Properties

Label 3024.2.t.d.1873.1
Level $3024$
Weight $2$
Character 3024.1873
Analytic conductor $24.147$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3024,2,Mod(289,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.t (of order \(3\), degree \(2\), not 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: \(24.1467615712\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1873.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1873
Dual form 3024.2.t.d.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +(-0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +(-0.500000 - 2.59808i) q^{7} +5.00000 q^{11} +(2.50000 - 4.33013i) q^{13} +(1.50000 - 2.59808i) q^{17} +(0.500000 + 0.866025i) q^{19} +3.00000 q^{23} -4.00000 q^{25} +(-0.500000 - 0.866025i) q^{29} +(-0.500000 - 2.59808i) q^{35} +(-1.50000 - 2.59808i) q^{37} +(-2.50000 + 4.33013i) q^{41} +(-0.500000 - 0.866025i) q^{43} +(-6.50000 + 2.59808i) q^{49} +(-4.50000 + 7.79423i) q^{53} +5.00000 q^{55} +(7.00000 - 12.1244i) q^{61} +(2.50000 - 4.33013i) q^{65} +(2.00000 + 3.46410i) q^{67} -12.0000 q^{71} +(-1.50000 + 2.59808i) q^{73} +(-2.50000 - 12.9904i) q^{77} +(4.00000 - 6.92820i) q^{79} +(4.50000 + 7.79423i) q^{83} +(1.50000 - 2.59808i) q^{85} +(-6.50000 - 11.2583i) q^{89} +(-12.5000 - 4.33013i) q^{91} +(0.500000 + 0.866025i) q^{95} +(4.50000 + 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - q^{7} + 10 q^{11} + 5 q^{13} + 3 q^{17} + q^{19} + 6 q^{23} - 8 q^{25} - q^{29} - q^{35} - 3 q^{37} - 5 q^{41} - q^{43} - 13 q^{49} - 9 q^{53} + 10 q^{55} + 14 q^{61} + 5 q^{65} + 4 q^{67} - 24 q^{71} - 3 q^{73} - 5 q^{77} + 8 q^{79} + 9 q^{83} + 3 q^{85} - 13 q^{89} - 25 q^{91} + q^{95} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −0.500000 2.59808i −0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 2.50000 4.33013i 0.693375 1.20096i −0.277350 0.960769i \(-0.589456\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.50000 2.59808i 0.363803 0.630126i −0.624780 0.780801i \(-0.714811\pi\)
0.988583 + 0.150675i \(0.0481447\pi\)
\(18\) 0 0
\(19\) 0.500000 + 0.866025i 0.114708 + 0.198680i 0.917663 0.397360i \(-0.130073\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.500000 0.866025i −0.0928477 0.160817i 0.815861 0.578249i \(-0.196264\pi\)
−0.908708 + 0.417432i \(0.862930\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.500000 2.59808i −0.0845154 0.439155i
\(36\) 0 0
\(37\) −1.50000 2.59808i −0.246598 0.427121i 0.715981 0.698119i \(-0.245980\pi\)
−0.962580 + 0.270998i \(0.912646\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.50000 + 4.33013i −0.390434 + 0.676252i −0.992507 0.122189i \(-0.961009\pi\)
0.602072 + 0.798441i \(0.294342\pi\)
\(42\) 0 0
\(43\) −0.500000 0.866025i −0.0762493 0.132068i 0.825380 0.564578i \(-0.190961\pi\)
−0.901629 + 0.432511i \(0.857628\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 + 7.79423i −0.618123 + 1.07062i 0.371706 + 0.928351i \(0.378773\pi\)
−0.989828 + 0.142269i \(0.954560\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 7.00000 12.1244i 0.896258 1.55236i 0.0640184 0.997949i \(-0.479608\pi\)
0.832240 0.554416i \(-0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.50000 4.33013i 0.310087 0.537086i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −1.50000 + 2.59808i −0.175562 + 0.304082i −0.940356 0.340193i \(-0.889507\pi\)
0.764794 + 0.644275i \(0.222841\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.50000 12.9904i −0.284901 1.48039i
\(78\) 0 0
\(79\) 4.00000 6.92820i 0.450035 0.779484i −0.548352 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.50000 + 7.79423i 0.493939 + 0.855528i 0.999976 0.00698436i \(-0.00222321\pi\)
−0.506036 + 0.862512i \(0.668890\pi\)
\(84\) 0 0
\(85\) 1.50000 2.59808i 0.162698 0.281801i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.50000 11.2583i −0.688999 1.19338i −0.972162 0.234309i \(-0.924717\pi\)
0.283164 0.959072i \(-0.408616\pi\)
\(90\) 0 0
