Properties

Label 7098.2.a.cm.1.3
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $3$
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] = [7098,2,Mod(1,7098)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7098, 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("7098.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.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: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 1.3
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.801938 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.801938 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.801938 q^{10} -6.04892 q^{11} +1.00000 q^{12} -1.00000 q^{14} +0.801938 q^{15} +1.00000 q^{16} -2.85086 q^{17} +1.00000 q^{18} +0.939001 q^{19} +0.801938 q^{20} -1.00000 q^{21} -6.04892 q^{22} +0.911854 q^{23} +1.00000 q^{24} -4.35690 q^{25} +1.00000 q^{27} -1.00000 q^{28} -3.35690 q^{29} +0.801938 q^{30} +2.15883 q^{31} +1.00000 q^{32} -6.04892 q^{33} -2.85086 q^{34} -0.801938 q^{35} +1.00000 q^{36} -5.91185 q^{37} +0.939001 q^{38} +0.801938 q^{40} -1.19806 q^{41} -1.00000 q^{42} +4.07606 q^{43} -6.04892 q^{44} +0.801938 q^{45} +0.911854 q^{46} +1.27413 q^{47} +1.00000 q^{48} +1.00000 q^{49} -4.35690 q^{50} -2.85086 q^{51} +6.15883 q^{53} +1.00000 q^{54} -4.85086 q^{55} -1.00000 q^{56} +0.939001 q^{57} -3.35690 q^{58} -7.29590 q^{59} +0.801938 q^{60} -1.00000 q^{61} +2.15883 q^{62} -1.00000 q^{63} +1.00000 q^{64} -6.04892 q^{66} +1.35690 q^{67} -2.85086 q^{68} +0.911854 q^{69} -0.801938 q^{70} -7.78017 q^{71} +1.00000 q^{72} -14.4547 q^{73} -5.91185 q^{74} -4.35690 q^{75} +0.939001 q^{76} +6.04892 q^{77} +0.131687 q^{79} +0.801938 q^{80} +1.00000 q^{81} -1.19806 q^{82} -16.9148 q^{83} -1.00000 q^{84} -2.28621 q^{85} +4.07606 q^{86} -3.35690 q^{87} -6.04892 q^{88} -12.7409 q^{89} +0.801938 q^{90} +0.911854 q^{92} +2.15883 q^{93} +1.27413 q^{94} +0.753020 q^{95} +1.00000 q^{96} -4.85623 q^{97} +1.00000 q^{98} -6.04892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 2 q^{5} + 3 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 2 q^{10} - 9 q^{11} + 3 q^{12} - 3 q^{14} - 2 q^{15} + 3 q^{16} + 5 q^{17} + 3 q^{18} - 7 q^{19} - 2 q^{20} - 3 q^{21} - 9 q^{22} - q^{23} + 3 q^{24} - 9 q^{25} + 3 q^{27} - 3 q^{28} - 6 q^{29} - 2 q^{30} - 2 q^{31} + 3 q^{32} - 9 q^{33} + 5 q^{34} + 2 q^{35} + 3 q^{36} - 14 q^{37} - 7 q^{38} - 2 q^{40} - 8 q^{41} - 3 q^{42} - 3 q^{43} - 9 q^{44} - 2 q^{45} - q^{46} - 7 q^{47} + 3 q^{48} + 3 q^{49} - 9 q^{50} + 5 q^{51} + 10 q^{53} + 3 q^{54} - q^{55} - 3 q^{56} - 7 q^{57} - 6 q^{58} - 8 q^{59} - 2 q^{60} - 3 q^{61} - 2 q^{62} - 3 q^{63} + 3 q^{64} - 9 q^{66} + 5 q^{68} - q^{69} + 2 q^{70} - 22 q^{71} + 3 q^{72} - 21 q^{73} - 14 q^{74} - 9 q^{75} - 7 q^{76} + 9 q^{77} - 2 q^{79} - 2 q^{80} + 3 q^{81} - 8 q^{82} - 3 q^{83} - 3 q^{84} - 15 q^{85} - 3 q^{86} - 6 q^{87} - 9 q^{88} - 24 q^{89} - 2 q^{90} - q^{92} - 2 q^{93} - 7 q^{94} + 7 q^{95} + 3 q^{96} + 2 q^{97} + 3 q^{98} - 9 q^{99}+O(q^{100}) \) 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
\(5\) 0.801938 0.358637 0.179319 0.983791i \(-0.442611\pi\)
0.179319 + 0.983791i \(0.442611\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.801938 0.253595
\(11\) −6.04892 −1.82382 −0.911909 0.410393i \(-0.865391\pi\)
−0.911909 + 0.410393i \(0.865391\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0.801938 0.207059
\(16\) 1.00000 0.250000
\(17\) −2.85086 −0.691434 −0.345717 0.938339i \(-0.612364\pi\)
−0.345717 + 0.938339i \(0.612364\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.939001 0.215422 0.107711 0.994182i \(-0.465648\pi\)
0.107711 + 0.994182i \(0.465648\pi\)
\(20\) 0.801938 0.179319
\(21\) −1.00000 −0.218218
\(22\) −6.04892 −1.28963
\(23\) 0.911854 0.190135 0.0950674 0.995471i \(-0.469693\pi\)
0.0950674 + 0.995471i \(0.469693\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.35690 −0.871379
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −3.35690 −0.623360 −0.311680 0.950187i \(-0.600892\pi\)
−0.311680 + 0.950187i \(0.600892\pi\)
\(30\) 0.801938 0.146413
\(31\) 2.15883 0.387738 0.193869 0.981027i \(-0.437896\pi\)
0.193869 + 0.981027i \(0.437896\pi\)
\(32\) 1.00000 0.176777
\(33\) −6.04892 −1.05298
\(34\) −2.85086 −0.488918
\(35\) −0.801938 −0.135552
\(36\) 1.00000 0.166667
\(37\) −5.91185 −0.971903 −0.485951 0.873986i \(-0.661527\pi\)
−0.485951 + 0.873986i \(0.661527\pi\)
\(38\) 0.939001 0.152326
\(39\) 0 0
\(40\) 0.801938 0.126797
\(41\) −1.19806 −0.187106 −0.0935529 0.995614i \(-0.529822\pi\)
−0.0935529 + 0.995614i \(0.529822\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.07606 0.621594 0.310797 0.950476i \(-0.399404\pi\)
0.310797 + 0.950476i \(0.399404\pi\)
\(44\) −6.04892 −0.911909
\(45\) 0.801938 0.119546
\(46\) 0.911854 0.134446
\(47\) 1.27413 0.185850 0.0929252 0.995673i \(-0.470378\pi\)
0.0929252 + 0.995673i \(0.470378\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −4.35690 −0.616158
\(51\) −2.85086 −0.399200
\(52\) 0 0
\(53\) 6.15883 0.845981 0.422990 0.906134i \(-0.360980\pi\)
0.422990 + 0.906134i \(0.360980\pi\)
\(54\) 1.00000 0.136083
