Properties

Label 8043.2.a.r.1.15
Level $8043$
Weight $2$
Character 8043.1
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $46$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(1\)
Dimension: \(46\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Character \(\chi\) \(=\) 8043.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.944841 q^{2} -1.00000 q^{3} -1.10728 q^{4} +3.73621 q^{5} +0.944841 q^{6} -1.00000 q^{7} +2.93588 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.944841 q^{2} -1.00000 q^{3} -1.10728 q^{4} +3.73621 q^{5} +0.944841 q^{6} -1.00000 q^{7} +2.93588 q^{8} +1.00000 q^{9} -3.53012 q^{10} +6.47318 q^{11} +1.10728 q^{12} -5.12465 q^{13} +0.944841 q^{14} -3.73621 q^{15} -0.559391 q^{16} -5.30213 q^{17} -0.944841 q^{18} +3.63286 q^{19} -4.13701 q^{20} +1.00000 q^{21} -6.11612 q^{22} -0.278452 q^{23} -2.93588 q^{24} +8.95926 q^{25} +4.84198 q^{26} -1.00000 q^{27} +1.10728 q^{28} +7.59321 q^{29} +3.53012 q^{30} -9.50320 q^{31} -5.34323 q^{32} -6.47318 q^{33} +5.00968 q^{34} -3.73621 q^{35} -1.10728 q^{36} -4.28310 q^{37} -3.43248 q^{38} +5.12465 q^{39} +10.9691 q^{40} +4.15413 q^{41} -0.944841 q^{42} -3.18694 q^{43} -7.16759 q^{44} +3.73621 q^{45} +0.263093 q^{46} +1.88454 q^{47} +0.559391 q^{48} +1.00000 q^{49} -8.46508 q^{50} +5.30213 q^{51} +5.67440 q^{52} -10.9878 q^{53} +0.944841 q^{54} +24.1851 q^{55} -2.93588 q^{56} -3.63286 q^{57} -7.17437 q^{58} -13.4972 q^{59} +4.13701 q^{60} +0.0371054 q^{61} +8.97902 q^{62} -1.00000 q^{63} +6.16728 q^{64} -19.1468 q^{65} +6.11612 q^{66} +14.1396 q^{67} +5.87092 q^{68} +0.278452 q^{69} +3.53012 q^{70} -16.3543 q^{71} +2.93588 q^{72} -3.07307 q^{73} +4.04685 q^{74} -8.95926 q^{75} -4.02258 q^{76} -6.47318 q^{77} -4.84198 q^{78} -10.8106 q^{79} -2.09000 q^{80} +1.00000 q^{81} -3.92499 q^{82} -11.4875 q^{83} -1.10728 q^{84} -19.8099 q^{85} +3.01115 q^{86} -7.59321 q^{87} +19.0045 q^{88} +1.46286 q^{89} -3.53012 q^{90} +5.12465 q^{91} +0.308323 q^{92} +9.50320 q^{93} -1.78059 q^{94} +13.5731 q^{95} +5.34323 q^{96} -16.2372 q^{97} -0.944841 q^{98} +6.47318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q + 3 q^{2} - 46 q^{3} + 45 q^{4} - 9 q^{5} - 3 q^{6} - 46 q^{7} + 6 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q + 3 q^{2} - 46 q^{3} + 45 q^{4} - 9 q^{5} - 3 q^{6} - 46 q^{7} + 6 q^{8} + 46 q^{9} - 10 q^{10} + 31 q^{11} - 45 q^{12} - 32 q^{13} - 3 q^{14} + 9 q^{15} + 43 q^{16} - 36 q^{17} + 3 q^{18} - 13 q^{19} - 19 q^{20} + 46 q^{21} - 13 q^{22} + 24 q^{23} - 6 q^{24} + 35 q^{25} - 11 q^{26} - 46 q^{27} - 45 q^{28} + 11 q^{29} + 10 q^{30} - 23 q^{31} + 5 q^{32} - 31 q^{33} - 35 q^{34} + 9 q^{35} + 45 q^{36} - 37 q^{37} - 32 q^{38} + 32 q^{39} - 28 q^{40} - 27 q^{41} + 3 q^{42} - 7 q^{43} + 46 q^{44} - 9 q^{45} + 16 q^{46} - 18 q^{47} - 43 q^{48} + 46 q^{49} + 10 q^{50} + 36 q^{51} - 62 q^{52} - 62 q^{53} - 3 q^{54} - 28 q^{55} - 6 q^{56} + 13 q^{57} - 36 q^{58} - 3 q^{59} + 19 q^{60} - 31 q^{61} - 41 q^{62} - 46 q^{63} + 42 q^{64} + 2 q^{65} + 13 q^{66} - 9 q^{67} - 70 q^{68} - 24 q^{69} + 10 q^{70} + 77 q^{71} + 6 q^{72} - 38 q^{73} + 14 q^{74} - 35 q^{75} - 41 q^{76} - 31 q^{77} + 11 q^{78} + 8 q^{79} - 59 q^{80} + 46 q^{81} - 53 q^{82} - 38 q^{83} + 45 q^{84} - 26 q^{85} + 37 q^{86} - 11 q^{87} - 26 q^{88} - 39 q^{89} - 10 q^{90} + 32 q^{91} + 2 q^{92} + 23 q^{93} - 55 q^{94} + 35 q^{95} - 5 q^{96} - 61 q^{97} + 3 q^{98} + 31 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.944841 −0.668104 −0.334052 0.942555i \(-0.608416\pi\)
−0.334052 + 0.942555i \(0.608416\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.10728 −0.553638
\(5\) 3.73621 1.67088 0.835442 0.549579i \(-0.185212\pi\)
0.835442 + 0.549579i \(0.185212\pi\)
\(6\) 0.944841 0.385730
\(7\) −1.00000 −0.377964
\(8\) 2.93588 1.03799
\(9\) 1.00000 0.333333
\(10\) −3.53012 −1.11632
\(11\) 6.47318 1.95174 0.975868 0.218361i \(-0.0700711\pi\)
0.975868 + 0.218361i \(0.0700711\pi\)
\(12\) 1.10728 0.319643
\(13\) −5.12465 −1.42132 −0.710661 0.703535i \(-0.751604\pi\)
−0.710661 + 0.703535i \(0.751604\pi\)
\(14\) 0.944841 0.252519
\(15\) −3.73621 −0.964685
\(16\) −0.559391 −0.139848
\(17\) −5.30213 −1.28596 −0.642978 0.765884i \(-0.722301\pi\)
−0.642978 + 0.765884i \(0.722301\pi\)
\(18\) −0.944841 −0.222701
\(19\) 3.63286 0.833436 0.416718 0.909036i \(-0.363180\pi\)
0.416718 + 0.909036i \(0.363180\pi\)
\(20\) −4.13701 −0.925064
\(21\) 1.00000 0.218218
\(22\) −6.11612 −1.30396
\(23\) −0.278452 −0.0580613 −0.0290306 0.999579i \(-0.509242\pi\)
−0.0290306 + 0.999579i \(0.509242\pi\)
\(24\) −2.93588 −0.599284
\(25\) 8.95926 1.79185
\(26\) 4.84198 0.949590
\(27\) −1.00000 −0.192450
\(28\) 1.10728 0.209255
\(29\) 7.59321 1.41002 0.705011 0.709196i \(-0.250942\pi\)
0.705011 + 0.709196i \(0.250942\pi\)
\(30\) 3.53012 0.644510
\(31\) −9.50320 −1.70683 −0.853413 0.521236i \(-0.825471\pi\)
−0.853413 + 0.521236i \(0.825471\pi\)
\(32\) −5.34323 −0.944558
\(33\) −6.47318 −1.12684
\(34\) 5.00968 0.859152
\(35\) −3.73621 −0.631535
\(36\) −1.10728 −0.184546
\(37\) −4.28310 −0.704138 −0.352069 0.935974i \(-0.614522\pi\)
−0.352069 + 0.935974i \(0.614522\pi\)
\(38\) −3.43248 −0.556822
\(39\) 5.12465 0.820601
\(40\) 10.9691 1.73436
\(41\) 4.15413 0.648766 0.324383 0.945926i \(-0.394843\pi\)
