Properties

Label 8034.2.a.u.1.6
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $11$
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4x^{10} - 18x^{9} + 64x^{8} + 85x^{7} - 249x^{6} - 109x^{5} + 230x^{4} + 97x^{3} - 53x^{2} - 32x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-3.29880\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.113923 q^{5} -1.00000 q^{6} +2.65133 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.113923 q^{5} -1.00000 q^{6} +2.65133 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.113923 q^{10} -3.17383 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.65133 q^{14} -0.113923 q^{15} +1.00000 q^{16} +3.51081 q^{17} +1.00000 q^{18} -2.25761 q^{19} +0.113923 q^{20} -2.65133 q^{21} -3.17383 q^{22} -4.90813 q^{23} -1.00000 q^{24} -4.98702 q^{25} -1.00000 q^{26} -1.00000 q^{27} +2.65133 q^{28} +0.991065 q^{29} -0.113923 q^{30} +5.57567 q^{31} +1.00000 q^{32} +3.17383 q^{33} +3.51081 q^{34} +0.302047 q^{35} +1.00000 q^{36} -7.31136 q^{37} -2.25761 q^{38} +1.00000 q^{39} +0.113923 q^{40} -8.22718 q^{41} -2.65133 q^{42} -6.12885 q^{43} -3.17383 q^{44} +0.113923 q^{45} -4.90813 q^{46} -0.711399 q^{47} -1.00000 q^{48} +0.0295321 q^{49} -4.98702 q^{50} -3.51081 q^{51} -1.00000 q^{52} +7.20535 q^{53} -1.00000 q^{54} -0.361572 q^{55} +2.65133 q^{56} +2.25761 q^{57} +0.991065 q^{58} -8.39566 q^{59} -0.113923 q^{60} -1.34766 q^{61} +5.57567 q^{62} +2.65133 q^{63} +1.00000 q^{64} -0.113923 q^{65} +3.17383 q^{66} -10.6030 q^{67} +3.51081 q^{68} +4.90813 q^{69} +0.302047 q^{70} +13.6456 q^{71} +1.00000 q^{72} -1.59020 q^{73} -7.31136 q^{74} +4.98702 q^{75} -2.25761 q^{76} -8.41487 q^{77} +1.00000 q^{78} +5.88236 q^{79} +0.113923 q^{80} +1.00000 q^{81} -8.22718 q^{82} -12.1432 q^{83} -2.65133 q^{84} +0.399961 q^{85} -6.12885 q^{86} -0.991065 q^{87} -3.17383 q^{88} -8.95858 q^{89} +0.113923 q^{90} -2.65133 q^{91} -4.90813 q^{92} -5.57567 q^{93} -0.711399 q^{94} -0.257193 q^{95} -1.00000 q^{96} +10.4001 q^{97} +0.0295321 q^{98} -3.17383 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 11 q^{2} - 11 q^{3} + 11 q^{4} - 2 q^{5} - 11 q^{6} - 2 q^{7} + 11 q^{8} + 11 q^{9} - 2 q^{10} - 10 q^{11} - 11 q^{12} - 11 q^{13} - 2 q^{14} + 2 q^{15} + 11 q^{16} + 6 q^{17} + 11 q^{18} - 3 q^{19} - 2 q^{20} + 2 q^{21} - 10 q^{22} + 3 q^{23} - 11 q^{24} - q^{25} - 11 q^{26} - 11 q^{27} - 2 q^{28} - 22 q^{29} + 2 q^{30} - 5 q^{31} + 11 q^{32} + 10 q^{33} + 6 q^{34} - 20 q^{35} + 11 q^{36} - 26 q^{37} - 3 q^{38} + 11 q^{39} - 2 q^{40} - 6 q^{41} + 2 q^{42} - 8 q^{43} - 10 q^{44} - 2 q^{45} + 3 q^{46} + 6 q^{47} - 11 q^{48} - 5 q^{49} - q^{50} - 6 q^{51} - 11 q^{52} - 25 q^{53} - 11 q^{54} - 2 q^{56} + 3 q^{57} - 22 q^{58} + 7 q^{59} + 2 q^{60} - 36 q^{61} - 5 q^{62} - 2 q^{63} + 11 q^{64} + 2 q^{65} + 10 q^{66} - 12 q^{67} + 6 q^{68} - 3 q^{69} - 20 q^{70} - 15 q^{71} + 11 q^{72} - 12 q^{73} - 26 q^{74} + q^{75} - 3 q^{76} - q^{77} + 11 q^{78} - 15 q^{79} - 2 q^{80} + 11 q^{81} - 6 q^{82} - 16 q^{83} + 2 q^{84} - 25 q^{85} - 8 q^{86} + 22 q^{87} - 10 q^{88} - 2 q^{89} - 2 q^{90} + 2 q^{91} + 3 q^{92} + 5 q^{93} + 6 q^{94} + 16 q^{95} - 11 q^{96} - 10 q^{97} - 5 q^{98} - 10 q^{99}+O(q^{100}) \) 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
\(5\) 0.113923 0.0509479 0.0254739 0.999675i \(-0.491891\pi\)
0.0254739 + 0.999675i \(0.491891\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.65133 1.00211 0.501054 0.865416i \(-0.332946\pi\)
0.501054 + 0.865416i \(0.332946\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.113923 0.0360256
\(11\) −3.17383 −0.956947 −0.478473 0.878102i \(-0.658810\pi\)
−0.478473 + 0.878102i \(0.658810\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.65133 0.708597
\(15\) −0.113923 −0.0294148
\(16\) 1.00000 0.250000
\(17\) 3.51081 0.851495 0.425748 0.904842i \(-0.360011\pi\)
0.425748 + 0.904842i \(0.360011\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.25761 −0.517930 −0.258965 0.965887i \(-0.583382\pi\)
−0.258965 + 0.965887i \(0.583382\pi\)
\(20\) 0.113923 0.0254739
\(21\) −2.65133 −0.578567
\(22\) −3.17383 −0.676663
\(23\) −4.90813 −1.02342 −0.511708 0.859159i \(-0.670987\pi\)
−0.511708 + 0.859159i \(0.670987\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.98702 −0.997404
