Properties

Label 847.2.a.i.1.2
Level $847$
Weight $2$
Character 847.1
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) 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)
 
\(f(q)\) \(=\) \(q-1.36333 q^{2} +0.363328 q^{3} -0.141336 q^{4} +3.14134 q^{5} -0.495336 q^{6} -1.00000 q^{7} +2.91934 q^{8} -2.86799 q^{9} +O(q^{10})\) \(q-1.36333 q^{2} +0.363328 q^{3} -0.141336 q^{4} +3.14134 q^{5} -0.495336 q^{6} -1.00000 q^{7} +2.91934 q^{8} -2.86799 q^{9} -4.28267 q^{10} -0.0513514 q^{12} -4.77801 q^{13} +1.36333 q^{14} +1.14134 q^{15} -3.69735 q^{16} -4.77801 q^{17} +3.91002 q^{18} -7.00933 q^{19} -0.443984 q^{20} -0.363328 q^{21} +5.14134 q^{23} +1.06068 q^{24} +4.86799 q^{25} +6.51399 q^{26} -2.13201 q^{27} +0.141336 q^{28} +7.00933 q^{29} -1.55602 q^{30} -3.63667 q^{31} -0.797984 q^{32} +6.51399 q^{34} -3.14134 q^{35} +0.405351 q^{36} -9.86799 q^{37} +9.55602 q^{38} -1.73599 q^{39} +9.17064 q^{40} -3.22199 q^{41} +0.495336 q^{42} +4.28267 q^{43} -9.00933 q^{45} -7.00933 q^{46} -0.778008 q^{47} -1.34335 q^{48} +1.00000 q^{49} -6.63667 q^{50} -1.73599 q^{51} +0.675305 q^{52} +2.28267 q^{53} +2.90663 q^{54} -2.91934 q^{56} -2.54669 q^{57} -9.55602 q^{58} -0.363328 q^{59} -0.161312 q^{60} +3.22199 q^{61} +4.95798 q^{62} +2.86799 q^{63} +8.48262 q^{64} -15.0093 q^{65} -6.59465 q^{67} +0.675305 q^{68} +1.86799 q^{69} +4.28267 q^{70} -15.1600 q^{71} -8.37266 q^{72} -3.22199 q^{73} +13.4533 q^{74} +1.76868 q^{75} +0.990671 q^{76} +2.36672 q^{78} -3.71733 q^{79} -11.6146 q^{80} +7.82936 q^{81} +4.39263 q^{82} +1.55602 q^{83} +0.0513514 q^{84} -15.0093 q^{85} -5.83869 q^{86} +2.54669 q^{87} -5.58532 q^{89} +12.2827 q^{90} +4.77801 q^{91} -0.726656 q^{92} -1.32131 q^{93} +1.06068 q^{94} -22.0187 q^{95} -0.289930 q^{96} +6.15066 q^{97} -1.36333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} + 4 q^{9} + 4 q^{10} + 2 q^{12} - 8 q^{13} + 2 q^{14} - 5 q^{15} + 10 q^{16} - 8 q^{17} + 18 q^{18} - 14 q^{20} + q^{21} + 7 q^{23} - 20 q^{24} + 2 q^{25} - 12 q^{26} - 19 q^{27} - 8 q^{28} + 8 q^{30} - 13 q^{31} - 34 q^{32} - 12 q^{34} - q^{35} + 18 q^{36} - 17 q^{37} + 16 q^{38} + 20 q^{39} + 36 q^{40} - 16 q^{41} + 12 q^{42} - 4 q^{43} - 6 q^{45} + 4 q^{47} + 36 q^{48} + 3 q^{49} - 22 q^{50} + 20 q^{51} - 10 q^{53} - 8 q^{54} + 6 q^{56} - 16 q^{57} - 16 q^{58} + q^{59} - 30 q^{60} + 16 q^{61} - 4 q^{62} - 4 q^{63} + 34 q^{64} - 24 q^{65} - 3 q^{67} - 7 q^{69} - 4 q^{70} + 5 q^{71} - 2 q^{72} - 16 q^{73} + 32 q^{74} + 20 q^{75} + 24 q^{76} + 28 q^{78} - 28 q^{79} - 56 q^{80} + 15 q^{81} + 28 q^{82} - 8 q^{83} - 2 q^{84} - 24 q^{85} + 12 q^{86} + 16 q^{87} - 21 q^{89} + 20 q^{90} + 8 q^{91} + 2 q^{92} + 17 q^{93} - 20 q^{94} - 24 q^{95} - 20 q^{96} - 11 q^{97} - 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36333 −0.964019 −0.482009 0.876166i \(-0.660093\pi\)
−0.482009 + 0.876166i \(0.660093\pi\)
\(3\) 0.363328 0.209768 0.104884 0.994484i \(-0.466553\pi\)
0.104884 + 0.994484i \(0.466553\pi\)
\(4\) −0.141336 −0.0706681
\(5\) 3.14134 1.40485 0.702424 0.711759i \(-0.252101\pi\)
0.702424 + 0.711759i \(0.252101\pi\)
\(6\) −0.495336 −0.202220
\(7\) −1.00000 −0.377964
\(8\) 2.91934 1.03214
\(9\) −2.86799 −0.955998
\(10\) −4.28267 −1.35430
\(11\) 0 0
\(12\) −0.0513514 −0.0148239
\(13\) −4.77801 −1.32518 −0.662590 0.748982i \(-0.730543\pi\)
−0.662590 + 0.748982i \(0.730543\pi\)
\(14\) 1.36333 0.364365
\(15\) 1.14134 0.294692
\(16\) −3.69735 −0.924338
\(17\) −4.77801 −1.15884 −0.579419 0.815030i \(-0.696720\pi\)
−0.579419 + 0.815030i \(0.696720\pi\)
\(18\) 3.91002 0.921599
\(19\) −7.00933 −1.60805 −0.804025 0.594595i \(-0.797312\pi\)
−0.804025 + 0.594595i \(0.797312\pi\)
\(20\) −0.443984 −0.0992779
\(21\) −0.363328 −0.0792847
\(22\) 0 0
\(23\) 5.14134 1.07204 0.536021 0.844204i \(-0.319927\pi\)
0.536021 + 0.844204i \(0.319927\pi\)
\(24\) 1.06068 0.216510
\(25\) 4.86799 0.973599
\(26\) 6.51399 1.27750
\(27\) −2.13201 −0.410305
\(28\) 0.141336 0.0267100
\(29\) 7.00933 1.30160 0.650800 0.759249i \(-0.274434\pi\)
0.650800 + 0.759249i \(0.274434\pi\)
\(30\) −1.55602 −0.284088
\(31\) −3.63667 −0.653166 −0.326583 0.945169i \(-0.605897\pi\)
−0.326583 + 0.945169i \(0.605897\pi\)
\(32\) −0.797984 −0.141065
\(33\) 0 0
\(34\) 6.51399 1.11714
\(35\) −3.14134 −0.530983
\(36\) 0.405351 0.0675585
\(37\) −9.86799 −1.62229 −0.811144 0.584846i \(-0.801155\pi\)
−0.811144 + 0.584846i \(0.801155\pi\)
\(38\) 9.55602 1.55019
\(39\) −1.73599 −0.277980
\(40\) 9.17064 1.45001
\(41\) −3.22199 −0.503191 −0.251595 0.967833i \(-0.580955\pi\)
−0.251595 + 0.967833i \(0.580955\pi\)
\(42\) 0.495336 0.0764319
\(43\) 4.28267 0.653101 0.326551 0.945180i \(-0.394114\pi\)
0.326551 + 0.945180i \(0.394114\pi\)
\(44\) 0 0
\(45\) −9.00933 −1.34303
\(46\) −7.00933 −1.03347
\(47\) −0.778008 −0.113484 −0.0567421 0.998389i \(-0.518071\pi\)
−0.0567421 + 0.998389i \(0.518071\pi\)
\(48\) −1.34335 −0.193896
\(49\) 1.00000 0.142857
\(50\) −6.63667 −0.938567
\(51\) −1.73599 −0.243087
\(52\) 0.675305 0.0936480
\(53\) 2.28267 0.313549 0.156775 0.987634i \(-0.449890\pi\)
0.156775 + 0.987634i \(0.449890\pi\)
\(54\) 2.90663 0.395542
\(55\) 0 0