0.324383 + 0.945926i \(0.394843\pi\)
\(42\) −0.944841 −0.145792
\(43\) −3.18694 −0.486004 −0.243002 0.970026i \(-0.578132\pi\)
−0.243002 + 0.970026i \(0.578132\pi\)
\(44\) −7.16759 −1.08055
\(45\) 3.73621 0.556961
\(46\) 0.263093 0.0387910
\(47\) 1.88454 0.274889 0.137444 0.990509i \(-0.456111\pi\)
0.137444 + 0.990509i \(0.456111\pi\)
\(48\) 0.559391 0.0807412
\(49\) 1.00000 0.142857
\(50\) −8.46508 −1.19714
\(51\) 5.30213 0.742447
\(52\) 5.67440 0.786897
\(53\) −10.9878 −1.50929 −0.754644 0.656134i \(-0.772191\pi\)
−0.754644 + 0.656134i \(0.772191\pi\)
\(54\) 0.944841 0.128577
\(55\) 24.1851 3.26112
\(56\) −2.93588 −0.392324
\(57\) −3.63286 −0.481185
\(58\) −7.17437 −0.942041
\(59\) −13.4972 −1.75718 −0.878590 0.477577i \(-0.841515\pi\)
−0.878590 + 0.477577i \(0.841515\pi\)
\(60\) 4.13701 0.534086
\(61\) 0.0371054 0.00475086 0.00237543 0.999997i \(-0.499244\pi\)
0.00237543 + 0.999997i \(0.499244\pi\)
\(62\) 8.97902 1.14034
\(63\) −1.00000 −0.125988
\(64\) 6.16728 0.770910
\(65\) −19.1468 −2.37486
\(66\) 6.11612 0.752843
\(67\) 14.1396 1.72742 0.863711 0.503987i \(-0.168134\pi\)
0.863711 + 0.503987i \(0.168134\pi\)
\(68\) 5.87092 0.711954
\(69\) 0.278452 0.0335217
\(70\) 3.53012 0.421931
\(71\) −16.3543 −1.94090 −0.970451 0.241297i \(-0.922427\pi\)
−0.970451 + 0.241297i \(0.922427\pi\)
\(72\) 2.93588 0.345997
\(73\) −3.07307 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(74\) 4.04685 0.470437
\(75\) −8.95926 −1.03453
\(76\) −4.02258 −0.461422
\(77\) −6.47318 −0.737687
\(78\) −4.84198 −0.548246
\(79\) −10.8106 −1.21629 −0.608144 0.793826i \(-0.708086\pi\)
−0.608144 + 0.793826i \(0.708086\pi\)
\(80\) −2.09000 −0.233670
\(81\) 1.00000 0.111111
\(82\) −3.92499 −0.433443
\(83\) −11.4875 −1.26091 −0.630457 0.776224i \(-0.717132\pi\)
−0.630457 + 0.776224i \(0.717132\pi\)
\(84\) −1.10728 −0.120814
\(85\) −19.8099 −2.14868
\(86\) 3.01115 0.324701
\(87\) −7.59321 −0.814077
\(88\) 19.0045 2.02588
\(89\) 1.46286 0.155063 0.0775315 0.996990i \(-0.475296\pi\)
0.0775315 + 0.996990i \(0.475296\pi\)
\(90\) −3.53012 −0.372108
\(91\) 5.12465 0.537209
\(92\) 0.308323 0.0321449
\(93\) 9.50320 0.985436
\(94\) −1.78059 −0.183654
\(95\) 13.5731 1.39257
\(96\) 5.34323 0.545341
\(97\) −16.2372 −1.64864 −0.824318 0.566127i \(-0.808441\pi\)
−0.824318 + 0.566127i \(0.808441\pi\)
\(98\) −0.944841 −0.0954434
\(99\) 6.47318 0.650579
\(100\) −9.92037 −0.992037
\(101\) −9.24335 −0.919748 −0.459874 0.887984i \(-0.652105\pi\)
−0.459874 + 0.887984i \(0.652105\pi\)
\(102\) −5.00968 −0.496032
\(103\) 1.52895 0.150652 0.0753258 0.997159i \(-0.476000\pi\)
0.0753258 + 0.997159i \(0.476000\pi\)
\(104\) −15.0454 −1.47532
\(105\) 3.73621 0.364617
\(106\) 10.3817 1.00836
\(107\) −9.14736 −0.884309 −0.442154 0.896939i \(-0.645786\pi\)
−0.442154 + 0.896939i \(0.645786\pi\)
\(108\) 1.10728 0.106548
\(109\) −19.5235 −1.87001 −0.935005 0.354635i \(-0.884605\pi\)
−0.935005 + 0.354635i \(0.884605\pi\)
\(110\) −22.8511 −2.17877
\(111\) 4.28310 0.406534
\(112\) 0.559391 0.0528575
\(113\) 6.82703 0.642233 0.321117 0.947040i \(-0.395942\pi\)
0.321117 + 0.947040i \(0.395942\pi\)
\(114\) 3.43248 0.321481
\(115\) −1.04036 −0.0970137
\(116\) −8.40777 −0.780642
\(117\) −5.12465 −0.473774
\(118\) 12.7527 1.17398
\(119\) 5.30213 0.486046
\(120\) −10.9691 −1.00133
\(121\) 30.9020 2.80927
\(122\) −0.0350587 −0.00317407
\(123\) −4.15413 −0.374565
\(124\) 10.5227 0.944963
\(125\) 14.7926 1.32309
\(126\) 0.944841 0.0841731
\(127\) 10.5712 0.938043 0.469021 0.883187i \(-0.344607\pi\)
0.469021 + 0.883187i \(0.344607\pi\)
\(128\) 4.85935 0.429510
\(129\) 3.18694 0.280594
\(130\) 18.0906 1.58665
\(131\) −9.04132 −0.789944 −0.394972 0.918693i \(-0.629246\pi\)
−0.394972 + 0.918693i \(0.629246\pi\)
\(132\) 7.16759 0.623858
\(133\) −3.63286 −0.315009
\(134\) −13.3596 −1.15410
\(135\) −3.73621 −0.321562
\(136\) −15.5664 −1.33481
\(137\) 9.10325 0.777743 0.388871 0.921292i \(-0.372865\pi\)
0.388871 + 0.921292i \(0.372865\pi\)
\(138\) −0.263093 −0.0223960
\(139\) −3.56848 −0.302674 −0.151337 0.988482i \(-0.548358\pi\)
−0.151337 + 0.988482i \(0.548358\pi\)
\(140\) 4.13701 0.349641
\(141\) −1.88454 −0.158707
\(142\) 15.4523 1.29672
\(143\) −33.1728 −2.77405
\(144\) −0.559391 −0.0466160
\(145\) 28.3698 2.35598
\(146\) 2.90356 0.240300
\(147\) −1.00000 −0.0824786
\(148\) 4.74257 0.389837
\(149\) 17.3492 1.42130 0.710652 0.703544i \(-0.248400\pi\)
0.710652 + 0.703544i \(0.248400\pi\)
\(150\) 8.46508 0.691171
\(151\) 5.91767 0.481573 0.240787 0.970578i \(-0.422595\pi\)
0.240787 + 0.970578i \(0.422595\pi\)
\(152\) 10.6657 0.865099
\(153\) −5.30213 −0.428652
\(154\) 6.11612 0.492851
\(155\) −35.5059 −2.85191
\(156\) −5.67440 −0.454315
\(157\) 6.46779 0.516186 0.258093 0.966120i \(-0.416906\pi\)
0.258093 + 0.966120i \(0.416906\pi\)
\(158\) 10.2143 0.812607
\(159\) 10.9878 0.871388
\(160\) −19.9634 −1.57825
\(161\) 0.278452 0.0219451
\(162\) −0.944841 −0.0742337
\(163\) 14.0864 1.10333 0.551664 0.834066i \(-0.313993\pi\)
0.551664 + 0.834066i \(0.313993\pi\)
\(164\) −4.59976 −0.359181
\(165\) −24.1851 −1.88281
\(166\) 10.8538 0.842421
\(167\) −15.9505 −1.23428 −0.617142 0.786852i \(-0.711710\pi\)
−0.617142 + 0.786852i \(0.711710\pi\)
\(168\) 2.93588 0.226508
\(169\) 13.2620 1.02016
