Properties

Label 483.2.a.e.1.1
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
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] = [483,2,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, 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("483.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.38197 q^{5} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.38197 q^{5} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -2.23607 q^{10} +1.00000 q^{11} +0.618034 q^{12} -1.61803 q^{13} -1.61803 q^{14} +1.38197 q^{15} -4.85410 q^{16} +3.47214 q^{17} -1.61803 q^{18} -0.236068 q^{19} +0.854102 q^{20} +1.00000 q^{21} -1.61803 q^{22} +1.00000 q^{23} +2.23607 q^{24} -3.09017 q^{25} +2.61803 q^{26} +1.00000 q^{27} +0.618034 q^{28} +6.23607 q^{29} -2.23607 q^{30} +4.70820 q^{31} +3.38197 q^{32} +1.00000 q^{33} -5.61803 q^{34} +1.38197 q^{35} +0.618034 q^{36} -4.23607 q^{37} +0.381966 q^{38} -1.61803 q^{39} +3.09017 q^{40} +5.47214 q^{41} -1.61803 q^{42} +2.85410 q^{43} +0.618034 q^{44} +1.38197 q^{45} -1.61803 q^{46} -1.70820 q^{47} -4.85410 q^{48} +1.00000 q^{49} +5.00000 q^{50} +3.47214 q^{51} -1.00000 q^{52} +11.0902 q^{53} -1.61803 q^{54} +1.38197 q^{55} +2.23607 q^{56} -0.236068 q^{57} -10.0902 q^{58} -1.38197 q^{59} +0.854102 q^{60} -1.14590 q^{61} -7.61803 q^{62} +1.00000 q^{63} +4.23607 q^{64} -2.23607 q^{65} -1.61803 q^{66} +3.09017 q^{67} +2.14590 q^{68} +1.00000 q^{69} -2.23607 q^{70} +5.32624 q^{71} +2.23607 q^{72} +6.23607 q^{73} +6.85410 q^{74} -3.09017 q^{75} -0.145898 q^{76} +1.00000 q^{77} +2.61803 q^{78} -9.76393 q^{79} -6.70820 q^{80} +1.00000 q^{81} -8.85410 q^{82} -0.0557281 q^{83} +0.618034 q^{84} +4.79837 q^{85} -4.61803 q^{86} +6.23607 q^{87} +2.23607 q^{88} -6.56231 q^{89} -2.23607 q^{90} -1.61803 q^{91} +0.618034 q^{92} +4.70820 q^{93} +2.76393 q^{94} -0.326238 q^{95} +3.38197 q^{96} -6.70820 q^{97} -1.61803 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} - q^{4} + 5 q^{5} - q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} - q^{4} + 5 q^{5} - q^{6} + 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{12} - q^{13} - q^{14} + 5 q^{15} - 3 q^{16} - 2 q^{17} - q^{18} + 4 q^{19} - 5 q^{20} + 2 q^{21} - q^{22} + 2 q^{23} + 5 q^{25} + 3 q^{26} + 2 q^{27} - q^{28} + 8 q^{29} - 4 q^{31} + 9 q^{32} + 2 q^{33} - 9 q^{34} + 5 q^{35} - q^{36} - 4 q^{37} + 3 q^{38} - q^{39} - 5 q^{40} + 2 q^{41} - q^{42} - q^{43} - q^{44} + 5 q^{45} - q^{46} + 10 q^{47} - 3 q^{48} + 2 q^{49} + 10 q^{50} - 2 q^{51} - 2 q^{52} + 11 q^{53} - q^{54} + 5 q^{55} + 4 q^{57} - 9 q^{58} - 5 q^{59} - 5 q^{60} - 9 q^{61} - 13 q^{62} + 2 q^{63} + 4 q^{64} - q^{66} - 5 q^{67} + 11 q^{68} + 2 q^{69} - 5 q^{71} + 8 q^{73} + 7 q^{74} + 5 q^{75} - 7 q^{76} + 2 q^{77} + 3 q^{78} - 24 q^{79} + 2 q^{81} - 11 q^{82} - 18 q^{83} - q^{84} - 15 q^{85} - 7 q^{86} + 8 q^{87} + 7 q^{89} - q^{91} - q^{92} - 4 q^{93} + 10 q^{94} + 15 q^{95} + 9 q^{96} - q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61803 −1.14412 −0.572061 0.820211i \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.618034 0.309017
\(5\) 1.38197 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) −2.23607 −0.707107
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0.618034 0.178411
\(13\) −1.61803 −0.448762 −0.224381 0.974502i \(-0.572036\pi\)
−0.224381 + 0.974502i \(0.572036\pi\)
\(14\) −1.61803 −0.432438
\(15\) 1.38197 0.356822
\(16\) −4.85410 −1.21353
\(17\) 3.47214 0.842117 0.421058 0.907034i \(-0.361659\pi\)
0.421058 + 0.907034i \(0.361659\pi\)
\(18\) −1.61803 −0.381374
\(19\) −0.236068 −0.0541577 −0.0270789 0.999633i \(-0.508621\pi\)
−0.0270789 + 0.999633i \(0.508621\pi\)
\(20\) 0.854102 0.190983
\(21\) 1.00000 0.218218
\(22\) −1.61803 −0.344966
\(23\) 1.00000 0.208514
\(24\) 2.23607 0.456435
\(25\) −3.09017 −0.618034
\(26\) 2.61803 0.513439
\(27\) 1.00000 0.192450
\(28\) 0.618034 0.116797
\(29\) 6.23607 1.15801 0.579004 0.815324i \(-0.303441\pi\)
0.579004 + 0.815324i \(0.303441\pi\)
\(30\) −2.23607 −0.408248
\(31\) 4.70820 0.845618 0.422809 0.906219i \(-0.361044\pi\)
0.422809 + 0.906219i \(0.361044\pi\)
\(32\) 3.38197 0.597853
\(33\) 1.00000 0.174078
\(34\) −5.61803 −0.963485
\(35\) 1.38197 0.233595
\(36\) 0.618034 0.103006
\(37\) −4.23607 −0.696405 −0.348203 0.937419i \(-0.613208\pi\)
−0.348203 + 0.937419i \(0.613208\pi\)
\(38\) 0.381966 0.0619631
\(39\) −1.61803 −0.259093
\(40\) 3.09017 0.488599
\(41\) 5.47214 0.854604 0.427302 0.904109i \(-0.359464\pi\)
0.427302 + 0.904109i \(0.359464\pi\)
\(42\) −1.61803 −0.249668
\(43\) 2.85410 0.435246 0.217623 0.976033i \(-0.430170\pi\)
0.217623 + 0.976033i \(0.430170\pi\)
\(44\) 0.618034 0.0931721
\(45\) 1.38197 0.206011
\(46\) −1.61803 −0.238566
\(47\) −1.70820 −0.249167 −0.124584 0.992209i \(-0.539759\pi\)
−0.124584 + 0.992209i \(0.539759\pi\)
\(48\) −4.85410 −0.700629
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) 3.47214 0.486196
\(52\) −1.00000 −0.138675
\(53\) 11.0902 1.52335 0.761676 0.647958i \(-0.224377\pi\)
0.761676 + 0.647958i \(0.224377\pi\)
\(54\) −1.61803 −0.220187
\(55\) 1.38197 0.186344
\(56\) 2.23607 0.298807
\(57\) −0.236068 −0.0312680
\(58\) −10.0902 −1.32490
\(59\) −1.38197 −0.179917 −0.0899583 0.995946i \(-0.528673\pi\)
