643599 + 0.765363i \(0.722559\pi\)
\(278\) 0 0
\(279\) 1.16366e27 1.45942
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −9.45168e25 1.63708e26i −0.101358 0.175557i 0.810886 0.585204i \(-0.198985\pi\)
−0.912244 + 0.409646i \(0.865652\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.87281e26 + 1.01720e27i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) −1.43390e26 2.48360e26i −0.113161 0.196001i
\(292\) −1.30311e27 + 2.25704e27i −0.990303 + 1.71526i
\(293\) 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
\(300\) −1.77147e27 −1.00000
\(301\) 1.26719e27 2.16461e27i 0.689621 1.17801i
\(302\) 0 0
\(303\) 0 0
\(304\) −1.81850e26 3.14973e26i −0.0887368 0.153697i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.12057e27 0.928816 0.464408 0.885621i \(-0.346267\pi\)
0.464408 + 0.885621i \(0.346267\pi\)
\(308\) 0 0
\(309\) 5.28079e26 0.215355
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 2.81044e27 4.86782e27i 0.994916 1.72325i 0.410245 0.911976i \(-0.365443\pi\)
0.584672 0.811270i \(-0.301223\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −4.35194e27 −1.38715
\(317\) 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
\(324\) −2.06521e27 + 3.57706e27i −0.500000 + 0.866025i
\(325\) −2.49662e27 4.32427e27i −0.584299 1.01204i
\(326\) 0 0
\(327\) 2.22171e27 3.84812e27i 0.486029 0.841828i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.29871e27 7.44558e27i 0.822678 1.42492i −0.0810032 0.996714i \(-0.525812\pi\)
0.903681 0.428206i \(-0.140854\pi\)
\(332\) 0 0
\(333\) −1.32517e27 2.29527e27i −0.237348 0.411099i
\(334\) 0 0
\(335\) 0 0
\(336\) −3.11318e27 + 5.31791e27i −0.505210 + 0.862996i
\(337\) 1.26159e28 1.98147 0.990737 0.135792i \(-0.0433579\pi\)
0.990737 + 0.135792i \(0.0433579\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −7.72973e27 1.39775e26i −0.999837 0.0180798i
\(344\) 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
\(349\) 1.04982e28 1.12211 0.561057 0.827777i \(-0.310395\pi\)
0.561057 + 0.827777i \(0.310395\pi\)
\(350\) 0 0
\(351\) −1.16424e28 −1.16860
\(352\) 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
\(361\) 6.57128e27 + 1.13818e28i 0.484252 + 0.838748i
\(362\) 0 0
\(363\) −1.44203e28 −1.00000
\(364\) −1.73689e28 1.04683e26i −1.16858 0.00704302i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.59218e28 + 2.75774e28i 0.978738 + 1.69522i 0.667002 + 0.745056i \(0.267577\pi\)
0.311736 + 0.950169i \(0.399089\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.75520e28 −1.45942
\(373\) 5.78887e27 1.00266e28i 0.297712 0.515653i −0.677900 0.735154i \(-0.737110\pi\)
0.975612 + 0.219501i \(0.0704430\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.59449e28 1.55098 0.775489 0.631361i \(-0.217503\pi\)
0.775489 + 0.631361i \(0.217503\pi\)
\(380\) 0 0
\(381\) 2.44955e28 + 4.24274e28i 0.997496 + 1.72771i
\(382\) 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
\(387\) −1.99035e28 + 3.44738e28i −0.682509 + 1.18214i
\(388\) 3.39505e27 + 5.88040e27i 0.113161 + 0.196001i
\(389\) 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
\(397\) 1.10151e27 1.90788e27i 0.0285295 0.0494145i −0.851408 0.524504i \(-0.824251\pi\)
0.879938 + 0.475089i \(0.157584\pi\)
\(398\) 0 0
\(399\) −3.65854e27 + 6.24949e27i −0.0896615 + 0.153159i
\(400\) 4.19430e28 1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) −3.88303e28 6.72561e28i −0.852738 1.47698i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.14863e28 7.18563e28i −0.774369 1.34125i −0.935148 0.354256i \(-0.884734\pi\)
0.160779 0.986990i \(-0.448599\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.25033e28 −0.215355
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6.23275e28 1.07954e29i −0.940119 1.62833i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 5.80624e28 0.788482 0.394241 0.919007i \(-0.371007\pi\)
