Properties

Label 560.2.q.e.401.1
Level $560$
Weight $2$
Character 560.401
Analytic conductor $4.472$
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] = [560,2,Mod(81,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("560.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.q (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: \(4.47162251319\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 401.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 560.401
Dual form 560.2.q.e.81.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(1.00000 - 1.73205i) q^{9} +(-1.00000 - 1.73205i) q^{11} +4.00000 q^{13} -1.00000 q^{15} +(3.00000 - 5.19615i) q^{19} +(2.50000 - 0.866025i) q^{21} +(1.50000 - 2.59808i) q^{23} +(-0.500000 - 0.866025i) q^{25} -5.00000 q^{27} -3.00000 q^{29} +(-1.00000 + 1.73205i) q^{33} +(2.00000 + 1.73205i) q^{35} +(6.00000 - 10.3923i) q^{37} +(-2.00000 - 3.46410i) q^{39} -7.00000 q^{41} +9.00000 q^{43} +(-1.00000 - 1.73205i) q^{45} +(-6.50000 - 2.59808i) q^{49} +(3.00000 + 5.19615i) q^{53} -2.00000 q^{55} -6.00000 q^{57} +(-5.00000 - 8.66025i) q^{59} +(-2.50000 + 4.33013i) q^{61} +(4.00000 + 3.46410i) q^{63} +(2.00000 - 3.46410i) q^{65} +(5.50000 + 9.52628i) q^{67} -3.00000 q^{69} +10.0000 q^{71} +(4.00000 + 6.92820i) q^{73} +(-0.500000 + 0.866025i) q^{75} +(5.00000 - 1.73205i) q^{77} +(3.00000 - 5.19615i) q^{79} +(-0.500000 - 0.866025i) q^{81} +3.00000 q^{83} +(1.50000 + 2.59808i) q^{87} +(-8.50000 + 14.7224i) q^{89} +(-2.00000 + 10.3923i) q^{91} +(-3.00000 - 5.19615i) q^{95} -2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{5} - q^{7} + 2 q^{9} - 2 q^{11} + 8 q^{13} - 2 q^{15} + 6 q^{19} + 5 q^{21} + 3 q^{23} - q^{25} - 10 q^{27} - 6 q^{29} - 2 q^{33} + 4 q^{35} + 12 q^{37} - 4 q^{39} - 14 q^{41} + 18 q^{43} - 2 q^{45} - 13 q^{49} + 6 q^{53} - 4 q^{55} - 12 q^{57} - 10 q^{59} - 5 q^{61} + 8 q^{63} + 4 q^{65} + 11 q^{67} - 6 q^{69} + 20 q^{71} + 8 q^{73} - q^{75} + 10 q^{77} + 6 q^{79} - q^{81} + 6 q^{83} + 3 q^{87} - 17 q^{89} - 4 q^{91} - 6 q^{95} - 4 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

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
\(3\) −0.500000 0.866025i −0.288675 0.500000i 0.684819 0.728714i \(-0.259881\pi\)
−0.973494 + 0.228714i \(0.926548\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 1.00000 1.73205i 0.333333 0.577350i
\(10\) 0 0
\(11\) −1.00000 1.73205i −0.301511 0.522233i 0.674967 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 3.00000 5.19615i 0.688247 1.19208i −0.284157 0.958778i \(-0.591714\pi\)
0.972404 0.233301i \(-0.0749529\pi\)
\(20\) 0 0
\(21\) 2.50000 0.866025i 0.545545 0.188982i
\(22\) 0 0
\(23\) 1.50000 2.59808i 0.312772 0.541736i −0.666190 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) −1.00000 + 1.73205i −0.174078 + 0.301511i
\(34\) 0 0
\(35\) 2.00000 + 1.73205i 0.338062 + 0.292770i
\(36\) 0 0
\(37\) 6.00000 10.3923i 0.986394 1.70848i 0.350823 0.936442i \(-0.385902\pi\)
0.635571 0.772043i \(-0.280765\pi\)
\(38\) 0 0
\(39\) −2.00000 3.46410i −0.320256 0.554700i
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 9.00000 1.37249 0.686244 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) 0 0
\(45\) −1.00000 1.73205i −0.149071 0.258199i
\(46\) 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
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) −5.00000 8.66025i −0.650945 1.12747i −0.982894 0.184172i \(-0.941040\pi\)
0.331949 0.943297i \(-0.392294\pi\)
\(60\) 0 0
\(61\) −2.50000 + 4.33013i −0.320092 + 0.554416i −0.980507 0.196485i \(-0.937047\pi\)
0.660415 + 0.750901i \(0.270381\pi\)
\(62\) 0 0
\(63\) 4.00000 + 3.46410i 0.503953 + 0.436436i
\(64\) 0 0
\(65\) 2.00000 3.46410i 0.248069 0.429669i
\(66\) 0 0
\(67\) 5.50000 + 9.52628i 0.671932 + 1.16382i 0.977356 + 0.211604i \(0.0678686\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) 4.00000 + 6.92820i 0.468165 + 0.810885i 0.999338 0.0363782i \(-0.0115821\pi\)
−0.531174 + 0.847263i \(0.678249\pi\)
\(74\) 0 0
\(75\) −0.500000 + 0.866025i −0.0577350 + 0.100000i
\(76\) 0 0
\(77\) 5.00000 1.73205i 0.569803 0.197386i
\(78\) 0 0
\(79\) 3.00000 5.19615i 0.337526 0.584613i −0.646440 0.762964i \(-0.723743\pi\)
0.983967 + 0.178352i \(0.0570765\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.50000 + 2.59808i 0.160817 + 0.278543i
\(88\) 0 0
