Properties

Label 1617.2.a.p.1.2
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
Dimension $2$
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] = [1617,2,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(12.9118100068\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.00000 q^{5} -1.61803 q^{6} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.00000 q^{5} -1.61803 q^{6} -2.23607 q^{8} +1.00000 q^{9} -1.61803 q^{10} +1.00000 q^{11} -0.618034 q^{12} +5.47214 q^{13} +1.00000 q^{15} -4.85410 q^{16} -0.763932 q^{17} +1.61803 q^{18} -6.70820 q^{19} -0.618034 q^{20} +1.61803 q^{22} -7.70820 q^{23} +2.23607 q^{24} -4.00000 q^{25} +8.85410 q^{26} -1.00000 q^{27} +5.00000 q^{29} +1.61803 q^{30} +0.763932 q^{31} -3.38197 q^{32} -1.00000 q^{33} -1.23607 q^{34} +0.618034 q^{36} -7.00000 q^{37} -10.8541 q^{38} -5.47214 q^{39} +2.23607 q^{40} -6.47214 q^{41} -7.70820 q^{43} +0.618034 q^{44} -1.00000 q^{45} -12.4721 q^{46} +4.23607 q^{47} +4.85410 q^{48} -6.47214 q^{50} +0.763932 q^{51} +3.38197 q^{52} +10.1803 q^{53} -1.61803 q^{54} -1.00000 q^{55} +6.70820 q^{57} +8.09017 q^{58} -11.1803 q^{59} +0.618034 q^{60} -2.00000 q^{61} +1.23607 q^{62} +4.23607 q^{64} -5.47214 q^{65} -1.61803 q^{66} -14.2361 q^{67} -0.472136 q^{68} +7.70820 q^{69} +6.47214 q^{71} -2.23607 q^{72} -13.4721 q^{73} -11.3262 q^{74} +4.00000 q^{75} -4.14590 q^{76} -8.85410 q^{78} -5.52786 q^{79} +4.85410 q^{80} +1.00000 q^{81} -10.4721 q^{82} -11.2361 q^{83} +0.763932 q^{85} -12.4721 q^{86} -5.00000 q^{87} -2.23607 q^{88} -4.47214 q^{89} -1.61803 q^{90} -4.76393 q^{92} -0.763932 q^{93} +6.85410 q^{94} +6.70820 q^{95} +3.38197 q^{96} +3.70820 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 2 q^{5} - q^{6} + 2 q^{9} - q^{10} + 2 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} - 6 q^{17} + q^{18} + q^{20} + q^{22} - 2 q^{23} - 8 q^{25} + 11 q^{26} - 2 q^{27} + 10 q^{29} + q^{30} + 6 q^{31} - 9 q^{32} - 2 q^{33} + 2 q^{34} - q^{36} - 14 q^{37} - 15 q^{38} - 2 q^{39} - 4 q^{41} - 2 q^{43} - q^{44} - 2 q^{45} - 16 q^{46} + 4 q^{47} + 3 q^{48} - 4 q^{50} + 6 q^{51} + 9 q^{52} - 2 q^{53} - q^{54} - 2 q^{55} + 5 q^{58} - q^{60} - 4 q^{61} - 2 q^{62} + 4 q^{64} - 2 q^{65} - q^{66} - 24 q^{67} + 8 q^{68} + 2 q^{69} + 4 q^{71} - 18 q^{73} - 7 q^{74} + 8 q^{75} - 15 q^{76} - 11 q^{78} - 20 q^{79} + 3 q^{80} + 2 q^{81} - 12 q^{82} - 18 q^{83} + 6 q^{85} - 16 q^{86} - 10 q^{87} - q^{90} - 14 q^{92} - 6 q^{93} + 7 q^{94} + 9 q^{96} - 6 q^{97} + 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.61803 −0.660560
\(7\) 0 0
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) −1.61803 −0.511667
\(11\) 1.00000 0.301511
\(12\) −0.618034 −0.178411
\(13\) 5.47214 1.51770 0.758849 0.651267i \(-0.225762\pi\)
0.758849 + 0.651267i \(0.225762\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) −4.85410 −1.21353
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) 1.61803 0.381374
\(19\) −6.70820 −1.53897 −0.769484 0.638666i \(-0.779486\pi\)
−0.769484 + 0.638666i \(0.779486\pi\)
\(20\) −0.618034 −0.138197
\(21\) 0 0
\(22\) 1.61803 0.344966
\(23\) −7.70820 −1.60727 −0.803636 0.595121i \(-0.797104\pi\)
−0.803636 + 0.595121i \(0.797104\pi\)
\(24\) 2.23607 0.456435
\(25\) −4.00000 −0.800000
\(26\) 8.85410 1.73643
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 1.61803 0.295411
\(31\) 0.763932 0.137206 0.0686031 0.997644i \(-0.478146\pi\)
0.0686031 + 0.997644i \(0.478146\pi\)
\(32\) −3.38197 −0.597853
\(33\) −1.00000 −0.174078
\(34\) −1.23607 −0.211984
\(35\) 0 0
\(36\) 0.618034 0.103006
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −10.8541 −1.76077
\(39\) −5.47214 −0.876243
\(40\) 2.23607 0.353553
\(41\) −6.47214 −1.01078 −0.505389 0.862892i \(-0.668651\pi\)
−0.505389 + 0.862892i \(0.668651\pi\)
\(42\) 0 0
\(43\) −7.70820 −1.17549 −0.587745 0.809046i \(-0.699984\pi\)
−0.587745 + 0.809046i \(0.699984\pi\)
\(44\) 0.618034 0.0931721
\(45\) −1.00000 −0.149071
\(46\) −12.4721 −1.83892
\(47\) 4.23607 0.617894 0.308947 0.951079i \(-0.400023\pi\)
0.308947 + 0.951079i \(0.400023\pi\)
\(48\) 4.85410 0.700629
\(49\) 0 0
\(50\) −6.47214 −0.915298
\(51\) 0.763932 0.106972
\(52\) 3.38197 0.468994
\(53\) 10.1803 1.39838 0.699189 0.714937i \(-0.253545\pi\)
0.699189 + 0.714937i \(0.253545\pi\)
\(54\) −1.61803 −0.220187
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 6.70820 0.888523
\(58\) 8.09017 1.06229
\(59\) −11.1803 −1.45556 −0.727778 0.685813i \(-0.759447\pi\)
−0.727778 + 0.685813i \(0.759447\pi\)
\(60\) 0.618034 0.0797878
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.23607 0.156981
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) −5.47214 −0.678735
\(66\) −1.61803 −0.199166
\(67\) −14.2361 −1.73921 −0.869606 0.493746i \(-0.835627\pi\)
−0.869606 + 0.493746i \(0.835627\pi\)
