Properties

Label 138.2.a.d.1.2
Level $138$
Weight $2$
Character 138.1
Self dual yes
Analytic conductor $1.102$
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] = [138,2,Mod(1,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, 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("138.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 138.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: \(1.10193554789\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 138.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} -4.47214 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} +1.00000 q^{6} -4.47214 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.23607 q^{10} -5.23607 q^{11} +1.00000 q^{12} +4.47214 q^{13} -4.47214 q^{14} +1.23607 q^{15} +1.00000 q^{16} -4.00000 q^{17} +1.00000 q^{18} +5.70820 q^{19} +1.23607 q^{20} -4.47214 q^{21} -5.23607 q^{22} +1.00000 q^{23} +1.00000 q^{24} -3.47214 q^{25} +4.47214 q^{26} +1.00000 q^{27} -4.47214 q^{28} -4.47214 q^{29} +1.23607 q^{30} -2.47214 q^{31} +1.00000 q^{32} -5.23607 q^{33} -4.00000 q^{34} -5.52786 q^{35} +1.00000 q^{36} +11.2361 q^{37} +5.70820 q^{38} +4.47214 q^{39} +1.23607 q^{40} -2.00000 q^{41} -4.47214 q^{42} -4.76393 q^{43} -5.23607 q^{44} +1.23607 q^{45} +1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +13.0000 q^{49} -3.47214 q^{50} -4.00000 q^{51} +4.47214 q^{52} +5.23607 q^{53} +1.00000 q^{54} -6.47214 q^{55} -4.47214 q^{56} +5.70820 q^{57} -4.47214 q^{58} -8.94427 q^{59} +1.23607 q^{60} +0.763932 q^{61} -2.47214 q^{62} -4.47214 q^{63} +1.00000 q^{64} +5.52786 q^{65} -5.23607 q^{66} +9.70820 q^{67} -4.00000 q^{68} +1.00000 q^{69} -5.52786 q^{70} +8.94427 q^{71} +1.00000 q^{72} -4.47214 q^{73} +11.2361 q^{74} -3.47214 q^{75} +5.70820 q^{76} +23.4164 q^{77} +4.47214 q^{78} +4.47214 q^{79} +1.23607 q^{80} +1.00000 q^{81} -2.00000 q^{82} +13.2361 q^{83} -4.47214 q^{84} -4.94427 q^{85} -4.76393 q^{86} -4.47214 q^{87} -5.23607 q^{88} -10.4721 q^{89} +1.23607 q^{90} -20.0000 q^{91} +1.00000 q^{92} -2.47214 q^{93} +4.00000 q^{94} +7.05573 q^{95} +1.00000 q^{96} +0.472136 q^{97} +13.0000 q^{98} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 6 q^{11} + 2 q^{12} - 2 q^{15} + 2 q^{16} - 8 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 6 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{27} - 2 q^{30} + 4 q^{31} + 2 q^{32} - 6 q^{33} - 8 q^{34} - 20 q^{35} + 2 q^{36} + 18 q^{37} - 2 q^{38} - 2 q^{40} - 4 q^{41} - 14 q^{43} - 6 q^{44} - 2 q^{45} + 2 q^{46} + 8 q^{47} + 2 q^{48} + 26 q^{49} + 2 q^{50} - 8 q^{51} + 6 q^{53} + 2 q^{54} - 4 q^{55} - 2 q^{57} - 2 q^{60} + 6 q^{61} + 4 q^{62} + 2 q^{64} + 20 q^{65} - 6 q^{66} + 6 q^{67} - 8 q^{68} + 2 q^{69} - 20 q^{70} + 2 q^{72} + 18 q^{74} + 2 q^{75} - 2 q^{76} + 20 q^{77} - 2 q^{80} + 2 q^{81} - 4 q^{82} + 22 q^{83} + 8 q^{85} - 14 q^{86} - 6 q^{88} - 12 q^{89} - 2 q^{90} - 40 q^{91} + 2 q^{92} + 4 q^{93} + 8 q^{94} + 32 q^{95} + 2 q^{96} - 8 q^{97} + 26 q^{98} - 6 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.47214 −1.69031 −0.845154 0.534522i \(-0.820491\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.23607 0.390879
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) −4.47214 −1.19523
\(15\) 1.23607 0.319151
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 1.00000 0.235702
\(19\) 5.70820 1.30955 0.654776 0.755823i \(-0.272763\pi\)
0.654776 + 0.755823i \(0.272763\pi\)
\(20\) 1.23607 0.276393
\(21\) −4.47214 −0.975900
\(22\) −5.23607 −1.11633
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −3.47214 −0.694427
\(26\) 4.47214 0.877058
\(27\) 1.00000 0.192450
\(28\) −4.47214 −0.845154
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 1.23607 0.225674
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.23607 −0.911482
\(34\) −4.00000 −0.685994
\(35\) −5.52786 −0.934380
\(36\) 1.00000 0.166667
\(37\) 11.2361 1.84720 0.923599 0.383360i \(-0.125233\pi\)
0.923599 + 0.383360i \(0.125233\pi\)
\(38\) 5.70820 0.925993
\(39\) 4.47214 0.716115
\(40\) 1.23607 0.195440
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) −4.47214 −0.690066
\(43\) −4.76393 −0.726493 −0.363246 0.931693i \(-0.618332\pi\)
−0.363246 + 0.931693i \(0.618332\pi\)
\(44\) −5.23607 −0.789367
\(45\) 1.23607 0.184262
\(46\) 1.00000 0.147442
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 13.0000 1.85714
\(50\) −3.47214 −0.491034
\(51\) −4.00000 −0.560112
\(52\) 4.47214 0.620174
\(53\) 5.23607 0.719229 0.359615 0.933101i \(-0.382908\pi\)
0.359615 + 0.933101i \(0.382908\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.47214 −0.872703
\(56\) −4.47214 −0.597614
\(57\) 5.70820 0.756070
\(58\) −4.47214 −0.587220
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 1.23607 0.159576
\(61\) 0.763932 0.0978115 0.0489057 0.998803i \(-0.484427\pi\)
0.0489057 + 0.998803i \(0.484427\pi\)
\(62\) −2.47214 −0.313962
\(63\) −4.47214 −0.563436
\(64\) 1.00000 0.125000
\(65\) 5.52786 0.685647
\(66\) −5.23607 −0.644515
\(67\) 9.70820 1.18605 0.593023 0.805186i \(-0.297934\pi\)
