Properties

Label 4018.2.a.bb.1.1
Level $4018$
Weight $2$
Character 4018.1
Self dual yes
Analytic conductor $32.084$
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] = [4018,2,Mod(1,4018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4018, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4018 = 2 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4018.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: \(32.0838915322\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.414214 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.414214 q^{6} +1.00000 q^{8} -2.82843 q^{9} -1.00000 q^{10} +0.585786 q^{11} -0.414214 q^{12} +6.24264 q^{13} +0.414214 q^{15} +1.00000 q^{16} +2.17157 q^{17} -2.82843 q^{18} -2.82843 q^{19} -1.00000 q^{20} +0.585786 q^{22} -4.00000 q^{23} -0.414214 q^{24} -4.00000 q^{25} +6.24264 q^{26} +2.41421 q^{27} +3.82843 q^{29} +0.414214 q^{30} +8.41421 q^{31} +1.00000 q^{32} -0.242641 q^{33} +2.17157 q^{34} -2.82843 q^{36} -1.41421 q^{37} -2.82843 q^{38} -2.58579 q^{39} -1.00000 q^{40} -1.00000 q^{41} -6.07107 q^{43} +0.585786 q^{44} +2.82843 q^{45} -4.00000 q^{46} +8.58579 q^{47} -0.414214 q^{48} -4.00000 q^{50} -0.899495 q^{51} +6.24264 q^{52} +8.65685 q^{53} +2.41421 q^{54} -0.585786 q^{55} +1.17157 q^{57} +3.82843 q^{58} -4.82843 q^{59} +0.414214 q^{60} +2.65685 q^{61} +8.41421 q^{62} +1.00000 q^{64} -6.24264 q^{65} -0.242641 q^{66} +3.17157 q^{67} +2.17157 q^{68} +1.65685 q^{69} +1.24264 q^{71} -2.82843 q^{72} +4.00000 q^{73} -1.41421 q^{74} +1.65685 q^{75} -2.82843 q^{76} -2.58579 q^{78} -3.24264 q^{79} -1.00000 q^{80} +7.48528 q^{81} -1.00000 q^{82} +1.07107 q^{83} -2.17157 q^{85} -6.07107 q^{86} -1.58579 q^{87} +0.585786 q^{88} +7.82843 q^{89} +2.82843 q^{90} -4.00000 q^{92} -3.48528 q^{93} +8.58579 q^{94} +2.82843 q^{95} -0.414214 q^{96} +12.1716 q^{97} -1.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} - 2 q^{10} + 4 q^{11} + 2 q^{12} + 4 q^{13} - 2 q^{15} + 2 q^{16} + 10 q^{17} - 2 q^{20} + 4 q^{22} - 8 q^{23} + 2 q^{24} - 8 q^{25} + 4 q^{26} + 2 q^{27} + 2 q^{29} - 2 q^{30} + 14 q^{31} + 2 q^{32} + 8 q^{33} + 10 q^{34} - 8 q^{39} - 2 q^{40} - 2 q^{41} + 2 q^{43} + 4 q^{44} - 8 q^{46} + 20 q^{47} + 2 q^{48} - 8 q^{50} + 18 q^{51} + 4 q^{52} + 6 q^{53} + 2 q^{54} - 4 q^{55} + 8 q^{57} + 2 q^{58} - 4 q^{59} - 2 q^{60} - 6 q^{61} + 14 q^{62} + 2 q^{64} - 4 q^{65} + 8 q^{66} + 12 q^{67} + 10 q^{68} - 8 q^{69} - 6 q^{71} + 8 q^{73} - 8 q^{75} - 8 q^{78} + 2 q^{79} - 2 q^{80} - 2 q^{81} - 2 q^{82} - 12 q^{83} - 10 q^{85} + 2 q^{86} - 6 q^{87} + 4 q^{88} + 10 q^{89} - 8 q^{92} + 10 q^{93} + 20 q^{94} + 2 q^{96} + 30 q^{97} + 8 q^{99}+O(q^{100}) \) 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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −0.414214 −0.169102
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −2.82843 −0.942809
\(10\) −1.00000 −0.316228
\(11\) 0.585786 0.176621 0.0883106 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) −0.414214 −0.119573
\(13\) 6.24264 1.73140 0.865699 0.500566i \(-0.166875\pi\)
0.865699 + 0.500566i \(0.166875\pi\)
\(14\) 0 0
\(15\) 0.414214 0.106949
\(16\) 1.00000 0.250000
\(17\) 2.17157 0.526684 0.263342 0.964703i \(-0.415175\pi\)
0.263342 + 0.964703i \(0.415175\pi\)
\(18\) −2.82843 −0.666667
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0.585786 0.124890
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −0.414214 −0.0845510
\(25\) −4.00000 −0.800000
\(26\) 6.24264 1.22428
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) 3.82843 0.710921 0.355461 0.934691i \(-0.384324\pi\)
0.355461 + 0.934691i \(0.384324\pi\)
\(30\) 0.414214 0.0756247
\(31\) 8.41421 1.51124 0.755619 0.655012i \(-0.227336\pi\)
0.755619 + 0.655012i \(0.227336\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.242641 −0.0422383
\(34\) 2.17157 0.372422
\(35\) 0 0
\(36\) −2.82843 −0.471405
\(37\) −1.41421 −0.232495 −0.116248 0.993220i \(-0.537087\pi\)
−0.116248 + 0.993220i \(0.537087\pi\)
\(38\) −2.82843 −0.458831
\(39\) −2.58579 −0.414057
\(40\) −1.00000 −0.158114
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −6.07107 −0.925829 −0.462915 0.886403i \(-0.653196\pi\)
−0.462915 + 0.886403i \(0.653196\pi\)
\(44\) 0.585786 0.0883106
\(45\) 2.82843 0.421637
\(46\) −4.00000 −0.589768
\(47\) 8.58579 1.25237 0.626183 0.779676i \(-0.284616\pi\)
0.626183 + 0.779676i \(0.284616\pi\)
\(48\) −0.414214 −0.0597866
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) −0.899495 −0.125954
\(52\) 6.24264 0.865699
\(53\) 8.65685 1.18911 0.594555 0.804055i \(-0.297328\pi\)
0.594555 + 0.804055i \(0.297328\pi\)
\(54\) 2.41421 0.328533
\(55\) −0.585786 −0.0789874
\(56\) 0 0
\(57\) 1.17157 0.155179
\(58\) 3.82843 0.502697
\(59\) −4.82843 −0.628608 −0.314304 0.949322i \(-0.601771\pi\)
−0.314304 + 0.949322i \(0.601771\pi\)
\(60\) 0.414214 0.0534747
