Properties

Label 847.2.a.p.1.1
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $0$
Dimension $8$
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] = [847,2,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("847.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.43045\) of defining polynomial
Character \(\chi\) \(=\) 847.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.43045 q^{2} +1.43421 q^{3} +3.90710 q^{4} +1.25863 q^{5} -3.48577 q^{6} +1.00000 q^{7} -4.63512 q^{8} -0.943053 q^{9} +O(q^{10})\) \(q-2.43045 q^{2} +1.43421 q^{3} +3.90710 q^{4} +1.25863 q^{5} -3.48577 q^{6} +1.00000 q^{7} -4.63512 q^{8} -0.943053 q^{9} -3.05904 q^{10} +5.60359 q^{12} -3.17831 q^{13} -2.43045 q^{14} +1.80514 q^{15} +3.45123 q^{16} +5.92418 q^{17} +2.29204 q^{18} +2.86222 q^{19} +4.91760 q^{20} +1.43421 q^{21} +6.76343 q^{23} -6.64771 q^{24} -3.41585 q^{25} +7.72473 q^{26} -5.65515 q^{27} +3.90710 q^{28} +4.49549 q^{29} -4.38730 q^{30} +9.72751 q^{31} +0.882184 q^{32} -14.3984 q^{34} +1.25863 q^{35} -3.68460 q^{36} -5.45495 q^{37} -6.95648 q^{38} -4.55835 q^{39} -5.83390 q^{40} -0.314484 q^{41} -3.48577 q^{42} -0.132562 q^{43} -1.18696 q^{45} -16.4382 q^{46} +9.37505 q^{47} +4.94977 q^{48} +1.00000 q^{49} +8.30206 q^{50} +8.49650 q^{51} -12.4180 q^{52} +4.35192 q^{53} +13.7446 q^{54} -4.63512 q^{56} +4.10501 q^{57} -10.9261 q^{58} -6.94685 q^{59} +7.05285 q^{60} -2.45215 q^{61} -23.6423 q^{62} -0.943053 q^{63} -9.04656 q^{64} -4.00032 q^{65} -9.41987 q^{67} +23.1464 q^{68} +9.70015 q^{69} -3.05904 q^{70} -0.116610 q^{71} +4.37116 q^{72} +0.615032 q^{73} +13.2580 q^{74} -4.89903 q^{75} +11.1830 q^{76} +11.0789 q^{78} +8.52928 q^{79} +4.34382 q^{80} -5.28149 q^{81} +0.764339 q^{82} +0.950532 q^{83} +5.60359 q^{84} +7.45636 q^{85} +0.322186 q^{86} +6.44746 q^{87} +10.0552 q^{89} +2.88484 q^{90} -3.17831 q^{91} +26.4254 q^{92} +13.9513 q^{93} -22.7856 q^{94} +3.60248 q^{95} +1.26523 q^{96} +17.5770 q^{97} -2.43045 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + 6 q^{10} + 9 q^{12} - 6 q^{13} + q^{14} + 11 q^{15} + q^{16} - 5 q^{17} + 8 q^{18} - 13 q^{19} + 23 q^{20} + 4 q^{21} + 16 q^{23} - 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} + 7 q^{28} + 9 q^{29} - 36 q^{30} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} + 13 q^{39} + 5 q^{40} - 10 q^{41} - q^{42} - 4 q^{43} + 35 q^{45} + 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} + 6 q^{50} + 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} + 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} + 14 q^{63} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} - 35 q^{72} - 25 q^{73} + 33 q^{74} - 13 q^{75} + 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} - 25 q^{83} + 9 q^{84} + 3 q^{85} + 4 q^{86} - 36 q^{87} + 37 q^{89} - 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} - 42 q^{94} + 21 q^{95} - 6 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.43045 −1.71859 −0.859295 0.511481i \(-0.829097\pi\)
−0.859295 + 0.511481i \(0.829097\pi\)
\(3\) 1.43421 0.828039 0.414020 0.910268i \(-0.364124\pi\)
0.414020 + 0.910268i \(0.364124\pi\)
\(4\) 3.90710 1.95355
\(5\) 1.25863 0.562877 0.281438 0.959579i \(-0.409188\pi\)
0.281438 + 0.959579i \(0.409188\pi\)
\(6\) −3.48577 −1.42306
\(7\) 1.00000 0.377964
\(8\) −4.63512 −1.63876
\(9\) −0.943053 −0.314351
\(10\) −3.05904 −0.967354
\(11\) 0 0
\(12\) 5.60359 1.61762
\(13\) −3.17831 −0.881504 −0.440752 0.897629i \(-0.645288\pi\)
−0.440752 + 0.897629i \(0.645288\pi\)
\(14\) −2.43045 −0.649566
\(15\) 1.80514 0.466084
\(16\) 3.45123 0.862807
\(17\) 5.92418 1.43682 0.718412 0.695617i \(-0.244869\pi\)
0.718412 + 0.695617i \(0.244869\pi\)
\(18\) 2.29204 0.540240
\(19\) 2.86222 0.656638 0.328319 0.944567i \(-0.393518\pi\)
0.328319 + 0.944567i \(0.393518\pi\)
\(20\) 4.91760 1.09961
\(21\) 1.43421 0.312969
\(22\) 0 0
\(23\) 6.76343 1.41027 0.705136 0.709072i \(-0.250886\pi\)
0.705136 + 0.709072i \(0.250886\pi\)
\(24\) −6.64771 −1.35696
\(25\) −3.41585 −0.683170
\(26\) 7.72473 1.51494
\(27\) −5.65515 −1.08833
\(28\) 3.90710 0.738372
\(29\) 4.49549 0.834792 0.417396 0.908725i \(-0.362943\pi\)
0.417396 + 0.908725i \(0.362943\pi\)
\(30\) −4.38730 −0.801008
\(31\) 9.72751 1.74711 0.873556 0.486723i \(-0.161808\pi\)
0.873556 + 0.486723i \(0.161808\pi\)
\(32\) 0.882184 0.155950
\(33\) 0 0
\(34\) −14.3984 −2.46931
\(35\) 1.25863 0.212747
\(36\) −3.68460 −0.614100
\(37\) −5.45495 −0.896789 −0.448394 0.893836i \(-0.648004\pi\)
−0.448394 + 0.893836i \(0.648004\pi\)
\(38\) −6.95648 −1.12849
\(39\) −4.55835 −0.729920
\(40\) −5.83390 −0.922421
\(41\) −0.314484 −0.0491142 −0.0245571 0.999698i \(-0.507818\pi\)
−0.0245571 + 0.999698i \(0.507818\pi\)
\(42\) −3.48577 −0.537866
\(43\) −0.132562 −0.0202155 −0.0101078 0.999949i \(-0.503217\pi\)
−0.0101078 + 0.999949i \(0.503217\pi\)
\(44\) 0 0
\(45\) −1.18696 −0.176941
\(46\) −16.4382 −2.42368
\(47\) 9.37505 1.36749 0.683746 0.729720i \(-0.260350\pi\)
0.683746 + 0.729720i \(0.260350\pi\)
\(48\) 4.94977 0.714438
\(49\) 1.00000 0.142857
\(50\) 8.30206 1.17409
\(51\) 8.49650 1.18975
\(52\) −12.4180 −1.72206
\(53\) 4.35192 0.597783 0.298891 0.954287i \(-0.403383\pi\)
0.298891 + 0.954287i \(0.403383\pi\)
\(54\) 13.7446 1.87040