0.593023 + 0.805186i \(0.297934\pi\)
\(68\) −4.00000 −0.485071
\(69\) 1.00000 0.120386
\(70\) −5.52786 −0.660706
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.47214 −0.523424 −0.261712 0.965146i \(-0.584287\pi\)
−0.261712 + 0.965146i \(0.584287\pi\)
\(74\) 11.2361 1.30617
\(75\) −3.47214 −0.400928
\(76\) 5.70820 0.654776
\(77\) 23.4164 2.66855
\(78\) 4.47214 0.506370
\(79\) 4.47214 0.503155 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(80\) 1.23607 0.138197
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 13.2361 1.45285 0.726424 0.687247i \(-0.241181\pi\)
0.726424 + 0.687247i \(0.241181\pi\)
\(84\) −4.47214 −0.487950
\(85\) −4.94427 −0.536282
\(86\) −4.76393 −0.513708
\(87\) −4.47214 −0.479463
\(88\) −5.23607 −0.558167
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 1.23607 0.130293
\(91\) −20.0000 −2.09657
\(92\) 1.00000 0.104257
\(93\) −2.47214 −0.256349
\(94\) 4.00000 0.412568
\(95\) 7.05573 0.723902
\(96\) 1.00000 0.102062
\(97\) 0.472136 0.0479381 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) 13.0000 1.31320
\(99\) −5.23607 −0.526245
\(100\) −3.47214 −0.347214
\(101\) 4.47214 0.444994 0.222497 0.974933i \(-0.428579\pi\)
0.222497 + 0.974933i \(0.428579\pi\)
\(102\) −4.00000 −0.396059
\(103\) −6.00000 −0.591198 −0.295599 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 4.47214 0.438529
\(105\) −5.52786 −0.539464
\(106\) 5.23607 0.508572
\(107\) −12.6525 −1.22316 −0.611581 0.791182i \(-0.709466\pi\)
−0.611581 + 0.791182i \(0.709466\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.76393 0.456302 0.228151 0.973626i \(-0.426732\pi\)
0.228151 + 0.973626i \(0.426732\pi\)
\(110\) −6.47214 −0.617094
\(111\) 11.2361 1.06648
\(112\) −4.47214 −0.422577
\(113\) −5.52786 −0.520018 −0.260009 0.965606i \(-0.583725\pi\)
−0.260009 + 0.965606i \(0.583725\pi\)
\(114\) 5.70820 0.534622
\(115\) 1.23607 0.115264
\(116\) −4.47214 −0.415227
\(117\) 4.47214 0.413449
\(118\) −8.94427 −0.823387
\(119\) 17.8885 1.63984
\(120\) 1.23607 0.112837
\(121\) 16.4164 1.49240
\(122\) 0.763932 0.0691632
\(123\) −2.00000 −0.180334
\(124\) −2.47214 −0.222004
\(125\) −10.4721 −0.936656
\(126\) −4.47214 −0.398410
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.76393 −0.419441
\(130\) 5.52786 0.484826
\(131\) −9.52786 −0.832453 −0.416227 0.909261i \(-0.636648\pi\)
−0.416227 + 0.909261i \(0.636648\pi\)
\(132\) −5.23607 −0.455741
\(133\) −25.5279 −2.21355
\(134\) 9.70820 0.838661
\(135\) 1.23607 0.106384
\(136\) −4.00000 −0.342997
\(137\) 3.05573 0.261068 0.130534 0.991444i \(-0.458331\pi\)
0.130534 + 0.991444i \(0.458331\pi\)
\(138\) 1.00000 0.0851257
\(139\) −16.9443 −1.43719 −0.718597 0.695427i \(-0.755215\pi\)
−0.718597 + 0.695427i \(0.755215\pi\)
\(140\) −5.52786 −0.467190
\(141\) 4.00000 0.336861
\(142\) 8.94427 0.750587
\(143\) −23.4164 −1.95818
\(144\) 1.00000 0.0833333
\(145\) −5.52786 −0.459064
\(146\) −4.47214 −0.370117
\(147\) 13.0000 1.07222
\(148\) 11.2361 0.923599
\(149\) −11.7082 −0.959173 −0.479587 0.877494i \(-0.659213\pi\)
−0.479587 + 0.877494i \(0.659213\pi\)
\(150\) −3.47214 −0.283499
\(151\) −14.4721 −1.17773 −0.588863 0.808233i \(-0.700424\pi\)
−0.588863 + 0.808233i \(0.700424\pi\)
\(152\) 5.70820 0.462996
\(153\) −4.00000 −0.323381
\(154\) 23.4164 1.88695
\(155\) −3.05573 −0.245442
\(156\) 4.47214 0.358057
\(157\) −6.65248 −0.530925 −0.265463 0.964121i \(-0.585525\pi\)
−0.265463 + 0.964121i \(0.585525\pi\)
\(158\) 4.47214 0.355784
\(159\) 5.23607 0.415247
\(160\) 1.23607 0.0977198
\(161\) −4.47214 −0.352454
\(162\) 1.00000 0.0785674
\(163\) −2.47214 −0.193633 −0.0968163 0.995302i \(-0.530866\pi\)
−0.0968163 + 0.995302i \(0.530866\pi\)
\(164\) −2.00000 −0.156174
\(165\) −6.47214 −0.503855
\(166\) 13.2361 1.02732
\(167\) 16.9443 1.31119 0.655594 0.755114i \(-0.272418\pi\)
0.655594 + 0.755114i \(0.272418\pi\)
\(168\) −4.47214 −0.345033
\(169\) 7.00000 0.538462
\(170\) −4.94427 −0.379208
\(171\) 5.70820 0.436517
\(172\) −4.76393 −0.363246
\(173\) −17.4164 −1.32414 −0.662072 0.749440i \(-0.730323\pi\)
−0.662072 + 0.749440i \(0.730323\pi\)
\(174\) −4.47214 −0.339032
\(175\) 15.5279 1.17380
\(176\) −5.23607 −0.394683
\(177\) −8.94427 −0.672293
\(178\) −10.4721 −0.784920
\(179\) 19.4164 1.45125 0.725625 0.688090i \(-0.241551\pi\)
0.725625 + 0.688090i \(0.241551\pi\)
\(180\) 1.23607 0.0921311
\(181\) −11.2361 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(182\) −20.0000 −1.48250
\(183\) 0.763932 0.0564715
\(184\) 1.00000 0.0737210
\(185\) 13.8885 1.02111
\(186\) −2.47214 −0.181266
\(187\) 20.9443 1.53160
\(188\) 4.00000 0.291730
\(189\) −4.47214 −0.325300
\(190\) 7.05573 0.511876
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) 1.00000 0.0721688
\(193\) 23.8885 1.71954 0.859768 0.510686i \(-0.170608\pi\)
0.859768 + 0.510686i \(0.170608\pi\)
\(194\) 0.472136 0.0338974
\(195\) 5.52786 0.395859
\(196\) 13.0000 0.928571
\(197\) 2.94427 0.209771 0.104885 0.994484i \(-0.466552\pi\)
0.104885 + 0.994484i \(0.466552\pi\)
\(198\) −5.23607 −0.372111
