Properties

Label 2001.2.a.f.1.1
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
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] = [2001,2,Mod(1,2001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, 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("2001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.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: \(15.9780654445\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -3.56155 q^{5} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -3.56155 q^{10} -3.56155 q^{11} +1.00000 q^{12} -3.56155 q^{13} +3.56155 q^{15} -1.00000 q^{16} +1.00000 q^{18} -5.12311 q^{19} +3.56155 q^{20} -3.56155 q^{22} +1.00000 q^{23} +3.00000 q^{24} +7.68466 q^{25} -3.56155 q^{26} -1.00000 q^{27} -1.00000 q^{29} +3.56155 q^{30} -2.43845 q^{31} +5.00000 q^{32} +3.56155 q^{33} -1.00000 q^{36} +2.43845 q^{37} -5.12311 q^{38} +3.56155 q^{39} +10.6847 q^{40} +7.56155 q^{41} -4.24621 q^{43} +3.56155 q^{44} -3.56155 q^{45} +1.00000 q^{46} +7.12311 q^{47} +1.00000 q^{48} -7.00000 q^{49} +7.68466 q^{50} +3.56155 q^{52} -4.24621 q^{53} -1.00000 q^{54} +12.6847 q^{55} +5.12311 q^{57} -1.00000 q^{58} +9.56155 q^{59} -3.56155 q^{60} -11.8078 q^{61} -2.43845 q^{62} +7.00000 q^{64} +12.6847 q^{65} +3.56155 q^{66} +6.43845 q^{67} -1.00000 q^{69} +10.4384 q^{71} -3.00000 q^{72} -5.12311 q^{73} +2.43845 q^{74} -7.68466 q^{75} +5.12311 q^{76} +3.56155 q^{78} -2.00000 q^{79} +3.56155 q^{80} +1.00000 q^{81} +7.56155 q^{82} -7.12311 q^{83} -4.24621 q^{86} +1.00000 q^{87} +10.6847 q^{88} +17.3693 q^{89} -3.56155 q^{90} -1.00000 q^{92} +2.43845 q^{93} +7.12311 q^{94} +18.2462 q^{95} -5.00000 q^{96} -13.3693 q^{97} -7.00000 q^{98} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} - 2 q^{4} - 3 q^{5} - 2 q^{6} - 6 q^{8} + 2 q^{9} - 3 q^{10} - 3 q^{11} + 2 q^{12} - 3 q^{13} + 3 q^{15} - 2 q^{16} + 2 q^{18} - 2 q^{19} + 3 q^{20} - 3 q^{22} + 2 q^{23} + 6 q^{24} + 3 q^{25} - 3 q^{26} - 2 q^{27} - 2 q^{29} + 3 q^{30} - 9 q^{31} + 10 q^{32} + 3 q^{33} - 2 q^{36} + 9 q^{37} - 2 q^{38} + 3 q^{39} + 9 q^{40} + 11 q^{41} + 8 q^{43} + 3 q^{44} - 3 q^{45} + 2 q^{46} + 6 q^{47} + 2 q^{48} - 14 q^{49} + 3 q^{50} + 3 q^{52} + 8 q^{53} - 2 q^{54} + 13 q^{55} + 2 q^{57} - 2 q^{58} + 15 q^{59} - 3 q^{60} - 3 q^{61} - 9 q^{62} + 14 q^{64} + 13 q^{65} + 3 q^{66} + 17 q^{67} - 2 q^{69} + 25 q^{71} - 6 q^{72} - 2 q^{73} + 9 q^{74} - 3 q^{75} + 2 q^{76} + 3 q^{78} - 4 q^{79} + 3 q^{80} + 2 q^{81} + 11 q^{82} - 6 q^{83} + 8 q^{86} + 2 q^{87} + 9 q^{88} + 10 q^{89} - 3 q^{90} - 2 q^{92} + 9 q^{93} + 6 q^{94} + 20 q^{95} - 10 q^{96} - 2 q^{97} - 14 q^{98} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −3.56155 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) −3.56155 −1.12626
\(11\) −3.56155 −1.07385 −0.536924 0.843630i \(-0.680414\pi\)
−0.536924 + 0.843630i \(0.680414\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.56155 −0.987797 −0.493899 0.869520i \(-0.664429\pi\)
−0.493899 + 0.869520i \(0.664429\pi\)
\(14\) 0 0
\(15\) 3.56155 0.919589
\(16\) −1.00000 −0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.12311 −1.17532 −0.587661 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 3.56155 0.796387
\(21\) 0 0
\(22\) −3.56155 −0.759326
\(23\) 1.00000 0.208514
\(24\) 3.00000 0.612372
\(25\) 7.68466 1.53693
\(26\) −3.56155 −0.698478
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 3.56155 0.650248
\(31\) −2.43845 −0.437958 −0.218979 0.975730i \(-0.570273\pi\)
−0.218979 + 0.975730i \(0.570273\pi\)
\(32\) 5.00000 0.883883
\(33\) 3.56155 0.619987
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 2.43845 0.400878 0.200439 0.979706i \(-0.435763\pi\)
0.200439 + 0.979706i \(0.435763\pi\)
\(38\) −5.12311 −0.831077
\(39\) 3.56155 0.570305
\(40\) 10.6847 1.68939
\(41\) 7.56155 1.18092 0.590458 0.807068i \(-0.298947\pi\)
0.590458 + 0.807068i \(0.298947\pi\)
\(42\) 0 0
\(43\) −4.24621 −0.647541 −0.323771 0.946136i \(-0.604951\pi\)
−0.323771 + 0.946136i \(0.604951\pi\)
\(44\) 3.56155 0.536924
\(45\) −3.56155 −0.530925
\(46\) 1.00000 0.147442
\(47\) 7.12311 1.03901 0.519506 0.854467i \(-0.326116\pi\)
0.519506 + 0.854467i \(0.326116\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 7.68466 1.08677
\(51\) 0 0
\(52\) 3.56155 0.493899
\(53\) −4.24621 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 12.6847 1.71040
\(56\) 0 0
\(57\) 5.12311 0.678572
\(58\) −1.00000 −0.131306
\(59\) 9.56155 1.24481 0.622404 0.782696i \(-0.286156\pi\)
0.622404 + 0.782696i \(0.286156\pi\)
\(60\) −3.56155 −0.459794
\(61\) −11.8078 −1.51183 −0.755915 0.654670i \(-0.772808\pi\)
−0.755915 + 0.654670i \(0.772808\pi\)
\(62\) −2.43845 −0.309683
