Properties

Label 980.2.i.b.961.1
Level $980$
Weight $2$
Character 980.961
Analytic conductor $7.825$
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] = [980,2,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (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: \(7.82533939809\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 980.961
Dual form 980.2.i.b.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 2.59808i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-3.00000 + 5.19615i) q^{9} +O(q^{10})\) \(q+(-1.50000 - 2.59808i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-3.00000 + 5.19615i) q^{9} +(2.50000 + 4.33013i) q^{11} -3.00000 q^{13} -3.00000 q^{15} +(0.500000 + 0.866025i) q^{17} +(-3.00000 + 5.19615i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +9.00000 q^{27} -9.00000 q^{29} +(2.00000 + 3.46410i) q^{31} +(7.50000 - 12.9904i) q^{33} +(-1.00000 + 1.73205i) q^{37} +(4.50000 + 7.79423i) q^{39} -4.00000 q^{41} +10.0000 q^{43} +(3.00000 + 5.19615i) q^{45} +(0.500000 - 0.866025i) q^{47} +(1.50000 - 2.59808i) q^{51} +(-2.00000 - 3.46410i) q^{53} +5.00000 q^{55} +18.0000 q^{57} +(4.00000 + 6.92820i) q^{59} +(4.00000 - 6.92820i) q^{61} +(-1.50000 + 2.59808i) q^{65} +(-6.00000 - 10.3923i) q^{67} +18.0000 q^{69} +8.00000 q^{71} +(-1.00000 - 1.73205i) q^{73} +(-1.50000 + 2.59808i) q^{75} +(-6.50000 + 11.2583i) q^{79} +(-4.50000 - 7.79423i) q^{81} -4.00000 q^{83} +1.00000 q^{85} +(13.5000 + 23.3827i) q^{87} +(-2.00000 + 3.46410i) q^{89} +(6.00000 - 10.3923i) q^{93} +(3.00000 + 5.19615i) q^{95} -13.0000 q^{97} -30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + q^{5} - 6 q^{9} + 5 q^{11} - 6 q^{13} - 6 q^{15} + q^{17} - 6 q^{19} - 6 q^{23} - q^{25} + 18 q^{27} - 18 q^{29} + 4 q^{31} + 15 q^{33} - 2 q^{37} + 9 q^{39} - 8 q^{41} + 20 q^{43} + 6 q^{45} + q^{47} + 3 q^{51} - 4 q^{53} + 10 q^{55} + 36 q^{57} + 8 q^{59} + 8 q^{61} - 3 q^{65} - 12 q^{67} + 36 q^{69} + 16 q^{71} - 2 q^{73} - 3 q^{75} - 13 q^{79} - 9 q^{81} - 8 q^{83} + 2 q^{85} + 27 q^{87} - 4 q^{89} + 12 q^{93} + 6 q^{95} - 26 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.50000 2.59808i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 + 5.19615i −1.00000 + 1.73205i
\(10\) 0 0
\(11\) 2.50000 + 4.33013i 0.753778 + 1.30558i 0.945979 + 0.324227i \(0.105104\pi\)
−0.192201 + 0.981356i \(0.561563\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) 0.500000 + 0.866025i 0.121268 + 0.210042i 0.920268 0.391289i \(-0.127971\pi\)
−0.799000 + 0.601331i \(0.794637\pi\)
\(18\) 0 0
\(19\) −3.00000 + 5.19615i −0.688247 + 1.19208i 0.284157 + 0.958778i \(0.408286\pi\)
−0.972404 + 0.233301i \(0.925047\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 7.50000 12.9904i 1.30558 2.26134i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 + 1.73205i −0.164399 + 0.284747i −0.936442 0.350823i \(-0.885902\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) 0 0
\(39\) 4.50000 + 7.79423i 0.720577 + 1.24808i
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 3.00000 + 5.19615i 0.447214 + 0.774597i
\(46\) 0 0
\(47\) 0.500000 0.866025i 0.0729325 0.126323i −0.827253 0.561830i \(-0.810098\pi\)
0.900185 + 0.435507i \(0.143431\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 1.50000 2.59808i 0.210042 0.363803i
\(52\) 0 0
\(53\) −2.00000 3.46410i −0.274721 0.475831i 0.695344 0.718677i \(-0.255252\pi\)
−0.970065 + 0.242846i \(0.921919\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 0 0
\(61\) 4.00000 6.92820i 0.512148 0.887066i −0.487753 0.872982i \(-0.662183\pi\)
0.999901 0.0140840i \(-0.00448323\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.50000 + 2.59808i −0.186052 + 0.322252i
\(66\) 0 0
\(67\) −6.00000 10.3923i −0.733017 1.26962i −0.955588 0.294706i \(-0.904778\pi\)
0.222571 0.974916i \(-0.428555\pi\)
\(68\) 0 0
\(69\) 18.0000 2.16695
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −1.00000 1.73205i −0.117041 0.202721i 0.801553 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) −1.50000 + 2.59808i −0.173205 + 0.300000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.50000 + 11.2583i −0.731307 + 1.26666i 0.225018 + 0.974355i \(0.427756\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) 13.5000 + 23.3827i 1.44735 + 2.50689i
\(88\) 0 0
\(89\) −2.00000 + 3.46410i −0.212000 + 0.367194i −0.952340 0.305038i \(-0.901331\pi\)
