Properties

Label 169.2.a.c.1.3
Level $169$
Weight $2$
Character 169.1
Self dual yes
Analytic conductor $1.349$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(1,169)]
 
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([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

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: yes
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24698 q^{2} -0.554958 q^{3} +3.04892 q^{4} +1.44504 q^{5} -1.24698 q^{6} -2.04892 q^{7} +2.35690 q^{8} -2.69202 q^{9} +O(q^{10})\) \(q+2.24698 q^{2} -0.554958 q^{3} +3.04892 q^{4} +1.44504 q^{5} -1.24698 q^{6} -2.04892 q^{7} +2.35690 q^{8} -2.69202 q^{9} +3.24698 q^{10} +2.55496 q^{11} -1.69202 q^{12} -4.60388 q^{14} -0.801938 q^{15} -0.801938 q^{16} -5.29590 q^{17} -6.04892 q^{18} +5.85086 q^{19} +4.40581 q^{20} +1.13706 q^{21} +5.74094 q^{22} -1.89008 q^{23} -1.30798 q^{24} -2.91185 q^{25} +3.15883 q^{27} -6.24698 q^{28} +2.26875 q^{29} -1.80194 q^{30} +4.26875 q^{31} -6.51573 q^{32} -1.41789 q^{33} -11.8998 q^{34} -2.96077 q^{35} -8.20775 q^{36} -5.35690 q^{37} +13.1468 q^{38} +3.40581 q^{40} -1.27413 q^{41} +2.55496 q^{42} +6.13706 q^{43} +7.78986 q^{44} -3.89008 q^{45} -4.24698 q^{46} +2.95108 q^{47} +0.445042 q^{48} -2.80194 q^{49} -6.54288 q^{50} +2.93900 q^{51} +5.52111 q^{53} +7.09783 q^{54} +3.69202 q^{55} -4.82908 q^{56} -3.24698 q^{57} +5.09783 q^{58} +12.2078 q^{59} -2.44504 q^{60} +8.56465 q^{61} +9.59179 q^{62} +5.51573 q^{63} -13.0368 q^{64} -3.18598 q^{66} -0.576728 q^{67} -16.1468 q^{68} +1.04892 q^{69} -6.65279 q^{70} +4.59419 q^{71} -6.34481 q^{72} +10.5526 q^{73} -12.0368 q^{74} +1.61596 q^{75} +17.8388 q^{76} -5.23490 q^{77} -15.7778 q^{79} -1.15883 q^{80} +6.32304 q^{81} -2.86294 q^{82} -7.72348 q^{83} +3.46681 q^{84} -7.65279 q^{85} +13.7899 q^{86} -1.25906 q^{87} +6.02177 q^{88} -6.61356 q^{89} -8.74094 q^{90} -5.76271 q^{92} -2.36898 q^{93} +6.63102 q^{94} +8.45473 q^{95} +3.61596 q^{96} -11.9269 q^{97} -6.29590 q^{98} -6.87800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} - 2 q^{3} + 4 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} - 3 q^{9} + 5 q^{10} + 8 q^{11} - 5 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 9 q^{18} + 4 q^{19} - 2 q^{21} + 3 q^{22} - 5 q^{23} - 9 q^{24} - 5 q^{25} + q^{27} - 14 q^{28} - q^{29} - q^{30} + 5 q^{31} - 7 q^{32} - 10 q^{33} - 13 q^{34} + 4 q^{35} - 7 q^{36} - 12 q^{37} + 12 q^{38} - 3 q^{40} + 7 q^{41} + 8 q^{42} + 13 q^{43} - 11 q^{45} - 8 q^{46} + 18 q^{47} + q^{48} - 4 q^{49} - q^{50} - q^{51} + q^{53} + 3 q^{54} + 6 q^{55} - 4 q^{56} - 5 q^{57} - 3 q^{58} + 19 q^{59} - 7 q^{60} + 4 q^{61} + q^{62} + 4 q^{63} - 11 q^{64} + 5 q^{66} + q^{67} - 21 q^{68} - 6 q^{69} - 2 q^{70} + 27 q^{71} + 4 q^{72} - 9 q^{73} - 8 q^{74} + 15 q^{75} + 21 q^{76} + 8 q^{77} - 5 q^{79} + 5 q^{80} - q^{81} - 14 q^{82} + 7 q^{83} + 7 q^{84} - 5 q^{85} + 18 q^{86} - 18 q^{87} + 15 q^{88} + 11 q^{89} - 12 q^{90} - 22 q^{93} + 5 q^{94} + 3 q^{95} + 21 q^{96} - 7 q^{97} - 5 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24698 1.58885 0.794427 0.607359i \(-0.207771\pi\)
0.794427 + 0.607359i \(0.207771\pi\)
\(3\) −0.554958 −0.320405 −0.160203 0.987084i \(-0.551215\pi\)
−0.160203 + 0.987084i \(0.551215\pi\)
\(4\) 3.04892 1.52446
\(5\) 1.44504 0.646242 0.323121 0.946358i \(-0.395268\pi\)
0.323121 + 0.946358i \(0.395268\pi\)
\(6\) −1.24698 −0.509077
\(7\) −2.04892 −0.774418 −0.387209 0.921992i \(-0.626561\pi\)
−0.387209 + 0.921992i \(0.626561\pi\)
\(8\) 2.35690 0.833289
\(9\) −2.69202 −0.897340
\(10\) 3.24698 1.02679
\(11\) 2.55496 0.770349 0.385174 0.922844i \(-0.374141\pi\)
0.385174 + 0.922844i \(0.374141\pi\)
\(12\) −1.69202 −0.488445
\(13\) 0 0
\(14\) −4.60388 −1.23044
\(15\) −0.801938 −0.207059
\(16\) −0.801938 −0.200484
\(17\) −5.29590 −1.28444 −0.642222 0.766519i \(-0.721987\pi\)
−0.642222 + 0.766519i \(0.721987\pi\)
\(18\) −6.04892 −1.42574
\(19\) 5.85086 1.34228 0.671139 0.741331i \(-0.265805\pi\)
0.671139 + 0.741331i \(0.265805\pi\)
\(20\) 4.40581 0.985170
\(21\) 1.13706 0.248128
\(22\) 5.74094 1.22397
\(23\) −1.89008 −0.394110 −0.197055 0.980392i \(-0.563138\pi\)
−0.197055 + 0.980392i \(0.563138\pi\)
\(24\) −1.30798 −0.266990
\(25\) −2.91185 −0.582371
\(26\) 0 0
\(27\) 3.15883 0.607918
\(28\) −6.24698 −1.18057
\(29\) 2.26875 0.421296 0.210648 0.977562i \(-0.432443\pi\)
0.210648 + 0.977562i \(0.432443\pi\)
\(30\) −1.80194 −0.328987
\(31\) 4.26875 0.766690 0.383345 0.923605i \(-0.374772\pi\)
0.383345 + 0.923605i \(0.374772\pi\)
\(32\) −6.51573 −1.15183
\(33\) −1.41789 −0.246824
\(34\) −11.8998 −2.04079
\(35\) −2.96077 −0.500462
\(36\) −8.20775 −1.36796
\(37\) −5.35690 −0.880668 −0.440334 0.897834i \(-0.645140\pi\)
−0.440334 + 0.897834i \(0.645140\pi\)
\(38\) 13.1468 2.13268
\(39\) 0 0
\(40\) 3.40581 0.538506
\(41\) −1.27413 −0.198985 −0.0994926 0.995038i \(-0.531722\pi\)
−0.0994926 + 0.995038i \(0.531722\pi\)
\(42\) 2.55496 0.394239
\(43\) 6.13706 0.935893 0.467947 0.883757i \(-0.344994\pi\)
0.467947 + 0.883757i \(0.344994\pi\)
\(44\) 7.78986 1.17437
\(45\) −3.89008 −0.579899
\(46\) −4.24698 −0.626183
\(47\) 2.95108 0.430460 0.215230 0.976563i \(-0.430950\pi\)
0.215230 + 0.976563i \(0.430950\pi\)
\(48\) 0.445042 0.0642363