\(56\) −2.91934 −0.390114
\(57\) −2.54669 −0.337317
\(58\) −9.55602 −1.25477
\(59\) −0.363328 −0.0473013 −0.0236507 0.999720i \(-0.507529\pi\)
−0.0236507 + 0.999720i \(0.507529\pi\)
\(60\) −0.161312 −0.0208253
\(61\) 3.22199 0.412534 0.206267 0.978496i \(-0.433868\pi\)
0.206267 + 0.978496i \(0.433868\pi\)
\(62\) 4.95798 0.629664
\(63\) 2.86799 0.361333
\(64\) 8.48262 1.06033
\(65\) −15.0093 −1.86168
\(66\) 0 0
\(67\) −6.59465 −0.805665 −0.402832 0.915274i \(-0.631974\pi\)
−0.402832 + 0.915274i \(0.631974\pi\)
\(68\) 0.675305 0.0818928
\(69\) 1.86799 0.224880
\(70\) 4.28267 0.511877
\(71\) −15.1600 −1.79916 −0.899580 0.436756i \(-0.856127\pi\)
−0.899580 + 0.436756i \(0.856127\pi\)
\(72\) −8.37266 −0.986727
\(73\) −3.22199 −0.377106 −0.188553 0.982063i \(-0.560380\pi\)
−0.188553 + 0.982063i \(0.560380\pi\)
\(74\) 13.4533 1.56392
\(75\) 1.76868 0.204229
\(76\) 0.990671 0.113638
\(77\) 0 0
\(78\) 2.36672 0.267978
\(79\) −3.71733 −0.418232 −0.209116 0.977891i \(-0.567059\pi\)
−0.209116 + 0.977891i \(0.567059\pi\)
\(80\) −11.6146 −1.29855
\(81\) 7.82936 0.869929
\(82\) 4.39263 0.485085
\(83\) 1.55602 0.170795 0.0853975 0.996347i \(-0.472784\pi\)
0.0853975 + 0.996347i \(0.472784\pi\)
\(84\) 0.0513514 0.00560290
\(85\) −15.0093 −1.62799
\(86\) −5.83869 −0.629602
\(87\) 2.54669 0.273034
\(88\) 0 0
\(89\) −5.58532 −0.592043 −0.296021 0.955181i \(-0.595660\pi\)
−0.296021 + 0.955181i \(0.595660\pi\)
\(90\) 12.2827 1.29471
\(91\) 4.77801 0.500871
\(92\) −0.726656 −0.0757592
\(93\) −1.32131 −0.137013
\(94\) 1.06068 0.109401
\(95\) −22.0187 −2.25907
\(96\) −0.289930 −0.0295909
\(97\) 6.15066 0.624505 0.312253 0.949999i \(-0.398916\pi\)
0.312253 + 0.949999i \(0.398916\pi\)
\(98\) −1.36333 −0.137717
\(99\) 0 0
\(100\) −0.688023 −0.0688023
\(101\) −10.7967 −1.07431 −0.537154 0.843484i \(-0.680501\pi\)
−0.537154 + 0.843484i \(0.680501\pi\)
\(102\) 2.36672 0.234340
\(103\) −13.2406 −1.30464 −0.652320 0.757944i \(-0.726204\pi\)
−0.652320 + 0.757944i \(0.726204\pi\)
\(104\) −13.9486 −1.36778
\(105\) −1.14134 −0.111383
\(106\) −3.11203 −0.302267
\(107\) 14.0187 1.35523 0.677617 0.735415i \(-0.263013\pi\)
0.677617 + 0.735415i \(0.263013\pi\)
\(108\) 0.301330 0.0289955
\(109\) −15.5747 −1.49178 −0.745892 0.666067i \(-0.767976\pi\)
−0.745892 + 0.666067i \(0.767976\pi\)
\(110\) 0 0
\(111\) −3.58532 −0.340304
\(112\) 3.69735 0.349367
\(113\) 4.13201 0.388707 0.194353 0.980932i \(-0.437739\pi\)
0.194353 + 0.980932i \(0.437739\pi\)
\(114\) 3.47197 0.325180
\(115\) 16.1507 1.50606
\(116\) −0.990671 −0.0919815
\(117\) 13.7033 1.26687
\(118\) 0.495336 0.0455993
\(119\) 4.77801 0.437999
\(120\) 3.33195 0.304164
\(121\) 0 0
\(122\) −4.39263 −0.397690
\(123\) −1.17064 −0.105553
\(124\) 0.513993 0.0461579
\(125\) −0.414680 −0.0370901
\(126\) −3.91002 −0.348332
\(127\) 22.0187 1.95384 0.976920 0.213606i \(-0.0685208\pi\)
0.976920 + 0.213606i \(0.0685208\pi\)
\(128\) −9.96862 −0.881110
\(129\) 1.55602 0.137000
\(130\) 20.4626 1.79469
\(131\) 15.0093 1.31137 0.655686 0.755034i \(-0.272380\pi\)
0.655686 + 0.755034i \(0.272380\pi\)
\(132\) 0 0
\(133\) 7.00933 0.607786
\(134\) 8.99067 0.776676
\(135\) −6.69735 −0.576416
\(136\) −13.9486 −1.19609
\(137\) −10.6974 −0.913936 −0.456968 0.889483i \(-0.651065\pi\)
−0.456968 + 0.889483i \(0.651065\pi\)
\(138\) −2.54669 −0.216788
\(139\) 4.10270 0.347987 0.173993 0.984747i \(-0.444333\pi\)
0.173993 + 0.984747i \(0.444333\pi\)
\(140\) 0.443984 0.0375235
\(141\) −0.282672 −0.0238053
\(142\) 20.6680 1.73442
\(143\) 0 0
\(144\) 10.6040 0.883665
\(145\) 22.0187 1.82855
\(146\) 4.39263 0.363537
\(147\) 0.363328 0.0299668
\(148\) 1.39470 0.114644
\(149\) 14.0187 1.14845 0.574227 0.818696i \(-0.305303\pi\)
0.574227 + 0.818696i \(0.305303\pi\)
\(150\) −2.41129 −0.196881
\(151\) −6.82936 −0.555765 −0.277883 0.960615i \(-0.589633\pi\)
−0.277883 + 0.960615i \(0.589633\pi\)
\(152\) −20.4626 −1.65974
\(153\) 13.7033 1.10785
\(154\) 0 0
\(155\) −11.4240 −0.917598
\(156\) 0.245357 0.0196443
\(157\) −23.4427 −1.87093 −0.935464 0.353421i \(-0.885018\pi\)
−0.935464 + 0.353421i \(0.885018\pi\)
\(158\) 5.06794 0.403183
\(159\) 0.829359 0.0657725
\(160\) −2.50674 −0.198175
\(161\) −5.14134 −0.405194
\(162\) −10.6740 −0.838628
\(163\) −0.990671 −0.0775954 −0.0387977 0.999247i \(-0.512353\pi\)
−0.0387977 + 0.999247i \(0.512353\pi\)
\(164\) 0.455384 0.0355595
\(165\) 0 0
\(166\) −2.12136 −0.164649
\(167\) 0.565344 0.0437477 0.0218738 0.999761i \(-0.493037\pi\)
0.0218738 + 0.999761i \(0.493037\pi\)
\(168\) −1.06068 −0.0818332
\(169\) 9.82936 0.756105
\(170\) 20.4626 1.56941
\(171\) 20.1027 1.53729
\(172\) −0.605296 −0.0461534
\(173\) −17.2406 −1.31078 −0.655391 0.755290i \(-0.727496\pi\)
−0.655391 + 0.755290i \(0.727496\pi\)
\(174\) −3.47197 −0.263209
\(175\) −4.86799 −0.367986
\(176\) 0 0
\(177\) −0.132007 −0.00992228
\(178\) 7.61462 0.570740
\(179\) 11.3213 0.846194 0.423097 0.906084i \(-0.360943\pi\)
0.423097 + 0.906084i \(0.360943\pi\)
\(180\) 1.27334 0.0949094
\(181\) 14.8773 1.10582 0.552911 0.833240i \(-0.313517\pi\)
0.552911 + 0.833240i \(0.313517\pi\)
\(182\) −6.51399 −0.482849
\(183\) 1.17064 0.0865363
