Properties

Label 6762.2.a.co.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $4$
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] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, 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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.16448.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.18398\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.18398 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.18398 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.18398 q^{10} -1.67440 q^{11} +1.00000 q^{12} +2.27259 q^{13} -3.18398 q^{15} +1.00000 q^{16} +0.367959 q^{17} +1.00000 q^{18} -3.41421 q^{19} -3.18398 q^{20} -1.67440 q^{22} -1.00000 q^{23} +1.00000 q^{24} +5.13772 q^{25} +2.27259 q^{26} +1.00000 q^{27} +3.44417 q^{29} -3.18398 q^{30} -3.41421 q^{31} +1.00000 q^{32} -1.67440 q^{33} +0.367959 q^{34} +1.00000 q^{36} -0.0953661 q^{37} -3.41421 q^{38} +2.27259 q^{39} -3.18398 q^{40} -0.681153 q^{41} -7.94699 q^{43} -1.67440 q^{44} -3.18398 q^{45} -1.00000 q^{46} -1.64445 q^{47} +1.00000 q^{48} +5.13772 q^{50} +0.367959 q^{51} +2.27259 q^{52} +9.69921 q^{53} +1.00000 q^{54} +5.33125 q^{55} -3.41421 q^{57} +3.44417 q^{58} -3.15403 q^{59} -3.18398 q^{60} -13.9594 q^{61} -3.41421 q^{62} +1.00000 q^{64} -7.23589 q^{65} -1.67440 q^{66} +5.19639 q^{67} +0.367959 q^{68} -1.00000 q^{69} +1.67440 q^{71} +1.00000 q^{72} -11.1310 q^{73} -0.0953661 q^{74} +5.13772 q^{75} -3.41421 q^{76} +2.27259 q^{78} -4.32560 q^{79} -3.18398 q^{80} +1.00000 q^{81} -0.681153 q^{82} +4.74547 q^{83} -1.17157 q^{85} -7.94699 q^{86} +3.44417 q^{87} -1.67440 q^{88} -11.3912 q^{89} -3.18398 q^{90} -1.00000 q^{92} -3.41421 q^{93} -1.64445 q^{94} +10.8708 q^{95} +1.00000 q^{96} -8.29014 q^{97} -1.67440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 6 q^{5} + 4 q^{6} + 4 q^{8} + 4 q^{9} - 6 q^{10} + 4 q^{12} - 10 q^{13} - 6 q^{15} + 4 q^{16} - 12 q^{17} + 4 q^{18} - 8 q^{19} - 6 q^{20} - 4 q^{23} + 4 q^{24} + 6 q^{25} - 10 q^{26} + 4 q^{27} + 6 q^{29} - 6 q^{30} - 8 q^{31} + 4 q^{32} - 12 q^{34} + 4 q^{36} - 6 q^{37} - 8 q^{38} - 10 q^{39} - 6 q^{40} - 14 q^{41} - 6 q^{43} - 6 q^{45} - 4 q^{46} - 2 q^{47} + 4 q^{48} + 6 q^{50} - 12 q^{51} - 10 q^{52} - 4 q^{53} + 4 q^{54} - 8 q^{55} - 8 q^{57} + 6 q^{58} - 8 q^{59} - 6 q^{60} - 12 q^{61} - 8 q^{62} + 4 q^{64} + 6 q^{65} - 4 q^{67} - 12 q^{68} - 4 q^{69} + 4 q^{72} - 12 q^{73} - 6 q^{74} + 6 q^{75} - 8 q^{76} - 10 q^{78} - 24 q^{79} - 6 q^{80} + 4 q^{81} - 14 q^{82} - 16 q^{83} - 16 q^{85} - 6 q^{86} + 6 q^{87} - 12 q^{89} - 6 q^{90} - 4 q^{92} - 8 q^{93} - 2 q^{94} + 12 q^{95} + 4 q^{96} - 30 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.18398 −1.42392 −0.711959 0.702221i \(-0.752192\pi\)
−0.711959 + 0.702221i \(0.752192\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.18398 −1.00686
\(11\) −1.67440 −0.504850 −0.252425 0.967616i \(-0.581228\pi\)
−0.252425 + 0.967616i \(0.581228\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.27259 0.630304 0.315152 0.949041i \(-0.397945\pi\)
0.315152 + 0.949041i \(0.397945\pi\)
\(14\) 0 0
\(15\) −3.18398 −0.822100
\(16\) 1.00000 0.250000
\(17\) 0.367959 0.0892431 0.0446215 0.999004i \(-0.485792\pi\)
0.0446215 + 0.999004i \(0.485792\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.41421 −0.783274 −0.391637 0.920120i \(-0.628091\pi\)
−0.391637 + 0.920120i \(0.628091\pi\)
\(20\) −3.18398 −0.711959
\(21\) 0 0
\(22\) −1.67440 −0.356983
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 5.13772 1.02754
\(26\) 2.27259 0.445692
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.44417 0.639565 0.319783 0.947491i \(-0.396390\pi\)
0.319783 + 0.947491i \(0.396390\pi\)
\(30\) −3.18398 −0.581312
\(31\) −3.41421 −0.613211 −0.306605 0.951837i \(-0.599193\pi\)
−0.306605 + 0.951837i \(0.599193\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.67440 −0.291476
\(34\) 0.367959 0.0631044
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −0.0953661 −0.0156781 −0.00783905 0.999969i \(-0.502495\pi\)
−0.00783905 + 0.999969i \(0.502495\pi\)
\(38\) −3.41421 −0.553859
\(39\) 2.27259 0.363906
\(40\) −3.18398 −0.503431
\(41\) −0.681153 −0.106378 −0.0531891 0.998584i \(-0.516939\pi\)
−0.0531891 + 0.998584i \(0.516939\pi\)
\(42\) 0 0
\(43\) −7.94699 −1.21190 −0.605952 0.795501i \(-0.707208\pi\)
−0.605952 + 0.795501i \(0.707208\pi\)
\(44\) −1.67440 −0.252425
\(45\) −3.18398 −0.474640
\(46\) −1.00000 −0.147442
\(47\) −1.64445 −0.239867 −0.119934 0.992782i \(-0.538268\pi\)
−0.119934 + 0.992782i \(0.538268\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 5.13772 0.726584
\(51\) 0.367959 0.0515245
\(52\) 2.27259 0.315152
\(53\) 9.69921 1.33229 0.666145 0.745823i \(-0.267943\pi\)
0.666145 + 0.745823i \(0.267943\pi\)
\(54\) 1.00000 0.136083
\(55\) 5.33125 0.718866
\(56\) 0 0
\(57\) −3.41421 −0.452224
\(58\) 3.44417 0.452241
\(59\) −3.15403 −0.410619 −0.205310 0.978697i \(-0.565820\pi\)