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 12.6847 1.57334
\(66\) 3.56155 0.438397
\(67\) 6.43845 0.786582 0.393291 0.919414i \(-0.371337\pi\)
0.393291 + 0.919414i \(0.371337\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.4384 1.23882 0.619408 0.785069i \(-0.287373\pi\)
0.619408 + 0.785069i \(0.287373\pi\)
\(72\) −3.00000 −0.353553
\(73\) −5.12311 −0.599614 −0.299807 0.954000i \(-0.596922\pi\)
−0.299807 + 0.954000i \(0.596922\pi\)
\(74\) 2.43845 0.283464
\(75\) −7.68466 −0.887348
\(76\) 5.12311 0.587661
\(77\) 0 0
\(78\) 3.56155 0.403266
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 3.56155 0.398194
\(81\) 1.00000 0.111111
\(82\) 7.56155 0.835034
\(83\) −7.12311 −0.781862 −0.390931 0.920420i \(-0.627847\pi\)
−0.390931 + 0.920420i \(0.627847\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.24621 −0.457881
\(87\) 1.00000 0.107211
\(88\) 10.6847 1.13899
\(89\) 17.3693 1.84114 0.920572 0.390573i \(-0.127723\pi\)
0.920572 + 0.390573i \(0.127723\pi\)
\(90\) −3.56155 −0.375421
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 2.43845 0.252855
\(94\) 7.12311 0.734692
\(95\) 18.2462 1.87202
\(96\) −5.00000 −0.510310
\(97\) −13.3693 −1.35745 −0.678724 0.734393i \(-0.737467\pi\)
−0.678724 + 0.734393i \(0.737467\pi\)
\(98\) −7.00000 −0.707107
\(99\) −3.56155 −0.357950
\(100\) −7.68466 −0.768466
\(101\) −7.56155 −0.752403 −0.376201 0.926538i \(-0.622770\pi\)
−0.376201 + 0.926538i \(0.622770\pi\)
\(102\) 0 0
\(103\) 1.56155 0.153864 0.0769322 0.997036i \(-0.475488\pi\)
0.0769322 + 0.997036i \(0.475488\pi\)
\(104\) 10.6847 1.04772
\(105\) 0 0
\(106\) −4.24621 −0.412428
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 12.6847 1.20943
\(111\) −2.43845 −0.231447
\(112\) 0 0
\(113\) 13.3693 1.25768 0.628840 0.777535i \(-0.283530\pi\)
0.628840 + 0.777535i \(0.283530\pi\)
\(114\) 5.12311 0.479823
\(115\) −3.56155 −0.332117
\(116\) 1.00000 0.0928477
\(117\) −3.56155 −0.329266
\(118\) 9.56155 0.880212
\(119\) 0 0
\(120\) −10.6847 −0.975371
\(121\) 1.68466 0.153151
\(122\) −11.8078 −1.06902
\(123\) −7.56155 −0.681802
\(124\) 2.43845 0.218979
\(125\) −9.56155 −0.855211
\(126\) 0 0
\(127\) 11.8078 1.04777 0.523885 0.851789i \(-0.324482\pi\)
0.523885 + 0.851789i \(0.324482\pi\)
\(128\) −3.00000 −0.265165
\(129\) 4.24621 0.373858
\(130\) 12.6847 1.11252
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −3.56155 −0.309993
\(133\) 0 0
\(134\) 6.43845 0.556197
\(135\) 3.56155 0.306530
\(136\) 0 0
\(137\) 1.75379 0.149836 0.0749181 0.997190i \(-0.476130\pi\)
0.0749181 + 0.997190i \(0.476130\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −19.6155 −1.66377 −0.831884 0.554950i \(-0.812737\pi\)
−0.831884 + 0.554950i \(0.812737\pi\)
\(140\) 0 0
\(141\) −7.12311 −0.599874
\(142\) 10.4384 0.875975
\(143\) 12.6847 1.06074
\(144\) −1.00000 −0.0833333
\(145\) 3.56155 0.295771
\(146\) −5.12311 −0.423991
\(147\) 7.00000 0.577350
\(148\) −2.43845 −0.200439
\(149\) −16.9309 −1.38703 −0.693515 0.720442i \(-0.743939\pi\)
−0.693515 + 0.720442i \(0.743939\pi\)
\(150\) −7.68466 −0.627450
\(151\) 6.24621 0.508309 0.254155 0.967164i \(-0.418203\pi\)
0.254155 + 0.967164i \(0.418203\pi\)
\(152\) 15.3693 1.24662
\(153\) 0 0
\(154\) 0 0
\(155\) 8.68466 0.697569
\(156\) −3.56155 −0.285152
\(157\) 19.1231 1.52619 0.763095 0.646286i \(-0.223679\pi\)
0.763095 + 0.646286i \(0.223679\pi\)
\(158\) −2.00000 −0.159111
\(159\) 4.24621 0.336746
\(160\) −17.8078 −1.40783
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 5.56155 0.435614 0.217807 0.975992i \(-0.430110\pi\)
0.217807 + 0.975992i \(0.430110\pi\)
\(164\) −7.56155 −0.590458
\(165\) −12.6847 −0.987499
\(166\) −7.12311 −0.552860
\(167\) −6.93087 −0.536327 −0.268163 0.963373i \(-0.586417\pi\)
−0.268163 + 0.963373i \(0.586417\pi\)
\(168\) 0 0
\(169\) −0.315342 −0.0242570
\(170\) 0 0
\(171\) −5.12311 −0.391774
\(172\) 4.24621 0.323771
\(173\) 10.4924 0.797724 0.398862 0.917011i \(-0.369405\pi\)
0.398862 + 0.917011i \(0.369405\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 3.56155 0.268462
\(177\) −9.56155 −0.718690
\(178\) 17.3693 1.30189
\(179\) −16.4924 −1.23270 −0.616351 0.787472i \(-0.711390\pi\)
−0.616351 + 0.787472i \(0.711390\pi\)
\(180\) 3.56155 0.265462
\(181\) 18.4924 1.37453 0.687265 0.726406i \(-0.258811\pi\)
0.687265 + 0.726406i \(0.258811\pi\)
\(182\) 0 0
\(183\) 11.8078 0.872855
\(184\) −3.00000 −0.221163
\(185\) −8.68466 −0.638509
\(186\) 2.43845 0.178796
\(187\) 0 0
\(188\) −7.12311 −0.519506
\(189\) 0 0
\(190\) 18.2462 1.32372
\(191\) −19.5616 −1.41542 −0.707712 0.706501i \(-0.750273\pi\)
−0.707712 + 0.706501i \(0.750273\pi\)
\(192\) −7.00000 −0.505181
\(193\) −6.87689 −0.495010 −0.247505 0.968887i \(-0.579611\pi\)
−0.247505 + 0.968887i \(0.579611\pi\)