\(55\) 0 0
\(56\) −4.63512 −0.619393
\(57\) 4.10501 0.543722
\(58\) −10.9261 −1.43466
\(59\) −6.94685 −0.904403 −0.452202 0.891916i \(-0.649361\pi\)
−0.452202 + 0.891916i \(0.649361\pi\)
\(60\) 7.05285 0.910519
\(61\) −2.45215 −0.313966 −0.156983 0.987601i \(-0.550177\pi\)
−0.156983 + 0.987601i \(0.550177\pi\)
\(62\) −23.6423 −3.00257
\(63\) −0.943053 −0.118813
\(64\) −9.04656 −1.13082
\(65\) −4.00032 −0.496179
\(66\) 0 0
\(67\) −9.41987 −1.15082 −0.575410 0.817865i \(-0.695158\pi\)
−0.575410 + 0.817865i \(0.695158\pi\)
\(68\) 23.1464 2.80691
\(69\) 9.70015 1.16776
\(70\) −3.05904 −0.365626
\(71\) −0.116610 −0.0138391 −0.00691954 0.999976i \(-0.502203\pi\)
−0.00691954 + 0.999976i \(0.502203\pi\)
\(72\) 4.37116 0.515146
\(73\) 0.615032 0.0719840 0.0359920 0.999352i \(-0.488541\pi\)
0.0359920 + 0.999352i \(0.488541\pi\)
\(74\) 13.2580 1.54121
\(75\) −4.89903 −0.565691
\(76\) 11.1830 1.28277
\(77\) 0 0
\(78\) 11.0789 1.25443
\(79\) 8.52928 0.959619 0.479810 0.877373i \(-0.340706\pi\)
0.479810 + 0.877373i \(0.340706\pi\)
\(80\) 4.34382 0.485654
\(81\) −5.28149 −0.586833
\(82\) 0.764339 0.0844072
\(83\) 0.950532 0.104334 0.0521672 0.998638i \(-0.483387\pi\)
0.0521672 + 0.998638i \(0.483387\pi\)
\(84\) 5.60359 0.611401
\(85\) 7.45636 0.808756
\(86\) 0.322186 0.0347422
\(87\) 6.44746 0.691241
\(88\) 0 0
\(89\) 10.0552 1.06585 0.532923 0.846164i \(-0.321094\pi\)
0.532923 + 0.846164i \(0.321094\pi\)
\(90\) 2.88484 0.304089
\(91\) −3.17831 −0.333177
\(92\) 26.4254 2.75504
\(93\) 13.9513 1.44668
\(94\) −22.7856 −2.35016
\(95\) 3.60248 0.369606
\(96\) 1.26523 0.129132
\(97\) 17.5770 1.78467 0.892337 0.451369i \(-0.149064\pi\)
0.892337 + 0.451369i \(0.149064\pi\)
\(98\) −2.43045 −0.245513
\(99\) 0 0
\(100\) −13.3461 −1.33461
\(101\) −12.4761 −1.24142 −0.620711 0.784039i \(-0.713156\pi\)
−0.620711 + 0.784039i \(0.713156\pi\)
\(102\) −20.6503 −2.04469
\(103\) 8.21802 0.809746 0.404873 0.914373i \(-0.367316\pi\)
0.404873 + 0.914373i \(0.367316\pi\)
\(104\) 14.7318 1.44457
\(105\) 1.80514 0.176163
\(106\) −10.5771 −1.02734
\(107\) 12.1885 1.17831 0.589154 0.808020i \(-0.299461\pi\)
0.589154 + 0.808020i \(0.299461\pi\)
\(108\) −22.0952 −2.12612
\(109\) 0.886088 0.0848718 0.0424359 0.999099i \(-0.486488\pi\)
0.0424359 + 0.999099i \(0.486488\pi\)
\(110\) 0 0
\(111\) −7.82353 −0.742577
\(112\) 3.45123 0.326110
\(113\) 4.54502 0.427559 0.213780 0.976882i \(-0.431423\pi\)
0.213780 + 0.976882i \(0.431423\pi\)
\(114\) −9.97703 −0.934435
\(115\) 8.51266 0.793810
\(116\) 17.5643 1.63081
\(117\) 2.99731 0.277102
\(118\) 16.8840 1.55430
\(119\) 5.92418 0.543069
\(120\) −8.36702 −0.763801
\(121\) 0 0
\(122\) 5.95984 0.539579
\(123\) −0.451035 −0.0406685
\(124\) 38.0064 3.41307
\(125\) −10.5924 −0.947417
\(126\) 2.29204 0.204192
\(127\) −8.02779 −0.712351 −0.356175 0.934419i \(-0.615919\pi\)
−0.356175 + 0.934419i \(0.615919\pi\)
\(128\) 20.2229 1.78747
\(129\) −0.190121 −0.0167392
\(130\) 9.72259 0.852727
\(131\) 0.101461 0.00886466 0.00443233 0.999990i \(-0.498589\pi\)
0.00443233 + 0.999990i \(0.498589\pi\)
\(132\) 0 0
\(133\) 2.86222 0.248186
\(134\) 22.8945 1.97779
\(135\) −7.11775 −0.612598
\(136\) −27.4593 −2.35461
\(137\) −4.56409 −0.389937 −0.194968 0.980810i \(-0.562460\pi\)
−0.194968 + 0.980810i \(0.562460\pi\)
\(138\) −23.5758 −2.00690
\(139\) −3.82533 −0.324460 −0.162230 0.986753i \(-0.551869\pi\)
−0.162230 + 0.986753i \(0.551869\pi\)
\(140\) 4.91760 0.415613
\(141\) 13.4458 1.13234
\(142\) 0.283415 0.0237837
\(143\) 0 0
\(144\) −3.25469 −0.271224
\(145\) 5.65817 0.469885
\(146\) −1.49480 −0.123711
\(147\) 1.43421 0.118291
\(148\) −21.3130 −1.75192
\(149\) −4.87223 −0.399149 −0.199574 0.979883i \(-0.563956\pi\)
−0.199574 + 0.979883i \(0.563956\pi\)
\(150\) 11.9069 0.972191
\(151\) −16.9739 −1.38131 −0.690657 0.723183i \(-0.742678\pi\)
−0.690657 + 0.723183i \(0.742678\pi\)
\(152\) −13.2667 −1.07607
\(153\) −5.58681 −0.451667
\(154\) 0 0
\(155\) 12.2434 0.983410
\(156\) −17.8099 −1.42594
\(157\) 2.24483 0.179157 0.0895785 0.995980i \(-0.471448\pi\)
0.0895785 + 0.995980i \(0.471448\pi\)
\(158\) −20.7300 −1.64919
\(159\) 6.24156 0.494988
\(160\) 1.11034 0.0877804
\(161\) 6.76343 0.533033
\(162\) 12.8364 1.00852
\(163\) −15.8372 −1.24047 −0.620235 0.784416i \(-0.712963\pi\)
−0.620235 + 0.784416i \(0.712963\pi\)
\(164\) −1.22872 −0.0959470
\(165\) 0 0
\(166\) −2.31022 −0.179308
\(167\) 20.7416 1.60503 0.802515 0.596632i \(-0.203495\pi\)
0.802515 + 0.596632i \(0.203495\pi\)
\(168\) −6.64771 −0.512882
\(169\) −2.89835 −0.222950
\(170\) −18.1223 −1.38992
\(171\) −2.69922 −0.206415
\(172\) −0.517933 −0.0394920
\(173\) −21.5554 −1.63883 −0.819414 0.573202i \(-0.805701\pi\)
−0.819414 + 0.573202i \(0.805701\pi\)
\(174\) −15.6703 −1.18796
\(175\) −3.41585 −0.258214
\(176\) 0 0
\(177\) −9.96322 −0.748881
\(178\) −24.4386 −1.83175
\(179\) 4.78161 0.357394 0.178697 0.983904i \(-0.442812\pi\)
0.178697 + 0.983904i \(0.442812\pi\)
\(180\) −4.63755 −0.345663
\(181\) −7.85284 −0.583697 −0.291849 0.956465i \(-0.594270\pi\)
−0.291849 + 0.956465i \(0.594270\pi\)
\(182\) 7.72473 0.572595
\(183\) −3.51689 −0.259976
\(184\) −31.3493 −2.31110
\(185\) −6.86578 −0.504782
\(186\) −33.9079 −2.48625
\(187\) 0 0
\(188\) 36.6293 2.67146