−0.0899583 + 0.995946i \(0.528673\pi\)
\(60\) 0.854102 0.110264
\(61\) −1.14590 −0.146717 −0.0733586 0.997306i \(-0.523372\pi\)
−0.0733586 + 0.997306i \(0.523372\pi\)
\(62\) −7.61803 −0.967491
\(63\) 1.00000 0.125988
\(64\) 4.23607 0.529508
\(65\) −2.23607 −0.277350
\(66\) −1.61803 −0.199166
\(67\) 3.09017 0.377524 0.188762 0.982023i \(-0.439552\pi\)
0.188762 + 0.982023i \(0.439552\pi\)
\(68\) 2.14590 0.260228
\(69\) 1.00000 0.120386
\(70\) −2.23607 −0.267261
\(71\) 5.32624 0.632108 0.316054 0.948741i \(-0.397642\pi\)
0.316054 + 0.948741i \(0.397642\pi\)
\(72\) 2.23607 0.263523
\(73\) 6.23607 0.729877 0.364938 0.931032i \(-0.381090\pi\)
0.364938 + 0.931032i \(0.381090\pi\)
\(74\) 6.85410 0.796773
\(75\) −3.09017 −0.356822
\(76\) −0.145898 −0.0167357
\(77\) 1.00000 0.113961
\(78\) 2.61803 0.296434
\(79\) −9.76393 −1.09853 −0.549264 0.835649i \(-0.685092\pi\)
−0.549264 + 0.835649i \(0.685092\pi\)
\(80\) −6.70820 −0.750000
\(81\) 1.00000 0.111111
\(82\) −8.85410 −0.977772
\(83\) −0.0557281 −0.00611695 −0.00305848 0.999995i \(-0.500974\pi\)
−0.00305848 + 0.999995i \(0.500974\pi\)
\(84\) 0.618034 0.0674330
\(85\) 4.79837 0.520457
\(86\) −4.61803 −0.497975
\(87\) 6.23607 0.668577
\(88\) 2.23607 0.238366
\(89\) −6.56231 −0.695603 −0.347802 0.937568i \(-0.613072\pi\)
−0.347802 + 0.937568i \(0.613072\pi\)
\(90\) −2.23607 −0.235702
\(91\) −1.61803 −0.169616
\(92\) 0.618034 0.0644345
\(93\) 4.70820 0.488218
\(94\) 2.76393 0.285078
\(95\) −0.326238 −0.0334713
\(96\) 3.38197 0.345170
\(97\) −6.70820 −0.681115 −0.340557 0.940224i \(-0.610616\pi\)
−0.340557 + 0.940224i \(0.610616\pi\)
\(98\) −1.61803 −0.163446
\(99\) 1.00000 0.100504
\(100\) −1.90983 −0.190983
\(101\) −9.09017 −0.904506 −0.452253 0.891890i \(-0.649380\pi\)
−0.452253 + 0.891890i \(0.649380\pi\)
\(102\) −5.61803 −0.556268
\(103\) −9.52786 −0.938808 −0.469404 0.882983i \(-0.655531\pi\)
−0.469404 + 0.882983i \(0.655531\pi\)
\(104\) −3.61803 −0.354777
\(105\) 1.38197 0.134866
\(106\) −17.9443 −1.74290
\(107\) −14.0902 −1.36215 −0.681074 0.732215i \(-0.738487\pi\)
−0.681074 + 0.732215i \(0.738487\pi\)
\(108\) 0.618034 0.0594703
\(109\) 4.61803 0.442327 0.221164 0.975237i \(-0.429014\pi\)
0.221164 + 0.975237i \(0.429014\pi\)
\(110\) −2.23607 −0.213201
\(111\) −4.23607 −0.402070
\(112\) −4.85410 −0.458670
\(113\) 2.90983 0.273734 0.136867 0.990589i \(-0.456297\pi\)
0.136867 + 0.990589i \(0.456297\pi\)
\(114\) 0.381966 0.0357744
\(115\) 1.38197 0.128869
\(116\) 3.85410 0.357844
\(117\) −1.61803 −0.149587
\(118\) 2.23607 0.205847
\(119\) 3.47214 0.318290
\(120\) 3.09017 0.282093
\(121\) −10.0000 −0.909091
\(122\) 1.85410 0.167863
\(123\) 5.47214 0.493406
\(124\) 2.90983 0.261310
\(125\) −11.1803 −1.00000
\(126\) −1.61803 −0.144146
\(127\) −1.14590 −0.101682 −0.0508410 0.998707i \(-0.516190\pi\)
−0.0508410 + 0.998707i \(0.516190\pi\)
\(128\) −13.6180 −1.20368
\(129\) 2.85410 0.251290
\(130\) 3.61803 0.317323
\(131\) −16.7082 −1.45980 −0.729901 0.683553i \(-0.760434\pi\)
−0.729901 + 0.683553i \(0.760434\pi\)
\(132\) 0.618034 0.0537930
\(133\) −0.236068 −0.0204697
\(134\) −5.00000 −0.431934
\(135\) 1.38197 0.118941
\(136\) 7.76393 0.665752
\(137\) −7.18034 −0.613458 −0.306729 0.951797i \(-0.599235\pi\)
−0.306729 + 0.951797i \(0.599235\pi\)
\(138\) −1.61803 −0.137736
\(139\) −8.03444 −0.681472 −0.340736 0.940159i \(-0.610676\pi\)
−0.340736 + 0.940159i \(0.610676\pi\)
\(140\) 0.854102 0.0721848
\(141\) −1.70820 −0.143857
\(142\) −8.61803 −0.723209
\(143\) −1.61803 −0.135307
\(144\) −4.85410 −0.404508
\(145\) 8.61803 0.715689
\(146\) −10.0902 −0.835068
\(147\) 1.00000 0.0824786
\(148\) −2.61803 −0.215201
\(149\) −9.70820 −0.795327 −0.397664 0.917531i \(-0.630179\pi\)
−0.397664 + 0.917531i \(0.630179\pi\)
\(150\) 5.00000 0.408248
\(151\) −22.6525 −1.84343 −0.921716 0.387865i \(-0.873213\pi\)
−0.921716 + 0.387865i \(0.873213\pi\)
\(152\) −0.527864 −0.0428154
\(153\) 3.47214 0.280706
\(154\) −1.61803 −0.130385
\(155\) 6.50658 0.522621
\(156\) −1.00000 −0.0800641
\(157\) 13.2361 1.05635 0.528177 0.849135i \(-0.322876\pi\)
0.528177 + 0.849135i \(0.322876\pi\)
\(158\) 15.7984 1.25685
\(159\) 11.0902 0.879508
\(160\) 4.67376 0.369493
\(161\) 1.00000 0.0788110
\(162\) −1.61803 −0.127125
\(163\) 13.2705 1.03943 0.519713 0.854341i \(-0.326039\pi\)
0.519713 + 0.854341i \(0.326039\pi\)
\(164\) 3.38197 0.264087
\(165\) 1.38197 0.107586
\(166\) 0.0901699 0.00699854
\(167\) −2.70820 −0.209567 −0.104784 0.994495i \(-0.533415\pi\)
−0.104784 + 0.994495i \(0.533415\pi\)
\(168\) 2.23607 0.172516
\(169\) −10.3820 −0.798613
\(170\) −7.76393 −0.595466
\(171\) −0.236068 −0.0180526
\(172\) 1.76393 0.134499
\(173\) 20.4164 1.55223 0.776115 0.630591i \(-0.217187\pi\)
0.776115 + 0.630591i \(0.217187\pi\)
\(174\) −10.0902 −0.764934
\(175\) −3.09017 −0.233595
\(176\) −4.85410 −0.365892
\(177\) −1.38197 −0.103875
\(178\) 10.6180 0.795855
\(179\) 10.7984 0.807108 0.403554 0.914956i \(-0.367775\pi\)
0.403554 + 0.914956i \(0.367775\pi\)
\(180\) 0.854102 0.0636610
\(181\) 19.1803 1.42566 0.712832 0.701335i \(-0.247412\pi\)
0.712832 + 0.701335i \(0.247412\pi\)
\(182\) 2.61803 0.194062
\(183\) −1.14590 −0.0847072
\(184\) 2.23607 0.164845
