Properties

Label 5265.2.a.bi.1.3
Level $5265$
Weight $2$
Character 5265.1
Self dual yes
Analytic conductor $42.041$
Analytic rank $1$
Dimension $14$
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] = [5265,2,Mod(1,5265)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5265, 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("5265.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5265.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: \(42.0412366642\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 20 x^{12} + 36 x^{11} + 156 x^{10} - 242 x^{9} - 601 x^{8} + 750 x^{7} + 1188 x^{6} - 1038 x^{5} - 1133 x^{4} + 438 x^{3} + 423 x^{2} + 90 x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.25591\) of defining polynomial
Character \(\chi\) \(=\) 5265.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.25591 q^{2} +3.08913 q^{4} -1.00000 q^{5} +4.21559 q^{7} -2.45697 q^{8} +O(q^{10})\) \(q-2.25591 q^{2} +3.08913 q^{4} -1.00000 q^{5} +4.21559 q^{7} -2.45697 q^{8} +2.25591 q^{10} -4.32629 q^{11} +1.00000 q^{13} -9.50998 q^{14} -0.635548 q^{16} -4.03439 q^{17} +1.09815 q^{19} -3.08913 q^{20} +9.75971 q^{22} +2.39337 q^{23} +1.00000 q^{25} -2.25591 q^{26} +13.0225 q^{28} +2.95802 q^{29} -9.70367 q^{31} +6.34768 q^{32} +9.10122 q^{34} -4.21559 q^{35} +6.76843 q^{37} -2.47733 q^{38} +2.45697 q^{40} +8.83954 q^{41} +0.799491 q^{43} -13.3645 q^{44} -5.39923 q^{46} -12.7882 q^{47} +10.7712 q^{49} -2.25591 q^{50} +3.08913 q^{52} +5.52360 q^{53} +4.32629 q^{55} -10.3576 q^{56} -6.67302 q^{58} +2.90495 q^{59} -13.0980 q^{61} +21.8906 q^{62} -13.0487 q^{64} -1.00000 q^{65} -9.66612 q^{67} -12.4627 q^{68} +9.50998 q^{70} -0.931785 q^{71} +13.6631 q^{73} -15.2690 q^{74} +3.39233 q^{76} -18.2378 q^{77} -11.9743 q^{79} +0.635548 q^{80} -19.9412 q^{82} -7.76044 q^{83} +4.03439 q^{85} -1.80358 q^{86} +10.6296 q^{88} +10.2430 q^{89} +4.21559 q^{91} +7.39343 q^{92} +28.8490 q^{94} -1.09815 q^{95} -3.96362 q^{97} -24.2988 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{2} + 16 q^{4} - 14 q^{5} + 4 q^{7} - 12 q^{8} + 2 q^{10} - 8 q^{11} + 14 q^{13} - 12 q^{14} + 16 q^{16} - 10 q^{17} + 10 q^{19} - 16 q^{20} - 10 q^{22} - 34 q^{23} + 14 q^{25} - 2 q^{26} + 2 q^{28} - 16 q^{29} + 6 q^{31} - 26 q^{32} + 8 q^{34} - 4 q^{35} + 4 q^{37} - 12 q^{38} + 12 q^{40} - 10 q^{41} - 8 q^{43} - 8 q^{44} - 16 q^{46} - 46 q^{47} + 2 q^{49} - 2 q^{50} + 16 q^{52} - 14 q^{53} + 8 q^{55} - 20 q^{56} - 28 q^{58} - 16 q^{59} - 30 q^{62} + 14 q^{64} - 14 q^{65} + 6 q^{67} - 4 q^{68} + 12 q^{70} - 34 q^{71} - 4 q^{73} - 8 q^{74} + 50 q^{76} - 24 q^{77} - 14 q^{79} - 16 q^{80} - 16 q^{82} - 4 q^{83} + 10 q^{85} - 6 q^{86} - 68 q^{88} - 18 q^{89} + 4 q^{91} - 90 q^{92} - 10 q^{95} + 12 q^{97} + 16 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.25591 −1.59517 −0.797584 0.603207i \(-0.793889\pi\)
−0.797584 + 0.603207i \(0.793889\pi\)
\(3\) 0 0
\(4\) 3.08913 1.54456
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.21559 1.59334 0.796671 0.604413i \(-0.206592\pi\)
0.796671 + 0.604413i \(0.206592\pi\)
\(8\) −2.45697 −0.868671
\(9\) 0 0
\(10\) 2.25591 0.713381
\(11\) −4.32629 −1.30442 −0.652212 0.758036i \(-0.726159\pi\)
−0.652212 + 0.758036i \(0.726159\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −9.50998 −2.54165
\(15\) 0 0
\(16\) −0.635548 −0.158887
\(17\) −4.03439 −0.978483 −0.489241 0.872148i \(-0.662726\pi\)
−0.489241 + 0.872148i \(0.662726\pi\)
\(18\) 0 0
\(19\) 1.09815 0.251933 0.125967 0.992034i \(-0.459797\pi\)
0.125967 + 0.992034i \(0.459797\pi\)
\(20\) −3.08913 −0.690750
\(21\) 0 0
\(22\) 9.75971 2.08078
\(23\) 2.39337 0.499052 0.249526 0.968368i \(-0.419725\pi\)
0.249526 + 0.968368i \(0.419725\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −2.25591 −0.442420
\(27\) 0 0
\(28\) 13.0225 2.46102
\(29\) 2.95802 0.549290 0.274645 0.961546i \(-0.411440\pi\)
0.274645 + 0.961546i \(0.411440\pi\)
\(30\) 0 0
\(31\) −9.70367 −1.74283 −0.871415 0.490546i \(-0.836797\pi\)
−0.871415 + 0.490546i \(0.836797\pi\)
\(32\) 6.34768 1.12212
\(33\) 0 0
\(34\) 9.10122 1.56085
\(35\) −4.21559 −0.712564
\(36\) 0 0
\(37\) 6.76843 1.11272 0.556361 0.830940i \(-0.312197\pi\)
0.556361 + 0.830940i \(0.312197\pi\)
\(38\) −2.47733 −0.401876
\(39\) 0 0
\(40\) 2.45697 0.388481
\(41\) 8.83954 1.38050 0.690252 0.723569i \(-0.257500\pi\)
0.690252 + 0.723569i \(0.257500\pi\)
\(42\) 0 0
\(43\) 0.799491 0.121921 0.0609606 0.998140i \(-0.480584\pi\)
0.0609606 + 0.998140i \(0.480584\pi\)
\(44\) −13.3645 −2.01477
\(45\) 0 0
\(46\) −5.39923 −0.796073
\(47\) −12.7882 −1.86535 −0.932675 0.360719i \(-0.882531\pi\)