\(55\) −4.85086 −0.654089
\(56\) −1.00000 −0.133631
\(57\) 0.939001 0.124374
\(58\) −3.35690 −0.440782
\(59\) −7.29590 −0.949845 −0.474922 0.880028i \(-0.657524\pi\)
−0.474922 + 0.880028i \(0.657524\pi\)
\(60\) 0.801938 0.103530
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 2.15883 0.274172
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.04892 −0.744570
\(67\) 1.35690 0.165771 0.0828856 0.996559i \(-0.473586\pi\)
0.0828856 + 0.996559i \(0.473586\pi\)
\(68\) −2.85086 −0.345717
\(69\) 0.911854 0.109774
\(70\) −0.801938 −0.0958499
\(71\) −7.78017 −0.923336 −0.461668 0.887053i \(-0.652749\pi\)
−0.461668 + 0.887053i \(0.652749\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.4547 −1.69180 −0.845899 0.533343i \(-0.820936\pi\)
−0.845899 + 0.533343i \(0.820936\pi\)
\(74\) −5.91185 −0.687239
\(75\) −4.35690 −0.503091
\(76\) 0.939001 0.107711
\(77\) 6.04892 0.689338
\(78\) 0 0
\(79\) 0.131687 0.0148159 0.00740795 0.999973i \(-0.497642\pi\)
0.00740795 + 0.999973i \(0.497642\pi\)
\(80\) 0.801938 0.0896594
\(81\) 1.00000 0.111111
\(82\) −1.19806 −0.132304
\(83\) −16.9148 −1.85664 −0.928322 0.371776i \(-0.878749\pi\)
−0.928322 + 0.371776i \(0.878749\pi\)
\(84\) −1.00000 −0.109109
\(85\) −2.28621 −0.247974
\(86\) 4.07606 0.439533
\(87\) −3.35690 −0.359897
\(88\) −6.04892 −0.644817
\(89\) −12.7409 −1.35054 −0.675268 0.737572i \(-0.735972\pi\)
−0.675268 + 0.737572i \(0.735972\pi\)
\(90\) 0.801938 0.0845317
\(91\) 0 0
\(92\) 0.911854 0.0950674
\(93\) 2.15883 0.223861
\(94\) 1.27413 0.131416
\(95\) 0.753020 0.0772583
\(96\) 1.00000 0.102062
\(97\) −4.85623 −0.493076 −0.246538 0.969133i \(-0.579293\pi\)
−0.246538 + 0.969133i \(0.579293\pi\)
\(98\) 1.00000 0.101015
\(99\) −6.04892 −0.607939
\(100\) −4.35690 −0.435690
\(101\) −4.92931 −0.490485 −0.245242 0.969462i \(-0.578868\pi\)
−0.245242 + 0.969462i \(0.578868\pi\)
\(102\) −2.85086 −0.282277
\(103\) −0.170915 −0.0168408 −0.00842039 0.999965i \(-0.502680\pi\)
−0.00842039 + 0.999965i \(0.502680\pi\)
\(104\) 0 0
\(105\) −0.801938 −0.0782611
\(106\) 6.15883 0.598199
\(107\) −6.30559 −0.609584 −0.304792 0.952419i \(-0.598587\pi\)
−0.304792 + 0.952419i \(0.598587\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.1739 1.35761 0.678807 0.734317i \(-0.262497\pi\)
0.678807 + 0.734317i \(0.262497\pi\)
\(110\) −4.85086 −0.462511
\(111\) −5.91185 −0.561128
\(112\) −1.00000 −0.0944911
\(113\) −0.198062 −0.0186321 −0.00931607 0.999957i \(-0.502965\pi\)
−0.00931607 + 0.999957i \(0.502965\pi\)
\(114\) 0.939001 0.0879455
\(115\) 0.731250 0.0681894
\(116\) −3.35690 −0.311680
\(117\) 0 0
\(118\) −7.29590 −0.671642
\(119\) 2.85086 0.261337
\(120\) 0.801938 0.0732066
\(121\) 25.5894 2.32631
\(122\) −1.00000 −0.0905357
\(123\) −1.19806 −0.108026
\(124\) 2.15883 0.193869
\(125\) −7.50365 −0.671147
\(126\) −1.00000 −0.0890871
\(127\) −17.0465 −1.51263 −0.756317 0.654205i \(-0.773003\pi\)
−0.756317 + 0.654205i \(0.773003\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.07606 0.358877
\(130\) 0 0
\(131\) −3.02715 −0.264483 −0.132242 0.991218i \(-0.542217\pi\)
−0.132242 + 0.991218i \(0.542217\pi\)
\(132\) −6.04892 −0.526491
\(133\) −0.939001 −0.0814217
\(134\) 1.35690 0.117218
\(135\) 0.801938 0.0690198
\(136\) −2.85086 −0.244459
\(137\) 17.3817 1.48501 0.742507 0.669838i \(-0.233636\pi\)
0.742507 + 0.669838i \(0.233636\pi\)
\(138\) 0.911854 0.0776222
\(139\) 7.99330 0.677982 0.338991 0.940790i \(-0.389914\pi\)
0.338991 + 0.940790i \(0.389914\pi\)
\(140\) −0.801938 −0.0677761
\(141\) 1.27413 0.107301
\(142\) −7.78017 −0.652897
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.69202 −0.223560
\(146\) −14.4547 −1.19628
\(147\) 1.00000 0.0824786
\(148\) −5.91185 −0.485951
\(149\) −4.74333 −0.388589 −0.194294 0.980943i \(-0.562242\pi\)
−0.194294 + 0.980943i \(0.562242\pi\)
\(150\) −4.35690 −0.355739
\(151\) 11.2687 0.917038 0.458519 0.888685i \(-0.348380\pi\)
0.458519 + 0.888685i \(0.348380\pi\)
\(152\) 0.939001 0.0761630
\(153\) −2.85086 −0.230478
\(154\) 6.04892 0.487436
\(155\) 1.73125 0.139057
\(156\) 0 0
\(157\) −21.5797 −1.72225 −0.861124 0.508395i \(-0.830239\pi\)
−0.861124 + 0.508395i \(0.830239\pi\)
\(158\) 0.131687 0.0104764
\(159\) 6.15883 0.488427
\(160\) 0.801938 0.0633987
\(161\) −0.911854 −0.0718642
\(162\) 1.00000 0.0785674
\(163\) 2.25667 0.176756 0.0883780 0.996087i \(-0.471832\pi\)
0.0883780 + 0.996087i \(0.471832\pi\)
\(164\) −1.19806 −0.0935529
\(165\) −4.85086 −0.377639
\(166\) −16.9148 −1.31285
\(167\) −0.302602 −0.0234160 −0.0117080 0.999931i \(-0.503727\pi\)
−0.0117080 + 0.999931i \(0.503727\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) −2.28621 −0.175344
\(171\) 0.939001 0.0718072
\(172\) 4.07606 0.310797
\(173\) 18.6896 1.42095 0.710473 0.703724i \(-0.248481\pi\)
0.710473 + 0.703724i \(0.248481\pi\)
\(174\) −3.35690 −0.254486
\(175\) 4.35690 0.329350
\(176\) −6.04892 −0.455954
\(177\) −7.29590 −0.548393
\(178\) −12.7409 −0.954974
\(179\) −22.5797 −1.68769 −0.843843 0.536589i \(-0.819712\pi\)
−0.843843 + 0.536589i \(0.819712\pi\)
\(180\) 0.801938 0.0597729
\(181\) 6.94331 0.516092 0.258046 0.966133i \(-0.416921\pi\)
