Properties

Label 209.2.a.c.1.4
Level $209$
Weight $2$
Character 209.1
Self dual yes
Analytic conductor $1.669$
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] = [209,2,Mod(1,209)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(209, 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("209.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 209 = 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 209.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: \(1.66887340224\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.51908\) of defining polynomial
Character \(\chi\) \(=\) 209.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.82669 q^{2} -0.563416 q^{3} +1.33679 q^{4} +2.34577 q^{5} -1.02918 q^{6} +1.69239 q^{7} -1.21147 q^{8} -2.68256 q^{9} +O(q^{10})\) \(q+1.82669 q^{2} -0.563416 q^{3} +1.33679 q^{4} +2.34577 q^{5} -1.02918 q^{6} +1.69239 q^{7} -1.21147 q^{8} -2.68256 q^{9} +4.28499 q^{10} +1.00000 q^{11} -0.753170 q^{12} +4.11168 q^{13} +3.09147 q^{14} -1.32164 q^{15} -4.88657 q^{16} -6.16499 q^{17} -4.90021 q^{18} -1.00000 q^{19} +3.13581 q^{20} -0.953520 q^{21} +1.82669 q^{22} +3.52199 q^{23} +0.682563 q^{24} +0.502638 q^{25} +7.51076 q^{26} +3.20164 q^{27} +2.26238 q^{28} -8.10336 q^{29} -2.41423 q^{30} -2.30144 q^{31} -6.50330 q^{32} -0.563416 q^{33} -11.2615 q^{34} +3.96996 q^{35} -3.58603 q^{36} -6.56016 q^{37} -1.82669 q^{38} -2.31659 q^{39} -2.84184 q^{40} +7.75013 q^{41} -1.74178 q^{42} +7.75102 q^{43} +1.33679 q^{44} -6.29268 q^{45} +6.43359 q^{46} -10.8969 q^{47} +2.75317 q^{48} -4.13581 q^{49} +0.918163 q^{50} +3.47345 q^{51} +5.49647 q^{52} +7.93511 q^{53} +5.84841 q^{54} +2.34577 q^{55} -2.05029 q^{56} +0.563416 q^{57} -14.8023 q^{58} +10.9247 q^{59} -1.76676 q^{60} -4.51162 q^{61} -4.20401 q^{62} -4.53995 q^{63} -2.10636 q^{64} +9.64506 q^{65} -1.02918 q^{66} +14.7201 q^{67} -8.24132 q^{68} -1.98435 q^{69} +7.25189 q^{70} +3.12026 q^{71} +3.24985 q^{72} +11.5827 q^{73} -11.9834 q^{74} -0.283194 q^{75} -1.33679 q^{76} +1.69239 q^{77} -4.23168 q^{78} +4.96184 q^{79} -11.4628 q^{80} +6.24383 q^{81} +14.1571 q^{82} -1.82905 q^{83} -1.27466 q^{84} -14.4617 q^{85} +14.1587 q^{86} +4.56556 q^{87} -1.21147 q^{88} -9.37496 q^{89} -11.4948 q^{90} +6.95858 q^{91} +4.70818 q^{92} +1.29666 q^{93} -19.9053 q^{94} -2.34577 q^{95} +3.66406 q^{96} -10.9937 q^{97} -7.55484 q^{98} -2.68256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} + q^{3} + 6 q^{4} - 5 q^{5} - 2 q^{6} + 6 q^{7} + 6 q^{8} + 4 q^{9} + 12 q^{10} + 5 q^{11} + 6 q^{12} + 4 q^{13} - 14 q^{14} + 3 q^{15} + 8 q^{16} - 4 q^{17} - 20 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{21} + 2 q^{22} + 3 q^{23} - 14 q^{24} + 6 q^{25} - 6 q^{26} - 11 q^{27} - 10 q^{28} + 10 q^{29} + 6 q^{30} + 11 q^{31} + 14 q^{32} + q^{33} - 4 q^{34} - 8 q^{35} - 26 q^{36} + q^{37} - 2 q^{38} + 2 q^{39} - 16 q^{40} + 2 q^{41} - 16 q^{42} + 20 q^{43} + 6 q^{44} - 28 q^{45} - 4 q^{46} - 20 q^{47} + 4 q^{48} + 3 q^{49} - 32 q^{50} + 24 q^{51} + 6 q^{52} - 14 q^{53} + 16 q^{54} - 5 q^{55} - 38 q^{56} - q^{57} - 6 q^{58} + 3 q^{59} - 40 q^{60} - 10 q^{61} - 6 q^{62} + 24 q^{63} - 2 q^{66} + 9 q^{67} + 24 q^{68} - 5 q^{69} + 50 q^{70} + 23 q^{71} - 12 q^{72} + 8 q^{74} - 18 q^{75} - 6 q^{76} + 6 q^{77} - 22 q^{78} + 44 q^{79} - 18 q^{80} + q^{81} - 30 q^{82} - 14 q^{83} + 14 q^{84} - 12 q^{85} + 52 q^{86} + 28 q^{87} + 6 q^{88} - 27 q^{89} + 26 q^{90} + 24 q^{91} + 58 q^{92} - 27 q^{93} - 8 q^{94} + 5 q^{95} + 50 q^{96} + 15 q^{97} - 10 q^{98} + 4 q^{99}+O(q^{100}) \) 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 \(-0.276511\pi\)
0.645832 + 0.763479i \(0.276511\pi\)
\(3\) −0.563416 −0.325288 −0.162644 0.986685i \(-0.552002\pi\)
−0.162644 + 0.986685i \(0.552002\pi\)
\(4\) 1.33679 0.668396
\(5\) 2.34577 1.04906 0.524530 0.851392i \(-0.324241\pi\)
0.524530 + 0.851392i \(0.324241\pi\)
\(6\) −1.02918 −0.420163
\(7\) 1.69239 0.639664 0.319832 0.947474i \(-0.396374\pi\)
0.319832 + 0.947474i \(0.396374\pi\)
\(8\) −1.21147 −0.428321
\(9\) −2.68256 −0.894188
\(10\) 4.28499 1.35503
\(11\) 1.00000 0.301511
\(12\) −0.753170 −0.217421
\(13\) 4.11168 1.14038 0.570188 0.821514i \(-0.306870\pi\)
0.570188 + 0.821514i \(0.306870\pi\)
\(14\) 3.09147 0.826231
\(15\) −1.32164 −0.341247
\(16\) −4.88657 −1.22164
\(17\) −6.16499 −1.49523 −0.747615 0.664132i \(-0.768801\pi\)
−0.747615 + 0.664132i \(0.768801\pi\)
