Properties

Label 169.2.a.b.1.1
Level $169$
Weight $2$
Character 169.1
Self dual yes
Analytic conductor $1.349$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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 1.1
Root \(1.80194\) 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 \(-0.792229\pi\)
−0.794427 + 0.607359i \(0.792229\pi\)
\(3\) −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 \(-0.604732\pi\)
−0.323121 + 0.946358i \(0.604732\pi\)
\(6\) 1.24698 0.509077
\(7\) 2.04892 0.774418 0.387209 0.921992i \(-0.373439\pi\)
0.387209 + 0.921992i \(0.373439\pi\)
\(8\) −2.35690 −0.833289
\(9\) −2.69202 −0.897340
\(10\) 3.24698 1.02679
\(11\) −2.55496 −0.770349 −0.385174 0.922844i \(-0.625859\pi\)
−0.385174 + 0.922844i \(0.625859\pi\)
\(12\) −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 \(-0.734195\pi\)
−0.671139 + 0.741331i \(0.734195\pi\)
\(20\) −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 \(-0.625228\pi\)
−0.383345 + 0.923605i \(0.625228\pi\)
\(32\) 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 \(-0.354860\pi\)
0.440334 + 0.897834i \(0.354860\pi\)
\(38\) 13.1468 2.13268
\(39\) 0 0
\(40\) 3.40581 0.538506
\(41\) 1.27413 0.198985 0.0994926 0.995038i \(-0.468278\pi\)
0.0994926 + 0.995038i \(0.468278\pi\)
\(42\) 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 \(-0.569050\pi\)
−0.215230 + 0.976563i \(0.569050\pi\)
\(48\) 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 \(-0.792349\pi\)
−0.794657 + 0.607059i \(0.792349\pi\)
\(60\) 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 \(-0.488784\pi\)
0.0352293 + 0.999379i \(0.488784\pi\)
\(68\) −16.1468 −1.95808
\(69\) 1.04892 0.126275
\(70\) 6.65279 0.795161
\(71\) −4.59419 −0.545230 −0.272615 0.962123i \(-0.587888\pi\)
−0.272615 + 0.962123i \(0.587888\pi\)
\(72\) 6.34481 0.747744
\(73\) −10.5526 −1.23508 −0.617542 0.786538i \(-0.711872\pi\)
−0.617542 + 0.786538i \(0.711872\pi\)
\(74\) −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 \(-0.360667\pi\)
0.423881 + 0.905718i \(0.360667\pi\)
\(84\) −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 \(-0.386005\pi\)
0.350518 + 0.936556i \(0.386005\pi\)
\(90\) −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 \(-0.292974\pi\)
0.605498 + 0.795847i \(0.292974\pi\)
\(98\) 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 \(-0.498140\pi\)
0.00584264 + 0.999983i \(0.498140\pi\)
\(110\) −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 \(-0.489225\pi\)
0.0338432 + 0.999427i \(0.489225\pi\)
\(138\) −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 \(-0.388111\pi\)
0.344316 + 0.938854i \(0.388111\pi\)
\(150\) −3.63102 −0.296472
\(151\) 14.1293 1.14983 0.574913 0.818215i \(-0.305036\pi\)
0.574913 + 0.818215i \(0.305036\pi\)
\(152\) 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 \(-0.389275\pi\)
0.340879 + 0.940107i \(0.389275\pi\)
\(164\) 3.88471 0.303345
\(165\) −2.04892 −0.159508
\(166\) −17.3545 −1.34697
\(167\) −23.8538 −1.84587 −0.922933 0.384961i \(-0.874215\pi\)
−0.922933 + 0.384961i \(0.874215\pi\)
\(168\) 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 \(-0.430080\pi\)
0.217899 + 0.975971i \(0.430080\pi\)
\(194\) −26.7995 −1.92410
