Properties

Label 5225.2.a.h.1.2
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
Dimension $5$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.51908\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.82669 q^{2} +0.563416 q^{3} +1.33679 q^{4} -1.02918 q^{6} -1.69239 q^{7} +1.21147 q^{8} -2.68256 q^{9} +O(q^{10})\) \(q-1.82669 q^{2} +0.563416 q^{3} +1.33679 q^{4} -1.02918 q^{6} -1.69239 q^{7} +1.21147 q^{8} -2.68256 q^{9} +1.00000 q^{11} +0.753170 q^{12} -4.11168 q^{13} +3.09147 q^{14} -4.88657 q^{16} +6.16499 q^{17} +4.90021 q^{18} -1.00000 q^{19} -0.953520 q^{21} -1.82669 q^{22} -3.52199 q^{23} +0.682563 q^{24} +7.51076 q^{26} -3.20164 q^{27} -2.26238 q^{28} -8.10336 q^{29} -2.30144 q^{31} +6.50330 q^{32} +0.563416 q^{33} -11.2615 q^{34} -3.58603 q^{36} +6.56016 q^{37} +1.82669 q^{38} -2.31659 q^{39} +7.75013 q^{41} +1.74178 q^{42} -7.75102 q^{43} +1.33679 q^{44} +6.43359 q^{46} +10.8969 q^{47} -2.75317 q^{48} -4.13581 q^{49} +3.47345 q^{51} -5.49647 q^{52} -7.93511 q^{53} +5.84841 q^{54} -2.05029 q^{56} -0.563416 q^{57} +14.8023 q^{58} +10.9247 q^{59} -4.51162 q^{61} +4.20401 q^{62} +4.53995 q^{63} -2.10636 q^{64} -1.02918 q^{66} -14.7201 q^{67} +8.24132 q^{68} -1.98435 q^{69} +3.12026 q^{71} -3.24985 q^{72} -11.5827 q^{73} -11.9834 q^{74} -1.33679 q^{76} -1.69239 q^{77} +4.23168 q^{78} +4.96184 q^{79} +6.24383 q^{81} -14.1571 q^{82} +1.82905 q^{83} -1.27466 q^{84} +14.1587 q^{86} -4.56556 q^{87} +1.21147 q^{88} -9.37496 q^{89} +6.95858 q^{91} -4.70818 q^{92} -1.29666 q^{93} -19.9053 q^{94} +3.66406 q^{96} +10.9937 q^{97} +7.55484 q^{98} -2.68256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} - q^{3} + 6 q^{4} - 2 q^{6} - 6 q^{7} - 6 q^{8} + 4 q^{9} + 5 q^{11} - 6 q^{12} - 4 q^{13} - 14 q^{14} + 8 q^{16} + 4 q^{17} + 20 q^{18} - 5 q^{19} + 10 q^{21} - 2 q^{22} - 3 q^{23} - 14 q^{24} - 6 q^{26} + 11 q^{27} + 10 q^{28} + 10 q^{29} + 11 q^{31} - 14 q^{32} - q^{33} - 4 q^{34} - 26 q^{36} - q^{37} + 2 q^{38} + 2 q^{39} + 2 q^{41} + 16 q^{42} - 20 q^{43} + 6 q^{44} - 4 q^{46} + 20 q^{47} - 4 q^{48} + 3 q^{49} + 24 q^{51} - 6 q^{52} + 14 q^{53} + 16 q^{54} - 38 q^{56} + q^{57} + 6 q^{58} + 3 q^{59} - 10 q^{61} + 6 q^{62} - 24 q^{63} - 2 q^{66} - 9 q^{67} - 24 q^{68} - 5 q^{69} + 23 q^{71} + 12 q^{72} + 8 q^{74} - 6 q^{76} - 6 q^{77} + 22 q^{78} + 44 q^{79} + q^{81} + 30 q^{82} + 14 q^{83} + 14 q^{84} + 52 q^{86} - 28 q^{87} - 6 q^{88} - 27 q^{89} + 24 q^{91} - 58 q^{92} + 27 q^{93} - 8 q^{94} + 50 q^{96} - 15 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.82669 −1.29166 −0.645832 0.763479i \(-0.723489\pi\)
−0.645832 + 0.763479i \(0.723489\pi\)
\(3\) 0.563416 0.325288 0.162644 0.986685i \(-0.447998\pi\)
0.162644 + 0.986685i \(0.447998\pi\)
\(4\) 1.33679 0.668396
\(5\) 0 0
\(6\) −1.02918 −0.420163
\(7\) −1.69239 −0.639664 −0.319832 0.947474i \(-0.603626\pi\)
−0.319832 + 0.947474i \(0.603626\pi\)
\(8\) 1.21147 0.428321
\(9\) −2.68256 −0.894188
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0.753170 0.217421
\(13\) −4.11168 −1.14038 −0.570188 0.821514i \(-0.693130\pi\)
−0.570188 + 0.821514i \(0.693130\pi\)
\(14\) 3.09147 0.826231
\(15\) 0 0
\(16\) −4.88657 −1.22164
\(17\) 6.16499 1.49523 0.747615 0.664132i \(-0.231199\pi\)
0.747615 + 0.664132i \(0.231199\pi\)
\(18\) 4.90021 1.15499
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.953520 −0.208075
\(22\) −1.82669 −0.389451
\(23\) −3.52199 −0.734386 −0.367193 0.930145i \(-0.619681\pi\)
−0.367193 + 0.930145i \(0.619681\pi\)
\(24\) 0.682563 0.139328
\(25\) 0 0
\(26\) 7.51076 1.47298
\(27\) −3.20164 −0.616157
\(28\) −2.26238 −0.427549
\(29\) −8.10336 −1.50476 −0.752378 0.658731i \(-0.771094\pi\)
−0.752378 + 0.658731i \(0.771094\pi\)
\(30\) 0 0
\(31\) −2.30144 −0.413350 −0.206675 0.978410i \(-0.566264\pi\)
−0.206675 + 0.978410i \(0.566264\pi\)
\(32\) 6.50330 1.14963
\(33\) 0.563416 0.0980781
\(34\) −11.2615 −1.93134
\(35\) 0 0
\(36\) −3.58603 −0.597672
\(37\) 6.56016 1.07848 0.539242 0.842151i \(-0.318711\pi\)
0.539242 + 0.842151i \(0.318711\pi\)
\(38\) 1.82669 0.296328
\(39\) −2.31659 −0.370951
\(40\) 0 0
\(41\) 7.75013 1.21037 0.605183 0.796086i \(-0.293100\pi\)
0.605183 + 0.796086i \(0.293100\pi\)
\(42\) 1.74178 0.268763
\(43\) −7.75102 −1.18202 −0.591010 0.806664i \(-0.701271\pi\)
−0.591010 + 0.806664i \(0.701271\pi\)
\(44\) 1.33679 0.201529
\(45\) 0 0
\(46\) 6.43359 0.948581
\(47\) 10.8969 1.58948 0.794742 0.606948i \(-0.207606\pi\)
0.794742 + 0.606948i \(0.207606\pi\)
\(48\) −2.75317 −0.397386
\(49\) −4.13581 −0.590830
\(50\) 0 0
\(51\) 3.47345 0.486381
\(52\) −5.49647 −0.762223
\(53\) −7.93511 −1.08997 −0.544986 0.838445i \(-0.683465\pi\)
−0.544986 + 0.838445i \(0.683465\pi\)
\(54\) 5.84841 0.795868
\(55\) 0 0
\(56\) −2.05029 −0.273981
\(57\) −0.563416 −0.0746262
\(58\) 14.8023 1.94364
