Properties

Label 4026.2.a.y.1.1
Level $4026$
Weight $2$
Character 4026.1
Self dual yes
Analytic conductor $32.148$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 21x^{5} + 39x^{4} + 89x^{3} - 100x^{2} - 96x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.80390\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.80390 q^{5} +1.00000 q^{6} -1.13011 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.80390 q^{5} +1.00000 q^{6} -1.13011 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.80390 q^{10} -1.00000 q^{11} -1.00000 q^{12} -4.24171 q^{13} +1.13011 q^{14} +3.80390 q^{15} +1.00000 q^{16} -0.128672 q^{17} -1.00000 q^{18} -0.265037 q^{19} -3.80390 q^{20} +1.13011 q^{21} +1.00000 q^{22} +4.37039 q^{23} +1.00000 q^{24} +9.46966 q^{25} +4.24171 q^{26} -1.00000 q^{27} -1.13011 q^{28} -3.55745 q^{29} -3.80390 q^{30} +7.85627 q^{31} -1.00000 q^{32} +1.00000 q^{33} +0.128672 q^{34} +4.29881 q^{35} +1.00000 q^{36} +11.0001 q^{37} +0.265037 q^{38} +4.24171 q^{39} +3.80390 q^{40} +2.09928 q^{41} -1.13011 q^{42} +1.99798 q^{43} -1.00000 q^{44} -3.80390 q^{45} -4.37039 q^{46} -0.338707 q^{47} -1.00000 q^{48} -5.72286 q^{49} -9.46966 q^{50} +0.128672 q^{51} -4.24171 q^{52} -2.22997 q^{53} +1.00000 q^{54} +3.80390 q^{55} +1.13011 q^{56} +0.265037 q^{57} +3.55745 q^{58} -4.93603 q^{59} +3.80390 q^{60} -1.00000 q^{61} -7.85627 q^{62} -1.13011 q^{63} +1.00000 q^{64} +16.1351 q^{65} -1.00000 q^{66} +13.4773 q^{67} -0.128672 q^{68} -4.37039 q^{69} -4.29881 q^{70} +2.75650 q^{71} -1.00000 q^{72} -11.1407 q^{73} -11.0001 q^{74} -9.46966 q^{75} -0.265037 q^{76} +1.13011 q^{77} -4.24171 q^{78} -4.18391 q^{79} -3.80390 q^{80} +1.00000 q^{81} -2.09928 q^{82} -5.50532 q^{83} +1.13011 q^{84} +0.489455 q^{85} -1.99798 q^{86} +3.55745 q^{87} +1.00000 q^{88} -0.489455 q^{89} +3.80390 q^{90} +4.79359 q^{91} +4.37039 q^{92} -7.85627 q^{93} +0.338707 q^{94} +1.00817 q^{95} +1.00000 q^{96} +16.4339 q^{97} +5.72286 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} - 2 q^{5} + 7 q^{6} + q^{7} - 7 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} - 7 q^{3} + 7 q^{4} - 2 q^{5} + 7 q^{6} + q^{7} - 7 q^{8} + 7 q^{9} + 2 q^{10} - 7 q^{11} - 7 q^{12} - q^{14} + 2 q^{15} + 7 q^{16} + 3 q^{17} - 7 q^{18} - 5 q^{19} - 2 q^{20} - q^{21} + 7 q^{22} - 3 q^{23} + 7 q^{24} + 11 q^{25} - 7 q^{27} + q^{28} - 14 q^{29} - 2 q^{30} + 5 q^{31} - 7 q^{32} + 7 q^{33} - 3 q^{34} - 9 q^{35} + 7 q^{36} + 14 q^{37} + 5 q^{38} + 2 q^{40} - 7 q^{41} + q^{42} + q^{43} - 7 q^{44} - 2 q^{45} + 3 q^{46} - 7 q^{48} - 11 q^{50} - 3 q^{51} - 3 q^{53} + 7 q^{54} + 2 q^{55} - q^{56} + 5 q^{57} + 14 q^{58} - 14 q^{59} + 2 q^{60} - 7 q^{61} - 5 q^{62} + q^{63} + 7 q^{64} - 10 q^{65} - 7 q^{66} + 3 q^{68} + 3 q^{69} + 9 q^{70} - 22 q^{71} - 7 q^{72} + q^{73} - 14 q^{74} - 11 q^{75} - 5 q^{76} - q^{77} + 10 q^{79} - 2 q^{80} + 7 q^{81} + 7 q^{82} - 17 q^{83} - q^{84} + 18 q^{85} - q^{86} + 14 q^{87} + 7 q^{88} - 18 q^{89} + 2 q^{90} + 21 q^{91} - 3 q^{92} - 5 q^{93} - 41 q^{95} + 7 q^{96} + 25 q^{97} - 7 q^{99}+O(q^{100}) \) 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
\(5\) −3.80390 −1.70116 −0.850578 0.525849i \(-0.823748\pi\)
−0.850578 + 0.525849i \(0.823748\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.13011 −0.427140 −0.213570 0.976928i \(-0.568509\pi\)
−0.213570 + 0.976928i \(0.568509\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.80390 1.20290
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −4.24171 −1.17644 −0.588220 0.808701i \(-0.700171\pi\)
−0.588220 + 0.808701i \(0.700171\pi\)
\(14\) 1.13011 0.302034
\(15\) 3.80390 0.982163
\(16\) 1.00000 0.250000
\(17\) −0.128672 −0.0312075 −0.0156037 0.999878i \(-0.504967\pi\)
−0.0156037 + 0.999878i \(0.504967\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.265037 −0.0608036 −0.0304018 0.999538i \(-0.509679\pi\)
−0.0304018 + 0.999538i \(0.509679\pi\)
\(20\) −3.80390 −0.850578
\(21\) 1.13011 0.246609
\(22\) 1.00000 0.213201
\(23\) 4.37039 0.911288 0.455644 0.890162i \(-0.349409\pi\)
0.455644 + 0.890162i \(0.349409\pi\)
\(24\) 1.00000 0.204124
\(25\) 9.46966 1.89393
\(26\) 4.24171 0.831869
\(27\) −1.00000 −0.192450
\(28\) −1.13011 −0.213570
\(29\) −3.55745 −0.660603 −0.330301 0.943876i \(-0.607150\pi\)
−0.330301 + 0.943876i \(0.607150\pi\)
\(30\) −3.80390 −0.694494
\(31\) 7.85627 1.41103 0.705514 0.708696i \(-0.250716\pi\)
0.705514 + 0.708696i \(0.250716\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 0.128672 0.0220670
\(35\) 4.29881 0.726632
\(36\) 1.00000 0.166667
\(37\) 11.0001 1.80841 0.904206 0.427097i \(-0.140464\pi\)
0.904206 + 0.427097i \(0.140464\pi\)
\(38\) 0.265037 0.0429947
\(39\) 4.24171 0.679218
\(40\) 3.80390 0.601450
\(41\) 2.09928 0.327852 0.163926 0.986473i \(-0.447584\pi\)
0.163926 + 0.986473i \(0.447584\pi\)
\(42\) −1.13011 −0.174379
\(43\) 1.99798 0.304689 0.152345 0.988327i \(-0.451318\pi\)
0.152345 + 0.988327i \(0.451318\pi\)
\(44\) −1.00000 −0.150756
\(45\) −3.80390 −0.567052
\(46\) −4.37039 −0.644378
\(47\) −0.338707 −0.0494055 −0.0247028 0.999695i \(-0.507864\pi\)
−0.0247028 + 0.999695i \(0.507864\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.72286 −0.817551
\(50\) −9.46966 −1.33921
\(51\) 0.128672 0.0180177
\(52\) −4.24171 −0.588220