−0.932675 + 0.360719i \(0.882531\pi\)
\(48\) 0 0
\(49\) 10.7712 1.53874
\(50\) −2.25591 −0.319034
\(51\) 0 0
\(52\) 3.08913 0.428385
\(53\) 5.52360 0.758724 0.379362 0.925248i \(-0.376143\pi\)
0.379362 + 0.925248i \(0.376143\pi\)
\(54\) 0 0
\(55\) 4.32629 0.583357
\(56\) −10.3576 −1.38409
\(57\) 0 0
\(58\) −6.67302 −0.876210
\(59\) 2.90495 0.378192 0.189096 0.981959i \(-0.439444\pi\)
0.189096 + 0.981959i \(0.439444\pi\)
\(60\) 0 0
\(61\) −13.0980 −1.67703 −0.838514 0.544880i \(-0.816575\pi\)
−0.838514 + 0.544880i \(0.816575\pi\)
\(62\) 21.8906 2.78011
\(63\) 0 0
\(64\) −13.0487 −1.63109
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −9.66612 −1.18090 −0.590452 0.807073i \(-0.701051\pi\)
−0.590452 + 0.807073i \(0.701051\pi\)
\(68\) −12.4627 −1.51133
\(69\) 0 0
\(70\) 9.50998 1.13666
\(71\) −0.931785 −0.110583 −0.0552913 0.998470i \(-0.517609\pi\)
−0.0552913 + 0.998470i \(0.517609\pi\)
\(72\) 0 0
\(73\) 13.6631 1.59914 0.799571 0.600571i \(-0.205060\pi\)
0.799571 + 0.600571i \(0.205060\pi\)
\(74\) −15.2690 −1.77498
\(75\) 0 0
\(76\) 3.39233 0.389127
\(77\) −18.2378 −2.07840
\(78\) 0 0
\(79\) −11.9743 −1.34722 −0.673610 0.739087i \(-0.735257\pi\)
−0.673610 + 0.739087i \(0.735257\pi\)
\(80\) 0.635548 0.0710564
\(81\) 0 0
\(82\) −19.9412 −2.20214
\(83\) −7.76044 −0.851819 −0.425910 0.904766i \(-0.640046\pi\)
−0.425910 + 0.904766i \(0.640046\pi\)
\(84\) 0 0
\(85\) 4.03439 0.437591
\(86\) −1.80358 −0.194485
\(87\) 0 0
\(88\) 10.6296 1.13312
\(89\) 10.2430 1.08575 0.542875 0.839813i \(-0.317336\pi\)
0.542875 + 0.839813i \(0.317336\pi\)
\(90\) 0 0
\(91\) 4.21559 0.441914
\(92\) 7.39343 0.770818
\(93\) 0 0
\(94\) 28.8490 2.97555
\(95\) −1.09815 −0.112668
\(96\) 0 0
\(97\) −3.96362 −0.402445 −0.201222 0.979546i \(-0.564491\pi\)
−0.201222 + 0.979546i \(0.564491\pi\)
\(98\) −24.2988 −2.45455
\(99\) 0 0
\(100\) 3.08913 0.308913
\(101\) −4.32854 −0.430706 −0.215353 0.976536i \(-0.569090\pi\)
−0.215353 + 0.976536i \(0.569090\pi\)
\(102\) 0 0
\(103\) −16.8309 −1.65840 −0.829198 0.558955i \(-0.811202\pi\)
−0.829198 + 0.558955i \(0.811202\pi\)
\(104\) −2.45697 −0.240926
\(105\) 0 0
\(106\) −12.4607 −1.21029
\(107\) −16.1393 −1.56025 −0.780125 0.625624i \(-0.784844\pi\)
−0.780125 + 0.625624i \(0.784844\pi\)
\(108\) 0 0
\(109\) −6.91526 −0.662362 −0.331181 0.943567i \(-0.607447\pi\)
−0.331181 + 0.943567i \(0.607447\pi\)
\(110\) −9.75971 −0.930552
\(111\) 0 0
\(112\) −2.67921 −0.253161
\(113\) 12.2408 1.15152 0.575761 0.817618i \(-0.304706\pi\)
0.575761 + 0.817618i \(0.304706\pi\)
\(114\) 0 0
\(115\) −2.39337 −0.223183
\(116\) 9.13769 0.848413
\(117\) 0 0
\(118\) −6.55329 −0.603280
\(119\) −17.0073 −1.55906
\(120\) 0 0
\(121\) 7.71677 0.701524
\(122\) 29.5479 2.67514
\(123\) 0 0
\(124\) −29.9759 −2.69191
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.07671 0.184279 0.0921393 0.995746i \(-0.470630\pi\)
0.0921393 + 0.995746i \(0.470630\pi\)
\(128\) 16.7413 1.47974
\(129\) 0 0
\(130\) 2.25591 0.197856
\(131\) 18.7352 1.63690 0.818450 0.574577i \(-0.194834\pi\)
0.818450 + 0.574577i \(0.194834\pi\)
\(132\) 0 0
\(133\) 4.62936 0.401416
\(134\) 21.8059 1.88374
\(135\) 0 0
\(136\) 9.91238 0.849980
\(137\) 0.478085 0.0408456 0.0204228 0.999791i \(-0.493499\pi\)
0.0204228 + 0.999791i \(0.493499\pi\)
\(138\) 0 0
\(139\) 10.0222 0.850075 0.425038 0.905176i \(-0.360261\pi\)
0.425038 + 0.905176i \(0.360261\pi\)
\(140\) −13.0225 −1.10060
\(141\) 0 0
\(142\) 2.10202 0.176398
\(143\) −4.32629 −0.361782
\(144\) 0 0
\(145\) −2.95802 −0.245650
\(146\) −30.8227 −2.55090
\(147\) 0 0
\(148\) 20.9085 1.71867
\(149\) −12.3025 −1.00786 −0.503931 0.863744i \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(150\) 0 0
\(151\) 9.77828 0.795745 0.397872 0.917441i \(-0.369749\pi\)
0.397872 + 0.917441i \(0.369749\pi\)
\(152\) −2.69813 −0.218847
\(153\) 0 0
\(154\) 41.1429 3.31539
\(155\) 9.70367 0.779417
\(156\) 0 0
\(157\) −3.48276 −0.277954 −0.138977 0.990296i \(-0.544381\pi\)
−0.138977 + 0.990296i \(0.544381\pi\)
\(158\) 27.0130 2.14904
\(159\) 0 0
\(160\) −6.34768 −0.501828
\(161\) 10.0895 0.795161
\(162\) 0 0
\(163\) 6.28319 0.492137 0.246069 0.969252i \(-0.420861\pi\)
0.246069 + 0.969252i \(0.420861\pi\)
\(164\) 27.3064 2.13228
\(165\) 0 0
\(166\) 17.5069 1.35880
\(167\) −19.1729 −1.48364 −0.741820 0.670599i \(-0.766037\pi\)
−0.741820 + 0.670599i \(0.766037\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −9.10122 −0.698031
\(171\) 0 0
\(172\) 2.46973 0.188315
\(173\) −25.4541 −1.93524 −0.967620 0.252413i \(-0.918776\pi\)
−0.967620 + 0.252413i \(0.918776\pi\)
\(174\) 0 0