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 2.65133 0.501054
\(29\) 0.991065 0.184036 0.0920181 0.995757i \(-0.470668\pi\)
0.0920181 + 0.995757i \(0.470668\pi\)
\(30\) −0.113923 −0.0207994
\(31\) 5.57567 1.00142 0.500710 0.865615i \(-0.333072\pi\)
0.500710 + 0.865615i \(0.333072\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.17383 0.552493
\(34\) 3.51081 0.602098
\(35\) 0.302047 0.0510552
\(36\) 1.00000 0.166667
\(37\) −7.31136 −1.20198 −0.600990 0.799256i \(-0.705227\pi\)
−0.600990 + 0.799256i \(0.705227\pi\)
\(38\) −2.25761 −0.366232
\(39\) 1.00000 0.160128
\(40\) 0.113923 0.0180128
\(41\) −8.22718 −1.28487 −0.642434 0.766341i \(-0.722076\pi\)
−0.642434 + 0.766341i \(0.722076\pi\)
\(42\) −2.65133 −0.409109
\(43\) −6.12885 −0.934641 −0.467320 0.884088i \(-0.654781\pi\)
−0.467320 + 0.884088i \(0.654781\pi\)
\(44\) −3.17383 −0.478473
\(45\) 0.113923 0.0169826
\(46\) −4.90813 −0.723665
\(47\) −0.711399 −0.103768 −0.0518841 0.998653i \(-0.516523\pi\)
−0.0518841 + 0.998653i \(0.516523\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0.0295321 0.00421887
\(50\) −4.98702 −0.705271
\(51\) −3.51081 −0.491611
\(52\) −1.00000 −0.138675
\(53\) 7.20535 0.989730 0.494865 0.868970i \(-0.335217\pi\)
0.494865 + 0.868970i \(0.335217\pi\)
\(54\) −1.00000 −0.136083
\(55\) −0.361572 −0.0487544
\(56\) 2.65133 0.354298
\(57\) 2.25761 0.299027
\(58\) 0.991065 0.130133
\(59\) −8.39566 −1.09302 −0.546511 0.837452i \(-0.684044\pi\)
−0.546511 + 0.837452i \(0.684044\pi\)
\(60\) −0.113923 −0.0147074
\(61\) −1.34766 −0.172550 −0.0862751 0.996271i \(-0.527496\pi\)
−0.0862751 + 0.996271i \(0.527496\pi\)
\(62\) 5.57567 0.708111
\(63\) 2.65133 0.334036
\(64\) 1.00000 0.125000
\(65\) −0.113923 −0.0141304
\(66\) 3.17383 0.390672
\(67\) −10.6030 −1.29536 −0.647679 0.761913i \(-0.724260\pi\)
−0.647679 + 0.761913i \(0.724260\pi\)
\(68\) 3.51081 0.425748
\(69\) 4.90813 0.590870
\(70\) 0.302047 0.0361015
\(71\) 13.6456 1.61943 0.809716 0.586822i \(-0.199621\pi\)
0.809716 + 0.586822i \(0.199621\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.59020 −0.186119 −0.0930593 0.995661i \(-0.529665\pi\)
−0.0930593 + 0.995661i \(0.529665\pi\)
\(74\) −7.31136 −0.849928
\(75\) 4.98702 0.575852
\(76\) −2.25761 −0.258965
\(77\) −8.41487 −0.958963
\(78\) 1.00000 0.113228
\(79\) 5.88236 0.661818 0.330909 0.943663i \(-0.392645\pi\)
0.330909 + 0.943663i \(0.392645\pi\)
\(80\) 0.113923 0.0127370
\(81\) 1.00000 0.111111
\(82\) −8.22718 −0.908540
\(83\) −12.1432 −1.33289 −0.666444 0.745555i \(-0.732185\pi\)
−0.666444 + 0.745555i \(0.732185\pi\)
\(84\) −2.65133 −0.289283
\(85\) 0.399961 0.0433819
\(86\) −6.12885 −0.660891
\(87\) −0.991065 −0.106253
\(88\) −3.17383 −0.338332
\(89\) −8.95858 −0.949607 −0.474804 0.880092i \(-0.657481\pi\)
−0.474804 + 0.880092i \(0.657481\pi\)
\(90\) 0.113923 0.0120085
\(91\) −2.65133 −0.277935
\(92\) −4.90813 −0.511708
\(93\) −5.57567 −0.578170
\(94\) −0.711399 −0.0733752
\(95\) −0.257193 −0.0263874
\(96\) −1.00000 −0.102062
\(97\) 10.4001 1.05597 0.527986 0.849253i \(-0.322947\pi\)
0.527986 + 0.849253i \(0.322947\pi\)
\(98\) 0.0295321 0.00298319
\(99\) −3.17383 −0.318982
\(100\) −4.98702 −0.498702
\(101\) −15.9846 −1.59053 −0.795264 0.606263i \(-0.792668\pi\)
−0.795264 + 0.606263i \(0.792668\pi\)
\(102\) −3.51081 −0.347622
\(103\) 1.00000 0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −0.302047 −0.0294767
\(106\) 7.20535 0.699845
\(107\) 14.4797 1.39980 0.699901 0.714240i \(-0.253227\pi\)
0.699901 + 0.714240i \(0.253227\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −17.6514 −1.69070 −0.845351 0.534211i \(-0.820609\pi\)
−0.845351 + 0.534211i \(0.820609\pi\)
\(110\) −0.361572 −0.0344745
\(111\) 7.31136 0.693964
\(112\) 2.65133 0.250527
\(113\) 17.4659 1.64305 0.821525 0.570173i \(-0.193124\pi\)
0.821525 + 0.570173i \(0.193124\pi\)
\(114\) 2.25761 0.211444
\(115\) −0.559148 −0.0521409
\(116\) 0.991065 0.0920181
\(117\) −1.00000 −0.0924500
\(118\) −8.39566 −0.772883
\(119\) 9.30829 0.853290
\(120\) −0.113923 −0.0103997
\(121\) −0.926787 −0.0842533
\(122\) −1.34766 −0.122011
\(123\) 8.22718 0.741819
\(124\) 5.57567 0.500710
\(125\) −1.13775 −0.101763
\(126\) 2.65133 0.236199