0.258046 + 0.966133i \(0.416921\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0.911854 0.0672228
\(185\) −4.74094 −0.348561
\(186\) 2.15883 0.158293
\(187\) 17.2446 1.26105
\(188\) 1.27413 0.0929252
\(189\) −1.00000 −0.0727393
\(190\) 0.753020 0.0546298
\(191\) 0.728857 0.0527383 0.0263691 0.999652i \(-0.491605\pi\)
0.0263691 + 0.999652i \(0.491605\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0411 −1.01070 −0.505352 0.862913i \(-0.668637\pi\)
−0.505352 + 0.862913i \(0.668637\pi\)
\(194\) −4.85623 −0.348657
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −7.94139 −0.565801 −0.282900 0.959149i \(-0.591297\pi\)
−0.282900 + 0.959149i \(0.591297\pi\)
\(198\) −6.04892 −0.429878
\(199\) 8.00538 0.567486 0.283743 0.958900i \(-0.408424\pi\)
0.283743 + 0.958900i \(0.408424\pi\)
\(200\) −4.35690 −0.308079
\(201\) 1.35690 0.0957081
\(202\) −4.92931 −0.346825
\(203\) 3.35690 0.235608
\(204\) −2.85086 −0.199600
\(205\) −0.960771 −0.0671032
\(206\) −0.170915 −0.0119082
\(207\) 0.911854 0.0633782
\(208\) 0 0
\(209\) −5.67994 −0.392890
\(210\) −0.801938 −0.0553390
\(211\) −16.2892 −1.12139 −0.560697 0.828021i \(-0.689467\pi\)
−0.560697 + 0.828021i \(0.689467\pi\)
\(212\) 6.15883 0.422990
\(213\) −7.78017 −0.533088
\(214\) −6.30559 −0.431041
\(215\) 3.26875 0.222927
\(216\) 1.00000 0.0680414
\(217\) −2.15883 −0.146551
\(218\) 14.1739 0.959978
\(219\) −14.4547 −0.976760
\(220\) −4.85086 −0.327045
\(221\) 0 0
\(222\) −5.91185 −0.396778
\(223\) 17.6256 1.18030 0.590150 0.807293i \(-0.299068\pi\)
0.590150 + 0.807293i \(0.299068\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.35690 −0.290460
\(226\) −0.198062 −0.0131749
\(227\) −11.1914 −0.742796 −0.371398 0.928474i \(-0.621122\pi\)
−0.371398 + 0.928474i \(0.621122\pi\)
\(228\) 0.939001 0.0621869
\(229\) 25.4131 1.67935 0.839673 0.543093i \(-0.182747\pi\)
0.839673 + 0.543093i \(0.182747\pi\)
\(230\) 0.731250 0.0482172
\(231\) 6.04892 0.397990
\(232\) −3.35690 −0.220391
\(233\) −4.20237 −0.275307 −0.137653 0.990480i \(-0.543956\pi\)
−0.137653 + 0.990480i \(0.543956\pi\)
\(234\) 0 0
\(235\) 1.02177 0.0666529
\(236\) −7.29590 −0.474922
\(237\) 0.131687 0.00855396
\(238\) 2.85086 0.184793
\(239\) −20.3177 −1.31424 −0.657120 0.753786i \(-0.728226\pi\)
−0.657120 + 0.753786i \(0.728226\pi\)
\(240\) 0.801938 0.0517649
\(241\) −15.9433 −1.02700 −0.513500 0.858090i \(-0.671651\pi\)
−0.513500 + 0.858090i \(0.671651\pi\)
\(242\) 25.5894 1.64495
\(243\) 1.00000 0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 0.801938 0.0512339
\(246\) −1.19806 −0.0763857
\(247\) 0 0
\(248\) 2.15883 0.137086
\(249\) −16.9148 −1.07193
\(250\) −7.50365 −0.474572
\(251\) 27.3207 1.72446 0.862232 0.506513i \(-0.169066\pi\)
0.862232 + 0.506513i \(0.169066\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −5.51573 −0.346771
\(254\) −17.0465 −1.06959
\(255\) −2.28621 −0.143168
\(256\) 1.00000 0.0625000
\(257\) 25.2446 1.57471 0.787357 0.616497i \(-0.211449\pi\)
0.787357 + 0.616497i \(0.211449\pi\)
\(258\) 4.07606 0.253765
\(259\) 5.91185 0.367345
\(260\) 0 0
\(261\) −3.35690 −0.207787
\(262\) −3.02715 −0.187018
\(263\) 11.8509 0.730755 0.365378 0.930859i \(-0.380940\pi\)
0.365378 + 0.930859i \(0.380940\pi\)
\(264\) −6.04892 −0.372285
\(265\) 4.93900 0.303400
\(266\) −0.939001 −0.0575738
\(267\) −12.7409 −0.779733
\(268\) 1.35690 0.0828856
\(269\) 8.56465 0.522196 0.261098 0.965312i \(-0.415915\pi\)
0.261098 + 0.965312i \(0.415915\pi\)
\(270\) 0.801938 0.0488044
\(271\) −3.57971 −0.217452 −0.108726 0.994072i \(-0.534677\pi\)
−0.108726 + 0.994072i \(0.534677\pi\)
\(272\) −2.85086 −0.172858
\(273\) 0 0
\(274\) 17.3817 1.05006
\(275\) 26.3545 1.58924
\(276\) 0.911854 0.0548872
\(277\) 12.1196 0.728196 0.364098 0.931361i \(-0.381377\pi\)
0.364098 + 0.931361i \(0.381377\pi\)
\(278\) 7.99330 0.479406
\(279\) 2.15883 0.129246
\(280\) −0.801938 −0.0479249
\(281\) −28.4330 −1.69617 −0.848084 0.529862i \(-0.822244\pi\)
−0.848084 + 0.529862i \(0.822244\pi\)
\(282\) 1.27413 0.0758731
\(283\) −2.67994 −0.159306 −0.0796529 0.996823i \(-0.525381\pi\)
−0.0796529 + 0.996823i \(0.525381\pi\)
\(284\) −7.78017 −0.461668
\(285\) 0.753020 0.0446051
\(286\) 0 0
\(287\) 1.19806 0.0707194
\(288\) 1.00000 0.0589256
\(289\) −8.87263 −0.521919
\(290\) −2.69202 −0.158081
\(291\) −4.85623 −0.284677
\(292\) −14.4547 −0.845899
\(293\) −28.2271 −1.64905 −0.824523 0.565828i \(-0.808557\pi\)
−0.824523 + 0.565828i \(0.808557\pi\)
\(294\) 1.00000 0.0583212
\(295\) −5.85086 −0.340650
\(296\) −5.91185 −0.343620
\(297\) −6.04892 −0.350994
\(298\) −4.74333 −0.274774
\(299\) 0 0
\(300\) −4.35690 −0.251546
\(301\) −4.07606 −0.234940
\(302\) 11.2687 0.648444
\(303\) −4.92931 −0.283182
\(304\) 0.939001 0.0538554
\(305\) −0.801938 −0.0459188
\(306\) −2.85086 −0.162973
\(307\) 9.16959 0.523336 0.261668 0.965158i \(-0.415727\pi\)
0.261668 + 0.965158i \(0.415727\pi\)
\(308\) 6.04892 0.344669
\(309\) −0.170915 −0.00972303
\(310\) 1.73125 0.0983284
\(311\) 26.2150 1.48652 0.743259 0.669003i \(-0.233279\pi\)
0.743259 + 0.669003i \(0.233279\pi\)
\(312\) 0 0
\(313\) 23.1782 1.31011 0.655055 0.755581i \(-0.272645\pi\)