\(59\) 10.9247 1.42228 0.711140 0.703051i \(-0.248179\pi\)
0.711140 + 0.703051i \(0.248179\pi\)
\(60\) 0 0
\(61\) −4.51162 −0.577653 −0.288827 0.957381i \(-0.593265\pi\)
−0.288827 + 0.957381i \(0.593265\pi\)
\(62\) 4.20401 0.533910
\(63\) 4.53995 0.571980
\(64\) −2.10636 −0.263295
\(65\) 0 0
\(66\) −1.02918 −0.126684
\(67\) −14.7201 −1.79835 −0.899173 0.437594i \(-0.855831\pi\)
−0.899173 + 0.437594i \(0.855831\pi\)
\(68\) 8.24132 0.999407
\(69\) −1.98435 −0.238887
\(70\) 0 0
\(71\) 3.12026 0.370307 0.185153 0.982710i \(-0.440722\pi\)
0.185153 + 0.982710i \(0.440722\pi\)
\(72\) −3.24985 −0.382999
\(73\) −11.5827 −1.35565 −0.677824 0.735224i \(-0.737077\pi\)
−0.677824 + 0.735224i \(0.737077\pi\)
\(74\) −11.9834 −1.39304
\(75\) 0 0
\(76\) −1.33679 −0.153341
\(77\) −1.69239 −0.192866
\(78\) 4.23168 0.479144
\(79\) 4.96184 0.558250 0.279125 0.960255i \(-0.409956\pi\)
0.279125 + 0.960255i \(0.409956\pi\)
\(80\) 0 0
\(81\) 6.24383 0.693759
\(82\) −14.1571 −1.56339
\(83\) 1.82905 0.200765 0.100382 0.994949i \(-0.467993\pi\)
0.100382 + 0.994949i \(0.467993\pi\)
\(84\) −1.27466 −0.139077
\(85\) 0 0
\(86\) 14.1587 1.52677
\(87\) −4.56556 −0.489480
\(88\) 1.21147 0.129144
\(89\) −9.37496 −0.993743 −0.496872 0.867824i \(-0.665518\pi\)
−0.496872 + 0.867824i \(0.665518\pi\)
\(90\) 0 0
\(91\) 6.95858 0.729457
\(92\) −4.70818 −0.490861
\(93\) −1.29666 −0.134458
\(94\) −19.9053 −2.05308
\(95\) 0 0
\(96\) 3.66406 0.373961
\(97\) 10.9937 1.11625 0.558123 0.829758i \(-0.311522\pi\)
0.558123 + 0.829758i \(0.311522\pi\)
\(98\) 7.55484 0.763154
\(99\) −2.68256 −0.269608
\(100\) 0 0
\(101\) 2.30621 0.229476 0.114738 0.993396i \(-0.463397\pi\)
0.114738 + 0.993396i \(0.463397\pi\)
\(102\) −6.34492 −0.628241
\(103\) −6.12000 −0.603021 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(104\) −4.98119 −0.488446
\(105\) 0 0
\(106\) 14.4950 1.40788
\(107\) −0.422947 −0.0408878 −0.0204439 0.999791i \(-0.506508\pi\)
−0.0204439 + 0.999791i \(0.506508\pi\)
\(108\) −4.27993 −0.411837
\(109\) 14.0180 1.34268 0.671338 0.741151i \(-0.265720\pi\)
0.671338 + 0.741151i \(0.265720\pi\)
\(110\) 0 0
\(111\) 3.69609 0.350818
\(112\) 8.26999 0.781441
\(113\) 2.53638 0.238602 0.119301 0.992858i \(-0.461935\pi\)
0.119301 + 0.992858i \(0.461935\pi\)
\(114\) 1.02918 0.0963920
\(115\) 0 0
\(116\) −10.8325 −1.00577
\(117\) 11.0298 1.01971
\(118\) −19.9561 −1.83711
\(119\) −10.4336 −0.956445
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.24132 0.746134
\(123\) 4.36654 0.393718
\(124\) −3.07654 −0.276282
\(125\) 0 0
\(126\) −8.29307 −0.738806
\(127\) −1.07275 −0.0951914 −0.0475957 0.998867i \(-0.515156\pi\)
−0.0475957 + 0.998867i \(0.515156\pi\)
\(128\) −9.15893 −0.809543
\(129\) −4.36705 −0.384497
\(130\) 0 0
\(131\) −7.82709 −0.683856 −0.341928 0.939726i \(-0.611080\pi\)
−0.341928 + 0.939726i \(0.611080\pi\)
\(132\) 0.753170 0.0655550
\(133\) 1.69239 0.146749
\(134\) 26.8890 2.32286
\(135\) 0 0
\(136\) 7.46873 0.640438
\(137\) 11.2473 0.960919 0.480460 0.877017i \(-0.340470\pi\)
0.480460 + 0.877017i \(0.340470\pi\)
\(138\) 3.62478 0.308562
\(139\) −0.905926 −0.0768397 −0.0384198 0.999262i \(-0.512232\pi\)
−0.0384198 + 0.999262i \(0.512232\pi\)
\(140\) 0 0
\(141\) 6.13951 0.517040
\(142\) −5.69975 −0.478312
\(143\) −4.11168 −0.343836
\(144\) 13.1085 1.09238
\(145\) 0 0
\(146\) 21.1579 1.75104
\(147\) −2.33018 −0.192190
\(148\) 8.76957 0.720854
\(149\) 10.5174 0.861622 0.430811 0.902442i \(-0.358228\pi\)
0.430811 + 0.902442i \(0.358228\pi\)
\(150\) 0 0
\(151\) 13.2436 1.07775 0.538873 0.842387i \(-0.318850\pi\)
0.538873 + 0.842387i \(0.318850\pi\)
\(152\) −1.21147 −0.0982635
\(153\) −16.5380 −1.33702
\(154\) 3.09147 0.249118
\(155\) 0 0
\(156\) −3.09679 −0.247942
\(157\) −1.99915 −0.159549 −0.0797747 0.996813i \(-0.525420\pi\)
−0.0797747 + 0.996813i \(0.525420\pi\)
\(158\) −9.06373 −0.721072
\(159\) −4.47076 −0.354555
\(160\) 0 0
\(161\) 5.96059 0.469761
\(162\) −11.4055 −0.896104
\(163\) 18.7557 1.46906 0.734531 0.678575i \(-0.237402\pi\)
0.734531 + 0.678575i \(0.237402\pi\)
\(164\) 10.3603 0.809005
\(165\) 0 0
\(166\) −3.34111 −0.259320
\(167\) 6.31203 0.488440 0.244220 0.969720i \(-0.421468\pi\)
0.244220 + 0.969720i \(0.421468\pi\)
\(168\) −1.15516 −0.0891229
\(169\) 3.90593 0.300456
\(170\) 0 0
\(171\) 2.68256 0.205141
\(172\) −10.3615 −0.790058
\(173\) 21.5269 1.63666 0.818328 0.574751i \(-0.194901\pi\)
0.818328 + 0.574751i \(0.194901\pi\)
\(174\) 8.33986 0.632243
\(175\) 0 0
\(176\) −4.88657 −0.368339
\(177\) 6.15516 0.462650
\(178\) 17.1251 1.28358
\(179\) −7.97018 −0.595719 −0.297860 0.954610i \(-0.596273\pi\)
−0.297860 + 0.954610i \(0.596273\pi\)
\(180\) 0 0
\(181\) −1.16955 −0.0869317 −0.0434659 0.999055i \(-0.513840\pi\)
−0.0434659 + 0.999055i \(0.513840\pi\)
\(182\) −12.7112 −0.942214
\(183\) −2.54191 −0.187904
