Properties

Label 21.18.g.a.5.1
Level $21$
Weight $18$
Character 21.5
Analytic conductor $38.477$
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] = [21,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("21.5");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 18 \)
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: \(38.4766383424\)
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 21.18.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)
 
\(f(q)\) \(=\) \(q+(9841.50 - 5681.99i) q^{3} +(-65536.0 - 113512. i) q^{4} +(-1.37909e6 + 1.51898e7i) q^{7} +(6.45701e7 - 1.11839e8i) q^{9} +O(q^{10})\) \(q+(9841.50 - 5681.99i) q^{3} +(-65536.0 - 113512. i) q^{4} +(-1.37909e6 + 1.51898e7i) q^{7} +(6.45701e7 - 1.11839e8i) q^{9} +(-1.28995e9 - 7.44750e8i) q^{12} -4.89177e9i q^{13} +(-8.58993e9 + 1.48782e10i) q^{16} +(-1.26967e11 - 7.33047e10i) q^{19} +(7.27358e10 + 1.57326e11i) q^{21} +(3.81470e11 + 6.60725e11i) q^{25} -1.46755e12i q^{27} +(1.81459e12 - 8.38933e11i) q^{28} +(-4.37326e12 + 2.52490e12i) q^{31} -1.69267e13 q^{36} +(-2.10510e13 + 3.64613e13i) q^{37} +(-2.77950e13 - 4.81424e13i) q^{39} -5.48977e13 q^{43} +1.95232e14i q^{48} +(-2.28827e14 - 4.18961e13i) q^{49} +(-5.55273e14 + 3.20587e14i) q^{52} -1.66607e15 q^{57} +(-2.59110e15 - 1.49597e15i) q^{61} +(1.60975e15 + 1.13504e15i) q^{63} +2.25180e15 q^{64} +(1.34670e14 + 2.33255e14i) q^{67} +(7.49410e15 - 4.32672e15i) q^{73} +(7.50847e15 + 4.33502e15i) q^{75} +1.92164e16i q^{76} +(1.32336e16 - 2.29213e16i) q^{79} +(-8.33859e15 - 1.44429e16i) q^{81} +(1.30915e16 - 1.85669e16i) q^{84} +(7.43048e16 + 6.74619e15i) q^{91} +(-2.86930e16 + 4.96977e16i) q^{93} +9.91902e16i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 19683 q^{3} - 131072 q^{4} - 2758181 q^{7} + 129140163 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 19683 q^{3} - 131072 q^{4} - 2758181 q^{7} + 129140163 q^{9} - 2579890176 q^{12} - 17179869184 q^{16} - 253934979759 q^{19} + 145471502394 q^{21} + 762939453125 q^{25} + 3629189169152 q^{28} - 8746519097955 q^{31} - 33853318889472 q^{36} - 42101902631989 q^{37} - 55590010927767 q^{39} - 109795409397670 q^{43} - 457653465545653 q^{49} - 11\!\cdots\!84 q^{52}+ \cdots - 57\!\cdots\!55 q^{93}+O(q^{100}) \) 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 \(0.666667\pi\)
\(3\) 9841.50 5681.99i 0.866025 0.500000i
\(4\) −65536.0 113512.i −0.500000 0.866025i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −1.37909e6 + 1.51898e7i −0.0904189 + 0.995904i
\(8\) 0 0
\(9\) 6.45701e7 1.11839e8i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −1.28995e9 7.44750e8i −0.866025 0.500000i
\(13\) 4.89177e9i 1.66321i −0.555366 0.831606i \(-0.687422\pi\)
0.555366 0.831606i \(-0.312578\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −8.58993e9 + 1.48782e10i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −1.26967e11 7.33047e10i −1.71509 0.990208i −0.927346 0.374205i \(-0.877916\pi\)
−0.787744 0.616003i \(-0.788751\pi\)
\(20\) 0 0
\(21\) 7.27358e10 + 1.57326e11i 0.419647 + 0.907687i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 3.81470e11 + 6.60725e11i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 1.46755e12i 1.00000i
\(28\) 1.81459e12 8.38933e11i 0.907687 0.419647i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.37326e12 + 2.52490e12i −0.920939 + 0.531704i −0.883935 0.467611i \(-0.845115\pi\)
−0.0370045 + 0.999315i \(0.511782\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.69267e13 −1.00000
\(37\) −2.10510e13 + 3.64613e13i −0.985274 + 1.70654i −0.344562 + 0.938764i \(0.611973\pi\)
−0.640712 + 0.767781i \(0.721361\pi\)
\(38\) 0 0
\(39\) −2.77950e13 4.81424e13i −0.831606 1.44038i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −5.48977e13 −0.716263 −0.358131 0.933671i \(-0.616586\pi\)