−0.205310 + 0.978697i \(0.565820\pi\)
\(60\) −3.18398 −0.411050
\(61\) −13.9594 −1.78732 −0.893659 0.448747i \(-0.851870\pi\)
−0.893659 + 0.448747i \(0.851870\pi\)
\(62\) −3.41421 −0.433606
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −7.23589 −0.897501
\(66\) −1.67440 −0.206104
\(67\) 5.19639 0.634840 0.317420 0.948285i \(-0.397184\pi\)
0.317420 + 0.948285i \(0.397184\pi\)
\(68\) 0.367959 0.0446215
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 1.67440 0.198715 0.0993573 0.995052i \(-0.468321\pi\)
0.0993573 + 0.995052i \(0.468321\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.1310 −1.30278 −0.651391 0.758742i \(-0.725814\pi\)
−0.651391 + 0.758742i \(0.725814\pi\)
\(74\) −0.0953661 −0.0110861
\(75\) 5.13772 0.593253
\(76\) −3.41421 −0.391637
\(77\) 0 0
\(78\) 2.27259 0.257320
\(79\) −4.32560 −0.486668 −0.243334 0.969943i \(-0.578241\pi\)
−0.243334 + 0.969943i \(0.578241\pi\)
\(80\) −3.18398 −0.355980
\(81\) 1.00000 0.111111
\(82\) −0.681153 −0.0752207
\(83\) 4.74547 0.520883 0.260441 0.965490i \(-0.416132\pi\)
0.260441 + 0.965490i \(0.416132\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) −7.94699 −0.856946
\(87\) 3.44417 0.369253
\(88\) −1.67440 −0.178492
\(89\) −11.3912 −1.20746 −0.603730 0.797189i \(-0.706320\pi\)
−0.603730 + 0.797189i \(0.706320\pi\)
\(90\) −3.18398 −0.335621
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −3.41421 −0.354037
\(94\) −1.64445 −0.169612
\(95\) 10.8708 1.11532
\(96\) 1.00000 0.102062
\(97\) −8.29014 −0.841736 −0.420868 0.907122i \(-0.638274\pi\)
−0.420868 + 0.907122i \(0.638274\pi\)
\(98\) 0 0
\(99\) −1.67440 −0.168283
\(100\) 5.13772 0.513772
\(101\) −10.1597 −1.01093 −0.505463 0.862848i \(-0.668678\pi\)
−0.505463 + 0.862848i \(0.668678\pi\)
\(102\) 0.367959 0.0364333
\(103\) −8.27259 −0.815123 −0.407561 0.913178i \(-0.633621\pi\)
−0.407561 + 0.913178i \(0.633621\pi\)
\(104\) 2.27259 0.222846
\(105\) 0 0
\(106\) 9.69921 0.942071
\(107\) −0.283242 −0.0273820 −0.0136910 0.999906i \(-0.504358\pi\)
−0.0136910 + 0.999906i \(0.504358\pi\)
\(108\) 1.00000 0.0962250
\(109\) −15.1857 −1.45453 −0.727265 0.686357i \(-0.759209\pi\)
−0.727265 + 0.686357i \(0.759209\pi\)
\(110\) 5.33125 0.508315
\(111\) −0.0953661 −0.00905175
\(112\) 0 0
\(113\) 7.57903 0.712975 0.356488 0.934300i \(-0.383974\pi\)
0.356488 + 0.934300i \(0.383974\pi\)
\(114\) −3.41421 −0.319770
\(115\) 3.18398 0.296908
\(116\) 3.44417 0.319783
\(117\) 2.27259 0.210101
\(118\) −3.15403 −0.290352
\(119\) 0 0
\(120\) −3.18398 −0.290656
\(121\) −8.19639 −0.745126
\(122\) −13.9594 −1.26382
\(123\) −0.681153 −0.0614175
\(124\) −3.41421 −0.306605
\(125\) −0.438512 −0.0392217
\(126\) 0 0
\(127\) 5.62139 0.498818 0.249409 0.968398i \(-0.419764\pi\)
0.249409 + 0.968398i \(0.419764\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.94699 −0.699694
\(130\) −7.23589 −0.634629
\(131\) 6.23309 0.544588 0.272294 0.962214i \(-0.412218\pi\)
0.272294 + 0.962214i \(0.412218\pi\)
\(132\) −1.67440 −0.145738
\(133\) 0 0
\(134\) 5.19639 0.448899
\(135\) −3.18398 −0.274033
\(136\) 0.367959 0.0315522
\(137\) 7.51913 0.642403 0.321201 0.947011i \(-0.395913\pi\)
0.321201 + 0.947011i \(0.395913\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −4.46736 −0.378917 −0.189458 0.981889i \(-0.560673\pi\)
−0.189458 + 0.981889i \(0.560673\pi\)
\(140\) 0 0
\(141\) −1.64445 −0.138487
\(142\) 1.67440 0.140512
\(143\) −3.80523 −0.318209
\(144\) 1.00000 0.0833333
\(145\) −10.9662 −0.910689
\(146\) −11.1310 −0.921206
\(147\) 0 0
\(148\) −0.0953661 −0.00783905
\(149\) −16.8956 −1.38414 −0.692071 0.721830i \(-0.743301\pi\)
−0.692071 + 0.721830i \(0.743301\pi\)
\(150\) 5.13772 0.419493
\(151\) −5.81212 −0.472984 −0.236492 0.971633i \(-0.575998\pi\)
−0.236492 + 0.971633i \(0.575998\pi\)
\(152\) −3.41421 −0.276929
\(153\) 0.367959 0.0297477
\(154\) 0 0
\(155\) 10.8708 0.873163
\(156\) 2.27259 0.181953
\(157\) −13.0136 −1.03860 −0.519301 0.854591i \(-0.673808\pi\)
−0.519301 + 0.854591i \(0.673808\pi\)
\(158\) −4.32560 −0.344126
\(159\) 9.69921 0.769198
\(160\) −3.18398 −0.251716
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) −0.681153 −0.0531891
\(165\) 5.33125 0.415038
\(166\) 4.74547 0.368320
\(167\) 0.685564 0.0530505 0.0265253 0.999648i \(-0.491556\pi\)
0.0265253 + 0.999648i \(0.491556\pi\)
\(168\) 0 0
\(169\) −7.83532 −0.602717
\(170\) −1.17157 −0.0898555
\(171\) −3.41421 −0.261091
\(172\) −7.94699 −0.605952
\(173\) 2.26974 0.172565 0.0862824 0.996271i \(-0.472501\pi\)
0.0862824 + 0.996271i \(0.472501\pi\)
\(174\) 3.44417 0.261102
\(175\) 0 0
\(176\) −1.67440 −0.126213
\(177\) −3.15403 −0.237071
\(178\) −11.3912 −0.853803
\(179\) 5.32840 0.398263 0.199132 0.979973i \(-0.436188\pi\)
0.199132 + 0.979973i \(0.436188\pi\)
\(180\) −3.18398 −0.237320
\(181\) −1.41017 −0.104817 −0.0524087 0.998626i \(-0.516690\pi\)
−0.0524087 + 0.998626i \(0.516690\pi\)
\(182\) 0 0
\(183\) −13.9594 −1.03191
\(184\) −1.00000 −0.0737210