\(68\) −0.472136 −0.0572549
\(69\) 7.70820 0.927959
\(70\) 0 0
\(71\) 6.47214 0.768101 0.384051 0.923312i \(-0.374529\pi\)
0.384051 + 0.923312i \(0.374529\pi\)
\(72\) −2.23607 −0.263523
\(73\) −13.4721 −1.57679 −0.788397 0.615167i \(-0.789089\pi\)
−0.788397 + 0.615167i \(0.789089\pi\)
\(74\) −11.3262 −1.31665
\(75\) 4.00000 0.461880
\(76\) −4.14590 −0.475567
\(77\) 0 0
\(78\) −8.85410 −1.00253
\(79\) −5.52786 −0.621933 −0.310967 0.950421i \(-0.600653\pi\)
−0.310967 + 0.950421i \(0.600653\pi\)
\(80\) 4.85410 0.542705
\(81\) 1.00000 0.111111
\(82\) −10.4721 −1.15645
\(83\) −11.2361 −1.23332 −0.616659 0.787230i \(-0.711514\pi\)
−0.616659 + 0.787230i \(0.711514\pi\)
\(84\) 0 0
\(85\) 0.763932 0.0828601
\(86\) −12.4721 −1.34491
\(87\) −5.00000 −0.536056
\(88\) −2.23607 −0.238366
\(89\) −4.47214 −0.474045 −0.237023 0.971504i \(-0.576172\pi\)
−0.237023 + 0.971504i \(0.576172\pi\)
\(90\) −1.61803 −0.170556
\(91\) 0 0
\(92\) −4.76393 −0.496674
\(93\) −0.763932 −0.0792161
\(94\) 6.85410 0.706947
\(95\) 6.70820 0.688247
\(96\) 3.38197 0.345170
\(97\) 3.70820 0.376511 0.188256 0.982120i \(-0.439717\pi\)
0.188256 + 0.982120i \(0.439717\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −2.47214 −0.247214
\(101\) 4.18034 0.415959 0.207980 0.978133i \(-0.433311\pi\)
0.207980 + 0.978133i \(0.433311\pi\)
\(102\) 1.23607 0.122389
\(103\) 9.41641 0.927826 0.463913 0.885881i \(-0.346445\pi\)
0.463913 + 0.885881i \(0.346445\pi\)
\(104\) −12.2361 −1.19985
\(105\) 0 0
\(106\) 16.4721 1.59992
\(107\) 0.236068 0.0228216 0.0114108 0.999935i \(-0.496368\pi\)
0.0114108 + 0.999935i \(0.496368\pi\)
\(108\) −0.618034 −0.0594703
\(109\) 7.23607 0.693090 0.346545 0.938033i \(-0.387355\pi\)
0.346545 + 0.938033i \(0.387355\pi\)
\(110\) −1.61803 −0.154273
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) 8.47214 0.796992 0.398496 0.917170i \(-0.369532\pi\)
0.398496 + 0.917170i \(0.369532\pi\)
\(114\) 10.8541 1.01658
\(115\) 7.70820 0.718794
\(116\) 3.09017 0.286915
\(117\) 5.47214 0.505899
\(118\) −18.0902 −1.66534
\(119\) 0 0
\(120\) −2.23607 −0.204124
\(121\) 1.00000 0.0909091
\(122\) −3.23607 −0.292980
\(123\) 6.47214 0.583573
\(124\) 0.472136 0.0423991
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 18.6525 1.65514 0.827570 0.561363i \(-0.189723\pi\)
0.827570 + 0.561363i \(0.189723\pi\)
\(128\) 13.6180 1.20368
\(129\) 7.70820 0.678670
\(130\) −8.85410 −0.776556
\(131\) 16.9443 1.48043 0.740214 0.672371i \(-0.234724\pi\)
0.740214 + 0.672371i \(0.234724\pi\)
\(132\) −0.618034 −0.0537930
\(133\) 0 0
\(134\) −23.0344 −1.98987
\(135\) 1.00000 0.0860663
\(136\) 1.70820 0.146477
\(137\) 6.29180 0.537544 0.268772 0.963204i \(-0.413382\pi\)
0.268772 + 0.963204i \(0.413382\pi\)
\(138\) 12.4721 1.06170
\(139\) −5.52786 −0.468867 −0.234434 0.972132i \(-0.575324\pi\)
−0.234434 + 0.972132i \(0.575324\pi\)
\(140\) 0 0
\(141\) −4.23607 −0.356741
\(142\) 10.4721 0.878802
\(143\) 5.47214 0.457603
\(144\) −4.85410 −0.404508
\(145\) −5.00000 −0.415227
\(146\) −21.7984 −1.80405
\(147\) 0 0
\(148\) −4.32624 −0.355615
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 6.47214 0.528448
\(151\) 8.18034 0.665707 0.332853 0.942979i \(-0.391989\pi\)
0.332853 + 0.942979i \(0.391989\pi\)
\(152\) 15.0000 1.21666
\(153\) −0.763932 −0.0617602
\(154\) 0 0
\(155\) −0.763932 −0.0613605
\(156\) −3.38197 −0.270774
\(157\) −11.4164 −0.911129 −0.455564 0.890203i \(-0.650562\pi\)
−0.455564 + 0.890203i \(0.650562\pi\)
\(158\) −8.94427 −0.711568
\(159\) −10.1803 −0.807353
\(160\) 3.38197 0.267368
\(161\) 0 0
\(162\) 1.61803 0.127125
\(163\) −9.29180 −0.727790 −0.363895 0.931440i \(-0.618553\pi\)
−0.363895 + 0.931440i \(0.618553\pi\)
\(164\) −4.00000 −0.312348
\(165\) 1.00000 0.0778499
\(166\) −18.1803 −1.41107
\(167\) −8.65248 −0.669549 −0.334774 0.942298i \(-0.608660\pi\)
−0.334774 + 0.942298i \(0.608660\pi\)
\(168\) 0 0
\(169\) 16.9443 1.30341
\(170\) 1.23607 0.0948021
\(171\) −6.70820 −0.512989
\(172\) −4.76393 −0.363246
\(173\) 10.4721 0.796182 0.398091 0.917346i \(-0.369673\pi\)
0.398091 + 0.917346i \(0.369673\pi\)
\(174\) −8.09017 −0.613314
\(175\) 0 0
\(176\) −4.85410 −0.365892
\(177\) 11.1803 0.840366
\(178\) −7.23607 −0.542366
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) −0.618034 −0.0460655
\(181\) 5.23607 0.389194 0.194597 0.980883i \(-0.437660\pi\)
0.194597 + 0.980883i \(0.437660\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 17.2361 1.27066
\(185\) 7.00000 0.514650
\(186\) −1.23607 −0.0906329
\(187\) −0.763932 −0.0558642
\(188\) 2.61803 0.190940
\(189\) 0 0
\(190\) 10.8541 0.787439
\(191\) −15.2361 −1.10244 −0.551222 0.834359i \(-0.685838\pi\)
−0.551222 + 0.834359i \(0.685838\pi\)
\(192\) −4.23607 −0.305712
\(193\) −16.6525 −1.19867 −0.599336 0.800498i \(-0.704569\pi\)
−0.599336 + 0.800498i \(0.704569\pi\)
