Properties

Label 847.2.a.p.1.7
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 1.7
Root \(1.98451\) 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)
 
\(f(q)\) \(=\) \(q+1.98451 q^{2} +2.78639 q^{3} +1.93830 q^{4} -0.0269243 q^{5} +5.52964 q^{6} +1.00000 q^{7} -0.122446 q^{8} +4.76399 q^{9} +O(q^{10})\) \(q+1.98451 q^{2} +2.78639 q^{3} +1.93830 q^{4} -0.0269243 q^{5} +5.52964 q^{6} +1.00000 q^{7} -0.122446 q^{8} +4.76399 q^{9} -0.0534317 q^{10} +5.40087 q^{12} -4.88112 q^{13} +1.98451 q^{14} -0.0750218 q^{15} -4.11959 q^{16} +1.67878 q^{17} +9.45421 q^{18} +1.37394 q^{19} -0.0521874 q^{20} +2.78639 q^{21} +8.06246 q^{23} -0.341182 q^{24} -4.99928 q^{25} -9.68665 q^{26} +4.91517 q^{27} +1.93830 q^{28} -6.39316 q^{29} -0.148882 q^{30} +4.01054 q^{31} -7.93050 q^{32} +3.33156 q^{34} -0.0269243 q^{35} +9.23404 q^{36} +0.521038 q^{37} +2.72661 q^{38} -13.6007 q^{39} +0.00329677 q^{40} -10.5987 q^{41} +5.52964 q^{42} +3.73968 q^{43} -0.128267 q^{45} +16.0001 q^{46} -8.95906 q^{47} -11.4788 q^{48} +1.00000 q^{49} -9.92114 q^{50} +4.67774 q^{51} -9.46107 q^{52} -3.96057 q^{53} +9.75422 q^{54} -0.122446 q^{56} +3.82834 q^{57} -12.6873 q^{58} +9.73022 q^{59} -0.145415 q^{60} +8.46723 q^{61} +7.95898 q^{62} +4.76399 q^{63} -7.49902 q^{64} +0.131421 q^{65} +2.81285 q^{67} +3.25398 q^{68} +22.4652 q^{69} -0.0534317 q^{70} +2.04551 q^{71} -0.583330 q^{72} -10.4693 q^{73} +1.03401 q^{74} -13.9299 q^{75} +2.66311 q^{76} -26.9908 q^{78} +5.85521 q^{79} +0.110917 q^{80} -0.596375 q^{81} -21.0334 q^{82} +2.60548 q^{83} +5.40087 q^{84} -0.0452000 q^{85} +7.42146 q^{86} -17.8139 q^{87} +1.21791 q^{89} -0.254548 q^{90} -4.88112 q^{91} +15.6275 q^{92} +11.1749 q^{93} -17.7794 q^{94} -0.0369924 q^{95} -22.0975 q^{96} +3.39670 q^{97} +1.98451 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.98451 1.40326 0.701632 0.712540i \(-0.252455\pi\)
0.701632 + 0.712540i \(0.252455\pi\)
\(3\) 2.78639 1.60873 0.804363 0.594139i \(-0.202507\pi\)
0.804363 + 0.594139i \(0.202507\pi\)
\(4\) 1.93830 0.969150
\(5\) −0.0269243 −0.0120409 −0.00602046 0.999982i \(-0.501916\pi\)
−0.00602046 + 0.999982i \(0.501916\pi\)
\(6\) 5.52964 2.25747
\(7\) 1.00000 0.377964
\(8\) −0.122446 −0.0432911
\(9\) 4.76399 1.58800
\(10\) −0.0534317 −0.0168966
\(11\) 0 0
\(12\) 5.40087 1.55910
\(13\) −4.88112 −1.35378 −0.676889 0.736085i \(-0.736672\pi\)
−0.676889 + 0.736085i \(0.736672\pi\)
\(14\) 1.98451 0.530384
\(15\) −0.0750218 −0.0193705
\(16\) −4.11959 −1.02990
\(17\) 1.67878 0.407164 0.203582 0.979058i \(-0.434742\pi\)
0.203582 + 0.979058i \(0.434742\pi\)
\(18\) 9.45421 2.22838
\(19\) 1.37394 0.315204 0.157602 0.987503i \(-0.449624\pi\)
0.157602 + 0.987503i \(0.449624\pi\)
\(20\) −0.0521874 −0.0116695
\(21\) 2.78639 0.608041
\(22\) 0 0
\(23\) 8.06246 1.68114 0.840570 0.541703i \(-0.182220\pi\)
0.840570 + 0.541703i \(0.182220\pi\)
\(24\) −0.341182 −0.0696435
\(25\) −4.99928 −0.999855
\(26\) −9.68665 −1.89971
\(27\) 4.91517 0.945925
\(28\) 1.93830 0.366304
\(29\) −6.39316 −1.18718 −0.593590 0.804768i \(-0.702290\pi\)
−0.593590 + 0.804768i \(0.702290\pi\)
\(30\) −0.148882 −0.0271820
\(31\) 4.01054 0.720315 0.360157 0.932892i \(-0.382723\pi\)
0.360157 + 0.932892i \(0.382723\pi\)
\(32\) −7.93050 −1.40193
\(33\) 0 0
\(34\) 3.33156 0.571358
\(35\) −0.0269243 −0.00455104
\(36\) 9.23404 1.53901
\(37\) 0.521038 0.0856581 0.0428290 0.999082i \(-0.486363\pi\)
0.0428290 + 0.999082i \(0.486363\pi\)
\(38\) 2.72661 0.442314
\(39\) −13.6007 −2.17786
\(40\) 0.00329677 0.000521265 0
\(41\) −10.5987 −1.65525 −0.827623 0.561285i \(-0.810307\pi\)
−0.827623 + 0.561285i \(0.810307\pi\)
\(42\) 5.52964 0.853242
\(43\) 3.73968 0.570296 0.285148 0.958483i \(-0.407957\pi\)
0.285148 + 0.958483i \(0.407957\pi\)
\(44\) 0 0
\(45\) −0.128267 −0.0191209
\(46\) 16.0001 2.35908
\(47\) −8.95906 −1.30681 −0.653407 0.757007i \(-0.726661\pi\)
−0.653407 + 0.757007i \(0.726661\pi\)
\(48\) −11.4788 −1.65682
\(49\) 1.00000 0.142857
\(50\) −9.92114 −1.40306
\(51\) 4.67774 0.655015
\(52\) −9.46107 −1.31201
\(53\) −3.96057 −0.544027 −0.272013 0.962293i \(-0.587689\pi\)
−0.272013 + 0.962293i \(0.587689\pi\)
\(54\) 9.75422 1.32738
\(55\) 0 0
\(56\) −0.122446 −0.0163625
\(57\) 3.82834 0.507076
\(58\) −12.6873 −1.66593
\(59\) 9.73022 1.26677 0.633383 0.773838i \(-0.281666\pi\)
0.633383 + 0.773838i \(0.281666\pi\)
\(60\) −0.145415 −0.0187730
\(61\) 8.46723 1.08412 0.542059 0.840341i \(-0.317645\pi\)
0.542059 + 0.840341i \(0.317645\pi\)
\(62\) 7.95898 1.01079
\(63\) 4.76399 0.600206
\(64\) −7.49902 −0.937377
\(65\) 0.131421 0.0163008
\(66\) 0 0
\(67\) 2.81285 0.343644 0.171822 0.985128i \(-0.445035\pi\)
0.171822 + 0.985128i \(0.445035\pi\)
\(68\) 3.25398 0.394603
\(69\) 22.4652 2.70449
\(70\) −0.0534317 −0.00638631
\(71\) 2.04551 0.242757 0.121379 0.992606i \(-0.461268\pi\)
0.121379 + 0.992606i \(0.461268\pi\)
\(72\) −0.583330 −0.0687461
\(73\) −10.4693 −1.22534 −0.612671 0.790338i \(-0.709905\pi\)
−0.612671 + 0.790338i \(0.709905\pi\)
\(74\) 1.03401 0.120201
\(75\) −13.9299 −1.60849
\(76\) 2.66311 0.305479
\(77\) 0 0
\(78\) −26.9908 −3.05611
\(79\) 5.85521 0.658763 0.329381 0.944197i \(-0.393160\pi\)
