Properties

Label 8034.2.a.s.1.3
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $10$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 15x^{8} + 72x^{7} - 27x^{6} - 115x^{5} + 54x^{4} + 68x^{3} - 15x^{2} - 15x - 2 \) 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 1.3
Root \(-0.671383\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.82988 q^{5} +1.00000 q^{6} +1.45707 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.82988 q^{5} +1.00000 q^{6} +1.45707 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.82988 q^{10} -0.513205 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.45707 q^{14} +1.82988 q^{15} +1.00000 q^{16} -0.785684 q^{17} -1.00000 q^{18} -1.12864 q^{19} -1.82988 q^{20} -1.45707 q^{21} +0.513205 q^{22} +5.13440 q^{23} +1.00000 q^{24} -1.65155 q^{25} +1.00000 q^{26} -1.00000 q^{27} +1.45707 q^{28} +3.60480 q^{29} -1.82988 q^{30} -8.18455 q^{31} -1.00000 q^{32} +0.513205 q^{33} +0.785684 q^{34} -2.66625 q^{35} +1.00000 q^{36} +2.66489 q^{37} +1.12864 q^{38} +1.00000 q^{39} +1.82988 q^{40} +5.68841 q^{41} +1.45707 q^{42} +5.80579 q^{43} -0.513205 q^{44} -1.82988 q^{45} -5.13440 q^{46} +4.47220 q^{47} -1.00000 q^{48} -4.87696 q^{49} +1.65155 q^{50} +0.785684 q^{51} -1.00000 q^{52} -9.91273 q^{53} +1.00000 q^{54} +0.939102 q^{55} -1.45707 q^{56} +1.12864 q^{57} -3.60480 q^{58} +10.2889 q^{59} +1.82988 q^{60} -9.58790 q^{61} +8.18455 q^{62} +1.45707 q^{63} +1.00000 q^{64} +1.82988 q^{65} -0.513205 q^{66} -11.1866 q^{67} -0.785684 q^{68} -5.13440 q^{69} +2.66625 q^{70} +4.34398 q^{71} -1.00000 q^{72} +0.434059 q^{73} -2.66489 q^{74} +1.65155 q^{75} -1.12864 q^{76} -0.747774 q^{77} -1.00000 q^{78} +7.70913 q^{79} -1.82988 q^{80} +1.00000 q^{81} -5.68841 q^{82} -0.451085 q^{83} -1.45707 q^{84} +1.43770 q^{85} -5.80579 q^{86} -3.60480 q^{87} +0.513205 q^{88} +0.196823 q^{89} +1.82988 q^{90} -1.45707 q^{91} +5.13440 q^{92} +8.18455 q^{93} -4.47220 q^{94} +2.06528 q^{95} +1.00000 q^{96} +0.774045 q^{97} +4.87696 q^{98} -0.513205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} - 10 q^{3} + 10 q^{4} + 6 q^{5} + 10 q^{6} - 9 q^{7} - 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} - 10 q^{3} + 10 q^{4} + 6 q^{5} + 10 q^{6} - 9 q^{7} - 10 q^{8} + 10 q^{9} - 6 q^{10} - q^{11} - 10 q^{12} - 10 q^{13} + 9 q^{14} - 6 q^{15} + 10 q^{16} + 5 q^{17} - 10 q^{18} - 9 q^{19} + 6 q^{20} + 9 q^{21} + q^{22} + q^{23} + 10 q^{24} + 20 q^{25} + 10 q^{26} - 10 q^{27} - 9 q^{28} - 22 q^{29} + 6 q^{30} - 13 q^{31} - 10 q^{32} + q^{33} - 5 q^{34} + 14 q^{35} + 10 q^{36} + 10 q^{37} + 9 q^{38} + 10 q^{39} - 6 q^{40} - 18 q^{41} - 9 q^{42} + 10 q^{43} - q^{44} + 6 q^{45} - q^{46} + 28 q^{47} - 10 q^{48} + 11 q^{49} - 20 q^{50} - 5 q^{51} - 10 q^{52} + 6 q^{53} + 10 q^{54} - 26 q^{55} + 9 q^{56} + 9 q^{57} + 22 q^{58} + 7 q^{59} - 6 q^{60} - 20 q^{61} + 13 q^{62} - 9 q^{63} + 10 q^{64} - 6 q^{65} - q^{66} - 21 q^{67} + 5 q^{68} - q^{69} - 14 q^{70} - 19 q^{71} - 10 q^{72} + 3 q^{73} - 10 q^{74} - 20 q^{75} - 9 q^{76} + 28 q^{77} - 10 q^{78} - 11 q^{79} + 6 q^{80} + 10 q^{81} + 18 q^{82} + 20 q^{83} + 9 q^{84} - q^{85} - 10 q^{86} + 22 q^{87} + q^{88} + 22 q^{89} - 6 q^{90} + 9 q^{91} + q^{92} + 13 q^{93} - 28 q^{94} + 10 q^{96} - 10 q^{97} - 11 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.82988 −0.818346 −0.409173 0.912457i \(-0.634183\pi\)
−0.409173 + 0.912457i \(0.634183\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.45707 0.550719 0.275360 0.961341i \(-0.411203\pi\)
0.275360 + 0.961341i \(0.411203\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.82988 0.578658
\(11\) −0.513205 −0.154737 −0.0773686 0.997003i \(-0.524652\pi\)
−0.0773686 + 0.997003i \(0.524652\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.45707 −0.389417
\(15\) 1.82988 0.472472
\(16\) 1.00000 0.250000
\(17\) −0.785684 −0.190556 −0.0952781 0.995451i \(-0.530374\pi\)
−0.0952781 + 0.995451i \(0.530374\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.12864 −0.258928 −0.129464 0.991584i \(-0.541326\pi\)
−0.129464 + 0.991584i \(0.541326\pi\)
\(20\) −1.82988 −0.409173
\(21\) −1.45707 −0.317958
\(22\) 0.513205 0.109416
\(23\) 5.13440 1.07060 0.535298 0.844663i \(-0.320199\pi\)
0.535298 + 0.844663i \(0.320199\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.65155 −0.330310
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.45707 0.275360
\(29\) 3.60480 0.669394 0.334697 0.942326i \(-0.391366\pi\)
0.334697 + 0.942326i \(0.391366\pi\)
\(30\) −1.82988 −0.334088
\(31\) −8.18455 −1.46999 −0.734994 0.678074i \(-0.762815\pi\)
−0.734994 + 0.678074i \(0.762815\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.513205 0.0893375
\(34\) 0.785684 0.134744
\(35\) −2.66625 −0.450679
\(36\) 1.00000 0.166667
\(37\) 2.66489 0.438106 0.219053 0.975713i \(-0.429703\pi\)
0.219053 + 0.975713i \(0.429703\pi\)
\(38\) 1.12864 0.183090
\(39\) 1.00000 0.160128
\(40\) 1.82988 0.289329
\(41\) 5.68841 0.888381 0.444190 0.895932i \(-0.353491\pi\)
0.444190 + 0.895932i \(0.353491\pi\)
\(42\) 1.45707 0.224830
\(43\) 5.80579 0.885375 0.442688 0.896676i \(-0.354025\pi\)
0.442688 + 0.896676i \(0.354025\pi\)
\(44\) −0.513205 −0.0773686
\(45\) −1.82988 −0.272782
\(46\) −5.13440 −0.757026
\(47\) 4.47220 0.652337 0.326168 0.945312i \(-0.394242\pi\)