\(49\) −2.80194 −0.400277
\(50\) −6.54288 −0.925302
\(51\) 2.93900 0.411542
\(52\) 0 0
\(53\) 5.52111 0.758382 0.379191 0.925318i \(-0.376202\pi\)
0.379191 + 0.925318i \(0.376202\pi\)
\(54\) 7.09783 0.965893
\(55\) 3.69202 0.497832
\(56\) −4.82908 −0.645314
\(57\) −3.24698 −0.430073
\(58\) 5.09783 0.669378
\(59\) 12.2078 1.58931 0.794657 0.607059i \(-0.207651\pi\)
0.794657 + 0.607059i \(0.207651\pi\)
\(60\) −2.44504 −0.315654
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) 9.59179 1.21816
\(63\) 5.51573 0.694917
\(64\) −13.0368 −1.62960
\(65\) 0 0
\(66\) −3.18598 −0.392167
\(67\) −0.576728 −0.0704586 −0.0352293 0.999379i \(-0.511216\pi\)
−0.0352293 + 0.999379i \(0.511216\pi\)
\(68\) −16.1468 −1.95808
\(69\) 1.04892 0.126275
\(70\) −6.65279 −0.795161
\(71\) 4.59419 0.545230 0.272615 0.962123i \(-0.412112\pi\)
0.272615 + 0.962123i \(0.412112\pi\)
\(72\) −6.34481 −0.747744
\(73\) 10.5526 1.23508 0.617542 0.786538i \(-0.288128\pi\)
0.617542 + 0.786538i \(0.288128\pi\)
\(74\) −12.0368 −1.39925
\(75\) 1.61596 0.186595
\(76\) 17.8388 2.04625
\(77\) −5.23490 −0.596572
\(78\) 0 0
\(79\) −15.7778 −1.77514 −0.887569 0.460674i \(-0.847608\pi\)
−0.887569 + 0.460674i \(0.847608\pi\)
\(80\) −1.15883 −0.129562
\(81\) 6.32304 0.702560
\(82\) −2.86294 −0.316158
\(83\) −7.72348 −0.847762 −0.423881 0.905718i \(-0.639333\pi\)
−0.423881 + 0.905718i \(0.639333\pi\)
\(84\) 3.46681 0.378260
\(85\) −7.65279 −0.830062
\(86\) 13.7899 1.48700
\(87\) −1.25906 −0.134986
\(88\) 6.02177 0.641923
\(89\) −6.61356 −0.701036 −0.350518 0.936556i \(-0.613995\pi\)
−0.350518 + 0.936556i \(0.613995\pi\)
\(90\) −8.74094 −0.921376
\(91\) 0 0
\(92\) −5.76271 −0.600804
\(93\) −2.36898 −0.245652
\(94\) 6.63102 0.683938
\(95\) 8.45473 0.867437
\(96\) 3.61596 0.369052
\(97\) −11.9269 −1.21100 −0.605498 0.795847i \(-0.707026\pi\)
−0.605498 + 0.795847i \(0.707026\pi\)
\(98\) −6.29590 −0.635982
\(99\) −6.87800 −0.691265
\(100\) −8.87800 −0.887800
\(101\) 13.0640 1.29991 0.649957 0.759971i \(-0.274787\pi\)
0.649957 + 0.759971i \(0.274787\pi\)
\(102\) 6.60388 0.653881
\(103\) 9.16852 0.903401 0.451701 0.892170i \(-0.350818\pi\)
0.451701 + 0.892170i \(0.350818\pi\)
\(104\) 0 0
\(105\) 1.64310 0.160351
\(106\) 12.4058 1.20496
\(107\) −6.89977 −0.667026 −0.333513 0.942745i \(-0.608234\pi\)
−0.333513 + 0.942745i \(0.608234\pi\)
\(108\) 9.63102 0.926746
\(109\) −0.121998 −0.0116853 −0.00584264 0.999983i \(-0.501860\pi\)
−0.00584264 + 0.999983i \(0.501860\pi\)
\(110\) 8.29590 0.790983
\(111\) 2.97285 0.282171
\(112\) 1.64310 0.155259
\(113\) 7.30798 0.687477 0.343738 0.939065i \(-0.388307\pi\)
0.343738 + 0.939065i \(0.388307\pi\)
\(114\) −7.29590 −0.683323
\(115\) −2.73125 −0.254690
\(116\) 6.91723 0.642249
\(117\) 0 0
\(118\) 27.4306 2.52519
\(119\) 10.8509 0.994696
\(120\) −1.89008 −0.172540
\(121\) −4.47219 −0.406563
\(122\) 19.2446 1.74232
\(123\) 0.707087 0.0637559
\(124\) 13.0151 1.16879
\(125\) −11.4330 −1.02260
\(126\) 12.3937 1.10412
\(127\) −18.9705 −1.68336 −0.841678 0.539980i \(-0.818432\pi\)
−0.841678 + 0.539980i \(0.818432\pi\)
\(128\) −16.2620 −1.43738
\(129\) −3.40581 −0.299865
\(130\) 0 0
\(131\) 3.25667 0.284536 0.142268 0.989828i \(-0.454560\pi\)
0.142268 + 0.989828i \(0.454560\pi\)
\(132\) −4.32304 −0.376273
\(133\) −11.9879 −1.03948
\(134\) −1.29590 −0.111948
\(135\) 4.56465 0.392862
\(136\) −12.4819 −1.07031
\(137\) −0.792249 −0.0676864 −0.0338432 0.999427i \(-0.510775\pi\)
−0.0338432 + 0.999427i \(0.510775\pi\)
\(138\) 2.35690 0.200632
\(139\) −11.3394 −0.961799 −0.480899 0.876776i \(-0.659690\pi\)
−0.480899 + 0.876776i \(0.659690\pi\)
\(140\) −9.02715 −0.762933
\(141\) −1.63773 −0.137922
\(142\) 10.3230 0.866291
\(143\) 0 0
\(144\) 2.15883 0.179903
\(145\) 3.27844 0.272260
\(146\) 23.7114 1.96237
\(147\) 1.55496 0.128251
\(148\) −16.3327 −1.34254
\(149\) −8.40581 −0.688631 −0.344316 0.938854i \(-0.611889\pi\)
−0.344316 + 0.938854i \(0.611889\pi\)
\(150\) 3.63102 0.296472
\(151\) −14.1293 −1.14983 −0.574913 0.818215i \(-0.694964\pi\)
−0.574913 + 0.818215i \(0.694964\pi\)
\(152\) 13.7899 1.11851
\(153\) 14.2567 1.15258
\(154\) −11.7627 −0.947866
\(155\) 6.16852 0.495468
\(156\) 0 0
\(157\) −9.43296 −0.752832 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(158\) −35.4523 −2.82044
\(159\) −3.06398 −0.242990
\(160\) −9.41550 −0.744361
\(161\) 3.87263 0.305206
\(162\) 14.2078 1.11627
\(163\) −8.70410 −0.681758 −0.340879 0.940107i \(-0.610725\pi\)
−0.340879 + 0.940107i \(0.610725\pi\)
\(164\) −3.88471 −0.303345
\(165\) −2.04892 −0.159508
\(166\) −17.3545 −1.34697
\(167\) 23.8538 1.84587 0.922933 0.384961i \(-0.125785\pi\)
0.922933 + 0.384961i \(0.125785\pi\)
\(168\) 2.67994 0.206762
\(169\) 0 0
\(170\) −17.1957 −1.31885
\(171\) −15.7506 −1.20448
\(172\) 18.7114 1.42673
\(173\) −18.8552 −1.43353 −0.716766 0.697314i \(-0.754378\pi\)
−0.716766 + 0.697314i \(0.754378\pi\)
\(174\) −2.82908 −0.214472
\(175\) 5.96615 0.450998
\(176\) −2.04892 −0.154443
\(177\) −6.77479 −0.509224
\(178\) −14.8605 −1.11384
\(179\) 6.02177 0.450088 0.225044 0.974349i \(-0.427747\pi\)
0.225044 + 0.974349i \(0.427747\pi\)
\(180\) −11.8605 −0.884033
\(181\) −4.77777 −0.355129 −0.177565 0.984109i \(-0.556822\pi\)
−0.177565 + 0.984109i \(0.556822\pi\)