\(194\) −13.3693 −0.959861
\(195\) −12.6847 −0.908367
\(196\) 7.00000 0.500000
\(197\) 12.2462 0.872506 0.436253 0.899824i \(-0.356305\pi\)
0.436253 + 0.899824i \(0.356305\pi\)
\(198\) −3.56155 −0.253109
\(199\) 3.31534 0.235018 0.117509 0.993072i \(-0.462509\pi\)
0.117509 + 0.993072i \(0.462509\pi\)
\(200\) −23.0540 −1.63016
\(201\) −6.43845 −0.454133
\(202\) −7.56155 −0.532029
\(203\) 0 0
\(204\) 0 0
\(205\) −26.9309 −1.88093
\(206\) 1.56155 0.108799
\(207\) 1.00000 0.0695048
\(208\) 3.56155 0.246949
\(209\) 18.2462 1.26212
\(210\) 0 0
\(211\) 21.1771 1.45789 0.728945 0.684572i \(-0.240011\pi\)
0.728945 + 0.684572i \(0.240011\pi\)
\(212\) 4.24621 0.291631
\(213\) −10.4384 −0.715231
\(214\) 0 0
\(215\) 15.1231 1.03139
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 5.12311 0.346187
\(220\) −12.6847 −0.855199
\(221\) 0 0
\(222\) −2.43845 −0.163658
\(223\) −28.4924 −1.90799 −0.953997 0.299817i \(-0.903075\pi\)
−0.953997 + 0.299817i \(0.903075\pi\)
\(224\) 0 0
\(225\) 7.68466 0.512311
\(226\) 13.3693 0.889314
\(227\) −29.3693 −1.94931 −0.974655 0.223713i \(-0.928182\pi\)
−0.974655 + 0.223713i \(0.928182\pi\)
\(228\) −5.12311 −0.339286
\(229\) 0.192236 0.0127033 0.00635165 0.999980i \(-0.497978\pi\)
0.00635165 + 0.999980i \(0.497978\pi\)
\(230\) −3.56155 −0.234842
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 9.12311 0.597675 0.298837 0.954304i \(-0.403401\pi\)
0.298837 + 0.954304i \(0.403401\pi\)
\(234\) −3.56155 −0.232826
\(235\) −25.3693 −1.65491
\(236\) −9.56155 −0.622404
\(237\) 2.00000 0.129914
\(238\) 0 0
\(239\) 10.4384 0.675207 0.337604 0.941288i \(-0.390384\pi\)
0.337604 + 0.941288i \(0.390384\pi\)
\(240\) −3.56155 −0.229897
\(241\) −17.6155 −1.13472 −0.567358 0.823471i \(-0.692034\pi\)
−0.567358 + 0.823471i \(0.692034\pi\)
\(242\) 1.68466 0.108294
\(243\) −1.00000 −0.0641500
\(244\) 11.8078 0.755915
\(245\) 24.9309 1.59277
\(246\) −7.56155 −0.482107
\(247\) 18.2462 1.16098
\(248\) 7.31534 0.464525
\(249\) 7.12311 0.451408
\(250\) −9.56155 −0.604726
\(251\) 10.6847 0.674410 0.337205 0.941431i \(-0.390519\pi\)
0.337205 + 0.941431i \(0.390519\pi\)
\(252\) 0 0
\(253\) −3.56155 −0.223913
\(254\) 11.8078 0.740885
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −4.24621 −0.264871 −0.132436 0.991192i \(-0.542280\pi\)
−0.132436 + 0.991192i \(0.542280\pi\)
\(258\) 4.24621 0.264358
\(259\) 0 0
\(260\) −12.6847 −0.786669
\(261\) −1.00000 −0.0618984
\(262\) −4.00000 −0.247121
\(263\) −23.3693 −1.44101 −0.720507 0.693448i \(-0.756091\pi\)
−0.720507 + 0.693448i \(0.756091\pi\)
\(264\) −10.6847 −0.657595
\(265\) 15.1231 0.929005
\(266\) 0 0
\(267\) −17.3693 −1.06298
\(268\) −6.43845 −0.393291
\(269\) −4.93087 −0.300640 −0.150320 0.988637i \(-0.548030\pi\)
−0.150320 + 0.988637i \(0.548030\pi\)
\(270\) 3.56155 0.216749
\(271\) −13.5616 −0.823806 −0.411903 0.911228i \(-0.635136\pi\)
−0.411903 + 0.911228i \(0.635136\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 1.75379 0.105950
\(275\) −27.3693 −1.65043
\(276\) 1.00000 0.0601929
\(277\) 12.9309 0.776941 0.388470 0.921461i \(-0.373004\pi\)
0.388470 + 0.921461i \(0.373004\pi\)
\(278\) −19.6155 −1.17646
\(279\) −2.43845 −0.145986
\(280\) 0 0
\(281\) −4.24621 −0.253308 −0.126654 0.991947i \(-0.540424\pi\)
−0.126654 + 0.991947i \(0.540424\pi\)
\(282\) −7.12311 −0.424175
\(283\) 0.684658 0.0406987 0.0203494 0.999793i \(-0.493522\pi\)
0.0203494 + 0.999793i \(0.493522\pi\)
\(284\) −10.4384 −0.619408
\(285\) −18.2462 −1.08081
\(286\) 12.6847 0.750060
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) −17.0000 −1.00000
\(290\) 3.56155 0.209142
\(291\) 13.3693 0.783723
\(292\) 5.12311 0.299807
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 7.00000 0.408248
\(295\) −34.0540 −1.98270
\(296\) −7.31534 −0.425196
\(297\) 3.56155 0.206662
\(298\) −16.9309 −0.980779
\(299\) −3.56155 −0.205970
\(300\) 7.68466 0.443674
\(301\) 0 0
\(302\) 6.24621 0.359429
\(303\) 7.56155 0.434400
\(304\) 5.12311 0.293830
\(305\) 42.0540 2.40800
\(306\) 0 0
\(307\) −7.31534 −0.417509 −0.208754 0.977968i \(-0.566941\pi\)
−0.208754 + 0.977968i \(0.566941\pi\)
\(308\) 0 0
\(309\) −1.56155 −0.0888336
\(310\) 8.68466 0.493255
\(311\) 16.8769 0.957001 0.478500 0.878087i \(-0.341181\pi\)
0.478500 + 0.878087i \(0.341181\pi\)
\(312\) −10.6847 −0.604900
\(313\) −3.75379 −0.212177 −0.106088 0.994357i \(-0.533833\pi\)
−0.106088 + 0.994357i \(0.533833\pi\)
\(314\) 19.1231 1.07918
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 8.93087 0.501608 0.250804 0.968038i \(-0.419305\pi\)
0.250804 + 0.968038i \(0.419305\pi\)
\(318\) 4.24621 0.238116
\(319\) 3.56155 0.199409
\(320\) −24.9309 −1.39368
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) −27.3693 −1.51818