\(89\) −8.50000 + 14.7224i −0.900998 + 1.56057i −0.0747975 + 0.997199i \(0.523831\pi\)
−0.826201 + 0.563376i \(0.809502\pi\)
\(90\) 0 0
\(91\) −2.00000 + 10.3923i −0.209657 + 1.08941i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.00000 5.19615i −0.307794 0.533114i
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 8.50000 + 14.7224i 0.845782 + 1.46494i 0.884941 + 0.465704i \(0.154199\pi\)
−0.0391591 + 0.999233i \(0.512468\pi\)
\(102\) 0 0
\(103\) −7.50000 + 12.9904i −0.738997 + 1.27998i 0.213950 + 0.976845i \(0.431367\pi\)
−0.952947 + 0.303136i \(0.901966\pi\)
\(104\) 0 0
\(105\) 0.500000 2.59808i 0.0487950 0.253546i
\(106\) 0 0
\(107\) −0.500000 + 0.866025i −0.0483368 + 0.0837218i −0.889182 0.457555i \(-0.848725\pi\)
0.840845 + 0.541276i \(0.182059\pi\)
\(108\) 0 0
\(109\) 2.50000 + 4.33013i 0.239457 + 0.414751i 0.960558 0.278078i \(-0.0896974\pi\)
−0.721102 + 0.692829i \(0.756364\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −1.50000 2.59808i −0.139876 0.242272i
\(116\) 0 0
\(117\) 4.00000 6.92820i 0.369800 0.640513i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 3.50000 + 6.06218i 0.315584 + 0.546608i
\(124\) 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
\(129\) −4.50000 7.79423i −0.396203 0.686244i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 12.0000 + 10.3923i 1.04053 + 0.901127i
\(134\) 0 0
\(135\) −2.50000 + 4.33013i −0.215166 + 0.372678i
\(136\) 0 0
\(137\) 2.00000 + 3.46410i 0.170872 + 0.295958i 0.938725 0.344668i \(-0.112008\pi\)
−0.767853 + 0.640626i \(0.778675\pi\)
\(138\) 0 0
\(139\) −18.0000 −1.52674 −0.763370 0.645961i \(-0.776457\pi\)
−0.763370 + 0.645961i \(0.776457\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 6.92820i −0.334497 0.579365i
\(144\) 0 0
\(145\) −1.50000 + 2.59808i −0.124568 + 0.215758i
\(146\) 0 0
\(147\) 1.00000 + 6.92820i 0.0824786 + 0.571429i
\(148\) 0 0
\(149\) 8.50000 14.7224i 0.696347 1.20611i −0.273377 0.961907i \(-0.588141\pi\)
0.969724 0.244202i \(-0.0785259\pi\)
\(150\) 0 0
\(151\) 10.0000 + 17.3205i 0.813788 + 1.40952i 0.910195 + 0.414181i \(0.135932\pi\)
−0.0964061 + 0.995342i \(0.530735\pi\)
\(152\) 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
\(159\) 3.00000 5.19615i 0.237915 0.412082i
\(160\) 0 0
\(161\) 6.00000 + 5.19615i 0.472866 + 0.409514i
\(162\) 0 0
\(163\) −6.00000 + 10.3923i −0.469956 + 0.813988i −0.999410 0.0343508i \(-0.989064\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(164\) 0 0
\(165\) 1.00000 + 1.73205i 0.0778499 + 0.134840i
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −6.00000 10.3923i −0.458831 0.794719i
\(172\) 0 0
\(173\) 4.00000 6.92820i 0.304114 0.526742i −0.672949 0.739689i \(-0.734973\pi\)
0.977064 + 0.212947i \(0.0683062\pi\)
\(174\) 0 0
\(175\) 2.50000 0.866025i 0.188982 0.0654654i
\(176\) 0 0
\(177\) −5.00000 + 8.66025i −0.375823 + 0.650945i
\(178\) 0 0
\(179\) −4.00000 6.92820i −0.298974 0.517838i 0.676927 0.736050i \(-0.263311\pi\)
−0.975901 + 0.218212i \(0.929978\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) −6.00000 10.3923i −0.441129 0.764057i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.50000 12.9904i 0.181848 0.944911i
\(190\) 0 0
\(191\) 7.00000 12.1244i 0.506502 0.877288i −0.493469 0.869763i \(-0.664272\pi\)
0.999972 0.00752447i \(-0.00239513\pi\)
\(192\) 0 0
\(193\) 11.0000 + 19.0526i 0.791797 + 1.37143i 0.924853 + 0.380325i \(0.124188\pi\)
−0.133056 + 0.991109i \(0.542479\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 6.00000 + 10.3923i 0.425329 + 0.736691i 0.996451 0.0841740i \(-0.0268252\pi\)
−0.571122 + 0.820865i \(0.693492\pi\)
\(200\) 0 0
\(201\) 5.50000 9.52628i 0.387940 0.671932i
\(202\) 0 0
\(203\) 1.50000 7.79423i 0.105279 0.547048i
\(204\) 0 0
\(205\) −3.50000 + 6.06218i −0.244451 + 0.423401i
\(206\) 0 0
\(207\) −3.00000 5.19615i −0.208514 0.361158i
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) −5.00000 8.66025i −0.342594 0.593391i
\(214\) 0 0
\(215\) 4.50000 7.79423i 0.306897 0.531562i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 4.00000 6.92820i 0.270295 0.468165i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −2.00000 3.46410i −0.132745 0.229920i 0.791989 0.610535i \(-0.209046\pi\)
−0.924734 + 0.380615i \(0.875712\pi\)
\(228\) 0 0
\(229\) 1.00000 1.73205i 0.0660819 0.114457i −0.831092 0.556136i \(-0.812283\pi\)
0.897173 + 0.441679i \(0.145617\pi\)
\(230\) 0 0