\(61\) 2.65685 0.340175 0.170088 0.985429i \(-0.445595\pi\)
0.170088 + 0.985429i \(0.445595\pi\)
\(62\) 8.41421 1.06861
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.24264 −0.774304
\(66\) −0.242641 −0.0298670
\(67\) 3.17157 0.387469 0.193735 0.981054i \(-0.437940\pi\)
0.193735 + 0.981054i \(0.437940\pi\)
\(68\) 2.17157 0.263342
\(69\) 1.65685 0.199462
\(70\) 0 0
\(71\) 1.24264 0.147474 0.0737372 0.997278i \(-0.476507\pi\)
0.0737372 + 0.997278i \(0.476507\pi\)
\(72\) −2.82843 −0.333333
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −1.41421 −0.164399
\(75\) 1.65685 0.191317
\(76\) −2.82843 −0.324443
\(77\) 0 0
\(78\) −2.58579 −0.292783
\(79\) −3.24264 −0.364826 −0.182413 0.983222i \(-0.558391\pi\)
−0.182413 + 0.983222i \(0.558391\pi\)
\(80\) −1.00000 −0.111803
\(81\) 7.48528 0.831698
\(82\) −1.00000 −0.110432
\(83\) 1.07107 0.117565 0.0587825 0.998271i \(-0.481278\pi\)
0.0587825 + 0.998271i \(0.481278\pi\)
\(84\) 0 0
\(85\) −2.17157 −0.235540
\(86\) −6.07107 −0.654660
\(87\) −1.58579 −0.170014
\(88\) 0.585786 0.0624450
\(89\) 7.82843 0.829812 0.414906 0.909864i \(-0.363815\pi\)
0.414906 + 0.909864i \(0.363815\pi\)
\(90\) 2.82843 0.298142
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −3.48528 −0.361407
\(94\) 8.58579 0.885556
\(95\) 2.82843 0.290191
\(96\) −0.414214 −0.0422755
\(97\) 12.1716 1.23584 0.617918 0.786243i \(-0.287976\pi\)
0.617918 + 0.786243i \(0.287976\pi\)
\(98\) 0 0
\(99\) −1.65685 −0.166520
\(100\) −4.00000 −0.400000
\(101\) 0.343146 0.0341443 0.0170721 0.999854i \(-0.494566\pi\)
0.0170721 + 0.999854i \(0.494566\pi\)
\(102\) −0.899495 −0.0890633
\(103\) 8.89949 0.876893 0.438447 0.898757i \(-0.355529\pi\)
0.438447 + 0.898757i \(0.355529\pi\)
\(104\) 6.24264 0.612141
\(105\) 0 0
\(106\) 8.65685 0.840828
\(107\) 12.8995 1.24704 0.623521 0.781807i \(-0.285702\pi\)
0.623521 + 0.781807i \(0.285702\pi\)
\(108\) 2.41421 0.232308
\(109\) 14.1421 1.35457 0.677285 0.735720i \(-0.263156\pi\)
0.677285 + 0.735720i \(0.263156\pi\)
\(110\) −0.585786 −0.0558525
\(111\) 0.585786 0.0556004
\(112\) 0 0
\(113\) −5.00000 −0.470360 −0.235180 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) 1.17157 0.109728
\(115\) 4.00000 0.373002
\(116\) 3.82843 0.355461
\(117\) −17.6569 −1.63238
\(118\) −4.82843 −0.444493
\(119\) 0 0
\(120\) 0.414214 0.0378124
\(121\) −10.6569 −0.968805
\(122\) 2.65685 0.240540
\(123\) 0.414214 0.0373484
\(124\) 8.41421 0.755619
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −5.75736 −0.510883 −0.255442 0.966825i \(-0.582221\pi\)
−0.255442 + 0.966825i \(0.582221\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.51472 0.221409
\(130\) −6.24264 −0.547516
\(131\) −0.242641 −0.0211996 −0.0105998 0.999944i \(-0.503374\pi\)
−0.0105998 + 0.999944i \(0.503374\pi\)
\(132\) −0.242641 −0.0211192
\(133\) 0 0
\(134\) 3.17157 0.273982
\(135\) −2.41421 −0.207782
\(136\) 2.17157 0.186211
\(137\) 1.65685 0.141555 0.0707773 0.997492i \(-0.477452\pi\)
0.0707773 + 0.997492i \(0.477452\pi\)
\(138\) 1.65685 0.141041
\(139\) 1.51472 0.128477 0.0642384 0.997935i \(-0.479538\pi\)
0.0642384 + 0.997935i \(0.479538\pi\)
\(140\) 0 0
\(141\) −3.55635 −0.299499
\(142\) 1.24264 0.104280
\(143\) 3.65685 0.305802
\(144\) −2.82843 −0.235702
\(145\) −3.82843 −0.317934
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −1.41421 −0.116248
\(149\) 20.6569 1.69228 0.846138 0.532964i \(-0.178922\pi\)
0.846138 + 0.532964i \(0.178922\pi\)
\(150\) 1.65685 0.135282
\(151\) 20.0711 1.63336 0.816680 0.577091i \(-0.195812\pi\)
0.816680 + 0.577091i \(0.195812\pi\)
\(152\) −2.82843 −0.229416
\(153\) −6.14214 −0.496562
\(154\) 0 0
\(155\) −8.41421 −0.675846
\(156\) −2.58579 −0.207029
\(157\) −13.6569 −1.08994 −0.544968 0.838457i \(-0.683458\pi\)
−0.544968 + 0.838457i \(0.683458\pi\)
\(158\) −3.24264 −0.257971
\(159\) −3.58579 −0.284371
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 7.48528 0.588099
\(163\) −25.3137 −1.98272 −0.991361 0.131159i \(-0.958130\pi\)
−0.991361 + 0.131159i \(0.958130\pi\)
\(164\) −1.00000 −0.0780869
\(165\) 0.242641 0.0188896
\(166\) 1.07107 0.0831310
\(167\) 24.9706 1.93228 0.966140 0.258018i \(-0.0830694\pi\)
0.966140 + 0.258018i \(0.0830694\pi\)
\(168\) 0 0
\(169\) 25.9706 1.99774
\(170\) −2.17157 −0.166552
\(171\) 8.00000 0.611775
\(172\) −6.07107 −0.462915
\(173\) −9.82843 −0.747241 −0.373621 0.927582i \(-0.621884\pi\)
−0.373621 + 0.927582i \(0.621884\pi\)
\(174\) −1.58579 −0.120218
\(175\) 0 0
\(176\) 0.585786 0.0441553
\(177\) 2.00000 0.150329
\(178\) 7.82843 0.586765
\(179\) 4.24264 0.317110 0.158555 0.987350i \(-0.449317\pi\)
0.158555 + 0.987350i \(0.449317\pi\)
\(180\) 2.82843 0.210819
\(181\) 9.89949 0.735824 0.367912 0.929861i \(-0.380073\pi\)
0.367912 + 0.929861i \(0.380073\pi\)
\(182\) 0 0
\(183\) −1.10051 −0.0813517
\(184\) −4.00000 −0.294884
\(185\) 1.41421 0.103975
\(186\) −3.48528 −0.255553
