Properties

Label 4026.2.a.r.1.1
Level $4026$
Weight $2$
Character 4026.1
Self dual yes
Analytic conductor $32.148$
Analytic rank $1$
Dimension $4$
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] = [4026,2,Mod(1,4026)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4026, 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("4026.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4026.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.1477718538\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.7537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.15976\) of defining polynomial
Character \(\chi\) \(=\) 4026.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.66454 q^{5} -1.00000 q^{6} -2.43525 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.66454 q^{5} -1.00000 q^{6} -2.43525 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.66454 q^{10} -1.00000 q^{11} +1.00000 q^{12} +6.41931 q^{13} +2.43525 q^{14} -1.66454 q^{15} +1.00000 q^{16} -0.265921 q^{17} -1.00000 q^{18} -4.82430 q^{19} -1.66454 q^{20} -2.43525 q^{21} +1.00000 q^{22} +5.37530 q^{23} -1.00000 q^{24} -2.22929 q^{25} -6.41931 q^{26} +1.00000 q^{27} -2.43525 q^{28} +2.75477 q^{29} +1.66454 q^{30} -4.15018 q^{31} -1.00000 q^{32} -1.00000 q^{33} +0.265921 q^{34} +4.05359 q^{35} +1.00000 q^{36} -6.27330 q^{37} +4.82430 q^{38} +6.41931 q^{39} +1.66454 q^{40} +7.91452 q^{41} +2.43525 q^{42} -10.7086 q^{43} -1.00000 q^{44} -1.66454 q^{45} -5.37530 q^{46} +0.474524 q^{47} +1.00000 q^{48} -1.06953 q^{49} +2.22929 q^{50} -0.265921 q^{51} +6.41931 q^{52} -10.2691 q^{53} -1.00000 q^{54} +1.66454 q^{55} +2.43525 q^{56} -4.82430 q^{57} -2.75477 q^{58} +10.8275 q^{59} -1.66454 q^{60} +1.00000 q^{61} +4.15018 q^{62} -2.43525 q^{63} +1.00000 q^{64} -10.6852 q^{65} +1.00000 q^{66} +1.15976 q^{67} -0.265921 q^{68} +5.37530 q^{69} -4.05359 q^{70} +6.34978 q^{71} -1.00000 q^{72} +14.8153 q^{73} +6.27330 q^{74} -2.22929 q^{75} -4.82430 q^{76} +2.43525 q^{77} -6.41931 q^{78} -13.9607 q^{79} -1.66454 q^{80} +1.00000 q^{81} -7.91452 q^{82} +7.07428 q^{83} -2.43525 q^{84} +0.442637 q^{85} +10.7086 q^{86} +2.75477 q^{87} +1.00000 q^{88} +4.10397 q^{89} +1.66454 q^{90} -15.6327 q^{91} +5.37530 q^{92} -4.15018 q^{93} -0.474524 q^{94} +8.03027 q^{95} -1.00000 q^{96} +4.25736 q^{97} +1.06953 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 4 q^{9} - q^{10} - 4 q^{11} + 4 q^{12} - 3 q^{13} + 4 q^{14} + q^{15} + 4 q^{16} - 3 q^{17} - 4 q^{18} - 4 q^{19} + q^{20} - 4 q^{21} + 4 q^{22} + 10 q^{23} - 4 q^{24} - 7 q^{25} + 3 q^{26} + 4 q^{27} - 4 q^{28} - 10 q^{29} - q^{30} - 9 q^{31} - 4 q^{32} - 4 q^{33} + 3 q^{34} - q^{35} + 4 q^{36} - 6 q^{37} + 4 q^{38} - 3 q^{39} - q^{40} + 3 q^{41} + 4 q^{42} - 18 q^{43} - 4 q^{44} + q^{45} - 10 q^{46} + 21 q^{47} + 4 q^{48} - 10 q^{49} + 7 q^{50} - 3 q^{51} - 3 q^{52} - 20 q^{53} - 4 q^{54} - q^{55} + 4 q^{56} - 4 q^{57} + 10 q^{58} + 5 q^{59} + q^{60} + 4 q^{61} + 9 q^{62} - 4 q^{63} + 4 q^{64} - 16 q^{65} + 4 q^{66} - 3 q^{67} - 3 q^{68} + 10 q^{69} + q^{70} - 9 q^{71} - 4 q^{72} + 6 q^{74} - 7 q^{75} - 4 q^{76} + 4 q^{77} + 3 q^{78} - 31 q^{79} + q^{80} + 4 q^{81} - 3 q^{82} - 8 q^{83} - 4 q^{84} - 25 q^{85} + 18 q^{86} - 10 q^{87} + 4 q^{88} + 5 q^{89} - q^{90} - 9 q^{91} + 10 q^{92} - 9 q^{93} - 21 q^{94} + 13 q^{95} - 4 q^{96} - 25 q^{97} + 10 q^{98} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.66454 −0.744407 −0.372204 0.928151i \(-0.621398\pi\)
−0.372204 + 0.928151i \(0.621398\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.43525 −0.920440 −0.460220 0.887805i \(-0.652229\pi\)
−0.460220 + 0.887805i \(0.652229\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.66454 0.526375
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) 6.41931 1.78040 0.890198 0.455573i \(-0.150566\pi\)
0.890198 + 0.455573i \(0.150566\pi\)
\(14\) 2.43525 0.650849
\(15\) −1.66454 −0.429784
\(16\) 1.00000 0.250000
\(17\) −0.265921 −0.0644952 −0.0322476 0.999480i \(-0.510267\pi\)
−0.0322476 + 0.999480i \(0.510267\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.82430 −1.10677 −0.553385 0.832925i \(-0.686664\pi\)
−0.553385 + 0.832925i \(0.686664\pi\)
\(20\) −1.66454 −0.372204
\(21\) −2.43525 −0.531416
\(22\) 1.00000 0.213201
\(23\) 5.37530 1.12083 0.560414 0.828213i \(-0.310642\pi\)
0.560414 + 0.828213i \(0.310642\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.22929 −0.445858
\(26\) −6.41931 −1.25893
\(27\) 1.00000 0.192450
\(28\) −2.43525 −0.460220
\(29\) 2.75477 0.511547 0.255774 0.966737i \(-0.417670\pi\)
0.255774 + 0.966737i \(0.417670\pi\)
\(30\) 1.66454 0.303903
\(31\) −4.15018 −0.745394 −0.372697 0.927953i \(-0.621567\pi\)
−0.372697 + 0.927953i \(0.621567\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.00000 −0.174078
\(34\) 0.265921 0.0456050
\(35\) 4.05359 0.685182
\(36\) 1.00000 0.166667
\(37\) −6.27330 −1.03132 −0.515662 0.856792i \(-0.672454\pi\)
−0.515662 + 0.856792i \(0.672454\pi\)
\(38\) 4.82430 0.782605
\(39\) 6.41931 1.02791
\(40\) 1.66454 0.263188
\(41\) 7.91452 1.23604 0.618020 0.786162i \(-0.287935\pi\)
0.618020 + 0.786162i \(0.287935\pi\)
\(42\) 2.43525 0.375768
\(43\) −10.7086 −1.63304 −0.816520 0.577317i \(-0.804100\pi\)
−0.816520 + 0.577317i \(0.804100\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.66454 −0.248136