\(326\) 5.56155 0.308026
\(327\) −2.00000 −0.110600
\(328\) −22.6847 −1.25255
\(329\) 0 0
\(330\) −12.6847 −0.698267
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 7.12311 0.390931
\(333\) 2.43845 0.133626
\(334\) −6.93087 −0.379240
\(335\) −22.9309 −1.25285
\(336\) 0 0
\(337\) −23.8078 −1.29689 −0.648446 0.761261i \(-0.724581\pi\)
−0.648446 + 0.761261i \(0.724581\pi\)
\(338\) −0.315342 −0.0171523
\(339\) −13.3693 −0.726122
\(340\) 0 0
\(341\) 8.68466 0.470301
\(342\) −5.12311 −0.277026
\(343\) 0 0
\(344\) 12.7386 0.686821
\(345\) 3.56155 0.191748
\(346\) 10.4924 0.564076
\(347\) 24.4924 1.31482 0.657411 0.753532i \(-0.271652\pi\)
0.657411 + 0.753532i \(0.271652\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 22.3002 1.19370 0.596851 0.802352i \(-0.296418\pi\)
0.596851 + 0.802352i \(0.296418\pi\)
\(350\) 0 0
\(351\) 3.56155 0.190102
\(352\) −17.8078 −0.949157
\(353\) 10.4924 0.558455 0.279228 0.960225i \(-0.409922\pi\)
0.279228 + 0.960225i \(0.409922\pi\)
\(354\) −9.56155 −0.508191
\(355\) −37.1771 −1.97315
\(356\) −17.3693 −0.920572
\(357\) 0 0
\(358\) −16.4924 −0.871652
\(359\) −5.12311 −0.270387 −0.135194 0.990819i \(-0.543166\pi\)
−0.135194 + 0.990819i \(0.543166\pi\)
\(360\) 10.6847 0.563131
\(361\) 7.24621 0.381380
\(362\) 18.4924 0.971940
\(363\) −1.68466 −0.0884216
\(364\) 0 0
\(365\) 18.2462 0.955050
\(366\) 11.8078 0.617202
\(367\) 24.7386 1.29135 0.645673 0.763614i \(-0.276577\pi\)
0.645673 + 0.763614i \(0.276577\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 7.56155 0.393639
\(370\) −8.68466 −0.451494
\(371\) 0 0
\(372\) −2.43845 −0.126428
\(373\) 2.49242 0.129053 0.0645264 0.997916i \(-0.479446\pi\)
0.0645264 + 0.997916i \(0.479446\pi\)
\(374\) 0 0
\(375\) 9.56155 0.493756
\(376\) −21.3693 −1.10204
\(377\) 3.56155 0.183429
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) −18.2462 −0.936011
\(381\) −11.8078 −0.604930
\(382\) −19.5616 −1.00086
\(383\) 8.87689 0.453588 0.226794 0.973943i \(-0.427176\pi\)
0.226794 + 0.973943i \(0.427176\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −6.87689 −0.350025
\(387\) −4.24621 −0.215847
\(388\) 13.3693 0.678724
\(389\) 38.2462 1.93916 0.969580 0.244775i \(-0.0787141\pi\)
0.969580 + 0.244775i \(0.0787141\pi\)
\(390\) −12.6847 −0.642313
\(391\) 0 0
\(392\) 21.0000 1.06066
\(393\) 4.00000 0.201773
\(394\) 12.2462 0.616955
\(395\) 7.12311 0.358402
\(396\) 3.56155 0.178975
\(397\) −15.7538 −0.790660 −0.395330 0.918539i \(-0.629370\pi\)
−0.395330 + 0.918539i \(0.629370\pi\)
\(398\) 3.31534 0.166183
\(399\) 0 0
\(400\) −7.68466 −0.384233
\(401\) −2.19224 −0.109475 −0.0547375 0.998501i \(-0.517432\pi\)
−0.0547375 + 0.998501i \(0.517432\pi\)
\(402\) −6.43845 −0.321121
\(403\) 8.68466 0.432614
\(404\) 7.56155 0.376201
\(405\) −3.56155 −0.176975
\(406\) 0 0
\(407\) −8.68466 −0.430483
\(408\) 0 0
\(409\) 20.2462 1.00111 0.500555 0.865705i \(-0.333129\pi\)
0.500555 + 0.865705i \(0.333129\pi\)
\(410\) −26.9309 −1.33002
\(411\) −1.75379 −0.0865080
\(412\) −1.56155 −0.0769322
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 25.3693 1.24533
\(416\) −17.8078 −0.873097
\(417\) 19.6155 0.960577
\(418\) 18.2462 0.892451
\(419\) 23.1231 1.12964 0.564819 0.825215i \(-0.308946\pi\)
0.564819 + 0.825215i \(0.308946\pi\)
\(420\) 0 0
\(421\) 19.8078 0.965371 0.482686 0.875794i \(-0.339661\pi\)
0.482686 + 0.875794i \(0.339661\pi\)
\(422\) 21.1771 1.03088
\(423\) 7.12311 0.346337
\(424\) 12.7386 0.618643
\(425\) 0 0
\(426\) −10.4384 −0.505744
\(427\) 0 0
\(428\) 0 0
\(429\) −12.6847 −0.612421
\(430\) 15.1231 0.729301
\(431\) 20.8769 1.00560 0.502802 0.864401i \(-0.332302\pi\)
0.502802 + 0.864401i \(0.332302\pi\)
\(432\) 1.00000 0.0481125
\(433\) −20.8769 −1.00328 −0.501640 0.865077i \(-0.667270\pi\)
−0.501640 + 0.865077i \(0.667270\pi\)
\(434\) 0 0
\(435\) −3.56155 −0.170763
\(436\) −2.00000 −0.0957826
\(437\) −5.12311 −0.245071
\(438\) 5.12311 0.244791
\(439\) 29.8617 1.42522 0.712612 0.701559i \(-0.247512\pi\)
0.712612 + 0.701559i \(0.247512\pi\)
\(440\) −38.0540 −1.81415
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 4.87689 0.231708 0.115854 0.993266i \(-0.463039\pi\)
0.115854 + 0.993266i \(0.463039\pi\)
\(444\) 2.43845 0.115724
\(445\) −61.8617 −2.93253
\(446\) −28.4924 −1.34916
\(447\) 16.9309 0.800803
\(448\) 0 0
\(449\) 17.3153 0.817161 0.408581 0.912722i \(-0.366024\pi\)
0.408581 + 0.912722i \(0.366024\pi\)
\(450\) 7.68466 0.362258
\(451\) −26.9309 −1.26813
\(452\) −13.3693 −0.628840
\(453\) −6.24621 −0.293473
\(454\) −29.3693 −1.37837
\(455\) 0 0
\(456\) −15.3693 −0.719734
\(457\) −1.12311 −0.0525367 −0.0262683 0.999655i \(-0.508362\pi\)
−0.0262683 + 0.999655i \(0.508362\pi\)