\(175\) 4.21559 0.318668
\(176\) 2.74956 0.207256
\(177\) 0 0
\(178\) −23.1072 −1.73196
\(179\) 0.0384041 0.00287046 0.00143523 0.999999i \(-0.499543\pi\)
0.00143523 + 0.999999i \(0.499543\pi\)
\(180\) 0 0
\(181\) 8.65492 0.643315 0.321657 0.946856i \(-0.395760\pi\)
0.321657 + 0.946856i \(0.395760\pi\)
\(182\) −9.50998 −0.704927
\(183\) 0 0
\(184\) −5.88044 −0.433512
\(185\) −6.76843 −0.497625
\(186\) 0 0
\(187\) 17.4539 1.27636
\(188\) −39.5043 −2.88115
\(189\) 0 0
\(190\) 2.47733 0.179725
\(191\) −0.842914 −0.0609911 −0.0304956 0.999535i \(-0.509709\pi\)
−0.0304956 + 0.999535i \(0.509709\pi\)
\(192\) 0 0
\(193\) 10.4549 0.752560 0.376280 0.926506i \(-0.377203\pi\)
0.376280 + 0.926506i \(0.377203\pi\)
\(194\) 8.94157 0.641967
\(195\) 0 0
\(196\) 33.2735 2.37668
\(197\) 6.60143 0.470333 0.235166 0.971955i \(-0.424436\pi\)
0.235166 + 0.971955i \(0.424436\pi\)
\(198\) 0 0
\(199\) −6.47208 −0.458794 −0.229397 0.973333i \(-0.573675\pi\)
−0.229397 + 0.973333i \(0.573675\pi\)
\(200\) −2.45697 −0.173734
\(201\) 0 0
\(202\) 9.76480 0.687049
\(203\) 12.4698 0.875207
\(204\) 0 0
\(205\) −8.83954 −0.617380
\(206\) 37.9689 2.64542
\(207\) 0 0
\(208\) −0.635548 −0.0440673
\(209\) −4.75092 −0.328628
\(210\) 0 0
\(211\) 15.1642 1.04395 0.521974 0.852961i \(-0.325196\pi\)
0.521974 + 0.852961i \(0.325196\pi\)
\(212\) 17.0631 1.17190
\(213\) 0 0
\(214\) 36.4089 2.48886
\(215\) −0.799491 −0.0545248
\(216\) 0 0
\(217\) −40.9067 −2.77692
\(218\) 15.6002 1.05658
\(219\) 0 0
\(220\) 13.3645 0.901031
\(221\) −4.03439 −0.271382
\(222\) 0 0
\(223\) −23.0022 −1.54034 −0.770171 0.637837i \(-0.779829\pi\)
−0.770171 + 0.637837i \(0.779829\pi\)
\(224\) 26.7592 1.78792
\(225\) 0 0
\(226\) −27.6142 −1.83687
\(227\) 16.5591 1.09906 0.549532 0.835473i \(-0.314806\pi\)
0.549532 + 0.835473i \(0.314806\pi\)
\(228\) 0 0
\(229\) 17.4916 1.15588 0.577938 0.816081i \(-0.303858\pi\)
0.577938 + 0.816081i \(0.303858\pi\)
\(230\) 5.39923 0.356014
\(231\) 0 0
\(232\) −7.26776 −0.477152
\(233\) −10.5425 −0.690660 −0.345330 0.938481i \(-0.612233\pi\)
−0.345330 + 0.938481i \(0.612233\pi\)
\(234\) 0 0
\(235\) 12.7882 0.834210
\(236\) 8.97375 0.584141
\(237\) 0 0
\(238\) 38.3670 2.48696
\(239\) −3.19160 −0.206447 −0.103224 0.994658i \(-0.532916\pi\)
−0.103224 + 0.994658i \(0.532916\pi\)
\(240\) 0 0
\(241\) 10.9488 0.705272 0.352636 0.935761i \(-0.385285\pi\)
0.352636 + 0.935761i \(0.385285\pi\)
\(242\) −17.4083 −1.11905
\(243\) 0 0
\(244\) −40.4614 −2.59028
\(245\) −10.7712 −0.688145
\(246\) 0 0
\(247\) 1.09815 0.0698737
\(248\) 23.8416 1.51395
\(249\) 0 0
\(250\) 2.25591 0.142676
\(251\) 1.57317 0.0992975 0.0496488 0.998767i \(-0.484190\pi\)
0.0496488 + 0.998767i \(0.484190\pi\)
\(252\) 0 0
\(253\) −10.3544 −0.650976
\(254\) −4.68488 −0.293955
\(255\) 0 0
\(256\) −11.6695 −0.729344
\(257\) −13.0843 −0.816178 −0.408089 0.912942i \(-0.633805\pi\)
−0.408089 + 0.912942i \(0.633805\pi\)
\(258\) 0 0
\(259\) 28.5329 1.77295
\(260\) −3.08913 −0.191580
\(261\) 0 0
\(262\) −42.2649 −2.61113
\(263\) −14.5499 −0.897182 −0.448591 0.893737i \(-0.648074\pi\)
−0.448591 + 0.893737i \(0.648074\pi\)
\(264\) 0 0
\(265\) −5.52360 −0.339312
\(266\) −10.4434 −0.640326
\(267\) 0 0
\(268\) −29.8599 −1.82398
\(269\) −12.2822 −0.748857 −0.374429 0.927256i \(-0.622161\pi\)
−0.374429 + 0.927256i \(0.622161\pi\)
\(270\) 0 0
\(271\) −30.7884 −1.87026 −0.935131 0.354303i \(-0.884718\pi\)
−0.935131 + 0.354303i \(0.884718\pi\)
\(272\) 2.56405 0.155468
\(273\) 0 0
\(274\) −1.07852 −0.0651556
\(275\) −4.32629 −0.260885
\(276\) 0 0
\(277\) 3.24859 0.195189 0.0975944 0.995226i \(-0.468885\pi\)
0.0975944 + 0.995226i \(0.468885\pi\)
\(278\) −22.6093 −1.35601
\(279\) 0 0
\(280\) 10.3576 0.618984
\(281\) −10.3407 −0.616871 −0.308436 0.951245i \(-0.599805\pi\)
−0.308436 + 0.951245i \(0.599805\pi\)
\(282\) 0 0
\(283\) −8.21779 −0.488497 −0.244248 0.969713i \(-0.578541\pi\)
−0.244248 + 0.969713i \(0.578541\pi\)
\(284\) −2.87840 −0.170802
\(285\) 0 0
\(286\) 9.75971 0.577104
\(287\) 37.2638 2.19961
\(288\) 0 0
\(289\) −0.723709 −0.0425711
\(290\) 6.67302 0.391853
\(291\) 0 0
\(292\) 42.2070 2.46998
\(293\) 3.66476 0.214098 0.107049 0.994254i \(-0.465860\pi\)
0.107049 + 0.994254i \(0.465860\pi\)
\(294\) 0 0
\(295\) −2.90495 −0.169132
\(296\) −16.6298 −0.966590
\(297\) 0 0
\(298\) 27.7534 1.60771
\(299\) 2.39337 0.138412
\(300\) 0 0
\(301\) 3.37032 0.194262
\(302\) −22.0589 −1.26935
\(303\) 0 0
\(304\) −0.697928 −0.0400289
\(305\) 13.0980 0.749990
\(306\) 0 0
\(307\) −7.85132 −0.448099 −0.224049 0.974578i \(-0.571928\pi\)
−0.224049 + 0.974578i \(0.571928\pi\)