\(199\) 17.4164 1.23462 0.617308 0.786721i \(-0.288223\pi\)
0.617308 + 0.786721i \(0.288223\pi\)
\(200\) −3.47214 −0.245517
\(201\) 9.70820 0.684764
\(202\) 4.47214 0.314658
\(203\) 20.0000 1.40372
\(204\) −4.00000 −0.280056
\(205\) −2.47214 −0.172661
\(206\) −6.00000 −0.418040
\(207\) 1.00000 0.0695048
\(208\) 4.47214 0.310087
\(209\) −29.8885 −2.06743
\(210\) −5.52786 −0.381459
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) 5.23607 0.359615
\(213\) 8.94427 0.612851
\(214\) −12.6525 −0.864905
\(215\) −5.88854 −0.401595
\(216\) 1.00000 0.0680414
\(217\) 11.0557 0.750512
\(218\) 4.76393 0.322654
\(219\) −4.47214 −0.302199
\(220\) −6.47214 −0.436351
\(221\) −17.8885 −1.20331
\(222\) 11.2361 0.754116
\(223\) 19.4164 1.30022 0.650109 0.759841i \(-0.274723\pi\)
0.650109 + 0.759841i \(0.274723\pi\)
\(224\) −4.47214 −0.298807
\(225\) −3.47214 −0.231476
\(226\) −5.52786 −0.367708
\(227\) −9.23607 −0.613019 −0.306510 0.951868i \(-0.599161\pi\)
−0.306510 + 0.951868i \(0.599161\pi\)
\(228\) 5.70820 0.378035
\(229\) 17.7082 1.17019 0.585096 0.810964i \(-0.301057\pi\)
0.585096 + 0.810964i \(0.301057\pi\)
\(230\) 1.23607 0.0815039
\(231\) 23.4164 1.54069
\(232\) −4.47214 −0.293610
\(233\) 19.8885 1.30294 0.651471 0.758674i \(-0.274152\pi\)
0.651471 + 0.758674i \(0.274152\pi\)
\(234\) 4.47214 0.292353
\(235\) 4.94427 0.322529
\(236\) −8.94427 −0.582223
\(237\) 4.47214 0.290496
\(238\) 17.8885 1.15954
\(239\) −4.94427 −0.319818 −0.159909 0.987132i \(-0.551120\pi\)
−0.159909 + 0.987132i \(0.551120\pi\)
\(240\) 1.23607 0.0797878
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) 16.4164 1.05529
\(243\) 1.00000 0.0641500
\(244\) 0.763932 0.0489057
\(245\) 16.0689 1.02660
\(246\) −2.00000 −0.127515
\(247\) 25.5279 1.62430
\(248\) −2.47214 −0.156981
\(249\) 13.2361 0.838802
\(250\) −10.4721 −0.662316
\(251\) −19.7082 −1.24397 −0.621985 0.783029i \(-0.713674\pi\)
−0.621985 + 0.783029i \(0.713674\pi\)
\(252\) −4.47214 −0.281718
\(253\) −5.23607 −0.329189
\(254\) −4.00000 −0.250982
\(255\) −4.94427 −0.309622
\(256\) 1.00000 0.0625000
\(257\) −11.8885 −0.741587 −0.370793 0.928715i \(-0.620914\pi\)
−0.370793 + 0.928715i \(0.620914\pi\)
\(258\) −4.76393 −0.296589
\(259\) −50.2492 −3.12233
\(260\) 5.52786 0.342824
\(261\) −4.47214 −0.276818
\(262\) −9.52786 −0.588633
\(263\) 24.9443 1.53813 0.769065 0.639171i \(-0.220722\pi\)
0.769065 + 0.639171i \(0.220722\pi\)
\(264\) −5.23607 −0.322258
\(265\) 6.47214 0.397580
\(266\) −25.5279 −1.56521
\(267\) −10.4721 −0.640884
\(268\) 9.70820 0.593023
\(269\) −13.0557 −0.796022 −0.398011 0.917381i \(-0.630299\pi\)
−0.398011 + 0.917381i \(0.630299\pi\)
\(270\) 1.23607 0.0752247
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) −4.00000 −0.242536
\(273\) −20.0000 −1.21046
\(274\) 3.05573 0.184603
\(275\) 18.1803 1.09632
\(276\) 1.00000 0.0601929
\(277\) −20.4721 −1.23005 −0.615026 0.788507i \(-0.710854\pi\)
−0.615026 + 0.788507i \(0.710854\pi\)
\(278\) −16.9443 −1.01625
\(279\) −2.47214 −0.148003
\(280\) −5.52786 −0.330353
\(281\) −13.5279 −0.807005 −0.403502 0.914979i \(-0.632207\pi\)
−0.403502 + 0.914979i \(0.632207\pi\)
\(282\) 4.00000 0.238197
\(283\) −3.81966 −0.227055 −0.113528 0.993535i \(-0.536215\pi\)
−0.113528 + 0.993535i \(0.536215\pi\)
\(284\) 8.94427 0.530745
\(285\) 7.05573 0.417945
\(286\) −23.4164 −1.38464
\(287\) 8.94427 0.527964
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −5.52786 −0.324607
\(291\) 0.472136 0.0276771
\(292\) −4.47214 −0.261712
\(293\) 0.291796 0.0170469 0.00852345 0.999964i \(-0.497287\pi\)
0.00852345 + 0.999964i \(0.497287\pi\)
\(294\) 13.0000 0.758175
\(295\) −11.0557 −0.643689
\(296\) 11.2361 0.653083
\(297\) −5.23607 −0.303827
\(298\) −11.7082 −0.678238
\(299\) 4.47214 0.258630
\(300\) −3.47214 −0.200464
\(301\) 21.3050 1.22800
\(302\) −14.4721 −0.832778
\(303\) 4.47214 0.256917
\(304\) 5.70820 0.327388
\(305\) 0.944272 0.0540689
\(306\) −4.00000 −0.228665
\(307\) 15.4164 0.879861 0.439930 0.898032i \(-0.355003\pi\)
0.439930 + 0.898032i \(0.355003\pi\)
\(308\) 23.4164 1.33427
\(309\) −6.00000 −0.341328
\(310\) −3.05573 −0.173554
\(311\) −20.9443 −1.18764 −0.593820 0.804598i \(-0.702381\pi\)
−0.593820 + 0.804598i \(0.702381\pi\)
\(312\) 4.47214 0.253185
\(313\) 15.5279 0.877687 0.438843 0.898564i \(-0.355388\pi\)
0.438843 + 0.898564i \(0.355388\pi\)
\(314\) −6.65248 −0.375421
\(315\) −5.52786 −0.311460
\(316\) 4.47214 0.251577
\(317\) 19.5279 1.09679 0.548397 0.836218i \(-0.315238\pi\)
0.548397 + 0.836218i \(0.315238\pi\)
\(318\) 5.23607 0.293624
\(319\) 23.4164 1.31107
\(320\) 1.23607 0.0690983
\(321\) −12.6525 −0.706192
\(322\) −4.47214 −0.249222
\(323\) −22.8328 −1.27045
\(324\) 1.00000 0.0555556
\(325\) −15.5279 −0.861331
\(326\) −2.47214 −0.136919
\(327\) 4.76393 0.263446
\(328\) −2.00000 −0.110432
\(329\) −17.8885 −0.986227
\(330\) −6.47214 −0.356279
\(331\) −10.4721 −0.575601 −0.287800 0.957690i \(-0.592924\pi\)
−0.287800 + 0.957690i \(0.592924\pi\)
\(332\) 13.2361 0.726424
\(333\) 11.2361 0.615733