0.326168 + 0.945312i \(0.394242\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.87696 −0.696708
\(50\) 1.65155 0.233564
\(51\) 0.785684 0.110018
\(52\) −1.00000 −0.138675
\(53\) −9.91273 −1.36162 −0.680809 0.732461i \(-0.738372\pi\)
−0.680809 + 0.732461i \(0.738372\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.939102 0.126629
\(56\) −1.45707 −0.194709
\(57\) 1.12864 0.149492
\(58\) −3.60480 −0.473333
\(59\) 10.2889 1.33951 0.669753 0.742584i \(-0.266400\pi\)
0.669753 + 0.742584i \(0.266400\pi\)
\(60\) 1.82988 0.236236
\(61\) −9.58790 −1.22760 −0.613802 0.789460i \(-0.710361\pi\)
−0.613802 + 0.789460i \(0.710361\pi\)
\(62\) 8.18455 1.03944
\(63\) 1.45707 0.183573
\(64\) 1.00000 0.125000
\(65\) 1.82988 0.226968
\(66\) −0.513205 −0.0631712
\(67\) −11.1866 −1.36667 −0.683333 0.730107i \(-0.739470\pi\)
−0.683333 + 0.730107i \(0.739470\pi\)
\(68\) −0.785684 −0.0952781
\(69\) −5.13440 −0.618109
\(70\) 2.66625 0.318678
\(71\) 4.34398 0.515536 0.257768 0.966207i \(-0.417013\pi\)
0.257768 + 0.966207i \(0.417013\pi\)
\(72\) −1.00000 −0.117851
\(73\) 0.434059 0.0508027 0.0254014 0.999677i \(-0.491914\pi\)
0.0254014 + 0.999677i \(0.491914\pi\)
\(74\) −2.66489 −0.309788
\(75\) 1.65155 0.190704
\(76\) −1.12864 −0.129464
\(77\) −0.747774 −0.0852167
\(78\) −1.00000 −0.113228
\(79\) 7.70913 0.867345 0.433672 0.901071i \(-0.357218\pi\)
0.433672 + 0.901071i \(0.357218\pi\)
\(80\) −1.82988 −0.204587
\(81\) 1.00000 0.111111
\(82\) −5.68841 −0.628180
\(83\) −0.451085 −0.0495130 −0.0247565 0.999694i \(-0.507881\pi\)
−0.0247565 + 0.999694i \(0.507881\pi\)
\(84\) −1.45707 −0.158979
\(85\) 1.43770 0.155941
\(86\) −5.80579 −0.626055
\(87\) −3.60480 −0.386475
\(88\) 0.513205 0.0547078
\(89\) 0.196823 0.0208632 0.0104316 0.999946i \(-0.496679\pi\)
0.0104316 + 0.999946i \(0.496679\pi\)
\(90\) 1.82988 0.192886
\(91\) −1.45707 −0.152742
\(92\) 5.13440 0.535298
\(93\) 8.18455 0.848698
\(94\) −4.47220 −0.461272
\(95\) 2.06528 0.211893
\(96\) 1.00000 0.102062
\(97\) 0.774045 0.0785923 0.0392962 0.999228i \(-0.487488\pi\)
0.0392962 + 0.999228i \(0.487488\pi\)
\(98\) 4.87696 0.492647
\(99\) −0.513205 −0.0515790
\(100\) −1.65155 −0.165155
\(101\) 13.7255 1.36574 0.682870 0.730540i \(-0.260731\pi\)
0.682870 + 0.730540i \(0.260731\pi\)
\(102\) −0.785684 −0.0777943
\(103\) −1.00000 −0.0985329
\(104\) 1.00000 0.0980581
\(105\) 2.66625 0.260200
\(106\) 9.91273 0.962809
\(107\) 11.3226 1.09460 0.547301 0.836936i \(-0.315655\pi\)
0.547301 + 0.836936i \(0.315655\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.23121 0.405276 0.202638 0.979254i \(-0.435049\pi\)
0.202638 + 0.979254i \(0.435049\pi\)
\(110\) −0.939102 −0.0895399
\(111\) −2.66489 −0.252940
\(112\) 1.45707 0.137680
\(113\) −17.2827 −1.62582 −0.812910 0.582389i \(-0.802118\pi\)
−0.812910 + 0.582389i \(0.802118\pi\)
\(114\) −1.12864 −0.105707
\(115\) −9.39533 −0.876119
\(116\) 3.60480 0.334697
\(117\) −1.00000 −0.0924500
\(118\) −10.2889 −0.947173
\(119\) −1.14479 −0.104943
\(120\) −1.82988 −0.167044
\(121\) −10.7366 −0.976056
\(122\) 9.58790 0.868048
\(123\) −5.68841 −0.512907
\(124\) −8.18455 −0.734994
\(125\) 12.1715 1.08865
\(126\) −1.45707 −0.129806
\(127\) −9.36539 −0.831044 −0.415522 0.909583i \(-0.636401\pi\)
−0.415522 + 0.909583i \(0.636401\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.80579 −0.511172
\(130\) −1.82988 −0.160491
\(131\) −11.5102 −1.00565 −0.502826 0.864388i \(-0.667706\pi\)
−0.502826 + 0.864388i \(0.667706\pi\)
\(132\) 0.513205 0.0446688
\(133\) −1.64451 −0.142597
\(134\) 11.1866 0.966378
\(135\) 1.82988 0.157491
\(136\) 0.785684 0.0673718
\(137\) −2.77252 −0.236872 −0.118436 0.992962i \(-0.537788\pi\)
−0.118436 + 0.992962i \(0.537788\pi\)
\(138\) 5.13440 0.437069
\(139\) −11.4546 −0.971568 −0.485784 0.874079i \(-0.661466\pi\)
−0.485784 + 0.874079i \(0.661466\pi\)
\(140\) −2.66625 −0.225340
\(141\) −4.47220 −0.376627
\(142\) −4.34398 −0.364539
\(143\) 0.513205 0.0429164
\(144\) 1.00000 0.0833333
\(145\) −6.59634 −0.547796
\(146\) −0.434059 −0.0359230
\(147\) 4.87696 0.402245
\(148\) 2.66489 0.219053
\(149\) −14.3197 −1.17311 −0.586556 0.809909i \(-0.699517\pi\)
−0.586556 + 0.809909i \(0.699517\pi\)
\(150\) −1.65155 −0.134848
\(151\) −9.19404 −0.748200 −0.374100 0.927388i \(-0.622048\pi\)
−0.374100 + 0.927388i \(0.622048\pi\)
\(152\) 1.12864 0.0915449
\(153\) −0.785684 −0.0635188
\(154\) 0.747774 0.0602573
\(155\) 14.9767 1.20296
\(156\) 1.00000 0.0800641
\(157\) 22.9121 1.82858 0.914291 0.405058i \(-0.132749\pi\)
0.914291 + 0.405058i \(0.132749\pi\)
\(158\) −7.70913 −0.613305
\(159\) 9.91273 0.786131
\(160\) 1.82988 0.144665
\(161\) 7.48116 0.589598
\(162\) −1.00000 −0.0785674
\(163\) −13.1323 −1.02860 −0.514301 0.857610i \(-0.671949\pi\)
−0.514301 + 0.857610i \(0.671949\pi\)
\(164\) 5.68841 0.444190
\(165\) −0.939102 −0.0731090
\(166\) 0.451085 0.0350110
\(167\) 12.9880 1.00504 0.502521 0.864565i \(-0.332406\pi\)
0.502521 + 0.864565i \(0.332406\pi\)
\(168\) 1.45707 0.112415
\(169\) 1.00000 0.0769231
\(170\) −1.43770 −0.110267
\(171\) −1.12864 −0.0863094
\(172\) 5.80579 0.442688
\(173\) 25.8509 1.96541 0.982705 0.185176i \(-0.0592855\pi\)
0.982705 + 0.185176i \(0.0592855\pi\)
\(174\) 3.60480 0.273279
\(175\) −2.40641 −0.181908
\(176\) −0.513205 −0.0386843
\(177\) −10.2889 −0.773364
\(178\) −0.196823 −0.0147525
