Properties

Label 4730.2.a.m.1.2
Level $4730$
Weight $2$
Character 4730.1
Self dual yes
Analytic conductor $37.769$
Analytic rank $1$
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] = [4730,2,Mod(1,4730)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4730, 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("4730.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4730 = 2 \cdot 5 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4730.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: \(37.7692401561\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 4730.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.56155 q^{6} +3.12311 q^{7} -1.00000 q^{8} -0.561553 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.56155 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.56155 q^{6} +3.12311 q^{7} -1.00000 q^{8} -0.561553 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.56155 q^{12} -4.00000 q^{13} -3.12311 q^{14} +1.56155 q^{15} +1.00000 q^{16} -4.00000 q^{17} +0.561553 q^{18} -7.12311 q^{19} +1.00000 q^{20} +4.87689 q^{21} +1.00000 q^{22} +6.00000 q^{23} -1.56155 q^{24} +1.00000 q^{25} +4.00000 q^{26} -5.56155 q^{27} +3.12311 q^{28} -5.56155 q^{29} -1.56155 q^{30} -1.00000 q^{32} -1.56155 q^{33} +4.00000 q^{34} +3.12311 q^{35} -0.561553 q^{36} -1.12311 q^{37} +7.12311 q^{38} -6.24621 q^{39} -1.00000 q^{40} -1.12311 q^{41} -4.87689 q^{42} +1.00000 q^{43} -1.00000 q^{44} -0.561553 q^{45} -6.00000 q^{46} -2.00000 q^{47} +1.56155 q^{48} +2.75379 q^{49} -1.00000 q^{50} -6.24621 q^{51} -4.00000 q^{52} -3.56155 q^{53} +5.56155 q^{54} -1.00000 q^{55} -3.12311 q^{56} -11.1231 q^{57} +5.56155 q^{58} +6.24621 q^{59} +1.56155 q^{60} -5.56155 q^{61} -1.75379 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.56155 q^{66} +2.00000 q^{67} -4.00000 q^{68} +9.36932 q^{69} -3.12311 q^{70} -1.12311 q^{71} +0.561553 q^{72} +1.31534 q^{73} +1.12311 q^{74} +1.56155 q^{75} -7.12311 q^{76} -3.12311 q^{77} +6.24621 q^{78} -13.5616 q^{79} +1.00000 q^{80} -7.00000 q^{81} +1.12311 q^{82} -2.43845 q^{83} +4.87689 q^{84} -4.00000 q^{85} -1.00000 q^{86} -8.68466 q^{87} +1.00000 q^{88} -7.36932 q^{89} +0.561553 q^{90} -12.4924 q^{91} +6.00000 q^{92} +2.00000 q^{94} -7.12311 q^{95} -1.56155 q^{96} -15.5616 q^{97} -2.75379 q^{98} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} + 3 q^{9} - 2 q^{10} - 2 q^{11} - q^{12} - 8 q^{13} + 2 q^{14} - q^{15} + 2 q^{16} - 8 q^{17} - 3 q^{18} - 6 q^{19} + 2 q^{20} + 18 q^{21} + 2 q^{22} + 12 q^{23} + q^{24} + 2 q^{25} + 8 q^{26} - 7 q^{27} - 2 q^{28} - 7 q^{29} + q^{30} - 2 q^{32} + q^{33} + 8 q^{34} - 2 q^{35} + 3 q^{36} + 6 q^{37} + 6 q^{38} + 4 q^{39} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 2 q^{43} - 2 q^{44} + 3 q^{45} - 12 q^{46} - 4 q^{47} - q^{48} + 22 q^{49} - 2 q^{50} + 4 q^{51} - 8 q^{52} - 3 q^{53} + 7 q^{54} - 2 q^{55} + 2 q^{56} - 14 q^{57} + 7 q^{58} - 4 q^{59} - q^{60} - 7 q^{61} - 20 q^{63} + 2 q^{64} - 8 q^{65} - q^{66} + 4 q^{67} - 8 q^{68} - 6 q^{69} + 2 q^{70} + 6 q^{71} - 3 q^{72} + 15 q^{73} - 6 q^{74} - q^{75} - 6 q^{76} + 2 q^{77} - 4 q^{78} - 23 q^{79} + 2 q^{80} - 14 q^{81} - 6 q^{82} - 9 q^{83} + 18 q^{84} - 8 q^{85} - 2 q^{86} - 5 q^{87} + 2 q^{88} + 10 q^{89} - 3 q^{90} + 8 q^{91} + 12 q^{92} + 4 q^{94} - 6 q^{95} + q^{96} - 27 q^{97} - 22 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.56155 −0.637501
\(7\) 3.12311 1.18042 0.590211 0.807249i \(-0.299044\pi\)
0.590211 + 0.807249i \(0.299044\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.561553 −0.187184
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 1.56155 0.450781
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −3.12311 −0.834685
\(15\) 1.56155 0.403191
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0.561553 0.132359
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 1.00000 0.223607
\(21\) 4.87689 1.06423
\(22\) 1.00000 0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −1.56155 −0.318751
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) −5.56155 −1.07032
\(28\) 3.12311 0.590211
\(29\) −5.56155 −1.03275 −0.516377 0.856361i \(-0.672720\pi\)
−0.516377 + 0.856361i \(0.672720\pi\)
\(30\) −1.56155 −0.285099
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.56155 −0.271831
\(34\) 4.00000 0.685994
\(35\) 3.12311 0.527901
\(36\) −0.561553 −0.0935921
\(37\) −1.12311 −0.184637 −0.0923187 0.995730i \(-0.529428\pi\)
−0.0923187 + 0.995730i \(0.529428\pi\)
\(38\) 7.12311 1.15552
\(39\) −6.24621 −1.00019
\(40\) −1.00000 −0.158114
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) −4.87689 −0.752521
\(43\) 1.00000 0.152499
\(44\) −1.00000 −0.150756
\(45\) −0.561553 −0.0837114
\(46\) −6.00000 −0.884652
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.56155 0.225391
\(49\) 2.75379 0.393398
\(50\) −1.00000 −0.141421
\(51\) −6.24621 −0.874645
\(52\) −4.00000 −0.554700
\(53\) −3.56155 −0.489217 −0.244608 0.969622i \(-0.578659\pi\)
−0.244608 + 0.969622i \(0.578659\pi\)
\(54\) 5.56155 0.756831
\(55\) −1.00000 −0.134840
\(56\) −3.12311 −0.417343