\(182\) 0 0
\(183\) −4.75302 −0.351353
\(184\) −4.45473 −0.328407
\(185\) −7.74094 −0.569125
\(186\) −5.32304 −0.390305
\(187\) −13.5308 −0.989470
\(188\) 8.99761 0.656218
\(189\) −6.47219 −0.470782
\(190\) 18.9976 1.37823
\(191\) 18.4306 1.33359 0.666795 0.745242i \(-0.267666\pi\)
0.666795 + 0.745242i \(0.267666\pi\)
\(192\) 7.23490 0.522134
\(193\) −6.05429 −0.435798 −0.217899 0.975971i \(-0.569920\pi\)
−0.217899 + 0.975971i \(0.569920\pi\)
\(194\) −26.7995 −1.92410
\(195\) 0 0
\(196\) −8.54288 −0.610205
\(197\) 11.4155 0.813321 0.406660 0.913579i \(-0.366693\pi\)
0.406660 + 0.913579i \(0.366693\pi\)
\(198\) −15.4547 −1.09832
\(199\) −13.9051 −0.985710 −0.492855 0.870111i \(-0.664047\pi\)
−0.492855 + 0.870111i \(0.664047\pi\)
\(200\) −6.86294 −0.485283
\(201\) 0.320060 0.0225753
\(202\) 29.3545 2.06538
\(203\) −4.64848 −0.326259
\(204\) 8.96077 0.627379
\(205\) −1.84117 −0.128593
\(206\) 20.6015 1.43537
\(207\) 5.08815 0.353651
\(208\) 0 0
\(209\) 14.9487 1.03402
\(210\) 3.69202 0.254774
\(211\) −13.2446 −0.911795 −0.455897 0.890032i \(-0.650682\pi\)
−0.455897 + 0.890032i \(0.650682\pi\)
\(212\) 16.8334 1.15612
\(213\) −2.54958 −0.174694
\(214\) −15.5036 −1.05981
\(215\) 8.86831 0.604814
\(216\) 7.44504 0.506571
\(217\) −8.74632 −0.593739
\(218\) −0.274127 −0.0185662
\(219\) −5.85623 −0.395727
\(220\) 11.2567 0.758924
\(221\) 0 0
\(222\) 6.67994 0.448328
\(223\) 7.33513 0.491196 0.245598 0.969372i \(-0.421016\pi\)
0.245598 + 0.969372i \(0.421016\pi\)
\(224\) 13.3502 0.891997
\(225\) 7.83877 0.522585
\(226\) 16.4209 1.09230
\(227\) 8.67456 0.575751 0.287875 0.957668i \(-0.407051\pi\)
0.287875 + 0.957668i \(0.407051\pi\)
\(228\) −9.89977 −0.655628
\(229\) 13.6866 0.904439 0.452219 0.891907i \(-0.350632\pi\)
0.452219 + 0.891907i \(0.350632\pi\)
\(230\) −6.13706 −0.404666
\(231\) 2.90515 0.191145
\(232\) 5.34721 0.351061
\(233\) −5.08815 −0.333336 −0.166668 0.986013i \(-0.553301\pi\)
−0.166668 + 0.986013i \(0.553301\pi\)
\(234\) 0 0
\(235\) 4.26444 0.278181
\(236\) 37.2204 2.42284
\(237\) 8.75600 0.568764
\(238\) 24.3817 1.58043
\(239\) 10.9239 0.706611 0.353305 0.935508i \(-0.385058\pi\)
0.353305 + 0.935508i \(0.385058\pi\)
\(240\) 0.643104 0.0415122
\(241\) −11.9148 −0.767502 −0.383751 0.923437i \(-0.625368\pi\)
−0.383751 + 0.923437i \(0.625368\pi\)
\(242\) −10.0489 −0.645969
\(243\) −12.9855 −0.833022
\(244\) 26.1129 1.67171
\(245\) −4.04892 −0.258676
\(246\) 1.58881 0.101299
\(247\) 0 0
\(248\) 10.0610 0.638874
\(249\) 4.28621 0.271627
\(250\) −25.6896 −1.62475
\(251\) 22.3478 1.41058 0.705290 0.708919i \(-0.250817\pi\)
0.705290 + 0.708919i \(0.250817\pi\)
\(252\) 16.8170 1.05937
\(253\) −4.82908 −0.303602
\(254\) −42.6262 −2.67461
\(255\) 4.24698 0.265956
\(256\) −10.4668 −0.654176
\(257\) −18.6601 −1.16398 −0.581992 0.813194i \(-0.697727\pi\)
−0.581992 + 0.813194i \(0.697727\pi\)
\(258\) −7.65279 −0.476442
\(259\) 10.9758 0.682005
\(260\) 0 0
\(261\) −6.10752 −0.378046
\(262\) 7.31767 0.452087
\(263\) 14.3991 0.887887 0.443944 0.896055i \(-0.353579\pi\)
0.443944 + 0.896055i \(0.353579\pi\)
\(264\) −3.34183 −0.205675
\(265\) 7.97823 0.490099
\(266\) −26.9366 −1.65159
\(267\) 3.67025 0.224616
\(268\) −1.75840 −0.107411
\(269\) 0.652793 0.0398015 0.0199007 0.999802i \(-0.493665\pi\)
0.0199007 + 0.999802i \(0.493665\pi\)
\(270\) 10.2567 0.624201
\(271\) 1.99569 0.121229 0.0606147 0.998161i \(-0.480694\pi\)
0.0606147 + 0.998161i \(0.480694\pi\)
\(272\) 4.24698 0.257511
\(273\) 0 0
\(274\) −1.78017 −0.107544
\(275\) −7.43967 −0.448629
\(276\) 3.19806 0.192501
\(277\) 11.7845 0.708061 0.354030 0.935234i \(-0.384811\pi\)
0.354030 + 0.935234i \(0.384811\pi\)
\(278\) −25.4795 −1.52816
\(279\) −11.4916 −0.687982
\(280\) −6.97823 −0.417029
\(281\) 6.47219 0.386098 0.193049 0.981189i \(-0.438162\pi\)
0.193049 + 0.981189i \(0.438162\pi\)
\(282\) −3.67994 −0.219137
\(283\) 6.58104 0.391202 0.195601 0.980684i \(-0.437334\pi\)
0.195601 + 0.980684i \(0.437334\pi\)
\(284\) 14.0073 0.831180
\(285\) −4.69202 −0.277931
\(286\) 0 0
\(287\) 2.61058 0.154098
\(288\) 17.5405 1.03358
\(289\) 11.0465 0.649796
\(290\) 7.36658 0.432581
\(291\) 6.61894 0.388009
\(292\) 32.1739 1.88284
\(293\) 24.3381 1.42185 0.710924 0.703269i \(-0.248277\pi\)
0.710924 + 0.703269i \(0.248277\pi\)
\(294\) 3.49396 0.203772
\(295\) 17.6407 1.02708
\(296\) −12.6256 −0.733851
\(297\) 8.07069 0.468309
\(298\) −18.8877 −1.09413
\(299\) 0 0
\(300\) 4.92692 0.284456
\(301\) −12.5743 −0.724773
\(302\) −31.7482 −1.82691
\(303\) −7.24996 −0.416500
\(304\) −4.69202 −0.269106
\(305\) 12.3763 0.708663
\(306\) 32.0344 1.83129
\(307\) 14.0737 0.803227 0.401613 0.915809i \(-0.368450\pi\)
0.401613 + 0.915809i \(0.368450\pi\)
\(308\) −15.9608 −0.909449
\(309\) −5.08815 −0.289455
\(310\) 13.8605 0.787226
\(311\) −29.7700 −1.68810 −0.844051 0.536263i \(-0.819836\pi\)
−0.844051 + 0.536263i \(0.819836\pi\)
\(312\) 0 0
\(313\) −7.47889 −0.422732 −0.211366 0.977407i \(-0.567791\pi\)
−0.211366 + 0.977407i \(0.567791\pi\)
\(314\) −21.1957 −1.19614
\(315\) 7.97046 0.449085
\(316\) −48.1051 −2.70613
\(317\) −30.0301 −1.68666 −0.843330 0.537396i \(-0.819408\pi\)
−0.843330 + 0.537396i \(0.819408\pi\)
\(318\) −6.88471 −0.386075
\(319\) 5.79656 0.324545
