Properties

Label 169.2.b.b.168.6
Level $169$
Weight $2$
Character 169.168
Analytic conductor $1.349$
Analytic rank $0$
Dimension $6$
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] = [169,2,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.b (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: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.6
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.2.b.b.168.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24698i q^{2} -0.554958 q^{3} -3.04892 q^{4} +1.44504i q^{5} -1.24698i q^{6} +2.04892i q^{7} -2.35690i q^{8} -2.69202 q^{9} +O(q^{10})\) \(q+2.24698i q^{2} -0.554958 q^{3} -3.04892 q^{4} +1.44504i q^{5} -1.24698i q^{6} +2.04892i q^{7} -2.35690i q^{8} -2.69202 q^{9} -3.24698 q^{10} -2.55496i q^{11} +1.69202 q^{12} -4.60388 q^{14} -0.801938i q^{15} -0.801938 q^{16} +5.29590 q^{17} -6.04892i q^{18} +5.85086i q^{19} -4.40581i q^{20} -1.13706i q^{21} +5.74094 q^{22} +1.89008 q^{23} +1.30798i q^{24} +2.91185 q^{25} +3.15883 q^{27} -6.24698i q^{28} +2.26875 q^{29} +1.80194 q^{30} +4.26875i q^{31} -6.51573i q^{32} +1.41789i q^{33} +11.8998i q^{34} -2.96077 q^{35} +8.20775 q^{36} +5.35690i q^{37} -13.1468 q^{38} +3.40581 q^{40} -1.27413i q^{41} +2.55496 q^{42} -6.13706 q^{43} +7.78986i q^{44} -3.89008i q^{45} +4.24698i q^{46} -2.95108i q^{47} +0.445042 q^{48} +2.80194 q^{49} +6.54288i q^{50} -2.93900 q^{51} +5.52111 q^{53} +7.09783i q^{54} +3.69202 q^{55} +4.82908 q^{56} -3.24698i q^{57} +5.09783i q^{58} -12.2078i q^{59} +2.44504i q^{60} +8.56465 q^{61} -9.59179 q^{62} -5.51573i q^{63} +13.0368 q^{64} -3.18598 q^{66} -0.576728i q^{67} -16.1468 q^{68} -1.04892 q^{69} -6.65279i q^{70} +4.59419i q^{71} +6.34481i q^{72} -10.5526i q^{73} -12.0368 q^{74} -1.61596 q^{75} -17.8388i q^{76} +5.23490 q^{77} -15.7778 q^{79} -1.15883i q^{80} +6.32304 q^{81} +2.86294 q^{82} -7.72348i q^{83} +3.46681i q^{84} +7.65279i q^{85} -13.7899i q^{86} -1.25906 q^{87} -6.02177 q^{88} +6.61356i q^{89} +8.74094 q^{90} -5.76271 q^{92} -2.36898i q^{93} +6.63102 q^{94} -8.45473 q^{95} +3.61596i q^{96} -11.9269i q^{97} +6.29590i q^{98} +6.87800i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24698i 1.58885i 0.607359 + 0.794427i \(0.292229\pi\)
−0.607359 + 0.794427i \(0.707771\pi\)
\(3\) −0.554958 −0.320405 −0.160203 0.987084i \(-0.551215\pi\)
−0.160203 + 0.987084i \(0.551215\pi\)
\(4\) −3.04892 −1.52446
\(5\) 1.44504i 0.646242i 0.946358 + 0.323121i \(0.104732\pi\)
−0.946358 + 0.323121i \(0.895268\pi\)
\(6\) − 1.24698i − 0.509077i
\(7\) 2.04892i 0.774418i 0.921992 + 0.387209i \(0.126561\pi\)
−0.921992 + 0.387209i \(0.873439\pi\)
\(8\) − 2.35690i − 0.833289i
\(9\) −2.69202 −0.897340
\(10\) −3.24698 −1.02679
\(11\) − 2.55496i − 0.770349i −0.922844 0.385174i \(-0.874141\pi\)
0.922844 0.385174i \(-0.125859\pi\)
\(12\) 1.69202 0.488445
\(13\) 0 0
\(14\) −4.60388 −1.23044
\(15\) − 0.801938i − 0.207059i
\(16\) −0.801938 −0.200484
\(17\) 5.29590 1.28444 0.642222 0.766519i \(-0.278013\pi\)
0.642222 + 0.766519i \(0.278013\pi\)
\(18\) − 6.04892i − 1.42574i
\(19\) 5.85086i 1.34228i 0.741331 + 0.671139i \(0.234195\pi\)
−0.741331 + 0.671139i \(0.765805\pi\)
\(20\) − 4.40581i − 0.985170i
\(21\) − 1.13706i − 0.248128i
\(22\) 5.74094 1.22397
\(23\) 1.89008 0.394110 0.197055 0.980392i \(-0.436862\pi\)
0.197055 + 0.980392i \(0.436862\pi\)
\(24\) 1.30798i 0.266990i
\(25\) 2.91185 0.582371
\(26\) 0 0
\(27\) 3.15883 0.607918
\(28\) − 6.24698i − 1.18057i
\(29\) 2.26875 0.421296 0.210648 0.977562i \(-0.432443\pi\)
0.210648 + 0.977562i \(0.432443\pi\)
\(30\) 1.80194 0.328987
\(31\) 4.26875i 0.766690i 0.923605 + 0.383345i \(0.125228\pi\)
−0.923605 + 0.383345i \(0.874772\pi\)
\(32\) − 6.51573i − 1.15183i
\(33\) 1.41789i 0.246824i
\(34\) 11.8998i 2.04079i
\(35\) −2.96077 −0.500462
\(36\) 8.20775 1.36796
\(37\) 5.35690i 0.880668i 0.897834 + 0.440334i \(0.145140\pi\)
−0.897834 + 0.440334i \(0.854860\pi\)
\(38\) −13.1468 −2.13268
\(39\) 0 0
\(40\) 3.40581 0.538506
\(41\) − 1.27413i − 0.198985i −0.995038 0.0994926i \(-0.968278\pi\)
0.995038 0.0994926i \(-0.0317220\pi\)
\(42\) 2.55496 0.394239
\(43\) −6.13706 −0.935893 −0.467947 0.883757i \(-0.655006\pi\)
−0.467947 + 0.883757i \(0.655006\pi\)
\(44\) 7.78986i 1.17437i
\(45\) − 3.89008i − 0.579899i
\(46\) 4.24698i 0.626183i
\(47\) − 2.95108i − 0.430460i −0.976563 0.215230i \(-0.930950\pi\)
0.976563 0.215230i \(-0.0690501\pi\)
\(48\) 0.445042 0.0642363
\(49\) 2.80194 0.400277
\(50\) 6.54288i 0.925302i
\(51\) −2.93900 −0.411542
\(52\) 0 0
\(53\) 5.52111 0.758382 0.379191 0.925318i \(-0.376202\pi\)
0.379191 + 0.925318i \(0.376202\pi\)
\(54\) 7.09783i 0.965893i
\(55\) 3.69202 0.497832
\(56\) 4.82908 0.645314
\(57\) − 3.24698i − 0.430073i
\(58\) 5.09783i 0.669378i
\(59\) − 12.2078i − 1.58931i −0.607059 0.794657i \(-0.707651\pi\)
0.607059 0.794657i \(-0.292349\pi\)
\(60\) 2.44504i 0.315654i
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) −9.59179 −1.21816
\(63\) − 5.51573i − 0.694917i
\(64\) 13.0368 1.62960
\(65\) 0 0
\(66\) −3.18598 −0.392167
\(67\) − 0.576728i − 0.0704586i −0.999379 0.0352293i \(-0.988784\pi\)
0.999379 0.0352293i \(-0.0112162\pi\)
\(68\) −16.1468 −1.95808
\(69\) −1.04892 −0.126275
\(70\) − 6.65279i − 0.795161i
\(71\) 4.59419i 0.545230i 0.962123 + 0.272615i \(0.0878885\pi\)
