Properties

Label 4600.2.e.m.4049.3
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), 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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.m.4049.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155i q^{3} -3.12311i q^{7} +0.561553 q^{9} +O(q^{10})\) \(q+1.56155i q^{3} -3.12311i q^{7} +0.561553 q^{9} -4.00000 q^{11} -3.56155i q^{13} +5.12311i q^{17} -4.00000 q^{19} +4.87689 q^{21} -1.00000i q^{23} +5.56155i q^{27} +4.43845 q^{29} +5.56155 q^{31} -6.24621i q^{33} +1.12311i q^{37} +5.56155 q^{39} -3.56155 q^{41} +0.876894i q^{43} +8.68466i q^{47} -2.75379 q^{49} -8.00000 q^{51} -12.2462i q^{53} -6.24621i q^{57} -10.2462 q^{59} +2.87689 q^{61} -1.75379i q^{63} -10.2462i q^{67} +1.56155 q^{69} -8.68466 q^{71} -12.4384i q^{73} +12.4924i q^{77} -6.24621 q^{79} -7.00000 q^{81} -12.0000i q^{83} +6.93087i q^{87} -10.0000 q^{89} -11.1231 q^{91} +8.68466i q^{93} +0.246211i q^{97} -2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} - 16 q^{11} - 16 q^{19} + 36 q^{21} + 26 q^{29} + 14 q^{31} + 14 q^{39} - 6 q^{41} - 44 q^{49} - 32 q^{51} - 8 q^{59} + 28 q^{61} - 2 q^{69} - 10 q^{71} + 8 q^{79} - 28 q^{81} - 40 q^{89} - 28 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.56155i 0.901563i 0.892634 + 0.450781i \(0.148855\pi\)
−0.892634 + 0.450781i \(0.851145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.12311i − 1.18042i −0.807249 0.590211i \(-0.799044\pi\)
0.807249 0.590211i \(-0.200956\pi\)
\(8\) 0 0
\(9\) 0.561553 0.187184
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) − 3.56155i − 0.987797i −0.869520 0.493899i \(-0.835571\pi\)
0.869520 0.493899i \(-0.164429\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.12311i 1.24254i 0.783598 + 0.621268i \(0.213382\pi\)
−0.783598 + 0.621268i \(0.786618\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 4.87689 1.06423
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.56155i 1.07032i
\(28\) 0 0
\(29\) 4.43845 0.824199 0.412099 0.911139i \(-0.364796\pi\)
0.412099 + 0.911139i \(0.364796\pi\)
\(30\) 0 0
\(31\) 5.56155 0.998884 0.499442 0.866347i \(-0.333538\pi\)
0.499442 + 0.866347i \(0.333538\pi\)
\(32\) 0 0
\(33\) − 6.24621i − 1.08733i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.12311i 0.184637i 0.995730 + 0.0923187i \(0.0294279\pi\)
−0.995730 + 0.0923187i \(0.970572\pi\)
\(38\) 0 0
\(39\) 5.56155 0.890561
\(40\) 0 0
\(41\) −3.56155 −0.556221 −0.278111 0.960549i \(-0.589708\pi\)
−0.278111 + 0.960549i \(0.589708\pi\)
\(42\) 0 0
\(43\) 0.876894i 0.133725i 0.997762 + 0.0668626i \(0.0212989\pi\)
−0.997762 + 0.0668626i \(0.978701\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.68466i 1.26679i 0.773830 + 0.633394i \(0.218339\pi\)
−0.773830 + 0.633394i \(0.781661\pi\)
\(48\) 0 0
\(49\) −2.75379 −0.393398
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) − 12.2462i − 1.68215i −0.540921 0.841073i \(-0.681924\pi\)
0.540921 0.841073i \(-0.318076\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.24621i − 0.827331i
\(58\) 0 0
\(59\) −10.2462 −1.33394 −0.666972 0.745083i \(-0.732410\pi\)
−0.666972 + 0.745083i \(0.732410\pi\)
\(60\) 0 0
\(61\) 2.87689 0.368349 0.184174 0.982894i \(-0.441039\pi\)
0.184174 + 0.982894i \(0.441039\pi\)
\(62\) 0 0
\(63\) − 1.75379i − 0.220957i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.2462i − 1.25177i −0.779914 0.625887i \(-0.784737\pi\)
0.779914 0.625887i \(-0.215263\pi\)
\(68\) 0 0
\(69\) 1.56155 0.187989
\(70\) 0 0
\(71\) −8.68466 −1.03068 −0.515340 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(72\) 0 0
\(73\) − 12.4384i − 1.45581i −0.685678 0.727905i \(-0.740494\pi\)
0.685678 0.727905i \(-0.259506\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.4924i 1.42364i
\(78\) 0 0
\(79\) −6.24621 −0.702754 −0.351377 0.936234i \(-0.614286\pi\)
−0.351377 + 0.936234i \(0.614286\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.93087i 0.743067i