\(231\) −4.00000 3.46410i −0.263181 0.227921i
\(232\) 0 0
\(233\) 2.00000 3.46410i 0.131024 0.226941i −0.793047 0.609160i \(-0.791507\pi\)
0.924072 + 0.382219i \(0.124840\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.00000 −0.389742
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −9.00000 15.5885i −0.579741 1.00414i −0.995509 0.0946700i \(-0.969820\pi\)
0.415768 0.909471i \(-0.363513\pi\)
\(242\) 0 0
\(243\) −8.00000 + 13.8564i −0.513200 + 0.888889i
\(244\) 0 0
\(245\) −5.50000 + 4.33013i −0.351382 + 0.276642i
\(246\) 0 0
\(247\) 12.0000 20.7846i 0.763542 1.32249i
\(248\) 0 0
\(249\) −1.50000 2.59808i −0.0950586 0.164646i
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.0000 + 20.7846i −0.748539 + 1.29651i 0.199983 + 0.979799i \(0.435911\pi\)
−0.948523 + 0.316709i \(0.897422\pi\)
\(258\) 0 0
\(259\) 24.0000 + 20.7846i 1.49129 + 1.29149i
\(260\) 0 0
\(261\) −3.00000 + 5.19615i −0.185695 + 0.321634i
\(262\) 0 0
\(263\) 10.5000 + 18.1865i 0.647458 + 1.12143i 0.983728 + 0.179664i \(0.0575011\pi\)
−0.336270 + 0.941766i \(0.609166\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 17.0000 1.04038
\(268\) 0 0
\(269\) −15.5000 26.8468i −0.945052 1.63688i −0.755648 0.654978i \(-0.772678\pi\)
−0.189404 0.981899i \(-0.560656\pi\)
\(270\) 0 0
\(271\) −1.00000 + 1.73205i −0.0607457 + 0.105215i −0.894799 0.446469i \(-0.852681\pi\)
0.834053 + 0.551684i \(0.186015\pi\)
\(272\) 0 0
\(273\) 10.0000 3.46410i 0.605228 0.209657i
\(274\) 0 0
\(275\) −1.00000 + 1.73205i −0.0603023 + 0.104447i
\(276\) 0 0
\(277\) −4.00000 6.92820i −0.240337 0.416275i 0.720473 0.693482i \(-0.243925\pi\)
−0.960810 + 0.277207i \(0.910591\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) −14.0000 24.2487i −0.832214 1.44144i −0.896279 0.443491i \(-0.853740\pi\)
0.0640654 0.997946i \(-0.479593\pi\)
\(284\) 0 0
\(285\) −3.00000 + 5.19615i −0.177705 + 0.307794i
\(286\) 0 0
\(287\) 3.50000 18.1865i 0.206598 1.07352i
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 1.00000 + 1.73205i 0.0586210 + 0.101535i
\(292\) 0 0
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) −10.0000 −0.582223
\(296\) 0 0
\(297\) 5.00000 + 8.66025i 0.290129 + 0.502519i
\(298\) 0 0
\(299\) 6.00000 10.3923i 0.346989 0.601003i
\(300\) 0 0
\(301\) −4.50000 + 23.3827i −0.259376 + 1.34776i
\(302\) 0 0
\(303\) 8.50000 14.7224i 0.488312 0.845782i
\(304\) 0 0
\(305\) 2.50000 + 4.33013i 0.143150 + 0.247942i
\(306\) 0 0
\(307\) 21.0000 1.19853 0.599267 0.800549i \(-0.295459\pi\)
0.599267 + 0.800549i \(0.295459\pi\)
\(308\) 0 0
\(309\) 15.0000 0.853320
\(310\) 0 0
\(311\) 5.00000 + 8.66025i 0.283524 + 0.491078i 0.972250 0.233944i \(-0.0751631\pi\)
−0.688726 + 0.725022i \(0.741830\pi\)
\(312\) 0 0
\(313\) 8.00000 13.8564i 0.452187 0.783210i −0.546335 0.837567i \(-0.683977\pi\)
0.998522 + 0.0543564i \(0.0173107\pi\)
\(314\) 0 0
\(315\) 5.00000 1.73205i 0.281718 0.0975900i
\(316\) 0 0
\(317\) 6.00000 10.3923i 0.336994 0.583690i −0.646872 0.762598i \(-0.723923\pi\)
0.983866 + 0.178908i \(0.0572566\pi\)
\(318\) 0 0
\(319\) 3.00000 + 5.19615i 0.167968 + 0.290929i
\(320\) 0 0
\(321\) 1.00000 0.0558146
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −2.00000 3.46410i −0.110940 0.192154i
\(326\) 0 0
\(327\) 2.50000 4.33013i 0.138250 0.239457i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 3.46410i 0.109930 0.190404i −0.805812 0.592172i \(-0.798271\pi\)
0.915742 + 0.401768i \(0.131604\pi\)
\(332\) 0 0
\(333\) −12.0000 20.7846i −0.657596 1.13899i
\(334\) 0 0
\(335\) 11.0000 0.600994
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 9.00000 + 15.5885i 0.488813 + 0.846649i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) −1.50000 + 2.59808i −0.0807573 + 0.139876i
\(346\) 0 0
\(347\) −12.5000 21.6506i −0.671035 1.16227i −0.977611 0.210421i \(-0.932517\pi\)
0.306576 0.951846i \(-0.400817\pi\)
\(348\) 0 0
\(349\) 17.0000 0.909989 0.454995 0.890494i \(-0.349641\pi\)
0.454995 + 0.890494i \(0.349641\pi\)
\(350\) 0 0
\(351\) −20.0000 −1.06752
\(352\) 0 0
\(353\) −9.00000 15.5885i −0.479022 0.829690i 0.520689 0.853746i \(-0.325675\pi\)
−0.999711 + 0.0240566i \(0.992342\pi\)
\(354\) 0 0
\(355\) 5.00000 8.66025i 0.265372 0.459639i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.0000 + 20.7846i −0.633336 + 1.09697i 0.353529 + 0.935423i \(0.384981\pi\)
−0.986865 + 0.161546i \(0.948352\pi\)
\(360\) 0 0