\(458\) 0.192236 0.00898260
\(459\) 0 0
\(460\) 3.56155 0.166058
\(461\) −38.3002 −1.78382 −0.891909 0.452215i \(-0.850634\pi\)
−0.891909 + 0.452215i \(0.850634\pi\)
\(462\) 0 0
\(463\) 20.4924 0.952364 0.476182 0.879347i \(-0.342020\pi\)
0.476182 + 0.879347i \(0.342020\pi\)
\(464\) 1.00000 0.0464238
\(465\) −8.68466 −0.402741
\(466\) 9.12311 0.422620
\(467\) 25.4233 1.17645 0.588225 0.808697i \(-0.299827\pi\)
0.588225 + 0.808697i \(0.299827\pi\)
\(468\) 3.56155 0.164633
\(469\) 0 0
\(470\) −25.3693 −1.17020
\(471\) −19.1231 −0.881146
\(472\) −28.6847 −1.32032
\(473\) 15.1231 0.695361
\(474\) 2.00000 0.0918630
\(475\) −39.3693 −1.80639
\(476\) 0 0
\(477\) −4.24621 −0.194421
\(478\) 10.4384 0.477443
\(479\) 21.3153 0.973923 0.486961 0.873423i \(-0.338105\pi\)
0.486961 + 0.873423i \(0.338105\pi\)
\(480\) 17.8078 0.812810
\(481\) −8.68466 −0.395986
\(482\) −17.6155 −0.802365
\(483\) 0 0
\(484\) −1.68466 −0.0765754
\(485\) 47.6155 2.16211
\(486\) −1.00000 −0.0453609
\(487\) −10.7386 −0.486614 −0.243307 0.969949i \(-0.578232\pi\)
−0.243307 + 0.969949i \(0.578232\pi\)
\(488\) 35.4233 1.60354
\(489\) −5.56155 −0.251502
\(490\) 24.9309 1.12626
\(491\) −2.63068 −0.118721 −0.0593605 0.998237i \(-0.518906\pi\)
−0.0593605 + 0.998237i \(0.518906\pi\)
\(492\) 7.56155 0.340901
\(493\) 0 0
\(494\) 18.2462 0.820936
\(495\) 12.6847 0.570133
\(496\) 2.43845 0.109490
\(497\) 0 0
\(498\) 7.12311 0.319194
\(499\) 5.36932 0.240364 0.120182 0.992752i \(-0.461652\pi\)
0.120182 + 0.992752i \(0.461652\pi\)
\(500\) 9.56155 0.427606
\(501\) 6.93087 0.309648
\(502\) 10.6847 0.476880
\(503\) 16.6307 0.741526 0.370763 0.928728i \(-0.379096\pi\)
0.370763 + 0.928728i \(0.379096\pi\)
\(504\) 0 0
\(505\) 26.9309 1.19841
\(506\) −3.56155 −0.158330
\(507\) 0.315342 0.0140048
\(508\) −11.8078 −0.523885
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 5.12311 0.226191
\(514\) −4.24621 −0.187292
\(515\) −5.56155 −0.245071
\(516\) −4.24621 −0.186929
\(517\) −25.3693 −1.11574
\(518\) 0 0
\(519\) −10.4924 −0.460566
\(520\) −38.0540 −1.66878
\(521\) −28.2462 −1.23749 −0.618744 0.785592i \(-0.712358\pi\)
−0.618744 + 0.785592i \(0.712358\pi\)
\(522\) −1.00000 −0.0437688
\(523\) −7.80776 −0.341410 −0.170705 0.985322i \(-0.554604\pi\)
−0.170705 + 0.985322i \(0.554604\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −23.3693 −1.01895
\(527\) 0 0
\(528\) −3.56155 −0.154997
\(529\) 1.00000 0.0434783
\(530\) 15.1231 0.656906
\(531\) 9.56155 0.414936
\(532\) 0 0
\(533\) −26.9309 −1.16651
\(534\) −17.3693 −0.751644
\(535\) 0 0
\(536\) −19.3153 −0.834296
\(537\) 16.4924 0.711701
\(538\) −4.93087 −0.212585
\(539\) 24.9309 1.07385
\(540\) −3.56155 −0.153265
\(541\) −4.24621 −0.182559 −0.0912794 0.995825i \(-0.529096\pi\)
−0.0912794 + 0.995825i \(0.529096\pi\)
\(542\) −13.5616 −0.582519
\(543\) −18.4924 −0.793586
\(544\) 0 0
\(545\) −7.12311 −0.305120
\(546\) 0 0
\(547\) 42.2462 1.80632 0.903159 0.429307i \(-0.141242\pi\)
0.903159 + 0.429307i \(0.141242\pi\)
\(548\) −1.75379 −0.0749181
\(549\) −11.8078 −0.503943
\(550\) −27.3693 −1.16703
\(551\) 5.12311 0.218252
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) 12.9309 0.549380
\(555\) 8.68466 0.368643
\(556\) 19.6155 0.831884
\(557\) −3.17708 −0.134617 −0.0673086 0.997732i \(-0.521441\pi\)
−0.0673086 + 0.997732i \(0.521441\pi\)
\(558\) −2.43845 −0.103228
\(559\) 15.1231 0.639639
\(560\) 0 0
\(561\) 0 0
\(562\) −4.24621 −0.179116
\(563\) 15.0691 0.635088 0.317544 0.948244i \(-0.397142\pi\)
0.317544 + 0.948244i \(0.397142\pi\)
\(564\) 7.12311 0.299937
\(565\) −47.6155 −2.00320
\(566\) 0.684658 0.0287783
\(567\) 0 0
\(568\) −31.3153 −1.31396
\(569\) −30.7386 −1.28863 −0.644315 0.764760i \(-0.722858\pi\)
−0.644315 + 0.764760i \(0.722858\pi\)
\(570\) −18.2462 −0.764250
\(571\) −13.1771 −0.551444 −0.275722 0.961237i \(-0.588917\pi\)
−0.275722 + 0.961237i \(0.588917\pi\)
\(572\) −12.6847 −0.530372
\(573\) 19.5616 0.817196
\(574\) 0 0
\(575\) 7.68466 0.320472
\(576\) 7.00000 0.291667
\(577\) −5.12311 −0.213278 −0.106639 0.994298i \(-0.534009\pi\)
−0.106639 + 0.994298i \(0.534009\pi\)
\(578\) −17.0000 −0.707107
\(579\) 6.87689 0.285794
\(580\) −3.56155 −0.147885
\(581\) 0 0
\(582\) 13.3693 0.554176
\(583\) 15.1231 0.626335
\(584\) 15.3693 0.635987
\(585\) 12.6847 0.524446
\(586\) −16.0000 −0.660954
\(587\) 35.2311 1.45414 0.727071 0.686563i \(-0.240881\pi\)
0.727071 + 0.686563i \(0.240881\pi\)
\(588\) −7.00000 −0.288675
\(589\) 12.4924 0.514741
\(590\) −34.0540 −1.40198
\(591\) −12.2462 −0.503742
\(592\) −2.43845 −0.100220
\(593\) 4.63068 0.190159 0.0950797 0.995470i \(-0.469689\pi\)
0.0950797 + 0.995470i \(0.469689\pi\)
\(594\) 3.56155 0.146132
\(595\) 0 0
\(596\) 16.9309 0.693515
