Properties

Label 2001.2.a.d.1.2
Level $2001$
Weight $2$
Character 2001.1
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(\Q(\zeta_{10})^+\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +2.47214 q^{11} +1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{15} -1.00000 q^{16} -4.47214 q^{17} -1.00000 q^{18} +2.47214 q^{19} +2.00000 q^{20} -2.47214 q^{22} -1.00000 q^{23} -3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +1.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} -2.47214 q^{33} +4.47214 q^{34} -1.00000 q^{36} +4.47214 q^{37} -2.47214 q^{38} +2.00000 q^{39} -6.00000 q^{40} +6.94427 q^{41} -2.47214 q^{43} -2.47214 q^{44} -2.00000 q^{45} +1.00000 q^{46} -4.94427 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +4.47214 q^{51} +2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} -4.94427 q^{55} -2.47214 q^{57} -1.00000 q^{58} -4.00000 q^{59} -2.00000 q^{60} -3.52786 q^{61} +8.00000 q^{62} +7.00000 q^{64} +4.00000 q^{65} +2.47214 q^{66} -4.00000 q^{67} +4.47214 q^{68} +1.00000 q^{69} +8.00000 q^{71} +3.00000 q^{72} +6.94427 q^{73} -4.47214 q^{74} +1.00000 q^{75} -2.47214 q^{76} -2.00000 q^{78} +1.52786 q^{79} +2.00000 q^{80} +1.00000 q^{81} -6.94427 q^{82} +4.00000 q^{83} +8.94427 q^{85} +2.47214 q^{86} -1.00000 q^{87} +7.41641 q^{88} +13.4164 q^{89} +2.00000 q^{90} +1.00000 q^{92} +8.00000 q^{93} +4.94427 q^{94} -4.94427 q^{95} +5.00000 q^{96} -9.41641 q^{97} +7.00000 q^{98} +2.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 4 q^{10} - 4 q^{11} + 2 q^{12} - 4 q^{13} + 4 q^{15} - 2 q^{16} - 2 q^{18} - 4 q^{19} + 4 q^{20} + 4 q^{22} - 2 q^{23} - 6 q^{24} - 2 q^{25} + 4 q^{26} - 2 q^{27} + 2 q^{29} - 4 q^{30} - 16 q^{31} - 10 q^{32} + 4 q^{33} - 2 q^{36} + 4 q^{38} + 4 q^{39} - 12 q^{40} - 4 q^{41} + 4 q^{43} + 4 q^{44} - 4 q^{45} + 2 q^{46} + 8 q^{47} + 2 q^{48} - 14 q^{49} + 2 q^{50} + 4 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{55} + 4 q^{57} - 2 q^{58} - 8 q^{59} - 4 q^{60} - 16 q^{61} + 16 q^{62} + 14 q^{64} + 8 q^{65} - 4 q^{66} - 8 q^{67} + 2 q^{69} + 16 q^{71} + 6 q^{72} - 4 q^{73} + 2 q^{75} + 4 q^{76} - 4 q^{78} + 12 q^{79} + 4 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 4 q^{86} - 2 q^{87} - 12 q^{88} + 4 q^{90} + 2 q^{92} + 16 q^{93} - 8 q^{94} + 8 q^{95} + 10 q^{96} + 8 q^{97} + 14 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 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
\(10\) 2.00000 0.632456
\(11\) 2.47214 0.745377 0.372689 0.927957i \(-0.378436\pi\)
0.372689 + 0.927957i \(0.378436\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) −1.00000 −0.250000
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −2.47214 −0.527061
\(23\) −1.00000 −0.208514
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) −2.00000 −0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.00000 −0.883883
\(33\) −2.47214 −0.430344
\(34\) 4.47214 0.766965
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 4.47214 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) −2.47214 −0.401033
\(39\) 2.00000 0.320256
\(40\) −6.00000 −0.948683
\(41\) 6.94427 1.08451 0.542257 0.840213i \(-0.317570\pi\)
0.542257 + 0.840213i \(0.317570\pi\)
\(42\) 0 0
\(43\) −2.47214 −0.376997 −0.188499 0.982073i \(-0.560362\pi\)
−0.188499 + 0.982073i \(0.560362\pi\)
\(44\) −2.47214 −0.372689
\(45\) −2.00000 −0.298142
\(46\) 1.00000 0.147442
\(47\) −4.94427 −0.721196 −0.360598 0.932721i \(-0.617427\pi\)
−0.360598 + 0.932721i \(0.617427\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) 4.47214 0.626224
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.94427 −0.666685
\(56\) 0 0
\(57\) −2.47214 −0.327442
\(58\) −1.00000 −0.131306
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) −2.00000 −0.258199
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 2.47214 0.304299
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 4.47214 0.542326
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000 0.353553
\(73\) 6.94427 0.812766 0.406383 0.913703i \(-0.366790\pi\)
0.406383 + 0.913703i \(0.366790\pi\)
\(74\) −4.47214 −0.519875
\(75\) 1.00000 0.115470
\(76\) −2.47214 −0.283573
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 1.52786 0.171898 0.0859491 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −6.94427 −0.766867
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 8.94427 0.970143
\(86\) 2.47214 0.266577
\(87\) −1.00000 −0.107211
\(88\) 7.41641 0.790592
\(89\) 13.4164 1.42214 0.711068 0.703123i \(-0.248212\pi\)
0.711068 + 0.703123i \(0.248212\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 4.94427 0.509963
