Properties

Label 2541.2.a.bq.1.6
Level $2541$
Weight $2$
Character 2541.1
Self dual yes
Analytic conductor $20.290$
Analytic rank $0$
Dimension $10$
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] = [2541,2,Mod(1,2541)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2541, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2541.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2541.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: \(20.2899871536\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.473713\) of defining polynomial
Character \(\chi\) \(=\) 2541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.473713 q^{2} +1.00000 q^{3} -1.77560 q^{4} +3.75881 q^{5} +0.473713 q^{6} -1.00000 q^{7} -1.78855 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.473713 q^{2} +1.00000 q^{3} -1.77560 q^{4} +3.75881 q^{5} +0.473713 q^{6} -1.00000 q^{7} -1.78855 q^{8} +1.00000 q^{9} +1.78060 q^{10} -1.77560 q^{12} +2.70774 q^{13} -0.473713 q^{14} +3.75881 q^{15} +2.70393 q^{16} +2.48283 q^{17} +0.473713 q^{18} -1.74466 q^{19} -6.67413 q^{20} -1.00000 q^{21} +3.79843 q^{23} -1.78855 q^{24} +9.12868 q^{25} +1.28269 q^{26} +1.00000 q^{27} +1.77560 q^{28} -5.80060 q^{29} +1.78060 q^{30} +2.24763 q^{31} +4.85799 q^{32} +1.17615 q^{34} -3.75881 q^{35} -1.77560 q^{36} -7.65311 q^{37} -0.826469 q^{38} +2.70774 q^{39} -6.72282 q^{40} -4.18500 q^{41} -0.473713 q^{42} +7.75162 q^{43} +3.75881 q^{45} +1.79937 q^{46} +12.4071 q^{47} +2.70393 q^{48} +1.00000 q^{49} +4.32437 q^{50} +2.48283 q^{51} -4.80786 q^{52} +8.83956 q^{53} +0.473713 q^{54} +1.78855 q^{56} -1.74466 q^{57} -2.74782 q^{58} +4.54166 q^{59} -6.67413 q^{60} -11.4825 q^{61} +1.06473 q^{62} -1.00000 q^{63} -3.10658 q^{64} +10.1779 q^{65} -12.2041 q^{67} -4.40849 q^{68} +3.79843 q^{69} -1.78060 q^{70} +7.75988 q^{71} -1.78855 q^{72} +15.6796 q^{73} -3.62538 q^{74} +9.12868 q^{75} +3.09781 q^{76} +1.28269 q^{78} -7.05380 q^{79} +10.1636 q^{80} +1.00000 q^{81} -1.98249 q^{82} +11.9004 q^{83} +1.77560 q^{84} +9.33248 q^{85} +3.67204 q^{86} -5.80060 q^{87} -1.69179 q^{89} +1.78060 q^{90} -2.70774 q^{91} -6.74448 q^{92} +2.24763 q^{93} +5.87739 q^{94} -6.55786 q^{95} +4.85799 q^{96} +6.99638 q^{97} +0.473713 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 18 q^{4} + 5 q^{5} - 10 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 18 q^{4} + 5 q^{5} - 10 q^{7} + 3 q^{8} + 10 q^{9} + 6 q^{10} + 18 q^{12} - 6 q^{13} + 5 q^{15} + 38 q^{16} - 8 q^{17} + 7 q^{20} - 10 q^{21} + 3 q^{24} + 31 q^{25} + q^{26} + 10 q^{27} - 18 q^{28} + 14 q^{29} + 6 q^{30} + 26 q^{31} + 41 q^{32} + 21 q^{34} - 5 q^{35} + 18 q^{36} + 24 q^{37} + 8 q^{38} - 6 q^{39} + 5 q^{40} - 19 q^{41} + 6 q^{43} + 5 q^{45} + q^{46} + 15 q^{47} + 38 q^{48} + 10 q^{49} + q^{50} - 8 q^{51} + 25 q^{52} - q^{53} - 3 q^{56} + 11 q^{58} + 23 q^{59} + 7 q^{60} - 11 q^{62} - 10 q^{63} + 53 q^{64} + 29 q^{65} + 38 q^{67} - 87 q^{68} - 6 q^{70} + 26 q^{71} + 3 q^{72} + q^{73} + 39 q^{74} + 31 q^{75} + 2 q^{76} + q^{78} - 5 q^{79} + 6 q^{80} + 10 q^{81} + 5 q^{82} - 6 q^{83} - 18 q^{84} + q^{85} - 41 q^{86} + 14 q^{87} - 9 q^{89} + 6 q^{90} + 6 q^{91} - 48 q^{92} + 26 q^{93} + 42 q^{95} + 41 q^{96} + 24 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.473713 0.334966 0.167483 0.985875i \(-0.446436\pi\)
0.167483 + 0.985875i \(0.446436\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.77560 −0.887798
\(5\) 3.75881 1.68099 0.840496 0.541818i \(-0.182264\pi\)
0.840496 + 0.541818i \(0.182264\pi\)
\(6\) 0.473713 0.193393
\(7\) −1.00000 −0.377964
\(8\) −1.78855 −0.632348
\(9\) 1.00000 0.333333
\(10\) 1.78060 0.563075
\(11\) 0 0
\(12\) −1.77560 −0.512570
\(13\) 2.70774 0.750993 0.375496 0.926824i \(-0.377472\pi\)
0.375496 + 0.926824i \(0.377472\pi\)
\(14\) −0.473713 −0.126605
\(15\) 3.75881 0.970521
\(16\) 2.70393 0.675983
\(17\) 2.48283 0.602174 0.301087 0.953597i \(-0.402651\pi\)
0.301087 + 0.953597i \(0.402651\pi\)
\(18\) 0.473713 0.111655
\(19\) −1.74466 −0.400253 −0.200126 0.979770i \(-0.564135\pi\)
−0.200126 + 0.979770i \(0.564135\pi\)
\(20\) −6.67413 −1.49238
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 3.79843 0.792028 0.396014 0.918244i \(-0.370393\pi\)
0.396014 + 0.918244i \(0.370393\pi\)
\(24\) −1.78855 −0.365086
\(25\) 9.12868 1.82574
\(26\) 1.28269 0.251557
\(27\) 1.00000 0.192450
\(28\) 1.77560 0.335556
\(29\) −5.80060 −1.07714 −0.538572 0.842580i \(-0.681036\pi\)
−0.538572 + 0.842580i \(0.681036\pi\)
\(30\) 1.78060 0.325091
\(31\) 2.24763 0.403686 0.201843 0.979418i \(-0.435307\pi\)
0.201843 + 0.979418i \(0.435307\pi\)
\(32\) 4.85799 0.858779
\(33\) 0 0
\(34\) 1.17615 0.201707
\(35\) −3.75881 −0.635355
\(36\) −1.77560 −0.295933
\(37\) −7.65311 −1.25816 −0.629082 0.777339i \(-0.716569\pi\)
−0.629082 + 0.777339i \(0.716569\pi\)
\(38\) −0.826469 −0.134071
\(39\) 2.70774 0.433586
\(40\) −6.72282 −1.06297
\(41\) −4.18500 −0.653587 −0.326793 0.945096i \(-0.605968\pi\)
−0.326793 + 0.945096i \(0.605968\pi\)
\(42\) −0.473713 −0.0730955
\(43\) 7.75162 1.18211 0.591056 0.806631i \(-0.298711\pi\)
0.591056 + 0.806631i \(0.298711\pi\)
\(44\) 0 0
\(45\) 3.75881 0.560331