\(320\) −18.8388 −1.05312
\(321\) 3.82908 0.213719
\(322\) 8.70171 0.484927
\(323\) −30.9855 −1.72408
\(324\) 19.2784 1.07102
\(325\) 0 0
\(326\) −19.5579 −1.08321
\(327\) 0.0677037 0.00374402
\(328\) −3.00298 −0.165812
\(329\) −6.04652 −0.333356
\(330\) −4.60388 −0.253435
\(331\) 15.7168 0.863872 0.431936 0.901904i \(-0.357831\pi\)
0.431936 + 0.901904i \(0.357831\pi\)
\(332\) −23.5483 −1.29238
\(333\) 14.4209 0.790259
\(334\) 53.5991 2.93281
\(335\) −0.833397 −0.0455333
\(336\) −0.911854 −0.0497457
\(337\) 1.95407 0.106445 0.0532224 0.998583i \(-0.483051\pi\)
0.0532224 + 0.998583i \(0.483051\pi\)
\(338\) 0 0
\(339\) −4.05562 −0.220271
\(340\) −23.3327 −1.26540
\(341\) 10.9065 0.590619
\(342\) −35.3913 −1.91374
\(343\) 20.0834 1.08440
\(344\) 14.4644 0.779869
\(345\) 1.51573 0.0816041
\(346\) −42.3672 −2.27767
\(347\) −17.1250 −0.919317 −0.459659 0.888096i \(-0.652028\pi\)
−0.459659 + 0.888096i \(0.652028\pi\)
\(348\) −3.83877 −0.205780
\(349\) 10.4668 0.560276 0.280138 0.959960i \(-0.409620\pi\)
0.280138 + 0.959960i \(0.409620\pi\)
\(350\) 13.4058 0.716571
\(351\) 0 0
\(352\) −16.6474 −0.887310
\(353\) −15.5308 −0.826621 −0.413310 0.910590i \(-0.635628\pi\)
−0.413310 + 0.910590i \(0.635628\pi\)
\(354\) −15.2228 −0.809084
\(355\) 6.63879 0.352351
\(356\) −20.1642 −1.06870
\(357\) −6.02177 −0.318706
\(358\) 13.5308 0.715125
\(359\) 21.4263 1.13083 0.565417 0.824805i \(-0.308715\pi\)
0.565417 + 0.824805i \(0.308715\pi\)
\(360\) −9.16852 −0.483224
\(361\) 15.2325 0.801711
\(362\) −10.7356 −0.564249
\(363\) 2.48188 0.130265
\(364\) 0 0
\(365\) 15.2489 0.798164
\(366\) −10.6799 −0.558249
\(367\) 34.3032 1.79061 0.895306 0.445452i \(-0.146957\pi\)
0.895306 + 0.445452i \(0.146957\pi\)
\(368\) 1.51573 0.0790129
\(369\) 3.42998 0.178557
\(370\) −17.3937 −0.904257
\(371\) −11.3123 −0.587305
\(372\) −7.22282 −0.374486
\(373\) −12.5961 −0.652202 −0.326101 0.945335i \(-0.605735\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(374\) −30.4034 −1.57212
\(375\) 6.34481 0.327645
\(376\) 6.95539 0.358697
\(377\) 0 0
\(378\) −14.5429 −0.748005
\(379\) −16.5386 −0.849529 −0.424765 0.905304i \(-0.639643\pi\)
−0.424765 + 0.905304i \(0.639643\pi\)
\(380\) 25.7778 1.32237
\(381\) 10.5278 0.539356
\(382\) 41.4131 2.11888
\(383\) −7.53617 −0.385080 −0.192540 0.981289i \(-0.561673\pi\)
−0.192540 + 0.981289i \(0.561673\pi\)
\(384\) 9.02475 0.460543
\(385\) −7.56465 −0.385530
\(386\) −13.6039 −0.692419
\(387\) −16.5211 −0.839815
\(388\) −36.3642 −1.84611
\(389\) 35.5555 1.80274 0.901369 0.433052i \(-0.142563\pi\)
0.901369 + 0.433052i \(0.142563\pi\)
\(390\) 0 0
\(391\) 10.0097 0.506212
\(392\) −6.60388 −0.333546
\(393\) −1.80731 −0.0911670
\(394\) 25.6504 1.29225
\(395\) −22.7995 −1.14717
\(396\) −20.9705 −1.05381
\(397\) −1.35152 −0.0678308 −0.0339154 0.999425i \(-0.510798\pi\)
−0.0339154 + 0.999425i \(0.510798\pi\)
\(398\) −31.2446 −1.56615
\(399\) 6.65279 0.333056
\(400\) 2.33513 0.116756
\(401\) −0.579121 −0.0289199 −0.0144600 0.999895i \(-0.504603\pi\)
−0.0144600 + 0.999895i \(0.504603\pi\)
\(402\) 0.719169 0.0358689
\(403\) 0 0
\(404\) 39.8310 1.98167
\(405\) 9.13706 0.454024
\(406\) −10.4450 −0.518379
\(407\) −13.6866 −0.678422
\(408\) 6.92692 0.342934
\(409\) 15.1575 0.749490 0.374745 0.927128i \(-0.377730\pi\)
0.374745 + 0.927128i \(0.377730\pi\)
\(410\) −4.13706 −0.204315
\(411\) 0.439665 0.0216871
\(412\) 27.9541 1.37720
\(413\) −25.0127 −1.23079
\(414\) 11.4330 0.561899
\(415\) −11.1608 −0.547860
\(416\) 0 0
\(417\) 6.29291 0.308165
\(418\) 33.5894 1.64291
\(419\) −35.7235 −1.74521 −0.872603 0.488430i \(-0.837570\pi\)
−0.872603 + 0.488430i \(0.837570\pi\)
\(420\) 5.00969 0.244448
\(421\) 35.0465 1.70806 0.854032 0.520221i \(-0.174151\pi\)
0.854032 + 0.520221i \(0.174151\pi\)
\(422\) −29.7603 −1.44871
\(423\) −7.94438 −0.386269
\(424\) 13.0127 0.631951
\(425\) 15.4209 0.748022
\(426\) −5.72886 −0.277564
\(427\) −17.5483 −0.849220
\(428\) −21.0368 −1.01685
\(429\) 0 0
\(430\) 19.9269 0.960961
\(431\) 34.2814 1.65128 0.825639 0.564199i \(-0.190815\pi\)
0.825639 + 0.564199i \(0.190815\pi\)
\(432\) −2.53319 −0.121878
\(433\) 13.7385 0.660232 0.330116 0.943940i \(-0.392912\pi\)
0.330116 + 0.943940i \(0.392912\pi\)
\(434\) −19.6528 −0.943364
\(435\) −1.81940 −0.0872334
\(436\) −0.371961 −0.0178137
\(437\) −11.0586 −0.529005
\(438\) −13.1588 −0.628753
\(439\) 10.2403 0.488742 0.244371 0.969682i \(-0.421419\pi\)
0.244371 + 0.969682i \(0.421419\pi\)
\(440\) 8.70171 0.414838
\(441\) 7.54288 0.359185
\(442\) 0 0
\(443\) 12.1763 0.578513 0.289257 0.957252i \(-0.406592\pi\)
0.289257 + 0.957252i \(0.406592\pi\)
\(444\) 9.06398 0.430158
\(445\) −9.55688 −0.453039
\(446\) 16.4819 0.780440
\(447\) 4.66487 0.220641
\(448\) 26.7114 1.26199
\(449\) 12.9051 0.609032 0.304516 0.952507i \(-0.401505\pi\)
0.304516 + 0.952507i \(0.401505\pi\)
\(450\) 17.6136 0.830311
\(451\) −3.25534 −0.153288
\(452\) 22.2814 1.04803
\(453\) 7.84117 0.368410
\(454\) 19.4916 0.914785
\(455\) 0 0
\(456\) −7.65279 −0.358375
\(457\) −4.65710 −0.217850 −0.108925 0.994050i \(-0.534741\pi\)
−0.108925 + 0.994050i \(0.534741\pi\)
\(458\) 30.7536 1.43702
\(459\) −16.7289 −0.780836
\(460\) −8.32736 −0.388265
\(461\) 31.5405 1.46899 0.734493 0.678616i \(-0.237420\pi\)