\(195\) 0 0
\(196\) −8.54288 −0.610205
\(197\) −11.4155 −0.813321 −0.406660 0.913579i \(-0.633307\pi\)
−0.406660 + 0.913579i \(0.633307\pi\)
\(198\) −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 \(-0.578984\pi\)
−0.245598 + 0.969372i \(0.578984\pi\)
\(224\) 13.3502 0.891997
\(225\) 7.83877 0.522585
\(226\) −16.4209 −1.09230
\(227\) −8.67456 −0.575751 −0.287875 0.957668i \(-0.592949\pi\)
−0.287875 + 0.957668i \(0.592949\pi\)
\(228\) 9.89977 0.655628
\(229\) −13.6866 −0.904439 −0.452219 0.891907i \(-0.649368\pi\)
−0.452219 + 0.891907i \(0.649368\pi\)
\(230\) −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 \(-0.614942\pi\)
−0.353305 + 0.935508i \(0.614942\pi\)
\(240\) −0.643104 −0.0415122
\(241\) 11.9148 0.767502 0.383751 0.923437i \(-0.374632\pi\)
0.383751 + 0.923437i \(0.374632\pi\)
\(242\) 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 \(-0.519306\pi\)
−0.0606147 + 0.998161i \(0.519306\pi\)
\(272\) 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 \(-0.561838\pi\)
−0.193049 + 0.981189i \(0.561838\pi\)
\(282\) −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 \(-0.751723\pi\)
−0.710924 + 0.703269i \(0.751723\pi\)
\(294\) −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 \(-0.631550\pi\)
−0.401613 + 0.915809i \(0.631550\pi\)
\(308\) −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 \(-0.180592\pi\)
0.843330 + 0.537396i \(0.180592\pi\)
\(318\) 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 \(-0.642169\pi\)
−0.431936 + 0.901904i \(0.642169\pi\)
\(332\) 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 \(-0.590380\pi\)
−0.280138 + 0.959960i \(0.590380\pi\)
\(350\) 13.4058 0.716571
\(351\) 0 0
\(352\) −16.6474 −0.887310
\(353\) 15.5308 0.826621 0.413310 0.910590i \(-0.364372\pi\)
0.413310 + 0.910590i \(0.364372\pi\)
\(354\) −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 \(-0.691285\pi\)
−0.565417 + 0.824805i \(0.691285\pi\)
\(360\) −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 \(-0.360357\pi\)
0.424765 + 0.905304i \(0.360357\pi\)
\(380\) 25.7778 1.32237
\(381\) 10.5278 0.539356
\(382\) −41.4131 −2.11888
\(383\) 7.53617 0.385080 0.192540 0.981289i \(-0.438327\pi\)
0.192540 + 0.981289i \(0.438327\pi\)
\(384\) −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 \(-0.489202\pi\)
0.0339154 + 0.999425i \(0.489202\pi\)
\(398\) 31.2446 1.56615
\(399\) 6.65279 0.333056
\(400\) 2.33513 0.116756
\(401\) 0.579121 0.0289199 0.0144600 0.999895i \(-0.495397\pi\)
0.0144600 + 0.999895i \(0.495397\pi\)
\(402\) 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 \(-0.622270\pi\)
−0.374745 + 0.927128i \(0.622270\pi\)
\(410\) 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 \(-0.825849\pi\)
−0.854032 + 0.520221i \(0.825849\pi\)
\(422\) 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 \(-0.809185\pi\)
−0.825639 + 0.564199i \(0.809185\pi\)
\(432\) −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 \(-0.598495\pi\)
−0.304516 + 0.952507i \(0.598495\pi\)
\(450\) −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 \(-0.465259\pi\)
0.108925 + 0.994050i \(0.465259\pi\)
\(458\) 30.7536 1.43702
\(459\) −16.7289 −0.780836
\(460\) 8.32736 0.388265