0.329381 + 0.944197i \(0.393160\pi\)
\(80\) 0.110917 0.0124009
\(81\) −0.596375 −0.0662639
\(82\) −21.0334 −2.32275
\(83\) 2.60548 0.285988 0.142994 0.989724i \(-0.454327\pi\)
0.142994 + 0.989724i \(0.454327\pi\)
\(84\) 5.40087 0.589283
\(85\) −0.0452000 −0.00490263
\(86\) 7.42146 0.800276
\(87\) −17.8139 −1.90985
\(88\) 0 0
\(89\) 1.21791 0.129099 0.0645493 0.997915i \(-0.479439\pi\)
0.0645493 + 0.997915i \(0.479439\pi\)
\(90\) −0.254548 −0.0268317
\(91\) −4.88112 −0.511680
\(92\) 15.6275 1.62928
\(93\) 11.1749 1.15879
\(94\) −17.7794 −1.83380
\(95\) −0.0369924 −0.00379534
\(96\) −22.0975 −2.25532
\(97\) 3.39670 0.344883 0.172441 0.985020i \(-0.444834\pi\)
0.172441 + 0.985020i \(0.444834\pi\)
\(98\) 1.98451 0.200466
\(99\) 0 0
\(100\) −9.69009 −0.969009
\(101\) −4.04836 −0.402826 −0.201413 0.979506i \(-0.564553\pi\)
−0.201413 + 0.979506i \(0.564553\pi\)
\(102\) 9.28304 0.919158
\(103\) −3.95297 −0.389498 −0.194749 0.980853i \(-0.562389\pi\)
−0.194749 + 0.980853i \(0.562389\pi\)
\(104\) 0.597672 0.0586066
\(105\) −0.0750218 −0.00732138
\(106\) −7.85982 −0.763413
\(107\) 10.3986 1.00527 0.502635 0.864498i \(-0.332364\pi\)
0.502635 + 0.864498i \(0.332364\pi\)
\(108\) 9.52707 0.916742
\(109\) −3.77697 −0.361768 −0.180884 0.983504i \(-0.557896\pi\)
−0.180884 + 0.983504i \(0.557896\pi\)
\(110\) 0 0
\(111\) 1.45182 0.137800
\(112\) −4.11959 −0.389265
\(113\) 6.34869 0.597234 0.298617 0.954373i \(-0.403475\pi\)
0.298617 + 0.954373i \(0.403475\pi\)
\(114\) 7.59740 0.711561
\(115\) −0.217076 −0.0202425
\(116\) −12.3919 −1.15056
\(117\) −23.2536 −2.14980
\(118\) 19.3098 1.77761
\(119\) 1.67878 0.153893
\(120\) 0.00918610 0.000838572 0
\(121\) 0 0
\(122\) 16.8033 1.52130
\(123\) −29.5323 −2.66283
\(124\) 7.77363 0.698093
\(125\) 0.269224 0.0240801
\(126\) 9.45421 0.842248
\(127\) −5.03128 −0.446454 −0.223227 0.974766i \(-0.571659\pi\)
−0.223227 + 0.974766i \(0.571659\pi\)
\(128\) 0.979100 0.0865410
\(129\) 10.4202 0.917450
\(130\) 0.260807 0.0228743
\(131\) −3.76357 −0.328825 −0.164412 0.986392i \(-0.552573\pi\)
−0.164412 + 0.986392i \(0.552573\pi\)
\(132\) 0 0
\(133\) 1.37394 0.119136
\(134\) 5.58214 0.482223
\(135\) −0.132338 −0.0113898
\(136\) −0.205559 −0.0176266
\(137\) 19.1379 1.63506 0.817529 0.575887i \(-0.195343\pi\)
0.817529 + 0.575887i \(0.195343\pi\)
\(138\) 44.5825 3.79512
\(139\) 3.51108 0.297806 0.148903 0.988852i \(-0.452426\pi\)
0.148903 + 0.988852i \(0.452426\pi\)
\(140\) −0.0521874 −0.00441064
\(141\) −24.9635 −2.10230
\(142\) 4.05934 0.340652
\(143\) 0 0
\(144\) −19.6257 −1.63548
\(145\) 0.172132 0.0142948
\(146\) −20.7766 −1.71948
\(147\) 2.78639 0.229818
\(148\) 1.00993 0.0830155
\(149\) −3.12643 −0.256127 −0.128063 0.991766i \(-0.540876\pi\)
−0.128063 + 0.991766i \(0.540876\pi\)
\(150\) −27.6442 −2.25714
\(151\) 19.7215 1.60491 0.802456 0.596712i \(-0.203526\pi\)
0.802456 + 0.596712i \(0.203526\pi\)
\(152\) −0.168233 −0.0136455
\(153\) 7.99769 0.646575
\(154\) 0 0
\(155\) −0.107981 −0.00867326
\(156\) −26.3623 −2.11067
\(157\) −22.9546 −1.83197 −0.915987 0.401207i \(-0.868591\pi\)
−0.915987 + 0.401207i \(0.868591\pi\)
\(158\) 11.6198 0.924418
\(159\) −11.0357 −0.875189
\(160\) 0.213524 0.0168805
\(161\) 8.06246 0.635411
\(162\) −1.18351 −0.0929857
\(163\) 18.9891 1.48734 0.743670 0.668547i \(-0.233083\pi\)
0.743670 + 0.668547i \(0.233083\pi\)
\(164\) −20.5435 −1.60418
\(165\) 0 0
\(166\) 5.17061 0.401317
\(167\) −3.02044 −0.233728 −0.116864 0.993148i \(-0.537284\pi\)
−0.116864 + 0.993148i \(0.537284\pi\)
\(168\) −0.341182 −0.0263228
\(169\) 10.8253 0.832717
\(170\) −0.0897001 −0.00687968
\(171\) 6.54544 0.500542
\(172\) 7.24863 0.552703
\(173\) 8.83629 0.671811 0.335905 0.941896i \(-0.390958\pi\)
0.335905 + 0.941896i \(0.390958\pi\)
\(174\) −35.3519 −2.68002
\(175\) −4.99928 −0.377910
\(176\) 0 0
\(177\) 27.1122 2.03788
\(178\) 2.41697 0.181159
\(179\) 8.46434 0.632654 0.316327 0.948650i \(-0.397550\pi\)
0.316327 + 0.948650i \(0.397550\pi\)
\(180\) −0.248620 −0.0185311
\(181\) −24.2008 −1.79883 −0.899415 0.437095i \(-0.856007\pi\)
−0.899415 + 0.437095i \(0.856007\pi\)
\(182\) −9.68665 −0.718022
\(183\) 23.5930 1.74405
\(184\) −0.987214 −0.0727784
\(185\) −0.0140286 −0.00103140
\(186\) 22.1769 1.62609
\(187\) 0 0
\(188\) −17.3653 −1.26650
\(189\) 4.91517 0.357526
\(190\) −0.0734120 −0.00532587
\(191\) −7.78718 −0.563460 −0.281730 0.959494i \(-0.590908\pi\)
−0.281730 + 0.959494i \(0.590908\pi\)
\(192\) −20.8952 −1.50798
\(193\) 17.0637 1.22827 0.614134 0.789201i \(-0.289505\pi\)
0.614134 + 0.789201i \(0.289505\pi\)
\(194\) 6.74081 0.483962
\(195\) 0.366190 0.0262234
\(196\) 1.93830 0.138450
\(197\) −6.05536 −0.431426 −0.215713 0.976457i \(-0.569208\pi\)
−0.215713 + 0.976457i \(0.569208\pi\)
\(198\) 0 0
\(199\) −13.7181 −0.972451 −0.486226 0.873833i \(-0.661627\pi\)
−0.486226 + 0.873833i \(0.661627\pi\)
\(200\) 0.612140 0.0432848
\(201\) 7.83770 0.552829
\(202\) −8.03402 −0.565272
\(203\) −6.39316 −0.448712
\(204\) 9.06686 0.634807
\(205\) 0.285364 0.0199307
\(206\) −7.84472 −0.546568
\(207\) 38.4095 2.66964
\(208\) 20.1082 1.39425
\(209\) 0 0
\(210\) −0.148882 −0.0102738