\(194\) 6.00000 0.430775
\(195\) 5.47214 0.391868
\(196\) 0 0
\(197\) −7.52786 −0.536338 −0.268169 0.963372i \(-0.586419\pi\)
−0.268169 + 0.963372i \(0.586419\pi\)
\(198\) 1.61803 0.114989
\(199\) 26.1803 1.85588 0.927938 0.372736i \(-0.121580\pi\)
0.927938 + 0.372736i \(0.121580\pi\)
\(200\) 8.94427 0.632456
\(201\) 14.2361 1.00413
\(202\) 6.76393 0.475909
\(203\) 0 0
\(204\) 0.472136 0.0330561
\(205\) 6.47214 0.452034
\(206\) 15.2361 1.06155
\(207\) −7.70820 −0.535757
\(208\) −26.5623 −1.84176
\(209\) −6.70820 −0.464016
\(210\) 0 0
\(211\) −21.4164 −1.47437 −0.737183 0.675693i \(-0.763845\pi\)
−0.737183 + 0.675693i \(0.763845\pi\)
\(212\) 6.29180 0.432122
\(213\) −6.47214 −0.443463
\(214\) 0.381966 0.0261107
\(215\) 7.70820 0.525695
\(216\) 2.23607 0.152145
\(217\) 0 0
\(218\) 11.7082 0.792980
\(219\) 13.4721 0.910363
\(220\) −0.618034 −0.0416678
\(221\) −4.18034 −0.281200
\(222\) 11.3262 0.760167
\(223\) 6.00000 0.401790 0.200895 0.979613i \(-0.435615\pi\)
0.200895 + 0.979613i \(0.435615\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 13.7082 0.911856
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 4.14590 0.274569
\(229\) −2.76393 −0.182646 −0.0913229 0.995821i \(-0.529110\pi\)
−0.0913229 + 0.995821i \(0.529110\pi\)
\(230\) 12.4721 0.822388
\(231\) 0 0
\(232\) −11.1803 −0.734025
\(233\) 2.94427 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(234\) 8.85410 0.578811
\(235\) −4.23607 −0.276331
\(236\) −6.90983 −0.449792
\(237\) 5.52786 0.359073
\(238\) 0 0
\(239\) −30.1246 −1.94860 −0.974300 0.225256i \(-0.927678\pi\)
−0.974300 + 0.225256i \(0.927678\pi\)
\(240\) −4.85410 −0.313331
\(241\) −8.05573 −0.518915 −0.259458 0.965755i \(-0.583544\pi\)
−0.259458 + 0.965755i \(0.583544\pi\)
\(242\) 1.61803 0.104011
\(243\) −1.00000 −0.0641500
\(244\) −1.23607 −0.0791311
\(245\) 0 0
\(246\) 10.4721 0.667679
\(247\) −36.7082 −2.33569
\(248\) −1.70820 −0.108471
\(249\) 11.2361 0.712057
\(250\) 14.5623 0.921001
\(251\) 28.1246 1.77521 0.887605 0.460606i \(-0.152368\pi\)
0.887605 + 0.460606i \(0.152368\pi\)
\(252\) 0 0
\(253\) −7.70820 −0.484611
\(254\) 30.1803 1.89368
\(255\) −0.763932 −0.0478393
\(256\) 13.5623 0.847644
\(257\) 7.00000 0.436648 0.218324 0.975876i \(-0.429941\pi\)
0.218324 + 0.975876i \(0.429941\pi\)
\(258\) 12.4721 0.776481
\(259\) 0 0
\(260\) −3.38197 −0.209741
\(261\) 5.00000 0.309492
\(262\) 27.4164 1.69379
\(263\) 14.1246 0.870961 0.435480 0.900198i \(-0.356579\pi\)
0.435480 + 0.900198i \(0.356579\pi\)
\(264\) 2.23607 0.137620
\(265\) −10.1803 −0.625373
\(266\) 0 0
\(267\) 4.47214 0.273690
\(268\) −8.79837 −0.537446
\(269\) −18.9443 −1.15505 −0.577526 0.816372i \(-0.695982\pi\)
−0.577526 + 0.816372i \(0.695982\pi\)
\(270\) 1.61803 0.0984704
\(271\) −18.7082 −1.13644 −0.568221 0.822876i \(-0.692368\pi\)
−0.568221 + 0.822876i \(0.692368\pi\)
\(272\) 3.70820 0.224843
\(273\) 0 0
\(274\) 10.1803 0.615017
\(275\) −4.00000 −0.241209
\(276\) 4.76393 0.286755
\(277\) 2.47214 0.148536 0.0742681 0.997238i \(-0.476338\pi\)
0.0742681 + 0.997238i \(0.476338\pi\)
\(278\) −8.94427 −0.536442
\(279\) 0.763932 0.0457354
\(280\) 0 0
\(281\) 2.52786 0.150800 0.0753999 0.997153i \(-0.475977\pi\)
0.0753999 + 0.997153i \(0.475977\pi\)
\(282\) −6.85410 −0.408156
\(283\) 18.2361 1.08402 0.542011 0.840371i \(-0.317663\pi\)
0.542011 + 0.840371i \(0.317663\pi\)
\(284\) 4.00000 0.237356
\(285\) −6.70820 −0.397360
\(286\) 8.85410 0.523554
\(287\) 0 0
\(288\) −3.38197 −0.199284
\(289\) −16.4164 −0.965671
\(290\) −8.09017 −0.475071
\(291\) −3.70820 −0.217379
\(292\) −8.32624 −0.487256
\(293\) 16.0000 0.934730 0.467365 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) 11.1803 0.650945
\(296\) 15.6525 0.909782
\(297\) −1.00000 −0.0580259
\(298\) 8.09017 0.468651
\(299\) −42.1803 −2.43935
\(300\) 2.47214 0.142729
\(301\) 0 0
\(302\) 13.2361 0.761650
\(303\) −4.18034 −0.240154
\(304\) 32.5623 1.86758
\(305\) 2.00000 0.114520
\(306\) −1.23607 −0.0706613
\(307\) 3.05573 0.174400 0.0871998 0.996191i \(-0.472208\pi\)
0.0871998 + 0.996191i \(0.472208\pi\)
\(308\) 0 0
\(309\) −9.41641 −0.535681
\(310\) −1.23607 −0.0702039
\(311\) 25.8885 1.46800 0.734002 0.679147i \(-0.237650\pi\)
0.734002 + 0.679147i \(0.237650\pi\)
\(312\) 12.2361 0.692731
\(313\) 6.65248 0.376020 0.188010 0.982167i \(-0.439796\pi\)
0.188010 + 0.982167i \(0.439796\pi\)
\(314\) −18.4721 −1.04244
\(315\) 0 0
\(316\) −3.41641 −0.192188
\(317\) 1.81966 0.102202 0.0511011 0.998693i \(-0.483727\pi\)
0.0511011 + 0.998693i \(0.483727\pi\)
\(318\) −16.4721 −0.923712
\(319\) 5.00000 0.279946
\(320\) −4.23607 −0.236803
\(321\) −0.236068 −0.0131760
\(322\) 0 0
\(323\) 5.12461 0.285141
\(324\) 0.618034 0.0343352
\(325\) −21.8885 −1.21416
\(326\) −15.0344 −0.832681
\(327\) −7.23607 −0.400155
\(328\) 14.4721 0.799090