\(461\) −31.5405 −1.46899 −0.734493 0.678616i \(-0.762580\pi\)
−0.734493 + 0.678616i \(0.762580\pi\)
\(462\) −6.52781 −0.303701
\(463\) 17.6504 0.820284 0.410142 0.912022i \(-0.365479\pi\)
0.410142 + 0.912022i \(0.365479\pi\)
\(464\) −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 \(-0.793742\pi\)
−0.797306 + 0.603576i \(0.793742\pi\)
\(480\) 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 \(-0.103238\pi\)
0.947864 + 0.318676i \(0.103238\pi\)
\(488\) −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 \(-0.323295\pi\)
0.527058 + 0.849829i \(0.323295\pi\)
\(500\) 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 \(-0.445994\pi\)
0.168852 + 0.985641i \(0.445994\pi\)
\(510\) 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 \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(542\) 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 \(-0.491391\pi\)
0.0270439 + 0.999634i \(0.491391\pi\)
\(558\) −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 \(-0.110877\pi\)
0.939944 + 0.341330i \(0.110877\pi\)
\(578\) −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 \(-0.733161\pi\)
−0.668728 + 0.743507i \(0.733161\pi\)
\(588\) 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 \(-0.228493\pi\)
0.753233 + 0.657754i \(0.228493\pi\)
\(594\) 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 \(-0.638657\pi\)
−0.421958 + 0.906615i \(0.638657\pi\)
\(614\) 31.6233 1.27621
\(615\) 1.02177 0.0412018
\(616\) 12.3381 0.497117
\(617\) 12.0992 0.487094 0.243547 0.969889i \(-0.421689\pi\)
0.243547 + 0.969889i \(0.421689\pi\)
\(618\) 11.4330 0.459901
\(619\) −10.5526 −0.424143 −0.212072 0.977254i \(-0.568021\pi\)
−0.212072 + 0.977254i \(0.568021\pi\)
\(620\) 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 \(-0.588913\pi\)
−0.275709 + 0.961241i \(0.588913\pi\)
\(632\) 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 \(-0.271172\pi\)
0.658545 + 0.752541i \(0.271172\pi\)
\(644\) −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 \(-0.586808\pi\)
−0.269347 + 0.963043i \(0.586808\pi\)
\(662\) 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 \(-0.412554\pi\)
0.271277 + 0.962501i \(0.412554\pi\)
\(684\) 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 \(-0.300799\pi\)
0.585753 + 0.810490i \(0.300799\pi\)
\(692\) −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 \(-0.852326\pi\)
−0.894300 + 0.447467i \(0.852326\pi\)
\(710\) −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 \(-0.267989\pi\)
0.666038 + 0.745918i \(0.267989\pi\)
\(734\) −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 \(-0.331024\pi\)
0.506269 + 0.862375i \(0.331024\pi\)
\(740\) −23.6015 −0.867608
\(741\) 0 0
\(742\) −25.4185 −0.933142
\(743\) −10.4692 −0.384078 −0.192039 0.981387i \(-0.561510\pi\)
−0.192039 + 0.981387i \(0.561510\pi\)
\(744\) −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 \(-0.662932\pi\)
−0.489804 + 0.871833i \(0.662932\pi\)
\(762\) −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 \(-0.739799\pi\)
−0.684088 + 0.729400i \(0.739799\pi\)
\(770\) −16.9976 −0.612551
\(771\) 10.3556 0.372947
\(772\) 18.4590 0.664355
\(773\) −16.3375 −0.587620 −0.293810 0.955864i \(-0.594923\pi\)
−0.293810 + 0.955864i \(0.594923\pi\)