\(127\) −16.0196 −1.42151 −0.710757 0.703438i \(-0.751647\pi\)
−0.710757 + 0.703438i \(0.751647\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.12885 0.539615
\(130\) −0.113923 −0.00999170
\(131\) 8.02984 0.701570 0.350785 0.936456i \(-0.385915\pi\)
0.350785 + 0.936456i \(0.385915\pi\)
\(132\) 3.17383 0.276247
\(133\) −5.98565 −0.519022
\(134\) −10.6030 −0.915957
\(135\) −0.113923 −0.00980492
\(136\) 3.51081 0.301049
\(137\) −0.471978 −0.0403238 −0.0201619 0.999797i \(-0.506418\pi\)
−0.0201619 + 0.999797i \(0.506418\pi\)
\(138\) 4.90813 0.417808
\(139\) 2.51841 0.213609 0.106805 0.994280i \(-0.465938\pi\)
0.106805 + 0.994280i \(0.465938\pi\)
\(140\) 0.302047 0.0255276
\(141\) 0.711399 0.0599106
\(142\) 13.6456 1.14511
\(143\) 3.17383 0.265409
\(144\) 1.00000 0.0833333
\(145\) 0.112905 0.00937624
\(146\) −1.59020 −0.131606
\(147\) −0.0295321 −0.00243576
\(148\) −7.31136 −0.600990
\(149\) −12.0279 −0.985367 −0.492684 0.870208i \(-0.663984\pi\)
−0.492684 + 0.870208i \(0.663984\pi\)
\(150\) 4.98702 0.407189
\(151\) 19.7757 1.60932 0.804662 0.593733i \(-0.202347\pi\)
0.804662 + 0.593733i \(0.202347\pi\)
\(152\) −2.25761 −0.183116
\(153\) 3.51081 0.283832
\(154\) −8.41487 −0.678089
\(155\) 0.635196 0.0510202
\(156\) 1.00000 0.0800641
\(157\) −3.75589 −0.299752 −0.149876 0.988705i \(-0.547887\pi\)
−0.149876 + 0.988705i \(0.547887\pi\)
\(158\) 5.88236 0.467976
\(159\) −7.20535 −0.571421
\(160\) 0.113923 0.00900639
\(161\) −13.0131 −1.02557
\(162\) 1.00000 0.0785674
\(163\) −11.0445 −0.865076 −0.432538 0.901616i \(-0.642382\pi\)
−0.432538 + 0.901616i \(0.642382\pi\)
\(164\) −8.22718 −0.642434
\(165\) 0.361572 0.0281483
\(166\) −12.1432 −0.942495
\(167\) 1.02005 0.0789340 0.0394670 0.999221i \(-0.487434\pi\)
0.0394670 + 0.999221i \(0.487434\pi\)
\(168\) −2.65133 −0.204554
\(169\) 1.00000 0.0769231
\(170\) 0.399961 0.0306756
\(171\) −2.25761 −0.172643
\(172\) −6.12885 −0.467320
\(173\) −2.77538 −0.211008 −0.105504 0.994419i \(-0.533646\pi\)
−0.105504 + 0.994419i \(0.533646\pi\)
\(174\) −0.991065 −0.0751324
\(175\) −13.2222 −0.999506
\(176\) −3.17383 −0.239237
\(177\) 8.39566 0.631056
\(178\) −8.95858 −0.671474
\(179\) 11.4232 0.853811 0.426906 0.904296i \(-0.359604\pi\)
0.426906 + 0.904296i \(0.359604\pi\)
\(180\) 0.113923 0.00849131
\(181\) −14.5444 −1.08108 −0.540540 0.841318i \(-0.681780\pi\)
−0.540540 + 0.841318i \(0.681780\pi\)
\(182\) −2.65133 −0.196529
\(183\) 1.34766 0.0996218
\(184\) −4.90813 −0.361832
\(185\) −0.832931 −0.0612383
\(186\) −5.57567 −0.408828
\(187\) −11.1427 −0.814836
\(188\) −0.711399 −0.0518841
\(189\) −2.65133 −0.192856
\(190\) −0.257193 −0.0186587
\(191\) −11.4688 −0.829851 −0.414925 0.909855i \(-0.636192\pi\)
−0.414925 + 0.909855i \(0.636192\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.05090 −0.507535 −0.253767 0.967265i \(-0.581670\pi\)
−0.253767 + 0.967265i \(0.581670\pi\)
\(194\) 10.4001 0.746684
\(195\) 0.113923 0.00815819
\(196\) 0.0295321 0.00210943
\(197\) 3.41934 0.243618 0.121809 0.992554i \(-0.461130\pi\)
0.121809 + 0.992554i \(0.461130\pi\)
\(198\) −3.17383 −0.225554
\(199\) 12.7201 0.901702 0.450851 0.892599i \(-0.351121\pi\)
0.450851 + 0.892599i \(0.351121\pi\)
\(200\) −4.98702 −0.352636
\(201\) 10.6030 0.747876
\(202\) −15.9846 −1.12467
\(203\) 2.62764 0.184424
\(204\) −3.51081 −0.245806
\(205\) −0.937263 −0.0654613
\(206\) 1.00000 0.0696733
\(207\) −4.90813 −0.341139
\(208\) −1.00000 −0.0693375
\(209\) 7.16527 0.495632
\(210\) −0.302047 −0.0208432
\(211\) −8.15531 −0.561435 −0.280717 0.959790i \(-0.590572\pi\)
−0.280717 + 0.959790i \(0.590572\pi\)
\(212\) 7.20535 0.494865
\(213\) −13.6456 −0.934980
\(214\) 14.4797 0.989810
\(215\) −0.698216 −0.0476179
\(216\) −1.00000 −0.0680414
\(217\) 14.7829 1.00353
\(218\) −17.6514 −1.19551
\(219\) 1.59020 0.107456
\(220\) −0.361572 −0.0243772
\(221\) −3.51081 −0.236162
\(222\) 7.31136 0.490706
\(223\) 1.92053 0.128608 0.0643041 0.997930i \(-0.479517\pi\)
0.0643041 + 0.997930i \(0.479517\pi\)
\(224\) 2.65133 0.177149
\(225\) −4.98702 −0.332468
\(226\) 17.4659 1.16181
\(227\) 7.73629 0.513475 0.256738 0.966481i \(-0.417352\pi\)
0.256738 + 0.966481i \(0.417352\pi\)
\(228\) 2.25761 0.149514
\(229\) 3.09858 0.204760 0.102380 0.994745i \(-0.467354\pi\)