−0.962123 + 0.272615i \(0.912112\pi\)
\(72\) 6.34481i 0.747744i
\(73\) − 10.5526i − 1.23508i −0.786538 0.617542i \(-0.788128\pi\)
0.786538 0.617542i \(-0.211872\pi\)
\(74\) −12.0368 −1.39925
\(75\) −1.61596 −0.186595
\(76\) − 17.8388i − 2.04625i
\(77\) 5.23490 0.596572
\(78\) 0 0
\(79\) −15.7778 −1.77514 −0.887569 0.460674i \(-0.847608\pi\)
−0.887569 + 0.460674i \(0.847608\pi\)
\(80\) − 1.15883i − 0.129562i
\(81\) 6.32304 0.702560
\(82\) 2.86294 0.316158
\(83\) − 7.72348i − 0.847762i −0.905718 0.423881i \(-0.860667\pi\)
0.905718 0.423881i \(-0.139333\pi\)
\(84\) 3.46681i 0.378260i
\(85\) 7.65279i 0.830062i
\(86\) − 13.7899i − 1.48700i
\(87\) −1.25906 −0.134986
\(88\) −6.02177 −0.641923
\(89\) 6.61356i 0.701036i 0.936556 + 0.350518i \(0.113995\pi\)
−0.936556 + 0.350518i \(0.886005\pi\)
\(90\) 8.74094 0.921376
\(91\) 0 0
\(92\) −5.76271 −0.600804
\(93\) − 2.36898i − 0.245652i
\(94\) 6.63102 0.683938
\(95\) −8.45473 −0.867437
\(96\) 3.61596i 0.369052i
\(97\) − 11.9269i − 1.21100i −0.795847 0.605498i \(-0.792974\pi\)
0.795847 0.605498i \(-0.207026\pi\)
\(98\) 6.29590i 0.635982i
\(99\) 6.87800i 0.691265i
\(100\) −8.87800 −0.887800
\(101\) −13.0640 −1.29991 −0.649957 0.759971i \(-0.725213\pi\)
−0.649957 + 0.759971i \(0.725213\pi\)
\(102\) − 6.60388i − 0.653881i
\(103\) −9.16852 −0.903401 −0.451701 0.892170i \(-0.649182\pi\)
−0.451701 + 0.892170i \(0.649182\pi\)
\(104\) 0 0
\(105\) 1.64310 0.160351
\(106\) 12.4058i 1.20496i
\(107\) −6.89977 −0.667026 −0.333513 0.942745i \(-0.608234\pi\)
−0.333513 + 0.942745i \(0.608234\pi\)
\(108\) −9.63102 −0.926746
\(109\) − 0.121998i − 0.0116853i −0.999983 0.00584264i \(-0.998140\pi\)
0.999983 0.00584264i \(-0.00185978\pi\)
\(110\) 8.29590i 0.790983i
\(111\) − 2.97285i − 0.282171i
\(112\) − 1.64310i − 0.155259i
\(113\) 7.30798 0.687477 0.343738 0.939065i \(-0.388307\pi\)
0.343738 + 0.939065i \(0.388307\pi\)
\(114\) 7.29590 0.683323
\(115\) 2.73125i 0.254690i
\(116\) −6.91723 −0.642249
\(117\) 0 0
\(118\) 27.4306 2.52519
\(119\) 10.8509i 0.994696i
\(120\) −1.89008 −0.172540
\(121\) 4.47219 0.406563
\(122\) 19.2446i 1.74232i
\(123\) 0.707087i 0.0637559i
\(124\) − 13.0151i − 1.16879i
\(125\) 11.4330i 1.02260i
\(126\) 12.3937 1.10412
\(127\) 18.9705 1.68336 0.841678 0.539980i \(-0.181568\pi\)
0.841678 + 0.539980i \(0.181568\pi\)
\(128\) 16.2620i 1.43738i
\(129\) 3.40581 0.299865
\(130\) 0 0
\(131\) 3.25667 0.284536 0.142268 0.989828i \(-0.454560\pi\)
0.142268 + 0.989828i \(0.454560\pi\)
\(132\) − 4.32304i − 0.376273i
\(133\) −11.9879 −1.03948
\(134\) 1.29590 0.111948
\(135\) 4.56465i 0.392862i
\(136\) − 12.4819i − 1.07031i
\(137\) 0.792249i 0.0676864i 0.999427 + 0.0338432i \(0.0107747\pi\)
−0.999427 + 0.0338432i \(0.989225\pi\)
\(138\) − 2.35690i − 0.200632i
\(139\) −11.3394 −0.961799 −0.480899 0.876776i \(-0.659690\pi\)
−0.480899 + 0.876776i \(0.659690\pi\)
\(140\) 9.02715 0.762933
\(141\) 1.63773i 0.137922i
\(142\) −10.3230 −0.866291
\(143\) 0 0
\(144\) 2.15883 0.179903
\(145\) 3.27844i 0.272260i
\(146\) 23.7114 1.96237
\(147\) −1.55496 −0.128251
\(148\) − 16.3327i − 1.34254i
\(149\) − 8.40581i − 0.688631i −0.938854 0.344316i \(-0.888111\pi\)
0.938854 0.344316i \(-0.111889\pi\)
\(150\) − 3.63102i − 0.296472i
\(151\) 14.1293i 1.14983i 0.818215 + 0.574913i \(0.194964\pi\)
−0.818215 + 0.574913i \(0.805036\pi\)
\(152\) 13.7899 1.11851
\(153\) −14.2567 −1.15258
\(154\) 11.7627i 0.947866i
\(155\) −6.16852 −0.495468
\(156\) 0 0
\(157\) −9.43296 −0.752832 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(158\) − 35.4523i − 2.82044i
\(159\) −3.06398 −0.242990
\(160\) 9.41550 0.744361
\(161\) 3.87263i 0.305206i
\(162\) 14.2078i 1.11627i
\(163\) 8.70410i 0.681758i 0.940107 + 0.340879i \(0.110725\pi\)
−0.940107 + 0.340879i \(0.889275\pi\)
\(164\) 3.88471i 0.303345i
\(165\) −2.04892 −0.159508
\(166\) 17.3545 1.34697
\(167\) − 23.8538i − 1.84587i −0.384961 0.922933i \(-0.625785\pi\)
0.384961 0.922933i \(-0.374215\pi\)
\(168\) −2.67994 −0.206762
\(169\) 0 0
\(170\) −17.1957 −1.31885
\(171\) − 15.7506i − 1.20448i
\(172\) 18.7114 1.42673
\(173\) 18.8552 1.43353 0.716766 0.697314i \(-0.245622\pi\)
0.716766 + 0.697314i \(0.245622\pi\)
\(174\) − 2.82908i − 0.214472i
\(175\) 5.96615i 0.450998i
\(176\) 2.04892i 0.154443i
\(177\) 6.77479i 0.509224i
\(178\) −14.8605 −1.11384
\(179\) −6.02177 −0.450088 −0.225044 0.974349i \(-0.572253\pi\)
−0.225044 + 0.974349i \(0.572253\pi\)
\(180\) 11.8605i 0.884033i
\(181\) 4.77777 0.355129 0.177565 0.984109i \(-0.443178\pi\)
0.177565 + 0.984109i \(0.443178\pi\)
\(182\) 0 0
\(183\) −4.75302 −0.351353
\(184\) − 4.45473i − 0.328407i
\(185\) −7.74094 −0.569125
\(186\) 5.32304 0.390305
\(187\) − 13.5308i − 0.989470i
\(188\) 8.99761i 0.656218i
\(189\) 6.47219i 0.470782i
\(190\) − 18.9976i − 1.37823i
\(191\) 18.4306 1.33359 0.666795 0.745242i \(-0.267666\pi\)
0.666795 + 0.745242i \(0.267666\pi\)
\(192\) −7.23490 −0.522134
\(193\) 6.05429i 0.435798i 0.975971 + 0.217899i \(0.0699203\pi\)
−0.975971 + 0.217899i \(0.930080\pi\)
\(194\) 26.7995 1.92410
\(195\) 0 0
\(196\) −8.54288 −0.610205
\(197\) 11.4155i 0.813321i 0.913579 + 0.406660i \(0.133307\pi\)
−0.913579 + 0.406660i \(0.866693\pi\)
\(198\) −15.4547 −1.09832
\(199\) 13.9051 0.985710 0.492855 0.870111i \(-0.335953\pi\)
0.492855 + 0.870111i \(0.335953\pi\)
\(200\) − 6.86294i − 0.485283i