\(361\) −8.50000 14.7224i −0.447368 0.774865i
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 6.50000 + 11.2583i 0.339297 + 0.587680i 0.984301 0.176500i \(-0.0564774\pi\)
−0.645003 + 0.764180i \(0.723144\pi\)
\(368\) 0 0
\(369\) −7.00000 + 12.1244i −0.364405 + 0.631169i
\(370\) 0 0
\(371\) −15.0000 + 5.19615i −0.778761 + 0.269771i
\(372\) 0 0
\(373\) −18.0000 + 31.1769i −0.932005 + 1.61428i −0.152115 + 0.988363i \(0.548608\pi\)
−0.779890 + 0.625917i \(0.784725\pi\)
\(374\) 0 0
\(375\) 0.500000 + 0.866025i 0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 0 0
\(381\) 4.00000 + 6.92820i 0.204926 + 0.354943i
\(382\) 0 0
\(383\) −16.5000 + 28.5788i −0.843111 + 1.46031i 0.0441413 + 0.999025i \(0.485945\pi\)
−0.887252 + 0.461285i \(0.847389\pi\)
\(384\) 0 0
\(385\) 1.00000 5.19615i 0.0509647 0.264820i
\(386\) 0 0
\(387\) 9.00000 15.5885i 0.457496 0.792406i
\(388\) 0 0
\(389\) 7.00000 + 12.1244i 0.354914 + 0.614729i 0.987103 0.160085i \(-0.0511768\pi\)
−0.632189 + 0.774814i \(0.717843\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.00000 5.19615i −0.150946 0.261447i
\(396\) 0 0
\(397\) 1.00000 1.73205i 0.0501886 0.0869291i −0.839840 0.542834i \(-0.817351\pi\)
0.890028 + 0.455905i \(0.150684\pi\)
\(398\) 0 0
\(399\) 3.00000 15.5885i 0.150188 0.780399i
\(400\) 0 0
\(401\) 16.5000 28.5788i 0.823971 1.42716i −0.0787327 0.996896i \(-0.525087\pi\)
0.902703 0.430263i \(-0.141579\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 6.50000 + 11.2583i 0.321404 + 0.556689i 0.980778 0.195127i \(-0.0625118\pi\)
−0.659374 + 0.751815i \(0.729178\pi\)
\(410\) 0 0
\(411\) 2.00000 3.46410i 0.0986527 0.170872i
\(412\) 0 0
\(413\) 25.0000 8.66025i 1.23017 0.426143i
\(414\) 0 0
\(415\) 1.50000 2.59808i 0.0736321 0.127535i
\(416\) 0 0
\(417\) 9.00000 + 15.5885i 0.440732 + 0.763370i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −31.0000 −1.51085 −0.755424 0.655237i \(-0.772569\pi\)
−0.755424 + 0.655237i \(0.772569\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 8.66025i −0.483934 0.419099i
\(428\) 0 0
\(429\) −4.00000 + 6.92820i −0.193122 + 0.334497i
\(430\) 0 0
\(431\) −11.0000 19.0526i −0.529851 0.917729i −0.999394 0.0348195i \(-0.988914\pi\)
0.469542 0.882910i \(-0.344419\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) −9.00000 15.5885i −0.430528 0.745697i
\(438\) 0 0
\(439\) −2.00000 + 3.46410i −0.0954548 + 0.165333i −0.909798 0.415051i \(-0.863764\pi\)
0.814344 + 0.580383i \(0.197097\pi\)
\(440\) 0 0
\(441\) −11.0000 + 8.66025i −0.523810 + 0.412393i
\(442\) 0 0
\(443\) −10.5000 + 18.1865i −0.498870 + 0.864068i −0.999999 0.00130426i \(-0.999585\pi\)
0.501129 + 0.865373i \(0.332918\pi\)
\(444\) 0 0
\(445\) 8.50000 + 14.7224i 0.402939 + 0.697910i
\(446\) 0 0
\(447\) −17.0000 −0.804072
\(448\) 0 0
\(449\) 39.0000 1.84052 0.920262 0.391303i \(-0.127976\pi\)
0.920262 + 0.391303i \(0.127976\pi\)
\(450\) 0 0
\(451\) 7.00000 + 12.1244i 0.329617 + 0.570914i
\(452\) 0 0
\(453\) 10.0000 17.3205i 0.469841 0.813788i
\(454\) 0 0
\(455\) 8.00000 + 6.92820i 0.375046 + 0.324799i
\(456\) 0 0
\(457\) −4.00000 + 6.92820i −0.187112 + 0.324088i −0.944286 0.329125i \(-0.893246\pi\)
0.757174 + 0.653213i \(0.226579\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −29.0000 −1.34774 −0.673872 0.738848i \(-0.735370\pi\)
−0.673872 + 0.738848i \(0.735370\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.5000 + 28.5788i −0.763529 + 1.32247i 0.177492 + 0.984122i \(0.443202\pi\)
−0.941021 + 0.338349i \(0.890132\pi\)
\(468\) 0 0
\(469\) −27.5000 + 9.52628i −1.26983 + 0.439883i
\(470\) 0 0
\(471\) 7.00000 12.1244i 0.322543 0.558661i
\(472\) 0 0
\(473\) −9.00000 15.5885i −0.413820 0.716758i
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 18.0000 + 31.1769i 0.822441 + 1.42451i 0.903859 + 0.427830i \(0.140722\pi\)
−0.0814184 + 0.996680i \(0.525945\pi\)
\(480\) 0 0
\(481\) 24.0000 41.5692i 1.09431 1.89539i
\(482\) 0 0
\(483\) 1.50000 7.79423i 0.0682524 0.354650i
\(484\) 0 0
\(485\) −1.00000 + 1.73205i −0.0454077 + 0.0786484i
\(486\) 0 0
\(487\) 16.0000 + 27.7128i 0.725029 + 1.25579i 0.958962 + 0.283535i \(0.0915071\pi\)
−0.233933 + 0.972253i \(0.575160\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.00000 + 3.46410i −0.0898933 + 0.155700i
\(496\) 0 0
\(497\) −5.00000 + 25.9808i −0.224281 + 1.16540i
\(498\) 0 0