\(184\) −4.26680 −0.314553
\(185\) 0 0
\(186\) 2.36860 0.173674
\(187\) 6.16499 0.450829
\(188\) 14.5670 1.06240
\(189\) 5.41844 0.394133
\(190\) 0 0
\(191\) −15.6673 −1.13365 −0.566824 0.823839i \(-0.691828\pi\)
−0.566824 + 0.823839i \(0.691828\pi\)
\(192\) −1.18676 −0.0856468
\(193\) −21.6769 −1.56034 −0.780169 0.625568i \(-0.784867\pi\)
−0.780169 + 0.625568i \(0.784867\pi\)
\(194\) −20.0821 −1.44181
\(195\) 0 0
\(196\) −5.52872 −0.394908
\(197\) −4.58794 −0.326877 −0.163439 0.986554i \(-0.552259\pi\)
−0.163439 + 0.986554i \(0.552259\pi\)
\(198\) 4.90021 0.348243
\(199\) 13.0619 0.925937 0.462968 0.886375i \(-0.346784\pi\)
0.462968 + 0.886375i \(0.346784\pi\)
\(200\) 0 0
\(201\) −8.29353 −0.584980
\(202\) −4.21272 −0.296406
\(203\) 13.7141 0.962539
\(204\) 4.64329 0.325095
\(205\) 0 0
\(206\) 11.1793 0.778901
\(207\) 9.44797 0.656679
\(208\) 20.0920 1.39313
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 2.91188 0.200462 0.100231 0.994964i \(-0.468042\pi\)
0.100231 + 0.994964i \(0.468042\pi\)
\(212\) −10.6076 −0.728533
\(213\) 1.75800 0.120456
\(214\) 0.772592 0.0528133
\(215\) 0 0
\(216\) −3.87871 −0.263913
\(217\) 3.89493 0.264405
\(218\) −25.6064 −1.73429
\(219\) −6.52585 −0.440976
\(220\) 0 0
\(221\) −25.3485 −1.70512
\(222\) −6.75161 −0.453139
\(223\) −6.34161 −0.424665 −0.212333 0.977197i \(-0.568106\pi\)
−0.212333 + 0.977197i \(0.568106\pi\)
\(224\) −11.0061 −0.735378
\(225\) 0 0
\(226\) −4.63317 −0.308194
\(227\) 2.00429 0.133029 0.0665147 0.997785i \(-0.478812\pi\)
0.0665147 + 0.997785i \(0.478812\pi\)
\(228\) −0.753170 −0.0498799
\(229\) −20.9893 −1.38701 −0.693507 0.720450i \(-0.743935\pi\)
−0.693507 + 0.720450i \(0.743935\pi\)
\(230\) 0 0
\(231\) −0.953520 −0.0627370
\(232\) −9.81701 −0.644518
\(233\) 17.6006 1.15305 0.576527 0.817078i \(-0.304407\pi\)
0.576527 + 0.817078i \(0.304407\pi\)
\(234\) −20.1481 −1.31712
\(235\) 0 0
\(236\) 14.6041 0.950646
\(237\) 2.79558 0.181592
\(238\) 19.0589 1.23541
\(239\) 26.6207 1.72195 0.860975 0.508647i \(-0.169854\pi\)
0.860975 + 0.508647i \(0.169854\pi\)
\(240\) 0 0
\(241\) 13.1342 0.846049 0.423024 0.906118i \(-0.360968\pi\)
0.423024 + 0.906118i \(0.360968\pi\)
\(242\) −1.82669 −0.117424
\(243\) 13.1228 0.841828
\(244\) −6.03109 −0.386101
\(245\) 0 0
\(246\) −7.97632 −0.508551
\(247\) 4.11168 0.261620
\(248\) −2.78813 −0.177046
\(249\) 1.03052 0.0653063
\(250\) 0 0
\(251\) −26.1636 −1.65143 −0.825715 0.564087i \(-0.809228\pi\)
−0.825715 + 0.564087i \(0.809228\pi\)
\(252\) 6.06897 0.382309
\(253\) −3.52199 −0.221426
\(254\) 1.95959 0.122955
\(255\) 0 0
\(256\) 20.9432 1.30895
\(257\) −12.6117 −0.786697 −0.393349 0.919389i \(-0.628683\pi\)
−0.393349 + 0.919389i \(0.628683\pi\)
\(258\) 7.97724 0.496641
\(259\) −11.1024 −0.689867
\(260\) 0 0
\(261\) 21.7378 1.34554
\(262\) 14.2977 0.883312
\(263\) 20.0550 1.23664 0.618322 0.785925i \(-0.287813\pi\)
0.618322 + 0.785925i \(0.287813\pi\)
\(264\) 0.682563 0.0420088
\(265\) 0 0
\(266\) −3.09147 −0.189550
\(267\) −5.28200 −0.323253
\(268\) −19.6777 −1.20201
\(269\) −20.1150 −1.22644 −0.613218 0.789914i \(-0.710125\pi\)
−0.613218 + 0.789914i \(0.710125\pi\)
\(270\) 0 0
\(271\) 15.5586 0.945117 0.472558 0.881299i \(-0.343331\pi\)
0.472558 + 0.881299i \(0.343331\pi\)
\(272\) −30.1257 −1.82664
\(273\) 3.92057 0.237284
\(274\) −20.5453 −1.24119
\(275\) 0 0
\(276\) −2.65266 −0.159671
\(277\) −1.89211 −0.113686 −0.0568429 0.998383i \(-0.518103\pi\)
−0.0568429 + 0.998383i \(0.518103\pi\)
\(278\) 1.65485 0.0992510
\(279\) 6.17375 0.369613
\(280\) 0 0
\(281\) 4.59299 0.273995 0.136997 0.990571i \(-0.456255\pi\)
0.136997 + 0.990571i \(0.456255\pi\)
\(282\) −11.2150 −0.667842
\(283\) 1.17798 0.0700234 0.0350117 0.999387i \(-0.488853\pi\)
0.0350117 + 0.999387i \(0.488853\pi\)
\(284\) 4.17114 0.247512
\(285\) 0 0
\(286\) 7.51076 0.444121
\(287\) −13.1163 −0.774228
\(288\) −17.4455 −1.02799
\(289\) 21.0071 1.23571
\(290\) 0 0
\(291\) 6.19405 0.363101
\(292\) −15.4836 −0.906110
\(293\) 11.2606 0.657851 0.328925 0.944356i \(-0.393314\pi\)
0.328925 + 0.944356i \(0.393314\pi\)
\(294\) 4.25651 0.248245
\(295\) 0 0
\(296\) 7.94745 0.461936
\(297\) −3.20164 −0.185778
\(298\) −19.2121 −1.11293
\(299\) 14.4813 0.837476
\(300\) 0 0
\(301\) 13.1178 0.756096
\(302\) −24.1919 −1.39209
\(303\) 1.29935 0.0746459
\(304\) 4.88657 0.280264
\(305\) 0 0
\(306\) 30.2098 1.72698
\(307\) −4.35245 −0.248407 −0.124204 0.992257i \(-0.539638\pi\)
−0.124204 + 0.992257i \(0.539638\pi\)
\(308\) −2.26238 −0.128911
\(309\) −3.44810 −0.196156
\(310\) 0 0
\(311\) 7.61725 0.431935 0.215967 0.976401i \(-0.430709\pi\)
0.215967 + 0.976401i \(0.430709\pi\)
\(312\) −2.80648 −0.158886
\(313\) 17.5654 0.992855 0.496427 0.868078i \(-0.334645\pi\)
0.496427 + 0.868078i \(0.334645\pi\)
\(314\) 3.65182 0.206084
\(315\) 0 0
\(316\) 6.63295 0.373132