0.102380 + 0.994745i \(0.467354\pi\)
\(230\) −0.559148 −0.0368692
\(231\) 8.41487 0.553658
\(232\) 0.991065 0.0650666
\(233\) 3.97487 0.260402 0.130201 0.991488i \(-0.458438\pi\)
0.130201 + 0.991488i \(0.458438\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −0.0810446 −0.00528677
\(236\) −8.39566 −0.546511
\(237\) −5.88236 −0.382101
\(238\) 9.30829 0.603367
\(239\) −17.6397 −1.14102 −0.570509 0.821291i \(-0.693254\pi\)
−0.570509 + 0.821291i \(0.693254\pi\)
\(240\) −0.113923 −0.00735369
\(241\) 24.3235 1.56682 0.783408 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(242\) −0.926787 −0.0595761
\(243\) −1.00000 −0.0641500
\(244\) −1.34766 −0.0862751
\(245\) 0.00336438 0.000214942 0
\(246\) 8.22718 0.524546
\(247\) 2.25761 0.143648
\(248\) 5.57567 0.354055
\(249\) 12.1432 0.769544
\(250\) −1.13775 −0.0719576
\(251\) 16.8621 1.06433 0.532163 0.846642i \(-0.321379\pi\)
0.532163 + 0.846642i \(0.321379\pi\)
\(252\) 2.65133 0.167018
\(253\) 15.5776 0.979355
\(254\) −16.0196 −1.00516
\(255\) −0.399961 −0.0250465
\(256\) 1.00000 0.0625000
\(257\) −19.0078 −1.18567 −0.592837 0.805322i \(-0.701992\pi\)
−0.592837 + 0.805322i \(0.701992\pi\)
\(258\) 6.12885 0.381565
\(259\) −19.3848 −1.20451
\(260\) −0.113923 −0.00706520
\(261\) 0.991065 0.0613454
\(262\) 8.02984 0.496085
\(263\) 16.5275 1.01913 0.509566 0.860432i \(-0.329806\pi\)
0.509566 + 0.860432i \(0.329806\pi\)
\(264\) 3.17383 0.195336
\(265\) 0.820854 0.0504246
\(266\) −5.98565 −0.367004
\(267\) 8.95858 0.548256
\(268\) −10.6030 −0.647679
\(269\) 1.06645 0.0650228 0.0325114 0.999471i \(-0.489649\pi\)
0.0325114 + 0.999471i \(0.489649\pi\)
\(270\) −0.113923 −0.00693312
\(271\) −28.1094 −1.70752 −0.853761 0.520664i \(-0.825684\pi\)
−0.853761 + 0.520664i \(0.825684\pi\)
\(272\) 3.51081 0.212874
\(273\) 2.65133 0.160466
\(274\) −0.471978 −0.0285133
\(275\) 15.8280 0.954463
\(276\) 4.90813 0.295435
\(277\) −8.95306 −0.537937 −0.268969 0.963149i \(-0.586683\pi\)
−0.268969 + 0.963149i \(0.586683\pi\)
\(278\) 2.51841 0.151044
\(279\) 5.57567 0.333807
\(280\) 0.302047 0.0180507
\(281\) 23.4140 1.39676 0.698381 0.715726i \(-0.253904\pi\)
0.698381 + 0.715726i \(0.253904\pi\)
\(282\) 0.711399 0.0423632
\(283\) −9.20239 −0.547025 −0.273513 0.961868i \(-0.588186\pi\)
−0.273513 + 0.961868i \(0.588186\pi\)
\(284\) 13.6456 0.809716
\(285\) 0.257193 0.0152348
\(286\) 3.17383 0.187673
\(287\) −21.8129 −1.28758
\(288\) 1.00000 0.0589256
\(289\) −4.67425 −0.274956
\(290\) 0.112905 0.00663001
\(291\) −10.4001 −0.609665
\(292\) −1.59020 −0.0930593
\(293\) −12.8741 −0.752111 −0.376055 0.926597i \(-0.622720\pi\)
−0.376055 + 0.926597i \(0.622720\pi\)
\(294\) −0.0295321 −0.00172234
\(295\) −0.956458 −0.0556871
\(296\) −7.31136 −0.424964
\(297\) 3.17383 0.184164
\(298\) −12.0279 −0.696760
\(299\) 4.90813 0.283845
\(300\) 4.98702 0.287926
\(301\) −16.2496 −0.936610
\(302\) 19.7757 1.13796
\(303\) 15.9846 0.918292
\(304\) −2.25761 −0.129483
\(305\) −0.153529 −0.00879106
\(306\) 3.51081 0.200699
\(307\) −7.06703 −0.403337 −0.201668 0.979454i \(-0.564636\pi\)
−0.201668 + 0.979454i \(0.564636\pi\)
\(308\) −8.41487 −0.479482
\(309\) −1.00000 −0.0568880
\(310\) 0.635196 0.0360767
\(311\) 8.92110 0.505869 0.252935 0.967483i \(-0.418604\pi\)
0.252935 + 0.967483i \(0.418604\pi\)
\(312\) 1.00000 0.0566139
\(313\) −13.7395 −0.776604 −0.388302 0.921532i \(-0.626938\pi\)
−0.388302 + 0.921532i \(0.626938\pi\)
\(314\) −3.75589 −0.211957
\(315\) 0.302047 0.0170184
\(316\) 5.88236 0.330909
\(317\) 21.1401 1.18734 0.593672 0.804707i \(-0.297678\pi\)
0.593672 + 0.804707i \(0.297678\pi\)
\(318\) −7.20535 −0.404056
\(319\) −3.14547 −0.176113
\(320\) 0.113923 0.00636848
\(321\) −14.4797 −0.808176
\(322\) −13.0131 −0.725189
\(323\) −7.92602 −0.441015
\(324\) 1.00000 0.0555556
\(325\) 4.98702 0.276630
\(326\) −11.0445 −0.611701
\(327\) 17.6514 0.976127
\(328\) −8.22718 −0.454270
\(329\) −1.88615 −0.103987
\(330\) 0.361572 0.0199039
\(331\) −19.2913 −1.06034 −0.530172 0.847890i \(-0.677873\pi\)
−0.530172 + 0.847890i \(0.677873\pi\)
\(332\) −12.1432 −0.666444
\(333\) −7.31136 −0.400660
\(334\) 1.02005 0.0558148
\(335\) −1.20792 −0.0659957
\(336\) −2.65133 −0.144642
\(337\) 14.3407 0.781189 0.390594 0.920563i \(-0.372270\pi\)