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −11.1231 −1.16602
\(92\) 0 0
\(93\) 8.68466i 0.900557i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.246211i 0.0249990i 0.999922 + 0.0124995i \(0.00397881\pi\)
−0.999922 + 0.0124995i \(0.996021\pi\)
\(98\) 0 0
\(99\) −2.24621 −0.225753
\(100\) 0 0
\(101\) −16.2462 −1.61656 −0.808279 0.588799i \(-0.799601\pi\)
−0.808279 + 0.588799i \(0.799601\pi\)
\(102\) 0 0
\(103\) − 14.2462i − 1.40372i −0.712314 0.701860i \(-0.752353\pi\)
0.712314 0.701860i \(-0.247647\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) 8.24621 0.789844 0.394922 0.918715i \(-0.370772\pi\)
0.394922 + 0.918715i \(0.370772\pi\)
\(110\) 0 0
\(111\) −1.75379 −0.166462
\(112\) 0 0
\(113\) − 3.75379i − 0.353127i −0.984289 0.176563i \(-0.943502\pi\)
0.984289 0.176563i \(-0.0564981\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) − 5.56155i − 0.501468i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.80776i 0.337884i 0.985626 + 0.168942i \(0.0540351\pi\)
−0.985626 + 0.168942i \(0.945965\pi\)
\(128\) 0 0
\(129\) −1.36932 −0.120562
\(130\) 0 0
\(131\) −18.9309 −1.65400 −0.826999 0.562204i \(-0.809954\pi\)
−0.826999 + 0.562204i \(0.809954\pi\)
\(132\) 0 0
\(133\) 12.4924i 1.08323i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.87689i 0.587533i 0.955877 + 0.293766i \(0.0949088\pi\)
−0.955877 + 0.293766i \(0.905091\pi\)
\(138\) 0 0
\(139\) −12.6847 −1.07590 −0.537949 0.842977i \(-0.680801\pi\)
−0.537949 + 0.842977i \(0.680801\pi\)
\(140\) 0 0
\(141\) −13.5616 −1.14209
\(142\) 0 0
\(143\) 14.2462i 1.19133i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 4.30019i − 0.354673i
\(148\) 0 0
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 0 0
\(151\) 3.80776 0.309871 0.154936 0.987925i \(-0.450483\pi\)
0.154936 + 0.987925i \(0.450483\pi\)
\(152\) 0 0
\(153\) 2.87689i 0.232583i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) 19.1231 1.51656
\(160\) 0 0
\(161\) −3.12311 −0.246135
\(162\) 0 0
\(163\) 12.6847i 0.993539i 0.867882 + 0.496770i \(0.165481\pi\)
−0.867882 + 0.496770i \(0.834519\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 0.315342 0.0242570
\(170\) 0 0
\(171\) −2.24621 −0.171772
\(172\) 0 0
\(173\) 22.4924i 1.71007i 0.518573 + 0.855034i \(0.326464\pi\)
−0.518573 + 0.855034i \(0.673536\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 16.0000i − 1.20263i
\(178\) 0 0
\(179\) −4.68466 −0.350148 −0.175074 0.984555i \(-0.556016\pi\)
−0.175074 + 0.984555i \(0.556016\pi\)
\(180\) 0 0
\(181\) −5.12311 −0.380797 −0.190399 0.981707i \(-0.560978\pi\)
−0.190399 + 0.981707i \(0.560978\pi\)
\(182\) 0 0
\(183\) 4.49242i 0.332089i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 20.4924i − 1.49855i
\(188\) 0 0
\(189\) 17.3693 1.26343
\(190\) 0 0
\(191\) −11.1231 −0.804840 −0.402420 0.915455i \(-0.631831\pi\)
−0.402420 + 0.915455i \(0.631831\pi\)
\(192\) 0 0
\(193\) − 20.4384i − 1.47119i −0.677421 0.735596i \(-0.736902\pi\)
0.677421 0.735596i \(-0.263098\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.5616i − 1.10871i −0.832279 0.554357i \(-0.812964\pi\)
0.832279 0.554357i \(-0.187036\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) − 13.8617i − 0.972903i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.561553i − 0.0390306i
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 2.24621 0.154636 0.0773178 0.997006i \(-0.475364\pi\)
0.0773178 + 0.997006i \(0.475364\pi\)
\(212\) 0 0
\(213\) − 13.5616i − 0.929222i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 17.3693i − 1.17911i
\(218\) 0 0
\(219\) 19.4233 1.31250
\(220\) 0 0
\(221\) 18.2462 1.22737
\(222\) 0 0
\(223\) − 12.4924i − 0.836554i −0.908319 0.418277i \(-0.862634\pi\)
0.908319 0.418277i \(-0.137366\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0.876894i 0.0582015i 0.999576 + 0.0291008i \(0.00926437\pi\)