\(187\) 1.27208 0.0930236
\(188\) 8.58579 0.626183
\(189\) 0 0
\(190\) 2.82843 0.205196
\(191\) −3.24264 −0.234629 −0.117315 0.993095i \(-0.537429\pi\)
−0.117315 + 0.993095i \(0.537429\pi\)
\(192\) −0.414214 −0.0298933
\(193\) 7.89949 0.568618 0.284309 0.958733i \(-0.408236\pi\)
0.284309 + 0.958733i \(0.408236\pi\)
\(194\) 12.1716 0.873868
\(195\) 2.58579 0.185172
\(196\) 0 0
\(197\) 23.3137 1.66103 0.830516 0.556994i \(-0.188045\pi\)
0.830516 + 0.556994i \(0.188045\pi\)
\(198\) −1.65685 −0.117748
\(199\) 11.3137 0.802008 0.401004 0.916076i \(-0.368661\pi\)
0.401004 + 0.916076i \(0.368661\pi\)
\(200\) −4.00000 −0.282843
\(201\) −1.31371 −0.0926619
\(202\) 0.343146 0.0241437
\(203\) 0 0
\(204\) −0.899495 −0.0629772
\(205\) 1.00000 0.0698430
\(206\) 8.89949 0.620057
\(207\) 11.3137 0.786357
\(208\) 6.24264 0.432849
\(209\) −1.65685 −0.114607
\(210\) 0 0
\(211\) −1.65685 −0.114063 −0.0570313 0.998372i \(-0.518163\pi\)
−0.0570313 + 0.998372i \(0.518163\pi\)
\(212\) 8.65685 0.594555
\(213\) −0.514719 −0.0352679
\(214\) 12.8995 0.881791
\(215\) 6.07107 0.414043
\(216\) 2.41421 0.164266
\(217\) 0 0
\(218\) 14.1421 0.957826
\(219\) −1.65685 −0.111960
\(220\) −0.585786 −0.0394937
\(221\) 13.5563 0.911899
\(222\) 0.585786 0.0393154
\(223\) −27.8701 −1.86632 −0.933159 0.359465i \(-0.882959\pi\)
−0.933159 + 0.359465i \(0.882959\pi\)
\(224\) 0 0
\(225\) 11.3137 0.754247
\(226\) −5.00000 −0.332595
\(227\) −7.72792 −0.512920 −0.256460 0.966555i \(-0.582556\pi\)
−0.256460 + 0.966555i \(0.582556\pi\)
\(228\) 1.17157 0.0775893
\(229\) 3.41421 0.225618 0.112809 0.993617i \(-0.464015\pi\)
0.112809 + 0.993617i \(0.464015\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 3.82843 0.251349
\(233\) −24.2426 −1.58819 −0.794094 0.607795i \(-0.792054\pi\)
−0.794094 + 0.607795i \(0.792054\pi\)
\(234\) −17.6569 −1.15426
\(235\) −8.58579 −0.560075
\(236\) −4.82843 −0.314304
\(237\) 1.34315 0.0872467
\(238\) 0 0
\(239\) 3.51472 0.227348 0.113674 0.993518i \(-0.463738\pi\)
0.113674 + 0.993518i \(0.463738\pi\)
\(240\) 0.414214 0.0267374
\(241\) −21.4142 −1.37941 −0.689705 0.724090i \(-0.742260\pi\)
−0.689705 + 0.724090i \(0.742260\pi\)
\(242\) −10.6569 −0.685049
\(243\) −10.3431 −0.663513
\(244\) 2.65685 0.170088
\(245\) 0 0
\(246\) 0.414214 0.0264093
\(247\) −17.6569 −1.12348
\(248\) 8.41421 0.534303
\(249\) −0.443651 −0.0281152
\(250\) 9.00000 0.569210
\(251\) −12.9289 −0.816067 −0.408033 0.912967i \(-0.633785\pi\)
−0.408033 + 0.912967i \(0.633785\pi\)
\(252\) 0 0
\(253\) −2.34315 −0.147312
\(254\) −5.75736 −0.361249
\(255\) 0.899495 0.0563286
\(256\) 1.00000 0.0625000
\(257\) 25.4853 1.58973 0.794864 0.606788i \(-0.207542\pi\)
0.794864 + 0.606788i \(0.207542\pi\)
\(258\) 2.51472 0.156560
\(259\) 0 0
\(260\) −6.24264 −0.387152
\(261\) −10.8284 −0.670263
\(262\) −0.242641 −0.0149904
\(263\) 8.97056 0.553149 0.276574 0.960993i \(-0.410801\pi\)
0.276574 + 0.960993i \(0.410801\pi\)
\(264\) −0.242641 −0.0149335
\(265\) −8.65685 −0.531786
\(266\) 0 0
\(267\) −3.24264 −0.198446
\(268\) 3.17157 0.193735
\(269\) −2.68629 −0.163786 −0.0818930 0.996641i \(-0.526097\pi\)
−0.0818930 + 0.996641i \(0.526097\pi\)
\(270\) −2.41421 −0.146924
\(271\) 6.00000 0.364474 0.182237 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) 2.17157 0.131671
\(273\) 0 0
\(274\) 1.65685 0.100094
\(275\) −2.34315 −0.141297
\(276\) 1.65685 0.0997309
\(277\) −12.3431 −0.741628 −0.370814 0.928707i \(-0.620921\pi\)
−0.370814 + 0.928707i \(0.620921\pi\)
\(278\) 1.51472 0.0908468
\(279\) −23.7990 −1.42481
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) −3.55635 −0.211778
\(283\) −22.4853 −1.33661 −0.668306 0.743887i \(-0.732980\pi\)
−0.668306 + 0.743887i \(0.732980\pi\)
\(284\) 1.24264 0.0737372
\(285\) −1.17157 −0.0693980
\(286\) 3.65685 0.216234
\(287\) 0 0
\(288\) −2.82843 −0.166667
\(289\) −12.2843 −0.722604
\(290\) −3.82843 −0.224813
\(291\) −5.04163 −0.295546
\(292\) 4.00000 0.234082
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 4.82843 0.281122
\(296\) −1.41421 −0.0821995
\(297\) 1.41421 0.0820610
\(298\) 20.6569 1.19662
\(299\) −24.9706 −1.44408
\(300\) 1.65685 0.0956585
\(301\) 0 0
\(302\) 20.0711 1.15496
\(303\) −0.142136 −0.00816548
\(304\) −2.82843 −0.162221
\(305\) −2.65685 −0.152131
\(306\) −6.14214 −0.351123
\(307\) 24.9706 1.42515 0.712573 0.701598i \(-0.247530\pi\)
0.712573 + 0.701598i \(0.247530\pi\)
\(308\) 0 0
\(309\) −3.68629 −0.209706
\(310\) −8.41421 −0.477895
\(311\) 5.51472 0.312711 0.156356 0.987701i \(-0.450025\pi\)
0.156356 + 0.987701i \(0.450025\pi\)
\(312\) −2.58579 −0.146391
\(313\) −4.97056 −0.280953 −0.140476 0.990084i \(-0.544863\pi\)
−0.140476 + 0.990084i \(0.544863\pi\)
\(314\) −13.6569 −0.770701
\(315\) 0 0
\(316\) −3.24264 −0.182413
\(317\) −21.6569 −1.21637 −0.608185 0.793795i \(-0.708102\pi\)
−0.608185 + 0.793795i \(0.708102\pi\)
\(318\) −3.58579 −0.201081