\(18\) −4.90021 −1.15499
\(19\) −1.00000 −0.229416
\(20\) 3.13581 0.701188
\(21\) −0.953520 −0.208075
\(22\) 1.82669 0.389451
\(23\) 3.52199 0.734386 0.367193 0.930145i \(-0.380319\pi\)
0.367193 + 0.930145i \(0.380319\pi\)
\(24\) 0.682563 0.139328
\(25\) 0.502638 0.100528
\(26\) 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 \(0.771094\pi\)
\(30\) −2.41423 −0.440776
\(31\) −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
\(35\) 3.96996 0.671046
\(36\) −3.58603 −0.597672
\(37\) −6.56016 −1.07848 −0.539242 0.842151i \(-0.681289\pi\)
−0.539242 + 0.842151i \(0.681289\pi\)
\(38\) −1.82669 −0.296328
\(39\) −2.31659 −0.370951
\(40\) −2.84184 −0.449334
\(41\) 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 \(-0.298729\pi\)
0.591010 + 0.806664i \(0.298729\pi\)
\(44\) 1.33679 0.201529
\(45\) −6.29268 −0.938057
\(46\) 6.43359 0.948581
\(47\) −10.8969 −1.58948 −0.794742 0.606948i \(-0.792394\pi\)
−0.794742 + 0.606948i \(0.792394\pi\)
\(48\) 2.75317 0.397386
\(49\) −4.13581 −0.590830
\(50\) 0.918163 0.129848
\(51\) 3.47345 0.486381
\(52\) 5.49647 0.762223
\(53\) 7.93511 1.08997 0.544986 0.838445i \(-0.316535\pi\)
0.544986 + 0.838445i \(0.316535\pi\)
\(54\) 5.84841 0.795868
\(55\) 2.34577 0.316304
\(56\) −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 \(0.248179\pi\)
\(60\) −1.76676 −0.228088
\(61\) −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
\(65\) 9.64506 1.19632
\(66\) −1.02918 −0.126684
\(67\) 14.7201 1.79835 0.899173 0.437594i \(-0.144169\pi\)
0.899173 + 0.437594i \(0.144169\pi\)
\(68\) −8.24132 −0.999407
\(69\) −1.98435 −0.238887
\(70\) 7.25189 0.866766
\(71\) 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 \(-0.262923\pi\)
0.677824 + 0.735224i \(0.262923\pi\)
\(74\) −11.9834 −1.39304
\(75\) −0.283194 −0.0327004
\(76\) −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 \(0.409956\pi\)
\(80\) −11.4628 −1.28158
\(81\) 6.24383 0.693759
\(82\) 14.1571 1.56339
\(83\) −1.82905 −0.200765 −0.100382 0.994949i \(-0.532007\pi\)
−0.100382 + 0.994949i \(0.532007\pi\)
\(84\) −1.27466 −0.139077
\(85\) −14.4617 −1.56859
\(86\) 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 \(0.665518\pi\)
\(90\) −11.4948 −1.21165
\(91\) 6.95858 0.729457
\(92\) 4.70818 0.490861
\(93\) 1.29666 0.134458
\(94\) −19.9053 −2.05308
\(95\) −2.34577 −0.240671
\(96\) 3.66406 0.373961
\(97\) −10.9937 −1.11625 −0.558123 0.829758i \(-0.688478\pi\)
−0.558123 + 0.829758i \(0.688478\pi\)
\(98\) −7.55484 −0.763154
\(99\) −2.68256 −0.269608
\(100\) 0.671923 0.0671923
\(101\) 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 \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(104\) −4.98119 −0.488446
\(105\) −2.23674 −0.218283
\(106\) 14.4950 1.40788
\(107\) 0.422947 0.0408878 0.0204439 0.999791i \(-0.493492\pi\)
0.0204439 + 0.999791i \(0.493492\pi\)
\(108\) 4.27993 0.411837
\(109\) 14.0180 1.34268 0.671338 0.741151i \(-0.265720\pi\)
0.671338 + 0.741151i \(0.265720\pi\)
\(110\) 4.28499 0.408558
\(111\) 3.69609 0.350818
\(112\) −8.26999 −0.781441
\(113\) −2.53638 −0.238602 −0.119301 0.992858i \(-0.538065\pi\)
−0.119301 + 0.992858i \(0.538065\pi\)
\(114\) 1.02918 0.0963920
\(115\) 8.26179 0.770416
\(116\) −10.8325 −1.00577
\(117\) −11.0298 −1.01971
\(118\) 19.9561 1.83711
\(119\) −10.4336 −0.956445
\(120\) 1.60114 0.146163
\(121\) 1.00000 0.0909091
\(122\) −8.24132 −0.746134
\(123\) −4.36654 −0.393718
\(124\) −3.07654 −0.276282
\(125\) −10.5498 −0.943601
\(126\) −8.29307 −0.738806
\(127\) 1.07275 0.0951914 0.0475957 0.998867i \(-0.484844\pi\)
0.0475957 + 0.998867i \(0.484844\pi\)
\(128\) 9.15893 0.809543
\(129\) −4.36705 −0.384497
\(130\) 17.6185 1.54525
\(131\) −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
\(135\) 7.51032 0.646386
\(136\) 7.46873 0.640438
\(137\) −11.2473 −0.960919 −0.480460 0.877017i \(-0.659530\pi\)
−0.480460 + 0.877017i \(0.659530\pi\)
\(138\) −3.62478 −0.308562
\(139\) −0.905926 −0.0768397 −0.0384198 0.999262i \(-0.512232\pi\)
−0.0384198 + 0.999262i \(0.512232\pi\)
\(140\) 5.30702 0.448525
\(141\) 6.13951 0.517040
\(142\) 5.69975 0.478312
\(143\) 4.11168 0.343836
\(144\) 13.1085 1.09238
\(145\) −19.0086 −1.57858
\(146\) 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 \(0.358228\pi\)
\(150\) −0.517307 −0.0422380
\(151\) 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
\(155\) −5.39864 −0.433629
\(156\) −3.09679 −0.247942