\(46\) −5.37530 −0.792545
\(47\) 0.474524 0.0692164 0.0346082 0.999401i \(-0.488982\pi\)
0.0346082 + 0.999401i \(0.488982\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.06953 −0.152791
\(50\) 2.22929 0.315269
\(51\) −0.265921 −0.0372363
\(52\) 6.41931 0.890198
\(53\) −10.2691 −1.41057 −0.705287 0.708922i \(-0.749182\pi\)
−0.705287 + 0.708922i \(0.749182\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.66454 0.224447
\(56\) 2.43525 0.325425
\(57\) −4.82430 −0.638994
\(58\) −2.75477 −0.361719
\(59\) 10.8275 1.40962 0.704811 0.709395i \(-0.251032\pi\)
0.704811 + 0.709395i \(0.251032\pi\)
\(60\) −1.66454 −0.214892
\(61\) 1.00000 0.128037
\(62\) 4.15018 0.527073
\(63\) −2.43525 −0.306813
\(64\) 1.00000 0.125000
\(65\) −10.6852 −1.32534
\(66\) 1.00000 0.123091
\(67\) 1.15976 0.141687 0.0708434 0.997487i \(-0.477431\pi\)
0.0708434 + 0.997487i \(0.477431\pi\)
\(68\) −0.265921 −0.0322476
\(69\) 5.37530 0.647110
\(70\) −4.05359 −0.484497
\(71\) 6.34978 0.753580 0.376790 0.926299i \(-0.377028\pi\)
0.376790 + 0.926299i \(0.377028\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.8153 1.73400 0.867000 0.498309i \(-0.166045\pi\)
0.867000 + 0.498309i \(0.166045\pi\)
\(74\) 6.27330 0.729257
\(75\) −2.22929 −0.257416
\(76\) −4.82430 −0.553385
\(77\) 2.43525 0.277523
\(78\) −6.41931 −0.726844
\(79\) −13.9607 −1.57070 −0.785352 0.619049i \(-0.787518\pi\)
−0.785352 + 0.619049i \(0.787518\pi\)
\(80\) −1.66454 −0.186102
\(81\) 1.00000 0.111111
\(82\) −7.91452 −0.874013
\(83\) 7.07428 0.776503 0.388251 0.921553i \(-0.373079\pi\)
0.388251 + 0.921553i \(0.373079\pi\)
\(84\) −2.43525 −0.265708
\(85\) 0.442637 0.0480107
\(86\) 10.7086 1.15473
\(87\) 2.75477 0.295342
\(88\) 1.00000 0.106600
\(89\) 4.10397 0.435020 0.217510 0.976058i \(-0.430207\pi\)
0.217510 + 0.976058i \(0.430207\pi\)
\(90\) 1.66454 0.175458
\(91\) −15.6327 −1.63875
\(92\) 5.37530 0.560414
\(93\) −4.15018 −0.430353
\(94\) −0.474524 −0.0489434
\(95\) 8.03027 0.823888
\(96\) −1.00000 −0.102062
\(97\) 4.25736 0.432269 0.216135 0.976364i \(-0.430655\pi\)
0.216135 + 0.976364i \(0.430655\pi\)
\(98\) 1.06953 0.108039
\(99\) −1.00000 −0.100504
\(100\) −2.22929 −0.222929
\(101\) −0.316298 −0.0314729 −0.0157364 0.999876i \(-0.505009\pi\)
−0.0157364 + 0.999876i \(0.505009\pi\)
\(102\) 0.265921 0.0263301
\(103\) 1.55297 0.153019 0.0765094 0.997069i \(-0.475622\pi\)
0.0765094 + 0.997069i \(0.475622\pi\)
\(104\) −6.41931 −0.629465
\(105\) 4.05359 0.395590
\(106\) 10.2691 0.997426
\(107\) −18.5998 −1.79811 −0.899053 0.437840i \(-0.855744\pi\)
−0.899053 + 0.437840i \(0.855744\pi\)
\(108\) 1.00000 0.0962250
\(109\) −5.78029 −0.553651 −0.276826 0.960920i \(-0.589282\pi\)
−0.276826 + 0.960920i \(0.589282\pi\)
\(110\) −1.66454 −0.158708
\(111\) −6.27330 −0.595436
\(112\) −2.43525 −0.230110
\(113\) 2.38905 0.224743 0.112371 0.993666i \(-0.464155\pi\)
0.112371 + 0.993666i \(0.464155\pi\)
\(114\) 4.82430 0.451837
\(115\) −8.94743 −0.834352
\(116\) 2.75477 0.255774
\(117\) 6.41931 0.593466
\(118\) −10.8275 −0.996753
\(119\) 0.647584 0.0593640
\(120\) 1.66454 0.151951
\(121\) 1.00000 0.0909091
\(122\) −1.00000 −0.0905357
\(123\) 7.91452 0.713628
\(124\) −4.15018 −0.372697
\(125\) 12.0335 1.07631
\(126\) 2.43525 0.216950
\(127\) −15.7341 −1.39617 −0.698087 0.716013i \(-0.745965\pi\)
−0.698087 + 0.716013i \(0.745965\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.7086 −0.942836
\(130\) 10.6852 0.937157
\(131\) −4.72771 −0.413062 −0.206531 0.978440i \(-0.566217\pi\)
−0.206531 + 0.978440i \(0.566217\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 11.7484 1.01872
\(134\) −1.15976 −0.100188
\(135\) −1.66454 −0.143261
\(136\) 0.265921 0.0228025
\(137\) −14.3137 −1.22290 −0.611452 0.791282i \(-0.709414\pi\)
−0.611452 + 0.791282i \(0.709414\pi\)
\(138\) −5.37530 −0.457576
\(139\) −18.7388 −1.58941 −0.794703 0.606998i \(-0.792373\pi\)
−0.794703 + 0.606998i \(0.792373\pi\)
\(140\) 4.05359 0.342591
\(141\) 0.474524 0.0399621
\(142\) −6.34978 −0.532861
\(143\) −6.41931 −0.536810
\(144\) 1.00000 0.0833333
\(145\) −4.58543 −0.380799
\(146\) −14.8153 −1.22612
\(147\) −1.06953 −0.0882137
\(148\) −6.27330 −0.515662
\(149\) −9.36352 −0.767090 −0.383545 0.923522i \(-0.625297\pi\)
−0.383545 + 0.923522i \(0.625297\pi\)
\(150\) 2.22929 0.182021
\(151\) −22.6721 −1.84503 −0.922513 0.385966i \(-0.873868\pi\)
−0.922513 + 0.385966i \(0.873868\pi\)
\(152\) 4.82430 0.391302
\(153\) −0.265921 −0.0214984
\(154\) −2.43525 −0.196238
\(155\) 6.90816 0.554877
\(156\) 6.41931 0.513956
\(157\) 8.87468 0.708276 0.354138 0.935193i \(-0.384774\pi\)
0.354138 + 0.935193i \(0.384774\pi\)
\(158\) 13.9607 1.11066
\(159\) −10.2691 −0.814395
\(160\) 1.66454 0.131594
\(161\) −13.0902 −1.03165
\(162\) −1.00000 −0.0785674
\(163\) −0.454411 −0.0355922 −0.0177961 0.999842i \(-0.505665\pi\)
−0.0177961 + 0.999842i \(0.505665\pi\)
\(164\) 7.91452 0.618020
\(165\) 1.66454 0.129585
\(166\) −7.07428 −0.549070
\(167\) −17.4592 −1.35103 −0.675515 0.737346i \(-0.736079\pi\)
−0.675515 + 0.737346i \(0.736079\pi\)
\(168\) 2.43525 0.187884
\(169\) 28.2076 2.16981
\(170\) −0.442637 −0.0339487
\(171\) −4.82430 −0.368923
\(172\) −10.7086 −0.816520
\(173\) −17.7516 −1.34963 −0.674815 0.737987i \(-0.735777\pi\)