\(46\) 1.79937 0.265302
\(47\) 12.4071 1.80976 0.904878 0.425670i \(-0.139962\pi\)
0.904878 + 0.425670i \(0.139962\pi\)
\(48\) 2.70393 0.390279
\(49\) 1.00000 0.142857
\(50\) 4.32437 0.611559
\(51\) 2.48283 0.347665
\(52\) −4.80786 −0.666730
\(53\) 8.83956 1.21421 0.607103 0.794623i \(-0.292331\pi\)
0.607103 + 0.794623i \(0.292331\pi\)
\(54\) 0.473713 0.0644642
\(55\) 0 0
\(56\) 1.78855 0.239005
\(57\) −1.74466 −0.231086
\(58\) −2.74782 −0.360806
\(59\) 4.54166 0.591274 0.295637 0.955300i \(-0.404468\pi\)
0.295637 + 0.955300i \(0.404468\pi\)
\(60\) −6.67413 −0.861627
\(61\) −11.4825 −1.47019 −0.735093 0.677967i \(-0.762861\pi\)
−0.735093 + 0.677967i \(0.762861\pi\)
\(62\) 1.06473 0.135221
\(63\) −1.00000 −0.125988
\(64\) −3.10658 −0.388322
\(65\) 10.1779 1.26241
\(66\) 0 0
\(67\) −12.2041 −1.49096 −0.745481 0.666527i \(-0.767780\pi\)
−0.745481 + 0.666527i \(0.767780\pi\)
\(68\) −4.40849 −0.534609
\(69\) 3.79843 0.457278
\(70\) −1.78060 −0.212822
\(71\) 7.75988 0.920928 0.460464 0.887678i \(-0.347683\pi\)
0.460464 + 0.887678i \(0.347683\pi\)
\(72\) −1.78855 −0.210783
\(73\) 15.6796 1.83516 0.917580 0.397552i \(-0.130140\pi\)
0.917580 + 0.397552i \(0.130140\pi\)
\(74\) −3.62538 −0.421441
\(75\) 9.12868 1.05409
\(76\) 3.09781 0.355344
\(77\) 0 0
\(78\) 1.28269 0.145236
\(79\) −7.05380 −0.793615 −0.396807 0.917902i \(-0.629882\pi\)
−0.396807 + 0.917902i \(0.629882\pi\)
\(80\) 10.1636 1.13632
\(81\) 1.00000 0.111111
\(82\) −1.98249 −0.218929
\(83\) 11.9004 1.30624 0.653122 0.757253i \(-0.273459\pi\)
0.653122 + 0.757253i \(0.273459\pi\)
\(84\) 1.77560 0.193733
\(85\) 9.33248 1.01225
\(86\) 3.67204 0.395967
\(87\) −5.80060 −0.621889
\(88\) 0 0
\(89\) −1.69179 −0.179329 −0.0896647 0.995972i \(-0.528580\pi\)
−0.0896647 + 0.995972i \(0.528580\pi\)
\(90\) 1.78060 0.187692
\(91\) −2.70774 −0.283848
\(92\) −6.74448 −0.703161
\(93\) 2.24763 0.233068
\(94\) 5.87739 0.606206
\(95\) −6.55786 −0.672822
\(96\) 4.85799 0.495816
\(97\) 6.99638 0.710375 0.355188 0.934795i \(-0.384417\pi\)
0.355188 + 0.934795i \(0.384417\pi\)
\(98\) 0.473713 0.0478522
\(99\) 0 0
\(100\) −16.2088 −1.62088
\(101\) 2.71683 0.270335 0.135167 0.990823i \(-0.456843\pi\)
0.135167 + 0.990823i \(0.456843\pi\)
\(102\) 1.17615 0.116456
\(103\) −0.255788 −0.0252036 −0.0126018 0.999921i \(-0.504011\pi\)
−0.0126018 + 0.999921i \(0.504011\pi\)
\(104\) −4.84293 −0.474888
\(105\) −3.75881 −0.366823
\(106\) 4.18741 0.406717
\(107\) 11.8608 1.14662 0.573311 0.819338i \(-0.305659\pi\)
0.573311 + 0.819338i \(0.305659\pi\)
\(108\) −1.77560 −0.170857
\(109\) −15.8720 −1.52026 −0.760129 0.649772i \(-0.774864\pi\)
−0.760129 + 0.649772i \(0.774864\pi\)
\(110\) 0 0
\(111\) −7.65311 −0.726401
\(112\) −2.70393 −0.255498
\(113\) −5.11650 −0.481320 −0.240660 0.970609i \(-0.577364\pi\)
−0.240660 + 0.970609i \(0.577364\pi\)
\(114\) −0.826469 −0.0774059
\(115\) 14.2776 1.33139
\(116\) 10.2995 0.956286
\(117\) 2.70774 0.250331
\(118\) 2.15144 0.198056
\(119\) −2.48283 −0.227600
\(120\) −6.72282 −0.613707
\(121\) 0 0
\(122\) −5.43942 −0.492462
\(123\) −4.18500 −0.377348
\(124\) −3.99088 −0.358392
\(125\) 15.5189 1.38805
\(126\) −0.473713 −0.0422017
\(127\) 0.203338 0.0180433 0.00902166 0.999959i \(-0.497128\pi\)
0.00902166 + 0.999959i \(0.497128\pi\)
\(128\) −11.1876 −0.988853
\(129\) 7.75162 0.682492
\(130\) 4.82140 0.422865
\(131\) −10.2788 −0.898067 −0.449034 0.893515i \(-0.648232\pi\)
−0.449034 + 0.893515i \(0.648232\pi\)
\(132\) 0 0
\(133\) 1.74466 0.151281
\(134\) −5.78122 −0.499421
\(135\) 3.75881 0.323507
\(136\) −4.44065 −0.380783
\(137\) −17.1096 −1.46177 −0.730887 0.682499i \(-0.760893\pi\)
−0.730887 + 0.682499i \(0.760893\pi\)
\(138\) 1.79937 0.153172
\(139\) 19.5009 1.65405 0.827023 0.562168i \(-0.190032\pi\)
0.827023 + 0.562168i \(0.190032\pi\)
\(140\) 6.67413 0.564067
\(141\) 12.4071 1.04486
\(142\) 3.67596 0.308479
\(143\) 0 0
\(144\) 2.70393 0.225328
\(145\) −21.8034 −1.81067
\(146\) 7.42763 0.614715
\(147\) 1.00000 0.0824786
\(148\) 13.5888 1.11699
\(149\) −9.89037 −0.810251 −0.405125 0.914261i \(-0.632772\pi\)
−0.405125 + 0.914261i \(0.632772\pi\)
\(150\) 4.32437 0.353084
\(151\) −15.5014 −1.26148 −0.630741 0.775993i \(-0.717249\pi\)
−0.630741 + 0.775993i \(0.717249\pi\)
\(152\) 3.12041 0.253099
\(153\) 2.48283 0.200725
\(154\) 0 0
\(155\) 8.44843 0.678594
\(156\) −4.80786 −0.384937
\(157\) 16.9536 1.35305 0.676523 0.736422i \(-0.263486\pi\)
0.676523 + 0.736422i \(0.263486\pi\)
\(158\) −3.34148 −0.265834
\(159\) 8.83956 0.701022
\(160\) 18.2603 1.44360
\(161\) −3.79843 −0.299359
\(162\) 0.473713 0.0372184
\(163\) 19.9739 1.56447 0.782237 0.622981i \(-0.214079\pi\)
0.782237 + 0.622981i \(0.214079\pi\)
\(164\) 7.43086 0.580253
\(165\) 0 0
\(166\) 5.63739 0.437547
\(167\) 17.4516 1.35044 0.675222 0.737615i \(-0.264048\pi\)
0.675222 + 0.737615i \(0.264048\pi\)
\(168\) 1.78855 0.137990
\(169\) −5.66813 −0.436010
\(170\) 4.42092 0.339069
\(171\) −1.74466 −0.133418
\(172\) −13.7637 −1.04948
\(173\) −9.35944 −0.711585 −0.355793 0.934565i \(-0.615789\pi\)
−0.355793 + 0.934565i \(0.615789\pi\)
\(174\) −2.74782 −0.208312
\(175\) −9.12868 −0.690063
\(176\) 0 0