\(308\) −56.3390 −3.21021
\(309\) 0 0
\(310\) −21.8906 −1.24330
\(311\) 4.60248 0.260983 0.130491 0.991449i \(-0.458345\pi\)
0.130491 + 0.991449i \(0.458345\pi\)
\(312\) 0 0
\(313\) −9.98950 −0.564640 −0.282320 0.959320i \(-0.591104\pi\)
−0.282320 + 0.959320i \(0.591104\pi\)
\(314\) 7.85679 0.443384
\(315\) 0 0
\(316\) −36.9903 −2.08087
\(317\) 27.2236 1.52903 0.764514 0.644607i \(-0.222979\pi\)
0.764514 + 0.644607i \(0.222979\pi\)
\(318\) 0 0
\(319\) −12.7972 −0.716507
\(320\) 13.0487 0.729445
\(321\) 0 0
\(322\) −22.7609 −1.26842
\(323\) −4.43037 −0.246513
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) −14.1743 −0.785042
\(327\) 0 0
\(328\) −21.7185 −1.19920
\(329\) −53.9097 −2.97214
\(330\) 0 0
\(331\) −29.3945 −1.61567 −0.807834 0.589411i \(-0.799360\pi\)
−0.807834 + 0.589411i \(0.799360\pi\)
\(332\) −23.9730 −1.31569
\(333\) 0 0
\(334\) 43.2522 2.36666
\(335\) 9.66612 0.528116
\(336\) 0 0
\(337\) −24.2391 −1.32039 −0.660194 0.751095i \(-0.729526\pi\)
−0.660194 + 0.751095i \(0.729526\pi\)
\(338\) −2.25591 −0.122705
\(339\) 0 0
\(340\) 12.4627 0.675887
\(341\) 41.9809 2.27339
\(342\) 0 0
\(343\) 15.8977 0.858395
\(344\) −1.96433 −0.105909
\(345\) 0 0
\(346\) 57.4221 3.08703
\(347\) −11.8083 −0.633903 −0.316951 0.948442i \(-0.602659\pi\)
−0.316951 + 0.948442i \(0.602659\pi\)
\(348\) 0 0
\(349\) 23.9748 1.28334 0.641670 0.766980i \(-0.278242\pi\)
0.641670 + 0.766980i \(0.278242\pi\)
\(350\) −9.50998 −0.508330
\(351\) 0 0
\(352\) −27.4619 −1.46372
\(353\) −14.4690 −0.770107 −0.385054 0.922894i \(-0.625817\pi\)
−0.385054 + 0.922894i \(0.625817\pi\)
\(354\) 0 0
\(355\) 0.931785 0.0494540
\(356\) 31.6418 1.67701
\(357\) 0 0
\(358\) −0.0866362 −0.00457887
\(359\) 23.2724 1.22827 0.614135 0.789201i \(-0.289505\pi\)
0.614135 + 0.789201i \(0.289505\pi\)
\(360\) 0 0
\(361\) −17.7941 −0.936530
\(362\) −19.5247 −1.02620
\(363\) 0 0
\(364\) 13.0225 0.682564
\(365\) −13.6631 −0.715158
\(366\) 0 0
\(367\) 32.6268 1.70311 0.851553 0.524268i \(-0.175661\pi\)
0.851553 + 0.524268i \(0.175661\pi\)
\(368\) −1.52110 −0.0792929
\(369\) 0 0
\(370\) 15.2690 0.793795
\(371\) 23.2852 1.20891
\(372\) 0 0
\(373\) −35.7273 −1.84989 −0.924945 0.380101i \(-0.875889\pi\)
−0.924945 + 0.380101i \(0.875889\pi\)
\(374\) −39.3745 −2.03601
\(375\) 0 0
\(376\) 31.4202 1.62037
\(377\) 2.95802 0.152346
\(378\) 0 0
\(379\) 5.26986 0.270695 0.135347 0.990798i \(-0.456785\pi\)
0.135347 + 0.990798i \(0.456785\pi\)
\(380\) −3.39233 −0.174023
\(381\) 0 0
\(382\) 1.90154 0.0972911
\(383\) −32.5075 −1.66105 −0.830527 0.556978i \(-0.811961\pi\)
−0.830527 + 0.556978i \(0.811961\pi\)
\(384\) 0 0
\(385\) 18.2378 0.929487
\(386\) −23.5853 −1.20046
\(387\) 0 0
\(388\) −12.2441 −0.621601
\(389\) 28.0499 1.42219 0.711093 0.703098i \(-0.248200\pi\)
0.711093 + 0.703098i \(0.248200\pi\)
\(390\) 0 0
\(391\) −9.65579 −0.488314
\(392\) −26.4645 −1.33666
\(393\) 0 0
\(394\) −14.8922 −0.750260
\(395\) 11.9743 0.602495
\(396\) 0 0
\(397\) 20.4381 1.02576 0.512880 0.858460i \(-0.328578\pi\)
0.512880 + 0.858460i \(0.328578\pi\)
\(398\) 14.6004 0.731854
\(399\) 0 0
\(400\) −0.635548 −0.0317774
\(401\) −15.4383 −0.770952 −0.385476 0.922718i \(-0.625963\pi\)
−0.385476 + 0.922718i \(0.625963\pi\)
\(402\) 0 0
\(403\) −9.70367 −0.483374
\(404\) −13.3714 −0.665253
\(405\) 0 0
\(406\) −28.1307 −1.39610
\(407\) −29.2822 −1.45146
\(408\) 0 0
\(409\) −9.01427 −0.445727 −0.222863 0.974850i \(-0.571540\pi\)
−0.222863 + 0.974850i \(0.571540\pi\)
\(410\) 19.9412 0.984825
\(411\) 0 0
\(412\) −51.9927 −2.56150
\(413\) 12.2461 0.602589
\(414\) 0 0
\(415\) 7.76044 0.380945
\(416\) 6.34768 0.311221
\(417\) 0 0
\(418\) 10.7176 0.524217
\(419\) 24.1705 1.18081 0.590403 0.807108i \(-0.298969\pi\)
0.590403 + 0.807108i \(0.298969\pi\)
\(420\) 0 0
\(421\) −14.6219 −0.712628 −0.356314 0.934366i \(-0.615967\pi\)
−0.356314 + 0.934366i \(0.615967\pi\)
\(422\) −34.2091 −1.66527
\(423\) 0 0
\(424\) −13.5713 −0.659082
\(425\) −4.03439 −0.195697
\(426\) 0 0
\(427\) −55.2158 −2.67208
\(428\) −49.8565 −2.40990
\(429\) 0 0
\(430\) 1.80358 0.0869763
\(431\) 19.1759 0.923672 0.461836 0.886965i \(-0.347191\pi\)
0.461836 + 0.886965i \(0.347191\pi\)
\(432\) 0 0
\(433\) −38.0916 −1.83056 −0.915282 0.402813i \(-0.868032\pi\)
−0.915282 + 0.402813i \(0.868032\pi\)
\(434\) 92.2817 4.42966
\(435\) 0 0
\(436\) −21.3621 −1.02306
\(437\) 2.62828 0.125728
\(438\) 0 0
\(439\) 8.16322 0.389609 0.194805 0.980842i \(-0.437593\pi\)
0.194805 + 0.980842i \(0.437593\pi\)
\(440\) −10.6296 −0.506745
\(441\) 0 0
\(442\) 9.10122 0.432901