−0.674815 + 0.737987i \(0.735777\pi\)
\(174\) −2.75477 −0.208838
\(175\) 5.42889 0.410385
\(176\) −1.00000 −0.0753778
\(177\) 10.8275 0.813846
\(178\) −4.10397 −0.307605
\(179\) −24.8418 −1.85677 −0.928383 0.371625i \(-0.878801\pi\)
−0.928383 + 0.371625i \(0.878801\pi\)
\(180\) −1.66454 −0.124068
\(181\) −25.7352 −1.91288 −0.956442 0.291922i \(-0.905705\pi\)
−0.956442 + 0.291922i \(0.905705\pi\)
\(182\) 15.6327 1.15877
\(183\) 1.00000 0.0739221
\(184\) −5.37530 −0.396272
\(185\) 10.4422 0.767725
\(186\) 4.15018 0.304306
\(187\) 0.265921 0.0194460
\(188\) 0.474524 0.0346082
\(189\) −2.43525 −0.177139
\(190\) −8.03027 −0.582577
\(191\) −22.6603 −1.63964 −0.819820 0.572621i \(-0.805927\pi\)
−0.819820 + 0.572621i \(0.805927\pi\)
\(192\) 1.00000 0.0721688
\(193\) −19.8646 −1.42988 −0.714942 0.699184i \(-0.753547\pi\)
−0.714942 + 0.699184i \(0.753547\pi\)
\(194\) −4.25736 −0.305661
\(195\) −10.6852 −0.765185
\(196\) −1.06953 −0.0763953
\(197\) −4.40776 −0.314040 −0.157020 0.987595i \(-0.550189\pi\)
−0.157020 + 0.987595i \(0.550189\pi\)
\(198\) 1.00000 0.0710669
\(199\) 9.74936 0.691114 0.345557 0.938398i \(-0.387690\pi\)
0.345557 + 0.938398i \(0.387690\pi\)
\(200\) 2.22929 0.157635
\(201\) 1.15976 0.0818029
\(202\) 0.316298 0.0222547
\(203\) −6.70856 −0.470848
\(204\) −0.265921 −0.0186182
\(205\) −13.1741 −0.920117
\(206\) −1.55297 −0.108201
\(207\) 5.37530 0.373609
\(208\) 6.41931 0.445099
\(209\) 4.82430 0.333704
\(210\) −4.05359 −0.279724
\(211\) 22.5478 1.55225 0.776126 0.630578i \(-0.217182\pi\)
0.776126 + 0.630578i \(0.217182\pi\)
\(212\) −10.2691 −0.705287
\(213\) 6.34978 0.435080
\(214\) 18.5998 1.27145
\(215\) 17.8249 1.21565
\(216\) −1.00000 −0.0680414
\(217\) 10.1067 0.686090
\(218\) 5.78029 0.391491
\(219\) 14.8153 1.00112
\(220\) 1.66454 0.112224
\(221\) −1.70703 −0.114827
\(222\) 6.27330 0.421037
\(223\) 22.9878 1.53938 0.769690 0.638418i \(-0.220411\pi\)
0.769690 + 0.638418i \(0.220411\pi\)
\(224\) 2.43525 0.162712
\(225\) −2.22929 −0.148619
\(226\) −2.38905 −0.158917
\(227\) −13.5329 −0.898207 −0.449104 0.893480i \(-0.648257\pi\)
−0.449104 + 0.893480i \(0.648257\pi\)
\(228\) −4.82430 −0.319497
\(229\) −4.39321 −0.290312 −0.145156 0.989409i \(-0.546368\pi\)
−0.145156 + 0.989409i \(0.546368\pi\)
\(230\) 8.94743 0.589976
\(231\) 2.43525 0.160228
\(232\) −2.75477 −0.180859
\(233\) −10.2601 −0.672163 −0.336082 0.941833i \(-0.609102\pi\)
−0.336082 + 0.941833i \(0.609102\pi\)
\(234\) −6.41931 −0.419643
\(235\) −0.789866 −0.0515252
\(236\) 10.8275 0.704811
\(237\) −13.9607 −0.906847
\(238\) −0.647584 −0.0419767
\(239\) 20.2952 1.31279 0.656395 0.754418i \(-0.272081\pi\)
0.656395 + 0.754418i \(0.272081\pi\)
\(240\) −1.66454 −0.107446
\(241\) −4.10674 −0.264538 −0.132269 0.991214i \(-0.542226\pi\)
−0.132269 + 0.991214i \(0.542226\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) 1.00000 0.0640184
\(245\) 1.78029 0.113738
\(246\) −7.91452 −0.504611
\(247\) −30.9687 −1.97049
\(248\) 4.15018 0.263537
\(249\) 7.07428 0.448314
\(250\) −12.0335 −0.761064
\(251\) −0.778315 −0.0491268 −0.0245634 0.999698i \(-0.507820\pi\)
−0.0245634 + 0.999698i \(0.507820\pi\)
\(252\) −2.43525 −0.153407
\(253\) −5.37530 −0.337942
\(254\) 15.7341 0.987244
\(255\) 0.442637 0.0277190
\(256\) 1.00000 0.0625000
\(257\) 16.1418 1.00690 0.503450 0.864024i \(-0.332064\pi\)
0.503450 + 0.864024i \(0.332064\pi\)
\(258\) 10.7086 0.666686
\(259\) 15.2771 0.949272
\(260\) −10.6852 −0.662670
\(261\) 2.75477 0.170516
\(262\) 4.72771 0.292079
\(263\) −6.31095 −0.389150 −0.194575 0.980888i \(-0.562333\pi\)
−0.194575 + 0.980888i \(0.562333\pi\)
\(264\) 1.00000 0.0615457
\(265\) 17.0934 1.05004
\(266\) −11.7484 −0.720341
\(267\) 4.10397 0.251159
\(268\) 1.15976 0.0708434
\(269\) 22.7531 1.38728 0.693642 0.720320i \(-0.256005\pi\)
0.693642 + 0.720320i \(0.256005\pi\)
\(270\) 1.66454 0.101301
\(271\) −7.33699 −0.445690 −0.222845 0.974854i \(-0.571534\pi\)
−0.222845 + 0.974854i \(0.571534\pi\)
\(272\) −0.265921 −0.0161238
\(273\) −15.6327 −0.946131
\(274\) 14.3137 0.864723
\(275\) 2.22929 0.134431
\(276\) 5.37530 0.323555
\(277\) −8.14645 −0.489473 −0.244736 0.969590i \(-0.578701\pi\)
−0.244736 + 0.969590i \(0.578701\pi\)
\(278\) 18.7388 1.12388
\(279\) −4.15018 −0.248465
\(280\) −4.05359 −0.242248
\(281\) −15.4404 −0.921099 −0.460550 0.887634i \(-0.652348\pi\)
−0.460550 + 0.887634i \(0.652348\pi\)
\(282\) −0.474524 −0.0282575
\(283\) 5.37947 0.319776 0.159888 0.987135i \(-0.448887\pi\)
0.159888 + 0.987135i \(0.448887\pi\)
\(284\) 6.34978 0.376790
\(285\) 8.03027 0.475672
\(286\) 6.41931 0.379582
\(287\) −19.2739 −1.13770
\(288\) −1.00000 −0.0589256
\(289\) −16.9293 −0.995840
\(290\) 4.58543 0.269266
\(291\) 4.25736 0.249571
\(292\) 14.8153 0.867000
\(293\) 25.2818 1.47698 0.738488 0.674266i \(-0.235540\pi\)
0.738488 + 0.674266i \(0.235540\pi\)
\(294\) 1.06953 0.0623765
\(295\) −18.0229 −1.04933
\(296\) 6.27330 0.364628
\(297\) −1.00000 −0.0580259
\(298\) 9.36352 0.542414
\(299\) 34.5057 1.99552
\(300\) −2.22929 −0.128708
\(301\) 26.0781 1.50311
\(302\) 22.6721 1.30463
\(303\) −0.316298 −0.0181709
\(304\) −4.82430 −0.276693
\(305\) −1.66454 −0.0953116
\(306\) 0.265921 0.0152017
\(307\) −19.0780 −1.08884 −0.544420 0.838813i \(-0.683250\pi\)
−0.544420 + 0.838813i \(0.683250\pi\)