0.740341 + 0.672232i \(0.234664\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 10.3923i 0.622171 1.07763i
\(94\) 0 0
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) −30.0000 −3.01511
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) −9.50000 + 16.4545i −0.936063 + 1.62131i −0.163335 + 0.986571i \(0.552225\pi\)
−0.772728 + 0.634738i \(0.781108\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 5.19615i 0.290021 0.502331i −0.683793 0.729676i \(-0.739671\pi\)
0.973814 + 0.227345i \(0.0730044\pi\)
\(108\) 0 0
\(109\) 1.50000 + 2.59808i 0.143674 + 0.248851i 0.928877 0.370387i \(-0.120775\pi\)
−0.785203 + 0.619238i \(0.787442\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 3.00000 + 5.19615i 0.279751 + 0.484544i
\(116\) 0 0
\(117\) 9.00000 15.5885i 0.832050 1.44115i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 6.00000 + 10.3923i 0.541002 + 0.937043i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −15.0000 25.9808i −1.32068 2.28748i
\(130\) 0 0
\(131\) 5.00000 8.66025i 0.436852 0.756650i −0.560593 0.828092i \(-0.689427\pi\)
0.997445 + 0.0714417i \(0.0227600\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 4.50000 7.79423i 0.387298 0.670820i
\(136\) 0 0
\(137\) 6.00000 + 10.3923i 0.512615 + 0.887875i 0.999893 + 0.0146279i \(0.00465636\pi\)
−0.487278 + 0.873247i \(0.662010\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −7.50000 12.9904i −0.627182 1.08631i
\(144\) 0 0
\(145\) −4.50000 + 7.79423i −0.373705 + 0.647275i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 5.19615i 0.245770 0.425685i −0.716578 0.697507i \(-0.754293\pi\)
0.962348 + 0.271821i \(0.0876260\pi\)
\(150\) 0 0
\(151\) −2.50000 4.33013i −0.203447 0.352381i 0.746190 0.665733i \(-0.231881\pi\)
−0.949637 + 0.313353i \(0.898548\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 1.00000 + 1.73205i 0.0798087 + 0.138233i 0.903167 0.429289i \(-0.141236\pi\)
−0.823359 + 0.567521i \(0.807902\pi\)
\(158\) 0 0
\(159\) −6.00000 + 10.3923i −0.475831 + 0.824163i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.00000 8.66025i 0.391630 0.678323i −0.601035 0.799223i \(-0.705245\pi\)
0.992665 + 0.120900i \(0.0385779\pi\)
\(164\) 0 0
\(165\) −7.50000 12.9904i −0.583874 1.01130i
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −18.0000 31.1769i −1.37649 2.38416i
\(172\) 0 0
\(173\) 0.500000 0.866025i 0.0380143 0.0658427i −0.846392 0.532560i \(-0.821230\pi\)
0.884407 + 0.466717i \(0.154563\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 20.7846i 0.901975 1.56227i
\(178\) 0 0
\(179\) 6.00000 + 10.3923i 0.448461 + 0.776757i 0.998286 0.0585225i \(-0.0186389\pi\)
−0.549825 + 0.835280i \(0.685306\pi\)
\(180\) 0 0
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) 1.00000 + 1.73205i 0.0735215 + 0.127343i
\(186\) 0 0
\(187\) −2.50000 + 4.33013i −0.182818 + 0.316650i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.50000 + 2.59808i −0.108536 + 0.187990i −0.915177 0.403051i \(-0.867950\pi\)
0.806641 + 0.591041i \(0.201283\pi\)
\(192\) 0 0
\(193\) 2.00000 + 3.46410i 0.143963 + 0.249351i 0.928986 0.370116i \(-0.120682\pi\)
−0.785022 + 0.619467i \(0.787349\pi\)
\(194\) 0 0
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −4.00000 6.92820i −0.283552 0.491127i 0.688705 0.725042i \(-0.258180\pi\)
−0.972257 + 0.233915i \(0.924846\pi\)
\(200\) 0 0
\(201\) −18.0000 + 31.1769i −1.26962 + 2.19905i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 + 3.46410i −0.139686 + 0.241943i
\(206\) 0 0
\(207\) −18.0000 31.1769i −1.25109 2.16695i
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −11.0000 −0.757271 −0.378636 0.925546i \(-0.623607\pi\)
−0.378636 + 0.925546i \(0.623607\pi\)
\(212\) 0 0
\(213\) −12.0000 20.7846i −0.822226 1.42414i
\(214\) 0 0
\(215\) 5.00000 8.66025i 0.340997 0.590624i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.00000 + 5.19615i −0.202721 + 0.351123i
\(220\) 0 0
\(221\) −1.50000 2.59808i −0.100901 0.174766i
\(222\) 0 0
\(223\) −5.00000 −0.334825 −0.167412 0.985887i \(-0.553541\pi\)
−0.167412 + 0.985887i \(0.553541\pi\)
\(224\) 0 0
\(225\) 6.00000 0.400000
\(226\) 0 0
\(227\) 0.500000 + 0.866025i 0.0331862 + 0.0574801i 0.882141 0.470985i \(-0.156101\pi\)
−0.848955 + 0.528465i \(0.822768\pi\)
\(228\) 0 0
\(229\) −2.00000 + 3.46410i −0.132164 + 0.228914i −0.924510 0.381157i \(-0.875526\pi\)