\(177\) 4.54166 0.341372
\(178\) −0.801423 −0.0600692
\(179\) 5.49347 0.410602 0.205301 0.978699i \(-0.434183\pi\)
0.205301 + 0.978699i \(0.434183\pi\)
\(180\) −6.67413 −0.497461
\(181\) −11.1844 −0.831332 −0.415666 0.909517i \(-0.636451\pi\)
−0.415666 + 0.909517i \(0.636451\pi\)
\(182\) −1.28269 −0.0950795
\(183\) −11.4825 −0.848812
\(184\) −6.79369 −0.500837
\(185\) −28.7666 −2.11496
\(186\) 1.06473 0.0780699
\(187\) 0 0
\(188\) −22.0299 −1.60670
\(189\) −1.00000 −0.0727393
\(190\) −3.10654 −0.225372
\(191\) −5.75822 −0.416650 −0.208325 0.978060i \(-0.566801\pi\)
−0.208325 + 0.978060i \(0.566801\pi\)
\(192\) −3.10658 −0.224198
\(193\) −14.8846 −1.07142 −0.535708 0.844404i \(-0.679955\pi\)
−0.535708 + 0.844404i \(0.679955\pi\)
\(194\) 3.31428 0.237951
\(195\) 10.1779 0.728854
\(196\) −1.77560 −0.126828
\(197\) 12.5476 0.893978 0.446989 0.894540i \(-0.352496\pi\)
0.446989 + 0.894540i \(0.352496\pi\)
\(198\) 0 0
\(199\) 0.834418 0.0591503 0.0295752 0.999563i \(-0.490585\pi\)
0.0295752 + 0.999563i \(0.490585\pi\)
\(200\) −16.3271 −1.15450
\(201\) −12.2041 −0.860808
\(202\) 1.28700 0.0905528
\(203\) 5.80060 0.407122
\(204\) −4.40849 −0.308656
\(205\) −15.7306 −1.09867
\(206\) −0.121170 −0.00844233
\(207\) 3.79843 0.264009
\(208\) 7.32155 0.507658
\(209\) 0 0
\(210\) −1.78060 −0.122873
\(211\) −26.8703 −1.84983 −0.924914 0.380176i \(-0.875863\pi\)
−0.924914 + 0.380176i \(0.875863\pi\)
\(212\) −15.6955 −1.07797
\(213\) 7.75988 0.531698
\(214\) 5.61859 0.384079
\(215\) 29.1369 1.98712
\(216\) −1.78855 −0.121695
\(217\) −2.24763 −0.152579
\(218\) −7.51876 −0.509234
\(219\) 15.6796 1.05953
\(220\) 0 0
\(221\) 6.72285 0.452228
\(222\) −3.62538 −0.243319
\(223\) 4.01918 0.269144 0.134572 0.990904i \(-0.457034\pi\)
0.134572 + 0.990904i \(0.457034\pi\)
\(224\) −4.85799 −0.324588
\(225\) 9.12868 0.608578
\(226\) −2.42375 −0.161226
\(227\) −11.7020 −0.776690 −0.388345 0.921514i \(-0.626953\pi\)
−0.388345 + 0.921514i \(0.626953\pi\)
\(228\) 3.09781 0.205158
\(229\) −6.54504 −0.432508 −0.216254 0.976337i \(-0.569384\pi\)
−0.216254 + 0.976337i \(0.569384\pi\)
\(230\) 6.76349 0.445971
\(231\) 0 0
\(232\) 10.3746 0.681129
\(233\) 0.172567 0.0113052 0.00565262 0.999984i \(-0.498201\pi\)
0.00565262 + 0.999984i \(0.498201\pi\)
\(234\) 1.28269 0.0838522
\(235\) 46.6358 3.04219
\(236\) −8.06415 −0.524932
\(237\) −7.05380 −0.458194
\(238\) −1.17615 −0.0762383
\(239\) −10.4504 −0.675982 −0.337991 0.941149i \(-0.609747\pi\)
−0.337991 + 0.941149i \(0.609747\pi\)
\(240\) 10.1636 0.656056
\(241\) −3.41821 −0.220186 −0.110093 0.993921i \(-0.535115\pi\)
−0.110093 + 0.993921i \(0.535115\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 20.3883 1.30523
\(245\) 3.75881 0.240142
\(246\) −1.98249 −0.126399
\(247\) −4.72409 −0.300587
\(248\) −4.02000 −0.255270
\(249\) 11.9004 0.754160
\(250\) 7.35151 0.464951
\(251\) 12.6751 0.800047 0.400024 0.916505i \(-0.369002\pi\)
0.400024 + 0.916505i \(0.369002\pi\)
\(252\) 1.77560 0.111852
\(253\) 0 0
\(254\) 0.0963238 0.00604389
\(255\) 9.33248 0.584422
\(256\) 0.913441 0.0570901
\(257\) −18.1852 −1.13436 −0.567182 0.823593i \(-0.691966\pi\)
−0.567182 + 0.823593i \(0.691966\pi\)
\(258\) 3.67204 0.228611
\(259\) 7.65311 0.475541
\(260\) −18.0718 −1.12077
\(261\) −5.80060 −0.359048
\(262\) −4.86922 −0.300822
\(263\) 15.7837 0.973264 0.486632 0.873607i \(-0.338225\pi\)
0.486632 + 0.873607i \(0.338225\pi\)
\(264\) 0 0
\(265\) 33.2262 2.04107
\(266\) 0.826469 0.0506741
\(267\) −1.69179 −0.103536
\(268\) 21.6695 1.32367
\(269\) 22.9896 1.40170 0.700851 0.713308i \(-0.252804\pi\)
0.700851 + 0.713308i \(0.252804\pi\)
\(270\) 1.78060 0.108364
\(271\) 9.34021 0.567377 0.283689 0.958916i \(-0.408442\pi\)
0.283689 + 0.958916i \(0.408442\pi\)
\(272\) 6.71339 0.407059
\(273\) −2.70774 −0.163880
\(274\) −8.10505 −0.489644
\(275\) 0 0
\(276\) −6.74448 −0.405970
\(277\) 19.4367 1.16784 0.583918 0.811813i \(-0.301519\pi\)
0.583918 + 0.811813i \(0.301519\pi\)
\(278\) 9.23784 0.554049
\(279\) 2.24763 0.134562
\(280\) 6.72282 0.401765
\(281\) −4.44358 −0.265082 −0.132541 0.991178i \(-0.542314\pi\)
−0.132541 + 0.991178i \(0.542314\pi\)
\(282\) 5.87739 0.349993
\(283\) 13.8486 0.823216 0.411608 0.911361i \(-0.364967\pi\)
0.411608 + 0.911361i \(0.364967\pi\)
\(284\) −13.7784 −0.817598
\(285\) −6.55786 −0.388454
\(286\) 0 0
\(287\) 4.18500 0.247033
\(288\) 4.85799 0.286260
\(289\) −10.8356 −0.637387
\(290\) −10.3285 −0.606512
\(291\) 6.99638 0.410135
\(292\) −27.8406 −1.62925
\(293\) −19.2359 −1.12377 −0.561887 0.827214i \(-0.689924\pi\)
−0.561887 + 0.827214i \(0.689924\pi\)
\(294\) 0.473713 0.0276275
\(295\) 17.0713 0.993927
\(296\) 13.6880 0.795596
\(297\) 0 0
\(298\) −4.68520 −0.271406
\(299\) 10.2852 0.594807
\(300\) −16.2088 −0.935818
\(301\) −7.75162 −0.446796
\(302\) −7.34319 −0.422553
\(303\) 2.71683 0.156078
\(304\) −4.71745 −0.270564
\(305\) −43.1606 −2.47137
\(306\) 1.17615 0.0672358
\(307\) 2.24474 0.128114 0.0640571 0.997946i \(-0.479596\pi\)
0.0640571 + 0.997946i \(0.479596\pi\)
\(308\) 0 0
\(309\) −0.255788 −0.0145513
\(310\) 4.00213 0.227306
\(311\) 12.6291 0.716129 0.358065 0.933697i \(-0.383437\pi\)