\(443\) −14.7454 −0.700573 −0.350286 0.936643i \(-0.613916\pi\)
−0.350286 + 0.936643i \(0.613916\pi\)
\(444\) 0 0
\(445\) −10.2430 −0.485562
\(446\) 51.8909 2.45711
\(447\) 0 0
\(448\) −55.0079 −2.59888
\(449\) 27.8207 1.31294 0.656469 0.754353i \(-0.272049\pi\)
0.656469 + 0.754353i \(0.272049\pi\)
\(450\) 0 0
\(451\) −38.2424 −1.80076
\(452\) 37.8135 1.77860
\(453\) 0 0
\(454\) −37.3557 −1.75319
\(455\) −4.21559 −0.197630
\(456\) 0 0
\(457\) 28.2011 1.31919 0.659597 0.751620i \(-0.270727\pi\)
0.659597 + 0.751620i \(0.270727\pi\)
\(458\) −39.4594 −1.84382
\(459\) 0 0
\(460\) −7.39343 −0.344720
\(461\) −12.2631 −0.571151 −0.285576 0.958356i \(-0.592185\pi\)
−0.285576 + 0.958356i \(0.592185\pi\)
\(462\) 0 0
\(463\) 7.68122 0.356977 0.178488 0.983942i \(-0.442879\pi\)
0.178488 + 0.983942i \(0.442879\pi\)
\(464\) −1.87996 −0.0872750
\(465\) 0 0
\(466\) 23.7829 1.10172
\(467\) 31.6557 1.46485 0.732426 0.680847i \(-0.238388\pi\)
0.732426 + 0.680847i \(0.238388\pi\)
\(468\) 0 0
\(469\) −40.7484 −1.88158
\(470\) −28.8490 −1.33071
\(471\) 0 0
\(472\) −7.13737 −0.328524
\(473\) −3.45883 −0.159037
\(474\) 0 0
\(475\) 1.09815 0.0503867
\(476\) −52.5378 −2.40806
\(477\) 0 0
\(478\) 7.19995 0.329318
\(479\) −21.5067 −0.982665 −0.491332 0.870972i \(-0.663490\pi\)
−0.491332 + 0.870972i \(0.663490\pi\)
\(480\) 0 0
\(481\) 6.76843 0.308614
\(482\) −24.6994 −1.12503
\(483\) 0 0
\(484\) 23.8381 1.08355
\(485\) 3.96362 0.179979
\(486\) 0 0
\(487\) 28.5371 1.29314 0.646571 0.762854i \(-0.276203\pi\)
0.646571 + 0.762854i \(0.276203\pi\)
\(488\) 32.1814 1.45679
\(489\) 0 0
\(490\) 24.2988 1.09771
\(491\) 15.4443 0.696991 0.348495 0.937310i \(-0.386693\pi\)
0.348495 + 0.937310i \(0.386693\pi\)
\(492\) 0 0
\(493\) −11.9338 −0.537471
\(494\) −2.47733 −0.111460
\(495\) 0 0
\(496\) 6.16715 0.276913
\(497\) −3.92802 −0.176196
\(498\) 0 0
\(499\) 7.52244 0.336751 0.168375 0.985723i \(-0.446148\pi\)
0.168375 + 0.985723i \(0.446148\pi\)
\(500\) −3.08913 −0.138150
\(501\) 0 0
\(502\) −3.54893 −0.158396
\(503\) −26.3229 −1.17368 −0.586839 0.809704i \(-0.699628\pi\)
−0.586839 + 0.809704i \(0.699628\pi\)
\(504\) 0 0
\(505\) 4.32854 0.192618
\(506\) 23.3586 1.03842
\(507\) 0 0
\(508\) 6.41523 0.284630
\(509\) −4.38464 −0.194346 −0.0971729 0.995268i \(-0.530980\pi\)
−0.0971729 + 0.995268i \(0.530980\pi\)
\(510\) 0 0
\(511\) 57.5979 2.54798
\(512\) −7.15730 −0.316311
\(513\) 0 0
\(514\) 29.5170 1.30194
\(515\) 16.8309 0.741657
\(516\) 0 0
\(517\) 55.3254 2.43321
\(518\) −64.3676 −2.82815
\(519\) 0 0
\(520\) 2.45697 0.107745
\(521\) −26.4071 −1.15692 −0.578458 0.815712i \(-0.696345\pi\)
−0.578458 + 0.815712i \(0.696345\pi\)
\(522\) 0 0
\(523\) −18.4053 −0.804807 −0.402404 0.915462i \(-0.631825\pi\)
−0.402404 + 0.915462i \(0.631825\pi\)
\(524\) 57.8754 2.52830
\(525\) 0 0
\(526\) 32.8231 1.43116
\(527\) 39.1484 1.70533
\(528\) 0 0
\(529\) −17.2718 −0.750947
\(530\) 12.4607 0.541260
\(531\) 0 0
\(532\) 14.3007 0.620013
\(533\) 8.83954 0.382883
\(534\) 0 0
\(535\) 16.1393 0.697765
\(536\) 23.7494 1.02582
\(537\) 0 0
\(538\) 27.7075 1.19455
\(539\) −46.5992 −2.00717
\(540\) 0 0
\(541\) −2.49220 −0.107148 −0.0535739 0.998564i \(-0.517061\pi\)
−0.0535739 + 0.998564i \(0.517061\pi\)
\(542\) 69.4558 2.98338
\(543\) 0 0
\(544\) −25.6090 −1.09798
\(545\) 6.91526 0.296217
\(546\) 0 0
\(547\) −21.0530 −0.900161 −0.450081 0.892988i \(-0.648605\pi\)
−0.450081 + 0.892988i \(0.648605\pi\)
\(548\) 1.47687 0.0630886
\(549\) 0 0
\(550\) 9.75971 0.416156
\(551\) 3.24835 0.138384
\(552\) 0 0
\(553\) −50.4789 −2.14658
\(554\) −7.32852 −0.311359
\(555\) 0 0
\(556\) 30.9600 1.31300
\(557\) −22.8244 −0.967102 −0.483551 0.875316i \(-0.660653\pi\)
−0.483551 + 0.875316i \(0.660653\pi\)
\(558\) 0 0
\(559\) 0.799491 0.0338149
\(560\) 2.67921 0.113217
\(561\) 0 0
\(562\) 23.3276 0.984014
\(563\) −15.4201 −0.649878 −0.324939 0.945735i \(-0.605344\pi\)
−0.324939 + 0.945735i \(0.605344\pi\)
\(564\) 0 0
\(565\) −12.2408 −0.514976
\(566\) 18.5386 0.779235
\(567\) 0 0
\(568\) 2.28937 0.0960598
\(569\) −4.76808 −0.199888 −0.0999442 0.994993i \(-0.531866\pi\)
−0.0999442 + 0.994993i \(0.531866\pi\)
\(570\) 0 0
\(571\) −28.3824 −1.18777 −0.593883 0.804552i \(-0.702406\pi\)
−0.593883 + 0.804552i \(0.702406\pi\)
\(572\) −13.3645 −0.558796
\(573\) 0 0
\(574\) −84.0638 −3.50876
\(575\) 2.39337 0.0998104
\(576\) 0 0
\(577\) −18.5884 −0.773844 −0.386922 0.922112i \(-0.626462\pi\)
−0.386922 + 0.922112i \(0.626462\pi\)
\(578\) 1.63262 0.0679081
\(579\) 0 0
\(580\) −9.13769 −0.379422
\(581\) −32.7148 −1.35724
\(582\) 0 0
\(583\) −23.8967 −0.989699
\(584\) −33.5698 −1.38913