\(189\) −5.65515 −0.411352
\(190\) −8.75565 −0.635201
\(191\) −8.83029 −0.638937 −0.319469 0.947597i \(-0.603504\pi\)
−0.319469 + 0.947597i \(0.603504\pi\)
\(192\) −12.9746 −0.936364
\(193\) 25.6023 1.84289 0.921446 0.388507i \(-0.127009\pi\)
0.921446 + 0.388507i \(0.127009\pi\)
\(194\) −42.7201 −3.06712
\(195\) −5.73728 −0.410855
\(196\) 3.90710 0.279079
\(197\) −11.1977 −0.797802 −0.398901 0.916994i \(-0.630608\pi\)
−0.398901 + 0.916994i \(0.630608\pi\)
\(198\) 0 0
\(199\) −12.2503 −0.868400 −0.434200 0.900817i \(-0.642969\pi\)
−0.434200 + 0.900817i \(0.642969\pi\)
\(200\) 15.8328 1.11955
\(201\) −13.5100 −0.952924
\(202\) 30.3227 2.13350
\(203\) 4.49549 0.315522
\(204\) 33.1967 2.32423
\(205\) −0.395820 −0.0276453
\(206\) −19.9735 −1.39162
\(207\) −6.37827 −0.443320
\(208\) −10.9691 −0.760568
\(209\) 0 0
\(210\) −4.38730 −0.302752
\(211\) −14.2636 −0.981946 −0.490973 0.871175i \(-0.663359\pi\)
−0.490973 + 0.871175i \(0.663359\pi\)
\(212\) 17.0034 1.16780
\(213\) −0.167243 −0.0114593
\(214\) −29.6236 −2.02503
\(215\) −0.166847 −0.0113788
\(216\) 26.2123 1.78352
\(217\) 9.72751 0.660347
\(218\) −2.15359 −0.145860
\(219\) 0.882082 0.0596056
\(220\) 0 0
\(221\) −18.8289 −1.26657
\(222\) 19.0147 1.27618
\(223\) −2.96400 −0.198484 −0.0992421 0.995063i \(-0.531642\pi\)
−0.0992421 + 0.995063i \(0.531642\pi\)
\(224\) 0.882184 0.0589434
\(225\) 3.22132 0.214755
\(226\) −11.0465 −0.734799
\(227\) 8.25605 0.547974 0.273987 0.961733i \(-0.411658\pi\)
0.273987 + 0.961733i \(0.411658\pi\)
\(228\) 16.0387 1.06219
\(229\) 13.1081 0.866208 0.433104 0.901344i \(-0.357418\pi\)
0.433104 + 0.901344i \(0.357418\pi\)
\(230\) −20.6896 −1.36423
\(231\) 0 0
\(232\) −20.8371 −1.36802
\(233\) −12.8277 −0.840369 −0.420185 0.907439i \(-0.638035\pi\)
−0.420185 + 0.907439i \(0.638035\pi\)
\(234\) −7.28483 −0.476224
\(235\) 11.7997 0.769730
\(236\) −27.1420 −1.76680
\(237\) 12.2328 0.794603
\(238\) −14.3984 −0.933312
\(239\) −4.98109 −0.322200 −0.161100 0.986938i \(-0.551504\pi\)
−0.161100 + 0.986938i \(0.551504\pi\)
\(240\) 6.22994 0.402141
\(241\) −2.62686 −0.169211 −0.0846053 0.996415i \(-0.526963\pi\)
−0.0846053 + 0.996415i \(0.526963\pi\)
\(242\) 0 0
\(243\) 9.39070 0.602414
\(244\) −9.58081 −0.613349
\(245\) 1.25863 0.0804110
\(246\) 1.09622 0.0698924
\(247\) −9.09701 −0.578829
\(248\) −45.0881 −2.86310
\(249\) 1.36326 0.0863930
\(250\) 25.7444 1.62822
\(251\) −26.2229 −1.65518 −0.827588 0.561337i \(-0.810287\pi\)
−0.827588 + 0.561337i \(0.810287\pi\)
\(252\) −3.68460 −0.232108
\(253\) 0 0
\(254\) 19.5112 1.22424
\(255\) 10.6940 0.669681
\(256\) −31.0576 −1.94110
\(257\) 29.9518 1.86834 0.934170 0.356827i \(-0.116141\pi\)
0.934170 + 0.356827i \(0.116141\pi\)
\(258\) 0.462081 0.0287679
\(259\) −5.45495 −0.338954
\(260\) −15.6296 −0.969309
\(261\) −4.23949 −0.262418
\(262\) −0.246595 −0.0152347
\(263\) 3.33709 0.205774 0.102887 0.994693i \(-0.467192\pi\)
0.102887 + 0.994693i \(0.467192\pi\)
\(264\) 0 0
\(265\) 5.47747 0.336478
\(266\) −6.95648 −0.426529
\(267\) 14.4212 0.882562
\(268\) −36.8044 −2.24818
\(269\) −1.73581 −0.105834 −0.0529172 0.998599i \(-0.516852\pi\)
−0.0529172 + 0.998599i \(0.516852\pi\)
\(270\) 17.2994 1.05280
\(271\) −2.47596 −0.150404 −0.0752019 0.997168i \(-0.523960\pi\)
−0.0752019 + 0.997168i \(0.523960\pi\)
\(272\) 20.4457 1.23970
\(273\) −4.55835 −0.275884
\(274\) 11.0928 0.670141
\(275\) 0 0
\(276\) 37.8995 2.28128
\(277\) 5.15571 0.309776 0.154888 0.987932i \(-0.450498\pi\)
0.154888 + 0.987932i \(0.450498\pi\)
\(278\) 9.29728 0.557614
\(279\) −9.17356 −0.549206
\(280\) −5.83390 −0.348642
\(281\) −22.5705 −1.34644 −0.673222 0.739441i \(-0.735090\pi\)
−0.673222 + 0.739441i \(0.735090\pi\)
\(282\) −32.6793 −1.94602
\(283\) −8.48757 −0.504534 −0.252267 0.967658i \(-0.581176\pi\)
−0.252267 + 0.967658i \(0.581176\pi\)
\(284\) −0.455608 −0.0270353
\(285\) 5.16669 0.306049
\(286\) 0 0
\(287\) −0.314484 −0.0185634
\(288\) −0.831946 −0.0490229
\(289\) 18.0959 1.06447
\(290\) −13.7519 −0.807540
\(291\) 25.2090 1.47778
\(292\) 2.40299 0.140624
\(293\) 24.6261 1.43867 0.719336 0.694662i \(-0.244446\pi\)
0.719336 + 0.694662i \(0.244446\pi\)
\(294\) −3.48577 −0.203294
\(295\) −8.74353 −0.509068
\(296\) 25.2843 1.46962
\(297\) 0 0
\(298\) 11.8417 0.685973
\(299\) −21.4963 −1.24316
\(300\) −19.1410 −1.10511
\(301\) −0.132562 −0.00764075
\(302\) 41.2542 2.37391
\(303\) −17.8934 −1.02795
\(304\) 9.87817 0.566552
\(305\) −3.08636 −0.176724
\(306\) 13.5785 0.776230
\(307\) 4.59391 0.262188 0.131094 0.991370i \(-0.458151\pi\)
0.131094 + 0.991370i \(0.458151\pi\)
\(308\) 0 0
\(309\) 11.7863 0.670502
\(310\) −29.7569 −1.69008
\(311\) −2.21073 −0.125359 −0.0626796 0.998034i \(-0.519965\pi\)
−0.0626796 + 0.998034i \(0.519965\pi\)
\(312\) 21.1285 1.19616
\(313\) −9.69928 −0.548236 −0.274118 0.961696i \(-0.588386\pi\)
−0.274118 + 0.961696i \(0.588386\pi\)
\(314\) −5.45595 −0.307897
\(315\) −1.18696 −0.0668774
\(316\) 33.3248 1.87466
\(317\) 6.45539 0.362571 0.181286 0.983431i \(-0.441974\pi\)
0.181286 + 0.983431i \(0.441974\pi\)
\(318\) −15.1698 −0.850680
\(319\) 0 0
\(320\) −11.3863 −0.636513
\(321\) 17.4809 0.975686
\(322\) −16.4382 −0.916065
\(323\) 16.9563 0.943474
\(324\) −20.6353 −1.14641
\(325\) 10.8566 0.602217