\(95\) −4.94427 −0.507272
\(96\) 5.00000 0.510310
\(97\) −9.41641 −0.956091 −0.478046 0.878335i \(-0.658655\pi\)
−0.478046 + 0.878335i \(0.658655\pi\)
\(98\) 7.00000 0.707107
\(99\) 2.47214 0.248459
\(100\) 1.00000 0.100000
\(101\) 18.9443 1.88503 0.942513 0.334170i \(-0.108456\pi\)
0.942513 + 0.334170i \(0.108456\pi\)
\(102\) −4.47214 −0.442807
\(103\) −12.9443 −1.27544 −0.637719 0.770270i \(-0.720122\pi\)
−0.637719 + 0.770270i \(0.720122\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −0.944272 −0.0912862 −0.0456431 0.998958i \(-0.514534\pi\)
−0.0456431 + 0.998958i \(0.514534\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.9443 1.04827 0.524136 0.851635i \(-0.324389\pi\)
0.524136 + 0.851635i \(0.324389\pi\)
\(110\) 4.94427 0.471418
\(111\) −4.47214 −0.424476
\(112\) 0 0
\(113\) 8.47214 0.796992 0.398496 0.917170i \(-0.369532\pi\)
0.398496 + 0.917170i \(0.369532\pi\)
\(114\) 2.47214 0.231537
\(115\) 2.00000 0.186501
\(116\) −1.00000 −0.0928477
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) −4.88854 −0.444413
\(122\) 3.52786 0.319398
\(123\) −6.94427 −0.626144
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 17.8885 1.58735 0.793676 0.608341i \(-0.208165\pi\)
0.793676 + 0.608341i \(0.208165\pi\)
\(128\) 3.00000 0.265165
\(129\) 2.47214 0.217659
\(130\) −4.00000 −0.350823
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 2.47214 0.215172
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 2.00000 0.172133
\(136\) −13.4164 −1.15045
\(137\) 3.52786 0.301406 0.150703 0.988579i \(-0.451846\pi\)
0.150703 + 0.988579i \(0.451846\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −0.944272 −0.0800921 −0.0400460 0.999198i \(-0.512750\pi\)
−0.0400460 + 0.999198i \(0.512750\pi\)
\(140\) 0 0
\(141\) 4.94427 0.416383
\(142\) −8.00000 −0.671345
\(143\) −4.94427 −0.413461
\(144\) −1.00000 −0.0833333
\(145\) −2.00000 −0.166091
\(146\) −6.94427 −0.574712
\(147\) 7.00000 0.577350
\(148\) −4.47214 −0.367607
\(149\) 10.9443 0.896590 0.448295 0.893886i \(-0.352031\pi\)
0.448295 + 0.893886i \(0.352031\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 20.9443 1.70442 0.852210 0.523199i \(-0.175262\pi\)
0.852210 + 0.523199i \(0.175262\pi\)
\(152\) 7.41641 0.601550
\(153\) −4.47214 −0.361551
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) −2.00000 −0.160128
\(157\) −3.52786 −0.281554 −0.140777 0.990041i \(-0.544960\pi\)
−0.140777 + 0.990041i \(0.544960\pi\)
\(158\) −1.52786 −0.121550
\(159\) 2.00000 0.158610
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −0.944272 −0.0739611 −0.0369805 0.999316i \(-0.511774\pi\)
−0.0369805 + 0.999316i \(0.511774\pi\)
\(164\) −6.94427 −0.542257
\(165\) 4.94427 0.384911
\(166\) −4.00000 −0.310460
\(167\) 20.9443 1.62072 0.810358 0.585935i \(-0.199273\pi\)
0.810358 + 0.585935i \(0.199273\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −8.94427 −0.685994
\(171\) 2.47214 0.189049
\(172\) 2.47214 0.188499
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) −2.47214 −0.186344
\(177\) 4.00000 0.300658
\(178\) −13.4164 −1.00560
\(179\) −8.94427 −0.668526 −0.334263 0.942480i \(-0.608487\pi\)
−0.334263 + 0.942480i \(0.608487\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 3.52786 0.260787
\(184\) −3.00000 −0.221163
\(185\) −8.94427 −0.657596
\(186\) −8.00000 −0.586588
\(187\) −11.0557 −0.808475
\(188\) 4.94427 0.360598
\(189\) 0 0
\(190\) 4.94427 0.358695
\(191\) −6.47214 −0.468307 −0.234154 0.972200i \(-0.575232\pi\)
−0.234154 + 0.972200i \(0.575232\pi\)
\(192\) −7.00000 −0.505181
\(193\) 5.05573 0.363919 0.181960 0.983306i \(-0.441756\pi\)
0.181960 + 0.983306i \(0.441756\pi\)
\(194\) 9.41641 0.676059
\(195\) −4.00000 −0.286446
\(196\) 7.00000 0.500000
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −2.47214 −0.175687
\(199\) −12.9443 −0.917595 −0.458798 0.888541i \(-0.651720\pi\)
−0.458798 + 0.888541i \(0.651720\pi\)
\(200\) −3.00000 −0.212132
\(201\) 4.00000 0.282138
\(202\) −18.9443 −1.33291
\(203\) 0 0
\(204\) −4.47214 −0.313112
\(205\) −13.8885 −0.970018
\(206\) 12.9443 0.901870
\(207\) −1.00000 −0.0695048
\(208\) 2.00000 0.138675
\(209\) 6.11146 0.422738
\(210\) 0 0
\(211\) −16.9443 −1.16649 −0.583246 0.812296i \(-0.698218\pi\)
−0.583246 + 0.812296i \(0.698218\pi\)
\(212\) 2.00000 0.137361
\(213\) −8.00000 −0.548151
\(214\) 0.944272 0.0645491
\(215\) 4.94427 0.337197
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) −10.9443 −0.741240
\(219\) −6.94427 −0.469250
\(220\) 4.94427 0.333343
\(221\) 8.94427 0.601657
\(222\) 4.47214 0.300150