\(317\) 1.94192 0.109069 0.0545345 0.998512i \(-0.482633\pi\)
0.0545345 + 0.998512i \(0.482633\pi\)
\(318\) 8.16670 0.457966
\(319\) −8.10336 −0.453701
\(320\) 0 0
\(321\) −0.238295 −0.0133003
\(322\) −10.8882 −0.606773
\(323\) −6.16499 −0.343029
\(324\) 8.34671 0.463706
\(325\) 0 0
\(326\) −34.2609 −1.89754
\(327\) 7.89793 0.436757
\(328\) 9.38907 0.518425
\(329\) −18.4419 −1.01674
\(330\) 0 0
\(331\) 13.3988 0.736462 0.368231 0.929734i \(-0.379964\pi\)
0.368231 + 0.929734i \(0.379964\pi\)
\(332\) 2.44506 0.134190
\(333\) −17.5980 −0.964366
\(334\) −11.5301 −0.630900
\(335\) 0 0
\(336\) 4.65944 0.254193
\(337\) −33.1631 −1.80651 −0.903254 0.429107i \(-0.858828\pi\)
−0.903254 + 0.429107i \(0.858828\pi\)
\(338\) −7.13491 −0.388088
\(339\) 1.42903 0.0776145
\(340\) 0 0
\(341\) −2.30144 −0.124630
\(342\) −4.90021 −0.264973
\(343\) 18.8462 1.01760
\(344\) −9.39016 −0.506283
\(345\) 0 0
\(346\) −39.3229 −2.11401
\(347\) −18.0268 −0.967730 −0.483865 0.875143i \(-0.660767\pi\)
−0.483865 + 0.875143i \(0.660767\pi\)
\(348\) −6.10321 −0.327166
\(349\) 8.81411 0.471808 0.235904 0.971776i \(-0.424195\pi\)
0.235904 + 0.971776i \(0.424195\pi\)
\(350\) 0 0
\(351\) 13.1641 0.702650
\(352\) 6.50330 0.346627
\(353\) 1.59249 0.0847599 0.0423799 0.999102i \(-0.486506\pi\)
0.0423799 + 0.999102i \(0.486506\pi\)
\(354\) −11.2436 −0.597589
\(355\) 0 0
\(356\) −12.5324 −0.664214
\(357\) −5.87844 −0.311120
\(358\) 14.5590 0.769469
\(359\) 36.3774 1.91993 0.959963 0.280126i \(-0.0903763\pi\)
0.959963 + 0.280126i \(0.0903763\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.13640 0.112287
\(363\) 0.563416 0.0295716
\(364\) 9.30218 0.487567
\(365\) 0 0
\(366\) 4.64329 0.242708
\(367\) 5.00276 0.261142 0.130571 0.991439i \(-0.458319\pi\)
0.130571 + 0.991439i \(0.458319\pi\)
\(368\) 17.2105 0.897158
\(369\) −20.7902 −1.08229
\(370\) 0 0
\(371\) 13.4293 0.697216
\(372\) −1.73337 −0.0898712
\(373\) 34.1313 1.76725 0.883625 0.468195i \(-0.155096\pi\)
0.883625 + 0.468195i \(0.155096\pi\)
\(374\) −11.2615 −0.582320
\(375\) 0 0
\(376\) 13.2014 0.680808
\(377\) 33.3185 1.71599
\(378\) −9.89780 −0.509088
\(379\) 15.7749 0.810302 0.405151 0.914250i \(-0.367219\pi\)
0.405151 + 0.914250i \(0.367219\pi\)
\(380\) 0 0
\(381\) −0.604405 −0.0309646
\(382\) 28.6193 1.46429
\(383\) 22.4260 1.14592 0.572958 0.819585i \(-0.305796\pi\)
0.572958 + 0.819585i \(0.305796\pi\)
\(384\) −5.16028 −0.263335
\(385\) 0 0
\(386\) 39.5970 2.01543
\(387\) 20.7926 1.05695
\(388\) 14.6964 0.746094
\(389\) −18.3488 −0.930321 −0.465160 0.885226i \(-0.654003\pi\)
−0.465160 + 0.885226i \(0.654003\pi\)
\(390\) 0 0
\(391\) −21.7131 −1.09808
\(392\) −5.01042 −0.253065
\(393\) −4.40990 −0.222450
\(394\) 8.38074 0.422216
\(395\) 0 0
\(396\) −3.58603 −0.180205
\(397\) 0.511483 0.0256706 0.0128353 0.999918i \(-0.495914\pi\)
0.0128353 + 0.999918i \(0.495914\pi\)
\(398\) −23.8601 −1.19600
\(399\) 0.953520 0.0477357
\(400\) 0 0
\(401\) 18.9824 0.947935 0.473967 0.880542i \(-0.342821\pi\)
0.473967 + 0.880542i \(0.342821\pi\)
\(402\) 15.1497 0.755598
\(403\) 9.46277 0.471374
\(404\) 3.08292 0.153381
\(405\) 0 0
\(406\) −25.0513 −1.24328
\(407\) 6.56016 0.325175
\(408\) 4.20800 0.208327
\(409\) 1.10089 0.0544355 0.0272177 0.999630i \(-0.491335\pi\)
0.0272177 + 0.999630i \(0.491335\pi\)
\(410\) 0 0
\(411\) 6.33689 0.312576
\(412\) −8.18117 −0.403057
\(413\) −18.4889 −0.909781
\(414\) −17.2585 −0.848209
\(415\) 0 0
\(416\) −26.7395 −1.31101
\(417\) −0.510413 −0.0249950
\(418\) 1.82669 0.0893463
\(419\) −18.7929 −0.918094 −0.459047 0.888412i \(-0.651809\pi\)
−0.459047 + 0.888412i \(0.651809\pi\)
\(420\) 0 0
\(421\) −24.7618 −1.20682 −0.603408 0.797433i \(-0.706191\pi\)
−0.603408 + 0.797433i \(0.706191\pi\)
\(422\) −5.31910 −0.258930
\(423\) −29.2318 −1.42130
\(424\) −9.61318 −0.466857
\(425\) 0 0
\(426\) −3.21133 −0.155589
\(427\) 7.63542 0.369504
\(428\) −0.565392 −0.0273293
\(429\) −2.31659 −0.111846
\(430\) 0 0
\(431\) −32.1985 −1.55095 −0.775474 0.631380i \(-0.782489\pi\)
−0.775474 + 0.631380i \(0.782489\pi\)
\(432\) 15.6451 0.752723
\(433\) 15.3042 0.735475 0.367737 0.929930i \(-0.380133\pi\)
0.367737 + 0.929930i \(0.380133\pi\)
\(434\) −7.11483 −0.341523
\(435\) 0 0
\(436\) 18.7391 0.897440
\(437\) 3.52199 0.168480
\(438\) 11.9207 0.569593
\(439\) 26.1812 1.24956 0.624781 0.780800i \(-0.285188\pi\)
0.624781 + 0.780800i \(0.285188\pi\)
\(440\) 0 0
\(441\) 11.0946 0.528313
\(442\) 46.3038 2.20245
\(443\) −21.5579 −1.02425 −0.512123 0.858912i \(-0.671141\pi\)
−0.512123 + 0.858912i \(0.671141\pi\)
\(444\) 4.94091 0.234485
\(445\) 0 0
\(446\) 11.5841 0.548525
\(447\) 5.92569 0.280275
\(448\) 3.56479 0.168420
\(449\) 11.7990 0.556829 0.278415 0.960461i \(-0.410191\pi\)
0.278415 + 0.960461i \(0.410191\pi\)
\(450\) 0 0
\(451\) 7.75013 0.364939
\(452\) 3.39061 0.159481