0.655055 + 0.755581i \(0.272645\pi\)
\(314\) −21.5797 −1.21781
\(315\) −0.801938 −0.0451841
\(316\) 0.131687 0.00740795
\(317\) 6.41252 0.360163 0.180081 0.983652i \(-0.442364\pi\)
0.180081 + 0.983652i \(0.442364\pi\)
\(318\) 6.15883 0.345370
\(319\) 20.3056 1.13689
\(320\) 0.801938 0.0448297
\(321\) −6.30559 −0.351943
\(322\) −0.911854 −0.0508156
\(323\) −2.67696 −0.148950
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 2.25667 0.124985
\(327\) 14.1739 0.783818
\(328\) −1.19806 −0.0661519
\(329\) −1.27413 −0.0702449
\(330\) −4.85086 −0.267031
\(331\) 31.2760 1.71909 0.859543 0.511063i \(-0.170748\pi\)
0.859543 + 0.511063i \(0.170748\pi\)
\(332\) −16.9148 −0.928322
\(333\) −5.91185 −0.323968
\(334\) −0.302602 −0.0165576
\(335\) 1.08815 0.0594518
\(336\) −1.00000 −0.0545545
\(337\) 15.8635 0.864141 0.432071 0.901840i \(-0.357783\pi\)
0.432071 + 0.901840i \(0.357783\pi\)
\(338\) 0 0
\(339\) −0.198062 −0.0107573
\(340\) −2.28621 −0.123987
\(341\) −13.0586 −0.707163
\(342\) 0.939001 0.0507754
\(343\) −1.00000 −0.0539949
\(344\) 4.07606 0.219767
\(345\) 0.731250 0.0393692
\(346\) 18.6896 1.00476
\(347\) 19.9594 1.07148 0.535740 0.844383i \(-0.320033\pi\)
0.535740 + 0.844383i \(0.320033\pi\)
\(348\) −3.35690 −0.179949
\(349\) −36.2127 −1.93842 −0.969209 0.246238i \(-0.920806\pi\)
−0.969209 + 0.246238i \(0.920806\pi\)
\(350\) 4.35690 0.232886
\(351\) 0 0
\(352\) −6.04892 −0.322408
\(353\) 27.3793 1.45725 0.728625 0.684912i \(-0.240160\pi\)
0.728625 + 0.684912i \(0.240160\pi\)
\(354\) −7.29590 −0.387773
\(355\) −6.23921 −0.331143
\(356\) −12.7409 −0.675268
\(357\) 2.85086 0.150883
\(358\) −22.5797 −1.19337
\(359\) 23.2989 1.22967 0.614834 0.788657i \(-0.289223\pi\)
0.614834 + 0.788657i \(0.289223\pi\)
\(360\) 0.801938 0.0422658
\(361\) −18.1183 −0.953594
\(362\) 6.94331 0.364932
\(363\) 25.5894 1.34310
\(364\) 0 0
\(365\) −11.5918 −0.606742
\(366\) −1.00000 −0.0522708
\(367\) 23.8780 1.24642 0.623211 0.782054i \(-0.285828\pi\)
0.623211 + 0.782054i \(0.285828\pi\)
\(368\) 0.911854 0.0475337
\(369\) −1.19806 −0.0623686
\(370\) −4.74094 −0.246470
\(371\) −6.15883 −0.319751
\(372\) 2.15883 0.111930
\(373\) −20.0271 −1.03697 −0.518483 0.855088i \(-0.673503\pi\)
−0.518483 + 0.855088i \(0.673503\pi\)
\(374\) 17.2446 0.891696
\(375\) −7.50365 −0.387487
\(376\) 1.27413 0.0657081
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) −0.295897 −0.0151992 −0.00759960 0.999971i \(-0.502419\pi\)
−0.00759960 + 0.999971i \(0.502419\pi\)
\(380\) 0.753020 0.0386291
\(381\) −17.0465 −0.873320
\(382\) 0.728857 0.0372916
\(383\) −22.2282 −1.13581 −0.567904 0.823095i \(-0.692245\pi\)
−0.567904 + 0.823095i \(0.692245\pi\)
\(384\) 1.00000 0.0510310
\(385\) 4.85086 0.247222
\(386\) −14.0411 −0.714676
\(387\) 4.07606 0.207198
\(388\) −4.85623 −0.246538
\(389\) 31.5840 1.60137 0.800687 0.599083i \(-0.204468\pi\)
0.800687 + 0.599083i \(0.204468\pi\)
\(390\) 0 0
\(391\) −2.59956 −0.131466
\(392\) 1.00000 0.0505076
\(393\) −3.02715 −0.152699
\(394\) −7.94139 −0.400082
\(395\) 0.105604 0.00531353
\(396\) −6.04892 −0.303970
\(397\) 11.6267 0.583528 0.291764 0.956490i \(-0.405758\pi\)
0.291764 + 0.956490i \(0.405758\pi\)
\(398\) 8.00538 0.401273
\(399\) −0.939001 −0.0470089
\(400\) −4.35690 −0.217845
\(401\) −6.49827 −0.324508 −0.162254 0.986749i \(-0.551876\pi\)
−0.162254 + 0.986749i \(0.551876\pi\)
\(402\) 1.35690 0.0676758
\(403\) 0 0
\(404\) −4.92931 −0.245242
\(405\) 0.801938 0.0398486
\(406\) 3.35690 0.166600
\(407\) 35.7603 1.77257
\(408\) −2.85086 −0.141138
\(409\) −35.8122 −1.77080 −0.885400 0.464830i \(-0.846116\pi\)
−0.885400 + 0.464830i \(0.846116\pi\)
\(410\) −0.960771 −0.0474491
\(411\) 17.3817 0.857374
\(412\) −0.170915 −0.00842039
\(413\) 7.29590 0.359008
\(414\) 0.911854 0.0448152
\(415\) −13.5646 −0.665862
\(416\) 0 0
\(417\) 7.99330 0.391433
\(418\) −5.67994 −0.277815
\(419\) −1.96376 −0.0959357 −0.0479679 0.998849i \(-0.515275\pi\)
−0.0479679 + 0.998849i \(0.515275\pi\)
\(420\) −0.801938 −0.0391306
\(421\) 8.76271 0.427068 0.213534 0.976936i \(-0.431503\pi\)
0.213534 + 0.976936i \(0.431503\pi\)
\(422\) −16.2892 −0.792945
\(423\) 1.27413 0.0619502
\(424\) 6.15883 0.299099
\(425\) 12.4209 0.602501
\(426\) −7.78017 −0.376950
\(427\) 1.00000 0.0483934
\(428\) −6.30559 −0.304792
\(429\) 0 0
\(430\) 3.26875 0.157633
\(431\) 33.5023 1.61375 0.806875 0.590722i \(-0.201157\pi\)
0.806875 + 0.590722i \(0.201157\pi\)
\(432\) 1.00000 0.0481125
\(433\) −0.928247 −0.0446087 −0.0223044 0.999751i \(-0.507100\pi\)
−0.0223044 + 0.999751i \(0.507100\pi\)
\(434\) −2.15883 −0.103627
\(435\) −2.69202 −0.129073
\(436\) 14.1739 0.678807
\(437\) 0.856232 0.0409591
\(438\) −14.4547 −0.690674
\(439\) −2.32736 −0.111079 −0.0555393 0.998457i \(-0.517688\pi\)
−0.0555393 + 0.998457i \(0.517688\pi\)
\(440\) −4.85086 −0.231255
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −18.8984 −0.897892 −0.448946 0.893559i \(-0.648200\pi\)
−0.448946 + 0.893559i \(0.648200\pi\)
\(444\) −5.91185 −0.280564
\(445\) −10.2174 −0.484353
\(446\) 17.6256 0.834599
\(447\) −4.74333 −0.224352
\(448\) −1.00000 −0.0472456