\(329\) 0 0
\(330\) 1.61803 0.0890698
\(331\) 15.4164 0.847362 0.423681 0.905811i \(-0.360738\pi\)
0.423681 + 0.905811i \(0.360738\pi\)
\(332\) −6.94427 −0.381116
\(333\) −7.00000 −0.383598
\(334\) −14.0000 −0.766046
\(335\) 14.2361 0.777799
\(336\) 0 0
\(337\) 14.1803 0.772452 0.386226 0.922404i \(-0.373778\pi\)
0.386226 + 0.922404i \(0.373778\pi\)
\(338\) 27.4164 1.49126
\(339\) −8.47214 −0.460143
\(340\) 0.472136 0.0256052
\(341\) 0.763932 0.0413692
\(342\) −10.8541 −0.586923
\(343\) 0 0
\(344\) 17.2361 0.929307
\(345\) −7.70820 −0.414996
\(346\) 16.9443 0.910930
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −3.09017 −0.165650
\(349\) −28.4164 −1.52110 −0.760548 0.649282i \(-0.775069\pi\)
−0.760548 + 0.649282i \(0.775069\pi\)
\(350\) 0 0
\(351\) −5.47214 −0.292081
\(352\) −3.38197 −0.180259
\(353\) −33.4721 −1.78154 −0.890771 0.454452i \(-0.849835\pi\)
−0.890771 + 0.454452i \(0.849835\pi\)
\(354\) 18.0902 0.961482
\(355\) −6.47214 −0.343505
\(356\) −2.76393 −0.146488
\(357\) 0 0
\(358\) −14.4721 −0.764876
\(359\) −3.41641 −0.180311 −0.0901556 0.995928i \(-0.528736\pi\)
−0.0901556 + 0.995928i \(0.528736\pi\)
\(360\) 2.23607 0.117851
\(361\) 26.0000 1.36842
\(362\) 8.47214 0.445286
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 13.4721 0.705164
\(366\) 3.23607 0.169152
\(367\) −15.8885 −0.829375 −0.414688 0.909964i \(-0.636109\pi\)
−0.414688 + 0.909964i \(0.636109\pi\)
\(368\) 37.4164 1.95047
\(369\) −6.47214 −0.336926
\(370\) 11.3262 0.588823
\(371\) 0 0
\(372\) −0.472136 −0.0244791
\(373\) −26.6525 −1.38001 −0.690006 0.723803i \(-0.742392\pi\)
−0.690006 + 0.723803i \(0.742392\pi\)
\(374\) −1.23607 −0.0639156
\(375\) −9.00000 −0.464758
\(376\) −9.47214 −0.488488
\(377\) 27.3607 1.40915
\(378\) 0 0
\(379\) −8.81966 −0.453036 −0.226518 0.974007i \(-0.572734\pi\)
−0.226518 + 0.974007i \(0.572734\pi\)
\(380\) 4.14590 0.212680
\(381\) −18.6525 −0.955595
\(382\) −24.6525 −1.26133
\(383\) −15.0557 −0.769312 −0.384656 0.923060i \(-0.625680\pi\)
−0.384656 + 0.923060i \(0.625680\pi\)
\(384\) −13.6180 −0.694942
\(385\) 0 0
\(386\) −26.9443 −1.37143
\(387\) −7.70820 −0.391830
\(388\) 2.29180 0.116348
\(389\) 28.9443 1.46753 0.733766 0.679402i \(-0.237761\pi\)
0.733766 + 0.679402i \(0.237761\pi\)
\(390\) 8.85410 0.448345
\(391\) 5.88854 0.297796
\(392\) 0 0
\(393\) −16.9443 −0.854725
\(394\) −12.1803 −0.613637
\(395\) 5.52786 0.278137
\(396\) 0.618034 0.0310574
\(397\) 17.1246 0.859460 0.429730 0.902958i \(-0.358609\pi\)
0.429730 + 0.902958i \(0.358609\pi\)
\(398\) 42.3607 2.12335
\(399\) 0 0
\(400\) 19.4164 0.970820
\(401\) −16.2918 −0.813573 −0.406787 0.913523i \(-0.633351\pi\)
−0.406787 + 0.913523i \(0.633351\pi\)
\(402\) 23.0344 1.14885
\(403\) 4.18034 0.208238
\(404\) 2.58359 0.128539
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −7.00000 −0.346977
\(408\) −1.70820 −0.0845687
\(409\) −38.9443 −1.92567 −0.962835 0.270090i \(-0.912947\pi\)
−0.962835 + 0.270090i \(0.912947\pi\)
\(410\) 10.4721 0.517182
\(411\) −6.29180 −0.310351
\(412\) 5.81966 0.286714
\(413\) 0 0
\(414\) −12.4721 −0.612972
\(415\) 11.2361 0.551557
\(416\) −18.5066 −0.907360
\(417\) 5.52786 0.270701
\(418\) −10.8541 −0.530891
\(419\) −21.1803 −1.03473 −0.517364 0.855766i \(-0.673087\pi\)
−0.517364 + 0.855766i \(0.673087\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) −34.6525 −1.68686
\(423\) 4.23607 0.205965
\(424\) −22.7639 −1.10551
\(425\) 3.05573 0.148225
\(426\) −10.4721 −0.507377
\(427\) 0 0
\(428\) 0.145898 0.00705225
\(429\) −5.47214 −0.264197
\(430\) 12.4721 0.601460
\(431\) −4.70820 −0.226786 −0.113393 0.993550i \(-0.536172\pi\)
−0.113393 + 0.993550i \(0.536172\pi\)
\(432\) 4.85410 0.233543
\(433\) 1.52786 0.0734245 0.0367122 0.999326i \(-0.488312\pi\)
0.0367122 + 0.999326i \(0.488312\pi\)
\(434\) 0 0
\(435\) 5.00000 0.239732
\(436\) 4.47214 0.214176
\(437\) 51.7082 2.47354
\(438\) 21.7984 1.04157
\(439\) 11.1803 0.533609 0.266804 0.963751i \(-0.414032\pi\)
0.266804 + 0.963751i \(0.414032\pi\)
\(440\) 2.23607 0.106600
\(441\) 0 0
\(442\) −6.76393 −0.321727
\(443\) 9.52786 0.452682 0.226341 0.974048i \(-0.427324\pi\)
0.226341 + 0.974048i \(0.427324\pi\)
\(444\) 4.32624 0.205314
\(445\) 4.47214 0.212000
\(446\) 9.70820 0.459697
\(447\) −5.00000 −0.236492
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) −6.47214 −0.305099
\(451\) −6.47214 −0.304761
\(452\) 5.23607 0.246284
\(453\) −8.18034 −0.384346
\(454\) 3.23607 0.151876
\(455\) 0 0
\(456\) −15.0000 −0.702439
\(457\) 10.7639 0.503516 0.251758 0.967790i \(-0.418991\pi\)
0.251758 + 0.967790i \(0.418991\pi\)
\(458\) −4.47214 −0.208969
\(459\) 0.763932 0.0356573
\(460\) 4.76393 0.222119
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) −27.1803 −1.26318 −0.631589 0.775304i \(-0.717597\pi\)
−0.631589 + 0.775304i \(0.717597\pi\)