−0.999576 + 0.0291008i \(0.990736\pi\)
\(228\) 0 0
\(229\) 16.2462 1.07358 0.536790 0.843716i \(-0.319637\pi\)
0.536790 + 0.843716i \(0.319637\pi\)
\(230\) 0 0
\(231\) −19.5076 −1.28350
\(232\) 0 0
\(233\) 8.43845i 0.552821i 0.961040 + 0.276411i \(0.0891449\pi\)
−0.961040 + 0.276411i \(0.910855\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 9.75379i − 0.633577i
\(238\) 0 0
\(239\) −11.8078 −0.763781 −0.381890 0.924208i \(-0.624727\pi\)
−0.381890 + 0.924208i \(0.624727\pi\)
\(240\) 0 0
\(241\) 14.4924 0.933539 0.466769 0.884379i \(-0.345418\pi\)
0.466769 + 0.884379i \(0.345418\pi\)
\(242\) 0 0
\(243\) 5.75379i 0.369106i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.2462i 0.906465i
\(248\) 0 0
\(249\) 18.7386 1.18751
\(250\) 0 0
\(251\) −10.2462 −0.646735 −0.323368 0.946273i \(-0.604815\pi\)
−0.323368 + 0.946273i \(0.604815\pi\)
\(252\) 0 0
\(253\) 4.00000i 0.251478i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.0540i 1.74996i 0.484160 + 0.874979i \(0.339125\pi\)
−0.484160 + 0.874979i \(0.660875\pi\)
\(258\) 0 0
\(259\) 3.50758 0.217950
\(260\) 0 0
\(261\) 2.49242 0.154277
\(262\) 0 0
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 15.6155i − 0.955655i
\(268\) 0 0
\(269\) −30.3002 −1.84743 −0.923717 0.383074i \(-0.874865\pi\)
−0.923717 + 0.383074i \(0.874865\pi\)
\(270\) 0 0
\(271\) 14.2462 0.865396 0.432698 0.901539i \(-0.357562\pi\)
0.432698 + 0.901539i \(0.357562\pi\)
\(272\) 0 0
\(273\) − 17.3693i − 1.05124i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 15.5616i − 0.935003i −0.883992 0.467502i \(-0.845154\pi\)
0.883992 0.467502i \(-0.154846\pi\)
\(278\) 0 0
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) −2.49242 −0.148685 −0.0743427 0.997233i \(-0.523686\pi\)
−0.0743427 + 0.997233i \(0.523686\pi\)
\(282\) 0 0
\(283\) − 31.1231i − 1.85008i −0.379874 0.925038i \(-0.624033\pi\)
0.379874 0.925038i \(-0.375967\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.1231i 0.656576i
\(288\) 0 0
\(289\) −9.24621 −0.543895
\(290\) 0 0
\(291\) −0.384472 −0.0225381
\(292\) 0 0
\(293\) 21.1231i 1.23403i 0.786953 + 0.617013i \(0.211657\pi\)
−0.786953 + 0.617013i \(0.788343\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 22.2462i − 1.29086i
\(298\) 0 0
\(299\) −3.56155 −0.205970
\(300\) 0 0
\(301\) 2.73863 0.157852
\(302\) 0 0
\(303\) − 25.3693i − 1.45743i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) 22.2462 1.26554
\(310\) 0 0
\(311\) −14.9309 −0.846652 −0.423326 0.905977i \(-0.639137\pi\)
−0.423326 + 0.905977i \(0.639137\pi\)
\(312\) 0 0
\(313\) − 6.87689i − 0.388705i −0.980932 0.194353i \(-0.937739\pi\)
0.980932 0.194353i \(-0.0622606\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 28.7386i − 1.61412i −0.590468 0.807061i \(-0.701057\pi\)
0.590468 0.807061i \(-0.298943\pi\)
\(318\) 0 0
\(319\) −17.7538 −0.994021
\(320\) 0 0
\(321\) 18.7386 1.04589
\(322\) 0 0
\(323\) − 20.4924i − 1.14023i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.8769i 0.712094i
\(328\) 0 0
\(329\) 27.1231 1.49535
\(330\) 0 0
\(331\) 6.43845 0.353889 0.176945 0.984221i \(-0.443379\pi\)
0.176945 + 0.984221i \(0.443379\pi\)
\(332\) 0 0
\(333\) 0.630683i 0.0345612i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 32.2462i 1.75656i 0.478144 + 0.878282i \(0.341310\pi\)
−0.478144 + 0.878282i \(0.658690\pi\)
\(338\) 0 0
\(339\) 5.86174 0.318366
\(340\) 0 0
\(341\) −22.2462 −1.20470
\(342\) 0 0
\(343\) − 13.2614i − 0.716046i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.4924i 0.885360i 0.896680 + 0.442680i \(0.145972\pi\)
−0.896680 + 0.442680i \(0.854028\pi\)
\(348\) 0 0
\(349\) −17.8078 −0.953228 −0.476614 0.879113i \(-0.658136\pi\)
−0.476614 + 0.879113i \(0.658136\pi\)
\(350\) 0 0
\(351\) 19.8078 1.05726
\(352\) 0 0
\(353\) − 14.1922i − 0.755377i −0.925933 0.377688i \(-0.876719\pi\)
0.925933 0.377688i \(-0.123281\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 24.9848i 1.32234i