\(449\) −2.64012 −0.124595 −0.0622975 0.998058i \(-0.519843\pi\)
−0.0622975 + 0.998058i \(0.519843\pi\)
\(450\) −4.35690 −0.205386
\(451\) 7.24698 0.341247
\(452\) −0.198062 −0.00931607
\(453\) 11.2687 0.529452
\(454\) −11.1914 −0.525236
\(455\) 0 0
\(456\) 0.939001 0.0439728
\(457\) 1.25368 0.0586449 0.0293224 0.999570i \(-0.490665\pi\)
0.0293224 + 0.999570i \(0.490665\pi\)
\(458\) 25.4131 1.18748
\(459\) −2.85086 −0.133067
\(460\) 0.731250 0.0340947
\(461\) 22.4198 1.04419 0.522097 0.852886i \(-0.325150\pi\)
0.522097 + 0.852886i \(0.325150\pi\)
\(462\) 6.04892 0.281421
\(463\) −8.94975 −0.415930 −0.207965 0.978136i \(-0.566684\pi\)
−0.207965 + 0.978136i \(0.566684\pi\)
\(464\) −3.35690 −0.155840
\(465\) 1.73125 0.0802848
\(466\) −4.20237 −0.194671
\(467\) 9.62996 0.445621 0.222811 0.974862i \(-0.428477\pi\)
0.222811 + 0.974862i \(0.428477\pi\)
\(468\) 0 0
\(469\) −1.35690 −0.0626556
\(470\) 1.02177 0.0471307
\(471\) −21.5797 −0.994341
\(472\) −7.29590 −0.335821
\(473\) −24.6558 −1.13367
\(474\) 0.131687 0.00604856
\(475\) −4.09113 −0.187714
\(476\) 2.85086 0.130669
\(477\) 6.15883 0.281994
\(478\) −20.3177 −0.929308
\(479\) 16.0586 0.733736 0.366868 0.930273i \(-0.380430\pi\)
0.366868 + 0.930273i \(0.380430\pi\)
\(480\) 0.801938 0.0366033
\(481\) 0 0
\(482\) −15.9433 −0.726198
\(483\) −0.911854 −0.0414908
\(484\) 25.5894 1.16315
\(485\) −3.89440 −0.176835
\(486\) 1.00000 0.0453609
\(487\) −30.4295 −1.37889 −0.689446 0.724337i \(-0.742146\pi\)
−0.689446 + 0.724337i \(0.742146\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 2.25667 0.102050
\(490\) 0.801938 0.0362279
\(491\) −22.4413 −1.01276 −0.506381 0.862310i \(-0.669017\pi\)
−0.506381 + 0.862310i \(0.669017\pi\)
\(492\) −1.19806 −0.0540128
\(493\) 9.57002 0.431012
\(494\) 0 0
\(495\) −4.85086 −0.218030
\(496\) 2.15883 0.0969345
\(497\) 7.78017 0.348988
\(498\) −16.9148 −0.757972
\(499\) 37.9909 1.70071 0.850353 0.526212i \(-0.176388\pi\)
0.850353 + 0.526212i \(0.176388\pi\)
\(500\) −7.50365 −0.335573
\(501\) −0.302602 −0.0135192
\(502\) 27.3207 1.21938
\(503\) 26.6950 1.19027 0.595136 0.803625i \(-0.297098\pi\)
0.595136 + 0.803625i \(0.297098\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −3.95300 −0.175906
\(506\) −5.51573 −0.245204
\(507\) 0 0
\(508\) −17.0465 −0.756317
\(509\) −42.2693 −1.87356 −0.936778 0.349925i \(-0.886207\pi\)
−0.936778 + 0.349925i \(0.886207\pi\)
\(510\) −2.28621 −0.101235
\(511\) 14.4547 0.639440
\(512\) 1.00000 0.0441942
\(513\) 0.939001 0.0414579
\(514\) 25.2446 1.11349
\(515\) −0.137063 −0.00603973
\(516\) 4.07606 0.179439
\(517\) −7.70709 −0.338957
\(518\) 5.91185 0.259752
\(519\) 18.6896 0.820384
\(520\) 0 0
\(521\) −12.2760 −0.537823 −0.268912 0.963165i \(-0.586664\pi\)
−0.268912 + 0.963165i \(0.586664\pi\)
\(522\) −3.35690 −0.146927
\(523\) −2.23921 −0.0979138 −0.0489569 0.998801i \(-0.515590\pi\)
−0.0489569 + 0.998801i \(0.515590\pi\)
\(524\) −3.02715 −0.132242
\(525\) 4.35690 0.190151
\(526\) 11.8509 0.516722
\(527\) −6.15452 −0.268095
\(528\) −6.04892 −0.263245
\(529\) −22.1685 −0.963849
\(530\) 4.93900 0.214536
\(531\) −7.29590 −0.316615
\(532\) −0.939001 −0.0407109
\(533\) 0 0
\(534\) −12.7409 −0.551354
\(535\) −5.05669 −0.218620
\(536\) 1.35690 0.0586090
\(537\) −22.5797 −0.974387
\(538\) 8.56465 0.369248
\(539\) −6.04892 −0.260545
\(540\) 0.801938 0.0345099
\(541\) −10.0519 −0.432165 −0.216082 0.976375i \(-0.569328\pi\)
−0.216082 + 0.976375i \(0.569328\pi\)
\(542\) −3.57971 −0.153762
\(543\) 6.94331 0.297966
\(544\) −2.85086 −0.122229
\(545\) 11.3666 0.486891
\(546\) 0 0
\(547\) 41.8745 1.79043 0.895213 0.445638i \(-0.147023\pi\)
0.895213 + 0.445638i \(0.147023\pi\)
\(548\) 17.3817 0.742507
\(549\) −1.00000 −0.0426790
\(550\) 26.3545 1.12376
\(551\) −3.15213 −0.134285
\(552\) 0.911854 0.0388111
\(553\) −0.131687 −0.00559988
\(554\) 12.1196 0.514913
\(555\) −4.74094 −0.201242
\(556\) 7.99330 0.338991
\(557\) 34.0847 1.44421 0.722107 0.691781i \(-0.243174\pi\)
0.722107 + 0.691781i \(0.243174\pi\)
\(558\) 2.15883 0.0913907
\(559\) 0 0
\(560\) −0.801938 −0.0338881
\(561\) 17.2446 0.728067
\(562\) −28.4330 −1.19937
\(563\) −7.10513 −0.299446 −0.149723 0.988728i \(-0.547838\pi\)
−0.149723 + 0.988728i \(0.547838\pi\)
\(564\) 1.27413 0.0536504
\(565\) −0.158834 −0.00668218
\(566\) −2.67994 −0.112646
\(567\) −1.00000 −0.0419961
\(568\) −7.78017 −0.326449
\(569\) −17.2959 −0.725082 −0.362541 0.931968i \(-0.618091\pi\)
−0.362541 + 0.931968i \(0.618091\pi\)
\(570\) 0.753020 0.0315406
\(571\) −1.25800 −0.0526455 −0.0263228 0.999653i \(-0.508380\pi\)
−0.0263228 + 0.999653i \(0.508380\pi\)
\(572\) 0 0
\(573\) 0.728857 0.0304484
\(574\) 1.19806 0.0500062
\(575\) −3.97285 −0.165679
\(576\) 1.00000 0.0416667
\(577\) 35.1062 1.46149 0.730745 0.682651i \(-0.239173\pi\)
0.730745 + 0.682651i \(0.239173\pi\)
\(578\) −8.87263 −0.369053
\(579\) −14.0411 −0.583530
\(580\) −2.69202 −0.111780
\(581\) 16.9148 0.701746
\(582\) −4.85623 −0.201297
\(583\) −37.2543 −1.54291
\(584\) −14.4547 −0.598141
\(585\) 0 0
\(586\) −28.2271 −1.16605
\(587\) −10.0532 −0.414941 −0.207471 0.978241i \(-0.566523\pi\)