\(211\) 18.0352 1.24160 0.620799 0.783970i \(-0.286808\pi\)
0.620799 + 0.783970i \(0.286808\pi\)
\(212\) −7.67678 −0.527243
\(213\) 5.69959 0.390530
\(214\) 20.6362 1.41066
\(215\) −0.100688 −0.00686690
\(216\) −0.601841 −0.0409501
\(217\) 4.01054 0.272253
\(218\) −7.49546 −0.507656
\(219\) −29.1717 −1.97124
\(220\) 0 0
\(221\) −8.19432 −0.551210
\(222\) 2.88115 0.193370
\(223\) −6.53314 −0.437491 −0.218746 0.975782i \(-0.570197\pi\)
−0.218746 + 0.975782i \(0.570197\pi\)
\(224\) −7.93050 −0.529879
\(225\) −23.8165 −1.58777
\(226\) 12.5991 0.838077
\(227\) 13.6077 0.903176 0.451588 0.892227i \(-0.350858\pi\)
0.451588 + 0.892227i \(0.350858\pi\)
\(228\) 7.42047 0.491433
\(229\) −19.4211 −1.28338 −0.641690 0.766964i \(-0.721767\pi\)
−0.641690 + 0.766964i \(0.721767\pi\)
\(230\) −0.430791 −0.0284055
\(231\) 0 0
\(232\) 0.782815 0.0513943
\(233\) 4.36150 0.285731 0.142866 0.989742i \(-0.454368\pi\)
0.142866 + 0.989742i \(0.454368\pi\)
\(234\) −46.1471 −3.01673
\(235\) 0.241217 0.0157353
\(236\) 18.8601 1.22769
\(237\) 16.3149 1.05977
\(238\) 3.33156 0.215953
\(239\) 10.5901 0.685018 0.342509 0.939515i \(-0.388723\pi\)
0.342509 + 0.939515i \(0.388723\pi\)
\(240\) 0.309059 0.0199497
\(241\) 18.6887 1.20384 0.601921 0.798555i \(-0.294402\pi\)
0.601921 + 0.798555i \(0.294402\pi\)
\(242\) 0 0
\(243\) −16.4072 −1.05252
\(244\) 16.4120 1.05067
\(245\) −0.0269243 −0.00172013
\(246\) −58.6072 −3.73666
\(247\) −6.70637 −0.426716
\(248\) −0.491074 −0.0311832
\(249\) 7.25988 0.460076
\(250\) 0.534279 0.0337907
\(251\) 18.9832 1.19821 0.599103 0.800672i \(-0.295524\pi\)
0.599103 + 0.800672i \(0.295524\pi\)
\(252\) 9.23404 0.581690
\(253\) 0 0
\(254\) −9.98465 −0.626493
\(255\) −0.125945 −0.00788698
\(256\) 16.9411 1.05882
\(257\) −13.7090 −0.855147 −0.427573 0.903981i \(-0.640631\pi\)
−0.427573 + 0.903981i \(0.640631\pi\)
\(258\) 20.6791 1.28742
\(259\) 0.521038 0.0323757
\(260\) 0.254733 0.0157979
\(261\) −30.4569 −1.88524
\(262\) −7.46887 −0.461428
\(263\) 16.5767 1.02217 0.511083 0.859531i \(-0.329244\pi\)
0.511083 + 0.859531i \(0.329244\pi\)
\(264\) 0 0
\(265\) 0.106636 0.00655059
\(266\) 2.72661 0.167179
\(267\) 3.39359 0.207684
\(268\) 5.45214 0.333043
\(269\) −8.54361 −0.520913 −0.260457 0.965486i \(-0.583873\pi\)
−0.260457 + 0.965486i \(0.583873\pi\)
\(270\) −0.262626 −0.0159829
\(271\) −6.34553 −0.385463 −0.192732 0.981251i \(-0.561735\pi\)
−0.192732 + 0.981251i \(0.561735\pi\)
\(272\) −6.91589 −0.419337
\(273\) −13.6007 −0.823153
\(274\) 37.9794 2.29442
\(275\) 0 0
\(276\) 43.5443 2.62106
\(277\) −24.0044 −1.44228 −0.721142 0.692788i \(-0.756382\pi\)
−0.721142 + 0.692788i \(0.756382\pi\)
\(278\) 6.96780 0.417901
\(279\) 19.1062 1.14386
\(280\) 0.00329677 0.000197020 0
\(281\) −3.21144 −0.191578 −0.0957891 0.995402i \(-0.530537\pi\)
−0.0957891 + 0.995402i \(0.530537\pi\)
\(282\) −49.5404 −2.95009
\(283\) −20.8034 −1.23663 −0.618317 0.785929i \(-0.712185\pi\)
−0.618317 + 0.785929i \(0.712185\pi\)
\(284\) 3.96481 0.235268
\(285\) −0.103075 −0.00610567
\(286\) 0 0
\(287\) −10.5987 −0.625624
\(288\) −37.7808 −2.22626
\(289\) −14.1817 −0.834218
\(290\) 0.341598 0.0200593
\(291\) 9.46455 0.554822
\(292\) −20.2927 −1.18754
\(293\) 5.51744 0.322332 0.161166 0.986927i \(-0.448474\pi\)
0.161166 + 0.986927i \(0.448474\pi\)
\(294\) 5.52964 0.322495
\(295\) −0.261980 −0.0152530
\(296\) −0.0637988 −0.00370823
\(297\) 0 0
\(298\) −6.20444 −0.359414
\(299\) −39.3538 −2.27589
\(300\) −27.0004 −1.55887
\(301\) 3.73968 0.215552
\(302\) 39.1376 2.25211
\(303\) −11.2803 −0.648037
\(304\) −5.66008 −0.324628
\(305\) −0.227974 −0.0130538
\(306\) 15.8715 0.907315
\(307\) 8.60991 0.491394 0.245697 0.969347i \(-0.420983\pi\)
0.245697 + 0.969347i \(0.420983\pi\)
\(308\) 0 0
\(309\) −11.0145 −0.626594
\(310\) −0.214290 −0.0121709
\(311\) 19.1258 1.08453 0.542263 0.840209i \(-0.317568\pi\)
0.542263 + 0.840209i \(0.317568\pi\)
\(312\) 1.66535 0.0942819
\(313\) −0.606755 −0.0342958 −0.0171479 0.999853i \(-0.505459\pi\)
−0.0171479 + 0.999853i \(0.505459\pi\)
\(314\) −45.5537 −2.57074
\(315\) −0.128267 −0.00722704
\(316\) 11.3492 0.638440
\(317\) −5.21283 −0.292782 −0.146391 0.989227i \(-0.546766\pi\)
−0.146391 + 0.989227i \(0.546766\pi\)
\(318\) −21.9005 −1.22812
\(319\) 0 0
\(320\) 0.201906 0.0112869
\(321\) 28.9746 1.61720
\(322\) 16.0001 0.891650
\(323\) 2.30654 0.128339
\(324\) −1.15595 −0.0642196
\(325\) 24.4021 1.35358
\(326\) 37.6841 2.08713
\(327\) −10.5241 −0.581986
\(328\) 1.29777 0.0716574
\(329\) −8.95906 −0.493929
\(330\) 0 0
\(331\) 0.669012 0.0367722 0.0183861 0.999831i \(-0.494147\pi\)
0.0183861 + 0.999831i \(0.494147\pi\)
\(332\) 5.05019 0.277165
\(333\) 2.48222 0.136025
\(334\) −5.99410 −0.327983
\(335\) −0.0757340 −0.00413779
\(336\) −11.4788 −0.626220
\(337\) −6.60215 −0.359642 −0.179821 0.983699i \(-0.557552\pi\)
−0.179821 + 0.983699i \(0.557552\pi\)
\(338\) 21.4830 1.16852
\(339\) 17.6899 0.960785
\(340\) −0.0876111 −0.00475138
\(341\) 0 0
\(342\) 12.9895 0.702393
\(343\) 1.00000 0.0539949
\(344\) −0.457908 −0.0246888
\(345\) −0.604860 −0.0325646
\(346\) 17.5358 0.942728
\(347\) 12.2001 0.654937 0.327469 0.944862i \(-0.393804\pi\)