0.358065 + 0.933697i \(0.383437\pi\)
\(312\) −4.84293 −0.274177
\(313\) 1.06646 0.0602797 0.0301398 0.999546i \(-0.490405\pi\)
0.0301398 + 0.999546i \(0.490405\pi\)
\(314\) 8.03115 0.453224
\(315\) −3.75881 −0.211785
\(316\) 12.5247 0.704569
\(317\) −25.1710 −1.41374 −0.706872 0.707342i \(-0.749894\pi\)
−0.706872 + 0.707342i \(0.749894\pi\)
\(318\) 4.18741 0.234818
\(319\) 0 0
\(320\) −11.6770 −0.652766
\(321\) 11.8608 0.662003
\(322\) −1.79937 −0.100275
\(323\) −4.33169 −0.241022
\(324\) −1.77560 −0.0986442
\(325\) 24.7181 1.37111
\(326\) 9.46188 0.524045
\(327\) −15.8720 −0.877722
\(328\) 7.48507 0.413294
\(329\) −12.4071 −0.684024
\(330\) 0 0
\(331\) −8.53640 −0.469203 −0.234601 0.972092i \(-0.575378\pi\)
−0.234601 + 0.972092i \(0.575378\pi\)
\(332\) −21.1304 −1.15968
\(333\) −7.65311 −0.419388
\(334\) 8.26704 0.452352
\(335\) −45.8728 −2.50630
\(336\) −2.70393 −0.147512
\(337\) −17.2565 −0.940020 −0.470010 0.882661i \(-0.655750\pi\)
−0.470010 + 0.882661i \(0.655750\pi\)
\(338\) −2.68507 −0.146048
\(339\) −5.11650 −0.277890
\(340\) −16.5707 −0.898673
\(341\) 0 0
\(342\) −0.826469 −0.0446903
\(343\) −1.00000 −0.0539949
\(344\) −13.8642 −0.747505
\(345\) 14.2776 0.768680
\(346\) −4.43369 −0.238357
\(347\) 4.15698 0.223159 0.111579 0.993756i \(-0.464409\pi\)
0.111579 + 0.993756i \(0.464409\pi\)
\(348\) 10.2995 0.552112
\(349\) 7.70785 0.412592 0.206296 0.978490i \(-0.433859\pi\)
0.206296 + 0.978490i \(0.433859\pi\)
\(350\) −4.32437 −0.231147
\(351\) 2.70774 0.144529
\(352\) 0 0
\(353\) −4.84242 −0.257736 −0.128868 0.991662i \(-0.541134\pi\)
−0.128868 + 0.991662i \(0.541134\pi\)
\(354\) 2.15144 0.114348
\(355\) 29.1679 1.54807
\(356\) 3.00393 0.159208
\(357\) −2.48283 −0.131405
\(358\) 2.60233 0.137537
\(359\) 4.41834 0.233191 0.116596 0.993179i \(-0.462802\pi\)
0.116596 + 0.993179i \(0.462802\pi\)
\(360\) −6.72282 −0.354324
\(361\) −15.9562 −0.839798
\(362\) −5.29821 −0.278468
\(363\) 0 0
\(364\) 4.80786 0.252000
\(365\) 58.9367 3.08489
\(366\) −5.43942 −0.284323
\(367\) −29.5315 −1.54153 −0.770766 0.637118i \(-0.780126\pi\)
−0.770766 + 0.637118i \(0.780126\pi\)
\(368\) 10.2707 0.535398
\(369\) −4.18500 −0.217862
\(370\) −13.6271 −0.708440
\(371\) −8.83956 −0.458927
\(372\) −3.99088 −0.206918
\(373\) 7.64965 0.396084 0.198042 0.980194i \(-0.436542\pi\)
0.198042 + 0.980194i \(0.436542\pi\)
\(374\) 0 0
\(375\) 15.5189 0.801394
\(376\) −22.1906 −1.14440
\(377\) −15.7065 −0.808927
\(378\) −0.473713 −0.0243652
\(379\) −11.2094 −0.575788 −0.287894 0.957662i \(-0.592955\pi\)
−0.287894 + 0.957662i \(0.592955\pi\)
\(380\) 11.6441 0.597330
\(381\) 0.203338 0.0104173
\(382\) −2.72774 −0.139563
\(383\) −24.7447 −1.26440 −0.632199 0.774806i \(-0.717847\pi\)
−0.632199 + 0.774806i \(0.717847\pi\)
\(384\) −11.1876 −0.570915
\(385\) 0 0
\(386\) −7.05102 −0.358887
\(387\) 7.75162 0.394037
\(388\) −12.4228 −0.630670
\(389\) −29.0347 −1.47212 −0.736060 0.676917i \(-0.763316\pi\)
−0.736060 + 0.676917i \(0.763316\pi\)
\(390\) 4.82140 0.244141
\(391\) 9.43085 0.476939
\(392\) −1.78855 −0.0903354
\(393\) −10.2788 −0.518499
\(394\) 5.94395 0.299452
\(395\) −26.5139 −1.33406
\(396\) 0 0
\(397\) −8.27461 −0.415291 −0.207645 0.978204i \(-0.566580\pi\)
−0.207645 + 0.978204i \(0.566580\pi\)
\(398\) 0.395275 0.0198133
\(399\) 1.74466 0.0873423
\(400\) 24.6833 1.23417
\(401\) −26.7390 −1.33528 −0.667642 0.744483i \(-0.732696\pi\)
−0.667642 + 0.744483i \(0.732696\pi\)
\(402\) −5.78122 −0.288341
\(403\) 6.08601 0.303166
\(404\) −4.82399 −0.240003
\(405\) 3.75881 0.186777
\(406\) 2.74782 0.136372
\(407\) 0 0
\(408\) −4.44065 −0.219845
\(409\) 34.6829 1.71496 0.857479 0.514519i \(-0.172030\pi\)
0.857479 + 0.514519i \(0.172030\pi\)
\(410\) −7.45180 −0.368018
\(411\) −17.1096 −0.843955
\(412\) 0.454177 0.0223757
\(413\) −4.54166 −0.223480
\(414\) 1.79937 0.0884341
\(415\) 44.7315 2.19578
\(416\) 13.1542 0.644936
\(417\) 19.5009 0.954964
\(418\) 0 0
\(419\) −4.34546 −0.212290 −0.106145 0.994351i \(-0.533851\pi\)
−0.106145 + 0.994351i \(0.533851\pi\)
\(420\) 6.67413 0.325664
\(421\) −4.73841 −0.230936 −0.115468 0.993311i \(-0.536837\pi\)
−0.115468 + 0.993311i \(0.536837\pi\)
\(422\) −12.7288 −0.619629
\(423\) 12.4071 0.603252
\(424\) −15.8100 −0.767800
\(425\) 22.6649 1.09941
\(426\) 3.67596 0.178101
\(427\) 11.4825 0.555678
\(428\) −21.0599 −1.01797
\(429\) 0 0
\(430\) 13.8025 0.665617
\(431\) −29.8622 −1.43841 −0.719206 0.694797i \(-0.755494\pi\)
−0.719206 + 0.694797i \(0.755494\pi\)
\(432\) 2.70393 0.130093
\(433\) −21.7239 −1.04399 −0.521993 0.852950i \(-0.674811\pi\)
−0.521993 + 0.852950i \(0.674811\pi\)
\(434\) −1.06473 −0.0511088
\(435\) −21.8034 −1.04539
\(436\) 28.1822 1.34968
\(437\) −6.62698 −0.317012
\(438\) 7.42763 0.354906
\(439\) −40.3447 −1.92555 −0.962773 0.270313i \(-0.912873\pi\)
−0.962773 + 0.270313i \(0.912873\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 3.18470 0.151481
\(443\) −13.8789 −0.659408 −0.329704 0.944084i \(-0.606949\pi\)
−0.329704 + 0.944084i \(0.606949\pi\)
\(444\) 13.5888 0.644897
\(445\) −6.35912 −0.301451
\(446\) 1.90394 0.0901540
\(447\) −9.89037 −0.467799
\(448\) 3.10658 0.146772