\(57\) −11.1231 −1.47329
\(58\) 5.56155 0.730268
\(59\) 6.24621 0.813187 0.406594 0.913609i \(-0.366716\pi\)
0.406594 + 0.913609i \(0.366716\pi\)
\(60\) 1.56155 0.201596
\(61\) −5.56155 −0.712084 −0.356042 0.934470i \(-0.615874\pi\)
−0.356042 + 0.934470i \(0.615874\pi\)
\(62\) 0 0
\(63\) −1.75379 −0.220957
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 1.56155 0.192214
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −4.00000 −0.485071
\(69\) 9.36932 1.12793
\(70\) −3.12311 −0.373283
\(71\) −1.12311 −0.133288 −0.0666441 0.997777i \(-0.521229\pi\)
−0.0666441 + 0.997777i \(0.521229\pi\)
\(72\) 0.561553 0.0661796
\(73\) 1.31534 0.153949 0.0769745 0.997033i \(-0.475474\pi\)
0.0769745 + 0.997033i \(0.475474\pi\)
\(74\) 1.12311 0.130558
\(75\) 1.56155 0.180313
\(76\) −7.12311 −0.817076
\(77\) −3.12311 −0.355911
\(78\) 6.24621 0.707244
\(79\) −13.5616 −1.52579 −0.762897 0.646520i \(-0.776224\pi\)
−0.762897 + 0.646520i \(0.776224\pi\)
\(80\) 1.00000 0.111803
\(81\) −7.00000 −0.777778
\(82\) 1.12311 0.124026
\(83\) −2.43845 −0.267654 −0.133827 0.991005i \(-0.542727\pi\)
−0.133827 + 0.991005i \(0.542727\pi\)
\(84\) 4.87689 0.532113
\(85\) −4.00000 −0.433861
\(86\) −1.00000 −0.107833
\(87\) −8.68466 −0.931093
\(88\) 1.00000 0.106600
\(89\) −7.36932 −0.781146 −0.390573 0.920572i \(-0.627723\pi\)
−0.390573 + 0.920572i \(0.627723\pi\)
\(90\) 0.561553 0.0591929
\(91\) −12.4924 −1.30956
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) −7.12311 −0.730815
\(96\) −1.56155 −0.159375
\(97\) −15.5616 −1.58004 −0.790018 0.613083i \(-0.789929\pi\)
−0.790018 + 0.613083i \(0.789929\pi\)
\(98\) −2.75379 −0.278175
\(99\) 0.561553 0.0564382
\(100\) 1.00000 0.100000
\(101\) −14.4924 −1.44205 −0.721025 0.692909i \(-0.756329\pi\)
−0.721025 + 0.692909i \(0.756329\pi\)
\(102\) 6.24621 0.618467
\(103\) 8.24621 0.812523 0.406262 0.913757i \(-0.366832\pi\)
0.406262 + 0.913757i \(0.366832\pi\)
\(104\) 4.00000 0.392232
\(105\) 4.87689 0.475936
\(106\) 3.56155 0.345929
\(107\) 3.80776 0.368110 0.184055 0.982916i \(-0.441077\pi\)
0.184055 + 0.982916i \(0.441077\pi\)
\(108\) −5.56155 −0.535161
\(109\) 1.12311 0.107574 0.0537870 0.998552i \(-0.482871\pi\)
0.0537870 + 0.998552i \(0.482871\pi\)
\(110\) 1.00000 0.0953463
\(111\) −1.75379 −0.166462
\(112\) 3.12311 0.295106
\(113\) 5.12311 0.481941 0.240971 0.970532i \(-0.422534\pi\)
0.240971 + 0.970532i \(0.422534\pi\)
\(114\) 11.1231 1.04177
\(115\) 6.00000 0.559503
\(116\) −5.56155 −0.516377
\(117\) 2.24621 0.207662
\(118\) −6.24621 −0.575010
\(119\) −12.4924 −1.14518
\(120\) −1.56155 −0.142550
\(121\) 1.00000 0.0909091
\(122\) 5.56155 0.503519
\(123\) −1.75379 −0.158134
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 1.75379 0.156240
\(127\) −3.80776 −0.337884 −0.168942 0.985626i \(-0.554035\pi\)
−0.168942 + 0.985626i \(0.554035\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.56155 0.137487
\(130\) 4.00000 0.350823
\(131\) 6.24621 0.545734 0.272867 0.962052i \(-0.412028\pi\)
0.272867 + 0.962052i \(0.412028\pi\)
\(132\) −1.56155 −0.135916
\(133\) −22.2462 −1.92899
\(134\) −2.00000 −0.172774
\(135\) −5.56155 −0.478662
\(136\) 4.00000 0.342997
\(137\) −5.12311 −0.437696 −0.218848 0.975759i \(-0.570230\pi\)
−0.218848 + 0.975759i \(0.570230\pi\)
\(138\) −9.36932 −0.797569
\(139\) −4.68466 −0.397348 −0.198674 0.980066i \(-0.563663\pi\)
−0.198674 + 0.980066i \(0.563663\pi\)
\(140\) 3.12311 0.263951
\(141\) −3.12311 −0.263013
\(142\) 1.12311 0.0942489
\(143\) 4.00000 0.334497
\(144\) −0.561553 −0.0467961
\(145\) −5.56155 −0.461862
\(146\) −1.31534 −0.108858
\(147\) 4.30019 0.354673
\(148\) −1.12311 −0.0923187
\(149\) 4.68466 0.383782 0.191891 0.981416i \(-0.438538\pi\)
0.191891 + 0.981416i \(0.438538\pi\)
\(150\) −1.56155 −0.127500
\(151\) 21.3693 1.73901 0.869505 0.493924i \(-0.164438\pi\)
0.869505 + 0.493924i \(0.164438\pi\)
\(152\) 7.12311 0.577760
\(153\) 2.24621 0.181595
\(154\) 3.12311 0.251667
\(155\) 0 0
\(156\) −6.24621 −0.500097
\(157\) 7.75379 0.618820 0.309410 0.950929i \(-0.399869\pi\)
0.309410 + 0.950929i \(0.399869\pi\)
\(158\) 13.5616 1.07890
\(159\) −5.56155 −0.441060
\(160\) −1.00000 −0.0790569
\(161\) 18.7386 1.47681
\(162\) 7.00000 0.549972
\(163\) 4.68466 0.366931 0.183465 0.983026i \(-0.441268\pi\)
0.183465 + 0.983026i \(0.441268\pi\)
\(164\) −1.12311 −0.0876998
\(165\) −1.56155 −0.121567
\(166\) 2.43845 0.189260
\(167\) 15.8078 1.22324 0.611621 0.791151i \(-0.290518\pi\)
0.611621 + 0.791151i \(0.290518\pi\)
\(168\) −4.87689 −0.376261
\(169\) 3.00000 0.230769
\(170\) 4.00000 0.306786
\(171\) 4.00000 0.305888
\(172\) 1.00000 0.0762493
\(173\) −2.24621 −0.170776 −0.0853881 0.996348i \(-0.527213\pi\)
−0.0853881 + 0.996348i \(0.527213\pi\)
\(174\) 8.68466 0.658382
\(175\) 3.12311 0.236085
\(176\) −1.00000 −0.0753778
\(177\) 9.75379 0.733140
\(178\) 7.36932 0.552354
\(179\) 8.43845 0.630719 0.315360 0.948972i \(-0.397875\pi\)
0.315360 + 0.948972i \(0.397875\pi\)
\(180\) −0.561553 −0.0418557
\(181\) −16.0540 −1.19328 −0.596641 0.802508i \(-0.703498\pi\)
−0.596641 + 0.802508i \(0.703498\pi\)