\(184\) 15.0093 1.10650
\(185\) −30.9987 −2.27907
\(186\) 1.80137 0.132083
\(187\) 0 0
\(188\) 0.109961 0.00801970
\(189\) 2.13201 0.155081
\(190\) 30.0187 2.17778
\(191\) 3.84934 0.278528 0.139264 0.990255i \(-0.455526\pi\)
0.139264 + 0.990255i \(0.455526\pi\)
\(192\) 3.08197 0.222422
\(193\) −2.54669 −0.183315 −0.0916573 0.995791i \(-0.529216\pi\)
−0.0916573 + 0.995791i \(0.529216\pi\)
\(194\) −8.38538 −0.602035
\(195\) −5.45331 −0.390520
\(196\) −0.141336 −0.0100954
\(197\) −10.5467 −0.751420 −0.375710 0.926737i \(-0.622601\pi\)
−0.375710 + 0.926737i \(0.622601\pi\)
\(198\) 0 0
\(199\) 11.6846 0.828302 0.414151 0.910208i \(-0.364079\pi\)
0.414151 + 0.910208i \(0.364079\pi\)
\(200\) 14.2113 1.00489
\(201\) −2.39602 −0.169002
\(202\) 14.7194 1.03565
\(203\) −7.00933 −0.491958
\(204\) 0.245357 0.0171785
\(205\) −10.1214 −0.706906
\(206\) 18.0514 1.25770
\(207\) −14.7453 −1.02487
\(208\) 17.6660 1.22492
\(209\) 0 0
\(210\) 1.55602 0.107375
\(211\) 11.1120 0.764984 0.382492 0.923959i \(-0.375066\pi\)
0.382492 + 0.923959i \(0.375066\pi\)
\(212\) −0.322624 −0.0221579
\(213\) −5.50805 −0.377406
\(214\) −19.1120 −1.30647
\(215\) 13.4533 0.917508
\(216\) −6.22406 −0.423494
\(217\) 3.63667 0.246873
\(218\) 21.2334 1.43811
\(219\) −1.17064 −0.0791046
\(220\) 0 0
\(221\) 22.8294 1.53567
\(222\) 4.88797 0.328059
\(223\) 28.0407 1.87774 0.938872 0.344266i \(-0.111872\pi\)
0.938872 + 0.344266i \(0.111872\pi\)
\(224\) 0.797984 0.0533175
\(225\) −13.9614 −0.930758
\(226\) −5.63328 −0.374720
\(227\) −23.0093 −1.52718 −0.763591 0.645700i \(-0.776566\pi\)
−0.763591 + 0.645700i \(0.776566\pi\)
\(228\) 0.359939 0.0238375
\(229\) 13.4240 0.887083 0.443542 0.896254i \(-0.353722\pi\)
0.443542 + 0.896254i \(0.353722\pi\)
\(230\) −22.0187 −1.45187
\(231\) 0 0
\(232\) 20.4626 1.34344
\(233\) −3.53736 −0.231740 −0.115870 0.993264i \(-0.536966\pi\)
−0.115870 + 0.993264i \(0.536966\pi\)
\(234\) −18.6821 −1.22129
\(235\) −2.44398 −0.159428
\(236\) 0.0513514 0.00334269
\(237\) −1.35061 −0.0877316
\(238\) −6.51399 −0.422239
\(239\) 22.0187 1.42427 0.712134 0.702043i \(-0.247729\pi\)
0.712134 + 0.702043i \(0.247729\pi\)
\(240\) −4.21992 −0.272395
\(241\) −0.315366 −0.0203145 −0.0101573 0.999948i \(-0.503233\pi\)
−0.0101573 + 0.999948i \(0.503233\pi\)
\(242\) 0 0
\(243\) 9.24065 0.592788
\(244\) −0.455384 −0.0291530
\(245\) 3.14134 0.200693
\(246\) 1.59597 0.101755
\(247\) 33.4906 2.13096
\(248\) −10.6167 −0.674161
\(249\) 0.565344 0.0358272
\(250\) 0.565344 0.0357555
\(251\) 18.6460 1.17693 0.588463 0.808524i \(-0.299733\pi\)
0.588463 + 0.808524i \(0.299733\pi\)
\(252\) −0.405351 −0.0255347
\(253\) 0 0
\(254\) −30.0187 −1.88354
\(255\) −5.45331 −0.341500
\(256\) −3.37473 −0.210920
\(257\) −8.54669 −0.533128 −0.266564 0.963817i \(-0.585888\pi\)
−0.266564 + 0.963817i \(0.585888\pi\)
\(258\) −2.12136 −0.132070
\(259\) 9.86799 0.613167
\(260\) 2.12136 0.131561
\(261\) −20.1027 −1.24433
\(262\) −20.4626 −1.26419
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 7.17064 0.440489
\(266\) −9.55602 −0.585917
\(267\) −2.02930 −0.124191
\(268\) 0.932062 0.0569348
\(269\) 7.55602 0.460698 0.230349 0.973108i \(-0.426013\pi\)
0.230349 + 0.973108i \(0.426013\pi\)
\(270\) 9.13069 0.555676
\(271\) 14.0187 0.851573 0.425786 0.904824i \(-0.359998\pi\)
0.425786 + 0.904824i \(0.359998\pi\)
\(272\) 17.6660 1.07116
\(273\) 1.73599 0.105067
\(274\) 14.5840 0.881052
\(275\) 0 0
\(276\) −0.264015 −0.0158918
\(277\) −6.01866 −0.361626 −0.180813 0.983517i \(-0.557873\pi\)
−0.180813 + 0.983517i \(0.557873\pi\)
\(278\) −5.59333 −0.335466
\(279\) 10.4299 0.624425
\(280\) −9.17064 −0.548051
\(281\) −20.6680 −1.23295 −0.616476 0.787374i \(-0.711440\pi\)
−0.616476 + 0.787374i \(0.711440\pi\)
\(282\) 0.385375 0.0229487
\(283\) 9.91595 0.589442 0.294721 0.955583i \(-0.404773\pi\)
0.294721 + 0.955583i \(0.404773\pi\)
\(284\) 2.14265 0.127143
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 3.22199 0.190188
\(288\) 2.28861 0.134858
\(289\) 5.82936 0.342903
\(290\) −30.0187 −1.76276
\(291\) 2.23471 0.131001
\(292\) 0.455384 0.0266493
\(293\) −26.7967 −1.56548 −0.782739 0.622350i \(-0.786178\pi\)
−0.782739 + 0.622350i \(0.786178\pi\)
\(294\) −0.495336 −0.0288886
\(295\) −1.14134 −0.0664512
\(296\) −28.8081 −1.67443
\(297\) 0 0
\(298\) −19.1120 −1.10713
\(299\) −24.5653 −1.42065
\(300\) −0.249978 −0.0144325
\(301\) −4.28267 −0.246849
\(302\) 9.31066 0.535768
\(303\) −3.92273 −0.225355
\(304\) 25.9160 1.48638
\(305\) 10.1214 0.579547
\(306\) −18.6821 −1.06798
\(307\) 11.8973 0.679015 0.339507 0.940603i \(-0.389740\pi\)
0.339507 + 0.940603i \(0.389740\pi\)
\(308\) 0 0
\(309\) −4.81070 −0.273671
\(310\) 15.5747 0.884582
\(311\) −21.2406 −1.20445 −0.602223 0.798328i \(-0.705718\pi\)
−0.602223 + 0.798328i \(0.705718\pi\)
\(312\) −5.06794 −0.286915
\(313\) −12.6974 −0.717697 −0.358848 0.933396i \(-0.616830\pi\)
−0.358848 + 0.933396i \(0.616830\pi\)
\(314\) 31.9600 1.80361
\(315\) 9.00933 0.507618
\(316\) 0.525393 0.0295556
\(317\) −11.8680 −0.666573 −0.333286 0.942826i \(-0.608158\pi\)
−0.333286 + 0.942826i \(0.608158\pi\)
\(318\) −1.13069 −0.0634059