\(597\) −3.31534 −0.135688
\(598\) −3.56155 −0.145643
\(599\) 20.9848 0.857418 0.428709 0.903443i \(-0.358969\pi\)
0.428709 + 0.903443i \(0.358969\pi\)
\(600\) 23.0540 0.941175
\(601\) 12.2462 0.499533 0.249767 0.968306i \(-0.419646\pi\)
0.249767 + 0.968306i \(0.419646\pi\)
\(602\) 0 0
\(603\) 6.43845 0.262194
\(604\) −6.24621 −0.254155
\(605\) −6.00000 −0.243935
\(606\) 7.56155 0.307167
\(607\) −35.2311 −1.42998 −0.714992 0.699132i \(-0.753570\pi\)
−0.714992 + 0.699132i \(0.753570\pi\)
\(608\) −25.6155 −1.03885
\(609\) 0 0
\(610\) 42.0540 1.70272
\(611\) −25.3693 −1.02633
\(612\) 0 0
\(613\) −10.4924 −0.423785 −0.211892 0.977293i \(-0.567963\pi\)
−0.211892 + 0.977293i \(0.567963\pi\)
\(614\) −7.31534 −0.295223
\(615\) 26.9309 1.08596
\(616\) 0 0
\(617\) 43.6155 1.75590 0.877948 0.478757i \(-0.158912\pi\)
0.877948 + 0.478757i \(0.158912\pi\)
\(618\) −1.56155 −0.0628149
\(619\) −8.24621 −0.331443 −0.165722 0.986173i \(-0.552995\pi\)
−0.165722 + 0.986173i \(0.552995\pi\)
\(620\) −8.68466 −0.348784
\(621\) −1.00000 −0.0401286
\(622\) 16.8769 0.676702
\(623\) 0 0
\(624\) −3.56155 −0.142576
\(625\) −4.36932 −0.174773
\(626\) −3.75379 −0.150032
\(627\) −18.2462 −0.728683
\(628\) −19.1231 −0.763095
\(629\) 0 0
\(630\) 0 0
\(631\) 30.4384 1.21174 0.605868 0.795565i \(-0.292826\pi\)
0.605868 + 0.795565i \(0.292826\pi\)
\(632\) 6.00000 0.238667
\(633\) −21.1771 −0.841713
\(634\) 8.93087 0.354690
\(635\) −42.0540 −1.66886
\(636\) −4.24621 −0.168373
\(637\) 24.9309 0.987797
\(638\) 3.56155 0.141003
\(639\) 10.4384 0.412939
\(640\) 10.6847 0.422348
\(641\) −38.7386 −1.53008 −0.765042 0.643980i \(-0.777282\pi\)
−0.765042 + 0.643980i \(0.777282\pi\)
\(642\) 0 0
\(643\) 16.4924 0.650398 0.325199 0.945646i \(-0.394569\pi\)
0.325199 + 0.945646i \(0.394569\pi\)
\(644\) 0 0
\(645\) −15.1231 −0.595472
\(646\) 0 0
\(647\) 24.6847 0.970454 0.485227 0.874388i \(-0.338737\pi\)
0.485227 + 0.874388i \(0.338737\pi\)
\(648\) −3.00000 −0.117851
\(649\) −34.0540 −1.33674
\(650\) −27.3693 −1.07351
\(651\) 0 0
\(652\) −5.56155 −0.217807
\(653\) 43.5616 1.70470 0.852348 0.522976i \(-0.175178\pi\)
0.852348 + 0.522976i \(0.175178\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 14.2462 0.556646
\(656\) −7.56155 −0.295229
\(657\) −5.12311 −0.199871
\(658\) 0 0
\(659\) 33.1231 1.29029 0.645147 0.764059i \(-0.276796\pi\)
0.645147 + 0.764059i \(0.276796\pi\)
\(660\) 12.6847 0.493750
\(661\) 17.6155 0.685165 0.342582 0.939488i \(-0.388698\pi\)
0.342582 + 0.939488i \(0.388698\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 21.3693 0.829290
\(665\) 0 0
\(666\) 2.43845 0.0944879
\(667\) −1.00000 −0.0387202
\(668\) 6.93087 0.268163
\(669\) 28.4924 1.10158
\(670\) −22.9309 −0.885897
\(671\) 42.0540 1.62348
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −23.8078 −0.917041
\(675\) −7.68466 −0.295783
\(676\) 0.315342 0.0121285
\(677\) −46.2462 −1.77739 −0.888693 0.458502i \(-0.848386\pi\)
−0.888693 + 0.458502i \(0.848386\pi\)
\(678\) −13.3693 −0.513446
\(679\) 0 0
\(680\) 0 0
\(681\) 29.3693 1.12543
\(682\) 8.68466 0.332553
\(683\) 9.56155 0.365863 0.182931 0.983126i \(-0.441441\pi\)
0.182931 + 0.983126i \(0.441441\pi\)
\(684\) 5.12311 0.195887
\(685\) −6.24621 −0.238655
\(686\) 0 0
\(687\) −0.192236 −0.00733426
\(688\) 4.24621 0.161885
\(689\) 15.1231 0.576144
\(690\) 3.56155 0.135586
\(691\) 24.1080 0.917110 0.458555 0.888666i \(-0.348367\pi\)
0.458555 + 0.888666i \(0.348367\pi\)
\(692\) −10.4924 −0.398862
\(693\) 0 0
\(694\) 24.4924 0.929720
\(695\) 69.8617 2.65001
\(696\) −3.00000 −0.113715
\(697\) 0 0
\(698\) 22.3002 0.844075
\(699\) −9.12311 −0.345068
\(700\) 0 0
\(701\) 16.0540 0.606350 0.303175 0.952935i \(-0.401953\pi\)
0.303175 + 0.952935i \(0.401953\pi\)
\(702\) 3.56155 0.134422
\(703\) −12.4924 −0.471161
\(704\) −24.9309 −0.939618
\(705\) 25.3693 0.955464
\(706\) 10.4924 0.394888
\(707\) 0 0
\(708\) 9.56155 0.359345
\(709\) 1.12311 0.0421791 0.0210896 0.999778i \(-0.493286\pi\)
0.0210896 + 0.999778i \(0.493286\pi\)
\(710\) −37.1771 −1.39523
\(711\) −2.00000 −0.0750059
\(712\) −52.1080 −1.95283
\(713\) −2.43845 −0.0913206
\(714\) 0 0
\(715\) −45.1771 −1.68953
\(716\) 16.4924 0.616351
\(717\) −10.4384 −0.389831
\(718\) −5.12311 −0.191193
\(719\) 40.3002 1.50294 0.751472 0.659765i \(-0.229344\pi\)
0.751472 + 0.659765i \(0.229344\pi\)
\(720\) 3.56155 0.132731
\(721\) 0 0
\(722\) 7.24621 0.269676
\(723\) 17.6155 0.655129
\(724\) −18.4924 −0.687265
\(725\) −7.68466 −0.285401
\(726\) −1.68466 −0.0625235
\(727\) −48.7386 −1.80762 −0.903808 0.427938i \(-0.859240\pi\)
−0.903808 + 0.427938i \(0.859240\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 18.2462 0.675323
\(731\) 0 0
\(732\) −11.8078 −0.436428