\(170\) 18.7172 1.43554
\(171\) 3.63286 0.277812
\(172\) 3.52882 0.269070
\(173\) 11.5900 0.881171 0.440585 0.897711i \(-0.354771\pi\)
0.440585 + 0.897711i \(0.354771\pi\)
\(174\) 7.17437 0.543888
\(175\) −8.95926 −0.677256
\(176\) −3.62104 −0.272946
\(177\) 13.4972 1.01451
\(178\) −1.38217 −0.103598
\(179\) −12.5110 −0.935116 −0.467558 0.883962i \(-0.654866\pi\)
−0.467558 + 0.883962i \(0.654866\pi\)
\(180\) −4.13701 −0.308355
\(181\) −23.7664 −1.76654 −0.883270 0.468865i \(-0.844663\pi\)
−0.883270 + 0.468865i \(0.844663\pi\)
\(182\) −4.84198 −0.358911
\(183\) −0.0371054 −0.00274291
\(184\) −0.817503 −0.0602671
\(185\) −16.0026 −1.17653
\(186\) −8.97902 −0.658373
\(187\) −34.3216 −2.50985
\(188\) −2.08671 −0.152189
\(189\) 1.00000 0.0727393
\(190\) −12.8245 −0.930384
\(191\) 15.7867 1.14229 0.571143 0.820851i \(-0.306500\pi\)
0.571143 + 0.820851i \(0.306500\pi\)
\(192\) −6.16728 −0.445085
\(193\) −9.25361 −0.666090 −0.333045 0.942911i \(-0.608076\pi\)
−0.333045 + 0.942911i \(0.608076\pi\)
\(194\) 15.3416 1.10146
\(195\) 19.1468 1.37113
\(196\) −1.10728 −0.0790911
\(197\) −25.7315 −1.83330 −0.916648 0.399695i \(-0.869116\pi\)
−0.916648 + 0.399695i \(0.869116\pi\)
\(198\) −6.11612 −0.434654
\(199\) 2.53649 0.179807 0.0899033 0.995950i \(-0.471344\pi\)
0.0899033 + 0.995950i \(0.471344\pi\)
\(200\) 26.3033 1.85993
\(201\) −14.1396 −0.997328
\(202\) 8.73350 0.614487
\(203\) −7.59321 −0.532939
\(204\) −5.87092 −0.411047
\(205\) 15.5207 1.08401
\(206\) −1.44461 −0.100651
\(207\) −0.278452 −0.0193538
\(208\) 2.86669 0.198769
\(209\) 23.5162 1.62665
\(210\) −3.53012 −0.243602
\(211\) 21.4603 1.47739 0.738694 0.674040i \(-0.235443\pi\)
0.738694 + 0.674040i \(0.235443\pi\)
\(212\) 12.1665 0.835599
\(213\) 16.3543 1.12058
\(214\) 8.64280 0.590810
\(215\) −11.9071 −0.812056
\(216\) −2.93588 −0.199761
\(217\) 9.50320 0.645119
\(218\) 18.4466 1.24936
\(219\) 3.07307 0.207659
\(220\) −26.7796 −1.80548
\(221\) 27.1716 1.82776
\(222\) −4.04685 −0.271607
\(223\) −19.0986 −1.27894 −0.639470 0.768816i \(-0.720846\pi\)
−0.639470 + 0.768816i \(0.720846\pi\)
\(224\) 5.34323 0.357009
\(225\) 8.95926 0.597284
\(226\) −6.45046 −0.429078
\(227\) −12.6892 −0.842214 −0.421107 0.907011i \(-0.638358\pi\)
−0.421107 + 0.907011i \(0.638358\pi\)
\(228\) 4.02258 0.266402
\(229\) 13.1299 0.867649 0.433825 0.900997i \(-0.357164\pi\)
0.433825 + 0.900997i \(0.357164\pi\)
\(230\) 0.982971 0.0648152
\(231\) 6.47318 0.425904
\(232\) 22.2928 1.46359
\(233\) 22.8584 1.49751 0.748753 0.662849i \(-0.230653\pi\)
0.748753 + 0.662849i \(0.230653\pi\)
\(234\) 4.84198 0.316530
\(235\) 7.04105 0.459307
\(236\) 14.9451 0.972841
\(237\) 10.8106 0.702225
\(238\) −5.00968 −0.324729
\(239\) −6.19886 −0.400971 −0.200485 0.979697i \(-0.564252\pi\)
−0.200485 + 0.979697i \(0.564252\pi\)
\(240\) 2.09000 0.134909
\(241\) 3.91505 0.252190 0.126095 0.992018i \(-0.459756\pi\)
0.126095 + 0.992018i \(0.459756\pi\)
\(242\) −29.1975 −1.87689
\(243\) −1.00000 −0.0641500
\(244\) −0.0410859 −0.00263025
\(245\) 3.73621 0.238698
\(246\) 3.92499 0.250248
\(247\) −18.6172 −1.18458
\(248\) −27.9003 −1.77167
\(249\) 11.4875 0.727989
\(250\) −13.9767 −0.883963
\(251\) 15.8405 0.999843 0.499921 0.866071i \(-0.333362\pi\)
0.499921 + 0.866071i \(0.333362\pi\)
\(252\) 1.10728 0.0697518
\(253\) −1.80247 −0.113320
\(254\) −9.98811 −0.626710
\(255\) 19.8099 1.24054
\(256\) −16.9259 −1.05787
\(257\) −10.6024 −0.661362 −0.330681 0.943743i \(-0.607278\pi\)
−0.330681 + 0.943743i \(0.607278\pi\)
\(258\) −3.01115 −0.187466
\(259\) 4.28310 0.266139
\(260\) 21.2007 1.31481
\(261\) 7.59321 0.470008
\(262\) 8.54261 0.527764
\(263\) 11.4422 0.705556 0.352778 0.935707i \(-0.385237\pi\)
0.352778 + 0.935707i \(0.385237\pi\)
\(264\) −19.0045 −1.16964
\(265\) −41.0527 −2.52185
\(266\) 3.43248 0.210459
\(267\) −1.46286 −0.0895257
\(268\) −15.6564 −0.956366
\(269\) −8.70422 −0.530705 −0.265353 0.964151i \(-0.585488\pi\)
−0.265353 + 0.964151i \(0.585488\pi\)
\(270\) 3.53012 0.214837
\(271\) 17.5628 1.06687 0.533434 0.845842i \(-0.320901\pi\)
0.533434 + 0.845842i \(0.320901\pi\)
\(272\) 2.96597 0.179838
\(273\) −5.12465 −0.310158
\(274\) −8.60112 −0.519613
\(275\) 57.9949 3.49722
\(276\) −0.308323 −0.0185589
\(277\) −12.4469 −0.747862 −0.373931 0.927457i \(-0.621990\pi\)
−0.373931 + 0.927457i \(0.621990\pi\)
\(278\) 3.37165 0.202218
\(279\) −9.50320 −0.568942
\(280\) −10.9691 −0.655527
\(281\) 17.3914 1.03748 0.518741 0.854931i \(-0.326401\pi\)
0.518741 + 0.854931i \(0.326401\pi\)
\(282\) 1.78059 0.106033
\(283\) 17.0134 1.01134 0.505670 0.862727i \(-0.331245\pi\)
0.505670 + 0.862727i \(0.331245\pi\)
\(284\) 18.1087 1.07456
\(285\) −13.5731 −0.804004
\(286\) 31.3430 1.85335
\(287\) −4.15413 −0.245211
\(288\) −5.34323 −0.314853
\(289\) 11.1126 0.653684
\(290\) −26.8050 −1.57404
\(291\) 16.2372 0.951840
\(292\) 3.40273 0.199130
\(293\) −4.69883 −0.274509 −0.137254 0.990536i \(-0.543828\pi\)
−0.137254 + 0.990536i \(0.543828\pi\)
\(294\) 0.944841 0.0551043
\(295\) −50.4282 −2.93604
\(296\) −12.5747 −0.730888
\(297\) −6.47318 −0.375612
\(298\) −16.3923 −0.949578
\(299\) 1.42697 0.0825238
\(300\) 9.92037 0.572753
\(301\) 3.18694 0.183692
\(302\) −5.59126 −0.321741
\(303\) 9.24335 0.531017
\(304\) −2.03219 −0.116554
\(305\) 0.138633 0.00793813