0.792347 + 0.610071i \(0.208859\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 + 20.7846i −0.786146 + 1.36165i 0.142166 + 0.989843i \(0.454593\pi\)
−0.928312 + 0.371802i \(0.878740\pi\)
\(234\) 0 0
\(235\) −0.500000 0.866025i −0.0326164 0.0564933i
\(236\) 0 0
\(237\) 39.0000 2.53332
\(238\) 0 0
\(239\) 1.00000 0.0646846 0.0323423 0.999477i \(-0.489703\pi\)
0.0323423 + 0.999477i \(0.489703\pi\)
\(240\) 0 0
\(241\) 13.0000 + 22.5167i 0.837404 + 1.45043i 0.892058 + 0.451920i \(0.149261\pi\)
−0.0546547 + 0.998505i \(0.517406\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.00000 15.5885i 0.572656 0.991870i
\(248\) 0 0
\(249\) 6.00000 + 10.3923i 0.380235 + 0.658586i
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) −30.0000 −1.88608
\(254\) 0 0
\(255\) −1.50000 2.59808i −0.0939336 0.162698i
\(256\) 0 0
\(257\) −7.00000 + 12.1244i −0.436648 + 0.756297i −0.997429 0.0716680i \(-0.977168\pi\)
0.560781 + 0.827964i \(0.310501\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 27.0000 46.7654i 1.67126 2.89470i
\(262\) 0 0
\(263\) −1.00000 1.73205i −0.0616626 0.106803i 0.833546 0.552450i \(-0.186307\pi\)
−0.895209 + 0.445647i \(0.852974\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −9.00000 15.5885i −0.548740 0.950445i −0.998361 0.0572259i \(-0.981774\pi\)
0.449622 0.893219i \(-0.351559\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.50000 4.33013i 0.150756 0.261116i
\(276\) 0 0
\(277\) −3.00000 5.19615i −0.180253 0.312207i 0.761714 0.647913i \(-0.224358\pi\)
−0.941966 + 0.335707i \(0.891025\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 11.0000 0.656205 0.328102 0.944642i \(-0.393591\pi\)
0.328102 + 0.944642i \(0.393591\pi\)
\(282\) 0 0
\(283\) 15.5000 + 26.8468i 0.921379 + 1.59588i 0.797283 + 0.603606i \(0.206270\pi\)
0.124096 + 0.992270i \(0.460397\pi\)
\(284\) 0 0
\(285\) 9.00000 15.5885i 0.533114 0.923381i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 13.8564i 0.470588 0.815083i
\(290\) 0 0
\(291\) 19.5000 + 33.7750i 1.14311 + 1.97993i
\(292\) 0 0
\(293\) −5.00000 −0.292103 −0.146052 0.989277i \(-0.546657\pi\)
−0.146052 + 0.989277i \(0.546657\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) 22.5000 + 38.9711i 1.30558 + 2.26134i
\(298\) 0 0
\(299\) 9.00000 15.5885i 0.520483 0.901504i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −9.00000 + 15.5885i −0.517036 + 0.895533i
\(304\) 0 0
\(305\) −4.00000 6.92820i −0.229039 0.396708i
\(306\) 0 0
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) 57.0000 3.24262
\(310\) 0 0
\(311\) 9.00000 + 15.5885i 0.510343 + 0.883940i 0.999928 + 0.0119847i \(0.00381495\pi\)
−0.489585 + 0.871956i \(0.662852\pi\)
\(312\) 0 0
\(313\) 3.50000 6.06218i 0.197832 0.342655i −0.749993 0.661445i \(-0.769943\pi\)
0.947825 + 0.318791i \(0.103277\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.0000 22.5167i 0.730153 1.26466i −0.226665 0.973973i \(-0.572782\pi\)
0.956818 0.290689i \(-0.0938844\pi\)
\(318\) 0 0
\(319\) −22.5000 38.9711i −1.25976 2.18197i
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 1.50000 + 2.59808i 0.0832050 + 0.144115i
\(326\) 0 0
\(327\) 4.50000 7.79423i 0.248851 0.431022i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.0000 + 24.2487i −0.769510 + 1.33283i 0.168320 + 0.985732i \(0.446166\pi\)
−0.937829 + 0.347097i \(0.887167\pi\)
\(332\) 0 0
\(333\) −6.00000 10.3923i −0.328798 0.569495i
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) −21.0000 36.3731i −1.14056 1.97551i
\(340\) 0 0
\(341\) −10.0000 + 17.3205i −0.541530 + 0.937958i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 9.00000 15.5885i 0.484544 0.839254i
\(346\) 0 0
\(347\) −5.00000 8.66025i −0.268414 0.464907i 0.700038 0.714105i \(-0.253166\pi\)
−0.968452 + 0.249198i \(0.919833\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −27.0000 −1.44115
\(352\) 0 0
\(353\) −1.50000 2.59808i −0.0798369 0.138282i 0.823343 0.567545i \(-0.192107\pi\)
−0.903179 + 0.429263i \(0.858773\pi\)
\(354\) 0 0
\(355\) 4.00000 6.92820i 0.212298 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.00000 13.8564i 0.422224 0.731313i −0.573933 0.818902i \(-0.694583\pi\)
0.996157 + 0.0875892i \(0.0279163\pi\)
\(360\) 0 0
\(361\) −8.50000 14.7224i −0.447368 0.774865i
\(362\) 0 0
\(363\) 42.0000 2.20443
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 8.50000 + 14.7224i 0.443696 + 0.768505i 0.997960 0.0638362i \(-0.0203335\pi\)