\(53\) −2.22997 −0.306309 −0.153155 0.988202i \(-0.548943\pi\)
−0.153155 + 0.988202i \(0.548943\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.80390 0.512918
\(56\) 1.13011 0.151017
\(57\) 0.265037 0.0351050
\(58\) 3.55745 0.467117
\(59\) −4.93603 −0.642616 −0.321308 0.946975i \(-0.604122\pi\)
−0.321308 + 0.946975i \(0.604122\pi\)
\(60\) 3.80390 0.491081
\(61\) −1.00000 −0.128037
\(62\) −7.85627 −0.997747
\(63\) −1.13011 −0.142380
\(64\) 1.00000 0.125000
\(65\) 16.1351 2.00131
\(66\) −1.00000 −0.123091
\(67\) 13.4773 1.64652 0.823259 0.567665i \(-0.192153\pi\)
0.823259 + 0.567665i \(0.192153\pi\)
\(68\) −0.128672 −0.0156037
\(69\) −4.37039 −0.526133
\(70\) −4.29881 −0.513806
\(71\) 2.75650 0.327136 0.163568 0.986532i \(-0.447700\pi\)
0.163568 + 0.986532i \(0.447700\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.1407 −1.30391 −0.651957 0.758256i \(-0.726052\pi\)
−0.651957 + 0.758256i \(0.726052\pi\)
\(74\) −11.0001 −1.27874
\(75\) −9.46966 −1.09346
\(76\) −0.265037 −0.0304018
\(77\) 1.13011 0.128788
\(78\) −4.24171 −0.480280
\(79\) −4.18391 −0.470726 −0.235363 0.971907i \(-0.575628\pi\)
−0.235363 + 0.971907i \(0.575628\pi\)
\(80\) −3.80390 −0.425289
\(81\) 1.00000 0.111111
\(82\) −2.09928 −0.231826
\(83\) −5.50532 −0.604287 −0.302143 0.953262i \(-0.597702\pi\)
−0.302143 + 0.953262i \(0.597702\pi\)
\(84\) 1.13011 0.123305
\(85\) 0.489455 0.0530888
\(86\) −1.99798 −0.215448
\(87\) 3.55745 0.381399
\(88\) 1.00000 0.106600
\(89\) −0.489455 −0.0518821 −0.0259410 0.999663i \(-0.508258\pi\)
−0.0259410 + 0.999663i \(0.508258\pi\)
\(90\) 3.80390 0.400966
\(91\) 4.79359 0.502505
\(92\) 4.37039 0.455644
\(93\) −7.85627 −0.814657
\(94\) 0.338707 0.0349350
\(95\) 1.00817 0.103436
\(96\) 1.00000 0.102062
\(97\) 16.4339 1.66861 0.834304 0.551305i \(-0.185870\pi\)
0.834304 + 0.551305i \(0.185870\pi\)
\(98\) 5.72286 0.578096
\(99\) −1.00000 −0.100504
\(100\) 9.46966 0.946966
\(101\) −5.19702 −0.517123 −0.258562 0.965995i \(-0.583249\pi\)
−0.258562 + 0.965995i \(0.583249\pi\)
\(102\) −0.128672 −0.0127404
\(103\) 16.9417 1.66932 0.834660 0.550766i \(-0.185664\pi\)
0.834660 + 0.550766i \(0.185664\pi\)
\(104\) 4.24171 0.415934
\(105\) −4.29881 −0.419521
\(106\) 2.22997 0.216593
\(107\) −0.918730 −0.0888170 −0.0444085 0.999013i \(-0.514140\pi\)
−0.0444085 + 0.999013i \(0.514140\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.41651 −0.423025 −0.211512 0.977375i \(-0.567839\pi\)
−0.211512 + 0.977375i \(0.567839\pi\)
\(110\) −3.80390 −0.362688
\(111\) −11.0001 −1.04409
\(112\) −1.13011 −0.106785
\(113\) 9.04088 0.850494 0.425247 0.905077i \(-0.360187\pi\)
0.425247 + 0.905077i \(0.360187\pi\)
\(114\) −0.265037 −0.0248230
\(115\) −16.6245 −1.55024
\(116\) −3.55745 −0.330301
\(117\) −4.24171 −0.392147
\(118\) 4.93603 0.454398
\(119\) 0.145413 0.0133300
\(120\) −3.80390 −0.347247
\(121\) 1.00000 0.0909091
\(122\) 1.00000 0.0905357
\(123\) −2.09928 −0.189285
\(124\) 7.85627 0.705514
\(125\) −17.0021 −1.52072
\(126\) 1.13011 0.100678
\(127\) 14.9447 1.32613 0.663065 0.748562i \(-0.269255\pi\)
0.663065 + 0.748562i \(0.269255\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.99798 −0.175912
\(130\) −16.1351 −1.41514
\(131\) −5.68475 −0.496679 −0.248340 0.968673i \(-0.579885\pi\)
−0.248340 + 0.968673i \(0.579885\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0.299520 0.0259717
\(134\) −13.4773 −1.16426
\(135\) 3.80390 0.327388
\(136\) 0.128672 0.0110335
\(137\) −13.2285 −1.13019 −0.565095 0.825026i \(-0.691160\pi\)
−0.565095 + 0.825026i \(0.691160\pi\)
\(138\) 4.37039 0.372032
\(139\) −14.6777 −1.24494 −0.622472 0.782642i \(-0.713872\pi\)
−0.622472 + 0.782642i \(0.713872\pi\)
\(140\) 4.29881 0.363316
\(141\) 0.338707 0.0285243
\(142\) −2.75650 −0.231320
\(143\) 4.24171 0.354710
\(144\) 1.00000 0.0833333
\(145\) 13.5322 1.12379
\(146\) 11.1407 0.922007
\(147\) 5.72286 0.472013
\(148\) 11.0001 0.904206
\(149\) 7.90661 0.647735 0.323868 0.946102i \(-0.395017\pi\)
0.323868 + 0.946102i \(0.395017\pi\)
\(150\) 9.46966 0.773195
\(151\) 1.15571 0.0940505 0.0470252 0.998894i \(-0.485026\pi\)
0.0470252 + 0.998894i \(0.485026\pi\)
\(152\) 0.265037 0.0214973
\(153\) −0.128672 −0.0104025
\(154\) −1.13011 −0.0910666
\(155\) −29.8845 −2.40038
\(156\) 4.24171 0.339609
\(157\) 9.07230 0.724048 0.362024 0.932169i \(-0.382086\pi\)
0.362024 + 0.932169i \(0.382086\pi\)
\(158\) 4.18391 0.332854
\(159\) 2.22997 0.176848
\(160\) 3.80390 0.300725
\(161\) −4.93900 −0.389248
\(162\) −1.00000 −0.0785674
\(163\) −8.13603 −0.637263 −0.318631 0.947879i \(-0.603223\pi\)
−0.318631 + 0.947879i \(0.603223\pi\)
\(164\) 2.09928 0.163926
\(165\) −3.80390 −0.296133
\(166\) 5.50532 0.427295
\(167\) −3.11680 −0.241186 −0.120593 0.992702i \(-0.538480\pi\)
−0.120593 + 0.992702i \(0.538480\pi\)
\(168\) −1.13011 −0.0871896
\(169\) 4.99214 0.384010
\(170\) −0.489455 −0.0375395
\(171\) −0.265037 −0.0202679
\(172\) 1.99798 0.152345
\(173\) −7.76547 −0.590398 −0.295199 0.955436i \(-0.595386\pi\)
−0.295199 + 0.955436i \(0.595386\pi\)
\(174\) −3.55745 −0.269690
\(175\) −10.7017 −0.808974
\(176\) −1.00000 −0.0753778
\(177\) 4.93603 0.371014
\(178\) 0.489455 0.0366862
\(179\) −3.25320 −0.243156 −0.121578 0.992582i \(-0.538795\pi\)
−0.121578 + 0.992582i \(0.538795\pi\)