\(179\) 17.2268 1.28759 0.643797 0.765196i \(-0.277358\pi\)
0.643797 + 0.765196i \(0.277358\pi\)
\(180\) −1.82988 −0.136391
\(181\) 9.57039 0.711361 0.355681 0.934608i \(-0.384249\pi\)
0.355681 + 0.934608i \(0.384249\pi\)
\(182\) 1.45707 0.108005
\(183\) 9.58790 0.708758
\(184\) −5.13440 −0.378513
\(185\) −4.87643 −0.358522
\(186\) −8.18455 −0.600120
\(187\) 0.403217 0.0294861
\(188\) 4.47220 0.326168
\(189\) −1.45707 −0.105986
\(190\) −2.06528 −0.149831
\(191\) 12.4477 0.900685 0.450343 0.892856i \(-0.351302\pi\)
0.450343 + 0.892856i \(0.351302\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 5.35638 0.385561 0.192780 0.981242i \(-0.438250\pi\)
0.192780 + 0.981242i \(0.438250\pi\)
\(194\) −0.774045 −0.0555732
\(195\) −1.82988 −0.131040
\(196\) −4.87696 −0.348354
\(197\) 5.23510 0.372985 0.186493 0.982456i \(-0.440288\pi\)
0.186493 + 0.982456i \(0.440288\pi\)
\(198\) 0.513205 0.0364719
\(199\) 10.4226 0.738836 0.369418 0.929263i \(-0.379557\pi\)
0.369418 + 0.929263i \(0.379557\pi\)
\(200\) 1.65155 0.116782
\(201\) 11.1866 0.789044
\(202\) −13.7255 −0.965725
\(203\) 5.25243 0.368648
\(204\) 0.785684 0.0550089
\(205\) −10.4091 −0.727003
\(206\) 1.00000 0.0696733
\(207\) 5.13440 0.356866
\(208\) −1.00000 −0.0693375
\(209\) 0.579225 0.0400658
\(210\) −2.66625 −0.183989
\(211\) 25.9138 1.78398 0.891990 0.452054i \(-0.149309\pi\)
0.891990 + 0.452054i \(0.149309\pi\)
\(212\) −9.91273 −0.680809
\(213\) −4.34398 −0.297645
\(214\) −11.3226 −0.774000
\(215\) −10.6239 −0.724543
\(216\) 1.00000 0.0680414
\(217\) −11.9254 −0.809551
\(218\) −4.23121 −0.286573
\(219\) −0.434059 −0.0293310
\(220\) 0.939102 0.0633143
\(221\) 0.785684 0.0528508
\(222\) 2.66489 0.178856
\(223\) −23.7842 −1.59271 −0.796355 0.604829i \(-0.793241\pi\)
−0.796355 + 0.604829i \(0.793241\pi\)
\(224\) −1.45707 −0.0973543
\(225\) −1.65155 −0.110103
\(226\) 17.2827 1.14963
\(227\) −12.9676 −0.860690 −0.430345 0.902664i \(-0.641608\pi\)
−0.430345 + 0.902664i \(0.641608\pi\)
\(228\) 1.12864 0.0747461
\(229\) −13.2509 −0.875642 −0.437821 0.899062i \(-0.644250\pi\)
−0.437821 + 0.899062i \(0.644250\pi\)
\(230\) 9.39533 0.619509
\(231\) 0.747774 0.0491999
\(232\) −3.60480 −0.236666
\(233\) −19.7166 −1.29168 −0.645839 0.763474i \(-0.723492\pi\)
−0.645839 + 0.763474i \(0.723492\pi\)
\(234\) 1.00000 0.0653720
\(235\) −8.18357 −0.533837
\(236\) 10.2889 0.669753
\(237\) −7.70913 −0.500762
\(238\) 1.14479 0.0742059
\(239\) −24.4023 −1.57846 −0.789228 0.614101i \(-0.789519\pi\)
−0.789228 + 0.614101i \(0.789519\pi\)
\(240\) 1.82988 0.118118
\(241\) −7.27230 −0.468450 −0.234225 0.972182i \(-0.575255\pi\)
−0.234225 + 0.972182i \(0.575255\pi\)
\(242\) 10.7366 0.690176
\(243\) −1.00000 −0.0641500
\(244\) −9.58790 −0.613802
\(245\) 8.92424 0.570149
\(246\) 5.68841 0.362680
\(247\) 1.12864 0.0718137
\(248\) 8.18455 0.519719
\(249\) 0.451085 0.0285864
\(250\) −12.1715 −0.769794
\(251\) −5.76786 −0.364064 −0.182032 0.983293i \(-0.558267\pi\)
−0.182032 + 0.983293i \(0.558267\pi\)
\(252\) 1.45707 0.0917865
\(253\) −2.63500 −0.165661
\(254\) 9.36539 0.587637
\(255\) −1.43770 −0.0900326
\(256\) 1.00000 0.0625000
\(257\) −7.06058 −0.440427 −0.220213 0.975452i \(-0.570675\pi\)
−0.220213 + 0.975452i \(0.570675\pi\)
\(258\) 5.80579 0.361453
\(259\) 3.88293 0.241273
\(260\) 1.82988 0.113484
\(261\) 3.60480 0.223131
\(262\) 11.5102 0.711103
\(263\) −27.5232 −1.69715 −0.848575 0.529075i \(-0.822539\pi\)
−0.848575 + 0.529075i \(0.822539\pi\)
\(264\) −0.513205 −0.0315856
\(265\) 18.1391 1.11428
\(266\) 1.64451 0.100831
\(267\) −0.196823 −0.0120454
\(268\) −11.1866 −0.683333
\(269\) 15.1132 0.921467 0.460733 0.887539i \(-0.347586\pi\)
0.460733 + 0.887539i \(0.347586\pi\)
\(270\) −1.82988 −0.111363
\(271\) −8.91417 −0.541497 −0.270749 0.962650i \(-0.587271\pi\)
−0.270749 + 0.962650i \(0.587271\pi\)
\(272\) −0.785684 −0.0476391
\(273\) 1.45707 0.0881857
\(274\) 2.77252 0.167494
\(275\) 0.847583 0.0511112
\(276\) −5.13440 −0.309055
\(277\) −2.29177 −0.137699 −0.0688495 0.997627i \(-0.521933\pi\)
−0.0688495 + 0.997627i \(0.521933\pi\)
\(278\) 11.4546 0.687003
\(279\) −8.18455 −0.489996
\(280\) 2.66625 0.159339
\(281\) 29.9684 1.78776 0.893882 0.448303i \(-0.147971\pi\)
0.893882 + 0.448303i \(0.147971\pi\)
\(282\) 4.47220 0.266315
\(283\) 27.0645 1.60882 0.804410 0.594075i \(-0.202482\pi\)
0.804410 + 0.594075i \(0.202482\pi\)
\(284\) 4.34398 0.257768
\(285\) −2.06528 −0.122336
\(286\) −0.513205 −0.0303465
\(287\) 8.28840 0.489248
\(288\) −1.00000 −0.0589256
\(289\) −16.3827 −0.963688
\(290\) 6.59634 0.387350
\(291\) −0.774045 −0.0453753
\(292\) 0.434059 0.0254014
\(293\) −16.0466 −0.937454 −0.468727 0.883343i \(-0.655287\pi\)
−0.468727 + 0.883343i \(0.655287\pi\)
\(294\) −4.87696 −0.284430
\(295\) −18.8275 −1.09618
\(296\) −2.66489 −0.154894
\(297\) 0.513205 0.0297792
\(298\) 14.3197 0.829515
\(299\) −5.13440 −0.296930
\(300\) 1.65155 0.0953522
\(301\) 8.45943 0.487593
\(302\) 9.19404 0.529057
\(303\) −13.7255 −0.788511
\(304\) −1.12864 −0.0647320
\(305\) 17.5447 1.00461
\(306\) 0.785684 0.0449145
\(307\) −0.570165 −0.0325410 −0.0162705 0.999868i \(-0.505179\pi\)
−0.0162705 + 0.999868i \(0.505179\pi\)
\(308\) −0.747774 −0.0426084
\(309\) 1.00000 0.0568880
\(310\) −14.9767 −0.850620