0.734493 + 0.678616i \(0.237420\pi\)
\(462\) 6.52781 0.303701
\(463\) −17.6504 −0.820284 −0.410142 0.912022i \(-0.634521\pi\)
−0.410142 + 0.912022i \(0.634521\pi\)
\(464\) −1.81940 −0.0844633
\(465\) −3.42327 −0.158750
\(466\) −11.4330 −0.529622
\(467\) −32.1726 −1.48877 −0.744385 0.667751i \(-0.767257\pi\)
−0.744385 + 0.667751i \(0.767257\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) 9.58211 0.441990
\(471\) 5.23490 0.241211
\(472\) 28.7724 1.32436
\(473\) 15.6799 0.720964
\(474\) 19.6746 0.903683
\(475\) −17.0368 −0.781704
\(476\) 33.0834 1.51637
\(477\) −14.8629 −0.680527
\(478\) 24.5459 1.12270
\(479\) 34.8998 1.59461 0.797306 0.603576i \(-0.206258\pi\)
0.797306 + 0.603576i \(0.206258\pi\)
\(480\) 5.22521 0.238497
\(481\) 0 0
\(482\) −26.7724 −1.21945
\(483\) −2.14914 −0.0977895
\(484\) −13.6353 −0.619788
\(485\) −17.2349 −0.782596
\(486\) −29.1782 −1.32355
\(487\) −41.8351 −1.89573 −0.947864 0.318676i \(-0.896762\pi\)
−0.947864 + 0.318676i \(0.896762\pi\)
\(488\) 20.1860 0.913776
\(489\) 4.83041 0.218439
\(490\) −9.09783 −0.410998
\(491\) 21.8455 0.985873 0.492936 0.870065i \(-0.335924\pi\)
0.492936 + 0.870065i \(0.335924\pi\)
\(492\) 2.15585 0.0971932
\(493\) −12.0151 −0.541131
\(494\) 0 0
\(495\) −9.93900 −0.446725
\(496\) −3.42327 −0.153709
\(497\) −9.41311 −0.422236
\(498\) 9.63102 0.431576
\(499\) −23.5472 −1.05412 −0.527058 0.849829i \(-0.676705\pi\)
−0.527058 + 0.849829i \(0.676705\pi\)
\(500\) −34.8582 −1.55890
\(501\) −13.2379 −0.591425
\(502\) 50.2150 2.24121
\(503\) −7.08682 −0.315986 −0.157993 0.987440i \(-0.550502\pi\)
−0.157993 + 0.987440i \(0.550502\pi\)
\(504\) 13.0000 0.579066
\(505\) 18.8780 0.840060
\(506\) −10.8509 −0.482379
\(507\) 0 0
\(508\) −57.8394 −2.56621
\(509\) −7.61894 −0.337704 −0.168852 0.985641i \(-0.554006\pi\)
−0.168852 + 0.985641i \(0.554006\pi\)
\(510\) 9.54288 0.422566
\(511\) −21.6213 −0.956471
\(512\) 9.00538 0.397985
\(513\) 18.4819 0.815995
\(514\) −41.9288 −1.84940
\(515\) 13.2489 0.583816
\(516\) −10.3840 −0.457132
\(517\) 7.53989 0.331604
\(518\) 24.6625 1.08361
\(519\) 10.4638 0.459311
\(520\) 0 0
\(521\) −39.5133 −1.73111 −0.865555 0.500813i \(-0.833034\pi\)
−0.865555 + 0.500813i \(0.833034\pi\)
\(522\) −13.7235 −0.600660
\(523\) −15.8194 −0.691734 −0.345867 0.938284i \(-0.612415\pi\)
−0.345867 + 0.938284i \(0.612415\pi\)
\(524\) 9.92931 0.433764
\(525\) −3.31096 −0.144502
\(526\) 32.3545 1.41072
\(527\) −22.6069 −0.984770
\(528\) 1.13706 0.0494843
\(529\) −19.4276 −0.844678
\(530\) 17.9269 0.778696
\(531\) −32.8635 −1.42616
\(532\) −36.5502 −1.58465
\(533\) 0 0
\(534\) 8.24698 0.356882
\(535\) −9.97046 −0.431061
\(536\) −1.35929 −0.0587123
\(537\) −3.34183 −0.144211
\(538\) 1.46681 0.0632388
\(539\) −7.15883 −0.308353
\(540\) 13.9172 0.598902
\(541\) −34.4819 −1.48249 −0.741246 0.671234i \(-0.765765\pi\)
−0.741246 + 0.671234i \(0.765765\pi\)
\(542\) 4.48427 0.192616
\(543\) 2.65146 0.113785
\(544\) 34.5066 1.47946
\(545\) −0.176292 −0.00755152
\(546\) 0 0
\(547\) 36.8582 1.57594 0.787970 0.615713i \(-0.211132\pi\)
0.787970 + 0.615713i \(0.211132\pi\)
\(548\) −2.41550 −0.103185
\(549\) −23.0562 −0.984015
\(550\) −16.7168 −0.712806
\(551\) 13.2741 0.565497
\(552\) 2.47219 0.105223
\(553\) 32.3274 1.37470
\(554\) 26.4795 1.12501
\(555\) 4.29590 0.182351
\(556\) −34.5730 −1.46622
\(557\) −1.27652 −0.0540879 −0.0270439 0.999634i \(-0.508609\pi\)
−0.0270439 + 0.999634i \(0.508609\pi\)
\(558\) −25.8213 −1.09310
\(559\) 0 0
\(560\) 2.37435 0.100335
\(561\) 7.50902 0.317031
\(562\) 14.5429 0.613454
\(563\) −9.12737 −0.384673 −0.192336 0.981329i \(-0.561607\pi\)
−0.192336 + 0.981329i \(0.561607\pi\)
\(564\) −4.99330 −0.210256
\(565\) 10.5603 0.444277
\(566\) 14.7875 0.621563
\(567\) −12.9554 −0.544075
\(568\) 10.8280 0.454334
\(569\) −5.72156 −0.239860 −0.119930 0.992782i \(-0.538267\pi\)
−0.119930 + 0.992782i \(0.538267\pi\)
\(570\) −10.5429 −0.441593
\(571\) 7.60148 0.318112 0.159056 0.987270i \(-0.449155\pi\)
0.159056 + 0.987270i \(0.449155\pi\)
\(572\) 0 0
\(573\) −10.2282 −0.427289
\(574\) 5.86592 0.244839
\(575\) 5.50365 0.229518
\(576\) 35.0954 1.46231
\(577\) −45.1564 −1.87989 −0.939944 0.341330i \(-0.889123\pi\)
−0.939944 + 0.341330i \(0.889123\pi\)
\(578\) 24.8213 1.03243
\(579\) 3.35988 0.139632
\(580\) 9.99569 0.415048
\(581\) 15.8248 0.656522
\(582\) 14.8726 0.616490
\(583\) 14.1062 0.584219
\(584\) 24.8713 1.02918
\(585\) 0 0
\(586\) 54.6872 2.25911
\(587\) 32.4040 1.33746 0.668728 0.743507i \(-0.266839\pi\)
0.668728 + 0.743507i \(0.266839\pi\)
\(588\) 4.74094 0.195513
\(589\) 24.9758 1.02911
\(590\) 39.6383 1.63188
\(591\) −6.33513 −0.260592
\(592\) 4.29590 0.176560
\(593\) −36.6848 −1.50647 −0.753233 0.657754i \(-0.771507\pi\)
−0.753233 + 0.657754i \(0.771507\pi\)
\(594\) 18.1347 0.744075
\(595\) 15.6799 0.642815
\(596\) −25.6286 −1.04979
\(597\) 7.71678 0.315827
\(598\) 0 0
\(599\) −9.99223 −0.408271 −0.204136 0.978943i \(-0.565438\pi\)
−0.204136 + 0.978943i \(0.565438\pi\)
\(600\) 3.80864 0.155487
\(601\) −1.81163 −0.0738978 −0.0369489 0.999317i \(-0.511764\pi\)
−0.0369489 + 0.999317i \(0.511764\pi\)
\(602\) −28.2543 −1.15156
\(603\) 1.55257 0.0632253
\(604\) −43.0790 −1.75286
\(605\) −6.46250 −0.262738