\(185\) −5.85410 −0.430402
\(186\) −7.61803 −0.558581
\(187\) 3.47214 0.253908
\(188\) −1.05573 −0.0769969
\(189\) 1.00000 0.0727393
\(190\) 0.527864 0.0382953
\(191\) −8.18034 −0.591909 −0.295954 0.955202i \(-0.595638\pi\)
−0.295954 + 0.955202i \(0.595638\pi\)
\(192\) 4.23607 0.305712
\(193\) 5.23607 0.376900 0.188450 0.982083i \(-0.439654\pi\)
0.188450 + 0.982083i \(0.439654\pi\)
\(194\) 10.8541 0.779279
\(195\) −2.23607 −0.160128
\(196\) 0.618034 0.0441453
\(197\) −8.56231 −0.610039 −0.305020 0.952346i \(-0.598663\pi\)
−0.305020 + 0.952346i \(0.598663\pi\)
\(198\) −1.61803 −0.114989
\(199\) 21.5623 1.52851 0.764256 0.644913i \(-0.223107\pi\)
0.764256 + 0.644913i \(0.223107\pi\)
\(200\) −6.90983 −0.488599
\(201\) 3.09017 0.217964
\(202\) 14.7082 1.03487
\(203\) 6.23607 0.437686
\(204\) 2.14590 0.150243
\(205\) 7.56231 0.528174
\(206\) 15.4164 1.07411
\(207\) 1.00000 0.0695048
\(208\) 7.85410 0.544584
\(209\) −0.236068 −0.0163292
\(210\) −2.23607 −0.154303
\(211\) −26.8885 −1.85108 −0.925542 0.378645i \(-0.876390\pi\)
−0.925542 + 0.378645i \(0.876390\pi\)
\(212\) 6.85410 0.470742
\(213\) 5.32624 0.364948
\(214\) 22.7984 1.55846
\(215\) 3.94427 0.268997
\(216\) 2.23607 0.152145
\(217\) 4.70820 0.319614
\(218\) −7.47214 −0.506077
\(219\) 6.23607 0.421394
\(220\) 0.854102 0.0575835
\(221\) −5.61803 −0.377910
\(222\) 6.85410 0.460017
\(223\) −0.618034 −0.0413866 −0.0206933 0.999786i \(-0.506587\pi\)
−0.0206933 + 0.999786i \(0.506587\pi\)
\(224\) 3.38197 0.225967
\(225\) −3.09017 −0.206011
\(226\) −4.70820 −0.313185
\(227\) −22.9787 −1.52515 −0.762575 0.646899i \(-0.776065\pi\)
−0.762575 + 0.646899i \(0.776065\pi\)
\(228\) −0.145898 −0.00966233
\(229\) 3.20163 0.211569 0.105785 0.994389i \(-0.466265\pi\)
0.105785 + 0.994389i \(0.466265\pi\)
\(230\) −2.23607 −0.147442
\(231\) 1.00000 0.0657952
\(232\) 13.9443 0.915486
\(233\) 24.3262 1.59366 0.796832 0.604200i \(-0.206507\pi\)
0.796832 + 0.604200i \(0.206507\pi\)
\(234\) 2.61803 0.171146
\(235\) −2.36068 −0.153994
\(236\) −0.854102 −0.0555973
\(237\) −9.76393 −0.634236
\(238\) −5.61803 −0.364163
\(239\) −8.79837 −0.569119 −0.284560 0.958658i \(-0.591847\pi\)
−0.284560 + 0.958658i \(0.591847\pi\)
\(240\) −6.70820 −0.433013
\(241\) −16.7082 −1.07627 −0.538135 0.842859i \(-0.680871\pi\)
−0.538135 + 0.842859i \(0.680871\pi\)
\(242\) 16.1803 1.04011
\(243\) 1.00000 0.0641500
\(244\) −0.708204 −0.0453381
\(245\) 1.38197 0.0882906
\(246\) −8.85410 −0.564517
\(247\) 0.381966 0.0243039
\(248\) 10.5279 0.668520
\(249\) −0.0557281 −0.00353162
\(250\) 18.0902 1.14412
\(251\) 4.29180 0.270896 0.135448 0.990784i \(-0.456753\pi\)
0.135448 + 0.990784i \(0.456753\pi\)
\(252\) 0.618034 0.0389325
\(253\) 1.00000 0.0628695
\(254\) 1.85410 0.116337
\(255\) 4.79837 0.300486
\(256\) 13.5623 0.847644
\(257\) 10.2918 0.641985 0.320992 0.947082i \(-0.395984\pi\)
0.320992 + 0.947082i \(0.395984\pi\)
\(258\) −4.61803 −0.287506
\(259\) −4.23607 −0.263216
\(260\) −1.38197 −0.0857059
\(261\) 6.23607 0.386003
\(262\) 27.0344 1.67019
\(263\) −9.65248 −0.595197 −0.297599 0.954691i \(-0.596186\pi\)
−0.297599 + 0.954691i \(0.596186\pi\)
\(264\) 2.23607 0.137620
\(265\) 15.3262 0.941483
\(266\) 0.381966 0.0234198
\(267\) −6.56231 −0.401607
\(268\) 1.90983 0.116661
\(269\) −26.7426 −1.63053 −0.815264 0.579090i \(-0.803408\pi\)
−0.815264 + 0.579090i \(0.803408\pi\)
\(270\) −2.23607 −0.136083
\(271\) −6.52786 −0.396540 −0.198270 0.980147i \(-0.563532\pi\)
−0.198270 + 0.980147i \(0.563532\pi\)
\(272\) −16.8541 −1.02193
\(273\) −1.61803 −0.0979279
\(274\) 11.6180 0.701871
\(275\) −3.09017 −0.186344
\(276\) 0.618034 0.0372013
\(277\) −8.67376 −0.521156 −0.260578 0.965453i \(-0.583913\pi\)
−0.260578 + 0.965453i \(0.583913\pi\)
\(278\) 13.0000 0.779688
\(279\) 4.70820 0.281873
\(280\) 3.09017 0.184673
\(281\) 16.6525 0.993403 0.496702 0.867921i \(-0.334544\pi\)
0.496702 + 0.867921i \(0.334544\pi\)
\(282\) 2.76393 0.164590
\(283\) 9.67376 0.575045 0.287523 0.957774i \(-0.407168\pi\)
0.287523 + 0.957774i \(0.407168\pi\)
\(284\) 3.29180 0.195332
\(285\) −0.326238 −0.0193247
\(286\) 2.61803 0.154808
\(287\) 5.47214 0.323010
\(288\) 3.38197 0.199284
\(289\) −4.94427 −0.290840
\(290\) −13.9443 −0.818836
\(291\) −6.70820 −0.393242
\(292\) 3.85410 0.225544
\(293\) 30.8328 1.80127 0.900636 0.434574i \(-0.143101\pi\)
0.900636 + 0.434574i \(0.143101\pi\)
\(294\) −1.61803 −0.0943657
\(295\) −1.90983 −0.111195
\(296\) −9.47214 −0.550557
\(297\) 1.00000 0.0580259
\(298\) 15.7082 0.909952
\(299\) −1.61803 −0.0935733
\(300\) −1.90983 −0.110264
\(301\) 2.85410 0.164508
\(302\) 36.6525 2.10911
\(303\) −9.09017 −0.522217
\(304\) 1.14590 0.0657218
\(305\) −1.58359 −0.0906762
\(306\) −5.61803 −0.321162
\(307\) 6.12461 0.349550 0.174775 0.984608i \(-0.444080\pi\)
0.174775 + 0.984608i \(0.444080\pi\)
\(308\) 0.618034 0.0352158
\(309\) −9.52786 −0.542021
\(310\) −10.5279 −0.597942
\(311\) 27.5066 1.55975 0.779877 0.625932i \(-0.215281\pi\)
0.779877 + 0.625932i \(0.215281\pi\)
\(312\) −3.61803 −0.204831
\(313\) −1.88854 −0.106747 −0.0533734 0.998575i \(-0.516997\pi\)
−0.0533734 + 0.998575i \(0.516997\pi\)
\(314\) −21.4164 −1.20860
\(315\) 1.38197 0.0778650
\(316\) −6.03444 −0.339464