\(499\) −5.00000 + 8.66025i −0.223831 + 0.387686i −0.955968 0.293471i \(-0.905190\pi\)
0.732137 + 0.681157i \(0.238523\pi\)
\(500\) 0 0
\(501\) 1.50000 + 2.59808i 0.0670151 + 0.116073i
\(502\) 0 0
\(503\) 37.0000 1.64975 0.824874 0.565316i \(-0.191246\pi\)
0.824874 + 0.565316i \(0.191246\pi\)
\(504\) 0 0
\(505\) 17.0000 0.756490
\(506\) 0 0
\(507\) −1.50000 2.59808i −0.0666173 0.115385i
\(508\) 0 0
\(509\) −10.5000 + 18.1865i −0.465404 + 0.806104i −0.999220 0.0394971i \(-0.987424\pi\)
0.533815 + 0.845601i \(0.320758\pi\)
\(510\) 0 0
\(511\) −20.0000 + 6.92820i −0.884748 + 0.306486i
\(512\) 0 0
\(513\) −15.0000 + 25.9808i −0.662266 + 1.14708i
\(514\) 0 0
\(515\) 7.50000 + 12.9904i 0.330489 + 0.572425i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −8.00000 −0.351161
\(520\) 0 0
\(521\) 9.00000 + 15.5885i 0.394297 + 0.682943i 0.993011 0.118020i \(-0.0376547\pi\)
−0.598714 + 0.800963i \(0.704321\pi\)
\(522\) 0 0
\(523\) 14.0000 24.2487i 0.612177 1.06032i −0.378695 0.925521i \(-0.623627\pi\)
0.990873 0.134801i \(-0.0430394\pi\)
\(524\) 0 0
\(525\) −2.00000 1.73205i −0.0872872 0.0755929i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 + 12.1244i 0.304348 + 0.527146i
\(530\) 0 0
\(531\) −20.0000 −0.867926
\(532\) 0 0
\(533\) −28.0000 −1.21281
\(534\) 0 0
\(535\) 0.500000 + 0.866025i 0.0216169 + 0.0374415i
\(536\) 0 0
\(537\) −4.00000 + 6.92820i −0.172613 + 0.298974i
\(538\) 0 0
\(539\) 2.00000 + 13.8564i 0.0861461 + 0.596838i
\(540\) 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
\(543\) 2.50000 + 4.33013i 0.107285 + 0.185824i
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 0 0
\(549\) 5.00000 + 8.66025i 0.213395 + 0.369611i
\(550\) 0 0
\(551\) −9.00000 + 15.5885i −0.383413 + 0.664091i
\(552\) 0 0
\(553\) 12.0000 + 10.3923i 0.510292 + 0.441926i
\(554\) 0 0
\(555\) −6.00000 + 10.3923i −0.254686 + 0.441129i
\(556\) 0 0
\(557\) −15.0000 25.9808i −0.635570 1.10084i −0.986394 0.164399i \(-0.947432\pi\)
0.350824 0.936442i \(-0.385902\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.50000 9.52628i −0.231797 0.401485i 0.726540 0.687124i \(-0.241127\pi\)
−0.958337 + 0.285640i \(0.907794\pi\)
\(564\) 0 0
\(565\) −9.00000 + 15.5885i −0.378633 + 0.655811i
\(566\) 0 0
\(567\) 2.50000 0.866025i 0.104990 0.0363696i
\(568\) 0 0
\(569\) −11.0000 + 19.0526i −0.461144 + 0.798725i −0.999018 0.0443003i \(-0.985894\pi\)
0.537874 + 0.843025i \(0.319228\pi\)
\(570\) 0 0
\(571\) −9.00000 15.5885i −0.376638 0.652357i 0.613933 0.789359i \(-0.289587\pi\)
−0.990571 + 0.137002i \(0.956253\pi\)
\(572\) 0 0
\(573\) −14.0000 −0.584858
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) −1.00000 1.73205i −0.0416305 0.0721062i 0.844459 0.535620i \(-0.179922\pi\)
−0.886090 + 0.463513i \(0.846589\pi\)
\(578\) 0 0
\(579\) 11.0000 19.0526i 0.457144 0.791797i
\(580\) 0 0
\(581\) −1.50000 + 7.79423i −0.0622305 + 0.323359i
\(582\) 0 0
\(583\) 6.00000 10.3923i 0.248495 0.430405i
\(584\) 0 0
\(585\) −4.00000 6.92820i −0.165380 0.286446i
\(586\) 0 0
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 3.00000 + 5.19615i 0.123404 + 0.213741i
\(592\) 0 0
\(593\) 5.00000 8.66025i 0.205325 0.355634i −0.744911 0.667164i \(-0.767508\pi\)
0.950236 + 0.311530i \(0.100841\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.00000 10.3923i 0.245564 0.425329i
\(598\) 0 0
\(599\) −6.00000 10.3923i −0.245153 0.424618i 0.717021 0.697051i \(-0.245505\pi\)
−0.962175 + 0.272433i \(0.912172\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 22.0000 0.895909
\(604\) 0 0
\(605\) −3.50000 6.06218i −0.142295 0.246463i
\(606\) 0 0
\(607\) −3.50000 + 6.06218i −0.142061 + 0.246056i −0.928272 0.371901i \(-0.878706\pi\)
0.786212 + 0.617957i \(0.212039\pi\)
\(608\) 0 0
\(609\) −7.50000 + 2.59808i −0.303915 + 0.105279i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −4.00000 6.92820i −0.161558 0.279827i 0.773869 0.633345i \(-0.218319\pi\)
−0.935428 + 0.353518i \(0.884985\pi\)
\(614\) 0 0
\(615\) 7.00000 0.282267
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) 0 0
\(619\) 13.0000 + 22.5167i 0.522514 + 0.905021i 0.999657 + 0.0261952i \(0.00833914\pi\)
−0.477143 + 0.878826i \(0.658328\pi\)
\(620\) 0 0
\(621\) −7.50000 + 12.9904i −0.300965 + 0.521286i
\(622\) 0 0
\(623\) −34.0000 29.4449i −1.36218 1.17968i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 6.00000 + 10.3923i 0.239617 + 0.415029i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 6.00000 0.238856 0.119428 0.992843i \(-0.461894\pi\)