\(180\) −3.80390 −0.283526
\(181\) −2.50450 −0.186158 −0.0930789 0.995659i \(-0.529671\pi\)
−0.0930789 + 0.995659i \(0.529671\pi\)
\(182\) −4.79359 −0.355324
\(183\) 1.00000 0.0739221
\(184\) −4.37039 −0.322189
\(185\) −41.8434 −3.07639
\(186\) 7.85627 0.576049
\(187\) 0.128672 0.00940941
\(188\) −0.338707 −0.0247028
\(189\) 1.13011 0.0822031
\(190\) −1.00817 −0.0731406
\(191\) −23.2019 −1.67883 −0.839415 0.543491i \(-0.817102\pi\)
−0.839415 + 0.543491i \(0.817102\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −19.7378 −1.42076 −0.710379 0.703819i \(-0.751477\pi\)
−0.710379 + 0.703819i \(0.751477\pi\)
\(194\) −16.4339 −1.17988
\(195\) −16.1351 −1.15546
\(196\) −5.72286 −0.408776
\(197\) 4.50236 0.320780 0.160390 0.987054i \(-0.448725\pi\)
0.160390 + 0.987054i \(0.448725\pi\)
\(198\) 1.00000 0.0710669
\(199\) 7.90054 0.560055 0.280027 0.959992i \(-0.409656\pi\)
0.280027 + 0.959992i \(0.409656\pi\)
\(200\) −9.46966 −0.669606
\(201\) −13.4773 −0.950618
\(202\) 5.19702 0.365661
\(203\) 4.02030 0.282170
\(204\) 0.128672 0.00900883
\(205\) −7.98544 −0.557727
\(206\) −16.9417 −1.18039
\(207\) 4.37039 0.303763
\(208\) −4.24171 −0.294110
\(209\) 0.265037 0.0183330
\(210\) 4.29881 0.296646
\(211\) 25.2862 1.74077 0.870387 0.492368i \(-0.163869\pi\)
0.870387 + 0.492368i \(0.163869\pi\)
\(212\) −2.22997 −0.153155
\(213\) −2.75650 −0.188872
\(214\) 0.918730 0.0628031
\(215\) −7.60012 −0.518324
\(216\) 1.00000 0.0680414
\(217\) −8.87842 −0.602706
\(218\) 4.41651 0.299124
\(219\) 11.1407 0.752815
\(220\) 3.80390 0.256459
\(221\) 0.545789 0.0367137
\(222\) 11.0001 0.738281
\(223\) −0.909127 −0.0608796 −0.0304398 0.999537i \(-0.509691\pi\)
−0.0304398 + 0.999537i \(0.509691\pi\)
\(224\) 1.13011 0.0755084
\(225\) 9.46966 0.631311
\(226\) −9.04088 −0.601390
\(227\) −6.58569 −0.437108 −0.218554 0.975825i \(-0.570134\pi\)
−0.218554 + 0.975825i \(0.570134\pi\)
\(228\) 0.265037 0.0175525
\(229\) −24.9926 −1.65156 −0.825780 0.563993i \(-0.809264\pi\)
−0.825780 + 0.563993i \(0.809264\pi\)
\(230\) 16.6245 1.09619
\(231\) −1.13011 −0.0743555
\(232\) 3.55745 0.233558
\(233\) −0.0101462 −0.000664701 0 −0.000332350 1.00000i \(-0.500106\pi\)
−0.000332350 1.00000i \(0.500106\pi\)
\(234\) 4.24171 0.277290
\(235\) 1.28841 0.0840465
\(236\) −4.93603 −0.321308
\(237\) 4.18391 0.271774
\(238\) −0.145413 −0.00942571
\(239\) −1.36601 −0.0883598 −0.0441799 0.999024i \(-0.514067\pi\)
−0.0441799 + 0.999024i \(0.514067\pi\)
\(240\) 3.80390 0.245541
\(241\) 23.3974 1.50716 0.753578 0.657358i \(-0.228326\pi\)
0.753578 + 0.657358i \(0.228326\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 21.7692 1.39078
\(246\) 2.09928 0.133845
\(247\) 1.12421 0.0715318
\(248\) −7.85627 −0.498873
\(249\) 5.50532 0.348885
\(250\) 17.0021 1.07531
\(251\) −13.0600 −0.824343 −0.412171 0.911106i \(-0.635229\pi\)
−0.412171 + 0.911106i \(0.635229\pi\)
\(252\) −1.13011 −0.0711900
\(253\) −4.37039 −0.274764
\(254\) −14.9447 −0.937715
\(255\) −0.489455 −0.0306508
\(256\) 1.00000 0.0625000
\(257\) 16.2686 1.01481 0.507404 0.861708i \(-0.330605\pi\)
0.507404 + 0.861708i \(0.330605\pi\)
\(258\) 1.99798 0.124389
\(259\) −12.4313 −0.772445
\(260\) 16.1351 1.00065
\(261\) −3.55745 −0.220201
\(262\) 5.68475 0.351205
\(263\) 6.16069 0.379884 0.189942 0.981795i \(-0.439170\pi\)
0.189942 + 0.981795i \(0.439170\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 8.48257 0.521080
\(266\) −0.299520 −0.0183647
\(267\) 0.489455 0.0299541
\(268\) 13.4773 0.823259
\(269\) −21.6824 −1.32200 −0.661000 0.750386i \(-0.729868\pi\)
−0.661000 + 0.750386i \(0.729868\pi\)
\(270\) −3.80390 −0.231498
\(271\) −19.0960 −1.16000 −0.579999 0.814617i \(-0.696947\pi\)
−0.579999 + 0.814617i \(0.696947\pi\)
\(272\) −0.128672 −0.00780187
\(273\) −4.79359 −0.290121
\(274\) 13.2285 0.799164
\(275\) −9.46966 −0.571042
\(276\) −4.37039 −0.263066
\(277\) −21.4653 −1.28973 −0.644863 0.764298i \(-0.723086\pi\)
−0.644863 + 0.764298i \(0.723086\pi\)
\(278\) 14.6777 0.880309
\(279\) 7.85627 0.470342
\(280\) −4.29881 −0.256903
\(281\) −20.1406 −1.20149 −0.600743 0.799442i \(-0.705128\pi\)
−0.600743 + 0.799442i \(0.705128\pi\)
\(282\) −0.338707 −0.0201697
\(283\) 6.60060 0.392365 0.196182 0.980567i \(-0.437146\pi\)
0.196182 + 0.980567i \(0.437146\pi\)
\(284\) 2.75650 0.163568
\(285\) −1.00817 −0.0597191
\(286\) −4.24171 −0.250818
\(287\) −2.37240 −0.140039
\(288\) −1.00000 −0.0589256
\(289\) −16.9834 −0.999026
\(290\) −13.5322 −0.794638
\(291\) −16.4339 −0.963371
\(292\) −11.1407 −0.651957
\(293\) 27.0042 1.57760 0.788801 0.614649i \(-0.210702\pi\)
0.788801 + 0.614649i \(0.210702\pi\)
\(294\) −5.72286 −0.333764
\(295\) 18.7762 1.09319
\(296\) −11.0001 −0.639370
\(297\) 1.00000 0.0580259
\(298\) −7.90661 −0.458018
\(299\) −18.5379 −1.07208
\(300\) −9.46966 −0.546731
\(301\) −2.25793 −0.130145
\(302\) −1.15571 −0.0665037
\(303\) 5.19702 0.298561
\(304\) −0.265037 −0.0152009
\(305\) 3.80390 0.217811
\(306\) 0.128672 0.00735568
\(307\) −21.1225 −1.20553 −0.602763 0.797921i \(-0.705933\pi\)
−0.602763 + 0.797921i \(0.705933\pi\)
\(308\) 1.13011 0.0643938
\(309\) −16.9417 −0.963782
\(310\) 29.8845 1.69732
\(311\) −25.0386 −1.41981 −0.709903 0.704299i \(-0.751261\pi\)
−0.709903 + 0.704299i \(0.751261\pi\)