\(319\) 0 0
\(320\) 26.6468 1.48960
\(321\) 5.09337 0.284284
\(322\) 7.00933 0.390615
\(323\) 33.4906 1.86347
\(324\) −1.10657 −0.0614762
\(325\) −23.2593 −1.29019
\(326\) 1.35061 0.0748034
\(327\) −5.65872 −0.312928
\(328\) −9.40610 −0.519365
\(329\) 0.778008 0.0428930
\(330\) 0 0
\(331\) −11.8867 −0.653349 −0.326675 0.945137i \(-0.605928\pi\)
−0.326675 + 0.945137i \(0.605928\pi\)
\(332\) −0.219921 −0.0120697
\(333\) 28.3013 1.55090
\(334\) −0.770750 −0.0421736
\(335\) −20.7160 −1.13184
\(336\) 1.34335 0.0732859
\(337\) 0.990671 0.0539653 0.0269827 0.999636i \(-0.491410\pi\)
0.0269827 + 0.999636i \(0.491410\pi\)
\(338\) −13.4006 −0.728899
\(339\) 1.50127 0.0815381
\(340\) 2.12136 0.115047
\(341\) 0 0
\(342\) −27.4066 −1.48198
\(343\) −1.00000 −0.0539949
\(344\) 12.5026 0.674095
\(345\) 5.86799 0.315922
\(346\) 23.5047 1.26362
\(347\) −15.1893 −0.815404 −0.407702 0.913115i \(-0.633670\pi\)
−0.407702 + 0.913115i \(0.633670\pi\)
\(348\) −0.359939 −0.0192947
\(349\) −29.7033 −1.58998 −0.794990 0.606622i \(-0.792524\pi\)
−0.794990 + 0.606622i \(0.792524\pi\)
\(350\) 6.63667 0.354745
\(351\) 10.1867 0.543728
\(352\) 0 0
\(353\) −6.71601 −0.357457 −0.178729 0.983898i \(-0.557198\pi\)
−0.178729 + 0.983898i \(0.557198\pi\)
\(354\) 0.179969 0.00956527
\(355\) −47.6226 −2.52755
\(356\) 0.789407 0.0418385
\(357\) 1.73599 0.0918781
\(358\) −15.4347 −0.815747
\(359\) 17.7360 0.936069 0.468035 0.883710i \(-0.344962\pi\)
0.468035 + 0.883710i \(0.344962\pi\)
\(360\) −26.3013 −1.38620
\(361\) 30.1307 1.58583
\(362\) −20.2827 −1.06603
\(363\) 0 0
\(364\) −0.675305 −0.0353956
\(365\) −10.1214 −0.529776
\(366\) −1.59597 −0.0834226
\(367\) −8.09931 −0.422781 −0.211390 0.977402i \(-0.567799\pi\)
−0.211390 + 0.977402i \(0.567799\pi\)
\(368\) −19.0093 −0.990930
\(369\) 9.24065 0.481049
\(370\) 42.2614 2.19706
\(371\) −2.28267 −0.118510
\(372\) 0.186748 0.00968244
\(373\) 0.565344 0.0292724 0.0146362 0.999893i \(-0.495341\pi\)
0.0146362 + 0.999893i \(0.495341\pi\)
\(374\) 0 0
\(375\) −0.150665 −0.00778030
\(376\) −2.27127 −0.117132
\(377\) −33.4906 −1.72486
\(378\) −2.90663 −0.149501
\(379\) −22.5360 −1.15760 −0.578799 0.815470i \(-0.696479\pi\)
−0.578799 + 0.815470i \(0.696479\pi\)
\(380\) 3.11203 0.159644
\(381\) 8.00000 0.409852
\(382\) −5.24791 −0.268506
\(383\) 21.3913 1.09305 0.546523 0.837444i \(-0.315951\pi\)
0.546523 + 0.837444i \(0.315951\pi\)
\(384\) −3.62188 −0.184828
\(385\) 0 0
\(386\) 3.47197 0.176719
\(387\) −12.2827 −0.624363
\(388\) −0.869311 −0.0441326
\(389\) 10.6787 0.541431 0.270716 0.962659i \(-0.412740\pi\)
0.270716 + 0.962659i \(0.412740\pi\)
\(390\) 7.43466 0.376468
\(391\) −24.5653 −1.24232
\(392\) 2.91934 0.147449
\(393\) 5.45331 0.275083
\(394\) 14.3786 0.724383
\(395\) −11.6774 −0.587553
\(396\) 0 0
\(397\) −23.9160 −1.20031 −0.600154 0.799885i \(-0.704894\pi\)
−0.600154 + 0.799885i \(0.704894\pi\)
\(398\) −15.9300 −0.798498
\(399\) 2.54669 0.127494
\(400\) −17.9987 −0.899934
\(401\) 11.6587 0.582209 0.291104 0.956691i \(-0.405977\pi\)
0.291104 + 0.956691i \(0.405977\pi\)
\(402\) 3.26656 0.162921
\(403\) 17.3760 0.865563
\(404\) 1.52596 0.0759193
\(405\) 24.5946 1.22212
\(406\) 9.55602 0.474257
\(407\) 0 0
\(408\) −5.06794 −0.250900
\(409\) 32.8153 1.62261 0.811307 0.584621i \(-0.198757\pi\)
0.811307 + 0.584621i \(0.198757\pi\)
\(410\) 13.7987 0.681471
\(411\) −3.88665 −0.191714
\(412\) 1.87138 0.0921964
\(413\) 0.363328 0.0178782
\(414\) 20.1027 0.987994
\(415\) 4.88797 0.239941
\(416\) 3.81277 0.186937
\(417\) 1.49063 0.0729964
\(418\) 0 0
\(419\) 10.7967 0.527452 0.263726 0.964598i \(-0.415049\pi\)
0.263726 + 0.964598i \(0.415049\pi\)
\(420\) 0.161312 0.00787122
\(421\) −1.15198 −0.0561442 −0.0280721 0.999606i \(-0.508937\pi\)
−0.0280721 + 0.999606i \(0.508937\pi\)
\(422\) −15.1493 −0.737459
\(423\) 2.23132 0.108491
\(424\) 6.66391 0.323628
\(425\) −23.2593 −1.12824
\(426\) 7.50929 0.363826
\(427\) −3.22199 −0.155923
\(428\) −1.98134 −0.0957718
\(429\) 0 0
\(430\) −18.3413 −0.884495
\(431\) −31.9600 −1.53946 −0.769731 0.638369i \(-0.779609\pi\)
−0.769731 + 0.638369i \(0.779609\pi\)
\(432\) 7.88278 0.379261
\(433\) −4.43334 −0.213053 −0.106526 0.994310i \(-0.533973\pi\)
−0.106526 + 0.994310i \(0.533973\pi\)
\(434\) −4.95798 −0.237991
\(435\) 8.00000 0.383571
\(436\) 2.20126 0.105421
\(437\) −36.0373 −1.72390
\(438\) 1.59597 0.0762583
\(439\) −27.4720 −1.31117 −0.655583 0.755123i \(-0.727577\pi\)
−0.655583 + 0.755123i \(0.727577\pi\)
\(440\) 0 0
\(441\) −2.86799 −0.136571
\(442\) −31.1239 −1.48041
\(443\) −7.79073 −0.370149 −0.185074 0.982725i \(-0.559253\pi\)
−0.185074 + 0.982725i \(0.559253\pi\)
\(444\) 0.506735 0.0240486
\(445\) −17.5454 −0.831730
\(446\) −38.2287 −1.81018
\(447\) 5.09337 0.240908
\(448\) −8.48262 −0.400766
\(449\) 20.5106 0.967955 0.483978 0.875080i \(-0.339192\pi\)
0.483978 + 0.875080i \(0.339192\pi\)
\(450\) 19.0339 0.897268
\(451\) 0 0
\(452\) −0.584002 −0.0274691
\(453\) −2.48130 −0.116582
\(454\) 31.3693 1.47223
\(455\) 15.0093 0.703648
\(456\) −7.43466 −0.348160
\(457\) −8.56534 −0.400670 −0.200335 0.979727i \(-0.564203\pi\)