−0.207471 + 0.978241i \(0.566523\pi\)
\(588\) 1.00000 0.0412393
\(589\) 2.02715 0.0835271
\(590\) −5.85086 −0.240876
\(591\) −7.94139 −0.326665
\(592\) −5.91185 −0.242976
\(593\) −23.2808 −0.956029 −0.478015 0.878352i \(-0.658643\pi\)
−0.478015 + 0.878352i \(0.658643\pi\)
\(594\) −6.04892 −0.248190
\(595\) 2.28621 0.0937254
\(596\) −4.74333 −0.194294
\(597\) 8.00538 0.327638
\(598\) 0 0
\(599\) −12.0285 −0.491470 −0.245735 0.969337i \(-0.579029\pi\)
−0.245735 + 0.969337i \(0.579029\pi\)
\(600\) −4.35690 −0.177870
\(601\) 8.67935 0.354038 0.177019 0.984207i \(-0.443355\pi\)
0.177019 + 0.984207i \(0.443355\pi\)
\(602\) −4.07606 −0.166128
\(603\) 1.35690 0.0552571
\(604\) 11.2687 0.458519
\(605\) 20.5211 0.834302
\(606\) −4.92931 −0.200240
\(607\) 38.6862 1.57022 0.785112 0.619354i \(-0.212606\pi\)
0.785112 + 0.619354i \(0.212606\pi\)
\(608\) 0.939001 0.0380815
\(609\) 3.35690 0.136028
\(610\) −0.801938 −0.0324695
\(611\) 0 0
\(612\) −2.85086 −0.115239
\(613\) −28.6601 −1.15757 −0.578785 0.815480i \(-0.696473\pi\)
−0.578785 + 0.815480i \(0.696473\pi\)
\(614\) 9.16959 0.370054
\(615\) −0.960771 −0.0387420
\(616\) 6.04892 0.243718
\(617\) −37.8756 −1.52481 −0.762407 0.647097i \(-0.775983\pi\)
−0.762407 + 0.647097i \(0.775983\pi\)
\(618\) −0.170915 −0.00687522
\(619\) −11.6256 −0.467274 −0.233637 0.972324i \(-0.575063\pi\)
−0.233637 + 0.972324i \(0.575063\pi\)
\(620\) 1.73125 0.0695287
\(621\) 0.911854 0.0365914
\(622\) 26.2150 1.05113
\(623\) 12.7409 0.510455
\(624\) 0 0
\(625\) 15.7670 0.630681
\(626\) 23.1782 0.926388
\(627\) −5.67994 −0.226835
\(628\) −21.5797 −0.861124
\(629\) 16.8538 0.672007
\(630\) −0.801938 −0.0319500
\(631\) 38.0853 1.51615 0.758076 0.652167i \(-0.226140\pi\)
0.758076 + 0.652167i \(0.226140\pi\)
\(632\) 0.131687 0.00523821
\(633\) −16.2892 −0.647437
\(634\) 6.41252 0.254674
\(635\) −13.6703 −0.542487
\(636\) 6.15883 0.244214
\(637\) 0 0
\(638\) 20.3056 0.803906
\(639\) −7.78017 −0.307779
\(640\) 0.801938 0.0316994
\(641\) −42.6359 −1.68402 −0.842009 0.539464i \(-0.818627\pi\)
−0.842009 + 0.539464i \(0.818627\pi\)
\(642\) −6.30559 −0.248862
\(643\) 8.86161 0.349468 0.174734 0.984616i \(-0.444093\pi\)
0.174734 + 0.984616i \(0.444093\pi\)
\(644\) −0.911854 −0.0359321
\(645\) 3.26875 0.128707
\(646\) −2.67696 −0.105323
\(647\) 6.01102 0.236317 0.118159 0.992995i \(-0.462301\pi\)
0.118159 + 0.992995i \(0.462301\pi\)
\(648\) 1.00000 0.0392837
\(649\) 44.1323 1.73234
\(650\) 0 0
\(651\) −2.15883 −0.0846114
\(652\) 2.25667 0.0883780
\(653\) −41.9221 −1.64054 −0.820270 0.571977i \(-0.806177\pi\)
−0.820270 + 0.571977i \(0.806177\pi\)
\(654\) 14.1739 0.554243
\(655\) −2.42758 −0.0948535
\(656\) −1.19806 −0.0467765
\(657\) −14.4547 −0.563933
\(658\) −1.27413 −0.0496706
\(659\) 18.1957 0.708803 0.354401 0.935093i \(-0.384685\pi\)
0.354401 + 0.935093i \(0.384685\pi\)
\(660\) −4.85086 −0.188819
\(661\) −28.6980 −1.11622 −0.558111 0.829766i \(-0.688474\pi\)
−0.558111 + 0.829766i \(0.688474\pi\)
\(662\) 31.2760 1.21558
\(663\) 0 0
\(664\) −16.9148 −0.656423
\(665\) −0.753020 −0.0292009
\(666\) −5.91185 −0.229080
\(667\) −3.06100 −0.118522
\(668\) −0.302602 −0.0117080
\(669\) 17.6256 0.681447
\(670\) 1.08815 0.0420387
\(671\) 6.04892 0.233516
\(672\) −1.00000 −0.0385758
\(673\) −36.2301 −1.39657 −0.698284 0.715821i \(-0.746053\pi\)
−0.698284 + 0.715821i \(0.746053\pi\)
\(674\) 15.8635 0.611040
\(675\) −4.35690 −0.167697
\(676\) 0 0
\(677\) −27.3045 −1.04940 −0.524699 0.851288i \(-0.675822\pi\)
−0.524699 + 0.851288i \(0.675822\pi\)
\(678\) −0.198062 −0.00760654
\(679\) 4.85623 0.186365
\(680\) −2.28621 −0.0876721
\(681\) −11.1914 −0.428854
\(682\) −13.0586 −0.500040
\(683\) −0.800610 −0.0306345 −0.0153172 0.999883i \(-0.504876\pi\)
−0.0153172 + 0.999883i \(0.504876\pi\)
\(684\) 0.939001 0.0359036
\(685\) 13.9390 0.532582
\(686\) −1.00000 −0.0381802
\(687\) 25.4131 0.969571
\(688\) 4.07606 0.155398
\(689\) 0 0
\(690\) 0.731250 0.0278382
\(691\) 26.7168 1.01635 0.508177 0.861253i \(-0.330320\pi\)
0.508177 + 0.861253i \(0.330320\pi\)
\(692\) 18.6896 0.710473
\(693\) 6.04892 0.229779
\(694\) 19.9594 0.757650
\(695\) 6.41013 0.243150
\(696\) −3.35690 −0.127243
\(697\) 3.41550 0.129371
\(698\) −36.2127 −1.37067
\(699\) −4.20237 −0.158948
\(700\) 4.35690 0.164675
\(701\) −11.8062 −0.445916 −0.222958 0.974828i \(-0.571571\pi\)
−0.222958 + 0.974828i \(0.571571\pi\)
\(702\) 0 0
\(703\) −5.55124 −0.209369
\(704\) −6.04892 −0.227977
\(705\) 1.02177 0.0384821
\(706\) 27.3793 1.03043
\(707\) 4.92931 0.185386
\(708\) −7.29590 −0.274197
\(709\) −34.9734 −1.31346 −0.656728 0.754128i \(-0.728060\pi\)
−0.656728 + 0.754128i \(0.728060\pi\)
\(710\) −6.23921 −0.234153
\(711\) 0.131687 0.00493863
\(712\) −12.7409 −0.477487
\(713\) 1.96854 0.0737224
\(714\) 2.85086 0.106691
\(715\) 0 0
\(716\) −22.5797 −0.843843
\(717\) −20.3177 −0.758777
\(718\) 23.2989 0.869507
\(719\) −19.6310 −0.732114 −0.366057 0.930593i \(-0.619292\pi\)
−0.366057 + 0.930593i \(0.619292\pi\)
\(720\) 0.801938 0.0298865
\(721\) 0.170915 0.00636521
\(722\) −18.1183 −0.674292
\(723\) −15.9433 −0.592938
\(724\) 6.94331 0.258046