0.390594 + 0.920563i \(0.372270\pi\)
\(338\) 1.00000 0.0543928
\(339\) −17.4659 −0.948615
\(340\) 0.399961 0.0216909
\(341\) −17.6962 −0.958305
\(342\) −2.25761 −0.122077
\(343\) −18.4810 −0.997879
\(344\) −6.12885 −0.330445
\(345\) 0.559148 0.0301035
\(346\) −2.77538 −0.149205
\(347\) 16.7458 0.898963 0.449482 0.893290i \(-0.351609\pi\)
0.449482 + 0.893290i \(0.351609\pi\)
\(348\) −0.991065 −0.0531266
\(349\) −13.2076 −0.706986 −0.353493 0.935437i \(-0.615006\pi\)
−0.353493 + 0.935437i \(0.615006\pi\)
\(350\) −13.2222 −0.706758
\(351\) 1.00000 0.0533761
\(352\) −3.17383 −0.169166
\(353\) −2.26811 −0.120719 −0.0603596 0.998177i \(-0.519225\pi\)
−0.0603596 + 0.998177i \(0.519225\pi\)
\(354\) 8.39566 0.446224
\(355\) 1.55454 0.0825066
\(356\) −8.95858 −0.474804
\(357\) −9.30829 −0.492647
\(358\) 11.4232 0.603736
\(359\) −2.18365 −0.115249 −0.0576243 0.998338i \(-0.518353\pi\)
−0.0576243 + 0.998338i \(0.518353\pi\)
\(360\) 0.113923 0.00600426
\(361\) −13.9032 −0.731748
\(362\) −14.5444 −0.764439
\(363\) 0.926787 0.0486437
\(364\) −2.65133 −0.138967
\(365\) −0.181160 −0.00948234
\(366\) 1.34766 0.0704433
\(367\) −18.7772 −0.980162 −0.490081 0.871677i \(-0.663033\pi\)
−0.490081 + 0.871677i \(0.663033\pi\)
\(368\) −4.90813 −0.255854
\(369\) −8.22718 −0.428290
\(370\) −0.832931 −0.0433020
\(371\) 19.1037 0.991816
\(372\) −5.57567 −0.289085
\(373\) −16.7478 −0.867168 −0.433584 0.901113i \(-0.642751\pi\)
−0.433584 + 0.901113i \(0.642751\pi\)
\(374\) −11.1427 −0.576176
\(375\) 1.13775 0.0587532
\(376\) −0.711399 −0.0366876
\(377\) −0.991065 −0.0510424
\(378\) −2.65133 −0.136370
\(379\) 13.4554 0.691160 0.345580 0.938389i \(-0.387682\pi\)
0.345580 + 0.938389i \(0.387682\pi\)
\(380\) −0.257193 −0.0131937
\(381\) 16.0196 0.820711
\(382\) −11.4688 −0.586793
\(383\) −18.2146 −0.930723 −0.465362 0.885121i \(-0.654076\pi\)
−0.465362 + 0.885121i \(0.654076\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.958646 −0.0488571
\(386\) −7.05090 −0.358881
\(387\) −6.12885 −0.311547
\(388\) 10.4001 0.527986
\(389\) −16.0164 −0.812064 −0.406032 0.913859i \(-0.633088\pi\)
−0.406032 + 0.913859i \(0.633088\pi\)
\(390\) 0.113923 0.00576871
\(391\) −17.2315 −0.871434
\(392\) 0.0295321 0.00149159
\(393\) −8.02984 −0.405052
\(394\) 3.41934 0.172264
\(395\) 0.670136 0.0337182
\(396\) −3.17383 −0.159491
\(397\) −7.31848 −0.367304 −0.183652 0.982991i \(-0.558792\pi\)
−0.183652 + 0.982991i \(0.558792\pi\)
\(398\) 12.7201 0.637599
\(399\) 5.98565 0.299657
\(400\) −4.98702 −0.249351
\(401\) 0.477356 0.0238380 0.0119190 0.999929i \(-0.496206\pi\)
0.0119190 + 0.999929i \(0.496206\pi\)
\(402\) 10.6030 0.528828
\(403\) −5.57567 −0.277744
\(404\) −15.9846 −0.795264
\(405\) 0.113923 0.00566087
\(406\) 2.62764 0.130407
\(407\) 23.2050 1.15023
\(408\) −3.51081 −0.173811
\(409\) −14.8337 −0.733480 −0.366740 0.930324i \(-0.619526\pi\)
−0.366740 + 0.930324i \(0.619526\pi\)
\(410\) −0.937263 −0.0462881
\(411\) 0.471978 0.0232810
\(412\) 1.00000 0.0492665
\(413\) −22.2596 −1.09533
\(414\) −4.90813 −0.241222
\(415\) −1.38339 −0.0679078
\(416\) −1.00000 −0.0490290
\(417\) −2.51841 −0.123327
\(418\) 7.16527 0.350465
\(419\) 6.72517 0.328546 0.164273 0.986415i \(-0.447472\pi\)
0.164273 + 0.986415i \(0.447472\pi\)
\(420\) −0.302047 −0.0147384
\(421\) 23.3325 1.13716 0.568579 0.822629i \(-0.307493\pi\)
0.568579 + 0.822629i \(0.307493\pi\)
\(422\) −8.15531 −0.396994
\(423\) −0.711399 −0.0345894
\(424\) 7.20535 0.349923
\(425\) −17.5085 −0.849285
\(426\) −13.6456 −0.661131
\(427\) −3.57309 −0.172914
\(428\) 14.4797 0.699901
\(429\) −3.17383 −0.153234
\(430\) −0.698216 −0.0336710
\(431\) 7.72227 0.371968 0.185984 0.982553i \(-0.440453\pi\)
0.185984 + 0.982553i \(0.440453\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.83059 0.136030 0.0680148 0.997684i \(-0.478333\pi\)
0.0680148 + 0.997684i \(0.478333\pi\)
\(434\) 14.7829 0.709603
\(435\) −0.112905 −0.00541338
\(436\) −17.6514 −0.845351
\(437\) 11.0806 0.530058
\(438\) 1.59020 0.0759826
\(439\) 12.0883 0.576943 0.288471 0.957489i \(-0.406853\pi\)
0.288471 + 0.957489i \(0.406853\pi\)
\(440\) −0.361572 −0.0172373
\(441\) 0.0295321 0.00140629
\(442\) −3.51081 −0.166992
\(443\) 10.8630 0.516115 0.258057 0.966130i \(-0.416918\pi\)