−0.200335 + 0.979727i \(0.564203\pi\)
\(458\) −18.3013 −0.855165
\(459\) 10.1867 0.475477
\(460\) −2.28267 −0.106430
\(461\) 9.66598 0.450189 0.225095 0.974337i \(-0.427731\pi\)
0.225095 + 0.974337i \(0.427731\pi\)
\(462\) 0 0
\(463\) 11.4240 0.530919 0.265459 0.964122i \(-0.414476\pi\)
0.265459 + 0.964122i \(0.414476\pi\)
\(464\) −25.9160 −1.20312
\(465\) −4.15066 −0.192482
\(466\) 4.82258 0.223402
\(467\) 18.2207 0.843152 0.421576 0.906793i \(-0.361477\pi\)
0.421576 + 0.906793i \(0.361477\pi\)
\(468\) −1.93677 −0.0895272
\(469\) 6.59465 0.304513
\(470\) 3.33195 0.153692
\(471\) −8.51738 −0.392460
\(472\) −1.06068 −0.0488218
\(473\) 0 0
\(474\) 1.84132 0.0845749
\(475\) −34.1214 −1.56560
\(476\) −0.675305 −0.0309526
\(477\) −6.54669 −0.299752
\(478\) −30.0187 −1.37302
\(479\) 3.47197 0.158638 0.0793192 0.996849i \(-0.474725\pi\)
0.0793192 + 0.996849i \(0.474725\pi\)
\(480\) −0.910768 −0.0415707
\(481\) 47.1493 2.14983
\(482\) 0.429948 0.0195836
\(483\) −1.86799 −0.0849966
\(484\) 0 0
\(485\) 19.3213 0.877335
\(486\) −12.5980 −0.571459
\(487\) −27.4613 −1.24439 −0.622196 0.782862i \(-0.713759\pi\)
−0.622196 + 0.782862i \(0.713759\pi\)
\(488\) 9.40610 0.425794
\(489\) −0.359939 −0.0162770
\(490\) −4.28267 −0.193471
\(491\) −6.26401 −0.282691 −0.141346 0.989960i \(-0.545143\pi\)
−0.141346 + 0.989960i \(0.545143\pi\)
\(492\) 0.165454 0.00745923
\(493\) −33.4906 −1.50834
\(494\) −45.6587 −2.05428
\(495\) 0 0
\(496\) 13.4461 0.603746
\(497\) 15.1600 0.680019
\(498\) −0.770750 −0.0345381
\(499\) −20.6680 −0.925229 −0.462614 0.886560i \(-0.653089\pi\)
−0.462614 + 0.886560i \(0.653089\pi\)
\(500\) 0.0586092 0.00262108
\(501\) 0.205406 0.00917685
\(502\) −25.4206 −1.13458
\(503\) 22.0187 0.981763 0.490882 0.871226i \(-0.336675\pi\)
0.490882 + 0.871226i \(0.336675\pi\)
\(504\) 8.37266 0.372948
\(505\) −33.9160 −1.50924
\(506\) 0 0
\(507\) 3.57128 0.158606
\(508\) −3.11203 −0.138074
\(509\) 21.4240 0.949602 0.474801 0.880093i \(-0.342520\pi\)
0.474801 + 0.880093i \(0.342520\pi\)
\(510\) 7.43466 0.329212
\(511\) 3.22199 0.142533
\(512\) 24.5381 1.08444
\(513\) 14.9439 0.659791
\(514\) 11.6519 0.513945
\(515\) −41.5933 −1.83282
\(516\) −0.219921 −0.00968149
\(517\) 0 0
\(518\) −13.4533 −0.591105
\(519\) −6.26401 −0.274960
\(520\) −43.8174 −1.92152
\(521\) 13.9453 0.610953 0.305476 0.952200i \(-0.401184\pi\)
0.305476 + 0.952200i \(0.401184\pi\)
\(522\) 27.4066 1.19955
\(523\) −18.4813 −0.808131 −0.404065 0.914730i \(-0.632403\pi\)
−0.404065 + 0.914730i \(0.632403\pi\)
\(524\) −2.12136 −0.0926721
\(525\) −1.76868 −0.0771915
\(526\) −21.8133 −0.951103
\(527\) 17.3760 0.756912
\(528\) 0 0
\(529\) 3.43334 0.149276
\(530\) −9.77594 −0.424640
\(531\) 1.04202 0.0452199
\(532\) −0.990671 −0.0429510
\(533\) 15.3947 0.666819
\(534\) 2.76661 0.119723
\(535\) 44.0373 1.90390
\(536\) −19.2520 −0.831562
\(537\) 4.11335 0.177504
\(538\) −10.3013 −0.444122
\(539\) 0 0
\(540\) 0.946578 0.0407342
\(541\) 37.4533 1.61024 0.805122 0.593109i \(-0.202100\pi\)
0.805122 + 0.593109i \(0.202100\pi\)
\(542\) −19.1120 −0.820932
\(543\) 5.40535 0.231966
\(544\) 3.81277 0.163471
\(545\) −48.9253 −2.09573
\(546\) −2.36672 −0.101286
\(547\) −17.1307 −0.732455 −0.366228 0.930525i \(-0.619351\pi\)
−0.366228 + 0.930525i \(0.619351\pi\)
\(548\) 1.51192 0.0645861
\(549\) −9.24065 −0.394381
\(550\) 0 0
\(551\) −49.1307 −2.09304
\(552\) 5.45331 0.232108
\(553\) 3.71733 0.158077
\(554\) 8.20541 0.348614
\(555\) −11.2627 −0.478075
\(556\) −0.579860 −0.0245915
\(557\) 24.5653 1.04087 0.520434 0.853902i \(-0.325770\pi\)
0.520434 + 0.853902i \(0.325770\pi\)
\(558\) −14.2194 −0.601957
\(559\) −20.4626 −0.865478
\(560\) 11.6146 0.490807
\(561\) 0 0
\(562\) 28.1773 1.18859
\(563\) −15.0093 −0.632568 −0.316284 0.948665i \(-0.602435\pi\)
−0.316284 + 0.948665i \(0.602435\pi\)
\(564\) 0.0399518 0.00168227
\(565\) 12.9800 0.546074
\(566\) −13.5187 −0.568233
\(567\) −7.82936 −0.328802
\(568\) −44.2572 −1.85699
\(569\) 12.1027 0.507372 0.253686 0.967287i \(-0.418357\pi\)
0.253686 + 0.967287i \(0.418357\pi\)
\(570\) 10.9066 0.456828
\(571\) 38.9439 1.62975 0.814877 0.579634i \(-0.196805\pi\)
0.814877 + 0.579634i \(0.196805\pi\)
\(572\) 0 0
\(573\) 1.39857 0.0584262
\(574\) −4.39263 −0.183345
\(575\) 25.0280 1.04374
\(576\) −24.3281 −1.01367
\(577\) 38.1507 1.58823 0.794116 0.607766i \(-0.207934\pi\)
0.794116 + 0.607766i \(0.207934\pi\)
\(578\) −7.94733 −0.330565
\(579\) −0.925283 −0.0384535
\(580\) −3.11203 −0.129220
\(581\) −1.55602 −0.0645544
\(582\) −3.04664 −0.126287
\(583\) 0 0
\(584\) −9.40610 −0.389227
\(585\) 43.0466 1.77976
\(586\) 36.5327 1.50915
\(587\) −34.7967 −1.43621 −0.718106 0.695934i \(-0.754991\pi\)
−0.718106 + 0.695934i \(0.754991\pi\)
\(588\) −0.0513514 −0.00211770
\(589\) 25.4906 1.05032
\(590\) 1.55602 0.0640602
\(591\) −3.83191 −0.157624
\(592\) 36.4854 1.49954
\(593\) −18.5913 −0.763452 −0.381726 0.924276i \(-0.624670\pi\)
−0.381726 + 0.924276i \(0.624670\pi\)
\(594\) 0 0
\(595\) 15.0093 0.615322
\(596\) −1.98134 −0.0811590
\(597\) 4.24536 0.173751
\(598\) 33.4906 1.36953
\(599\) 24.6680 1.00791 0.503955 0.863730i \(-0.331878\pi\)