\(606\) −16.2905 −0.661757
\(607\) 11.2161 0.455248 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(608\) −38.1226 −1.54608
\(609\) 2.57971 0.104535
\(610\) 27.8092 1.12596
\(611\) 0 0
\(612\) 43.4674 1.75707
\(613\) 20.8944 0.843917 0.421958 0.906615i \(-0.361343\pi\)
0.421958 + 0.906615i \(0.361343\pi\)
\(614\) 31.6233 1.27621
\(615\) 1.02177 0.0412018
\(616\) −12.3381 −0.497117
\(617\) −12.0992 −0.487094 −0.243547 0.969889i \(-0.578311\pi\)
−0.243547 + 0.969889i \(0.578311\pi\)
\(618\) −11.4330 −0.459901
\(619\) 10.5526 0.424143 0.212072 0.977254i \(-0.431979\pi\)
0.212072 + 0.977254i \(0.431979\pi\)
\(620\) 18.8073 0.755320
\(621\) −5.97046 −0.239586
\(622\) −66.8926 −2.68215
\(623\) 13.5506 0.542895
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) −16.8049 −0.671660
\(627\) −8.29590 −0.331306
\(628\) −28.7603 −1.14766
\(629\) 28.3696 1.13117
\(630\) 17.9095 0.713530
\(631\) 13.8514 0.551417 0.275709 0.961241i \(-0.411087\pi\)
0.275709 + 0.961241i \(0.411087\pi\)
\(632\) −37.1866 −1.47920
\(633\) 7.35019 0.292144
\(634\) −67.4771 −2.67986
\(635\) −27.4131 −1.08786
\(636\) −9.34183 −0.370428
\(637\) 0 0
\(638\) 13.0248 0.515655
\(639\) −12.3676 −0.489257
\(640\) −23.4993 −0.928893
\(641\) 34.9608 1.38087 0.690434 0.723396i \(-0.257420\pi\)
0.690434 + 0.723396i \(0.257420\pi\)
\(642\) 8.60388 0.339568
\(643\) −33.3980 −1.31709 −0.658545 0.752541i \(-0.728828\pi\)
−0.658545 + 0.752541i \(0.728828\pi\)
\(644\) 11.8073 0.465273
\(645\) −4.92154 −0.193786
\(646\) −69.6238 −2.73931
\(647\) 2.32842 0.0915397 0.0457698 0.998952i \(-0.485426\pi\)
0.0457698 + 0.998952i \(0.485426\pi\)
\(648\) 14.9028 0.585436
\(649\) 31.1903 1.22433
\(650\) 0 0
\(651\) 4.85384 0.190237
\(652\) −26.5381 −1.03931
\(653\) 14.5714 0.570221 0.285111 0.958495i \(-0.407970\pi\)
0.285111 + 0.958495i \(0.407970\pi\)
\(654\) 0.152129 0.00594871
\(655\) 4.70602 0.183879
\(656\) 1.02177 0.0398934
\(657\) −28.4077 −1.10829
\(658\) −13.5864 −0.529654
\(659\) 11.1395 0.433932 0.216966 0.976179i \(-0.430384\pi\)
0.216966 + 0.976179i \(0.430384\pi\)
\(660\) −6.24698 −0.243163
\(661\) 13.8498 0.538694 0.269347 0.963043i \(-0.413192\pi\)
0.269347 + 0.963043i \(0.413192\pi\)
\(662\) 35.3153 1.37257
\(663\) 0 0
\(664\) −18.2034 −0.706430
\(665\) −17.3230 −0.671759
\(666\) 32.4034 1.25561
\(667\) −4.28813 −0.166037
\(668\) 72.7284 2.81395
\(669\) −4.07069 −0.157382
\(670\) −1.87263 −0.0723458
\(671\) 21.8823 0.844757
\(672\) −7.40880 −0.285801
\(673\) −6.52973 −0.251703 −0.125851 0.992049i \(-0.540166\pi\)
−0.125851 + 0.992049i \(0.540166\pi\)
\(674\) 4.39075 0.169125
\(675\) −9.19806 −0.354034
\(676\) 0 0
\(677\) −11.3104 −0.434693 −0.217346 0.976095i \(-0.569740\pi\)
−0.217346 + 0.976095i \(0.569740\pi\)
\(678\) −9.11290 −0.349979
\(679\) 24.4373 0.937816
\(680\) −18.0368 −0.691681
\(681\) −4.81402 −0.184474
\(682\) 24.5066 0.938407
\(683\) −14.1793 −0.542555 −0.271277 0.962501i \(-0.587446\pi\)
−0.271277 + 0.962501i \(0.587446\pi\)
\(684\) −48.0224 −1.83618
\(685\) −1.14483 −0.0437418
\(686\) 45.1269 1.72295
\(687\) −7.59551 −0.289787
\(688\) −4.92154 −0.187632
\(689\) 0 0
\(690\) 3.40581 0.129657
\(691\) −30.7952 −1.17151 −0.585753 0.810490i \(-0.699201\pi\)
−0.585753 + 0.810490i \(0.699201\pi\)
\(692\) −57.4878 −2.18536
\(693\) 14.0925 0.535328
\(694\) −38.4795 −1.46066
\(695\) −16.3860 −0.621555
\(696\) −2.96748 −0.112482
\(697\) 6.74764 0.255585
\(698\) 23.5187 0.890196
\(699\) 2.82371 0.106802
\(700\) 18.1903 0.687528
\(701\) 6.73184 0.254258 0.127129 0.991886i \(-0.459424\pi\)
0.127129 + 0.991886i \(0.459424\pi\)
\(702\) 0 0
\(703\) −31.3424 −1.18210
\(704\) −33.3086 −1.25536
\(705\) −2.36658 −0.0891307
\(706\) −34.8974 −1.31338
\(707\) −26.7670 −1.00668
\(708\) −20.6558 −0.776292
\(709\) 47.6252 1.78860 0.894300 0.447467i \(-0.147674\pi\)
0.894300 + 0.447467i \(0.147674\pi\)
\(710\) 14.9172 0.559834
\(711\) 42.4741 1.59290
\(712\) −15.5875 −0.584166
\(713\) −8.06829 −0.302160
\(714\) −13.5308 −0.506377
\(715\) 0 0
\(716\) 18.3599 0.686141
\(717\) −6.06233 −0.226402
\(718\) 48.1444 1.79673
\(719\) −5.99330 −0.223512 −0.111756 0.993736i \(-0.535648\pi\)
−0.111756 + 0.993736i \(0.535648\pi\)
\(720\) 3.11960 0.116261
\(721\) −18.7855 −0.699610
\(722\) 34.2271 1.27380
\(723\) 6.61224 0.245912
\(724\) −14.5670 −0.541380
\(725\) −6.60627 −0.245351
\(726\) 5.57673 0.206972
\(727\) −24.1226 −0.894657 −0.447329 0.894370i \(-0.647625\pi\)
−0.447329 + 0.894370i \(0.647625\pi\)
\(728\) 0 0
\(729\) −11.7627 −0.435656
\(730\) 34.2640 1.26817
\(731\) −32.5013 −1.20210
\(732\) −14.4916 −0.535624
\(733\) −36.0646 −1.33208 −0.666038 0.745918i \(-0.732011\pi\)
−0.666038 + 0.745918i \(0.732011\pi\)
\(734\) 77.0786 2.84502
\(735\) 2.24698 0.0828811
\(736\) 12.3153 0.453947
\(737\) −1.47352 −0.0542777
\(738\) 7.70709 0.283702
\(739\) −27.5254 −1.01254 −0.506269 0.862375i \(-0.668976\pi\)
−0.506269 + 0.862375i \(0.668976\pi\)
\(740\) −23.6015 −0.867608
\(741\) 0 0
\(742\) −25.4185 −0.933142
\(743\) 10.4692 0.384078 0.192039 0.981387i \(-0.438490\pi\)
0.192039 + 0.981387i \(0.438490\pi\)
\(744\) −5.58343 −0.204699
\(745\) −12.1468 −0.445023
\(746\) −28.3032 −1.03625
\(747\) 20.7918 0.760731
\(748\) −41.2543 −1.50841
\(749\) 14.1371 0.516557