\(308\) 2.43525 0.138762
\(309\) 1.55297 0.0883454
\(310\) −6.90816 −0.392357
\(311\) 32.0099 1.81512 0.907558 0.419926i \(-0.137944\pi\)
0.907558 + 0.419926i \(0.137944\pi\)
\(312\) −6.41931 −0.363422
\(313\) −23.9824 −1.35557 −0.677784 0.735261i \(-0.737059\pi\)
−0.677784 + 0.735261i \(0.737059\pi\)
\(314\) −8.87468 −0.500827
\(315\) 4.05359 0.228394
\(316\) −13.9607 −0.785352
\(317\) 3.15237 0.177055 0.0885275 0.996074i \(-0.471784\pi\)
0.0885275 + 0.996074i \(0.471784\pi\)
\(318\) 10.2691 0.575864
\(319\) −2.75477 −0.154237
\(320\) −1.66454 −0.0930509
\(321\) −18.5998 −1.03814
\(322\) 13.0902 0.729490
\(323\) 1.28288 0.0713814
\(324\) 1.00000 0.0555556
\(325\) −14.3105 −0.793804
\(326\) 0.454411 0.0251675
\(327\) −5.78029 −0.319651
\(328\) −7.91452 −0.437006
\(329\) −1.15559 −0.0637096
\(330\) −1.66454 −0.0916302
\(331\) −35.5079 −1.95169 −0.975846 0.218461i \(-0.929896\pi\)
−0.975846 + 0.218461i \(0.929896\pi\)
\(332\) 7.07428 0.388251
\(333\) −6.27330 −0.343775
\(334\) 17.4592 0.955322
\(335\) −1.93047 −0.105473
\(336\) −2.43525 −0.132854
\(337\) −10.3992 −0.566480 −0.283240 0.959049i \(-0.591409\pi\)
−0.283240 + 0.959049i \(0.591409\pi\)
\(338\) −28.2076 −1.53429
\(339\) 2.38905 0.129755
\(340\) 0.442637 0.0240054
\(341\) 4.15018 0.224745
\(342\) 4.82430 0.260868
\(343\) 19.6514 1.06107
\(344\) 10.7086 0.577367
\(345\) −8.94743 −0.481713
\(346\) 17.7516 0.954333
\(347\) −19.5319 −1.04853 −0.524264 0.851556i \(-0.675659\pi\)
−0.524264 + 0.851556i \(0.675659\pi\)
\(348\) 2.75477 0.147671
\(349\) −9.23667 −0.494428 −0.247214 0.968961i \(-0.579515\pi\)
−0.247214 + 0.968961i \(0.579515\pi\)
\(350\) −5.42889 −0.290186
\(351\) 6.41931 0.342637
\(352\) 1.00000 0.0533002
\(353\) −8.02126 −0.426929 −0.213464 0.976951i \(-0.568475\pi\)
−0.213464 + 0.976951i \(0.568475\pi\)
\(354\) −10.8275 −0.575476
\(355\) −10.5695 −0.560970
\(356\) 4.10397 0.217510
\(357\) 0.647584 0.0342738
\(358\) 24.8418 1.31293
\(359\) 32.7129 1.72652 0.863261 0.504758i \(-0.168418\pi\)
0.863261 + 0.504758i \(0.168418\pi\)
\(360\) 1.66454 0.0877292
\(361\) 4.27388 0.224941
\(362\) 25.7352 1.35261
\(363\) 1.00000 0.0524864
\(364\) −15.6327 −0.819374
\(365\) −24.6607 −1.29080
\(366\) −1.00000 −0.0522708
\(367\) 11.7532 0.613514 0.306757 0.951788i \(-0.400756\pi\)
0.306757 + 0.951788i \(0.400756\pi\)
\(368\) 5.37530 0.280207
\(369\) 7.91452 0.412014
\(370\) −10.4422 −0.542864
\(371\) 25.0080 1.29835
\(372\) −4.15018 −0.215177
\(373\) 21.7119 1.12420 0.562100 0.827069i \(-0.309994\pi\)
0.562100 + 0.827069i \(0.309994\pi\)
\(374\) −0.265921 −0.0137504
\(375\) 12.0335 0.621406
\(376\) −0.474524 −0.0244717
\(377\) 17.6837 0.910757
\(378\) 2.43525 0.125256
\(379\) −28.2596 −1.45160 −0.725799 0.687906i \(-0.758530\pi\)
−0.725799 + 0.687906i \(0.758530\pi\)
\(380\) 8.03027 0.411944
\(381\) −15.7341 −0.806081
\(382\) 22.6603 1.15940
\(383\) −30.0875 −1.53740 −0.768700 0.639609i \(-0.779096\pi\)
−0.768700 + 0.639609i \(0.779096\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.05359 −0.206590
\(386\) 19.8646 1.01108
\(387\) −10.7086 −0.544347
\(388\) 4.25736 0.216135
\(389\) 3.01375 0.152803 0.0764015 0.997077i \(-0.475657\pi\)
0.0764015 + 0.997077i \(0.475657\pi\)
\(390\) 10.6852 0.541068
\(391\) −1.42940 −0.0722880
\(392\) 1.06953 0.0540196
\(393\) −4.72771 −0.238482
\(394\) 4.40776 0.222060
\(395\) 23.2383 1.16924
\(396\) −1.00000 −0.0502519
\(397\) 31.0546 1.55859 0.779293 0.626660i \(-0.215578\pi\)
0.779293 + 0.626660i \(0.215578\pi\)
\(398\) −9.74936 −0.488691
\(399\) 11.7484 0.588156
\(400\) −2.22929 −0.111465
\(401\) 17.0466 0.851265 0.425632 0.904896i \(-0.360052\pi\)
0.425632 + 0.904896i \(0.360052\pi\)
\(402\) −1.15976 −0.0578434
\(403\) −26.6413 −1.32710
\(404\) −0.316298 −0.0157364
\(405\) −1.66454 −0.0827119
\(406\) 6.70856 0.332940
\(407\) 6.27330 0.310956
\(408\) 0.265921 0.0131650
\(409\) −2.40082 −0.118713 −0.0593565 0.998237i \(-0.518905\pi\)
−0.0593565 + 0.998237i \(0.518905\pi\)
\(410\) 13.1741 0.650621
\(411\) −14.3137 −0.706044
\(412\) 1.55297 0.0765094
\(413\) −26.3678 −1.29747
\(414\) −5.37530 −0.264182
\(415\) −11.7755 −0.578034
\(416\) −6.41931 −0.314733
\(417\) −18.7388 −0.917644
\(418\) −4.82430 −0.235964
\(419\) 10.8918 0.532098 0.266049 0.963960i \(-0.414282\pi\)
0.266049 + 0.963960i \(0.414282\pi\)
\(420\) 4.05359 0.197795
\(421\) 39.8498 1.94216 0.971080 0.238754i \(-0.0767391\pi\)
0.971080 + 0.238754i \(0.0767391\pi\)
\(422\) −22.5478 −1.09761
\(423\) 0.474524 0.0230721
\(424\) 10.2691 0.498713
\(425\) 0.592814 0.0287557
\(426\) −6.34978 −0.307648
\(427\) −2.43525 −0.117850
\(428\) −18.5998 −0.899053
\(429\) −6.41931 −0.309927
\(430\) −17.8249 −0.859592
\(431\) −24.3923 −1.17493 −0.587467 0.809248i \(-0.699875\pi\)
−0.587467 + 0.809248i \(0.699875\pi\)
\(432\) 1.00000 0.0481125
\(433\) −17.4973 −0.840867 −0.420434 0.907323i \(-0.638122\pi\)
−0.420434 + 0.907323i \(0.638122\pi\)
\(434\) −10.1067 −0.485139
\(435\) −4.58543 −0.219855
\(436\) −5.78029 −0.276826
\(437\) −25.9321 −1.24050
\(438\) −14.8153 −0.707902
\(439\) 21.4162 1.02214 0.511069 0.859540i \(-0.329250\pi\)
0.511069 + 0.859540i \(0.329250\pi\)
\(440\) −1.66454 −0.0793541
\(441\) −1.06953 −0.0509302
\(442\) 1.70703 0.0811950
\(443\) −5.25612 −0.249726 −0.124863 0.992174i \(-0.539849\pi\)