\(182\) 12.4924 0.926000
\(183\) −8.68466 −0.641988
\(184\) −6.00000 −0.442326
\(185\) −1.12311 −0.0825724
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) −2.00000 −0.145865
\(189\) −17.3693 −1.26343
\(190\) 7.12311 0.516764
\(191\) −9.31534 −0.674034 −0.337017 0.941498i \(-0.609418\pi\)
−0.337017 + 0.941498i \(0.609418\pi\)
\(192\) 1.56155 0.112695
\(193\) −14.2462 −1.02546 −0.512732 0.858548i \(-0.671367\pi\)
−0.512732 + 0.858548i \(0.671367\pi\)
\(194\) 15.5616 1.11725
\(195\) −6.24621 −0.447300
\(196\) 2.75379 0.196699
\(197\) −1.75379 −0.124952 −0.0624761 0.998046i \(-0.519900\pi\)
−0.0624761 + 0.998046i \(0.519900\pi\)
\(198\) −0.561553 −0.0399078
\(199\) −1.12311 −0.0796148 −0.0398074 0.999207i \(-0.512674\pi\)
−0.0398074 + 0.999207i \(0.512674\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 3.12311 0.220287
\(202\) 14.4924 1.01968
\(203\) −17.3693 −1.21909
\(204\) −6.24621 −0.437322
\(205\) −1.12311 −0.0784411
\(206\) −8.24621 −0.574541
\(207\) −3.36932 −0.234184
\(208\) −4.00000 −0.277350
\(209\) 7.12311 0.492716
\(210\) −4.87689 −0.336538
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −3.56155 −0.244608
\(213\) −1.75379 −0.120168
\(214\) −3.80776 −0.260293
\(215\) 1.00000 0.0681994
\(216\) 5.56155 0.378416
\(217\) 0 0
\(218\) −1.12311 −0.0760663
\(219\) 2.05398 0.138795
\(220\) −1.00000 −0.0674200
\(221\) 16.0000 1.07628
\(222\) 1.75379 0.117707
\(223\) −18.9309 −1.26770 −0.633852 0.773454i \(-0.718527\pi\)
−0.633852 + 0.773454i \(0.718527\pi\)
\(224\) −3.12311 −0.208671
\(225\) −0.561553 −0.0374369
\(226\) −5.12311 −0.340784
\(227\) −5.36932 −0.356374 −0.178187 0.983997i \(-0.557023\pi\)
−0.178187 + 0.983997i \(0.557023\pi\)
\(228\) −11.1231 −0.736646
\(229\) −3.56155 −0.235354 −0.117677 0.993052i \(-0.537545\pi\)
−0.117677 + 0.993052i \(0.537545\pi\)
\(230\) −6.00000 −0.395628
\(231\) −4.87689 −0.320876
\(232\) 5.56155 0.365134
\(233\) 3.56155 0.233325 0.116663 0.993172i \(-0.462780\pi\)
0.116663 + 0.993172i \(0.462780\pi\)
\(234\) −2.24621 −0.146839
\(235\) −2.00000 −0.130466
\(236\) 6.24621 0.406594
\(237\) −21.1771 −1.37560
\(238\) 12.4924 0.809763
\(239\) 0.684658 0.0442869 0.0221434 0.999755i \(-0.492951\pi\)
0.0221434 + 0.999755i \(0.492951\pi\)
\(240\) 1.56155 0.100798
\(241\) 16.8769 1.08714 0.543568 0.839365i \(-0.317073\pi\)
0.543568 + 0.839365i \(0.317073\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 5.75379 0.369106
\(244\) −5.56155 −0.356042
\(245\) 2.75379 0.175933
\(246\) 1.75379 0.111817
\(247\) 28.4924 1.81293
\(248\) 0 0
\(249\) −3.80776 −0.241307
\(250\) −1.00000 −0.0632456
\(251\) 25.3693 1.60130 0.800649 0.599134i \(-0.204488\pi\)
0.800649 + 0.599134i \(0.204488\pi\)
\(252\) −1.75379 −0.110478
\(253\) −6.00000 −0.377217
\(254\) 3.80776 0.238920
\(255\) −6.24621 −0.391153
\(256\) 1.00000 0.0625000
\(257\) 10.4924 0.654499 0.327250 0.944938i \(-0.393878\pi\)
0.327250 + 0.944938i \(0.393878\pi\)
\(258\) −1.56155 −0.0972180
\(259\) −3.50758 −0.217950
\(260\) −4.00000 −0.248069
\(261\) 3.12311 0.193315
\(262\) −6.24621 −0.385892
\(263\) 22.2462 1.37176 0.685880 0.727715i \(-0.259417\pi\)
0.685880 + 0.727715i \(0.259417\pi\)
\(264\) 1.56155 0.0961069
\(265\) −3.56155 −0.218784
\(266\) 22.2462 1.36400
\(267\) −11.5076 −0.704252
\(268\) 2.00000 0.122169
\(269\) 8.05398 0.491060 0.245530 0.969389i \(-0.421038\pi\)
0.245530 + 0.969389i \(0.421038\pi\)
\(270\) 5.56155 0.338465
\(271\) −9.75379 −0.592500 −0.296250 0.955110i \(-0.595736\pi\)
−0.296250 + 0.955110i \(0.595736\pi\)
\(272\) −4.00000 −0.242536
\(273\) −19.5076 −1.18065
\(274\) 5.12311 0.309498
\(275\) −1.00000 −0.0603023
\(276\) 9.36932 0.563967
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 4.68466 0.280967
\(279\) 0 0
\(280\) −3.12311 −0.186641
\(281\) 21.6155 1.28947 0.644737 0.764404i \(-0.276967\pi\)
0.644737 + 0.764404i \(0.276967\pi\)
\(282\) 3.12311 0.185978
\(283\) 9.56155 0.568375 0.284188 0.958769i \(-0.408276\pi\)
0.284188 + 0.958769i \(0.408276\pi\)
\(284\) −1.12311 −0.0666441
\(285\) −11.1231 −0.658876
\(286\) −4.00000 −0.236525
\(287\) −3.50758 −0.207046
\(288\) 0.561553 0.0330898
\(289\) −1.00000 −0.0588235
\(290\) 5.56155 0.326586
\(291\) −24.3002 −1.42450
\(292\) 1.31534 0.0769745
\(293\) −2.24621 −0.131225 −0.0656125 0.997845i \(-0.520900\pi\)
−0.0656125 + 0.997845i \(0.520900\pi\)
\(294\) −4.30019 −0.250792
\(295\) 6.24621 0.363668
\(296\) 1.12311 0.0652792
\(297\) 5.56155 0.322714
\(298\) −4.68466 −0.271375
\(299\) −24.0000 −1.38796
\(300\) 1.56155 0.0901563
\(301\) 3.12311 0.180013
\(302\) −21.3693 −1.22967
\(303\) −22.6307 −1.30010
\(304\) −7.12311 −0.408538
\(305\) −5.56155 −0.318454
\(306\) −2.24621 −0.128407
\(307\) 27.8078 1.58707 0.793536 0.608523i \(-0.208238\pi\)
0.793536 + 0.608523i \(0.208238\pi\)
\(308\) −3.12311 −0.177955
\(309\) 12.8769 0.732541
\(310\) 0 0
\(311\) −7.12311 −0.403914 −0.201957 0.979394i \(-0.564730\pi\)
−0.201957 + 0.979394i \(0.564730\pi\)
\(312\) 6.24621 0.353622
\(313\) −13.1231 −0.741762 −0.370881 0.928680i \(-0.620944\pi\)