−0.358131 + 0.933671i \(0.616586\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 1.95232e14i 1.00000i
\(49\) −2.28827e14 4.18961e13i −0.983649 0.180097i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.55273e14 + 3.20587e14i −1.44038 + 0.831606i
\(53\) 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
\(57\) −1.66607e15 −1.98042
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −2.59110e15 1.49597e15i −1.73053 0.999124i −0.886191 0.463320i \(-0.846658\pi\)
−0.844343 0.535804i \(-0.820009\pi\)
\(62\) 0 0
\(63\) 1.60975e15 + 1.13504e15i 0.817269 + 0.576257i
\(64\) 2.25180e15 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.34670e14 + 2.33255e14i 0.0405167 + 0.0701769i 0.885573 0.464501i \(-0.153766\pi\)
−0.845056 + 0.534678i \(0.820433\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 7.49410e15 4.32672e15i 1.08762 0.627935i 0.154675 0.987965i \(-0.450567\pi\)
0.932941 + 0.360030i \(0.117234\pi\)
\(74\) 0 0
\(75\) 7.50847e15 + 4.33502e15i 0.866025 + 0.500000i
\(76\) 1.92164e16i 1.98042i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.32336e16 2.29213e16i 0.981405 1.69984i 0.324470 0.945896i \(-0.394814\pi\)
0.656935 0.753948i \(-0.271853\pi\)
\(80\) 0 0
\(81\) −8.33859e15 1.44429e16i −0.500000 0.866025i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 1.30915e16 1.85669e16i 0.576257 0.817269i
\(85\) 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
\(91\) 7.43048e16 + 6.74619e15i 1.65640 + 0.150386i
\(92\) 0 0
\(93\) −2.86930e16 + 4.96977e16i −0.531704 + 0.920939i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.91902e16i 1.28502i 0.766278 + 0.642509i \(0.222106\pi\)
−0.766278 + 0.642509i \(0.777894\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000e16 8.66025e16i 0.500000 0.866025i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −2.12892e17 1.22913e17i −1.65593 0.956053i −0.974563 0.224112i \(-0.928052\pi\)
−0.681369 0.731940i \(-0.738615\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −1.66584e17 + 9.61772e16i −0.866025 + 0.500000i
\(109\) −9.30230e16 1.61121e17i −0.447162 0.774508i 0.551038 0.834480i \(-0.314232\pi\)
−0.998200 + 0.0599724i \(0.980899\pi\)
\(110\) 0 0
\(111\) 4.78445e17i 1.97055i
\(112\) −2.14150e17 1.50997e17i −0.817269 0.576257i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.47089e17 3.15862e17i −1.44038 0.831606i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.52724e17 + 4.37730e17i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0 0
\(124\) 5.73212e17 + 3.30944e17i 0.920939 + 0.531704i
\(125\) 0 0
\(126\) 0 0
\(127\) 1.48484e18 1.94692 0.973460 0.228856i \(-0.0734984\pi\)
0.973460 + 0.228856i \(0.0734984\pi\)
\(128\) 0 0
\(129\) −5.40276e17 + 3.11928e17i −0.620302 + 0.358131i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 1.28858e18 1.82751e18i 1.14123 1.61853i
\(134\) 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
\(139\) 1.31451e18i 0.800091i −0.916495 0.400046i \(-0.868994\pi\)
0.916495 0.400046i \(-0.131006\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.10931e18 + 1.92137e18i 0.500000 + 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −2.49005e18 + 8.87871e17i −0.941913 + 0.335856i
\(148\) 5.51838e18 1.97055
\(149\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) −1.17729e18 2.03913e18i −0.354471 0.613962i 0.632556 0.774514i \(-0.282006\pi\)
−0.987027 + 0.160553i \(0.948672\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −3.64315e18 + 6.31012e18i −0.831606 + 1.44038i
\(157\) 6.36649e18 3.67569e18i 1.37643 0.794680i 0.384698 0.923042i \(-0.374306\pi\)