\(733\) −33.5616 −1.23962 −0.619812 0.784750i \(-0.712791\pi\)
−0.619812 + 0.784750i \(0.712791\pi\)
\(734\) 24.7386 0.913120
\(735\) −24.9309 −0.919589
\(736\) 5.00000 0.184302
\(737\) −22.9309 −0.844669
\(738\) 7.56155 0.278345
\(739\) −30.5464 −1.12367 −0.561834 0.827250i \(-0.689904\pi\)
−0.561834 + 0.827250i \(0.689904\pi\)
\(740\) 8.68466 0.319254
\(741\) −18.2462 −0.670291
\(742\) 0 0
\(743\) −22.3002 −0.818114 −0.409057 0.912509i \(-0.634142\pi\)
−0.409057 + 0.912509i \(0.634142\pi\)
\(744\) −7.31534 −0.268193
\(745\) 60.3002 2.20923
\(746\) 2.49242 0.0912541
\(747\) −7.12311 −0.260621
\(748\) 0 0
\(749\) 0 0
\(750\) 9.56155 0.349139
\(751\) 19.8617 0.724765 0.362383 0.932029i \(-0.381963\pi\)
0.362383 + 0.932029i \(0.381963\pi\)
\(752\) −7.12311 −0.259753
\(753\) −10.6847 −0.389371
\(754\) 3.56155 0.129704
\(755\) −22.2462 −0.809623
\(756\) 0 0
\(757\) 20.6847 0.751797 0.375898 0.926661i \(-0.377334\pi\)
0.375898 + 0.926661i \(0.377334\pi\)
\(758\) 30.0000 1.08965
\(759\) 3.56155 0.129276
\(760\) −54.7386 −1.98558
\(761\) −18.9848 −0.688200 −0.344100 0.938933i \(-0.611816\pi\)
−0.344100 + 0.938933i \(0.611816\pi\)
\(762\) −11.8078 −0.427750
\(763\) 0 0
\(764\) 19.5616 0.707712
\(765\) 0 0
\(766\) 8.87689 0.320735
\(767\) −34.0540 −1.22962
\(768\) 17.0000 0.613435
\(769\) −5.56155 −0.200555 −0.100277 0.994960i \(-0.531973\pi\)
−0.100277 + 0.994960i \(0.531973\pi\)
\(770\) 0 0
\(771\) 4.24621 0.152924
\(772\) 6.87689 0.247505
\(773\) −44.0000 −1.58257 −0.791285 0.611448i \(-0.790588\pi\)
−0.791285 + 0.611448i \(0.790588\pi\)
\(774\) −4.24621 −0.152627
\(775\) −18.7386 −0.673112
\(776\) 40.1080 1.43979
\(777\) 0 0
\(778\) 38.2462 1.37119
\(779\) −38.7386 −1.38796
\(780\) 12.6847 0.454184
\(781\) −37.1771 −1.33030
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 7.00000 0.250000
\(785\) −68.1080 −2.43088
\(786\) 4.00000 0.142675
\(787\) −17.9460 −0.639707 −0.319853 0.947467i \(-0.603634\pi\)
−0.319853 + 0.947467i \(0.603634\pi\)
\(788\) −12.2462 −0.436253
\(789\) 23.3693 0.831970
\(790\) 7.12311 0.253429
\(791\) 0 0
\(792\) 10.6847 0.379663
\(793\) 42.0540 1.49338
\(794\) −15.7538 −0.559081
\(795\) −15.1231 −0.536361
\(796\) −3.31534 −0.117509
\(797\) 32.1080 1.13732 0.568661 0.822572i \(-0.307462\pi\)
0.568661 + 0.822572i \(0.307462\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 38.4233 1.35847
\(801\) 17.3693 0.613715
\(802\) −2.19224 −0.0774105
\(803\) 18.2462 0.643895
\(804\) 6.43845 0.227067
\(805\) 0 0
\(806\) 8.68466 0.305904
\(807\) 4.93087 0.173575
\(808\) 22.6847 0.798043
\(809\) −2.30019 −0.0808703 −0.0404351 0.999182i \(-0.512874\pi\)
−0.0404351 + 0.999182i \(0.512874\pi\)
\(810\) −3.56155 −0.125140
\(811\) −18.2462 −0.640711 −0.320356 0.947297i \(-0.603802\pi\)
−0.320356 + 0.947297i \(0.603802\pi\)
\(812\) 0 0
\(813\) 13.5616 0.475624
\(814\) −8.68466 −0.304397
\(815\) −19.8078 −0.693836
\(816\) 0 0
\(817\) 21.7538 0.761069
\(818\) 20.2462 0.707892
\(819\) 0 0
\(820\) 26.9309 0.940467
\(821\) 3.36932 0.117590 0.0587950 0.998270i \(-0.481274\pi\)
0.0587950 + 0.998270i \(0.481274\pi\)
\(822\) −1.75379 −0.0611704
\(823\) 28.3002 0.986482 0.493241 0.869893i \(-0.335812\pi\)
0.493241 + 0.869893i \(0.335812\pi\)
\(824\) −4.68466 −0.163198
\(825\) 27.3693 0.952877
\(826\) 0 0
\(827\) −24.9309 −0.866931 −0.433466 0.901170i \(-0.642709\pi\)
−0.433466 + 0.901170i \(0.642709\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −40.3542 −1.40156 −0.700779 0.713378i \(-0.747164\pi\)
−0.700779 + 0.713378i \(0.747164\pi\)
\(830\) 25.3693 0.880582
\(831\) −12.9309 −0.448567
\(832\) −24.9309 −0.864322
\(833\) 0 0
\(834\) 19.6155 0.679230
\(835\) 24.6847 0.854248
\(836\) −18.2462 −0.631058
\(837\) 2.43845 0.0842851
\(838\) 23.1231 0.798774
\(839\) 49.8078 1.71955 0.859777 0.510669i \(-0.170602\pi\)
0.859777 + 0.510669i \(0.170602\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 19.8078 0.682621
\(843\) 4.24621 0.146247
\(844\) −21.1771 −0.728945
\(845\) 1.12311 0.0386360
\(846\) 7.12311 0.244897
\(847\) 0 0
\(848\) 4.24621 0.145815
\(849\) −0.684658 −0.0234974
\(850\) 0 0
\(851\) 2.43845 0.0835889
\(852\) 10.4384 0.357615
\(853\) −19.3693 −0.663193 −0.331596 0.943421i \(-0.607587\pi\)
−0.331596 + 0.943421i \(0.607587\pi\)
\(854\) 0 0
\(855\) 18.2462 0.624007
\(856\) 0 0
\(857\) 15.7538 0.538139 0.269070 0.963121i \(-0.413284\pi\)
0.269070 + 0.963121i \(0.413284\pi\)
\(858\) −12.6847 −0.433047
\(859\) 12.9848 0.443037 0.221519 0.975156i \(-0.428899\pi\)
0.221519 + 0.975156i \(0.428899\pi\)
\(860\) −15.1231 −0.515694
\(861\) 0 0
\(862\) 20.8769 0.711070
\(863\) −22.9309 −0.780576 −0.390288 0.920693i \(-0.627625\pi\)
−0.390288 + 0.920693i \(0.627625\pi\)