\(464\) −24.2705 −1.12673
\(465\) 0.763932 0.0354265
\(466\) 4.76393 0.220685
\(467\) −6.81966 −0.315576 −0.157788 0.987473i \(-0.550436\pi\)
−0.157788 + 0.987473i \(0.550436\pi\)
\(468\) 3.38197 0.156331
\(469\) 0 0
\(470\) −6.85410 −0.316156
\(471\) 11.4164 0.526040
\(472\) 25.0000 1.15072
\(473\) −7.70820 −0.354424
\(474\) 8.94427 0.410824
\(475\) 26.8328 1.23117
\(476\) 0 0
\(477\) 10.1803 0.466126
\(478\) −48.7426 −2.22944
\(479\) 1.70820 0.0780498 0.0390249 0.999238i \(-0.487575\pi\)
0.0390249 + 0.999238i \(0.487575\pi\)
\(480\) −3.38197 −0.154365
\(481\) −38.3050 −1.74656
\(482\) −13.0344 −0.593703
\(483\) 0 0
\(484\) 0.618034 0.0280925
\(485\) −3.70820 −0.168381
\(486\) −1.61803 −0.0733955
\(487\) −0.944272 −0.0427890 −0.0213945 0.999771i \(-0.506811\pi\)
−0.0213945 + 0.999771i \(0.506811\pi\)
\(488\) 4.47214 0.202444
\(489\) 9.29180 0.420190
\(490\) 0 0
\(491\) 22.1246 0.998470 0.499235 0.866467i \(-0.333614\pi\)
0.499235 + 0.866467i \(0.333614\pi\)
\(492\) 4.00000 0.180334
\(493\) −3.81966 −0.172029
\(494\) −59.3951 −2.67231
\(495\) −1.00000 −0.0449467
\(496\) −3.70820 −0.166503
\(497\) 0 0
\(498\) 18.1803 0.814681
\(499\) 2.23607 0.100100 0.0500501 0.998747i \(-0.484062\pi\)
0.0500501 + 0.998747i \(0.484062\pi\)
\(500\) 5.56231 0.248754
\(501\) 8.65248 0.386564
\(502\) 45.5066 2.03106
\(503\) −15.7082 −0.700394 −0.350197 0.936676i \(-0.613885\pi\)
−0.350197 + 0.936676i \(0.613885\pi\)
\(504\) 0 0
\(505\) −4.18034 −0.186023
\(506\) −12.4721 −0.554454
\(507\) −16.9443 −0.752522
\(508\) 11.5279 0.511466
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) −1.23607 −0.0547340
\(511\) 0 0
\(512\) −5.29180 −0.233867
\(513\) 6.70820 0.296174
\(514\) 11.3262 0.499579
\(515\) −9.41641 −0.414937
\(516\) 4.76393 0.209720
\(517\) 4.23607 0.186302
\(518\) 0 0
\(519\) −10.4721 −0.459676
\(520\) 12.2361 0.536587
\(521\) 24.3050 1.06482 0.532410 0.846487i \(-0.321287\pi\)
0.532410 + 0.846487i \(0.321287\pi\)
\(522\) 8.09017 0.354097
\(523\) −9.65248 −0.422073 −0.211037 0.977478i \(-0.567684\pi\)
−0.211037 + 0.977478i \(0.567684\pi\)
\(524\) 10.4721 0.457477
\(525\) 0 0
\(526\) 22.8541 0.996486
\(527\) −0.583592 −0.0254217
\(528\) 4.85410 0.211248
\(529\) 36.4164 1.58332
\(530\) −16.4721 −0.715504
\(531\) −11.1803 −0.485185
\(532\) 0 0
\(533\) −35.4164 −1.53405
\(534\) 7.23607 0.313135
\(535\) −0.236068 −0.0102061
\(536\) 31.8328 1.37497
\(537\) 8.94427 0.385974
\(538\) −30.6525 −1.32152
\(539\) 0 0
\(540\) 0.618034 0.0265959
\(541\) −19.0557 −0.819270 −0.409635 0.912250i \(-0.634344\pi\)
−0.409635 + 0.912250i \(0.634344\pi\)
\(542\) −30.2705 −1.30023
\(543\) −5.23607 −0.224701
\(544\) 2.58359 0.110771
\(545\) −7.23607 −0.309959
\(546\) 0 0
\(547\) −38.8328 −1.66037 −0.830186 0.557487i \(-0.811766\pi\)
−0.830186 + 0.557487i \(0.811766\pi\)
\(548\) 3.88854 0.166110
\(549\) −2.00000 −0.0853579
\(550\) −6.47214 −0.275973
\(551\) −33.5410 −1.42890
\(552\) −17.2361 −0.733616
\(553\) 0 0
\(554\) 4.00000 0.169944
\(555\) −7.00000 −0.297133
\(556\) −3.41641 −0.144888
\(557\) −21.4721 −0.909804 −0.454902 0.890542i \(-0.650326\pi\)
−0.454902 + 0.890542i \(0.650326\pi\)
\(558\) 1.23607 0.0523269
\(559\) −42.1803 −1.78404
\(560\) 0 0
\(561\) 0.763932 0.0322532
\(562\) 4.09017 0.172533
\(563\) −3.34752 −0.141081 −0.0705407 0.997509i \(-0.522472\pi\)
−0.0705407 + 0.997509i \(0.522472\pi\)
\(564\) −2.61803 −0.110239
\(565\) −8.47214 −0.356425
\(566\) 29.5066 1.24025
\(567\) 0 0
\(568\) −14.4721 −0.607237
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −10.8541 −0.454628
\(571\) 26.4721 1.10782 0.553912 0.832575i \(-0.313134\pi\)
0.553912 + 0.832575i \(0.313134\pi\)
\(572\) 3.38197 0.141407
\(573\) 15.2361 0.636496
\(574\) 0 0
\(575\) 30.8328 1.28582
\(576\) 4.23607 0.176503
\(577\) −38.6525 −1.60912 −0.804562 0.593869i \(-0.797600\pi\)
−0.804562 + 0.593869i \(0.797600\pi\)
\(578\) −26.5623 −1.10485
\(579\) 16.6525 0.692053
\(580\) −3.09017 −0.128312
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) 10.1803 0.421627
\(584\) 30.1246 1.24657
\(585\) −5.47214 −0.226245
\(586\) 25.8885 1.06945
\(587\) 30.0132 1.23878 0.619388 0.785085i \(-0.287381\pi\)
0.619388 + 0.785085i \(0.287381\pi\)
\(588\) 0 0
\(589\) −5.12461 −0.211156
\(590\) 18.0902 0.744761
\(591\) 7.52786 0.309655
\(592\) 33.9787 1.39652
\(593\) −30.8328 −1.26615 −0.633076 0.774090i \(-0.718208\pi\)
−0.633076 + 0.774090i \(0.718208\pi\)
\(594\) −1.61803 −0.0663887
\(595\) 0 0
\(596\) 3.09017 0.126578
\(597\) −26.1803 −1.07149
\(598\) −68.2492 −2.79092
\(599\) −34.4721 −1.40849 −0.704247 0.709955i \(-0.748715\pi\)
−0.704247 + 0.709955i \(0.748715\pi\)
\(600\) −8.94427 −0.365148
\(601\) −27.0000 −1.10135 −0.550676 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) 0 0
\(603\) −14.2361 −0.579738
\(604\) 5.05573 0.205715