\(312\) −4.24171 −0.240140
\(313\) −24.5109 −1.38544 −0.692719 0.721208i \(-0.743587\pi\)
−0.692719 + 0.721208i \(0.743587\pi\)
\(314\) −9.07230 −0.511980
\(315\) 4.29881 0.242211
\(316\) −4.18391 −0.235363
\(317\) 29.1713 1.63842 0.819210 0.573493i \(-0.194412\pi\)
0.819210 + 0.573493i \(0.194412\pi\)
\(318\) −2.22997 −0.125050
\(319\) 3.55745 0.199179
\(320\) −3.80390 −0.212645
\(321\) 0.918730 0.0512785
\(322\) 4.93900 0.275240
\(323\) 0.0341028 0.00189753
\(324\) 1.00000 0.0555556
\(325\) −40.1676 −2.22810
\(326\) 8.13603 0.450613
\(327\) 4.41651 0.244234
\(328\) −2.09928 −0.115913
\(329\) 0.382775 0.0211031
\(330\) 3.80390 0.209398
\(331\) 11.8658 0.652204 0.326102 0.945335i \(-0.394265\pi\)
0.326102 + 0.945335i \(0.394265\pi\)
\(332\) −5.50532 −0.302143
\(333\) 11.0001 0.602804
\(334\) 3.11680 0.170544
\(335\) −51.2665 −2.80099
\(336\) 1.13011 0.0616524
\(337\) 18.0097 0.981051 0.490525 0.871427i \(-0.336805\pi\)
0.490525 + 0.871427i \(0.336805\pi\)
\(338\) −4.99214 −0.271536
\(339\) −9.04088 −0.491033
\(340\) 0.489455 0.0265444
\(341\) −7.85627 −0.425441
\(342\) 0.265037 0.0143316
\(343\) 14.3782 0.776349
\(344\) −1.99798 −0.107724
\(345\) 16.6245 0.895034
\(346\) 7.76547 0.417474
\(347\) −12.1166 −0.650454 −0.325227 0.945636i \(-0.605441\pi\)
−0.325227 + 0.945636i \(0.605441\pi\)
\(348\) 3.55745 0.190700
\(349\) −21.1577 −1.13255 −0.566274 0.824217i \(-0.691616\pi\)
−0.566274 + 0.824217i \(0.691616\pi\)
\(350\) 10.7017 0.572031
\(351\) 4.24171 0.226406
\(352\) 1.00000 0.0533002
\(353\) −2.82372 −0.150292 −0.0751458 0.997173i \(-0.523942\pi\)
−0.0751458 + 0.997173i \(0.523942\pi\)
\(354\) −4.93603 −0.262347
\(355\) −10.4854 −0.556510
\(356\) −0.489455 −0.0259410
\(357\) −0.145413 −0.00769606
\(358\) 3.25320 0.171937
\(359\) 18.0948 0.955008 0.477504 0.878630i \(-0.341542\pi\)
0.477504 + 0.878630i \(0.341542\pi\)
\(360\) 3.80390 0.200483
\(361\) −18.9298 −0.996303
\(362\) 2.50450 0.131633
\(363\) −1.00000 −0.0524864
\(364\) 4.79359 0.251252
\(365\) 42.3779 2.21816
\(366\) −1.00000 −0.0522708
\(367\) −20.6125 −1.07596 −0.537982 0.842956i \(-0.680813\pi\)
−0.537982 + 0.842956i \(0.680813\pi\)
\(368\) 4.37039 0.227822
\(369\) 2.09928 0.109284
\(370\) 41.8434 2.17534
\(371\) 2.52010 0.130837
\(372\) −7.85627 −0.407328
\(373\) 25.1657 1.30303 0.651516 0.758634i \(-0.274133\pi\)
0.651516 + 0.758634i \(0.274133\pi\)
\(374\) −0.128672 −0.00665346
\(375\) 17.0021 0.877987
\(376\) 0.338707 0.0174675
\(377\) 15.0897 0.777159
\(378\) −1.13011 −0.0581264
\(379\) 2.60869 0.134000 0.0669998 0.997753i \(-0.478657\pi\)
0.0669998 + 0.997753i \(0.478657\pi\)
\(380\) 1.00817 0.0517182
\(381\) −14.9447 −0.765641
\(382\) 23.2019 1.18711
\(383\) 10.1177 0.516989 0.258495 0.966013i \(-0.416774\pi\)
0.258495 + 0.966013i \(0.416774\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.29881 −0.219088
\(386\) 19.7378 1.00463
\(387\) 1.99798 0.101563
\(388\) 16.4339 0.834304
\(389\) 12.2480 0.621000 0.310500 0.950573i \(-0.399504\pi\)
0.310500 + 0.950573i \(0.399504\pi\)
\(390\) 16.1351 0.817030
\(391\) −0.562345 −0.0284390
\(392\) 5.72286 0.289048
\(393\) 5.68475 0.286758
\(394\) −4.50236 −0.226826
\(395\) 15.9152 0.800779
\(396\) −1.00000 −0.0502519
\(397\) −21.9214 −1.10020 −0.550101 0.835098i \(-0.685411\pi\)
−0.550101 + 0.835098i \(0.685411\pi\)
\(398\) −7.90054 −0.396018
\(399\) −0.299520 −0.0149948
\(400\) 9.46966 0.473483
\(401\) −21.0450 −1.05094 −0.525468 0.850813i \(-0.676110\pi\)
−0.525468 + 0.850813i \(0.676110\pi\)
\(402\) 13.4773 0.672189
\(403\) −33.3240 −1.65999
\(404\) −5.19702 −0.258562
\(405\) −3.80390 −0.189017
\(406\) −4.02030 −0.199524
\(407\) −11.0001 −0.545257
\(408\) −0.128672 −0.00637020
\(409\) 26.3348 1.30217 0.651086 0.759004i \(-0.274314\pi\)
0.651086 + 0.759004i \(0.274314\pi\)
\(410\) 7.98544 0.394373
\(411\) 13.2285 0.652515
\(412\) 16.9417 0.834660
\(413\) 5.57824 0.274487
\(414\) −4.37039 −0.214793
\(415\) 20.9417 1.02799
\(416\) 4.24171 0.207967
\(417\) 14.6777 0.718769
\(418\) −0.265037 −0.0129634
\(419\) 16.1013 0.786602 0.393301 0.919410i \(-0.371333\pi\)
0.393301 + 0.919410i \(0.371333\pi\)
\(420\) −4.29881 −0.209761
\(421\) −17.0787 −0.832363 −0.416181 0.909282i \(-0.636632\pi\)
−0.416181 + 0.909282i \(0.636632\pi\)
\(422\) −25.2862 −1.23091
\(423\) −0.338707 −0.0164685
\(424\) 2.22997 0.108297
\(425\) −1.21848 −0.0591049
\(426\) 2.75650 0.133553
\(427\) 1.13011 0.0546897
\(428\) −0.918730 −0.0444085
\(429\) −4.24171 −0.204792
\(430\) 7.60012 0.366510
\(431\) 25.0261 1.20546 0.602732 0.797944i \(-0.294079\pi\)
0.602732 + 0.797944i \(0.294079\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.76689 −0.229082 −0.114541 0.993419i \(-0.536540\pi\)
−0.114541 + 0.993419i \(0.536540\pi\)
\(434\) 8.87842 0.426178
\(435\) −13.5322 −0.648819
\(436\) −4.41651 −0.211512
\(437\) −1.15831 −0.0554097
\(438\) −11.1407 −0.532321
\(439\) −19.2883 −0.920582 −0.460291 0.887768i \(-0.652255\pi\)
−0.460291 + 0.887768i \(0.652255\pi\)
\(440\) −3.80390 −0.181344
\(441\) −5.72286 −0.272517
\(442\) −0.545789 −0.0259605
\(443\) −21.1686 −1.00575 −0.502876 0.864359i \(-0.667725\pi\)
−0.502876 + 0.864359i \(0.667725\pi\)
\(444\) −11.0001 −0.522044
\(445\) 1.86184 0.0882595
\(446\) 0.909127 0.0430484