\(319\) 2.24264 0.125564
\(320\) −1.00000 −0.0559017
\(321\) −5.34315 −0.298225
\(322\) 0 0
\(323\) −6.14214 −0.341758
\(324\) 7.48528 0.415849
\(325\) −24.9706 −1.38512
\(326\) −25.3137 −1.40200
\(327\) −5.85786 −0.323941
\(328\) −1.00000 −0.0552158
\(329\) 0 0
\(330\) 0.242641 0.0133569
\(331\) −16.0416 −0.881728 −0.440864 0.897574i \(-0.645328\pi\)
−0.440864 + 0.897574i \(0.645328\pi\)
\(332\) 1.07107 0.0587825
\(333\) 4.00000 0.219199
\(334\) 24.9706 1.36633
\(335\) −3.17157 −0.173282
\(336\) 0 0
\(337\) −8.65685 −0.471569 −0.235784 0.971805i \(-0.575766\pi\)
−0.235784 + 0.971805i \(0.575766\pi\)
\(338\) 25.9706 1.41261
\(339\) 2.07107 0.112485
\(340\) −2.17157 −0.117770
\(341\) 4.92893 0.266917
\(342\) 8.00000 0.432590
\(343\) 0 0
\(344\) −6.07107 −0.327330
\(345\) −1.65685 −0.0892020
\(346\) −9.82843 −0.528380
\(347\) −5.65685 −0.303676 −0.151838 0.988405i \(-0.548519\pi\)
−0.151838 + 0.988405i \(0.548519\pi\)
\(348\) −1.58579 −0.0850071
\(349\) 19.7990 1.05982 0.529908 0.848055i \(-0.322227\pi\)
0.529908 + 0.848055i \(0.322227\pi\)
\(350\) 0 0
\(351\) 15.0711 0.804434
\(352\) 0.585786 0.0312225
\(353\) −6.34315 −0.337612 −0.168806 0.985649i \(-0.553991\pi\)
−0.168806 + 0.985649i \(0.553991\pi\)
\(354\) 2.00000 0.106299
\(355\) −1.24264 −0.0659525
\(356\) 7.82843 0.414906
\(357\) 0 0
\(358\) 4.24264 0.224231
\(359\) 18.3848 0.970311 0.485156 0.874428i \(-0.338763\pi\)
0.485156 + 0.874428i \(0.338763\pi\)
\(360\) 2.82843 0.149071
\(361\) −11.0000 −0.578947
\(362\) 9.89949 0.520306
\(363\) 4.41421 0.231686
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) −1.10051 −0.0575243
\(367\) 3.24264 0.169264 0.0846322 0.996412i \(-0.473028\pi\)
0.0846322 + 0.996412i \(0.473028\pi\)
\(368\) −4.00000 −0.208514
\(369\) 2.82843 0.147242
\(370\) 1.41421 0.0735215
\(371\) 0 0
\(372\) −3.48528 −0.180703
\(373\) −11.7574 −0.608773 −0.304386 0.952549i \(-0.598451\pi\)
−0.304386 + 0.952549i \(0.598451\pi\)
\(374\) 1.27208 0.0657776
\(375\) −3.72792 −0.192509
\(376\) 8.58579 0.442778
\(377\) 23.8995 1.23089
\(378\) 0 0
\(379\) −27.3848 −1.40666 −0.703331 0.710863i \(-0.748305\pi\)
−0.703331 + 0.710863i \(0.748305\pi\)
\(380\) 2.82843 0.145095
\(381\) 2.38478 0.122176
\(382\) −3.24264 −0.165908
\(383\) 10.7279 0.548171 0.274086 0.961705i \(-0.411625\pi\)
0.274086 + 0.961705i \(0.411625\pi\)
\(384\) −0.414214 −0.0211377
\(385\) 0 0
\(386\) 7.89949 0.402074
\(387\) 17.1716 0.872880
\(388\) 12.1716 0.617918
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 2.58579 0.130936
\(391\) −8.68629 −0.439285
\(392\) 0 0
\(393\) 0.100505 0.00506981
\(394\) 23.3137 1.17453
\(395\) 3.24264 0.163155
\(396\) −1.65685 −0.0832601
\(397\) 8.58579 0.430908 0.215454 0.976514i \(-0.430877\pi\)
0.215454 + 0.976514i \(0.430877\pi\)
\(398\) 11.3137 0.567105
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 13.9706 0.697657 0.348828 0.937187i \(-0.386580\pi\)
0.348828 + 0.937187i \(0.386580\pi\)
\(402\) −1.31371 −0.0655218
\(403\) 52.5269 2.61655
\(404\) 0.343146 0.0170721
\(405\) −7.48528 −0.371947
\(406\) 0 0
\(407\) −0.828427 −0.0410636
\(408\) −0.899495 −0.0445316
\(409\) −14.1005 −0.697225 −0.348613 0.937267i \(-0.613347\pi\)
−0.348613 + 0.937267i \(0.613347\pi\)
\(410\) 1.00000 0.0493865
\(411\) −0.686292 −0.0338523
\(412\) 8.89949 0.438447
\(413\) 0 0
\(414\) 11.3137 0.556038
\(415\) −1.07107 −0.0525767
\(416\) 6.24264 0.306071
\(417\) −0.627417 −0.0307247
\(418\) −1.65685 −0.0810394
\(419\) −20.2426 −0.988918 −0.494459 0.869201i \(-0.664634\pi\)
−0.494459 + 0.869201i \(0.664634\pi\)
\(420\) 0 0
\(421\) 36.1127 1.76003 0.880013 0.474950i \(-0.157534\pi\)
0.880013 + 0.474950i \(0.157534\pi\)
\(422\) −1.65685 −0.0806544
\(423\) −24.2843 −1.18074
\(424\) 8.65685 0.420414
\(425\) −8.68629 −0.421347
\(426\) −0.514719 −0.0249382
\(427\) 0 0
\(428\) 12.8995 0.623521
\(429\) −1.51472 −0.0731313
\(430\) 6.07107 0.292773
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 2.41421 0.116154
\(433\) 24.9706 1.20001 0.600004 0.799997i \(-0.295166\pi\)
0.600004 + 0.799997i \(0.295166\pi\)
\(434\) 0 0
\(435\) 1.58579 0.0760326
\(436\) 14.1421 0.677285
\(437\) 11.3137 0.541208
\(438\) −1.65685 −0.0791676
\(439\) −9.51472 −0.454113 −0.227056 0.973882i \(-0.572910\pi\)
−0.227056 + 0.973882i \(0.572910\pi\)
\(440\) −0.585786 −0.0279263
\(441\) 0 0
\(442\) 13.5563 0.644810
\(443\) 9.24264 0.439131 0.219566 0.975598i \(-0.429536\pi\)
0.219566 + 0.975598i \(0.429536\pi\)
\(444\) 0.585786 0.0278002
\(445\) −7.82843 −0.371103
\(446\) −27.8701 −1.31969
\(447\) −8.55635 −0.404701
\(448\) 0 0
\(449\) 14.1716 0.668798 0.334399 0.942432i \(-0.391467\pi\)
0.334399 + 0.942432i \(0.391467\pi\)
\(450\) 11.3137 0.533333
\(451\) −0.585786 −0.0275836
\(452\) −5.00000 −0.235180
\(453\) −8.31371 −0.390612
\(454\) −7.72792 −0.362689
\(455\) 0 0
\(456\) 1.17157 0.0548639