\(326\) 38.4917 2.13186
\(327\) 1.27083 0.0702772
\(328\) 1.45767 0.0804864
\(329\) 9.37505 0.516863
\(330\) 0 0
\(331\) 3.62076 0.199015 0.0995075 0.995037i \(-0.468273\pi\)
0.0995075 + 0.995037i \(0.468273\pi\)
\(332\) 3.71382 0.203823
\(333\) 5.14431 0.281906
\(334\) −50.4114 −2.75839
\(335\) −11.8561 −0.647770
\(336\) 4.94977 0.270032
\(337\) −6.40510 −0.348908 −0.174454 0.984665i \(-0.555816\pi\)
−0.174454 + 0.984665i \(0.555816\pi\)
\(338\) 7.04430 0.383159
\(339\) 6.51849 0.354036
\(340\) 29.1327 1.57994
\(341\) 0 0
\(342\) 6.56033 0.354742
\(343\) 1.00000 0.0539949
\(344\) 0.614440 0.0331284
\(345\) 12.2089 0.657306
\(346\) 52.3894 2.81647
\(347\) −17.4600 −0.937302 −0.468651 0.883383i \(-0.655260\pi\)
−0.468651 + 0.883383i \(0.655260\pi\)
\(348\) 25.1909 1.35037
\(349\) −21.6249 −1.15755 −0.578777 0.815486i \(-0.696470\pi\)
−0.578777 + 0.815486i \(0.696470\pi\)
\(350\) 8.30206 0.443764
\(351\) 17.9738 0.959371
\(352\) 0 0
\(353\) 4.40150 0.234268 0.117134 0.993116i \(-0.462629\pi\)
0.117134 + 0.993116i \(0.462629\pi\)
\(354\) 24.2151 1.28702
\(355\) −0.146769 −0.00778970
\(356\) 39.2865 2.08218
\(357\) 8.49650 0.449682
\(358\) −11.6215 −0.614214
\(359\) −10.6817 −0.563760 −0.281880 0.959450i \(-0.590958\pi\)
−0.281880 + 0.959450i \(0.590958\pi\)
\(360\) 5.50167 0.289964
\(361\) −10.8077 −0.568827
\(362\) 19.0860 1.00314
\(363\) 0 0
\(364\) −12.4180 −0.650879
\(365\) 0.774098 0.0405181
\(366\) 8.54765 0.446793
\(367\) 10.2167 0.533309 0.266655 0.963792i \(-0.414082\pi\)
0.266655 + 0.963792i \(0.414082\pi\)
\(368\) 23.3421 1.21679
\(369\) 0.296575 0.0154391
\(370\) 16.6869 0.867513
\(371\) 4.35192 0.225941
\(372\) 54.5090 2.82616
\(373\) −27.8851 −1.44383 −0.721917 0.691980i \(-0.756739\pi\)
−0.721917 + 0.691980i \(0.756739\pi\)
\(374\) 0 0
\(375\) −15.1918 −0.784499
\(376\) −43.4544 −2.24099
\(377\) −14.2881 −0.735873
\(378\) 13.7446 0.706945
\(379\) 1.14803 0.0589703 0.0294852 0.999565i \(-0.490613\pi\)
0.0294852 + 0.999565i \(0.490613\pi\)
\(380\) 14.0752 0.722044
\(381\) −11.5135 −0.589855
\(382\) 21.4616 1.09807
\(383\) −19.9516 −1.01948 −0.509740 0.860328i \(-0.670258\pi\)
−0.509740 + 0.860328i \(0.670258\pi\)
\(384\) 29.0038 1.48009
\(385\) 0 0
\(386\) −62.2251 −3.16717
\(387\) 0.125013 0.00635476
\(388\) 68.6751 3.48645
\(389\) 2.60000 0.131825 0.0659127 0.997825i \(-0.479004\pi\)
0.0659127 + 0.997825i \(0.479004\pi\)
\(390\) 13.9442 0.706092
\(391\) 40.0678 2.02631
\(392\) −4.63512 −0.234109
\(393\) 0.145516 0.00734029
\(394\) 27.2154 1.37109
\(395\) 10.7352 0.540148
\(396\) 0 0
\(397\) 8.77237 0.440272 0.220136 0.975469i \(-0.429350\pi\)
0.220136 + 0.975469i \(0.429350\pi\)
\(398\) 29.7737 1.49242
\(399\) 4.10501 0.205508
\(400\) −11.7889 −0.589444
\(401\) 28.3535 1.41591 0.707953 0.706260i \(-0.249619\pi\)
0.707953 + 0.706260i \(0.249619\pi\)
\(402\) 32.8355 1.63769
\(403\) −30.9170 −1.54009
\(404\) −48.7455 −2.42518
\(405\) −6.64745 −0.330315
\(406\) −10.9261 −0.542252
\(407\) 0 0
\(408\) −39.3822 −1.94971
\(409\) 19.9055 0.984264 0.492132 0.870521i \(-0.336218\pi\)
0.492132 + 0.870521i \(0.336218\pi\)
\(410\) 0.962021 0.0475108
\(411\) −6.54585 −0.322883
\(412\) 32.1086 1.58188
\(413\) −6.94685 −0.341832
\(414\) 15.5021 0.761886
\(415\) 1.19637 0.0587275
\(416\) −2.80385 −0.137470
\(417\) −5.48631 −0.268666
\(418\) 0 0
\(419\) −30.8957 −1.50935 −0.754676 0.656097i \(-0.772206\pi\)
−0.754676 + 0.656097i \(0.772206\pi\)
\(420\) 7.05285 0.344144
\(421\) −24.1931 −1.17910 −0.589551 0.807732i \(-0.700695\pi\)
−0.589551 + 0.807732i \(0.700695\pi\)
\(422\) 34.6670 1.68756
\(423\) −8.84117 −0.429872
\(424\) −20.1717 −0.979623
\(425\) −20.2361 −0.981595
\(426\) 0.406476 0.0196938
\(427\) −2.45215 −0.118668
\(428\) 47.6218 2.30188
\(429\) 0 0
\(430\) 0.405513 0.0195556
\(431\) 19.4001 0.934468 0.467234 0.884134i \(-0.345251\pi\)
0.467234 + 0.884134i \(0.345251\pi\)
\(432\) −19.5172 −0.939022
\(433\) −21.3877 −1.02783 −0.513914 0.857842i \(-0.671805\pi\)
−0.513914 + 0.857842i \(0.671805\pi\)
\(434\) −23.6423 −1.13486
\(435\) 8.11498 0.389083
\(436\) 3.46203 0.165801
\(437\) 19.3584 0.926038
\(438\) −2.14386 −0.102438
\(439\) −31.7315 −1.51446 −0.757232 0.653146i \(-0.773449\pi\)
−0.757232 + 0.653146i \(0.773449\pi\)
\(440\) 0 0
\(441\) −0.943053 −0.0449073
\(442\) 45.7627 2.17671
\(443\) −1.66610 −0.0791590 −0.0395795 0.999216i \(-0.512602\pi\)
−0.0395795 + 0.999216i \(0.512602\pi\)
\(444\) −30.5673 −1.45066
\(445\) 12.6557 0.599940
\(446\) 7.20386 0.341113
\(447\) −6.98779 −0.330511
\(448\) −9.04656 −0.427410
\(449\) −17.1858 −0.811047 −0.405524 0.914085i \(-0.632911\pi\)
−0.405524 + 0.914085i \(0.632911\pi\)
\(450\) −7.82927 −0.369076
\(451\) 0 0
\(452\) 17.7578 0.835258
\(453\) −24.3440 −1.14378
\(454\) −20.0659 −0.941742
\(455\) −4.00032 −0.187538
\(456\) −19.0272 −0.891030
\(457\) 10.2206 0.478101 0.239051 0.971007i \(-0.423164\pi\)
0.239051 + 0.971007i \(0.423164\pi\)
\(458\) −31.8586 −1.48866
\(459\) −33.5021 −1.56375
\(460\) 33.2598 1.55075
\(461\) −22.1160 −1.03004 −0.515022 0.857177i \(-0.672216\pi\)
−0.515022 + 0.857177i \(0.672216\pi\)
\(462\) 0 0
\(463\) −30.3717 −1.41149 −0.705747 0.708464i \(-0.749389\pi\)
−0.705747 + 0.708464i \(0.749389\pi\)