\(453\) 7.46163 0.350578
\(454\) −3.66122 −0.171829
\(455\) 0 0
\(456\) −0.682563 −0.0319639
\(457\) 15.6863 0.733773 0.366886 0.930266i \(-0.380424\pi\)
0.366886 + 0.930266i \(0.380424\pi\)
\(458\) 38.3410 1.79156
\(459\) −19.7381 −0.921296
\(460\) 0 0
\(461\) 27.2451 1.26893 0.634466 0.772951i \(-0.281220\pi\)
0.634466 + 0.772951i \(0.281220\pi\)
\(462\) 1.74178 0.0810352
\(463\) 17.9364 0.833573 0.416787 0.909004i \(-0.363156\pi\)
0.416787 + 0.909004i \(0.363156\pi\)
\(464\) 39.5977 1.83828
\(465\) 0 0
\(466\) −32.1509 −1.48936
\(467\) −29.0100 −1.34242 −0.671211 0.741266i \(-0.734226\pi\)
−0.671211 + 0.741266i \(0.734226\pi\)
\(468\) 14.7446 0.681570
\(469\) 24.9122 1.15034
\(470\) 0 0
\(471\) −1.12635 −0.0518995
\(472\) 13.2350 0.609191
\(473\) −7.75102 −0.356392
\(474\) −5.10665 −0.234556
\(475\) 0 0
\(476\) −13.9475 −0.639285
\(477\) 21.2864 0.974639
\(478\) −48.6277 −2.22418
\(479\) 34.0896 1.55759 0.778797 0.627276i \(-0.215830\pi\)
0.778797 + 0.627276i \(0.215830\pi\)
\(480\) 0 0
\(481\) −26.9733 −1.22988
\(482\) −23.9921 −1.09281
\(483\) 3.35829 0.152808
\(484\) 1.33679 0.0607633
\(485\) 0 0
\(486\) −23.9713 −1.08736
\(487\) −6.38349 −0.289264 −0.144632 0.989486i \(-0.546200\pi\)
−0.144632 + 0.989486i \(0.546200\pi\)
\(488\) −5.46570 −0.247421
\(489\) 10.5673 0.477869
\(490\) 0 0
\(491\) −31.4552 −1.41956 −0.709778 0.704426i \(-0.751205\pi\)
−0.709778 + 0.704426i \(0.751205\pi\)
\(492\) 5.83716 0.263160
\(493\) −49.9572 −2.24996
\(494\) −7.51076 −0.337925
\(495\) 0 0
\(496\) 11.2461 0.504966
\(497\) −5.28071 −0.236872
\(498\) −1.88243 −0.0843539
\(499\) −31.9667 −1.43103 −0.715514 0.698599i \(-0.753807\pi\)
−0.715514 + 0.698599i \(0.753807\pi\)
\(500\) 0 0
\(501\) 3.55630 0.158884
\(502\) 47.7927 2.13309
\(503\) 34.0522 1.51831 0.759157 0.650907i \(-0.225611\pi\)
0.759157 + 0.650907i \(0.225611\pi\)
\(504\) 5.50003 0.244991
\(505\) 0 0
\(506\) 6.43359 0.286008
\(507\) 2.20066 0.0977347
\(508\) −1.43405 −0.0636256
\(509\) 25.7904 1.14314 0.571569 0.820554i \(-0.306335\pi\)
0.571569 + 0.820554i \(0.306335\pi\)
\(510\) 0 0
\(511\) 19.6024 0.867160
\(512\) −19.9389 −0.881184
\(513\) 3.20164 0.141356
\(514\) 23.0377 1.01615
\(515\) 0 0
\(516\) −5.83784 −0.256996
\(517\) 10.8969 0.479247
\(518\) 20.2806 0.891076
\(519\) 12.1286 0.532385
\(520\) 0 0
\(521\) 31.3774 1.37467 0.687334 0.726342i \(-0.258781\pi\)
0.687334 + 0.726342i \(0.258781\pi\)
\(522\) −39.7082 −1.73798
\(523\) 23.2388 1.01616 0.508082 0.861309i \(-0.330355\pi\)
0.508082 + 0.861309i \(0.330355\pi\)
\(524\) −10.4632 −0.457087
\(525\) 0 0
\(526\) −36.6343 −1.59733
\(527\) −14.1883 −0.618054
\(528\) −2.75317 −0.119816
\(529\) −10.5956 −0.460677
\(530\) 0 0
\(531\) −29.3063 −1.27178
\(532\) 2.26238 0.0980865
\(533\) −31.8661 −1.38027
\(534\) 9.64856 0.417534
\(535\) 0 0
\(536\) −17.8330 −0.770268
\(537\) −4.49052 −0.193780
\(538\) 36.7439 1.58414
\(539\) −4.13581 −0.178142
\(540\) 0 0
\(541\) 36.0085 1.54812 0.774062 0.633109i \(-0.218222\pi\)
0.774062 + 0.633109i \(0.218222\pi\)
\(542\) −28.4207 −1.22077
\(543\) −0.658941 −0.0282779
\(544\) 40.0928 1.71896
\(545\) 0 0
\(546\) −7.16166 −0.306491
\(547\) 28.1855 1.20513 0.602563 0.798071i \(-0.294146\pi\)
0.602563 + 0.798071i \(0.294146\pi\)
\(548\) 15.0353 0.642275
\(549\) 12.1027 0.516530
\(550\) 0 0
\(551\) 8.10336 0.345215
\(552\) −2.40398 −0.102320
\(553\) −8.39738 −0.357093
\(554\) 3.45629 0.146844
\(555\) 0 0
\(556\) −1.21104 −0.0513594
\(557\) −21.6248 −0.916274 −0.458137 0.888882i \(-0.651483\pi\)
−0.458137 + 0.888882i \(0.651483\pi\)
\(558\) −11.2775 −0.477415
\(559\) 31.8697 1.34795
\(560\) 0 0
\(561\) 3.47345 0.146649
\(562\) −8.38997 −0.353909
\(563\) −13.1791 −0.555434 −0.277717 0.960663i \(-0.589578\pi\)
−0.277717 + 0.960663i \(0.589578\pi\)
\(564\) 8.20725 0.345588
\(565\) 0 0
\(566\) −2.15179 −0.0904467
\(567\) −10.5670 −0.443773
\(568\) 3.78011 0.158610
\(569\) 8.61143 0.361010 0.180505 0.983574i \(-0.442227\pi\)
0.180505 + 0.983574i \(0.442227\pi\)
\(570\) 0 0
\(571\) −17.6546 −0.738821 −0.369410 0.929266i \(-0.620440\pi\)
−0.369410 + 0.929266i \(0.620440\pi\)
\(572\) −5.49647 −0.229819
\(573\) −8.82722 −0.368762
\(574\) 23.9593 1.00004
\(575\) 0 0
\(576\) 5.65045 0.235435
\(577\) 31.0907 1.29432 0.647162 0.762352i \(-0.275956\pi\)
0.647162 + 0.762352i \(0.275956\pi\)
\(578\) −38.3735 −1.59613
\(579\) −12.2131 −0.507560
\(580\) 0 0
\(581\) −3.09547 −0.128422
\(582\) −11.3146 −0.469005
\(583\) −7.93511 −0.328639
\(584\) −14.0321 −0.580652
\(585\) 0 0
\(586\) −20.5696 −0.849722
\(587\) 26.7211 1.10290 0.551450 0.834208i \(-0.314075\pi\)
0.551450 + 0.834208i \(0.314075\pi\)
\(588\) −3.11497 −0.128459
\(589\) 2.30144 0.0948290
\(590\) 0 0
\(591\) −2.58492 −0.106329
\(592\) −32.0567 −1.31752
\(593\) 24.2460 0.995666 0.497833 0.867273i \(-0.334129\pi\)