\(358\) 0 0
\(359\) 30.2462 1.59633 0.798167 0.602436i \(-0.205803\pi\)
0.798167 + 0.602436i \(0.205803\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 7.80776i 0.409801i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 12.4924i − 0.652099i −0.945353 0.326050i \(-0.894282\pi\)
0.945353 0.326050i \(-0.105718\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −38.2462 −1.98564
\(372\) 0 0
\(373\) − 24.7386i − 1.28092i −0.767992 0.640459i \(-0.778744\pi\)
0.767992 0.640459i \(-0.221256\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 15.8078i − 0.814141i
\(378\) 0 0
\(379\) −2.24621 −0.115380 −0.0576901 0.998335i \(-0.518374\pi\)
−0.0576901 + 0.998335i \(0.518374\pi\)
\(380\) 0 0
\(381\) −5.94602 −0.304624
\(382\) 0 0
\(383\) 7.61553i 0.389135i 0.980889 + 0.194568i \(0.0623304\pi\)
−0.980889 + 0.194568i \(0.937670\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.492423i 0.0250312i
\(388\) 0 0
\(389\) 33.6155 1.70437 0.852187 0.523237i \(-0.175276\pi\)
0.852187 + 0.523237i \(0.175276\pi\)
\(390\) 0 0
\(391\) 5.12311 0.259087
\(392\) 0 0
\(393\) − 29.5616i − 1.49118i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 24.9309i − 1.25124i −0.780126 0.625622i \(-0.784845\pi\)
0.780126 0.625622i \(-0.215155\pi\)
\(398\) 0 0
\(399\) −19.5076 −0.976600
\(400\) 0 0
\(401\) 20.7386 1.03564 0.517819 0.855490i \(-0.326744\pi\)
0.517819 + 0.855490i \(0.326744\pi\)
\(402\) 0 0
\(403\) − 19.8078i − 0.986695i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.49242i − 0.222681i
\(408\) 0 0
\(409\) 14.6847 0.726110 0.363055 0.931768i \(-0.381734\pi\)
0.363055 + 0.931768i \(0.381734\pi\)
\(410\) 0 0
\(411\) −10.7386 −0.529698
\(412\) 0 0
\(413\) 32.0000i 1.57462i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 19.8078i − 0.969990i
\(418\) 0 0
\(419\) 0.876894 0.0428391 0.0214195 0.999771i \(-0.493181\pi\)
0.0214195 + 0.999771i \(0.493181\pi\)
\(420\) 0 0
\(421\) −6.49242 −0.316421 −0.158211 0.987405i \(-0.550573\pi\)
−0.158211 + 0.987405i \(0.550573\pi\)
\(422\) 0 0
\(423\) 4.87689i 0.237123i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.98485i − 0.434807i
\(428\) 0 0
\(429\) −22.2462 −1.07406
\(430\) 0 0
\(431\) 7.61553 0.366827 0.183414 0.983036i \(-0.441285\pi\)
0.183414 + 0.983036i \(0.441285\pi\)
\(432\) 0 0
\(433\) 24.7386i 1.18886i 0.804146 + 0.594431i \(0.202623\pi\)
−0.804146 + 0.594431i \(0.797377\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.00000i 0.191346i
\(438\) 0 0
\(439\) 32.3002 1.54160 0.770802 0.637075i \(-0.219856\pi\)
0.770802 + 0.637075i \(0.219856\pi\)
\(440\) 0 0
\(441\) −1.54640 −0.0736380
\(442\) 0 0
\(443\) 3.31534i 0.157517i 0.996894 + 0.0787583i \(0.0250955\pi\)
−0.996894 + 0.0787583i \(0.974904\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 19.1231i − 0.904492i
\(448\) 0 0
\(449\) 32.7386 1.54503 0.772516 0.634996i \(-0.218998\pi\)
0.772516 + 0.634996i \(0.218998\pi\)
\(450\) 0 0
\(451\) 14.2462 0.670828
\(452\) 0 0
\(453\) 5.94602i 0.279369i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 9.12311i − 0.426761i −0.976969 0.213380i \(-0.931553\pi\)
0.976969 0.213380i \(-0.0684474\pi\)
\(458\) 0 0
\(459\) −28.4924 −1.32991
\(460\) 0 0
\(461\) 27.5616 1.28367 0.641835 0.766843i \(-0.278174\pi\)
0.641835 + 0.766843i \(0.278174\pi\)
\(462\) 0 0
\(463\) 6.24621i 0.290286i 0.989411 + 0.145143i \(0.0463642\pi\)
−0.989411 + 0.145143i \(0.953636\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.6307i 0.491929i 0.969279 + 0.245965i \(0.0791047\pi\)
−0.969279 + 0.245965i \(0.920895\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 3.12311 0.143905
\(472\) 0 0
\(473\) − 3.50758i − 0.161279i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.87689i − 0.314871i
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) − 4.87689i − 0.221906i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0540i 1.54313i 0.636149 + 0.771566i \(0.280526\pi\)