0.991727 + 0.128363i \(0.0409722\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.84535e18 8.39239e18i 0.761606 1.31914i −0.180416 0.983590i \(-0.557744\pi\)
0.942022 0.335551i \(-0.108922\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1.52790e19 −1.76627
\(170\) 0 0
\(171\) −1.63966e19 + 9.46658e18i −1.71509 + 0.990208i
\(172\) 3.59778e18 + 6.23153e18i 0.358131 + 0.620302i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) −1.05623e19 + 4.88323e18i −0.907687 + 0.419647i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 7.31707e18i 0.472138i −0.971736 0.236069i \(-0.924141\pi\)
0.971736 0.236069i \(-0.0758592\pi\)
\(182\) 0 0
\(183\) −3.40004e19 −1.99825
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.22917e19 + 2.02388e18i 0.995904 + 0.0904189i
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 2.21611e19 1.27947e19i 0.866025 0.500000i
\(193\) −4.72356e18 8.18145e18i −0.176617 0.305910i 0.764103 0.645095i \(-0.223182\pi\)
−0.940720 + 0.339185i \(0.889849\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.02407e19 + 2.87202e19i 0.335856 + 0.941913i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 5.20953e19 3.00772e19i 1.50158 0.866936i 0.501579 0.865112i \(-0.332753\pi\)
0.999998 0.00182370i \(-0.000580503\pi\)
\(200\) 0 0
\(201\) 2.65070e18 + 1.53038e18i 0.0701769 + 0.0405167i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 7.27808e19 + 4.20200e19i 1.44038 + 0.831606i
\(209\) 0 0
\(210\) 0 0
\(211\) −5.37704e19 −0.942199 −0.471099 0.882080i \(-0.656143\pi\)
−0.471099 + 0.882080i \(0.656143\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3.23215e19 6.99108e19i −0.446256 0.965243i
\(218\) 0 0
\(219\) 4.91688e19 8.51629e19i 0.627935 1.08762i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.36793e19i 0.478282i 0.970985 + 0.239141i \(0.0768659\pi\)
−0.970985 + 0.239141i \(0.923134\pi\)
\(224\) 0 0
\(225\) 9.85261e19 1.00000
\(226\) 0 0
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 1.09187e20 + 1.89118e20i 0.990208 + 1.71509i
\(229\) −1.34333e20 7.75570e19i −1.17376 0.677672i −0.219198 0.975680i \(-0.570344\pi\)
−0.954563 + 0.298009i \(0.903678\pi\)
\(230\) 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
\(237\) 3.00773e20i 1.96281i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −3.00454e19 + 1.73467e19i −0.170072 + 0.0981913i −0.582620 0.812745i \(-0.697972\pi\)
0.412548 + 0.910936i \(0.364639\pi\)
\(242\) 0 0
\(243\) −1.64128e20 9.47596e19i −0.866025 0.500000i
\(244\) 3.92160e20i 1.99825i
\(245\) 0 0
\(246\) 0 0
\(247\) −3.58590e20 + 6.21096e20i −1.64693 + 2.85256i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 2.33434e19 2.57112e20i 0.0904189 0.995904i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −1.47574e20 2.55606e20i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −5.24807e20 3.70042e20i −1.61047 1.13554i
\(260\) 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
\(268\) 1.76514e19 3.05731e19i 0.0405167 0.0701769i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 5.95646e18 + 3.43897e18i 0.0124380 + 0.00718106i 0.506206 0.862413i \(-0.331048\pi\)
−0.493768 + 0.869594i \(0.664381\pi\)
\(272\) 0 0
\(273\) 7.69602e20 3.55807e20i 1.50968 0.697962i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.76007e20 + 9.97673e20i 0.998504 + 1.72946i 0.546612 + 0.837386i \(0.315917\pi\)
0.451891 + 0.892073i \(0.350750\pi\)
\(278\) 0 0
\(279\) 6.52133e20i 1.06341i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.09395e21 6.31593e20i 1.58057 0.912541i 0.585791 0.810462i \(-0.300784\pi\)
0.994776 0.102079i \(-0.0325493\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 4.13620e20 7.16411e20i 0.500000 0.866025i
\(290\) 0 0
\(291\) 5.63598e20 + 9.76180e20i 0.642509 + 1.11286i
\(292\) −9.82267e20 5.67112e20i −1.08762 0.627935i