−0.370881 + 0.928680i \(0.620944\pi\)
\(314\) −7.75379 −0.437572
\(315\) −1.75379 −0.0988148
\(316\) −13.5616 −0.762897
\(317\) −7.56155 −0.424699 −0.212350 0.977194i \(-0.568112\pi\)
−0.212350 + 0.977194i \(0.568112\pi\)
\(318\) 5.56155 0.311876
\(319\) 5.56155 0.311387
\(320\) 1.00000 0.0559017
\(321\) 5.94602 0.331875
\(322\) −18.7386 −1.04426
\(323\) 28.4924 1.58536
\(324\) −7.00000 −0.388889
\(325\) −4.00000 −0.221880
\(326\) −4.68466 −0.259459
\(327\) 1.75379 0.0969847
\(328\) 1.12311 0.0620131
\(329\) −6.24621 −0.344365
\(330\) 1.56155 0.0859607
\(331\) −14.8769 −0.817708 −0.408854 0.912600i \(-0.634071\pi\)
−0.408854 + 0.912600i \(0.634071\pi\)
\(332\) −2.43845 −0.133827
\(333\) 0.630683 0.0345612
\(334\) −15.8078 −0.864962
\(335\) 2.00000 0.109272
\(336\) 4.87689 0.266056
\(337\) −29.3693 −1.59985 −0.799924 0.600101i \(-0.795127\pi\)
−0.799924 + 0.600101i \(0.795127\pi\)
\(338\) −3.00000 −0.163178
\(339\) 8.00000 0.434500
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −13.2614 −0.716046
\(344\) −1.00000 −0.0539164
\(345\) 9.36932 0.504427
\(346\) 2.24621 0.120757
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −8.68466 −0.465547
\(349\) 3.12311 0.167176 0.0835880 0.996500i \(-0.473362\pi\)
0.0835880 + 0.996500i \(0.473362\pi\)
\(350\) −3.12311 −0.166937
\(351\) 22.2462 1.18741
\(352\) 1.00000 0.0533002
\(353\) 12.4384 0.662032 0.331016 0.943625i \(-0.392609\pi\)
0.331016 + 0.943625i \(0.392609\pi\)
\(354\) −9.75379 −0.518408
\(355\) −1.12311 −0.0596083
\(356\) −7.36932 −0.390573
\(357\) −19.5076 −1.03245
\(358\) −8.43845 −0.445986
\(359\) −16.6847 −0.880583 −0.440291 0.897855i \(-0.645125\pi\)
−0.440291 + 0.897855i \(0.645125\pi\)
\(360\) 0.561553 0.0295964
\(361\) 31.7386 1.67045
\(362\) 16.0540 0.843778
\(363\) 1.56155 0.0819603
\(364\) −12.4924 −0.654781
\(365\) 1.31534 0.0688481
\(366\) 8.68466 0.453954
\(367\) 30.4924 1.59169 0.795846 0.605499i \(-0.207027\pi\)
0.795846 + 0.605499i \(0.207027\pi\)
\(368\) 6.00000 0.312772
\(369\) 0.630683 0.0328321
\(370\) 1.12311 0.0583875
\(371\) −11.1231 −0.577483
\(372\) 0 0
\(373\) −20.9309 −1.08376 −0.541880 0.840456i \(-0.682287\pi\)
−0.541880 + 0.840456i \(0.682287\pi\)
\(374\) −4.00000 −0.206835
\(375\) 1.56155 0.0806382
\(376\) 2.00000 0.103142
\(377\) 22.2462 1.14574
\(378\) 17.3693 0.893381
\(379\) 23.6155 1.21305 0.606524 0.795065i \(-0.292563\pi\)
0.606524 + 0.795065i \(0.292563\pi\)
\(380\) −7.12311 −0.365408
\(381\) −5.94602 −0.304624
\(382\) 9.31534 0.476614
\(383\) 33.1771 1.69527 0.847635 0.530580i \(-0.178026\pi\)
0.847635 + 0.530580i \(0.178026\pi\)
\(384\) −1.56155 −0.0796877
\(385\) −3.12311 −0.159168
\(386\) 14.2462 0.725113
\(387\) −0.561553 −0.0285453
\(388\) −15.5616 −0.790018
\(389\) −20.7386 −1.05149 −0.525745 0.850642i \(-0.676213\pi\)
−0.525745 + 0.850642i \(0.676213\pi\)
\(390\) 6.24621 0.316289
\(391\) −24.0000 −1.21373
\(392\) −2.75379 −0.139087
\(393\) 9.75379 0.492014
\(394\) 1.75379 0.0883546
\(395\) −13.5616 −0.682356
\(396\) 0.561553 0.0282191
\(397\) −20.4384 −1.02578 −0.512888 0.858455i \(-0.671424\pi\)
−0.512888 + 0.858455i \(0.671424\pi\)
\(398\) 1.12311 0.0562962
\(399\) −34.7386 −1.73911
\(400\) 1.00000 0.0500000
\(401\) 22.6847 1.13282 0.566409 0.824124i \(-0.308332\pi\)
0.566409 + 0.824124i \(0.308332\pi\)
\(402\) −3.12311 −0.155766
\(403\) 0 0
\(404\) −14.4924 −0.721025
\(405\) −7.00000 −0.347833
\(406\) 17.3693 0.862025
\(407\) 1.12311 0.0556703
\(408\) 6.24621 0.309234
\(409\) −21.5616 −1.06615 −0.533075 0.846068i \(-0.678964\pi\)
−0.533075 + 0.846068i \(0.678964\pi\)
\(410\) 1.12311 0.0554662
\(411\) −8.00000 −0.394611
\(412\) 8.24621 0.406262
\(413\) 19.5076 0.959905
\(414\) 3.36932 0.165593
\(415\) −2.43845 −0.119699
\(416\) 4.00000 0.196116
\(417\) −7.31534 −0.358234
\(418\) −7.12311 −0.348402
\(419\) −20.0540 −0.979701 −0.489850 0.871807i \(-0.662949\pi\)
−0.489850 + 0.871807i \(0.662949\pi\)
\(420\) 4.87689 0.237968
\(421\) −30.8769 −1.50485 −0.752424 0.658679i \(-0.771115\pi\)
−0.752424 + 0.658679i \(0.771115\pi\)
\(422\) 0 0
\(423\) 1.12311 0.0546073
\(424\) 3.56155 0.172964
\(425\) −4.00000 −0.194029
\(426\) 1.75379 0.0849713
\(427\) −17.3693 −0.840560
\(428\) 3.80776 0.184055
\(429\) 6.24621 0.301570
\(430\) −1.00000 −0.0482243
\(431\) 26.7386 1.28795 0.643977 0.765045i \(-0.277283\pi\)
0.643977 + 0.765045i \(0.277283\pi\)
\(432\) −5.56155 −0.267580
\(433\) 31.3693 1.50751 0.753757 0.657154i \(-0.228240\pi\)
0.753757 + 0.657154i \(0.228240\pi\)
\(434\) 0 0
\(435\) −8.68466 −0.416398
\(436\) 1.12311 0.0537870
\(437\) −42.7386 −2.04447
\(438\) −2.05398 −0.0981427
\(439\) 20.4924 0.978050 0.489025 0.872270i \(-0.337353\pi\)
0.489025 + 0.872270i \(0.337353\pi\)
\(440\) 1.00000 0.0476731
\(441\) −1.54640 −0.0736380
\(442\) −16.0000 −0.761042
\(443\) 16.2462 0.771881 0.385940 0.922524i \(-0.373877\pi\)
0.385940 + 0.922524i \(0.373877\pi\)
\(444\) −1.75379 −0.0832311
\(445\) −7.36932 −0.349339
\(446\) 18.9309 0.896403
\(447\) 7.31534 0.346004
\(448\) 3.12311 0.147553
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0.561553 0.0264719