−0.636149 + 0.771566i \(0.719474\pi\)
\(488\) 0 0
\(489\) −19.8078 −0.895738
\(490\) 0 0
\(491\) 6.43845 0.290563 0.145282 0.989390i \(-0.453591\pi\)
0.145282 + 0.989390i \(0.453591\pi\)
\(492\) 0 0
\(493\) 22.7386i 1.02410i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.1231i 1.21664i
\(498\) 0 0
\(499\) −38.0540 −1.70353 −0.851765 0.523924i \(-0.824468\pi\)
−0.851765 + 0.523924i \(0.824468\pi\)
\(500\) 0 0
\(501\) 12.4924 0.558120
\(502\) 0 0
\(503\) 15.6155i 0.696262i 0.937446 + 0.348131i \(0.113184\pi\)
−0.937446 + 0.348131i \(0.886816\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.492423i 0.0218693i
\(508\) 0 0
\(509\) 4.43845 0.196731 0.0983654 0.995150i \(-0.468639\pi\)
0.0983654 + 0.995150i \(0.468639\pi\)
\(510\) 0 0
\(511\) −38.8466 −1.71847
\(512\) 0 0
\(513\) − 22.2462i − 0.982194i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 34.7386i − 1.52780i
\(518\) 0 0
\(519\) −35.1231 −1.54173
\(520\) 0 0
\(521\) 39.8617 1.74637 0.873187 0.487385i \(-0.162049\pi\)
0.873187 + 0.487385i \(0.162049\pi\)
\(522\) 0 0
\(523\) − 33.8617i − 1.48067i −0.672238 0.740335i \(-0.734667\pi\)
0.672238 0.740335i \(-0.265333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 28.4924i 1.24115i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −5.75379 −0.249693
\(532\) 0 0
\(533\) 12.6847i 0.549434i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 7.31534i − 0.315680i
\(538\) 0 0
\(539\) 11.0152 0.474456
\(540\) 0 0
\(541\) 21.3153 0.916418 0.458209 0.888844i \(-0.348491\pi\)
0.458209 + 0.888844i \(0.348491\pi\)
\(542\) 0 0
\(543\) − 8.00000i − 0.343313i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.0540i 1.62707i 0.581516 + 0.813535i \(0.302460\pi\)
−0.581516 + 0.813535i \(0.697540\pi\)
\(548\) 0 0
\(549\) 1.61553 0.0689491
\(550\) 0 0
\(551\) −17.7538 −0.756337
\(552\) 0 0
\(553\) 19.5076i 0.829547i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.2462i − 1.02734i −0.857986 0.513672i \(-0.828285\pi\)
0.857986 0.513672i \(-0.171715\pi\)
\(558\) 0 0
\(559\) 3.12311 0.132093
\(560\) 0 0
\(561\) 32.0000 1.35104
\(562\) 0 0
\(563\) − 43.2311i − 1.82197i −0.412438 0.910986i \(-0.635322\pi\)
0.412438 0.910986i \(-0.364678\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 21.8617i 0.918107i
\(568\) 0 0
\(569\) −28.7386 −1.20479 −0.602393 0.798200i \(-0.705786\pi\)
−0.602393 + 0.798200i \(0.705786\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) − 17.3693i − 0.725614i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.4233i 1.22491i 0.790506 + 0.612454i \(0.209817\pi\)
−0.790506 + 0.612454i \(0.790183\pi\)
\(578\) 0 0
\(579\) 31.9157 1.32637
\(580\) 0 0
\(581\) −37.4773 −1.55482
\(582\) 0 0
\(583\) 48.9848i 2.02874i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.17708i 0.378779i 0.981902 + 0.189389i \(0.0606508\pi\)
−0.981902 + 0.189389i \(0.939349\pi\)
\(588\) 0 0
\(589\) −22.2462 −0.916639
\(590\) 0 0
\(591\) 24.3002 0.999576
\(592\) 0 0
\(593\) 38.9848i 1.60092i 0.599389 + 0.800458i \(0.295410\pi\)
−0.599389 + 0.800458i \(0.704590\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 37.4773i − 1.53384i
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −27.1771 −1.10858 −0.554288 0.832325i \(-0.687009\pi\)
−0.554288 + 0.832325i \(0.687009\pi\)
\(602\) 0 0
\(603\) − 5.75379i − 0.234312i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 21.6458 0.877134
\(610\) 0 0
\(611\) 30.9309 1.25133
\(612\) 0 0
\(613\) 5.12311i 0.206920i 0.994634 + 0.103460i \(0.0329914\pi\)
−0.994634 + 0.103460i \(0.967009\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5.61553i − 0.226073i −0.993591 0.113036i \(-0.963942\pi\)
0.993591 0.113036i \(-0.0360576\pi\)
\(618\) 0 0
\(619\) −36.9848 −1.48655 −0.743273 0.668988i \(-0.766728\pi\)
−0.743273 + 0.668988i \(0.766728\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) 31.2311i 1.25125i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 24.9848i 0.997799i