0.327469 + 0.944862i \(0.393804\pi\)
\(348\) −34.5286 −1.85093
\(349\) −0.885800 −0.0474158 −0.0237079 0.999719i \(-0.507547\pi\)
−0.0237079 + 0.999719i \(0.507547\pi\)
\(350\) −9.92114 −0.530307
\(351\) −23.9915 −1.28057
\(352\) 0 0
\(353\) 5.36012 0.285291 0.142645 0.989774i \(-0.454439\pi\)
0.142645 + 0.989774i \(0.454439\pi\)
\(354\) 53.8046 2.85968
\(355\) −0.0550740 −0.00292302
\(356\) 2.36068 0.125116
\(357\) 4.67774 0.247572
\(358\) 16.7976 0.887781
\(359\) 25.8450 1.36405 0.682023 0.731330i \(-0.261100\pi\)
0.682023 + 0.731330i \(0.261100\pi\)
\(360\) 0.0157058 0.000827767 0
\(361\) −17.1123 −0.900647
\(362\) −48.0268 −2.52423
\(363\) 0 0
\(364\) −9.46107 −0.495895
\(365\) 0.281880 0.0147543
\(366\) 46.8207 2.44736
\(367\) −20.4504 −1.06750 −0.533752 0.845641i \(-0.679218\pi\)
−0.533752 + 0.845641i \(0.679218\pi\)
\(368\) −33.2141 −1.73140
\(369\) −50.4923 −2.62852
\(370\) −0.0278400 −0.00144733
\(371\) −3.96057 −0.205623
\(372\) 21.6604 1.12304
\(373\) 9.39109 0.486252 0.243126 0.969995i \(-0.421827\pi\)
0.243126 + 0.969995i \(0.421827\pi\)
\(374\) 0 0
\(375\) 0.750163 0.0387383
\(376\) 1.09700 0.0565734
\(377\) 31.2058 1.60718
\(378\) 9.75422 0.501703
\(379\) −5.63593 −0.289498 −0.144749 0.989468i \(-0.546238\pi\)
−0.144749 + 0.989468i \(0.546238\pi\)
\(380\) −0.0717024 −0.00367826
\(381\) −14.0191 −0.718222
\(382\) −15.4538 −0.790684
\(383\) −1.25944 −0.0643544 −0.0321772 0.999482i \(-0.510244\pi\)
−0.0321772 + 0.999482i \(0.510244\pi\)
\(384\) 2.72816 0.139221
\(385\) 0 0
\(386\) 33.8631 1.72359
\(387\) 17.8158 0.905629
\(388\) 6.58383 0.334243
\(389\) −5.88409 −0.298335 −0.149168 0.988812i \(-0.547659\pi\)
−0.149168 + 0.988812i \(0.547659\pi\)
\(390\) 0.726710 0.0367984
\(391\) 13.5351 0.684499
\(392\) −0.122446 −0.00618444
\(393\) −10.4868 −0.528989
\(394\) −12.0169 −0.605405
\(395\) −0.157648 −0.00793212
\(396\) 0 0
\(397\) 4.86018 0.243925 0.121963 0.992535i \(-0.461081\pi\)
0.121963 + 0.992535i \(0.461081\pi\)
\(398\) −27.2238 −1.36461
\(399\) 3.82834 0.191657
\(400\) 20.5950 1.02975
\(401\) −25.5322 −1.27502 −0.637508 0.770444i \(-0.720035\pi\)
−0.637508 + 0.770444i \(0.720035\pi\)
\(402\) 15.5540 0.775765
\(403\) −19.5759 −0.975147
\(404\) −7.84692 −0.390399
\(405\) 0.0160570 0.000797878 0
\(406\) −12.6873 −0.629661
\(407\) 0 0
\(408\) −0.572769 −0.0283563
\(409\) 21.0267 1.03970 0.519851 0.854257i \(-0.325988\pi\)
0.519851 + 0.854257i \(0.325988\pi\)
\(410\) 0.566309 0.0279680
\(411\) 53.3256 2.63036
\(412\) −7.66204 −0.377481
\(413\) 9.73022 0.478793
\(414\) 76.2242 3.74622
\(415\) −0.0701507 −0.00344356
\(416\) 38.7097 1.89790
\(417\) 9.78326 0.479088
\(418\) 0 0
\(419\) −31.3141 −1.52980 −0.764898 0.644151i \(-0.777211\pi\)
−0.764898 + 0.644151i \(0.777211\pi\)
\(420\) −0.145415 −0.00709551
\(421\) 12.7886 0.623280 0.311640 0.950200i \(-0.399122\pi\)
0.311640 + 0.950200i \(0.399122\pi\)
\(422\) 35.7912 1.74229
\(423\) −42.6809 −2.07522
\(424\) 0.484955 0.0235515
\(425\) −8.39268 −0.407105
\(426\) 11.3109 0.548016
\(427\) 8.46723 0.409758
\(428\) 20.1556 0.974258
\(429\) 0 0
\(430\) −0.199818 −0.00963607
\(431\) 21.0679 1.01480 0.507402 0.861709i \(-0.330606\pi\)
0.507402 + 0.861709i \(0.330606\pi\)
\(432\) −20.2485 −0.974206
\(433\) 38.0030 1.82631 0.913155 0.407613i \(-0.133639\pi\)
0.913155 + 0.407613i \(0.133639\pi\)
\(434\) 7.95898 0.382043
\(435\) 0.479626 0.0229963
\(436\) −7.32090 −0.350608
\(437\) 11.0773 0.529901
\(438\) −57.8916 −2.76617
\(439\) −37.7677 −1.80256 −0.901278 0.433241i \(-0.857370\pi\)
−0.901278 + 0.433241i \(0.857370\pi\)
\(440\) 0 0
\(441\) 4.76399 0.226857
\(442\) −16.2618 −0.773493
\(443\) −1.02792 −0.0488381 −0.0244190 0.999702i \(-0.507774\pi\)
−0.0244190 + 0.999702i \(0.507774\pi\)
\(444\) 2.81405 0.133549
\(445\) −0.0327915 −0.00155447
\(446\) −12.9651 −0.613916
\(447\) −8.71146 −0.412038
\(448\) −7.49902 −0.354295
\(449\) 20.1333 0.950149 0.475074 0.879946i \(-0.342421\pi\)
0.475074 + 0.879946i \(0.342421\pi\)
\(450\) −47.2642 −2.22806
\(451\) 0 0
\(452\) 12.3057 0.578809
\(453\) 54.9518 2.58186
\(454\) 27.0047 1.26739
\(455\) 0.131421 0.00616111
\(456\) −0.468764 −0.0219519
\(457\) −24.1456 −1.12948 −0.564742 0.825268i \(-0.691024\pi\)
−0.564742 + 0.825268i \(0.691024\pi\)
\(458\) −38.5414 −1.80092
\(459\) 8.25148 0.385146
\(460\) −0.420759 −0.0196180
\(461\) 11.5885 0.539731 0.269865 0.962898i \(-0.413021\pi\)
0.269865 + 0.962898i \(0.413021\pi\)
\(462\) 0 0
\(463\) 21.6077 1.00419 0.502097 0.864811i \(-0.332562\pi\)
0.502097 + 0.864811i \(0.332562\pi\)
\(464\) 26.3372 1.22268
\(465\) −0.300878 −0.0139529
\(466\) 8.65547 0.400957
\(467\) 15.9149 0.736456 0.368228 0.929736i \(-0.379965\pi\)
0.368228 + 0.929736i \(0.379965\pi\)
\(468\) −45.0724 −2.08347
\(469\) 2.81285 0.129885
\(470\) 0.478698 0.0220807
\(471\) −63.9605 −2.94714
\(472\) −1.19142 −0.0548397
\(473\) 0 0
\(474\) 32.3772 1.48713
\(475\) −6.86871 −0.315158
\(476\) 3.25398 0.149146
\(477\) −18.8681 −0.863913
\(478\) 21.0162 0.961261
\(479\) 5.28372 0.241419 0.120710 0.992688i \(-0.461483\pi\)
0.120710 + 0.992688i \(0.461483\pi\)
\(480\) 0.594961 0.0271561