\(334\) 16.9443 0.927149
\(335\) 12.0000 0.655630
\(336\) −4.47214 −0.243975
\(337\) −19.8885 −1.08340 −0.541699 0.840573i \(-0.682219\pi\)
−0.541699 + 0.840573i \(0.682219\pi\)
\(338\) 7.00000 0.380750
\(339\) −5.52786 −0.300232
\(340\) −4.94427 −0.268141
\(341\) 12.9443 0.700972
\(342\) 5.70820 0.308664
\(343\) −26.8328 −1.44884
\(344\) −4.76393 −0.256854
\(345\) 1.23607 0.0665477
\(346\) −17.4164 −0.936312
\(347\) 30.4721 1.63583 0.817915 0.575339i \(-0.195130\pi\)
0.817915 + 0.575339i \(0.195130\pi\)
\(348\) −4.47214 −0.239732
\(349\) 3.88854 0.208149 0.104074 0.994570i \(-0.466812\pi\)
0.104074 + 0.994570i \(0.466812\pi\)
\(350\) 15.5279 0.829999
\(351\) 4.47214 0.238705
\(352\) −5.23607 −0.279083
\(353\) −3.88854 −0.206966 −0.103483 0.994631i \(-0.532999\pi\)
−0.103483 + 0.994631i \(0.532999\pi\)
\(354\) −8.94427 −0.475383
\(355\) 11.0557 0.586777
\(356\) −10.4721 −0.555022
\(357\) 17.8885 0.946762
\(358\) 19.4164 1.02619
\(359\) 29.3050 1.54666 0.773328 0.634006i \(-0.218591\pi\)
0.773328 + 0.634006i \(0.218591\pi\)
\(360\) 1.23607 0.0651465
\(361\) 13.5836 0.714926
\(362\) −11.2361 −0.590555
\(363\) 16.4164 0.861638
\(364\) −20.0000 −1.04828
\(365\) −5.52786 −0.289342
\(366\) 0.763932 0.0399314
\(367\) 9.41641 0.491532 0.245766 0.969329i \(-0.420960\pi\)
0.245766 + 0.969329i \(0.420960\pi\)
\(368\) 1.00000 0.0521286
\(369\) −2.00000 −0.104116
\(370\) 13.8885 0.722031
\(371\) −23.4164 −1.21572
\(372\) −2.47214 −0.128174
\(373\) −35.5967 −1.84313 −0.921565 0.388224i \(-0.873089\pi\)
−0.921565 + 0.388224i \(0.873089\pi\)
\(374\) 20.9443 1.08300
\(375\) −10.4721 −0.540779
\(376\) 4.00000 0.206284
\(377\) −20.0000 −1.03005
\(378\) −4.47214 −0.230022
\(379\) 13.7082 0.704143 0.352072 0.935973i \(-0.385477\pi\)
0.352072 + 0.935973i \(0.385477\pi\)
\(380\) 7.05573 0.361951
\(381\) −4.00000 −0.204926
\(382\) 6.47214 0.331143
\(383\) 7.05573 0.360531 0.180265 0.983618i \(-0.442304\pi\)
0.180265 + 0.983618i \(0.442304\pi\)
\(384\) 1.00000 0.0510310
\(385\) 28.9443 1.47514
\(386\) 23.8885 1.21589
\(387\) −4.76393 −0.242164
\(388\) 0.472136 0.0239691
\(389\) 23.7082 1.20205 0.601027 0.799229i \(-0.294758\pi\)
0.601027 + 0.799229i \(0.294758\pi\)
\(390\) 5.52786 0.279914
\(391\) −4.00000 −0.202289
\(392\) 13.0000 0.656599
\(393\) −9.52786 −0.480617
\(394\) 2.94427 0.148330
\(395\) 5.52786 0.278137
\(396\) −5.23607 −0.263122
\(397\) 26.9443 1.35229 0.676147 0.736767i \(-0.263648\pi\)
0.676147 + 0.736767i \(0.263648\pi\)
\(398\) 17.4164 0.873006
\(399\) −25.5279 −1.27799
\(400\) −3.47214 −0.173607
\(401\) 8.94427 0.446656 0.223328 0.974743i \(-0.428308\pi\)
0.223328 + 0.974743i \(0.428308\pi\)
\(402\) 9.70820 0.484201
\(403\) −11.0557 −0.550725
\(404\) 4.47214 0.222497
\(405\) 1.23607 0.0614207
\(406\) 20.0000 0.992583
\(407\) −58.8328 −2.91623
\(408\) −4.00000 −0.198030
\(409\) −35.8885 −1.77457 −0.887287 0.461217i \(-0.847413\pi\)
−0.887287 + 0.461217i \(0.847413\pi\)
\(410\) −2.47214 −0.122090
\(411\) 3.05573 0.150728
\(412\) −6.00000 −0.295599
\(413\) 40.0000 1.96827
\(414\) 1.00000 0.0491473
\(415\) 16.3607 0.803114
\(416\) 4.47214 0.219265
\(417\) −16.9443 −0.829765
\(418\) −29.8885 −1.46190
\(419\) 28.0689 1.37125 0.685627 0.727953i \(-0.259528\pi\)
0.685627 + 0.727953i \(0.259528\pi\)
\(420\) −5.52786 −0.269732
\(421\) −0.763932 −0.0372318 −0.0186159 0.999827i \(-0.505926\pi\)
−0.0186159 + 0.999827i \(0.505926\pi\)
\(422\) −23.4164 −1.13989
\(423\) 4.00000 0.194487
\(424\) 5.23607 0.254286
\(425\) 13.8885 0.673693
\(426\) 8.94427 0.433351
\(427\) −3.41641 −0.165332
\(428\) −12.6525 −0.611581
\(429\) −23.4164 −1.13055
\(430\) −5.88854 −0.283971
\(431\) −23.4164 −1.12793 −0.563964 0.825799i \(-0.690724\pi\)
−0.563964 + 0.825799i \(0.690724\pi\)
\(432\) 1.00000 0.0481125
\(433\) −21.4164 −1.02921 −0.514603 0.857428i \(-0.672061\pi\)
−0.514603 + 0.857428i \(0.672061\pi\)
\(434\) 11.0557 0.530692
\(435\) −5.52786 −0.265041
\(436\) 4.76393 0.228151
\(437\) 5.70820 0.273060
\(438\) −4.47214 −0.213687
\(439\) 19.0557 0.909480 0.454740 0.890624i \(-0.349732\pi\)
0.454740 + 0.890624i \(0.349732\pi\)
\(440\) −6.47214 −0.308547
\(441\) 13.0000 0.619048
\(442\) −17.8885 −0.850871
\(443\) −15.0557 −0.715319 −0.357660 0.933852i \(-0.616425\pi\)
−0.357660 + 0.933852i \(0.616425\pi\)
\(444\) 11.2361 0.533240
\(445\) −12.9443 −0.613617
\(446\) 19.4164 0.919394
\(447\) −11.7082 −0.553779
\(448\) −4.47214 −0.211289
\(449\) 15.8885 0.749827 0.374913 0.927060i \(-0.377672\pi\)
0.374913 + 0.927060i \(0.377672\pi\)
\(450\) −3.47214 −0.163678
\(451\) 10.4721 0.493114
\(452\) −5.52786 −0.260009
\(453\) −14.4721 −0.679960
\(454\) −9.23607 −0.433470
\(455\) −24.7214 −1.15896
\(456\) 5.70820 0.267311
\(457\) 27.5279 1.28770 0.643850 0.765152i \(-0.277336\pi\)
0.643850 + 0.765152i \(0.277336\pi\)
\(458\) 17.7082 0.827450
\(459\) −4.00000 −0.186704
\(460\) 1.23607 0.0576320
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) 23.4164 1.08943
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −4.47214 −0.207614