\(293\) 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
\(300\) 1.13640e21i 1.00000i
\(301\) 7.57089e19 8.33883e20i 0.0647637 0.713329i
\(302\) 0 0
\(303\) 0 0
\(304\) 2.18128e21 1.25937e21i 1.71509 0.990208i
\(305\) 0 0
\(306\) 0 0
\(307\) 2.06009e21i 1.49008i 0.667019 + 0.745041i \(0.267570\pi\)
−0.667019 + 0.745041i \(0.732430\pi\)
\(308\) 0 0
\(309\) −2.79357e21 −1.91211
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 1.67683e21 + 9.68119e20i 1.02887 + 0.594021i 0.916662 0.399663i \(-0.130873\pi\)
0.112213 + 0.993684i \(0.464206\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −3.46911e21 −1.96281
\(317\) 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
\(324\) −1.09296e21 + 1.89305e21i −0.500000 + 0.866025i
\(325\) 3.23211e21 1.86606e21i 1.44038 0.831606i
\(326\) 0 0
\(327\) −1.83097e21 1.05711e21i −0.774508 0.447162i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.18080e21 + 2.04521e21i −0.450443 + 0.780190i −0.998413 0.0563079i \(-0.982067\pi\)
0.547971 + 0.836497i \(0.315400\pi\)
\(332\) 0 0
\(333\) 2.71852e21 + 4.70862e21i 0.985274 + 1.70654i
\(334\) 0 0
\(335\) 0 0
\(336\) −2.96552e21 2.69242e20i −0.995904 0.0904189i
\(337\) −4.46257e21 −1.46127 −0.730636 0.682767i \(-0.760777\pi\)
−0.730636 + 0.682767i \(0.760777\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 9.51964e20 3.41804e21i 0.268300 0.963335i
\(344\) 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
\(349\) 7.97556e21i 1.93975i −0.243608 0.969874i \(-0.578331\pi\)
0.243608 0.969874i \(-0.421669\pi\)
\(350\) 0 0
\(351\) −7.17890e21 −1.66321
\(352\) 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
\(361\) 8.00697e21 + 1.38685e22i 1.46102 + 2.53057i
\(362\) 0 0
\(363\) 5.74389e21i 1.00000i
\(364\) −4.10387e21 8.87658e21i −0.697962 1.50968i
\(365\) 0 0
\(366\) 0 0
\(367\) 8.13126e21 4.69459e21i 1.28972 0.744622i 0.311118 0.950371i \(-0.399296\pi\)
0.978605 + 0.205749i \(0.0659631\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 7.52169e21 1.06341
\(373\) −3.13320e21 + 5.42687e21i −0.432976 + 0.749936i −0.997128 0.0757347i \(-0.975870\pi\)
0.564152 + 0.825671i \(0.309203\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.03036e22 1.24324 0.621620 0.783319i \(-0.286475\pi\)
0.621620 + 0.783319i \(0.286475\pi\)
\(380\) 0 0
\(381\) 1.46131e22 8.43685e21i 1.68608 0.973460i
\(382\) 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
\(387\) −3.54475e21 + 6.13969e21i −0.358131 + 0.620302i
\(388\) 1.12592e22 6.50053e21i 1.11286 0.642509i
\(389\) 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
\(397\) −1.73248e22 1.00025e22i −1.40913 0.813559i −0.413821 0.910358i \(-0.635806\pi\)
−0.995304 + 0.0967994i \(0.969139\pi\)
\(398\) 0 0
\(399\) 2.29766e21 2.53072e22i 0.179067 1.97230i
\(400\) −1.31072e22 −1.00000
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 1.23512e22 + 2.13930e22i 0.884337 + 1.53172i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2.69628e22 + 1.55670e22i −1.70262 + 0.983006i −0.759522 + 0.650481i \(0.774567\pi\)
−0.943094 + 0.332525i \(0.892099\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 3.22209e22i 1.91211i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.46906e21 1.29368e22i −0.400046 0.692899i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −3.83194e22 −1.89244 −0.946218 0.323529i \(-0.895131\pi\)
−0.946218 + 0.323529i \(0.895131\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.62968e22 3.72950e22i 1.15150 1.63311i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(432\) 2.18345e22 + 1.26061e22i 0.866025 + 0.500000i
\(433\) 1.48866e22i 0.578961i 0.957184 + 0.289480i \(0.0934825\pi\)