\(185\) 0.303644 0.0223243
\(186\) −3.41421 −0.250342
\(187\) −0.616110 −0.0450544
\(188\) −1.64445 −0.119934
\(189\) 0 0
\(190\) 10.8708 0.788650
\(191\) −10.3329 −0.747660 −0.373830 0.927497i \(-0.621956\pi\)
−0.373830 + 0.927497i \(0.621956\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.70986 0.123079 0.0615393 0.998105i \(-0.480399\pi\)
0.0615393 + 0.998105i \(0.480399\pi\)
\(194\) −8.29014 −0.595197
\(195\) −7.23589 −0.518173
\(196\) 0 0
\(197\) −6.81778 −0.485747 −0.242873 0.970058i \(-0.578090\pi\)
−0.242873 + 0.970058i \(0.578090\pi\)
\(198\) −1.67440 −0.118994
\(199\) −13.4690 −0.954791 −0.477395 0.878689i \(-0.658419\pi\)
−0.477395 + 0.878689i \(0.658419\pi\)
\(200\) 5.13772 0.363292
\(201\) 5.19639 0.366525
\(202\) −10.1597 −0.714833
\(203\) 0 0
\(204\) 0.367959 0.0257623
\(205\) 2.16878 0.151474
\(206\) −8.27259 −0.576379
\(207\) −1.00000 −0.0695048
\(208\) 2.27259 0.157576
\(209\) 5.71676 0.395436
\(210\) 0 0
\(211\) −6.44292 −0.443549 −0.221775 0.975098i \(-0.571185\pi\)
−0.221775 + 0.975098i \(0.571185\pi\)
\(212\) 9.69921 0.666145
\(213\) 1.67440 0.114728
\(214\) −0.283242 −0.0193620
\(215\) 25.3031 1.72565
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −15.1857 −1.02851
\(219\) −11.1310 −0.752161
\(220\) 5.33125 0.359433
\(221\) 0.836220 0.0562503
\(222\) −0.0953661 −0.00640056
\(223\) 3.85962 0.258459 0.129230 0.991615i \(-0.458750\pi\)
0.129230 + 0.991615i \(0.458750\pi\)
\(224\) 0 0
\(225\) 5.13772 0.342515
\(226\) 7.57903 0.504150
\(227\) −6.05476 −0.401869 −0.200934 0.979605i \(-0.564398\pi\)
−0.200934 + 0.979605i \(0.564398\pi\)
\(228\) −3.41421 −0.226112
\(229\) −14.0943 −0.931375 −0.465688 0.884949i \(-0.654193\pi\)
−0.465688 + 0.884949i \(0.654193\pi\)
\(230\) 3.18398 0.209945
\(231\) 0 0
\(232\) 3.44417 0.226121
\(233\) −1.69921 −0.111319 −0.0556596 0.998450i \(-0.517726\pi\)
−0.0556596 + 0.998450i \(0.517726\pi\)
\(234\) 2.27259 0.148564
\(235\) 5.23589 0.341552
\(236\) −3.15403 −0.205310
\(237\) −4.32560 −0.280978
\(238\) 0 0
\(239\) −0.0982225 −0.00635349 −0.00317674 0.999995i \(-0.501011\pi\)
−0.00317674 + 0.999995i \(0.501011\pi\)
\(240\) −3.18398 −0.205525
\(241\) 22.9775 1.48011 0.740054 0.672547i \(-0.234800\pi\)
0.740054 + 0.672547i \(0.234800\pi\)
\(242\) −8.19639 −0.526884
\(243\) 1.00000 0.0641500
\(244\) −13.9594 −0.893659
\(245\) 0 0
\(246\) −0.681153 −0.0434287
\(247\) −7.75912 −0.493701
\(248\) −3.41421 −0.216803
\(249\) 4.74547 0.300732
\(250\) −0.438512 −0.0277339
\(251\) 0.128174 0.00809030 0.00404515 0.999992i \(-0.498712\pi\)
0.00404515 + 0.999992i \(0.498712\pi\)
\(252\) 0 0
\(253\) 1.67440 0.105269
\(254\) 5.62139 0.352717
\(255\) −1.17157 −0.0733667
\(256\) 1.00000 0.0625000
\(257\) 10.4757 0.653458 0.326729 0.945118i \(-0.394054\pi\)
0.326729 + 0.945118i \(0.394054\pi\)
\(258\) −7.94699 −0.494758
\(259\) 0 0
\(260\) −7.23589 −0.448751
\(261\) 3.44417 0.213188
\(262\) 6.23309 0.385082
\(263\) 4.48814 0.276750 0.138375 0.990380i \(-0.455812\pi\)
0.138375 + 0.990380i \(0.455812\pi\)
\(264\) −1.67440 −0.103052
\(265\) −30.8821 −1.89707
\(266\) 0 0
\(267\) −11.3912 −0.697128
\(268\) 5.19639 0.317420
\(269\) 30.0594 1.83275 0.916376 0.400318i \(-0.131100\pi\)
0.916376 + 0.400318i \(0.131100\pi\)
\(270\) −3.18398 −0.193771
\(271\) 8.14442 0.494738 0.247369 0.968921i \(-0.420434\pi\)
0.247369 + 0.968921i \(0.420434\pi\)
\(272\) 0.367959 0.0223108
\(273\) 0 0
\(274\) 7.51913 0.454247
\(275\) −8.60260 −0.518757
\(276\) −1.00000 −0.0601929
\(277\) 23.3561 1.40333 0.701665 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(278\) −4.46736 −0.267935
\(279\) −3.41421 −0.204404
\(280\) 0 0
\(281\) −11.1434 −0.664758 −0.332379 0.943146i \(-0.607851\pi\)
−0.332379 + 0.943146i \(0.607851\pi\)
\(282\) −1.64445 −0.0979254
\(283\) 29.9707 1.78157 0.890787 0.454422i \(-0.150154\pi\)
0.890787 + 0.454422i \(0.150154\pi\)
\(284\) 1.67440 0.0993573
\(285\) 10.8708 0.643930
\(286\) −3.80523 −0.225008
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.8646 −0.992036
\(290\) −10.9662 −0.643955
\(291\) −8.29014 −0.485976
\(292\) −11.1310 −0.651391
\(293\) −2.47002 −0.144300 −0.0721500 0.997394i \(-0.522986\pi\)
−0.0721500 + 0.997394i \(0.522986\pi\)
\(294\) 0 0
\(295\) 10.0424 0.584689
\(296\) −0.0953661 −0.00554304
\(297\) −1.67440 −0.0971585
\(298\) −16.8956 −0.978736
\(299\) −2.27259 −0.131427
\(300\) 5.13772 0.296627
\(301\) 0 0
\(302\) −5.81212 −0.334450
\(303\) −10.1597 −0.583658
\(304\) −3.41421 −0.195819
\(305\) 44.4464 2.54500
\(306\) 0.367959 0.0210348
\(307\) −34.4787 −1.96780 −0.983901 0.178715i \(-0.942806\pi\)
−0.983901 + 0.178715i \(0.942806\pi\)
\(308\) 0 0
\(309\) −8.27259 −0.470611
\(310\) 10.8708 0.617419
\(311\) 7.85962 0.445678 0.222839 0.974855i \(-0.428468\pi\)
0.222839 + 0.974855i \(0.428468\pi\)
\(312\) 2.27259 0.128660
\(313\) −13.0982 −0.740352 −0.370176 0.928962i \(-0.620703\pi\)
−0.370176 + 0.928962i \(0.620703\pi\)
\(314\) −13.0136 −0.734403
\(315\) 0 0