\(465\) −3.05573 −0.141706
\(466\) 19.8885 0.921319
\(467\) −17.8197 −0.824596 −0.412298 0.911049i \(-0.635274\pi\)
−0.412298 + 0.911049i \(0.635274\pi\)
\(468\) 4.47214 0.206725
\(469\) −43.4164 −2.00478
\(470\) 4.94427 0.228062
\(471\) −6.65248 −0.306530
\(472\) −8.94427 −0.411693
\(473\) 24.9443 1.14694
\(474\) 4.47214 0.205412
\(475\) −19.8197 −0.909388
\(476\) 17.8885 0.819920
\(477\) 5.23607 0.239743
\(478\) −4.94427 −0.226146
\(479\) −28.9443 −1.32250 −0.661249 0.750167i \(-0.729973\pi\)
−0.661249 + 0.750167i \(0.729973\pi\)
\(480\) 1.23607 0.0564185
\(481\) 50.2492 2.29117
\(482\) −12.4721 −0.568090
\(483\) −4.47214 −0.203489
\(484\) 16.4164 0.746200
\(485\) 0.583592 0.0264996
\(486\) 1.00000 0.0453609
\(487\) 9.88854 0.448093 0.224046 0.974578i \(-0.428073\pi\)
0.224046 + 0.974578i \(0.428073\pi\)
\(488\) 0.763932 0.0345816
\(489\) −2.47214 −0.111794
\(490\) 16.0689 0.725918
\(491\) 29.3050 1.32251 0.661257 0.750159i \(-0.270023\pi\)
0.661257 + 0.750159i \(0.270023\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 17.8885 0.805659
\(494\) 25.5279 1.14855
\(495\) −6.47214 −0.290901
\(496\) −2.47214 −0.111002
\(497\) −40.0000 −1.79425
\(498\) 13.2361 0.593122
\(499\) 0.583592 0.0261252 0.0130626 0.999915i \(-0.495842\pi\)
0.0130626 + 0.999915i \(0.495842\pi\)
\(500\) −10.4721 −0.468328
\(501\) 16.9443 0.757014
\(502\) −19.7082 −0.879620
\(503\) 10.4721 0.466929 0.233465 0.972365i \(-0.424994\pi\)
0.233465 + 0.972365i \(0.424994\pi\)
\(504\) −4.47214 −0.199205
\(505\) 5.52786 0.245987
\(506\) −5.23607 −0.232772
\(507\) 7.00000 0.310881
\(508\) −4.00000 −0.177471
\(509\) −33.4164 −1.48116 −0.740578 0.671970i \(-0.765448\pi\)
−0.740578 + 0.671970i \(0.765448\pi\)
\(510\) −4.94427 −0.218936
\(511\) 20.0000 0.884748
\(512\) 1.00000 0.0441942
\(513\) 5.70820 0.252023
\(514\) −11.8885 −0.524381
\(515\) −7.41641 −0.326806
\(516\) −4.76393 −0.209720
\(517\) −20.9443 −0.921128
\(518\) −50.2492 −2.20782
\(519\) −17.4164 −0.764495
\(520\) 5.52786 0.242413
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) −4.47214 −0.195740
\(523\) 25.7082 1.12414 0.562071 0.827089i \(-0.310005\pi\)
0.562071 + 0.827089i \(0.310005\pi\)
\(524\) −9.52786 −0.416227
\(525\) 15.5279 0.677692
\(526\) 24.9443 1.08762
\(527\) 9.88854 0.430752
\(528\) −5.23607 −0.227871
\(529\) 1.00000 0.0434783
\(530\) 6.47214 0.281132
\(531\) −8.94427 −0.388148
\(532\) −25.5279 −1.10677
\(533\) −8.94427 −0.387419
\(534\) −10.4721 −0.453174
\(535\) −15.6393 −0.676147
\(536\) 9.70820 0.419331
\(537\) 19.4164 0.837880
\(538\) −13.0557 −0.562872
\(539\) −68.0689 −2.93193
\(540\) 1.23607 0.0531919
\(541\) 8.11146 0.348739 0.174369 0.984680i \(-0.444211\pi\)
0.174369 + 0.984680i \(0.444211\pi\)
\(542\) −16.9443 −0.727819
\(543\) −11.2361 −0.482186
\(544\) −4.00000 −0.171499
\(545\) 5.88854 0.252238
\(546\) −20.0000 −0.855921
\(547\) 41.3050 1.76607 0.883036 0.469305i \(-0.155496\pi\)
0.883036 + 0.469305i \(0.155496\pi\)
\(548\) 3.05573 0.130534
\(549\) 0.763932 0.0326038
\(550\) 18.1803 0.775212
\(551\) −25.5279 −1.08752
\(552\) 1.00000 0.0425628
\(553\) −20.0000 −0.850487
\(554\) −20.4721 −0.869778
\(555\) 13.8885 0.589536
\(556\) −16.9443 −0.718597
\(557\) −15.1246 −0.640850 −0.320425 0.947274i \(-0.603826\pi\)
−0.320425 + 0.947274i \(0.603826\pi\)
\(558\) −2.47214 −0.104654
\(559\) −21.3050 −0.901103
\(560\) −5.52786 −0.233595
\(561\) 20.9443 0.884268
\(562\) −13.5279 −0.570639
\(563\) 24.6525 1.03898 0.519489 0.854477i \(-0.326122\pi\)
0.519489 + 0.854477i \(0.326122\pi\)
\(564\) 4.00000 0.168430
\(565\) −6.83282 −0.287459
\(566\) −3.81966 −0.160552
\(567\) −4.47214 −0.187812
\(568\) 8.94427 0.375293
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 7.05573 0.295532
\(571\) −14.2918 −0.598093 −0.299047 0.954239i \(-0.596669\pi\)
−0.299047 + 0.954239i \(0.596669\pi\)
\(572\) −23.4164 −0.979089
\(573\) 6.47214 0.270377
\(574\) 8.94427 0.373327
\(575\) −3.47214 −0.144798
\(576\) 1.00000 0.0416667
\(577\) −34.3607 −1.43045 −0.715227 0.698892i \(-0.753677\pi\)
−0.715227 + 0.698892i \(0.753677\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 23.8885 0.992774
\(580\) −5.52786 −0.229532
\(581\) −59.1935 −2.45576
\(582\) 0.472136 0.0195707
\(583\) −27.4164 −1.13547
\(584\) −4.47214 −0.185058
\(585\) 5.52786 0.228549
\(586\) 0.291796 0.0120540
\(587\) 6.47214 0.267134 0.133567 0.991040i \(-0.457357\pi\)
0.133567 + 0.991040i \(0.457357\pi\)
\(588\) 13.0000 0.536111
\(589\) −14.1115 −0.581452
\(590\) −11.0557 −0.455157
\(591\) 2.94427 0.121111
\(592\) 11.2361 0.461800
\(593\) 33.7771 1.38706 0.693529 0.720428i \(-0.256055\pi\)
0.693529 + 0.720428i \(0.256055\pi\)
\(594\) −5.23607 −0.214838
\(595\) 22.1115 0.906481
\(596\) −11.7082 −0.479587
\(597\) 17.4164 0.712806
\(598\) 4.47214 0.182879
\(599\) 33.8885 1.38465 0.692324 0.721587i \(-0.256587\pi\)
0.692324 + 0.721587i \(0.256587\pi\)
\(600\) −3.47214 −0.141749
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) 21.3050 0.868325
\(603\) 9.70820 0.395349