\(464\) 15.5150 0.720265
\(465\) 17.5595 0.814302
\(466\) 31.1771 1.44425
\(467\) 25.1909 1.16570 0.582848 0.812581i \(-0.301938\pi\)
0.582848 + 0.812581i \(0.301938\pi\)
\(468\) 11.7108 0.541332
\(469\) −9.41987 −0.434969
\(470\) −28.6787 −1.32285
\(471\) 3.21955 0.148349
\(472\) 32.1995 1.48210
\(473\) 0 0
\(474\) −29.7311 −1.36560
\(475\) −9.77690 −0.448595
\(476\) 23.1464 1.06091
\(477\) −4.10409 −0.187913
\(478\) 12.1063 0.553729
\(479\) −21.2040 −0.968835 −0.484417 0.874837i \(-0.660968\pi\)
−0.484417 + 0.874837i \(0.660968\pi\)
\(480\) 1.59246 0.0726856
\(481\) 17.3375 0.790523
\(482\) 6.38445 0.290804
\(483\) 9.70015 0.441372
\(484\) 0 0
\(485\) 22.1230 1.00455
\(486\) −22.8237 −1.03530
\(487\) −16.8198 −0.762176 −0.381088 0.924539i \(-0.624451\pi\)
−0.381088 + 0.924539i \(0.624451\pi\)
\(488\) 11.3660 0.514515
\(489\) −22.7139 −1.02716
\(490\) −3.05904 −0.138193
\(491\) 4.83804 0.218338 0.109169 0.994023i \(-0.465181\pi\)
0.109169 + 0.994023i \(0.465181\pi\)
\(492\) −1.76224 −0.0794479
\(493\) 26.6321 1.19945
\(494\) 22.1099 0.994770
\(495\) 0 0
\(496\) 33.5719 1.50742
\(497\) −0.116610 −0.00523068
\(498\) −3.31334 −0.148474
\(499\) 30.5836 1.36911 0.684554 0.728962i \(-0.259997\pi\)
0.684554 + 0.728962i \(0.259997\pi\)
\(500\) −41.3858 −1.85083
\(501\) 29.7477 1.32903
\(502\) 63.7335 2.84457
\(503\) −28.2407 −1.25919 −0.629594 0.776924i \(-0.716779\pi\)
−0.629594 + 0.776924i \(0.716779\pi\)
\(504\) 4.37116 0.194707
\(505\) −15.7029 −0.698768
\(506\) 0 0
\(507\) −4.15683 −0.184611
\(508\) −31.3654 −1.39161
\(509\) −4.20065 −0.186190 −0.0930952 0.995657i \(-0.529676\pi\)
−0.0930952 + 0.995657i \(0.529676\pi\)
\(510\) −25.9912 −1.15091
\(511\) 0.615032 0.0272074
\(512\) 35.0383 1.54849
\(513\) −16.1863 −0.714641
\(514\) −72.7964 −3.21091
\(515\) 10.3435 0.455787
\(516\) −0.742823 −0.0327009
\(517\) 0 0
\(518\) 13.2580 0.582523
\(519\) −30.9149 −1.35701
\(520\) 18.5419 0.813118
\(521\) −20.6470 −0.904560 −0.452280 0.891876i \(-0.649389\pi\)
−0.452280 + 0.891876i \(0.649389\pi\)
\(522\) 10.3039 0.450988
\(523\) −22.6341 −0.989719 −0.494859 0.868973i \(-0.664780\pi\)
−0.494859 + 0.868973i \(0.664780\pi\)
\(524\) 0.396417 0.0173176
\(525\) −4.89903 −0.213811
\(526\) −8.11063 −0.353640
\(527\) 57.6275 2.51030
\(528\) 0 0
\(529\) 22.7440 0.988869
\(530\) −13.3127 −0.578268
\(531\) 6.55125 0.284300
\(532\) 11.1830 0.484843
\(533\) 0.999529 0.0432944
\(534\) −35.0500 −1.51676
\(535\) 15.3409 0.663243
\(536\) 43.6622 1.88592
\(537\) 6.85781 0.295936
\(538\) 4.21881 0.181886
\(539\) 0 0
\(540\) −27.8098 −1.19674
\(541\) 42.7163 1.83652 0.918258 0.395982i \(-0.129596\pi\)
0.918258 + 0.395982i \(0.129596\pi\)
\(542\) 6.01770 0.258482
\(543\) −11.2626 −0.483324
\(544\) 5.22622 0.224072
\(545\) 1.11526 0.0477724
\(546\) 11.0789 0.474131
\(547\) 44.4827 1.90194 0.950972 0.309278i \(-0.100087\pi\)
0.950972 + 0.309278i \(0.100087\pi\)
\(548\) −17.8324 −0.761761
\(549\) 2.31251 0.0986955
\(550\) 0 0
\(551\) 12.8671 0.548156
\(552\) −44.9613 −1.91368
\(553\) 8.52928 0.362702
\(554\) −12.5307 −0.532379
\(555\) −9.84694 −0.417979
\(556\) −14.9459 −0.633849
\(557\) −24.9911 −1.05891 −0.529454 0.848339i \(-0.677603\pi\)
−0.529454 + 0.848339i \(0.677603\pi\)
\(558\) 22.2959 0.943860
\(559\) 0.421323 0.0178201
\(560\) 4.34382 0.183560
\(561\) 0 0
\(562\) 54.8565 2.31398
\(563\) 9.01005 0.379729 0.189864 0.981810i \(-0.439195\pi\)
0.189864 + 0.981810i \(0.439195\pi\)
\(564\) 52.5339 2.21208
\(565\) 5.72050 0.240663
\(566\) 20.6286 0.867086
\(567\) −5.28149 −0.221802
\(568\) 0.540502 0.0226789
\(569\) 29.1738 1.22303 0.611516 0.791232i \(-0.290560\pi\)
0.611516 + 0.791232i \(0.290560\pi\)
\(570\) −12.5574 −0.525972
\(571\) 1.78994 0.0749067 0.0374533 0.999298i \(-0.488075\pi\)
0.0374533 + 0.999298i \(0.488075\pi\)
\(572\) 0 0
\(573\) −12.6645 −0.529065
\(574\) 0.764339 0.0319029
\(575\) −23.1028 −0.963455
\(576\) 8.53138 0.355474
\(577\) −21.5850 −0.898594 −0.449297 0.893382i \(-0.648325\pi\)
−0.449297 + 0.893382i \(0.648325\pi\)
\(578\) −43.9813 −1.82938
\(579\) 36.7189 1.52599
\(580\) 22.1070 0.917944
\(581\) 0.950532 0.0394347
\(582\) −61.2694 −2.53970
\(583\) 0 0
\(584\) −2.85074 −0.117965
\(585\) 3.77251 0.155974
\(586\) −59.8526 −2.47249
\(587\) 5.52355 0.227981 0.113991 0.993482i \(-0.463637\pi\)
0.113991 + 0.993482i \(0.463637\pi\)
\(588\) 5.60359 0.231088
\(589\) 27.8423 1.14722
\(590\) 21.2507 0.874879
\(591\) −16.0598 −0.660611
\(592\) −18.8263 −0.773756
\(593\) −40.7867 −1.67491 −0.837454 0.546508i \(-0.815957\pi\)
−0.837454 + 0.546508i \(0.815957\pi\)
\(594\) 0 0
\(595\) 7.45636 0.305681
\(596\) −19.0363 −0.779757
\(597\) −17.5694 −0.719069
\(598\) 52.2457 2.13648
\(599\) −26.5149 −1.08337 −0.541685 0.840582i \(-0.682213\pi\)
−0.541685 + 0.840582i \(0.682213\pi\)
\(600\) 22.7076 0.927033
\(601\) −37.7699 −1.54067 −0.770333 0.637641i \(-0.779910\pi\)
−0.770333 + 0.637641i \(0.779910\pi\)
\(602\) 0.322186 0.0131313
\(603\) 8.88343 0.361761
\(604\) −66.3186 −2.69847
\(605\) 0 0
\(606\) 43.4890 1.76662
\(607\) −27.7244 −1.12530 −0.562649 0.826696i \(-0.690218\pi\)
−0.562649 + 0.826696i \(0.690218\pi\)
\(608\) 2.52500 0.102402
\(609\) 6.44746 0.261264
\(610\) 7.50125 0.303717
\(611\) −29.7968 −1.20545