\(306\) 5.00968 0.286384
\(307\) 1.94371 0.110933 0.0554667 0.998461i \(-0.482335\pi\)
0.0554667 + 0.998461i \(0.482335\pi\)
\(308\) 7.16759 0.408411
\(309\) −1.52895 −0.0869787
\(310\) 33.5475 1.90537
\(311\) 21.5224 1.22042 0.610212 0.792238i \(-0.291084\pi\)
0.610212 + 0.792238i \(0.291084\pi\)
\(312\) 15.0454 0.851776
\(313\) −23.9927 −1.35615 −0.678073 0.734994i \(-0.737185\pi\)
−0.678073 + 0.734994i \(0.737185\pi\)
\(314\) −6.11104 −0.344866
\(315\) −3.73621 −0.210512
\(316\) 11.9703 0.673383
\(317\) −3.80509 −0.213715 −0.106858 0.994274i \(-0.534079\pi\)
−0.106858 + 0.994274i \(0.534079\pi\)
\(318\) −10.3817 −0.582178
\(319\) 49.1522 2.75199
\(320\) 23.0423 1.28810
\(321\) 9.14736 0.510556
\(322\) −0.263093 −0.0146616
\(323\) −19.2619 −1.07176
\(324\) −1.10728 −0.0615153
\(325\) −45.9131 −2.54680
\(326\) −13.3094 −0.737138
\(327\) 19.5235 1.07965
\(328\) 12.1960 0.673413
\(329\) −1.88454 −0.103898
\(330\) 22.8511 1.25791
\(331\) 24.3553 1.33869 0.669345 0.742952i \(-0.266575\pi\)
0.669345 + 0.742952i \(0.266575\pi\)
\(332\) 12.7198 0.698089
\(333\) −4.28310 −0.234713
\(334\) 15.0707 0.824630
\(335\) 52.8284 2.88632
\(336\) −0.559391 −0.0305173
\(337\) −0.254338 −0.0138547 −0.00692733 0.999976i \(-0.502205\pi\)
−0.00692733 + 0.999976i \(0.502205\pi\)
\(338\) −12.5305 −0.681570
\(339\) −6.82703 −0.370793
\(340\) 21.9350 1.18959
\(341\) −61.5159 −3.33127
\(342\) −3.43248 −0.185607
\(343\) −1.00000 −0.0539949
\(344\) −9.35648 −0.504468
\(345\) 1.04036 0.0560109
\(346\) −10.9507 −0.588713
\(347\) 7.73657 0.415321 0.207660 0.978201i \(-0.433415\pi\)
0.207660 + 0.978201i \(0.433415\pi\)
\(348\) 8.40777 0.450704
\(349\) 27.9850 1.49800 0.749000 0.662570i \(-0.230534\pi\)
0.749000 + 0.662570i \(0.230534\pi\)
\(350\) 8.46508 0.452477
\(351\) 5.12465 0.273534
\(352\) −34.5876 −1.84353
\(353\) −12.7351 −0.677822 −0.338911 0.940818i \(-0.610059\pi\)
−0.338911 + 0.940818i \(0.610059\pi\)
\(354\) −12.7527 −0.677797
\(355\) −61.1032 −3.24302
\(356\) −1.61979 −0.0858487
\(357\) −5.30213 −0.280619
\(358\) 11.8209 0.624755
\(359\) −24.3489 −1.28509 −0.642543 0.766249i \(-0.722121\pi\)
−0.642543 + 0.766249i \(0.722121\pi\)
\(360\) 10.9691 0.578121
\(361\) −5.80230 −0.305384
\(362\) 22.4554 1.18023
\(363\) −30.9020 −1.62193
\(364\) −5.67440 −0.297419
\(365\) −11.4816 −0.600976
\(366\) 0.0350587 0.00183255
\(367\) 24.3239 1.26970 0.634850 0.772636i \(-0.281062\pi\)
0.634850 + 0.772636i \(0.281062\pi\)
\(368\) 0.155764 0.00811975
\(369\) 4.15413 0.216255
\(370\) 15.1199 0.786045
\(371\) 10.9878 0.570457
\(372\) −10.5227 −0.545574
\(373\) −25.4732 −1.31895 −0.659476 0.751725i \(-0.729222\pi\)
−0.659476 + 0.751725i \(0.729222\pi\)
\(374\) 32.4285 1.67684
\(375\) −14.7926 −0.763888
\(376\) 5.53280 0.285332
\(377\) −38.9125 −2.00410
\(378\) −0.944841 −0.0485974
\(379\) 17.4613 0.896928 0.448464 0.893801i \(-0.351971\pi\)
0.448464 + 0.893801i \(0.351971\pi\)
\(380\) −15.0292 −0.770982
\(381\) −10.5712 −0.541579
\(382\) −14.9159 −0.763165
\(383\) −1.00000 −0.0510976
\(384\) −4.85935 −0.247978
\(385\) −24.1851 −1.23259
\(386\) 8.74319 0.445017
\(387\) −3.18694 −0.162001
\(388\) 17.9790 0.912747
\(389\) 11.6344 0.589888 0.294944 0.955515i \(-0.404699\pi\)
0.294944 + 0.955515i \(0.404699\pi\)
\(390\) −18.0906 −0.916056
\(391\) 1.47639 0.0746643
\(392\) 2.93588 0.148284
\(393\) 9.04132 0.456074
\(394\) 24.3122 1.22483
\(395\) −40.3907 −2.03228
\(396\) −7.16759 −0.360185
\(397\) 0.102318 0.00513518 0.00256759 0.999997i \(-0.499183\pi\)
0.00256759 + 0.999997i \(0.499183\pi\)
\(398\) −2.39658 −0.120129
\(399\) 3.63286 0.181871
\(400\) −5.01173 −0.250587
\(401\) −26.2737 −1.31204 −0.656022 0.754741i \(-0.727762\pi\)
−0.656022 + 0.754741i \(0.727762\pi\)
\(402\) 13.3596 0.666318
\(403\) 48.7006 2.42595
\(404\) 10.2349 0.509207
\(405\) 3.73621 0.185654
\(406\) 7.17437 0.356058
\(407\) −27.7253 −1.37429
\(408\) 15.5664 0.770654
\(409\) −15.7495 −0.778761 −0.389381 0.921077i \(-0.627311\pi\)
−0.389381 + 0.921077i \(0.627311\pi\)
\(410\) −14.6646 −0.724233
\(411\) −9.10325 −0.449030
\(412\) −1.69296 −0.0834063
\(413\) 13.4972 0.664152
\(414\) 0.263093 0.0129303
\(415\) −42.9196 −2.10684
\(416\) 27.3822 1.34252
\(417\) 3.56848 0.174749
\(418\) −22.2190 −1.08677
\(419\) 3.93006 0.191996 0.0959980 0.995382i \(-0.469396\pi\)
0.0959980 + 0.995382i \(0.469396\pi\)
\(420\) −4.13701 −0.201866
\(421\) 14.5684 0.710018 0.355009 0.934863i \(-0.384478\pi\)
0.355009 + 0.934863i \(0.384478\pi\)
\(422\) −20.2766 −0.987049
\(423\) 1.88454 0.0916296
\(424\) −32.2588 −1.56663
\(425\) −47.5032 −2.30424
\(426\) −15.4523 −0.748664
\(427\) −0.0371054 −0.00179566
\(428\) 10.1286 0.489587
\(429\) 33.1728 1.60160
\(430\) 11.2503 0.542537
\(431\) −28.2912 −1.36274 −0.681369 0.731940i \(-0.738615\pi\)
−0.681369 + 0.731940i \(0.738615\pi\)
\(432\) 0.559391 0.0269137
\(433\) −32.7988 −1.57621 −0.788106 0.615540i \(-0.788938\pi\)
−0.788106 + 0.615540i \(0.788938\pi\)
\(434\) −8.97902 −0.431007
\(435\) −28.3698 −1.36023
\(436\) 21.6179 1.03531
\(437\) −1.01158 −0.0483904
\(438\) −2.90356 −0.138737
\(439\) −9.95764 −0.475252 −0.237626 0.971357i \(-0.576369\pi\)
−0.237626 + 0.971357i \(0.576369\pi\)
\(440\) 71.0047 3.38502
\(441\) 1.00000 0.0476190
\(442\) −25.6728 −1.22113