\(585\) 0 0
\(586\) −8.26737 −0.341522
\(587\) −10.2852 −0.424515 −0.212257 0.977214i \(-0.568082\pi\)
−0.212257 + 0.977214i \(0.568082\pi\)
\(588\) 0 0
\(589\) −10.6561 −0.439077
\(590\) 6.55329 0.269795
\(591\) 0 0
\(592\) −4.30166 −0.176797
\(593\) −29.9608 −1.23034 −0.615171 0.788394i \(-0.710913\pi\)
−0.615171 + 0.788394i \(0.710913\pi\)
\(594\) 0 0
\(595\) 17.0073 0.697232
\(596\) −38.0040 −1.55671
\(597\) 0 0
\(598\) −5.39923 −0.220791
\(599\) −34.1173 −1.39399 −0.696997 0.717074i \(-0.745481\pi\)
−0.696997 + 0.717074i \(0.745481\pi\)
\(600\) 0 0
\(601\) −37.7701 −1.54068 −0.770338 0.637636i \(-0.779913\pi\)
−0.770338 + 0.637636i \(0.779913\pi\)
\(602\) −7.60314 −0.309881
\(603\) 0 0
\(604\) 30.2063 1.22908
\(605\) −7.71677 −0.313731
\(606\) 0 0
\(607\) −34.3129 −1.39272 −0.696358 0.717694i \(-0.745198\pi\)
−0.696358 + 0.717694i \(0.745198\pi\)
\(608\) 6.97072 0.282700
\(609\) 0 0
\(610\) −29.5479 −1.19636
\(611\) −12.7882 −0.517355
\(612\) 0 0
\(613\) 17.6306 0.712095 0.356048 0.934468i \(-0.384124\pi\)
0.356048 + 0.934468i \(0.384124\pi\)
\(614\) 17.7119 0.714793
\(615\) 0 0
\(616\) 44.8099 1.80544
\(617\) 3.05417 0.122956 0.0614782 0.998108i \(-0.480419\pi\)
0.0614782 + 0.998108i \(0.480419\pi\)
\(618\) 0 0
\(619\) −18.6857 −0.751041 −0.375521 0.926814i \(-0.622536\pi\)
−0.375521 + 0.926814i \(0.622536\pi\)
\(620\) 29.9759 1.20386
\(621\) 0 0
\(622\) −10.3828 −0.416311
\(623\) 43.1800 1.72997
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 22.5354 0.900696
\(627\) 0 0
\(628\) −10.7587 −0.429318
\(629\) −27.3065 −1.08878
\(630\) 0 0
\(631\) 23.1057 0.919823 0.459911 0.887965i \(-0.347881\pi\)
0.459911 + 0.887965i \(0.347881\pi\)
\(632\) 29.4206 1.17029
\(633\) 0 0
\(634\) −61.4139 −2.43906
\(635\) −2.07671 −0.0824119
\(636\) 0 0
\(637\) 10.7712 0.426769
\(638\) 28.8694 1.14295
\(639\) 0 0
\(640\) −16.7413 −0.661759
\(641\) 36.0663 1.42454 0.712268 0.701908i \(-0.247668\pi\)
0.712268 + 0.701908i \(0.247668\pi\)
\(642\) 0 0
\(643\) −4.19636 −0.165488 −0.0827441 0.996571i \(-0.526368\pi\)
−0.0827441 + 0.996571i \(0.526368\pi\)
\(644\) 31.1676 1.22818
\(645\) 0 0
\(646\) 9.99452 0.393229
\(647\) 0.325332 0.0127901 0.00639506 0.999980i \(-0.497964\pi\)
0.00639506 + 0.999980i \(0.497964\pi\)
\(648\) 0 0
\(649\) −12.5676 −0.493323
\(650\) −2.25591 −0.0884840
\(651\) 0 0
\(652\) 19.4096 0.760137
\(653\) 17.5931 0.688471 0.344235 0.938883i \(-0.388138\pi\)
0.344235 + 0.938883i \(0.388138\pi\)
\(654\) 0 0
\(655\) −18.7352 −0.732044
\(656\) −5.61795 −0.219344
\(657\) 0 0
\(658\) 121.615 4.74106
\(659\) −41.6237 −1.62143 −0.810714 0.585442i \(-0.800921\pi\)
−0.810714 + 0.585442i \(0.800921\pi\)
\(660\) 0 0
\(661\) −5.29887 −0.206102 −0.103051 0.994676i \(-0.532860\pi\)
−0.103051 + 0.994676i \(0.532860\pi\)
\(662\) 66.3113 2.57726
\(663\) 0 0
\(664\) 19.0672 0.739950
\(665\) −4.62936 −0.179519
\(666\) 0 0
\(667\) 7.07963 0.274124
\(668\) −59.2274 −2.29158
\(669\) 0 0
\(670\) −21.8059 −0.842435
\(671\) 56.6658 2.18756
\(672\) 0 0
\(673\) 5.91548 0.228025 0.114012 0.993479i \(-0.463630\pi\)
0.114012 + 0.993479i \(0.463630\pi\)
\(674\) 54.6812 2.10624
\(675\) 0 0
\(676\) 3.08913 0.118813
\(677\) 1.13425 0.0435928 0.0217964 0.999762i \(-0.493061\pi\)
0.0217964 + 0.999762i \(0.493061\pi\)
\(678\) 0 0
\(679\) −16.7090 −0.641232
\(680\) −9.91238 −0.380122
\(681\) 0 0
\(682\) −94.7050 −3.62644
\(683\) 36.6543 1.40254 0.701268 0.712898i \(-0.252618\pi\)
0.701268 + 0.712898i \(0.252618\pi\)
\(684\) 0 0
\(685\) −0.478085 −0.0182667
\(686\) −35.8638 −1.36929
\(687\) 0 0
\(688\) −0.508115 −0.0193717
\(689\) 5.52360 0.210432
\(690\) 0 0
\(691\) 0.106753 0.00406106 0.00203053 0.999998i \(-0.499354\pi\)
0.00203053 + 0.999998i \(0.499354\pi\)
\(692\) −78.6309 −2.98910
\(693\) 0 0
\(694\) 26.6384 1.01118
\(695\) −10.0222 −0.380165
\(696\) 0 0
\(697\) −35.6621 −1.35080
\(698\) −54.0849 −2.04715
\(699\) 0 0
\(700\) 13.0225 0.492204
\(701\) −32.7446 −1.23674 −0.618372 0.785885i \(-0.712208\pi\)
−0.618372 + 0.785885i \(0.712208\pi\)
\(702\) 0 0
\(703\) 7.43276 0.280332
\(704\) 56.4524 2.12763
\(705\) 0 0
\(706\) 32.6408 1.22845
\(707\) −18.2473 −0.686262
\(708\) 0 0
\(709\) −5.10454 −0.191705 −0.0958524 0.995396i \(-0.530558\pi\)
−0.0958524 + 0.995396i \(0.530558\pi\)
\(710\) −2.10202 −0.0788875
\(711\) 0 0
\(712\) −25.1666 −0.943160
\(713\) −23.2245 −0.869763
\(714\) 0 0
\(715\) 4.32629 0.161794
\(716\) 0.118635 0.00443361
\(717\) 0 0
\(718\) −52.5004 −1.95930
\(719\) −35.4870 −1.32344 −0.661722 0.749750i \(-0.730174\pi\)
−0.661722 + 0.749750i \(0.730174\pi\)
\(720\) 0 0