\(774\) 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 \(-0.608103\pi\)
−0.333126 + 0.942882i \(0.608103\pi\)
\(788\) −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 \(-0.478232\pi\)
0.0683333 + 0.997663i \(0.478232\pi\)
\(812\) 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 \(-0.203410\pi\)
0.802674 + 0.596418i \(0.203410\pi\)
\(822\) 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 \(-0.338377\pi\)
0.486216 + 0.873839i \(0.338377\pi\)
\(828\) 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 \(-0.335111\pi\)
0.495155 + 0.868804i \(0.335111\pi\)
\(840\) −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 \(-0.765034\pi\)
−0.739703 + 0.672934i \(0.765034\pi\)
\(854\) −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 \(-0.748824\pi\)
−0.704489 + 0.709715i \(0.748824\pi\)
\(864\) 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 \(-0.637208\pi\)
−0.417826 + 0.908527i \(0.637208\pi\)
\(878\) −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 \(-0.369887\pi\)
0.397472 + 0.917614i \(0.369887\pi\)
\(930\) 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 \(-0.477461\pi\)
0.0707482 + 0.997494i \(0.477461\pi\)
\(942\) −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 \(-0.761128\pi\)
−0.731389 + 0.681961i \(0.761128\pi\)
\(948\) 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 \(-0.467729\pi\)
0.101210 + 0.994865i \(0.467729\pi\)
\(968\) 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 \(-0.375937\pi\)
0.379963 + 0.925002i \(0.375937\pi\)
\(978\) 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 \(-0.848375\pi\)
−0.888678 + 0.458532i \(0.848375\pi\)
\(984\) 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.b.1.1 3
3.2 odd 2 1521.2.a.r.1.3 3
4.3 odd 2 2704.2.a.z.1.2 3
5.4 even 2 4225.2.a.bg.1.3 3
7.6 odd 2 8281.2.a.bf.1.1 3
13.2 odd 12 169.2.e.b.147.1 12
13.3 even 3 169.2.c.c.22.3 6
13.4 even 6 169.2.c.b.146.1 6
13.5 odd 4 169.2.b.b.168.6 6
13.6 odd 12 169.2.e.b.23.6 12
13.7 odd 12 169.2.e.b.23.1 12
13.8 odd 4 169.2.b.b.168.1 6
13.9 even 3 169.2.c.c.146.3 6
13.10 even 6 169.2.c.b.22.1 6
13.11 odd 12 169.2.e.b.147.6 12
13.12 even 2 169.2.a.c.1.3 yes 3
39.5 even 4 1521.2.b.l.1351.1 6
39.8 even 4 1521.2.b.l.1351.6 6
39.38 odd 2 1521.2.a.o.1.1 3
52.31 even 4 2704.2.f.o.337.4 6
52.47 even 4 2704.2.f.o.337.3 6
52.51 odd 2 2704.2.a.ba.1.2 3
65.64 even 2 4225.2.a.bb.1.1 3
91.90 odd 2 8281.2.a.bj.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 1.1 even 1 trivial
169.2.a.c.1.3 yes 3 13.12 even 2
169.2.b.b.168.1 6 13.8 odd 4
169.2.b.b.168.6 6 13.5 odd 4
169.2.c.b.22.1 6 13.10 even 6
169.2.c.b.146.1 6 13.4 even 6
169.2.c.c.22.3 6 13.3 even 3
169.2.c.c.146.3 6 13.9 even 3
169.2.e.b.23.1 12 13.7 odd 12
169.2.e.b.23.6 12 13.6 odd 12
169.2.e.b.147.1 12 13.2 odd 12
169.2.e.b.147.6 12 13.11 odd 12
1521.2.a.o.1.1 3 39.38 odd 2
1521.2.a.r.1.3 3 3.2 odd 2
1521.2.b.l.1351.1 6 39.5 even 4
1521.2.b.l.1351.6 6 39.8 even 4
2704.2.a.z.1.2 3 4.3 odd 2
2704.2.a.ba.1.2 3 52.51 odd 2
2704.2.f.o.337.3 6 52.47 even 4
2704.2.f.o.337.4 6 52.31 even 4
4225.2.a.bb.1.1 3 65.64 even 2
4225.2.a.bg.1.3 3 5.4 even 2
8281.2.a.bf.1.1 3 7.6 odd 2
8281.2.a.bj.1.3 3 91.90 odd 2