\(443\) −19.8469 −0.942953 −0.471477 0.881879i \(-0.656279\pi\)
−0.471477 + 0.881879i \(0.656279\pi\)
\(444\) −4.74257 −0.225073
\(445\) 5.46556 0.259092
\(446\) 18.0452 0.854464
\(447\) −17.3492 −0.820590
\(448\) −6.16728 −0.291377
\(449\) −9.56848 −0.451564 −0.225782 0.974178i \(-0.572494\pi\)
−0.225782 + 0.974178i \(0.572494\pi\)
\(450\) −8.46508 −0.399048
\(451\) 26.8904 1.26622
\(452\) −7.55940 −0.355564
\(453\) −5.91767 −0.278037
\(454\) 11.9893 0.562687
\(455\) 19.1468 0.897614
\(456\) −10.6657 −0.499465
\(457\) −13.9465 −0.652392 −0.326196 0.945302i \(-0.605767\pi\)
−0.326196 + 0.945302i \(0.605767\pi\)
\(458\) −12.4057 −0.579680
\(459\) 5.30213 0.247482
\(460\) 1.15196 0.0537104
\(461\) −5.98078 −0.278553 −0.139276 0.990254i \(-0.544478\pi\)
−0.139276 + 0.990254i \(0.544478\pi\)
\(462\) −6.11612 −0.284548
\(463\) −18.8469 −0.875890 −0.437945 0.899002i \(-0.644294\pi\)
−0.437945 + 0.899002i \(0.644294\pi\)
\(464\) −4.24757 −0.197189
\(465\) 35.5059 1.64655
\(466\) −21.5976 −1.00049
\(467\) 22.6056 1.04606 0.523032 0.852313i \(-0.324801\pi\)
0.523032 + 0.852313i \(0.324801\pi\)
\(468\) 5.67440 0.262299
\(469\) −14.1396 −0.652904
\(470\) −6.65267 −0.306865
\(471\) −6.46779 −0.298020
\(472\) −39.6260 −1.82394
\(473\) −20.6296 −0.948551
\(474\) −10.2143 −0.469159
\(475\) 32.5478 1.49339
\(476\) −5.87092 −0.269093
\(477\) −10.9878 −0.503096
\(478\) 5.85693 0.267890
\(479\) 17.7002 0.808742 0.404371 0.914595i \(-0.367491\pi\)
0.404371 + 0.914595i \(0.367491\pi\)
\(480\) 19.9634 0.911201
\(481\) 21.9494 1.00081
\(482\) −3.69910 −0.168489
\(483\) −0.278452 −0.0126700
\(484\) −34.2170 −1.55532
\(485\) −60.6655 −2.75468
\(486\) 0.944841 0.0428589
\(487\) −19.8867 −0.901153 −0.450576 0.892738i \(-0.648781\pi\)
−0.450576 + 0.892738i \(0.648781\pi\)
\(488\) 0.108937 0.00493135
\(489\) −14.0864 −0.637007
\(490\) −3.53012 −0.159475
\(491\) 8.01984 0.361930 0.180965 0.983489i \(-0.442078\pi\)
0.180965 + 0.983489i \(0.442078\pi\)
\(492\) 4.59976 0.207373
\(493\) −40.2602 −1.81323
\(494\) 17.5903 0.791423
\(495\) 24.1851 1.08704
\(496\) 5.31601 0.238696
\(497\) 16.3543 0.733592
\(498\) −10.8538 −0.486372
\(499\) 4.88699 0.218772 0.109386 0.993999i \(-0.465112\pi\)
0.109386 + 0.993999i \(0.465112\pi\)
\(500\) −16.3795 −0.732514
\(501\) 15.9505 0.712614
\(502\) −14.9667 −0.667999
\(503\) 22.9160 1.02178 0.510888 0.859647i \(-0.329317\pi\)
0.510888 + 0.859647i \(0.329317\pi\)
\(504\) −2.93588 −0.130775
\(505\) −34.5351 −1.53679
\(506\) 1.70305 0.0757097
\(507\) −13.2620 −0.588987
\(508\) −11.7052 −0.519336
\(509\) −4.50555 −0.199705 −0.0998524 0.995002i \(-0.531837\pi\)
−0.0998524 + 0.995002i \(0.531837\pi\)
\(510\) −18.7172 −0.828811
\(511\) 3.07307 0.135944
\(512\) 6.27357 0.277255
\(513\) −3.63286 −0.160395
\(514\) 10.0176 0.441859
\(515\) 5.71246 0.251721
\(516\) −3.52882 −0.155348
\(517\) 12.1990 0.536511
\(518\) −4.04685 −0.177808
\(519\) −11.5900 −0.508744
\(520\) −56.2126 −2.46509
\(521\) 11.8206 0.517872 0.258936 0.965894i \(-0.416628\pi\)
0.258936 + 0.965894i \(0.416628\pi\)
\(522\) −7.17437 −0.314014
\(523\) −37.2939 −1.63075 −0.815374 0.578934i \(-0.803469\pi\)
−0.815374 + 0.578934i \(0.803469\pi\)
\(524\) 10.0112 0.437342
\(525\) 8.95926 0.391014
\(526\) −10.8111 −0.471384
\(527\) 50.3873 2.19490
\(528\) 3.62104 0.157586
\(529\) −22.9225 −0.996629
\(530\) 38.7883 1.68485
\(531\) −13.4972 −0.585727
\(532\) 4.02258 0.174401
\(533\) −21.2885 −0.922106
\(534\) 1.38217 0.0598124
\(535\) −34.1765 −1.47758
\(536\) 41.5121 1.79305
\(537\) 12.5110 0.539890
\(538\) 8.22410 0.354566
\(539\) 6.47318 0.278819
\(540\) 4.13701 0.178029
\(541\) −23.6543 −1.01698 −0.508488 0.861069i \(-0.669795\pi\)
−0.508488 + 0.861069i \(0.669795\pi\)
\(542\) −16.5941 −0.712778
\(543\) 23.7664 1.01991
\(544\) 28.3305 1.21466
\(545\) −72.9438 −3.12457
\(546\) 4.84198 0.207218
\(547\) −19.9032 −0.850998 −0.425499 0.904959i \(-0.639901\pi\)
−0.425499 + 0.904959i \(0.639901\pi\)
\(548\) −10.0798 −0.430588
\(549\) 0.0371054 0.00158362
\(550\) −54.7959 −2.33651
\(551\) 27.5851 1.17516
\(552\) 0.817503 0.0347952
\(553\) 10.8106 0.459714
\(554\) 11.7603 0.499649
\(555\) 16.0026 0.679271
\(556\) 3.95129 0.167572
\(557\) −2.21512 −0.0938576 −0.0469288 0.998898i \(-0.514943\pi\)
−0.0469288 + 0.998898i \(0.514943\pi\)
\(558\) 8.97902 0.380112
\(559\) 16.3320 0.690768
\(560\) 2.09000 0.0883188
\(561\) 34.3216 1.44906
\(562\) −16.4321 −0.693146
\(563\) 14.0710 0.593022 0.296511 0.955029i \(-0.404177\pi\)
0.296511 + 0.955029i \(0.404177\pi\)
\(564\) 2.08671 0.0878663
\(565\) 25.5072 1.07310
\(566\) −16.0749 −0.675679
\(567\) −1.00000 −0.0419961
\(568\) −48.0144 −2.01464
\(569\) 11.0651 0.463874 0.231937 0.972731i \(-0.425494\pi\)
0.231937 + 0.972731i \(0.425494\pi\)
\(570\) 12.8245 0.537158
\(571\) −15.5384 −0.650262 −0.325131 0.945669i \(-0.605408\pi\)
−0.325131 + 0.945669i \(0.605408\pi\)
\(572\) 36.7314 1.53582
\(573\) −15.7867 −0.659499
\(574\) 3.92499 0.163826
\(575\) −2.49473 −0.104037
\(576\) 6.16728 0.256970
\(577\) 33.1582 1.38039 0.690196 0.723622i \(-0.257524\pi\)
0.690196 + 0.723622i \(0.257524\pi\)
\(578\) −10.4997 −0.436729
\(579\) 9.25361 0.384567
\(580\) −31.4132 −1.30436
\(581\) 11.4875 0.476581
\(582\) −15.3416 −0.635928