\(223\) 25.8885 1.73363 0.866813 0.498634i \(-0.166165\pi\)
0.866813 + 0.498634i \(0.166165\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −8.47214 −0.563558
\(227\) −21.8885 −1.45279 −0.726397 0.687276i \(-0.758807\pi\)
−0.726397 + 0.687276i \(0.758807\pi\)
\(228\) 2.47214 0.163721
\(229\) 17.4164 1.15091 0.575454 0.817834i \(-0.304825\pi\)
0.575454 + 0.817834i \(0.304825\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 2.00000 0.130744
\(235\) 9.88854 0.645057
\(236\) 4.00000 0.260378
\(237\) −1.52786 −0.0992454
\(238\) 0 0
\(239\) 12.9443 0.837295 0.418648 0.908149i \(-0.362504\pi\)
0.418648 + 0.908149i \(0.362504\pi\)
\(240\) −2.00000 −0.129099
\(241\) −26.9443 −1.73563 −0.867817 0.496885i \(-0.834477\pi\)
−0.867817 + 0.496885i \(0.834477\pi\)
\(242\) 4.88854 0.314247
\(243\) −1.00000 −0.0641500
\(244\) 3.52786 0.225848
\(245\) 14.0000 0.894427
\(246\) 6.94427 0.442751
\(247\) −4.94427 −0.314596
\(248\) −24.0000 −1.52400
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) 28.3607 1.79011 0.895055 0.445956i \(-0.147136\pi\)
0.895055 + 0.445956i \(0.147136\pi\)
\(252\) 0 0
\(253\) −2.47214 −0.155422
\(254\) −17.8885 −1.12243
\(255\) −8.94427 −0.560112
\(256\) −17.0000 −1.06250
\(257\) −23.8885 −1.49013 −0.745063 0.666994i \(-0.767581\pi\)
−0.745063 + 0.666994i \(0.767581\pi\)
\(258\) −2.47214 −0.153908
\(259\) 0 0
\(260\) −4.00000 −0.248069
\(261\) 1.00000 0.0618984
\(262\) −12.0000 −0.741362
\(263\) 1.52786 0.0942121 0.0471061 0.998890i \(-0.485000\pi\)
0.0471061 + 0.998890i \(0.485000\pi\)
\(264\) −7.41641 −0.456448
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) −13.4164 −0.821071
\(268\) 4.00000 0.244339
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) −2.00000 −0.121716
\(271\) 17.8885 1.08665 0.543326 0.839522i \(-0.317165\pi\)
0.543326 + 0.839522i \(0.317165\pi\)
\(272\) 4.47214 0.271163
\(273\) 0 0
\(274\) −3.52786 −0.213126
\(275\) −2.47214 −0.149075
\(276\) −1.00000 −0.0601929
\(277\) 15.8885 0.954650 0.477325 0.878727i \(-0.341606\pi\)
0.477325 + 0.878727i \(0.341606\pi\)
\(278\) 0.944272 0.0566337
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 30.9443 1.84598 0.922990 0.384823i \(-0.125738\pi\)
0.922990 + 0.384823i \(0.125738\pi\)
\(282\) −4.94427 −0.294427
\(283\) 16.9443 1.00723 0.503616 0.863927i \(-0.332003\pi\)
0.503616 + 0.863927i \(0.332003\pi\)
\(284\) −8.00000 −0.474713
\(285\) 4.94427 0.292873
\(286\) 4.94427 0.292361
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) 3.00000 0.176471
\(290\) 2.00000 0.117444
\(291\) 9.41641 0.552000
\(292\) −6.94427 −0.406383
\(293\) −13.4164 −0.783795 −0.391897 0.920009i \(-0.628181\pi\)
−0.391897 + 0.920009i \(0.628181\pi\)
\(294\) −7.00000 −0.408248
\(295\) 8.00000 0.465778
\(296\) 13.4164 0.779813
\(297\) −2.47214 −0.143448
\(298\) −10.9443 −0.633985
\(299\) 2.00000 0.115663
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −20.9443 −1.20521
\(303\) −18.9443 −1.08832
\(304\) −2.47214 −0.141787
\(305\) 7.05573 0.404010
\(306\) 4.47214 0.255655
\(307\) −7.05573 −0.402692 −0.201346 0.979520i \(-0.564532\pi\)
−0.201346 + 0.979520i \(0.564532\pi\)
\(308\) 0 0
\(309\) 12.9443 0.736374
\(310\) −16.0000 −0.908739
\(311\) −12.9443 −0.734002 −0.367001 0.930220i \(-0.619615\pi\)
−0.367001 + 0.930220i \(0.619615\pi\)
\(312\) 6.00000 0.339683
\(313\) 22.9443 1.29689 0.648443 0.761263i \(-0.275420\pi\)
0.648443 + 0.761263i \(0.275420\pi\)
\(314\) 3.52786 0.199089
\(315\) 0 0
\(316\) −1.52786 −0.0859491
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) −2.00000 −0.112154
\(319\) 2.47214 0.138413
\(320\) −14.0000 −0.782624
\(321\) 0.944272 0.0527041
\(322\) 0 0
\(323\) −11.0557 −0.615157
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) 0.944272 0.0522984
\(327\) −10.9443 −0.605220
\(328\) 20.8328 1.15030
\(329\) 0 0
\(330\) −4.94427 −0.272173
\(331\) 29.8885 1.64282 0.821411 0.570336i \(-0.193187\pi\)
0.821411 + 0.570336i \(0.193187\pi\)
\(332\) −4.00000 −0.219529
\(333\) 4.47214 0.245072
\(334\) −20.9443 −1.14602
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 0.472136 0.0257189 0.0128594 0.999917i \(-0.495907\pi\)
0.0128594 + 0.999917i \(0.495907\pi\)
\(338\) 9.00000 0.489535
\(339\) −8.47214 −0.460143
\(340\) −8.94427 −0.485071
\(341\) −19.7771 −1.07099
\(342\) −2.47214 −0.133678
\(343\) 0 0
\(344\) −7.41641 −0.399866
\(345\) −2.00000 −0.107676
\(346\) 2.00000 0.107521
\(347\) 5.88854 0.316114 0.158057 0.987430i \(-0.449477\pi\)
0.158057 + 0.987430i \(0.449477\pi\)
\(348\) 1.00000 0.0536056
\(349\) −27.8885 −1.49284 −0.746420 0.665475i \(-0.768229\pi\)