\(612\) −21.8282 −0.882354
\(613\) 36.2264 1.46317 0.731586 0.681749i \(-0.238780\pi\)
0.731586 + 0.681749i \(0.238780\pi\)
\(614\) −11.1653 −0.450594
\(615\) −0.567687 −0.0228914
\(616\) 0 0
\(617\) 41.1920 1.65833 0.829163 0.559007i \(-0.188817\pi\)
0.829163 + 0.559007i \(0.188817\pi\)
\(618\) −28.6461 −1.15232
\(619\) −35.7395 −1.43649 −0.718247 0.695789i \(-0.755055\pi\)
−0.718247 + 0.695789i \(0.755055\pi\)
\(620\) 47.8360 1.92114
\(621\) −38.2482 −1.53485
\(622\) 5.37308 0.215441
\(623\) 10.0552 0.402852
\(624\) −15.7319 −0.629780
\(625\) 3.74725 0.149890
\(626\) 23.5736 0.942192
\(627\) 0 0
\(628\) 8.77077 0.349992
\(629\) −32.3161 −1.28853
\(630\) 2.88484 0.114935
\(631\) 2.76508 0.110076 0.0550380 0.998484i \(-0.482472\pi\)
0.0550380 + 0.998484i \(0.482472\pi\)
\(632\) −39.5342 −1.57259
\(633\) −20.4569 −0.813090
\(634\) −15.6895 −0.623111
\(635\) −10.1040 −0.400966
\(636\) 24.3864 0.966983
\(637\) −3.17831 −0.125929
\(638\) 0 0
\(639\) 0.109970 0.00435033
\(640\) 25.4531 1.00612
\(641\) 38.4864 1.52012 0.760061 0.649851i \(-0.225169\pi\)
0.760061 + 0.649851i \(0.225169\pi\)
\(642\) −42.4864 −1.67680
\(643\) −26.3781 −1.04025 −0.520126 0.854089i \(-0.674115\pi\)
−0.520126 + 0.854089i \(0.674115\pi\)
\(644\) 26.4254 1.04131
\(645\) −0.239293 −0.00942213
\(646\) −41.2115 −1.62144
\(647\) −27.3211 −1.07410 −0.537051 0.843549i \(-0.680462\pi\)
−0.537051 + 0.843549i \(0.680462\pi\)
\(648\) 24.4803 0.961678
\(649\) 0 0
\(650\) −26.3865 −1.03496
\(651\) 13.9513 0.546793
\(652\) −61.8777 −2.42332
\(653\) −8.46165 −0.331130 −0.165565 0.986199i \(-0.552945\pi\)
−0.165565 + 0.986199i \(0.552945\pi\)
\(654\) −3.08870 −0.120778
\(655\) 0.127702 0.00498971
\(656\) −1.08536 −0.0423761
\(657\) −0.580007 −0.0226282
\(658\) −22.7856 −0.888276
\(659\) −5.29247 −0.206165 −0.103083 0.994673i \(-0.532871\pi\)
−0.103083 + 0.994673i \(0.532871\pi\)
\(660\) 0 0
\(661\) −19.2700 −0.749517 −0.374759 0.927122i \(-0.622274\pi\)
−0.374759 + 0.927122i \(0.622274\pi\)
\(662\) −8.80008 −0.342025
\(663\) −27.0045 −1.04877
\(664\) −4.40583 −0.170979
\(665\) 3.60248 0.139698
\(666\) −12.5030 −0.484481
\(667\) 30.4050 1.17728
\(668\) 81.0393 3.13551
\(669\) −4.25099 −0.164353
\(670\) 28.8158 1.11325
\(671\) 0 0
\(672\) 1.26523 0.0488074
\(673\) −18.9922 −0.732096 −0.366048 0.930596i \(-0.619289\pi\)
−0.366048 + 0.930596i \(0.619289\pi\)
\(674\) 15.5673 0.599629
\(675\) 19.3171 0.743517
\(676\) −11.3241 −0.435544
\(677\) 32.4219 1.24607 0.623037 0.782192i \(-0.285899\pi\)
0.623037 + 0.782192i \(0.285899\pi\)
\(678\) −15.8429 −0.608442
\(679\) 17.5770 0.674544
\(680\) −34.5611 −1.32536
\(681\) 11.8409 0.453744
\(682\) 0 0
\(683\) 15.1260 0.578779 0.289389 0.957211i \(-0.406548\pi\)
0.289389 + 0.957211i \(0.406548\pi\)
\(684\) −10.5461 −0.403241
\(685\) −5.74451 −0.219486
\(686\) −2.43045 −0.0927951
\(687\) 18.7997 0.717255
\(688\) −0.457502 −0.0174421
\(689\) −13.8318 −0.526948
\(690\) −29.6732 −1.12964
\(691\) −31.4108 −1.19492 −0.597461 0.801898i \(-0.703824\pi\)
−0.597461 + 0.801898i \(0.703824\pi\)
\(692\) −84.2192 −3.20153
\(693\) 0 0
\(694\) 42.4357 1.61084
\(695\) −4.81468 −0.182631
\(696\) −29.8847 −1.13278
\(697\) −1.86306 −0.0705685
\(698\) 52.5583 1.98936
\(699\) −18.3975 −0.695859
\(700\) −13.3461 −0.504434
\(701\) −16.0647 −0.606755 −0.303377 0.952870i \(-0.598114\pi\)
−0.303377 + 0.952870i \(0.598114\pi\)
\(702\) −43.6845 −1.64877
\(703\) −15.6133 −0.588865
\(704\) 0 0
\(705\) 16.9232 0.637366
\(706\) −10.6976 −0.402611
\(707\) −12.4761 −0.469214
\(708\) −38.9273 −1.46298
\(709\) 39.6978 1.49088 0.745441 0.666572i \(-0.232239\pi\)
0.745441 + 0.666572i \(0.232239\pi\)
\(710\) 0.356716 0.0133873
\(711\) −8.04356 −0.301657
\(712\) −46.6069 −1.74667
\(713\) 65.7913 2.46391
\(714\) −20.6503 −0.772819
\(715\) 0 0
\(716\) 18.6822 0.698187
\(717\) −7.14391 −0.266794
\(718\) 25.9614 0.968873
\(719\) −9.86241 −0.367806 −0.183903 0.982944i \(-0.558873\pi\)
−0.183903 + 0.982944i \(0.558873\pi\)
\(720\) −4.09645 −0.152666
\(721\) 8.21802 0.306055
\(722\) 26.2676 0.977580
\(723\) −3.76745 −0.140113
\(724\) −30.6818 −1.14028
\(725\) −15.3559 −0.570305
\(726\) 0 0
\(727\) −31.5764 −1.17111 −0.585553 0.810634i \(-0.699122\pi\)
−0.585553 + 0.810634i \(0.699122\pi\)
\(728\) 14.7318 0.545998
\(729\) 29.3127 1.08565
\(730\) −1.88141 −0.0696340
\(731\) −0.785321 −0.0290462
\(732\) −13.7409 −0.507877
\(733\) 41.2926 1.52518 0.762588 0.646885i \(-0.223928\pi\)
0.762588 + 0.646885i \(0.223928\pi\)
\(734\) −24.8313 −0.916540
\(735\) 1.80514 0.0665835
\(736\) 5.96659 0.219931
\(737\) 0 0
\(738\) −0.720812 −0.0265335
\(739\) 35.4095 1.30256 0.651281 0.758837i \(-0.274232\pi\)
0.651281 + 0.758837i \(0.274232\pi\)
\(740\) −26.8253 −0.986116
\(741\) −13.0470 −0.479293
\(742\) −10.5771 −0.388299
\(743\) −4.02408 −0.147629 −0.0738146 0.997272i \(-0.523517\pi\)
−0.0738146 + 0.997272i \(0.523517\pi\)
\(744\) −64.6657 −2.37076
\(745\) −6.13235 −0.224672
\(746\) 67.7733 2.48136
\(747\) −0.896402 −0.0327976
\(748\) 0 0
\(749\) 12.1885 0.445359
\(750\) 36.9228 1.34823
\(751\) 24.4190 0.891062 0.445531 0.895267i \(-0.353015\pi\)
0.445531 + 0.895267i \(0.353015\pi\)
\(752\) 32.3554 1.17988
\(753\) −37.6091 −1.37055
\(754\) 34.7265 1.26466