\(317\) 11.0902 0.622886 0.311443 0.950265i \(-0.399188\pi\)
0.311443 + 0.950265i \(0.399188\pi\)
\(318\) −17.9443 −1.00626
\(319\) 6.23607 0.349153
\(320\) 5.85410 0.327254
\(321\) −14.0902 −0.786437
\(322\) −1.61803 −0.0901695
\(323\) −0.819660 −0.0456071
\(324\) 0.618034 0.0343352
\(325\) 5.00000 0.277350
\(326\) −21.4721 −1.18923
\(327\) 4.61803 0.255378
\(328\) 12.2361 0.675624
\(329\) −1.70820 −0.0941763
\(330\) −2.23607 −0.123091
\(331\) −27.8885 −1.53289 −0.766447 0.642308i \(-0.777977\pi\)
−0.766447 + 0.642308i \(0.777977\pi\)
\(332\) −0.0344419 −0.00189024
\(333\) −4.23607 −0.232135
\(334\) 4.38197 0.239771
\(335\) 4.27051 0.233323
\(336\) −4.85410 −0.264813
\(337\) −19.0344 −1.03687 −0.518436 0.855116i \(-0.673486\pi\)
−0.518436 + 0.855116i \(0.673486\pi\)
\(338\) 16.7984 0.913711
\(339\) 2.90983 0.158040
\(340\) 2.96556 0.160830
\(341\) 4.70820 0.254964
\(342\) 0.381966 0.0206544
\(343\) 1.00000 0.0539949
\(344\) 6.38197 0.344093
\(345\) 1.38197 0.0744025
\(346\) −33.0344 −1.77594
\(347\) −20.7082 −1.11167 −0.555837 0.831291i \(-0.687602\pi\)
−0.555837 + 0.831291i \(0.687602\pi\)
\(348\) 3.85410 0.206602
\(349\) 4.79837 0.256851 0.128426 0.991719i \(-0.459008\pi\)
0.128426 + 0.991719i \(0.459008\pi\)
\(350\) 5.00000 0.267261
\(351\) −1.61803 −0.0863643
\(352\) 3.38197 0.180259
\(353\) −20.4164 −1.08666 −0.543328 0.839521i \(-0.682836\pi\)
−0.543328 + 0.839521i \(0.682836\pi\)
\(354\) 2.23607 0.118846
\(355\) 7.36068 0.390664
\(356\) −4.05573 −0.214953
\(357\) 3.47214 0.183765
\(358\) −17.4721 −0.923431
\(359\) −11.5623 −0.610235 −0.305118 0.952315i \(-0.598696\pi\)
−0.305118 + 0.952315i \(0.598696\pi\)
\(360\) 3.09017 0.162866
\(361\) −18.9443 −0.997067
\(362\) −31.0344 −1.63113
\(363\) −10.0000 −0.524864
\(364\) −1.00000 −0.0524142
\(365\) 8.61803 0.451089
\(366\) 1.85410 0.0969155
\(367\) −20.2705 −1.05811 −0.529056 0.848587i \(-0.677454\pi\)
−0.529056 + 0.848587i \(0.677454\pi\)
\(368\) −4.85410 −0.253038
\(369\) 5.47214 0.284868
\(370\) 9.47214 0.492433
\(371\) 11.0902 0.575773
\(372\) 2.90983 0.150868
\(373\) −6.12461 −0.317120 −0.158560 0.987349i \(-0.550685\pi\)
−0.158560 + 0.987349i \(0.550685\pi\)
\(374\) −5.61803 −0.290502
\(375\) −11.1803 −0.577350
\(376\) −3.81966 −0.196984
\(377\) −10.0902 −0.519670
\(378\) −1.61803 −0.0832227
\(379\) −34.4721 −1.77071 −0.885357 0.464911i \(-0.846086\pi\)
−0.885357 + 0.464911i \(0.846086\pi\)
\(380\) −0.201626 −0.0103432
\(381\) −1.14590 −0.0587061
\(382\) 13.2361 0.677216
\(383\) 13.9443 0.712519 0.356260 0.934387i \(-0.384052\pi\)
0.356260 + 0.934387i \(0.384052\pi\)
\(384\) −13.6180 −0.694942
\(385\) 1.38197 0.0704315
\(386\) −8.47214 −0.431220
\(387\) 2.85410 0.145082
\(388\) −4.14590 −0.210476
\(389\) −2.05573 −0.104230 −0.0521148 0.998641i \(-0.516596\pi\)
−0.0521148 + 0.998641i \(0.516596\pi\)
\(390\) 3.61803 0.183206
\(391\) 3.47214 0.175593
\(392\) 2.23607 0.112938
\(393\) −16.7082 −0.842817
\(394\) 13.8541 0.697960
\(395\) −13.4934 −0.678928
\(396\) 0.618034 0.0310574
\(397\) −0.180340 −0.00905100 −0.00452550 0.999990i \(-0.501441\pi\)
−0.00452550 + 0.999990i \(0.501441\pi\)
\(398\) −34.8885 −1.74880
\(399\) −0.236068 −0.0118182
\(400\) 15.0000 0.750000
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) −5.00000 −0.249377
\(403\) −7.61803 −0.379481
\(404\) −5.61803 −0.279508
\(405\) 1.38197 0.0686704
\(406\) −10.0902 −0.500767
\(407\) −4.23607 −0.209974
\(408\) 7.76393 0.384372
\(409\) 10.8885 0.538404 0.269202 0.963084i \(-0.413240\pi\)
0.269202 + 0.963084i \(0.413240\pi\)
\(410\) −12.2361 −0.604296
\(411\) −7.18034 −0.354180
\(412\) −5.88854 −0.290108
\(413\) −1.38197 −0.0680021
\(414\) −1.61803 −0.0795220
\(415\) −0.0770143 −0.00378048
\(416\) −5.47214 −0.268294
\(417\) −8.03444 −0.393448
\(418\) 0.381966 0.0186826
\(419\) 25.0902 1.22574 0.612868 0.790186i \(-0.290016\pi\)
0.612868 + 0.790186i \(0.290016\pi\)
\(420\) 0.854102 0.0416759
\(421\) −9.43769 −0.459965 −0.229983 0.973195i \(-0.573867\pi\)
−0.229983 + 0.973195i \(0.573867\pi\)
\(422\) 43.5066 2.11787
\(423\) −1.70820 −0.0830557
\(424\) 24.7984 1.20432
\(425\) −10.7295 −0.520457
\(426\) −8.61803 −0.417545
\(427\) −1.14590 −0.0554539
\(428\) −8.70820 −0.420927
\(429\) −1.61803 −0.0781194
\(430\) −6.38197 −0.307766
\(431\) −22.2705 −1.07273 −0.536366 0.843985i \(-0.680203\pi\)
−0.536366 + 0.843985i \(0.680203\pi\)
\(432\) −4.85410 −0.233543
\(433\) 25.2361 1.21277 0.606384 0.795172i \(-0.292619\pi\)
0.606384 + 0.795172i \(0.292619\pi\)
\(434\) −7.61803 −0.365677
\(435\) 8.61803 0.413203
\(436\) 2.85410 0.136687
\(437\) −0.236068 −0.0112927
\(438\) −10.0902 −0.482127
\(439\) −13.1803 −0.629063 −0.314532 0.949247i \(-0.601847\pi\)
−0.314532 + 0.949247i \(0.601847\pi\)
\(440\) 3.09017 0.147318
\(441\) 1.00000 0.0476190
\(442\) 9.09017 0.432375
\(443\) 26.8328 1.27487 0.637433 0.770506i \(-0.279996\pi\)
0.637433 + 0.770506i \(0.279996\pi\)
\(444\) −2.61803 −0.124246
\(445\) −9.06888 −0.429906
\(446\) 1.00000 0.0473514
\(447\) −9.70820 −0.459182
\(448\) 4.23607 0.200135
\(449\) 6.72949 0.317584 0.158792 0.987312i \(-0.449240\pi\)
0.158792 + 0.987312i \(0.449240\pi\)
\(450\) 5.00000 0.235702
\(451\) 5.47214 0.257673
\(452\) 1.79837 0.0845884