−0.957184 + 0.289480i \(0.906518\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.21927e22 + 2.11184e22i −0.447162 + 0.774508i
\(437\) 0 0
\(438\) 0 0
\(439\) −4.96629e22 2.86729e22i −1.71824 0.992026i −0.922137 0.386862i \(-0.873559\pi\)
−0.796101 0.605163i \(-0.793108\pi\)
\(440\) 0 0
\(441\) −1.94610e22 + 2.28864e22i −0.647793 + 0.761816i
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 5.43091e22 3.13554e22i 1.70654 0.985274i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −3.10544e21 + 3.42043e22i −0.0904189 + 0.995904i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.31727e22 1.33787e22i −0.613962 0.354471i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.10055e21 1.22985e22i 0.174584 0.302388i −0.765433 0.643515i \(-0.777475\pi\)
0.940017 + 0.341127i \(0.110809\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −1.45502e22 −0.320207 −0.160104 0.987100i \(-0.551183\pi\)
−0.160104 + 0.987100i \(0.551183\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 8.28013e22i 1.66321i
\(469\) −3.72880e21 + 1.72392e21i −0.0735529 + 0.0340054i
\(470\) 0 0
\(471\) 4.17705e22 7.23486e22i 0.794680 1.37643i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.11854e23i 1.98042i
\(476\) 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
\(481\) 1.78360e23 + 1.02976e23i 2.83835 + 1.63872i
\(482\) 0 0
\(483\) 0 0
\(484\) 6.62500e22 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 5.06702e22 + 8.77633e22i 0.725698 + 1.25695i 0.958686 + 0.284467i \(0.0918166\pi\)
−0.232987 + 0.972480i \(0.574850\pi\)
\(488\) 0 0
\(489\) 1.10125e23i 1.52321i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.67550e22i 1.06341i
\(497\) 0 0
\(498\) 0 0
\(499\) 3.46243e21 5.99710e21i 0.0403205 0.0698372i −0.845161 0.534512i \(-0.820495\pi\)
0.885481 + 0.464675i \(0.153829\pi\)
\(500\) 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
\(507\) −1.50368e23 + 8.68152e22i −1.52964 + 0.883137i
\(508\) −9.73105e22 1.68547e23i −0.973460 1.68608i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 5.53868e22 + 1.19801e23i 0.527022 + 1.13994i
\(512\) 0 0
\(513\) −1.07578e23 + 1.86331e23i −0.990208 + 1.71509i
\(514\) 0 0
\(515\) 0 0
\(516\) 7.08150e22 + 4.08851e22i 0.620302 + 0.358131i
\(517\) 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
\(523\) −1.38798e23 8.01353e22i −1.08423 0.625978i −0.152193 0.988351i \(-0.548634\pi\)
−0.932033 + 0.362372i \(0.881967\pi\)
\(524\) 0 0
\(525\) −7.62027e22 + 1.08073e23i −0.576257 + 0.817269i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.05250e22 1.22153e23i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −2.91892e23 2.65011e22i −1.97230 0.179067i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.53914e23 2.66586e23i 0.901780 1.56193i 0.0765970 0.997062i \(-0.475595\pi\)
0.825183 0.564866i \(-0.191072\pi\)
\(542\) 0 0
\(543\) −4.15755e22 7.20109e22i −0.236069 0.408884i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.70109e23 1.97441 0.987205 0.159454i \(-0.0509733\pi\)
0.987205 + 0.159454i \(0.0509733\pi\)
\(548\) 0 0
\(549\) −3.34615e23 + 1.93190e23i −1.73053 + 0.999124i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 3.29919e23 + 2.32626e23i 1.60414 + 1.13108i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.49213e23 + 8.61480e22i −0.692899 + 0.400046i
\(557\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(558\) 0 0
\(559\) 2.68547e23i 1.19130i
\(560\) 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
\(567\) 2.30883e23 1.06743e23i 0.907687 0.419647i
\(568\) 0 0
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −1.45147e23 2.51403e23i −0.537530 0.931029i −0.999036 0.0438919i \(-0.986024\pi\)
0.461507 0.887137i \(-0.347309\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.45399e23 2.51838e23i 0.500000 0.866025i