\(449\) −14.3023 −0.674966 −0.337483 0.941332i \(-0.609575\pi\)
−0.337483 + 0.941332i \(0.609575\pi\)
\(450\) 4.32437 0.203853
\(451\) 0 0
\(452\) 9.08484 0.427315
\(453\) −15.5014 −0.728317
\(454\) −5.54340 −0.260164
\(455\) −10.1779 −0.477147
\(456\) 3.12041 0.146127
\(457\) −14.1278 −0.660869 −0.330435 0.943829i \(-0.607195\pi\)
−0.330435 + 0.943829i \(0.607195\pi\)
\(458\) −3.10047 −0.144875
\(459\) 2.48283 0.115888
\(460\) −25.3513 −1.18201
\(461\) −0.966736 −0.0450254 −0.0225127 0.999747i \(-0.507167\pi\)
−0.0225127 + 0.999747i \(0.507167\pi\)
\(462\) 0 0
\(463\) 31.8119 1.47842 0.739212 0.673473i \(-0.235198\pi\)
0.739212 + 0.673473i \(0.235198\pi\)
\(464\) −15.6844 −0.728131
\(465\) 8.44843 0.391786
\(466\) 0.0817473 0.00378687
\(467\) −29.6133 −1.37034 −0.685169 0.728384i \(-0.740272\pi\)
−0.685169 + 0.728384i \(0.740272\pi\)
\(468\) −4.80786 −0.222243
\(469\) 12.2041 0.563531
\(470\) 22.0920 1.01903
\(471\) 16.9536 0.781181
\(472\) −8.12298 −0.373891
\(473\) 0 0
\(474\) −3.34148 −0.153479
\(475\) −15.9264 −0.730756
\(476\) 4.40849 0.202063
\(477\) 8.83956 0.404735
\(478\) −4.95051 −0.226431
\(479\) −26.5853 −1.21471 −0.607356 0.794430i \(-0.707770\pi\)
−0.607356 + 0.794430i \(0.707770\pi\)
\(480\) 18.2603 0.833463
\(481\) −20.7226 −0.944871
\(482\) −1.61925 −0.0737548
\(483\) −3.79843 −0.172835
\(484\) 0 0
\(485\) 26.2981 1.19414
\(486\) 0.473713 0.0214881
\(487\) 14.1343 0.640486 0.320243 0.947335i \(-0.396235\pi\)
0.320243 + 0.947335i \(0.396235\pi\)
\(488\) 20.5370 0.929668
\(489\) 19.9739 0.903249
\(490\) 1.78060 0.0804392
\(491\) 15.4186 0.695831 0.347916 0.937526i \(-0.386890\pi\)
0.347916 + 0.937526i \(0.386890\pi\)
\(492\) 7.43086 0.335009
\(493\) −14.4019 −0.648627
\(494\) −2.23786 −0.100686
\(495\) 0 0
\(496\) 6.07744 0.272885
\(497\) −7.75988 −0.348078
\(498\) 5.63739 0.252618
\(499\) 30.7024 1.37443 0.687214 0.726455i \(-0.258833\pi\)
0.687214 + 0.726455i \(0.258833\pi\)
\(500\) −27.5553 −1.23231
\(501\) 17.4516 0.779679
\(502\) 6.00438 0.267988
\(503\) −18.1890 −0.811005 −0.405503 0.914094i \(-0.632904\pi\)
−0.405503 + 0.914094i \(0.632904\pi\)
\(504\) 1.78855 0.0796683
\(505\) 10.2121 0.454431
\(506\) 0 0
\(507\) −5.66813 −0.251731
\(508\) −0.361046 −0.0160188
\(509\) −32.1744 −1.42611 −0.713054 0.701110i \(-0.752688\pi\)
−0.713054 + 0.701110i \(0.752688\pi\)
\(510\) 4.42092 0.195761
\(511\) −15.6796 −0.693625
\(512\) 22.8079 1.00798
\(513\) −1.74466 −0.0770287
\(514\) −8.61458 −0.379973
\(515\) −0.961461 −0.0423670
\(516\) −13.7637 −0.605915
\(517\) 0 0
\(518\) 3.62538 0.159290
\(519\) −9.35944 −0.410834
\(520\) −18.2037 −0.798284
\(521\) −27.8238 −1.21898 −0.609490 0.792793i \(-0.708626\pi\)
−0.609490 + 0.792793i \(0.708626\pi\)
\(522\) −2.74782 −0.120269
\(523\) −19.1282 −0.836419 −0.418209 0.908351i \(-0.637342\pi\)
−0.418209 + 0.908351i \(0.637342\pi\)
\(524\) 18.2511 0.797302
\(525\) −9.12868 −0.398408
\(526\) 7.47694 0.326010
\(527\) 5.58048 0.243089
\(528\) 0 0
\(529\) −8.57190 −0.372691
\(530\) 15.7397 0.683689
\(531\) 4.54166 0.197091
\(532\) −3.09781 −0.134307
\(533\) −11.3319 −0.490839
\(534\) −0.801423 −0.0346810
\(535\) 44.5824 1.92746
\(536\) 21.8275 0.942806
\(537\) 5.49347 0.237061
\(538\) 10.8905 0.469522
\(539\) 0 0
\(540\) −6.67413 −0.287209
\(541\) 42.6820 1.83504 0.917521 0.397688i \(-0.130187\pi\)
0.917521 + 0.397688i \(0.130187\pi\)
\(542\) 4.42458 0.190052
\(543\) −11.1844 −0.479970
\(544\) 12.0615 0.517134
\(545\) −59.6597 −2.55554
\(546\) −1.28269 −0.0548942
\(547\) −25.1162 −1.07389 −0.536946 0.843617i \(-0.680422\pi\)
−0.536946 + 0.843617i \(0.680422\pi\)
\(548\) 30.3798 1.29776
\(549\) −11.4825 −0.490062
\(550\) 0 0
\(551\) 10.1201 0.431130
\(552\) −6.79369 −0.289158
\(553\) 7.05380 0.299958
\(554\) 9.20740 0.391185
\(555\) −28.7666 −1.22107
\(556\) −34.6258 −1.46846
\(557\) −4.25720 −0.180383 −0.0901917 0.995924i \(-0.528748\pi\)
−0.0901917 + 0.995924i \(0.528748\pi\)
\(558\) 1.06473 0.0450737
\(559\) 20.9894 0.887757
\(560\) −10.1636 −0.429490
\(561\) 0 0
\(562\) −2.10498 −0.0887933
\(563\) −7.43714 −0.313438 −0.156719 0.987643i \(-0.550092\pi\)
−0.156719 + 0.987643i \(0.550092\pi\)
\(564\) −22.0299 −0.927628
\(565\) −19.2320 −0.809096
\(566\) 6.56028 0.275749
\(567\) −1.00000 −0.0419961
\(568\) −13.8789 −0.582347
\(569\) 43.6293 1.82904 0.914518 0.404546i \(-0.132570\pi\)
0.914518 + 0.404546i \(0.132570\pi\)
\(570\) −3.10654 −0.130119
\(571\) −34.1074 −1.42735 −0.713676 0.700476i \(-0.752971\pi\)
−0.713676 + 0.700476i \(0.752971\pi\)
\(572\) 0 0
\(573\) −5.75822 −0.240553
\(574\) 1.98249 0.0827474
\(575\) 34.6747 1.44603
\(576\) −3.10658 −0.129441
\(577\) 28.0804 1.16900 0.584501 0.811393i \(-0.301290\pi\)
0.584501 + 0.811393i \(0.301290\pi\)
\(578\) −5.13295 −0.213503
\(579\) −14.8846 −0.618582
\(580\) 38.7140 1.60751
\(581\) −11.9004 −0.493713
\(582\) 3.31428 0.137381
\(583\) 0 0
\(584\) −28.0437 −1.16046
\(585\) 10.1779 0.420804
\(586\) −9.11230 −0.376426
\(587\) −6.72735 −0.277668 −0.138834 0.990316i \(-0.544335\pi\)
−0.138834 + 0.990316i \(0.544335\pi\)
\(588\) −1.77560 −0.0732243
\(589\) −3.92136 −0.161577
\(590\) 8.08687 0.332931
\(591\) 12.5476 0.516138