\(453\) −22.6525 −1.06431
\(454\) 37.1803 1.74496
\(455\) −2.23607 −0.104828
\(456\) −0.527864 −0.0247195
\(457\) 39.7426 1.85908 0.929541 0.368718i \(-0.120203\pi\)
0.929541 + 0.368718i \(0.120203\pi\)
\(458\) −5.18034 −0.242061
\(459\) 3.47214 0.162065
\(460\) 0.854102 0.0398227
\(461\) 10.3262 0.480941 0.240470 0.970656i \(-0.422698\pi\)
0.240470 + 0.970656i \(0.422698\pi\)
\(462\) −1.61803 −0.0752778
\(463\) −26.1246 −1.21411 −0.607057 0.794658i \(-0.707650\pi\)
−0.607057 + 0.794658i \(0.707650\pi\)
\(464\) −30.2705 −1.40527
\(465\) 6.50658 0.301735
\(466\) −39.3607 −1.82335
\(467\) 2.88854 0.133666 0.0668329 0.997764i \(-0.478711\pi\)
0.0668329 + 0.997764i \(0.478711\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 3.09017 0.142691
\(470\) 3.81966 0.176188
\(471\) 13.2361 0.609886
\(472\) −3.09017 −0.142237
\(473\) 2.85410 0.131232
\(474\) 15.7984 0.725643
\(475\) 0.729490 0.0334713
\(476\) 2.14590 0.0983571
\(477\) 11.0902 0.507784
\(478\) 14.2361 0.651143
\(479\) −9.29180 −0.424553 −0.212276 0.977210i \(-0.568088\pi\)
−0.212276 + 0.977210i \(0.568088\pi\)
\(480\) 4.67376 0.213327
\(481\) 6.85410 0.312520
\(482\) 27.0344 1.23139
\(483\) 1.00000 0.0455016
\(484\) −6.18034 −0.280925
\(485\) −9.27051 −0.420952
\(486\) −1.61803 −0.0733955
\(487\) 10.1246 0.458790 0.229395 0.973333i \(-0.426325\pi\)
0.229395 + 0.973333i \(0.426325\pi\)
\(488\) −2.56231 −0.115990
\(489\) 13.2705 0.600113
\(490\) −2.23607 −0.101015
\(491\) −22.5623 −1.01822 −0.509111 0.860701i \(-0.670026\pi\)
−0.509111 + 0.860701i \(0.670026\pi\)
\(492\) 3.38197 0.152471
\(493\) 21.6525 0.975178
\(494\) −0.618034 −0.0278067
\(495\) 1.38197 0.0621148
\(496\) −22.8541 −1.02618
\(497\) 5.32624 0.238914
\(498\) 0.0901699 0.00404061
\(499\) −3.56231 −0.159471 −0.0797354 0.996816i \(-0.525408\pi\)
−0.0797354 + 0.996816i \(0.525408\pi\)
\(500\) −6.90983 −0.309017
\(501\) −2.70820 −0.120994
\(502\) −6.94427 −0.309938
\(503\) 34.6869 1.54661 0.773307 0.634032i \(-0.218601\pi\)
0.773307 + 0.634032i \(0.218601\pi\)
\(504\) 2.23607 0.0996024
\(505\) −12.5623 −0.559015
\(506\) −1.61803 −0.0719304
\(507\) −10.3820 −0.461079
\(508\) −0.708204 −0.0314215
\(509\) 35.2361 1.56181 0.780906 0.624649i \(-0.214758\pi\)
0.780906 + 0.624649i \(0.214758\pi\)
\(510\) −7.76393 −0.343793
\(511\) 6.23607 0.275867
\(512\) 5.29180 0.233867
\(513\) −0.236068 −0.0104227
\(514\) −16.6525 −0.734509
\(515\) −13.1672 −0.580215
\(516\) 1.76393 0.0776528
\(517\) −1.70820 −0.0751267
\(518\) 6.85410 0.301152
\(519\) 20.4164 0.896181
\(520\) −5.00000 −0.219265
\(521\) 5.41641 0.237297 0.118649 0.992936i \(-0.462144\pi\)
0.118649 + 0.992936i \(0.462144\pi\)
\(522\) −10.0902 −0.441635
\(523\) −37.4164 −1.63611 −0.818053 0.575143i \(-0.804946\pi\)
−0.818053 + 0.575143i \(0.804946\pi\)
\(524\) −10.3262 −0.451104
\(525\) −3.09017 −0.134866
\(526\) 15.6180 0.680979
\(527\) 16.3475 0.712109
\(528\) −4.85410 −0.211248
\(529\) 1.00000 0.0434783
\(530\) −24.7984 −1.07717
\(531\) −1.38197 −0.0599722
\(532\) −0.145898 −0.00632548
\(533\) −8.85410 −0.383514
\(534\) 10.6180 0.459487
\(535\) −19.4721 −0.841854
\(536\) 6.90983 0.298459
\(537\) 10.7984 0.465984
\(538\) 43.2705 1.86552
\(539\) 1.00000 0.0430730
\(540\) 0.854102 0.0367547
\(541\) −2.29180 −0.0985320 −0.0492660 0.998786i \(-0.515688\pi\)
−0.0492660 + 0.998786i \(0.515688\pi\)
\(542\) 10.5623 0.453690
\(543\) 19.1803 0.823107
\(544\) 11.7426 0.503462
\(545\) 6.38197 0.273373
\(546\) 2.61803 0.112042
\(547\) −32.1459 −1.37446 −0.687230 0.726440i \(-0.741173\pi\)
−0.687230 + 0.726440i \(0.741173\pi\)
\(548\) −4.43769 −0.189569
\(549\) −1.14590 −0.0489057
\(550\) 5.00000 0.213201
\(551\) −1.47214 −0.0627151
\(552\) 2.23607 0.0951734
\(553\) −9.76393 −0.415205
\(554\) 14.0344 0.596266
\(555\) −5.85410 −0.248493
\(556\) −4.96556 −0.210587
\(557\) 17.2361 0.730316 0.365158 0.930946i \(-0.381015\pi\)
0.365158 + 0.930946i \(0.381015\pi\)
\(558\) −7.61803 −0.322497
\(559\) −4.61803 −0.195322
\(560\) −6.70820 −0.283473
\(561\) 3.47214 0.146594
\(562\) −26.9443 −1.13658
\(563\) −15.2148 −0.641227 −0.320613 0.947210i \(-0.603889\pi\)
−0.320613 + 0.947210i \(0.603889\pi\)
\(564\) −1.05573 −0.0444542
\(565\) 4.02129 0.169177
\(566\) −15.6525 −0.657923
\(567\) 1.00000 0.0419961
\(568\) 11.9098 0.499725
\(569\) −9.88854 −0.414549 −0.207275 0.978283i \(-0.566459\pi\)
−0.207275 + 0.978283i \(0.566459\pi\)
\(570\) 0.527864 0.0221098
\(571\) 24.5967 1.02934 0.514671 0.857388i \(-0.327914\pi\)
0.514671 + 0.857388i \(0.327914\pi\)
\(572\) −1.00000 −0.0418121
\(573\) −8.18034 −0.341739
\(574\) −8.85410 −0.369563
\(575\) −3.09017 −0.128869
\(576\) 4.23607 0.176503
\(577\) 40.2492 1.67560 0.837799 0.545979i \(-0.183842\pi\)
0.837799 + 0.545979i \(0.183842\pi\)
\(578\) 8.00000 0.332756
\(579\) 5.23607 0.217604
\(580\) 5.32624 0.221160
\(581\) −0.0557281 −0.00231199
\(582\) 10.8541 0.449917
\(583\) 11.0902 0.459308
\(584\) 13.9443 0.577018
\(585\) −2.23607 −0.0924500
\(586\) −49.8885 −2.06088
\(587\) 20.2016 0.833810 0.416905 0.908950i \(-0.363115\pi\)
0.416905 + 0.908950i \(0.363115\pi\)
\(588\) 0.618034 0.0254873
\(589\) −1.11146 −0.0457968
\(590\) 3.09017 0.127220
\(591\) −8.56231 −0.352206
\(592\) 20.5623 0.845106