\(604\) −14.4721 −0.588863
\(605\) 20.2918 0.824979
\(606\) 4.47214 0.181668
\(607\) −17.5279 −0.711434 −0.355717 0.934594i \(-0.615763\pi\)
−0.355717 + 0.934594i \(0.615763\pi\)
\(608\) 5.70820 0.231498
\(609\) 20.0000 0.810441
\(610\) 0.944272 0.0382325
\(611\) 17.8885 0.723693
\(612\) −4.00000 −0.161690
\(613\) −25.1246 −1.01477 −0.507387 0.861718i \(-0.669388\pi\)
−0.507387 + 0.861718i \(0.669388\pi\)
\(614\) 15.4164 0.622156
\(615\) −2.47214 −0.0996861
\(616\) 23.4164 0.943474
\(617\) 20.3607 0.819690 0.409845 0.912155i \(-0.365583\pi\)
0.409845 + 0.912155i \(0.365583\pi\)
\(618\) −6.00000 −0.241355
\(619\) 18.2918 0.735209 0.367605 0.929982i \(-0.380178\pi\)
0.367605 + 0.929982i \(0.380178\pi\)
\(620\) −3.05573 −0.122721
\(621\) 1.00000 0.0401286
\(622\) −20.9443 −0.839789
\(623\) 46.8328 1.87632
\(624\) 4.47214 0.179029
\(625\) 4.41641 0.176656
\(626\) 15.5279 0.620618
\(627\) −29.8885 −1.19363
\(628\) −6.65248 −0.265463
\(629\) −44.9443 −1.79205
\(630\) −5.52786 −0.220235
\(631\) −6.00000 −0.238856 −0.119428 0.992843i \(-0.538106\pi\)
−0.119428 + 0.992843i \(0.538106\pi\)
\(632\) 4.47214 0.177892
\(633\) −23.4164 −0.930719
\(634\) 19.5279 0.775551
\(635\) −4.94427 −0.196207
\(636\) 5.23607 0.207624
\(637\) 58.1378 2.30350
\(638\) 23.4164 0.927064
\(639\) 8.94427 0.353830
\(640\) 1.23607 0.0488599
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) −12.6525 −0.499353
\(643\) −32.5410 −1.28329 −0.641646 0.767001i \(-0.721748\pi\)
−0.641646 + 0.767001i \(0.721748\pi\)
\(644\) −4.47214 −0.176227
\(645\) −5.88854 −0.231861
\(646\) −22.8328 −0.898345
\(647\) 12.9443 0.508892 0.254446 0.967087i \(-0.418107\pi\)
0.254446 + 0.967087i \(0.418107\pi\)
\(648\) 1.00000 0.0392837
\(649\) 46.8328 1.83835
\(650\) −15.5279 −0.609053
\(651\) 11.0557 0.433308
\(652\) −2.47214 −0.0968163
\(653\) 9.41641 0.368493 0.184246 0.982880i \(-0.441016\pi\)
0.184246 + 0.982880i \(0.441016\pi\)
\(654\) 4.76393 0.186284
\(655\) −11.7771 −0.460169
\(656\) −2.00000 −0.0780869
\(657\) −4.47214 −0.174475
\(658\) −17.8885 −0.697368
\(659\) −32.6525 −1.27196 −0.635980 0.771706i \(-0.719404\pi\)
−0.635980 + 0.771706i \(0.719404\pi\)
\(660\) −6.47214 −0.251928
\(661\) 3.81966 0.148568 0.0742838 0.997237i \(-0.476333\pi\)
0.0742838 + 0.997237i \(0.476333\pi\)
\(662\) −10.4721 −0.407011
\(663\) −17.8885 −0.694733
\(664\) 13.2361 0.513659
\(665\) −31.5542 −1.22362
\(666\) 11.2361 0.435389
\(667\) −4.47214 −0.173162
\(668\) 16.9443 0.655594
\(669\) 19.4164 0.750682
\(670\) 12.0000 0.463600
\(671\) −4.00000 −0.154418
\(672\) −4.47214 −0.172516
\(673\) 4.47214 0.172388 0.0861941 0.996278i \(-0.472529\pi\)
0.0861941 + 0.996278i \(0.472529\pi\)
\(674\) −19.8885 −0.766078
\(675\) −3.47214 −0.133643
\(676\) 7.00000 0.269231
\(677\) 19.1246 0.735019 0.367509 0.930020i \(-0.380211\pi\)
0.367509 + 0.930020i \(0.380211\pi\)
\(678\) −5.52786 −0.212296
\(679\) −2.11146 −0.0810303
\(680\) −4.94427 −0.189604
\(681\) −9.23607 −0.353927
\(682\) 12.9443 0.495662
\(683\) 14.4721 0.553761 0.276880 0.960904i \(-0.410699\pi\)
0.276880 + 0.960904i \(0.410699\pi\)
\(684\) 5.70820 0.218259
\(685\) 3.77709 0.144315
\(686\) −26.8328 −1.02448
\(687\) 17.7082 0.675610
\(688\) −4.76393 −0.181623
\(689\) 23.4164 0.892094
\(690\) 1.23607 0.0470563
\(691\) 16.5836 0.630870 0.315435 0.948947i \(-0.397850\pi\)
0.315435 + 0.948947i \(0.397850\pi\)
\(692\) −17.4164 −0.662072
\(693\) 23.4164 0.889516
\(694\) 30.4721 1.15671
\(695\) −20.9443 −0.794462
\(696\) −4.47214 −0.169516
\(697\) 8.00000 0.303022
\(698\) 3.88854 0.147184
\(699\) 19.8885 0.752254
\(700\) 15.5279 0.586898
\(701\) −3.12461 −0.118015 −0.0590075 0.998258i \(-0.518794\pi\)
−0.0590075 + 0.998258i \(0.518794\pi\)
\(702\) 4.47214 0.168790
\(703\) 64.1378 2.41900
\(704\) −5.23607 −0.197342
\(705\) 4.94427 0.186212
\(706\) −3.88854 −0.146347
\(707\) −20.0000 −0.752177
\(708\) −8.94427 −0.336146
\(709\) 35.0132 1.31495 0.657473 0.753478i \(-0.271625\pi\)
0.657473 + 0.753478i \(0.271625\pi\)
\(710\) 11.0557 0.414914
\(711\) 4.47214 0.167718
\(712\) −10.4721 −0.392460
\(713\) −2.47214 −0.0925822
\(714\) 17.8885 0.669462
\(715\) −28.9443 −1.08245
\(716\) 19.4164 0.725625
\(717\) −4.94427 −0.184647
\(718\) 29.3050 1.09365
\(719\) −3.05573 −0.113959 −0.0569797 0.998375i \(-0.518147\pi\)
−0.0569797 + 0.998375i \(0.518147\pi\)
\(720\) 1.23607 0.0460655
\(721\) 26.8328 0.999306
\(722\) 13.5836 0.505529
\(723\) −12.4721 −0.463844
\(724\) −11.2361 −0.417585
\(725\) 15.5279 0.576690
\(726\) 16.4164 0.609270
\(727\) −19.3050 −0.715981 −0.357991 0.933725i \(-0.616538\pi\)
−0.357991 + 0.933725i \(0.616538\pi\)
\(728\) −20.0000 −0.741249
\(729\) 1.00000 0.0370370
\(730\) −5.52786 −0.204595
\(731\) 19.0557 0.704802
\(732\) 0.763932 0.0282357
\(733\) −21.7082 −0.801811 −0.400905 0.916119i \(-0.631304\pi\)
−0.400905 + 0.916119i \(0.631304\pi\)
\(734\) 9.41641 0.347566
\(735\) 16.0689 0.592710
\(736\) 1.00000 0.0368605
\(737\) −50.8328 −1.87245
\(738\) −2.00000 −0.0736210
\(739\) −32.9443 −1.21187 −0.605937 0.795512i \(-0.707202\pi\)