−0.746420 + 0.665475i \(0.768229\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −12.3607 −0.658826
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −4.00000 −0.212598
\(355\) −16.0000 −0.849192
\(356\) −13.4164 −0.711068
\(357\) 0 0
\(358\) 8.94427 0.472719
\(359\) 30.4721 1.60826 0.804129 0.594455i \(-0.202632\pi\)
0.804129 + 0.594455i \(0.202632\pi\)
\(360\) −6.00000 −0.316228
\(361\) −12.8885 −0.678344
\(362\) 10.0000 0.525588
\(363\) 4.88854 0.256582
\(364\) 0 0
\(365\) −13.8885 −0.726960
\(366\) −3.52786 −0.184404
\(367\) 1.52786 0.0797539 0.0398769 0.999205i \(-0.487303\pi\)
0.0398769 + 0.999205i \(0.487303\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.94427 0.361504
\(370\) 8.94427 0.464991
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 11.0557 0.571678
\(375\) −12.0000 −0.619677
\(376\) −14.8328 −0.764944
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) 10.4721 0.537917 0.268959 0.963152i \(-0.413320\pi\)
0.268959 + 0.963152i \(0.413320\pi\)
\(380\) 4.94427 0.253636
\(381\) −17.8885 −0.916458
\(382\) 6.47214 0.331143
\(383\) 19.0557 0.973702 0.486851 0.873485i \(-0.338145\pi\)
0.486851 + 0.873485i \(0.338145\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) −5.05573 −0.257330
\(387\) −2.47214 −0.125666
\(388\) 9.41641 0.478046
\(389\) −13.4164 −0.680239 −0.340119 0.940382i \(-0.610468\pi\)
−0.340119 + 0.940382i \(0.610468\pi\)
\(390\) 4.00000 0.202548
\(391\) 4.47214 0.226166
\(392\) −21.0000 −1.06066
\(393\) −12.0000 −0.605320
\(394\) 10.0000 0.503793
\(395\) −3.05573 −0.153750
\(396\) −2.47214 −0.124230
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 12.9443 0.648838
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.94427 0.346780 0.173390 0.984853i \(-0.444528\pi\)
0.173390 + 0.984853i \(0.444528\pi\)
\(402\) −4.00000 −0.199502
\(403\) 16.0000 0.797017
\(404\) −18.9443 −0.942513
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 11.0557 0.548012
\(408\) 13.4164 0.664211
\(409\) −9.05573 −0.447777 −0.223888 0.974615i \(-0.571875\pi\)
−0.223888 + 0.974615i \(0.571875\pi\)
\(410\) 13.8885 0.685906
\(411\) −3.52786 −0.174017
\(412\) 12.9443 0.637719
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −8.00000 −0.392705
\(416\) 10.0000 0.490290
\(417\) 0.944272 0.0462412
\(418\) −6.11146 −0.298921
\(419\) −21.8885 −1.06933 −0.534663 0.845066i \(-0.679561\pi\)
−0.534663 + 0.845066i \(0.679561\pi\)
\(420\) 0 0
\(421\) −11.5279 −0.561834 −0.280917 0.959732i \(-0.590639\pi\)
−0.280917 + 0.959732i \(0.590639\pi\)
\(422\) 16.9443 0.824834
\(423\) −4.94427 −0.240399
\(424\) −6.00000 −0.291386
\(425\) 4.47214 0.216930
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) 0.944272 0.0456431
\(429\) 4.94427 0.238712
\(430\) −4.94427 −0.238434
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −6.36068 −0.305675 −0.152837 0.988251i \(-0.548841\pi\)
−0.152837 + 0.988251i \(0.548841\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) −10.9443 −0.524136
\(437\) −2.47214 −0.118258
\(438\) 6.94427 0.331810
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −14.8328 −0.707127
\(441\) −7.00000 −0.333333
\(442\) −8.94427 −0.425436
\(443\) 26.8328 1.27487 0.637433 0.770506i \(-0.279996\pi\)
0.637433 + 0.770506i \(0.279996\pi\)
\(444\) 4.47214 0.212238
\(445\) −26.8328 −1.27200
\(446\) −25.8885 −1.22586
\(447\) −10.9443 −0.517646
\(448\) 0 0
\(449\) −26.9443 −1.27158 −0.635789 0.771863i \(-0.719325\pi\)
−0.635789 + 0.771863i \(0.719325\pi\)
\(450\) 1.00000 0.0471405
\(451\) 17.1672 0.808371
\(452\) −8.47214 −0.398496
\(453\) −20.9443 −0.984048
\(454\) 21.8885 1.02728
\(455\) 0 0
\(456\) −7.41641 −0.347305
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) −17.4164 −0.813815
\(459\) 4.47214 0.208741
\(460\) −2.00000 −0.0932505
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 12.9443 0.601571 0.300786 0.953692i \(-0.402751\pi\)
0.300786 + 0.953692i \(0.402751\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −16.0000 −0.741982
\(466\) −19.8885 −0.921319
\(467\) −34.4721 −1.59518 −0.797590 0.603200i \(-0.793892\pi\)
−0.797590 + 0.603200i \(0.793892\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) −9.88854 −0.456125
\(471\) 3.52786 0.162555
\(472\) −12.0000 −0.552345
\(473\) −6.11146 −0.281005
\(474\) 1.52786 0.0701771
\(475\) −2.47214 −0.113429
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −12.9443 −0.592057
\(479\) 42.2492 1.93042 0.965208 0.261483i \(-0.0842115\pi\)
0.965208 + 0.261483i \(0.0842115\pi\)
\(480\) −10.0000 −0.456435
\(481\) −8.94427 −0.407824
\(482\) 26.9443 1.22728