\(593\) −4.87539 −0.200208 −0.100104 0.994977i \(-0.531918\pi\)
−0.100104 + 0.994977i \(0.531918\pi\)
\(594\) −1.61803 −0.0663887
\(595\) 4.79837 0.196714
\(596\) −6.00000 −0.245770
\(597\) 21.5623 0.882486
\(598\) 2.61803 0.107059
\(599\) −9.61803 −0.392982 −0.196491 0.980506i \(-0.562955\pi\)
−0.196491 + 0.980506i \(0.562955\pi\)
\(600\) −6.90983 −0.282093
\(601\) −28.6869 −1.17016 −0.585082 0.810974i \(-0.698938\pi\)
−0.585082 + 0.810974i \(0.698938\pi\)
\(602\) −4.61803 −0.188217
\(603\) 3.09017 0.125841
\(604\) −14.0000 −0.569652
\(605\) −13.8197 −0.561849
\(606\) 14.7082 0.597480
\(607\) −31.5066 −1.27881 −0.639406 0.768869i \(-0.720820\pi\)
−0.639406 + 0.768869i \(0.720820\pi\)
\(608\) −0.798374 −0.0323783
\(609\) 6.23607 0.252698
\(610\) 2.56231 0.103745
\(611\) 2.76393 0.111817
\(612\) 2.14590 0.0867428
\(613\) 31.8328 1.28572 0.642858 0.765986i \(-0.277749\pi\)
0.642858 + 0.765986i \(0.277749\pi\)
\(614\) −9.90983 −0.399928
\(615\) 7.56231 0.304942
\(616\) 2.23607 0.0900937
\(617\) 34.7426 1.39869 0.699343 0.714786i \(-0.253476\pi\)
0.699343 + 0.714786i \(0.253476\pi\)
\(618\) 15.4164 0.620139
\(619\) 16.0902 0.646719 0.323359 0.946276i \(-0.395188\pi\)
0.323359 + 0.946276i \(0.395188\pi\)
\(620\) 4.02129 0.161499
\(621\) 1.00000 0.0401286
\(622\) −44.5066 −1.78455
\(623\) −6.56231 −0.262913
\(624\) 7.85410 0.314416
\(625\) 0 0
\(626\) 3.05573 0.122131
\(627\) −0.236068 −0.00942765
\(628\) 8.18034 0.326431
\(629\) −14.7082 −0.586454
\(630\) −2.23607 −0.0890871
\(631\) 5.00000 0.199047 0.0995234 0.995035i \(-0.468268\pi\)
0.0995234 + 0.995035i \(0.468268\pi\)
\(632\) −21.8328 −0.868463
\(633\) −26.8885 −1.06872
\(634\) −17.9443 −0.712658
\(635\) −1.58359 −0.0628429
\(636\) 6.85410 0.271783
\(637\) −1.61803 −0.0641088
\(638\) −10.0902 −0.399474
\(639\) 5.32624 0.210703
\(640\) −18.8197 −0.743912
\(641\) −0.201626 −0.00796375 −0.00398188 0.999992i \(-0.501267\pi\)
−0.00398188 + 0.999992i \(0.501267\pi\)
\(642\) 22.7984 0.899780
\(643\) 33.0344 1.30275 0.651376 0.758755i \(-0.274192\pi\)
0.651376 + 0.758755i \(0.274192\pi\)
\(644\) 0.618034 0.0243540
\(645\) 3.94427 0.155306
\(646\) 1.32624 0.0521801
\(647\) 6.72949 0.264564 0.132282 0.991212i \(-0.457770\pi\)
0.132282 + 0.991212i \(0.457770\pi\)
\(648\) 2.23607 0.0878410
\(649\) −1.38197 −0.0542469
\(650\) −8.09017 −0.317323
\(651\) 4.70820 0.184529
\(652\) 8.20163 0.321200
\(653\) −7.09017 −0.277460 −0.138730 0.990330i \(-0.544302\pi\)
−0.138730 + 0.990330i \(0.544302\pi\)
\(654\) −7.47214 −0.292184
\(655\) −23.0902 −0.902208
\(656\) −26.5623 −1.03708
\(657\) 6.23607 0.243292
\(658\) 2.76393 0.107749
\(659\) −18.0557 −0.703351 −0.351676 0.936122i \(-0.614388\pi\)
−0.351676 + 0.936122i \(0.614388\pi\)
\(660\) 0.854102 0.0332459
\(661\) −7.65248 −0.297647 −0.148823 0.988864i \(-0.547549\pi\)
−0.148823 + 0.988864i \(0.547549\pi\)
\(662\) 45.1246 1.75382
\(663\) −5.61803 −0.218186
\(664\) −0.124612 −0.00483588
\(665\) −0.326238 −0.0126510
\(666\) 6.85410 0.265591
\(667\) 6.23607 0.241462
\(668\) −1.67376 −0.0647598
\(669\) −0.618034 −0.0238946
\(670\) −6.90983 −0.266950
\(671\) −1.14590 −0.0442369
\(672\) 3.38197 0.130462
\(673\) −7.58359 −0.292326 −0.146163 0.989261i \(-0.546692\pi\)
−0.146163 + 0.989261i \(0.546692\pi\)
\(674\) 30.7984 1.18631
\(675\) −3.09017 −0.118941
\(676\) −6.41641 −0.246785
\(677\) 18.6738 0.717691 0.358845 0.933397i \(-0.383170\pi\)
0.358845 + 0.933397i \(0.383170\pi\)
\(678\) −4.70820 −0.180817
\(679\) −6.70820 −0.257437
\(680\) 10.7295 0.411457
\(681\) −22.9787 −0.880546
\(682\) −7.61803 −0.291710
\(683\) −26.1246 −0.999630 −0.499815 0.866132i \(-0.666599\pi\)
−0.499815 + 0.866132i \(0.666599\pi\)
\(684\) −0.145898 −0.00557855
\(685\) −9.92299 −0.379138
\(686\) −1.61803 −0.0617768
\(687\) 3.20163 0.122150
\(688\) −13.8541 −0.528183
\(689\) −17.9443 −0.683622
\(690\) −2.23607 −0.0851257
\(691\) −18.3820 −0.699283 −0.349641 0.936884i \(-0.613697\pi\)
−0.349641 + 0.936884i \(0.613697\pi\)
\(692\) 12.6180 0.479666
\(693\) 1.00000 0.0379869
\(694\) 33.5066 1.27189
\(695\) −11.1033 −0.421173
\(696\) 13.9443 0.528556
\(697\) 19.0000 0.719676
\(698\) −7.76393 −0.293869
\(699\) 24.3262 0.920103
\(700\) −1.90983 −0.0721848
\(701\) −3.09017 −0.116714 −0.0583571 0.998296i \(-0.518586\pi\)
−0.0583571 + 0.998296i \(0.518586\pi\)
\(702\) 2.61803 0.0988113
\(703\) 1.00000 0.0377157
\(704\) 4.23607 0.159653
\(705\) −2.36068 −0.0889083
\(706\) 33.0344 1.24327
\(707\) −9.09017 −0.341871
\(708\) −0.854102 −0.0320991
\(709\) 42.2705 1.58750 0.793751 0.608243i \(-0.208125\pi\)
0.793751 + 0.608243i \(0.208125\pi\)
\(710\) −11.9098 −0.446968
\(711\) −9.76393 −0.366176
\(712\) −14.6738 −0.549922
\(713\) 4.70820 0.176324
\(714\) −5.61803 −0.210250
\(715\) −2.23607 −0.0836242
\(716\) 6.67376 0.249410
\(717\) −8.79837 −0.328581
\(718\) 18.7082 0.698184
\(719\) −2.88854 −0.107725 −0.0538623 0.998548i \(-0.517153\pi\)
−0.0538623 + 0.998548i \(0.517153\pi\)
\(720\) −6.70820 −0.250000
\(721\) −9.52786 −0.354836
\(722\) 30.6525 1.14077
\(723\) −16.7082 −0.621385
\(724\) 11.8541 0.440554
\(725\) −19.2705 −0.715689
\(726\) 16.1803 0.600509
\(727\) −15.5410 −0.576385 −0.288192 0.957573i \(-0.593054\pi\)