0.503955 + 0.863730i \(0.331878\pi\)
\(600\) 5.16338 0.210794
\(601\) −30.1286 −1.22897 −0.614486 0.788928i \(-0.710637\pi\)
−0.614486 + 0.788928i \(0.710637\pi\)
\(602\) 5.83869 0.237967
\(603\) 18.9134 0.770213
\(604\) 0.965235 0.0392749
\(605\) 0 0
\(606\) 5.34797 0.217247
\(607\) −13.6587 −0.554390 −0.277195 0.960814i \(-0.589405\pi\)
−0.277195 + 0.960814i \(0.589405\pi\)
\(608\) 5.59333 0.226840
\(609\) −2.54669 −0.103197
\(610\) −13.7987 −0.558694
\(611\) 3.71733 0.150387
\(612\) −1.93677 −0.0782893
\(613\) −45.0280 −1.81866 −0.909332 0.416072i \(-0.863406\pi\)
−0.909332 + 0.416072i \(0.863406\pi\)
\(614\) −16.2199 −0.654583
\(615\) −3.67738 −0.148286
\(616\) 0 0
\(617\) −8.26401 −0.332697 −0.166348 0.986067i \(-0.553198\pi\)
−0.166348 + 0.986067i \(0.553198\pi\)
\(618\) 6.55857 0.263824
\(619\) 43.0500 1.73033 0.865163 0.501490i \(-0.167215\pi\)
0.865163 + 0.501490i \(0.167215\pi\)
\(620\) 1.61462 0.0648449
\(621\) −10.9614 −0.439864
\(622\) 28.9580 1.16111
\(623\) 5.58532 0.223771
\(624\) 6.41855 0.256948
\(625\) −25.6426 −1.02570
\(626\) 17.3107 0.691873
\(627\) 0 0
\(628\) 3.31330 0.132215
\(629\) 47.1493 1.87997
\(630\) −12.2827 −0.489353
\(631\) 1.03863 0.0413473 0.0206737 0.999786i \(-0.493419\pi\)
0.0206737 + 0.999786i \(0.493419\pi\)
\(632\) −10.8522 −0.431676
\(633\) 4.03731 0.160469
\(634\) 16.1800 0.642589
\(635\) 69.1680 2.74485
\(636\) −0.117218 −0.00464801
\(637\) −4.77801 −0.189312
\(638\) 0 0
\(639\) 43.4787 1.71999
\(640\) −31.3148 −1.23783
\(641\) 33.6413 1.32875 0.664376 0.747399i \(-0.268698\pi\)
0.664376 + 0.747399i \(0.268698\pi\)
\(642\) −6.94394 −0.274055
\(643\) −21.0314 −0.829396 −0.414698 0.909959i \(-0.636113\pi\)
−0.414698 + 0.909959i \(0.636113\pi\)
\(644\) 0.726656 0.0286343
\(645\) 4.88797 0.192464
\(646\) −45.6587 −1.79642
\(647\) 25.2300 0.991894 0.495947 0.868353i \(-0.334821\pi\)
0.495947 + 0.868353i \(0.334821\pi\)
\(648\) 22.8566 0.897892
\(649\) 0 0
\(650\) 31.7101 1.24377
\(651\) 1.32131 0.0517860
\(652\) 0.140018 0.00548351
\(653\) 20.6387 0.807656 0.403828 0.914835i \(-0.367679\pi\)
0.403828 + 0.914835i \(0.367679\pi\)
\(654\) 7.71469 0.301668
\(655\) 47.1493 1.84228
\(656\) 11.9128 0.465118
\(657\) 9.24065 0.360512
\(658\) −1.06068 −0.0413496
\(659\) −18.5067 −0.720920 −0.360460 0.932775i \(-0.617380\pi\)
−0.360460 + 0.932775i \(0.617380\pi\)
\(660\) 0 0
\(661\) 16.8001 0.653446 0.326723 0.945120i \(-0.394056\pi\)
0.326723 + 0.945120i \(0.394056\pi\)
\(662\) 16.2054 0.629841
\(663\) 8.29455 0.322134
\(664\) 4.54255 0.176285
\(665\) 22.0187 0.853847
\(666\) −38.5840 −1.49510
\(667\) 36.0373 1.39537
\(668\) −0.0799036 −0.00309156
\(669\) 10.1880 0.393890
\(670\) 28.2427 1.09111
\(671\) 0 0
\(672\) 0.289930 0.0111843
\(673\) 17.5560 0.676735 0.338367 0.941014i \(-0.390125\pi\)
0.338367 + 0.941014i \(0.390125\pi\)
\(674\) −1.35061 −0.0520236
\(675\) −10.3786 −0.399472
\(676\) −1.38924 −0.0534324
\(677\) −3.64732 −0.140178 −0.0700889 0.997541i \(-0.522328\pi\)
−0.0700889 + 0.997541i \(0.522328\pi\)
\(678\) −2.04673 −0.0786042
\(679\) −6.15066 −0.236041
\(680\) −43.8174 −1.68032
\(681\) −8.35994 −0.320354
\(682\) 0 0
\(683\) 27.4720 1.05119 0.525593 0.850736i \(-0.323844\pi\)
0.525593 + 0.850736i \(0.323844\pi\)
\(684\) −2.84124 −0.108637
\(685\) −33.6040 −1.28394
\(686\) 1.36333 0.0520521
\(687\) 4.87732 0.186081
\(688\) −15.8345 −0.603686
\(689\) −10.9066 −0.415509
\(690\) −8.00000 −0.304555
\(691\) −5.95666 −0.226602 −0.113301 0.993561i \(-0.536142\pi\)
−0.113301 + 0.993561i \(0.536142\pi\)
\(692\) 2.43673 0.0926304
\(693\) 0 0
\(694\) 20.7080 0.786065
\(695\) 12.8880 0.488869
\(696\) 7.43466 0.281810
\(697\) 15.3947 0.583116
\(698\) 40.4953 1.53277
\(699\) −1.28522 −0.0486116
\(700\) 0.688023 0.0260048
\(701\) 7.36927 0.278333 0.139167 0.990269i \(-0.455558\pi\)
0.139167 + 0.990269i \(0.455558\pi\)
\(702\) −13.8879 −0.524164
\(703\) 69.1680 2.60872
\(704\) 0 0
\(705\) −0.887968 −0.0334428
\(706\) 9.15613 0.344595
\(707\) 10.7967 0.406050
\(708\) 0.0186574 0.000701189 0
\(709\) −34.7160 −1.30379 −0.651894 0.758310i \(-0.726025\pi\)
−0.651894 + 0.758310i \(0.726025\pi\)
\(710\) 64.9253 2.43660
\(711\) 10.6613 0.399829
\(712\) −16.3055 −0.611073
\(713\) −18.6974 −0.700221
\(714\) −2.36672 −0.0885722
\(715\) 0 0
\(716\) −1.60011 −0.0597989
\(717\) 8.00000 0.298765
\(718\) −24.1800 −0.902388
\(719\) −4.06200 −0.151487 −0.0757435 0.997127i \(-0.524133\pi\)
−0.0757435 + 0.997127i \(0.524133\pi\)
\(720\) 33.3107 1.24141
\(721\) 13.2406 0.493108
\(722\) −41.0780 −1.52877
\(723\) −0.114581 −0.00426133
\(724\) −2.10270 −0.0781463
\(725\) 34.1214 1.26724
\(726\) 0 0
\(727\) −30.5433 −1.13279 −0.566394 0.824135i \(-0.691662\pi\)
−0.566394 + 0.824135i \(0.691662\pi\)
\(728\) 13.9486 0.516971
\(729\) −20.1307 −0.745581
\(730\) 13.7987 0.510714
\(731\) −20.4626 −0.756838
\(732\) −0.165454 −0.00611535
\(733\) −30.8340 −1.13888 −0.569440 0.822033i \(-0.692840\pi\)
−0.569440 + 0.822033i \(0.692840\pi\)
\(734\) 11.0420 0.407568
\(735\) 1.14134 0.0420988
\(736\) −4.10270 −0.151228
\(737\) 0 0
\(738\) −12.5980 −0.463740
\(739\) 22.2241 0.817525 0.408763 0.912641i \(-0.365960\pi\)