0.258057 + 0.966130i \(0.416918\pi\)
\(444\) 7.31136 0.346982
\(445\) −1.02059 −0.0483805
\(446\) 1.92053 0.0909397
\(447\) 12.0279 0.568902
\(448\) 2.65133 0.125263
\(449\) −18.7203 −0.883467 −0.441733 0.897146i \(-0.645636\pi\)
−0.441733 + 0.897146i \(0.645636\pi\)
\(450\) −4.98702 −0.235090
\(451\) 26.1117 1.22955
\(452\) 17.4659 0.821525
\(453\) −19.7757 −0.929144
\(454\) 7.73629 0.363082
\(455\) −0.302047 −0.0141602
\(456\) 2.25761 0.105722
\(457\) −1.82995 −0.0856014 −0.0428007 0.999084i \(-0.513628\pi\)
−0.0428007 + 0.999084i \(0.513628\pi\)
\(458\) 3.09858 0.144787
\(459\) −3.51081 −0.163870
\(460\) −0.559148 −0.0260704
\(461\) 5.02597 0.234083 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(462\) 8.41487 0.391495
\(463\) −22.5592 −1.04842 −0.524208 0.851590i \(-0.675638\pi\)
−0.524208 + 0.851590i \(0.675638\pi\)
\(464\) 0.991065 0.0460090
\(465\) −0.635196 −0.0294565
\(466\) 3.97487 0.184132
\(467\) 31.3275 1.44966 0.724832 0.688926i \(-0.241917\pi\)
0.724832 + 0.688926i \(0.241917\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −28.1119 −1.29809
\(470\) −0.0810446 −0.00373831
\(471\) 3.75589 0.173062
\(472\) −8.39566 −0.386442
\(473\) 19.4519 0.894401
\(474\) −5.88236 −0.270186
\(475\) 11.2587 0.516586
\(476\) 9.30829 0.426645
\(477\) 7.20535 0.329910
\(478\) −17.6397 −0.806822
\(479\) −9.71552 −0.443913 −0.221957 0.975057i \(-0.571244\pi\)
−0.221957 + 0.975057i \(0.571244\pi\)
\(480\) −0.113923 −0.00519984
\(481\) 7.31136 0.333369
\(482\) 24.3235 1.10791
\(483\) 13.0131 0.592115
\(484\) −0.926787 −0.0421267
\(485\) 1.18481 0.0537995
\(486\) −1.00000 −0.0453609
\(487\) −22.4933 −1.01927 −0.509634 0.860391i \(-0.670219\pi\)
−0.509634 + 0.860391i \(0.670219\pi\)
\(488\) −1.34766 −0.0610057
\(489\) 11.0445 0.499452
\(490\) 0.00336438 0.000151987 0
\(491\) −13.9549 −0.629778 −0.314889 0.949129i \(-0.601967\pi\)
−0.314889 + 0.949129i \(0.601967\pi\)
\(492\) 8.22718 0.370910
\(493\) 3.47944 0.156706
\(494\) 2.25761 0.101575
\(495\) −0.361572 −0.0162515
\(496\) 5.57567 0.250355
\(497\) 36.1789 1.62285
\(498\) 12.1432 0.544150
\(499\) 22.1708 0.992502 0.496251 0.868179i \(-0.334710\pi\)
0.496251 + 0.868179i \(0.334710\pi\)
\(500\) −1.13775 −0.0508817
\(501\) −1.02005 −0.0455726
\(502\) 16.8621 0.752592
\(503\) −23.8656 −1.06411 −0.532057 0.846709i \(-0.678581\pi\)
−0.532057 + 0.846709i \(0.678581\pi\)
\(504\) 2.65133 0.118099
\(505\) −1.82101 −0.0810340
\(506\) 15.5776 0.692508
\(507\) −1.00000 −0.0444116
\(508\) −16.0196 −0.710757
\(509\) 6.89132 0.305452 0.152726 0.988269i \(-0.451195\pi\)
0.152726 + 0.988269i \(0.451195\pi\)
\(510\) −0.399961 −0.0177106
\(511\) −4.21613 −0.186511
\(512\) 1.00000 0.0441942
\(513\) 2.25761 0.0996758
\(514\) −19.0078 −0.838399
\(515\) 0.113923 0.00502004
\(516\) 6.12885 0.269808
\(517\) 2.25786 0.0993007
\(518\) −19.3848 −0.851719
\(519\) 2.77538 0.121825
\(520\) −0.113923 −0.00499585
\(521\) −14.4424 −0.632734 −0.316367 0.948637i \(-0.602463\pi\)
−0.316367 + 0.948637i \(0.602463\pi\)
\(522\) 0.991065 0.0433777
\(523\) −21.0139 −0.918874 −0.459437 0.888210i \(-0.651949\pi\)
−0.459437 + 0.888210i \(0.651949\pi\)
\(524\) 8.02984 0.350785
\(525\) 13.2222 0.577065
\(526\) 16.5275 0.720635
\(527\) 19.5751 0.852704
\(528\) 3.17383 0.138123
\(529\) 1.08976 0.0473808
\(530\) 0.820854 0.0356556
\(531\) −8.39566 −0.364341
\(532\) −5.98565 −0.259511
\(533\) 8.22718 0.356359
\(534\) 8.95858 0.387676
\(535\) 1.64956 0.0713169
\(536\) −10.6030 −0.457979
\(537\) −11.4232 −0.492948
\(538\) 1.06645 0.0459781
\(539\) −0.0937298 −0.00403723
\(540\) −0.113923 −0.00490246
\(541\) 10.5619 0.454093 0.227046 0.973884i \(-0.427093\pi\)
0.227046 + 0.973884i \(0.427093\pi\)
\(542\) −28.1094 −1.20740
\(543\) 14.5444 0.624162
\(544\) 3.51081 0.150525
\(545\) −2.01090 −0.0861376
\(546\) 2.65133 0.113466
\(547\) 25.2712 1.08052 0.540259 0.841499i \(-0.318326\pi\)
0.540259 + 0.841499i \(0.318326\pi\)
\(548\) −0.471978 −0.0201619
\(549\) −1.34766 −0.0575167
\(550\) 15.8280 0.674907
\(551\) −2.23743 −0.0953179
\(552\) 4.90813 0.208904
\(553\) 15.5961 0.663212
\(554\) −8.95306 −0.380379
\(555\) 0.832931 0.0353560
\(556\) 2.51841 0.106805
\(557\) −34.1539 −1.44715 −0.723574 0.690247i \(-0.757502\pi\)