\(451\) 1.12311 0.0528850
\(452\) 5.12311 0.240971
\(453\) 33.3693 1.56783
\(454\) 5.36932 0.251995
\(455\) −12.4924 −0.585654
\(456\) 11.1231 0.520887
\(457\) −42.4924 −1.98771 −0.993856 0.110682i \(-0.964696\pi\)
−0.993856 + 0.110682i \(0.964696\pi\)
\(458\) 3.56155 0.166420
\(459\) 22.2462 1.03836
\(460\) 6.00000 0.279751
\(461\) −7.36932 −0.343223 −0.171612 0.985165i \(-0.554897\pi\)
−0.171612 + 0.985165i \(0.554897\pi\)
\(462\) 4.87689 0.226894
\(463\) −2.24621 −0.104390 −0.0521951 0.998637i \(-0.516622\pi\)
−0.0521951 + 0.998637i \(0.516622\pi\)
\(464\) −5.56155 −0.258189
\(465\) 0 0
\(466\) −3.56155 −0.164986
\(467\) −18.7386 −0.867121 −0.433560 0.901125i \(-0.642743\pi\)
−0.433560 + 0.901125i \(0.642743\pi\)
\(468\) 2.24621 0.103831
\(469\) 6.24621 0.288423
\(470\) 2.00000 0.0922531
\(471\) 12.1080 0.557905
\(472\) −6.24621 −0.287505
\(473\) −1.00000 −0.0459800
\(474\) 21.1771 0.972696
\(475\) −7.12311 −0.326831
\(476\) −12.4924 −0.572589
\(477\) 2.00000 0.0915737
\(478\) −0.684658 −0.0313155
\(479\) 8.68466 0.396812 0.198406 0.980120i \(-0.436424\pi\)
0.198406 + 0.980120i \(0.436424\pi\)
\(480\) −1.56155 −0.0712748
\(481\) 4.49242 0.204837
\(482\) −16.8769 −0.768721
\(483\) 29.2614 1.33144
\(484\) 1.00000 0.0454545
\(485\) −15.5616 −0.706614
\(486\) −5.75379 −0.260997
\(487\) −6.49242 −0.294200 −0.147100 0.989122i \(-0.546994\pi\)
−0.147100 + 0.989122i \(0.546994\pi\)
\(488\) 5.56155 0.251760
\(489\) 7.31534 0.330811
\(490\) −2.75379 −0.124403
\(491\) 3.12311 0.140944 0.0704719 0.997514i \(-0.477549\pi\)
0.0704719 + 0.997514i \(0.477549\pi\)
\(492\) −1.75379 −0.0790669
\(493\) 22.2462 1.00192
\(494\) −28.4924 −1.28193
\(495\) 0.561553 0.0252399
\(496\) 0 0
\(497\) −3.50758 −0.157336
\(498\) 3.80776 0.170630
\(499\) −3.56155 −0.159437 −0.0797185 0.996817i \(-0.525402\pi\)
−0.0797185 + 0.996817i \(0.525402\pi\)
\(500\) 1.00000 0.0447214
\(501\) 24.6847 1.10283
\(502\) −25.3693 −1.13229
\(503\) −26.7386 −1.19222 −0.596108 0.802904i \(-0.703287\pi\)
−0.596108 + 0.802904i \(0.703287\pi\)
\(504\) 1.75379 0.0781200
\(505\) −14.4924 −0.644904
\(506\) 6.00000 0.266733
\(507\) 4.68466 0.208053
\(508\) −3.80776 −0.168942
\(509\) −16.7386 −0.741927 −0.370963 0.928647i \(-0.620972\pi\)
−0.370963 + 0.928647i \(0.620972\pi\)
\(510\) 6.24621 0.276587
\(511\) 4.10795 0.181725
\(512\) −1.00000 −0.0441942
\(513\) 39.6155 1.74907
\(514\) −10.4924 −0.462801
\(515\) 8.24621 0.363371
\(516\) 1.56155 0.0687435
\(517\) 2.00000 0.0879599
\(518\) 3.50758 0.154114
\(519\) −3.50758 −0.153966
\(520\) 4.00000 0.175412
\(521\) 22.9848 1.00698 0.503492 0.864000i \(-0.332048\pi\)
0.503492 + 0.864000i \(0.332048\pi\)
\(522\) −3.12311 −0.136695
\(523\) −10.2462 −0.448036 −0.224018 0.974585i \(-0.571917\pi\)
−0.224018 + 0.974585i \(0.571917\pi\)
\(524\) 6.24621 0.272867
\(525\) 4.87689 0.212845
\(526\) −22.2462 −0.969981
\(527\) 0 0
\(528\) −1.56155 −0.0679579
\(529\) 13.0000 0.565217
\(530\) 3.56155 0.154704
\(531\) −3.50758 −0.152216
\(532\) −22.2462 −0.964496
\(533\) 4.49242 0.194588
\(534\) 11.5076 0.497982
\(535\) 3.80776 0.164624
\(536\) −2.00000 −0.0863868
\(537\) 13.1771 0.568633
\(538\) −8.05398 −0.347232
\(539\) −2.75379 −0.118614
\(540\) −5.56155 −0.239331
\(541\) 20.7386 0.891624 0.445812 0.895127i \(-0.352915\pi\)
0.445812 + 0.895127i \(0.352915\pi\)
\(542\) 9.75379 0.418961
\(543\) −25.0691 −1.07582
\(544\) 4.00000 0.171499
\(545\) 1.12311 0.0481086
\(546\) 19.5076 0.834847
\(547\) −45.5616 −1.94807 −0.974036 0.226395i \(-0.927306\pi\)
−0.974036 + 0.226395i \(0.927306\pi\)
\(548\) −5.12311 −0.218848
\(549\) 3.12311 0.133291
\(550\) 1.00000 0.0426401
\(551\) 39.6155 1.68768
\(552\) −9.36932 −0.398785
\(553\) −42.3542 −1.80108
\(554\) 22.0000 0.934690
\(555\) −1.75379 −0.0744442
\(556\) −4.68466 −0.198674
\(557\) −15.1231 −0.640787 −0.320393 0.947285i \(-0.603815\pi\)
−0.320393 + 0.947285i \(0.603815\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 3.12311 0.131975
\(561\) 6.24621 0.263715
\(562\) −21.6155 −0.911796
\(563\) −1.75379 −0.0739134 −0.0369567 0.999317i \(-0.511766\pi\)
−0.0369567 + 0.999317i \(0.511766\pi\)
\(564\) −3.12311 −0.131506
\(565\) 5.12311 0.215531
\(566\) −9.56155 −0.401902
\(567\) −21.8617 −0.918107
\(568\) 1.12311 0.0471245
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 11.1231 0.465896
\(571\) −6.63068 −0.277485 −0.138743 0.990328i \(-0.544306\pi\)
−0.138743 + 0.990328i \(0.544306\pi\)
\(572\) 4.00000 0.167248
\(573\) −14.5464 −0.607684
\(574\) 3.50758 0.146403
\(575\) 6.00000 0.250217
\(576\) −0.561553 −0.0233980
\(577\) 5.61553 0.233777 0.116889 0.993145i \(-0.462708\pi\)
0.116889 + 0.993145i \(0.462708\pi\)
\(578\) 1.00000 0.0415945
\(579\) −22.2462 −0.924521
\(580\) −5.56155 −0.230931
\(581\) −7.61553 −0.315945
\(582\) 24.3002 1.00728
\(583\) 3.56155 0.147504
\(584\) −1.31534 −0.0544292
\(585\) 2.24621 0.0928694
\(586\) 2.24621 0.0927901
\(587\) −46.2462 −1.90879 −0.954393 0.298554i \(-0.903496\pi\)
−0.954393 + 0.298554i \(0.903496\pi\)
\(588\) 4.30019 0.177337
\(589\) 0 0