\(628\) 0 0
\(629\) −5.75379 −0.229419
\(630\) 0 0
\(631\) −6.63068 −0.263963 −0.131982 0.991252i \(-0.542134\pi\)
−0.131982 + 0.991252i \(0.542134\pi\)
\(632\) 0 0
\(633\) 3.50758i 0.139414i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 9.80776i 0.388598i
\(638\) 0 0
\(639\) −4.87689 −0.192927
\(640\) 0 0
\(641\) 40.2462 1.58963 0.794815 0.606852i \(-0.207568\pi\)
0.794815 + 0.606852i \(0.207568\pi\)
\(642\) 0 0
\(643\) 46.3542i 1.82803i 0.405681 + 0.914015i \(0.367034\pi\)
−0.405681 + 0.914015i \(0.632966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 5.56155i − 0.218647i −0.994006 0.109324i \(-0.965132\pi\)
0.994006 0.109324i \(-0.0348685\pi\)
\(648\) 0 0
\(649\) 40.9848 1.60880
\(650\) 0 0
\(651\) 27.1231 1.06304
\(652\) 0 0
\(653\) − 50.7926i − 1.98767i −0.110876 0.993834i \(-0.535366\pi\)
0.110876 0.993834i \(-0.464634\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 6.98485i − 0.272505i
\(658\) 0 0
\(659\) 2.24621 0.0875000 0.0437500 0.999043i \(-0.486070\pi\)
0.0437500 + 0.999043i \(0.486070\pi\)
\(660\) 0 0
\(661\) −22.4924 −0.874854 −0.437427 0.899254i \(-0.644110\pi\)
−0.437427 + 0.899254i \(0.644110\pi\)
\(662\) 0 0
\(663\) 28.4924i 1.10655i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 4.43845i − 0.171857i
\(668\) 0 0
\(669\) 19.5076 0.754207
\(670\) 0 0
\(671\) −11.5076 −0.444245
\(672\) 0 0
\(673\) 0.438447i 0.0169009i 0.999964 + 0.00845045i \(0.00268989\pi\)
−0.999964 + 0.00845045i \(0.997310\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 28.7386i − 1.10452i −0.833673 0.552258i \(-0.813766\pi\)
0.833673 0.552258i \(-0.186234\pi\)
\(678\) 0 0
\(679\) 0.768944 0.0295094
\(680\) 0 0
\(681\) −1.36932 −0.0524723
\(682\) 0 0
\(683\) 14.4384i 0.552472i 0.961090 + 0.276236i \(0.0890871\pi\)
−0.961090 + 0.276236i \(0.910913\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 25.3693i 0.967900i
\(688\) 0 0
\(689\) −43.6155 −1.66162
\(690\) 0 0
\(691\) −10.2462 −0.389784 −0.194892 0.980825i \(-0.562436\pi\)
−0.194892 + 0.980825i \(0.562436\pi\)
\(692\) 0 0
\(693\) 7.01515i 0.266484i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 18.2462i − 0.691125i
\(698\) 0 0
\(699\) −13.1771 −0.498403
\(700\) 0 0
\(701\) −26.9848 −1.01920 −0.509602 0.860410i \(-0.670207\pi\)
−0.509602 + 0.860410i \(0.670207\pi\)
\(702\) 0 0
\(703\) − 4.49242i − 0.169435i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 50.7386i 1.90822i
\(708\) 0 0
\(709\) −8.73863 −0.328186 −0.164093 0.986445i \(-0.552470\pi\)
−0.164093 + 0.986445i \(0.552470\pi\)
\(710\) 0 0
\(711\) −3.50758 −0.131544
\(712\) 0 0
\(713\) − 5.56155i − 0.208282i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 18.4384i − 0.688596i
\(718\) 0 0
\(719\) 10.7386 0.400483 0.200242 0.979747i \(-0.435827\pi\)
0.200242 + 0.979747i \(0.435827\pi\)
\(720\) 0 0
\(721\) −44.4924 −1.65698
\(722\) 0 0
\(723\) 22.6307i 0.841644i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 28.8769i 1.07098i 0.844540 + 0.535492i \(0.179874\pi\)
−0.844540 + 0.535492i \(0.820126\pi\)
\(728\) 0 0
\(729\) −29.9848 −1.11055
\(730\) 0 0
\(731\) −4.49242 −0.166158
\(732\) 0 0
\(733\) − 2.87689i − 0.106261i −0.998588 0.0531303i \(-0.983080\pi\)
0.998588 0.0531303i \(-0.0169199\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.9848i 1.50970i
\(738\) 0 0
\(739\) 42.5464 1.56509 0.782547 0.622591i \(-0.213920\pi\)
0.782547 + 0.622591i \(0.213920\pi\)
\(740\) 0 0
\(741\) −22.2462 −0.817235
\(742\) 0 0
\(743\) 30.2462i 1.10963i 0.831975 + 0.554813i \(0.187210\pi\)
−0.831975 + 0.554813i \(0.812790\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.73863i − 0.246554i
\(748\) 0 0
\(749\) −37.4773 −1.36939
\(750\) 0 0
\(751\) 8.38447 0.305954 0.152977 0.988230i \(-0.451114\pi\)
0.152977 + 0.988230i \(0.451114\pi\)
\(752\) 0 0