\(583\) −71.1259 −2.94573
\(584\) −9.02216 −0.373340
\(585\) −19.1468 −0.791621
\(586\) 4.43965 0.183400
\(587\) −20.1697 −0.832491 −0.416245 0.909252i \(-0.636654\pi\)
−0.416245 + 0.909252i \(0.636654\pi\)
\(588\) 1.10728 0.0456633
\(589\) −34.5238 −1.42253
\(590\) 47.6466 1.96158
\(591\) 25.7315 1.05845
\(592\) 2.39593 0.0984721
\(593\) 5.68382 0.233407 0.116703 0.993167i \(-0.462767\pi\)
0.116703 + 0.993167i \(0.462767\pi\)
\(594\) 6.11612 0.250948
\(595\) 19.8099 0.812126
\(596\) −19.2104 −0.786887
\(597\) −2.53649 −0.103811
\(598\) −1.34826 −0.0551344
\(599\) 17.2044 0.702954 0.351477 0.936196i \(-0.385679\pi\)
0.351477 + 0.936196i \(0.385679\pi\)
\(600\) −26.3033 −1.07383
\(601\) −2.43307 −0.0992469 −0.0496234 0.998768i \(-0.515802\pi\)
−0.0496234 + 0.998768i \(0.515802\pi\)
\(602\) −3.01115 −0.122725
\(603\) 14.1396 0.575807
\(604\) −6.55249 −0.266617
\(605\) 115.456 4.69397
\(606\) −8.73350 −0.354774
\(607\) 20.3386 0.825518 0.412759 0.910840i \(-0.364565\pi\)
0.412759 + 0.910840i \(0.364565\pi\)
\(608\) −19.4112 −0.787229
\(609\) 7.59321 0.307692
\(610\) −0.130987 −0.00530349
\(611\) −9.65763 −0.390706
\(612\) 5.87092 0.237318
\(613\) 19.0284 0.768551 0.384275 0.923219i \(-0.374451\pi\)
0.384275 + 0.923219i \(0.374451\pi\)
\(614\) −1.83650 −0.0741149
\(615\) −15.5207 −0.625855
\(616\) −19.0045 −0.765712
\(617\) 8.61548 0.346846 0.173423 0.984847i \(-0.444517\pi\)
0.173423 + 0.984847i \(0.444517\pi\)
\(618\) 1.44461 0.0581108
\(619\) −2.27320 −0.0913675 −0.0456838 0.998956i \(-0.514547\pi\)
−0.0456838 + 0.998956i \(0.514547\pi\)
\(620\) 39.3149 1.57892
\(621\) 0.278452 0.0111739
\(622\) −20.3353 −0.815370
\(623\) −1.46286 −0.0586083
\(624\) −2.86669 −0.114759
\(625\) 10.4720 0.418882
\(626\) 22.6693 0.906046
\(627\) −23.5162 −0.939145
\(628\) −7.16163 −0.285780
\(629\) 22.7096 0.905490
\(630\) 3.53012 0.140644
\(631\) −3.46104 −0.137782 −0.0688909 0.997624i \(-0.521946\pi\)
−0.0688909 + 0.997624i \(0.521946\pi\)
\(632\) −31.7387 −1.26250
\(633\) −21.4603 −0.852971
\(634\) 3.59521 0.142784
\(635\) 39.4962 1.56736
\(636\) −12.1665 −0.482433
\(637\) −5.12465 −0.203046
\(638\) −46.4410 −1.83862
\(639\) −16.3543 −0.646968
\(640\) 18.1555 0.717661
\(641\) −22.6478 −0.894534 −0.447267 0.894401i \(-0.647603\pi\)
−0.447267 + 0.894401i \(0.647603\pi\)
\(642\) −8.64280 −0.341104
\(643\) 4.47086 0.176313 0.0881567 0.996107i \(-0.471902\pi\)
0.0881567 + 0.996107i \(0.471902\pi\)
\(644\) −0.308323 −0.0121496
\(645\) 11.9071 0.468841
\(646\) 18.1995 0.716049
\(647\) −18.6291 −0.732385 −0.366192 0.930539i \(-0.619339\pi\)
−0.366192 + 0.930539i \(0.619339\pi\)
\(648\) 2.93588 0.115332
\(649\) −87.3694 −3.42955
\(650\) 43.3806 1.70153
\(651\) −9.50320 −0.372460
\(652\) −15.5975 −0.610844
\(653\) −39.6339 −1.55099 −0.775497 0.631351i \(-0.782501\pi\)
−0.775497 + 0.631351i \(0.782501\pi\)
\(654\) −18.4466 −0.721319
\(655\) −33.7803 −1.31990
\(656\) −2.32378 −0.0907286
\(657\) −3.07307 −0.119892
\(658\) 1.78059 0.0694148
\(659\) −13.3293 −0.519236 −0.259618 0.965711i \(-0.583597\pi\)
−0.259618 + 0.965711i \(0.583597\pi\)
\(660\) 26.7796 1.04239
\(661\) −12.8878 −0.501277 −0.250639 0.968081i \(-0.580641\pi\)
−0.250639 + 0.968081i \(0.580641\pi\)
\(662\) −23.0119 −0.894383
\(663\) −27.1716 −1.05526
\(664\) −33.7259 −1.30882
\(665\) −13.5731 −0.526344
\(666\) 4.04685 0.156812
\(667\) −2.11434 −0.0818678
\(668\) 17.6616 0.683346
\(669\) 19.0986 0.738396
\(670\) −49.9144 −1.92836
\(671\) 0.240190 0.00927242
\(672\) −5.34323 −0.206119
\(673\) 33.5509 1.29329 0.646646 0.762791i \(-0.276171\pi\)
0.646646 + 0.762791i \(0.276171\pi\)
\(674\) 0.240309 0.00925635
\(675\) −8.95926 −0.344842
\(676\) −14.6847 −0.564797
\(677\) −6.79747 −0.261248 −0.130624 0.991432i \(-0.541698\pi\)
−0.130624 + 0.991432i \(0.541698\pi\)
\(678\) 6.45046 0.247728
\(679\) 16.2372 0.623126
\(680\) −58.1595 −2.23031
\(681\) 12.6892 0.486253
\(682\) 58.1227 2.22564
\(683\) 27.5708 1.05497 0.527484 0.849565i \(-0.323136\pi\)
0.527484 + 0.849565i \(0.323136\pi\)
\(684\) −4.02258 −0.153807
\(685\) 34.0116 1.29952
\(686\) 0.944841 0.0360742
\(687\) −13.1299 −0.500938
\(688\) 1.78275 0.0679666
\(689\) 56.3085 2.14518
\(690\) −0.982971 −0.0374211
\(691\) 4.79103 0.182259 0.0911296 0.995839i \(-0.470952\pi\)
0.0911296 + 0.995839i \(0.470952\pi\)
\(692\) −12.8333 −0.487849
\(693\) −6.47318 −0.245896
\(694\) −7.30983 −0.277477
\(695\) −13.3326 −0.505734
\(696\) −22.2928 −0.845005
\(697\) −22.0258 −0.834285
\(698\) −26.4413 −1.00082
\(699\) −22.8584 −0.864586
\(700\) 9.92037 0.374955
\(701\) 30.8363 1.16467 0.582335 0.812949i \(-0.302139\pi\)
0.582335 + 0.812949i \(0.302139\pi\)
\(702\) −4.84198 −0.182749
\(703\) −15.5599 −0.586854
\(704\) 39.9219 1.50461
\(705\) −7.04105 −0.265181
\(706\) 12.0327 0.452855
\(707\) 9.24335 0.347632
\(708\) −14.9451 −0.561670
\(709\) −13.3569 −0.501630 −0.250815 0.968035i \(-0.580699\pi\)
−0.250815 + 0.968035i \(0.580699\pi\)
\(710\) 57.7328 2.16667
\(711\) −10.8106 −0.405430
\(712\) 4.29479 0.160954
\(713\) 2.64619 0.0991005
\(714\) 5.00968 0.187482
\(715\) −123.940 −4.63511
\(716\) 13.8531 0.517716
\(717\) 6.19886 0.231501
\(718\) 23.0059 0.858571
\(719\) −24.8389 −0.926336 −0.463168 0.886271i \(-0.653287\pi\)
−0.463168 + 0.886271i \(0.653287\pi\)