\(590\) −6.24621 −0.257152
\(591\) −2.73863 −0.112652
\(592\) −1.12311 −0.0461594
\(593\) 26.1922 1.07559 0.537793 0.843077i \(-0.319258\pi\)
0.537793 + 0.843077i \(0.319258\pi\)
\(594\) −5.56155 −0.228193
\(595\) −12.4924 −0.512139
\(596\) 4.68466 0.191891
\(597\) −1.75379 −0.0717778
\(598\) 24.0000 0.981433
\(599\) 5.75379 0.235093 0.117547 0.993067i \(-0.462497\pi\)
0.117547 + 0.993067i \(0.462497\pi\)
\(600\) −1.56155 −0.0637501
\(601\) −0.876894 −0.0357693 −0.0178846 0.999840i \(-0.505693\pi\)
−0.0178846 + 0.999840i \(0.505693\pi\)
\(602\) −3.12311 −0.127288
\(603\) −1.12311 −0.0457364
\(604\) 21.3693 0.869505
\(605\) 1.00000 0.0406558
\(606\) 22.6307 0.919309
\(607\) −1.36932 −0.0555789 −0.0277894 0.999614i \(-0.508847\pi\)
−0.0277894 + 0.999614i \(0.508847\pi\)
\(608\) 7.12311 0.288880
\(609\) −27.1231 −1.09908
\(610\) 5.56155 0.225181
\(611\) 8.00000 0.323645
\(612\) 2.24621 0.0907977
\(613\) −47.1231 −1.90328 −0.951642 0.307209i \(-0.900605\pi\)
−0.951642 + 0.307209i \(0.900605\pi\)
\(614\) −27.8078 −1.12223
\(615\) −1.75379 −0.0707196
\(616\) 3.12311 0.125834
\(617\) −46.4924 −1.87171 −0.935857 0.352379i \(-0.885373\pi\)
−0.935857 + 0.352379i \(0.885373\pi\)
\(618\) −12.8769 −0.517985
\(619\) 18.2462 0.733377 0.366689 0.930344i \(-0.380491\pi\)
0.366689 + 0.930344i \(0.380491\pi\)
\(620\) 0 0
\(621\) −33.3693 −1.33906
\(622\) 7.12311 0.285611
\(623\) −23.0152 −0.922083
\(624\) −6.24621 −0.250049
\(625\) 1.00000 0.0400000
\(626\) 13.1231 0.524505
\(627\) 11.1231 0.444214
\(628\) 7.75379 0.309410
\(629\) 4.49242 0.179125
\(630\) 1.75379 0.0698726
\(631\) −46.6847 −1.85849 −0.929243 0.369468i \(-0.879540\pi\)
−0.929243 + 0.369468i \(0.879540\pi\)
\(632\) 13.5616 0.539450
\(633\) 0 0
\(634\) 7.56155 0.300308
\(635\) −3.80776 −0.151107
\(636\) −5.56155 −0.220530
\(637\) −11.0152 −0.436436
\(638\) −5.56155 −0.220184
\(639\) 0.630683 0.0249494
\(640\) −1.00000 −0.0395285
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) −5.94602 −0.234671
\(643\) 2.49242 0.0982915 0.0491458 0.998792i \(-0.484350\pi\)
0.0491458 + 0.998792i \(0.484350\pi\)
\(644\) 18.7386 0.738406
\(645\) 1.56155 0.0614861
\(646\) −28.4924 −1.12102
\(647\) 21.5616 0.847672 0.423836 0.905739i \(-0.360683\pi\)
0.423836 + 0.905739i \(0.360683\pi\)
\(648\) 7.00000 0.274986
\(649\) −6.24621 −0.245185
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 4.68466 0.183465
\(653\) 22.9848 0.899466 0.449733 0.893163i \(-0.351519\pi\)
0.449733 + 0.893163i \(0.351519\pi\)
\(654\) −1.75379 −0.0685786
\(655\) 6.24621 0.244060
\(656\) −1.12311 −0.0438499
\(657\) −0.738634 −0.0288168
\(658\) 6.24621 0.243503
\(659\) −30.4384 −1.18571 −0.592857 0.805308i \(-0.702000\pi\)
−0.592857 + 0.805308i \(0.702000\pi\)
\(660\) −1.56155 −0.0607834
\(661\) 12.4384 0.483800 0.241900 0.970301i \(-0.422229\pi\)
0.241900 + 0.970301i \(0.422229\pi\)
\(662\) 14.8769 0.578207
\(663\) 24.9848 0.970331
\(664\) 2.43845 0.0946301
\(665\) −22.2462 −0.862671
\(666\) −0.630683 −0.0244385
\(667\) −33.3693 −1.29207
\(668\) 15.8078 0.611621
\(669\) −29.5616 −1.14292
\(670\) −2.00000 −0.0772667
\(671\) 5.56155 0.214701
\(672\) −4.87689 −0.188130
\(673\) −28.0540 −1.08140 −0.540701 0.841215i \(-0.681841\pi\)
−0.540701 + 0.841215i \(0.681841\pi\)
\(674\) 29.3693 1.13126
\(675\) −5.56155 −0.214064
\(676\) 3.00000 0.115385
\(677\) 2.68466 0.103180 0.0515899 0.998668i \(-0.483571\pi\)
0.0515899 + 0.998668i \(0.483571\pi\)
\(678\) −8.00000 −0.307238
\(679\) −48.6004 −1.86511
\(680\) 4.00000 0.153393
\(681\) −8.38447 −0.321294
\(682\) 0 0
\(683\) −5.61553 −0.214872 −0.107436 0.994212i \(-0.534264\pi\)
−0.107436 + 0.994212i \(0.534264\pi\)
\(684\) 4.00000 0.152944
\(685\) −5.12311 −0.195744
\(686\) 13.2614 0.506321
\(687\) −5.56155 −0.212186
\(688\) 1.00000 0.0381246
\(689\) 14.2462 0.542737
\(690\) −9.36932 −0.356684
\(691\) 26.3002 1.00051 0.500253 0.865879i \(-0.333240\pi\)
0.500253 + 0.865879i \(0.333240\pi\)
\(692\) −2.24621 −0.0853881
\(693\) 1.75379 0.0666209
\(694\) 12.0000 0.455514
\(695\) −4.68466 −0.177699
\(696\) 8.68466 0.329191
\(697\) 4.49242 0.170163
\(698\) −3.12311 −0.118211
\(699\) 5.56155 0.210357
\(700\) 3.12311 0.118042
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) −22.2462 −0.839629
\(703\) 8.00000 0.301726
\(704\) −1.00000 −0.0376889
\(705\) −3.12311 −0.117623
\(706\) −12.4384 −0.468127
\(707\) −45.2614 −1.70223
\(708\) 9.75379 0.366570
\(709\) 8.05398 0.302473 0.151237 0.988498i \(-0.451674\pi\)
0.151237 + 0.988498i \(0.451674\pi\)
\(710\) 1.12311 0.0421494
\(711\) 7.61553 0.285605
\(712\) 7.36932 0.276177
\(713\) 0 0
\(714\) 19.5076 0.730053
\(715\) 4.00000 0.149592
\(716\) 8.43845 0.315360
\(717\) 1.06913 0.0399274
\(718\) 16.6847 0.622666
\(719\) 0.876894 0.0327026 0.0163513 0.999866i \(-0.494795\pi\)
0.0163513 + 0.999866i \(0.494795\pi\)
\(720\) −0.561553 −0.0209278
\(721\) 25.7538 0.959121
\(722\) −31.7386 −1.18119
\(723\) 26.3542 0.980122
\(724\) −16.0540 −0.596641
\(725\) −5.56155 −0.206551
\(726\) −1.56155 −0.0579547
\(727\) 17.1771 0.637063 0.318531 0.947912i \(-0.396810\pi\)