−0.288192 + 0.957573i \(0.593054\pi\)
\(728\) −3.61803 −0.134093
\(729\) 1.00000 0.0370370
\(730\) −13.9443 −0.516101
\(731\) 9.90983 0.366528
\(732\) −0.708204 −0.0261760
\(733\) 42.7639 1.57952 0.789761 0.613415i \(-0.210205\pi\)
0.789761 + 0.613415i \(0.210205\pi\)
\(734\) 32.7984 1.21061
\(735\) 1.38197 0.0509746
\(736\) 3.38197 0.124661
\(737\) 3.09017 0.113828
\(738\) −8.85410 −0.325924
\(739\) −22.5279 −0.828701 −0.414350 0.910117i \(-0.635991\pi\)
−0.414350 + 0.910117i \(0.635991\pi\)
\(740\) −3.61803 −0.133002
\(741\) 0.381966 0.0140319
\(742\) −17.9443 −0.658755
\(743\) 13.3820 0.490937 0.245468 0.969405i \(-0.421058\pi\)
0.245468 + 0.969405i \(0.421058\pi\)
\(744\) 10.5279 0.385970
\(745\) −13.4164 −0.491539
\(746\) 9.90983 0.362825
\(747\) −0.0557281 −0.00203898
\(748\) 2.14590 0.0784618
\(749\) −14.0902 −0.514844
\(750\) 18.0902 0.660560
\(751\) −45.4508 −1.65853 −0.829263 0.558859i \(-0.811239\pi\)
−0.829263 + 0.558859i \(0.811239\pi\)
\(752\) 8.29180 0.302371
\(753\) 4.29180 0.156402
\(754\) 16.3262 0.594567
\(755\) −31.3050 −1.13930
\(756\) 0.618034 0.0224777
\(757\) 22.3475 0.812235 0.406117 0.913821i \(-0.366882\pi\)
0.406117 + 0.913821i \(0.366882\pi\)
\(758\) 55.7771 2.02592
\(759\) 1.00000 0.0362977
\(760\) −0.729490 −0.0264614
\(761\) 6.11146 0.221540 0.110770 0.993846i \(-0.464668\pi\)
0.110770 + 0.993846i \(0.464668\pi\)
\(762\) 1.85410 0.0671670
\(763\) 4.61803 0.167184
\(764\) −5.05573 −0.182910
\(765\) 4.79837 0.173486
\(766\) −22.5623 −0.815209
\(767\) 2.23607 0.0807397
\(768\) 13.5623 0.489388
\(769\) 20.6525 0.744747 0.372374 0.928083i \(-0.378544\pi\)
0.372374 + 0.928083i \(0.378544\pi\)
\(770\) −2.23607 −0.0805823
\(771\) 10.2918 0.370650
\(772\) 3.23607 0.116469
\(773\) 28.5967 1.02855 0.514277 0.857624i \(-0.328060\pi\)
0.514277 + 0.857624i \(0.328060\pi\)
\(774\) −4.61803 −0.165992
\(775\) −14.5492 −0.522621
\(776\) −15.0000 −0.538469
\(777\) −4.23607 −0.151968
\(778\) 3.32624 0.119251
\(779\) −1.29180 −0.0462834
\(780\) −1.38197 −0.0494823
\(781\) 5.32624 0.190588
\(782\) −5.61803 −0.200900
\(783\) 6.23607 0.222859
\(784\) −4.85410 −0.173361
\(785\) 18.2918 0.652862
\(786\) 27.0344 0.964287
\(787\) −10.7984 −0.384920 −0.192460 0.981305i \(-0.561647\pi\)
−0.192460 + 0.981305i \(0.561647\pi\)
\(788\) −5.29180 −0.188512
\(789\) −9.65248 −0.343637
\(790\) 21.8328 0.776777
\(791\) 2.90983 0.103462
\(792\) 2.23607 0.0794552
\(793\) 1.85410 0.0658411
\(794\) 0.291796 0.0103555
\(795\) 15.3262 0.543566
\(796\) 13.3262 0.472336
\(797\) −33.1246 −1.17333 −0.586667 0.809828i \(-0.699560\pi\)
−0.586667 + 0.809828i \(0.699560\pi\)
\(798\) 0.381966 0.0135215
\(799\) −5.93112 −0.209828
\(800\) −10.4508 −0.369493
\(801\) −6.56231 −0.231868
\(802\) 8.09017 0.285674
\(803\) 6.23607 0.220066
\(804\) 1.90983 0.0673545
\(805\) 1.38197 0.0487079
\(806\) 12.3262 0.434173
\(807\) −26.7426 −0.941386
\(808\) −20.3262 −0.715075
\(809\) −40.2705 −1.41584 −0.707918 0.706295i \(-0.750365\pi\)
−0.707918 + 0.706295i \(0.750365\pi\)
\(810\) −2.23607 −0.0785674
\(811\) 50.3050 1.76645 0.883223 0.468953i \(-0.155369\pi\)
0.883223 + 0.468953i \(0.155369\pi\)
\(812\) 3.85410 0.135252
\(813\) −6.52786 −0.228942
\(814\) 6.85410 0.240236
\(815\) 18.3394 0.642401
\(816\) −16.8541 −0.590012
\(817\) −0.673762 −0.0235720
\(818\) −17.6180 −0.616000
\(819\) −1.61803 −0.0565387
\(820\) 4.67376 0.163215
\(821\) 36.5410 1.27529 0.637645 0.770330i \(-0.279909\pi\)
0.637645 + 0.770330i \(0.279909\pi\)
\(822\) 11.6180 0.405225
\(823\) −8.21478 −0.286349 −0.143175 0.989697i \(-0.545731\pi\)
−0.143175 + 0.989697i \(0.545731\pi\)
\(824\) −21.3050 −0.742193
\(825\) −3.09017 −0.107586
\(826\) 2.23607 0.0778028
\(827\) 5.85410 0.203567 0.101784 0.994807i \(-0.467545\pi\)
0.101784 + 0.994807i \(0.467545\pi\)
\(828\) 0.618034 0.0214782
\(829\) −32.7771 −1.13840 −0.569198 0.822201i \(-0.692746\pi\)
−0.569198 + 0.822201i \(0.692746\pi\)
\(830\) 0.124612 0.00432534
\(831\) −8.67376 −0.300889
\(832\) −6.85410 −0.237623
\(833\) 3.47214 0.120302
\(834\) 13.0000 0.450153
\(835\) −3.74265 −0.129520
\(836\) −0.145898 −0.00504599
\(837\) 4.70820 0.162739
\(838\) −40.5967 −1.40239
\(839\) 45.9787 1.58736 0.793681 0.608335i \(-0.208162\pi\)
0.793681 + 0.608335i \(0.208162\pi\)
\(840\) 3.09017 0.106621
\(841\) 9.88854 0.340984
\(842\) 15.2705 0.526257
\(843\) 16.6525 0.573542
\(844\) −16.6180 −0.572016
\(845\) −14.3475 −0.493570
\(846\) 2.76393 0.0950259
\(847\) −10.0000 −0.343604
\(848\) −53.8328 −1.84863
\(849\) 9.67376 0.332003
\(850\) 17.3607 0.595466
\(851\) −4.23607 −0.145211
\(852\) 3.29180 0.112775
\(853\) −36.0689 −1.23498 −0.617488 0.786581i \(-0.711849\pi\)
−0.617488 + 0.786581i \(0.711849\pi\)
\(854\) 1.85410 0.0634461
\(855\) −0.326238 −0.0111571
\(856\) −31.5066 −1.07687
\(857\) −39.1246 −1.33647 −0.668236 0.743950i \(-0.732950\pi\)
−0.668236 + 0.743950i \(0.732950\pi\)
\(858\) 2.61803 0.0893782
\(859\) −46.3607 −1.58181 −0.790903 0.611942i \(-0.790389\pi\)
−0.790903 + 0.611942i \(0.790389\pi\)
\(860\) 2.43769 0.0831247
\(861\) 5.47214 0.186490
\(862\) 36.0344 1.22734
\(863\) 30.1803 1.02735 0.513675 0.857985i \(-0.328284\pi\)
0.513675 + 0.857985i \(0.328284\pi\)
\(864\) 3.38197 0.115057