\(577\) −1.32448e23 + 7.64688e22i −0.448797 + 0.259113i −0.707322 0.706891i \(-0.750097\pi\)
0.258525 + 0.966005i \(0.416764\pi\)
\(578\) 0 0
\(579\) −9.29739e22 5.36785e22i −0.305910 0.176617i
\(580\) 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 \(0.5\pi\)
\(588\) 2.63972e23 + 2.24462e23i 0.761816 + 0.647793i
\(589\) 7.40349e23 2.10599
\(590\) 0 0
\(591\) 0 0
\(592\) −3.61653e23 6.26401e23i −0.985274 1.70654i
\(593\) 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
\(597\) 3.41797e23 5.92010e23i 0.866936 1.50158i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) 0 0
\(601\) 4.71692e23i 1.13038i 0.824960 + 0.565192i \(0.191198\pi\)
−0.824960 + 0.565192i \(0.808802\pi\)
\(602\) 0 0
\(603\) 3.47825e22 0.0810333
\(604\) −1.54310e23 + 2.67273e23i −0.354471 + 0.613962i
\(605\) 0 0
\(606\) 0 0
\(607\) 7.66573e23 + 4.42581e23i 1.68830 + 0.974741i 0.955820 + 0.293954i \(0.0949711\pi\)
0.732481 + 0.680787i \(0.238362\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.53080e23 2.65142e23i −0.310101 0.537111i 0.668283 0.743907i \(-0.267030\pi\)
−0.978384 + 0.206796i \(0.933696\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.55058e23 + 8.95225e22i −0.289149 + 0.166940i −0.637558 0.770402i \(-0.720055\pi\)
0.348409 + 0.937343i \(0.386722\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 9.55029e23 1.66321
\(625\) −2.91038e23 + 5.04093e23i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) −8.34468e23 4.81780e23i −1.37643 0.794680i
\(629\) 0 0
\(630\) 0 0
\(631\) −8.21877e23 −1.30184 −0.650920 0.759146i \(-0.725617\pi\)
−0.650920 + 0.759146i \(0.725617\pi\)
\(632\) 0 0
\(633\) −5.29181e23 + 3.05523e23i −0.815968 + 0.471099i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.04946e23 + 1.11937e24i −0.299540 + 1.63602i
\(638\) 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
\(643\) 1.04553e24i 1.41106i −0.708681 0.705529i \(-0.750709\pi\)
0.708681 0.705529i \(-0.249291\pi\)
\(644\) 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
\(651\) −7.15325e23 5.04377e23i −0.869090 0.612797i
\(652\) −1.27018e24 −1.52321
\(653\) 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
\(657\) 1.11751e24i 1.25587i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.40530e24 + 8.11351e23i −1.49988 + 0.865957i −1.00000 0.000137079i \(-0.999956\pi\)
−0.499881 + 0.866094i \(0.666623\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 2.48185e23 + 4.29870e23i 0.239141 + 0.414205i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.75121e24 1.60402 0.802009 0.597312i \(-0.203765\pi\)
0.802009 + 0.597312i \(0.203765\pi\)
\(674\) 0 0
\(675\) 9.69645e23 5.59825e23i 0.866025 0.500000i
\(676\) 1.00132e24 + 1.73435e24i 0.883137 + 1.52964i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) −1.50667e24 1.36792e23i −1.27975 0.116190i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(684\) 2.14914e24 + 1.24080e24i 1.71509 + 0.990208i
\(685\) 0 0
\(686\) 0 0
\(687\) −1.76271e24 −1.35534
\(688\) 4.71568e23 8.16779e23i 0.358131 0.620302i
\(689\) 0 0
\(690\) 0 0
\(691\) −4.98451e22 2.87781e22i −0.0364804 0.0210619i 0.481649 0.876364i \(-0.340038\pi\)
−0.518129 + 0.855302i \(0.673371\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.24652e24 + 8.78920e23i 0.817269 + 0.576257i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 5.34557e24 3.08627e24i 3.37967 1.95125i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.38946e24 + 2.40661e24i −0.817244 + 1.41551i 0.0904608 + 0.995900i \(0.471166\pi\)
−0.907705 + 0.419609i \(0.862167\pi\)
\(710\) 0 0
\(711\) −1.70899e24 2.96006e24i −0.981405 1.69984i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 2.16062e24 3.06427e24i 1.10186 1.56270i