\(157\) 1.99915 0.159549 0.0797747 0.996813i \(-0.474580\pi\)
0.0797747 + 0.996813i \(0.474580\pi\)
\(158\) 9.06373 0.721072
\(159\) −4.47076 −0.354555
\(160\) −15.2552 −1.20603
\(161\) 5.96059 0.469761
\(162\) 11.4055 0.896104
\(163\) −18.7557 −1.46906 −0.734531 0.678575i \(-0.762598\pi\)
−0.734531 + 0.678575i \(0.762598\pi\)
\(164\) 10.3603 0.809005
\(165\) −1.32164 −0.102890
\(166\) −3.34111 −0.259320
\(167\) −6.31203 −0.488440 −0.244220 0.969720i \(-0.578532\pi\)
−0.244220 + 0.969720i \(0.578532\pi\)
\(168\) 1.15516 0.0891229
\(169\) 3.90593 0.300456
\(170\) −26.4170 −2.02609
\(171\) 2.68256 0.205141
\(172\) 10.3615 0.790058
\(173\) −21.5269 −1.63666 −0.818328 0.574751i \(-0.805099\pi\)
−0.818328 + 0.574751i \(0.805099\pi\)
\(174\) 8.33986 0.632243
\(175\) 0.850661 0.0643039
\(176\) −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 \(0.596273\pi\)
\(180\) −8.41200 −0.626994
\(181\) −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
\(185\) −15.3886 −1.13139
\(186\) 2.36860 0.173674
\(187\) −6.16499 −0.450829
\(188\) −14.5670 −1.06240
\(189\) 5.41844 0.394133
\(190\) −4.28499 −0.310866
\(191\) −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 \(-0.215133\pi\)
0.780169 + 0.625568i \(0.215133\pi\)
\(194\) −20.0821 −1.44181
\(195\) −5.43418 −0.389149
\(196\) −5.52872 −0.394908
\(197\) 4.58794 0.326877 0.163439 0.986554i \(-0.447741\pi\)
0.163439 + 0.986554i \(0.447741\pi\)
\(198\) −4.90021 −0.348243
\(199\) 13.0619 0.925937 0.462968 0.886375i \(-0.346784\pi\)
0.462968 + 0.886375i \(0.346784\pi\)
\(200\) −0.608933 −0.0430580
\(201\) −8.29353 −0.584980
\(202\) 4.21272 0.296406
\(203\) −13.7141 −0.962539
\(204\) 4.64329 0.325095
\(205\) 18.1800 1.26975
\(206\) 11.1793 0.778901
\(207\) −9.44797 −0.656679
\(208\) −20.0920 −1.39313
\(209\) −1.00000 −0.0691714
\(210\) −4.08583 −0.281949
\(211\) 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
\(215\) 18.1821 1.24001
\(216\) −3.87871 −0.263913
\(217\) −3.89493 −0.264405
\(218\) 25.6064 1.73429
\(219\) −6.52585 −0.440976
\(220\) 3.13581 0.211416
\(221\) −25.3485 −1.70512
\(222\) 6.75161 0.453139
\(223\) 6.34161 0.424665 0.212333 0.977197i \(-0.431894\pi\)
0.212333 + 0.977197i \(0.431894\pi\)
\(224\) −11.0061 −0.735378
\(225\) −1.34836 −0.0898905
\(226\) −4.63317 −0.308194
\(227\) −2.00429 −0.133029 −0.0665147 0.997785i \(-0.521188\pi\)
−0.0665147 + 0.997785i \(0.521188\pi\)
\(228\) 0.753170 0.0498799
\(229\) −20.9893 −1.38701 −0.693507 0.720450i \(-0.743935\pi\)
−0.693507 + 0.720450i \(0.743935\pi\)
\(230\) 15.0917 0.995118
\(231\) −0.953520 −0.0627370
\(232\) 9.81701 0.644518
\(233\) −17.6006 −1.15305 −0.576527 0.817078i \(-0.695593\pi\)
−0.576527 + 0.817078i \(0.695593\pi\)
\(234\) −20.1481 −1.31712
\(235\) −25.5617 −1.66746
\(236\) 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 \(0.169854\pi\)
\(240\) 6.45830 0.416882
\(241\) 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
\(245\) −9.70166 −0.619816
\(246\) −7.97632 −0.508551
\(247\) −4.11168 −0.261620
\(248\) 2.78813 0.177046
\(249\) 1.03052 0.0653063
\(250\) −19.2712 −1.21882
\(251\) −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
\(255\) 8.14792 0.510243
\(256\) 20.9432 1.30895
\(257\) 12.6117 0.786697 0.393349 0.919389i \(-0.371317\pi\)
0.393349 + 0.919389i \(0.371317\pi\)
\(258\) −7.97724 −0.496641
\(259\) −11.1024 −0.689867
\(260\) 12.8934 0.799618
\(261\) 21.7378 1.34554
\(262\) −14.2977 −0.883312
\(263\) −20.0550 −1.23664 −0.618322 0.785925i \(-0.712187\pi\)
−0.618322 + 0.785925i \(0.712187\pi\)
\(264\) 0.682563 0.0420088
\(265\) 18.6139 1.14345
\(266\) −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 \(0.710125\pi\)
\(270\) 13.7190 0.834913
\(271\) 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
\(275\) 0.502638 0.0303102
\(276\) −2.65266 −0.159671
\(277\) 1.89211 0.113686 0.0568429 0.998383i \(-0.481897\pi\)
0.0568429 + 0.998383i \(0.481897\pi\)
\(278\) −1.65485 −0.0992510
\(279\) 6.17375 0.369613
\(280\) −4.80951 −0.287423
\(281\) 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 \(-0.511147\pi\)
−0.0350117 + 0.999387i \(0.511147\pi\)
\(284\) 4.17114 0.247512
\(285\) 1.32164 0.0782874
\(286\) 7.51076 0.444121
\(287\) 13.1163 0.774228
\(288\) 17.4455 1.02799
\(289\) 21.0071 1.23571
\(290\) −34.7229 −2.03900
\(291\) 6.19405 0.363101
\(292\) 15.4836 0.906110
\(293\) −11.2606 −0.657851 −0.328925 0.944356i \(-0.606686\pi\)