−0.554264 + 0.832341i \(0.687000\pi\)
\(368\) 0 0
\(369\) 12.0000 20.7846i 0.624695 1.08200i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 3.46410i 0.103556 0.179364i −0.809591 0.586994i \(-0.800311\pi\)
0.913147 + 0.407630i \(0.133645\pi\)
\(374\) 0 0
\(375\) 1.50000 + 2.59808i 0.0774597 + 0.134164i
\(376\) 0 0
\(377\) 27.0000 1.39057
\(378\) 0 0
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 12.0000 + 20.7846i 0.614779 + 1.06483i
\(382\) 0 0
\(383\) −18.0000 + 31.1769i −0.919757 + 1.59307i −0.119974 + 0.992777i \(0.538281\pi\)
−0.799783 + 0.600289i \(0.795052\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −30.0000 + 51.9615i −1.52499 + 2.64135i
\(388\) 0 0
\(389\) 15.5000 + 26.8468i 0.785881 + 1.36119i 0.928471 + 0.371404i \(0.121124\pi\)
−0.142590 + 0.989782i \(0.545543\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) −30.0000 −1.51330
\(394\) 0 0
\(395\) 6.50000 + 11.2583i 0.327050 + 0.566468i
\(396\) 0 0
\(397\) −3.50000 + 6.06218i −0.175660 + 0.304252i −0.940389 0.340099i \(-0.889539\pi\)
0.764730 + 0.644351i \(0.222873\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.5000 + 21.6506i −0.624220 + 1.08118i 0.364471 + 0.931215i \(0.381250\pi\)
−0.988691 + 0.149966i \(0.952083\pi\)
\(402\) 0 0
\(403\) −6.00000 10.3923i −0.298881 0.517678i
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −3.00000 5.19615i −0.148340 0.256933i 0.782274 0.622935i \(-0.214060\pi\)
−0.930614 + 0.366002i \(0.880726\pi\)
\(410\) 0 0
\(411\) 18.0000 31.1769i 0.887875 1.53784i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −2.00000 + 3.46410i −0.0981761 + 0.170046i
\(416\) 0 0
\(417\) 21.0000 + 36.3731i 1.02837 + 1.78120i
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 3.00000 + 5.19615i 0.145865 + 0.252646i
\(424\) 0 0
\(425\) 0.500000 0.866025i 0.0242536 0.0420084i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −22.5000 + 38.9711i −1.08631 + 1.88154i
\(430\) 0 0
\(431\) 4.50000 + 7.79423i 0.216757 + 0.375435i 0.953815 0.300395i \(-0.0971186\pi\)
−0.737057 + 0.675830i \(0.763785\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 27.0000 1.29455
\(436\) 0 0
\(437\) −18.0000 31.1769i −0.861057 1.49139i
\(438\) 0 0
\(439\) 1.00000 1.73205i 0.0477274 0.0826663i −0.841175 0.540763i \(-0.818135\pi\)
0.888902 + 0.458097i \(0.151469\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.00000 12.1244i 0.332580 0.576046i −0.650437 0.759560i \(-0.725414\pi\)
0.983017 + 0.183515i \(0.0587475\pi\)
\(444\) 0 0
\(445\) 2.00000 + 3.46410i 0.0948091 + 0.164214i
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) −10.0000 17.3205i −0.470882 0.815591i
\(452\) 0 0
\(453\) −7.50000 + 12.9904i −0.352381 + 0.610341i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 + 13.8564i −0.374224 + 0.648175i −0.990211 0.139581i \(-0.955424\pi\)
0.615986 + 0.787757i \(0.288758\pi\)
\(458\) 0 0
\(459\) 4.50000 + 7.79423i 0.210042 + 0.363803i
\(460\) 0 0
\(461\) 40.0000 1.86299 0.931493 0.363760i \(-0.118507\pi\)
0.931493 + 0.363760i \(0.118507\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) −6.00000 10.3923i −0.278243 0.481932i
\(466\) 0 0
\(467\) −2.50000 + 4.33013i −0.115686 + 0.200374i −0.918054 0.396456i \(-0.870240\pi\)
0.802368 + 0.596830i \(0.203573\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.00000 5.19615i 0.138233 0.239426i
\(472\) 0 0
\(473\) 25.0000 + 43.3013i 1.14950 + 1.99099i
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) 24.0000 1.09888
\(478\) 0 0
\(479\) −3.00000 5.19615i −0.137073 0.237418i 0.789314 0.613990i \(-0.210436\pi\)
−0.926388 + 0.376571i \(0.877103\pi\)
\(480\) 0 0
\(481\) 3.00000 5.19615i 0.136788 0.236924i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.50000 + 11.2583i −0.295150 + 0.511214i
\(486\) 0 0
\(487\) −13.0000 22.5167i −0.589086 1.02033i −0.994352 0.106129i \(-0.966154\pi\)
0.405266 0.914199i \(-0.367179\pi\)
\(488\) 0 0
\(489\) −30.0000 −1.35665
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) −4.50000 7.79423i −0.202670 0.351034i
\(494\) 0 0
\(495\) −15.0000 + 25.9808i −0.674200 + 1.16775i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.50000 7.79423i 0.201448 0.348918i −0.747547 0.664208i \(-0.768769\pi\)
0.948995 + 0.315291i \(0.102102\pi\)
\(500\) 0 0
\(501\) −4.50000 7.79423i −0.201045 0.348220i
\(502\) 0 0