−0.124863 + 0.992174i \(0.539849\pi\)
\(444\) −6.27330 −0.297718
\(445\) −6.83124 −0.323832
\(446\) −22.9878 −1.08851
\(447\) −9.36352 −0.442879
\(448\) −2.43525 −0.115055
\(449\) 5.87430 0.277225 0.138613 0.990347i \(-0.455736\pi\)
0.138613 + 0.990347i \(0.455736\pi\)
\(450\) 2.22929 0.105090
\(451\) −7.91452 −0.372680
\(452\) 2.38905 0.112371
\(453\) −22.6721 −1.06523
\(454\) 13.5329 0.635129
\(455\) 26.0213 1.21990
\(456\) 4.82430 0.225919
\(457\) 13.4282 0.628146 0.314073 0.949399i \(-0.398306\pi\)
0.314073 + 0.949399i \(0.398306\pi\)
\(458\) 4.39321 0.205281
\(459\) −0.265921 −0.0124121
\(460\) −8.94743 −0.417176
\(461\) −38.8023 −1.80721 −0.903603 0.428372i \(-0.859087\pi\)
−0.903603 + 0.428372i \(0.859087\pi\)
\(462\) −2.43525 −0.113298
\(463\) −15.2240 −0.707520 −0.353760 0.935336i \(-0.615097\pi\)
−0.353760 + 0.935336i \(0.615097\pi\)
\(464\) 2.75477 0.127887
\(465\) 6.90816 0.320358
\(466\) 10.2601 0.475291
\(467\) 36.6004 1.69366 0.846832 0.531860i \(-0.178507\pi\)
0.846832 + 0.531860i \(0.178507\pi\)
\(468\) 6.41931 0.296733
\(469\) −2.82430 −0.130414
\(470\) 0.789866 0.0364338
\(471\) 8.87468 0.408924
\(472\) −10.8275 −0.498377
\(473\) 10.7086 0.492380
\(474\) 13.9607 0.641238
\(475\) 10.7548 0.493463
\(476\) 0.647584 0.0296820
\(477\) −10.2691 −0.470191
\(478\) −20.2952 −0.928282
\(479\) 15.5333 0.709735 0.354867 0.934917i \(-0.384526\pi\)
0.354867 + 0.934917i \(0.384526\pi\)
\(480\) 1.66454 0.0759757
\(481\) −40.2703 −1.83617
\(482\) 4.10674 0.187057
\(483\) −13.0902 −0.595626
\(484\) 1.00000 0.0454545
\(485\) −7.08657 −0.321784
\(486\) −1.00000 −0.0453609
\(487\) 2.55632 0.115838 0.0579189 0.998321i \(-0.481554\pi\)
0.0579189 + 0.998321i \(0.481554\pi\)
\(488\) −1.00000 −0.0452679
\(489\) −0.454411 −0.0205492
\(490\) −1.78029 −0.0804252
\(491\) −18.8074 −0.848766 −0.424383 0.905483i \(-0.639509\pi\)
−0.424383 + 0.905483i \(0.639509\pi\)
\(492\) 7.91452 0.356814
\(493\) −0.732549 −0.0329924
\(494\) 30.9687 1.39335
\(495\) 1.66454 0.0748157
\(496\) −4.15018 −0.186348
\(497\) −15.4633 −0.693625
\(498\) −7.07428 −0.317006
\(499\) 26.3269 1.17855 0.589277 0.807931i \(-0.299413\pi\)
0.589277 + 0.807931i \(0.299413\pi\)
\(500\) 12.0335 0.538154
\(501\) −17.4592 −0.780018
\(502\) 0.778315 0.0347379
\(503\) −23.5590 −1.05044 −0.525221 0.850966i \(-0.676017\pi\)
−0.525221 + 0.850966i \(0.676017\pi\)
\(504\) 2.43525 0.108475
\(505\) 0.526493 0.0234286
\(506\) 5.37530 0.238961
\(507\) 28.2076 1.25274
\(508\) −15.7341 −0.698087
\(509\) −30.1737 −1.33743 −0.668713 0.743521i \(-0.733154\pi\)
−0.668713 + 0.743521i \(0.733154\pi\)
\(510\) −0.442637 −0.0196003
\(511\) −36.0790 −1.59604
\(512\) −1.00000 −0.0441942
\(513\) −4.82430 −0.212998
\(514\) −16.1418 −0.711986
\(515\) −2.58499 −0.113908
\(516\) −10.7086 −0.471418
\(517\) −0.474524 −0.0208695
\(518\) −15.2771 −0.671237
\(519\) −17.7516 −0.779209
\(520\) 10.6852 0.468578
\(521\) −12.9022 −0.565254 −0.282627 0.959230i \(-0.591206\pi\)
−0.282627 + 0.959230i \(0.591206\pi\)
\(522\) −2.75477 −0.120573
\(523\) −31.8933 −1.39460 −0.697298 0.716781i \(-0.745615\pi\)
−0.697298 + 0.716781i \(0.745615\pi\)
\(524\) −4.72771 −0.206531
\(525\) 5.42889 0.236936
\(526\) 6.31095 0.275170
\(527\) 1.10362 0.0480743
\(528\) −1.00000 −0.0435194
\(529\) 5.89384 0.256254
\(530\) −17.0934 −0.742491
\(531\) 10.8275 0.469874
\(532\) 11.7484 0.509358
\(533\) 50.8058 2.20064
\(534\) −4.10397 −0.177596
\(535\) 30.9601 1.33852
\(536\) −1.15976 −0.0500938
\(537\) −24.8418 −1.07200
\(538\) −22.7531 −0.980958
\(539\) 1.06953 0.0460681
\(540\) −1.66454 −0.0716306
\(541\) −7.86994 −0.338355 −0.169177 0.985586i \(-0.554111\pi\)
−0.169177 + 0.985586i \(0.554111\pi\)
\(542\) 7.33699 0.315151
\(543\) −25.7352 −1.10440
\(544\) 0.265921 0.0114013
\(545\) 9.62155 0.412142
\(546\) 15.6327 0.669016
\(547\) 13.7814 0.589252 0.294626 0.955613i \(-0.404805\pi\)
0.294626 + 0.955613i \(0.404805\pi\)
\(548\) −14.3137 −0.611452
\(549\) 1.00000 0.0426790
\(550\) −2.22929 −0.0950573
\(551\) −13.2898 −0.566165
\(552\) −5.37530 −0.228788
\(553\) 33.9979 1.44574
\(554\) 8.14645 0.346110
\(555\) 10.4422 0.443246
\(556\) −18.7388 −0.794703
\(557\) 29.1736 1.23613 0.618063 0.786128i \(-0.287918\pi\)
0.618063 + 0.786128i \(0.287918\pi\)
\(558\) 4.15018 0.175691
\(559\) −68.7416 −2.90746
\(560\) 4.05359 0.171295
\(561\) 0.265921 0.0112272
\(562\) 15.4404 0.651316
\(563\) 28.9527 1.22021 0.610106 0.792320i \(-0.291127\pi\)
0.610106 + 0.792320i \(0.291127\pi\)
\(564\) 0.474524 0.0199811
\(565\) −3.97667 −0.167300
\(566\) −5.37947 −0.226116
\(567\) −2.43525 −0.102271
\(568\) −6.34978 −0.266431
\(569\) 31.3358 1.31367 0.656833 0.754036i \(-0.271896\pi\)
0.656833 + 0.754036i \(0.271896\pi\)
\(570\) −8.03027 −0.336351
\(571\) −12.3865 −0.518359 −0.259179 0.965829i \(-0.583452\pi\)
−0.259179 + 0.965829i \(0.583452\pi\)
\(572\) −6.41931 −0.268405
\(573\) −22.6603 −0.946647
\(574\) 19.2739 0.804476
\(575\) −11.9831 −0.499730
\(576\) 1.00000 0.0416667
\(577\) −34.5015 −1.43632 −0.718159 0.695879i \(-0.755015\pi\)
−0.718159 + 0.695879i \(0.755015\pi\)
\(578\) 16.9293 0.704165
\(579\) −19.8646 −0.825544
\(580\) −4.58543 −0.190400
\(581\) −17.2277 −0.714724
\(582\) −4.25736 −0.176473
\(583\) 10.2691 0.425304