\(457\) −36.7279 −1.71806 −0.859030 0.511925i \(-0.828932\pi\)
−0.859030 + 0.511925i \(0.828932\pi\)
\(458\) 3.41421 0.159536
\(459\) 5.24264 0.244706
\(460\) 4.00000 0.186501
\(461\) −31.9706 −1.48902 −0.744509 0.667613i \(-0.767316\pi\)
−0.744509 + 0.667613i \(0.767316\pi\)
\(462\) 0 0
\(463\) 5.65685 0.262896 0.131448 0.991323i \(-0.458037\pi\)
0.131448 + 0.991323i \(0.458037\pi\)
\(464\) 3.82843 0.177730
\(465\) 3.48528 0.161626
\(466\) −24.2426 −1.12302
\(467\) −34.4853 −1.59579 −0.797894 0.602797i \(-0.794053\pi\)
−0.797894 + 0.602797i \(0.794053\pi\)
\(468\) −17.6569 −0.816188
\(469\) 0 0
\(470\) −8.58579 −0.396033
\(471\) 5.65685 0.260654
\(472\) −4.82843 −0.222246
\(473\) −3.55635 −0.163521
\(474\) 1.34315 0.0616927
\(475\) 11.3137 0.519109
\(476\) 0 0
\(477\) −24.4853 −1.12110
\(478\) 3.51472 0.160759
\(479\) 35.2132 1.60893 0.804466 0.593998i \(-0.202451\pi\)
0.804466 + 0.593998i \(0.202451\pi\)
\(480\) 0.414214 0.0189062
\(481\) −8.82843 −0.402542
\(482\) −21.4142 −0.975391
\(483\) 0 0
\(484\) −10.6569 −0.484402
\(485\) −12.1716 −0.552683
\(486\) −10.3431 −0.469175
\(487\) −18.5858 −0.842202 −0.421101 0.907014i \(-0.638356\pi\)
−0.421101 + 0.907014i \(0.638356\pi\)
\(488\) 2.65685 0.120270
\(489\) 10.4853 0.474161
\(490\) 0 0
\(491\) 15.8701 0.716206 0.358103 0.933682i \(-0.383424\pi\)
0.358103 + 0.933682i \(0.383424\pi\)
\(492\) 0.414214 0.0186742
\(493\) 8.31371 0.374431
\(494\) −17.6569 −0.794419
\(495\) 1.65685 0.0744701
\(496\) 8.41421 0.377809
\(497\) 0 0
\(498\) −0.443651 −0.0198805
\(499\) 22.4853 1.00658 0.503290 0.864118i \(-0.332123\pi\)
0.503290 + 0.864118i \(0.332123\pi\)
\(500\) 9.00000 0.402492
\(501\) −10.3431 −0.462098
\(502\) −12.9289 −0.577046
\(503\) −10.3431 −0.461178 −0.230589 0.973051i \(-0.574065\pi\)
−0.230589 + 0.973051i \(0.574065\pi\)
\(504\) 0 0
\(505\) −0.343146 −0.0152698
\(506\) −2.34315 −0.104166
\(507\) −10.7574 −0.477751
\(508\) −5.75736 −0.255442
\(509\) 27.3137 1.21066 0.605329 0.795975i \(-0.293041\pi\)
0.605329 + 0.795975i \(0.293041\pi\)
\(510\) 0.899495 0.0398303
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −6.82843 −0.301482
\(514\) 25.4853 1.12411
\(515\) −8.89949 −0.392159
\(516\) 2.51472 0.110704
\(517\) 5.02944 0.221194
\(518\) 0 0
\(519\) 4.07107 0.178700
\(520\) −6.24264 −0.273758
\(521\) 9.17157 0.401814 0.200907 0.979610i \(-0.435611\pi\)
0.200907 + 0.979610i \(0.435611\pi\)
\(522\) −10.8284 −0.473947
\(523\) −15.3137 −0.669622 −0.334811 0.942285i \(-0.608672\pi\)
−0.334811 + 0.942285i \(0.608672\pi\)
\(524\) −0.242641 −0.0105998
\(525\) 0 0
\(526\) 8.97056 0.391135
\(527\) 18.2721 0.795944
\(528\) −0.242641 −0.0105596
\(529\) −7.00000 −0.304348
\(530\) −8.65685 −0.376030
\(531\) 13.6569 0.592657
\(532\) 0 0
\(533\) −6.24264 −0.270399
\(534\) −3.24264 −0.140323
\(535\) −12.8995 −0.557694
\(536\) 3.17157 0.136991
\(537\) −1.75736 −0.0758357
\(538\) −2.68629 −0.115814
\(539\) 0 0
\(540\) −2.41421 −0.103891
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 6.00000 0.257722
\(543\) −4.10051 −0.175970
\(544\) 2.17157 0.0931054
\(545\) −14.1421 −0.605783
\(546\) 0 0
\(547\) 21.7990 0.932058 0.466029 0.884770i \(-0.345684\pi\)
0.466029 + 0.884770i \(0.345684\pi\)
\(548\) 1.65685 0.0707773
\(549\) −7.51472 −0.320720
\(550\) −2.34315 −0.0999121
\(551\) −10.8284 −0.461307
\(552\) 1.65685 0.0705204
\(553\) 0 0
\(554\) −12.3431 −0.524410
\(555\) −0.585786 −0.0248652
\(556\) 1.51472 0.0642384
\(557\) −0.798990 −0.0338543 −0.0169271 0.999857i \(-0.505388\pi\)
−0.0169271 + 0.999857i \(0.505388\pi\)
\(558\) −23.7990 −1.00749
\(559\) −37.8995 −1.60298
\(560\) 0 0
\(561\) −0.526912 −0.0222462
\(562\) −26.0000 −1.09674
\(563\) −31.9411 −1.34616 −0.673079 0.739571i \(-0.735029\pi\)
−0.673079 + 0.739571i \(0.735029\pi\)
\(564\) −3.55635 −0.149749
\(565\) 5.00000 0.210352
\(566\) −22.4853 −0.945127
\(567\) 0 0
\(568\) 1.24264 0.0521400
\(569\) 22.1716 0.929481 0.464740 0.885447i \(-0.346148\pi\)
0.464740 + 0.885447i \(0.346148\pi\)
\(570\) −1.17157 −0.0490718
\(571\) −45.2548 −1.89386 −0.946928 0.321446i \(-0.895831\pi\)
−0.946928 + 0.321446i \(0.895831\pi\)
\(572\) 3.65685 0.152901
\(573\) 1.34315 0.0561107
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) −2.82843 −0.117851
\(577\) −6.34315 −0.264069 −0.132034 0.991245i \(-0.542151\pi\)
−0.132034 + 0.991245i \(0.542151\pi\)
\(578\) −12.2843 −0.510958
\(579\) −3.27208 −0.135983
\(580\) −3.82843 −0.158967
\(581\) 0 0
\(582\) −5.04163 −0.208982
\(583\) 5.07107 0.210022
\(584\) 4.00000 0.165521
\(585\) 17.6569 0.730021
\(586\) 14.0000 0.578335
\(587\) 11.1005 0.458167 0.229083 0.973407i \(-0.426427\pi\)
0.229083 + 0.973407i \(0.426427\pi\)
\(588\) 0 0
\(589\) −23.7990 −0.980620
\(590\) 4.82843 0.198783
\(591\) −9.65685 −0.397230
\(592\) −1.41421 −0.0581238
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 1.41421 0.0580259
\(595\) 0 0