\(201\) 0.320060i 0.0225753i
\(202\) − 29.3545i − 2.06538i
\(203\) 4.64848i 0.326259i
\(204\) 8.96077 0.627379
\(205\) 1.84117 0.128593
\(206\) − 20.6015i − 1.43537i
\(207\) −5.08815 −0.353651
\(208\) 0 0
\(209\) 14.9487 1.03402
\(210\) 3.69202i 0.254774i
\(211\) −13.2446 −0.911795 −0.455897 0.890032i \(-0.650682\pi\)
−0.455897 + 0.890032i \(0.650682\pi\)
\(212\) −16.8334 −1.15612
\(213\) − 2.54958i − 0.174694i
\(214\) − 15.5036i − 1.05981i
\(215\) − 8.86831i − 0.604814i
\(216\) − 7.44504i − 0.506571i
\(217\) −8.74632 −0.593739
\(218\) 0.274127 0.0185662
\(219\) 5.85623i 0.395727i
\(220\) −11.2567 −0.758924
\(221\) 0 0
\(222\) 6.67994 0.448328
\(223\) 7.33513i 0.491196i 0.969372 + 0.245598i \(0.0789844\pi\)
−0.969372 + 0.245598i \(0.921016\pi\)
\(224\) 13.3502 0.891997
\(225\) −7.83877 −0.522585
\(226\) 16.4209i 1.09230i
\(227\) 8.67456i 0.575751i 0.957668 + 0.287875i \(0.0929489\pi\)
−0.957668 + 0.287875i \(0.907051\pi\)
\(228\) 9.89977i 0.655628i
\(229\) − 13.6866i − 0.904439i −0.891907 0.452219i \(-0.850632\pi\)
0.891907 0.452219i \(-0.149368\pi\)
\(230\) −6.13706 −0.404666
\(231\) −2.90515 −0.191145
\(232\) − 5.34721i − 0.351061i
\(233\) 5.08815 0.333336 0.166668 0.986013i \(-0.446699\pi\)
0.166668 + 0.986013i \(0.446699\pi\)
\(234\) 0 0
\(235\) 4.26444 0.278181
\(236\) 37.2204i 2.42284i
\(237\) 8.75600 0.568764
\(238\) −24.3817 −1.58043
\(239\) 10.9239i 0.706611i 0.935508 + 0.353305i \(0.114942\pi\)
−0.935508 + 0.353305i \(0.885058\pi\)
\(240\) 0.643104i 0.0415122i
\(241\) 11.9148i 0.767502i 0.923437 + 0.383751i \(0.125368\pi\)
−0.923437 + 0.383751i \(0.874632\pi\)
\(242\) 10.0489i 0.645969i
\(243\) −12.9855 −0.833022
\(244\) −26.1129 −1.67171
\(245\) 4.04892i 0.258676i
\(246\) −1.58881 −0.101299
\(247\) 0 0
\(248\) 10.0610 0.638874
\(249\) 4.28621i 0.271627i
\(250\) −25.6896 −1.62475
\(251\) −22.3478 −1.41058 −0.705290 0.708919i \(-0.749183\pi\)
−0.705290 + 0.708919i \(0.749183\pi\)
\(252\) 16.8170i 1.05937i
\(253\) − 4.82908i − 0.303602i
\(254\) 42.6262i 2.67461i
\(255\) − 4.24698i − 0.265956i
\(256\) −10.4668 −0.654176
\(257\) 18.6601 1.16398 0.581992 0.813194i \(-0.302273\pi\)
0.581992 + 0.813194i \(0.302273\pi\)
\(258\) 7.65279i 0.476442i
\(259\) −10.9758 −0.682005
\(260\) 0 0
\(261\) −6.10752 −0.378046
\(262\) 7.31767i 0.452087i
\(263\) 14.3991 0.887887 0.443944 0.896055i \(-0.353579\pi\)
0.443944 + 0.896055i \(0.353579\pi\)
\(264\) 3.34183 0.205675
\(265\) 7.97823i 0.490099i
\(266\) − 26.9366i − 1.65159i
\(267\) − 3.67025i − 0.224616i
\(268\) 1.75840i 0.107411i
\(269\) 0.652793 0.0398015 0.0199007 0.999802i \(-0.493665\pi\)
0.0199007 + 0.999802i \(0.493665\pi\)
\(270\) −10.2567 −0.624201
\(271\) − 1.99569i − 0.121229i −0.998161 0.0606147i \(-0.980694\pi\)
0.998161 0.0606147i \(-0.0193061\pi\)
\(272\) −4.24698 −0.257511
\(273\) 0 0
\(274\) −1.78017 −0.107544
\(275\) − 7.43967i − 0.448629i
\(276\) 3.19806 0.192501
\(277\) −11.7845 −0.708061 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(278\) − 25.4795i − 1.52816i
\(279\) − 11.4916i − 0.687982i
\(280\) 6.97823i 0.417029i
\(281\) − 6.47219i − 0.386098i −0.981189 0.193049i \(-0.938162\pi\)
0.981189 0.193049i \(-0.0618377\pi\)
\(282\) −3.67994 −0.219137
\(283\) −6.58104 −0.391202 −0.195601 0.980684i \(-0.562666\pi\)
−0.195601 + 0.980684i \(0.562666\pi\)
\(284\) − 14.0073i − 0.831180i
\(285\) 4.69202 0.277931
\(286\) 0 0
\(287\) 2.61058 0.154098
\(288\) 17.5405i 1.03358i
\(289\) 11.0465 0.649796
\(290\) −7.36658 −0.432581
\(291\) 6.61894i 0.388009i
\(292\) 32.1739i 1.88284i
\(293\) − 24.3381i − 1.42185i −0.703269 0.710924i \(-0.748277\pi\)
0.703269 0.710924i \(-0.251723\pi\)
\(294\) − 3.49396i − 0.203772i
\(295\) 17.6407 1.02708
\(296\) 12.6256 0.733851
\(297\) − 8.07069i − 0.468309i
\(298\) 18.8877 1.09413
\(299\) 0 0
\(300\) 4.92692 0.284456
\(301\) − 12.5743i − 0.724773i
\(302\) −31.7482 −1.82691
\(303\) 7.24996 0.416500
\(304\) − 4.69202i − 0.269106i
\(305\) 12.3763i 0.708663i
\(306\) − 32.0344i − 1.83129i
\(307\) − 14.0737i − 0.803227i −0.915809 0.401613i \(-0.868450\pi\)
0.915809 0.401613i \(-0.131550\pi\)
\(308\) −15.9608 −0.909449
\(309\) 5.08815 0.289455
\(310\) − 13.8605i − 0.787226i
\(311\) 29.7700 1.68810 0.844051 0.536263i \(-0.180164\pi\)
0.844051 + 0.536263i \(0.180164\pi\)
\(312\) 0 0
\(313\) −7.47889 −0.422732 −0.211366 0.977407i \(-0.567791\pi\)
−0.211366 + 0.977407i \(0.567791\pi\)
\(314\) − 21.1957i − 1.19614i
\(315\) 7.97046 0.449085
\(316\) 48.1051 2.70613
\(317\) − 30.0301i − 1.68666i −0.537396 0.843330i \(-0.680592\pi\)
0.537396 0.843330i \(-0.319408\pi\)
\(318\) − 6.88471i − 0.386075i
\(319\) − 5.79656i − 0.324545i
\(320\) 18.8388i 1.05312i
\(321\) 3.82908 0.213719
\(322\) −8.70171 −0.484927
\(323\) 30.9855i 1.72408i
\(324\) −19.2784 −1.07102
\(325\) 0 0
\(326\) −19.5579 −1.08321
\(327\) 0.0677037i 0.00374402i
\(328\) −3.00298 −0.165812
\(329\) 6.04652 0.333356
\(330\) − 4.60388i − 0.253435i
\(331\) 15.7168i 0.863872i 0.901904 + 0.431936i \(0.142169\pi\)
−0.901904 + 0.431936i \(0.857831\pi\)
\(332\) 23.5483i 1.29238i
\(333\) − 14.4209i − 0.790259i
\(334\) 53.5991 2.93281
\(335\) 0.833397 0.0455333
\(336\) 0.911854i 0.0497457i
\(337\) −1.95407 −0.106445 −0.0532224 0.998583i \(-0.516949\pi\)
−0.0532224 + 0.998583i \(0.516949\pi\)
\(338\) 0 0
\(339\) −4.05562 −0.220271