\(311\) −13.7186 −0.777909 −0.388954 0.921257i \(-0.627164\pi\)
−0.388954 + 0.921257i \(0.627164\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −8.93255 −0.504898 −0.252449 0.967610i \(-0.581236\pi\)
−0.252449 + 0.967610i \(0.581236\pi\)
\(314\) −22.9121 −1.29300
\(315\) −2.66625 −0.150226
\(316\) 7.70913 0.433672
\(317\) 2.51336 0.141164 0.0705821 0.997506i \(-0.477514\pi\)
0.0705821 + 0.997506i \(0.477514\pi\)
\(318\) −9.91273 −0.555878
\(319\) −1.85000 −0.103580
\(320\) −1.82988 −0.102293
\(321\) −11.3226 −0.631968
\(322\) −7.48116 −0.416909
\(323\) 0.886755 0.0493404
\(324\) 1.00000 0.0555556
\(325\) 1.65155 0.0916114
\(326\) 13.1323 0.727332
\(327\) −4.23121 −0.233986
\(328\) −5.68841 −0.314090
\(329\) 6.51629 0.359254
\(330\) 0.939102 0.0516959
\(331\) −6.17555 −0.339439 −0.169719 0.985492i \(-0.554286\pi\)
−0.169719 + 0.985492i \(0.554286\pi\)
\(332\) −0.451085 −0.0247565
\(333\) 2.66489 0.146035
\(334\) −12.9880 −0.710672
\(335\) 20.4702 1.11841
\(336\) −1.45707 −0.0794895
\(337\) 22.9070 1.24782 0.623911 0.781496i \(-0.285543\pi\)
0.623911 + 0.781496i \(0.285543\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 17.2827 0.938668
\(340\) 1.43770 0.0779705
\(341\) 4.20035 0.227462
\(342\) 1.12864 0.0610299
\(343\) −17.3055 −0.934410
\(344\) −5.80579 −0.313027
\(345\) 9.39533 0.505827
\(346\) −25.8509 −1.38976
\(347\) 20.6315 1.10756 0.553779 0.832663i \(-0.313185\pi\)
0.553779 + 0.832663i \(0.313185\pi\)
\(348\) −3.60480 −0.193237
\(349\) 10.8998 0.583452 0.291726 0.956502i \(-0.405770\pi\)
0.291726 + 0.956502i \(0.405770\pi\)
\(350\) 2.40641 0.128628
\(351\) 1.00000 0.0533761
\(352\) 0.513205 0.0273539
\(353\) −1.25568 −0.0668332 −0.0334166 0.999442i \(-0.510639\pi\)
−0.0334166 + 0.999442i \(0.510639\pi\)
\(354\) 10.2889 0.546851
\(355\) −7.94896 −0.421887
\(356\) 0.196823 0.0104316
\(357\) 1.14479 0.0605889
\(358\) −17.2268 −0.910467
\(359\) −19.3745 −1.02255 −0.511274 0.859418i \(-0.670826\pi\)
−0.511274 + 0.859418i \(0.670826\pi\)
\(360\) 1.82988 0.0964430
\(361\) −17.7262 −0.932956
\(362\) −9.57039 −0.503009
\(363\) 10.7366 0.563526
\(364\) −1.45707 −0.0763710
\(365\) −0.794274 −0.0415742
\(366\) −9.58790 −0.501168
\(367\) 12.8242 0.669417 0.334709 0.942322i \(-0.391362\pi\)
0.334709 + 0.942322i \(0.391362\pi\)
\(368\) 5.13440 0.267649
\(369\) 5.68841 0.296127
\(370\) 4.87643 0.253513
\(371\) −14.4435 −0.749869
\(372\) 8.18455 0.424349
\(373\) 14.3607 0.743570 0.371785 0.928319i \(-0.378746\pi\)
0.371785 + 0.928319i \(0.378746\pi\)
\(374\) −0.403217 −0.0208498
\(375\) −12.1715 −0.628535
\(376\) −4.47220 −0.230636
\(377\) −3.60480 −0.185656
\(378\) 1.45707 0.0749434
\(379\) 9.96185 0.511706 0.255853 0.966716i \(-0.417644\pi\)
0.255853 + 0.966716i \(0.417644\pi\)
\(380\) 2.06528 0.105946
\(381\) 9.36539 0.479803
\(382\) −12.4477 −0.636881
\(383\) −5.93297 −0.303161 −0.151580 0.988445i \(-0.548436\pi\)
−0.151580 + 0.988445i \(0.548436\pi\)
\(384\) 1.00000 0.0510310
\(385\) 1.36833 0.0697368
\(386\) −5.35638 −0.272633
\(387\) 5.80579 0.295125
\(388\) 0.774045 0.0392962
\(389\) −25.3451 −1.28505 −0.642523 0.766267i \(-0.722112\pi\)
−0.642523 + 0.766267i \(0.722112\pi\)
\(390\) 1.82988 0.0926595
\(391\) −4.03401 −0.204009
\(392\) 4.87696 0.246324
\(393\) 11.5102 0.580613
\(394\) −5.23510 −0.263740
\(395\) −14.1068 −0.709788
\(396\) −0.513205 −0.0257895
\(397\) −27.8148 −1.39599 −0.697993 0.716104i \(-0.745923\pi\)
−0.697993 + 0.716104i \(0.745923\pi\)
\(398\) −10.4226 −0.522436
\(399\) 1.64451 0.0823283
\(400\) −1.65155 −0.0825774
\(401\) −37.3383 −1.86459 −0.932293 0.361704i \(-0.882195\pi\)
−0.932293 + 0.361704i \(0.882195\pi\)
\(402\) −11.1866 −0.557939
\(403\) 8.18455 0.407701
\(404\) 13.7255 0.682870
\(405\) −1.82988 −0.0909273
\(406\) −5.25243 −0.260674
\(407\) −1.36764 −0.0677912
\(408\) −0.785684 −0.0388971
\(409\) −18.2282 −0.901326 −0.450663 0.892694i \(-0.648812\pi\)
−0.450663 + 0.892694i \(0.648812\pi\)
\(410\) 10.4091 0.514069
\(411\) 2.77252 0.136758
\(412\) −1.00000 −0.0492665
\(413\) 14.9917 0.737691
\(414\) −5.13440 −0.252342
\(415\) 0.825431 0.0405188
\(416\) 1.00000 0.0490290
\(417\) 11.4546 0.560935
\(418\) −0.579225 −0.0283308
\(419\) 23.0355 1.12536 0.562680 0.826675i \(-0.309770\pi\)
0.562680 + 0.826675i \(0.309770\pi\)
\(420\) 2.66625 0.130100
\(421\) −37.1791 −1.81200 −0.905999 0.423279i \(-0.860879\pi\)
−0.905999 + 0.423279i \(0.860879\pi\)
\(422\) −25.9138 −1.26146
\(423\) 4.47220 0.217446
\(424\) 9.91273 0.481405
\(425\) 1.29759 0.0629426
\(426\) 4.34398 0.210467
\(427\) −13.9702 −0.676066
\(428\) 11.3226 0.547301
\(429\) −0.513205 −0.0247778
\(430\) 10.6239 0.512330
\(431\) −9.91314 −0.477499 −0.238750 0.971081i \(-0.576738\pi\)
−0.238750 + 0.971081i \(0.576738\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 0.898037 0.0431569 0.0215785 0.999767i \(-0.493131\pi\)
0.0215785 + 0.999767i \(0.493131\pi\)
\(434\) 11.9254 0.572439
\(435\) 6.59634 0.316270
\(436\) 4.23121 0.202638
\(437\) −5.79490 −0.277208
\(438\) 0.434059 0.0207401
\(439\) −20.9744 −1.00105 −0.500527 0.865721i \(-0.666860\pi\)
−0.500527 + 0.865721i \(0.666860\pi\)
\(440\) −0.939102 −0.0447700
\(441\) −4.87696 −0.232236
\(442\) −0.785684 −0.0373712
\(443\) −12.8861 −0.612237 −0.306118 0.951993i \(-0.599030\pi\)
−0.306118 + 0.951993i \(0.599030\pi\)
\(444\) −2.66489 −0.126470