0.318531 + 0.947912i \(0.396810\pi\)
\(728\) 12.4924 0.463000
\(729\) 29.9848 1.11055
\(730\) −1.31534 −0.0486830
\(731\) −4.00000 −0.147945
\(732\) −8.68466 −0.320994
\(733\) 11.0691 0.408848 0.204424 0.978882i \(-0.434468\pi\)
0.204424 + 0.978882i \(0.434468\pi\)
\(734\) −30.4924 −1.12550
\(735\) 4.30019 0.158615
\(736\) −6.00000 −0.221163
\(737\) −2.00000 −0.0736709
\(738\) −0.630683 −0.0232158
\(739\) −14.7386 −0.542169 −0.271085 0.962555i \(-0.587382\pi\)
−0.271085 + 0.962555i \(0.587382\pi\)
\(740\) −1.12311 −0.0412862
\(741\) 44.4924 1.63447
\(742\) 11.1231 0.408342
\(743\) −42.3542 −1.55382 −0.776912 0.629610i \(-0.783215\pi\)
−0.776912 + 0.629610i \(0.783215\pi\)
\(744\) 0 0
\(745\) 4.68466 0.171633
\(746\) 20.9309 0.766334
\(747\) 1.36932 0.0501007
\(748\) 4.00000 0.146254
\(749\) 11.8920 0.434526
\(750\) −1.56155 −0.0570198
\(751\) −19.1771 −0.699782 −0.349891 0.936790i \(-0.613781\pi\)
−0.349891 + 0.936790i \(0.613781\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 39.6155 1.44367
\(754\) −22.2462 −0.810159
\(755\) 21.3693 0.777709
\(756\) −17.3693 −0.631716
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) −23.6155 −0.857755
\(759\) −9.36932 −0.340085
\(760\) 7.12311 0.258382
\(761\) 22.9309 0.831243 0.415622 0.909538i \(-0.363564\pi\)
0.415622 + 0.909538i \(0.363564\pi\)
\(762\) 5.94602 0.215402
\(763\) 3.50758 0.126983
\(764\) −9.31534 −0.337017
\(765\) 2.24621 0.0812119
\(766\) −33.1771 −1.19874
\(767\) −24.9848 −0.902150
\(768\) 1.56155 0.0563477
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 3.12311 0.112549
\(771\) 16.3845 0.590072
\(772\) −14.2462 −0.512732
\(773\) −41.6155 −1.49681 −0.748403 0.663244i \(-0.769179\pi\)
−0.748403 + 0.663244i \(0.769179\pi\)
\(774\) 0.561553 0.0201846
\(775\) 0 0
\(776\) 15.5616 0.558627
\(777\) −5.47727 −0.196496
\(778\) 20.7386 0.743516
\(779\) 8.00000 0.286630
\(780\) −6.24621 −0.223650
\(781\) 1.12311 0.0401879
\(782\) 24.0000 0.858238
\(783\) 30.9309 1.10538
\(784\) 2.75379 0.0983496
\(785\) 7.75379 0.276745
\(786\) −9.75379 −0.347906
\(787\) 42.2462 1.50591 0.752957 0.658069i \(-0.228627\pi\)
0.752957 + 0.658069i \(0.228627\pi\)
\(788\) −1.75379 −0.0624761
\(789\) 34.7386 1.23673
\(790\) 13.5616 0.482498
\(791\) 16.0000 0.568895
\(792\) −0.561553 −0.0199539
\(793\) 22.2462 0.789986
\(794\) 20.4384 0.725333
\(795\) −5.56155 −0.197248
\(796\) −1.12311 −0.0398074
\(797\) −4.73863 −0.167851 −0.0839255 0.996472i \(-0.526746\pi\)
−0.0839255 + 0.996472i \(0.526746\pi\)
\(798\) 34.7386 1.22973
\(799\) 8.00000 0.283020
\(800\) −1.00000 −0.0353553
\(801\) 4.13826 0.146218
\(802\) −22.6847 −0.801023
\(803\) −1.31534 −0.0464174
\(804\) 3.12311 0.110143
\(805\) 18.7386 0.660450
\(806\) 0 0
\(807\) 12.5767 0.442721
\(808\) 14.4924 0.509842
\(809\) −7.36932 −0.259091 −0.129546 0.991573i \(-0.541352\pi\)
−0.129546 + 0.991573i \(0.541352\pi\)
\(810\) 7.00000 0.245955
\(811\) 24.8769 0.873546 0.436773 0.899572i \(-0.356121\pi\)
0.436773 + 0.899572i \(0.356121\pi\)
\(812\) −17.3693 −0.609544
\(813\) −15.2311 −0.534176
\(814\) −1.12311 −0.0393648
\(815\) 4.68466 0.164096
\(816\) −6.24621 −0.218661
\(817\) −7.12311 −0.249206
\(818\) 21.5616 0.753882
\(819\) 7.01515 0.245129
\(820\) −1.12311 −0.0392205
\(821\) −33.6155 −1.17319 −0.586595 0.809880i \(-0.699532\pi\)
−0.586595 + 0.809880i \(0.699532\pi\)
\(822\) 8.00000 0.279032
\(823\) −29.2311 −1.01893 −0.509465 0.860491i \(-0.670157\pi\)
−0.509465 + 0.860491i \(0.670157\pi\)
\(824\) −8.24621 −0.287270
\(825\) −1.56155 −0.0543663
\(826\) −19.5076 −0.678755
\(827\) 14.4384 0.502074 0.251037 0.967977i \(-0.419228\pi\)
0.251037 + 0.967977i \(0.419228\pi\)
\(828\) −3.36932 −0.117092
\(829\) 51.8617 1.80123 0.900616 0.434615i \(-0.143116\pi\)
0.900616 + 0.434615i \(0.143116\pi\)
\(830\) 2.43845 0.0846397
\(831\) −34.3542 −1.19173
\(832\) −4.00000 −0.138675
\(833\) −11.0152 −0.381653
\(834\) 7.31534 0.253310
\(835\) 15.8078 0.547050
\(836\) 7.12311 0.246358
\(837\) 0 0
\(838\) 20.0540 0.692753
\(839\) −22.1922 −0.766161 −0.383081 0.923715i \(-0.625137\pi\)
−0.383081 + 0.923715i \(0.625137\pi\)
\(840\) −4.87689 −0.168269
\(841\) 1.93087 0.0665817
\(842\) 30.8769 1.06409
\(843\) 33.7538 1.16254
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) −1.12311 −0.0386132
\(847\) 3.12311 0.107311
\(848\) −3.56155 −0.122304
\(849\) 14.9309 0.512426
\(850\) 4.00000 0.137199
\(851\) −6.73863 −0.230997
\(852\) −1.75379 −0.0600838
\(853\) 19.1231 0.654763 0.327381 0.944892i \(-0.393834\pi\)
0.327381 + 0.944892i \(0.393834\pi\)
\(854\) 17.3693 0.594366
\(855\) 4.00000 0.136797
\(856\) −3.80776 −0.130147
\(857\) −44.4924 −1.51983 −0.759916 0.650021i \(-0.774760\pi\)
−0.759916 + 0.650021i \(0.774760\pi\)
\(858\) −6.24621 −0.213242
\(859\) −19.8617 −0.677674 −0.338837 0.940845i \(-0.610033\pi\)
−0.338837 + 0.940845i \(0.610033\pi\)
\(860\) 1.00000 0.0340997
\(861\) −5.47727 −0.186665
\(862\) −26.7386 −0.910721
\(863\) 22.5464 0.767488 0.383744 0.923439i \(-0.374634\pi\)
0.383744 + 0.923439i \(0.374634\pi\)