\(340\) − 23.3327i − 1.26540i
\(341\) 10.9065 0.590619
\(342\) 35.3913 1.91374
\(343\) 20.0834i 1.08440i
\(344\) 14.4644i 0.779869i
\(345\) − 1.51573i − 0.0816041i
\(346\) 42.3672i 2.27767i
\(347\) −17.1250 −0.919317 −0.459659 0.888096i \(-0.652028\pi\)
−0.459659 + 0.888096i \(0.652028\pi\)
\(348\) 3.83877 0.205780
\(349\) − 10.4668i − 0.560276i −0.959960 0.280138i \(-0.909620\pi\)
0.959960 0.280138i \(-0.0903802\pi\)
\(350\) −13.4058 −0.716571
\(351\) 0 0
\(352\) −16.6474 −0.887310
\(353\) − 15.5308i − 0.826621i −0.910590 0.413310i \(-0.864372\pi\)
0.910590 0.413310i \(-0.135628\pi\)
\(354\) −15.2228 −0.809084
\(355\) −6.63879 −0.352351
\(356\) − 20.1642i − 1.06870i
\(357\) − 6.02177i − 0.318706i
\(358\) − 13.5308i − 0.715125i
\(359\) − 21.4263i − 1.13083i −0.824805 0.565417i \(-0.808715\pi\)
0.824805 0.565417i \(-0.191285\pi\)
\(360\) −9.16852 −0.483224
\(361\) −15.2325 −0.801711
\(362\) 10.7356i 0.564249i
\(363\) −2.48188 −0.130265
\(364\) 0 0
\(365\) 15.2489 0.798164
\(366\) − 10.6799i − 0.558249i
\(367\) 34.3032 1.79061 0.895306 0.445452i \(-0.146957\pi\)
0.895306 + 0.445452i \(0.146957\pi\)
\(368\) −1.51573 −0.0790129
\(369\) 3.42998i 0.178557i
\(370\) − 17.3937i − 0.904257i
\(371\) 11.3123i 0.587305i
\(372\) 7.22282i 0.374486i
\(373\) −12.5961 −0.652202 −0.326101 0.945335i \(-0.605735\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(374\) 30.4034 1.57212
\(375\) − 6.34481i − 0.327645i
\(376\) −6.95539 −0.358697
\(377\) 0 0
\(378\) −14.5429 −0.748005
\(379\) − 16.5386i − 0.849529i −0.905304 0.424765i \(-0.860357\pi\)
0.905304 0.424765i \(-0.139643\pi\)
\(380\) 25.7778 1.32237
\(381\) −10.5278 −0.539356
\(382\) 41.4131i 2.11888i
\(383\) − 7.53617i − 0.385080i −0.981289 0.192540i \(-0.938327\pi\)
0.981289 0.192540i \(-0.0616726\pi\)
\(384\) − 9.02475i − 0.460543i
\(385\) 7.56465i 0.385530i
\(386\) −13.6039 −0.692419
\(387\) 16.5211 0.839815
\(388\) 36.3642i 1.84611i
\(389\) −35.5555 −1.80274 −0.901369 0.433052i \(-0.857437\pi\)
−0.901369 + 0.433052i \(0.857437\pi\)
\(390\) 0 0
\(391\) 10.0097 0.506212
\(392\) − 6.60388i − 0.333546i
\(393\) −1.80731 −0.0911670
\(394\) −25.6504 −1.29225
\(395\) − 22.7995i − 1.14717i
\(396\) − 20.9705i − 1.05381i
\(397\) 1.35152i 0.0678308i 0.999425 + 0.0339154i \(0.0107977\pi\)
−0.999425 + 0.0339154i \(0.989202\pi\)
\(398\) 31.2446i 1.56615i
\(399\) 6.65279 0.333056
\(400\) −2.33513 −0.116756
\(401\) 0.579121i 0.0289199i 0.999895 + 0.0144600i \(0.00460291\pi\)
−0.999895 + 0.0144600i \(0.995397\pi\)
\(402\) −0.719169 −0.0358689
\(403\) 0 0
\(404\) 39.8310 1.98167
\(405\) 9.13706i 0.454024i
\(406\) −10.4450 −0.518379
\(407\) 13.6866 0.678422
\(408\) 6.92692i 0.342934i
\(409\) 15.1575i 0.749490i 0.927128 + 0.374745i \(0.122270\pi\)
−0.927128 + 0.374745i \(0.877730\pi\)
\(410\) 4.13706i 0.204315i
\(411\) − 0.439665i − 0.0216871i
\(412\) 27.9541 1.37720
\(413\) 25.0127 1.23079
\(414\) − 11.4330i − 0.561899i
\(415\) 11.1608 0.547860
\(416\) 0 0
\(417\) 6.29291 0.308165
\(418\) 33.5894i 1.64291i
\(419\) −35.7235 −1.74521 −0.872603 0.488430i \(-0.837570\pi\)
−0.872603 + 0.488430i \(0.837570\pi\)
\(420\) −5.00969 −0.244448
\(421\) 35.0465i 1.70806i 0.520221 + 0.854032i \(0.325849\pi\)
−0.520221 + 0.854032i \(0.674151\pi\)
\(422\) − 29.7603i − 1.44871i
\(423\) 7.94438i 0.386269i
\(424\) − 13.0127i − 0.631951i
\(425\) 15.4209 0.748022
\(426\) 5.72886 0.277564
\(427\) 17.5483i 0.849220i
\(428\) 21.0368 1.01685
\(429\) 0 0
\(430\) 19.9269 0.960961
\(431\) 34.2814i 1.65128i 0.564199 + 0.825639i \(0.309185\pi\)
−0.564199 + 0.825639i \(0.690815\pi\)
\(432\) −2.53319 −0.121878
\(433\) −13.7385 −0.660232 −0.330116 0.943940i \(-0.607088\pi\)
−0.330116 + 0.943940i \(0.607088\pi\)
\(434\) − 19.6528i − 0.943364i
\(435\) − 1.81940i − 0.0872334i
\(436\) 0.371961i 0.0178137i
\(437\) 11.0586i 0.529005i
\(438\) −13.1588 −0.628753
\(439\) −10.2403 −0.488742 −0.244371 0.969682i \(-0.578581\pi\)
−0.244371 + 0.969682i \(0.578581\pi\)
\(440\) − 8.70171i − 0.414838i
\(441\) −7.54288 −0.359185
\(442\) 0 0
\(443\) 12.1763 0.578513 0.289257 0.957252i \(-0.406592\pi\)
0.289257 + 0.957252i \(0.406592\pi\)
\(444\) 9.06398i 0.430158i
\(445\) −9.55688 −0.453039
\(446\) −16.4819 −0.780440
\(447\) 4.66487i 0.220641i
\(448\) 26.7114i 1.26199i
\(449\) − 12.9051i − 0.609032i −0.952507 0.304516i \(-0.901505\pi\)
0.952507 0.304516i \(-0.0984947\pi\)
\(450\) − 17.6136i − 0.830311i
\(451\) −3.25534 −0.153288
\(452\) −22.2814 −1.04803
\(453\) − 7.84117i − 0.368410i
\(454\) −19.4916 −0.914785
\(455\) 0 0
\(456\) −7.65279 −0.358375
\(457\) − 4.65710i − 0.217850i −0.994050 0.108925i \(-0.965259\pi\)
0.994050 0.108925i \(-0.0347409\pi\)
\(458\) 30.7536 1.43702
\(459\) 16.7289 0.780836
\(460\) − 8.32736i − 0.388265i
\(461\) 31.5405i 1.46899i 0.678616 + 0.734493i \(0.262580\pi\)
−0.678616 + 0.734493i \(0.737420\pi\)
\(462\) − 6.52781i − 0.303701i
\(463\) 17.6504i 0.820284i 0.912022 + 0.410142i \(0.134521\pi\)
−0.912022 + 0.410142i \(0.865479\pi\)
\(464\) −1.81940 −0.0844633
\(465\) 3.42327 0.158750
\(466\) 11.4330i 0.529622i
\(467\) 32.1726 1.48877 0.744385 0.667751i \(-0.232743\pi\)
0.744385 + 0.667751i \(0.232743\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) 9.58211i 0.441990i
\(471\) 5.23490 0.241211
\(472\) −28.7724 −1.32436
\(473\) 15.6799i 0.720964i