\(605\) −1.00000 −0.0406558
\(606\) −6.76393 −0.274766
\(607\) −29.1803 −1.18439 −0.592197 0.805793i \(-0.701739\pi\)
−0.592197 + 0.805793i \(0.701739\pi\)
\(608\) 22.6869 0.920076
\(609\) 0 0
\(610\) 3.23607 0.131025
\(611\) 23.1803 0.937776
\(612\) −0.472136 −0.0190850
\(613\) −35.5967 −1.43774 −0.718870 0.695145i \(-0.755340\pi\)
−0.718870 + 0.695145i \(0.755340\pi\)
\(614\) 4.94427 0.199535
\(615\) −6.47214 −0.260982
\(616\) 0 0
\(617\) 23.5279 0.947196 0.473598 0.880741i \(-0.342955\pi\)
0.473598 + 0.880741i \(0.342955\pi\)
\(618\) −15.2361 −0.612885
\(619\) −14.0689 −0.565476 −0.282738 0.959197i \(-0.591243\pi\)
−0.282738 + 0.959197i \(0.591243\pi\)
\(620\) −0.472136 −0.0189614
\(621\) 7.70820 0.309320
\(622\) 41.8885 1.67958
\(623\) 0 0
\(624\) 26.5623 1.06334
\(625\) 11.0000 0.440000
\(626\) 10.7639 0.430213
\(627\) 6.70820 0.267900
\(628\) −7.05573 −0.281554
\(629\) 5.34752 0.213220
\(630\) 0 0
\(631\) −0.360680 −0.0143584 −0.00717922 0.999974i \(-0.502285\pi\)
−0.00717922 + 0.999974i \(0.502285\pi\)
\(632\) 12.3607 0.491681
\(633\) 21.4164 0.851226
\(634\) 2.94427 0.116932
\(635\) −18.6525 −0.740201
\(636\) −6.29180 −0.249486
\(637\) 0 0
\(638\) 8.09017 0.320293
\(639\) 6.47214 0.256034
\(640\) −13.6180 −0.538300
\(641\) 20.5410 0.811321 0.405661 0.914024i \(-0.367041\pi\)
0.405661 + 0.914024i \(0.367041\pi\)
\(642\) −0.381966 −0.0150750
\(643\) −45.9574 −1.81238 −0.906192 0.422866i \(-0.861024\pi\)
−0.906192 + 0.422866i \(0.861024\pi\)
\(644\) 0 0
\(645\) −7.70820 −0.303510
\(646\) 8.29180 0.326236
\(647\) −43.6525 −1.71616 −0.858078 0.513519i \(-0.828341\pi\)
−0.858078 + 0.513519i \(0.828341\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −11.1803 −0.438867
\(650\) −35.4164 −1.38915
\(651\) 0 0
\(652\) −5.74265 −0.224899
\(653\) −27.0557 −1.05877 −0.529386 0.848381i \(-0.677578\pi\)
−0.529386 + 0.848381i \(0.677578\pi\)
\(654\) −11.7082 −0.457827
\(655\) −16.9443 −0.662067
\(656\) 31.4164 1.22660
\(657\) −13.4721 −0.525598
\(658\) 0 0
\(659\) −43.5410 −1.69612 −0.848059 0.529902i \(-0.822229\pi\)
−0.848059 + 0.529902i \(0.822229\pi\)
\(660\) 0.618034 0.0240569
\(661\) 26.5410 1.03233 0.516163 0.856490i \(-0.327360\pi\)
0.516163 + 0.856490i \(0.327360\pi\)
\(662\) 24.9443 0.969487
\(663\) 4.18034 0.162351
\(664\) 25.1246 0.975024
\(665\) 0 0
\(666\) −11.3262 −0.438883
\(667\) −38.5410 −1.49231
\(668\) −5.34752 −0.206902
\(669\) −6.00000 −0.231973
\(670\) 23.0344 0.889898
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −45.5967 −1.75763 −0.878813 0.477167i \(-0.841664\pi\)
−0.878813 + 0.477167i \(0.841664\pi\)
\(674\) 22.9443 0.883780
\(675\) 4.00000 0.153960
\(676\) 10.4721 0.402774
\(677\) 43.3050 1.66434 0.832172 0.554517i \(-0.187097\pi\)
0.832172 + 0.554517i \(0.187097\pi\)
\(678\) −13.7082 −0.526460
\(679\) 0 0
\(680\) −1.70820 −0.0655066
\(681\) −2.00000 −0.0766402
\(682\) 1.23607 0.0473315
\(683\) −39.0132 −1.49280 −0.746398 0.665499i \(-0.768219\pi\)
−0.746398 + 0.665499i \(0.768219\pi\)
\(684\) −4.14590 −0.158522
\(685\) −6.29180 −0.240397
\(686\) 0 0
\(687\) 2.76393 0.105451
\(688\) 37.4164 1.42649
\(689\) 55.7082 2.12231
\(690\) −12.4721 −0.474806
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) 6.47214 0.246034
\(693\) 0 0
\(694\) 45.3050 1.71975
\(695\) 5.52786 0.209684
\(696\) 11.1803 0.423790
\(697\) 4.94427 0.187278
\(698\) −45.9787 −1.74032
\(699\) −2.94427 −0.111363
\(700\) 0 0
\(701\) 39.8885 1.50657 0.753285 0.657695i \(-0.228468\pi\)
0.753285 + 0.657695i \(0.228468\pi\)
\(702\) −8.85410 −0.334177
\(703\) 46.9574 1.77103
\(704\) 4.23607 0.159653
\(705\) 4.23607 0.159540
\(706\) −54.1591 −2.03830
\(707\) 0 0
\(708\) 6.90983 0.259687
\(709\) 39.7214 1.49177 0.745883 0.666076i \(-0.232028\pi\)
0.745883 + 0.666076i \(0.232028\pi\)
\(710\) −10.4721 −0.393012
\(711\) −5.52786 −0.207311
\(712\) 10.0000 0.374766
\(713\) −5.88854 −0.220528
\(714\) 0 0
\(715\) −5.47214 −0.204646
\(716\) −5.52786 −0.206586
\(717\) 30.1246 1.12502
\(718\) −5.52786 −0.206298
\(719\) −12.2361 −0.456328 −0.228164 0.973623i \(-0.573272\pi\)
−0.228164 + 0.973623i \(0.573272\pi\)
\(720\) 4.85410 0.180902
\(721\) 0 0
\(722\) 42.0689 1.56564
\(723\) 8.05573 0.299596
\(724\) 3.23607 0.120268
\(725\) −20.0000 −0.742781
\(726\) −1.61803 −0.0600509
\(727\) −1.81966 −0.0674875 −0.0337437 0.999431i \(-0.510743\pi\)
−0.0337437 + 0.999431i \(0.510743\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 21.7984 0.806794
\(731\) 5.88854 0.217796
\(732\) 1.23607 0.0456864
\(733\) 43.8885 1.62106 0.810530 0.585697i \(-0.199179\pi\)
0.810530 + 0.585697i \(0.199179\pi\)
\(734\) −25.7082 −0.948907
\(735\) 0 0
\(736\) 26.0689 0.960912
\(737\) −14.2361 −0.524392
\(738\) −10.4721 −0.385485
\(739\) −24.0689 −0.885388 −0.442694 0.896673i \(-0.645977\pi\)