\(596\) 20.6569 0.846138
\(597\) −4.68629 −0.191797
\(598\) −24.9706 −1.02112
\(599\) −47.8406 −1.95471 −0.977357 0.211595i \(-0.932134\pi\)
−0.977357 + 0.211595i \(0.932134\pi\)
\(600\) 1.65685 0.0676408
\(601\) −0.313708 −0.0127964 −0.00639822 0.999980i \(-0.502037\pi\)
−0.00639822 + 0.999980i \(0.502037\pi\)
\(602\) 0 0
\(603\) −8.97056 −0.365310
\(604\) 20.0711 0.816680
\(605\) 10.6569 0.433263
\(606\) −0.142136 −0.00577387
\(607\) 7.38478 0.299739 0.149869 0.988706i \(-0.452115\pi\)
0.149869 + 0.988706i \(0.452115\pi\)
\(608\) −2.82843 −0.114708
\(609\) 0 0
\(610\) −2.65685 −0.107573
\(611\) 53.5980 2.16834
\(612\) −6.14214 −0.248281
\(613\) 38.0416 1.53649 0.768243 0.640158i \(-0.221131\pi\)
0.768243 + 0.640158i \(0.221131\pi\)
\(614\) 24.9706 1.00773
\(615\) −0.414214 −0.0167027
\(616\) 0 0
\(617\) −31.7990 −1.28018 −0.640090 0.768300i \(-0.721103\pi\)
−0.640090 + 0.768300i \(0.721103\pi\)
\(618\) −3.68629 −0.148284
\(619\) −11.0711 −0.444984 −0.222492 0.974935i \(-0.571419\pi\)
−0.222492 + 0.974935i \(0.571419\pi\)
\(620\) −8.41421 −0.337923
\(621\) −9.65685 −0.387516
\(622\) 5.51472 0.221120
\(623\) 0 0
\(624\) −2.58579 −0.103514
\(625\) 11.0000 0.440000
\(626\) −4.97056 −0.198664
\(627\) 0.686292 0.0274078
\(628\) −13.6569 −0.544968
\(629\) −3.07107 −0.122451
\(630\) 0 0
\(631\) 19.2132 0.764866 0.382433 0.923983i \(-0.375086\pi\)
0.382433 + 0.923983i \(0.375086\pi\)
\(632\) −3.24264 −0.128985
\(633\) 0.686292 0.0272776
\(634\) −21.6569 −0.860104
\(635\) 5.75736 0.228474
\(636\) −3.58579 −0.142186
\(637\) 0 0
\(638\) 2.24264 0.0887870
\(639\) −3.51472 −0.139040
\(640\) −1.00000 −0.0395285
\(641\) 18.8701 0.745322 0.372661 0.927967i \(-0.378445\pi\)
0.372661 + 0.927967i \(0.378445\pi\)
\(642\) −5.34315 −0.210877
\(643\) 6.55635 0.258557 0.129279 0.991608i \(-0.458734\pi\)
0.129279 + 0.991608i \(0.458734\pi\)
\(644\) 0 0
\(645\) −2.51472 −0.0990169
\(646\) −6.14214 −0.241659
\(647\) −11.3848 −0.447582 −0.223791 0.974637i \(-0.571843\pi\)
−0.223791 + 0.974637i \(0.571843\pi\)
\(648\) 7.48528 0.294050
\(649\) −2.82843 −0.111025
\(650\) −24.9706 −0.979426
\(651\) 0 0
\(652\) −25.3137 −0.991361
\(653\) −25.6274 −1.00288 −0.501439 0.865193i \(-0.667196\pi\)
−0.501439 + 0.865193i \(0.667196\pi\)
\(654\) −5.85786 −0.229061
\(655\) 0.242641 0.00948076
\(656\) −1.00000 −0.0390434
\(657\) −11.3137 −0.441390
\(658\) 0 0
\(659\) 1.37258 0.0534682 0.0267341 0.999643i \(-0.491489\pi\)
0.0267341 + 0.999643i \(0.491489\pi\)
\(660\) 0.242641 0.00944478
\(661\) −33.4558 −1.30128 −0.650641 0.759386i \(-0.725500\pi\)
−0.650641 + 0.759386i \(0.725500\pi\)
\(662\) −16.0416 −0.623476
\(663\) −5.61522 −0.218077
\(664\) 1.07107 0.0415655
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −15.3137 −0.592949
\(668\) 24.9706 0.966140
\(669\) 11.5442 0.446323
\(670\) −3.17157 −0.122529
\(671\) 1.55635 0.0600822
\(672\) 0 0
\(673\) 15.5563 0.599653 0.299827 0.953994i \(-0.403071\pi\)
0.299827 + 0.953994i \(0.403071\pi\)
\(674\) −8.65685 −0.333450
\(675\) −9.65685 −0.371692
\(676\) 25.9706 0.998868
\(677\) 11.7990 0.453472 0.226736 0.973956i \(-0.427195\pi\)
0.226736 + 0.973956i \(0.427195\pi\)
\(678\) 2.07107 0.0795389
\(679\) 0 0
\(680\) −2.17157 −0.0832760
\(681\) 3.20101 0.122663
\(682\) 4.92893 0.188739
\(683\) 1.65685 0.0633978 0.0316989 0.999497i \(-0.489908\pi\)
0.0316989 + 0.999497i \(0.489908\pi\)
\(684\) 8.00000 0.305888
\(685\) −1.65685 −0.0633051
\(686\) 0 0
\(687\) −1.41421 −0.0539556
\(688\) −6.07107 −0.231457
\(689\) 54.0416 2.05882
\(690\) −1.65685 −0.0630754
\(691\) −19.7279 −0.750486 −0.375243 0.926927i \(-0.622441\pi\)
−0.375243 + 0.926927i \(0.622441\pi\)
\(692\) −9.82843 −0.373621
\(693\) 0 0
\(694\) −5.65685 −0.214731
\(695\) −1.51472 −0.0574566
\(696\) −1.58579 −0.0601091
\(697\) −2.17157 −0.0822542
\(698\) 19.7990 0.749403
\(699\) 10.0416 0.379809
\(700\) 0 0
\(701\) −8.97056 −0.338813 −0.169407 0.985546i \(-0.554185\pi\)
−0.169407 + 0.985546i \(0.554185\pi\)
\(702\) 15.0711 0.568821
\(703\) 4.00000 0.150863
\(704\) 0.585786 0.0220777
\(705\) 3.55635 0.133940
\(706\) −6.34315 −0.238727
\(707\) 0 0
\(708\) 2.00000 0.0751646
\(709\) 22.7990 0.856234 0.428117 0.903723i \(-0.359177\pi\)
0.428117 + 0.903723i \(0.359177\pi\)
\(710\) −1.24264 −0.0466355
\(711\) 9.17157 0.343961
\(712\) 7.82843 0.293383
\(713\) −33.6569 −1.26046
\(714\) 0 0
\(715\) −3.65685 −0.136759
\(716\) 4.24264 0.158555
\(717\) −1.45584 −0.0543695
\(718\) 18.3848 0.686114
\(719\) −27.3553 −1.02018 −0.510091 0.860120i \(-0.670388\pi\)
−0.510091 + 0.860120i \(0.670388\pi\)
\(720\) 2.82843 0.105409
\(721\) 0 0
\(722\) −11.0000 −0.409378
\(723\) 8.87006 0.329881
\(724\) 9.89949 0.367912
\(725\) −15.3137 −0.568737
\(726\) 4.41421 0.163827
\(727\) −43.0711 −1.59742 −0.798709 0.601718i \(-0.794483\pi\)
−0.798709 + 0.601718i \(0.794483\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) −4.00000 −0.148047