\(750\) 14.2567 0.520580
\(751\) 4.06770 0.148433 0.0742163 0.997242i \(-0.476354\pi\)
0.0742163 + 0.997242i \(0.476354\pi\)
\(752\) −2.36658 −0.0863005
\(753\) −12.4021 −0.451957
\(754\) 0 0
\(755\) −20.4174 −0.743066
\(756\) −19.7332 −0.717688
\(757\) 20.4336 0.742670 0.371335 0.928499i \(-0.378900\pi\)
0.371335 + 0.928499i \(0.378900\pi\)
\(758\) −37.1618 −1.34978
\(759\) 2.67994 0.0972757
\(760\) 19.9269 0.722825
\(761\) 27.0237 0.979608 0.489804 0.871833i \(-0.337068\pi\)
0.489804 + 0.871833i \(0.337068\pi\)
\(762\) 23.6558 0.856958
\(763\) 0.249964 0.00904929
\(764\) 56.1933 2.03300
\(765\) 20.6015 0.744848
\(766\) −16.9336 −0.611837
\(767\) 0 0
\(768\) 5.80864 0.209601
\(769\) 37.9407 1.36818 0.684088 0.729400i \(-0.260201\pi\)
0.684088 + 0.729400i \(0.260201\pi\)
\(770\) −16.9976 −0.612551
\(771\) 10.3556 0.372947
\(772\) −18.4590 −0.664355
\(773\) 16.3375 0.587620 0.293810 0.955864i \(-0.405077\pi\)
0.293810 + 0.955864i \(0.405077\pi\)
\(774\) −37.1226 −1.33434
\(775\) −12.4300 −0.446498
\(776\) −28.1105 −1.00911
\(777\) −6.09113 −0.218518
\(778\) 79.8926 2.86429
\(779\) −7.45473 −0.267093
\(780\) 0 0
\(781\) 11.7380 0.420017
\(782\) 22.4916 0.804297
\(783\) 7.16660 0.256114
\(784\) 2.24698 0.0802493
\(785\) −13.6310 −0.486512
\(786\) −4.06100 −0.144851
\(787\) 18.6907 0.666251 0.333126 0.942882i \(-0.391897\pi\)
0.333126 + 0.942882i \(0.391897\pi\)
\(788\) 34.8049 1.23987
\(789\) −7.99090 −0.284484
\(790\) −51.2301 −1.82269
\(791\) −14.9734 −0.532394
\(792\) −16.2107 −0.576023
\(793\) 0 0
\(794\) −3.03684 −0.107773
\(795\) −4.42758 −0.157030
\(796\) −42.3957 −1.50267
\(797\) 29.2519 1.03615 0.518077 0.855334i \(-0.326648\pi\)
0.518077 + 0.855334i \(0.326648\pi\)
\(798\) 14.9487 0.529178
\(799\) −15.6286 −0.552901
\(800\) 18.9729 0.670792
\(801\) 17.8039 0.629068
\(802\) −1.30127 −0.0459496
\(803\) 26.9614 0.951446
\(804\) 0.975837 0.0344151
\(805\) 5.59611 0.197237
\(806\) 0 0
\(807\) −0.362273 −0.0127526
\(808\) 30.7904 1.08320
\(809\) −6.65087 −0.233832 −0.116916 0.993142i \(-0.537301\pi\)
−0.116916 + 0.993142i \(0.537301\pi\)
\(810\) 20.5308 0.721379
\(811\) −3.89200 −0.136667 −0.0683333 0.997663i \(-0.521768\pi\)
−0.0683333 + 0.997663i \(0.521768\pi\)
\(812\) −14.1728 −0.497369
\(813\) −1.10752 −0.0388425
\(814\) −30.7536 −1.07791
\(815\) −12.5778 −0.440581
\(816\) −2.35690 −0.0825079
\(817\) 35.9071 1.25623
\(818\) 34.0586 1.19083
\(819\) 0 0
\(820\) −5.61356 −0.196034
\(821\) −45.9982 −1.60535 −0.802674 0.596418i \(-0.796590\pi\)
−0.802674 + 0.596418i \(0.796590\pi\)
\(822\) 0.987918 0.0344576
\(823\) 7.95300 0.277224 0.138612 0.990347i \(-0.455736\pi\)
0.138612 + 0.990347i \(0.455736\pi\)
\(824\) 21.6093 0.752794
\(825\) 4.12870 0.143743
\(826\) −56.2030 −1.95555
\(827\) −27.9648 −0.972432 −0.486216 0.873839i \(-0.661623\pi\)
−0.486216 + 0.873839i \(0.661623\pi\)
\(828\) 15.5133 0.539126
\(829\) 27.6310 0.959665 0.479833 0.877360i \(-0.340697\pi\)
0.479833 + 0.877360i \(0.340697\pi\)
\(830\) −25.0780 −0.870470
\(831\) −6.53989 −0.226866
\(832\) 0 0
\(833\) 14.8388 0.514133
\(834\) 14.1400 0.489630
\(835\) 34.4698 1.19288
\(836\) 45.5773 1.57632
\(837\) 13.4843 0.466085
\(838\) −80.2699 −2.77288
\(839\) −28.6848 −0.990311 −0.495155 0.868804i \(-0.664889\pi\)
−0.495155 + 0.868804i \(0.664889\pi\)
\(840\) 3.87263 0.133618
\(841\) −23.8528 −0.822509
\(842\) 78.7488 2.71386
\(843\) −3.59179 −0.123708
\(844\) −40.3817 −1.38999
\(845\) 0 0
\(846\) −17.8509 −0.613725
\(847\) 9.16315 0.314849
\(848\) −4.42758 −0.152044
\(849\) −3.65220 −0.125343
\(850\) 34.6504 1.18850
\(851\) 10.1250 0.347080
\(852\) −7.77346 −0.266314
\(853\) 43.2078 1.47941 0.739703 0.672934i \(-0.234966\pi\)
0.739703 + 0.672934i \(0.234966\pi\)
\(854\) −39.4306 −1.34929
\(855\) −22.7603 −0.778386
\(856\) −16.2620 −0.555825
\(857\) 35.1685 1.20133 0.600667 0.799499i \(-0.294902\pi\)
0.600667 + 0.799499i \(0.294902\pi\)
\(858\) 0 0
\(859\) 27.3793 0.934168 0.467084 0.884213i \(-0.345305\pi\)
0.467084 + 0.884213i \(0.345305\pi\)
\(860\) 27.0388 0.922014
\(861\) −1.44876 −0.0493737
\(862\) 77.0297 2.62364
\(863\) 41.3913 1.40898 0.704489 0.709715i \(-0.251176\pi\)
0.704489 + 0.709715i \(0.251176\pi\)
\(864\) −20.5821 −0.700217
\(865\) −27.2465 −0.926409
\(866\) 30.8702 1.04901
\(867\) −6.13036 −0.208198
\(868\) −26.6668 −0.905130
\(869\) −40.3116 −1.36748
\(870\) −4.08815 −0.138601
\(871\) 0 0
\(872\) −0.287536 −0.00973721
\(873\) 32.1075 1.08668
\(874\) −24.8485 −0.840512
\(875\) 23.4252 0.791916
\(876\) −17.8552 −0.603270
\(877\) 24.7472 0.835653 0.417826 0.908527i \(-0.362792\pi\)
0.417826 + 0.908527i \(0.362792\pi\)
\(878\) 23.0097 0.776539
\(879\) −13.5066 −0.455567
\(880\) −2.96077 −0.0998076
\(881\) −28.5875 −0.963137 −0.481568 0.876409i \(-0.659933\pi\)
−0.481568 + 0.876409i \(0.659933\pi\)
\(882\) 16.9487 0.570692
\(883\) 9.61702 0.323639 0.161819 0.986820i \(-0.448264\pi\)
0.161819 + 0.986820i \(0.448264\pi\)
\(884\) 0 0
\(885\) −9.78986 −0.329082
\(886\) 27.3599 0.919173
\(887\) 15.9661 0.536091 0.268045 0.963406i \(-0.413622\pi\)
0.268045 + 0.963406i \(0.413622\pi\)
\(888\) 7.00670 0.235130
\(889\) 38.8689 1.30362
\(890\) −21.4741 −0.719814
\(891\) 16.1551 0.541217
\(892\) 22.3642 0.748809