\(445\) −0.360162 −0.0170733
\(446\) 23.7842 1.12622
\(447\) 14.3197 0.677296
\(448\) 1.45707 0.0688399
\(449\) −1.17918 −0.0556489 −0.0278245 0.999613i \(-0.508858\pi\)
−0.0278245 + 0.999613i \(0.508858\pi\)
\(450\) 1.65155 0.0778547
\(451\) −2.91932 −0.137466
\(452\) −17.2827 −0.812910
\(453\) 9.19404 0.431973
\(454\) 12.9676 0.608600
\(455\) 2.66625 0.124996
\(456\) −1.12864 −0.0528535
\(457\) −22.1846 −1.03775 −0.518876 0.854849i \(-0.673649\pi\)
−0.518876 + 0.854849i \(0.673649\pi\)
\(458\) 13.2509 0.619172
\(459\) 0.785684 0.0366726
\(460\) −9.39533 −0.438059
\(461\) −1.97709 −0.0920822 −0.0460411 0.998940i \(-0.514661\pi\)
−0.0460411 + 0.998940i \(0.514661\pi\)
\(462\) −0.747774 −0.0347896
\(463\) 29.1462 1.35454 0.677268 0.735736i \(-0.263164\pi\)
0.677268 + 0.735736i \(0.263164\pi\)
\(464\) 3.60480 0.167348
\(465\) −14.9767 −0.694529
\(466\) 19.7166 0.913354
\(467\) −1.07865 −0.0499138 −0.0249569 0.999689i \(-0.507945\pi\)
−0.0249569 + 0.999689i \(0.507945\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −16.2997 −0.752649
\(470\) 8.18357 0.377480
\(471\) −22.9121 −1.05573
\(472\) −10.2889 −0.473587
\(473\) −2.97956 −0.137000
\(474\) 7.70913 0.354092
\(475\) 1.86401 0.0855264
\(476\) −1.14479 −0.0524715
\(477\) −9.91273 −0.453873
\(478\) 24.4023 1.11614
\(479\) 43.5747 1.99098 0.995488 0.0948823i \(-0.0302475\pi\)
0.995488 + 0.0948823i \(0.0302475\pi\)
\(480\) −1.82988 −0.0835221
\(481\) −2.66489 −0.121509
\(482\) 7.27230 0.331244
\(483\) −7.48116 −0.340405
\(484\) −10.7366 −0.488028
\(485\) −1.41641 −0.0643157
\(486\) 1.00000 0.0453609
\(487\) 2.28651 0.103612 0.0518059 0.998657i \(-0.483502\pi\)
0.0518059 + 0.998657i \(0.483502\pi\)
\(488\) 9.58790 0.434024
\(489\) 13.1323 0.593864
\(490\) −8.92424 −0.403156
\(491\) −2.07996 −0.0938672 −0.0469336 0.998898i \(-0.514945\pi\)
−0.0469336 + 0.998898i \(0.514945\pi\)
\(492\) −5.68841 −0.256453
\(493\) −2.83223 −0.127557
\(494\) −1.12864 −0.0507800
\(495\) 0.939102 0.0422095
\(496\) −8.18455 −0.367497
\(497\) 6.32947 0.283916
\(498\) −0.451085 −0.0202136
\(499\) −5.05021 −0.226078 −0.113039 0.993591i \(-0.536059\pi\)
−0.113039 + 0.993591i \(0.536059\pi\)
\(500\) 12.1715 0.544327
\(501\) −12.9880 −0.580261
\(502\) 5.76786 0.257432
\(503\) −25.8794 −1.15391 −0.576953 0.816777i \(-0.695759\pi\)
−0.576953 + 0.816777i \(0.695759\pi\)
\(504\) −1.45707 −0.0649029
\(505\) −25.1160 −1.11765
\(506\) 2.63500 0.117140
\(507\) −1.00000 −0.0444116
\(508\) −9.36539 −0.415522
\(509\) −35.6770 −1.58135 −0.790677 0.612233i \(-0.790271\pi\)
−0.790677 + 0.612233i \(0.790271\pi\)
\(510\) 1.43770 0.0636626
\(511\) 0.632452 0.0279780
\(512\) −1.00000 −0.0441942
\(513\) 1.12864 0.0498307
\(514\) 7.06058 0.311429
\(515\) 1.82988 0.0806340
\(516\) −5.80579 −0.255586
\(517\) −2.29515 −0.100941
\(518\) −3.88293 −0.170606
\(519\) −25.8509 −1.13473
\(520\) −1.82988 −0.0802454
\(521\) −19.1843 −0.840481 −0.420240 0.907413i \(-0.638054\pi\)
−0.420240 + 0.907413i \(0.638054\pi\)
\(522\) −3.60480 −0.157778
\(523\) 27.2070 1.18968 0.594840 0.803844i \(-0.297215\pi\)
0.594840 + 0.803844i \(0.297215\pi\)
\(524\) −11.5102 −0.502826
\(525\) 2.40641 0.105025
\(526\) 27.5232 1.20007
\(527\) 6.43046 0.280115
\(528\) 0.513205 0.0223344
\(529\) 3.36207 0.146177
\(530\) −18.1391 −0.787911
\(531\) 10.2889 0.446502
\(532\) −1.64451 −0.0712984
\(533\) −5.68841 −0.246393
\(534\) 0.196823 0.00851736
\(535\) −20.7191 −0.895763
\(536\) 11.1866 0.483189
\(537\) −17.2268 −0.743393
\(538\) −15.1132 −0.651575
\(539\) 2.50288 0.107807
\(540\) 1.82988 0.0787454
\(541\) 10.1326 0.435633 0.217816 0.975990i \(-0.430107\pi\)
0.217816 + 0.975990i \(0.430107\pi\)
\(542\) 8.91417 0.382896
\(543\) −9.57039 −0.410705
\(544\) 0.785684 0.0336859
\(545\) −7.74259 −0.331656
\(546\) −1.45707 −0.0623567
\(547\) −2.67032 −0.114175 −0.0570874 0.998369i \(-0.518181\pi\)
−0.0570874 + 0.998369i \(0.518181\pi\)
\(548\) −2.77252 −0.118436
\(549\) −9.58790 −0.409202
\(550\) −0.847583 −0.0361410
\(551\) −4.06852 −0.173325
\(552\) 5.13440 0.218535
\(553\) 11.2327 0.477663
\(554\) 2.29177 0.0973679
\(555\) 4.87643 0.206993
\(556\) −11.4546 −0.485784
\(557\) −36.9005 −1.56352 −0.781762 0.623577i \(-0.785679\pi\)
−0.781762 + 0.623577i \(0.785679\pi\)
\(558\) 8.18455 0.346479
\(559\) −5.80579 −0.245559
\(560\) −2.66625 −0.112670
\(561\) −0.403217 −0.0170238
\(562\) −29.9684 −1.26414
\(563\) −12.5724 −0.529865 −0.264933 0.964267i \(-0.585350\pi\)
−0.264933 + 0.964267i \(0.585350\pi\)
\(564\) −4.47220 −0.188313
\(565\) 31.6252 1.33048
\(566\) −27.0645 −1.13761
\(567\) 1.45707 0.0611910
\(568\) −4.34398 −0.182270
\(569\) −37.7583 −1.58291 −0.791455 0.611228i \(-0.790676\pi\)
−0.791455 + 0.611228i \(0.790676\pi\)
\(570\) 2.06528 0.0865049
\(571\) −17.5934 −0.736259 −0.368130 0.929774i \(-0.620002\pi\)
−0.368130 + 0.929774i \(0.620002\pi\)
\(572\) 0.513205 0.0214582
\(573\) −12.4477 −0.520011
\(574\) −8.28840 −0.345951
\(575\) −8.47971 −0.353628
\(576\) 1.00000 0.0416667
\(577\) 26.8868 1.11931 0.559656 0.828725i \(-0.310933\pi\)
0.559656 + 0.828725i \(0.310933\pi\)
\(578\) 16.3827 0.681431
\(579\) −5.35638 −0.222604
\(580\) −6.59634 −0.273898
\(581\) −0.657261 −0.0272678
\(582\) 0.774045 0.0320852
\(583\) 5.08726 0.210693
\(584\) −0.434059 −0.0179615
\(585\) 1.82988 0.0756561