0.119428 + 0.992843i \(0.461894\pi\)
\(632\) 0 0
\(633\) −7.00000 12.1244i −0.278225 0.481900i
\(634\) 0 0
\(635\) −4.00000 + 6.92820i −0.158735 + 0.274937i
\(636\) 0 0
\(637\) −26.0000 10.3923i −1.03016 0.411758i
\(638\) 0 0
\(639\) 10.0000 17.3205i 0.395594 0.685189i
\(640\) 0 0
\(641\) −6.50000 11.2583i −0.256735 0.444677i 0.708631 0.705580i \(-0.249313\pi\)
−0.965365 + 0.260902i \(0.915980\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) −9.00000 −0.354375
\(646\) 0 0
\(647\) 22.5000 + 38.9711i 0.884566 + 1.53211i 0.846210 + 0.532850i \(0.178879\pi\)
0.0383563 + 0.999264i \(0.487788\pi\)
\(648\) 0 0
\(649\) −10.0000 + 17.3205i −0.392534 + 0.679889i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) 46.0000 1.79191 0.895953 0.444149i \(-0.146494\pi\)
0.895953 + 0.444149i \(0.146494\pi\)
\(660\) 0 0
\(661\) 3.50000 + 6.06218i 0.136134 + 0.235791i 0.926030 0.377450i \(-0.123199\pi\)
−0.789896 + 0.613241i \(0.789865\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.0000 5.19615i 0.581675 0.201498i
\(666\) 0 0
\(667\) −4.50000 + 7.79423i −0.174241 + 0.301794i
\(668\) 0 0
\(669\) −2.00000 3.46410i −0.0773245 0.133930i
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) 0 0
\(675\) 2.50000 + 4.33013i 0.0962250 + 0.166667i
\(676\) 0 0
\(677\) 18.0000 31.1769i 0.691796 1.19823i −0.279453 0.960159i \(-0.590153\pi\)
0.971249 0.238067i \(-0.0765137\pi\)
\(678\) 0 0
\(679\) 1.00000 5.19615i 0.0383765 0.199410i
\(680\) 0 0
\(681\) −2.00000 + 3.46410i −0.0766402 + 0.132745i
\(682\) 0 0
\(683\) 3.50000 + 6.06218i 0.133924 + 0.231963i 0.925186 0.379514i \(-0.123909\pi\)
−0.791262 + 0.611477i \(0.790576\pi\)
\(684\) 0 0
\(685\) 4.00000 0.152832
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) 12.0000 + 20.7846i 0.457164 + 0.791831i
\(690\) 0 0
\(691\) −5.00000 + 8.66025i −0.190209 + 0.329452i −0.945319 0.326146i \(-0.894250\pi\)
0.755110 + 0.655598i \(0.227583\pi\)
\(692\) 0 0
\(693\) 2.00000 10.3923i 0.0759737 0.394771i
\(694\) 0 0
\(695\) −9.00000 + 15.5885i −0.341389 + 0.591304i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −4.00000 −0.151294
\(700\) 0 0
\(701\) 45.0000 1.69963 0.849813 0.527084i \(-0.176715\pi\)
0.849813 + 0.527084i \(0.176715\pi\)
\(702\) 0 0
\(703\) −36.0000 62.3538i −1.35777 2.35172i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −42.5000 + 14.7224i −1.59838 + 0.553694i
\(708\) 0 0
\(709\) −0.500000 + 0.866025i −0.0187779 + 0.0325243i −0.875262 0.483650i \(-0.839311\pi\)
0.856484 + 0.516174i \(0.172644\pi\)
\(710\) 0 0
\(711\) −6.00000 10.3923i −0.225018 0.389742i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) 10.0000 + 17.3205i 0.373457 + 0.646846i
\(718\) 0 0
\(719\) 13.0000 22.5167i 0.484818 0.839730i −0.515030 0.857172i \(-0.672219\pi\)
0.999848 + 0.0174426i \(0.00555244\pi\)
\(720\) 0 0
\(721\) −30.0000 25.9808i −1.11726 0.967574i
\(722\) 0 0
\(723\) −9.00000 + 15.5885i −0.334714 + 0.579741i
\(724\) 0 0
\(725\) 1.50000 + 2.59808i 0.0557086 + 0.0964901i
\(726\) 0 0
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.00000 + 12.1244i −0.258551 + 0.447823i −0.965854 0.259087i \(-0.916578\pi\)
0.707303 + 0.706910i \(0.249912\pi\)
\(734\) 0 0
\(735\) 6.50000 + 2.59808i 0.239756 + 0.0958315i
\(736\) 0 0
\(737\) 11.0000 19.0526i 0.405190 0.701810i
\(738\) 0 0
\(739\) −5.00000 8.66025i −0.183928 0.318573i 0.759287 0.650756i \(-0.225548\pi\)
−0.943215 + 0.332184i \(0.892215\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 0 0
\(745\) −8.50000 14.7224i −0.311416 0.539388i
\(746\) 0 0
\(747\) 3.00000 5.19615i 0.109764 0.190117i
\(748\) 0 0
\(749\) −2.00000 1.73205i −0.0730784 0.0632878i
\(750\) 0 0
\(751\) 14.0000 24.2487i 0.510867 0.884848i −0.489053 0.872254i \(-0.662658\pi\)
0.999921 0.0125942i \(-0.00400897\pi\)
\(752\) 0 0
\(753\) −2.00000 3.46410i −0.0728841 0.126239i
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) 3.00000 + 5.19615i 0.108893 + 0.188608i
\(760\) 0 0
\(761\) 11.0000 19.0526i 0.398750 0.690655i −0.594822 0.803857i \(-0.702778\pi\)
0.993572 + 0.113203i \(0.0361109\pi\)
\(762\) 0 0
\(763\) −12.5000 + 4.33013i −0.452530 + 0.156761i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −20.0000 34.6410i −0.722158 1.25081i
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 24.0000 0.864339