0.394241 + 0.919007i \(0.371007\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −6.59338e28 1.15807e29i −0.766305 1.34595i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 4.88980e28 8.46938e28i 0.500000 0.866025i
\(433\) 2.00619e29 1.99989 0.999945 0.0104622i \(-0.00333027\pi\)
0.999945 + 0.0104622i \(0.00333027\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.26034e28 + 9.11118e28i −0.486029 + 0.841828i
\(437\) 0 0
\(438\) 0 0
\(439\) −1.07908e29 + 1.86903e29i −0.924582 + 1.60142i −0.132351 + 0.991203i \(0.542253\pi\)
−0.792231 + 0.610221i \(0.791081\pi\)
\(440\) 0 0
\(441\) 1.22685e29 + 1.47890e27i 0.999927 + 0.0120536i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 3.13761e28 + 5.43450e28i 0.237348 + 0.411099i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 7.37107e28 1.25912e29i 0.505210 0.862996i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.48070e29 2.56464e29i 0.898227 1.55578i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.81422e29 + 3.14232e29i −0.999106 + 1.73050i −0.462934 + 0.886393i \(0.653203\pi\)
−0.536172 + 0.844109i \(0.680130\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −4.13744e29 −1.97397 −0.986985 0.160810i \(-0.948590\pi\)
−0.986985 + 0.160810i \(0.948590\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) 2.75657e29 1.16860
\(469\) −4.23397e29 2.55181e27i −1.75326 0.0105669i
\(470\) 0 0
\(471\) −2.25097e29 + 3.89879e29i −0.889487 + 1.54064i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.92905e28 0.177474
\(476\) 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
\(481\) −8.84397e28 + 1.53182e29i −0.277365 + 0.480410i
\(482\) 0 0
\(483\) 0 0
\(484\) 3.41428e29 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 2.83862e29 + 4.91663e29i 0.776762 + 1.34539i 0.933799 + 0.357799i \(0.116473\pi\)
−0.157036 + 0.987593i \(0.550194\pi\)
\(488\) 0 0
\(489\) 5.70992e29 1.49359
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 6.52347e29 1.45942
\(497\) 0 0
\(498\) 0 0
\(499\) −4.64295e29 + 8.04182e29i −0.972048 + 1.68364i −0.282698 + 0.959209i \(0.591229\pi\)
−0.689350 + 0.724428i \(0.742104\pi\)
\(500\) 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
\(507\) 1.04014e29 + 1.80157e29i 0.182811 + 0.316638i
\(508\) −5.79979e29 1.00455e30i −0.997496 1.72771i
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) −1.22863e30 7.40494e27i −1.98057 0.0119369i
\(512\) 0 0
\(513\) 5.74638e28 9.95303e28i 0.0887368 0.153697i
\(514\) 0 0
\(515\) 0 0
\(516\) 4.71254e29 8.16236e29i 0.682509 1.18214i
\(517\) 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
\(523\) 7.59905e29 1.31619e30i 0.948944 1.64362i 0.201290 0.979532i \(-0.435487\pi\)
0.747654 0.664088i \(-0.231180\pi\)
\(524\) 0 0
\(525\) −4.13197e29 7.25744e29i −0.494771 0.869023i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −4.53923e29 7.86218e29i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 8.66231e28 1.47969e29i 0.0896615 0.153159i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.22232e29 + 1.25094e30i −0.621594 + 1.07663i 0.367595 + 0.929986i \(0.380181\pi\)
−0.989189 + 0.146646i \(0.953152\pi\)
\(542\) 0 0
\(543\) −1.06582e30 1.84605e30i −0.880817 1.52562i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.19905e30 1.67639 0.838193 0.545374i \(-0.183612\pi\)
0.838193 + 0.545374i \(0.183612\pi\)
\(548\) 0 0
\(549\) 1.05746e30 + 1.83157e30i 0.774403 + 1.34131i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.01509e30 1.78292e30i −0.686324 1.20547i
\(554\) 0 0
\(555\) 0 0
\(556\) 1.47573e30 + 2.55603e30i 0.940119 + 1.62833i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 2.65665e30 1.59516
\(560\) 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