\(584\) −14.8153 −0.613061
\(585\) −10.6852 −0.441780
\(586\) −25.2818 −1.04438
\(587\) 8.09913 0.334287 0.167144 0.985933i \(-0.446546\pi\)
0.167144 + 0.985933i \(0.446546\pi\)
\(588\) −1.06953 −0.0441069
\(589\) 20.0217 0.824980
\(590\) 18.0229 0.741990
\(591\) −4.40776 −0.181311
\(592\) −6.27330 −0.257831
\(593\) 43.5483 1.78832 0.894158 0.447752i \(-0.147775\pi\)
0.894158 + 0.447752i \(0.147775\pi\)
\(594\) 1.00000 0.0410305
\(595\) −1.07793 −0.0441910
\(596\) −9.36352 −0.383545
\(597\) 9.74936 0.399015
\(598\) −34.5057 −1.41104
\(599\) −9.14433 −0.373627 −0.186814 0.982395i \(-0.559816\pi\)
−0.186814 + 0.982395i \(0.559816\pi\)
\(600\) 2.22929 0.0910104
\(601\) −0.590710 −0.0240956 −0.0120478 0.999927i \(-0.503835\pi\)
−0.0120478 + 0.999927i \(0.503835\pi\)
\(602\) −26.0781 −1.06286
\(603\) 1.15976 0.0472289
\(604\) −22.6721 −0.922513
\(605\) −1.66454 −0.0676734
\(606\) 0.316298 0.0128487
\(607\) −7.74423 −0.314329 −0.157164 0.987572i \(-0.550235\pi\)
−0.157164 + 0.987572i \(0.550235\pi\)
\(608\) 4.82430 0.195651
\(609\) −6.70856 −0.271844
\(610\) 1.66454 0.0673955
\(611\) 3.04612 0.123233
\(612\) −0.265921 −0.0107492
\(613\) 42.0390 1.69794 0.848970 0.528441i \(-0.177223\pi\)
0.848970 + 0.528441i \(0.177223\pi\)
\(614\) 19.0780 0.769926
\(615\) −13.1741 −0.531230
\(616\) −2.43525 −0.0981192
\(617\) 13.2644 0.534004 0.267002 0.963696i \(-0.413967\pi\)
0.267002 + 0.963696i \(0.413967\pi\)
\(618\) −1.55297 −0.0624696
\(619\) 7.29399 0.293170 0.146585 0.989198i \(-0.453172\pi\)
0.146585 + 0.989198i \(0.453172\pi\)
\(620\) 6.90816 0.277438
\(621\) 5.37530 0.215703
\(622\) −32.0099 −1.28348
\(623\) −9.99421 −0.400410
\(624\) 6.41931 0.256978
\(625\) −8.88381 −0.355353
\(626\) 23.9824 0.958531
\(627\) 4.82430 0.192664
\(628\) 8.87468 0.354138
\(629\) 1.66820 0.0665155
\(630\) −4.05359 −0.161499
\(631\) 27.8402 1.10830 0.554149 0.832417i \(-0.313044\pi\)
0.554149 + 0.832417i \(0.313044\pi\)
\(632\) 13.9607 0.555328
\(633\) 22.5478 0.896193
\(634\) −3.15237 −0.125197
\(635\) 26.1901 1.03932
\(636\) −10.2691 −0.407198
\(637\) −6.86567 −0.272028
\(638\) 2.75477 0.109062
\(639\) 6.34978 0.251193
\(640\) 1.66454 0.0657969
\(641\) −24.8992 −0.983457 −0.491729 0.870748i \(-0.663635\pi\)
−0.491729 + 0.870748i \(0.663635\pi\)
\(642\) 18.5998 0.734074
\(643\) 14.6561 0.577981 0.288990 0.957332i \(-0.406680\pi\)
0.288990 + 0.957332i \(0.406680\pi\)
\(644\) −13.0902 −0.515827
\(645\) 17.8249 0.701854
\(646\) −1.28288 −0.0504743
\(647\) 39.5453 1.55468 0.777342 0.629078i \(-0.216567\pi\)
0.777342 + 0.629078i \(0.216567\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −10.8275 −0.425017
\(650\) 14.3105 0.561304
\(651\) 10.1067 0.396114
\(652\) −0.454411 −0.0177961
\(653\) 16.2690 0.636654 0.318327 0.947981i \(-0.396879\pi\)
0.318327 + 0.947981i \(0.396879\pi\)
\(654\) 5.78029 0.226027
\(655\) 7.86949 0.307487
\(656\) 7.91452 0.309010
\(657\) 14.8153 0.578000
\(658\) 1.15559 0.0450495
\(659\) 34.8015 1.35567 0.677836 0.735213i \(-0.262918\pi\)
0.677836 + 0.735213i \(0.262918\pi\)
\(660\) 1.66454 0.0647923
\(661\) −7.62155 −0.296444 −0.148222 0.988954i \(-0.547355\pi\)
−0.148222 + 0.988954i \(0.547355\pi\)
\(662\) 35.5079 1.38005
\(663\) −1.70703 −0.0662954
\(664\) −7.07428 −0.274535
\(665\) −19.5557 −0.758339
\(666\) 6.27330 0.243086
\(667\) 14.8077 0.573356
\(668\) −17.4592 −0.675515
\(669\) 22.9878 0.888761
\(670\) 1.93047 0.0745804
\(671\) −1.00000 −0.0386046
\(672\) 2.43525 0.0939420
\(673\) 35.6135 1.37280 0.686400 0.727224i \(-0.259190\pi\)
0.686400 + 0.727224i \(0.259190\pi\)
\(674\) 10.3992 0.400562
\(675\) −2.22929 −0.0858054
\(676\) 28.2076 1.08491
\(677\) −43.2410 −1.66189 −0.830944 0.556356i \(-0.812199\pi\)
−0.830944 + 0.556356i \(0.812199\pi\)
\(678\) −2.38905 −0.0917508
\(679\) −10.3678 −0.397878
\(680\) −0.442637 −0.0169743
\(681\) −13.5329 −0.518580
\(682\) −4.15018 −0.158919
\(683\) −21.4393 −0.820353 −0.410177 0.912006i \(-0.634533\pi\)
−0.410177 + 0.912006i \(0.634533\pi\)
\(684\) −4.82430 −0.184462
\(685\) 23.8258 0.910338
\(686\) −19.6514 −0.750293
\(687\) −4.39321 −0.167612
\(688\) −10.7086 −0.408260
\(689\) −65.9208 −2.51138
\(690\) 8.94743 0.340623
\(691\) 37.9995 1.44557 0.722785 0.691073i \(-0.242862\pi\)
0.722785 + 0.691073i \(0.242862\pi\)
\(692\) −17.7516 −0.674815
\(693\) 2.43525 0.0925077
\(694\) 19.5319 0.741421
\(695\) 31.1916 1.18317
\(696\) −2.75477 −0.104419
\(697\) −2.10463 −0.0797187
\(698\) 9.23667 0.349613
\(699\) −10.2601 −0.388074
\(700\) 5.42889 0.205193
\(701\) −10.7427 −0.405746 −0.202873 0.979205i \(-0.565028\pi\)
−0.202873 + 0.979205i \(0.565028\pi\)
\(702\) −6.41931 −0.242281
\(703\) 30.2643 1.14144
\(704\) −1.00000 −0.0376889
\(705\) −0.789866 −0.0297481
\(706\) 8.02126 0.301884
\(707\) 0.770267 0.0289689
\(708\) 10.8275 0.406923
\(709\) 26.5299 0.996352 0.498176 0.867076i \(-0.334003\pi\)
0.498176 + 0.867076i \(0.334003\pi\)
\(710\) 10.5695 0.396666
\(711\) −13.9607 −0.523568
\(712\) −4.10397 −0.153803
\(713\) −22.3084 −0.835458
\(714\) −0.647584 −0.0242352
\(715\) 10.6852 0.399605
\(716\) −24.8418 −0.928383
\(717\) 20.2952 0.757939
\(718\) −32.7129 −1.22084
\(719\) −1.14521 −0.0427092 −0.0213546 0.999772i \(-0.506798\pi\)
−0.0213546 + 0.999772i \(0.506798\pi\)
\(720\) −1.66454 −0.0620339