\(447\) −7.90661 −0.373970
\(448\) −1.13011 −0.0533925
\(449\) 4.72877 0.223164 0.111582 0.993755i \(-0.464408\pi\)
0.111582 + 0.993755i \(0.464408\pi\)
\(450\) −9.46966 −0.446404
\(451\) −2.09928 −0.0988510
\(452\) 9.04088 0.425247
\(453\) −1.15571 −0.0543001
\(454\) 6.58569 0.309082
\(455\) −18.2343 −0.854839
\(456\) −0.265037 −0.0124115
\(457\) 11.3987 0.533210 0.266605 0.963806i \(-0.414098\pi\)
0.266605 + 0.963806i \(0.414098\pi\)
\(458\) 24.9926 1.16783
\(459\) 0.128672 0.00600588
\(460\) −16.6245 −0.775122
\(461\) −2.01992 −0.0940772 −0.0470386 0.998893i \(-0.514978\pi\)
−0.0470386 + 0.998893i \(0.514978\pi\)
\(462\) 1.13011 0.0525773
\(463\) 1.91078 0.0888014 0.0444007 0.999014i \(-0.485862\pi\)
0.0444007 + 0.999014i \(0.485862\pi\)
\(464\) −3.55745 −0.165151
\(465\) 29.8845 1.38586
\(466\) 0.0101462 0.000470014 0
\(467\) 15.1715 0.702052 0.351026 0.936366i \(-0.385833\pi\)
0.351026 + 0.936366i \(0.385833\pi\)
\(468\) −4.24171 −0.196073
\(469\) −15.2308 −0.703294
\(470\) −1.28841 −0.0594299
\(471\) −9.07230 −0.418030
\(472\) 4.93603 0.227199
\(473\) −1.99798 −0.0918673
\(474\) −4.18391 −0.192173
\(475\) −2.50981 −0.115158
\(476\) 0.145413 0.00666499
\(477\) −2.22997 −0.102103
\(478\) 1.36601 0.0624798
\(479\) 32.5688 1.48810 0.744052 0.668121i \(-0.232901\pi\)
0.744052 + 0.668121i \(0.232901\pi\)
\(480\) −3.80390 −0.173624
\(481\) −46.6594 −2.12749
\(482\) −23.3974 −1.06572
\(483\) 4.93900 0.224732
\(484\) 1.00000 0.0454545
\(485\) −62.5129 −2.83856
\(486\) 1.00000 0.0453609
\(487\) 1.71068 0.0775184 0.0387592 0.999249i \(-0.487659\pi\)
0.0387592 + 0.999249i \(0.487659\pi\)
\(488\) 1.00000 0.0452679
\(489\) 8.13603 0.367924
\(490\) −21.7692 −0.983432
\(491\) −6.63332 −0.299357 −0.149679 0.988735i \(-0.547824\pi\)
−0.149679 + 0.988735i \(0.547824\pi\)
\(492\) −2.09928 −0.0946426
\(493\) 0.457744 0.0206158
\(494\) −1.12421 −0.0505806
\(495\) 3.80390 0.170973
\(496\) 7.85627 0.352757
\(497\) −3.11514 −0.139733
\(498\) −5.50532 −0.246699
\(499\) −14.8590 −0.665180 −0.332590 0.943072i \(-0.607922\pi\)
−0.332590 + 0.943072i \(0.607922\pi\)
\(500\) −17.0021 −0.760359
\(501\) 3.11680 0.139249
\(502\) 13.0600 0.582898
\(503\) 3.36636 0.150099 0.0750493 0.997180i \(-0.476089\pi\)
0.0750493 + 0.997180i \(0.476089\pi\)
\(504\) 1.13011 0.0503389
\(505\) 19.7690 0.879707
\(506\) 4.37039 0.194287
\(507\) −4.99214 −0.221709
\(508\) 14.9447 0.663065
\(509\) 11.0387 0.489281 0.244641 0.969614i \(-0.421330\pi\)
0.244641 + 0.969614i \(0.421330\pi\)
\(510\) 0.489455 0.0216734
\(511\) 12.5901 0.556954
\(512\) −1.00000 −0.0441942
\(513\) 0.265037 0.0117017
\(514\) −16.2686 −0.717578
\(515\) −64.4447 −2.83977
\(516\) −1.99798 −0.0879562
\(517\) 0.338707 0.0148963
\(518\) 12.4313 0.546201
\(519\) 7.76547 0.340866
\(520\) −16.1351 −0.707569
\(521\) 34.7021 1.52033 0.760163 0.649732i \(-0.225119\pi\)
0.760163 + 0.649732i \(0.225119\pi\)
\(522\) 3.55745 0.155706
\(523\) −22.1747 −0.969633 −0.484817 0.874616i \(-0.661114\pi\)
−0.484817 + 0.874616i \(0.661114\pi\)
\(524\) −5.68475 −0.248340
\(525\) 10.7017 0.467062
\(526\) −6.16069 −0.268619
\(527\) −1.01088 −0.0440346
\(528\) 1.00000 0.0435194
\(529\) −3.89973 −0.169553
\(530\) −8.48257 −0.368459
\(531\) −4.93603 −0.214205
\(532\) 0.299520 0.0129858
\(533\) −8.90453 −0.385698
\(534\) −0.489455 −0.0211808
\(535\) 3.49476 0.151092
\(536\) −13.4773 −0.582132
\(537\) 3.25320 0.140386
\(538\) 21.6824 0.934795
\(539\) 5.72286 0.246501
\(540\) 3.80390 0.163694
\(541\) 19.2666 0.828336 0.414168 0.910200i \(-0.364073\pi\)
0.414168 + 0.910200i \(0.364073\pi\)
\(542\) 19.0960 0.820242
\(543\) 2.50450 0.107478
\(544\) 0.128672 0.00551676
\(545\) 16.8000 0.719631
\(546\) 4.79359 0.205147
\(547\) −34.1442 −1.45990 −0.729951 0.683499i \(-0.760457\pi\)
−0.729951 + 0.683499i \(0.760457\pi\)
\(548\) −13.2285 −0.565095
\(549\) −1.00000 −0.0426790
\(550\) 9.46966 0.403788
\(551\) 0.942857 0.0401671
\(552\) 4.37039 0.186016
\(553\) 4.72826 0.201066
\(554\) 21.4653 0.911974
\(555\) 41.8434 1.77616
\(556\) −14.6777 −0.622472
\(557\) −21.0969 −0.893903 −0.446952 0.894558i \(-0.647490\pi\)
−0.446952 + 0.894558i \(0.647490\pi\)
\(558\) −7.85627 −0.332582
\(559\) −8.47486 −0.358448
\(560\) 4.29881 0.181658
\(561\) −0.128672 −0.00543253
\(562\) 20.1406 0.849578
\(563\) −15.6125 −0.657990 −0.328995 0.944332i \(-0.606710\pi\)
−0.328995 + 0.944332i \(0.606710\pi\)
\(564\) 0.338707 0.0142621
\(565\) −34.3906 −1.44682
\(566\) −6.60060 −0.277444
\(567\) −1.13011 −0.0474600
\(568\) −2.75650 −0.115660
\(569\) 1.28802 0.0539966 0.0269983 0.999635i \(-0.491405\pi\)
0.0269983 + 0.999635i \(0.491405\pi\)
\(570\) 1.00817 0.0422278
\(571\) −15.7671 −0.659834 −0.329917 0.944010i \(-0.607021\pi\)
−0.329917 + 0.944010i \(0.607021\pi\)
\(572\) 4.24171 0.177355
\(573\) 23.2019 0.969273
\(574\) 2.37240 0.0990223
\(575\) 41.3861 1.72592
\(576\) 1.00000 0.0416667
\(577\) 1.85705 0.0773099 0.0386549 0.999253i \(-0.487693\pi\)
0.0386549 + 0.999253i \(0.487693\pi\)
\(578\) 16.9834 0.706418
\(579\) 19.7378 0.820276
\(580\) 13.5322 0.561894
\(581\) 6.22159 0.258115
\(582\) 16.4339 0.681206
\(583\) 2.22997 0.0923558
\(584\) 11.1407 0.461003
\(585\) 16.1351 0.667103
\(586\) −27.0042 −1.11553
\(587\) −31.2101 −1.28818 −0.644088 0.764951i \(-0.722763\pi\)