0.497833 + 0.867273i \(0.334129\pi\)
\(594\) 5.84841 0.239963
\(595\) 0 0
\(596\) 14.0596 0.575905
\(597\) 7.35930 0.301196
\(598\) −26.4529 −1.08174
\(599\) −18.2095 −0.744019 −0.372009 0.928229i \(-0.621331\pi\)
−0.372009 + 0.928229i \(0.621331\pi\)
\(600\) 0 0
\(601\) 37.9824 1.54934 0.774668 0.632368i \(-0.217917\pi\)
0.774668 + 0.632368i \(0.217917\pi\)
\(602\) −23.9621 −0.976622
\(603\) 39.4876 1.60806
\(604\) 17.7039 0.720362
\(605\) 0 0
\(606\) −2.37351 −0.0964174
\(607\) 20.0130 0.812301 0.406151 0.913806i \(-0.366871\pi\)
0.406151 + 0.913806i \(0.366871\pi\)
\(608\) −6.50330 −0.263744
\(609\) 7.72672 0.313103
\(610\) 0 0
\(611\) −44.8048 −1.81261
\(612\) −22.1079 −0.893657
\(613\) 42.1414 1.70208 0.851038 0.525105i \(-0.175974\pi\)
0.851038 + 0.525105i \(0.175974\pi\)
\(614\) 7.95057 0.320859
\(615\) 0 0
\(616\) −2.05029 −0.0826085
\(617\) −16.9044 −0.680545 −0.340273 0.940327i \(-0.610519\pi\)
−0.340273 + 0.940327i \(0.610519\pi\)
\(618\) 6.29861 0.253367
\(619\) 5.31177 0.213498 0.106749 0.994286i \(-0.465956\pi\)
0.106749 + 0.994286i \(0.465956\pi\)
\(620\) 0 0
\(621\) 11.2762 0.452497
\(622\) −13.9144 −0.557915
\(623\) 15.8661 0.635662
\(624\) 11.3202 0.453169
\(625\) 0 0
\(626\) −32.0865 −1.28243
\(627\) −0.563416 −0.0225006
\(628\) −2.67245 −0.106642
\(629\) 40.4433 1.61258
\(630\) 0 0
\(631\) 34.0425 1.35521 0.677604 0.735427i \(-0.263018\pi\)
0.677604 + 0.735427i \(0.263018\pi\)
\(632\) 6.01113 0.239110
\(633\) 1.64060 0.0652079
\(634\) −3.54728 −0.140880
\(635\) 0 0
\(636\) −5.97649 −0.236983
\(637\) 17.0051 0.673768
\(638\) 14.8023 0.586030
\(639\) −8.37030 −0.331124
\(640\) 0 0
\(641\) −15.0405 −0.594063 −0.297031 0.954868i \(-0.595997\pi\)
−0.297031 + 0.954868i \(0.595997\pi\)
\(642\) 0.435291 0.0171795
\(643\) −32.3618 −1.27622 −0.638112 0.769943i \(-0.720285\pi\)
−0.638112 + 0.769943i \(0.720285\pi\)
\(644\) 7.96808 0.313986
\(645\) 0 0
\(646\) 11.2615 0.443079
\(647\) 49.0042 1.92655 0.963276 0.268512i \(-0.0865319\pi\)
0.963276 + 0.268512i \(0.0865319\pi\)
\(648\) 7.56424 0.297151
\(649\) 10.9247 0.428833
\(650\) 0 0
\(651\) 2.19447 0.0860079
\(652\) 25.0725 0.981916
\(653\) −31.1992 −1.22092 −0.610460 0.792047i \(-0.709016\pi\)
−0.610460 + 0.792047i \(0.709016\pi\)
\(654\) −14.4271 −0.564143
\(655\) 0 0
\(656\) −37.8715 −1.47864
\(657\) 31.0712 1.21220
\(658\) 33.6876 1.31328
\(659\) −35.7237 −1.39160 −0.695799 0.718237i \(-0.744950\pi\)
−0.695799 + 0.718237i \(0.744950\pi\)
\(660\) 0 0
\(661\) 33.8677 1.31730 0.658651 0.752449i \(-0.271128\pi\)
0.658651 + 0.752449i \(0.271128\pi\)
\(662\) −24.4754 −0.951262
\(663\) −14.2817 −0.554657
\(664\) 2.21585 0.0859916
\(665\) 0 0
\(666\) 32.1461 1.24564
\(667\) 28.5400 1.10507
\(668\) 8.43788 0.326471
\(669\) −3.57296 −0.138139
\(670\) 0 0
\(671\) −4.51162 −0.174169
\(672\) −6.20103 −0.239210
\(673\) −21.1648 −0.815842 −0.407921 0.913017i \(-0.633746\pi\)
−0.407921 + 0.913017i \(0.633746\pi\)
\(674\) 60.5786 2.33340
\(675\) 0 0
\(676\) 5.22141 0.200824
\(677\) 34.5239 1.32686 0.663431 0.748238i \(-0.269100\pi\)
0.663431 + 0.748238i \(0.269100\pi\)
\(678\) −2.61040 −0.100252
\(679\) −18.6057 −0.714022
\(680\) 0 0
\(681\) 1.12925 0.0432729
\(682\) 4.20401 0.160980
\(683\) −12.8022 −0.489862 −0.244931 0.969540i \(-0.578765\pi\)
−0.244931 + 0.969540i \(0.578765\pi\)
\(684\) 3.58603 0.137115
\(685\) 0 0
\(686\) −34.4261 −1.31439
\(687\) −11.8257 −0.451179
\(688\) 37.8759 1.44401
\(689\) 32.6267 1.24298
\(690\) 0 0
\(691\) 39.5406 1.50419 0.752097 0.659052i \(-0.229042\pi\)
0.752097 + 0.659052i \(0.229042\pi\)
\(692\) 28.7769 1.09394
\(693\) 4.53995 0.172458
\(694\) 32.9294 1.24998
\(695\) 0 0
\(696\) −5.53106 −0.209654
\(697\) 47.7795 1.80978
\(698\) −16.1006 −0.609418
\(699\) 9.91646 0.375075
\(700\) 0 0
\(701\) 22.4546 0.848097 0.424049 0.905639i \(-0.360609\pi\)
0.424049 + 0.905639i \(0.360609\pi\)
\(702\) −24.0468 −0.907588
\(703\) −6.56016 −0.247421
\(704\) −2.10636 −0.0793865
\(705\) 0 0
\(706\) −2.90899 −0.109481
\(707\) −3.90301 −0.146788
\(708\) 8.22818 0.309234
\(709\) −1.41244 −0.0530451 −0.0265226 0.999648i \(-0.508443\pi\)
−0.0265226 + 0.999648i \(0.508443\pi\)
\(710\) 0 0
\(711\) −13.3104 −0.499181
\(712\) −11.3575 −0.425641
\(713\) 8.10564 0.303559
\(714\) 10.7381 0.401863
\(715\) 0 0
\(716\) −10.6545 −0.398177
\(717\) 14.9985 0.560130
\(718\) −66.4502 −2.47990
\(719\) −39.0879 −1.45773 −0.728867 0.684655i \(-0.759953\pi\)
−0.728867 + 0.684655i \(0.759953\pi\)
\(720\) 0 0
\(721\) 10.3574 0.385731
\(722\) −1.82669 −0.0679823
\(723\) 7.40002 0.275210
\(724\) −1.56344 −0.0581049
\(725\) 0 0
\(726\) −1.02918 −0.0381966
\(727\) −9.42098 −0.349405 −0.174702 0.984621i \(-0.555896\pi\)
−0.174702 + 0.984621i \(0.555896\pi\)
\(728\) 8.43013 0.312442
\(729\) −11.3379 −0.419922
\(730\) 0 0
\(731\) −47.7850 −1.76739