0.408763 + 0.912641i \(0.365960\pi\)
\(740\) 4.38123 0.161057
\(741\) 12.1681 0.447006
\(742\) 3.11203 0.114246
\(743\) −31.1493 −1.14276 −0.571379 0.820686i \(-0.693591\pi\)
−0.571379 + 0.820686i \(0.693591\pi\)
\(744\) −3.85735 −0.141417
\(745\) 44.0373 1.61340
\(746\) −0.770750 −0.0282192
\(747\) −4.46264 −0.163280
\(748\) 0 0
\(749\) −14.0187 −0.512231
\(750\) 0.205406 0.00750035
\(751\) 11.9894 0.437498 0.218749 0.975781i \(-0.429802\pi\)
0.218749 + 0.975781i \(0.429802\pi\)
\(752\) 2.87657 0.104898
\(753\) 6.77462 0.246881
\(754\) 45.6587 1.66279
\(755\) −21.4533 −0.780766
\(756\) −0.301330 −0.0109593
\(757\) −11.4533 −0.416278 −0.208139 0.978099i \(-0.566741\pi\)
−0.208139 + 0.978099i \(0.566741\pi\)
\(758\) 30.7240 1.11595
\(759\) 0 0
\(760\) −64.2800 −2.33168
\(761\) 7.89004 0.286014 0.143007 0.989722i \(-0.454323\pi\)
0.143007 + 0.989722i \(0.454323\pi\)
\(762\) −10.9066 −0.395105
\(763\) 15.5747 0.563841
\(764\) −0.544050 −0.0196830
\(765\) 43.0466 1.55635
\(766\) −29.1634 −1.05372
\(767\) 1.73599 0.0626828
\(768\) −1.22613 −0.0442443
\(769\) −48.8153 −1.76033 −0.880163 0.474672i \(-0.842567\pi\)
−0.880163 + 0.474672i \(0.842567\pi\)
\(770\) 0 0
\(771\) −3.10525 −0.111833
\(772\) 0.359939 0.0129545
\(773\) −3.19608 −0.114955 −0.0574774 0.998347i \(-0.518306\pi\)
−0.0574774 + 0.998347i \(0.518306\pi\)
\(774\) 16.7453 0.601898
\(775\) −17.7033 −0.635921
\(776\) 17.9559 0.644579
\(777\) 3.58532 0.128623
\(778\) −14.5586 −0.521950
\(779\) 22.5840 0.809156
\(780\) 0.770750 0.0275973
\(781\) 0 0
\(782\) 33.4906 1.19762
\(783\) −14.9439 −0.534053
\(784\) −3.69735 −0.132048
\(785\) −73.6413 −2.62837
\(786\) −7.43466 −0.265185
\(787\) −42.4813 −1.51429 −0.757147 0.653244i \(-0.773408\pi\)
−0.757147 + 0.653244i \(0.773408\pi\)
\(788\) 1.49063 0.0531014
\(789\) 5.81325 0.206957
\(790\) 15.9201 0.566412
\(791\) −4.13201 −0.146917
\(792\) 0 0
\(793\) −15.3947 −0.546682
\(794\) 32.6053 1.15712
\(795\) 2.60530 0.0924003
\(796\) −1.65146 −0.0585345
\(797\) −28.8587 −1.02223 −0.511113 0.859513i \(-0.670767\pi\)
−0.511113 + 0.859513i \(0.670767\pi\)
\(798\) −3.47197 −0.122906
\(799\) 3.71733 0.131510
\(800\) −3.88458 −0.137341
\(801\) 16.0187 0.565991
\(802\) −15.8947 −0.561260
\(803\) 0 0
\(804\) 0.338644 0.0119431
\(805\) −16.1507 −0.569236
\(806\) −23.6893 −0.834418
\(807\) 2.74531 0.0966396
\(808\) −31.5192 −1.10884
\(809\) −29.4533 −1.03552 −0.517762 0.855525i \(-0.673235\pi\)
−0.517762 + 0.855525i \(0.673235\pi\)
\(810\) −33.5306 −1.17814
\(811\) −2.48130 −0.0871302 −0.0435651 0.999051i \(-0.513872\pi\)
−0.0435651 + 0.999051i \(0.513872\pi\)
\(812\) 0.990671 0.0347657
\(813\) 5.09337 0.178632
\(814\) 0 0
\(815\) −3.11203 −0.109010
\(816\) 6.41855 0.224694
\(817\) −30.0187 −1.05022
\(818\) −44.7381 −1.56423
\(819\) −13.7033 −0.478832
\(820\) 1.43051 0.0499557
\(821\) −14.0187 −0.489255 −0.244627 0.969617i \(-0.578666\pi\)
−0.244627 + 0.969617i \(0.578666\pi\)
\(822\) 5.29878 0.184816
\(823\) −42.7933 −1.49168 −0.745840 0.666125i \(-0.767952\pi\)
−0.745840 + 0.666125i \(0.767952\pi\)
\(824\) −38.6540 −1.34658
\(825\) 0 0
\(826\) −0.495336 −0.0172349
\(827\) 9.17064 0.318894 0.159447 0.987206i \(-0.449029\pi\)
0.159447 + 0.987206i \(0.449029\pi\)
\(828\) 2.08405 0.0724256
\(829\) 29.1973 1.01406 0.507032 0.861927i \(-0.330743\pi\)
0.507032 + 0.861927i \(0.330743\pi\)
\(830\) −6.66391 −0.231308
\(831\) −2.18675 −0.0758575
\(832\) −40.5300 −1.40513
\(833\) −4.77801 −0.165548
\(834\) −2.03221 −0.0703698
\(835\) 1.77594 0.0614588
\(836\) 0 0
\(837\) 7.75341 0.267997
\(838\) −14.7194 −0.508473
\(839\) 19.7326 0.681245 0.340622 0.940200i \(-0.389362\pi\)
0.340622 + 0.940200i \(0.389362\pi\)
\(840\) −3.33195 −0.114963
\(841\) 20.1307 0.694162
\(842\) 1.57053 0.0541241
\(843\) −7.50929 −0.258634
\(844\) −1.57053 −0.0540599
\(845\) 30.8773 1.06221
\(846\) −3.04202 −0.104587
\(847\) 0 0
\(848\) −8.43984 −0.289825
\(849\) 3.60275 0.123646
\(850\) 31.7101 1.08765
\(851\) −50.7347 −1.73916
\(852\) 0.778487 0.0266705
\(853\) 28.3527 0.970777 0.485389 0.874298i \(-0.338678\pi\)
0.485389 + 0.874298i \(0.338678\pi\)
\(854\) 4.39263 0.150313
\(855\) 63.1493 2.15966
\(856\) 40.9253 1.39880
\(857\) −13.2033 −0.451017 −0.225509 0.974241i \(-0.572404\pi\)
−0.225509 + 0.974241i \(0.572404\pi\)
\(858\) 0 0
\(859\) −20.8260 −0.710573 −0.355286 0.934757i \(-0.615617\pi\)
−0.355286 + 0.934757i \(0.615617\pi\)
\(860\) −1.90144 −0.0648385
\(861\) 1.17064 0.0398953
\(862\) 43.5720 1.48407
\(863\) 34.9439 1.18951 0.594753 0.803909i \(-0.297250\pi\)
0.594753 + 0.803909i \(0.297250\pi\)
\(864\) 1.70131 0.0578797
\(865\) −54.1587 −1.84145
\(866\) 6.04409 0.205387
\(867\) 2.11797 0.0719301
\(868\) −0.513993 −0.0174461
\(869\) 0 0
\(870\) −10.9066 −0.369769
\(871\) 31.5093 1.06765
\(872\) −45.4678 −1.53973
\(873\) −17.6401 −0.597026
\(874\) 49.1307 1.66187
\(875\) 0.414680 0.0140187
\(876\) 0.165454 0.00559017
\(877\) 26.4813 0.894210 0.447105 0.894481i \(-0.352455\pi\)
0.447105 + 0.894481i \(0.352455\pi\)
\(878\) 37.4533 1.26399
\(879\) −9.73599 −0.328387
\(880\) 0 0