\(316\) −4.32560 −0.243334
\(317\) −17.3956 −0.977032 −0.488516 0.872555i \(-0.662462\pi\)
−0.488516 + 0.872555i \(0.662462\pi\)
\(318\) 9.69921 0.543905
\(319\) −5.76691 −0.322885
\(320\) −3.18398 −0.177990
\(321\) −0.283242 −0.0158090
\(322\) 0 0
\(323\) −1.25629 −0.0699018
\(324\) 1.00000 0.0555556
\(325\) 11.6760 0.647665
\(326\) 8.48528 0.469956
\(327\) −15.1857 −0.839773
\(328\) −0.681153 −0.0376104
\(329\) 0 0
\(330\) 5.33125 0.293476
\(331\) −7.96491 −0.437791 −0.218896 0.975748i \(-0.570245\pi\)
−0.218896 + 0.975748i \(0.570245\pi\)
\(332\) 4.74547 0.260441
\(333\) −0.0953661 −0.00522603
\(334\) 0.685564 0.0375124
\(335\) −16.5452 −0.903960
\(336\) 0 0
\(337\) 2.71514 0.147903 0.0739517 0.997262i \(-0.476439\pi\)
0.0739517 + 0.997262i \(0.476439\pi\)
\(338\) −7.83532 −0.426185
\(339\) 7.57903 0.411637
\(340\) −1.17157 −0.0635375
\(341\) 5.71676 0.309580
\(342\) −3.41421 −0.184620
\(343\) 0 0
\(344\) −7.94699 −0.428473
\(345\) 3.18398 0.171420
\(346\) 2.26974 0.122022
\(347\) 0.459285 0.0246557 0.0123279 0.999924i \(-0.496076\pi\)
0.0123279 + 0.999924i \(0.496076\pi\)
\(348\) 3.44417 0.184627
\(349\) −3.49313 −0.186983 −0.0934916 0.995620i \(-0.529803\pi\)
−0.0934916 + 0.995620i \(0.529803\pi\)
\(350\) 0 0
\(351\) 2.27259 0.121302
\(352\) −1.67440 −0.0892458
\(353\) −33.8811 −1.80331 −0.901653 0.432460i \(-0.857646\pi\)
−0.901653 + 0.432460i \(0.857646\pi\)
\(354\) −3.15403 −0.167635
\(355\) −5.33125 −0.282954
\(356\) −11.3912 −0.603730
\(357\) 0 0
\(358\) 5.32840 0.281615
\(359\) −5.85044 −0.308775 −0.154387 0.988010i \(-0.549340\pi\)
−0.154387 + 0.988010i \(0.549340\pi\)
\(360\) −3.18398 −0.167810
\(361\) −7.34315 −0.386481
\(362\) −1.41017 −0.0741171
\(363\) −8.19639 −0.430199
\(364\) 0 0
\(365\) 35.4408 1.85506
\(366\) −13.9594 −0.729669
\(367\) 16.5072 0.861671 0.430835 0.902431i \(-0.358219\pi\)
0.430835 + 0.902431i \(0.358219\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −0.681153 −0.0354594
\(370\) 0.303644 0.0157857
\(371\) 0 0
\(372\) −3.41421 −0.177019
\(373\) 5.31619 0.275262 0.137631 0.990484i \(-0.456051\pi\)
0.137631 + 0.990484i \(0.456051\pi\)
\(374\) −0.616110 −0.0318583
\(375\) −0.438512 −0.0226447
\(376\) −1.64445 −0.0848059
\(377\) 7.82718 0.403121
\(378\) 0 0
\(379\) −13.0722 −0.671472 −0.335736 0.941956i \(-0.608985\pi\)
−0.335736 + 0.941956i \(0.608985\pi\)
\(380\) 10.8708 0.557660
\(381\) 5.62139 0.287993
\(382\) −10.3329 −0.528676
\(383\) 1.81654 0.0928206 0.0464103 0.998922i \(-0.485222\pi\)
0.0464103 + 0.998922i \(0.485222\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 1.70986 0.0870297
\(387\) −7.94699 −0.403968
\(388\) −8.29014 −0.420868
\(389\) 24.5101 1.24271 0.621356 0.783529i \(-0.286582\pi\)
0.621356 + 0.783529i \(0.286582\pi\)
\(390\) −7.23589 −0.366403
\(391\) −0.367959 −0.0186085
\(392\) 0 0
\(393\) 6.23309 0.314418
\(394\) −6.81778 −0.343475
\(395\) 13.7726 0.692976
\(396\) −1.67440 −0.0841417
\(397\) −29.3385 −1.47246 −0.736229 0.676732i \(-0.763395\pi\)
−0.736229 + 0.676732i \(0.763395\pi\)
\(398\) −13.4690 −0.675139
\(399\) 0 0
\(400\) 5.13772 0.256886
\(401\) −7.65114 −0.382080 −0.191040 0.981582i \(-0.561186\pi\)
−0.191040 + 0.981582i \(0.561186\pi\)
\(402\) 5.19639 0.259172
\(403\) −7.75912 −0.386509
\(404\) −10.1597 −0.505463
\(405\) −3.18398 −0.158213
\(406\) 0 0
\(407\) 0.159681 0.00791509
\(408\) 0.367959 0.0182167
\(409\) 14.4798 0.715979 0.357989 0.933726i \(-0.383462\pi\)
0.357989 + 0.933726i \(0.383462\pi\)
\(410\) 2.16878 0.107108
\(411\) 7.51913 0.370891
\(412\) −8.27259 −0.407561
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −15.1095 −0.741695
\(416\) 2.27259 0.111423
\(417\) −4.46736 −0.218768
\(418\) 5.71676 0.279616
\(419\) −12.4838 −0.609874 −0.304937 0.952373i \(-0.598635\pi\)
−0.304937 + 0.952373i \(0.598635\pi\)
\(420\) 0 0
\(421\) −25.0301 −1.21989 −0.609946 0.792443i \(-0.708809\pi\)
−0.609946 + 0.792443i \(0.708809\pi\)
\(422\) −6.44292 −0.313637
\(423\) −1.64445 −0.0799558
\(424\) 9.69921 0.471035
\(425\) 1.89047 0.0917013
\(426\) 1.67440 0.0811249
\(427\) 0 0
\(428\) −0.283242 −0.0136910
\(429\) −3.80523 −0.183718
\(430\) 25.3031 1.22022
\(431\) −35.8866 −1.72859 −0.864297 0.502981i \(-0.832237\pi\)
−0.864297 + 0.502981i \(0.832237\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.7538 −1.28571 −0.642854 0.765989i \(-0.722250\pi\)
−0.642854 + 0.765989i \(0.722250\pi\)
\(434\) 0 0
\(435\) −10.9662 −0.525787
\(436\) −15.1857 −0.727265
\(437\) 3.41421 0.163324
\(438\) −11.1310 −0.531858
\(439\) 20.5046 0.978630 0.489315 0.872107i \(-0.337247\pi\)
0.489315 + 0.872107i \(0.337247\pi\)
\(440\) 5.33125 0.254158
\(441\) 0 0
\(442\) 0.836220 0.0397749
\(443\) 16.0916 0.764536 0.382268 0.924052i \(-0.375143\pi\)
0.382268 + 0.924052i \(0.375143\pi\)
\(444\) −0.0953661 −0.00452588
\(445\) 36.2692 1.71933
\(446\) 3.85962 0.182758
\(447\) −16.8956 −0.799134
\(448\) 0 0
\(449\) 37.7122 1.77975 0.889874 0.456206i \(-0.150792\pi\)