\(732\) −3.39801 −0.125594
\(733\) −4.63361 −0.171146 −0.0855731 0.996332i \(-0.527272\pi\)
−0.0855731 + 0.996332i \(0.527272\pi\)
\(734\) −9.13849 −0.337308
\(735\) 0 0
\(736\) −22.9046 −0.844274
\(737\) −14.7201 −0.542222
\(738\) 37.9772 1.39796
\(739\) −13.1654 −0.484298 −0.242149 0.970239i \(-0.577852\pi\)
−0.242149 + 0.970239i \(0.577852\pi\)
\(740\) 0 0
\(741\) 2.31659 0.0851019
\(742\) −24.5312 −0.900568
\(743\) 14.7689 0.541817 0.270909 0.962605i \(-0.412676\pi\)
0.270909 + 0.962605i \(0.412676\pi\)
\(744\) −1.57087 −0.0575911
\(745\) 0 0
\(746\) −62.3472 −2.28269
\(747\) −4.90655 −0.179521
\(748\) 8.24132 0.301332
\(749\) 0.715792 0.0261545
\(750\) 0 0
\(751\) 35.0415 1.27868 0.639341 0.768924i \(-0.279207\pi\)
0.639341 + 0.768924i \(0.279207\pi\)
\(752\) −53.2487 −1.94178
\(753\) −14.7410 −0.537191
\(754\) −60.8625 −2.21648
\(755\) 0 0
\(756\) 7.24333 0.263437
\(757\) 23.8005 0.865044 0.432522 0.901623i \(-0.357624\pi\)
0.432522 + 0.901623i \(0.357624\pi\)
\(758\) −28.8158 −1.04664
\(759\) −1.98435 −0.0720272
\(760\) 0 0
\(761\) −39.0282 −1.41477 −0.707386 0.706827i \(-0.750126\pi\)
−0.707386 + 0.706827i \(0.750126\pi\)
\(762\) 1.10406 0.0399959
\(763\) −23.7239 −0.858862
\(764\) −20.9440 −0.757726
\(765\) 0 0
\(766\) −40.9654 −1.48014
\(767\) −44.9190 −1.62193
\(768\) 11.7997 0.425787
\(769\) 9.84757 0.355113 0.177556 0.984111i \(-0.443181\pi\)
0.177556 + 0.984111i \(0.443181\pi\)
\(770\) 0 0
\(771\) −7.10564 −0.255903
\(772\) −28.9776 −1.04292
\(773\) 42.9837 1.54602 0.773008 0.634396i \(-0.218751\pi\)
0.773008 + 0.634396i \(0.218751\pi\)
\(774\) −37.9816 −1.36522
\(775\) 0 0
\(776\) 13.3186 0.478111
\(777\) −6.25524 −0.224405
\(778\) 33.5175 1.20166
\(779\) −7.75013 −0.277677
\(780\) 0 0
\(781\) 3.12026 0.111652
\(782\) 39.6630 1.41835
\(783\) 25.9441 0.927166
\(784\) 20.2099 0.721783
\(785\) 0 0
\(786\) 8.05552 0.287331
\(787\) 25.8709 0.922199 0.461099 0.887349i \(-0.347455\pi\)
0.461099 + 0.887349i \(0.347455\pi\)
\(788\) −6.13313 −0.218484
\(789\) 11.2993 0.402266
\(790\) 0 0
\(791\) −4.29254 −0.152625
\(792\) −3.24985 −0.115479
\(793\) 18.5503 0.658741
\(794\) −0.934320 −0.0331578
\(795\) 0 0
\(796\) 17.4611 0.618893
\(797\) −3.62989 −0.128577 −0.0642886 0.997931i \(-0.520478\pi\)
−0.0642886 + 0.997931i \(0.520478\pi\)
\(798\) −1.74178 −0.0616585
\(799\) 67.1796 2.37664
\(800\) 0 0
\(801\) 25.1489 0.888593
\(802\) −34.6749 −1.22441
\(803\) −11.5827 −0.408743
\(804\) −11.0867 −0.390999
\(805\) 0 0
\(806\) −17.2855 −0.608857
\(807\) −11.3331 −0.398945
\(808\) 2.79391 0.0982894
\(809\) −13.1327 −0.461720 −0.230860 0.972987i \(-0.574154\pi\)
−0.230860 + 0.972987i \(0.574154\pi\)
\(810\) 0 0
\(811\) 9.40230 0.330160 0.165080 0.986280i \(-0.447212\pi\)
0.165080 + 0.986280i \(0.447212\pi\)
\(812\) 18.3329 0.643358
\(813\) 8.76595 0.307435
\(814\) −11.9834 −0.420017
\(815\) 0 0
\(816\) −16.9733 −0.594183
\(817\) 7.75102 0.271174
\(818\) −2.01098 −0.0703124
\(819\) −18.6668 −0.652272
\(820\) 0 0
\(821\) −21.1701 −0.738843 −0.369422 0.929262i \(-0.620444\pi\)
−0.369422 + 0.929262i \(0.620444\pi\)
\(822\) −11.5755 −0.403743
\(823\) −25.2277 −0.879384 −0.439692 0.898149i \(-0.644912\pi\)
−0.439692 + 0.898149i \(0.644912\pi\)
\(824\) −7.41422 −0.258286
\(825\) 0 0
\(826\) 33.7735 1.17513
\(827\) 39.4460 1.37167 0.685836 0.727756i \(-0.259437\pi\)
0.685836 + 0.727756i \(0.259437\pi\)
\(828\) 12.6300 0.438922
\(829\) −36.5068 −1.26793 −0.633967 0.773360i \(-0.718574\pi\)
−0.633967 + 0.773360i \(0.718574\pi\)
\(830\) 0 0
\(831\) −1.06604 −0.0369806
\(832\) 8.66069 0.300255
\(833\) −25.4972 −0.883427
\(834\) 0.932366 0.0322852
\(835\) 0 0
\(836\) −1.33679 −0.0462339
\(837\) 7.36838 0.254688
\(838\) 34.3288 1.18587
\(839\) 42.4670 1.46612 0.733061 0.680163i \(-0.238091\pi\)
0.733061 + 0.680163i \(0.238091\pi\)
\(840\) 0 0
\(841\) 36.6645 1.26429
\(842\) 45.2321 1.55880
\(843\) 2.58776 0.0891273
\(844\) 3.89258 0.133988
\(845\) 0 0
\(846\) 53.3973 1.83584
\(847\) −1.69239 −0.0581513
\(848\) 38.7755 1.33156
\(849\) 0.663690 0.0227778
\(850\) 0 0
\(851\) −23.1048 −0.792023
\(852\) 2.35009 0.0805126
\(853\) −21.5088 −0.736446 −0.368223 0.929738i \(-0.620034\pi\)
−0.368223 + 0.929738i \(0.620034\pi\)
\(854\) −13.9475 −0.477275
\(855\) 0 0
\(856\) −0.512389 −0.0175131
\(857\) 12.6370 0.431670 0.215835 0.976430i \(-0.430753\pi\)
0.215835 + 0.976430i \(0.430753\pi\)
\(858\) 4.23168 0.144467
\(859\) −3.39720 −0.115911 −0.0579555 0.998319i \(-0.518458\pi\)
−0.0579555 + 0.998319i \(0.518458\pi\)
\(860\) 0 0
\(861\) −7.38990 −0.251847
\(862\) 58.8167 2.00330
\(863\) 10.9308 0.372088 0.186044 0.982541i \(-0.440433\pi\)
0.186044 + 0.982541i \(0.440433\pi\)
\(864\) −20.8212 −0.708353
\(865\) 0 0
\(866\) −27.9561 −0.949987
\(867\) 11.8358 0.401963
\(868\) 5.20672 0.176727
\(869\) 4.96184 0.168319