\(722\) 0 0
\(723\) −1.97128e23 + 3.41436e23i −0.0981913 + 0.170072i
\(724\) −8.30573e23 + 4.79531e23i −0.408884 + 0.236069i
\(725\) 0 0
\(726\) 0 0
\(727\) 3.62788e24i 1.72429i −0.506661 0.862145i \(-0.669120\pi\)
0.506661 0.862145i \(-0.330880\pi\)
\(728\) 0 0
\(729\) −2.15369e24 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 2.22825e24 + 3.85944e24i 0.999124 + 1.73053i
\(733\) −2.31196e24 1.33481e24i −1.02470 0.591610i −0.109237 0.994016i \(-0.534841\pi\)
−0.915461 + 0.402406i \(0.868174\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.87788e24 + 3.25258e24i 0.776587 + 1.34509i 0.933898 + 0.357539i \(0.116384\pi\)
−0.157312 + 0.987549i \(0.550283\pi\)
\(740\) 0 0
\(741\) 8.15002e24i 3.29385i
\(742\) 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
\(751\) 2.69991e24 4.67639e24i 0.973667 1.68644i 0.289401 0.957208i \(-0.406544\pi\)
0.684266 0.729233i \(-0.260123\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −1.23117e24 2.66300e24i −0.419647 0.907687i
\(757\) −5.17299e24 −1.74352 −0.871760 0.489933i \(-0.837021\pi\)
−0.871760 + 0.489933i \(0.837021\pi\)
\(758\) 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
\(763\) 2.57567e24 1.19080e24i 0.811767 0.375301i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −2.90470e24 1.67703e24i −0.866025 0.500000i
\(769\) 6.15382e24i 1.81456i 0.420530 + 0.907279i \(0.361844\pi\)
−0.420530 + 0.907279i \(0.638156\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.19127e23 + 1.07236e24i −0.176617 + 0.305910i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) −3.33653e24 1.92635e24i −0.920939 0.531704i
\(776\) 0 0
\(777\) −7.26747e24 6.59820e23i −1.96248 0.178175i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.58895e24 3.04465e24i 0.647793 0.761816i
\(785\) 0 0
\(786\) 0 0
\(787\) 6.20358e24 3.58164e24i 1.50265 0.867555i 0.502653 0.864488i \(-0.332357\pi\)
0.999995 0.00306646i \(-0.000976087\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.31794e24 + 1.26751e25i −1.66175 + 2.87824i
\(794\) 0 0
\(795\) 0 0
\(796\) −6.82824e24 3.94229e24i −1.50158 0.866936i
\(797\) 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
\(804\) 4.01181e23i 0.0810333i
\(805\) 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
\(811\) 2.37365e23i 0.0445389i −0.999752 0.0222695i \(-0.992911\pi\)
0.999752 0.0222695i \(-0.00708918\pi\)
\(812\) 0 0
\(813\) 7.81607e22 0.0143621
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.97022e24 + 4.02426e24i 1.22845 + 0.709249i
\(818\) 0 0
\(819\) 5.55235e24 7.87455e24i 0.958438 1.35929i
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) 2.55075e24 + 4.41803e24i 0.422445 + 0.731696i 0.996178 0.0873465i \(-0.0278387\pi\)
−0.573733 + 0.819042i \(0.694505\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 5.34156e24 3.08395e24i 0.831678 0.480169i −0.0227491 0.999741i \(-0.507242\pi\)
0.854427 + 0.519572i \(0.173909\pi\)
\(830\) 0 0
\(831\) 1.13375e25 + 6.54573e24i 1.72946 + 0.998504i
\(832\) 1.10153e25i 1.66321i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.70541e24 + 6.41796e24i 0.531704 + 0.920939i
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.25715e24 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 3.52390e24 + 6.10357e24i 0.471099 + 0.815968i
\(845\) 0 0
\(846\) 0 0
\(847\) −6.30048e24 4.44248e24i −0.817269 0.576257i
\(848\) 0 0
\(849\) 7.17741e24 1.24316e25i 0.912541 1.58057i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.19884e24i 0.146452i 0.997315 + 0.0732258i \(0.0233294\pi\)
−0.997315 + 0.0732258i \(0.976671\pi\)
\(854\) 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
\(859\) −1.39361e25 8.04600e24i −1.60398 0.926058i −0.990680 0.136206i \(-0.956509\pi\)