0.889874 + 0.456206i \(0.150792\pi\)
\(450\) 5.13772 0.242195
\(451\) 1.14052 0.0537051
\(452\) 7.57903 0.356488
\(453\) −5.81212 −0.273077
\(454\) −6.05476 −0.284164
\(455\) 0 0
\(456\) −3.41421 −0.159885
\(457\) 1.65934 0.0776206 0.0388103 0.999247i \(-0.487643\pi\)
0.0388103 + 0.999247i \(0.487643\pi\)
\(458\) −14.0943 −0.658582
\(459\) 0.367959 0.0171748
\(460\) 3.18398 0.148454
\(461\) 15.0455 0.700740 0.350370 0.936611i \(-0.386056\pi\)
0.350370 + 0.936611i \(0.386056\pi\)
\(462\) 0 0
\(463\) 5.98935 0.278349 0.139174 0.990268i \(-0.455555\pi\)
0.139174 + 0.990268i \(0.455555\pi\)
\(464\) 3.44417 0.159891
\(465\) 10.8708 0.504121
\(466\) −1.69921 −0.0787145
\(467\) 4.07635 0.188631 0.0943155 0.995542i \(-0.469934\pi\)
0.0943155 + 0.995542i \(0.469934\pi\)
\(468\) 2.27259 0.105051
\(469\) 0 0
\(470\) 5.23589 0.241513
\(471\) −13.0136 −0.599637
\(472\) −3.15403 −0.145176
\(473\) 13.3064 0.611831
\(474\) −4.32560 −0.198681
\(475\) −17.5413 −0.804850
\(476\) 0 0
\(477\) 9.69921 0.444096
\(478\) −0.0982225 −0.00449259
\(479\) −29.1999 −1.33418 −0.667089 0.744978i \(-0.732460\pi\)
−0.667089 + 0.744978i \(0.732460\pi\)
\(480\) −3.18398 −0.145328
\(481\) −0.216728 −0.00988196
\(482\) 22.9775 1.04659
\(483\) 0 0
\(484\) −8.19639 −0.372563
\(485\) 26.3956 1.19856
\(486\) 1.00000 0.0453609
\(487\) 28.3413 1.28427 0.642134 0.766593i \(-0.278049\pi\)
0.642134 + 0.766593i \(0.278049\pi\)
\(488\) −13.9594 −0.631912
\(489\) 8.48528 0.383718
\(490\) 0 0
\(491\) 34.3680 1.55100 0.775502 0.631345i \(-0.217497\pi\)
0.775502 + 0.631345i \(0.217497\pi\)
\(492\) −0.681153 −0.0307087
\(493\) 1.26731 0.0570768
\(494\) −7.75912 −0.349099
\(495\) 5.33125 0.239622
\(496\) −3.41421 −0.153303
\(497\) 0 0
\(498\) 4.74547 0.212649
\(499\) 14.8071 0.662858 0.331429 0.943480i \(-0.392469\pi\)
0.331429 + 0.943480i \(0.392469\pi\)
\(500\) −0.438512 −0.0196108
\(501\) 0.685564 0.0306287
\(502\) 0.128174 0.00572070
\(503\) 14.4319 0.643487 0.321743 0.946827i \(-0.395731\pi\)
0.321743 + 0.946827i \(0.395731\pi\)
\(504\) 0 0
\(505\) 32.3482 1.43948
\(506\) 1.67440 0.0744361
\(507\) −7.83532 −0.347979
\(508\) 5.62139 0.249409
\(509\) −4.19229 −0.185820 −0.0929099 0.995675i \(-0.529617\pi\)
−0.0929099 + 0.995675i \(0.529617\pi\)
\(510\) −1.17157 −0.0518781
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −3.41421 −0.150741
\(514\) 10.4757 0.462065
\(515\) 26.3398 1.16067
\(516\) −7.94699 −0.349847
\(517\) 2.75346 0.121097
\(518\) 0 0
\(519\) 2.26974 0.0996304
\(520\) −7.23589 −0.317315
\(521\) −12.1116 −0.530619 −0.265310 0.964163i \(-0.585474\pi\)
−0.265310 + 0.964163i \(0.585474\pi\)
\(522\) 3.44417 0.150747
\(523\) 37.0187 1.61872 0.809358 0.587316i \(-0.199815\pi\)
0.809358 + 0.587316i \(0.199815\pi\)
\(524\) 6.23309 0.272294
\(525\) 0 0
\(526\) 4.48814 0.195692
\(527\) −1.25629 −0.0547248
\(528\) −1.67440 −0.0728689
\(529\) 1.00000 0.0434783
\(530\) −30.8821 −1.34143
\(531\) −3.15403 −0.136873
\(532\) 0 0
\(533\) −1.54798 −0.0670506
\(534\) −11.3912 −0.492944
\(535\) 0.901837 0.0389898
\(536\) 5.19639 0.224450
\(537\) 5.32840 0.229937
\(538\) 30.0594 1.29595
\(539\) 0 0
\(540\) −3.18398 −0.137017
\(541\) 38.0634 1.63647 0.818237 0.574881i \(-0.194952\pi\)
0.818237 + 0.574881i \(0.194952\pi\)
\(542\) 8.14442 0.349833
\(543\) −1.41017 −0.0605164
\(544\) 0.367959 0.0157761
\(545\) 48.3511 2.07113
\(546\) 0 0
\(547\) −12.9999 −0.555837 −0.277919 0.960605i \(-0.589645\pi\)
−0.277919 + 0.960605i \(0.589645\pi\)
\(548\) 7.51913 0.321201
\(549\) −13.9594 −0.595773
\(550\) −8.60260 −0.366816
\(551\) −11.7591 −0.500955
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 23.3561 0.992304
\(555\) 0.303644 0.0128890
\(556\) −4.46736 −0.189458
\(557\) −1.65685 −0.0702032 −0.0351016 0.999384i \(-0.511175\pi\)
−0.0351016 + 0.999384i \(0.511175\pi\)
\(558\) −3.41421 −0.144535
\(559\) −18.0603 −0.763868
\(560\) 0 0
\(561\) −0.616110 −0.0260122
\(562\) −11.1434 −0.470055
\(563\) −5.67901 −0.239342 −0.119671 0.992814i \(-0.538184\pi\)
−0.119671 + 0.992814i \(0.538184\pi\)
\(564\) −1.64445 −0.0692437
\(565\) −24.1315 −1.01522
\(566\) 29.9707 1.25976
\(567\) 0 0
\(568\) 1.67440 0.0702562
\(569\) 0.205421 0.00861171 0.00430585 0.999991i \(-0.498629\pi\)
0.00430585 + 0.999991i \(0.498629\pi\)
\(570\) 10.8708 0.455327
\(571\) 23.3137 0.975648 0.487824 0.872942i \(-0.337791\pi\)
0.487824 + 0.872942i \(0.337791\pi\)
\(572\) −3.80523 −0.159105
\(573\) −10.3329 −0.431662
\(574\) 0 0
\(575\) −5.13772 −0.214258
\(576\) 1.00000 0.0416667
\(577\) −25.8574 −1.07646 −0.538229 0.842798i \(-0.680907\pi\)
−0.538229 + 0.842798i \(0.680907\pi\)
\(578\) −16.8646 −0.701475
\(579\) 1.70986 0.0710594
\(580\) −10.9662 −0.455345
\(581\) 0 0
\(582\) −8.29014 −0.343637
\(583\) −16.2404 −0.672607
\(584\) −11.1310 −0.460603
\(585\) −7.23589 −0.299167
\(586\) −2.47002 −0.102036
\(587\) 14.4078 0.594675 0.297337 0.954772i \(-0.403901\pi\)
0.297337 + 0.954772i \(0.403901\pi\)
\(588\) 0 0
\(589\) 11.6569 0.480312