\(586\) 16.0466 0.662880
\(587\) 13.7189 0.566242 0.283121 0.959084i \(-0.408630\pi\)
0.283121 + 0.959084i \(0.408630\pi\)
\(588\) 4.87696 0.201122
\(589\) 9.23742 0.380621
\(590\) 18.8275 0.775116
\(591\) −5.23510 −0.215343
\(592\) 2.66489 0.109526
\(593\) 16.5470 0.679504 0.339752 0.940515i \(-0.389657\pi\)
0.339752 + 0.940515i \(0.389657\pi\)
\(594\) −0.513205 −0.0210571
\(595\) 2.09483 0.0858797
\(596\) −14.3197 −0.586556
\(597\) −10.4226 −0.426567
\(598\) 5.13440 0.209961
\(599\) −44.4217 −1.81502 −0.907511 0.420028i \(-0.862020\pi\)
−0.907511 + 0.420028i \(0.862020\pi\)
\(600\) −1.65155 −0.0674242
\(601\) 0.0157036 0.000640562 0 0.000320281 1.00000i \(-0.499898\pi\)
0.000320281 1.00000i \(0.499898\pi\)
\(602\) −8.45943 −0.344780
\(603\) −11.1866 −0.455555
\(604\) −9.19404 −0.374100
\(605\) 19.6467 0.798752
\(606\) 13.7255 0.557561
\(607\) −34.7387 −1.41000 −0.705001 0.709207i \(-0.749053\pi\)
−0.705001 + 0.709207i \(0.749053\pi\)
\(608\) 1.12864 0.0457725
\(609\) −5.25243 −0.212839
\(610\) −17.5447 −0.710363
\(611\) −4.47220 −0.180926
\(612\) −0.785684 −0.0317594
\(613\) −38.7813 −1.56636 −0.783181 0.621794i \(-0.786404\pi\)
−0.783181 + 0.621794i \(0.786404\pi\)
\(614\) 0.570165 0.0230100
\(615\) 10.4091 0.419735
\(616\) 0.747774 0.0301287
\(617\) −3.46954 −0.139679 −0.0698393 0.997558i \(-0.522249\pi\)
−0.0698393 + 0.997558i \(0.522249\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −34.6311 −1.39194 −0.695971 0.718069i \(-0.745026\pi\)
−0.695971 + 0.718069i \(0.745026\pi\)
\(620\) 14.9767 0.601479
\(621\) −5.13440 −0.206036
\(622\) 13.7186 0.550065
\(623\) 0.286784 0.0114898
\(624\) 1.00000 0.0400320
\(625\) −14.0146 −0.560586
\(626\) 8.93255 0.357017
\(627\) −0.579225 −0.0231320
\(628\) 22.9121 0.914291
\(629\) −2.09376 −0.0834838
\(630\) 2.66625 0.106226
\(631\) −5.31616 −0.211633 −0.105816 0.994386i \(-0.533746\pi\)
−0.105816 + 0.994386i \(0.533746\pi\)
\(632\) −7.70913 −0.306653
\(633\) −25.9138 −1.02998
\(634\) −2.51336 −0.0998182
\(635\) 17.1375 0.680081
\(636\) 9.91273 0.393065
\(637\) 4.87696 0.193232
\(638\) 1.85000 0.0732422
\(639\) 4.34398 0.171845
\(640\) 1.82988 0.0723323
\(641\) −34.6319 −1.36788 −0.683940 0.729539i \(-0.739735\pi\)
−0.683940 + 0.729539i \(0.739735\pi\)
\(642\) 11.3226 0.446869
\(643\) 14.2684 0.562690 0.281345 0.959607i \(-0.409220\pi\)
0.281345 + 0.959607i \(0.409220\pi\)
\(644\) 7.48116 0.294799
\(645\) 10.6239 0.418315
\(646\) −0.886755 −0.0348889
\(647\) −32.7461 −1.28738 −0.643690 0.765286i \(-0.722597\pi\)
−0.643690 + 0.765286i \(0.722597\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.28033 −0.207271
\(650\) −1.65155 −0.0647790
\(651\) 11.9254 0.467394
\(652\) −13.1323 −0.514301
\(653\) −42.3825 −1.65855 −0.829277 0.558838i \(-0.811247\pi\)
−0.829277 + 0.558838i \(0.811247\pi\)
\(654\) 4.23121 0.165453
\(655\) 21.0623 0.822971
\(656\) 5.68841 0.222095
\(657\) 0.434059 0.0169342
\(658\) −6.51629 −0.254031
\(659\) −14.8469 −0.578352 −0.289176 0.957276i \(-0.593381\pi\)
−0.289176 + 0.957276i \(0.593381\pi\)
\(660\) −0.939102 −0.0365545
\(661\) 2.53869 0.0987436 0.0493718 0.998780i \(-0.484278\pi\)
0.0493718 + 0.998780i \(0.484278\pi\)
\(662\) 6.17555 0.240019
\(663\) −0.785684 −0.0305134
\(664\) 0.451085 0.0175055
\(665\) 3.00924 0.116693
\(666\) −2.66489 −0.103263
\(667\) 18.5085 0.716651
\(668\) 12.9880 0.502521
\(669\) 23.7842 0.919552
\(670\) −20.4702 −0.790832
\(671\) 4.92056 0.189956
\(672\) 1.45707 0.0562076
\(673\) 38.2772 1.47548 0.737739 0.675086i \(-0.235893\pi\)
0.737739 + 0.675086i \(0.235893\pi\)
\(674\) −22.9070 −0.882343
\(675\) 1.65155 0.0635681
\(676\) 1.00000 0.0384615
\(677\) −4.80497 −0.184670 −0.0923350 0.995728i \(-0.529433\pi\)
−0.0923350 + 0.995728i \(0.529433\pi\)
\(678\) −17.2827 −0.663738
\(679\) 1.12783 0.0432823
\(680\) −1.43770 −0.0551335
\(681\) 12.9676 0.496920
\(682\) −4.20035 −0.160840
\(683\) −41.1100 −1.57303 −0.786514 0.617572i \(-0.788116\pi\)
−0.786514 + 0.617572i \(0.788116\pi\)
\(684\) −1.12864 −0.0431547
\(685\) 5.07337 0.193843
\(686\) 17.3055 0.660728
\(687\) 13.2509 0.505552
\(688\) 5.80579 0.221344
\(689\) 9.91273 0.377645
\(690\) −9.39533 −0.357674
\(691\) −36.1423 −1.37492 −0.687458 0.726224i \(-0.741274\pi\)
−0.687458 + 0.726224i \(0.741274\pi\)
\(692\) 25.8509 0.982705
\(693\) −0.747774 −0.0284056
\(694\) −20.6315 −0.783162
\(695\) 20.9606 0.795079
\(696\) 3.60480 0.136639
\(697\) −4.46929 −0.169287
\(698\) −10.8998 −0.412563
\(699\) 19.7166 0.745750
\(700\) −2.40641 −0.0909539
\(701\) 42.6017 1.60904 0.804522 0.593923i \(-0.202422\pi\)
0.804522 + 0.593923i \(0.202422\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −3.00771 −0.113438
\(704\) −0.513205 −0.0193421
\(705\) 8.18357 0.308211
\(706\) 1.25568 0.0472582
\(707\) 19.9990 0.752140
\(708\) −10.2889 −0.386682
\(709\) −12.9681 −0.487027 −0.243513 0.969898i \(-0.578300\pi\)
−0.243513 + 0.969898i \(0.578300\pi\)
\(710\) 7.94896 0.298319
\(711\) 7.70913 0.289115
\(712\) −0.196823 −0.00737625
\(713\) −42.0227 −1.57376
\(714\) −1.14479 −0.0428428
\(715\) −0.939102 −0.0351204
\(716\) 17.2268 0.643797
\(717\) 24.4023 0.911322
\(718\) 19.3745 0.723050
\(719\) 28.7639 1.07271 0.536356 0.843992i \(-0.319800\pi\)
0.536356 + 0.843992i \(0.319800\pi\)
\(720\) −1.82988 −0.0681955