\(91\) −12.5000 4.33013i −1.31036 0.453921i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.500000 + 0.866025i 0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) 4.50000 + 7.79423i 0.456906 + 0.791384i 0.998796 0.0490655i \(-0.0156243\pi\)
−0.541890 + 0.840450i \(0.682291\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.0000 1.69156 0.845782 0.533529i \(-0.179135\pi\)
0.845782 + 0.533529i \(0.179135\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.50000 14.7224i −0.821726 1.42327i −0.904396 0.426694i \(-0.859678\pi\)
0.0826699 0.996577i \(-0.473655\pi\)
\(108\) 0 0
\(109\) 4.50000 7.79423i 0.431022 0.746552i −0.565940 0.824447i \(-0.691487\pi\)
0.996962 + 0.0778949i \(0.0248199\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.500000 + 0.866025i −0.0470360 + 0.0814688i −0.888585 0.458712i \(-0.848311\pi\)
0.841549 + 0.540181i \(0.181644\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.50000 2.59808i −0.687524 0.238165i
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 −0.0873704 −0.0436852 0.999045i \(-0.513910\pi\)
−0.0436852 + 0.999045i \(0.513910\pi\)
\(132\) 0 0
\(133\) 2.00000 1.73205i 0.173422 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 4.50000 7.79423i 0.381685 0.661098i −0.609618 0.792695i \(-0.708677\pi\)
0.991303 + 0.131597i \(0.0420106\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.5000 21.6506i 1.04530 1.81052i
\(144\) 0 0
\(145\) −0.500000 0.866025i −0.0415227 0.0719195i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.00000 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000 + 12.1244i 0.558661 + 0.967629i 0.997609 + 0.0691164i \(0.0220180\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.50000 7.79423i −0.118217 0.614271i
\(162\) 0 0
\(163\) −5.50000 9.52628i −0.430793 0.746156i 0.566149 0.824303i \(-0.308433\pi\)
−0.996942 + 0.0781474i \(0.975100\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.50000 16.4545i 0.735132 1.27329i −0.219533 0.975605i \(-0.570453\pi\)
0.954665 0.297681i \(-0.0962132\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.00000 + 12.1244i −0.532200 + 0.921798i 0.467093 + 0.884208i \(0.345301\pi\)
−0.999293 + 0.0375896i \(0.988032\pi\)
\(174\) 0 0
\(175\) 2.00000 + 10.3923i 0.151186 + 0.785584i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.50000 + 16.4545i −0.710063 + 1.22987i 0.254770 + 0.967002i \(0.418000\pi\)
−0.964833 + 0.262864i \(0.915333\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.50000 2.59808i −0.110282 0.191014i
\(186\) 0 0
\(187\) 7.50000 12.9904i 0.548454 0.949951i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 + 6.92820i −0.289430 + 0.501307i −0.973674 0.227946i \(-0.926799\pi\)
0.684244 + 0.729253i \(0.260132\pi\)
\(192\) 0 0
\(193\) 5.00000 + 8.66025i 0.359908 + 0.623379i 0.987945 0.154805i \(-0.0494748\pi\)
−0.628037 + 0.778183i \(0.716141\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 1.50000 2.59808i 0.106332 0.184173i −0.807950 0.589252i \(-0.799423\pi\)
0.914282 + 0.405079i \(0.132756\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.00000 + 1.73205i −0.140372 + 0.121566i
\(204\) 0 0
\(205\) −2.50000 + 4.33013i −0.174608 + 0.302429i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.50000 + 4.33013i 0.172929 + 0.299521i
\(210\) 0 0
\(211\) 6.50000 11.2583i 0.447478 0.775055i −0.550743 0.834675i \(-0.685655\pi\)
0.998221 + 0.0596196i \(0.0189888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.500000 0.866025i −0.0340997 0.0590624i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.50000 12.9904i −0.504505 0.873828i
\(222\) 0 0
\(223\) 9.50000 + 16.4545i 0.636167 + 1.10187i 0.986267 + 0.165161i \(0.0528144\pi\)
−0.350100 + 0.936713i \(0.613852\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.50000 + 2.59808i 0.0982683 + 0.170206i 0.910968 0.412477i \(-0.135336\pi\)
−0.812700 + 0.582683i \(0.802003\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.50000 12.9904i 0.485135 0.840278i −0.514719 0.857359i \(-0.672104\pi\)