\(721\) −70.9520 −2.64239
\(722\) 40.1418 1.49392
\(723\) 0 0
\(724\) 26.7361 0.993641
\(725\) 2.95802 0.109858
\(726\) 0 0
\(727\) −29.1780 −1.08215 −0.541076 0.840974i \(-0.681983\pi\)
−0.541076 + 0.840974i \(0.681983\pi\)
\(728\) −10.3576 −0.383877
\(729\) 0 0
\(730\) 30.8227 1.14080
\(731\) −3.22546 −0.119298
\(732\) 0 0
\(733\) 34.5000 1.27429 0.637144 0.770745i \(-0.280116\pi\)
0.637144 + 0.770745i \(0.280116\pi\)
\(734\) −73.6031 −2.71674
\(735\) 0 0
\(736\) 15.1924 0.559998
\(737\) 41.8184 1.54040
\(738\) 0 0
\(739\) −20.1371 −0.740756 −0.370378 0.928881i \(-0.620772\pi\)
−0.370378 + 0.928881i \(0.620772\pi\)
\(740\) −20.9085 −0.768613
\(741\) 0 0
\(742\) −52.5293 −1.92841
\(743\) −11.8986 −0.436518 −0.218259 0.975891i \(-0.570038\pi\)
−0.218259 + 0.975891i \(0.570038\pi\)
\(744\) 0 0
\(745\) 12.3025 0.450729
\(746\) 80.5976 2.95089
\(747\) 0 0
\(748\) 53.9174 1.97142
\(749\) −68.0368 −2.48601
\(750\) 0 0
\(751\) 3.42721 0.125061 0.0625303 0.998043i \(-0.480083\pi\)
0.0625303 + 0.998043i \(0.480083\pi\)
\(752\) 8.12751 0.296380
\(753\) 0 0
\(754\) −6.67302 −0.243017
\(755\) −9.77828 −0.355868
\(756\) 0 0
\(757\) −15.8498 −0.576070 −0.288035 0.957620i \(-0.593002\pi\)
−0.288035 + 0.957620i \(0.593002\pi\)
\(758\) −11.8883 −0.431804
\(759\) 0 0
\(760\) 2.69813 0.0978714
\(761\) 1.25854 0.0456222 0.0228111 0.999740i \(-0.492738\pi\)
0.0228111 + 0.999740i \(0.492738\pi\)
\(762\) 0 0
\(763\) −29.1519 −1.05537
\(764\) −2.60387 −0.0942047
\(765\) 0 0
\(766\) 73.3339 2.64966
\(767\) 2.90495 0.104892
\(768\) 0 0
\(769\) 36.8522 1.32892 0.664462 0.747322i \(-0.268661\pi\)
0.664462 + 0.747322i \(0.268661\pi\)
\(770\) −41.1429 −1.48269
\(771\) 0 0
\(772\) 32.2965 1.16238
\(773\) −46.8403 −1.68473 −0.842364 0.538909i \(-0.818837\pi\)
−0.842364 + 0.538909i \(0.818837\pi\)
\(774\) 0 0
\(775\) −9.70367 −0.348566
\(776\) 9.73850 0.349592
\(777\) 0 0
\(778\) −63.2780 −2.26863
\(779\) 9.70715 0.347795
\(780\) 0 0
\(781\) 4.03117 0.144247
\(782\) 21.7826 0.778943
\(783\) 0 0
\(784\) −6.84560 −0.244486
\(785\) 3.48276 0.124305
\(786\) 0 0
\(787\) −37.5720 −1.33930 −0.669649 0.742678i \(-0.733555\pi\)
−0.669649 + 0.742678i \(0.733555\pi\)
\(788\) 20.3927 0.726459
\(789\) 0 0
\(790\) −27.0130 −0.961081
\(791\) 51.6024 1.83477
\(792\) 0 0
\(793\) −13.0980 −0.465124
\(794\) −46.1066 −1.63626
\(795\) 0 0
\(796\) −19.9931 −0.708636
\(797\) −10.4391 −0.369771 −0.184886 0.982760i \(-0.559191\pi\)
−0.184886 + 0.982760i \(0.559191\pi\)
\(798\) 0 0
\(799\) 51.5925 1.82521
\(800\) 6.34768 0.224424
\(801\) 0 0
\(802\) 34.8274 1.22980
\(803\) −59.1104 −2.08596
\(804\) 0 0
\(805\) −10.0895 −0.355607
\(806\) 21.8906 0.771063
\(807\) 0 0
\(808\) 10.6351 0.374142
\(809\) −37.5611 −1.32058 −0.660289 0.751012i \(-0.729566\pi\)
−0.660289 + 0.751012i \(0.729566\pi\)
\(810\) 0 0
\(811\) 43.2183 1.51760 0.758800 0.651324i \(-0.225786\pi\)
0.758800 + 0.651324i \(0.225786\pi\)
\(812\) 38.5207 1.35181
\(813\) 0 0
\(814\) 66.0579 2.31533
\(815\) −6.28319 −0.220091
\(816\) 0 0
\(817\) 0.877962 0.0307160
\(818\) 20.3354 0.711010
\(819\) 0 0
\(820\) −27.3064 −0.953583
\(821\) −3.68430 −0.128583 −0.0642914 0.997931i \(-0.520479\pi\)
−0.0642914 + 0.997931i \(0.520479\pi\)
\(822\) 0 0
\(823\) −51.3253 −1.78909 −0.894544 0.446981i \(-0.852499\pi\)
−0.894544 + 0.446981i \(0.852499\pi\)
\(824\) 41.3530 1.44060
\(825\) 0 0
\(826\) −27.6260 −0.961231
\(827\) −32.3392 −1.12454 −0.562272 0.826953i \(-0.690072\pi\)
−0.562272 + 0.826953i \(0.690072\pi\)
\(828\) 0 0
\(829\) −47.9258 −1.66453 −0.832265 0.554378i \(-0.812956\pi\)
−0.832265 + 0.554378i \(0.812956\pi\)
\(830\) −17.5069 −0.607672
\(831\) 0 0
\(832\) −13.0487 −0.452382
\(833\) −43.4551 −1.50563
\(834\) 0 0
\(835\) 19.1729 0.663504
\(836\) −14.6762 −0.507587
\(837\) 0 0
\(838\) −54.5265 −1.88359
\(839\) 6.54467 0.225947 0.112974 0.993598i \(-0.463962\pi\)
0.112974 + 0.993598i \(0.463962\pi\)
\(840\) 0 0
\(841\) −20.2501 −0.698281
\(842\) 32.9857 1.13676
\(843\) 0 0
\(844\) 46.8442 1.61244
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 32.5307 1.11777
\(848\) −3.51051 −0.120551
\(849\) 0 0
\(850\) 9.10122 0.312169
\(851\) 16.1994 0.555307
\(852\) 0 0
\(853\) 30.5312 1.04537 0.522684 0.852527i \(-0.324931\pi\)
0.522684 + 0.852527i \(0.324931\pi\)
\(854\) 124.562 4.26242
\(855\) 0 0
\(856\) 39.6539 1.35534
\(857\) 51.7653 1.76827 0.884134 0.467234i \(-0.154749\pi\)
0.884134 + 0.467234i \(0.154749\pi\)
\(858\) 0 0
\(859\) 7.73672 0.263973 0.131987 0.991251i \(-0.457864\pi\)
0.131987 + 0.991251i \(0.457864\pi\)
\(860\) −2.46973 −0.0842170
\(861\) 0 0