−0.328925 + 0.944356i \(0.606686\pi\)
\(294\) 4.25651 0.248245
\(295\) 25.6269 1.49206
\(296\) 7.94745 0.461936
\(297\) 3.20164 0.185778
\(298\) 19.2121 1.11293
\(299\) 14.4813 0.837476
\(300\) −0.378572 −0.0218568
\(301\) 13.1178 0.756096
\(302\) 24.1919 1.39209
\(303\) −1.29935 −0.0746459
\(304\) 4.88657 0.280264
\(305\) −10.5832 −0.605993
\(306\) 30.2098 1.72698
\(307\) 4.35245 0.248407 0.124204 0.992257i \(-0.460362\pi\)
0.124204 + 0.992257i \(0.460362\pi\)
\(308\) 2.26238 0.128911
\(309\) −3.44810 −0.196156
\(310\) −9.86164 −0.560103
\(311\) 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 \(-0.665355\pi\)
−0.496427 + 0.868078i \(0.665355\pi\)
\(314\) 3.65182 0.206084
\(315\) −10.6497 −0.600041
\(316\) 6.63295 0.373132
\(317\) −1.94192 −0.109069 −0.0545345 0.998512i \(-0.517367\pi\)
−0.0545345 + 0.998512i \(0.517367\pi\)
\(318\) −8.16670 −0.457966
\(319\) −8.10336 −0.453701
\(320\) −4.94104 −0.276213
\(321\) −0.238295 −0.0133003
\(322\) 10.8882 0.606773
\(323\) 6.16499 0.343029
\(324\) 8.34671 0.463706
\(325\) 2.06669 0.114639
\(326\) −34.2609 −1.89754
\(327\) −7.89793 −0.436757
\(328\) −9.38907 −0.518425
\(329\) −18.4419 −1.01674
\(330\) −2.41423 −0.132899
\(331\) 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
\(335\) 34.5300 1.88657
\(336\) 4.65944 0.254193
\(337\) 33.1631 1.80651 0.903254 0.429107i \(-0.141172\pi\)
0.903254 + 0.429107i \(0.141172\pi\)
\(338\) 7.13491 0.388088
\(339\) 1.42903 0.0776145
\(340\) −19.3322 −1.04844
\(341\) −2.30144 −0.124630
\(342\) 4.90021 0.264973
\(343\) −18.8462 −1.01760
\(344\) −9.39016 −0.506283
\(345\) −4.65482 −0.250607
\(346\) −39.3229 −2.11401
\(347\) 18.0268 0.967730 0.483865 0.875143i \(-0.339233\pi\)
0.483865 + 0.875143i \(0.339233\pi\)
\(348\) 6.10321 0.327166
\(349\) 8.81411 0.471808 0.235904 0.971776i \(-0.424195\pi\)
0.235904 + 0.971776i \(0.424195\pi\)
\(350\) 1.55389 0.0830590
\(351\) 13.1641 0.702650
\(352\) −6.50330 −0.346627
\(353\) −1.59249 −0.0847599 −0.0423799 0.999102i \(-0.513494\pi\)
−0.0423799 + 0.999102i \(0.513494\pi\)
\(354\) −11.2436 −0.597589
\(355\) 7.31942 0.388474
\(356\) −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 \(0.0903763\pi\)
\(360\) 7.62341 0.401789
\(361\) 1.00000 0.0526316
\(362\) −2.13640 −0.112287
\(363\) −0.563416 −0.0295716
\(364\) 9.30218 0.487567
\(365\) 27.1703 1.42216
\(366\) 4.64329 0.242708
\(367\) −5.00276 −0.261142 −0.130571 0.991439i \(-0.541681\pi\)
−0.130571 + 0.991439i \(0.541681\pi\)
\(368\) −17.2105 −0.897158
\(369\) −20.7902 −1.08229
\(370\) −28.1102 −1.46138
\(371\) 13.4293 0.697216
\(372\) 1.73337 0.0898712
\(373\) −34.1313 −1.76725 −0.883625 0.468195i \(-0.844904\pi\)
−0.883625 + 0.468195i \(0.844904\pi\)
\(374\) −11.2615 −0.582320
\(375\) 5.94391 0.306942
\(376\) 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 \(0.367219\pi\)
\(380\) −3.13581 −0.160864
\(381\) −0.604405 −0.0309646
\(382\) −28.6193 −1.46429
\(383\) −22.4260 −1.14592 −0.572958 0.819585i \(-0.694204\pi\)
−0.572958 + 0.819585i \(0.694204\pi\)
\(384\) −5.16028 −0.263335
\(385\) 3.96996 0.202328
\(386\) 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 \(0.654003\pi\)
\(390\) −9.92655 −0.502650
\(391\) −21.7131 −1.09808
\(392\) 5.01042 0.253065
\(393\) 4.40990 0.222450
\(394\) 8.38074 0.422216
\(395\) 11.6393 0.585638
\(396\) −3.58603 −0.180205
\(397\) −0.511483 −0.0256706 −0.0128353 0.999918i \(-0.504086\pi\)
−0.0128353 + 0.999918i \(0.504086\pi\)
\(398\) 23.8601 1.19600
\(399\) 0.953520 0.0477357
\(400\) −2.45618 −0.122809
\(401\) 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
\(405\) 14.6466 0.727795
\(406\) −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 \(0.491335\pi\)
\(410\) 33.2092 1.64009
\(411\) 6.33689 0.312576
\(412\) 8.18117 0.403057
\(413\) 18.4889 0.909781
\(414\) −17.2585 −0.848209
\(415\) −4.29054 −0.210614
\(416\) −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 \(0.651809\pi\)
\(420\) −2.99006 −0.145900
\(421\) −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
\(425\) −3.09876 −0.150312
\(426\) −3.21133 −0.155589
\(427\) −7.63542 −0.369504
\(428\) 0.565392 0.0273293
\(429\) −2.31659 −0.111846
\(430\) 33.2131 1.60168
\(431\) −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 \(-0.619867\pi\)
−0.367737 + 0.929930i \(0.619867\pi\)
\(434\) −7.11483 −0.341523
\(435\) 10.7098 0.513494