−0.644088 + 0.764951i \(0.722763\pi\)
\(588\) 5.72286 0.236007
\(589\) −2.08220 −0.0857956
\(590\) −18.7762 −0.773002
\(591\) −4.50236 −0.185202
\(592\) 11.0001 0.452103
\(593\) −39.1914 −1.60940 −0.804699 0.593683i \(-0.797673\pi\)
−0.804699 + 0.593683i \(0.797673\pi\)
\(594\) −1.00000 −0.0410305
\(595\) −0.553136 −0.0226764
\(596\) 7.90661 0.323868
\(597\) −7.90054 −0.323348
\(598\) 18.5379 0.758072
\(599\) −44.7511 −1.82848 −0.914241 0.405172i \(-0.867212\pi\)
−0.914241 + 0.405172i \(0.867212\pi\)
\(600\) 9.46966 0.386597
\(601\) 26.7372 1.09063 0.545316 0.838231i \(-0.316410\pi\)
0.545316 + 0.838231i \(0.316410\pi\)
\(602\) 2.25793 0.0920264
\(603\) 13.4773 0.548840
\(604\) 1.15571 0.0470252
\(605\) −3.80390 −0.154651
\(606\) −5.19702 −0.211115
\(607\) −13.2882 −0.539350 −0.269675 0.962951i \(-0.586916\pi\)
−0.269675 + 0.962951i \(0.586916\pi\)
\(608\) 0.265037 0.0107487
\(609\) −4.02030 −0.162911
\(610\) −3.80390 −0.154015
\(611\) 1.43670 0.0581226
\(612\) −0.128672 −0.00520125
\(613\) 29.6723 1.19845 0.599227 0.800579i \(-0.295475\pi\)
0.599227 + 0.800579i \(0.295475\pi\)
\(614\) 21.1225 0.852435
\(615\) 7.98544 0.322004
\(616\) −1.13011 −0.0455333
\(617\) 13.6157 0.548146 0.274073 0.961709i \(-0.411629\pi\)
0.274073 + 0.961709i \(0.411629\pi\)
\(618\) 16.9417 0.681497
\(619\) −14.7681 −0.593580 −0.296790 0.954943i \(-0.595916\pi\)
−0.296790 + 0.954943i \(0.595916\pi\)
\(620\) −29.8845 −1.20019
\(621\) −4.37039 −0.175378
\(622\) 25.0386 1.00395
\(623\) 0.553136 0.0221609
\(624\) 4.24171 0.169804
\(625\) 17.3262 0.693047
\(626\) 24.5109 0.979652
\(627\) −0.265037 −0.0105846
\(628\) 9.07230 0.362024
\(629\) −1.41541 −0.0564360
\(630\) −4.29881 −0.171269
\(631\) −19.9567 −0.794465 −0.397232 0.917718i \(-0.630029\pi\)
−0.397232 + 0.917718i \(0.630029\pi\)
\(632\) 4.18391 0.166427
\(633\) −25.2862 −1.00504
\(634\) −29.1713 −1.15854
\(635\) −56.8482 −2.25595
\(636\) 2.22997 0.0884239
\(637\) 24.2747 0.961800
\(638\) −3.55745 −0.140841
\(639\) 2.75650 0.109045
\(640\) 3.80390 0.150362
\(641\) 31.5939 1.24789 0.623943 0.781470i \(-0.285530\pi\)
0.623943 + 0.781470i \(0.285530\pi\)
\(642\) −0.918730 −0.0362594
\(643\) 11.2434 0.443398 0.221699 0.975115i \(-0.428840\pi\)
0.221699 + 0.975115i \(0.428840\pi\)
\(644\) −4.93900 −0.194624
\(645\) 7.60012 0.299254
\(646\) −0.0341028 −0.00134176
\(647\) 0.806620 0.0317115 0.0158558 0.999874i \(-0.494953\pi\)
0.0158558 + 0.999874i \(0.494953\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 4.93603 0.193756
\(650\) 40.1676 1.57550
\(651\) 8.87842 0.347973
\(652\) −8.13603 −0.318631
\(653\) 31.5647 1.23522 0.617611 0.786484i \(-0.288101\pi\)
0.617611 + 0.786484i \(0.288101\pi\)
\(654\) −4.41651 −0.172699
\(655\) 21.6242 0.844929
\(656\) 2.09928 0.0819629
\(657\) −11.1407 −0.434638
\(658\) −0.382775 −0.0149221
\(659\) −27.7659 −1.08161 −0.540804 0.841149i \(-0.681880\pi\)
−0.540804 + 0.841149i \(0.681880\pi\)
\(660\) −3.80390 −0.148067
\(661\) 30.8385 1.19948 0.599740 0.800195i \(-0.295271\pi\)
0.599740 + 0.800195i \(0.295271\pi\)
\(662\) −11.8658 −0.461178
\(663\) −0.545789 −0.0211967
\(664\) 5.50532 0.213648
\(665\) −1.13934 −0.0441819
\(666\) −11.0001 −0.426247
\(667\) −15.5474 −0.602000
\(668\) −3.11680 −0.120593
\(669\) 0.909127 0.0351489
\(670\) 51.2665 1.98060
\(671\) 1.00000 0.0386046
\(672\) −1.13011 −0.0435948
\(673\) 34.6630 1.33616 0.668080 0.744089i \(-0.267116\pi\)
0.668080 + 0.744089i \(0.267116\pi\)
\(674\) −18.0097 −0.693708
\(675\) −9.46966 −0.364487
\(676\) 4.99214 0.192005
\(677\) −29.8345 −1.14663 −0.573317 0.819334i \(-0.694344\pi\)
−0.573317 + 0.819334i \(0.694344\pi\)
\(678\) 9.04088 0.347213
\(679\) −18.5720 −0.712729
\(680\) −0.489455 −0.0187697
\(681\) 6.58569 0.252364
\(682\) 7.85627 0.300832
\(683\) 22.5557 0.863068 0.431534 0.902097i \(-0.357972\pi\)
0.431534 + 0.902097i \(0.357972\pi\)
\(684\) −0.265037 −0.0101339
\(685\) 50.3200 1.92263
\(686\) −14.3782 −0.548962
\(687\) 24.9926 0.953528
\(688\) 1.99798 0.0761723
\(689\) 9.45888 0.360355
\(690\) −16.6245 −0.632884
\(691\) −11.0236 −0.419356 −0.209678 0.977771i \(-0.567242\pi\)
−0.209678 + 0.977771i \(0.567242\pi\)
\(692\) −7.76547 −0.295199
\(693\) 1.13011 0.0429292
\(694\) 12.1166 0.459941
\(695\) 55.8324 2.11784
\(696\) −3.55745 −0.134845
\(697\) −0.270118 −0.0102314
\(698\) 21.1577 0.800832
\(699\) 0.0101462 0.000383765 0
\(700\) −10.7017 −0.404487
\(701\) −1.35614 −0.0512207 −0.0256103 0.999672i \(-0.508153\pi\)
−0.0256103 + 0.999672i \(0.508153\pi\)
\(702\) −4.24171 −0.160093
\(703\) −2.91544 −0.109958
\(704\) −1.00000 −0.0376889
\(705\) −1.28841 −0.0485243
\(706\) 2.82372 0.106272
\(707\) 5.87319 0.220884
\(708\) 4.93603 0.185507
\(709\) −27.9823 −1.05090 −0.525449 0.850825i \(-0.676103\pi\)
−0.525449 + 0.850825i \(0.676103\pi\)
\(710\) 10.4854 0.393512
\(711\) −4.18391 −0.156909
\(712\) 0.489455 0.0183431
\(713\) 34.3349 1.28585
\(714\) 0.145413 0.00544194
\(715\) −16.1351 −0.603417
\(716\) −3.25320 −0.121578
\(717\) 1.36601 0.0510145
\(718\) −18.0948 −0.675293
\(719\) −8.95333 −0.333903 −0.166951 0.985965i \(-0.553392\pi\)
−0.166951 + 0.985965i \(0.553392\pi\)
\(720\) −3.80390 −0.141763
\(721\) −19.1460 −0.713033
\(722\) 18.9298 0.704493
\(723\) −23.3974 −0.870157