\(481\) −2.54325 −0.115962
\(482\) 37.0879 1.68931
\(483\) 22.4652 1.02220
\(484\) 0 0
\(485\) −0.0914540 −0.00415271
\(486\) −32.5604 −1.47697
\(487\) −27.0656 −1.22646 −0.613229 0.789905i \(-0.710130\pi\)
−0.613229 + 0.789905i \(0.710130\pi\)
\(488\) −1.03678 −0.0469326
\(489\) 52.9111 2.39272
\(490\) −0.0534317 −0.00241380
\(491\) 18.8000 0.848431 0.424215 0.905561i \(-0.360550\pi\)
0.424215 + 0.905561i \(0.360550\pi\)
\(492\) −57.2424 −2.58069
\(493\) −10.7327 −0.483377
\(494\) −13.3089 −0.598795
\(495\) 0 0
\(496\) −16.5218 −0.741851
\(497\) 2.04551 0.0917536
\(498\) 14.4073 0.645608
\(499\) −39.7667 −1.78020 −0.890100 0.455765i \(-0.849366\pi\)
−0.890100 + 0.455765i \(0.849366\pi\)
\(500\) 0.521836 0.0233372
\(501\) −8.41612 −0.376005
\(502\) 37.6724 1.68140
\(503\) 40.5624 1.80859 0.904294 0.426910i \(-0.140398\pi\)
0.904294 + 0.426910i \(0.140398\pi\)
\(504\) −0.583330 −0.0259836
\(505\) 0.108999 0.00485040
\(506\) 0 0
\(507\) 30.1636 1.33961
\(508\) −9.75213 −0.432681
\(509\) −15.4809 −0.686179 −0.343090 0.939303i \(-0.611473\pi\)
−0.343090 + 0.939303i \(0.611473\pi\)
\(510\) −0.249940 −0.0110675
\(511\) −10.4693 −0.463136
\(512\) 31.6616 1.39926
\(513\) 6.75315 0.298159
\(514\) −27.2058 −1.20000
\(515\) 0.106431 0.00468991
\(516\) 20.1975 0.889147
\(517\) 0 0
\(518\) 1.03401 0.0454317
\(519\) 24.6214 1.08076
\(520\) −0.0160919 −0.000705677 0
\(521\) 37.4738 1.64176 0.820878 0.571103i \(-0.193484\pi\)
0.820878 + 0.571103i \(0.193484\pi\)
\(522\) −60.4423 −2.64549
\(523\) −20.8115 −0.910024 −0.455012 0.890485i \(-0.650365\pi\)
−0.455012 + 0.890485i \(0.650365\pi\)
\(524\) −7.29493 −0.318681
\(525\) −13.9299 −0.607953
\(526\) 32.8968 1.43437
\(527\) 6.73281 0.293286
\(528\) 0 0
\(529\) 42.0033 1.82623
\(530\) 0.211620 0.00919220
\(531\) 46.3547 2.01162
\(532\) 2.66311 0.115460
\(533\) 51.7337 2.24084
\(534\) 6.73463 0.291436
\(535\) −0.279975 −0.0121044
\(536\) −0.344421 −0.0148767
\(537\) 23.5850 1.01777
\(538\) −16.9549 −0.730979
\(539\) 0 0
\(540\) −0.256510 −0.0110384
\(541\) −17.4259 −0.749196 −0.374598 0.927187i \(-0.622219\pi\)
−0.374598 + 0.927187i \(0.622219\pi\)
\(542\) −12.5928 −0.540907
\(543\) −67.4329 −2.89382
\(544\) −13.3136 −0.570814
\(545\) 0.101692 0.00435602
\(546\) −26.9908 −1.15510
\(547\) −15.7960 −0.675386 −0.337693 0.941256i \(-0.609647\pi\)
−0.337693 + 0.941256i \(0.609647\pi\)
\(548\) 37.0949 1.58462
\(549\) 40.3378 1.72157
\(550\) 0 0
\(551\) −8.78382 −0.374203
\(552\) −2.75077 −0.117080
\(553\) 5.85521 0.248989
\(554\) −47.6371 −2.02390
\(555\) −0.0390892 −0.00165924
\(556\) 6.80553 0.288619
\(557\) −16.8351 −0.713325 −0.356663 0.934233i \(-0.616085\pi\)
−0.356663 + 0.934233i \(0.616085\pi\)
\(558\) 37.9165 1.60513
\(559\) −18.2538 −0.772055
\(560\) 0.110917 0.00468711
\(561\) 0 0
\(562\) −6.37315 −0.268835
\(563\) −37.4304 −1.57750 −0.788751 0.614713i \(-0.789272\pi\)
−0.788751 + 0.614713i \(0.789272\pi\)
\(564\) −48.3867 −2.03745
\(565\) −0.170934 −0.00719125
\(566\) −41.2846 −1.73532
\(567\) −0.596375 −0.0250454
\(568\) −0.250464 −0.0105092
\(569\) −2.76910 −0.116087 −0.0580433 0.998314i \(-0.518486\pi\)
−0.0580433 + 0.998314i \(0.518486\pi\)
\(570\) −0.204555 −0.00856786
\(571\) −8.85289 −0.370482 −0.185241 0.982693i \(-0.559307\pi\)
−0.185241 + 0.982693i \(0.559307\pi\)
\(572\) 0 0
\(573\) −21.6981 −0.906453
\(574\) −21.0334 −0.877916
\(575\) −40.3065 −1.68090
\(576\) −35.7252 −1.48855
\(577\) 37.1682 1.54733 0.773667 0.633593i \(-0.218420\pi\)
0.773667 + 0.633593i \(0.218420\pi\)
\(578\) −28.1438 −1.17063
\(579\) 47.5461 1.97595
\(580\) 0.333643 0.0138538
\(581\) 2.60548 0.108093
\(582\) 18.7825 0.778562
\(583\) 0 0
\(584\) 1.28193 0.0530464
\(585\) 0.626088 0.0258855
\(586\) 10.9494 0.452317
\(587\) −35.3189 −1.45777 −0.728884 0.684638i \(-0.759960\pi\)
−0.728884 + 0.684638i \(0.759960\pi\)
\(588\) 5.40087 0.222728
\(589\) 5.51025 0.227046
\(590\) −0.519902 −0.0214040
\(591\) −16.8726 −0.694046
\(592\) −2.14646 −0.0882191
\(593\) −8.09224 −0.332308 −0.166154 0.986100i \(-0.553135\pi\)
−0.166154 + 0.986100i \(0.553135\pi\)
\(594\) 0 0
\(595\) −0.0452000 −0.00185302
\(596\) −6.05995 −0.248225
\(597\) −38.2241 −1.56441
\(598\) −78.0983 −3.19368
\(599\) 3.83616 0.156741 0.0783706 0.996924i \(-0.475028\pi\)
0.0783706 + 0.996924i \(0.475028\pi\)
\(600\) 1.70566 0.0696334
\(601\) −14.3366 −0.584804 −0.292402 0.956296i \(-0.594455\pi\)
−0.292402 + 0.956296i \(0.594455\pi\)
\(602\) 7.42146 0.302476
\(603\) 13.4004 0.545705
\(604\) 38.2261 1.55540
\(605\) 0 0
\(606\) −22.3859 −0.909367
\(607\) 13.7786 0.559255 0.279627 0.960109i \(-0.409789\pi\)
0.279627 + 0.960109i \(0.409789\pi\)
\(608\) −10.8960 −0.441893
\(609\) −17.8139 −0.721854
\(610\) −0.452419 −0.0183179
\(611\) 43.7303 1.76914
\(612\) 15.5019 0.626628
\(613\) −9.54435 −0.385493 −0.192746 0.981249i \(-0.561739\pi\)
−0.192746 + 0.981249i \(0.561739\pi\)
\(614\) 17.0865 0.689555
\(615\) 0.795137 0.0320630
\(616\) 0 0
\(617\) −31.5153 −1.26876 −0.634379 0.773023i \(-0.718744\pi\)
−0.634379 + 0.773023i \(0.718744\pi\)
\(618\) −21.8585 −0.879277
\(619\) 20.5297 0.825159 0.412579 0.910922i \(-0.364628\pi\)