\(483\) 0 0
\(484\) 4.88854 0.222207
\(485\) 18.8328 0.855154
\(486\) 1.00000 0.0453609
\(487\) −11.0557 −0.500983 −0.250491 0.968119i \(-0.580592\pi\)
−0.250491 + 0.968119i \(0.580592\pi\)
\(488\) −10.5836 −0.479097
\(489\) 0.944272 0.0427015
\(490\) −14.0000 −0.632456
\(491\) −2.11146 −0.0952887 −0.0476443 0.998864i \(-0.515171\pi\)
−0.0476443 + 0.998864i \(0.515171\pi\)
\(492\) 6.94427 0.313072
\(493\) −4.47214 −0.201415
\(494\) 4.94427 0.222453
\(495\) −4.94427 −0.222228
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 32.9443 1.47479 0.737394 0.675463i \(-0.236056\pi\)
0.737394 + 0.675463i \(0.236056\pi\)
\(500\) −12.0000 −0.536656
\(501\) −20.9443 −0.935721
\(502\) −28.3607 −1.26580
\(503\) 17.5279 0.781529 0.390764 0.920491i \(-0.372211\pi\)
0.390764 + 0.920491i \(0.372211\pi\)
\(504\) 0 0
\(505\) −37.8885 −1.68602
\(506\) 2.47214 0.109900
\(507\) 9.00000 0.399704
\(508\) −17.8885 −0.793676
\(509\) 33.7771 1.49714 0.748572 0.663054i \(-0.230740\pi\)
0.748572 + 0.663054i \(0.230740\pi\)
\(510\) 8.94427 0.396059
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) −2.47214 −0.109147
\(514\) 23.8885 1.05368
\(515\) 25.8885 1.14079
\(516\) −2.47214 −0.108830
\(517\) −12.2229 −0.537563
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 12.0000 0.526235
\(521\) −10.9443 −0.479477 −0.239739 0.970837i \(-0.577062\pi\)
−0.239739 + 0.970837i \(0.577062\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 13.8885 0.607304 0.303652 0.952783i \(-0.401794\pi\)
0.303652 + 0.952783i \(0.401794\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −1.52786 −0.0666180
\(527\) 35.7771 1.55847
\(528\) 2.47214 0.107586
\(529\) 1.00000 0.0434783
\(530\) −4.00000 −0.173749
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −13.8885 −0.601580
\(534\) 13.4164 0.580585
\(535\) 1.88854 0.0816489
\(536\) −12.0000 −0.518321
\(537\) 8.94427 0.385974
\(538\) −14.0000 −0.603583
\(539\) −17.3050 −0.745377
\(540\) −2.00000 −0.0860663
\(541\) −8.83282 −0.379752 −0.189876 0.981808i \(-0.560809\pi\)
−0.189876 + 0.981808i \(0.560809\pi\)
\(542\) −17.8885 −0.768379
\(543\) 10.0000 0.429141
\(544\) 22.3607 0.958706
\(545\) −21.8885 −0.937602
\(546\) 0 0
\(547\) −5.88854 −0.251776 −0.125888 0.992044i \(-0.540178\pi\)
−0.125888 + 0.992044i \(0.540178\pi\)
\(548\) −3.52786 −0.150703
\(549\) −3.52786 −0.150566
\(550\) 2.47214 0.105412
\(551\) 2.47214 0.105317
\(552\) 3.00000 0.127688
\(553\) 0 0
\(554\) −15.8885 −0.675040
\(555\) 8.94427 0.379663
\(556\) 0.944272 0.0400460
\(557\) 15.8885 0.673219 0.336610 0.941644i \(-0.390720\pi\)
0.336610 + 0.941644i \(0.390720\pi\)
\(558\) 8.00000 0.338667
\(559\) 4.94427 0.209120
\(560\) 0 0
\(561\) 11.0557 0.466773
\(562\) −30.9443 −1.30531
\(563\) 7.41641 0.312564 0.156282 0.987712i \(-0.450049\pi\)
0.156282 + 0.987712i \(0.450049\pi\)
\(564\) −4.94427 −0.208191
\(565\) −16.9443 −0.712851
\(566\) −16.9443 −0.712221
\(567\) 0 0
\(568\) 24.0000 1.00702
\(569\) 32.4721 1.36130 0.680651 0.732607i \(-0.261697\pi\)
0.680651 + 0.732607i \(0.261697\pi\)
\(570\) −4.94427 −0.207093
\(571\) −24.9443 −1.04389 −0.521943 0.852981i \(-0.674793\pi\)
−0.521943 + 0.852981i \(0.674793\pi\)
\(572\) 4.94427 0.206730
\(573\) 6.47214 0.270377
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 7.00000 0.291667
\(577\) −26.9443 −1.12170 −0.560852 0.827916i \(-0.689526\pi\)
−0.560852 + 0.827916i \(0.689526\pi\)
\(578\) −3.00000 −0.124784
\(579\) −5.05573 −0.210109
\(580\) 2.00000 0.0830455
\(581\) 0 0
\(582\) −9.41641 −0.390323
\(583\) −4.94427 −0.204771
\(584\) 20.8328 0.862068
\(585\) 4.00000 0.165380
\(586\) 13.4164 0.554227
\(587\) 8.94427 0.369170 0.184585 0.982817i \(-0.440906\pi\)
0.184585 + 0.982817i \(0.440906\pi\)
\(588\) −7.00000 −0.288675
\(589\) −19.7771 −0.814901
\(590\) −8.00000 −0.329355
\(591\) 10.0000 0.411345
\(592\) −4.47214 −0.183804
\(593\) −23.8885 −0.980985 −0.490492 0.871445i \(-0.663183\pi\)
−0.490492 + 0.871445i \(0.663183\pi\)
\(594\) 2.47214 0.101433
\(595\) 0 0
\(596\) −10.9443 −0.448295
\(597\) 12.9443 0.529774
\(598\) −2.00000 −0.0817861
\(599\) −3.05573 −0.124854 −0.0624268 0.998050i \(-0.519884\pi\)
−0.0624268 + 0.998050i \(0.519884\pi\)
\(600\) 3.00000 0.122474
\(601\) −2.94427 −0.120099 −0.0600497 0.998195i \(-0.519126\pi\)
−0.0600497 + 0.998195i \(0.519126\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) −20.9443 −0.852210
\(605\) 9.77709 0.397495
\(606\) 18.9443 0.769558
\(607\) −30.8328 −1.25147 −0.625733 0.780038i \(-0.715200\pi\)
−0.625733 + 0.780038i \(0.715200\pi\)
\(608\) −12.3607 −0.501292