\(720\) −2.09000 −0.0778898
\(721\) −1.52895 −0.0569409
\(722\) 5.48225 0.204028
\(723\) −3.91505 −0.145602
\(724\) 26.3159 0.978023
\(725\) 68.0295 2.52655
\(726\) 29.1975 1.08362
\(727\) −24.4015 −0.905000 −0.452500 0.891764i \(-0.649468\pi\)
−0.452500 + 0.891764i \(0.649468\pi\)
\(728\) 15.0454 0.557618
\(729\) 1.00000 0.0370370
\(730\) 10.8483 0.401514
\(731\) 16.8976 0.624980
\(732\) 0.0410859 0.00151858
\(733\) −5.48668 −0.202655 −0.101328 0.994853i \(-0.532309\pi\)
−0.101328 + 0.994853i \(0.532309\pi\)
\(734\) −22.9823 −0.848291
\(735\) −3.73621 −0.137812
\(736\) 1.48783 0.0548423
\(737\) 91.5279 3.37147
\(738\) −3.92499 −0.144481
\(739\) −8.18683 −0.301157 −0.150579 0.988598i \(-0.548114\pi\)
−0.150579 + 0.988598i \(0.548114\pi\)
\(740\) 17.7192 0.651372
\(741\) 18.6172 0.683918
\(742\) −10.3817 −0.381125
\(743\) −13.4114 −0.492017 −0.246009 0.969268i \(-0.579119\pi\)
−0.246009 + 0.969268i \(0.579119\pi\)
\(744\) 27.9003 1.02287
\(745\) 64.8203 2.37483
\(746\) 24.0681 0.881197
\(747\) −11.4875 −0.420305
\(748\) 38.0035 1.38955
\(749\) 9.14736 0.334237
\(750\) 13.9767 0.510356
\(751\) 32.1013 1.17139 0.585697 0.810530i \(-0.300821\pi\)
0.585697 + 0.810530i \(0.300821\pi\)
\(752\) −1.05420 −0.0384426
\(753\) −15.8405 −0.577260
\(754\) 36.7661 1.33894
\(755\) 22.1097 0.804653
\(756\) −1.10728 −0.0402712
\(757\) 52.5913 1.91147 0.955733 0.294237i \(-0.0950655\pi\)
0.955733 + 0.294237i \(0.0950655\pi\)
\(758\) −16.4982 −0.599241
\(759\) 1.80247 0.0654255
\(760\) 39.8491 1.44548
\(761\) −34.8173 −1.26213 −0.631064 0.775731i \(-0.717381\pi\)
−0.631064 + 0.775731i \(0.717381\pi\)
\(762\) 9.98811 0.361831
\(763\) 19.5235 0.706797
\(764\) −17.4802 −0.632412
\(765\) −19.8099 −0.716228
\(766\) 0.944841 0.0341385
\(767\) 69.1682 2.49752
\(768\) 16.9259 0.610760
\(769\) −32.4849 −1.17143 −0.585717 0.810516i \(-0.699187\pi\)
−0.585717 + 0.810516i \(0.699187\pi\)
\(770\) 22.8511 0.823497
\(771\) 10.6024 0.381838
\(772\) 10.2463 0.368772
\(773\) 21.9140 0.788192 0.394096 0.919069i \(-0.371058\pi\)
0.394096 + 0.919069i \(0.371058\pi\)
\(774\) 3.01115 0.108234
\(775\) −85.1417 −3.05838
\(776\) −47.6704 −1.71127
\(777\) −4.28310 −0.153655
\(778\) −10.9927 −0.394106
\(779\) 15.0914 0.540705
\(780\) −21.2007 −0.759108
\(781\) −105.864 −3.78813
\(782\) −1.39495 −0.0498835
\(783\) −7.59321 −0.271359
\(784\) −0.559391 −0.0199783
\(785\) 24.1650 0.862487
\(786\) −8.54261 −0.304705
\(787\) 21.2378 0.757047 0.378523 0.925592i \(-0.376432\pi\)
0.378523 + 0.925592i \(0.376432\pi\)
\(788\) 28.4919 1.01498
\(789\) −11.4422 −0.407353
\(790\) 38.1628 1.35777
\(791\) −6.82703 −0.242741
\(792\) 19.0045 0.675295
\(793\) −0.190152 −0.00675250
\(794\) −0.0966740 −0.00343083
\(795\) 41.0527 1.45599
\(796\) −2.80859 −0.0995477
\(797\) −52.7511 −1.86854 −0.934271 0.356564i \(-0.883948\pi\)
−0.934271 + 0.356564i \(0.883948\pi\)
\(798\) −3.43248 −0.121508
\(799\) −9.99210 −0.353495
\(800\) −47.8714 −1.69251
\(801\) 1.46286 0.0516877
\(802\) 24.8245 0.876582
\(803\) −19.8925 −0.701991
\(804\) 15.6564 0.552158
\(805\) 1.04036 0.0366677
\(806\) −46.0143 −1.62078
\(807\) 8.70422 0.306403
\(808\) −27.1374 −0.954690
\(809\) −51.0058 −1.79327 −0.896634 0.442772i \(-0.853995\pi\)
−0.896634 + 0.442772i \(0.853995\pi\)
\(810\) −3.53012 −0.124036
\(811\) 36.2768 1.27385 0.636925 0.770926i \(-0.280206\pi\)
0.636925 + 0.770926i \(0.280206\pi\)
\(812\) 8.40777 0.295055
\(813\) −17.5628 −0.615956
\(814\) 26.1960 0.918169
\(815\) 52.6296 1.84353
\(816\) −2.96597 −0.103830
\(817\) −11.5777 −0.405053
\(818\) 14.8807 0.520293
\(819\) 5.12465 0.179070
\(820\) −17.1857 −0.600150
\(821\) 30.4544 1.06286 0.531432 0.847101i \(-0.321654\pi\)
0.531432 + 0.847101i \(0.321654\pi\)
\(822\) 8.60112 0.299999
\(823\) 20.0670 0.699493 0.349747 0.936844i \(-0.386268\pi\)
0.349747 + 0.936844i \(0.386268\pi\)
\(824\) 4.48880 0.156375
\(825\) −57.9949 −2.01912
\(826\) −12.7527 −0.443722
\(827\) 32.0817 1.11559 0.557795 0.829979i \(-0.311648\pi\)
0.557795 + 0.829979i \(0.311648\pi\)
\(828\) 0.308323 0.0107150
\(829\) −16.1028 −0.559274 −0.279637 0.960106i \(-0.590214\pi\)
−0.279637 + 0.960106i \(0.590214\pi\)
\(830\) 40.5522 1.40759
\(831\) 12.4469 0.431778
\(832\) −31.6052 −1.09571
\(833\) −5.30213 −0.183708
\(834\) −3.37165 −0.116751
\(835\) −59.5943 −2.06235
\(836\) −26.0389 −0.900573
\(837\) 9.50320 0.328479
\(838\) −3.71328 −0.128273
\(839\) 15.6199 0.539258 0.269629 0.962964i \(-0.413099\pi\)
0.269629 + 0.962964i \(0.413099\pi\)
\(840\) 10.9691 0.378469
\(841\) 28.6568 0.988165
\(842\) −13.7648 −0.474366
\(843\) −17.3914 −0.598991
\(844\) −23.7625 −0.817938
\(845\) 49.5497 1.70456
\(846\) −1.78059 −0.0612181
\(847\) −30.9020 −1.06181
\(848\) 6.14647 0.211071
\(849\) −17.0134 −0.583897
\(850\) 44.8830 1.53947
\(851\) 1.19264 0.0408831
\(852\) −18.1087 −0.620396
\(853\) −52.7653 −1.80665 −0.903325 0.428957i \(-0.858881\pi\)
−0.903325 + 0.428957i \(0.858881\pi\)
\(854\) 0.0350587 0.00119968
\(855\) 13.5731 0.464192
\(856\) −26.8556 −0.917905
\(857\) −22.6476 −0.773627 −0.386814 0.922158i \(-0.626424\pi\)
−0.386814 + 0.922158i \(0.626424\pi\)
\(858\) −31.3430 −1.07003
\(859\) −25.0295 −0.853995 −0.426998 0.904253i \(-0.640429\pi\)
−0.426998 + 0.904253i \(0.640429\pi\)
\(860\) 13.1844 0.449585