0.412579 + 0.910922i \(0.364628\pi\)
\(620\) −0.209300 −0.00840568
\(621\) 39.6284 1.59023
\(622\) 37.9555 1.52188
\(623\) 1.21791 0.0487947
\(624\) 56.0294 2.24297
\(625\) 24.9891 0.999565
\(626\) −1.20411 −0.0481261
\(627\) 0 0
\(628\) −44.4928 −1.77546
\(629\) 0.874707 0.0348769
\(630\) −0.254548 −0.0101414
\(631\) 35.9658 1.43178 0.715889 0.698214i \(-0.246022\pi\)
0.715889 + 0.698214i \(0.246022\pi\)
\(632\) −0.716946 −0.0285186
\(633\) 50.2533 1.99739
\(634\) −10.3449 −0.410850
\(635\) 0.135464 0.00537572
\(636\) −21.3905 −0.848190
\(637\) −4.88112 −0.193397
\(638\) 0 0
\(639\) 9.74478 0.385498
\(640\) −0.0263616 −0.00104203
\(641\) 13.6205 0.537976 0.268988 0.963144i \(-0.413311\pi\)
0.268988 + 0.963144i \(0.413311\pi\)
\(642\) 57.5005 2.26937
\(643\) 21.9891 0.867166 0.433583 0.901114i \(-0.357249\pi\)
0.433583 + 0.901114i \(0.357249\pi\)
\(644\) 15.6275 0.615809
\(645\) −0.280558 −0.0110470
\(646\) 4.57737 0.180094
\(647\) −14.8203 −0.582644 −0.291322 0.956625i \(-0.594095\pi\)
−0.291322 + 0.956625i \(0.594095\pi\)
\(648\) 0.0730235 0.00286863
\(649\) 0 0
\(650\) 48.4262 1.89943
\(651\) 11.1749 0.437981
\(652\) 36.8065 1.44145
\(653\) 43.3776 1.69750 0.848748 0.528798i \(-0.177357\pi\)
0.848748 + 0.528798i \(0.177357\pi\)
\(654\) −20.8853 −0.816679
\(655\) 0.101332 0.00395936
\(656\) 43.6625 1.70473
\(657\) −49.8758 −1.94584
\(658\) −17.7794 −0.693113
\(659\) 42.2093 1.64424 0.822121 0.569313i \(-0.192791\pi\)
0.822121 + 0.569313i \(0.192791\pi\)
\(660\) 0 0
\(661\) 16.2794 0.633197 0.316599 0.948560i \(-0.397459\pi\)
0.316599 + 0.948560i \(0.397459\pi\)
\(662\) 1.32766 0.0516012
\(663\) −22.8326 −0.886745
\(664\) −0.319029 −0.0123807
\(665\) −0.0369924 −0.00143451
\(666\) 4.92600 0.190879
\(667\) −51.5446 −1.99582
\(668\) −5.85451 −0.226518
\(669\) −18.2039 −0.703803
\(670\) −0.150295 −0.00580642
\(671\) 0 0
\(672\) −22.0975 −0.852430
\(673\) −5.31047 −0.204704 −0.102352 0.994748i \(-0.532637\pi\)
−0.102352 + 0.994748i \(0.532637\pi\)
\(674\) −13.1021 −0.504673
\(675\) −24.5723 −0.945787
\(676\) 20.9827 0.807027
\(677\) −33.7947 −1.29884 −0.649419 0.760431i \(-0.724988\pi\)
−0.649419 + 0.760431i \(0.724988\pi\)
\(678\) 35.1059 1.34824
\(679\) 3.39670 0.130354
\(680\) 0.00553455 0.000212240 0
\(681\) 37.9164 1.45296
\(682\) 0 0
\(683\) −15.8834 −0.607761 −0.303880 0.952710i \(-0.598282\pi\)
−0.303880 + 0.952710i \(0.598282\pi\)
\(684\) 12.6870 0.485100
\(685\) −0.515274 −0.0196876
\(686\) 1.98451 0.0757691
\(687\) −54.1148 −2.06461
\(688\) −15.4060 −0.587347
\(689\) 19.3320 0.736492
\(690\) −1.20035 −0.0456967
\(691\) 25.5375 0.971490 0.485745 0.874100i \(-0.338548\pi\)
0.485745 + 0.874100i \(0.338548\pi\)
\(692\) 17.1274 0.651085
\(693\) 0 0
\(694\) 24.2113 0.919050
\(695\) −0.0945336 −0.00358586
\(696\) 2.18123 0.0826794
\(697\) −17.7929 −0.673956
\(698\) −1.75788 −0.0665369
\(699\) 12.1529 0.459663
\(700\) −9.69009 −0.366251
\(701\) −3.29535 −0.124464 −0.0622319 0.998062i \(-0.519822\pi\)
−0.0622319 + 0.998062i \(0.519822\pi\)
\(702\) −47.6115 −1.79698
\(703\) 0.715875 0.0269997
\(704\) 0 0
\(705\) 0.672125 0.0253137
\(706\) 10.6372 0.400338
\(707\) −4.04836 −0.152254
\(708\) 52.5516 1.97501
\(709\) −44.9026 −1.68635 −0.843176 0.537638i \(-0.819317\pi\)
−0.843176 + 0.537638i \(0.819317\pi\)
\(710\) −0.109295 −0.00410177
\(711\) 27.8942 1.04611
\(712\) −0.149128 −0.00558882
\(713\) 32.3348 1.21095
\(714\) 9.28304 0.347409
\(715\) 0 0
\(716\) 16.4064 0.613137
\(717\) 29.5082 1.10201
\(718\) 51.2898 1.91412
\(719\) 3.19264 0.119065 0.0595327 0.998226i \(-0.481039\pi\)
0.0595327 + 0.998226i \(0.481039\pi\)
\(720\) 0.528409 0.0196926
\(721\) −3.95297 −0.147216
\(722\) −33.9596 −1.26385
\(723\) 52.0740 1.93665
\(724\) −46.9084 −1.74334
\(725\) 31.9612 1.18701
\(726\) 0 0
\(727\) 20.1654 0.747891 0.373946 0.927451i \(-0.378005\pi\)
0.373946 + 0.927451i \(0.378005\pi\)
\(728\) 0.597672 0.0221512
\(729\) −43.9279 −1.62696
\(730\) 0.559395 0.0207041
\(731\) 6.27810 0.232204
\(732\) 45.7303 1.69024
\(733\) −19.7094 −0.727985 −0.363992 0.931402i \(-0.618587\pi\)
−0.363992 + 0.931402i \(0.618587\pi\)
\(734\) −40.5842 −1.49799
\(735\) −0.0750218 −0.00276722
\(736\) −63.9394 −2.35684
\(737\) 0 0
\(738\) −100.203 −3.68851
\(739\) −34.5575 −1.27122 −0.635610 0.772010i \(-0.719251\pi\)
−0.635610 + 0.772010i \(0.719251\pi\)
\(740\) −0.0271916 −0.000999584 0
\(741\) −18.6866 −0.686469
\(742\) −7.85982 −0.288543
\(743\) −25.8381 −0.947910 −0.473955 0.880549i \(-0.657174\pi\)
−0.473955 + 0.880549i \(0.657174\pi\)
\(744\) −1.36832 −0.0501652
\(745\) 0.0841770 0.00308401
\(746\) 18.6368 0.682340
\(747\) 12.4125 0.454148
\(748\) 0 0
\(749\) 10.3986 0.379957
\(750\) 1.48871 0.0543600
\(751\) −35.7199 −1.30344 −0.651719 0.758460i \(-0.725952\pi\)
−0.651719 + 0.758460i \(0.725952\pi\)
\(752\) 36.9077 1.34589
\(753\) 52.8946 1.92758
\(754\) 61.9283 2.25530
\(755\) −0.530988 −0.0193246
\(756\) 9.52707 0.346496
\(757\) −14.8197 −0.538629 −0.269315 0.963052i \(-0.586797\pi\)
−0.269315 + 0.963052i \(0.586797\pi\)
\(758\) −11.1846 −0.406242
\(759\) 0 0
\(760\) 0.00452957 0.000164305 0
\(761\) −21.7759 −0.789377 −0.394689 0.918815i \(-0.629148\pi\)