\(731\) −13.1838 −0.487619
\(732\) −1.10051 −0.0406758
\(733\) −23.8284 −0.880123 −0.440062 0.897968i \(-0.645043\pi\)
−0.440062 + 0.897968i \(0.645043\pi\)
\(734\) 3.24264 0.119688
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 1.85786 0.0684353
\(738\) 2.82843 0.104116
\(739\) 18.5563 0.682606 0.341303 0.939953i \(-0.389132\pi\)
0.341303 + 0.939953i \(0.389132\pi\)
\(740\) 1.41421 0.0519875
\(741\) 7.31371 0.268676
\(742\) 0 0
\(743\) −33.9411 −1.24518 −0.622590 0.782549i \(-0.713919\pi\)
−0.622590 + 0.782549i \(0.713919\pi\)
\(744\) −3.48528 −0.127777
\(745\) −20.6569 −0.756809
\(746\) −11.7574 −0.430468
\(747\) −3.02944 −0.110841
\(748\) 1.27208 0.0465118
\(749\) 0 0
\(750\) −3.72792 −0.136124
\(751\) 20.3431 0.742332 0.371166 0.928567i \(-0.378958\pi\)
0.371166 + 0.928567i \(0.378958\pi\)
\(752\) 8.58579 0.313091
\(753\) 5.35534 0.195159
\(754\) 23.8995 0.870368
\(755\) −20.0711 −0.730461
\(756\) 0 0
\(757\) 4.45584 0.161950 0.0809752 0.996716i \(-0.474197\pi\)
0.0809752 + 0.996716i \(0.474197\pi\)
\(758\) −27.3848 −0.994660
\(759\) 0.970563 0.0352292
\(760\) 2.82843 0.102598
\(761\) −7.55635 −0.273917 −0.136959 0.990577i \(-0.543733\pi\)
−0.136959 + 0.990577i \(0.543733\pi\)
\(762\) 2.38478 0.0863913
\(763\) 0 0
\(764\) −3.24264 −0.117315
\(765\) 6.14214 0.222069
\(766\) 10.7279 0.387616
\(767\) −30.1421 −1.08837
\(768\) −0.414214 −0.0149466
\(769\) 10.9706 0.395609 0.197804 0.980242i \(-0.436619\pi\)
0.197804 + 0.980242i \(0.436619\pi\)
\(770\) 0 0
\(771\) −10.5563 −0.380178
\(772\) 7.89949 0.284309
\(773\) −23.5563 −0.847263 −0.423631 0.905835i \(-0.639245\pi\)
−0.423631 + 0.905835i \(0.639245\pi\)
\(774\) 17.1716 0.617219
\(775\) −33.6569 −1.20899
\(776\) 12.1716 0.436934
\(777\) 0 0
\(778\) −2.00000 −0.0717035
\(779\) 2.82843 0.101339
\(780\) 2.58579 0.0925860
\(781\) 0.727922 0.0260471
\(782\) −8.68629 −0.310621
\(783\) 9.24264 0.330305
\(784\) 0 0
\(785\) 13.6569 0.487434
\(786\) 0.100505 0.00358490
\(787\) 14.3431 0.511278 0.255639 0.966772i \(-0.417714\pi\)
0.255639 + 0.966772i \(0.417714\pi\)
\(788\) 23.3137 0.830516
\(789\) −3.71573 −0.132283
\(790\) 3.24264 0.115368
\(791\) 0 0
\(792\) −1.65685 −0.0588738
\(793\) 16.5858 0.588979
\(794\) 8.58579 0.304698
\(795\) 3.58579 0.127175
\(796\) 11.3137 0.401004
\(797\) −7.62742 −0.270177 −0.135088 0.990834i \(-0.543132\pi\)
−0.135088 + 0.990834i \(0.543132\pi\)
\(798\) 0 0
\(799\) 18.6447 0.659601
\(800\) −4.00000 −0.141421
\(801\) −22.1421 −0.782354
\(802\) 13.9706 0.493318
\(803\) 2.34315 0.0826878
\(804\) −1.31371 −0.0463309
\(805\) 0 0
\(806\) 52.5269 1.85018
\(807\) 1.11270 0.0391688
\(808\) 0.343146 0.0120718
\(809\) 4.24264 0.149163 0.0745817 0.997215i \(-0.476238\pi\)
0.0745817 + 0.997215i \(0.476238\pi\)
\(810\) −7.48528 −0.263006
\(811\) −55.5563 −1.95085 −0.975424 0.220338i \(-0.929284\pi\)
−0.975424 + 0.220338i \(0.929284\pi\)
\(812\) 0 0
\(813\) −2.48528 −0.0871626
\(814\) −0.828427 −0.0290364
\(815\) 25.3137 0.886701
\(816\) −0.899495 −0.0314886
\(817\) 17.1716 0.600757
\(818\) −14.1005 −0.493013
\(819\) 0 0
\(820\) 1.00000 0.0349215
\(821\) −10.2426 −0.357471 −0.178735 0.983897i \(-0.557201\pi\)
−0.178735 + 0.983897i \(0.557201\pi\)
\(822\) −0.686292 −0.0239372
\(823\) −51.1838 −1.78415 −0.892077 0.451883i \(-0.850752\pi\)
−0.892077 + 0.451883i \(0.850752\pi\)
\(824\) 8.89949 0.310029
\(825\) 0.970563 0.0337907
\(826\) 0 0
\(827\) −35.4558 −1.23292 −0.616460 0.787386i \(-0.711434\pi\)
−0.616460 + 0.787386i \(0.711434\pi\)
\(828\) 11.3137 0.393179
\(829\) 22.8579 0.793886 0.396943 0.917843i \(-0.370071\pi\)
0.396943 + 0.917843i \(0.370071\pi\)
\(830\) −1.07107 −0.0371773
\(831\) 5.11270 0.177358
\(832\) 6.24264 0.216425
\(833\) 0 0
\(834\) −0.627417 −0.0217257
\(835\) −24.9706 −0.864142
\(836\) −1.65685 −0.0573035
\(837\) 20.3137 0.702144
\(838\) −20.2426 −0.699270
\(839\) −36.8284 −1.27146 −0.635729 0.771912i \(-0.719301\pi\)
−0.635729 + 0.771912i \(0.719301\pi\)
\(840\) 0 0
\(841\) −14.3431 −0.494591
\(842\) 36.1127 1.24453
\(843\) 10.7696 0.370923
\(844\) −1.65685 −0.0570313
\(845\) −25.9706 −0.893415
\(846\) −24.2843 −0.834910
\(847\) 0 0
\(848\) 8.65685 0.297278
\(849\) 9.31371 0.319646
\(850\) −8.68629 −0.297937
\(851\) 5.65685 0.193914
\(852\) −0.514719 −0.0176340
\(853\) −21.0000 −0.719026 −0.359513 0.933140i \(-0.617057\pi\)
−0.359513 + 0.933140i \(0.617057\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 12.8995 0.440896
\(857\) 39.9411 1.36436 0.682181 0.731183i \(-0.261031\pi\)
0.682181 + 0.731183i \(0.261031\pi\)
\(858\) −1.51472 −0.0517116
\(859\) −26.1005 −0.890538 −0.445269 0.895397i \(-0.646892\pi\)
−0.445269 + 0.895397i \(0.646892\pi\)
\(860\) 6.07107 0.207022
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 23.5563 0.801867 0.400934 0.916107i \(-0.368686\pi\)
0.400934 + 0.916107i \(0.368686\pi\)