\(755\) −21.3638 −0.777510
\(756\) −22.0952 −0.803596
\(757\) −1.31723 −0.0478755 −0.0239377 0.999713i \(-0.507620\pi\)
−0.0239377 + 0.999713i \(0.507620\pi\)
\(758\) −2.79023 −0.101346
\(759\) 0 0
\(760\) −16.6979 −0.605696
\(761\) −7.24404 −0.262596 −0.131298 0.991343i \(-0.541915\pi\)
−0.131298 + 0.991343i \(0.541915\pi\)
\(762\) 27.9830 1.01372
\(763\) 0.886088 0.0320785
\(764\) −34.5008 −1.24820
\(765\) −7.03174 −0.254233
\(766\) 48.4915 1.75207
\(767\) 22.0792 0.797235
\(768\) −44.5430 −1.60731
\(769\) 44.3139 1.59800 0.798999 0.601332i \(-0.205363\pi\)
0.798999 + 0.601332i \(0.205363\pi\)
\(770\) 0 0
\(771\) 42.9570 1.54706
\(772\) 100.031 3.60018
\(773\) 19.0559 0.685395 0.342697 0.939446i \(-0.388659\pi\)
0.342697 + 0.939446i \(0.388659\pi\)
\(774\) −0.303838 −0.0109212
\(775\) −33.2277 −1.19357
\(776\) −81.4714 −2.92465
\(777\) −7.82353 −0.280668
\(778\) −6.31918 −0.226554
\(779\) −0.900123 −0.0322502
\(780\) −22.4161 −0.802626
\(781\) 0 0
\(782\) −97.3828 −3.48240
\(783\) −25.4227 −0.908533
\(784\) 3.45123 0.123258
\(785\) 2.82541 0.100843
\(786\) −0.353669 −0.0126149
\(787\) 31.4600 1.12143 0.560714 0.828009i \(-0.310527\pi\)
0.560714 + 0.828009i \(0.310527\pi\)
\(788\) −43.7505 −1.55855
\(789\) 4.78607 0.170389
\(790\) −26.0914 −0.928292
\(791\) 4.54502 0.161602
\(792\) 0 0
\(793\) 7.79370 0.276763
\(794\) −21.3208 −0.756648
\(795\) 7.85582 0.278617
\(796\) −47.8631 −1.69646
\(797\) −12.5453 −0.444376 −0.222188 0.975004i \(-0.571320\pi\)
−0.222188 + 0.975004i \(0.571320\pi\)
\(798\) −9.97703 −0.353183
\(799\) 55.5395 1.96485
\(800\) −3.01340 −0.106540
\(801\) −9.48255 −0.335049
\(802\) −68.9118 −2.43336
\(803\) 0 0
\(804\) −52.7850 −1.86159
\(805\) 8.51266 0.300032
\(806\) 75.1424 2.64678
\(807\) −2.48951 −0.0876350
\(808\) 57.8284 2.03440
\(809\) −5.38024 −0.189159 −0.0945796 0.995517i \(-0.530151\pi\)
−0.0945796 + 0.995517i \(0.530151\pi\)
\(810\) 16.1563 0.567675
\(811\) 13.6999 0.481070 0.240535 0.970641i \(-0.422677\pi\)
0.240535 + 0.970641i \(0.422677\pi\)
\(812\) 17.5643 0.616387
\(813\) −3.55103 −0.124540
\(814\) 0 0
\(815\) −19.9333 −0.698231
\(816\) 29.3234 1.02652
\(817\) −0.379421 −0.0132743
\(818\) −48.3794 −1.69155
\(819\) 2.99731 0.104735
\(820\) −1.54651 −0.0540064
\(821\) −9.23116 −0.322170 −0.161085 0.986941i \(-0.551499\pi\)
−0.161085 + 0.986941i \(0.551499\pi\)
\(822\) 15.9094 0.554903
\(823\) −12.4427 −0.433724 −0.216862 0.976202i \(-0.569582\pi\)
−0.216862 + 0.976202i \(0.569582\pi\)
\(824\) −38.0915 −1.32698
\(825\) 0 0
\(826\) 16.8840 0.587469
\(827\) −4.48387 −0.155919 −0.0779597 0.996957i \(-0.524841\pi\)
−0.0779597 + 0.996957i \(0.524841\pi\)
\(828\) −24.9205 −0.866048
\(829\) −36.8776 −1.28081 −0.640405 0.768037i \(-0.721234\pi\)
−0.640405 + 0.768037i \(0.721234\pi\)
\(830\) −2.90772 −0.100928
\(831\) 7.39435 0.256507
\(832\) 28.7528 0.996823
\(833\) 5.92418 0.205261
\(834\) 13.3342 0.461726
\(835\) 26.1060 0.903435
\(836\) 0 0
\(837\) −55.0105 −1.90144
\(838\) 75.0905 2.59396
\(839\) 9.16407 0.316379 0.158189 0.987409i \(-0.449434\pi\)
0.158189 + 0.987409i \(0.449434\pi\)
\(840\) −8.36702 −0.288689
\(841\) −8.79054 −0.303122
\(842\) 58.8003 2.02639
\(843\) −32.3708 −1.11491
\(844\) −55.7293 −1.91828
\(845\) −3.64795 −0.125493
\(846\) 21.4880 0.738774
\(847\) 0 0
\(848\) 15.0195 0.515771
\(849\) −12.1729 −0.417774
\(850\) 49.1829 1.68696
\(851\) −36.8942 −1.26472
\(852\) −0.653435 −0.0223863
\(853\) 6.52049 0.223257 0.111629 0.993750i \(-0.464393\pi\)
0.111629 + 0.993750i \(0.464393\pi\)
\(854\) 5.95984 0.203942
\(855\) −3.39732 −0.116186
\(856\) −56.4952 −1.93097
\(857\) 48.0736 1.64216 0.821082 0.570810i \(-0.193371\pi\)
0.821082 + 0.570810i \(0.193371\pi\)
\(858\) 0 0
\(859\) −0.316298 −0.0107920 −0.00539598 0.999985i \(-0.501718\pi\)
−0.00539598 + 0.999985i \(0.501718\pi\)
\(860\) −0.651887 −0.0222291
\(861\) −0.451035 −0.0153712
\(862\) −47.1509 −1.60597
\(863\) 3.62693 0.123462 0.0617311 0.998093i \(-0.480338\pi\)
0.0617311 + 0.998093i \(0.480338\pi\)
\(864\) −4.98888 −0.169725
\(865\) −27.1303 −0.922458
\(866\) 51.9818 1.76641
\(867\) 25.9533 0.881420
\(868\) 38.0064 1.29002
\(869\) 0 0
\(870\) −19.7231 −0.668675
\(871\) 29.9393 1.01445
\(872\) −4.10712 −0.139085
\(873\) −16.5760 −0.561014
\(874\) −47.0497 −1.59148
\(875\) −10.5924 −0.358090
\(876\) 3.44638 0.116442
\(877\) 26.7076 0.901852 0.450926 0.892561i \(-0.351094\pi\)
0.450926 + 0.892561i \(0.351094\pi\)
\(878\) 77.1220 2.60274
\(879\) 35.3189 1.19128
\(880\) 0 0
\(881\) 2.91937 0.0983560 0.0491780 0.998790i \(-0.484340\pi\)
0.0491780 + 0.998790i \(0.484340\pi\)
\(882\) 2.29204 0.0771772
\(883\) −45.1574 −1.51967 −0.759834 0.650117i \(-0.774720\pi\)
−0.759834 + 0.650117i \(0.774720\pi\)
\(884\) −73.5663 −2.47430
\(885\) −12.5400 −0.421528
\(886\) 4.04939 0.136042
\(887\) −24.0249 −0.806676 −0.403338 0.915051i \(-0.632150\pi\)
−0.403338 + 0.915051i \(0.632150\pi\)
\(888\) 36.2630 1.21691
\(889\) −8.02779 −0.269243
\(890\) −30.7592 −1.03105
\(891\) 0 0
\(892\) −11.5806 −0.387749
\(893\) 26.8334 0.897947
\(894\) 16.9835 0.568013
\(895\) 6.01828 0.201169
\(896\) 20.2229 0.675599
\(897\) −30.8301 −1.02939
\(898\) 41.7692 1.39386
\(899\) 43.7300 1.45848
\(900\) 12.5860 0.419534