\(590\) 10.0424 0.413437
\(591\) −6.81778 −0.280446
\(592\) −0.0953661 −0.00391952
\(593\) 27.0987 1.11281 0.556406 0.830911i \(-0.312180\pi\)
0.556406 + 0.830911i \(0.312180\pi\)
\(594\) −1.67440 −0.0687014
\(595\) 0 0
\(596\) −16.8956 −0.692071
\(597\) −13.4690 −0.551249
\(598\) −2.27259 −0.0929332
\(599\) −0.895598 −0.0365932 −0.0182966 0.999833i \(-0.505824\pi\)
−0.0182966 + 0.999833i \(0.505824\pi\)
\(600\) 5.13772 0.209747
\(601\) 7.77813 0.317277 0.158638 0.987337i \(-0.449290\pi\)
0.158638 + 0.987337i \(0.449290\pi\)
\(602\) 0 0
\(603\) 5.19639 0.211613
\(604\) −5.81212 −0.236492
\(605\) 26.0971 1.06100
\(606\) −10.1597 −0.412709
\(607\) 9.07731 0.368436 0.184218 0.982885i \(-0.441025\pi\)
0.184218 + 0.982885i \(0.441025\pi\)
\(608\) −3.41421 −0.138465
\(609\) 0 0
\(610\) 44.4464 1.79958
\(611\) −3.73716 −0.151189
\(612\) 0.367959 0.0148738
\(613\) 29.6632 1.19809 0.599043 0.800717i \(-0.295548\pi\)
0.599043 + 0.800717i \(0.295548\pi\)
\(614\) −34.4787 −1.39145
\(615\) 2.16878 0.0874535
\(616\) 0 0
\(617\) 32.6274 1.31353 0.656765 0.754095i \(-0.271924\pi\)
0.656765 + 0.754095i \(0.271924\pi\)
\(618\) −8.27259 −0.332772
\(619\) 41.7967 1.67995 0.839976 0.542624i \(-0.182569\pi\)
0.839976 + 0.542624i \(0.182569\pi\)
\(620\) 10.8708 0.436581
\(621\) −1.00000 −0.0401286
\(622\) 7.85962 0.315142
\(623\) 0 0
\(624\) 2.27259 0.0909765
\(625\) −24.2924 −0.971696
\(626\) −13.0982 −0.523508
\(627\) 5.71676 0.228305
\(628\) −13.0136 −0.519301
\(629\) −0.0350908 −0.00139916
\(630\) 0 0
\(631\) −49.8551 −1.98470 −0.992351 0.123449i \(-0.960604\pi\)
−0.992351 + 0.123449i \(0.960604\pi\)
\(632\) −4.32560 −0.172063
\(633\) −6.44292 −0.256083
\(634\) −17.3956 −0.690866
\(635\) −17.8984 −0.710276
\(636\) 9.69921 0.384599
\(637\) 0 0
\(638\) −5.76691 −0.228314
\(639\) 1.67440 0.0662382
\(640\) −3.18398 −0.125858
\(641\) 19.0644 0.752998 0.376499 0.926417i \(-0.377128\pi\)
0.376499 + 0.926417i \(0.377128\pi\)
\(642\) −0.283242 −0.0111787
\(643\) 17.7862 0.701420 0.350710 0.936484i \(-0.385940\pi\)
0.350710 + 0.936484i \(0.385940\pi\)
\(644\) 0 0
\(645\) 25.3031 0.996307
\(646\) −1.25629 −0.0494281
\(647\) 5.02092 0.197393 0.0986963 0.995118i \(-0.468533\pi\)
0.0986963 + 0.995118i \(0.468533\pi\)
\(648\) 1.00000 0.0392837
\(649\) 5.28110 0.207301
\(650\) 11.6760 0.457969
\(651\) 0 0
\(652\) 8.48528 0.332309
\(653\) −34.1205 −1.33524 −0.667618 0.744504i \(-0.732686\pi\)
−0.667618 + 0.744504i \(0.732686\pi\)
\(654\) −15.1857 −0.593809
\(655\) −19.8460 −0.775449
\(656\) −0.681153 −0.0265945
\(657\) −11.1310 −0.434261
\(658\) 0 0
\(659\) 2.22738 0.0867663 0.0433832 0.999059i \(-0.486186\pi\)
0.0433832 + 0.999059i \(0.486186\pi\)
\(660\) 5.33125 0.207519
\(661\) −18.0730 −0.702957 −0.351478 0.936196i \(-0.614321\pi\)
−0.351478 + 0.936196i \(0.614321\pi\)
\(662\) −7.96491 −0.309565
\(663\) 0.836220 0.0324761
\(664\) 4.74547 0.184160
\(665\) 0 0
\(666\) −0.0953661 −0.00369536
\(667\) −3.44417 −0.133359
\(668\) 0.685564 0.0265253
\(669\) 3.85962 0.149222
\(670\) −16.5452 −0.639196
\(671\) 23.3736 0.902328
\(672\) 0 0
\(673\) −4.32523 −0.166725 −0.0833627 0.996519i \(-0.526566\pi\)
−0.0833627 + 0.996519i \(0.526566\pi\)
\(674\) 2.71514 0.104583
\(675\) 5.13772 0.197751
\(676\) −7.83532 −0.301359
\(677\) −18.9553 −0.728511 −0.364256 0.931299i \(-0.618677\pi\)
−0.364256 + 0.931299i \(0.618677\pi\)
\(678\) 7.57903 0.291071
\(679\) 0 0
\(680\) −1.17157 −0.0449278
\(681\) −6.05476 −0.232019
\(682\) 5.71676 0.218906
\(683\) −45.9659 −1.75884 −0.879419 0.476049i \(-0.842069\pi\)
−0.879419 + 0.476049i \(0.842069\pi\)
\(684\) −3.41421 −0.130546
\(685\) −23.9408 −0.914729
\(686\) 0 0
\(687\) −14.0943 −0.537730
\(688\) −7.94699 −0.302976
\(689\) 22.0424 0.839747
\(690\) 3.18398 0.121212
\(691\) 33.3093 1.26715 0.633573 0.773683i \(-0.281588\pi\)
0.633573 + 0.773683i \(0.281588\pi\)
\(692\) 2.26974 0.0862824
\(693\) 0 0
\(694\) 0.459285 0.0174342
\(695\) 14.2240 0.539547
\(696\) 3.44417 0.130551
\(697\) −0.250636 −0.00949352
\(698\) −3.49313 −0.132217
\(699\) −1.69921 −0.0642701
\(700\) 0 0
\(701\) −30.9763 −1.16996 −0.584979 0.811049i \(-0.698897\pi\)
−0.584979 + 0.811049i \(0.698897\pi\)
\(702\) 2.27259 0.0857735
\(703\) 0.325600 0.0122802
\(704\) −1.67440 −0.0631063
\(705\) 5.23589 0.197195
\(706\) −33.8811 −1.27513
\(707\) 0 0
\(708\) −3.15403 −0.118536
\(709\) 18.8908 0.709459 0.354730 0.934969i \(-0.384573\pi\)
0.354730 + 0.934969i \(0.384573\pi\)
\(710\) −5.33125 −0.200078
\(711\) −4.32560 −0.162223
\(712\) −11.3912 −0.426902
\(713\) 3.41421 0.127863
\(714\) 0 0
\(715\) 12.1158 0.453104
\(716\) 5.32840 0.199132
\(717\) −0.0982225 −0.00366819
\(718\) −5.85044 −0.218337
\(719\) −19.2609 −0.718310 −0.359155 0.933278i \(-0.616935\pi\)
−0.359155 + 0.933278i \(0.616935\pi\)
\(720\) −3.18398 −0.118660
\(721\) 0 0
\(722\) −7.34315 −0.273284
\(723\) 22.9775 0.854541
\(724\) −1.41017 −0.0524087
\(725\) 17.6952 0.657182
\(726\) −8.19639 −0.304196