\(881\) 17.2627 0.581595 0.290798 0.956785i \(-0.406079\pi\)
0.290798 + 0.956785i \(0.406079\pi\)
\(882\) 3.91002 0.131657
\(883\) −44.6027 −1.50100 −0.750499 0.660871i \(-0.770187\pi\)
−0.750499 + 0.660871i \(0.770187\pi\)
\(884\) −3.22661 −0.108523
\(885\) −0.414680 −0.0139393
\(886\) 10.6213 0.356830
\(887\) −30.7894 −1.03381 −0.516904 0.856044i \(-0.672915\pi\)
−0.516904 + 0.856044i \(0.672915\pi\)
\(888\) −10.4668 −0.351242
\(889\) −22.0187 −0.738482
\(890\) 23.9201 0.801803
\(891\) 0 0
\(892\) −3.96316 −0.132697
\(893\) 5.45331 0.182488
\(894\) −6.94394 −0.232240
\(895\) 35.5640 1.18877
\(896\) 9.96862 0.333028
\(897\) −8.92528 −0.298007
\(898\) −27.9627 −0.933127
\(899\) −25.4906 −0.850160
\(900\) 1.97325 0.0657748
\(901\) −10.9066 −0.363352
\(902\) 0 0
\(903\) −1.55602 −0.0517810
\(904\) 12.0628 0.401201
\(905\) 46.7347 1.55351
\(906\) 3.38283 0.112387
\(907\) 23.7801 0.789605 0.394802 0.918766i \(-0.370813\pi\)
0.394802 + 0.918766i \(0.370813\pi\)
\(908\) 3.25205 0.107923
\(909\) 30.9648 1.02704
\(910\) −20.4626 −0.678330
\(911\) 49.0280 1.62437 0.812185 0.583400i \(-0.198278\pi\)
0.812185 + 0.583400i \(0.198278\pi\)
\(912\) 9.41600 0.311795
\(913\) 0 0
\(914\) 11.6774 0.386253
\(915\) 3.67738 0.121570
\(916\) −1.89730 −0.0626885
\(917\) −15.0093 −0.495652
\(918\) −13.8879 −0.458368
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 47.1493 1.55447
\(921\) 4.32262 0.142435
\(922\) −13.1779 −0.433991
\(923\) 72.4346 2.38421
\(924\) 0 0
\(925\) −48.0373 −1.57946
\(926\) −15.5747 −0.511816
\(927\) 37.9741 1.24723
\(928\) −5.59333 −0.183610
\(929\) −56.5840 −1.85646 −0.928230 0.372006i \(-0.878670\pi\)
−0.928230 + 0.372006i \(0.878670\pi\)
\(930\) 5.65872 0.185557
\(931\) −7.00933 −0.229721
\(932\) 0.499956 0.0163766
\(933\) −7.71733 −0.252654
\(934\) −24.8408 −0.812814
\(935\) 0 0
\(936\) 40.0046 1.30759
\(937\) −16.8153 −0.549333 −0.274666 0.961540i \(-0.588567\pi\)
−0.274666 + 0.961540i \(0.588567\pi\)
\(938\) −8.99067 −0.293556
\(939\) −4.61331 −0.150550
\(940\) 0.345423 0.0112665
\(941\) 7.25931 0.236647 0.118323 0.992975i \(-0.462248\pi\)
0.118323 + 0.992975i \(0.462248\pi\)
\(942\) 11.6120 0.378339
\(943\) −16.5653 −0.539442
\(944\) 1.34335 0.0437224
\(945\) 6.69735 0.217865
\(946\) 0 0
\(947\) −25.6040 −0.832017 −0.416009 0.909361i \(-0.636571\pi\)
−0.416009 + 0.909361i \(0.636571\pi\)
\(948\) 0.190890 0.00619982
\(949\) 15.3947 0.499733
\(950\) 46.5186 1.50926
\(951\) −4.31198 −0.139825
\(952\) 13.9486 0.452078
\(953\) 7.14935 0.231590 0.115795 0.993273i \(-0.463058\pi\)
0.115795 + 0.993273i \(0.463058\pi\)
\(954\) 8.92528 0.288967
\(955\) 12.0921 0.391290
\(956\) −3.11203 −0.100650
\(957\) 0 0
\(958\) −4.73344 −0.152930
\(959\) 10.6974 0.345436
\(960\) 9.68152 0.312470
\(961\) −17.7746 −0.573375
\(962\) −64.2800 −2.07247
\(963\) −40.2054 −1.29560
\(964\) 0.0445726 0.00143559
\(965\) −8.00000 −0.257529
\(966\) 2.54669 0.0819383
\(967\) 15.9600 0.513241 0.256620 0.966512i \(-0.417391\pi\)
0.256620 + 0.966512i \(0.417391\pi\)
\(968\) 0 0
\(969\) 12.1681 0.390895
\(970\) −26.3413 −0.845768
\(971\) −30.4779 −0.978083 −0.489041 0.872261i \(-0.662653\pi\)
−0.489041 + 0.872261i \(0.662653\pi\)
\(972\) −1.30604 −0.0418912
\(973\) −4.10270 −0.131527
\(974\) 37.4388 1.19962
\(975\) −8.45076 −0.270641
\(976\) −11.9128 −0.381321
\(977\) 4.13201 0.132195 0.0660973 0.997813i \(-0.478945\pi\)
0.0660973 + 0.997813i \(0.478945\pi\)
\(978\) 0.490715 0.0156913
\(979\) 0 0
\(980\) −0.443984 −0.0141826
\(981\) 44.6680 1.42614
\(982\) 8.53991 0.272519
\(983\) −25.0246 −0.798161 −0.399080 0.916916i \(-0.630671\pi\)
−0.399080 + 0.916916i \(0.630671\pi\)
\(984\) −3.41750 −0.108946
\(985\) −33.1307 −1.05563
\(986\) 45.6587 1.45407
\(987\) 0.282672 0.00899756
\(988\) −4.73344 −0.150591
\(989\) 22.0187 0.700153
\(990\) 0 0
\(991\) −27.5747 −0.875938 −0.437969 0.898990i \(-0.644302\pi\)
−0.437969 + 0.898990i \(0.644302\pi\)
\(992\) 2.90201 0.0921388
\(993\) −4.31876 −0.137052
\(994\) −20.6680 −0.655551
\(995\) 36.7054 1.16364
\(996\) −0.0799036 −0.00253184
\(997\) 23.6846 0.750100 0.375050 0.927005i \(-0.377626\pi\)
0.375050 + 0.927005i \(0.377626\pi\)
\(998\) 28.1773 0.891938
\(999\) 21.0386 0.665633
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.i.1.2 3
3.2 odd 2 7623.2.a.ce.1.2 3
7.6 odd 2 5929.2.a.t.1.2 3
11.2 odd 10 847.2.f.t.323.2 12
11.3 even 5 847.2.f.u.372.2 12
11.4 even 5 847.2.f.u.148.2 12
11.5 even 5 847.2.f.u.729.2 12
11.6 odd 10 847.2.f.t.729.2 12
11.7 odd 10 847.2.f.t.148.2 12
11.8 odd 10 847.2.f.t.372.2 12
11.9 even 5 847.2.f.u.323.2 12
11.10 odd 2 847.2.a.j.1.2 yes 3
33.32 even 2 7623.2.a.bz.1.2 3
77.76 even 2 5929.2.a.y.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.i.1.2 3 1.1 even 1 trivial
847.2.a.j.1.2 yes 3 11.10 odd 2
847.2.f.t.148.2 12 11.7 odd 10
847.2.f.t.323.2 12 11.2 odd 10
847.2.f.t.372.2 12 11.8 odd 10
847.2.f.t.729.2 12 11.6 odd 10
847.2.f.u.148.2 12 11.4 even 5
847.2.f.u.323.2 12 11.9 even 5
847.2.f.u.372.2 12 11.3 even 5
847.2.f.u.729.2 12 11.5 even 5
5929.2.a.t.1.2 3 7.6 odd 2
5929.2.a.y.1.2 3 77.76 even 2
7623.2.a.bz.1.2 3 33.32 even 2
7623.2.a.ce.1.2 3 3.2 odd 2