\(503\) −3.00000 −0.133763 −0.0668817 0.997761i \(-0.521305\pi\)
−0.0668817 + 0.997761i \(0.521305\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 6.00000 + 10.3923i 0.266469 + 0.461538i
\(508\) 0 0
\(509\) 7.00000 12.1244i 0.310270 0.537403i −0.668151 0.744026i \(-0.732914\pi\)
0.978421 + 0.206623i \(0.0662474\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −27.0000 + 46.7654i −1.19208 + 2.06474i
\(514\) 0 0
\(515\) 9.50000 + 16.4545i 0.418620 + 0.725071i
\(516\) 0 0
\(517\) 5.00000 0.219900
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) −7.00000 12.1244i −0.306676 0.531178i 0.670957 0.741496i \(-0.265883\pi\)
−0.977633 + 0.210318i \(0.932550\pi\)
\(522\) 0 0
\(523\) −2.00000 + 3.46410i −0.0874539 + 0.151475i −0.906434 0.422347i \(-0.861206\pi\)
0.818980 + 0.573822i \(0.194540\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.00000 + 3.46410i −0.0871214 + 0.150899i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) −48.0000 −2.08302
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −3.00000 5.19615i −0.129701 0.224649i
\(536\) 0 0
\(537\) 18.0000 31.1769i 0.776757 1.34538i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −19.5000 + 33.7750i −0.838370 + 1.45210i 0.0528859 + 0.998601i \(0.483158\pi\)
−0.891256 + 0.453500i \(0.850175\pi\)
\(542\) 0 0
\(543\) 30.0000 + 51.9615i 1.28742 + 2.22988i
\(544\) 0 0
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) 0 0
\(549\) 24.0000 + 41.5692i 1.02430 + 1.77413i
\(550\) 0 0
\(551\) 27.0000 46.7654i 1.15024 1.99227i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3.00000 5.19615i 0.127343 0.220564i
\(556\) 0 0
\(557\) −20.0000 34.6410i −0.847427 1.46779i −0.883497 0.468438i \(-0.844817\pi\)
0.0360693 0.999349i \(-0.488516\pi\)
\(558\) 0 0
\(559\) −30.0000 −1.26886
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 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
\(565\) 7.00000 12.1244i 0.294492 0.510075i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.0000 25.9808i 0.628833 1.08917i −0.358954 0.933355i \(-0.616866\pi\)
0.987786 0.155815i \(-0.0498003\pi\)
\(570\) 0 0
\(571\) 6.00000 + 10.3923i 0.251092 + 0.434904i 0.963827 0.266529i \(-0.0858769\pi\)
−0.712735 + 0.701434i \(0.752544\pi\)
\(572\) 0 0
\(573\) 9.00000 0.375980
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 0 0
\(577\) −6.50000 11.2583i −0.270599 0.468690i 0.698417 0.715691i \(-0.253888\pi\)
−0.969015 + 0.247001i \(0.920555\pi\)
\(578\) 0 0
\(579\) 6.00000 10.3923i 0.249351 0.431889i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 10.0000 17.3205i 0.414158 0.717342i
\(584\) 0 0
\(585\) −9.00000 15.5885i −0.372104 0.644503i
\(586\) 0 0
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −12.0000 20.7846i −0.493614 0.854965i
\(592\) 0 0
\(593\) 13.5000 23.3827i 0.554379 0.960212i −0.443573 0.896238i \(-0.646289\pi\)
0.997952 0.0639736i \(-0.0203773\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −12.0000 + 20.7846i −0.491127 + 0.850657i
\(598\) 0 0
\(599\) −7.50000 12.9904i −0.306442 0.530773i 0.671140 0.741331i \(-0.265805\pi\)
−0.977581 + 0.210558i \(0.932472\pi\)
\(600\) 0 0
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 72.0000 2.93207
\(604\) 0 0
\(605\) 7.00000 + 12.1244i 0.284590 + 0.492925i
\(606\) 0 0
\(607\) −6.50000 + 11.2583i −0.263827 + 0.456962i −0.967256 0.253804i \(-0.918318\pi\)
0.703429 + 0.710766i \(0.251651\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.50000 + 2.59808i −0.0606835 + 0.105107i
\(612\) 0 0
\(613\) −21.0000 36.3731i −0.848182 1.46909i −0.882829 0.469695i \(-0.844364\pi\)
0.0346469 0.999400i \(-0.488969\pi\)
\(614\) 0 0
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −46.0000 −1.85189 −0.925945 0.377658i \(-0.876729\pi\)
−0.925945 + 0.377658i \(0.876729\pi\)
\(618\) 0 0
\(619\) 5.00000 + 8.66025i 0.200967 + 0.348085i 0.948840 0.315757i \(-0.102258\pi\)
−0.747873 + 0.663842i \(0.768925\pi\)
\(620\) 0 0
\(621\) −27.0000 + 46.7654i −1.08347 + 1.87663i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 45.0000 + 77.9423i 1.79713 + 3.11272i
\(628\) 0 0
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) 47.0000 1.87104 0.935520 0.353273i \(-0.114931\pi\)
0.935520 + 0.353273i \(0.114931\pi\)
\(632\) 0 0
\(633\) 16.5000 + 28.5788i 0.655816 + 1.13591i
\(634\) 0 0