\(474\) 19.6746i 0.903683i
\(475\) 17.0368i 0.781704i
\(476\) − 33.0834i − 1.51637i
\(477\) −14.8629 −0.680527
\(478\) −24.5459 −1.12270
\(479\) − 34.8998i − 1.59461i −0.603576 0.797306i \(-0.706258\pi\)
0.603576 0.797306i \(-0.293742\pi\)
\(480\) −5.22521 −0.238497
\(481\) 0 0
\(482\) −26.7724 −1.21945
\(483\) − 2.14914i − 0.0977895i
\(484\) −13.6353 −0.619788
\(485\) 17.2349 0.782596
\(486\) − 29.1782i − 1.32355i
\(487\) − 41.8351i − 1.89573i −0.318676 0.947864i \(-0.603238\pi\)
0.318676 0.947864i \(-0.396762\pi\)
\(488\) − 20.1860i − 0.913776i
\(489\) − 4.83041i − 0.218439i
\(490\) −9.09783 −0.410998
\(491\) −21.8455 −0.985873 −0.492936 0.870065i \(-0.664076\pi\)
−0.492936 + 0.870065i \(0.664076\pi\)
\(492\) − 2.15585i − 0.0971932i
\(493\) 12.0151 0.541131
\(494\) 0 0
\(495\) −9.93900 −0.446725
\(496\) − 3.42327i − 0.153709i
\(497\) −9.41311 −0.422236
\(498\) −9.63102 −0.431576
\(499\) − 23.5472i − 1.05412i −0.849829 0.527058i \(-0.823295\pi\)
0.849829 0.527058i \(-0.176705\pi\)
\(500\) − 34.8582i − 1.55890i
\(501\) 13.2379i 0.591425i
\(502\) − 50.2150i − 2.24121i
\(503\) −7.08682 −0.315986 −0.157993 0.987440i \(-0.550502\pi\)
−0.157993 + 0.987440i \(0.550502\pi\)
\(504\) −13.0000 −0.579066
\(505\) − 18.8780i − 0.840060i
\(506\) 10.8509 0.482379
\(507\) 0 0
\(508\) −57.8394 −2.56621
\(509\) − 7.61894i − 0.337704i −0.985641 0.168852i \(-0.945994\pi\)
0.985641 0.168852i \(-0.0540059\pi\)
\(510\) 9.54288 0.422566
\(511\) 21.6213 0.956471
\(512\) 9.00538i 0.397985i
\(513\) 18.4819i 0.815995i
\(514\) 41.9288i 1.84940i
\(515\) − 13.2489i − 0.583816i
\(516\) −10.3840 −0.457132
\(517\) −7.53989 −0.331604
\(518\) − 24.6625i − 1.08361i
\(519\) −10.4638 −0.459311
\(520\) 0 0
\(521\) −39.5133 −1.73111 −0.865555 0.500813i \(-0.833034\pi\)
−0.865555 + 0.500813i \(0.833034\pi\)
\(522\) − 13.7235i − 0.600660i
\(523\) −15.8194 −0.691734 −0.345867 0.938284i \(-0.612415\pi\)
−0.345867 + 0.938284i \(0.612415\pi\)
\(524\) −9.92931 −0.433764
\(525\) − 3.31096i − 0.144502i
\(526\) 32.3545i 1.41072i
\(527\) 22.6069i 0.984770i
\(528\) − 1.13706i − 0.0494843i
\(529\) −19.4276 −0.844678
\(530\) −17.9269 −0.778696
\(531\) 32.8635i 1.42616i
\(532\) 36.5502 1.58465
\(533\) 0 0
\(534\) 8.24698 0.356882
\(535\) − 9.97046i − 0.431061i
\(536\) −1.35929 −0.0587123
\(537\) 3.34183 0.144211
\(538\) 1.46681i 0.0632388i
\(539\) − 7.15883i − 0.308353i
\(540\) − 13.9172i − 0.598902i
\(541\) 34.4819i 1.48249i 0.671234 + 0.741246i \(0.265765\pi\)
−0.671234 + 0.741246i \(0.734235\pi\)
\(542\) 4.48427 0.192616
\(543\) −2.65146 −0.113785
\(544\) − 34.5066i − 1.47946i
\(545\) 0.176292 0.00755152
\(546\) 0 0
\(547\) 36.8582 1.57594 0.787970 0.615713i \(-0.211132\pi\)
0.787970 + 0.615713i \(0.211132\pi\)
\(548\) − 2.41550i − 0.103185i
\(549\) −23.0562 −0.984015
\(550\) 16.7168 0.712806
\(551\) 13.2741i 0.565497i
\(552\) 2.47219i 0.105223i
\(553\) − 32.3274i − 1.37470i
\(554\) − 26.4795i − 1.12501i
\(555\) 4.29590 0.182351
\(556\) 34.5730 1.46622
\(557\) 1.27652i 0.0540879i 0.999634 + 0.0270439i \(0.00860940\pi\)
−0.999634 + 0.0270439i \(0.991391\pi\)
\(558\) 25.8213 1.09310
\(559\) 0 0
\(560\) 2.37435 0.100335
\(561\) 7.50902i 0.317031i
\(562\) 14.5429 0.613454
\(563\) 9.12737 0.384673 0.192336 0.981329i \(-0.438393\pi\)
0.192336 + 0.981329i \(0.438393\pi\)
\(564\) − 4.99330i − 0.210256i
\(565\) 10.5603i 0.444277i
\(566\) − 14.7875i − 0.621563i
\(567\) 12.9554i 0.544075i
\(568\) 10.8280 0.454334
\(569\) 5.72156 0.239860 0.119930 0.992782i \(-0.461733\pi\)
0.119930 + 0.992782i \(0.461733\pi\)
\(570\) 10.5429i 0.441593i
\(571\) −7.60148 −0.318112 −0.159056 0.987270i \(-0.550845\pi\)
−0.159056 + 0.987270i \(0.550845\pi\)
\(572\) 0 0
\(573\) −10.2282 −0.427289
\(574\) 5.86592i 0.244839i
\(575\) 5.50365 0.229518
\(576\) −35.0954 −1.46231
\(577\) − 45.1564i − 1.87989i −0.341330 0.939944i \(-0.610877\pi\)
0.341330 0.939944i \(-0.389123\pi\)
\(578\) 24.8213i 1.03243i
\(579\) − 3.35988i − 0.139632i
\(580\) − 9.99569i − 0.415048i
\(581\) 15.8248 0.656522
\(582\) −14.8726 −0.616490
\(583\) − 14.1062i − 0.584219i
\(584\) −24.8713 −1.02918
\(585\) 0 0
\(586\) 54.6872 2.25911
\(587\) 32.4040i 1.33746i 0.743507 + 0.668728i \(0.233161\pi\)
−0.743507 + 0.668728i \(0.766839\pi\)
\(588\) 4.74094 0.195513
\(589\) −24.9758 −1.02911
\(590\) 39.6383i 1.63188i
\(591\) − 6.33513i − 0.260592i
\(592\) − 4.29590i − 0.176560i
\(593\) 36.6848i 1.50647i 0.657754 + 0.753233i \(0.271507\pi\)
−0.657754 + 0.753233i \(0.728493\pi\)
\(594\) 18.1347 0.744075
\(595\) −15.6799 −0.642815
\(596\) 25.6286i 1.04979i
\(597\) −7.71678 −0.315827
\(598\) 0 0
\(599\) −9.99223 −0.408271 −0.204136 0.978943i \(-0.565438\pi\)
−0.204136 + 0.978943i \(0.565438\pi\)
\(600\) 3.80864i 0.155487i
\(601\) −1.81163 −0.0738978 −0.0369489 0.999317i \(-0.511764\pi\)
−0.0369489 + 0.999317i \(0.511764\pi\)
\(602\) 28.2543 1.15156
\(603\) 1.55257i 0.0632253i
\(604\) − 43.0790i − 1.75286i
\(605\) 6.46250i 0.262738i
\(606\) 16.2905i 0.661757i
\(607\) 11.2161 0.455248 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(608\) 38.1226 1.54608
\(609\) − 2.57971i − 0.104535i
\(610\) −27.8092 −1.12596
\(611\) 0 0
\(612\) 43.4674 1.75707
\(613\) 20.8944i 0.843917i 0.906615 + 0.421958i \(0.138657\pi\)
−0.906615 + 0.421958i \(0.861343\pi\)
\(614\) 31.6233 1.27621