−0.723574 + 0.690247i \(0.757502\pi\)
\(558\) 5.57567 0.236037
\(559\) 6.12885 0.259223
\(560\) 0.302047 0.0127638
\(561\) 11.1427 0.470446
\(562\) 23.4140 0.987660
\(563\) −13.9801 −0.589191 −0.294596 0.955622i \(-0.595185\pi\)
−0.294596 + 0.955622i \(0.595185\pi\)
\(564\) 0.711399 0.0299553
\(565\) 1.98976 0.0837098
\(566\) −9.20239 −0.386805
\(567\) 2.65133 0.111345
\(568\) 13.6456 0.572556
\(569\) −31.7065 −1.32920 −0.664602 0.747197i \(-0.731399\pi\)
−0.664602 + 0.747197i \(0.731399\pi\)
\(570\) 0.257193 0.0107726
\(571\) −40.9651 −1.71433 −0.857167 0.515038i \(-0.827778\pi\)
−0.857167 + 0.515038i \(0.827778\pi\)
\(572\) 3.17383 0.132705
\(573\) 11.4688 0.479114
\(574\) −21.8129 −0.910454
\(575\) 24.4770 1.02076
\(576\) 1.00000 0.0416667
\(577\) 19.1531 0.797353 0.398677 0.917092i \(-0.369470\pi\)
0.398677 + 0.917092i \(0.369470\pi\)
\(578\) −4.67425 −0.194423
\(579\) 7.05090 0.293025
\(580\) 0.112905 0.00468812
\(581\) −32.1956 −1.33570
\(582\) −10.4001 −0.431098
\(583\) −22.8686 −0.947119
\(584\) −1.59020 −0.0658029
\(585\) −0.113923 −0.00471013
\(586\) −12.8741 −0.531823
\(587\) 9.44305 0.389757 0.194878 0.980827i \(-0.437569\pi\)
0.194878 + 0.980827i \(0.437569\pi\)
\(588\) −0.0295321 −0.00121788
\(589\) −12.5877 −0.518666
\(590\) −0.956458 −0.0393767
\(591\) −3.41934 −0.140653
\(592\) −7.31136 −0.300495
\(593\) 3.05451 0.125434 0.0627169 0.998031i \(-0.480023\pi\)
0.0627169 + 0.998031i \(0.480023\pi\)
\(594\) 3.17383 0.130224
\(595\) 1.06043 0.0434733
\(596\) −12.0279 −0.492684
\(597\) −12.7201 −0.520598
\(598\) 4.90813 0.200708
\(599\) 20.8674 0.852621 0.426310 0.904577i \(-0.359813\pi\)
0.426310 + 0.904577i \(0.359813\pi\)
\(600\) 4.98702 0.203594
\(601\) 11.1295 0.453981 0.226991 0.973897i \(-0.427111\pi\)
0.226991 + 0.973897i \(0.427111\pi\)
\(602\) −16.2496 −0.662283
\(603\) −10.6030 −0.431786
\(604\) 19.7757 0.804662
\(605\) −0.105582 −0.00429253
\(606\) 15.9846 0.649330
\(607\) 48.9659 1.98747 0.993733 0.111783i \(-0.0356563\pi\)
0.993733 + 0.111783i \(0.0356563\pi\)
\(608\) −2.25761 −0.0915580
\(609\) −2.62764 −0.106477
\(610\) −0.153529 −0.00621622
\(611\) 0.711399 0.0287801
\(612\) 3.51081 0.141916
\(613\) −29.6137 −1.19609 −0.598043 0.801464i \(-0.704055\pi\)
−0.598043 + 0.801464i \(0.704055\pi\)
\(614\) −7.06703 −0.285202
\(615\) 0.937263 0.0377941
\(616\) −8.41487 −0.339045
\(617\) −31.9205 −1.28507 −0.642536 0.766255i \(-0.722118\pi\)
−0.642536 + 0.766255i \(0.722118\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 34.9784 1.40590 0.702950 0.711240i \(-0.251866\pi\)
0.702950 + 0.711240i \(0.251866\pi\)
\(620\) 0.635196 0.0255101
\(621\) 4.90813 0.196957
\(622\) 8.92110 0.357704
\(623\) −23.7521 −0.951608
\(624\) 1.00000 0.0400320
\(625\) 24.8055 0.992220
\(626\) −13.7395 −0.549142
\(627\) −7.16527 −0.286153
\(628\) −3.75589 −0.149876
\(629\) −25.6688 −1.02348
\(630\) 0.302047 0.0120338
\(631\) −1.28189 −0.0510311 −0.0255156 0.999674i \(-0.508123\pi\)
−0.0255156 + 0.999674i \(0.508123\pi\)
\(632\) 5.88236 0.233988
\(633\) 8.15531 0.324144
\(634\) 21.1401 0.839579
\(635\) −1.82500 −0.0724230
\(636\) −7.20535 −0.285711
\(637\) −0.0295321 −0.00117010
\(638\) −3.14547 −0.124530
\(639\) 13.6456 0.539811
\(640\) 0.113923 0.00450320
\(641\) −16.0646 −0.634514 −0.317257 0.948340i \(-0.602762\pi\)
−0.317257 + 0.948340i \(0.602762\pi\)
\(642\) −14.4797 −0.571467
\(643\) 12.2700 0.483881 0.241940 0.970291i \(-0.422216\pi\)
0.241940 + 0.970291i \(0.422216\pi\)
\(644\) −13.0131 −0.512786
\(645\) 0.698216 0.0274922
\(646\) −7.92602 −0.311845
\(647\) 36.5059 1.43519 0.717597 0.696459i \(-0.245242\pi\)
0.717597 + 0.696459i \(0.245242\pi\)
\(648\) 1.00000 0.0392837
\(649\) 26.6464 1.04596
\(650\) 4.98702 0.195607
\(651\) −14.7829 −0.579388
\(652\) −11.0445 −0.432538
\(653\) −40.3637 −1.57955 −0.789777 0.613394i \(-0.789804\pi\)
−0.789777 + 0.613394i \(0.789804\pi\)
\(654\) 17.6514 0.690226
\(655\) 0.914782 0.0357435
\(656\) −8.22718 −0.321217
\(657\) −1.59020 −0.0620395
\(658\) −1.88615 −0.0735299
\(659\) −1.13751 −0.0443112 −0.0221556 0.999755i \(-0.507053\pi\)
−0.0221556 + 0.999755i \(0.507053\pi\)
\(660\) 0.361572 0.0140742
\(661\) 22.7149 0.883506 0.441753 0.897137i \(-0.354357\pi\)
0.441753 + 0.897137i \(0.354357\pi\)