−0.394689 + 0.918815i \(0.629148\pi\)
\(762\) −27.8212 −1.00785
\(763\) −3.77697 −0.136736
\(764\) −15.0939 −0.546077
\(765\) −0.215332 −0.00778536
\(766\) −2.49938 −0.0903062
\(767\) −47.4943 −1.71492
\(768\) 47.2045 1.70335
\(769\) −35.6991 −1.28734 −0.643672 0.765302i \(-0.722590\pi\)
−0.643672 + 0.765302i \(0.722590\pi\)
\(770\) 0 0
\(771\) −38.1988 −1.37570
\(772\) 33.0745 1.19038
\(773\) 42.6229 1.53304 0.766520 0.642220i \(-0.221987\pi\)
0.766520 + 0.642220i \(0.221987\pi\)
\(774\) 35.3557 1.27084
\(775\) −20.0498 −0.720210
\(776\) −0.415912 −0.0149304
\(777\) 1.45182 0.0520836
\(778\) −11.6771 −0.418643
\(779\) −14.5620 −0.521739
\(780\) 0.709786 0.0254144
\(781\) 0 0
\(782\) 26.8606 0.960533
\(783\) −31.4235 −1.12298
\(784\) −4.11959 −0.147128
\(785\) 0.618037 0.0220587
\(786\) −20.8112 −0.742311
\(787\) −47.0867 −1.67846 −0.839230 0.543777i \(-0.816994\pi\)
−0.839230 + 0.543777i \(0.816994\pi\)
\(788\) −11.7371 −0.418117
\(789\) 46.1893 1.64438
\(790\) −0.312854 −0.0111309
\(791\) 6.34869 0.225733
\(792\) 0 0
\(793\) −41.3295 −1.46765
\(794\) 9.64509 0.342291
\(795\) 0.297129 0.0105381
\(796\) −26.5898 −0.942451
\(797\) −7.48263 −0.265048 −0.132524 0.991180i \(-0.542308\pi\)
−0.132524 + 0.991180i \(0.542308\pi\)
\(798\) 7.59740 0.268945
\(799\) −15.0403 −0.532087
\(800\) 39.6468 1.40173
\(801\) 5.80213 0.205008
\(802\) −50.6690 −1.78918
\(803\) 0 0
\(804\) 15.1918 0.535774
\(805\) −0.217076 −0.00765094
\(806\) −38.8487 −1.36839
\(807\) −23.8059 −0.838006
\(808\) 0.495704 0.0174388
\(809\) 34.7490 1.22171 0.610855 0.791743i \(-0.290826\pi\)
0.610855 + 0.791743i \(0.290826\pi\)
\(810\) 0.0318653 0.00111963
\(811\) 12.8133 0.449935 0.224968 0.974366i \(-0.427772\pi\)
0.224968 + 0.974366i \(0.427772\pi\)
\(812\) −12.3919 −0.434869
\(813\) −17.6811 −0.620105
\(814\) 0 0
\(815\) −0.511268 −0.0179089
\(816\) −19.2704 −0.674599
\(817\) 5.13810 0.179759
\(818\) 41.7277 1.45898
\(819\) −23.2536 −0.812546
\(820\) 0.553121 0.0193158
\(821\) −54.3225 −1.89587 −0.947934 0.318466i \(-0.896832\pi\)
−0.947934 + 0.318466i \(0.896832\pi\)
\(822\) 105.825 3.69109
\(823\) −19.7697 −0.689129 −0.344565 0.938763i \(-0.611973\pi\)
−0.344565 + 0.938763i \(0.611973\pi\)
\(824\) 0.484024 0.0168618
\(825\) 0 0
\(826\) 19.3098 0.671873
\(827\) 12.3098 0.428055 0.214028 0.976828i \(-0.431342\pi\)
0.214028 + 0.976828i \(0.431342\pi\)
\(828\) 74.4491 2.58728
\(829\) −25.7259 −0.893496 −0.446748 0.894660i \(-0.647418\pi\)
−0.446748 + 0.894660i \(0.647418\pi\)
\(830\) −0.139215 −0.00483223
\(831\) −66.8857 −2.32024
\(832\) 36.6036 1.26900
\(833\) 1.67878 0.0581662
\(834\) 19.4150 0.672288
\(835\) 0.0813232 0.00281431
\(836\) 0 0
\(837\) 19.7125 0.681363
\(838\) −62.1434 −2.14671
\(839\) −46.6136 −1.60928 −0.804640 0.593762i \(-0.797642\pi\)
−0.804640 + 0.593762i \(0.797642\pi\)
\(840\) 0.00918610 0.000316950 0
\(841\) 11.8725 0.409397
\(842\) 25.3793 0.874627
\(843\) −8.94833 −0.308197
\(844\) 34.9577 1.20329
\(845\) −0.291465 −0.0100267
\(846\) −84.7009 −2.91208
\(847\) 0 0
\(848\) 16.3160 0.560292
\(849\) −57.9664 −1.98940
\(850\) −16.6554 −0.571275
\(851\) 4.20085 0.144003
\(852\) 11.0475 0.378482
\(853\) −9.80881 −0.335847 −0.167924 0.985800i \(-0.553706\pi\)
−0.167924 + 0.985800i \(0.553706\pi\)
\(854\) 16.8033 0.574998
\(855\) −0.176232 −0.00602699
\(856\) −1.27326 −0.0435193
\(857\) −32.9168 −1.12442 −0.562208 0.826996i \(-0.690048\pi\)
−0.562208 + 0.826996i \(0.690048\pi\)
\(858\) 0 0
\(859\) −28.4747 −0.971543 −0.485772 0.874086i \(-0.661461\pi\)
−0.485772 + 0.874086i \(0.661461\pi\)
\(860\) −0.195164 −0.00665505
\(861\) −29.5323 −1.00646
\(862\) 41.8095 1.42404
\(863\) 29.2770 0.996600 0.498300 0.867005i \(-0.333958\pi\)
0.498300 + 0.867005i \(0.333958\pi\)
\(864\) −38.9798 −1.32612
\(865\) −0.237911 −0.00808922
\(866\) 75.4176 2.56279
\(867\) −39.5158 −1.34203
\(868\) 7.77363 0.263854
\(869\) 0 0
\(870\) 0.951826 0.0322699
\(871\) −13.7298 −0.465218
\(872\) 0.462474 0.0156613
\(873\) 16.1819 0.547673
\(874\) 21.9832 0.743591
\(875\) 0.269224 0.00910143
\(876\) −56.5435 −1.91043
\(877\) −22.6025 −0.763233 −0.381616 0.924321i \(-0.624632\pi\)
−0.381616 + 0.924321i \(0.624632\pi\)
\(878\) −74.9506 −2.52946
\(879\) 15.3738 0.518544
\(880\) 0 0
\(881\) −35.1102 −1.18289 −0.591447 0.806344i \(-0.701443\pi\)
−0.591447 + 0.806344i \(0.701443\pi\)
\(882\) 9.45421 0.318340
\(883\) 15.2032 0.511628 0.255814 0.966726i \(-0.417657\pi\)
0.255814 + 0.966726i \(0.417657\pi\)
\(884\) −15.8830 −0.534205
\(885\) −0.729978 −0.0245380
\(886\) −2.03993 −0.0685327
\(887\) −38.5316 −1.29377 −0.646883 0.762589i \(-0.723928\pi\)
−0.646883 + 0.762589i \(0.723928\pi\)
\(888\) −0.177769 −0.00596553
\(889\) −5.03128 −0.168744
\(890\) −0.0650753 −0.00218133
\(891\) 0 0
\(892\) −12.6632 −0.423995
\(893\) −12.3092 −0.411912
\(894\) −17.2880 −0.578198
\(895\) −0.227897 −0.00761774
\(896\) 0.979100 0.0327094
\(897\) −109.655 −3.66128
\(898\) 39.9548 1.33331
\(899\) −25.6400 −0.855143
\(900\) −46.1635 −1.53878
\(901\) −6.64893 −0.221508
\(902\) 0 0
\(903\) 10.4202 0.346764
\(904\) −0.777369 −0.0258549