\(721\) −1.45707 −0.0542640
\(722\) 17.7262 0.659700
\(723\) 7.27230 0.270460
\(724\) 9.57039 0.355681
\(725\) −5.95349 −0.221107
\(726\) −10.7366 −0.398473
\(727\) −0.295380 −0.0109550 −0.00547751 0.999985i \(-0.501744\pi\)
−0.00547751 + 0.999985i \(0.501744\pi\)
\(728\) 1.45707 0.0540025
\(729\) 1.00000 0.0370370
\(730\) 0.794274 0.0293974
\(731\) −4.56152 −0.168714
\(732\) 9.58790 0.354379
\(733\) 8.63759 0.319037 0.159518 0.987195i \(-0.449006\pi\)
0.159518 + 0.987195i \(0.449006\pi\)
\(734\) −12.8242 −0.473350
\(735\) −8.92424 −0.329175
\(736\) −5.13440 −0.189257
\(737\) 5.74104 0.211474
\(738\) −5.68841 −0.209393
\(739\) −13.6062 −0.500513 −0.250256 0.968180i \(-0.580515\pi\)
−0.250256 + 0.968180i \(0.580515\pi\)
\(740\) −4.87643 −0.179261
\(741\) −1.12864 −0.0414617
\(742\) 14.4435 0.530238
\(743\) 5.71623 0.209708 0.104854 0.994488i \(-0.466562\pi\)
0.104854 + 0.994488i \(0.466562\pi\)
\(744\) −8.18455 −0.300060
\(745\) 26.2032 0.960012
\(746\) −14.3607 −0.525784
\(747\) −0.451085 −0.0165043
\(748\) 0.403217 0.0147431
\(749\) 16.4978 0.602818
\(750\) 12.1715 0.444441
\(751\) −10.6900 −0.390084 −0.195042 0.980795i \(-0.562484\pi\)
−0.195042 + 0.980795i \(0.562484\pi\)
\(752\) 4.47220 0.163084
\(753\) 5.76786 0.210192
\(754\) 3.60480 0.131279
\(755\) 16.8240 0.612287
\(756\) −1.45707 −0.0529930
\(757\) −20.4338 −0.742678 −0.371339 0.928497i \(-0.621101\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(758\) −9.96185 −0.361831
\(759\) 2.63500 0.0956445
\(760\) −2.06528 −0.0749154
\(761\) 15.5349 0.563140 0.281570 0.959541i \(-0.409145\pi\)
0.281570 + 0.959541i \(0.409145\pi\)
\(762\) −9.36539 −0.339272
\(763\) 6.16515 0.223193
\(764\) 12.4477 0.450343
\(765\) 1.43770 0.0519803
\(766\) 5.93297 0.214367
\(767\) −10.2889 −0.371512
\(768\) −1.00000 −0.0360844
\(769\) −20.4423 −0.737167 −0.368583 0.929595i \(-0.620157\pi\)
−0.368583 + 0.929595i \(0.620157\pi\)
\(770\) −1.36833 −0.0493114
\(771\) 7.06058 0.254281
\(772\) 5.35638 0.192780
\(773\) 36.3765 1.30837 0.654186 0.756334i \(-0.273011\pi\)
0.654186 + 0.756334i \(0.273011\pi\)
\(774\) −5.80579 −0.208685
\(775\) 13.5172 0.485551
\(776\) −0.774045 −0.0277866
\(777\) −3.88293 −0.139299
\(778\) 25.3451 0.908665
\(779\) −6.42018 −0.230027
\(780\) −1.82988 −0.0655201
\(781\) −2.22935 −0.0797726
\(782\) 4.03401 0.144256
\(783\) −3.60480 −0.128825
\(784\) −4.87696 −0.174177
\(785\) −41.9263 −1.49641
\(786\) −11.5102 −0.410556
\(787\) −47.6154 −1.69730 −0.848652 0.528951i \(-0.822585\pi\)
−0.848652 + 0.528951i \(0.822585\pi\)
\(788\) 5.23510 0.186493
\(789\) 27.5232 0.979850
\(790\) 14.1068 0.501896
\(791\) −25.1820 −0.895370
\(792\) 0.513205 0.0182359
\(793\) 9.58790 0.340476
\(794\) 27.8148 0.987111
\(795\) −18.1391 −0.643327
\(796\) 10.4226 0.369418
\(797\) 4.78644 0.169544 0.0847722 0.996400i \(-0.472984\pi\)
0.0847722 + 0.996400i \(0.472984\pi\)
\(798\) −1.64451 −0.0582149
\(799\) −3.51373 −0.124307
\(800\) 1.65155 0.0583910
\(801\) 0.196823 0.00695439
\(802\) 37.3383 1.31846
\(803\) −0.222761 −0.00786107
\(804\) 11.1866 0.394522
\(805\) −13.6896 −0.482495
\(806\) −8.18455 −0.288288
\(807\) −15.1132 −0.532009
\(808\) −13.7255 −0.482862
\(809\) 33.5579 1.17983 0.589916 0.807465i \(-0.299161\pi\)
0.589916 + 0.807465i \(0.299161\pi\)
\(810\) 1.82988 0.0642953
\(811\) −6.88017 −0.241596 −0.120798 0.992677i \(-0.538545\pi\)
−0.120798 + 0.992677i \(0.538545\pi\)
\(812\) 5.25243 0.184324
\(813\) 8.91417 0.312633
\(814\) 1.36764 0.0479356
\(815\) 24.0305 0.841753
\(816\) 0.785684 0.0275044
\(817\) −6.55266 −0.229249
\(818\) 18.2282 0.637334
\(819\) −1.45707 −0.0509140
\(820\) −10.4091 −0.363502
\(821\) 8.51374 0.297131 0.148566 0.988903i \(-0.452534\pi\)
0.148566 + 0.988903i \(0.452534\pi\)
\(822\) −2.77252 −0.0967027
\(823\) −50.3331 −1.75450 −0.877252 0.480031i \(-0.840625\pi\)
−0.877252 + 0.480031i \(0.840625\pi\)
\(824\) 1.00000 0.0348367
\(825\) −0.847583 −0.0295090
\(826\) −14.9917 −0.521627
\(827\) −0.219786 −0.00764271 −0.00382135 0.999993i \(-0.501216\pi\)
−0.00382135 + 0.999993i \(0.501216\pi\)
\(828\) 5.13440 0.178433
\(829\) 3.17717 0.110348 0.0551739 0.998477i \(-0.482429\pi\)
0.0551739 + 0.998477i \(0.482429\pi\)
\(830\) −0.825431 −0.0286511
\(831\) 2.29177 0.0795005
\(832\) −1.00000 −0.0346688
\(833\) 3.83175 0.132762
\(834\) −11.4546 −0.396641
\(835\) −23.7664 −0.822472
\(836\) 0.579225 0.0200329
\(837\) 8.18455 0.282899
\(838\) −23.0355 −0.795750
\(839\) 40.7319 1.40622 0.703111 0.711080i \(-0.251794\pi\)
0.703111 + 0.711080i \(0.251794\pi\)
\(840\) −2.66625 −0.0919945
\(841\) −16.0054 −0.551912
\(842\) 37.1791 1.28128
\(843\) −29.9684 −1.03217
\(844\) 25.9138 0.891990
\(845\) −1.82988 −0.0629497
\(846\) −4.47220 −0.153757
\(847\) −15.6440 −0.537533
\(848\) −9.91273 −0.340405
\(849\) −27.0645 −0.928852
\(850\) −1.29759 −0.0445071
\(851\) 13.6826 0.469035
\(852\) −4.34398 −0.148822
\(853\) 26.0115 0.890616 0.445308 0.895378i \(-0.353094\pi\)
0.445308 + 0.895378i \(0.353094\pi\)
\(854\) 13.9702 0.478051
\(855\) 2.06528 0.0706309
\(856\) −11.3226 −0.387000
\(857\) 7.99227 0.273011 0.136505 0.990639i \(-0.456413\pi\)
0.136505 + 0.990639i \(0.456413\pi\)
\(858\) 0.513205 0.0175205
\(859\) −14.4445 −0.492841 −0.246421 0.969163i \(-0.579254\pi\)
−0.246421 + 0.969163i \(0.579254\pi\)