0.999854 + 0.0170808i \(0.00543724\pi\)
\(240\) 0 0
\(241\) 11.0000 0.708572 0.354286 0.935137i \(-0.384724\pi\)
0.354286 + 0.935137i \(0.384724\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.50000 + 2.59808i −0.415270 + 0.165985i
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 29.0000 1.80897 0.904485 0.426505i \(-0.140255\pi\)
0.904485 + 0.426505i \(0.140255\pi\)
\(258\) 0 0
\(259\) −6.00000 + 5.19615i −0.372822 + 0.322873i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.00000 0.308313 0.154157 0.988046i \(-0.450734\pi\)
0.154157 + 0.988046i \(0.450734\pi\)
\(264\) 0 0
\(265\) −4.50000 + 7.79423i −0.276433 + 0.478796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.50000 2.59808i 0.0914566 0.158408i −0.816668 0.577108i \(-0.804181\pi\)
0.908124 + 0.418701i \(0.137514\pi\)
\(270\) 0 0
\(271\) 0.500000 + 0.866025i 0.0303728 + 0.0526073i 0.880812 0.473466i \(-0.156997\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −20.0000 −1.20605
\(276\) 0 0
\(277\) 19.0000 1.14160 0.570800 0.821089i \(-0.306633\pi\)
0.570800 + 0.821089i \(0.306633\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.5000 25.1147i −0.864997 1.49822i −0.867050 0.498222i \(-0.833987\pi\)
0.00205220 0.999998i \(-0.499347\pi\)
\(282\) 0 0
\(283\) 14.0000 + 24.2487i 0.832214 + 1.44144i 0.896279 + 0.443491i \(0.146260\pi\)
−0.0640654 + 0.997946i \(0.520407\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.5000 + 4.33013i 0.737852 + 0.255599i
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.50000 + 4.33013i −0.146052 + 0.252969i −0.929765 0.368154i \(-0.879990\pi\)
0.783713 + 0.621123i \(0.213323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.50000 12.9904i 0.433736 0.751253i
\(300\) 0 0
\(301\) −2.00000 + 1.73205i −0.115278 + 0.0998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.00000 12.1244i 0.400819 0.694239i
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −7.00000 + 12.1244i −0.395663 + 0.685309i −0.993186 0.116543i \(-0.962819\pi\)
0.597522 + 0.801852i \(0.296152\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 + 5.19615i −0.168497 + 0.291845i −0.937892 0.346929i \(-0.887225\pi\)
0.769395 + 0.638774i \(0.220558\pi\)
\(318\) 0 0
\(319\) −2.50000 4.33013i −0.139973 0.242441i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −10.0000 + 17.3205i −0.554700 + 0.960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 6.92820i 0.219860 0.380808i −0.734905 0.678170i \(-0.762773\pi\)
0.954765 + 0.297361i \(0.0961066\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.00000 + 3.46410i 0.109272 + 0.189264i
\(336\) 0 0
\(337\) 14.5000 25.1147i 0.789865 1.36809i −0.136184 0.990684i \(-0.543484\pi\)
0.926049 0.377403i \(-0.123183\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.00000 3.46410i −0.107366 0.185963i 0.807337 0.590091i \(-0.200908\pi\)
−0.914702 + 0.404128i \(0.867575\pi\)
\(348\) 0 0
\(349\) −9.50000 16.4545i −0.508523 0.880788i −0.999951 0.00987003i \(-0.996858\pi\)
0.491428 0.870918i \(-0.336475\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.0000 −0.585471 −0.292735 0.956193i \(-0.594566\pi\)
−0.292735 + 0.956193i \(0.594566\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.50000 + 9.52628i 0.290279 + 0.502778i 0.973876 0.227082i \(-0.0729186\pi\)
−0.683597 + 0.729860i \(0.739585\pi\)
\(360\) 0 0
\(361\) 9.00000 15.5885i 0.473684 0.820445i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.50000 + 2.59808i −0.0785136 + 0.135990i
\(366\) 0 0
\(367\) 3.00000 0.156599 0.0782994 0.996930i \(-0.475051\pi\)
0.0782994 + 0.996930i \(0.475051\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.5000 + 7.79423i 1.16814 + 0.404656i
\(372\) 0 0
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.0000 1.37964 0.689818 0.723983i \(-0.257691\pi\)
0.689818 + 0.723983i \(0.257691\pi\)
\(384\) 0 0