\(905\) 0.651590 0.0216596
\(906\) 109.053 3.62303
\(907\) −56.4214 −1.87344 −0.936721 0.350076i \(-0.886156\pi\)
−0.936721 + 0.350076i \(0.886156\pi\)
\(908\) 26.3758 0.875312
\(909\) −19.2863 −0.639687
\(910\) 0.260807 0.00864566
\(911\) −30.5904 −1.01350 −0.506752 0.862092i \(-0.669154\pi\)
−0.506752 + 0.862092i \(0.669154\pi\)
\(912\) −15.7712 −0.522237
\(913\) 0 0
\(914\) −47.9173 −1.58496
\(915\) −0.635226 −0.0209999
\(916\) −37.6439 −1.24379
\(917\) −3.76357 −0.124284
\(918\) 16.3752 0.540462
\(919\) 32.8904 1.08495 0.542477 0.840071i \(-0.317487\pi\)
0.542477 + 0.840071i \(0.317487\pi\)
\(920\) 0.0265801 0.000876319 0
\(921\) 23.9906 0.790517
\(922\) 22.9976 0.757385
\(923\) −9.98437 −0.328640
\(924\) 0 0
\(925\) −2.60481 −0.0856457
\(926\) 42.8808 1.40915
\(927\) −18.8319 −0.618521
\(928\) 50.7010 1.66434
\(929\) 21.8326 0.716305 0.358152 0.933663i \(-0.383407\pi\)
0.358152 + 0.933663i \(0.383407\pi\)
\(930\) −0.597097 −0.0195796
\(931\) 1.37394 0.0450291
\(932\) 8.45390 0.276917
\(933\) 53.2921 1.74471
\(934\) 31.5835 1.03344
\(935\) 0 0
\(936\) 2.84730 0.0930670
\(937\) −34.9523 −1.14184 −0.570921 0.821005i \(-0.693414\pi\)
−0.570921 + 0.821005i \(0.693414\pi\)
\(938\) 5.58214 0.182263
\(939\) −1.69066 −0.0551726
\(940\) 0.467550 0.0152498
\(941\) −20.4205 −0.665689 −0.332845 0.942982i \(-0.608008\pi\)
−0.332845 + 0.942982i \(0.608008\pi\)
\(942\) −126.931 −4.13562
\(943\) −85.4520 −2.78270
\(944\) −40.0845 −1.30464
\(945\) −0.132338 −0.00430494
\(946\) 0 0
\(947\) −11.3122 −0.367597 −0.183799 0.982964i \(-0.558839\pi\)
−0.183799 + 0.982964i \(0.558839\pi\)
\(948\) 31.6232 1.02707
\(949\) 51.1021 1.65884
\(950\) −13.6311 −0.442250
\(951\) −14.5250 −0.471005
\(952\) −0.205559 −0.00666222
\(953\) −4.55919 −0.147687 −0.0738434 0.997270i \(-0.523526\pi\)
−0.0738434 + 0.997270i \(0.523526\pi\)
\(954\) −37.4441 −1.21230
\(955\) 0.209665 0.00678459
\(956\) 20.5268 0.663885
\(957\) 0 0
\(958\) 10.4856 0.338775
\(959\) 19.1379 0.617994
\(960\) 0.562590 0.0181575
\(961\) −14.9156 −0.481147
\(962\) −5.04711 −0.162725
\(963\) 49.5388 1.59637
\(964\) 36.2242 1.16670
\(965\) −0.459428 −0.0147895
\(966\) 44.5825 1.43442
\(967\) −38.9153 −1.25143 −0.625715 0.780052i \(-0.715193\pi\)
−0.625715 + 0.780052i \(0.715193\pi\)
\(968\) 0 0
\(969\) 6.42694 0.206463
\(970\) −0.181492 −0.00582735
\(971\) 42.3409 1.35878 0.679392 0.733775i \(-0.262243\pi\)
0.679392 + 0.733775i \(0.262243\pi\)
\(972\) −31.8021 −1.02005
\(973\) 3.51108 0.112560
\(974\) −53.7120 −1.72104
\(975\) 67.9937 2.17754
\(976\) −34.8815 −1.11653
\(977\) 35.6575 1.14078 0.570392 0.821372i \(-0.306791\pi\)
0.570392 + 0.821372i \(0.306791\pi\)
\(978\) 105.003 3.35762
\(979\) 0 0
\(980\) −0.0521874 −0.00166707
\(981\) −17.9934 −0.574487
\(982\) 37.3088 1.19057
\(983\) −29.5213 −0.941583 −0.470791 0.882245i \(-0.656032\pi\)
−0.470791 + 0.882245i \(0.656032\pi\)
\(984\) 3.61610 0.115277
\(985\) 0.163036 0.00519477
\(986\) −21.2992 −0.678305
\(987\) −24.9635 −0.794596
\(988\) −12.9989 −0.413552
\(989\) 30.1511 0.958748
\(990\) 0 0
\(991\) −30.7292 −0.976145 −0.488072 0.872803i \(-0.662300\pi\)
−0.488072 + 0.872803i \(0.662300\pi\)
\(992\) −31.8056 −1.00983
\(993\) 1.86413 0.0591564
\(994\) 4.05934 0.128755
\(995\) 0.369351 0.0117092
\(996\) 14.0718 0.445883
\(997\) 49.9958 1.58338 0.791691 0.610922i \(-0.209201\pi\)
0.791691 + 0.610922i \(0.209201\pi\)
\(998\) −78.9175 −2.49809
\(999\) 2.56099 0.0810261
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.1.7 8
3.2 odd 2 7623.2.a.ct.1.2 8
7.6 odd 2 5929.2.a.bt.1.7 8
11.2 odd 10 847.2.f.x.323.4 16
11.3 even 5 847.2.f.w.372.4 16
11.4 even 5 847.2.f.w.148.4 16
11.5 even 5 77.2.f.b.36.1 yes 16
11.6 odd 10 847.2.f.x.729.4 16
11.7 odd 10 847.2.f.v.148.1 16
11.8 odd 10 847.2.f.v.372.1 16
11.9 even 5 77.2.f.b.15.1 16
11.10 odd 2 847.2.a.o.1.2 8
33.5 odd 10 693.2.m.i.190.4 16
33.20 odd 10 693.2.m.i.631.4 16
33.32 even 2 7623.2.a.cw.1.7 8
77.5 odd 30 539.2.q.f.410.4 32
77.9 even 15 539.2.q.g.312.1 32
77.16 even 15 539.2.q.g.410.4 32
77.20 odd 10 539.2.f.e.246.1 16
77.27 odd 10 539.2.f.e.344.1 16
77.31 odd 30 539.2.q.f.422.4 32
77.38 odd 30 539.2.q.f.520.1 32
77.53 even 15 539.2.q.g.422.4 32
77.60 even 15 539.2.q.g.520.1 32
77.75 odd 30 539.2.q.f.312.1 32
77.76 even 2 5929.2.a.bs.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.1 16 11.9 even 5
77.2.f.b.36.1 yes 16 11.5 even 5
539.2.f.e.246.1 16 77.20 odd 10
539.2.f.e.344.1 16 77.27 odd 10
539.2.q.f.312.1 32 77.75 odd 30
539.2.q.f.410.4 32 77.5 odd 30
539.2.q.f.422.4 32 77.31 odd 30
539.2.q.f.520.1 32 77.38 odd 30
539.2.q.g.312.1 32 77.9 even 15
539.2.q.g.410.4 32 77.16 even 15
539.2.q.g.422.4 32 77.53 even 15
539.2.q.g.520.1 32 77.60 even 15
693.2.m.i.190.4 16 33.5 odd 10
693.2.m.i.631.4 16 33.20 odd 10
847.2.a.o.1.2 8 11.10 odd 2
847.2.a.p.1.7 8 1.1 even 1 trivial
847.2.f.v.148.1 16 11.7 odd 10
847.2.f.v.372.1 16 11.8 odd 10
847.2.f.w.148.4 16 11.4 even 5
847.2.f.w.372.4 16 11.3 even 5
847.2.f.x.323.4 16 11.2 odd 10
847.2.f.x.729.4 16 11.6 odd 10
5929.2.a.bs.1.2 8 77.76 even 2
5929.2.a.bt.1.7 8 7.6 odd 2
7623.2.a.ct.1.2 8 3.2 odd 2
7623.2.a.cw.1.7 8 33.32 even 2