\(862\) −43.2592 −1.47341
\(863\) 33.8502 1.15228 0.576138 0.817353i \(-0.304559\pi\)
0.576138 + 0.817353i \(0.304559\pi\)
\(864\) 0 0
\(865\) 25.4541 0.865465
\(866\) 85.9312 2.92006
\(867\) 0 0
\(868\) −126.366 −4.28914
\(869\) 51.8045 1.75735
\(870\) 0 0
\(871\) −9.66612 −0.327524
\(872\) 16.9906 0.575375
\(873\) 0 0
\(874\) −5.92917 −0.200557
\(875\) −4.21559 −0.142513
\(876\) 0 0
\(877\) 40.7722 1.37678 0.688389 0.725341i \(-0.258318\pi\)
0.688389 + 0.725341i \(0.258318\pi\)
\(878\) −18.4155 −0.621493
\(879\) 0 0
\(880\) −2.74956 −0.0926877
\(881\) 29.1285 0.981363 0.490681 0.871339i \(-0.336748\pi\)
0.490681 + 0.871339i \(0.336748\pi\)
\(882\) 0 0
\(883\) −22.4971 −0.757086 −0.378543 0.925584i \(-0.623575\pi\)
−0.378543 + 0.925584i \(0.623575\pi\)
\(884\) −12.4627 −0.419167
\(885\) 0 0
\(886\) 33.2642 1.11753
\(887\) −25.8907 −0.869323 −0.434662 0.900594i \(-0.643132\pi\)
−0.434662 + 0.900594i \(0.643132\pi\)
\(888\) 0 0
\(889\) 8.75457 0.293619
\(890\) 23.1072 0.774554
\(891\) 0 0
\(892\) −71.0568 −2.37916
\(893\) −14.0434 −0.469944
\(894\) 0 0
\(895\) −0.0384041 −0.00128371
\(896\) 70.5745 2.35773
\(897\) 0 0
\(898\) −62.7609 −2.09436
\(899\) −28.7036 −0.957319
\(900\) 0 0
\(901\) −22.2843 −0.742399
\(902\) 86.2713 2.87252
\(903\) 0 0
\(904\) −30.0754 −1.00029
\(905\) −8.65492 −0.287699
\(906\) 0 0
\(907\) 5.90315 0.196011 0.0980055 0.995186i \(-0.468754\pi\)
0.0980055 + 0.995186i \(0.468754\pi\)
\(908\) 51.1531 1.69757
\(909\) 0 0
\(910\) 9.50998 0.315253
\(911\) 17.8384 0.591012 0.295506 0.955341i \(-0.404512\pi\)
0.295506 + 0.955341i \(0.404512\pi\)
\(912\) 0 0
\(913\) 33.5739 1.11113
\(914\) −63.6192 −2.10434
\(915\) 0 0
\(916\) 54.0337 1.78532
\(917\) 78.9798 2.60814
\(918\) 0 0
\(919\) −50.3670 −1.66145 −0.830727 0.556680i \(-0.812075\pi\)
−0.830727 + 0.556680i \(0.812075\pi\)
\(920\) 5.88044 0.193873
\(921\) 0 0
\(922\) 27.6645 0.911082
\(923\) −0.931785 −0.0306701
\(924\) 0 0
\(925\) 6.76843 0.222545
\(926\) −17.3281 −0.569438
\(927\) 0 0
\(928\) 18.7765 0.616370
\(929\) −30.6296 −1.00492 −0.502462 0.864599i \(-0.667572\pi\)
−0.502462 + 0.864599i \(0.667572\pi\)
\(930\) 0 0
\(931\) 11.8284 0.387660
\(932\) −32.5670 −1.06677
\(933\) 0 0
\(934\) −71.4124 −2.33669
\(935\) −17.4539 −0.570804
\(936\) 0 0
\(937\) −23.3784 −0.763740 −0.381870 0.924216i \(-0.624720\pi\)
−0.381870 + 0.924216i \(0.624720\pi\)
\(938\) 91.9246 3.00144
\(939\) 0 0
\(940\) 39.5043 1.28849
\(941\) −23.5653 −0.768208 −0.384104 0.923290i \(-0.625490\pi\)
−0.384104 + 0.923290i \(0.625490\pi\)
\(942\) 0 0
\(943\) 21.1563 0.688943
\(944\) −1.84623 −0.0600897
\(945\) 0 0
\(946\) 7.80280 0.253691
\(947\) −4.17712 −0.135738 −0.0678690 0.997694i \(-0.521620\pi\)
−0.0678690 + 0.997694i \(0.521620\pi\)
\(948\) 0 0
\(949\) 13.6631 0.443522
\(950\) −2.47733 −0.0803753
\(951\) 0 0
\(952\) 41.7865 1.35431
\(953\) 26.3851 0.854697 0.427348 0.904087i \(-0.359448\pi\)
0.427348 + 0.904087i \(0.359448\pi\)
\(954\) 0 0
\(955\) 0.842914 0.0272761
\(956\) −9.85925 −0.318871
\(957\) 0 0
\(958\) 48.5171 1.56752
\(959\) 2.01541 0.0650810
\(960\) 0 0
\(961\) 63.1612 2.03746
\(962\) −15.2690 −0.492291
\(963\) 0 0
\(964\) 33.8221 1.08934
\(965\) −10.4549 −0.336555
\(966\) 0 0
\(967\) 10.2763 0.330464 0.165232 0.986255i \(-0.447163\pi\)
0.165232 + 0.986255i \(0.447163\pi\)
\(968\) −18.9599 −0.609394
\(969\) 0 0
\(970\) −8.94157 −0.287096
\(971\) 50.4571 1.61925 0.809623 0.586951i \(-0.199672\pi\)
0.809623 + 0.586951i \(0.199672\pi\)
\(972\) 0 0
\(973\) 42.2496 1.35446
\(974\) −64.3772 −2.06278
\(975\) 0 0
\(976\) 8.32441 0.266458
\(977\) 21.5111 0.688202 0.344101 0.938933i \(-0.388184\pi\)
0.344101 + 0.938933i \(0.388184\pi\)
\(978\) 0 0
\(979\) −44.3140 −1.41628
\(980\) −33.2735 −1.06288
\(981\) 0 0
\(982\) −34.8409 −1.11182
\(983\) 3.34360 0.106644 0.0533221 0.998577i \(-0.483019\pi\)
0.0533221 + 0.998577i \(0.483019\pi\)
\(984\) 0 0
\(985\) −6.60143 −0.210339
\(986\) 26.9215 0.857357
\(987\) 0 0
\(988\) 3.39233 0.107924
\(989\) 1.91348 0.0608450
\(990\) 0 0
\(991\) 11.2253 0.356584 0.178292 0.983978i \(-0.442943\pi\)
0.178292 + 0.983978i \(0.442943\pi\)
\(992\) −61.5958 −1.95567
\(993\) 0 0
\(994\) 8.86126 0.281062
\(995\) 6.47208 0.205179
\(996\) 0 0
\(997\) −40.8375 −1.29334 −0.646669 0.762771i \(-0.723838\pi\)
−0.646669 + 0.762771i \(0.723838\pi\)
\(998\) −16.9699 −0.537174
\(999\) 0 0
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 5265.2.a.bi.1.3 14
3.2 odd 2 5265.2.a.bj.1.12 yes 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5265.2.a.bi.1.3 14 1.1 even 1 trivial
5265.2.a.bj.1.12 yes 14 3.2 odd 2