\(436\) 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 \(0.285188\pi\)
\(440\) −2.84184 −0.135479
\(441\) 11.0946 0.528313
\(442\) −46.3038 −2.20245
\(443\) 21.5579 1.02425 0.512123 0.858912i \(-0.328859\pi\)
0.512123 + 0.858912i \(0.328859\pi\)
\(444\) 4.94091 0.234485
\(445\) −21.9915 −1.04250
\(446\) 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 \(0.410191\pi\)
\(450\) −2.46303 −0.116108
\(451\) 7.75013 0.364939
\(452\) −3.39061 −0.159481
\(453\) −7.46163 −0.350578
\(454\) −3.66122 −0.171829
\(455\) 16.3232 0.765245
\(456\) −0.682563 −0.0319639
\(457\) −15.6863 −0.733773 −0.366886 0.930266i \(-0.619576\pi\)
−0.366886 + 0.930266i \(0.619576\pi\)
\(458\) −38.3410 −1.79156
\(459\) −19.7381 −0.921296
\(460\) 11.0443 0.514943
\(461\) 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 \(-0.636844\pi\)
−0.416787 + 0.909004i \(0.636844\pi\)
\(464\) 39.5977 1.83828
\(465\) 3.04168 0.141054
\(466\) −32.1509 −1.48936
\(467\) 29.0100 1.34242 0.671211 0.741266i \(-0.265774\pi\)
0.671211 + 0.741266i \(0.265774\pi\)
\(468\) −14.7446 −0.681570
\(469\) 24.9122 1.15034
\(470\) −46.6933 −2.15380
\(471\) −1.12635 −0.0518995
\(472\) −13.2350 −0.609191
\(473\) 7.75102 0.356392
\(474\) −5.10665 −0.234556
\(475\) −0.502638 −0.0230626
\(476\) −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 \(0.215830\pi\)
\(480\) 8.59504 0.392308
\(481\) −26.9733 −1.22988
\(482\) 23.9921 1.09281
\(483\) −3.35829 −0.152808
\(484\) 1.33679 0.0607633
\(485\) −25.7888 −1.17101
\(486\) −23.9713 −1.08736
\(487\) 6.38349 0.289264 0.144632 0.989486i \(-0.453800\pi\)
0.144632 + 0.989486i \(0.453800\pi\)
\(488\) 5.46570 0.247421
\(489\) 10.5673 0.477869
\(490\) −17.7219 −0.800594
\(491\) −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
\(495\) −6.29268 −0.282835
\(496\) 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 \(0.753807\pi\)
\(500\) −14.1029 −0.630699
\(501\) 3.55630 0.158884
\(502\) −47.7927 −2.13309
\(503\) −34.0522 −1.51831 −0.759157 0.650907i \(-0.774389\pi\)
−0.759157 + 0.650907i \(0.774389\pi\)
\(504\) 5.50003 0.244991
\(505\) 5.40983 0.240734
\(506\) 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 \(0.306335\pi\)
\(510\) 14.8837 0.659062
\(511\) 19.6024 0.867160
\(512\) 19.9389 0.881184
\(513\) −3.20164 −0.141356
\(514\) 23.0377 1.01615
\(515\) 14.3561 0.632606
\(516\) −5.83784 −0.256996
\(517\) −10.8969 −0.479247
\(518\) −20.2806 −0.891076
\(519\) 12.1286 0.532385
\(520\) −11.6847 −0.512410
\(521\) 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 \(-0.669645\pi\)
−0.508082 + 0.861309i \(0.669645\pi\)
\(524\) −10.4632 −0.457087
\(525\) −0.479275 −0.0209173
\(526\) −36.6343 −1.59733
\(527\) 14.1883 0.618054
\(528\) 2.75317 0.119816
\(529\) −10.5956 −0.460677
\(530\) 34.0019 1.47695
\(531\) −29.3063 −1.27178
\(532\) −2.26238 −0.0980865
\(533\) 31.8661 1.38027
\(534\) 9.64856 0.417534
\(535\) 0.992136 0.0428938
\(536\) −17.8330 −0.770268
\(537\) 4.49052 0.193780
\(538\) −36.7439 −1.58414
\(539\) −4.13581 −0.178142
\(540\) 10.0397 0.432042
\(541\) 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
\(545\) 32.8829 1.40855
\(546\) −7.16166 −0.306491
\(547\) −28.1855 −1.20513 −0.602563 0.798071i \(-0.705854\pi\)
−0.602563 + 0.798071i \(0.705854\pi\)
\(548\) −15.0353 −0.642275
\(549\) 12.1027 0.516530
\(550\) 0.918163 0.0391506
\(551\) 8.10336 0.345215
\(552\) 2.40398 0.102320
\(553\) 8.39738 0.357093
\(554\) 3.45629 0.146844
\(555\) 8.67019 0.368029
\(556\) −1.21104 −0.0513594
\(557\) 21.6248 0.916274 0.458137 0.888882i \(-0.348517\pi\)
0.458137 + 0.888882i \(0.348517\pi\)
\(558\) 11.2775 0.477415
\(559\) 31.8697 1.34795
\(560\) −19.3995 −0.819779
\(561\) 3.47345 0.146649
\(562\) 8.38997 0.353909
\(563\) 13.1791 0.555434 0.277717 0.960663i \(-0.410422\pi\)
0.277717 + 0.960663i \(0.410422\pi\)
\(564\) 8.20725 0.345588
\(565\) −5.94976 −0.250308
\(566\) −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 \(0.442227\pi\)
\(570\) 2.41423 0.101121
\(571\) −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
\(575\) 1.77029 0.0738261
\(576\) 5.65045 0.235435
\(577\) −31.0907 −1.29432 −0.647162 0.762352i \(-0.724044\pi\)
−0.647162 + 0.762352i \(0.724044\pi\)
\(578\) 38.3735 1.59613
\(579\) −12.2131 −0.507560
\(580\) −25.4106 −1.05512
\(581\) −3.09547 −0.128422
\(582\) 11.3146 0.469005
\(583\) 7.93511 0.328639
\(584\) −14.0321 −0.580652
\(585\) −25.8735 −1.06974