\(721\) −3.78188 −0.140845
\(722\) −4.27388 −0.159057
\(723\) −4.10674 −0.152731
\(724\) −25.7352 −0.956442
\(725\) −6.14117 −0.228077
\(726\) −1.00000 −0.0371135
\(727\) 26.7828 0.993318 0.496659 0.867946i \(-0.334560\pi\)
0.496659 + 0.867946i \(0.334560\pi\)
\(728\) 15.6327 0.579385
\(729\) 1.00000 0.0370370
\(730\) 24.6607 0.912734
\(731\) 2.84763 0.105323
\(732\) 1.00000 0.0369611
\(733\) −5.28082 −0.195051 −0.0975257 0.995233i \(-0.531093\pi\)
−0.0975257 + 0.995233i \(0.531093\pi\)
\(734\) −11.7532 −0.433820
\(735\) 1.78029 0.0656669
\(736\) −5.37530 −0.198136
\(737\) −1.15976 −0.0427202
\(738\) −7.91452 −0.291338
\(739\) 9.85355 0.362469 0.181234 0.983440i \(-0.441991\pi\)
0.181234 + 0.983440i \(0.441991\pi\)
\(740\) 10.4422 0.383863
\(741\) −30.9687 −1.13766
\(742\) −25.0080 −0.918071
\(743\) 11.5351 0.423182 0.211591 0.977358i \(-0.432135\pi\)
0.211591 + 0.977358i \(0.432135\pi\)
\(744\) 4.15018 0.152153
\(745\) 15.5860 0.571027
\(746\) −21.7119 −0.794929
\(747\) 7.07428 0.258834
\(748\) 0.265921 0.00972302
\(749\) 45.2951 1.65505
\(750\) −12.0335 −0.439401
\(751\) −3.37895 −0.123300 −0.0616499 0.998098i \(-0.519636\pi\)
−0.0616499 + 0.998098i \(0.519636\pi\)
\(752\) 0.474524 0.0173041
\(753\) −0.778315 −0.0283634
\(754\) −17.6837 −0.644002
\(755\) 37.7387 1.37345
\(756\) −2.43525 −0.0885694
\(757\) −10.5487 −0.383401 −0.191700 0.981454i \(-0.561400\pi\)
−0.191700 + 0.981454i \(0.561400\pi\)
\(758\) 28.2596 1.02644
\(759\) −5.37530 −0.195111
\(760\) −8.03027 −0.291288
\(761\) −26.6125 −0.964702 −0.482351 0.875978i \(-0.660217\pi\)
−0.482351 + 0.875978i \(0.660217\pi\)
\(762\) 15.7341 0.569986
\(763\) 14.0765 0.509603
\(764\) −22.6603 −0.819820
\(765\) 0.442637 0.0160036
\(766\) 30.0875 1.08711
\(767\) 69.5052 2.50969
\(768\) 1.00000 0.0360844
\(769\) −46.8000 −1.68765 −0.843825 0.536618i \(-0.819702\pi\)
−0.843825 + 0.536618i \(0.819702\pi\)
\(770\) 4.05359 0.146081
\(771\) 16.1418 0.581334
\(772\) −19.8646 −0.714942
\(773\) 34.6050 1.24466 0.622328 0.782756i \(-0.286187\pi\)
0.622328 + 0.782756i \(0.286187\pi\)
\(774\) 10.7086 0.384911
\(775\) 9.25195 0.332340
\(776\) −4.25736 −0.152830
\(777\) 15.2771 0.548063
\(778\) −3.01375 −0.108048
\(779\) −38.1820 −1.36801
\(780\) −10.6852 −0.382593
\(781\) −6.34978 −0.227213
\(782\) 1.42940 0.0511153
\(783\) 2.75477 0.0984473
\(784\) −1.06953 −0.0381977
\(785\) −14.7723 −0.527246
\(786\) 4.72771 0.168632
\(787\) −46.4840 −1.65698 −0.828488 0.560007i \(-0.810799\pi\)
−0.828488 + 0.560007i \(0.810799\pi\)
\(788\) −4.40776 −0.157020
\(789\) −6.31095 −0.224676
\(790\) −23.2383 −0.826780
\(791\) −5.81794 −0.206862
\(792\) 1.00000 0.0355335
\(793\) 6.41931 0.227956
\(794\) −31.0546 −1.10209
\(795\) 17.0934 0.606242
\(796\) 9.74936 0.345557
\(797\) 10.7058 0.379218 0.189609 0.981860i \(-0.439278\pi\)
0.189609 + 0.981860i \(0.439278\pi\)
\(798\) −11.7484 −0.415889
\(799\) −0.126186 −0.00446413
\(800\) 2.22929 0.0788173
\(801\) 4.10397 0.145007
\(802\) −17.0466 −0.601935
\(803\) −14.8153 −0.522820
\(804\) 1.15976 0.0409014
\(805\) 21.7893 0.767971
\(806\) 26.6413 0.938399
\(807\) 22.7531 0.800949
\(808\) 0.316298 0.0111273
\(809\) −6.21321 −0.218445 −0.109223 0.994017i \(-0.534836\pi\)
−0.109223 + 0.994017i \(0.534836\pi\)
\(810\) 1.66454 0.0584861
\(811\) −6.31006 −0.221576 −0.110788 0.993844i \(-0.535338\pi\)
−0.110788 + 0.993844i \(0.535338\pi\)
\(812\) −6.70856 −0.235424
\(813\) −7.33699 −0.257319
\(814\) −6.27330 −0.219879
\(815\) 0.756388 0.0264951
\(816\) −0.265921 −0.00930908
\(817\) 51.6613 1.80740
\(818\) 2.40082 0.0839427
\(819\) −15.6327 −0.546249
\(820\) −13.1741 −0.460059
\(821\) 18.9399 0.661007 0.330503 0.943805i \(-0.392781\pi\)
0.330503 + 0.943805i \(0.392781\pi\)
\(822\) 14.3137 0.499248
\(823\) −35.7429 −1.24592 −0.622960 0.782254i \(-0.714070\pi\)
−0.622960 + 0.782254i \(0.714070\pi\)
\(824\) −1.55297 −0.0541003
\(825\) 2.22929 0.0776139
\(826\) 26.3678 0.917452
\(827\) −31.1136 −1.08193 −0.540963 0.841047i \(-0.681940\pi\)
−0.540963 + 0.841047i \(0.681940\pi\)
\(828\) 5.37530 0.186805
\(829\) 0.559046 0.0194165 0.00970824 0.999953i \(-0.496910\pi\)
0.00970824 + 0.999953i \(0.496910\pi\)
\(830\) 11.7755 0.408732
\(831\) −8.14645 −0.282597
\(832\) 6.41931 0.222550
\(833\) 0.284411 0.00985427
\(834\) 18.7388 0.648872
\(835\) 29.0615 1.00572
\(836\) 4.82430 0.166852
\(837\) −4.15018 −0.143451
\(838\) −10.8918 −0.376250
\(839\) −19.8577 −0.685563 −0.342782 0.939415i \(-0.611369\pi\)
−0.342782 + 0.939415i \(0.611369\pi\)
\(840\) −4.05359 −0.139862
\(841\) −21.4113 −0.738319
\(842\) −39.8498 −1.37331
\(843\) −15.4404 −0.531797
\(844\) 22.5478 0.776126
\(845\) −46.9527 −1.61522
\(846\) −0.474524 −0.0163145
\(847\) −2.43525 −0.0836763
\(848\) −10.2691 −0.352644
\(849\) 5.37947 0.184623
\(850\) −0.592814 −0.0203334
\(851\) −33.7209 −1.15594
\(852\) 6.34978 0.217540
\(853\) 29.8486 1.02200 0.510999 0.859581i \(-0.329276\pi\)
0.510999 + 0.859581i \(0.329276\pi\)
\(854\) 2.43525 0.0833327
\(855\) 8.03027 0.274629
\(856\) 18.5998 0.635726
\(857\) −28.8335 −0.984934 −0.492467 0.870331i \(-0.663905\pi\)
−0.492467 + 0.870331i \(0.663905\pi\)
\(858\) 6.41931 0.219152
\(859\) 2.04481 0.0697680 0.0348840 0.999391i \(-0.488894\pi\)
0.0348840 + 0.999391i \(0.488894\pi\)
\(860\) 17.8249 0.607823