\(615\) −1.02177 −0.0412018
\(616\) − 12.3381i − 0.497117i
\(617\) − 12.0992i − 0.487094i −0.969889 0.243547i \(-0.921689\pi\)
0.969889 0.243547i \(-0.0783110\pi\)
\(618\) 11.4330i 0.459901i
\(619\) − 10.5526i − 0.424143i −0.977254 0.212072i \(-0.931979\pi\)
0.977254 0.212072i \(-0.0680210\pi\)
\(620\) 18.8073 0.755320
\(621\) 5.97046 0.239586
\(622\) 66.8926i 2.68215i
\(623\) −13.5506 −0.542895
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) − 16.8049i − 0.671660i
\(627\) −8.29590 −0.331306
\(628\) 28.7603 1.14766
\(629\) 28.3696i 1.13117i
\(630\) 17.9095i 0.713530i
\(631\) − 13.8514i − 0.551417i −0.961241 0.275709i \(-0.911087\pi\)
0.961241 0.275709i \(-0.0889125\pi\)
\(632\) 37.1866i 1.47920i
\(633\) 7.35019 0.292144
\(634\) 67.4771 2.67986
\(635\) 27.4131i 1.08786i
\(636\) 9.34183 0.370428
\(637\) 0 0
\(638\) 13.0248 0.515655
\(639\) − 12.3676i − 0.489257i
\(640\) −23.4993 −0.928893
\(641\) −34.9608 −1.38087 −0.690434 0.723396i \(-0.742580\pi\)
−0.690434 + 0.723396i \(0.742580\pi\)
\(642\) 8.60388i 0.339568i
\(643\) − 33.3980i − 1.31709i −0.752541 0.658545i \(-0.771172\pi\)
0.752541 0.658545i \(-0.228828\pi\)
\(644\) − 11.8073i − 0.465273i
\(645\) 4.92154i 0.193786i
\(646\) −69.6238 −2.73931
\(647\) −2.32842 −0.0915397 −0.0457698 0.998952i \(-0.514574\pi\)
−0.0457698 + 0.998952i \(0.514574\pi\)
\(648\) − 14.9028i − 0.585436i
\(649\) −31.1903 −1.22433
\(650\) 0 0
\(651\) 4.85384 0.190237
\(652\) − 26.5381i − 1.03931i
\(653\) 14.5714 0.570221 0.285111 0.958495i \(-0.407970\pi\)
0.285111 + 0.958495i \(0.407970\pi\)
\(654\) −0.152129 −0.00594871
\(655\) 4.70602i 0.183879i
\(656\) 1.02177i 0.0398934i
\(657\) 28.4077i 1.10829i
\(658\) 13.5864i 0.529654i
\(659\) 11.1395 0.433932 0.216966 0.976179i \(-0.430384\pi\)
0.216966 + 0.976179i \(0.430384\pi\)
\(660\) 6.24698 0.243163
\(661\) − 13.8498i − 0.538694i −0.963043 0.269347i \(-0.913192\pi\)
0.963043 0.269347i \(-0.0868079\pi\)
\(662\) −35.3153 −1.37257
\(663\) 0 0
\(664\) −18.2034 −0.706430
\(665\) − 17.3230i − 0.671759i
\(666\) 32.4034 1.25561
\(667\) 4.28813 0.166037
\(668\) 72.7284i 2.81395i
\(669\) − 4.07069i − 0.157382i
\(670\) 1.87263i 0.0723458i
\(671\) − 21.8823i − 0.844757i
\(672\) −7.40880 −0.285801
\(673\) 6.52973 0.251703 0.125851 0.992049i \(-0.459834\pi\)
0.125851 + 0.992049i \(0.459834\pi\)
\(674\) − 4.39075i − 0.169125i
\(675\) 9.19806 0.354034
\(676\) 0 0
\(677\) −11.3104 −0.434693 −0.217346 0.976095i \(-0.569740\pi\)
−0.217346 + 0.976095i \(0.569740\pi\)
\(678\) − 9.11290i − 0.349979i
\(679\) 24.4373 0.937816
\(680\) 18.0368 0.691681
\(681\) − 4.81402i − 0.184474i
\(682\) 24.5066i 0.938407i
\(683\) 14.1793i 0.542555i 0.962501 + 0.271277i \(0.0874461\pi\)
−0.962501 + 0.271277i \(0.912554\pi\)
\(684\) 48.0224i 1.83618i
\(685\) −1.14483 −0.0437418
\(686\) −45.1269 −1.72295
\(687\) 7.59551i 0.289787i
\(688\) 4.92154 0.187632
\(689\) 0 0
\(690\) 3.40581 0.129657
\(691\) − 30.7952i − 1.17151i −0.810490 0.585753i \(-0.800799\pi\)
0.810490 0.585753i \(-0.199201\pi\)
\(692\) −57.4878 −2.18536
\(693\) −14.0925 −0.535328
\(694\) − 38.4795i − 1.46066i
\(695\) − 16.3860i − 0.621555i
\(696\) 2.96748i 0.112482i
\(697\) − 6.74764i − 0.255585i
\(698\) 23.5187 0.890196
\(699\) −2.82371 −0.106802
\(700\) − 18.1903i − 0.687528i
\(701\) −6.73184 −0.254258 −0.127129 0.991886i \(-0.540576\pi\)
−0.127129 + 0.991886i \(0.540576\pi\)
\(702\) 0 0
\(703\) −31.3424 −1.18210
\(704\) − 33.3086i − 1.25536i
\(705\) −2.36658 −0.0891307
\(706\) 34.8974 1.31338
\(707\) − 26.7670i − 1.00668i
\(708\) − 20.6558i − 0.776292i
\(709\) − 47.6252i − 1.78860i −0.447467 0.894300i \(-0.647674\pi\)
0.447467 0.894300i \(-0.352326\pi\)
\(710\) − 14.9172i − 0.559834i
\(711\) 42.4741 1.59290
\(712\) 15.5875 0.584166
\(713\) 8.06829i 0.302160i
\(714\) 13.5308 0.506377
\(715\) 0 0
\(716\) 18.3599 0.686141
\(717\) − 6.06233i − 0.226402i
\(718\) 48.1444 1.79673
\(719\) 5.99330 0.223512 0.111756 0.993736i \(-0.464352\pi\)
0.111756 + 0.993736i \(0.464352\pi\)
\(720\) 3.11960i 0.116261i
\(721\) − 18.7855i − 0.699610i
\(722\) − 34.2271i − 1.27380i
\(723\) − 6.61224i − 0.245912i
\(724\) −14.5670 −0.541380
\(725\) 6.60627 0.245351
\(726\) − 5.57673i − 0.206972i
\(727\) 24.1226 0.894657 0.447329 0.894370i \(-0.352375\pi\)
0.447329 + 0.894370i \(0.352375\pi\)
\(728\) 0 0
\(729\) −11.7627 −0.435656
\(730\) 34.2640i 1.26817i
\(731\) −32.5013 −1.20210
\(732\) 14.4916 0.535624
\(733\) − 36.0646i − 1.33208i −0.745918 0.666038i \(-0.767989\pi\)
0.745918 0.666038i \(-0.232011\pi\)
\(734\) 77.0786i 2.84502i
\(735\) − 2.24698i − 0.0828811i
\(736\) − 12.3153i − 0.453947i
\(737\) −1.47352 −0.0542777
\(738\) −7.70709 −0.283702
\(739\) 27.5254i 1.01254i 0.862375 + 0.506269i \(0.168976\pi\)
−0.862375 + 0.506269i \(0.831024\pi\)
\(740\) 23.6015 0.867608
\(741\) 0 0
\(742\) −25.4185 −0.933142
\(743\) 10.4692i 0.384078i 0.981387 + 0.192039i \(0.0615100\pi\)
−0.981387 + 0.192039i \(0.938490\pi\)
\(744\) −5.58343 −0.204699
\(745\) 12.1468 0.445023
\(746\) − 28.3032i − 1.03625i
\(747\) 20.7918i 0.760731i
\(748\) 41.2543i 1.50841i
\(749\) − 14.1371i − 0.516557i
\(750\) 14.2567 0.520580
\(751\) −4.06770 −0.148433 −0.0742163 0.997242i \(-0.523646\pi\)
−0.0742163 + 0.997242i \(0.523646\pi\)
\(752\) 2.36658i 0.0863005i
\(753\) 12.4021 0.451957
\(754\) 0 0
\(755\) −20.4174 −0.743066