\(864\) 2.41421 0.0821332
\(865\) 9.82843 0.334177
\(866\) 24.9706 0.848534
\(867\) 5.08831 0.172808
\(868\) 0 0
\(869\) −1.89949 −0.0644360
\(870\) 1.58579 0.0537632
\(871\) 19.7990 0.670863
\(872\) 14.1421 0.478913
\(873\) −34.4264 −1.16516
\(874\) 11.3137 0.382692
\(875\) 0 0
\(876\) −1.65685 −0.0559799
\(877\) 34.3431 1.15969 0.579843 0.814728i \(-0.303114\pi\)
0.579843 + 0.814728i \(0.303114\pi\)
\(878\) −9.51472 −0.321106
\(879\) −5.79899 −0.195595
\(880\) −0.585786 −0.0197469
\(881\) 53.1543 1.79081 0.895407 0.445248i \(-0.146884\pi\)
0.895407 + 0.445248i \(0.146884\pi\)
\(882\) 0 0
\(883\) −35.4558 −1.19318 −0.596592 0.802545i \(-0.703479\pi\)
−0.596592 + 0.802545i \(0.703479\pi\)
\(884\) 13.5563 0.455949
\(885\) −2.00000 −0.0672293
\(886\) 9.24264 0.310513
\(887\) −37.2548 −1.25089 −0.625447 0.780266i \(-0.715084\pi\)
−0.625447 + 0.780266i \(0.715084\pi\)
\(888\) 0.585786 0.0196577
\(889\) 0 0
\(890\) −7.82843 −0.262409
\(891\) 4.38478 0.146896
\(892\) −27.8701 −0.933159
\(893\) −24.2843 −0.812642
\(894\) −8.55635 −0.286167
\(895\) −4.24264 −0.141816
\(896\) 0 0
\(897\) 10.3431 0.345348
\(898\) 14.1716 0.472912
\(899\) 32.2132 1.07437
\(900\) 11.3137 0.377124
\(901\) 18.7990 0.626285
\(902\) −0.585786 −0.0195046
\(903\) 0 0
\(904\) −5.00000 −0.166298
\(905\) −9.89949 −0.329070
\(906\) −8.31371 −0.276204
\(907\) 53.2426 1.76789 0.883946 0.467588i \(-0.154877\pi\)
0.883946 + 0.467588i \(0.154877\pi\)
\(908\) −7.72792 −0.256460
\(909\) −0.970563 −0.0321915
\(910\) 0 0
\(911\) 18.7696 0.621863 0.310932 0.950432i \(-0.399359\pi\)
0.310932 + 0.950432i \(0.399359\pi\)
\(912\) 1.17157 0.0387947
\(913\) 0.627417 0.0207645
\(914\) −36.7279 −1.21485
\(915\) 1.10051 0.0363816
\(916\) 3.41421 0.112809
\(917\) 0 0
\(918\) 5.24264 0.173033
\(919\) −0.213203 −0.00703293 −0.00351647 0.999994i \(-0.501119\pi\)
−0.00351647 + 0.999994i \(0.501119\pi\)
\(920\) 4.00000 0.131876
\(921\) −10.3431 −0.340818
\(922\) −31.9706 −1.05289
\(923\) 7.75736 0.255337
\(924\) 0 0
\(925\) 5.65685 0.185996
\(926\) 5.65685 0.185896
\(927\) −25.1716 −0.826743
\(928\) 3.82843 0.125674
\(929\) 25.1716 0.825853 0.412926 0.910764i \(-0.364507\pi\)
0.412926 + 0.910764i \(0.364507\pi\)
\(930\) 3.48528 0.114287
\(931\) 0 0
\(932\) −24.2426 −0.794094
\(933\) −2.28427 −0.0747837
\(934\) −34.4853 −1.12839
\(935\) −1.27208 −0.0416014
\(936\) −17.6569 −0.577132
\(937\) 3.97056 0.129713 0.0648563 0.997895i \(-0.479341\pi\)
0.0648563 + 0.997895i \(0.479341\pi\)
\(938\) 0 0
\(939\) 2.05887 0.0671888
\(940\) −8.58579 −0.280037
\(941\) 25.9411 0.845657 0.422828 0.906210i \(-0.361037\pi\)
0.422828 + 0.906210i \(0.361037\pi\)
\(942\) 5.65685 0.184310
\(943\) 4.00000 0.130258
\(944\) −4.82843 −0.157152
\(945\) 0 0
\(946\) −3.55635 −0.115627
\(947\) 40.7696 1.32483 0.662416 0.749136i \(-0.269531\pi\)
0.662416 + 0.749136i \(0.269531\pi\)
\(948\) 1.34315 0.0436233
\(949\) 24.9706 0.810579
\(950\) 11.3137 0.367065
\(951\) 8.97056 0.290890
\(952\) 0 0
\(953\) 3.51472 0.113853 0.0569265 0.998378i \(-0.481870\pi\)
0.0569265 + 0.998378i \(0.481870\pi\)
\(954\) −24.4853 −0.792740
\(955\) 3.24264 0.104929
\(956\) 3.51472 0.113674
\(957\) −0.928932 −0.0300281
\(958\) 35.2132 1.13769
\(959\) 0 0
\(960\) 0.414214 0.0133687
\(961\) 39.7990 1.28384
\(962\) −8.82843 −0.284640
\(963\) −36.4853 −1.17572
\(964\) −21.4142 −0.689705
\(965\) −7.89949 −0.254294
\(966\) 0 0
\(967\) −48.0711 −1.54586 −0.772931 0.634490i \(-0.781210\pi\)
−0.772931 + 0.634490i \(0.781210\pi\)
\(968\) −10.6569 −0.342524
\(969\) 2.54416 0.0817301
\(970\) −12.1716 −0.390806
\(971\) 5.24264 0.168244 0.0841222 0.996455i \(-0.473191\pi\)
0.0841222 + 0.996455i \(0.473191\pi\)
\(972\) −10.3431 −0.331757
\(973\) 0 0
\(974\) −18.5858 −0.595527
\(975\) 10.3431 0.331246
\(976\) 2.65685 0.0850438
\(977\) −6.28427 −0.201052 −0.100526 0.994934i \(-0.532053\pi\)
−0.100526 + 0.994934i \(0.532053\pi\)
\(978\) 10.4853 0.335282
\(979\) 4.58579 0.146562
\(980\) 0 0
\(981\) −40.0000 −1.27710
\(982\) 15.8701 0.506434
\(983\) −23.2426 −0.741325 −0.370663 0.928768i \(-0.620869\pi\)
−0.370663 + 0.928768i \(0.620869\pi\)
\(984\) 0.414214 0.0132046
\(985\) −23.3137 −0.742837
\(986\) 8.31371 0.264762
\(987\) 0 0
\(988\) −17.6569 −0.561739
\(989\) 24.2843 0.772195
\(990\) 1.65685 0.0526583
\(991\) 8.34315 0.265029 0.132514 0.991181i \(-0.457695\pi\)
0.132514 + 0.991181i \(0.457695\pi\)
\(992\) 8.41421 0.267152
\(993\) 6.64466 0.210862
\(994\) 0 0
\(995\) −11.3137 −0.358669
\(996\) −0.443651 −0.0140576
\(997\) 36.2843 1.14913 0.574567 0.818457i \(-0.305170\pi\)
0.574567 + 0.818457i \(0.305170\pi\)
\(998\) 22.4853 0.711759
\(999\) −3.41421 −0.108021
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 4018.2.a.bb.1.1 yes 2
7.6 odd 2 4018.2.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4018.2.a.z.1.2 2 7.6 odd 2
4018.2.a.bb.1.1 yes 2 1.1 even 1 trivial