\(724\) −2.50450 −0.0930789
\(725\) −33.6879 −1.25114
\(726\) 1.00000 0.0371135
\(727\) 21.7053 0.805006 0.402503 0.915419i \(-0.368140\pi\)
0.402503 + 0.915419i \(0.368140\pi\)
\(728\) −4.79359 −0.177662
\(729\) 1.00000 0.0370370
\(730\) −42.3779 −1.56848
\(731\) −0.257084 −0.00950859
\(732\) 1.00000 0.0369611
\(733\) −40.4633 −1.49455 −0.747274 0.664517i \(-0.768638\pi\)
−0.747274 + 0.664517i \(0.768638\pi\)
\(734\) 20.6125 0.760821
\(735\) −21.7692 −0.802969
\(736\) −4.37039 −0.161095
\(737\) −13.4773 −0.496444
\(738\) −2.09928 −0.0772754
\(739\) −22.9213 −0.843173 −0.421586 0.906788i \(-0.638527\pi\)
−0.421586 + 0.906788i \(0.638527\pi\)
\(740\) −41.8434 −1.53820
\(741\) −1.12421 −0.0412989
\(742\) −2.52010 −0.0925158
\(743\) 9.71416 0.356378 0.178189 0.983996i \(-0.442976\pi\)
0.178189 + 0.983996i \(0.442976\pi\)
\(744\) 7.85627 0.288025
\(745\) −30.0760 −1.10190
\(746\) −25.1657 −0.921383
\(747\) −5.50532 −0.201429
\(748\) 0.128672 0.00470471
\(749\) 1.03826 0.0379373
\(750\) −17.0021 −0.620831
\(751\) 29.3354 1.07046 0.535231 0.844706i \(-0.320224\pi\)
0.535231 + 0.844706i \(0.320224\pi\)
\(752\) −0.338707 −0.0123514
\(753\) 13.0600 0.475934
\(754\) −15.0897 −0.549535
\(755\) −4.39621 −0.159995
\(756\) 1.13011 0.0411016
\(757\) 37.8532 1.37580 0.687900 0.725806i \(-0.258533\pi\)
0.687900 + 0.725806i \(0.258533\pi\)
\(758\) −2.60869 −0.0947520
\(759\) 4.37039 0.158635
\(760\) −1.00817 −0.0365703
\(761\) 27.3474 0.991344 0.495672 0.868510i \(-0.334922\pi\)
0.495672 + 0.868510i \(0.334922\pi\)
\(762\) 14.9447 0.541390
\(763\) 4.99113 0.180691
\(764\) −23.2019 −0.839415
\(765\) 0.489455 0.0176963
\(766\) −10.1177 −0.365567
\(767\) 20.9372 0.755999
\(768\) −1.00000 −0.0360844
\(769\) −12.3128 −0.444011 −0.222005 0.975045i \(-0.571260\pi\)
−0.222005 + 0.975045i \(0.571260\pi\)
\(770\) 4.29881 0.154918
\(771\) −16.2686 −0.585900
\(772\) −19.7378 −0.710379
\(773\) 38.5443 1.38634 0.693171 0.720773i \(-0.256213\pi\)
0.693171 + 0.720773i \(0.256213\pi\)
\(774\) −1.99798 −0.0718159
\(775\) 74.3962 2.67239
\(776\) −16.4339 −0.589942
\(777\) 12.4313 0.445971
\(778\) −12.2480 −0.439113
\(779\) −0.556386 −0.0199346
\(780\) −16.1351 −0.577728
\(781\) −2.75650 −0.0986353
\(782\) 0.562345 0.0201094
\(783\) 3.55745 0.127133
\(784\) −5.72286 −0.204388
\(785\) −34.5101 −1.23172
\(786\) −5.68475 −0.202768
\(787\) 0.241105 0.00859445 0.00429723 0.999991i \(-0.498632\pi\)
0.00429723 + 0.999991i \(0.498632\pi\)
\(788\) 4.50236 0.160390
\(789\) −6.16069 −0.219326
\(790\) −15.9152 −0.566236
\(791\) −10.2172 −0.363280
\(792\) 1.00000 0.0355335
\(793\) 4.24171 0.150628
\(794\) 21.9214 0.777960
\(795\) −8.48257 −0.300846
\(796\) 7.90054 0.280027
\(797\) −8.85215 −0.313559 −0.156780 0.987634i \(-0.550111\pi\)
−0.156780 + 0.987634i \(0.550111\pi\)
\(798\) 0.299520 0.0106029
\(799\) 0.0435821 0.00154182
\(800\) −9.46966 −0.334803
\(801\) −0.489455 −0.0172940
\(802\) 21.0450 0.743124
\(803\) 11.1407 0.393145
\(804\) −13.4773 −0.475309
\(805\) 18.7875 0.662171
\(806\) 33.3240 1.17379
\(807\) 21.6824 0.763257
\(808\) 5.19702 0.182831
\(809\) −19.2003 −0.675048 −0.337524 0.941317i \(-0.609589\pi\)
−0.337524 + 0.941317i \(0.609589\pi\)
\(810\) 3.80390 0.133655
\(811\) −51.1276 −1.79533 −0.897667 0.440675i \(-0.854739\pi\)
−0.897667 + 0.440675i \(0.854739\pi\)
\(812\) 4.02030 0.141085
\(813\) 19.0960 0.669725
\(814\) 11.0001 0.385555
\(815\) 30.9486 1.08408
\(816\) 0.128672 0.00450441
\(817\) −0.529539 −0.0185262
\(818\) −26.3348 −0.920774
\(819\) 4.79359 0.167502
\(820\) −7.98544 −0.278864
\(821\) 16.4508 0.574136 0.287068 0.957910i \(-0.407319\pi\)
0.287068 + 0.957910i \(0.407319\pi\)
\(822\) −13.2285 −0.461398
\(823\) 17.6294 0.614523 0.307262 0.951625i \(-0.400587\pi\)
0.307262 + 0.951625i \(0.400587\pi\)
\(824\) −16.9417 −0.590194
\(825\) 9.46966 0.329691
\(826\) −5.57824 −0.194092
\(827\) −41.0075 −1.42597 −0.712986 0.701179i \(-0.752658\pi\)
−0.712986 + 0.701179i \(0.752658\pi\)
\(828\) 4.37039 0.151881
\(829\) 23.5411 0.817616 0.408808 0.912620i \(-0.365945\pi\)
0.408808 + 0.912620i \(0.365945\pi\)
\(830\) −20.9417 −0.726896
\(831\) 21.4653 0.744624
\(832\) −4.24171 −0.147055
\(833\) 0.736371 0.0255137
\(834\) −14.6777 −0.508246
\(835\) 11.8560 0.410294
\(836\) 0.265037 0.00916649
\(837\) −7.85627 −0.271552
\(838\) −16.1013 −0.556211
\(839\) 11.1533 0.385053 0.192527 0.981292i \(-0.438332\pi\)
0.192527 + 0.981292i \(0.438332\pi\)
\(840\) 4.29881 0.148323
\(841\) −16.3445 −0.563604
\(842\) 17.0787 0.588569
\(843\) 20.1406 0.693678
\(844\) 25.2862 0.870387
\(845\) −18.9896 −0.653262
\(846\) 0.338707 0.0116450
\(847\) −1.13011 −0.0388309
\(848\) −2.22997 −0.0765774
\(849\) −6.60060 −0.226532
\(850\) 1.21848 0.0417935
\(851\) 48.0749 1.64798
\(852\) −2.75650 −0.0944361
\(853\) −32.4544 −1.11122 −0.555608 0.831444i \(-0.687515\pi\)
−0.555608 + 0.831444i \(0.687515\pi\)
\(854\) −1.13011 −0.0386714
\(855\) 1.00817 0.0344788
\(856\) 0.918730 0.0314016
\(857\) −5.83593 −0.199352 −0.0996758 0.995020i \(-0.531781\pi\)
−0.0996758 + 0.995020i \(0.531781\pi\)
\(858\) 4.24171 0.144810
\(859\) −10.8604 −0.370551 −0.185276 0.982687i \(-0.559318\pi\)
−0.185276 + 0.982687i \(0.559318\pi\)
\(860\) −7.60012 −0.259162
\(861\) 2.37240 0.0808513