\(586\) −20.5696 −0.849722
\(587\) −26.7211 −1.10290 −0.551450 0.834208i \(-0.685925\pi\)
−0.551450 + 0.834208i \(0.685925\pi\)
\(588\) 3.11497 0.128459
\(589\) 2.30144 0.0948290
\(590\) 46.8124 1.92724
\(591\) −2.58492 −0.106329
\(592\) 32.0567 1.31752
\(593\) −24.2460 −0.995666 −0.497833 0.867273i \(-0.665871\pi\)
−0.497833 + 0.867273i \(0.665871\pi\)
\(594\) 5.84841 0.239963
\(595\) −24.4748 −1.00337
\(596\) 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 \(0.621331\pi\)
\(600\) 0.343082 0.0140063
\(601\) 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
\(605\) 2.34577 0.0953691
\(606\) −2.37351 −0.0964174
\(607\) −20.0130 −0.812301 −0.406151 0.913806i \(-0.633129\pi\)
−0.406151 + 0.913806i \(0.633129\pi\)
\(608\) 6.50330 0.263744
\(609\) 7.72672 0.313103
\(610\) −19.3322 −0.782739
\(611\) −44.8048 −1.81261
\(612\) 22.1079 0.893657
\(613\) −42.1414 −1.70208 −0.851038 0.525105i \(-0.824026\pi\)
−0.851038 + 0.525105i \(0.824026\pi\)
\(614\) 7.95057 0.320859
\(615\) −10.2429 −0.413034
\(616\) −2.05029 −0.0826085
\(617\) 16.9044 0.680545 0.340273 0.940327i \(-0.389481\pi\)
0.340273 + 0.940327i \(0.389481\pi\)
\(618\) −6.29861 −0.253367
\(619\) 5.31177 0.213498 0.106749 0.994286i \(-0.465956\pi\)
0.106749 + 0.994286i \(0.465956\pi\)
\(620\) −7.21686 −0.289836
\(621\) 11.2762 0.452497
\(622\) 13.9144 0.557915
\(623\) −15.8661 −0.635662
\(624\) 11.3202 0.453169
\(625\) −27.2605 −1.09042
\(626\) −32.0865 −1.28243
\(627\) 0.563416 0.0225006
\(628\) 2.67245 0.106642
\(629\) 40.4433 1.61258
\(630\) −19.4536 −0.775052
\(631\) 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
\(635\) 2.51643 0.0998615
\(636\) −5.97649 −0.236983
\(637\) −17.0051 −0.673768
\(638\) −14.8023 −0.586030
\(639\) −8.37030 −0.331124
\(640\) 21.4847 0.849259
\(641\) −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 \(-0.279715\pi\)
0.638112 + 0.769943i \(0.279715\pi\)
\(644\) 7.96808 0.313986
\(645\) −10.2441 −0.403361
\(646\) 11.2615 0.443079
\(647\) −49.0042 −1.92655 −0.963276 0.268512i \(-0.913468\pi\)
−0.963276 + 0.268512i \(0.913468\pi\)
\(648\) −7.56424 −0.297151
\(649\) 10.9247 0.428833
\(650\) 3.77520 0.148075
\(651\) 2.19447 0.0860079
\(652\) −25.0725 −0.981916
\(653\) 31.1992 1.22092 0.610460 0.792047i \(-0.290984\pi\)
0.610460 + 0.792047i \(0.290984\pi\)
\(654\) −14.4271 −0.564143
\(655\) −18.3605 −0.717406
\(656\) −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 \(0.744950\pi\)
\(660\) −1.76676 −0.0687712
\(661\) 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
\(665\) −3.96996 −0.153949
\(666\) 32.1461 1.24564
\(667\) −28.5400 −1.10507
\(668\) −8.43788 −0.326471
\(669\) −3.57296 −0.138139
\(670\) 63.0755 2.43682
\(671\) −4.51162 −0.174169
\(672\) 6.20103 0.239210
\(673\) 21.1648 0.815842 0.407921 0.913017i \(-0.366254\pi\)
0.407921 + 0.913017i \(0.366254\pi\)
\(674\) 60.5786 2.33340
\(675\) 1.60927 0.0619408
\(676\) 5.22141 0.200824
\(677\) −34.5239 −1.32686 −0.663431 0.748238i \(-0.730900\pi\)
−0.663431 + 0.748238i \(0.730900\pi\)
\(678\) 2.61040 0.100252
\(679\) −18.6057 −0.714022
\(680\) 17.5199 0.671858
\(681\) 1.12925 0.0432729
\(682\) −4.20401 −0.160980
\(683\) 12.8022 0.489862 0.244931 0.969540i \(-0.421235\pi\)
0.244931 + 0.969540i \(0.421235\pi\)
\(684\) 3.58603 0.137115
\(685\) −26.3835 −1.00806
\(686\) −34.4261 −1.31439
\(687\) 11.8257 0.451179
\(688\) −37.8759 −1.44401
\(689\) 32.6267 1.24298
\(690\) −8.50291 −0.323700
\(691\) 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
\(695\) −2.12510 −0.0806094
\(696\) −5.53106 −0.209654
\(697\) −47.7795 −1.80978
\(698\) 16.1006 0.609418
\(699\) 9.91646 0.375075
\(700\) 1.13716 0.0429805
\(701\) 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
\(705\) 14.4019 0.542406
\(706\) −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 \(0.508443\pi\)
\(710\) 13.3703 0.501778
\(711\) −13.3104 −0.499181
\(712\) 11.3575 0.425641
\(713\) −8.10564 −0.303559
\(714\) 10.7381 0.401863
\(715\) 9.64506 0.360705
\(716\) −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 \(0.759953\pi\)
\(720\) 30.7496 1.14597
\(721\) 10.3574 0.385731
\(722\) 1.82669 0.0679823
\(723\) −7.40002 −0.275210
\(724\) −1.56344 −0.0581049
\(725\) −4.07306 −0.151270
\(726\) −1.02918 −0.0381966
\(727\) 9.42098 0.349405 0.174702 0.984621i \(-0.444104\pi\)
0.174702 + 0.984621i \(0.444104\pi\)
\(728\) −8.43013 −0.312442