\(901\) 25.7816 0.858909
\(902\) 0 0
\(903\) −0.190121 −0.00632684
\(904\) −21.0667 −0.700667
\(905\) −9.88384 −0.328550
\(906\) 59.1670 1.96569
\(907\) −16.0270 −0.532166 −0.266083 0.963950i \(-0.585730\pi\)
−0.266083 + 0.963950i \(0.585730\pi\)
\(908\) 32.2572 1.07049
\(909\) 11.7657 0.390242
\(910\) 9.72259 0.322301
\(911\) −16.4043 −0.543500 −0.271750 0.962368i \(-0.587602\pi\)
−0.271750 + 0.962368i \(0.587602\pi\)
\(912\) 14.1673 0.469127
\(913\) 0 0
\(914\) −24.8408 −0.821660
\(915\) −4.42647 −0.146335
\(916\) 51.2147 1.69218
\(917\) 0.101461 0.00335053
\(918\) 81.4253 2.68744
\(919\) 32.3201 1.06614 0.533071 0.846070i \(-0.321038\pi\)
0.533071 + 0.846070i \(0.321038\pi\)
\(920\) −39.4572 −1.30086
\(921\) 6.58861 0.217102
\(922\) 53.7518 1.77022
\(923\) 0.370623 0.0121992
\(924\) 0 0
\(925\) 18.6333 0.612659
\(926\) 73.8171 2.42578
\(927\) −7.75003 −0.254544
\(928\) 3.96585 0.130185
\(929\) −8.19531 −0.268879 −0.134440 0.990922i \(-0.542923\pi\)
−0.134440 + 0.990922i \(0.542923\pi\)
\(930\) −42.6775 −1.39945
\(931\) 2.86222 0.0938054
\(932\) −50.1190 −1.64170
\(933\) −3.17065 −0.103802
\(934\) −61.2253 −2.00335
\(935\) 0 0
\(936\) −13.8929 −0.454103
\(937\) 34.1374 1.11522 0.557610 0.830103i \(-0.311718\pi\)
0.557610 + 0.830103i \(0.311718\pi\)
\(938\) 22.8945 0.747533
\(939\) −13.9108 −0.453961
\(940\) 46.1027 1.50371
\(941\) 0.979371 0.0319266 0.0159633 0.999873i \(-0.494919\pi\)
0.0159633 + 0.999873i \(0.494919\pi\)
\(942\) −7.82496 −0.254951
\(943\) −2.12699 −0.0692644
\(944\) −23.9752 −0.780326
\(945\) −7.11775 −0.231540
\(946\) 0 0
\(947\) 0.935599 0.0304029 0.0152014 0.999884i \(-0.495161\pi\)
0.0152014 + 0.999884i \(0.495161\pi\)
\(948\) 47.7946 1.55230
\(949\) −1.95476 −0.0634542
\(950\) 23.7623 0.770950
\(951\) 9.25837 0.300223
\(952\) −27.4593 −0.889960
\(953\) 16.7321 0.542004 0.271002 0.962579i \(-0.412645\pi\)
0.271002 + 0.962579i \(0.412645\pi\)
\(954\) 9.97480 0.322946
\(955\) −11.1141 −0.359643
\(956\) −19.4616 −0.629434
\(957\) 0 0
\(958\) 51.5353 1.66503
\(959\) −4.56409 −0.147382
\(960\) −16.3303 −0.527058
\(961\) 63.6245 2.05240
\(962\) −42.1380 −1.35859
\(963\) −11.4944 −0.370402
\(964\) −10.2634 −0.330561
\(965\) 32.2238 1.03732
\(966\) −23.5758 −0.758538
\(967\) 36.4439 1.17196 0.585978 0.810327i \(-0.300710\pi\)
0.585978 + 0.810327i \(0.300710\pi\)
\(968\) 0 0
\(969\) 24.3188 0.781233
\(970\) −53.7688 −1.72641
\(971\) −33.2417 −1.06678 −0.533389 0.845870i \(-0.679082\pi\)
−0.533389 + 0.845870i \(0.679082\pi\)
\(972\) 36.6904 1.17685
\(973\) −3.82533 −0.122634
\(974\) 40.8796 1.30987
\(975\) 15.5706 0.498659
\(976\) −8.46294 −0.270892
\(977\) 22.5666 0.721970 0.360985 0.932572i \(-0.382441\pi\)
0.360985 + 0.932572i \(0.382441\pi\)
\(978\) 55.2050 1.76526
\(979\) 0 0
\(980\) 4.91760 0.157087
\(981\) −0.835628 −0.0266795
\(982\) −11.7586 −0.375233
\(983\) −15.1048 −0.481767 −0.240884 0.970554i \(-0.577437\pi\)
−0.240884 + 0.970554i \(0.577437\pi\)
\(984\) 2.09060 0.0666459
\(985\) −14.0938 −0.449064
\(986\) −64.7281 −2.06136
\(987\) 13.4458 0.427983
\(988\) −35.5429 −1.13077
\(989\) −0.896574 −0.0285094
\(990\) 0 0
\(991\) −55.1534 −1.75201 −0.876003 0.482305i \(-0.839800\pi\)
−0.876003 + 0.482305i \(0.839800\pi\)
\(992\) 8.58145 0.272461
\(993\) 5.19292 0.164792
\(994\) 0.283415 0.00898939
\(995\) −15.4186 −0.488802
\(996\) 5.32639 0.168773
\(997\) 49.7048 1.57417 0.787083 0.616847i \(-0.211590\pi\)
0.787083 + 0.616847i \(0.211590\pi\)
\(998\) −74.3319 −2.35293
\(999\) 30.8486 0.976006
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 847.2.a.p.1.1 8
3.2 odd 2 7623.2.a.ct.1.8 8
7.6 odd 2 5929.2.a.bt.1.1 8
11.2 odd 10 847.2.f.v.323.1 16
11.3 even 5 77.2.f.b.64.1 16
11.4 even 5 77.2.f.b.71.1 yes 16
11.5 even 5 847.2.f.w.729.4 16
11.6 odd 10 847.2.f.v.729.1 16
11.7 odd 10 847.2.f.x.148.4 16
11.8 odd 10 847.2.f.x.372.4 16
11.9 even 5 847.2.f.w.323.4 16
11.10 odd 2 847.2.a.o.1.8 8
33.14 odd 10 693.2.m.i.64.4 16
33.26 odd 10 693.2.m.i.379.4 16
33.32 even 2 7623.2.a.cw.1.1 8
77.3 odd 30 539.2.q.f.471.1 32
77.4 even 15 539.2.q.g.324.4 32
77.25 even 15 539.2.q.g.471.1 32
77.26 odd 30 539.2.q.f.214.1 32
77.37 even 15 539.2.q.g.214.1 32
77.47 odd 30 539.2.q.f.361.4 32
77.48 odd 10 539.2.f.e.148.1 16
77.58 even 15 539.2.q.g.361.4 32
77.59 odd 30 539.2.q.f.324.4 32
77.69 odd 10 539.2.f.e.295.1 16
77.76 even 2 5929.2.a.bs.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.64.1 16 11.3 even 5
77.2.f.b.71.1 yes 16 11.4 even 5
539.2.f.e.148.1 16 77.48 odd 10
539.2.f.e.295.1 16 77.69 odd 10
539.2.q.f.214.1 32 77.26 odd 30
539.2.q.f.324.4 32 77.59 odd 30
539.2.q.f.361.4 32 77.47 odd 30
539.2.q.f.471.1 32 77.3 odd 30
539.2.q.g.214.1 32 77.37 even 15
539.2.q.g.324.4 32 77.4 even 15
539.2.q.g.361.4 32 77.58 even 15
539.2.q.g.471.1 32 77.25 even 15
693.2.m.i.64.4 16 33.14 odd 10
693.2.m.i.379.4 16 33.26 odd 10
847.2.a.o.1.8 8 11.10 odd 2
847.2.a.p.1.1 8 1.1 even 1 trivial
847.2.f.v.323.1 16 11.2 odd 10
847.2.f.v.729.1 16 11.6 odd 10
847.2.f.w.323.4 16 11.9 even 5
847.2.f.w.729.4 16 11.5 even 5
847.2.f.x.148.4 16 11.7 odd 10
847.2.f.x.372.4 16 11.8 odd 10
5929.2.a.bs.1.8 8 77.76 even 2
5929.2.a.bt.1.1 8 7.6 odd 2
7623.2.a.ct.1.8 8 3.2 odd 2
7623.2.a.cw.1.1 8 33.32 even 2