\(567\) −1.94718e30 1.17357e28i −0.999982 0.00602689i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −2.06977e30 3.58494e30i −0.983840 1.70406i −0.646981 0.762506i \(-0.723969\pi\)
−0.336860 0.941555i \(-0.609365\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −1.15776e30 + 2.00529e30i −0.500000 + 0.866025i
\(577\) 1.47889e30 + 2.56151e30i 0.626617 + 1.08533i 0.988226 + 0.153002i \(0.0488941\pi\)
−0.361609 + 0.932330i \(0.617773\pi\)
\(578\) 0 0
\(579\) −2.14037e30 + 3.70724e30i −0.873023 + 1.51212i
\(580\) 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 \(\pi\)
\(588\) −2.90482e30 3.50160e28i −0.999927 0.0120536i
\(589\) 7.66623e29 0.259008
\(590\) 0 0
\(591\) 0 0
\(592\) −7.42891e29 1.28672e30i −0.237348 0.411099i
\(593\) 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
\(597\) −3.11568e30 + 5.39652e30i −0.907476 + 1.57180i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 5.26756e30 1.42557 0.712787 0.701380i \(-0.247432\pi\)
0.712787 + 0.701380i \(0.247432\pi\)
\(602\) 0 0
\(603\) 6.71962e30 1.75329
\(604\) −3.50584e30 + 6.07230e30i −0.898227 + 1.55578i
\(605\) 0 0
\(606\) 0 0
\(607\) 2.81999e30 4.88437e30i 0.684184 1.18504i −0.289509 0.957175i \(-0.593492\pi\)
0.973693 0.227865i \(-0.0731746\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.08617e30 1.88131e30i −0.236502 0.409633i 0.723206 0.690632i \(-0.242668\pi\)
−0.959708 + 0.280999i \(0.909334\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 3.88698e30 + 6.73244e30i 0.760352 + 1.31697i 0.942669 + 0.333729i \(0.108307\pi\)
−0.182317 + 0.983240i \(0.558360\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −6.52673e30 −1.16860
\(625\) −2.84217e30 + 4.92278e30i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 5.32961e30 9.23116e30i 0.889487 1.54064i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.08052e30 0.171124 0.0855620 0.996333i \(-0.472731\pi\)
0.0855620 + 0.996333i \(0.472731\pi\)
\(632\) 0 0
\(633\) −6.30935e30 1.09281e31i −0.965037 1.67149i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00843e30 7.14020e30i −0.572058 1.01901i
\(638\) 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
\(643\) 1.54659e31 1.99093 0.995464 0.0951395i \(-0.0303297\pi\)
0.995464 + 0.0951395i \(0.0303297\pi\)
\(644\) 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
\(651\) −6.42652e30 1.12876e31i −0.722079 1.26827i
\(652\) −1.35194e31 −1.49359
\(653\) 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
\(657\) 1.94992e31 1.98061
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −9.27164e30 1.60590e31i −0.880928 1.52581i −0.850311 0.526281i \(-0.823586\pi\)
−0.0306177 0.999531i \(-0.509747\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 4.24076e29 + 7.34521e29i 0.0352985 + 0.0611388i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −4.79437e30 −0.373737 −0.186868 0.982385i \(-0.559834\pi\)
−0.186868 + 0.982385i \(0.559834\pi\)
\(674\) 0 0
\(675\) 6.62692e30 + 1.14782e31i 0.500000 + 0.866025i
\(676\) −2.46273e30 4.26557e30i −0.182811 0.316638i
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) −1.61721e30 + 2.76251e30i −0.114340 + 0.195315i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) −1.36057e30 + 2.35658e30i −0.0887368 + 0.153697i
\(685\) 0 0
\(686\) 0 0
\(687\) 1.31915e31 0.819915
\(688\) −1.11579e31 + 1.93260e31i −0.682509 + 1.18214i
\(689\) 0 0
\(690\) 0 0
\(691\) 5.14109e30 8.90464e30i 0.299776 0.519228i −0.676308 0.736619i \(-0.736421\pi\)
0.976085 + 0.217391i \(0.0697547\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 9.78325e30 + 1.71834e31i 0.494771 + 0.869023i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −8.73028e29 1.51213e30i −0.0421230 0.0729592i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.27215e31 + 3.93547e31i −0.998453 + 1.72937i −0.451077 + 0.892485i \(0.648960\pi\)
−0.547376 + 0.836887i \(0.684373\pi\)
\(710\) 0 0