\(609\) 0 0
\(610\) −7.05573 −0.285678
\(611\) 9.88854 0.400048
\(612\) 4.47214 0.180775
\(613\) 44.8328 1.81078 0.905390 0.424581i \(-0.139578\pi\)
0.905390 + 0.424581i \(0.139578\pi\)
\(614\) 7.05573 0.284746
\(615\) 13.8885 0.560040
\(616\) 0 0
\(617\) 0.472136 0.0190075 0.00950374 0.999955i \(-0.496975\pi\)
0.00950374 + 0.999955i \(0.496975\pi\)
\(618\) −12.9443 −0.520695
\(619\) 0.583592 0.0234565 0.0117283 0.999931i \(-0.496267\pi\)
0.0117283 + 0.999931i \(0.496267\pi\)
\(620\) −16.0000 −0.642575
\(621\) 1.00000 0.0401286
\(622\) 12.9443 0.519018
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) −22.9443 −0.917038
\(627\) −6.11146 −0.244068
\(628\) 3.52786 0.140777
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 19.0557 0.758597 0.379298 0.925274i \(-0.376165\pi\)
0.379298 + 0.925274i \(0.376165\pi\)
\(632\) 4.58359 0.182326
\(633\) 16.9443 0.673474
\(634\) 2.00000 0.0794301
\(635\) −35.7771 −1.41977
\(636\) −2.00000 −0.0793052
\(637\) 14.0000 0.554700
\(638\) −2.47214 −0.0978728
\(639\) 8.00000 0.316475
\(640\) −6.00000 −0.237171
\(641\) −23.5279 −0.929295 −0.464647 0.885496i \(-0.653819\pi\)
−0.464647 + 0.885496i \(0.653819\pi\)
\(642\) −0.944272 −0.0372674
\(643\) −13.8885 −0.547711 −0.273855 0.961771i \(-0.588299\pi\)
−0.273855 + 0.961771i \(0.588299\pi\)
\(644\) 0 0
\(645\) −4.94427 −0.194681
\(646\) 11.0557 0.434982
\(647\) −27.7771 −1.09203 −0.546015 0.837775i \(-0.683856\pi\)
−0.546015 + 0.837775i \(0.683856\pi\)
\(648\) 3.00000 0.117851
\(649\) −9.88854 −0.388159
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) 0.944272 0.0369805
\(653\) −30.9443 −1.21094 −0.605471 0.795867i \(-0.707015\pi\)
−0.605471 + 0.795867i \(0.707015\pi\)
\(654\) 10.9443 0.427955
\(655\) −24.0000 −0.937758
\(656\) −6.94427 −0.271128
\(657\) 6.94427 0.270922
\(658\) 0 0
\(659\) −8.58359 −0.334369 −0.167185 0.985926i \(-0.553468\pi\)
−0.167185 + 0.985926i \(0.553468\pi\)
\(660\) −4.94427 −0.192456
\(661\) 34.9443 1.35917 0.679587 0.733595i \(-0.262159\pi\)
0.679587 + 0.733595i \(0.262159\pi\)
\(662\) −29.8885 −1.16165
\(663\) −8.94427 −0.347367
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −4.47214 −0.173292
\(667\) −1.00000 −0.0387202
\(668\) −20.9443 −0.810358
\(669\) −25.8885 −1.00091
\(670\) −8.00000 −0.309067
\(671\) −8.72136 −0.336684
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −0.472136 −0.0181860
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) 28.4721 1.09427 0.547137 0.837043i \(-0.315718\pi\)
0.547137 + 0.837043i \(0.315718\pi\)
\(678\) 8.47214 0.325370
\(679\) 0 0
\(680\) 26.8328 1.02899
\(681\) 21.8885 0.838771
\(682\) 19.7771 0.757304
\(683\) 15.0557 0.576091 0.288046 0.957617i \(-0.406994\pi\)
0.288046 + 0.957617i \(0.406994\pi\)
\(684\) −2.47214 −0.0945245
\(685\) −7.05573 −0.269586
\(686\) 0 0
\(687\) −17.4164 −0.664477
\(688\) 2.47214 0.0942493
\(689\) 4.00000 0.152388
\(690\) 2.00000 0.0761387
\(691\) 10.8328 0.412100 0.206050 0.978541i \(-0.433939\pi\)
0.206050 + 0.978541i \(0.433939\pi\)
\(692\) 2.00000 0.0760286
\(693\) 0 0
\(694\) −5.88854 −0.223526
\(695\) 1.88854 0.0716366
\(696\) −3.00000 −0.113715
\(697\) −31.0557 −1.17632
\(698\) 27.8885 1.05560
\(699\) −19.8885 −0.752254
\(700\) 0 0
\(701\) 25.0557 0.946342 0.473171 0.880971i \(-0.343109\pi\)
0.473171 + 0.880971i \(0.343109\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 11.0557 0.416975
\(704\) 17.3050 0.652205
\(705\) −9.88854 −0.372424
\(706\) −2.00000 −0.0752710
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −0.111456 −0.00418582 −0.00209291 0.999998i \(-0.500666\pi\)
−0.00209291 + 0.999998i \(0.500666\pi\)
\(710\) 16.0000 0.600469
\(711\) 1.52786 0.0572994
\(712\) 40.2492 1.50840
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 9.88854 0.369811
\(716\) 8.94427 0.334263
\(717\) −12.9443 −0.483413
\(718\) −30.4721 −1.13721
\(719\) −28.9443 −1.07944 −0.539720 0.841845i \(-0.681470\pi\)
−0.539720 + 0.841845i \(0.681470\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 12.8885 0.479662
\(723\) 26.9443 1.00207
\(724\) 10.0000 0.371647
\(725\) −1.00000 −0.0371391
\(726\) −4.88854 −0.181431
\(727\) 48.3607 1.79360 0.896799 0.442438i \(-0.145886\pi\)
0.896799 + 0.442438i \(0.145886\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 13.8885 0.514038
\(731\) 11.0557 0.408911
\(732\) −3.52786 −0.130394
\(733\) −42.3607 −1.56463 −0.782314 0.622885i \(-0.785961\pi\)
−0.782314 + 0.622885i \(0.785961\pi\)
\(734\) −1.52786 −0.0563945
\(735\) −14.0000 −0.516398
\(736\) 5.00000 0.184302
\(737\) −9.88854 −0.364249
\(738\) −6.94427 −0.255622