\(592\) −20.6935 −0.850497
\(593\) 20.8442 0.855969 0.427984 0.903786i \(-0.359224\pi\)
0.427984 + 0.903786i \(0.359224\pi\)
\(594\) 0 0
\(595\) −9.33248 −0.382594
\(596\) 17.5613 0.719339
\(597\) 0.834418 0.0341504
\(598\) 4.87222 0.199240
\(599\) 0.180675 0.00738218 0.00369109 0.999993i \(-0.498825\pi\)
0.00369109 + 0.999993i \(0.498825\pi\)
\(600\) −16.3271 −0.666550
\(601\) 27.6043 1.12600 0.563001 0.826456i \(-0.309647\pi\)
0.563001 + 0.826456i \(0.309647\pi\)
\(602\) −3.67204 −0.149661
\(603\) −12.2041 −0.496988
\(604\) 27.5241 1.11994
\(605\) 0 0
\(606\) 1.28700 0.0522807
\(607\) 23.3280 0.946854 0.473427 0.880833i \(-0.343017\pi\)
0.473427 + 0.880833i \(0.343017\pi\)
\(608\) −8.47554 −0.343729
\(609\) 5.80060 0.235052
\(610\) −20.4457 −0.827824
\(611\) 33.5951 1.35911
\(612\) −4.40849 −0.178203
\(613\) −19.5003 −0.787608 −0.393804 0.919194i \(-0.628841\pi\)
−0.393804 + 0.919194i \(0.628841\pi\)
\(614\) 1.06336 0.0429139
\(615\) −15.7306 −0.634320
\(616\) 0 0
\(617\) −23.2566 −0.936276 −0.468138 0.883655i \(-0.655075\pi\)
−0.468138 + 0.883655i \(0.655075\pi\)
\(618\) −0.121170 −0.00487418
\(619\) −28.5491 −1.14749 −0.573743 0.819035i \(-0.694509\pi\)
−0.573743 + 0.819035i \(0.694509\pi\)
\(620\) −15.0010 −0.602454
\(621\) 3.79843 0.152426
\(622\) 5.98256 0.239879
\(623\) 1.69179 0.0677801
\(624\) 7.32155 0.293097
\(625\) 12.6893 0.507574
\(626\) 0.505194 0.0201916
\(627\) 0 0
\(628\) −30.1028 −1.20123
\(629\) −19.0013 −0.757633
\(630\) −1.78060 −0.0709407
\(631\) 5.30112 0.211034 0.105517 0.994417i \(-0.466350\pi\)
0.105517 + 0.994417i \(0.466350\pi\)
\(632\) 12.6161 0.501840
\(633\) −26.8703 −1.06800
\(634\) −11.9238 −0.473556
\(635\) 0.764309 0.0303307
\(636\) −15.6955 −0.622366
\(637\) 2.70774 0.107285
\(638\) 0 0
\(639\) 7.75988 0.306976
\(640\) −42.0521 −1.66225
\(641\) −29.8881 −1.18051 −0.590255 0.807217i \(-0.700973\pi\)
−0.590255 + 0.807217i \(0.700973\pi\)
\(642\) 5.61859 0.221748
\(643\) 38.4403 1.51594 0.757969 0.652291i \(-0.226192\pi\)
0.757969 + 0.652291i \(0.226192\pi\)
\(644\) 6.74448 0.265770
\(645\) 29.1369 1.14726
\(646\) −2.05198 −0.0807340
\(647\) −32.3602 −1.27221 −0.636105 0.771603i \(-0.719455\pi\)
−0.636105 + 0.771603i \(0.719455\pi\)
\(648\) −1.78855 −0.0702608
\(649\) 0 0
\(650\) 11.7093 0.459276
\(651\) −2.24763 −0.0880916
\(652\) −35.4655 −1.38894
\(653\) 23.9046 0.935459 0.467729 0.883872i \(-0.345072\pi\)
0.467729 + 0.883872i \(0.345072\pi\)
\(654\) −7.51876 −0.294007
\(655\) −38.6363 −1.50964
\(656\) −11.3160 −0.441814
\(657\) 15.6796 0.611720
\(658\) −5.87739 −0.229124
\(659\) 34.5124 1.34441 0.672205 0.740365i \(-0.265347\pi\)
0.672205 + 0.740365i \(0.265347\pi\)
\(660\) 0 0
\(661\) −2.56237 −0.0996647 −0.0498324 0.998758i \(-0.515869\pi\)
−0.0498324 + 0.998758i \(0.515869\pi\)
\(662\) −4.04380 −0.157167
\(663\) 6.72285 0.261094
\(664\) −21.2845 −0.826000
\(665\) 6.55786 0.254303
\(666\) −3.62538 −0.140480
\(667\) −22.0332 −0.853128
\(668\) −30.9869 −1.19892
\(669\) 4.01918 0.155390
\(670\) −21.7305 −0.839523
\(671\) 0 0
\(672\) −4.85799 −0.187401
\(673\) 36.6670 1.41341 0.706704 0.707509i \(-0.250181\pi\)
0.706704 + 0.707509i \(0.250181\pi\)
\(674\) −8.17462 −0.314874
\(675\) 9.12868 0.351363
\(676\) 10.0643 0.387089
\(677\) 31.9812 1.22914 0.614568 0.788864i \(-0.289330\pi\)
0.614568 + 0.788864i \(0.289330\pi\)
\(678\) −2.42375 −0.0930837
\(679\) −6.99638 −0.268497
\(680\) −16.6916 −0.640093
\(681\) −11.7020 −0.448422
\(682\) 0 0
\(683\) 6.96871 0.266650 0.133325 0.991072i \(-0.457435\pi\)
0.133325 + 0.991072i \(0.457435\pi\)
\(684\) 3.09781 0.118448
\(685\) −64.3119 −2.45723
\(686\) −0.473713 −0.0180864
\(687\) −6.54504 −0.249709
\(688\) 20.9599 0.799087
\(689\) 23.9352 0.911860
\(690\) 6.76349 0.257482
\(691\) −5.07203 −0.192949 −0.0964746 0.995335i \(-0.530757\pi\)
−0.0964746 + 0.995335i \(0.530757\pi\)
\(692\) 16.6186 0.631744
\(693\) 0 0
\(694\) 1.96922 0.0747505
\(695\) 73.3003 2.78044
\(696\) 10.3746 0.393250
\(697\) −10.3906 −0.393573
\(698\) 3.65131 0.138204
\(699\) 0.172567 0.00652709
\(700\) 16.2088 0.612637
\(701\) 0.418223 0.0157961 0.00789803 0.999969i \(-0.497486\pi\)
0.00789803 + 0.999969i \(0.497486\pi\)
\(702\) 1.28269 0.0484121
\(703\) 13.3521 0.503583
\(704\) 0 0
\(705\) 46.6358 1.75641
\(706\) −2.29392 −0.0863328
\(707\) −2.71683 −0.102177
\(708\) −8.06415 −0.303069
\(709\) −3.60664 −0.135450 −0.0677251 0.997704i \(-0.521574\pi\)
−0.0677251 + 0.997704i \(0.521574\pi\)
\(710\) 13.8172 0.518551
\(711\) −7.05380 −0.264538
\(712\) 3.02585 0.113398
\(713\) 8.53748 0.319731
\(714\) −1.17615 −0.0440162
\(715\) 0 0
\(716\) −9.75419 −0.364531
\(717\) −10.4504 −0.390279
\(718\) 2.09303 0.0781110
\(719\) −17.9697 −0.670155 −0.335078 0.942190i \(-0.608763\pi\)
−0.335078 + 0.942190i \(0.608763\pi\)
\(720\) 10.1636 0.378774
\(721\) 0.255788 0.00952606
\(722\) −7.55864 −0.281303
\(723\) −3.41821 −0.127124
\(724\) 19.8590 0.738055
\(725\) −52.9518 −1.96658
\(726\) 0 0
\(727\) 10.0774 0.373752 0.186876 0.982384i \(-0.440164\pi\)
0.186876 + 0.982384i \(0.440164\pi\)
\(728\) 4.84293 0.179491
\(729\) 1.00000 0.0370370
\(730\) 27.9191 1.03333