\(711\) 1.62802e31 + 2.81982e31i 0.693577 + 1.20131i
\(712\) 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
\(721\) −2.91641e30 5.12241e30i −0.106552 0.187149i
\(722\) 0 0
\(723\) 1.82714e31 3.16470e31i 0.647517 1.12153i
\(724\) 2.52354e31 + 4.37090e31i 0.880817 + 1.52562i
\(725\) 0 0
\(726\) 0 0
\(727\) −5.98866e31 −1.99734 −0.998668 0.0516010i \(-0.983568\pi\)
−0.998668 + 0.0516010i \(0.983568\pi\)
\(728\) 0 0
\(729\) 3.09032e31 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −2.50374e31 4.33660e31i −0.774403 1.34131i
\(733\) −3.14541e31 + 5.44802e31i −0.958372 + 1.65995i −0.231917 + 0.972736i \(0.574500\pi\)
−0.726455 + 0.687214i \(0.758834\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.36695e30 1.44920e31i −0.233066 0.403682i 0.725643 0.688071i \(-0.241542\pi\)
−0.958709 + 0.284390i \(0.908209\pi\)
\(740\) 0 0
\(741\) −7.67007e30 −0.207395
\(742\) 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
\(751\) 4.14538e31 7.18001e31i 0.967219 1.67527i 0.263689 0.964608i \(-0.415061\pi\)
0.703531 0.710665i \(-0.251606\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 4.61033e31 + 2.77865e29i 0.999982 + 0.00602689i
\(757\) 1.57356e31 0.336379 0.168189 0.985755i \(-0.446208\pi\)
0.168189 + 0.985755i \(0.446208\pi\)
\(758\) 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
\(763\) −4.95969e31 2.98921e29i −0.972041 0.00585850i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 2.74122e31 4.74793e31i 0.500000 0.866025i
\(769\) −2.96591e31 −0.533296 −0.266648 0.963794i \(-0.585916\pi\)
−0.266648 + 0.963794i \(0.585916\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.06776e31 8.77761e31i 0.873023 1.51212i
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0 0
\(775\) −4.42047e31 + 7.65649e31i −0.729710 + 1.26389i
\(776\) 0 0
\(777\) −1.49458e31 + 2.55303e31i −0.239822 + 0.409661i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 6.87773e31 + 8.29072e29i 0.999927 + 0.0120536i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.50680e31 2.60985e31i −0.210055 0.363826i 0.741677 0.670758i \(-0.234031\pi\)
−0.951731 + 0.306932i \(0.900698\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.05728e31 1.22236e32i 0.904966 1.56745i
\(794\) 0 0
\(795\) 0 0
\(796\) 7.37699e31 1.27773e32i 0.907476 1.57180i
\(797\) 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
\(804\) −1.59100e32 −1.75329
\(805\) 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
\(811\) −1.24855e32 −1.25076 −0.625381 0.780319i \(-0.715057\pi\)
−0.625381 + 0.780319i \(0.715057\pi\)
\(812\) 0 0
\(813\) −1.90580e32 −1.85815
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.31125e31 + 2.27115e31i −0.121127 + 0.209799i
\(818\) 0 0
\(819\) 6.42973e31 + 1.12933e32i 0.578189 + 1.01554i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 1.15540e32 + 2.00120e32i 0.984765 + 1.70566i 0.642977 + 0.765885i \(0.277699\pi\)
0.341787 + 0.939777i \(0.388968\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.13840e32 1.97177e32i −0.895767 1.55151i −0.832852 0.553496i \(-0.813293\pi\)
−0.0629156 0.998019i \(-0.520040\pi\)
\(830\) 0 0
\(831\) −4.45038e31 + 7.70828e31i −0.341025 + 0.590672i
\(832\) 1.54533e32 1.16860
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.03069e32 + 1.78522e32i 0.729710 + 1.26389i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.48852e32 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 1.49386e32 + 2.58745e32i 0.965037 + 1.67149i
\(845\) 0 0
\(846\) 0 0
\(847\) 7.96383e31 + 1.39878e32i 0.494771 + 0.869023i
\(848\) 0 0
\(849\) 1.67434e31 2.90004e31i 0.101358 0.175557i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.51656e29 −0.00489581 −0.00244791 0.999997i \(-0.500779\pi\)
−0.00244791 + 0.999997i \(0.500779\pi\)
\(854\) 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