\(729\) −11.3379 −0.419922
\(730\) 49.6316 1.83695
\(731\) −47.7850 −1.76739
\(732\) 3.39801 0.125594
\(733\) 4.63361 0.171146 0.0855731 0.996332i \(-0.472728\pi\)
0.0855731 + 0.996332i \(0.472728\pi\)
\(734\) −9.13849 −0.337308
\(735\) 5.46606 0.201619
\(736\) −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 \(0.577852\pi\)
\(740\) −20.5714 −0.756219
\(741\) 2.31659 0.0851019
\(742\) 24.5312 0.900568
\(743\) −14.7689 −0.541817 −0.270909 0.962605i \(-0.587324\pi\)
−0.270909 + 0.962605i \(0.587324\pi\)
\(744\) −1.57087 −0.0575911
\(745\) 24.6715 0.903894
\(746\) −62.3472 −2.28269
\(747\) 4.90655 0.179521
\(748\) −8.24132 −0.301332
\(749\) 0.715792 0.0261545
\(750\) 10.8577 0.396466
\(751\) 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
\(755\) 31.0664 1.13062
\(756\) 7.24333 0.263437
\(757\) −23.8005 −0.865044 −0.432522 0.901623i \(-0.642376\pi\)
−0.432522 + 0.901623i \(0.642376\pi\)
\(758\) 28.8158 1.04664
\(759\) −1.98435 −0.0720272
\(760\) 2.84184 0.103084
\(761\) −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
\(765\) 38.7943 1.40261
\(766\) −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 \(0.443181\pi\)
\(770\) 7.25189 0.261340
\(771\) −7.10564 −0.255903
\(772\) 28.9776 1.04292
\(773\) −42.9837 −1.54602 −0.773008 0.634396i \(-0.781249\pi\)
−0.773008 + 0.634396i \(0.781249\pi\)
\(774\) −37.9816 −1.36522
\(775\) −1.15679 −0.0415531
\(776\) 13.3186 0.478111
\(777\) 6.25524 0.224405
\(778\) −33.5175 −1.20166
\(779\) −7.75013 −0.277677
\(780\) −7.26437 −0.260106
\(781\) 3.12026 0.111652
\(782\) −39.6630 −1.41835
\(783\) −25.9441 −0.927166
\(784\) 20.2099 0.721783
\(785\) 4.68954 0.167377
\(786\) 8.05552 0.287331
\(787\) −25.8709 −0.922199 −0.461099 0.887349i \(-0.652545\pi\)
−0.461099 + 0.887349i \(0.652545\pi\)
\(788\) 6.13313 0.218484
\(789\) 11.2993 0.402266
\(790\) 21.2614 0.756448
\(791\) −4.29254 −0.152625
\(792\) 3.24985 0.115479
\(793\) −18.5503 −0.658741
\(794\) −0.934320 −0.0331578
\(795\) −10.4874 −0.371949
\(796\) 17.4611 0.618893
\(797\) 3.62989 0.128577 0.0642886 0.997931i \(-0.479522\pi\)
0.0642886 + 0.997931i \(0.479522\pi\)
\(798\) 1.74178 0.0616585
\(799\) 67.1796 2.37664
\(800\) −3.26880 −0.115570
\(801\) 25.1489 0.888593
\(802\) 34.6749 1.22441
\(803\) 11.5827 0.408743
\(804\) −11.0867 −0.390999
\(805\) 13.9822 0.492807
\(806\) −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 \(0.574154\pi\)
\(810\) 26.7548 0.940067
\(811\) 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
\(815\) −43.9966 −1.54114
\(816\) −16.9733 −0.594183
\(817\) −7.75102 −0.271174
\(818\) 2.01098 0.0703124
\(819\) −18.6668 −0.652272
\(820\) 24.3029 0.848695
\(821\) −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 \(-0.355088\pi\)
0.439692 + 0.898149i \(0.355088\pi\)
\(824\) −7.41422 −0.258286
\(825\) −0.283194 −0.00985955
\(826\) 33.7735 1.17513
\(827\) −39.4460 −1.37167 −0.685836 0.727756i \(-0.740563\pi\)
−0.685836 + 0.727756i \(0.740563\pi\)
\(828\) −12.6300 −0.438922
\(829\) −36.5068 −1.26793 −0.633967 0.773360i \(-0.718574\pi\)
−0.633967 + 0.773360i \(0.718574\pi\)
\(830\) −7.83748 −0.272043
\(831\) −1.06604 −0.0369806
\(832\) −8.66069 −0.300255
\(833\) 25.4972 0.883427
\(834\) 0.932366 0.0322852
\(835\) −14.8066 −0.512403
\(836\) −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 \(0.238091\pi\)
\(840\) 2.70975 0.0934953
\(841\) 36.6645 1.26429
\(842\) −45.2321 −1.55880
\(843\) −2.58776 −0.0891273
\(844\) 3.89258 0.133988
\(845\) 9.16241 0.315196
\(846\) 53.3973 1.83584
\(847\) 1.69239 0.0581513
\(848\) −38.7755 −1.33156
\(849\) 0.663690 0.0227778
\(850\) −5.66047 −0.194153
\(851\) −23.1048 −0.792023
\(852\) −2.35009 −0.0805126
\(853\) 21.5088 0.736446 0.368223 0.929738i \(-0.379966\pi\)
0.368223 + 0.929738i \(0.379966\pi\)
\(854\) −13.9475 −0.477275
\(855\) 6.29268 0.215205
\(856\) −0.512389 −0.0175131
\(857\) −12.6370 −0.431670 −0.215835 0.976430i \(-0.569247\pi\)
−0.215835 + 0.976430i \(0.569247\pi\)
\(858\) −4.23168 −0.144467
\(859\) −3.39720 −0.115911 −0.0579555 0.998319i \(-0.518458\pi\)
−0.0579555 + 0.998319i \(0.518458\pi\)
\(860\) 24.3057 0.828818
\(861\) −7.38990 −0.251847
\(862\) −58.8167 −2.00330
\(863\) −10.9308 −0.372088 −0.186044 0.982541i \(-0.559567\pi\)
−0.186044 + 0.982541i \(0.559567\pi\)
\(864\) −20.8212 −0.708353
\(865\) −50.4971 −1.71695
\(866\) −27.9561 −0.949987
\(867\) −11.8358 −0.401963