\(727\) 19.7281 0.731673 0.365837 0.930679i \(-0.380783\pi\)
0.365837 + 0.930679i \(0.380783\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 35.4408 1.31172
\(731\) −2.92416 −0.108154
\(732\) −13.9594 −0.515954
\(733\) 11.8070 0.436101 0.218050 0.975937i \(-0.430030\pi\)
0.218050 + 0.975937i \(0.430030\pi\)
\(734\) 16.5072 0.609293
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −8.70083 −0.320499
\(738\) −0.681153 −0.0250736
\(739\) −19.4273 −0.714644 −0.357322 0.933981i \(-0.616310\pi\)
−0.357322 + 0.933981i \(0.616310\pi\)
\(740\) 0.303644 0.0111622
\(741\) −7.75912 −0.285038
\(742\) 0 0
\(743\) 51.9765 1.90683 0.953416 0.301657i \(-0.0975398\pi\)
0.953416 + 0.301657i \(0.0975398\pi\)
\(744\) −3.41421 −0.125171
\(745\) 53.7952 1.97090
\(746\) 5.31619 0.194640
\(747\) 4.74547 0.173628
\(748\) −0.616110 −0.0225272
\(749\) 0 0
\(750\) −0.438512 −0.0160122
\(751\) −5.61698 −0.204967 −0.102483 0.994735i \(-0.532679\pi\)
−0.102483 + 0.994735i \(0.532679\pi\)
\(752\) −1.64445 −0.0599668
\(753\) 0.128174 0.00467093
\(754\) 7.82718 0.285049
\(755\) 18.5057 0.673491
\(756\) 0 0
\(757\) −26.2810 −0.955201 −0.477600 0.878577i \(-0.658493\pi\)
−0.477600 + 0.878577i \(0.658493\pi\)
\(758\) −13.0722 −0.474802
\(759\) 1.67440 0.0607769
\(760\) 10.8708 0.394325
\(761\) 5.14280 0.186427 0.0932133 0.995646i \(-0.470286\pi\)
0.0932133 + 0.995646i \(0.470286\pi\)
\(762\) 5.62139 0.203642
\(763\) 0 0
\(764\) −10.3329 −0.373830
\(765\) −1.17157 −0.0423583
\(766\) 1.81654 0.0656341
\(767\) −7.16782 −0.258815
\(768\) 1.00000 0.0360844
\(769\) 29.0158 1.04634 0.523168 0.852230i \(-0.324750\pi\)
0.523168 + 0.852230i \(0.324750\pi\)
\(770\) 0 0
\(771\) 10.4757 0.377274
\(772\) 1.70986 0.0615393
\(773\) −7.93092 −0.285255 −0.142628 0.989776i \(-0.545555\pi\)
−0.142628 + 0.989776i \(0.545555\pi\)
\(774\) −7.94699 −0.285649
\(775\) −17.5413 −0.630102
\(776\) −8.29014 −0.297599
\(777\) 0 0
\(778\) 24.5101 0.878729
\(779\) 2.32560 0.0833233
\(780\) −7.23589 −0.259086
\(781\) −2.80361 −0.100321
\(782\) −0.367959 −0.0131582
\(783\) 3.44417 0.123084
\(784\) 0 0
\(785\) 41.4352 1.47889
\(786\) 6.23309 0.222327
\(787\) 38.7584 1.38159 0.690795 0.723051i \(-0.257261\pi\)
0.690795 + 0.723051i \(0.257261\pi\)
\(788\) −6.81778 −0.242873
\(789\) 4.48814 0.159782
\(790\) 13.7726 0.490008
\(791\) 0 0
\(792\) −1.67440 −0.0594972
\(793\) −31.7240 −1.12655
\(794\) −29.3385 −1.04119
\(795\) −30.8821 −1.09527
\(796\) −13.4690 −0.477395
\(797\) 45.8149 1.62285 0.811424 0.584458i \(-0.198693\pi\)
0.811424 + 0.584458i \(0.198693\pi\)
\(798\) 0 0
\(799\) −0.605089 −0.0214065
\(800\) 5.13772 0.181646
\(801\) −11.3912 −0.402487
\(802\) −7.65114 −0.270171
\(803\) 18.6377 0.657710
\(804\) 5.19639 0.183262
\(805\) 0 0
\(806\) −7.75912 −0.273303
\(807\) 30.0594 1.05814
\(808\) −10.1597 −0.357416
\(809\) 42.9477 1.50996 0.754981 0.655747i \(-0.227646\pi\)
0.754981 + 0.655747i \(0.227646\pi\)
\(810\) −3.18398 −0.111874
\(811\) 53.0023 1.86116 0.930581 0.366087i \(-0.119303\pi\)
0.930581 + 0.366087i \(0.119303\pi\)
\(812\) 0 0
\(813\) 8.14442 0.285637
\(814\) 0.159681 0.00559682
\(815\) −27.0170 −0.946363
\(816\) 0.367959 0.0128811
\(817\) 27.1327 0.949254
\(818\) 14.4798 0.506273
\(819\) 0 0
\(820\) 2.16878 0.0757369
\(821\) −21.1741 −0.738980 −0.369490 0.929235i \(-0.620468\pi\)
−0.369490 + 0.929235i \(0.620468\pi\)
\(822\) 7.51913 0.262260
\(823\) −7.06490 −0.246267 −0.123133 0.992390i \(-0.539294\pi\)
−0.123133 + 0.992390i \(0.539294\pi\)
\(824\) −8.27259 −0.288189
\(825\) −8.60260 −0.299504
\(826\) 0 0
\(827\) −13.5994 −0.472899 −0.236449 0.971644i \(-0.575984\pi\)
−0.236449 + 0.971644i \(0.575984\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 15.9075 0.552490 0.276245 0.961087i \(-0.410910\pi\)
0.276245 + 0.961087i \(0.410910\pi\)
\(830\) −15.1095 −0.524457
\(831\) 23.3561 0.810213
\(832\) 2.27259 0.0787880
\(833\) 0 0
\(834\) −4.46736 −0.154692
\(835\) −2.18282 −0.0755396
\(836\) 5.71676 0.197718
\(837\) −3.41421 −0.118012
\(838\) −12.4838 −0.431246
\(839\) 29.2794 1.01084 0.505419 0.862874i \(-0.331338\pi\)
0.505419 + 0.862874i \(0.331338\pi\)
\(840\) 0 0
\(841\) −17.1377 −0.590956
\(842\) −25.0301 −0.862594
\(843\) −11.1434 −0.383798
\(844\) −6.44292 −0.221775
\(845\) 24.9475 0.858220
\(846\) −1.64445 −0.0565373
\(847\) 0 0
\(848\) 9.69921 0.333072
\(849\) 29.9707 1.02859
\(850\) 1.89047 0.0648426
\(851\) 0.0953661 0.00326911
\(852\) 1.67440 0.0573640
\(853\) −6.23750 −0.213568 −0.106784 0.994282i \(-0.534055\pi\)
−0.106784 + 0.994282i \(0.534055\pi\)
\(854\) 0 0
\(855\) 10.8708 0.371773
\(856\) −0.283242 −0.00968101
\(857\) 3.57976 0.122282 0.0611412 0.998129i \(-0.480526\pi\)
0.0611412 + 0.998129i \(0.480526\pi\)
\(858\) −3.80523 −0.129908
\(859\) 31.3206 1.06865 0.534323 0.845281i \(-0.320567\pi\)
0.534323 + 0.845281i \(0.320567\pi\)
\(860\) 25.3031 0.862827
\(861\) 0 0
\(862\) −35.8866 −1.22230
\(863\) 25.4232 0.865417 0.432709 0.901534i \(-0.357558\pi\)