\(870\) 0 0
\(871\) 60.5243 2.05079
\(872\) 16.9824 0.575096
\(873\) −29.4914 −0.998133
\(874\) −6.43359 −0.217619
\(875\) 0 0
\(876\) −8.72371 −0.294747
\(877\) −31.7230 −1.07121 −0.535605 0.844468i \(-0.679917\pi\)
−0.535605 + 0.844468i \(0.679917\pi\)
\(878\) −47.8250 −1.61402
\(879\) 6.34439 0.213991
\(880\) 0 0
\(881\) −6.45839 −0.217589 −0.108794 0.994064i \(-0.534699\pi\)
−0.108794 + 0.994064i \(0.534699\pi\)
\(882\) −20.2663 −0.682403
\(883\) 48.5242 1.63297 0.816485 0.577367i \(-0.195920\pi\)
0.816485 + 0.577367i \(0.195920\pi\)
\(884\) −33.8857 −1.13970
\(885\) 0 0
\(886\) 39.3796 1.32298
\(887\) 44.0097 1.47770 0.738851 0.673869i \(-0.235369\pi\)
0.738851 + 0.673869i \(0.235369\pi\)
\(888\) 4.47772 0.150262
\(889\) 1.81552 0.0608905
\(890\) 0 0
\(891\) 6.24383 0.209176
\(892\) −8.47742 −0.283845
\(893\) −10.8969 −0.364652
\(894\) −10.8244 −0.362022
\(895\) 0 0
\(896\) 15.5005 0.517835
\(897\) 8.15900 0.272421
\(898\) −21.5531 −0.719236
\(899\) 18.6494 0.621991
\(900\) 0 0
\(901\) −48.9199 −1.62976
\(902\) −14.1571 −0.471379
\(903\) 7.39076 0.245949
\(904\) 3.07275 0.102198
\(905\) 0 0
\(906\) −13.6301 −0.452829
\(907\) 2.05084 0.0680971 0.0340485 0.999420i \(-0.489160\pi\)
0.0340485 + 0.999420i \(0.489160\pi\)
\(908\) 2.67932 0.0889164
\(909\) −6.18655 −0.205195
\(910\) 0 0
\(911\) −16.8417 −0.557990 −0.278995 0.960293i \(-0.590001\pi\)
−0.278995 + 0.960293i \(0.590001\pi\)
\(912\) 2.75317 0.0911666
\(913\) 1.82905 0.0605328
\(914\) −28.6539 −0.947788
\(915\) 0 0
\(916\) −28.0584 −0.927075
\(917\) 13.2465 0.437438
\(918\) 36.0554 1.19001
\(919\) 8.01221 0.264299 0.132149 0.991230i \(-0.457812\pi\)
0.132149 + 0.991230i \(0.457812\pi\)
\(920\) 0 0
\(921\) −2.45224 −0.0808039
\(922\) −49.7683 −1.63903
\(923\) −12.8295 −0.422289
\(924\) −1.27466 −0.0419332
\(925\) 0 0
\(926\) −32.7641 −1.07670
\(927\) 16.4173 0.539214
\(928\) −52.6986 −1.72992
\(929\) 16.7897 0.550853 0.275426 0.961322i \(-0.411181\pi\)
0.275426 + 0.961322i \(0.411181\pi\)
\(930\) 0 0
\(931\) 4.13581 0.135546
\(932\) 23.5284 0.770698
\(933\) 4.29168 0.140503
\(934\) 52.9922 1.73396
\(935\) 0 0
\(936\) 13.3624 0.436763
\(937\) −21.8471 −0.713713 −0.356856 0.934159i \(-0.616151\pi\)
−0.356856 + 0.934159i \(0.616151\pi\)
\(938\) −45.5068 −1.48585
\(939\) 9.89661 0.322964
\(940\) 0 0
\(941\) 17.8421 0.581635 0.290817 0.956779i \(-0.406073\pi\)
0.290817 + 0.956779i \(0.406073\pi\)
\(942\) 2.05749 0.0670368
\(943\) −27.2959 −0.888877
\(944\) −53.3845 −1.73752
\(945\) 0 0
\(946\) 14.1587 0.460339
\(947\) −41.9736 −1.36396 −0.681980 0.731371i \(-0.738881\pi\)
−0.681980 + 0.731371i \(0.738881\pi\)
\(948\) 3.73711 0.121376
\(949\) 47.6242 1.54595
\(950\) 0 0
\(951\) 1.09411 0.0354788
\(952\) −12.6400 −0.409665
\(953\) 8.06945 0.261395 0.130697 0.991422i \(-0.458278\pi\)
0.130697 + 0.991422i \(0.458278\pi\)
\(954\) −38.8837 −1.25891
\(955\) 0 0
\(956\) 35.5864 1.15094
\(957\) −4.56556 −0.147584
\(958\) −62.2711 −2.01189
\(959\) −19.0348 −0.614666
\(960\) 0 0
\(961\) −25.7034 −0.829142
\(962\) 49.2718 1.58859
\(963\) 1.13458 0.0365614
\(964\) 17.5577 0.565496
\(965\) 0 0
\(966\) −6.13455 −0.197376
\(967\) 3.86360 0.124245 0.0621225 0.998069i \(-0.480213\pi\)
0.0621225 + 0.998069i \(0.480213\pi\)
\(968\) 1.21147 0.0389382
\(969\) −3.47345 −0.111583
\(970\) 0 0
\(971\) 42.7787 1.37283 0.686417 0.727208i \(-0.259182\pi\)
0.686417 + 0.727208i \(0.259182\pi\)
\(972\) 17.5425 0.562675
\(973\) 1.53318 0.0491516
\(974\) 11.6607 0.373631
\(975\) 0 0
\(976\) 22.0463 0.705686
\(977\) −54.5146 −1.74408 −0.872039 0.489437i \(-0.837202\pi\)
−0.872039 + 0.489437i \(0.837202\pi\)
\(978\) −19.3031 −0.617246
\(979\) −9.37496 −0.299625
\(980\) 0 0
\(981\) −37.6040 −1.20060
\(982\) 57.4590 1.83359
\(983\) −33.5422 −1.06983 −0.534916 0.844905i \(-0.679657\pi\)
−0.534916 + 0.844905i \(0.679657\pi\)
\(984\) 5.28995 0.168637
\(985\) 0 0
\(986\) 91.2562 2.90619
\(987\) −10.3905 −0.330732
\(988\) 5.49647 0.174866
\(989\) 27.2991 0.868060
\(990\) 0 0
\(991\) 31.7767 1.00942 0.504710 0.863289i \(-0.331600\pi\)
0.504710 + 0.863289i \(0.331600\pi\)
\(992\) −14.9669 −0.475200
\(993\) 7.54907 0.239562
\(994\) 9.64621 0.305959
\(995\) 0 0
\(996\) 1.37759 0.0436505
\(997\) 47.5668 1.50646 0.753228 0.657759i \(-0.228496\pi\)
0.753228 + 0.657759i \(0.228496\pi\)
\(998\) 58.3933 1.84841
\(999\) −21.0033 −0.664515
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 5225.2.a.h.1.2 5
5.4 even 2 209.2.a.c.1.4 5
15.14 odd 2 1881.2.a.k.1.2 5
20.19 odd 2 3344.2.a.t.1.4 5
55.54 odd 2 2299.2.a.n.1.2 5
95.94 odd 2 3971.2.a.h.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
209.2.a.c.1.4 5 5.4 even 2
1881.2.a.k.1.2 5 15.14 odd 2
2299.2.a.n.1.2 5 55.54 odd 2
3344.2.a.t.1.4 5 20.19 odd 2
3971.2.a.h.1.2 5 95.94 odd 2
5225.2.a.h.1.2 5 1.1 even 1 trivial