\(385\) −2.50000 12.9904i −0.127412 0.662051i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) 4.50000 7.79423i 0.227575 0.394171i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.00000 6.92820i 0.201262 0.348596i
\(396\) 0 0
\(397\) −7.50000 12.9904i −0.376414 0.651969i 0.614123 0.789210i \(-0.289510\pi\)
−0.990538 + 0.137241i \(0.956176\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.50000 12.9904i −0.371761 0.643909i
\(408\) 0 0
\(409\) −7.00000 12.1244i −0.346128 0.599511i 0.639430 0.768849i \(-0.279170\pi\)
−0.985558 + 0.169338i \(0.945837\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.50000 + 7.79423i 0.220896 + 0.382604i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.50000 + 7.79423i −0.219839 + 0.380773i −0.954759 0.297382i \(-0.903887\pi\)
0.734919 + 0.678155i \(0.237220\pi\)
\(420\) 0 0
\(421\) 0.500000 + 0.866025i 0.0243685 + 0.0422075i 0.877952 0.478748i \(-0.158909\pi\)
−0.853584 + 0.520955i \(0.825576\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.00000 + 10.3923i −0.291043 + 0.504101i
\(426\) 0 0
\(427\) −35.0000 12.1244i −1.69377 0.586739i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.50000 7.79423i 0.216757 0.375435i −0.737057 0.675830i \(-0.763785\pi\)
0.953815 + 0.300395i \(0.0971186\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.50000 + 2.59808i 0.0717547 + 0.124283i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.0000 + 31.1769i −0.855206 + 1.48126i 0.0212481 + 0.999774i \(0.493236\pi\)
−0.876454 + 0.481486i \(0.840097\pi\)
\(444\) 0 0
\(445\) −6.50000 11.2583i −0.308130 0.533696i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −12.5000 + 21.6506i −0.588602 + 1.01949i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −12.5000 4.33013i −0.586009 0.202999i
\(456\) 0 0
\(457\) −11.0000 + 19.0526i −0.514558 + 0.891241i 0.485299 + 0.874348i \(0.338711\pi\)
−0.999857 + 0.0168929i \(0.994623\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.50000 + 16.4545i 0.442459 + 0.766362i 0.997871 0.0652135i \(-0.0207728\pi\)
−0.555412 + 0.831575i \(0.687440\pi\)
\(462\) 0 0
\(463\) 6.50000 11.2583i 0.302081 0.523219i −0.674526 0.738251i \(-0.735652\pi\)
0.976607 + 0.215032i \(0.0689855\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.5000 + 23.3827i 0.624705 + 1.08202i 0.988598 + 0.150581i \(0.0481143\pi\)
−0.363892 + 0.931441i \(0.618552\pi\)
\(468\) 0 0
\(469\) 8.00000 6.92820i 0.369406 0.319915i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.50000 4.33013i −0.114950 0.199099i
\(474\) 0 0
\(475\) −2.00000 3.46410i −0.0917663 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 25.0000 1.14228 0.571140 0.820853i \(-0.306501\pi\)
0.571140 + 0.820853i \(0.306501\pi\)
\(480\) 0 0
\(481\) −15.0000 −0.683941
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.50000 + 7.79423i 0.204334 + 0.353918i
\(486\) 0 0
\(487\) 9.50000 16.4545i 0.430486 0.745624i −0.566429 0.824110i \(-0.691675\pi\)
0.996915 + 0.0784867i \(0.0250088\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.50000 + 11.2583i −0.293341 + 0.508081i −0.974598 0.223963i \(-0.928100\pi\)
0.681257 + 0.732045i \(0.261434\pi\)
\(492\) 0 0
\(493\) −3.00000 −0.135113
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 + 31.1769i 0.269137 + 1.39848i
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 17.0000 0.756490
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.0000 1.28540 0.642701 0.766117i \(-0.277814\pi\)
0.642701 + 0.766117i \(0.277814\pi\)
\(510\) 0 0
\(511\) 7.50000 + 2.59808i 0.331780 + 0.114932i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.00000 0.0440653
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.50000 2.59808i 0.0657162 0.113824i −0.831295 0.555831i \(-0.812400\pi\)
0.897011 + 0.442007i \(0.145733\pi\)
\(522\) 0 0
\(523\) 0.500000 + 0.866025i 0.0218635 + 0.0378686i 0.876750 0.480946i \(-0.159707\pi\)
−0.854887 + 0.518815i \(0.826373\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.5000 + 21.6506i 0.541435 + 0.937793i
\(534\) 0 0