−0.442694 + 0.896673i \(0.645977\pi\)
\(740\) 4.32624 0.159036
\(741\) 36.7082 1.34851
\(742\) 0 0
\(743\) 25.1803 0.923777 0.461889 0.886938i \(-0.347172\pi\)
0.461889 + 0.886938i \(0.347172\pi\)
\(744\) 1.70820 0.0626258
\(745\) −5.00000 −0.183186
\(746\) −43.1246 −1.57890
\(747\) −11.2361 −0.411106
\(748\) −0.472136 −0.0172630
\(749\) 0 0
\(750\) −14.5623 −0.531740
\(751\) 44.2361 1.61420 0.807099 0.590417i \(-0.201037\pi\)
0.807099 + 0.590417i \(0.201037\pi\)
\(752\) −20.5623 −0.749830
\(753\) −28.1246 −1.02492
\(754\) 44.2705 1.61224
\(755\) −8.18034 −0.297713
\(756\) 0 0
\(757\) 37.7214 1.37101 0.685503 0.728070i \(-0.259582\pi\)
0.685503 + 0.728070i \(0.259582\pi\)
\(758\) −14.2705 −0.518328
\(759\) 7.70820 0.279790
\(760\) −15.0000 −0.544107
\(761\) 43.7771 1.58692 0.793459 0.608624i \(-0.208278\pi\)
0.793459 + 0.608624i \(0.208278\pi\)
\(762\) −30.1803 −1.09332
\(763\) 0 0
\(764\) −9.41641 −0.340674
\(765\) 0.763932 0.0276200
\(766\) −24.3607 −0.880187
\(767\) −61.1803 −2.20909
\(768\) −13.5623 −0.489388
\(769\) 3.94427 0.142234 0.0711170 0.997468i \(-0.477344\pi\)
0.0711170 + 0.997468i \(0.477344\pi\)
\(770\) 0 0
\(771\) −7.00000 −0.252099
\(772\) −10.2918 −0.370410
\(773\) −3.47214 −0.124884 −0.0624420 0.998049i \(-0.519889\pi\)
−0.0624420 + 0.998049i \(0.519889\pi\)
\(774\) −12.4721 −0.448302
\(775\) −3.05573 −0.109765
\(776\) −8.29180 −0.297658
\(777\) 0 0
\(778\) 46.8328 1.67904
\(779\) 43.4164 1.55555
\(780\) 3.38197 0.121094
\(781\) 6.47214 0.231591
\(782\) 9.52786 0.340716
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 11.4164 0.407469
\(786\) −27.4164 −0.977911
\(787\) 27.6525 0.985704 0.492852 0.870113i \(-0.335954\pi\)
0.492852 + 0.870113i \(0.335954\pi\)
\(788\) −4.65248 −0.165738
\(789\) −14.1246 −0.502849
\(790\) 8.94427 0.318223
\(791\) 0 0
\(792\) −2.23607 −0.0794552
\(793\) −10.9443 −0.388642
\(794\) 27.7082 0.983327
\(795\) 10.1803 0.361059
\(796\) 16.1803 0.573497
\(797\) 11.4721 0.406364 0.203182 0.979141i \(-0.434872\pi\)
0.203182 + 0.979141i \(0.434872\pi\)
\(798\) 0 0
\(799\) −3.23607 −0.114484
\(800\) 13.5279 0.478282
\(801\) −4.47214 −0.158015
\(802\) −26.3607 −0.930828
\(803\) −13.4721 −0.475421
\(804\) 8.79837 0.310295
\(805\) 0 0
\(806\) 6.76393 0.238249
\(807\) 18.9443 0.666870
\(808\) −9.34752 −0.328845
\(809\) 6.30495 0.221670 0.110835 0.993839i \(-0.464647\pi\)
0.110835 + 0.993839i \(0.464647\pi\)
\(810\) −1.61803 −0.0568519
\(811\) −28.7082 −1.00808 −0.504041 0.863680i \(-0.668154\pi\)
−0.504041 + 0.863680i \(0.668154\pi\)
\(812\) 0 0
\(813\) 18.7082 0.656125
\(814\) −11.3262 −0.396984
\(815\) 9.29180 0.325477
\(816\) −3.70820 −0.129813
\(817\) 51.7082 1.80904
\(818\) −63.0132 −2.20320
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 1.47214 0.0513779 0.0256889 0.999670i \(-0.491822\pi\)
0.0256889 + 0.999670i \(0.491822\pi\)
\(822\) −10.1803 −0.355080
\(823\) 5.18034 0.180575 0.0902876 0.995916i \(-0.471221\pi\)
0.0902876 + 0.995916i \(0.471221\pi\)
\(824\) −21.0557 −0.733511
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 43.6525 1.51795 0.758973 0.651122i \(-0.225702\pi\)
0.758973 + 0.651122i \(0.225702\pi\)
\(828\) −4.76393 −0.165558
\(829\) 35.7771 1.24259 0.621295 0.783577i \(-0.286607\pi\)
0.621295 + 0.783577i \(0.286607\pi\)
\(830\) 18.1803 0.631049
\(831\) −2.47214 −0.0857574
\(832\) 23.1803 0.803634
\(833\) 0 0
\(834\) 8.94427 0.309715
\(835\) 8.65248 0.299431
\(836\) −4.14590 −0.143389
\(837\) −0.763932 −0.0264054
\(838\) −34.2705 −1.18386
\(839\) −43.5410 −1.50320 −0.751601 0.659617i \(-0.770718\pi\)
−0.751601 + 0.659617i \(0.770718\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −21.0344 −0.724895
\(843\) −2.52786 −0.0870643
\(844\) −13.2361 −0.455604
\(845\) −16.9443 −0.582901
\(846\) 6.85410 0.235649
\(847\) 0 0
\(848\) −49.4164 −1.69697
\(849\) −18.2361 −0.625860
\(850\) 4.94427 0.169587
\(851\) 53.9574 1.84964
\(852\) −4.00000 −0.137038
\(853\) 2.58359 0.0884605 0.0442303 0.999021i \(-0.485916\pi\)
0.0442303 + 0.999021i \(0.485916\pi\)
\(854\) 0 0
\(855\) 6.70820 0.229416
\(856\) −0.527864 −0.0180420
\(857\) −35.8885 −1.22593 −0.612965 0.790110i \(-0.710023\pi\)
−0.612965 + 0.790110i \(0.710023\pi\)
\(858\) −8.85410 −0.302274
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 4.76393 0.162449
\(861\) 0 0
\(862\) −7.61803 −0.259471
\(863\) −38.7639 −1.31954 −0.659770 0.751468i \(-0.729346\pi\)
−0.659770 + 0.751468i \(0.729346\pi\)
\(864\) 3.38197 0.115057
\(865\) −10.4721 −0.356063
\(866\) 2.47214 0.0840066
\(867\) 16.4164 0.557530
\(868\) 0 0
\(869\) −5.52786 −0.187520
\(870\) 8.09017 0.274282
\(871\) −77.9017 −2.63960
\(872\) −16.1803 −0.547935
\(873\) 3.70820 0.125504
\(874\) 83.6656 2.83003
\(875\) 0 0
\(876\) 8.32624 0.281318
\(877\) 31.4164 1.06086 0.530428 0.847730i \(-0.322031\pi\)