\(731\) 19.2459 0.711836
\(732\) 20.3883 0.753574
\(733\) −12.2275 −0.451631 −0.225816 0.974170i \(-0.572505\pi\)
−0.225816 + 0.974170i \(0.572505\pi\)
\(734\) −13.9895 −0.516360
\(735\) 3.75881 0.138646
\(736\) 18.4527 0.680177
\(737\) 0 0
\(738\) −1.98249 −0.0729764
\(739\) 3.49821 0.128684 0.0643419 0.997928i \(-0.479505\pi\)
0.0643419 + 0.997928i \(0.479505\pi\)
\(740\) 51.0779 1.87766
\(741\) −4.72409 −0.173544
\(742\) −4.18741 −0.153725
\(743\) 2.73642 0.100389 0.0501947 0.998739i \(-0.484016\pi\)
0.0501947 + 0.998739i \(0.484016\pi\)
\(744\) −4.02000 −0.147380
\(745\) −37.1761 −1.36203
\(746\) 3.62374 0.132674
\(747\) 11.9004 0.435414
\(748\) 0 0
\(749\) −11.8608 −0.433382
\(750\) 7.35151 0.268439
\(751\) −10.0480 −0.366655 −0.183327 0.983052i \(-0.558687\pi\)
−0.183327 + 0.983052i \(0.558687\pi\)
\(752\) 33.5479 1.22337
\(753\) 12.6751 0.461908
\(754\) −7.44038 −0.270963
\(755\) −58.2667 −2.12054
\(756\) 1.77560 0.0645778
\(757\) 15.1490 0.550600 0.275300 0.961358i \(-0.411223\pi\)
0.275300 + 0.961358i \(0.411223\pi\)
\(758\) −5.31004 −0.192869
\(759\) 0 0
\(760\) 11.7290 0.425457
\(761\) 35.4413 1.28475 0.642373 0.766392i \(-0.277950\pi\)
0.642373 + 0.766392i \(0.277950\pi\)
\(762\) 0.0963238 0.00348944
\(763\) 15.8720 0.574604
\(764\) 10.2243 0.369901
\(765\) 9.33248 0.337416
\(766\) −11.7219 −0.423530
\(767\) 12.2976 0.444042
\(768\) 0.913441 0.0329610
\(769\) 38.7592 1.39769 0.698845 0.715273i \(-0.253698\pi\)
0.698845 + 0.715273i \(0.253698\pi\)
\(770\) 0 0
\(771\) −18.1852 −0.654925
\(772\) 26.4290 0.951200
\(773\) 26.5419 0.954646 0.477323 0.878728i \(-0.341607\pi\)
0.477323 + 0.878728i \(0.341607\pi\)
\(774\) 3.67204 0.131989
\(775\) 20.5179 0.737025
\(776\) −12.5134 −0.449204
\(777\) 7.65311 0.274554
\(778\) −13.7541 −0.493109
\(779\) 7.30140 0.261600
\(780\) −18.0718 −0.647075
\(781\) 0 0
\(782\) 4.46752 0.159758
\(783\) −5.80060 −0.207296
\(784\) 2.70393 0.0965690
\(785\) 63.7255 2.27446
\(786\) −4.86922 −0.173679
\(787\) 19.6802 0.701524 0.350762 0.936465i \(-0.385923\pi\)
0.350762 + 0.936465i \(0.385923\pi\)
\(788\) −22.2794 −0.793672
\(789\) 15.7837 0.561914
\(790\) −12.5600 −0.446864
\(791\) 5.11650 0.181922
\(792\) 0 0
\(793\) −31.0917 −1.10410
\(794\) −3.91979 −0.139108
\(795\) 33.2262 1.17841
\(796\) −1.48159 −0.0525135
\(797\) −25.2797 −0.895451 −0.447726 0.894171i \(-0.647766\pi\)
−0.447726 + 0.894171i \(0.647766\pi\)
\(798\) 0.826469 0.0292567
\(799\) 30.8046 1.08979
\(800\) 44.3470 1.56790
\(801\) −1.69179 −0.0597764
\(802\) −12.6666 −0.447274
\(803\) 0 0
\(804\) 21.6695 0.764223
\(805\) −14.2776 −0.503219
\(806\) 2.88302 0.101550
\(807\) 22.9896 0.809273
\(808\) −4.85918 −0.170945
\(809\) −35.7891 −1.25828 −0.629138 0.777293i \(-0.716592\pi\)
−0.629138 + 0.777293i \(0.716592\pi\)
\(810\) 1.78060 0.0625639
\(811\) 20.9415 0.735355 0.367677 0.929953i \(-0.380153\pi\)
0.367677 + 0.929953i \(0.380153\pi\)
\(812\) −10.2995 −0.361442
\(813\) 9.34021 0.327575
\(814\) 0 0
\(815\) 75.0780 2.62987
\(816\) 6.71339 0.235016
\(817\) −13.5240 −0.473143
\(818\) 16.4297 0.574452
\(819\) −2.70774 −0.0946162
\(820\) 27.9312 0.975401
\(821\) −8.06663 −0.281527 −0.140764 0.990043i \(-0.544956\pi\)
−0.140764 + 0.990043i \(0.544956\pi\)
\(822\) −8.10505 −0.282696
\(823\) −20.1359 −0.701895 −0.350948 0.936395i \(-0.614140\pi\)
−0.350948 + 0.936395i \(0.614140\pi\)
\(824\) 0.457490 0.0159374
\(825\) 0 0
\(826\) −2.15144 −0.0748583
\(827\) −21.4181 −0.744780 −0.372390 0.928076i \(-0.621462\pi\)
−0.372390 + 0.928076i \(0.621462\pi\)
\(828\) −6.74448 −0.234387
\(829\) −27.3960 −0.951504 −0.475752 0.879580i \(-0.657824\pi\)
−0.475752 + 0.879580i \(0.657824\pi\)
\(830\) 21.1899 0.735512
\(831\) 19.4367 0.674251
\(832\) −8.41181 −0.291627
\(833\) 2.48283 0.0860248
\(834\) 9.23784 0.319880
\(835\) 65.5972 2.27008
\(836\) 0 0
\(837\) 2.24763 0.0776895
\(838\) −2.05850 −0.0711097
\(839\) −20.3757 −0.703446 −0.351723 0.936104i \(-0.614404\pi\)
−0.351723 + 0.936104i \(0.614404\pi\)
\(840\) 6.72282 0.231959
\(841\) 4.64691 0.160238
\(842\) −2.24465 −0.0773556
\(843\) −4.44358 −0.153045
\(844\) 47.7108 1.64227
\(845\) −21.3055 −0.732930
\(846\) 5.87739 0.202069
\(847\) 0 0
\(848\) 23.9016 0.820783
\(849\) 13.8486 0.475284
\(850\) 10.7367 0.368264
\(851\) −29.0698 −0.996501
\(852\) −13.7784 −0.472041
\(853\) −8.28686 −0.283737 −0.141868 0.989886i \(-0.545311\pi\)
−0.141868 + 0.989886i \(0.545311\pi\)
\(854\) 5.43942 0.186133
\(855\) −6.55786 −0.224274
\(856\) −21.2135 −0.725064
\(857\) −31.0459 −1.06051 −0.530254 0.847839i \(-0.677903\pi\)
−0.530254 + 0.847839i \(0.677903\pi\)
\(858\) 0 0
\(859\) −1.63124 −0.0556573 −0.0278287 0.999613i \(-0.508859\pi\)
−0.0278287 + 0.999613i \(0.508859\pi\)
\(860\) −51.7354 −1.76416
\(861\) 4.18500 0.142624
\(862\) −14.1461 −0.481819
\(863\) 5.32862 0.181388 0.0906942 0.995879i \(-0.471091\pi\)
0.0906942 + 0.995879i \(0.471091\pi\)
\(864\) 4.85799 0.165272
\(865\) −35.1804 −1.19617
\(866\) −10.2909 −0.349699
\(867\) −10.8356 −0.367996
\(868\) 3.99088 0.135459
\(869\) 0 0
\(870\) −10.3285 −0.350170
\(871\) −33.0454 −1.11970
\(872\) 28.3878 0.961332
\(873\) 6.99638 0.236792