\(868\) −5.20672 −0.176727
\(869\) 4.96184 0.168319
\(870\) 19.5634 0.663261
\(871\) 60.5243 2.05079
\(872\) −16.9824 −0.575096
\(873\) 29.4914 0.998133
\(874\) −6.43359 −0.217619
\(875\) −17.8544 −0.603588
\(876\) −8.72371 −0.294747
\(877\) 31.7230 1.07121 0.535605 0.844468i \(-0.320083\pi\)
0.535605 + 0.844468i \(0.320083\pi\)
\(878\) 47.8250 1.61402
\(879\) 6.34439 0.213991
\(880\) −11.4628 −0.386410
\(881\) −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 \(-0.804080\pi\)
−0.816485 + 0.577367i \(0.804080\pi\)
\(884\) −33.8857 −1.13970
\(885\) −14.4386 −0.485348
\(886\) 39.3796 1.32298
\(887\) −44.0097 −1.47770 −0.738851 0.673869i \(-0.764631\pi\)
−0.738851 + 0.673869i \(0.764631\pi\)
\(888\) −4.47772 −0.150262
\(889\) 1.81552 0.0608905
\(890\) −40.1716 −1.34656
\(891\) 6.24383 0.209176
\(892\) 8.47742 0.283845
\(893\) 10.8969 0.364652
\(894\) −10.8244 −0.362022
\(895\) −18.6962 −0.624946
\(896\) 15.5005 0.517835
\(897\) −8.15900 −0.272421
\(898\) 21.5531 0.719236
\(899\) 18.6494 0.621991
\(900\) −1.80248 −0.0600825
\(901\) −48.9199 −1.62976
\(902\) 14.1571 0.471379
\(903\) −7.39076 −0.245949
\(904\) 3.07275 0.102198
\(905\) −2.74349 −0.0911966
\(906\) −13.6301 −0.452829
\(907\) −2.05084 −0.0680971 −0.0340485 0.999420i \(-0.510840\pi\)
−0.0340485 + 0.999420i \(0.510840\pi\)
\(908\) −2.67932 −0.0889164
\(909\) −6.18655 −0.205195
\(910\) 29.8175 0.988439
\(911\) −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
\(915\) 5.96275 0.197122
\(916\) −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 \(0.457812\pi\)
\(920\) −10.0089 −0.329985
\(921\) −2.45224 −0.0808039
\(922\) 49.7683 1.63903
\(923\) 12.8295 0.422289
\(924\) −1.27466 −0.0419332
\(925\) −3.29738 −0.108417
\(926\) −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 \(0.411181\pi\)
\(930\) 5.55620 0.182195
\(931\) 4.13581 0.135546
\(932\) −23.5284 −0.770698
\(933\) −4.29168 −0.140503
\(934\) 52.9922 1.73396
\(935\) −14.4617 −0.472947
\(936\) 13.3624 0.436763
\(937\) 21.8471 0.713713 0.356856 0.934159i \(-0.383849\pi\)
0.356856 + 0.934159i \(0.383849\pi\)
\(938\) 45.5068 1.48585
\(939\) 9.89661 0.322964
\(940\) −34.1707 −1.11453
\(941\) 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
\(945\) 12.7104 0.413470
\(946\) 14.1587 0.460339
\(947\) 41.9736 1.36396 0.681980 0.731371i \(-0.261119\pi\)
0.681980 + 0.731371i \(0.261119\pi\)
\(948\) −3.73711 −0.121376
\(949\) 47.6242 1.54595
\(950\) −0.918163 −0.0297892
\(951\) 1.09411 0.0354788
\(952\) 12.6400 0.409665
\(953\) −8.06945 −0.261395 −0.130697 0.991422i \(-0.541722\pi\)
−0.130697 + 0.991422i \(0.541722\pi\)
\(954\) −38.8837 −1.25891
\(955\) −36.7520 −1.18927
\(956\) 35.5864 1.15094
\(957\) 4.56556 0.147584
\(958\) 62.2711 2.01189
\(959\) −19.0348 −0.614666
\(960\) 2.78386 0.0898487
\(961\) −25.7034 −0.829142
\(962\) −49.2718 −1.58859
\(963\) −1.13458 −0.0365614
\(964\) 17.5577 0.565496
\(965\) 50.8491 1.63689
\(966\) −6.13455 −0.197376
\(967\) −3.86360 −0.124245 −0.0621225 0.998069i \(-0.519787\pi\)
−0.0621225 + 0.998069i \(0.519787\pi\)
\(968\) −1.21147 −0.0389382
\(969\) −3.47345 −0.111583
\(970\) −47.1081 −1.51255
\(971\) 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
\(975\) −1.16440 −0.0372908
\(976\) 22.0463 0.705686
\(977\) 54.5146 1.74408 0.872039 0.489437i \(-0.162798\pi\)
0.872039 + 0.489437i \(0.162798\pi\)
\(978\) 19.3031 0.617246
\(979\) −9.37496 −0.299625
\(980\) −12.9691 −0.414283
\(981\) −37.6040 −1.20060
\(982\) −57.4590 −1.83359
\(983\) 33.5422 1.06983 0.534916 0.844905i \(-0.320343\pi\)
0.534916 + 0.844905i \(0.320343\pi\)
\(984\) 5.28995 0.168637
\(985\) 10.7623 0.342914
\(986\) 91.2562 2.90619
\(987\) 10.3905 0.330732
\(988\) −5.49647 −0.174866
\(989\) 27.2991 0.868060
\(990\) −11.4948 −0.365328
\(991\) 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
\(995\) 30.6403 0.971363
\(996\) 1.37759 0.0436505
\(997\) −47.5668 −1.50646 −0.753228 0.657759i \(-0.771504\pi\)
−0.753228 + 0.657759i \(0.771504\pi\)
\(998\) −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 209.2.a.c.1.4 5
3.2 odd 2 1881.2.a.k.1.2 5
4.3 odd 2 3344.2.a.t.1.4 5
5.4 even 2 5225.2.a.h.1.2 5
11.10 odd 2 2299.2.a.n.1.2 5
19.18 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 1.1 even 1 trivial
1881.2.a.k.1.2 5 3.2 odd 2
2299.2.a.n.1.2 5 11.10 odd 2
3344.2.a.t.1.4 5 4.3 odd 2
3971.2.a.h.1.2 5 19.18 odd 2
5225.2.a.h.1.2 5 5.4 even 2