\(859\) −1.78597e32 + 3.09338e32i −0.950492 + 1.64630i −0.206128 + 0.978525i \(0.566086\pi\)
−0.744363 + 0.667775i \(0.767247\pi\)
\(860\) 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
\(867\) −2.08070e32 −1.00000
\(868\) 1.52160e32 + 2.67257e32i 0.722079 + 1.26827i
\(869\) 0 0
\(870\) 0 0
\(871\) −2.24228e32 3.88374e32i −1.02445 1.77439i
\(872\) 0 0
\(873\) 2.54012e31 4.39962e31i 0.113161 0.196001i
\(874\) 0 0
\(875\) 0 0
\(876\) −4.61682e32 −1.98061
\(877\) 1.49418e32 2.58800e32i 0.633007 1.09640i −0.353926 0.935273i \(-0.615154\pi\)
0.986934 0.161128i \(-0.0515130\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.75662e32 −0.690414 −0.345207 0.938527i \(-0.612191\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(884\) 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
\(889\) 2.76269e32 4.71921e32i 1.00789 1.72167i
\(890\) 0 0
\(891\) 0 0
\(892\) −1.00408e31 1.73912e31i −0.0352985 0.0611388i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.56905e32 2.71768e32i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) 4.44320e32 + 2.67792e30i 1.36499 + 0.00822682i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.57085e31 + 1.48451e32i 0.250809 + 0.434414i 0.963749 0.266811i \(-0.0859700\pi\)
−0.712940 + 0.701225i \(0.752637\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 3.22142e31 5.57966e31i 0.0887368 0.153697i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −3.12334e32 −0.819915
\(917\) 0 0
\(918\) 0 0
\(919\) 3.69170e32 6.39422e32i 0.934881 1.61926i 0.160035 0.987111i \(-0.448839\pi\)
0.774846 0.632150i \(-0.217827\pi\)
\(920\) 0 0
\(921\) 1.87826e32 + 3.25324e32i 0.464408 + 0.804378i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.01361e32 0.474697
\(926\) 0 0
\(927\) 4.67738e31 + 8.10145e31i 0.107678 + 0.186503i
\(928\) 0 0
\(929\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 8.08255e31 + 9.74307e29i 0.177461 + 0.00213919i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.65292e32 −1.36106 −0.680529 0.732721i \(-0.738250\pi\)
−0.680529 + 0.732721i \(0.738250\pi\)
\(938\) 0 0
\(939\) 9.95722e32 1.98983
\(940\) 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 \(0.666667\pi\)
\(948\) −3.85466e32 6.67647e32i −0.693577 1.20131i
\(949\) −6.50671e32 1.12700e33i −1.15727 2.00444i
\(950\) 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
\(961\) −3.64728e32 + 6.31728e32i −0.564952 + 0.978526i
\(962\) 0 0
\(963\) 0 0
\(964\) −4.32612e32 + 7.49306e32i −0.647517 + 1.12153i
\(965\) 0 0
\(966\) 0 0
\(967\) −5.33247e32 −0.771324 −0.385662 0.922640i \(-0.626027\pi\)
−0.385662 + 0.922640i \(0.626027\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) −7.31693e32 −1.00000
\(973\) −7.02953e32 + 1.20078e33i −0.949916 + 1.62264i
\(974\) 0 0
\(975\) 4.42269e32 7.66032e32i 0.584299 1.01204i
\(976\) 5.92809e32 + 1.02678e33i 0.774403 + 1.34131i
\(977\) 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
\(981\) 7.87139e32 0.972059
\(982\) 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
\(988\) 1.81604e32 0.207395
\(989\) 0 0
\(990\) 0 0
\(991\) 8.72952e32 + 1.51200e33i 0.964229 + 1.67009i 0.711673 + 0.702511i \(0.247938\pi\)
0.252556 + 0.967582i \(0.418729\pi\)
\(992\) 0 0
\(993\) 1.52301e33 1.64536
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 7.45970e32 + 1.29206e33i 0.771036 + 1.33547i 0.936995 + 0.349342i \(0.113595\pi\)
−0.165959 + 0.986133i \(0.553072\pi\)
\(998\) 0 0
\(999\) 2.34750e32 4.06600e32i 0.237348 0.411099i
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 21.23.h.a.2.1 2
3.2 odd 2 CM 21.23.h.a.2.1 2
7.4 even 3 inner 21.23.h.a.11.1 yes 2
21.11 odd 6 inner 21.23.h.a.11.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.23.h.a.2.1 2 1.1 even 1 trivial
21.23.h.a.2.1 2 3.2 odd 2 CM
21.23.h.a.11.1 yes 2 7.4 even 3 inner
21.23.h.a.11.1 yes 2 21.11 odd 6 inner