\(864\) 5.56155 0.189208
\(865\) −2.24621 −0.0763735
\(866\) −31.3693 −1.06597
\(867\) −1.56155 −0.0530331
\(868\) 0 0
\(869\) 13.5616 0.460044
\(870\) 8.68466 0.294437
\(871\) −8.00000 −0.271070
\(872\) −1.12311 −0.0380332
\(873\) 8.73863 0.295758
\(874\) 42.7386 1.44566
\(875\) 3.12311 0.105580
\(876\) 2.05398 0.0693974
\(877\) 54.7386 1.84839 0.924196 0.381918i \(-0.124736\pi\)
0.924196 + 0.381918i \(0.124736\pi\)
\(878\) −20.4924 −0.691586
\(879\) −3.50758 −0.118308
\(880\) −1.00000 −0.0337100
\(881\) 36.4384 1.22764 0.613821 0.789445i \(-0.289632\pi\)
0.613821 + 0.789445i \(0.289632\pi\)
\(882\) 1.54640 0.0520699
\(883\) −19.3693 −0.651829 −0.325915 0.945399i \(-0.605672\pi\)
−0.325915 + 0.945399i \(0.605672\pi\)
\(884\) 16.0000 0.538138
\(885\) 9.75379 0.327870
\(886\) −16.2462 −0.545802
\(887\) −26.7386 −0.897795 −0.448898 0.893583i \(-0.648183\pi\)
−0.448898 + 0.893583i \(0.648183\pi\)
\(888\) 1.75379 0.0588533
\(889\) −11.8920 −0.398847
\(890\) 7.36932 0.247020
\(891\) 7.00000 0.234509
\(892\) −18.9309 −0.633852
\(893\) 14.2462 0.476731
\(894\) −7.31534 −0.244662
\(895\) 8.43845 0.282066
\(896\) −3.12311 −0.104336
\(897\) −37.4773 −1.25133
\(898\) −34.0000 −1.13459
\(899\) 0 0
\(900\) −0.561553 −0.0187184
\(901\) 14.2462 0.474610
\(902\) −1.12311 −0.0373953
\(903\) 4.87689 0.162293
\(904\) −5.12311 −0.170392
\(905\) −16.0540 −0.533652
\(906\) −33.3693 −1.10862
\(907\) 18.9848 0.630381 0.315191 0.949028i \(-0.397932\pi\)
0.315191 + 0.949028i \(0.397932\pi\)
\(908\) −5.36932 −0.178187
\(909\) 8.13826 0.269929
\(910\) 12.4924 0.414120
\(911\) 33.8078 1.12010 0.560051 0.828458i \(-0.310782\pi\)
0.560051 + 0.828458i \(0.310782\pi\)
\(912\) −11.1231 −0.368323
\(913\) 2.43845 0.0807008
\(914\) 42.4924 1.40552
\(915\) −8.68466 −0.287106
\(916\) −3.56155 −0.117677
\(917\) 19.5076 0.644197
\(918\) −22.2462 −0.734234
\(919\) 17.0691 0.563059 0.281529 0.959553i \(-0.409158\pi\)
0.281529 + 0.959553i \(0.409158\pi\)
\(920\) −6.00000 −0.197814
\(921\) 43.4233 1.43085
\(922\) 7.36932 0.242696
\(923\) 4.49242 0.147870
\(924\) −4.87689 −0.160438
\(925\) −1.12311 −0.0369275
\(926\) 2.24621 0.0738151
\(927\) −4.63068 −0.152092
\(928\) 5.56155 0.182567
\(929\) −8.63068 −0.283164 −0.141582 0.989927i \(-0.545219\pi\)
−0.141582 + 0.989927i \(0.545219\pi\)
\(930\) 0 0
\(931\) −19.6155 −0.642873
\(932\) 3.56155 0.116663
\(933\) −11.1231 −0.364154
\(934\) 18.7386 0.613147
\(935\) 4.00000 0.130814
\(936\) −2.24621 −0.0734197
\(937\) 2.68466 0.0877040 0.0438520 0.999038i \(-0.486037\pi\)
0.0438520 + 0.999038i \(0.486037\pi\)
\(938\) −6.24621 −0.203946
\(939\) −20.4924 −0.668745
\(940\) −2.00000 −0.0652328
\(941\) 57.6155 1.87821 0.939106 0.343627i \(-0.111656\pi\)
0.939106 + 0.343627i \(0.111656\pi\)
\(942\) −12.1080 −0.394498
\(943\) −6.73863 −0.219440
\(944\) 6.24621 0.203297
\(945\) −17.3693 −0.565024
\(946\) 1.00000 0.0325128
\(947\) −3.26137 −0.105980 −0.0529901 0.998595i \(-0.516875\pi\)
−0.0529901 + 0.998595i \(0.516875\pi\)
\(948\) −21.1771 −0.687800
\(949\) −5.26137 −0.170791
\(950\) 7.12311 0.231104
\(951\) −11.8078 −0.382893
\(952\) 12.4924 0.404882
\(953\) 49.8078 1.61343 0.806716 0.590940i \(-0.201243\pi\)
0.806716 + 0.590940i \(0.201243\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −9.31534 −0.301437
\(956\) 0.684658 0.0221434
\(957\) 8.68466 0.280735
\(958\) −8.68466 −0.280589
\(959\) −16.0000 −0.516667
\(960\) 1.56155 0.0503989
\(961\) −31.0000 −1.00000
\(962\) −4.49242 −0.144842
\(963\) −2.13826 −0.0689045
\(964\) 16.8769 0.543568
\(965\) −14.2462 −0.458602
\(966\) −29.2614 −0.941469
\(967\) −39.4233 −1.26777 −0.633884 0.773428i \(-0.718540\pi\)
−0.633884 + 0.773428i \(0.718540\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 44.4924 1.42930
\(970\) 15.5616 0.499651
\(971\) 24.9848 0.801802 0.400901 0.916121i \(-0.368697\pi\)
0.400901 + 0.916121i \(0.368697\pi\)
\(972\) 5.75379 0.184553
\(973\) −14.6307 −0.469038
\(974\) 6.49242 0.208031
\(975\) −6.24621 −0.200039
\(976\) −5.56155 −0.178021
\(977\) −31.5616 −1.00974 −0.504872 0.863194i \(-0.668460\pi\)
−0.504872 + 0.863194i \(0.668460\pi\)
\(978\) −7.31534 −0.233919
\(979\) 7.36932 0.235524
\(980\) 2.75379 0.0879666
\(981\) −0.630683 −0.0201362
\(982\) −3.12311 −0.0996623
\(983\) 1.26137 0.0402313 0.0201157 0.999798i \(-0.493597\pi\)
0.0201157 + 0.999798i \(0.493597\pi\)
\(984\) 1.75379 0.0559087
\(985\) −1.75379 −0.0558804
\(986\) −22.2462 −0.708464
\(987\) −9.75379 −0.310467
\(988\) 28.4924 0.906465
\(989\) 6.00000 0.190789
\(990\) −0.561553 −0.0178473
\(991\) −59.1771 −1.87982 −0.939911 0.341420i \(-0.889092\pi\)
−0.939911 + 0.341420i \(0.889092\pi\)
\(992\) 0 0
\(993\) −23.2311 −0.737215
\(994\) 3.50758 0.111254
\(995\) −1.12311 −0.0356048
\(996\) −3.80776 −0.120654
\(997\) −36.7386 −1.16352 −0.581762 0.813359i \(-0.697637\pi\)
−0.581762 + 0.813359i \(0.697637\pi\)
\(998\) 3.56155 0.112739
\(999\) 6.24621 0.197621
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 4730.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4730.2.a.m.1.2 2 1.1 even 1 trivial