\(893\) 17.2664 0.577797
\(894\) 10.4819 0.350566
\(895\) 8.70171 0.290866
\(896\) 33.3196 1.11313
\(897\) 0 0
\(898\) 28.9976 0.967663
\(899\) 9.68473 0.323004
\(900\) 23.8998 0.796659
\(901\) −29.2392 −0.974099
\(902\) −7.31468 −0.243552
\(903\) 6.97823 0.232221
\(904\) 17.2241 0.572867
\(905\) −6.90408 −0.229500
\(906\) 17.6189 0.585350
\(907\) −28.8364 −0.957496 −0.478748 0.877952i \(-0.658909\pi\)
−0.478748 + 0.877952i \(0.658909\pi\)
\(908\) 26.4480 0.877709
\(909\) −35.1685 −1.16647
\(910\) 0 0
\(911\) 38.5633 1.27766 0.638830 0.769348i \(-0.279419\pi\)
0.638830 + 0.769348i \(0.279419\pi\)
\(912\) 2.60388 0.0862229
\(913\) −19.7332 −0.653073
\(914\) −10.4644 −0.346132
\(915\) −6.86831 −0.227059
\(916\) 41.7294 1.37878
\(917\) −6.67264 −0.220350
\(918\) −37.5894 −1.24064
\(919\) 8.87502 0.292760 0.146380 0.989228i \(-0.453238\pi\)
0.146380 + 0.989228i \(0.453238\pi\)
\(920\) −6.43727 −0.212231
\(921\) −7.81030 −0.257358
\(922\) 70.8708 2.33401
\(923\) 0 0
\(924\) 8.85756 0.291392
\(925\) 15.5985 0.512875
\(926\) −39.6601 −1.30331
\(927\) −24.6819 −0.810659
\(928\) −14.7826 −0.485261
\(929\) −24.2295 −0.794945 −0.397472 0.917614i \(-0.630113\pi\)
−0.397472 + 0.917614i \(0.630113\pi\)
\(930\) −7.69202 −0.252231
\(931\) −16.3937 −0.537283
\(932\) −15.5133 −0.508156
\(933\) 16.5211 0.540877
\(934\) −72.2911 −2.36544
\(935\) −19.5526 −0.639437
\(936\) 0 0
\(937\) 17.2644 0.564005 0.282002 0.959414i \(-0.409001\pi\)
0.282002 + 0.959414i \(0.409001\pi\)
\(938\) 2.65519 0.0866949
\(939\) 4.15047 0.135446
\(940\) 13.0019 0.424076
\(941\) −4.34050 −0.141496 −0.0707482 0.997494i \(-0.522539\pi\)
−0.0707482 + 0.997494i \(0.522539\pi\)
\(942\) 11.7627 0.383250
\(943\) 2.40821 0.0784220
\(944\) −9.78986 −0.318633
\(945\) −9.35258 −0.304240
\(946\) 35.2325 1.14551
\(947\) 45.0146 1.46278 0.731389 0.681961i \(-0.238872\pi\)
0.731389 + 0.681961i \(0.238872\pi\)
\(948\) 26.6963 0.867057
\(949\) 0 0
\(950\) −38.2814 −1.24201
\(951\) 16.6655 0.540415
\(952\) 25.5743 0.828869
\(953\) −46.8859 −1.51878 −0.759391 0.650634i \(-0.774503\pi\)
−0.759391 + 0.650634i \(0.774503\pi\)
\(954\) −33.3967 −1.08126
\(955\) 26.6329 0.861822
\(956\) 33.3062 1.07720
\(957\) −3.21685 −0.103986
\(958\) 78.4191 2.53361
\(959\) 1.62325 0.0524176
\(960\) 10.4547 0.337425
\(961\) −12.7778 −0.412186
\(962\) 0 0
\(963\) 18.5743 0.598550
\(964\) −36.3274 −1.17003
\(965\) −8.74871 −0.281631
\(966\) −4.82908 −0.155373
\(967\) −6.29457 −0.202420 −0.101210 0.994865i \(-0.532271\pi\)
−0.101210 + 0.994865i \(0.532271\pi\)
\(968\) −10.5405 −0.338784
\(969\) 17.1957 0.552404
\(970\) −38.7265 −1.24343
\(971\) −41.8068 −1.34165 −0.670823 0.741618i \(-0.734059\pi\)
−0.670823 + 0.741618i \(0.734059\pi\)
\(972\) −39.5918 −1.26991
\(973\) 23.2336 0.744834
\(974\) −94.0025 −3.01203
\(975\) 0 0
\(976\) −6.86831 −0.219849
\(977\) −23.7530 −0.759926 −0.379963 0.925002i \(-0.624063\pi\)
−0.379963 + 0.925002i \(0.624063\pi\)
\(978\) 10.8538 0.347067
\(979\) −16.8974 −0.540043
\(980\) −12.3448 −0.394341
\(981\) 0.328421 0.0104857
\(982\) 49.0863 1.56641
\(983\) 55.7251 1.77736 0.888678 0.458532i \(-0.151625\pi\)
0.888678 + 0.458532i \(0.151625\pi\)
\(984\) 1.66653 0.0531270
\(985\) 16.4959 0.525602
\(986\) −26.9976 −0.859779
\(987\) 3.35557 0.106809
\(988\) 0 0
\(989\) −11.5996 −0.368845
\(990\) −22.3327 −0.709781
\(991\) −35.5512 −1.12932 −0.564661 0.825323i \(-0.690993\pi\)
−0.564661 + 0.825323i \(0.690993\pi\)
\(992\) −27.8140 −0.883096
\(993\) −8.72215 −0.276789
\(994\) −21.1511 −0.670871
\(995\) −20.0935 −0.637007
\(996\) 13.0683 0.414085
\(997\) 6.61058 0.209359 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(998\) −52.9101 −1.67484
\(999\) −16.9215 −0.535374
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.a.c.1.3 yes 3
3.2 odd 2 1521.2.a.o.1.1 3
4.3 odd 2 2704.2.a.ba.1.2 3
5.4 even 2 4225.2.a.bb.1.1 3
7.6 odd 2 8281.2.a.bj.1.3 3
13.2 odd 12 169.2.e.b.147.6 12
13.3 even 3 169.2.c.b.22.1 6
13.4 even 6 169.2.c.c.146.3 6
13.5 odd 4 169.2.b.b.168.1 6
13.6 odd 12 169.2.e.b.23.1 12
13.7 odd 12 169.2.e.b.23.6 12
13.8 odd 4 169.2.b.b.168.6 6
13.9 even 3 169.2.c.b.146.1 6
13.10 even 6 169.2.c.c.22.3 6
13.11 odd 12 169.2.e.b.147.1 12
13.12 even 2 169.2.a.b.1.1 3
39.5 even 4 1521.2.b.l.1351.6 6
39.8 even 4 1521.2.b.l.1351.1 6
39.38 odd 2 1521.2.a.r.1.3 3
52.31 even 4 2704.2.f.o.337.3 6
52.47 even 4 2704.2.f.o.337.4 6
52.51 odd 2 2704.2.a.z.1.2 3
65.64 even 2 4225.2.a.bg.1.3 3
91.90 odd 2 8281.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 13.12 even 2
169.2.a.c.1.3 yes 3 1.1 even 1 trivial
169.2.b.b.168.1 6 13.5 odd 4
169.2.b.b.168.6 6 13.8 odd 4
169.2.c.b.22.1 6 13.3 even 3
169.2.c.b.146.1 6 13.9 even 3
169.2.c.c.22.3 6 13.10 even 6
169.2.c.c.146.3 6 13.4 even 6
169.2.e.b.23.1 12 13.6 odd 12
169.2.e.b.23.6 12 13.7 odd 12
169.2.e.b.147.1 12 13.11 odd 12
169.2.e.b.147.6 12 13.2 odd 12
1521.2.a.o.1.1 3 3.2 odd 2
1521.2.a.r.1.3 3 39.38 odd 2
1521.2.b.l.1351.1 6 39.8 even 4
1521.2.b.l.1351.6 6 39.5 even 4
2704.2.a.z.1.2 3 52.51 odd 2
2704.2.a.ba.1.2 3 4.3 odd 2
2704.2.f.o.337.3 6 52.31 even 4
2704.2.f.o.337.4 6 52.47 even 4
4225.2.a.bb.1.1 3 5.4 even 2
4225.2.a.bg.1.3 3 65.64 even 2
8281.2.a.bf.1.1 3 91.90 odd 2
8281.2.a.bj.1.3 3 7.6 odd 2