\(635\) −4.00000 + 6.92820i −0.158735 + 0.274937i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −24.0000 + 41.5692i −0.949425 + 1.64445i
\(640\) 0 0
\(641\) −1.00000 1.73205i −0.0394976 0.0684119i 0.845601 0.533816i \(-0.179242\pi\)
−0.885098 + 0.465404i \(0.845909\pi\)
\(642\) 0 0
\(643\) 19.0000 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) −30.0000 −1.18125
\(646\) 0 0
\(647\) 4.00000 + 6.92820i 0.157256 + 0.272376i 0.933878 0.357591i \(-0.116402\pi\)
−0.776622 + 0.629967i \(0.783068\pi\)
\(648\) 0 0
\(649\) −20.0000 + 34.6410i −0.785069 + 1.35978i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00000 + 1.73205i −0.0391330 + 0.0677804i −0.884929 0.465727i \(-0.845793\pi\)
0.845796 + 0.533507i \(0.179126\pi\)
\(654\) 0 0
\(655\) −5.00000 8.66025i −0.195366 0.338384i
\(656\) 0 0
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) 31.0000 1.20759 0.603794 0.797140i \(-0.293655\pi\)
0.603794 + 0.797140i \(0.293655\pi\)
\(660\) 0 0
\(661\) 16.0000 + 27.7128i 0.622328 + 1.07790i 0.989051 + 0.147573i \(0.0471463\pi\)
−0.366723 + 0.930330i \(0.619520\pi\)
\(662\) 0 0
\(663\) −4.50000 + 7.79423i −0.174766 + 0.302703i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 27.0000 46.7654i 1.04544 1.81076i
\(668\) 0 0
\(669\) 7.50000 + 12.9904i 0.289967 + 0.502237i
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) −4.50000 7.79423i −0.173205 0.300000i
\(676\) 0 0
\(677\) 13.5000 23.3827i 0.518847 0.898670i −0.480913 0.876768i \(-0.659695\pi\)
0.999760 0.0219013i \(-0.00697196\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1.50000 2.59808i 0.0574801 0.0995585i
\(682\) 0 0
\(683\) −4.00000 6.92820i −0.153056 0.265100i 0.779294 0.626659i \(-0.215578\pi\)
−0.932349 + 0.361559i \(0.882245\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) 6.00000 + 10.3923i 0.228582 + 0.395915i
\(690\) 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
\(695\) −7.00000 + 12.1244i −0.265525 + 0.459903i
\(696\) 0 0
\(697\) −2.00000 3.46410i −0.0757554 0.131212i
\(698\) 0 0
\(699\) 72.0000 2.72329
\(700\) 0 0
\(701\) −5.00000 −0.188847 −0.0944237 0.995532i \(-0.530101\pi\)
−0.0944237 + 0.995532i \(0.530101\pi\)
\(702\) 0 0
\(703\) −6.00000 10.3923i −0.226294 0.391953i
\(704\) 0 0
\(705\) −1.50000 + 2.59808i −0.0564933 + 0.0978492i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.5000 + 26.8468i −0.582115 + 1.00825i 0.413114 + 0.910679i \(0.364441\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 0 0
\(711\) −39.0000 67.5500i −1.46261 2.53332i
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) −15.0000 −0.560968
\(716\) 0 0
\(717\) −1.50000 2.59808i −0.0560185 0.0970269i
\(718\) 0 0
\(719\) −3.00000 + 5.19615i −0.111881 + 0.193784i −0.916529 0.399969i \(-0.869021\pi\)
0.804648 + 0.593753i \(0.202354\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 39.0000 67.5500i 1.45043 2.51221i
\(724\) 0 0
\(725\) 4.50000 + 7.79423i 0.167126 + 0.289470i
\(726\) 0 0
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 5.00000 + 8.66025i 0.184932 + 0.320311i
\(732\) 0 0
\(733\) 23.5000 40.7032i 0.867992 1.50341i 0.00394730 0.999992i \(-0.498744\pi\)
0.864045 0.503415i \(-0.167923\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 30.0000 51.9615i 1.10506 1.91403i
\(738\) 0 0
\(739\) 2.50000 + 4.33013i 0.0919640 + 0.159286i 0.908337 0.418238i \(-0.137352\pi\)
−0.816373 + 0.577524i \(0.804019\pi\)
\(740\) 0 0
\(741\) −54.0000 −1.98374
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −3.00000 5.19615i −0.109911 0.190372i
\(746\) 0 0
\(747\) 12.0000 20.7846i 0.439057 0.760469i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 9.50000 16.4545i 0.346660 0.600433i −0.638994 0.769212i \(-0.720649\pi\)
0.985654 + 0.168779i \(0.0539825\pi\)
\(752\) 0 0
\(753\) −45.0000 77.9423i −1.63989 2.84037i
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) −32.0000 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) 0 0
\(759\) 45.0000 + 77.9423i 1.63340 + 2.82913i
\(760\) 0 0
\(761\) 9.00000 15.5885i 0.326250 0.565081i −0.655515 0.755182i \(-0.727548\pi\)
0.981764 + 0.190101i \(0.0608816\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −3.00000 + 5.19615i −0.108465 + 0.187867i
\(766\) 0 0
\(767\) −12.0000 20.7846i −0.433295 0.750489i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 42.0000 1.51259
\(772\) 0 0
\(773\) −22.5000 38.9711i −0.809269 1.40169i −0.913371 0.407128i \(-0.866530\pi\)