\(535\) −8.50000 14.7224i −0.367487 0.636506i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −32.5000 + 12.9904i −1.39987 + 0.559535i
\(540\) 0 0
\(541\) 12.5000 + 21.6506i 0.537417 + 0.930834i 0.999042 + 0.0437584i \(0.0139332\pi\)
−0.461625 + 0.887075i \(0.652733\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.50000 7.79423i 0.192759 0.333868i
\(546\) 0 0
\(547\) −14.5000 25.1147i −0.619975 1.07383i −0.989490 0.144604i \(-0.953809\pi\)
0.369514 0.929225i \(-0.379524\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0.500000 0.866025i 0.0213007 0.0368939i
\(552\) 0 0
\(553\) −20.0000 6.92820i −0.850487 0.294617i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.5000 + 32.0429i −0.783870 + 1.35770i 0.145802 + 0.989314i \(0.453424\pi\)
−0.929672 + 0.368389i \(0.879909\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.0000 + 24.2487i 0.590030 + 1.02196i 0.994228 + 0.107290i \(0.0342173\pi\)
−0.404198 + 0.914671i \(0.632449\pi\)
\(564\) 0 0
\(565\) −0.500000 + 0.866025i −0.0210352 + 0.0364340i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17.0000 + 29.4449i −0.712677 + 1.23439i 0.251172 + 0.967943i \(0.419184\pi\)
−0.963849 + 0.266450i \(0.914149\pi\)
\(570\) 0 0
\(571\) 16.0000 + 27.7128i 0.669579 + 1.15975i 0.978022 + 0.208502i \(0.0668588\pi\)
−0.308443 + 0.951243i \(0.599808\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −15.5000 + 26.8468i −0.645273 + 1.11765i 0.338965 + 0.940799i \(0.389923\pi\)
−0.984238 + 0.176847i \(0.943410\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 18.0000 15.5885i 0.746766 0.646718i
\(582\) 0 0
\(583\) −22.5000 + 38.9711i −0.931855 + 1.61402i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.5000 + 32.0429i 0.763577 + 1.32255i 0.940996 + 0.338418i \(0.109892\pi\)
−0.177419 + 0.984135i \(0.556775\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.50000 + 12.9904i 0.307988 + 0.533451i 0.977922 0.208970i \(-0.0670110\pi\)
−0.669934 + 0.742421i \(0.733678\pi\)
\(594\) 0 0
\(595\) −7.50000 2.59808i −0.307470 0.106511i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 + 20.7846i 0.490307 + 0.849236i 0.999938 0.0111569i \(-0.00355143\pi\)
−0.509631 + 0.860393i \(0.670218\pi\)
\(600\) 0 0
\(601\) 4.50000 + 7.79423i 0.183559 + 0.317933i 0.943090 0.332538i \(-0.107905\pi\)
−0.759531 + 0.650471i \(0.774572\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −9.50000 + 16.4545i −0.383701 + 0.664590i −0.991588 0.129433i \(-0.958684\pi\)
0.607887 + 0.794024i \(0.292017\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.5000 23.3827i 0.543490 0.941351i −0.455211 0.890384i \(-0.650436\pi\)
0.998700 0.0509678i \(-0.0162306\pi\)
\(618\) 0 0
\(619\) −25.0000 −1.00483 −0.502417 0.864625i \(-0.667556\pi\)
−0.502417 + 0.864625i \(0.667556\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −26.0000 + 22.5167i −1.04167 + 0.902111i
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.00000 −0.358854
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) −5.00000 + 34.6410i −0.198107 + 1.37253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) −9.50000 + 16.4545i −0.374643 + 0.648901i −0.990274 0.139134i \(-0.955568\pi\)
0.615630 + 0.788035i \(0.288902\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.5000 + 26.8468i −0.609368 + 1.05546i 0.381977 + 0.924172i \(0.375243\pi\)
−0.991345 + 0.131284i \(0.958090\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.00000 −0.117399 −0.0586995 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) 0 0
\(655\) −1.00000 −0.0390732
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13.5000 23.3827i −0.525885 0.910860i −0.999545 0.0301523i \(-0.990401\pi\)
0.473660 0.880708i \(-0.342933\pi\)
\(660\) 0 0
\(661\) 7.00000 + 12.1244i 0.272268 + 0.471583i 0.969442 0.245319i \(-0.0788928\pi\)
−0.697174 + 0.716902i \(0.745559\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.00000 1.73205i 0.0775567 0.0671660i
\(666\) 0 0