−0.613298 0.789851i \(-0.710158\pi\)
\(860\) 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
\(867\) 9.40075e24i 1.00000i
\(868\) −5.81747e24 + 8.25055e24i −0.612797 + 0.869090i
\(869\) 0 0
\(870\) 0 0
\(871\) 1.14103e24 6.58773e23i 0.116719 0.0673878i
\(872\) 0 0
\(873\) 1.10933e25 + 6.40472e24i 1.11286 + 0.642509i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.28893e25 −1.25587
\(877\) −8.54701e24 + 1.48039e25i −0.824741 + 1.42849i 0.0773755 + 0.997002i \(0.475346\pi\)
−0.902117 + 0.431492i \(0.857987\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −4.38397e24 −0.399210 −0.199605 0.979876i \(-0.563966\pi\)
−0.199605 + 0.979876i \(0.563966\pi\)
\(884\) 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
\(889\) −2.04773e24 + 2.25544e25i −0.176039 + 1.93895i
\(890\) 0 0
\(891\) 0 0
\(892\) 4.95811e24 2.86257e24i 0.414205 0.239141i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −6.45701e24 1.11839e25i −0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) −3.99303e24 8.63683e24i −0.300577 0.650143i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.31611e25 2.27956e25i −0.954177 1.65268i −0.736241 0.676719i \(-0.763401\pi\)
−0.217936 0.975963i \(-0.569932\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.43114e25 2.47881e25i 0.990208 1.71509i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 2.03311e25i 1.35534i
\(917\) 0 0
\(918\) 0 0
\(919\) −4.26832e24 + 7.39294e24i −0.276742 + 0.479331i −0.970573 0.240807i \(-0.922588\pi\)
0.693831 + 0.720138i \(0.255921\pi\)
\(920\) 0 0
\(921\) 1.17054e25 + 2.02744e25i 0.745041 + 1.29045i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.21212e25 −1.97055
\(926\) 0 0
\(927\) −2.74929e25 + 1.58730e25i −1.65593 + 0.956053i
\(928\) 0 0
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) 2.59824e25 + 2.20935e25i 1.50871 + 1.28290i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.20078e25i 1.21001i −0.796220 0.605007i \(-0.793170\pi\)
0.796220 0.605007i \(-0.206830\pi\)
\(938\) 0 0
\(939\) 2.20034e25 1.18804
\(940\) 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 \(0.666667\pi\)
\(948\) −3.41413e25 + 1.97115e25i −1.69984 + 0.981405i
\(949\) −2.11653e25 3.66594e25i −1.04439 1.80893i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.47521e24 2.55513e24i 0.0654191 0.113309i
\(962\) 0 0
\(963\) 0 0
\(964\) 3.93812e24 + 2.27367e24i 0.170072 + 0.0981913i
\(965\) 0 0
\(966\) 0 0
\(967\) 4.66598e25 1.96254 0.981269 0.192645i \(-0.0617065\pi\)
0.981269 + 0.192645i \(0.0617065\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 2.48407e25i 1.00000i
\(973\) 1.99671e25 + 1.81283e24i 0.796814 + 0.0723434i
\(974\) 0 0
\(975\) 2.12059e25 3.67297e25i 0.831606 1.44038i
\(976\) 4.45147e25 2.57006e25i 1.73053 0.999124i
\(977\) 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
\(981\) −2.40260e25 −0.894325
\(982\) 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
\(988\) 9.40022e25 3.29385
\(989\) 0 0
\(990\) 0 0
\(991\) −2.92228e25 5.06154e25i −0.997920 1.72845i −0.554785 0.831994i \(-0.687200\pi\)
−0.443135 0.896455i \(-0.646134\pi\)
\(992\) 0 0
\(993\) 2.68373e25i 0.900885i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.37937e25 + 2.52843e25i −1.42070 + 0.820243i −0.996359 0.0852587i \(-0.972828\pi\)
−0.424343 + 0.905501i \(0.639495\pi\)
\(998\) 0 0
\(999\) 5.35087e25 + 3.08933e25i 1.70654 + 0.985274i
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 21.18.g.a.5.1 2
3.2 odd 2 CM 21.18.g.a.5.1 2
7.3 odd 6 inner 21.18.g.a.17.1 yes 2
21.17 even 6 inner 21.18.g.a.17.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.18.g.a.5.1 2 1.1 even 1 trivial
21.18.g.a.5.1 2 3.2 odd 2 CM
21.18.g.a.17.1 yes 2 7.3 odd 6 inner
21.18.g.a.17.1 yes 2 21.17 even 6 inner