0.104102 0.994567i \(-0.466803\pi\)
\(774\) 0 0
\(775\) 2.00000 3.46410i 0.0718421 0.124434i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 20.7846i 0.429945 0.744686i
\(780\) 0 0
\(781\) 20.0000 + 34.6410i 0.715656 + 1.23955i
\(782\) 0 0
\(783\) −81.0000 −2.89470
\(784\) 0 0
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −15.5000 26.8468i −0.552515 0.956985i −0.998092 0.0617409i \(-0.980335\pi\)
0.445577 0.895244i \(-0.352999\pi\)
\(788\) 0 0
\(789\) −3.00000 + 5.19615i −0.106803 + 0.184988i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12.0000 + 20.7846i −0.426132 + 0.738083i
\(794\) 0 0
\(795\) 6.00000 + 10.3923i 0.212798 + 0.368577i
\(796\) 0 0
\(797\) −33.0000 −1.16892 −0.584460 0.811423i \(-0.698694\pi\)
−0.584460 + 0.811423i \(0.698694\pi\)
\(798\) 0 0
\(799\) 1.00000 0.0353775
\(800\) 0 0
\(801\) −12.0000 20.7846i −0.423999 0.734388i
\(802\) 0 0
\(803\) 5.00000 8.66025i 0.176446 0.305614i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −27.0000 + 46.7654i −0.950445 + 1.64622i
\(808\) 0 0
\(809\) −4.50000 7.79423i −0.158212 0.274030i 0.776012 0.630718i \(-0.217239\pi\)
−0.934224 + 0.356687i \(0.883906\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.00000 8.66025i −0.175142 0.303355i
\(816\) 0 0
\(817\) −30.0000 + 51.9615i −1.04957 + 1.81790i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.5000 33.7750i 0.680555 1.17876i −0.294257 0.955726i \(-0.595072\pi\)
0.974812 0.223029i \(-0.0715945\pi\)
\(822\) 0 0
\(823\) −14.0000 24.2487i −0.488009 0.845257i 0.511896 0.859048i \(-0.328943\pi\)
−0.999905 + 0.0137907i \(0.995610\pi\)
\(824\) 0 0
\(825\) −15.0000 −0.522233
\(826\) 0 0
\(827\) 50.0000 1.73867 0.869335 0.494223i \(-0.164547\pi\)
0.869335 + 0.494223i \(0.164547\pi\)
\(828\) 0 0
\(829\) 14.0000 + 24.2487i 0.486240 + 0.842193i 0.999875 0.0158163i \(-0.00503471\pi\)
−0.513635 + 0.858009i \(0.671701\pi\)
\(830\) 0 0
\(831\) −9.00000 + 15.5885i −0.312207 + 0.540758i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.50000 2.59808i 0.0519096 0.0899101i
\(836\) 0 0
\(837\) 18.0000 + 31.1769i 0.622171 + 1.07763i
\(838\) 0 0
\(839\) 2.00000 0.0690477 0.0345238 0.999404i \(-0.489009\pi\)
0.0345238 + 0.999404i \(0.489009\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) −16.5000 28.5788i −0.568290 0.984307i
\(844\) 0 0
\(845\) −2.00000 + 3.46410i −0.0688021 + 0.119169i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 46.5000 80.5404i 1.59588 2.76414i
\(850\) 0 0
\(851\) −6.00000 10.3923i −0.205677 0.356244i
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) −36.0000 −1.23117
\(856\) 0 0
\(857\) −9.00000 15.5885i −0.307434 0.532492i 0.670366 0.742030i \(-0.266137\pi\)
−0.977800 + 0.209539i \(0.932804\pi\)
\(858\) 0 0
\(859\) −18.0000 + 31.1769i −0.614152 + 1.06374i 0.376381 + 0.926465i \(0.377169\pi\)
−0.990533 + 0.137277i \(0.956165\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.00000 + 13.8564i −0.272323 + 0.471678i −0.969456 0.245264i \(-0.921125\pi\)
0.697133 + 0.716942i \(0.254459\pi\)
\(864\) 0 0
\(865\) −0.500000 0.866025i −0.0170005 0.0294457i
\(866\) 0 0
\(867\) −48.0000 −1.63017
\(868\) 0 0
\(869\) −65.0000 −2.20497
\(870\) 0 0
\(871\) 18.0000 + 31.1769i 0.609907 + 1.05639i
\(872\) 0 0
\(873\) 39.0000 67.5500i 1.31995 2.28622i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −13.0000 + 22.5167i −0.438979 + 0.760334i −0.997611 0.0690819i \(-0.977993\pi\)
0.558632 + 0.829416i \(0.311326\pi\)
\(878\) 0 0
\(879\) 7.50000 + 12.9904i 0.252969 + 0.438155i
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) −12.0000 20.7846i −0.403376 0.698667i
\(886\) 0 0
\(887\) −4.00000 + 6.92820i −0.134307 + 0.232626i −0.925332 0.379157i \(-0.876214\pi\)
0.791026 + 0.611783i \(0.209547\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 22.5000 38.9711i 0.753778 1.30558i
\(892\) 0 0
\(893\) 3.00000 + 5.19615i 0.100391 + 0.173883i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) −54.0000 −1.80301
\(898\) 0 0
\(899\) −18.0000 31.1769i −0.600334 1.03981i
\(900\) 0 0
\(901\) 2.00000 3.46410i 0.0666297 0.115406i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.0000 + 17.3205i −0.332411 + 0.575753i
\(906\) 0 0
\(907\) −7.00000 12.1244i −0.232431 0.402583i 0.726092 0.687598i \(-0.241335\pi\)
−0.958523 + 0.285015i \(0.908001\pi\)
\(908\) 0 0