\(756\) − 19.7332i − 0.717688i
\(757\) 20.4336 0.742670 0.371335 0.928499i \(-0.378900\pi\)
0.371335 + 0.928499i \(0.378900\pi\)
\(758\) 37.1618 1.34978
\(759\) 2.67994i 0.0972757i
\(760\) 19.9269i 0.722825i
\(761\) − 27.0237i − 0.979608i −0.871833 0.489804i \(-0.837068\pi\)
0.871833 0.489804i \(-0.162932\pi\)
\(762\) − 23.6558i − 0.856958i
\(763\) 0.249964 0.00904929
\(764\) −56.1933 −2.03300
\(765\) − 20.6015i − 0.744848i
\(766\) 16.9336 0.611837
\(767\) 0 0
\(768\) 5.80864 0.209601
\(769\) 37.9407i 1.36818i 0.729400 + 0.684088i \(0.239799\pi\)
−0.729400 + 0.684088i \(0.760201\pi\)
\(770\) −16.9976 −0.612551
\(771\) −10.3556 −0.372947
\(772\) − 18.4590i − 0.664355i
\(773\) 16.3375i 0.587620i 0.955864 + 0.293810i \(0.0949232\pi\)
−0.955864 + 0.293810i \(0.905077\pi\)
\(774\) 37.1226i 1.33434i
\(775\) 12.4300i 0.446498i
\(776\) −28.1105 −1.00911
\(777\) 6.09113 0.218518
\(778\) − 79.8926i − 2.86429i
\(779\) 7.45473 0.267093
\(780\) 0 0
\(781\) 11.7380 0.420017
\(782\) 22.4916i 0.804297i
\(783\) 7.16660 0.256114
\(784\) −2.24698 −0.0802493
\(785\) − 13.6310i − 0.486512i
\(786\) − 4.06100i − 0.144851i
\(787\) − 18.6907i − 0.666251i −0.942882 0.333126i \(-0.891897\pi\)
0.942882 0.333126i \(-0.108103\pi\)
\(788\) − 34.8049i − 1.23987i
\(789\) −7.99090 −0.284484
\(790\) 51.2301 1.82269
\(791\) 14.9734i 0.532394i
\(792\) 16.2107 0.576023
\(793\) 0 0
\(794\) −3.03684 −0.107773
\(795\) − 4.42758i − 0.157030i
\(796\) −42.3957 −1.50267
\(797\) −29.2519 −1.03615 −0.518077 0.855334i \(-0.673352\pi\)
−0.518077 + 0.855334i \(0.673352\pi\)
\(798\) 14.9487i 0.529178i
\(799\) − 15.6286i − 0.552901i
\(800\) − 18.9729i − 0.670792i
\(801\) − 17.8039i − 0.629068i
\(802\) −1.30127 −0.0459496
\(803\) −26.9614 −0.951446
\(804\) − 0.975837i − 0.0344151i
\(805\) −5.59611 −0.197237
\(806\) 0 0
\(807\) −0.362273 −0.0127526
\(808\) 30.7904i 1.08320i
\(809\) −6.65087 −0.233832 −0.116916 0.993142i \(-0.537301\pi\)
−0.116916 + 0.993142i \(0.537301\pi\)
\(810\) −20.5308 −0.721379
\(811\) − 3.89200i − 0.136667i −0.997663 0.0683333i \(-0.978232\pi\)
0.997663 0.0683333i \(-0.0217681\pi\)
\(812\) − 14.1728i − 0.497369i
\(813\) 1.10752i 0.0388425i
\(814\) 30.7536i 1.07791i
\(815\) −12.5778 −0.440581
\(816\) 2.35690 0.0825079
\(817\) − 35.9071i − 1.25623i
\(818\) −34.0586 −1.19083
\(819\) 0 0
\(820\) −5.61356 −0.196034
\(821\) − 45.9982i − 1.60535i −0.596418 0.802674i \(-0.703410\pi\)
0.596418 0.802674i \(-0.296590\pi\)
\(822\) 0.987918 0.0344576
\(823\) −7.95300 −0.277224 −0.138612 0.990347i \(-0.544264\pi\)
−0.138612 + 0.990347i \(0.544264\pi\)
\(824\) 21.6093i 0.752794i
\(825\) 4.12870i 0.143743i
\(826\) 56.2030i 1.95555i
\(827\) 27.9648i 0.972432i 0.873839 + 0.486216i \(0.161623\pi\)
−0.873839 + 0.486216i \(0.838377\pi\)
\(828\) 15.5133 0.539126
\(829\) −27.6310 −0.959665 −0.479833 0.877360i \(-0.659303\pi\)
−0.479833 + 0.877360i \(0.659303\pi\)
\(830\) 25.0780i 0.870470i
\(831\) 6.53989 0.226866
\(832\) 0 0
\(833\) 14.8388 0.514133
\(834\) 14.1400i 0.489630i
\(835\) 34.4698 1.19288
\(836\) −45.5773 −1.57632
\(837\) 13.4843i 0.466085i
\(838\) − 80.2699i − 2.77288i
\(839\) 28.6848i 0.990311i 0.868804 + 0.495155i \(0.164889\pi\)
−0.868804 + 0.495155i \(0.835111\pi\)
\(840\) − 3.87263i − 0.133618i
\(841\) −23.8528 −0.822509
\(842\) −78.7488 −2.71386
\(843\) 3.59179i 0.123708i
\(844\) 40.3817 1.38999
\(845\) 0 0
\(846\) −17.8509 −0.613725
\(847\) 9.16315i 0.314849i
\(848\) −4.42758 −0.152044
\(849\) 3.65220 0.125343
\(850\) 34.6504i 1.18850i
\(851\) 10.1250i 0.347080i
\(852\) 7.77346i 0.266314i
\(853\) − 43.2078i − 1.47941i −0.672934 0.739703i \(-0.734966\pi\)
0.672934 0.739703i \(-0.265034\pi\)
\(854\) −39.4306 −1.34929
\(855\) 22.7603 0.778386
\(856\) 16.2620i 0.555825i
\(857\) −35.1685 −1.20133 −0.600667 0.799499i \(-0.705098\pi\)
−0.600667 + 0.799499i \(0.705098\pi\)
\(858\) 0 0
\(859\) 27.3793 0.934168 0.467084 0.884213i \(-0.345305\pi\)
0.467084 + 0.884213i \(0.345305\pi\)
\(860\) 27.0388i 0.922014i
\(861\) −1.44876 −0.0493737
\(862\) −77.0297 −2.62364
\(863\) 41.3913i 1.40898i 0.709715 + 0.704489i \(0.248824\pi\)
−0.709715 + 0.704489i \(0.751176\pi\)
\(864\) − 20.5821i − 0.700217i
\(865\) 27.2465i 0.926409i
\(866\) − 30.8702i − 1.04901i
\(867\) −6.13036 −0.208198
\(868\) 26.6668 0.905130
\(869\) 40.3116i 1.36748i
\(870\) 4.08815 0.138601
\(871\) 0 0
\(872\) −0.287536 −0.00973721
\(873\) 32.1075i 1.08668i
\(874\) −24.8485 −0.840512
\(875\) −23.4252 −0.791916
\(876\) − 17.8552i − 0.603270i
\(877\) 24.7472i 0.835653i 0.908527 + 0.417826i \(0.137208\pi\)
−0.908527 + 0.417826i \(0.862792\pi\)
\(878\) − 23.0097i − 0.776539i
\(879\) 13.5066i 0.455567i
\(880\) −2.96077 −0.0998076
\(881\) 28.5875 0.963137 0.481568 0.876409i \(-0.340067\pi\)
0.481568 + 0.876409i \(0.340067\pi\)
\(882\) − 16.9487i − 0.570692i
\(883\) −9.61702 −0.323639 −0.161819 0.986820i \(-0.551736\pi\)
−0.161819 + 0.986820i \(0.551736\pi\)
\(884\) 0 0
\(885\) −9.78986 −0.329082
\(886\) 27.3599i 0.919173i
\(887\) 15.9661 0.536091 0.268045 0.963406i \(-0.413622\pi\)
0.268045 + 0.963406i \(0.413622\pi\)
\(888\) −7.00670 −0.235130
\(889\) 38.8689i 1.30362i
\(890\) − 21.4741i − 0.719814i
\(891\) − 16.1551i − 0.541217i
\(892\) − 22.3642i − 0.748809i
\(893\) 17.2664 0.577797
\(894\) −10.4819 −0.350566
\(895\) − 8.70171i − 0.290866i