\(725\) 14.6256 0.543183
\(726\) 25.5894 0.949712
\(727\) −24.4523 −0.906887 −0.453444 0.891285i \(-0.649805\pi\)
−0.453444 + 0.891285i \(0.649805\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −11.5918 −0.429032
\(731\) −11.6203 −0.429791
\(732\) −1.00000 −0.0369611
\(733\) 38.0441 1.40519 0.702596 0.711589i \(-0.252024\pi\)
0.702596 + 0.711589i \(0.252024\pi\)
\(734\) 23.8780 0.881353
\(735\) 0.801938 0.0295799
\(736\) 0.911854 0.0336114
\(737\) −8.20775 −0.302336
\(738\) −1.19806 −0.0441013
\(739\) 23.3045 0.857271 0.428635 0.903478i \(-0.358995\pi\)
0.428635 + 0.903478i \(0.358995\pi\)
\(740\) −4.74094 −0.174280
\(741\) 0 0
\(742\) −6.15883 −0.226098
\(743\) −20.5714 −0.754690 −0.377345 0.926073i \(-0.623163\pi\)
−0.377345 + 0.926073i \(0.623163\pi\)
\(744\) 2.15883 0.0791467
\(745\) −3.80386 −0.139363
\(746\) −20.0271 −0.733246
\(747\) −16.9148 −0.618882
\(748\) 17.2446 0.630525
\(749\) 6.30559 0.230401
\(750\) −7.50365 −0.273994
\(751\) −8.68366 −0.316871 −0.158436 0.987369i \(-0.550645\pi\)
−0.158436 + 0.987369i \(0.550645\pi\)
\(752\) 1.27413 0.0464626
\(753\) 27.3207 0.995620
\(754\) 0 0
\(755\) 9.03684 0.328884
\(756\) −1.00000 −0.0363696
\(757\) 32.4969 1.18112 0.590561 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(758\) −0.295897 −0.0107475
\(759\) −5.51573 −0.200208
\(760\) 0.753020 0.0273149
\(761\) 12.6243 0.457631 0.228816 0.973470i \(-0.426515\pi\)
0.228816 + 0.973470i \(0.426515\pi\)
\(762\) −17.0465 −0.617530
\(763\) −14.1739 −0.513130
\(764\) 0.728857 0.0263691
\(765\) −2.28621 −0.0826580
\(766\) −22.2282 −0.803137
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 15.0271 0.541893 0.270946 0.962594i \(-0.412663\pi\)
0.270946 + 0.962594i \(0.412663\pi\)
\(770\) 4.85086 0.174813
\(771\) 25.2446 0.909162
\(772\) −14.0411 −0.505352
\(773\) −44.7096 −1.60809 −0.804046 0.594567i \(-0.797324\pi\)
−0.804046 + 0.594567i \(0.797324\pi\)
\(774\) 4.07606 0.146511
\(775\) −9.40581 −0.337867
\(776\) −4.85623 −0.174329
\(777\) 5.91185 0.212087
\(778\) 31.5840 1.13234
\(779\) −1.12498 −0.0403067
\(780\) 0 0
\(781\) 47.0616 1.68400
\(782\) −2.59956 −0.0929602
\(783\) −3.35690 −0.119966
\(784\) 1.00000 0.0357143
\(785\) −17.3056 −0.617663
\(786\) −3.02715 −0.107975
\(787\) 33.0140 1.17682 0.588411 0.808562i \(-0.299754\pi\)
0.588411 + 0.808562i \(0.299754\pi\)
\(788\) −7.94139 −0.282900
\(789\) 11.8509 0.421902
\(790\) 0.105604 0.00375724
\(791\) 0.198062 0.00704228
\(792\) −6.04892 −0.214939
\(793\) 0 0
\(794\) 11.6267 0.412617
\(795\) 4.93900 0.175168
\(796\) 8.00538 0.283743
\(797\) −32.4644 −1.14995 −0.574974 0.818171i \(-0.694988\pi\)
−0.574974 + 0.818171i \(0.694988\pi\)
\(798\) −0.939001 −0.0332403
\(799\) −3.63235 −0.128503
\(800\) −4.35690 −0.154040
\(801\) −12.7409 −0.450179
\(802\) −6.49827 −0.229462
\(803\) 87.4355 3.08553
\(804\) 1.35690 0.0478540
\(805\) −0.731250 −0.0257732
\(806\) 0 0
\(807\) 8.56465 0.301490
\(808\) −4.92931 −0.173413
\(809\) 33.2597 1.16935 0.584674 0.811269i \(-0.301223\pi\)
0.584674 + 0.811269i \(0.301223\pi\)
\(810\) 0.801938 0.0281772
\(811\) 0.508434 0.0178535 0.00892676 0.999960i \(-0.497158\pi\)
0.00892676 + 0.999960i \(0.497158\pi\)
\(812\) 3.35690 0.117804
\(813\) −3.57971 −0.125546
\(814\) 35.7603 1.25340
\(815\) 1.80971 0.0633913
\(816\) −2.85086 −0.0997999
\(817\) 3.82743 0.133905
\(818\) −35.8122 −1.25214
\(819\) 0 0
\(820\) −0.960771 −0.0335516
\(821\) 24.8713 0.868014 0.434007 0.900909i \(-0.357099\pi\)
0.434007 + 0.900909i \(0.357099\pi\)
\(822\) 17.3817 0.606255
\(823\) −43.8001 −1.52678 −0.763388 0.645940i \(-0.776466\pi\)
−0.763388 + 0.645940i \(0.776466\pi\)
\(824\) −0.170915 −0.00595411
\(825\) 26.3545 0.917546
\(826\) 7.29590 0.253857
\(827\) −36.3900 −1.26540 −0.632702 0.774395i \(-0.718054\pi\)
−0.632702 + 0.774395i \(0.718054\pi\)
\(828\) 0.911854 0.0316891
\(829\) −29.1976 −1.01407 −0.507037 0.861924i \(-0.669259\pi\)
−0.507037 + 0.861924i \(0.669259\pi\)
\(830\) −13.5646 −0.470836
\(831\) 12.1196 0.420424
\(832\) 0 0
\(833\) −2.85086 −0.0987763
\(834\) 7.99330 0.276785
\(835\) −0.242668 −0.00839786
\(836\) −5.67994 −0.196445
\(837\) 2.15883 0.0746202
\(838\) −1.96376 −0.0678368
\(839\) 13.2446 0.457254 0.228627 0.973514i \(-0.426576\pi\)
0.228627 + 0.973514i \(0.426576\pi\)
\(840\) −0.801938 −0.0276695
\(841\) −17.7313 −0.611422
\(842\) 8.76271 0.301983
\(843\) −28.4330 −0.979283
\(844\) −16.2892 −0.560697
\(845\) 0 0
\(846\) 1.27413 0.0438054
\(847\) −25.5894 −0.879262
\(848\) 6.15883 0.211495
\(849\) −2.67994 −0.0919753
\(850\) 12.4209 0.426033
\(851\) −5.39075 −0.184792
\(852\) −7.78017 −0.266544
\(853\) 56.7047 1.94153 0.970766 0.240028i \(-0.0771568\pi\)
0.970766 + 0.240028i \(0.0771568\pi\)
\(854\) 1.00000 0.0342193
\(855\) 0.753020 0.0257528
\(856\) −6.30559 −0.215520
\(857\) 21.0737 0.719863 0.359931 0.932979i \(-0.382800\pi\)
0.359931 + 0.932979i \(0.382800\pi\)
\(858\) 0 0
\(859\) 16.4494 0.561245 0.280622 0.959818i \(-0.409459\pi\)
0.280622 + 0.959818i \(0.409459\pi\)
\(860\) 3.26875 0.111463
\(861\) 1.19806 0.0408299
\(862\) 33.5023 1.14109
\(863\) −46.0743 −1.56839 −0.784193 0.620517i \(-0.786923\pi\)