\(862\) −25.0261 −0.852392
\(863\) −19.5059 −0.663988 −0.331994 0.943281i \(-0.607721\pi\)
−0.331994 + 0.943281i \(0.607721\pi\)
\(864\) 1.00000 0.0340207
\(865\) 29.5391 1.00436
\(866\) 4.76689 0.161985
\(867\) 16.9834 0.576788
\(868\) −8.87842 −0.301353
\(869\) 4.18391 0.141929
\(870\) 13.5322 0.458785
\(871\) −57.1670 −1.93703
\(872\) 4.41651 0.149562
\(873\) 16.4339 0.556203
\(874\) 1.15831 0.0391805
\(875\) 19.2142 0.649560
\(876\) 11.1407 0.376408
\(877\) 34.3417 1.15964 0.579818 0.814746i \(-0.303124\pi\)
0.579818 + 0.814746i \(0.303124\pi\)
\(878\) 19.2883 0.650950
\(879\) −27.0042 −0.910828
\(880\) 3.80390 0.128229
\(881\) −17.8374 −0.600958 −0.300479 0.953788i \(-0.597146\pi\)
−0.300479 + 0.953788i \(0.597146\pi\)
\(882\) 5.72286 0.192699
\(883\) 34.2420 1.15234 0.576168 0.817332i \(-0.304548\pi\)
0.576168 + 0.817332i \(0.304548\pi\)
\(884\) 0.545789 0.0183569
\(885\) −18.7762 −0.631154
\(886\) 21.1686 0.711174
\(887\) 2.99807 0.100665 0.0503327 0.998733i \(-0.483972\pi\)
0.0503327 + 0.998733i \(0.483972\pi\)
\(888\) 11.0001 0.369141
\(889\) −16.8891 −0.566443
\(890\) −1.86184 −0.0624089
\(891\) −1.00000 −0.0335013
\(892\) −0.909127 −0.0304398
\(893\) 0.0897699 0.00300404
\(894\) 7.90661 0.264437
\(895\) 12.3749 0.413646
\(896\) 1.13011 0.0377542
\(897\) 18.5379 0.618963
\(898\) −4.72877 −0.157801
\(899\) −27.9483 −0.932128
\(900\) 9.46966 0.315655
\(901\) 0.286934 0.00955915
\(902\) 2.09928 0.0698982
\(903\) 2.25793 0.0751392
\(904\) −9.04088 −0.300695
\(905\) 9.52686 0.316683
\(906\) 1.15571 0.0383959
\(907\) −53.2408 −1.76783 −0.883916 0.467645i \(-0.845103\pi\)
−0.883916 + 0.467645i \(0.845103\pi\)
\(908\) −6.58569 −0.218554
\(909\) −5.19702 −0.172374
\(910\) 18.2343 0.604462
\(911\) −18.8795 −0.625507 −0.312753 0.949834i \(-0.601251\pi\)
−0.312753 + 0.949834i \(0.601251\pi\)
\(912\) 0.265037 0.00877625
\(913\) 5.50532 0.182199
\(914\) −11.3987 −0.377036
\(915\) −3.80390 −0.125753
\(916\) −24.9926 −0.825780
\(917\) 6.42438 0.212152
\(918\) −0.128672 −0.00424680
\(919\) 27.9575 0.922231 0.461116 0.887340i \(-0.347449\pi\)
0.461116 + 0.887340i \(0.347449\pi\)
\(920\) 16.6245 0.548094
\(921\) 21.1225 0.696010
\(922\) 2.01992 0.0665227
\(923\) −11.6923 −0.384856
\(924\) −1.13011 −0.0371778
\(925\) 104.168 3.42501
\(926\) −1.91078 −0.0627921
\(927\) 16.9417 0.556440
\(928\) 3.55745 0.116779
\(929\) 7.92958 0.260161 0.130081 0.991503i \(-0.458476\pi\)
0.130081 + 0.991503i \(0.458476\pi\)
\(930\) −29.8845 −0.979950
\(931\) 1.51677 0.0497101
\(932\) −0.0101462 −0.000332350 0
\(933\) 25.0386 0.819726
\(934\) −15.1715 −0.496426
\(935\) −0.489455 −0.0160069
\(936\) 4.24171 0.138645
\(937\) 19.0547 0.622490 0.311245 0.950330i \(-0.399254\pi\)
0.311245 + 0.950330i \(0.399254\pi\)
\(938\) 15.2308 0.497304
\(939\) 24.5109 0.799882
\(940\) 1.28841 0.0420233
\(941\) 33.9522 1.10681 0.553405 0.832912i \(-0.313328\pi\)
0.553405 + 0.832912i \(0.313328\pi\)
\(942\) 9.07230 0.295592
\(943\) 9.17464 0.298767
\(944\) −4.93603 −0.160654
\(945\) −4.29881 −0.139840
\(946\) 1.99798 0.0649600
\(947\) 41.8174 1.35888 0.679441 0.733730i \(-0.262222\pi\)
0.679441 + 0.733730i \(0.262222\pi\)
\(948\) 4.18391 0.135887
\(949\) 47.2555 1.53398
\(950\) 2.50981 0.0814290
\(951\) −29.1713 −0.945943
\(952\) −0.145413 −0.00471286
\(953\) −16.0917 −0.521262 −0.260631 0.965439i \(-0.583931\pi\)
−0.260631 + 0.965439i \(0.583931\pi\)
\(954\) 2.22997 0.0721978
\(955\) 88.2577 2.85595
\(956\) −1.36601 −0.0441799
\(957\) −3.55745 −0.114996
\(958\) −32.5688 −1.05225
\(959\) 14.9496 0.482749
\(960\) 3.80390 0.122770
\(961\) 30.7209 0.990998
\(962\) 46.6594 1.50436
\(963\) −0.918730 −0.0296057
\(964\) 23.3974 0.753578
\(965\) 75.0807 2.41693
\(966\) −4.93900 −0.158910
\(967\) 33.6948 1.08355 0.541777 0.840523i \(-0.317752\pi\)
0.541777 + 0.840523i \(0.317752\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −0.0341028 −0.00109554
\(970\) 62.5129 2.00717
\(971\) −4.90939 −0.157550 −0.0787749 0.996892i \(-0.525101\pi\)
−0.0787749 + 0.996892i \(0.525101\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 16.5873 0.531766
\(974\) −1.71068 −0.0548138
\(975\) 40.1676 1.28639
\(976\) −1.00000 −0.0320092
\(977\) 9.45469 0.302482 0.151241 0.988497i \(-0.451673\pi\)
0.151241 + 0.988497i \(0.451673\pi\)
\(978\) −8.13603 −0.260161
\(979\) 0.489455 0.0156430
\(980\) 21.7692 0.695391
\(981\) −4.41651 −0.141008
\(982\) 6.63332 0.211678
\(983\) 8.19968 0.261529 0.130765 0.991413i \(-0.458257\pi\)
0.130765 + 0.991413i \(0.458257\pi\)
\(984\) 2.09928 0.0669225
\(985\) −17.1265 −0.545696
\(986\) −0.457744 −0.0145775
\(987\) −0.382775 −0.0121839
\(988\) 1.12421 0.0357659
\(989\) 8.73195 0.277660
\(990\) −3.80390 −0.120896
\(991\) −0.622074 −0.0197608 −0.00988042 0.999951i \(-0.503145\pi\)
−0.00988042 + 0.999951i \(0.503145\pi\)
\(992\) −7.85627 −0.249437
\(993\) −11.8658 −0.376550
\(994\) 3.11514 0.0988061
\(995\) −30.0529 −0.952740
\(996\) 5.50532 0.174443
\(997\) 11.0385 0.349593 0.174796 0.984605i \(-0.444073\pi\)
0.174796 + 0.984605i \(0.444073\pi\)
\(998\) 14.8590 0.470353
\(999\) −11.0001 −0.348029
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.y.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4026.2.a.y.1.1 7 1.1 even 1 trivial