\(772\) 0 0
\(773\) −2.00000 3.46410i −0.0719350 0.124595i 0.827814 0.561002i \(-0.189584\pi\)
−0.899749 + 0.436407i \(0.856251\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.00000 31.1769i 0.215249 1.11847i
\(778\) 0 0
\(779\) −21.0000 + 36.3731i −0.752403 + 1.30320i
\(780\) 0 0
\(781\) −10.0000 17.3205i −0.357828 0.619777i
\(782\) 0 0
\(783\) 15.0000 0.536056
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −18.5000 32.0429i −0.659454 1.14221i −0.980757 0.195231i \(-0.937454\pi\)
0.321303 0.946976i \(-0.395879\pi\)
\(788\) 0 0
\(789\) 10.5000 18.1865i 0.373810 0.647458i
\(790\) 0 0
\(791\) 9.00000 46.7654i 0.320003 1.66279i
\(792\) 0 0
\(793\) −10.0000 + 17.3205i −0.355110 + 0.615069i
\(794\) 0 0
\(795\) −3.00000 5.19615i −0.106399 0.184289i
\(796\) 0 0
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 17.0000 + 29.4449i 0.600665 + 1.04038i
\(802\) 0 0
\(803\) 8.00000 13.8564i 0.282314 0.488982i
\(804\) 0 0
\(805\) 7.50000 2.59808i 0.264340 0.0915702i
\(806\) 0 0
\(807\) −15.5000 + 26.8468i −0.545626 + 0.945052i
\(808\) 0 0
\(809\) 5.50000 + 9.52628i 0.193370 + 0.334926i 0.946365 0.323100i \(-0.104725\pi\)
−0.752995 + 0.658026i \(0.771392\pi\)
\(810\) 0 0
\(811\) −32.0000 −1.12367 −0.561836 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) 6.00000 + 10.3923i 0.210171 + 0.364027i
\(816\) 0 0
\(817\) 27.0000 46.7654i 0.944610 1.63611i
\(818\) 0 0
\(819\) 16.0000 + 13.8564i 0.559085 + 0.484182i
\(820\) 0 0
\(821\) 17.0000 29.4449i 0.593304 1.02763i −0.400480 0.916306i \(-0.631157\pi\)
0.993784 0.111327i \(-0.0355102\pi\)
\(822\) 0 0
\(823\) −6.50000 11.2583i −0.226576 0.392441i 0.730215 0.683217i \(-0.239420\pi\)
−0.956791 + 0.290776i \(0.906086\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 31.0000 1.07798 0.538988 0.842314i \(-0.318807\pi\)
0.538988 + 0.842314i \(0.318807\pi\)
\(828\) 0 0
\(829\) 7.00000 + 12.1244i 0.243120 + 0.421096i 0.961601 0.274450i \(-0.0884958\pi\)
−0.718481 + 0.695546i \(0.755162\pi\)
\(830\) 0 0
\(831\) −4.00000 + 6.92820i −0.138758 + 0.240337i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −1.50000 + 2.59808i −0.0519096 + 0.0899101i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −46.0000 −1.58810 −0.794048 0.607855i \(-0.792030\pi\)
−0.794048 + 0.607855i \(0.792030\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −13.0000 22.5167i −0.447744 0.775515i
\(844\) 0 0
\(845\) 1.50000 2.59808i 0.0516016 0.0893765i
\(846\) 0 0
\(847\) 14.0000 + 12.1244i 0.481046 + 0.416598i
\(848\) 0 0
\(849\) −14.0000 + 24.2487i −0.480479 + 0.832214i
\(850\) 0 0
\(851\) −18.0000 31.1769i −0.617032 1.06873i
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) 13.0000 + 22.5167i 0.444072 + 0.769154i 0.997987 0.0634184i \(-0.0202003\pi\)
−0.553915 + 0.832573i \(0.686867\pi\)
\(858\) 0 0
\(859\) 28.0000 48.4974i 0.955348 1.65471i 0.221777 0.975097i \(-0.428814\pi\)
0.733571 0.679613i \(-0.237852\pi\)
\(860\) 0 0
\(861\) −17.5000 + 6.06218i −0.596398 + 0.206598i
\(862\) 0 0
\(863\) −21.5000 + 37.2391i −0.731869 + 1.26763i 0.224215 + 0.974540i \(0.428018\pi\)
−0.956084 + 0.293094i \(0.905315\pi\)
\(864\) 0 0
\(865\) −4.00000 6.92820i −0.136004 0.235566i
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) 22.0000 + 38.1051i 0.745442 + 1.29114i
\(872\) 0 0
\(873\) −2.00000 + 3.46410i −0.0676897 + 0.117242i
\(874\) 0 0
\(875\) 0.500000 2.59808i 0.0169031 0.0878310i
\(876\) 0 0
\(877\) 11.0000 19.0526i 0.371444 0.643359i −0.618344 0.785907i \(-0.712196\pi\)
0.989788 + 0.142548i \(0.0455296\pi\)
\(878\) 0 0
\(879\) 2.00000 + 3.46410i 0.0674583 + 0.116841i
\(880\) 0 0
\(881\) 41.0000 1.38133 0.690663 0.723177i \(-0.257319\pi\)
0.690663 + 0.723177i \(0.257319\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 5.00000 + 8.66025i 0.168073 + 0.291111i
\(886\) 0 0
\(887\) 18.5000 32.0429i 0.621169 1.07590i −0.368099 0.929787i \(-0.619991\pi\)
0.989268 0.146110i \(-0.0466754\pi\)
\(888\) 0 0
\(889\) 4.00000 20.7846i 0.134156 0.697093i
\(890\) 0 0
\(891\) −1.00000 + 1.73205i −0.0335013 + 0.0580259i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −12.0000 −0.400668
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 22.5000 7.79423i 0.748753 0.259376i
\(904\) 0 0
\(905\) −2.50000 + 4.33013i −0.0831028 + 0.143938i
\(906\) 0 0
\(907\) 4.50000 + 7.79423i 0.149420 + 0.258803i 0.931013 0.364985i \(-0.118926\pi\)