0.432709 + 0.901534i \(0.357558\pi\)
\(864\) 1.00000 0.0340207
\(865\) −7.22679 −0.245718
\(866\) −26.7538 −0.909132
\(867\) −16.8646 −0.572752
\(868\) 0 0
\(869\) 7.24278 0.245695
\(870\) −10.9662 −0.371787
\(871\) 11.8093 0.400142
\(872\) −15.1857 −0.514254
\(873\) −8.29014 −0.280579
\(874\) 3.41421 0.115487
\(875\) 0 0
\(876\) −11.1310 −0.376081
\(877\) 25.4583 0.859667 0.429833 0.902908i \(-0.358572\pi\)
0.429833 + 0.902908i \(0.358572\pi\)
\(878\) 20.5046 0.691996
\(879\) −2.47002 −0.0833117
\(880\) 5.33125 0.179717
\(881\) −41.3523 −1.39319 −0.696597 0.717462i \(-0.745304\pi\)
−0.696597 + 0.717462i \(0.745304\pi\)
\(882\) 0 0
\(883\) −28.9342 −0.973713 −0.486857 0.873482i \(-0.661857\pi\)
−0.486857 + 0.873482i \(0.661857\pi\)
\(884\) 0.836220 0.0281251
\(885\) 10.0424 0.337570
\(886\) 16.0916 0.540608
\(887\) −27.6880 −0.929672 −0.464836 0.885397i \(-0.653887\pi\)
−0.464836 + 0.885397i \(0.653887\pi\)
\(888\) −0.0953661 −0.00320028
\(889\) 0 0
\(890\) 36.2692 1.21575
\(891\) −1.67440 −0.0560945
\(892\) 3.85962 0.129230
\(893\) 5.61450 0.187882
\(894\) −16.8956 −0.565073
\(895\) −16.9655 −0.567094
\(896\) 0 0
\(897\) −2.27259 −0.0758797
\(898\) 37.7122 1.25847
\(899\) −11.7591 −0.392188
\(900\) 5.13772 0.171257
\(901\) 3.56891 0.118898
\(902\) 1.14052 0.0379752
\(903\) 0 0
\(904\) 7.57903 0.252075
\(905\) 4.48996 0.149251
\(906\) −5.81212 −0.193095
\(907\) 31.9831 1.06198 0.530991 0.847378i \(-0.321820\pi\)
0.530991 + 0.847378i \(0.321820\pi\)
\(908\) −6.05476 −0.200934
\(909\) −10.1597 −0.336975
\(910\) 0 0
\(911\) −38.7589 −1.28414 −0.642070 0.766646i \(-0.721924\pi\)
−0.642070 + 0.766646i \(0.721924\pi\)
\(912\) −3.41421 −0.113056
\(913\) −7.94581 −0.262968
\(914\) 1.65934 0.0548861
\(915\) 44.4464 1.46935
\(916\) −14.0943 −0.465688
\(917\) 0 0
\(918\) 0.367959 0.0121444
\(919\) −52.0882 −1.71823 −0.859116 0.511781i \(-0.828986\pi\)
−0.859116 + 0.511781i \(0.828986\pi\)
\(920\) 3.18398 0.104973
\(921\) −34.4787 −1.13611
\(922\) 15.0455 0.495498
\(923\) 3.80523 0.125251
\(924\) 0 0
\(925\) −0.489965 −0.0161099
\(926\) 5.98935 0.196822
\(927\) −8.27259 −0.271708
\(928\) 3.44417 0.113060
\(929\) −33.3337 −1.09364 −0.546821 0.837249i \(-0.684162\pi\)
−0.546821 + 0.837249i \(0.684162\pi\)
\(930\) 10.8708 0.356467
\(931\) 0 0
\(932\) −1.69921 −0.0556596
\(933\) 7.85962 0.257312
\(934\) 4.07635 0.133382
\(935\) 1.96168 0.0641538
\(936\) 2.27259 0.0742820
\(937\) −5.66127 −0.184945 −0.0924727 0.995715i \(-0.529477\pi\)
−0.0924727 + 0.995715i \(0.529477\pi\)
\(938\) 0 0
\(939\) −13.0982 −0.427442
\(940\) 5.23589 0.170776
\(941\) −24.5250 −0.799492 −0.399746 0.916626i \(-0.630902\pi\)
−0.399746 + 0.916626i \(0.630902\pi\)
\(942\) −13.0136 −0.424008
\(943\) 0.681153 0.0221814
\(944\) −3.15403 −0.102655
\(945\) 0 0
\(946\) 13.3064 0.432630
\(947\) 21.9392 0.712928 0.356464 0.934309i \(-0.383982\pi\)
0.356464 + 0.934309i \(0.383982\pi\)
\(948\) −4.32560 −0.140489
\(949\) −25.2962 −0.821148
\(950\) −17.5413 −0.569115
\(951\) −17.3956 −0.564090
\(952\) 0 0
\(953\) 27.6897 0.896959 0.448479 0.893793i \(-0.351966\pi\)
0.448479 + 0.893793i \(0.351966\pi\)
\(954\) 9.69921 0.314024
\(955\) 32.8996 1.06461
\(956\) −0.0982225 −0.00317674
\(957\) −5.76691 −0.186418
\(958\) −29.1999 −0.943406
\(959\) 0 0
\(960\) −3.18398 −0.102762
\(961\) −19.3431 −0.623972
\(962\) −0.216728 −0.00698760
\(963\) −0.283242 −0.00912735
\(964\) 22.9775 0.740054
\(965\) −5.44417 −0.175254
\(966\) 0 0
\(967\) −3.84510 −0.123650 −0.0618251 0.998087i \(-0.519692\pi\)
−0.0618251 + 0.998087i \(0.519692\pi\)
\(968\) −8.19639 −0.263442
\(969\) −1.25629 −0.0403578
\(970\) 26.3956 0.847513
\(971\) −23.3583 −0.749605 −0.374803 0.927105i \(-0.622289\pi\)
−0.374803 + 0.927105i \(0.622289\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 28.3413 0.908114
\(975\) 11.6760 0.373930
\(976\) −13.9594 −0.446829
\(977\) 15.8034 0.505596 0.252798 0.967519i \(-0.418649\pi\)
0.252798 + 0.967519i \(0.418649\pi\)
\(978\) 8.48528 0.271329
\(979\) 19.0734 0.609587
\(980\) 0 0
\(981\) −15.1857 −0.484843
\(982\) 34.3680 1.09673
\(983\) −17.6528 −0.563037 −0.281519 0.959556i \(-0.590838\pi\)
−0.281519 + 0.959556i \(0.590838\pi\)
\(984\) −0.681153 −0.0217144
\(985\) 21.7077 0.691664
\(986\) 1.26731 0.0403594
\(987\) 0 0
\(988\) −7.75912 −0.246850
\(989\) 7.94699 0.252700
\(990\) 5.33125 0.169438
\(991\) −15.0224 −0.477202 −0.238601 0.971118i \(-0.576689\pi\)
−0.238601 + 0.971118i \(0.576689\pi\)
\(992\) −3.41421 −0.108401
\(993\) −7.96491 −0.252759
\(994\) 0 0
\(995\) 42.8849 1.35954
\(996\) 4.74547 0.150366
\(997\) 33.9954 1.07665 0.538323 0.842739i \(-0.319058\pi\)
0.538323 + 0.842739i \(0.319058\pi\)
\(998\) 14.8071 0.468711
\(999\) −0.0953661 −0.00301725
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 6762.2.a.co.1.2 yes 4
7.6 odd 2 6762.2.a.cn.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.cn.1.3 4 7.6 odd 2
6762.2.a.co.1.2 yes 4 1.1 even 1 trivial