−0.784193 + 0.620517i \(0.786923\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.9879 0.509605
\(866\) −0.928247 −0.0315431
\(867\) −8.87263 −0.301330
\(868\) −2.15883 −0.0732756
\(869\) −0.796561 −0.0270215
\(870\) −2.69202 −0.0912681
\(871\) 0 0
\(872\) 14.1739 0.479989
\(873\) −4.85623 −0.164359
\(874\) 0.856232 0.0289625
\(875\) 7.50365 0.253670
\(876\) −14.4547 −0.488380
\(877\) 9.46548 0.319627 0.159813 0.987147i \(-0.448911\pi\)
0.159813 + 0.987147i \(0.448911\pi\)
\(878\) −2.32736 −0.0785445
\(879\) −28.2271 −0.952077
\(880\) −4.85086 −0.163522
\(881\) −20.3840 −0.686756 −0.343378 0.939197i \(-0.611571\pi\)
−0.343378 + 0.939197i \(0.611571\pi\)
\(882\) 1.00000 0.0336718
\(883\) −28.4446 −0.957236 −0.478618 0.878023i \(-0.658862\pi\)
−0.478618 + 0.878023i \(0.658862\pi\)
\(884\) 0 0
\(885\) −5.85086 −0.196674
\(886\) −18.8984 −0.634906
\(887\) −30.2881 −1.01698 −0.508488 0.861069i \(-0.669795\pi\)
−0.508488 + 0.861069i \(0.669795\pi\)
\(888\) −5.91185 −0.198389
\(889\) 17.0465 0.571722
\(890\) −10.2174 −0.342489
\(891\) −6.04892 −0.202646
\(892\) 17.6256 0.590150
\(893\) 1.19641 0.0400362
\(894\) −4.74333 −0.158641
\(895\) −18.1075 −0.605268
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −2.64012 −0.0881020
\(899\) −7.24698 −0.241700
\(900\) −4.35690 −0.145230
\(901\) −17.5579 −0.584940
\(902\) 7.24698 0.241298
\(903\) −4.07606 −0.135643
\(904\) −0.198062 −0.00658745
\(905\) 5.56810 0.185090
\(906\) 11.2687 0.374379
\(907\) 59.5478 1.97725 0.988626 0.150396i \(-0.0480550\pi\)
0.988626 + 0.150396i \(0.0480550\pi\)
\(908\) −11.1914 −0.371398
\(909\) −4.92931 −0.163495
\(910\) 0 0
\(911\) 25.9957 0.861276 0.430638 0.902525i \(-0.358289\pi\)
0.430638 + 0.902525i \(0.358289\pi\)
\(912\) 0.939001 0.0310934
\(913\) 102.316 3.38618
\(914\) 1.25368 0.0414682
\(915\) −0.801938 −0.0265112
\(916\) 25.4131 0.839673
\(917\) 3.02715 0.0999652
\(918\) −2.85086 −0.0940922
\(919\) 17.6088 0.580860 0.290430 0.956896i \(-0.406202\pi\)
0.290430 + 0.956896i \(0.406202\pi\)
\(920\) 0.731250 0.0241086
\(921\) 9.16959 0.302148
\(922\) 22.4198 0.738357
\(923\) 0 0
\(924\) 6.04892 0.198995
\(925\) 25.7573 0.846896
\(926\) −8.94975 −0.294107
\(927\) −0.170915 −0.00561359
\(928\) −3.35690 −0.110196
\(929\) −15.6324 −0.512881 −0.256440 0.966560i \(-0.582550\pi\)
−0.256440 + 0.966560i \(0.582550\pi\)
\(930\) 1.73125 0.0567699
\(931\) 0.939001 0.0307745
\(932\) −4.20237 −0.137653
\(933\) 26.2150 0.858242
\(934\) 9.62996 0.315102
\(935\) 13.8291 0.452259
\(936\) 0 0
\(937\) 32.1849 1.05144 0.525718 0.850659i \(-0.323797\pi\)
0.525718 + 0.850659i \(0.323797\pi\)
\(938\) −1.35690 −0.0443042
\(939\) 23.1782 0.756392
\(940\) 1.02177 0.0333265
\(941\) 48.9952 1.59720 0.798599 0.601863i \(-0.205575\pi\)
0.798599 + 0.601863i \(0.205575\pi\)
\(942\) −21.5797 −0.703105
\(943\) −1.09246 −0.0355753
\(944\) −7.29590 −0.237461
\(945\) −0.801938 −0.0260870
\(946\) −24.6558 −0.801628
\(947\) 7.05190 0.229156 0.114578 0.993414i \(-0.463448\pi\)
0.114578 + 0.993414i \(0.463448\pi\)
\(948\) 0.131687 0.00427698
\(949\) 0 0
\(950\) −4.09113 −0.132734
\(951\) 6.41252 0.207940
\(952\) 2.85086 0.0923967
\(953\) 6.99164 0.226481 0.113241 0.993568i \(-0.463877\pi\)
0.113241 + 0.993568i \(0.463877\pi\)
\(954\) 6.15883 0.199400
\(955\) 0.584498 0.0189139
\(956\) −20.3177 −0.657120
\(957\) 20.3056 0.656386
\(958\) 16.0586 0.518830
\(959\) −17.3817 −0.561283
\(960\) 0.801938 0.0258824
\(961\) −26.3394 −0.849659
\(962\) 0 0
\(963\) −6.30559 −0.203195
\(964\) −15.9433 −0.513500
\(965\) −11.2601 −0.362476
\(966\) −0.911854 −0.0293384
\(967\) −9.68473 −0.311440 −0.155720 0.987801i \(-0.549770\pi\)
−0.155720 + 0.987801i \(0.549770\pi\)
\(968\) 25.5894 0.822474
\(969\) −2.67696 −0.0859962
\(970\) −3.89440 −0.125042
\(971\) −42.8340 −1.37461 −0.687304 0.726370i \(-0.741206\pi\)
−0.687304 + 0.726370i \(0.741206\pi\)
\(972\) 1.00000 0.0320750
\(973\) −7.99330 −0.256253
\(974\) −30.4295 −0.975024
\(975\) 0 0
\(976\) −1.00000 −0.0320092
\(977\) −37.3510 −1.19497 −0.597483 0.801882i \(-0.703832\pi\)
−0.597483 + 0.801882i \(0.703832\pi\)
\(978\) 2.25667 0.0721603
\(979\) 77.0689 2.46313
\(980\) 0.801938 0.0256170
\(981\) 14.1739 0.452538
\(982\) −22.4413 −0.716131
\(983\) −5.61058 −0.178950 −0.0894749 0.995989i \(-0.528519\pi\)
−0.0894749 + 0.995989i \(0.528519\pi\)
\(984\) −1.19806 −0.0381928
\(985\) −6.36850 −0.202917
\(986\) 9.57002 0.304772
\(987\) −1.27413 −0.0405559
\(988\) 0 0
\(989\) 3.71678 0.118187
\(990\) −4.85086 −0.154170
\(991\) 8.12605 0.258132 0.129066 0.991636i \(-0.458802\pi\)
0.129066 + 0.991636i \(0.458802\pi\)
\(992\) 2.15883 0.0685430
\(993\) 31.2760 0.992515
\(994\) 7.78017 0.246772
\(995\) 6.41981 0.203522
\(996\) −16.9148 −0.535967
\(997\) 7.72050 0.244511 0.122255 0.992499i \(-0.460987\pi\)
0.122255 + 0.992499i \(0.460987\pi\)
\(998\) 37.9909 1.20258
\(999\) −5.91185 −0.187043
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 7098.2.a.cm.1.3 yes 3
13.12 even 2 7098.2.a.cf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cf.1.1 3 13.12 even 2
7098.2.a.cm.1.3 yes 3 1.1 even 1 trivial