\(667\) −1.50000 2.59808i −0.0580802 0.100598i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 35.0000 60.6218i 1.35116 2.34028i
\(672\) 0 0
\(673\) 14.5000 + 25.1147i 0.558934 + 0.968102i 0.997586 + 0.0694449i \(0.0221228\pi\)
−0.438652 + 0.898657i \(0.644544\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.0000 36.3731i 0.807096 1.39793i −0.107772 0.994176i \(-0.534372\pi\)
0.914867 0.403755i \(-0.132295\pi\)
\(678\) 0 0
\(679\) 18.0000 15.5885i 0.690777 0.598230i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.50000 7.79423i 0.172188 0.298238i −0.766997 0.641651i \(-0.778250\pi\)
0.939184 + 0.343413i \(0.111583\pi\)
\(684\) 0 0
\(685\) 9.00000 0.343872
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 22.5000 + 38.9711i 0.857182 + 1.48468i
\(690\) 0 0
\(691\) −14.0000 + 24.2487i −0.532585 + 0.922464i 0.466691 + 0.884420i \(0.345446\pi\)
−0.999276 + 0.0380440i \(0.987887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.50000 7.79423i 0.170695 0.295652i
\(696\) 0 0
\(697\) 7.50000 + 12.9904i 0.284083 + 0.492046i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 1.50000 2.59808i 0.0565736 0.0979883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −8.50000 44.1673i −0.319675 1.66108i
\(708\) 0 0
\(709\) 3.00000 5.19615i 0.112667 0.195146i −0.804178 0.594389i \(-0.797394\pi\)
0.916845 + 0.399244i \(0.130727\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 12.5000 21.6506i 0.467473 0.809688i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 13.5000 + 23.3827i 0.503465 + 0.872027i 0.999992 + 0.00400572i \(0.00127506\pi\)
−0.496527 + 0.868021i \(0.665392\pi\)
\(720\) 0 0
\(721\) −0.500000 2.59808i −0.0186210 0.0967574i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000 + 3.46410i 0.0742781 + 0.128654i
\(726\) 0 0
\(727\) 23.5000 + 40.7032i 0.871567 + 1.50960i 0.860376 + 0.509661i \(0.170229\pi\)
0.0111912 + 0.999937i \(0.496438\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 27.0000 0.997268 0.498634 0.866813i \(-0.333835\pi\)
0.498634 + 0.866813i \(0.333835\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 + 17.3205i 0.368355 + 0.638009i
\(738\) 0 0
\(739\) −4.50000 + 7.79423i −0.165535 + 0.286715i −0.936845 0.349744i \(-0.886268\pi\)
0.771310 + 0.636460i \(0.219602\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.50000 12.9904i 0.275148 0.476571i −0.695024 0.718986i \(-0.744606\pi\)
0.970173 + 0.242415i \(0.0779397\pi\)
\(744\) 0 0
\(745\) −3.00000 −0.109911
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −34.0000 + 29.4449i −1.24233 + 1.07589i
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) −22.5000 7.79423i −0.814555 0.282170i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −11.5000 + 19.9186i −0.414701 + 0.718283i −0.995397 0.0958377i \(-0.969447\pi\)
0.580696 + 0.814120i \(0.302780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.5000 26.8468i 0.557496 0.965612i −0.440208 0.897896i \(-0.645095\pi\)
0.997705 0.0677162i \(-0.0215712\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.00000 −0.179144
\(780\) 0 0
\(781\) −60.0000 −2.14697
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.00000 + 12.1244i 0.249841 + 0.432737i
\(786\) 0 0
\(787\) 14.0000 + 24.2487i 0.499046 + 0.864373i 0.999999 0.00110111i \(-0.000350496\pi\)
−0.500953 + 0.865474i \(0.667017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.50000 + 0.866025i 0.0888898 + 0.0307923i
\(792\) 0 0
\(793\) −35.0000 60.6218i −1.24289 2.15274i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.5000 19.9186i 0.407351 0.705552i −0.587241 0.809412i \(-0.699786\pi\)
0.994592 + 0.103860i \(0.0331193\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −7.50000 + 12.9904i −0.264669 + 0.458421i
\(804\) 0 0
\(805\) −1.50000 7.79423i −0.0528681 0.274710i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.50000 + 7.79423i −0.158212 + 0.274030i −0.934224 0.356687i \(-0.883906\pi\)
0.776012 + 0.630718i \(0.217239\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.50000 9.52628i −0.192657 0.333691i