−0.781593 + 0.623788i \(0.785593\pi\)
\(908\) 0 0
\(909\) 34.0000 1.12771
\(910\) 0 0
\(911\) 34.0000 1.12647 0.563235 0.826297i \(-0.309557\pi\)
0.563235 + 0.826297i \(0.309557\pi\)
\(912\) 0 0
\(913\) −3.00000 5.19615i −0.0992855 0.171968i
\(914\) 0 0
\(915\) 2.50000 4.33013i 0.0826475 0.143150i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 17.0000 29.4449i 0.560778 0.971296i −0.436650 0.899631i \(-0.643835\pi\)
0.997429 0.0716652i \(-0.0228313\pi\)
\(920\) 0 0
\(921\) −10.5000 18.1865i −0.345987 0.599267i
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) −12.0000 −0.394558
\(926\) 0 0
\(927\) 15.0000 + 25.9808i 0.492665 + 0.853320i
\(928\) 0 0
\(929\) −13.5000 + 23.3827i −0.442921 + 0.767161i −0.997905 0.0646999i \(-0.979391\pi\)
0.554984 + 0.831861i \(0.312724\pi\)
\(930\) 0 0
\(931\) −33.0000 + 25.9808i −1.08153 + 0.851485i
\(932\) 0 0
\(933\) 5.00000 8.66025i 0.163693 0.283524i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 19.0000 + 32.9090i 0.619382 + 1.07280i 0.989599 + 0.143856i \(0.0459502\pi\)
−0.370216 + 0.928946i \(0.620716\pi\)
\(942\) 0 0
\(943\) −10.5000 + 18.1865i −0.341927 + 0.592235i
\(944\) 0 0
\(945\) −10.0000 8.66025i −0.325300 0.281718i
\(946\) 0 0
\(947\) −6.50000 + 11.2583i −0.211222 + 0.365847i −0.952097 0.305796i \(-0.901078\pi\)
0.740875 + 0.671642i \(0.234411\pi\)
\(948\) 0 0
\(949\) 16.0000 + 27.7128i 0.519382 + 0.899596i
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −28.0000 −0.907009 −0.453504 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) 0 0
\(955\) −7.00000 12.1244i −0.226515 0.392335i
\(956\) 0 0
\(957\) 3.00000 5.19615i 0.0969762 0.167968i
\(958\) 0 0
\(959\) −10.0000 + 3.46410i −0.322917 + 0.111862i
\(960\) 0 0
\(961\) 15.5000 26.8468i 0.500000 0.866025i
\(962\) 0 0
\(963\) 1.00000 + 1.73205i 0.0322245 + 0.0558146i
\(964\) 0 0
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 34.6410i 0.641831 1.11168i −0.343193 0.939265i \(-0.611509\pi\)
0.985024 0.172418i \(-0.0551581\pi\)
\(972\) 0 0
\(973\) 9.00000 46.7654i 0.288527 1.49923i
\(974\) 0 0
\(975\) −2.00000 + 3.46410i −0.0640513 + 0.110940i
\(976\) 0 0
\(977\) −9.00000 15.5885i −0.287936 0.498719i 0.685381 0.728184i \(-0.259636\pi\)
−0.973317 + 0.229465i \(0.926302\pi\)
\(978\) 0 0
\(979\) 34.0000 1.08664
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 4.50000 + 7.79423i 0.143528 + 0.248597i 0.928823 0.370525i \(-0.120822\pi\)
−0.785295 + 0.619122i \(0.787489\pi\)
\(984\) 0 0
\(985\) −3.00000 + 5.19615i −0.0955879 + 0.165563i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.5000 23.3827i 0.429275 0.743526i
\(990\) 0 0
\(991\) 10.0000 + 17.3205i 0.317660 + 0.550204i 0.979999 0.199000i \(-0.0637695\pi\)
−0.662339 + 0.749204i \(0.730436\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) −7.00000 12.1244i −0.221692 0.383982i 0.733630 0.679549i \(-0.237825\pi\)
−0.955322 + 0.295567i \(0.904491\pi\)
\(998\) 0 0
\(999\) −30.0000 + 51.9615i −0.949158 + 1.64399i
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 560.2.q.e.401.1 2
4.3 odd 2 280.2.q.b.121.1 yes 2
7.2 even 3 3920.2.a.v.1.1 1
7.4 even 3 inner 560.2.q.e.81.1 2
7.5 odd 6 3920.2.a.q.1.1 1
12.11 even 2 2520.2.bi.d.1801.1 2
20.3 even 4 1400.2.bh.c.849.1 4
20.7 even 4 1400.2.bh.c.849.2 4
20.19 odd 2 1400.2.q.c.401.1 2
28.3 even 6 1960.2.q.d.361.1 2
28.11 odd 6 280.2.q.b.81.1 2
28.19 even 6 1960.2.a.l.1.1 1
28.23 odd 6 1960.2.a.c.1.1 1
28.27 even 2 1960.2.q.d.961.1 2
84.11 even 6 2520.2.bi.d.361.1 2
140.19 even 6 9800.2.a.o.1.1 1
140.39 odd 6 1400.2.q.c.1201.1 2
140.67 even 12 1400.2.bh.c.249.1 4
140.79 odd 6 9800.2.a.z.1.1 1
140.123 even 12 1400.2.bh.c.249.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.b.81.1 2 28.11 odd 6
280.2.q.b.121.1 yes 2 4.3 odd 2
560.2.q.e.81.1 2 7.4 even 3 inner
560.2.q.e.401.1 2 1.1 even 1 trivial
1400.2.q.c.401.1 2 20.19 odd 2
1400.2.q.c.1201.1 2 140.39 odd 6
1400.2.bh.c.249.1 4 140.67 even 12
1400.2.bh.c.249.2 4 140.123 even 12
1400.2.bh.c.849.1 4 20.3 even 4
1400.2.bh.c.849.2 4 20.7 even 4
1960.2.a.c.1.1 1 28.23 odd 6
1960.2.a.l.1.1 1 28.19 even 6
1960.2.q.d.361.1 2 28.3 even 6
1960.2.q.d.961.1 2 28.27 even 2
2520.2.bi.d.361.1 2 84.11 even 6
2520.2.bi.d.1801.1 2 12.11 even 2
3920.2.a.q.1.1 1 7.5 odd 6
3920.2.a.v.1.1 1 7.2 even 3
9800.2.a.o.1.1 1 140.19 even 6
9800.2.a.z.1.1 1 140.79 odd 6