0.530428 + 0.847730i \(0.322031\pi\)
\(878\) 18.0902 0.610514
\(879\) −16.0000 −0.539667
\(880\) 4.85410 0.163632
\(881\) −19.1115 −0.643881 −0.321941 0.946760i \(-0.604335\pi\)
−0.321941 + 0.946760i \(0.604335\pi\)
\(882\) 0 0
\(883\) −37.1803 −1.25122 −0.625609 0.780137i \(-0.715149\pi\)
−0.625609 + 0.780137i \(0.715149\pi\)
\(884\) −2.58359 −0.0868956
\(885\) −11.1803 −0.375823
\(886\) 15.4164 0.517924
\(887\) −15.2361 −0.511577 −0.255789 0.966733i \(-0.582335\pi\)
−0.255789 + 0.966733i \(0.582335\pi\)
\(888\) −15.6525 −0.525263
\(889\) 0 0
\(890\) 7.23607 0.242554
\(891\) 1.00000 0.0335013
\(892\) 3.70820 0.124160
\(893\) −28.4164 −0.950919
\(894\) −8.09017 −0.270576
\(895\) 8.94427 0.298974
\(896\) 0 0
\(897\) 42.1803 1.40836
\(898\) 32.3607 1.07989
\(899\) 3.81966 0.127393
\(900\) −2.47214 −0.0824045
\(901\) −7.77709 −0.259092
\(902\) −10.4721 −0.348684
\(903\) 0 0
\(904\) −18.9443 −0.630077
\(905\) −5.23607 −0.174053
\(906\) −13.2361 −0.439739
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 1.23607 0.0410204
\(909\) 4.18034 0.138653
\(910\) 0 0
\(911\) −41.4164 −1.37219 −0.686093 0.727513i \(-0.740676\pi\)
−0.686093 + 0.727513i \(0.740676\pi\)
\(912\) −32.5623 −1.07825
\(913\) −11.2361 −0.371860
\(914\) 17.4164 0.576084
\(915\) −2.00000 −0.0661180
\(916\) −1.70820 −0.0564406
\(917\) 0 0
\(918\) 1.23607 0.0407963
\(919\) −9.59675 −0.316567 −0.158284 0.987394i \(-0.550596\pi\)
−0.158284 + 0.987394i \(0.550596\pi\)
\(920\) −17.2361 −0.568256
\(921\) −3.05573 −0.100690
\(922\) 45.3050 1.49204
\(923\) 35.4164 1.16575
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) −43.9787 −1.44523
\(927\) 9.41641 0.309275
\(928\) −16.9098 −0.555092
\(929\) 22.8885 0.750949 0.375474 0.926833i \(-0.377480\pi\)
0.375474 + 0.926833i \(0.377480\pi\)
\(930\) 1.23607 0.0405323
\(931\) 0 0
\(932\) 1.81966 0.0596049
\(933\) −25.8885 −0.847553
\(934\) −11.0344 −0.361058
\(935\) 0.763932 0.0249832
\(936\) −12.2361 −0.399948
\(937\) 4.11146 0.134315 0.0671577 0.997742i \(-0.478607\pi\)
0.0671577 + 0.997742i \(0.478607\pi\)
\(938\) 0 0
\(939\) −6.65248 −0.217095
\(940\) −2.61803 −0.0853909
\(941\) 15.6393 0.509827 0.254914 0.966964i \(-0.417953\pi\)
0.254914 + 0.966964i \(0.417953\pi\)
\(942\) 18.4721 0.601855
\(943\) 49.8885 1.62459
\(944\) 54.2705 1.76635
\(945\) 0 0
\(946\) −12.4721 −0.405504
\(947\) 21.4164 0.695940 0.347970 0.937506i \(-0.386871\pi\)
0.347970 + 0.937506i \(0.386871\pi\)
\(948\) 3.41641 0.110960
\(949\) −73.7214 −2.39310
\(950\) 43.4164 1.40861
\(951\) −1.81966 −0.0590065
\(952\) 0 0
\(953\) −26.7771 −0.867395 −0.433697 0.901059i \(-0.642791\pi\)
−0.433697 + 0.901059i \(0.642791\pi\)
\(954\) 16.4721 0.533305
\(955\) 15.2361 0.493028
\(956\) −18.6180 −0.602150
\(957\) −5.00000 −0.161627
\(958\) 2.76393 0.0892986
\(959\) 0 0
\(960\) 4.23607 0.136719
\(961\) −30.4164 −0.981174
\(962\) −61.9787 −1.99827
\(963\) 0.236068 0.00760718
\(964\) −4.97871 −0.160354
\(965\) 16.6525 0.536062
\(966\) 0 0
\(967\) 37.1935 1.19606 0.598031 0.801473i \(-0.295950\pi\)
0.598031 + 0.801473i \(0.295950\pi\)
\(968\) −2.23607 −0.0718699
\(969\) −5.12461 −0.164626
\(970\) −6.00000 −0.192648
\(971\) 8.12461 0.260731 0.130366 0.991466i \(-0.458385\pi\)
0.130366 + 0.991466i \(0.458385\pi\)
\(972\) −0.618034 −0.0198234
\(973\) 0 0
\(974\) −1.52786 −0.0489559
\(975\) 21.8885 0.700994
\(976\) 9.70820 0.310752
\(977\) −18.1803 −0.581641 −0.290820 0.956778i \(-0.593928\pi\)
−0.290820 + 0.956778i \(0.593928\pi\)
\(978\) 15.0344 0.480748
\(979\) −4.47214 −0.142930
\(980\) 0 0
\(981\) 7.23607 0.231030
\(982\) 35.7984 1.14237
\(983\) −0.583592 −0.0186137 −0.00930685 0.999957i \(-0.502963\pi\)
−0.00930685 + 0.999957i \(0.502963\pi\)
\(984\) −14.4721 −0.461355
\(985\) 7.52786 0.239858
\(986\) −6.18034 −0.196822
\(987\) 0 0
\(988\) −22.6869 −0.721767
\(989\) 59.4164 1.88933
\(990\) −1.61803 −0.0514245
\(991\) −6.81966 −0.216634 −0.108317 0.994116i \(-0.534546\pi\)
−0.108317 + 0.994116i \(0.534546\pi\)
\(992\) −2.58359 −0.0820291
\(993\) −15.4164 −0.489225
\(994\) 0 0
\(995\) −26.1803 −0.829973
\(996\) 6.94427 0.220038
\(997\) −9.05573 −0.286798 −0.143399 0.989665i \(-0.545803\pi\)
−0.143399 + 0.989665i \(0.545803\pi\)
\(998\) 3.61803 0.114527
\(999\) 7.00000 0.221470
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 1617.2.a.p.1.2 2
3.2 odd 2 4851.2.a.w.1.1 2
7.6 odd 2 231.2.a.c.1.2 2
21.20 even 2 693.2.a.f.1.1 2
28.27 even 2 3696.2.a.be.1.1 2
35.34 odd 2 5775.2.a.be.1.1 2
77.76 even 2 2541.2.a.t.1.1 2
231.230 odd 2 7623.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.a.c.1.2 2 7.6 odd 2
693.2.a.f.1.1 2 21.20 even 2
1617.2.a.p.1.2 2 1.1 even 1 trivial
2541.2.a.t.1.1 2 77.76 even 2
3696.2.a.be.1.1 2 28.27 even 2
4851.2.a.w.1.1 2 3.2 odd 2
5775.2.a.be.1.1 2 35.34 odd 2
7623.2.a.bm.1.2 2 231.230 odd 2