\(864\) −5.00000 −0.170103
\(865\) −37.3693 −1.27059
\(866\) −20.8769 −0.709426
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 7.12311 0.241635
\(870\) −3.56155 −0.120748
\(871\) −22.9309 −0.776983
\(872\) −6.00000 −0.203186
\(873\) −13.3693 −0.452483
\(874\) −5.12311 −0.173292
\(875\) 0 0
\(876\) −5.12311 −0.173094
\(877\) 33.2311 1.12213 0.561067 0.827771i \(-0.310391\pi\)
0.561067 + 0.827771i \(0.310391\pi\)
\(878\) 29.8617 1.00778
\(879\) 16.0000 0.539667
\(880\) −12.6847 −0.427600
\(881\) 36.1080 1.21651 0.608254 0.793743i \(-0.291870\pi\)
0.608254 + 0.793743i \(0.291870\pi\)
\(882\) −7.00000 −0.235702
\(883\) 54.7386 1.84210 0.921051 0.389442i \(-0.127332\pi\)
0.921051 + 0.389442i \(0.127332\pi\)
\(884\) 0 0
\(885\) 34.0540 1.14471
\(886\) 4.87689 0.163842
\(887\) 4.49242 0.150841 0.0754204 0.997152i \(-0.475970\pi\)
0.0754204 + 0.997152i \(0.475970\pi\)
\(888\) 7.31534 0.245487
\(889\) 0 0
\(890\) −61.8617 −2.07361
\(891\) −3.56155 −0.119317
\(892\) 28.4924 0.953997
\(893\) −36.4924 −1.22117
\(894\) 16.9309 0.566253
\(895\) 58.7386 1.96342
\(896\) 0 0
\(897\) 3.56155 0.118917
\(898\) 17.3153 0.577820
\(899\) 2.43845 0.0813268
\(900\) −7.68466 −0.256155
\(901\) 0 0
\(902\) −26.9309 −0.896700
\(903\) 0 0
\(904\) −40.1080 −1.33397
\(905\) −65.8617 −2.18932
\(906\) −6.24621 −0.207516
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 29.3693 0.974655
\(909\) −7.56155 −0.250801
\(910\) 0 0
\(911\) −29.3153 −0.971261 −0.485630 0.874164i \(-0.661410\pi\)
−0.485630 + 0.874164i \(0.661410\pi\)
\(912\) −5.12311 −0.169643
\(913\) 25.3693 0.839602
\(914\) −1.12311 −0.0371490
\(915\) −42.0540 −1.39026
\(916\) −0.192236 −0.00635165
\(917\) 0 0
\(918\) 0 0
\(919\) 11.8078 0.389502 0.194751 0.980853i \(-0.437610\pi\)
0.194751 + 0.980853i \(0.437610\pi\)
\(920\) 10.6847 0.352263
\(921\) 7.31534 0.241049
\(922\) −38.3002 −1.26135
\(923\) −37.1771 −1.22370
\(924\) 0 0
\(925\) 18.7386 0.616122
\(926\) 20.4924 0.673423
\(927\) 1.56155 0.0512881
\(928\) −5.00000 −0.164133
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) −8.68466 −0.284781
\(931\) 35.8617 1.17532
\(932\) −9.12311 −0.298837
\(933\) −16.8769 −0.552525
\(934\) 25.4233 0.831876
\(935\) 0 0
\(936\) 10.6847 0.349239
\(937\) 18.9848 0.620208 0.310104 0.950703i \(-0.399636\pi\)
0.310104 + 0.950703i \(0.399636\pi\)
\(938\) 0 0
\(939\) 3.75379 0.122500
\(940\) 25.3693 0.827456
\(941\) −44.5464 −1.45217 −0.726086 0.687604i \(-0.758662\pi\)
−0.726086 + 0.687604i \(0.758662\pi\)
\(942\) −19.1231 −0.623064
\(943\) 7.56155 0.246238
\(944\) −9.56155 −0.311202
\(945\) 0 0
\(946\) 15.1231 0.491695
\(947\) −49.4773 −1.60780 −0.803898 0.594768i \(-0.797244\pi\)
−0.803898 + 0.594768i \(0.797244\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 18.2462 0.592297
\(950\) −39.3693 −1.27731
\(951\) −8.93087 −0.289603
\(952\) 0 0
\(953\) 26.9848 0.874125 0.437062 0.899431i \(-0.356019\pi\)
0.437062 + 0.899431i \(0.356019\pi\)
\(954\) −4.24621 −0.137476
\(955\) 69.6695 2.25445
\(956\) −10.4384 −0.337604
\(957\) −3.56155 −0.115129
\(958\) 21.3153 0.688667
\(959\) 0 0
\(960\) 24.9309 0.804640
\(961\) −25.0540 −0.808193
\(962\) −8.68466 −0.280005
\(963\) 0 0
\(964\) 17.6155 0.567358
\(965\) 24.4924 0.788439
\(966\) 0 0
\(967\) 37.7538 1.21408 0.607040 0.794671i \(-0.292357\pi\)
0.607040 + 0.794671i \(0.292357\pi\)
\(968\) −5.05398 −0.162441
\(969\) 0 0
\(970\) 47.6155 1.52884
\(971\) 17.8078 0.571478 0.285739 0.958307i \(-0.407761\pi\)
0.285739 + 0.958307i \(0.407761\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −10.7386 −0.344088
\(975\) 27.3693 0.876520
\(976\) 11.8078 0.377957
\(977\) −15.1771 −0.485558 −0.242779 0.970082i \(-0.578059\pi\)
−0.242779 + 0.970082i \(0.578059\pi\)
\(978\) −5.56155 −0.177839
\(979\) −61.8617 −1.97711
\(980\) −24.9309 −0.796387
\(981\) 2.00000 0.0638551
\(982\) −2.63068 −0.0839485
\(983\) −7.56155 −0.241176 −0.120588 0.992703i \(-0.538478\pi\)
−0.120588 + 0.992703i \(0.538478\pi\)
\(984\) 22.6847 0.723160
\(985\) −43.6155 −1.38971
\(986\) 0 0
\(987\) 0 0
\(988\) −18.2462 −0.580489
\(989\) −4.24621 −0.135022
\(990\) 12.6847 0.403145
\(991\) −11.5076 −0.365550 −0.182775 0.983155i \(-0.558508\pi\)
−0.182775 + 0.983155i \(0.558508\pi\)
\(992\) −12.1922 −0.387104
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −11.8078 −0.374331
\(996\) −7.12311 −0.225704
\(997\) −24.7386 −0.783480 −0.391740 0.920076i \(-0.628127\pi\)
−0.391740 + 0.920076i \(0.628127\pi\)
\(998\) 5.36932 0.169963
\(999\) −2.43845 −0.0771491
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 2001.2.a.f.1.1 2
3.2 odd 2 6003.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.f.1.1 2 1.1 even 1 trivial
6003.2.a.d.1.2 2 3.2 odd 2