\(865\) 28.2148 0.959331
\(866\) −40.8328 −1.38756
\(867\) −4.94427 −0.167916
\(868\) 2.90983 0.0987661
\(869\) −9.76393 −0.331219
\(870\) −13.9443 −0.472755
\(871\) −5.00000 −0.169419
\(872\) 10.3262 0.349691
\(873\) −6.70820 −0.227038
\(874\) 0.381966 0.0129202
\(875\) −11.1803 −0.377964
\(876\) 3.85410 0.130218
\(877\) 20.0000 0.675352 0.337676 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 21.3262 0.719726
\(879\) 30.8328 1.03997
\(880\) −6.70820 −0.226134
\(881\) −47.7082 −1.60733 −0.803665 0.595082i \(-0.797120\pi\)
−0.803665 + 0.595082i \(0.797120\pi\)
\(882\) −1.61803 −0.0544820
\(883\) 16.0902 0.541477 0.270739 0.962653i \(-0.412732\pi\)
0.270739 + 0.962653i \(0.412732\pi\)
\(884\) −3.47214 −0.116781
\(885\) −1.90983 −0.0641982
\(886\) −43.4164 −1.45860
\(887\) 49.9787 1.67812 0.839060 0.544038i \(-0.183105\pi\)
0.839060 + 0.544038i \(0.183105\pi\)
\(888\) −9.47214 −0.317864
\(889\) −1.14590 −0.0384322
\(890\) 14.6738 0.491866
\(891\) 1.00000 0.0335013
\(892\) −0.381966 −0.0127892
\(893\) 0.403252 0.0134943
\(894\) 15.7082 0.525361
\(895\) 14.9230 0.498820
\(896\) −13.6180 −0.454947
\(897\) −1.61803 −0.0540246
\(898\) −10.8885 −0.363355
\(899\) 29.3607 0.979233
\(900\) −1.90983 −0.0636610
\(901\) 38.5066 1.28284
\(902\) −8.85410 −0.294809
\(903\) 2.85410 0.0949786
\(904\) 6.50658 0.216406
\(905\) 26.5066 0.881108
\(906\) 36.6525 1.21770
\(907\) 31.2705 1.03832 0.519160 0.854677i \(-0.326245\pi\)
0.519160 + 0.854677i \(0.326245\pi\)
\(908\) −14.2016 −0.471298
\(909\) −9.09017 −0.301502
\(910\) 3.61803 0.119937
\(911\) 35.0132 1.16004 0.580019 0.814603i \(-0.303045\pi\)
0.580019 + 0.814603i \(0.303045\pi\)
\(912\) 1.14590 0.0379445
\(913\) −0.0557281 −0.00184433
\(914\) −64.3050 −2.12702
\(915\) −1.58359 −0.0523519
\(916\) 1.97871 0.0653785
\(917\) −16.7082 −0.551754
\(918\) −5.61803 −0.185423
\(919\) 56.1935 1.85365 0.926826 0.375491i \(-0.122526\pi\)
0.926826 + 0.375491i \(0.122526\pi\)
\(920\) 3.09017 0.101880
\(921\) 6.12461 0.201813
\(922\) −16.7082 −0.550255
\(923\) −8.61803 −0.283666
\(924\) 0.618034 0.0203318
\(925\) 13.0902 0.430402
\(926\) 42.2705 1.38910
\(927\) −9.52786 −0.312936
\(928\) 21.0902 0.692319
\(929\) 29.9230 0.981741 0.490871 0.871232i \(-0.336679\pi\)
0.490871 + 0.871232i \(0.336679\pi\)
\(930\) −10.5279 −0.345222
\(931\) −0.236068 −0.00773682
\(932\) 15.0344 0.492470
\(933\) 27.5066 0.900525
\(934\) −4.67376 −0.152930
\(935\) 4.79837 0.156924
\(936\) −3.61803 −0.118259
\(937\) −23.3607 −0.763160 −0.381580 0.924336i \(-0.624620\pi\)
−0.381580 + 0.924336i \(0.624620\pi\)
\(938\) −5.00000 −0.163256
\(939\) −1.88854 −0.0616303
\(940\) −1.45898 −0.0475867
\(941\) 44.3607 1.44612 0.723058 0.690787i \(-0.242736\pi\)
0.723058 + 0.690787i \(0.242736\pi\)
\(942\) −21.4164 −0.697784
\(943\) 5.47214 0.178197
\(944\) 6.70820 0.218333
\(945\) 1.38197 0.0449554
\(946\) −4.61803 −0.150145
\(947\) −53.4164 −1.73580 −0.867900 0.496739i \(-0.834531\pi\)
−0.867900 + 0.496739i \(0.834531\pi\)
\(948\) −6.03444 −0.195990
\(949\) −10.0902 −0.327541
\(950\) −1.18034 −0.0382953
\(951\) 11.0902 0.359623
\(952\) 7.76393 0.251630
\(953\) −33.9787 −1.10068 −0.550339 0.834941i \(-0.685502\pi\)
−0.550339 + 0.834941i \(0.685502\pi\)
\(954\) −17.9443 −0.580967
\(955\) −11.3050 −0.365820
\(956\) −5.43769 −0.175868
\(957\) 6.23607 0.201583
\(958\) 15.0344 0.485741
\(959\) −7.18034 −0.231865
\(960\) 5.85410 0.188940
\(961\) −8.83282 −0.284930
\(962\) −11.0902 −0.357561
\(963\) −14.0902 −0.454049
\(964\) −10.3262 −0.332586
\(965\) 7.23607 0.232937
\(966\) −1.61803 −0.0520594
\(967\) −7.88854 −0.253678 −0.126839 0.991923i \(-0.540483\pi\)
−0.126839 + 0.991923i \(0.540483\pi\)
\(968\) −22.3607 −0.718699
\(969\) −0.819660 −0.0263313
\(970\) 15.0000 0.481621
\(971\) 20.9656 0.672817 0.336408 0.941716i \(-0.390788\pi\)
0.336408 + 0.941716i \(0.390788\pi\)
\(972\) 0.618034 0.0198234
\(973\) −8.03444 −0.257572
\(974\) −16.3820 −0.524912
\(975\) 5.00000 0.160128
\(976\) 5.56231 0.178045
\(977\) 39.2148 1.25459 0.627296 0.778781i \(-0.284162\pi\)
0.627296 + 0.778781i \(0.284162\pi\)
\(978\) −21.4721 −0.686603
\(979\) −6.56231 −0.209732
\(980\) 0.854102 0.0272833
\(981\) 4.61803 0.147442
\(982\) 36.5066 1.16497
\(983\) 5.00000 0.159475 0.0797376 0.996816i \(-0.474592\pi\)
0.0797376 + 0.996816i \(0.474592\pi\)
\(984\) 12.2361 0.390072
\(985\) −11.8328 −0.377025
\(986\) −35.0344 −1.11572
\(987\) −1.70820 −0.0543727
\(988\) 0.236068 0.00751032
\(989\) 2.85410 0.0907552
\(990\) −2.23607 −0.0710669
\(991\) −48.3820 −1.53690 −0.768452 0.639908i \(-0.778973\pi\)
−0.768452 + 0.639908i \(0.778973\pi\)
\(992\) 15.9230 0.505555
\(993\) −27.8885 −0.885016
\(994\) −8.61803 −0.273347
\(995\) 29.7984 0.944672
\(996\) −0.0344419 −0.00109133
\(997\) −22.3050 −0.706405 −0.353202 0.935547i \(-0.614907\pi\)
−0.353202 + 0.935547i \(0.614907\pi\)
\(998\) 5.76393 0.182454
\(999\) −4.23607 −0.134023
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 483.2.a.e.1.1 2
3.2 odd 2 1449.2.a.g.1.2 2
4.3 odd 2 7728.2.a.be.1.1 2
7.6 odd 2 3381.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.e.1.1 2 1.1 even 1 trivial
1449.2.a.g.1.2 2 3.2 odd 2
3381.2.a.o.1.1 2 7.6 odd 2
7728.2.a.be.1.1 2 4.3 odd 2