\(860\) −10.6239 −0.362272
\(861\) −8.28840 −0.282468
\(862\) 9.91314 0.337643
\(863\) 3.73344 0.127088 0.0635439 0.997979i \(-0.479760\pi\)
0.0635439 + 0.997979i \(0.479760\pi\)
\(864\) 1.00000 0.0340207
\(865\) −47.3041 −1.60839
\(866\) −0.898037 −0.0305165
\(867\) 16.3827 0.556386
\(868\) −11.9254 −0.404775
\(869\) −3.95636 −0.134210
\(870\) −6.59634 −0.223637
\(871\) 11.1866 0.379045
\(872\) −4.23121 −0.143287
\(873\) 0.774045 0.0261974
\(874\) 5.79490 0.196015
\(875\) 17.7347 0.599543
\(876\) −0.434059 −0.0146655
\(877\) 53.4114 1.80357 0.901787 0.432181i \(-0.142256\pi\)
0.901787 + 0.432181i \(0.142256\pi\)
\(878\) 20.9744 0.707852
\(879\) 16.0466 0.541239
\(880\) 0.939102 0.0316571
\(881\) 6.04164 0.203548 0.101774 0.994808i \(-0.467548\pi\)
0.101774 + 0.994808i \(0.467548\pi\)
\(882\) 4.87696 0.164216
\(883\) −57.7805 −1.94447 −0.972234 0.234012i \(-0.924814\pi\)
−0.972234 + 0.234012i \(0.924814\pi\)
\(884\) 0.785684 0.0264254
\(885\) 18.8275 0.632879
\(886\) 12.8861 0.432917
\(887\) −40.6851 −1.36607 −0.683037 0.730384i \(-0.739341\pi\)
−0.683037 + 0.730384i \(0.739341\pi\)
\(888\) 2.66489 0.0894280
\(889\) −13.6460 −0.457672
\(890\) 0.360162 0.0120727
\(891\) −0.513205 −0.0171930
\(892\) −23.7842 −0.796355
\(893\) −5.04751 −0.168908
\(894\) −14.3197 −0.478921
\(895\) −31.5230 −1.05370
\(896\) −1.45707 −0.0486772
\(897\) 5.13440 0.171433
\(898\) 1.17918 0.0393497
\(899\) −29.5036 −0.984001
\(900\) −1.65155 −0.0550516
\(901\) 7.78827 0.259465
\(902\) 2.91932 0.0972028
\(903\) −8.45943 −0.281512
\(904\) 17.2827 0.574814
\(905\) −17.5126 −0.582140
\(906\) −9.19404 −0.305451
\(907\) 26.4114 0.876975 0.438487 0.898737i \(-0.355514\pi\)
0.438487 + 0.898737i \(0.355514\pi\)
\(908\) −12.9676 −0.430345
\(909\) 13.7255 0.455247
\(910\) −2.66625 −0.0883854
\(911\) −5.31146 −0.175977 −0.0879883 0.996122i \(-0.528044\pi\)
−0.0879883 + 0.996122i \(0.528044\pi\)
\(912\) 1.12864 0.0373731
\(913\) 0.231499 0.00766151
\(914\) 22.1846 0.733802
\(915\) −17.5447 −0.580009
\(916\) −13.2509 −0.437821
\(917\) −16.7711 −0.553832
\(918\) −0.785684 −0.0259314
\(919\) 11.1230 0.366915 0.183457 0.983028i \(-0.441271\pi\)
0.183457 + 0.983028i \(0.441271\pi\)
\(920\) 9.39533 0.309755
\(921\) 0.570165 0.0187876
\(922\) 1.97709 0.0651119
\(923\) −4.34398 −0.142984
\(924\) 0.747774 0.0246000
\(925\) −4.40120 −0.144711
\(926\) −29.1462 −0.957802
\(927\) −1.00000 −0.0328443
\(928\) −3.60480 −0.118333
\(929\) −13.4206 −0.440315 −0.220158 0.975464i \(-0.570657\pi\)
−0.220158 + 0.975464i \(0.570657\pi\)
\(930\) 14.9767 0.491106
\(931\) 5.50434 0.180397
\(932\) −19.7166 −0.645839
\(933\) 13.7186 0.449126
\(934\) 1.07865 0.0352944
\(935\) −0.737837 −0.0241299
\(936\) 1.00000 0.0326860
\(937\) −15.0557 −0.491847 −0.245923 0.969289i \(-0.579091\pi\)
−0.245923 + 0.969289i \(0.579091\pi\)
\(938\) 16.2997 0.532203
\(939\) 8.93255 0.291503
\(940\) −8.18357 −0.266919
\(941\) 5.11672 0.166800 0.0834001 0.996516i \(-0.473422\pi\)
0.0834001 + 0.996516i \(0.473422\pi\)
\(942\) 22.9121 0.746516
\(943\) 29.2066 0.951098
\(944\) 10.2889 0.334876
\(945\) 2.66625 0.0867332
\(946\) 2.97956 0.0968739
\(947\) 3.79903 0.123452 0.0617260 0.998093i \(-0.480340\pi\)
0.0617260 + 0.998093i \(0.480340\pi\)
\(948\) −7.70913 −0.250381
\(949\) −0.434059 −0.0140901
\(950\) −1.86401 −0.0604763
\(951\) −2.51336 −0.0815012
\(952\) 1.14479 0.0371030
\(953\) 47.8946 1.55146 0.775729 0.631066i \(-0.217382\pi\)
0.775729 + 0.631066i \(0.217382\pi\)
\(954\) 9.91273 0.320936
\(955\) −22.7778 −0.737072
\(956\) −24.4023 −0.789228
\(957\) 1.85000 0.0598020
\(958\) −43.5747 −1.40783
\(959\) −4.03974 −0.130450
\(960\) 1.82988 0.0590590
\(961\) 35.9868 1.16086
\(962\) 2.66489 0.0859196
\(963\) 11.3226 0.364867
\(964\) −7.27230 −0.234225
\(965\) −9.80152 −0.315522
\(966\) 7.48116 0.240702
\(967\) 33.3006 1.07087 0.535437 0.844575i \(-0.320147\pi\)
0.535437 + 0.844575i \(0.320147\pi\)
\(968\) 10.7366 0.345088
\(969\) −0.886755 −0.0284867
\(970\) 1.41641 0.0454781
\(971\) −22.6344 −0.726371 −0.363186 0.931717i \(-0.618311\pi\)
−0.363186 + 0.931717i \(0.618311\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.6901 −0.535061
\(974\) −2.28651 −0.0732646
\(975\) −1.65155 −0.0528919
\(976\) −9.58790 −0.306901
\(977\) −55.6987 −1.78196 −0.890979 0.454044i \(-0.849981\pi\)
−0.890979 + 0.454044i \(0.849981\pi\)
\(978\) −13.1323 −0.419925
\(979\) −0.101010 −0.00322831
\(980\) 8.92424 0.285074
\(981\) 4.23121 0.135092
\(982\) 2.07996 0.0663741
\(983\) 32.7830 1.04562 0.522808 0.852450i \(-0.324884\pi\)
0.522808 + 0.852450i \(0.324884\pi\)
\(984\) 5.68841 0.181340
\(985\) −9.57959 −0.305231
\(986\) 2.83223 0.0901966
\(987\) −6.51629 −0.207416
\(988\) 1.12864 0.0359069
\(989\) 29.8093 0.947880
\(990\) −0.939102 −0.0298466
\(991\) −26.8844 −0.854011 −0.427005 0.904249i \(-0.640431\pi\)
−0.427005 + 0.904249i \(0.640431\pi\)
\(992\) 8.18455 0.259860
\(993\) 6.17555 0.195975
\(994\) −6.32947 −0.200759
\(995\) −19.0720 −0.604623
\(996\) 0.451085 0.0142932
\(997\) 38.3685 1.21514 0.607571 0.794266i \(-0.292144\pi\)
0.607571 + 0.794266i \(0.292144\pi\)
\(998\) 5.05021 0.159862
\(999\) −2.66489 −0.0843135
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 8034.2.a.s.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.s.1.3 10 1.1 even 1 trivial