\(896\) −33.3196 −1.11313
\(897\) 0 0
\(898\) 28.9976 0.967663
\(899\) 9.68473i 0.323004i
\(900\) 23.8998 0.796659
\(901\) 29.2392 0.974099
\(902\) − 7.31468i − 0.243552i
\(903\) 6.97823i 0.232221i
\(904\) − 17.2241i − 0.572867i
\(905\) 6.90408i 0.229500i
\(906\) 17.6189 0.585350
\(907\) 28.8364 0.957496 0.478748 0.877952i \(-0.341091\pi\)
0.478748 + 0.877952i \(0.341091\pi\)
\(908\) − 26.4480i − 0.877709i
\(909\) 35.1685 1.16647
\(910\) 0 0
\(911\) 38.5633 1.27766 0.638830 0.769348i \(-0.279419\pi\)
0.638830 + 0.769348i \(0.279419\pi\)
\(912\) 2.60388i 0.0862229i
\(913\) −19.7332 −0.653073
\(914\) 10.4644 0.346132
\(915\) − 6.86831i − 0.227059i
\(916\) 41.7294i 1.37878i
\(917\) 6.67264i 0.220350i
\(918\) 37.5894i 1.24064i
\(919\) 8.87502 0.292760 0.146380 0.989228i \(-0.453238\pi\)
0.146380 + 0.989228i \(0.453238\pi\)
\(920\) 6.43727 0.212231
\(921\) 7.81030i 0.257358i
\(922\) −70.8708 −2.33401
\(923\) 0 0
\(924\) 8.85756 0.291392
\(925\) 15.5985i 0.512875i
\(926\) −39.6601 −1.30331
\(927\) 24.6819 0.810659
\(928\) − 14.7826i − 0.485261i
\(929\) − 24.2295i − 0.794945i −0.917614 0.397472i \(-0.869887\pi\)
0.917614 0.397472i \(-0.130113\pi\)
\(930\) 7.69202i 0.252231i
\(931\) 16.3937i 0.537283i
\(932\) −15.5133 −0.508156
\(933\) −16.5211 −0.540877
\(934\) 72.2911i 2.36544i
\(935\) 19.5526 0.639437
\(936\) 0 0
\(937\) 17.2644 0.564005 0.282002 0.959414i \(-0.409001\pi\)
0.282002 + 0.959414i \(0.409001\pi\)
\(938\) 2.65519i 0.0866949i
\(939\) 4.15047 0.135446
\(940\) −13.0019 −0.424076
\(941\) − 4.34050i − 0.141496i −0.997494 0.0707482i \(-0.977461\pi\)
0.997494 0.0707482i \(-0.0225387\pi\)
\(942\) 11.7627i 0.383250i
\(943\) − 2.40821i − 0.0784220i
\(944\) 9.78986i 0.318633i
\(945\) −9.35258 −0.304240
\(946\) −35.2325 −1.14551
\(947\) − 45.0146i − 1.46278i −0.681961 0.731389i \(-0.738872\pi\)
0.681961 0.731389i \(-0.261128\pi\)
\(948\) −26.6963 −0.867057
\(949\) 0 0
\(950\) −38.2814 −1.24201
\(951\) 16.6655i 0.540415i
\(952\) 25.5743 0.828869
\(953\) 46.8859 1.51878 0.759391 0.650634i \(-0.225497\pi\)
0.759391 + 0.650634i \(0.225497\pi\)
\(954\) − 33.3967i − 1.08126i
\(955\) 26.6329i 0.861822i
\(956\) − 33.3062i − 1.07720i
\(957\) 3.21685i 0.103986i
\(958\) 78.4191 2.53361
\(959\) −1.62325 −0.0524176
\(960\) − 10.4547i − 0.337425i
\(961\) 12.7778 0.412186
\(962\) 0 0
\(963\) 18.5743 0.598550
\(964\) − 36.3274i − 1.17003i
\(965\) −8.74871 −0.281631
\(966\) 4.82908 0.155373
\(967\) − 6.29457i − 0.202420i −0.994865 0.101210i \(-0.967729\pi\)
0.994865 0.101210i \(-0.0322714\pi\)
\(968\) − 10.5405i − 0.338784i
\(969\) − 17.1957i − 0.552404i
\(970\) 38.7265i 1.24343i
\(971\) −41.8068 −1.34165 −0.670823 0.741618i \(-0.734059\pi\)
−0.670823 + 0.741618i \(0.734059\pi\)
\(972\) 39.5918 1.26991
\(973\) − 23.2336i − 0.744834i
\(974\) 94.0025 3.01203
\(975\) 0 0
\(976\) −6.86831 −0.219849
\(977\) − 23.7530i − 0.759926i −0.925002 0.379963i \(-0.875937\pi\)
0.925002 0.379963i \(-0.124063\pi\)
\(978\) 10.8538 0.347067
\(979\) 16.8974 0.540043
\(980\) − 12.3448i − 0.394341i
\(981\) 0.328421i 0.0104857i
\(982\) − 49.0863i − 1.56641i
\(983\) − 55.7251i − 1.77736i −0.458532 0.888678i \(-0.651625\pi\)
0.458532 0.888678i \(-0.348375\pi\)
\(984\) 1.66653 0.0531270
\(985\) −16.4959 −0.525602
\(986\) 26.9976i 0.859779i
\(987\) −3.35557 −0.106809
\(988\) 0 0
\(989\) −11.5996 −0.368845
\(990\) − 22.3327i − 0.709781i
\(991\) −35.5512 −1.12932 −0.564661 0.825323i \(-0.690993\pi\)
−0.564661 + 0.825323i \(0.690993\pi\)
\(992\) 27.8140 0.883096
\(993\) − 8.72215i − 0.276789i
\(994\) − 21.1511i − 0.670871i
\(995\) 20.0935i 0.637007i
\(996\) − 13.0683i − 0.414085i
\(997\) 6.61058 0.209359 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(998\) 52.9101 1.67484
\(999\) 16.9215i 0.535374i
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 169.2.b.b.168.6 6
3.2 odd 2 1521.2.b.l.1351.1 6
4.3 odd 2 2704.2.f.o.337.4 6
13.2 odd 12 169.2.c.b.22.1 6
13.3 even 3 169.2.e.b.147.1 12
13.4 even 6 169.2.e.b.23.1 12
13.5 odd 4 169.2.a.c.1.3 yes 3
13.6 odd 12 169.2.c.b.146.1 6
13.7 odd 12 169.2.c.c.146.3 6
13.8 odd 4 169.2.a.b.1.1 3
13.9 even 3 169.2.e.b.23.6 12
13.10 even 6 169.2.e.b.147.6 12
13.11 odd 12 169.2.c.c.22.3 6
13.12 even 2 inner 169.2.b.b.168.1 6
39.5 even 4 1521.2.a.o.1.1 3
39.8 even 4 1521.2.a.r.1.3 3
39.38 odd 2 1521.2.b.l.1351.6 6
52.31 even 4 2704.2.a.ba.1.2 3
52.47 even 4 2704.2.a.z.1.2 3
52.51 odd 2 2704.2.f.o.337.3 6
65.34 odd 4 4225.2.a.bg.1.3 3
65.44 odd 4 4225.2.a.bb.1.1 3
91.34 even 4 8281.2.a.bf.1.1 3
91.83 even 4 8281.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 13.8 odd 4
169.2.a.c.1.3 yes 3 13.5 odd 4
169.2.b.b.168.1 6 13.12 even 2 inner
169.2.b.b.168.6 6 1.1 even 1 trivial
169.2.c.b.22.1 6 13.2 odd 12
169.2.c.b.146.1 6 13.6 odd 12
169.2.c.c.22.3 6 13.11 odd 12
169.2.c.c.146.3 6 13.7 odd 12
169.2.e.b.23.1 12 13.4 even 6
169.2.e.b.23.6 12 13.9 even 3
169.2.e.b.147.1 12 13.3 even 3
169.2.e.b.147.6 12 13.10 even 6
1521.2.a.o.1.1 3 39.5 even 4
1521.2.a.r.1.3 3 39.8 even 4
1521.2.b.l.1351.1 6 3.2 odd 2
1521.2.b.l.1351.6 6 39.38 odd 2
2704.2.a.z.1.2 3 52.47 even 4
2704.2.a.ba.1.2 3 52.31 even 4
2704.2.f.o.337.3 6 52.51 odd 2
2704.2.f.o.337.4 6 4.3 odd 2
4225.2.a.bb.1.1 3 65.44 odd 4
4225.2.a.bg.1.3 3 65.34 odd 4
8281.2.a.bf.1.1 3 91.34 even 4
8281.2.a.bj.1.3 3 91.83 even 4