−0.605937 + 0.795512i \(0.707202\pi\)
\(740\) 13.8885 0.510553
\(741\) 25.5279 0.937790
\(742\) −23.4164 −0.859643
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) −2.47214 −0.0906329
\(745\) −14.4721 −0.530218
\(746\) −35.5967 −1.30329
\(747\) 13.2361 0.484282
\(748\) 20.9443 0.765798
\(749\) 56.5836 2.06752
\(750\) −10.4721 −0.382388
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) 4.00000 0.145865
\(753\) −19.7082 −0.718207
\(754\) −20.0000 −0.728357
\(755\) −17.8885 −0.651031
\(756\) −4.47214 −0.162650
\(757\) −1.34752 −0.0489766 −0.0244883 0.999700i \(-0.507796\pi\)
−0.0244883 + 0.999700i \(0.507796\pi\)
\(758\) 13.7082 0.497904
\(759\) −5.23607 −0.190057
\(760\) 7.05573 0.255938
\(761\) −5.05573 −0.183270 −0.0916350 0.995793i \(-0.529209\pi\)
−0.0916350 + 0.995793i \(0.529209\pi\)
\(762\) −4.00000 −0.144905
\(763\) −21.3050 −0.771291
\(764\) 6.47214 0.234154
\(765\) −4.94427 −0.178761
\(766\) 7.05573 0.254934
\(767\) −40.0000 −1.44432
\(768\) 1.00000 0.0360844
\(769\) 12.4721 0.449757 0.224878 0.974387i \(-0.427802\pi\)
0.224878 + 0.974387i \(0.427802\pi\)
\(770\) 28.9443 1.04308
\(771\) −11.8885 −0.428155
\(772\) 23.8885 0.859768
\(773\) −38.1803 −1.37325 −0.686626 0.727011i \(-0.740909\pi\)
−0.686626 + 0.727011i \(0.740909\pi\)
\(774\) −4.76393 −0.171236
\(775\) 8.58359 0.308332
\(776\) 0.472136 0.0169487
\(777\) −50.2492 −1.80268
\(778\) 23.7082 0.849980
\(779\) −11.4164 −0.409035
\(780\) 5.52786 0.197929
\(781\) −46.8328 −1.67581
\(782\) −4.00000 −0.143040
\(783\) −4.47214 −0.159821
\(784\) 13.0000 0.464286
\(785\) −8.22291 −0.293488
\(786\) −9.52786 −0.339848
\(787\) 35.2361 1.25603 0.628015 0.778201i \(-0.283868\pi\)
0.628015 + 0.778201i \(0.283868\pi\)
\(788\) 2.94427 0.104885
\(789\) 24.9443 0.888040
\(790\) 5.52786 0.196673
\(791\) 24.7214 0.878990
\(792\) −5.23607 −0.186056
\(793\) 3.41641 0.121320
\(794\) 26.9443 0.956216
\(795\) 6.47214 0.229543
\(796\) 17.4164 0.617308
\(797\) 41.5967 1.47343 0.736716 0.676202i \(-0.236375\pi\)
0.736716 + 0.676202i \(0.236375\pi\)
\(798\) −25.5279 −0.903677
\(799\) −16.0000 −0.566039
\(800\) −3.47214 −0.122759
\(801\) −10.4721 −0.370015
\(802\) 8.94427 0.315833
\(803\) 23.4164 0.826347
\(804\) 9.70820 0.342382
\(805\) −5.52786 −0.194832
\(806\) −11.0557 −0.389421
\(807\) −13.0557 −0.459583
\(808\) 4.47214 0.157329
\(809\) 42.9443 1.50984 0.754920 0.655817i \(-0.227676\pi\)
0.754920 + 0.655817i \(0.227676\pi\)
\(810\) 1.23607 0.0434310
\(811\) 41.3050 1.45041 0.725207 0.688531i \(-0.241744\pi\)
0.725207 + 0.688531i \(0.241744\pi\)
\(812\) 20.0000 0.701862
\(813\) −16.9443 −0.594262
\(814\) −58.8328 −2.06209
\(815\) −3.05573 −0.107037
\(816\) −4.00000 −0.140028
\(817\) −27.1935 −0.951380
\(818\) −35.8885 −1.25481
\(819\) −20.0000 −0.698857
\(820\) −2.47214 −0.0863307
\(821\) 39.5279 1.37953 0.689766 0.724032i \(-0.257713\pi\)
0.689766 + 0.724032i \(0.257713\pi\)
\(822\) 3.05573 0.106581
\(823\) −52.3607 −1.82518 −0.912589 0.408877i \(-0.865920\pi\)
−0.912589 + 0.408877i \(0.865920\pi\)
\(824\) −6.00000 −0.209020
\(825\) 18.1803 0.632958
\(826\) 40.0000 1.39178
\(827\) −21.5967 −0.750993 −0.375496 0.926824i \(-0.622528\pi\)
−0.375496 + 0.926824i \(0.622528\pi\)
\(828\) 1.00000 0.0347524
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 16.3607 0.567887
\(831\) −20.4721 −0.710171
\(832\) 4.47214 0.155043
\(833\) −52.0000 −1.80169
\(834\) −16.9443 −0.586732
\(835\) 20.9443 0.724806
\(836\) −29.8885 −1.03372
\(837\) −2.47214 −0.0854495
\(838\) 28.0689 0.969623
\(839\) −45.8885 −1.58425 −0.792124 0.610360i \(-0.791025\pi\)
−0.792124 + 0.610360i \(0.791025\pi\)
\(840\) −5.52786 −0.190729
\(841\) −9.00000 −0.310345
\(842\) −0.763932 −0.0263268
\(843\) −13.5279 −0.465924
\(844\) −23.4164 −0.806026
\(845\) 8.65248 0.297654
\(846\) 4.00000 0.137523
\(847\) −73.4164 −2.52262
\(848\) 5.23607 0.179807
\(849\) −3.81966 −0.131090
\(850\) 13.8885 0.476373
\(851\) 11.2361 0.385167
\(852\) 8.94427 0.306426
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −3.41641 −0.116907
\(855\) 7.05573 0.241301
\(856\) −12.6525 −0.432453
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −23.4164 −0.799423
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −5.88854 −0.200798
\(861\) 8.94427 0.304820
\(862\) −23.4164 −0.797566
\(863\) 14.8328 0.504915 0.252457 0.967608i \(-0.418761\pi\)
0.252457 + 0.967608i \(0.418761\pi\)
\(864\) 1.00000 0.0340207
\(865\) −21.5279 −0.731969
\(866\) −21.4164 −0.727759
\(867\) −1.00000 −0.0339618
\(868\) 11.0557 0.375256
\(869\) −23.4164 −0.794347
\(870\) −5.52786 −0.187412
\(871\) 43.4164 1.47111
\(872\) 4.76393 0.161327
\(873\) 0.472136 0.0159794
\(874\) 5.70820 0.193083
\(875\) 46.8328 1.58324
\(876\) −4.47214 −0.151099
\(877\) −48.2492 −1.62926 −0.814630 0.579981i \(-0.803060\pi\)
−0.814630 + 0.579981i \(0.803060\pi\)
\(878\) 19.0557 0.643100
\(879\) 0.291796 0.00984204
\(880\) −6.47214 −0.218176
\(881\) 39.4164 1.32797 0.663986 0.747745i \(-0.268863\pi\)
0.663986 + 0.747745i \(0.268863\pi\)