\(753\) − 16.0000i − 0.583072i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 33.1231i 1.20388i 0.798541 + 0.601940i \(0.205605\pi\)
−0.798541 + 0.601940i \(0.794395\pi\)
\(758\) 0 0
\(759\) −6.24621 −0.226723
\(760\) 0 0
\(761\) 42.3002 1.53338 0.766690 0.642017i \(-0.221902\pi\)
0.766690 + 0.642017i \(0.221902\pi\)
\(762\) 0 0
\(763\) − 25.7538i − 0.932350i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.4924i 1.31767i
\(768\) 0 0
\(769\) −5.50758 −0.198608 −0.0993042 0.995057i \(-0.531662\pi\)
−0.0993042 + 0.995057i \(0.531662\pi\)
\(770\) 0 0
\(771\) −43.8078 −1.57770
\(772\) 0 0
\(773\) − 12.2462i − 0.440466i −0.975447 0.220233i \(-0.929318\pi\)
0.975447 0.220233i \(-0.0706817\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5.47727i 0.196496i
\(778\) 0 0
\(779\) 14.2462 0.510423
\(780\) 0 0
\(781\) 34.7386 1.24305
\(782\) 0 0
\(783\) 24.6847i 0.882158i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 29.7538i − 1.06061i −0.847808 0.530304i \(-0.822078\pi\)
0.847808 0.530304i \(-0.177922\pi\)
\(788\) 0 0
\(789\) −12.4924 −0.444742
\(790\) 0 0
\(791\) −11.7235 −0.416839
\(792\) 0 0
\(793\) − 10.2462i − 0.363854i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.8769i 0.668654i 0.942457 + 0.334327i \(0.108509\pi\)
−0.942457 + 0.334327i \(0.891491\pi\)
\(798\) 0 0
\(799\) −44.4924 −1.57403
\(800\) 0 0
\(801\) −5.61553 −0.198415
\(802\) 0 0
\(803\) 49.7538i 1.75577i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 47.3153i − 1.66558i
\(808\) 0 0
\(809\) −22.4924 −0.790791 −0.395396 0.918511i \(-0.629393\pi\)
−0.395396 + 0.918511i \(0.629393\pi\)
\(810\) 0 0
\(811\) −20.6847 −0.726337 −0.363168 0.931724i \(-0.618305\pi\)
−0.363168 + 0.931724i \(0.618305\pi\)
\(812\) 0 0
\(813\) 22.2462i 0.780209i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3.50758i − 0.122715i
\(818\) 0 0
\(819\) −6.24621 −0.218260
\(820\) 0 0
\(821\) −48.2462 −1.68380 −0.841902 0.539630i \(-0.818564\pi\)
−0.841902 + 0.539630i \(0.818564\pi\)
\(822\) 0 0
\(823\) 21.5616i 0.751588i 0.926703 + 0.375794i \(0.122630\pi\)
−0.926703 + 0.375794i \(0.877370\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) −17.5076 −0.608063 −0.304032 0.952662i \(-0.598333\pi\)
−0.304032 + 0.952662i \(0.598333\pi\)
\(830\) 0 0
\(831\) 24.3002 0.842964
\(832\) 0 0
\(833\) − 14.1080i − 0.488812i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 30.9309i 1.06913i
\(838\) 0 0
\(839\) 31.6155 1.09149 0.545745 0.837952i \(-0.316247\pi\)
0.545745 + 0.837952i \(0.316247\pi\)
\(840\) 0 0
\(841\) −9.30019 −0.320696
\(842\) 0 0
\(843\) − 3.89205i − 0.134049i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 15.6155i − 0.536556i
\(848\) 0 0
\(849\) 48.6004 1.66796
\(850\) 0 0
\(851\) 1.12311 0.0384996
\(852\) 0 0
\(853\) 42.9848i 1.47177i 0.677105 + 0.735887i \(0.263234\pi\)
−0.677105 + 0.735887i \(0.736766\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 11.1771i − 0.381802i −0.981609 0.190901i \(-0.938859\pi\)
0.981609 0.190901i \(-0.0611409\pi\)
\(858\) 0 0
\(859\) 23.4233 0.799192 0.399596 0.916691i \(-0.369150\pi\)
0.399596 + 0.916691i \(0.369150\pi\)
\(860\) 0 0
\(861\) −17.3693 −0.591945
\(862\) 0 0
\(863\) 43.4233i 1.47815i 0.673625 + 0.739073i \(0.264736\pi\)
−0.673625 + 0.739073i \(0.735264\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 14.4384i − 0.490355i
\(868\) 0 0
\(869\) 24.9848 0.847553
\(870\) 0 0
\(871\) −36.4924 −1.23650
\(872\) 0 0
\(873\) 0.138261i 0.00467941i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.73863i 0.295083i 0.989056 + 0.147541i \(0.0471359\pi\)
−0.989056 + 0.147541i \(0.952864\pi\)
\(878\) 0 0
\(879\) −32.9848 −1.11255
\(880\) 0 0
\(881\) −50.1080 −1.68818 −0.844090 0.536202i \(-0.819859\pi\)
−0.844090 + 0.536202i \(0.819859\pi\)
\(882\) 0 0
\(883\) 7.50758i 0.252650i 0.991989 + 0.126325i \(0.0403182\pi\)
−0.991989 + 0.126325i \(0.959682\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 37.5616i − 1.26119i −0.776111 0.630597i \(-0.782810\pi\)