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −10.0000 17.3205i −0.330952 0.573225i
\(914\) 0 0
\(915\) −12.0000 + 20.7846i −0.396708 + 0.687118i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.5000 26.8468i 0.511298 0.885594i −0.488616 0.872499i \(-0.662498\pi\)
0.999914 0.0130951i \(-0.00416842\pi\)
\(920\) 0 0
\(921\) 34.5000 + 59.7558i 1.13681 + 1.96902i
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −57.0000 98.7269i −1.87213 3.24262i
\(928\) 0 0
\(929\) −14.0000 + 24.2487i −0.459325 + 0.795574i −0.998925 0.0463469i \(-0.985242\pi\)
0.539600 + 0.841921i \(0.318575\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 27.0000 46.7654i 0.883940 1.53103i
\(934\) 0 0
\(935\) 2.50000 + 4.33013i 0.0817587 + 0.141610i
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) −21.0000 −0.685309
\(940\) 0 0
\(941\) −16.0000 27.7128i −0.521585 0.903412i −0.999685 0.0251063i \(-0.992008\pi\)
0.478100 0.878306i \(-0.341326\pi\)
\(942\) 0 0
\(943\) 12.0000 20.7846i 0.390774 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 + 6.92820i −0.129983 + 0.225136i −0.923670 0.383190i \(-0.874825\pi\)
0.793687 + 0.608326i \(0.208159\pi\)
\(948\) 0 0
\(949\) 3.00000 + 5.19615i 0.0973841 + 0.168674i
\(950\) 0 0
\(951\) −78.0000 −2.52932
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) 1.50000 + 2.59808i 0.0485389 + 0.0840718i
\(956\) 0 0
\(957\) −67.5000 + 116.913i −2.18197 + 3.77927i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 18.0000 + 31.1769i 0.580042 + 1.00466i
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 0 0
\(969\) 9.00000 + 15.5885i 0.289122 + 0.500773i
\(970\) 0 0
\(971\) 12.0000 20.7846i 0.385098 0.667010i −0.606685 0.794943i \(-0.707501\pi\)
0.991783 + 0.127933i \(0.0408342\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 4.50000 7.79423i 0.144115 0.249615i
\(976\) 0 0
\(977\) 21.0000 + 36.3731i 0.671850 + 1.16368i 0.977379 + 0.211495i \(0.0678332\pi\)
−0.305530 + 0.952183i \(0.598833\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) −18.0000 −0.574696
\(982\) 0 0
\(983\) 21.5000 + 37.2391i 0.685744 + 1.18774i 0.973203 + 0.229950i \(0.0738562\pi\)
−0.287459 + 0.957793i \(0.592811\pi\)
\(984\) 0 0
\(985\) 4.00000 6.92820i 0.127451 0.220751i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.0000 + 51.9615i −0.953945 + 1.65228i
\(990\) 0 0
\(991\) −8.00000 13.8564i −0.254128 0.440163i 0.710530 0.703667i \(-0.248455\pi\)
−0.964658 + 0.263504i \(0.915122\pi\)
\(992\) 0 0
\(993\) 84.0000 2.66566
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −5.50000 9.52628i −0.174187 0.301700i 0.765693 0.643206i \(-0.222396\pi\)
−0.939880 + 0.341506i \(0.889063\pi\)
\(998\) 0 0
\(999\) −9.00000 + 15.5885i −0.284747 + 0.493197i
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 980.2.i.b.961.1 2
7.2 even 3 140.2.a.b.1.1 1
7.3 odd 6 980.2.i.j.361.1 2
7.4 even 3 inner 980.2.i.b.361.1 2
7.5 odd 6 980.2.a.b.1.1 1
7.6 odd 2 980.2.i.j.961.1 2
21.2 odd 6 1260.2.a.h.1.1 1
21.5 even 6 8820.2.a.n.1.1 1
28.19 even 6 3920.2.a.bl.1.1 1
28.23 odd 6 560.2.a.a.1.1 1
35.2 odd 12 700.2.e.a.449.1 2
35.9 even 6 700.2.a.b.1.1 1
35.12 even 12 4900.2.e.a.2549.2 2
35.19 odd 6 4900.2.a.u.1.1 1
35.23 odd 12 700.2.e.a.449.2 2
35.33 even 12 4900.2.e.a.2549.1 2
56.37 even 6 2240.2.a.c.1.1 1
56.51 odd 6 2240.2.a.bb.1.1 1
84.23 even 6 5040.2.a.bd.1.1 1
105.2 even 12 6300.2.k.p.6049.1 2
105.23 even 12 6300.2.k.p.6049.2 2
105.44 odd 6 6300.2.a.bf.1.1 1
140.23 even 12 2800.2.g.c.449.1 2
140.79 odd 6 2800.2.a.be.1.1 1
140.107 even 12 2800.2.g.c.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.a.b.1.1 1 7.2 even 3
560.2.a.a.1.1 1 28.23 odd 6
700.2.a.b.1.1 1 35.9 even 6
700.2.e.a.449.1 2 35.2 odd 12
700.2.e.a.449.2 2 35.23 odd 12
980.2.a.b.1.1 1 7.5 odd 6
980.2.i.b.361.1 2 7.4 even 3 inner
980.2.i.b.961.1 2 1.1 even 1 trivial
980.2.i.j.361.1 2 7.3 odd 6
980.2.i.j.961.1 2 7.6 odd 2
1260.2.a.h.1.1 1 21.2 odd 6
2240.2.a.c.1.1 1 56.37 even 6
2240.2.a.bb.1.1 1 56.51 odd 6
2800.2.a.be.1.1 1 140.79 odd 6
2800.2.g.c.449.1 2 140.23 even 12
2800.2.g.c.449.2 2 140.107 even 12
3920.2.a.bl.1.1 1 28.19 even 6
4900.2.a.u.1.1 1 35.19 odd 6
4900.2.e.a.2549.1 2 35.33 even 12
4900.2.e.a.2549.2 2 35.12 even 12
5040.2.a.bd.1.1 1 84.23 even 6
6300.2.a.bf.1.1 1 105.44 odd 6
6300.2.k.p.6049.1 2 105.2 even 12
6300.2.k.p.6049.2 2 105.23 even 12
8820.2.a.n.1.1 1 21.5 even 6