\(739\) 31.0557 1.14240 0.571202 0.820810i \(-0.306477\pi\)
0.571202 + 0.820810i \(0.306477\pi\)
\(740\) 8.94427 0.328798
\(741\) 4.94427 0.181632
\(742\) 0 0
\(743\) −34.2492 −1.25648 −0.628241 0.778019i \(-0.716225\pi\)
−0.628241 + 0.778019i \(0.716225\pi\)
\(744\) 24.0000 0.879883
\(745\) −21.8885 −0.801934
\(746\) −6.00000 −0.219676
\(747\) 4.00000 0.146352
\(748\) 11.0557 0.404237
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 33.5279 1.22345 0.611725 0.791071i \(-0.290476\pi\)
0.611725 + 0.791071i \(0.290476\pi\)
\(752\) 4.94427 0.180299
\(753\) −28.3607 −1.03352
\(754\) 2.00000 0.0728357
\(755\) −41.8885 −1.52448
\(756\) 0 0
\(757\) −24.4721 −0.889455 −0.444727 0.895666i \(-0.646699\pi\)
−0.444727 + 0.895666i \(0.646699\pi\)
\(758\) −10.4721 −0.380365
\(759\) 2.47214 0.0897329
\(760\) −14.8328 −0.538043
\(761\) 45.7771 1.65942 0.829709 0.558196i \(-0.188506\pi\)
0.829709 + 0.558196i \(0.188506\pi\)
\(762\) 17.8885 0.648034
\(763\) 0 0
\(764\) 6.47214 0.234154
\(765\) 8.94427 0.323381
\(766\) −19.0557 −0.688511
\(767\) 8.00000 0.288863
\(768\) 17.0000 0.613435
\(769\) 23.3050 0.840398 0.420199 0.907432i \(-0.361960\pi\)
0.420199 + 0.907432i \(0.361960\pi\)
\(770\) 0 0
\(771\) 23.8885 0.860325
\(772\) −5.05573 −0.181960
\(773\) −6.58359 −0.236795 −0.118398 0.992966i \(-0.537776\pi\)
−0.118398 + 0.992966i \(0.537776\pi\)
\(774\) 2.47214 0.0888591
\(775\) 8.00000 0.287368
\(776\) −28.2492 −1.01409
\(777\) 0 0
\(778\) 13.4164 0.481002
\(779\) 17.1672 0.615078
\(780\) 4.00000 0.143223
\(781\) 19.7771 0.707680
\(782\) −4.47214 −0.159923
\(783\) −1.00000 −0.0357371
\(784\) 7.00000 0.250000
\(785\) 7.05573 0.251830
\(786\) 12.0000 0.428026
\(787\) −10.1115 −0.360434 −0.180217 0.983627i \(-0.557680\pi\)
−0.180217 + 0.983627i \(0.557680\pi\)
\(788\) 10.0000 0.356235
\(789\) −1.52786 −0.0543934
\(790\) 3.05573 0.108718
\(791\) 0 0
\(792\) 7.41641 0.263531
\(793\) 7.05573 0.250556
\(794\) 2.00000 0.0709773
\(795\) −4.00000 −0.141865
\(796\) 12.9443 0.458798
\(797\) −50.3607 −1.78387 −0.891933 0.452167i \(-0.850651\pi\)
−0.891933 + 0.452167i \(0.850651\pi\)
\(798\) 0 0
\(799\) 22.1115 0.782247
\(800\) 5.00000 0.176777
\(801\) 13.4164 0.474045
\(802\) −6.94427 −0.245211
\(803\) 17.1672 0.605817
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −14.0000 −0.492823
\(808\) 56.8328 1.99937
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 2.00000 0.0702728
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) −17.8885 −0.627379
\(814\) −11.0557 −0.387503
\(815\) 1.88854 0.0661528
\(816\) −4.47214 −0.156556
\(817\) −6.11146 −0.213813
\(818\) 9.05573 0.316626
\(819\) 0 0
\(820\) 13.8885 0.485009
\(821\) 31.8885 1.11292 0.556459 0.830875i \(-0.312160\pi\)
0.556459 + 0.830875i \(0.312160\pi\)
\(822\) 3.52786 0.123048
\(823\) 44.9443 1.56666 0.783329 0.621607i \(-0.213520\pi\)
0.783329 + 0.621607i \(0.213520\pi\)
\(824\) −38.8328 −1.35281
\(825\) 2.47214 0.0860687
\(826\) 0 0
\(827\) −30.2492 −1.05187 −0.525934 0.850525i \(-0.676284\pi\)
−0.525934 + 0.850525i \(0.676284\pi\)
\(828\) 1.00000 0.0347524
\(829\) −14.9443 −0.519036 −0.259518 0.965738i \(-0.583564\pi\)
−0.259518 + 0.965738i \(0.583564\pi\)
\(830\) 8.00000 0.277684
\(831\) −15.8885 −0.551167
\(832\) −14.0000 −0.485363
\(833\) 31.3050 1.08465
\(834\) −0.944272 −0.0326975
\(835\) −41.8885 −1.44961
\(836\) −6.11146 −0.211369
\(837\) 8.00000 0.276520
\(838\) 21.8885 0.756127
\(839\) 40.3607 1.39341 0.696703 0.717360i \(-0.254650\pi\)
0.696703 + 0.717360i \(0.254650\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 11.5279 0.397276
\(843\) −30.9443 −1.06578
\(844\) 16.9443 0.583246
\(845\) 18.0000 0.619219
\(846\) 4.94427 0.169988
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) −16.9443 −0.581526
\(850\) −4.47214 −0.153393
\(851\) −4.47214 −0.153303
\(852\) 8.00000 0.274075
\(853\) −45.7771 −1.56738 −0.783689 0.621154i \(-0.786664\pi\)
−0.783689 + 0.621154i \(0.786664\pi\)
\(854\) 0 0
\(855\) −4.94427 −0.169091
\(856\) −2.83282 −0.0968236
\(857\) −41.7771 −1.42708 −0.713539 0.700615i \(-0.752909\pi\)
−0.713539 + 0.700615i \(0.752909\pi\)
\(858\) −4.94427 −0.168795
\(859\) −2.11146 −0.0720420 −0.0360210 0.999351i \(-0.511468\pi\)
−0.0360210 + 0.999351i \(0.511468\pi\)
\(860\) −4.94427 −0.168598
\(861\) 0 0
\(862\) 0 0
\(863\) 22.8328 0.777238 0.388619 0.921399i \(-0.372952\pi\)
0.388619 + 0.921399i \(0.372952\pi\)
\(864\) 5.00000 0.170103
\(865\) 4.00000 0.136004
\(866\) 6.36068 0.216145
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 3.77709 0.128129
\(870\) −2.00000 −0.0678064