0.776111 0.630597i \(-0.217190\pi\)
\(888\) 0 0
\(889\) 11.8920 0.398847
\(890\) 0 0
\(891\) 28.0000 0.938035
\(892\) 0 0
\(893\) − 34.7386i − 1.16248i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 5.56155i − 0.185695i
\(898\) 0 0
\(899\) 24.6847 0.823279
\(900\) 0 0
\(901\) 62.7386 2.09013
\(902\) 0 0
\(903\) 4.27652i 0.142314i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000i 0.664089i 0.943264 + 0.332045i \(0.107738\pi\)
−0.943264 + 0.332045i \(0.892262\pi\)
\(908\) 0 0
\(909\) −9.12311 −0.302594
\(910\) 0 0
\(911\) 12.4924 0.413892 0.206946 0.978352i \(-0.433647\pi\)
0.206946 + 0.978352i \(0.433647\pi\)
\(912\) 0 0
\(913\) 48.0000i 1.58857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 59.1231i 1.95242i
\(918\) 0 0
\(919\) −21.8617 −0.721152 −0.360576 0.932730i \(-0.617420\pi\)
−0.360576 + 0.932730i \(0.617420\pi\)
\(920\) 0 0
\(921\) −18.7386 −0.617459
\(922\) 0 0
\(923\) 30.9309i 1.01810i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 8.00000i − 0.262754i
\(928\) 0 0
\(929\) 24.0540 0.789185 0.394593 0.918856i \(-0.370886\pi\)
0.394593 + 0.918856i \(0.370886\pi\)
\(930\) 0 0
\(931\) 11.0152 0.361007
\(932\) 0 0
\(933\) − 23.3153i − 0.763310i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.6307i 0.543301i 0.962396 + 0.271650i \(0.0875694\pi\)
−0.962396 + 0.271650i \(0.912431\pi\)
\(938\) 0 0
\(939\) 10.7386 0.350442
\(940\) 0 0
\(941\) 46.6004 1.51913 0.759564 0.650432i \(-0.225412\pi\)
0.759564 + 0.650432i \(0.225412\pi\)
\(942\) 0 0
\(943\) 3.56155i 0.115980i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 55.8078i − 1.81351i −0.421659 0.906754i \(-0.638552\pi\)
0.421659 0.906754i \(-0.361448\pi\)
\(948\) 0 0
\(949\) −44.3002 −1.43804
\(950\) 0 0
\(951\) 44.8769 1.45523
\(952\) 0 0
\(953\) − 46.4924i − 1.50604i −0.657999 0.753019i \(-0.728597\pi\)
0.657999 0.753019i \(-0.271403\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 27.7235i − 0.896173i
\(958\) 0 0
\(959\) 21.4773 0.693537
\(960\) 0 0
\(961\) −0.0691303 −0.00223001
\(962\) 0 0
\(963\) − 6.73863i − 0.217149i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 45.1771i − 1.45280i −0.687274 0.726398i \(-0.741193\pi\)
0.687274 0.726398i \(-0.258807\pi\)
\(968\) 0 0
\(969\) 32.0000 1.02799
\(970\) 0 0
\(971\) −21.3693 −0.685774 −0.342887 0.939377i \(-0.611405\pi\)
−0.342887 + 0.939377i \(0.611405\pi\)
\(972\) 0 0
\(973\) 39.6155i 1.27002i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 26.1080i − 0.835267i −0.908615 0.417634i \(-0.862860\pi\)
0.908615 0.417634i \(-0.137140\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 4.63068 0.147846
\(982\) 0 0
\(983\) − 28.8769i − 0.921030i −0.887652 0.460515i \(-0.847665\pi\)
0.887652 0.460515i \(-0.152335\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 42.3542i 1.34815i
\(988\) 0 0
\(989\) 0.876894 0.0278836
\(990\) 0 0
\(991\) 20.4924 0.650963 0.325482 0.945548i \(-0.394474\pi\)
0.325482 + 0.945548i \(0.394474\pi\)
\(992\) 0 0
\(993\) 10.0540i 0.319053i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.738634i 0.0233928i 0.999932 + 0.0116964i \(0.00372316\pi\)
−0.999932 + 0.0116964i \(0.996277\pi\)
\(998\) 0 0
\(999\) −6.24621 −0.197621
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 4600.2.e.m.4049.3 4
5.2 odd 4 4600.2.a.r.1.2 2
5.3 odd 4 920.2.a.f.1.1 2
5.4 even 2 inner 4600.2.e.m.4049.2 4
15.8 even 4 8280.2.a.bb.1.1 2
20.3 even 4 1840.2.a.k.1.2 2
20.7 even 4 9200.2.a.bx.1.1 2
40.3 even 4 7360.2.a.bm.1.1 2
40.13 odd 4 7360.2.a.bj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.f.1.1 2 5.3 odd 4
1840.2.a.k.1.2 2 20.3 even 4
4600.2.a.r.1.2 2 5.2 odd 4
4600.2.e.m.4049.2 4 5.4 even 2 inner
4600.2.e.m.4049.3 4 1.1 even 1 trivial
7360.2.a.bj.1.2 2 40.13 odd 4
7360.2.a.bm.1.1 2 40.3 even 4
8280.2.a.bb.1.1 2 15.8 even 4
9200.2.a.bx.1.1 2 20.7 even 4