\(816\) 0 0
\(817\) 0.500000 0.866025i 0.0174928 0.0302984i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.0000 19.0526i 0.383903 0.664939i −0.607714 0.794156i \(-0.707913\pi\)
0.991616 + 0.129217i \(0.0412465\pi\)
\(822\) 0 0
\(823\) −12.0000 20.7846i −0.418294 0.724506i 0.577474 0.816409i \(-0.304038\pi\)
−0.995768 + 0.0919029i \(0.970705\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 12.5000 21.6506i 0.434143 0.751958i −0.563082 0.826401i \(-0.690385\pi\)
0.997225 + 0.0744432i \(0.0237179\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.00000 + 20.7846i −0.103944 + 0.720144i
\(834\) 0 0
\(835\) 9.50000 16.4545i 0.328761 0.569431i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 18.5000 + 32.0429i 0.638691 + 1.10625i 0.985720 + 0.168391i \(0.0538571\pi\)
−0.347029 + 0.937854i \(0.612810\pi\)
\(840\) 0 0
\(841\) 14.0000 24.2487i 0.482759 0.836162i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.00000 10.3923i −0.206406 0.357506i
\(846\) 0 0
\(847\) −7.00000 36.3731i −0.240523 1.24979i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.50000 7.79423i −0.154258 0.267183i
\(852\) 0 0
\(853\) 18.5000 + 32.0429i 0.633428 + 1.09713i 0.986846 + 0.161664i \(0.0516860\pi\)
−0.353418 + 0.935466i \(0.614981\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −11.0000 −0.375753 −0.187876 0.982193i \(-0.560160\pi\)
−0.187876 + 0.982193i \(0.560160\pi\)
\(858\) 0 0
\(859\) 1.00000 0.0341196 0.0170598 0.999854i \(-0.494569\pi\)
0.0170598 + 0.999854i \(0.494569\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 19.5000 + 33.7750i 0.663788 + 1.14971i 0.979612 + 0.200897i \(0.0643855\pi\)
−0.315825 + 0.948818i \(0.602281\pi\)
\(864\) 0 0
\(865\) −7.00000 + 12.1244i −0.238007 + 0.412240i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 20.0000 34.6410i 0.678454 1.17512i
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.50000 + 23.3827i 0.152128 + 0.790479i
\(876\) 0 0
\(877\) −53.0000 −1.78968 −0.894841 0.446384i \(-0.852711\pi\)
−0.894841 + 0.446384i \(0.852711\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29.0000 −0.973725 −0.486862 0.873479i \(-0.661859\pi\)
−0.486862 + 0.873479i \(0.661859\pi\)
\(888\) 0 0
\(889\) −6.00000 31.1769i −0.201234 1.04564i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −9.50000 + 16.4545i −0.317550 + 0.550013i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 13.5000 + 23.3827i 0.449750 + 0.778990i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −5.00000 −0.166022 −0.0830111 0.996549i \(-0.526454\pi\)
−0.0830111 + 0.996549i \(0.526454\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −13.5000 23.3827i −0.447275 0.774703i 0.550933 0.834550i \(-0.314272\pi\)
−0.998208 + 0.0598468i \(0.980939\pi\)
\(912\) 0 0
\(913\) 22.5000 + 38.9711i 0.744641 + 1.28976i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.500000 + 2.59808i 0.0165115 + 0.0857960i
\(918\) 0 0
\(919\) 8.50000 + 14.7224i 0.280389 + 0.485648i 0.971481 0.237119i \(-0.0762032\pi\)
−0.691091 + 0.722767i \(0.742870\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −30.0000 + 51.9615i −0.987462 + 1.71033i
\(924\) 0 0
\(925\) 6.00000 + 10.3923i 0.197279 + 0.341697i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7.00000 + 12.1244i −0.229663 + 0.397787i −0.957708 0.287742i \(-0.907096\pi\)
0.728046 + 0.685529i \(0.240429\pi\)
\(930\) 0 0
\(931\) −5.50000 4.33013i −0.180255 0.141914i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.50000 12.9904i 0.245276 0.424831i
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.00000 12.1244i −0.228193 0.395243i 0.729079 0.684429i \(-0.239949\pi\)
−0.957273 + 0.289187i \(0.906615\pi\)
\(942\) 0 0
\(943\) −7.50000 + 12.9904i −0.244234 + 0.423025i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 10.0000 17.3205i 0.324956 0.562841i −0.656547 0.754285i \(-0.727984\pi\)
0.981504 + 0.191444i \(0.0613171\pi\)
\(948\) 0 0