\(882\) 13.0000 0.437733
\(883\) 13.5279 0.455249 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(884\) −17.8885 −0.601657
\(885\) −11.0557 −0.371634
\(886\) −15.0557 −0.505807
\(887\) −16.9443 −0.568933 −0.284466 0.958686i \(-0.591816\pi\)
−0.284466 + 0.958686i \(0.591816\pi\)
\(888\) 11.2361 0.377058
\(889\) 17.8885 0.599963
\(890\) −12.9443 −0.433893
\(891\) −5.23607 −0.175415
\(892\) 19.4164 0.650109
\(893\) 22.8328 0.764071
\(894\) −11.7082 −0.391581
\(895\) 24.0000 0.802232
\(896\) −4.47214 −0.149404
\(897\) 4.47214 0.149320
\(898\) 15.8885 0.530208
\(899\) 11.0557 0.368729
\(900\) −3.47214 −0.115738
\(901\) −20.9443 −0.697755
\(902\) 10.4721 0.348684
\(903\) 21.3050 0.708984
\(904\) −5.52786 −0.183854
\(905\) −13.8885 −0.461671
\(906\) −14.4721 −0.480805
\(907\) 4.18034 0.138806 0.0694030 0.997589i \(-0.477891\pi\)
0.0694030 + 0.997589i \(0.477891\pi\)
\(908\) −9.23607 −0.306510
\(909\) 4.47214 0.148331
\(910\) −24.7214 −0.819505
\(911\) 16.5836 0.549439 0.274719 0.961524i \(-0.411415\pi\)
0.274719 + 0.961524i \(0.411415\pi\)
\(912\) 5.70820 0.189018
\(913\) −69.3050 −2.29366
\(914\) 27.5279 0.910541
\(915\) 0.944272 0.0312167
\(916\) 17.7082 0.585096
\(917\) 42.6099 1.40710
\(918\) −4.00000 −0.132020
\(919\) −16.4721 −0.543366 −0.271683 0.962387i \(-0.587580\pi\)
−0.271683 + 0.962387i \(0.587580\pi\)
\(920\) 1.23607 0.0407520
\(921\) 15.4164 0.507988
\(922\) 22.0000 0.724531
\(923\) 40.0000 1.31662
\(924\) 23.4164 0.770343
\(925\) −39.0132 −1.28274
\(926\) 24.0000 0.788689
\(927\) −6.00000 −0.197066
\(928\) −4.47214 −0.146805
\(929\) −24.8328 −0.814738 −0.407369 0.913264i \(-0.633554\pi\)
−0.407369 + 0.913264i \(0.633554\pi\)
\(930\) −3.05573 −0.100201
\(931\) 74.2067 2.43202
\(932\) 19.8885 0.651471
\(933\) −20.9443 −0.685685
\(934\) −17.8197 −0.583077
\(935\) 25.8885 0.846646
\(936\) 4.47214 0.146176
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) −43.4164 −1.41760
\(939\) 15.5279 0.506733
\(940\) 4.94427 0.161264
\(941\) 38.1803 1.24464 0.622322 0.782762i \(-0.286190\pi\)
0.622322 + 0.782762i \(0.286190\pi\)
\(942\) −6.65248 −0.216749
\(943\) −2.00000 −0.0651290
\(944\) −8.94427 −0.291111
\(945\) −5.52786 −0.179821
\(946\) 24.9443 0.811008
\(947\) −47.1935 −1.53358 −0.766791 0.641897i \(-0.778148\pi\)
−0.766791 + 0.641897i \(0.778148\pi\)
\(948\) 4.47214 0.145248
\(949\) −20.0000 −0.649227
\(950\) −19.8197 −0.643035
\(951\) 19.5279 0.633234
\(952\) 17.8885 0.579771
\(953\) −47.7771 −1.54765 −0.773826 0.633398i \(-0.781659\pi\)
−0.773826 + 0.633398i \(0.781659\pi\)
\(954\) 5.23607 0.169524
\(955\) 8.00000 0.258874
\(956\) −4.94427 −0.159909
\(957\) 23.4164 0.756945
\(958\) −28.9443 −0.935147
\(959\) −13.6656 −0.441286
\(960\) 1.23607 0.0398939
\(961\) −24.8885 −0.802856
\(962\) 50.2492 1.62010
\(963\) −12.6525 −0.407720
\(964\) −12.4721 −0.401700
\(965\) 29.5279 0.950536
\(966\) −4.47214 −0.143889
\(967\) 15.4164 0.495758 0.247879 0.968791i \(-0.420266\pi\)
0.247879 + 0.968791i \(0.420266\pi\)
\(968\) 16.4164 0.527643
\(969\) −22.8328 −0.733496
\(970\) 0.583592 0.0187380
\(971\) −19.1246 −0.613738 −0.306869 0.951752i \(-0.599281\pi\)
−0.306869 + 0.951752i \(0.599281\pi\)
\(972\) 1.00000 0.0320750
\(973\) 75.7771 2.42930
\(974\) 9.88854 0.316849
\(975\) −15.5279 −0.497290
\(976\) 0.763932 0.0244529
\(977\) 1.16718 0.0373415 0.0186708 0.999826i \(-0.494057\pi\)
0.0186708 + 0.999826i \(0.494057\pi\)
\(978\) −2.47214 −0.0790502
\(979\) 54.8328 1.75246
\(980\) 16.0689 0.513302
\(981\) 4.76393 0.152101
\(982\) 29.3050 0.935159
\(983\) 0.583592 0.0186137 0.00930685 0.999957i \(-0.497037\pi\)
0.00930685 + 0.999957i \(0.497037\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 3.63932 0.115958
\(986\) 17.8885 0.569687
\(987\) −17.8885 −0.569399
\(988\) 25.5279 0.812150
\(989\) −4.76393 −0.151484
\(990\) −6.47214 −0.205698
\(991\) −10.4721 −0.332658 −0.166329 0.986070i \(-0.553191\pi\)
−0.166329 + 0.986070i \(0.553191\pi\)
\(992\) −2.47214 −0.0784904
\(993\) −10.4721 −0.332323
\(994\) −40.0000 −1.26872
\(995\) 21.5279 0.682479
\(996\) 13.2361 0.419401
\(997\) 5.05573 0.160117 0.0800583 0.996790i \(-0.474489\pi\)
0.0800583 + 0.996790i \(0.474489\pi\)
\(998\) 0.583592 0.0184733
\(999\) 11.2361 0.355493
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 138.2.a.d.1.2 2
3.2 odd 2 414.2.a.f.1.1 2
4.3 odd 2 1104.2.a.j.1.2 2
5.2 odd 4 3450.2.d.x.2899.3 4
5.3 odd 4 3450.2.d.x.2899.2 4
5.4 even 2 3450.2.a.be.1.2 2
7.6 odd 2 6762.2.a.cb.1.1 2
8.3 odd 2 4416.2.a.bl.1.1 2
8.5 even 2 4416.2.a.bh.1.1 2
12.11 even 2 3312.2.a.bc.1.1 2
23.22 odd 2 3174.2.a.s.1.1 2
69.68 even 2 9522.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.2 2 1.1 even 1 trivial
414.2.a.f.1.1 2 3.2 odd 2
1104.2.a.j.1.2 2 4.3 odd 2
3174.2.a.s.1.1 2 23.22 odd 2
3312.2.a.bc.1.1 2 12.11 even 2
3450.2.a.be.1.2 2 5.4 even 2
3450.2.d.x.2899.2 4 5.3 odd 4
3450.2.d.x.2899.3 4 5.2 odd 4
4416.2.a.bh.1.1 2 8.5 even 2
4416.2.a.bl.1.1 2 8.3 odd 2
6762.2.a.cb.1.1 2 7.6 odd 2
9522.2.a.q.1.2 2 69.68 even 2