\(861\) 4.15413 0.141572
\(862\) 26.7307 0.910450
\(863\) 6.12003 0.208328 0.104164 0.994560i \(-0.466783\pi\)
0.104164 + 0.994560i \(0.466783\pi\)
\(864\) 5.34323 0.181780
\(865\) 43.3026 1.47233
\(866\) 30.9897 1.05307
\(867\) −11.1126 −0.377405
\(868\) −10.5227 −0.357162
\(869\) −69.9790 −2.37387
\(870\) 26.8050 0.908773
\(871\) −72.4603 −2.45522
\(872\) −57.3186 −1.94105
\(873\) −16.2372 −0.549545
\(874\) 0.955782 0.0323298
\(875\) −14.7926 −0.500082
\(876\) −3.40273 −0.114968
\(877\) 33.8899 1.14438 0.572190 0.820121i \(-0.306094\pi\)
0.572190 + 0.820121i \(0.306094\pi\)
\(878\) 9.40839 0.317518
\(879\) 4.69883 0.158488
\(880\) −13.5290 −0.456061
\(881\) −7.63399 −0.257196 −0.128598 0.991697i \(-0.541048\pi\)
−0.128598 + 0.991697i \(0.541048\pi\)
\(882\) −0.944841 −0.0318145
\(883\) −3.95306 −0.133031 −0.0665155 0.997785i \(-0.521188\pi\)
−0.0665155 + 0.997785i \(0.521188\pi\)
\(884\) −30.0864 −1.01192
\(885\) 50.4282 1.69513
\(886\) 18.7521 0.629991
\(887\) −47.3071 −1.58842 −0.794209 0.607645i \(-0.792114\pi\)
−0.794209 + 0.607645i \(0.792114\pi\)
\(888\) 12.5747 0.421979
\(889\) −10.5712 −0.354547
\(890\) −5.16408 −0.173101
\(891\) 6.47318 0.216860
\(892\) 21.1474 0.708069
\(893\) 6.84629 0.229102
\(894\) 16.3923 0.548239
\(895\) −46.7437 −1.56247
\(896\) −4.85935 −0.162339
\(897\) −1.42697 −0.0476451
\(898\) 9.04069 0.301692
\(899\) −72.1598 −2.40666
\(900\) −9.92037 −0.330679
\(901\) 58.2587 1.94088
\(902\) −25.4072 −0.845966
\(903\) −3.18694 −0.106055
\(904\) 20.0434 0.666632
\(905\) −88.7961 −2.95168
\(906\) 5.59126 0.185757
\(907\) 21.2732 0.706365 0.353182 0.935555i \(-0.385100\pi\)
0.353182 + 0.935555i \(0.385100\pi\)
\(908\) 14.0505 0.466282
\(909\) −9.24335 −0.306583
\(910\) −18.0906 −0.599699
\(911\) −11.4264 −0.378574 −0.189287 0.981922i \(-0.560618\pi\)
−0.189287 + 0.981922i \(0.560618\pi\)
\(912\) 2.03219 0.0672926
\(913\) −74.3604 −2.46097
\(914\) 13.1773 0.435865
\(915\) −0.138633 −0.00458308
\(916\) −14.5384 −0.480363
\(917\) 9.04132 0.298571
\(918\) −5.00968 −0.165344
\(919\) −26.8004 −0.884062 −0.442031 0.897000i \(-0.645742\pi\)
−0.442031 + 0.897000i \(0.645742\pi\)
\(920\) −3.05436 −0.100699
\(921\) −1.94371 −0.0640474
\(922\) 5.65089 0.186102
\(923\) 83.8102 2.75865
\(924\) −7.16759 −0.235796
\(925\) −38.3734 −1.26171
\(926\) 17.8073 0.585185
\(927\) 1.52895 0.0502172
\(928\) −40.5722 −1.33185
\(929\) −36.9136 −1.21110 −0.605548 0.795808i \(-0.707046\pi\)
−0.605548 + 0.795808i \(0.707046\pi\)
\(930\) −33.5475 −1.10007
\(931\) 3.63286 0.119062
\(932\) −25.3106 −0.829076
\(933\) −21.5224 −0.704612
\(934\) −21.3587 −0.698879
\(935\) −128.233 −4.19366
\(936\) −15.0454 −0.491773
\(937\) 35.8406 1.17086 0.585431 0.810722i \(-0.300925\pi\)
0.585431 + 0.810722i \(0.300925\pi\)
\(938\) 13.3596 0.436208
\(939\) 23.9927 0.782971
\(940\) −7.79638 −0.254290
\(941\) 9.52369 0.310464 0.155232 0.987878i \(-0.450388\pi\)
0.155232 + 0.987878i \(0.450388\pi\)
\(942\) 6.11104 0.199108
\(943\) −1.15673 −0.0376682
\(944\) 7.55019 0.245738
\(945\) 3.73621 0.121539
\(946\) 19.4917 0.633730
\(947\) −23.2013 −0.753941 −0.376971 0.926225i \(-0.623034\pi\)
−0.376971 + 0.926225i \(0.623034\pi\)
\(948\) −11.9703 −0.388778
\(949\) 15.7484 0.511214
\(950\) −30.7525 −0.997742
\(951\) 3.80509 0.123389
\(952\) 15.5664 0.504511
\(953\) −21.3002 −0.689982 −0.344991 0.938606i \(-0.612118\pi\)
−0.344991 + 0.938606i \(0.612118\pi\)
\(954\) 10.3817 0.336120
\(955\) 58.9824 1.90863
\(956\) 6.86384 0.221992
\(957\) −49.1522 −1.58886
\(958\) −16.7239 −0.540323
\(959\) −9.10325 −0.293959
\(960\) −23.0423 −0.743686
\(961\) 59.3108 1.91325
\(962\) −20.7387 −0.668642
\(963\) −9.14736 −0.294770
\(964\) −4.33504 −0.139622
\(965\) −34.5734 −1.11296
\(966\) 0.263093 0.00846488
\(967\) −33.8778 −1.08944 −0.544718 0.838619i \(-0.683363\pi\)
−0.544718 + 0.838619i \(0.683363\pi\)
\(968\) 90.7246 2.91600
\(969\) 19.2619 0.618783
\(970\) 57.3193 1.84041
\(971\) 31.6379 1.01531 0.507654 0.861561i \(-0.330513\pi\)
0.507654 + 0.861561i \(0.330513\pi\)
\(972\) 1.10728 0.0355159
\(973\) 3.56848 0.114400
\(974\) 18.7898 0.602063
\(975\) 45.9131 1.47039
\(976\) −0.0207564 −0.000664397 0
\(977\) −1.30470 −0.0417412 −0.0208706 0.999782i \(-0.506644\pi\)
−0.0208706 + 0.999782i \(0.506644\pi\)
\(978\) 13.3094 0.425587
\(979\) 9.46936 0.302642
\(980\) −4.13701 −0.132152
\(981\) −19.5235 −0.623337
\(982\) −7.57748 −0.241807
\(983\) −2.15632 −0.0687760 −0.0343880 0.999409i \(-0.510948\pi\)
−0.0343880 + 0.999409i \(0.510948\pi\)
\(984\) −12.1960 −0.388795
\(985\) −96.1384 −3.06323
\(986\) 38.0395 1.21142
\(987\) 1.88454 0.0599857
\(988\) 20.6143 0.655829
\(989\) 0.887410 0.0282180
\(990\) −22.8511 −0.726256
\(991\) −24.3042 −0.772049 −0.386025 0.922488i \(-0.626152\pi\)
−0.386025 + 0.922488i \(0.626152\pi\)
\(992\) 50.7778 1.61220
\(993\) −24.3553 −0.772893
\(994\) −15.4523 −0.490116
\(995\) 9.47684 0.300436
\(996\) −12.7198 −0.403042
\(997\) −42.7177 −1.35288 −0.676441 0.736497i \(-0.736479\pi\)
−0.676441 + 0.736497i \(0.736479\pi\)
\(998\) −4.61743 −0.146162
\(999\) 4.28310 0.135511
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 8043.2.a.r.1.15 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8043.2.a.r.1.15 46 1.1 even 1 trivial