\(861\) −19.2739 −0.656852
\(862\) 24.3923 0.830804
\(863\) 32.2334 1.09724 0.548618 0.836073i \(-0.315154\pi\)
0.548618 + 0.836073i \(0.315154\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 29.5484 1.00467
\(866\) 17.4973 0.594583
\(867\) −16.9293 −0.574949
\(868\) 10.1067 0.343045
\(869\) 13.9607 0.473585
\(870\) 4.58543 0.155461
\(871\) 7.44483 0.252259
\(872\) 5.78029 0.195745
\(873\) 4.25736 0.144090
\(874\) 25.9321 0.877165
\(875\) −29.3046 −0.990676
\(876\) 14.8153 0.500562
\(877\) −25.8087 −0.871498 −0.435749 0.900068i \(-0.643516\pi\)
−0.435749 + 0.900068i \(0.643516\pi\)
\(878\) −21.4162 −0.722760
\(879\) 25.2818 0.852733
\(880\) 1.66454 0.0561118
\(881\) 39.7123 1.33794 0.668971 0.743288i \(-0.266735\pi\)
0.668971 + 0.743288i \(0.266735\pi\)
\(882\) 1.06953 0.0360131
\(883\) −33.9378 −1.14210 −0.571049 0.820916i \(-0.693463\pi\)
−0.571049 + 0.820916i \(0.693463\pi\)
\(884\) −1.70703 −0.0574135
\(885\) −18.0229 −0.605833
\(886\) 5.25612 0.176583
\(887\) −4.86619 −0.163391 −0.0816953 0.996657i \(-0.526033\pi\)
−0.0816953 + 0.996657i \(0.526033\pi\)
\(888\) 6.27330 0.210518
\(889\) 38.3165 1.28509
\(890\) 6.83124 0.228984
\(891\) −1.00000 −0.0335013
\(892\) 22.9878 0.769690
\(893\) −2.28925 −0.0766067
\(894\) 9.36352 0.313163
\(895\) 41.3504 1.38219
\(896\) 2.43525 0.0813562
\(897\) 34.5057 1.15211
\(898\) −5.87430 −0.196028
\(899\) −11.4328 −0.381304
\(900\) −2.22929 −0.0743097
\(901\) 2.73077 0.0909753
\(902\) 7.91452 0.263525
\(903\) 26.0781 0.867824
\(904\) −2.38905 −0.0794585
\(905\) 42.8374 1.42396
\(906\) 22.6721 0.753229
\(907\) 9.01696 0.299403 0.149702 0.988731i \(-0.452169\pi\)
0.149702 + 0.988731i \(0.452169\pi\)
\(908\) −13.5329 −0.449104
\(909\) −0.316298 −0.0104910
\(910\) −26.0213 −0.862596
\(911\) 0.412595 0.0136699 0.00683494 0.999977i \(-0.497824\pi\)
0.00683494 + 0.999977i \(0.497824\pi\)
\(912\) −4.82430 −0.159749
\(913\) −7.07428 −0.234124
\(914\) −13.4282 −0.444166
\(915\) −1.66454 −0.0550282
\(916\) −4.39321 −0.145156
\(917\) 11.5132 0.380199
\(918\) 0.265921 0.00877669
\(919\) −37.1635 −1.22591 −0.612955 0.790118i \(-0.710019\pi\)
−0.612955 + 0.790118i \(0.710019\pi\)
\(920\) 8.94743 0.294988
\(921\) −19.0780 −0.628642
\(922\) 38.8023 1.27789
\(923\) 40.7612 1.34167
\(924\) 2.43525 0.0801140
\(925\) 13.9850 0.459824
\(926\) 15.2240 0.500292
\(927\) 1.55297 0.0510063
\(928\) −2.75477 −0.0904296
\(929\) −39.4005 −1.29269 −0.646344 0.763046i \(-0.723703\pi\)
−0.646344 + 0.763046i \(0.723703\pi\)
\(930\) −6.90816 −0.226527
\(931\) 5.15976 0.169104
\(932\) −10.2601 −0.336082
\(933\) 32.0099 1.04796
\(934\) −36.6004 −1.19760
\(935\) −0.442637 −0.0144758
\(936\) −6.41931 −0.209822
\(937\) −32.6053 −1.06517 −0.532584 0.846377i \(-0.678779\pi\)
−0.532584 + 0.846377i \(0.678779\pi\)
\(938\) 2.82430 0.0922167
\(939\) −23.9824 −0.782637
\(940\) −0.789866 −0.0257626
\(941\) −45.5906 −1.48621 −0.743106 0.669174i \(-0.766648\pi\)
−0.743106 + 0.669174i \(0.766648\pi\)
\(942\) −8.87468 −0.289153
\(943\) 42.5429 1.38539
\(944\) 10.8275 0.352406
\(945\) 4.05359 0.131863
\(946\) −10.7086 −0.348165
\(947\) 32.2802 1.04897 0.524483 0.851421i \(-0.324258\pi\)
0.524483 + 0.851421i \(0.324258\pi\)
\(948\) −13.9607 −0.453423
\(949\) 95.1040 3.08721
\(950\) −10.7548 −0.348931
\(951\) 3.15237 0.102223
\(952\) −0.647584 −0.0209883
\(953\) 33.3741 1.08109 0.540547 0.841314i \(-0.318217\pi\)
0.540547 + 0.841314i \(0.318217\pi\)
\(954\) 10.2691 0.332475
\(955\) 37.7191 1.22056
\(956\) 20.2952 0.656395
\(957\) −2.75477 −0.0890489
\(958\) −15.5333 −0.501858
\(959\) 34.8576 1.12561
\(960\) −1.66454 −0.0537230
\(961\) −13.7760 −0.444388
\(962\) 40.2703 1.29837
\(963\) −18.5998 −0.599369
\(964\) −4.10674 −0.132269
\(965\) 33.0655 1.06442
\(966\) 13.0902 0.421171
\(967\) −28.1252 −0.904446 −0.452223 0.891905i \(-0.649369\pi\)
−0.452223 + 0.891905i \(0.649369\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 1.28288 0.0412121
\(970\) 7.08657 0.227536
\(971\) −37.7925 −1.21282 −0.606409 0.795153i \(-0.707391\pi\)
−0.606409 + 0.795153i \(0.707391\pi\)
\(972\) 1.00000 0.0320750
\(973\) 45.6338 1.46295
\(974\) −2.55632 −0.0819096
\(975\) −14.3105 −0.458303
\(976\) 1.00000 0.0320092
\(977\) −41.0126 −1.31211 −0.656055 0.754713i \(-0.727776\pi\)
−0.656055 + 0.754713i \(0.727776\pi\)
\(978\) 0.454411 0.0145305
\(979\) −4.10397 −0.131163
\(980\) 1.78029 0.0568692
\(981\) −5.78029 −0.184550
\(982\) 18.8074 0.600168
\(983\) 13.3748 0.426589 0.213295 0.976988i \(-0.431581\pi\)
0.213295 + 0.976988i \(0.431581\pi\)
\(984\) −7.91452 −0.252306
\(985\) 7.33691 0.233774
\(986\) 0.732549 0.0233291
\(987\) −1.15559 −0.0367827
\(988\) −30.9687 −0.985245
\(989\) −57.5617 −1.83036
\(990\) −1.66454 −0.0529027
\(991\) 29.8651 0.948695 0.474348 0.880338i \(-0.342684\pi\)
0.474348 + 0.880338i \(0.342684\pi\)
\(992\) 4.15018 0.131768
\(993\) −35.5079 −1.12681
\(994\) 15.4633 0.490467
\(995\) −16.2282 −0.514470
\(996\) 7.07428 0.224157
\(997\) −21.2637 −0.673428 −0.336714 0.941607i \(-0.609316\pi\)
−0.336714 + 0.941607i \(0.609316\pi\)
\(998\) −26.3269 −0.833363
\(999\) −6.27330 −0.198479
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 4026.2.a.r.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4026.2.a.r.1.1 4 1.1 even 1 trivial