\(949\) 7.50000 + 12.9904i 0.243460 + 0.421686i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) −4.00000 + 6.92820i −0.129437 + 0.224191i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.50000 23.3827i −0.145313 0.755066i
\(960\) 0 0
\(961\) 15.5000 26.8468i 0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.00000 + 8.66025i 0.160956 + 0.278783i
\(966\) 0 0
\(967\) 6.50000 11.2583i 0.209026 0.362043i −0.742382 0.669977i \(-0.766304\pi\)
0.951408 + 0.307933i \(0.0996374\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.5000 49.3634i −0.914609 1.58415i −0.807473 0.589904i \(-0.799166\pi\)
−0.107135 0.994244i \(-0.534168\pi\)
\(972\) 0 0
\(973\) −22.5000 7.79423i −0.721317 0.249871i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.00000 + 15.5885i 0.287936 + 0.498719i 0.973317 0.229465i \(-0.0736978\pi\)
−0.685381 + 0.728184i \(0.740364\pi\)
\(978\) 0 0
\(979\) −32.5000 56.2917i −1.03870 1.79909i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.50000 2.59808i −0.0476972 0.0826140i
\(990\) 0 0
\(991\) −18.5000 + 32.0429i −0.587672 + 1.01788i 0.406865 + 0.913488i \(0.366622\pi\)
−0.994537 + 0.104389i \(0.966711\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.50000 2.59808i 0.0475532 0.0823646i
\(996\) 0 0
\(997\) −17.0000 −0.538395 −0.269198 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(998\) 0 0
\(999\) 0 0
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 3024.2.t.d.1873.1 2
3.2 odd 2 1008.2.t.d.193.1 2
4.3 odd 2 189.2.g.a.172.1 2
7.2 even 3 3024.2.q.b.2305.1 2
9.2 odd 6 1008.2.q.c.529.1 2
9.7 even 3 3024.2.q.b.2881.1 2
12.11 even 2 63.2.g.a.4.1 2
21.2 odd 6 1008.2.q.c.625.1 2
28.3 even 6 1323.2.f.b.442.1 2
28.11 odd 6 1323.2.f.a.442.1 2
28.19 even 6 1323.2.h.a.226.1 2
28.23 odd 6 189.2.h.a.37.1 2
28.27 even 2 1323.2.g.a.361.1 2
36.7 odd 6 189.2.h.a.46.1 2
36.11 even 6 63.2.h.a.25.1 yes 2
36.23 even 6 567.2.e.a.487.1 2
36.31 odd 6 567.2.e.b.487.1 2
63.2 odd 6 1008.2.t.d.961.1 2
63.16 even 3 inner 3024.2.t.d.289.1 2
84.11 even 6 441.2.f.b.148.1 2
84.23 even 6 63.2.h.a.58.1 yes 2
84.47 odd 6 441.2.h.a.373.1 2
84.59 odd 6 441.2.f.a.148.1 2
84.83 odd 2 441.2.g.a.67.1 2
252.11 even 6 441.2.f.b.295.1 2
252.23 even 6 567.2.e.a.163.1 2
252.31 even 6 3969.2.a.a.1.1 1
252.47 odd 6 441.2.g.a.79.1 2
252.59 odd 6 3969.2.a.f.1.1 1
252.67 odd 6 3969.2.a.c.1.1 1
252.79 odd 6 189.2.g.a.100.1 2
252.83 odd 6 441.2.h.a.214.1 2
252.95 even 6 3969.2.a.d.1.1 1
252.115 even 6 1323.2.f.b.883.1 2
252.151 odd 6 1323.2.f.a.883.1 2
252.187 even 6 1323.2.g.a.667.1 2
252.191 even 6 63.2.g.a.16.1 yes 2
252.223 even 6 1323.2.h.a.802.1 2
252.227 odd 6 441.2.f.a.295.1 2
252.247 odd 6 567.2.e.b.163.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.g.a.4.1 2 12.11 even 2
63.2.g.a.16.1 yes 2 252.191 even 6
63.2.h.a.25.1 yes 2 36.11 even 6
63.2.h.a.58.1 yes 2 84.23 even 6
189.2.g.a.100.1 2 252.79 odd 6
189.2.g.a.172.1 2 4.3 odd 2
189.2.h.a.37.1 2 28.23 odd 6
189.2.h.a.46.1 2 36.7 odd 6
441.2.f.a.148.1 2 84.59 odd 6
441.2.f.a.295.1 2 252.227 odd 6
441.2.f.b.148.1 2 84.11 even 6
441.2.f.b.295.1 2 252.11 even 6
441.2.g.a.67.1 2 84.83 odd 2
441.2.g.a.79.1 2 252.47 odd 6
441.2.h.a.214.1 2 252.83 odd 6
441.2.h.a.373.1 2 84.47 odd 6
567.2.e.a.163.1 2 252.23 even 6
567.2.e.a.487.1 2 36.23 even 6
567.2.e.b.163.1 2 252.247 odd 6
567.2.e.b.487.1 2 36.31 odd 6
1008.2.q.c.529.1 2 9.2 odd 6
1008.2.q.c.625.1 2 21.2 odd 6
1008.2.t.d.193.1 2 3.2 odd 2
1008.2.t.d.961.1 2 63.2 odd 6
1323.2.f.a.442.1 2 28.11 odd 6
1323.2.f.a.883.1 2 252.151 odd 6
1323.2.f.b.442.1 2 28.3 even 6
1323.2.f.b.883.1 2 252.115 even 6
1323.2.g.a.361.1 2 28.27 even 2
1323.2.g.a.667.1 2 252.187 even 6
1323.2.h.a.226.1 2 28.19 even 6
1323.2.h.a.802.1 2 252.223 even 6
3024.2.q.b.2305.1 2 7.2 even 3
3024.2.q.b.2881.1 2 9.7 even 3
3024.2.t.d.289.1 2 63.16 even 3 inner
3024.2.t.d.1873.1 2 1.1 even 1 trivial
3969.2.a.a.1.1 1 252.31 even 6
3969.2.a.c.1.1 1 252.67 odd 6
3969.2.a.d.1.1 1 252.95 even 6
3969.2.a.f.1.1 1 252.59 odd 6