\(871\) 8.00000 0.271070
\(872\) 32.8328 1.11186
\(873\) −9.41641 −0.318697
\(874\) 2.47214 0.0836212
\(875\) 0 0
\(876\) 6.94427 0.234625
\(877\) 4.11146 0.138834 0.0694170 0.997588i \(-0.477886\pi\)
0.0694170 + 0.997588i \(0.477886\pi\)
\(878\) −8.00000 −0.269987
\(879\) 13.4164 0.452524
\(880\) 4.94427 0.166671
\(881\) 11.5279 0.388384 0.194192 0.980964i \(-0.437792\pi\)
0.194192 + 0.980964i \(0.437792\pi\)
\(882\) 7.00000 0.235702
\(883\) 0.944272 0.0317773 0.0158886 0.999874i \(-0.494942\pi\)
0.0158886 + 0.999874i \(0.494942\pi\)
\(884\) −8.94427 −0.300828
\(885\) −8.00000 −0.268917
\(886\) −26.8328 −0.901466
\(887\) 9.88854 0.332025 0.166012 0.986124i \(-0.446911\pi\)
0.166012 + 0.986124i \(0.446911\pi\)
\(888\) −13.4164 −0.450225
\(889\) 0 0
\(890\) 26.8328 0.899438
\(891\) 2.47214 0.0828197
\(892\) −25.8885 −0.866813
\(893\) −12.2229 −0.409024
\(894\) 10.9443 0.366031
\(895\) 17.8885 0.597948
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 26.9443 0.899142
\(899\) −8.00000 −0.266815
\(900\) 1.00000 0.0333333
\(901\) 8.94427 0.297977
\(902\) −17.1672 −0.571605
\(903\) 0 0
\(904\) 25.4164 0.845337
\(905\) 20.0000 0.664822
\(906\) 20.9443 0.695827
\(907\) −34.4721 −1.14463 −0.572314 0.820034i \(-0.693954\pi\)
−0.572314 + 0.820034i \(0.693954\pi\)
\(908\) 21.8885 0.726397
\(909\) 18.9443 0.628342
\(910\) 0 0
\(911\) −16.3607 −0.542054 −0.271027 0.962572i \(-0.587363\pi\)
−0.271027 + 0.962572i \(0.587363\pi\)
\(912\) 2.47214 0.0818606
\(913\) 9.88854 0.327263
\(914\) 6.00000 0.198462
\(915\) −7.05573 −0.233255
\(916\) −17.4164 −0.575454
\(917\) 0 0
\(918\) −4.47214 −0.147602
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 6.00000 0.197814
\(921\) 7.05573 0.232494
\(922\) 18.0000 0.592798
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −4.47214 −0.147043
\(926\) −12.9443 −0.425375
\(927\) −12.9443 −0.425146
\(928\) −5.00000 −0.164133
\(929\) −33.7771 −1.10819 −0.554095 0.832453i \(-0.686936\pi\)
−0.554095 + 0.832453i \(0.686936\pi\)
\(930\) 16.0000 0.524661
\(931\) −17.3050 −0.567147
\(932\) −19.8885 −0.651471
\(933\) 12.9443 0.423776
\(934\) 34.4721 1.12796
\(935\) 22.1115 0.723122
\(936\) −6.00000 −0.196116
\(937\) 35.8885 1.17243 0.586214 0.810156i \(-0.300618\pi\)
0.586214 + 0.810156i \(0.300618\pi\)
\(938\) 0 0
\(939\) −22.9443 −0.748758
\(940\) −9.88854 −0.322529
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) −3.52786 −0.114944
\(943\) −6.94427 −0.226137
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 6.11146 0.198701
\(947\) −10.1115 −0.328578 −0.164289 0.986412i \(-0.552533\pi\)
−0.164289 + 0.986412i \(0.552533\pi\)
\(948\) 1.52786 0.0496227
\(949\) −13.8885 −0.450841
\(950\) 2.47214 0.0802067
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 40.8328 1.32270 0.661352 0.750075i \(-0.269983\pi\)
0.661352 + 0.750075i \(0.269983\pi\)
\(954\) 2.00000 0.0647524
\(955\) 12.9443 0.418867
\(956\) −12.9443 −0.418648
\(957\) −2.47214 −0.0799128
\(958\) −42.2492 −1.36501
\(959\) 0 0
\(960\) 14.0000 0.451848
\(961\) 33.0000 1.06452
\(962\) 8.94427 0.288375
\(963\) −0.944272 −0.0304287
\(964\) 26.9443 0.867817
\(965\) −10.1115 −0.325499
\(966\) 0 0
\(967\) 32.7214 1.05225 0.526124 0.850408i \(-0.323645\pi\)
0.526124 + 0.850408i \(0.323645\pi\)
\(968\) −14.6656 −0.471371
\(969\) 11.0557 0.355161
\(970\) −18.8328 −0.604685
\(971\) −36.3607 −1.16687 −0.583435 0.812160i \(-0.698292\pi\)
−0.583435 + 0.812160i \(0.698292\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 11.0557 0.354248
\(975\) −2.00000 −0.0640513
\(976\) 3.52786 0.112924
\(977\) 0.111456 0.00356580 0.00178290 0.999998i \(-0.499432\pi\)
0.00178290 + 0.999998i \(0.499432\pi\)
\(978\) −0.944272 −0.0301945
\(979\) 33.1672 1.06003
\(980\) −14.0000 −0.447214
\(981\) 10.9443 0.349424
\(982\) 2.11146 0.0673793
\(983\) 8.36068 0.266664 0.133332 0.991071i \(-0.457432\pi\)
0.133332 + 0.991071i \(0.457432\pi\)
\(984\) −20.8328 −0.664126
\(985\) 20.0000 0.637253
\(986\) 4.47214 0.142422
\(987\) 0 0
\(988\) 4.94427 0.157298
\(989\) 2.47214 0.0786094
\(990\) 4.94427 0.157139
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 40.0000 1.27000
\(993\) −29.8885 −0.948484
\(994\) 0 0
\(995\) 25.8885 0.820722
\(996\) 4.00000 0.126745
\(997\) 18.9443 0.599971 0.299986 0.953944i \(-0.403018\pi\)
0.299986 + 0.953944i \(0.403018\pi\)
\(998\) −32.9443 −1.04283
\(999\) −4.47214 −0.141492
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.d.1.2 2
3.2 odd 2 6003.2.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.d.1.2 2 1.1 even 1 trivial
6003.2.a.f.1.1 2 3.2 odd 2