\(874\) −3.13929 −0.106188
\(875\) −15.5189 −0.524635
\(876\) −27.8406 −0.940648
\(877\) −16.1267 −0.544561 −0.272281 0.962218i \(-0.587778\pi\)
−0.272281 + 0.962218i \(0.587778\pi\)
\(878\) −19.1118 −0.644992
\(879\) −19.2359 −0.648811
\(880\) 0 0
\(881\) −35.1173 −1.18313 −0.591565 0.806257i \(-0.701490\pi\)
−0.591565 + 0.806257i \(0.701490\pi\)
\(882\) 0.473713 0.0159507
\(883\) 0.480232 0.0161611 0.00808054 0.999967i \(-0.497428\pi\)
0.00808054 + 0.999967i \(0.497428\pi\)
\(884\) −11.9371 −0.401487
\(885\) 17.0713 0.573844
\(886\) −6.57463 −0.220879
\(887\) 36.7238 1.23306 0.616532 0.787330i \(-0.288537\pi\)
0.616532 + 0.787330i \(0.288537\pi\)
\(888\) 13.6880 0.459338
\(889\) −0.203338 −0.00681973
\(890\) −3.01240 −0.100976
\(891\) 0 0
\(892\) −7.13644 −0.238946
\(893\) −21.6461 −0.724360
\(894\) −4.68520 −0.156696
\(895\) 20.6489 0.690218
\(896\) 11.1876 0.373751
\(897\) 10.2852 0.343412
\(898\) −6.77517 −0.226090
\(899\) −13.0376 −0.434828
\(900\) −16.2088 −0.540295
\(901\) 21.9471 0.731163
\(902\) 0 0
\(903\) −7.75162 −0.257958
\(904\) 9.15112 0.304362
\(905\) −42.0402 −1.39746
\(906\) −7.34319 −0.243961
\(907\) 12.1917 0.404818 0.202409 0.979301i \(-0.435123\pi\)
0.202409 + 0.979301i \(0.435123\pi\)
\(908\) 20.7781 0.689544
\(909\) 2.71683 0.0901116
\(910\) −4.82140 −0.159828
\(911\) −8.11848 −0.268977 −0.134489 0.990915i \(-0.542939\pi\)
−0.134489 + 0.990915i \(0.542939\pi\)
\(912\) −4.71745 −0.156210
\(913\) 0 0
\(914\) −6.69251 −0.221369
\(915\) −43.1606 −1.42685
\(916\) 11.6213 0.383980
\(917\) 10.2788 0.339438
\(918\) 1.17615 0.0388186
\(919\) −49.6539 −1.63793 −0.818965 0.573843i \(-0.805452\pi\)
−0.818965 + 0.573843i \(0.805452\pi\)
\(920\) −25.5362 −0.841903
\(921\) 2.24474 0.0739668
\(922\) −0.457955 −0.0150820
\(923\) 21.0118 0.691610
\(924\) 0 0
\(925\) −69.8627 −2.29707
\(926\) 15.0697 0.495221
\(927\) −0.255788 −0.00840119
\(928\) −28.1792 −0.925028
\(929\) −0.373776 −0.0122632 −0.00613160 0.999981i \(-0.501952\pi\)
−0.00613160 + 0.999981i \(0.501952\pi\)
\(930\) 4.00213 0.131235
\(931\) −1.74466 −0.0571790
\(932\) −0.306409 −0.0100368
\(933\) 12.6291 0.413458
\(934\) −14.0282 −0.459016
\(935\) 0 0
\(936\) −4.84293 −0.158296
\(937\) −46.0667 −1.50493 −0.752467 0.658630i \(-0.771136\pi\)
−0.752467 + 0.658630i \(0.771136\pi\)
\(938\) 5.78122 0.188763
\(939\) 1.06646 0.0348025
\(940\) −82.8064 −2.70085
\(941\) −19.3374 −0.630380 −0.315190 0.949029i \(-0.602068\pi\)
−0.315190 + 0.949029i \(0.602068\pi\)
\(942\) 8.03115 0.261669
\(943\) −15.8964 −0.517659
\(944\) 12.2803 0.399691
\(945\) −3.75881 −0.122274
\(946\) 0 0
\(947\) −8.06969 −0.262230 −0.131115 0.991367i \(-0.541856\pi\)
−0.131115 + 0.991367i \(0.541856\pi\)
\(948\) 12.5247 0.406783
\(949\) 42.4563 1.37819
\(950\) −7.54457 −0.244778
\(951\) −25.1710 −0.816225
\(952\) 4.44065 0.143922
\(953\) 34.6358 1.12196 0.560982 0.827828i \(-0.310424\pi\)
0.560982 + 0.827828i \(0.310424\pi\)
\(954\) 4.18741 0.135572
\(955\) −21.6441 −0.700385
\(956\) 18.5557 0.600136
\(957\) 0 0
\(958\) −12.5938 −0.406887
\(959\) 17.1096 0.552498
\(960\) −11.6770 −0.376875
\(961\) −25.9482 −0.837037
\(962\) −9.81659 −0.316499
\(963\) 11.8608 0.382207
\(964\) 6.06935 0.195481
\(965\) −55.9483 −1.80104
\(966\) −1.79937 −0.0578937
\(967\) 8.15074 0.262110 0.131055 0.991375i \(-0.458164\pi\)
0.131055 + 0.991375i \(0.458164\pi\)
\(968\) 0 0
\(969\) −4.33169 −0.139154
\(970\) 12.4578 0.399994
\(971\) 33.4330 1.07292 0.536458 0.843927i \(-0.319762\pi\)
0.536458 + 0.843927i \(0.319762\pi\)
\(972\) −1.77560 −0.0569523
\(973\) −19.5009 −0.625171
\(974\) 6.69560 0.214541
\(975\) 24.7181 0.791613
\(976\) −31.0480 −0.993821
\(977\) −5.48798 −0.175576 −0.0877880 0.996139i \(-0.527980\pi\)
−0.0877880 + 0.996139i \(0.527980\pi\)
\(978\) 9.46188 0.302557
\(979\) 0 0
\(980\) −6.67413 −0.213197
\(981\) −15.8720 −0.506753
\(982\) 7.30398 0.233080
\(983\) 14.5404 0.463767 0.231884 0.972744i \(-0.425511\pi\)
0.231884 + 0.972744i \(0.425511\pi\)
\(984\) 7.48507 0.238615
\(985\) 47.1640 1.50277
\(986\) −6.82235 −0.217268
\(987\) −12.4071 −0.394921
\(988\) 8.38808 0.266860
\(989\) 29.4440 0.936266
\(990\) 0 0
\(991\) −9.68326 −0.307599 −0.153799 0.988102i \(-0.549151\pi\)
−0.153799 + 0.988102i \(0.549151\pi\)
\(992\) 10.9190 0.346677
\(993\) −8.53640 −0.270894
\(994\) −3.67596 −0.116594
\(995\) 3.13642 0.0994312
\(996\) −21.1304 −0.669542
\(997\) −25.2216 −0.798775 −0.399388 0.916782i \(-0.630777\pi\)
−0.399388 + 0.916782i \(0.630777\pi\)
\(998\) 14.5441 0.460387
\(999\) −7.65311 −0.242134
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 2541.2.a.bq.1.6 10
3.2 odd 2 7623.2.a.cx.1.5 10
11.5 even 5 231.2.j.g.190.3 yes 20
11.9 even 5 231.2.j.g.169.3 20
11.10 odd 2 2541.2.a.br.1.5 10
33.5 odd 10 693.2.m.j.190.3 20
33.20 odd 10 693.2.m.j.631.3 20
33.32 even 2 7623.2.a.cy.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.g.169.3 20 11.9 even 5
231.2.j.g.190.3 yes 20 11.5 even 5
693.2.m.j.190.3 20 33.5 odd 10
693.2.m.j.631.3 20 33.20 odd 10
2541.2.a.bq.1.6 10 1.1 even 1 trivial
2541.2.a.br.1.5 10 11.10 odd 2
7623.2.a.cx.1.5 10 3.2 odd 2
7623.2.a.cy.1.6 10 33.32 even 2