\(662\) −19.2913 −0.749776
\(663\) 3.51081 0.136348
\(664\) −12.1432 −0.471247
\(665\) −0.681903 −0.0264430
\(666\) −7.31136 −0.283309
\(667\) −4.86428 −0.188346
\(668\) 1.02005 0.0394670
\(669\) −1.92053 −0.0742520
\(670\) −1.20792 −0.0466660
\(671\) 4.27725 0.165121
\(672\) −2.65133 −0.102277
\(673\) 5.46060 0.210491 0.105245 0.994446i \(-0.466437\pi\)
0.105245 + 0.994446i \(0.466437\pi\)
\(674\) 14.3407 0.552384
\(675\) 4.98702 0.191951
\(676\) 1.00000 0.0384615
\(677\) −12.4615 −0.478936 −0.239468 0.970904i \(-0.576973\pi\)
−0.239468 + 0.970904i \(0.576973\pi\)
\(678\) −17.4659 −0.670772
\(679\) 27.5741 1.05820
\(680\) 0.399961 0.0153378
\(681\) −7.73629 −0.296455
\(682\) −17.6962 −0.677624
\(683\) −3.23234 −0.123682 −0.0618410 0.998086i \(-0.519697\pi\)
−0.0618410 + 0.998086i \(0.519697\pi\)
\(684\) −2.25761 −0.0863217
\(685\) −0.0537691 −0.00205441
\(686\) −18.4810 −0.705607
\(687\) −3.09858 −0.118218
\(688\) −6.12885 −0.233660
\(689\) −7.20535 −0.274502
\(690\) 0.559148 0.0212864
\(691\) −19.9201 −0.757798 −0.378899 0.925438i \(-0.623697\pi\)
−0.378899 + 0.925438i \(0.623697\pi\)
\(692\) −2.77538 −0.105504
\(693\) −8.41487 −0.319654
\(694\) 16.7458 0.635663
\(695\) 0.286905 0.0108829
\(696\) −0.991065 −0.0375662
\(697\) −28.8840 −1.09406
\(698\) −13.2076 −0.499914
\(699\) −3.97487 −0.150343
\(700\) −13.2222 −0.499753
\(701\) 42.0697 1.58895 0.794476 0.607295i \(-0.207746\pi\)
0.794476 + 0.607295i \(0.207746\pi\)
\(702\) 1.00000 0.0377426
\(703\) 16.5062 0.622542
\(704\) −3.17383 −0.119618
\(705\) 0.0810446 0.00305232
\(706\) −2.26811 −0.0853613
\(707\) −42.3804 −1.59388
\(708\) 8.39566 0.315528
\(709\) −13.6687 −0.513340 −0.256670 0.966499i \(-0.582625\pi\)
−0.256670 + 0.966499i \(0.582625\pi\)
\(710\) 1.55454 0.0583410
\(711\) 5.88236 0.220606
\(712\) −8.95858 −0.335737
\(713\) −27.3661 −1.02487
\(714\) −9.30829 −0.348354
\(715\) 0.361572 0.0135220
\(716\) 11.4232 0.426906
\(717\) 17.6397 0.658768
\(718\) −2.18365 −0.0814931
\(719\) 31.2150 1.16412 0.582061 0.813145i \(-0.302247\pi\)
0.582061 + 0.813145i \(0.302247\pi\)
\(720\) 0.113923 0.00424565
\(721\) 2.65133 0.0987406
\(722\) −13.9032 −0.517424
\(723\) −24.3235 −0.904601
\(724\) −14.5444 −0.540540
\(725\) −4.94246 −0.183558
\(726\) 0.926787 0.0343963
\(727\) −44.5784 −1.65332 −0.826661 0.562700i \(-0.809762\pi\)
−0.826661 + 0.562700i \(0.809762\pi\)
\(728\) −2.65133 −0.0982647
\(729\) 1.00000 0.0370370
\(730\) −0.181160 −0.00670503
\(731\) −21.5172 −0.795842
\(732\) 1.34766 0.0498109
\(733\) 21.7393 0.802961 0.401480 0.915868i \(-0.368496\pi\)
0.401480 + 0.915868i \(0.368496\pi\)
\(734\) −18.7772 −0.693079
\(735\) −0.00336438 −0.000124097 0
\(736\) −4.90813 −0.180916
\(737\) 33.6520 1.23959
\(738\) −8.22718 −0.302847
\(739\) −4.53334 −0.166762 −0.0833808 0.996518i \(-0.526572\pi\)
−0.0833808 + 0.996518i \(0.526572\pi\)
\(740\) −0.832931 −0.0306192
\(741\) −2.25761 −0.0829353
\(742\) 19.1037 0.701320
\(743\) −39.9930 −1.46720 −0.733601 0.679581i \(-0.762162\pi\)
−0.733601 + 0.679581i \(0.762162\pi\)
\(744\) −5.57567 −0.204414
\(745\) −1.37026 −0.0502024
\(746\) −16.7478 −0.613180
\(747\) −12.1432 −0.444296
\(748\) −11.1427 −0.407418
\(749\) 38.3903 1.40275
\(750\) 1.13775 0.0415448
\(751\) −3.51993 −0.128444 −0.0642221 0.997936i \(-0.520457\pi\)
−0.0642221 + 0.997936i \(0.520457\pi\)
\(752\) −0.711399 −0.0259421
\(753\) −16.8621 −0.614489
\(754\) −0.991065 −0.0360924
\(755\) 2.25290 0.0819916
\(756\) −2.65133 −0.0964278
\(757\) −21.0596 −0.765425 −0.382712 0.923868i \(-0.625010\pi\)
−0.382712 + 0.923868i \(0.625010\pi\)
\(758\) 13.4554 0.488724
\(759\) −15.5776 −0.565431
\(760\) −0.257193 −0.00932937
\(761\) −32.5443 −1.17973 −0.589865 0.807502i \(-0.700819\pi\)
−0.589865 + 0.807502i \(0.700819\pi\)
\(762\) 16.0196 0.580330
\(763\) −46.7998 −1.69426
\(764\) −11.4688 −0.414925
\(765\) 0.399961 0.0144606
\(766\) −18.2146 −0.658121
\(767\) 8.39566 0.303150
\(768\) −1.00000 −0.0360844
\(769\) −32.0823 −1.15692 −0.578460 0.815711i \(-0.696346\pi\)
−0.578460 + 0.815711i \(0.696346\pi\)
\(770\) −0.958646 −0.0345472
\(771\) 19.0078 0.684550
\(772\) −7.05090 −0.253767
\(773\) 46.5934 1.67585 0.837923 0.545788i \(-0.183770\pi\)
0.837923 + 0.545788i \(0.183770\pi\)
\(774\) −6.12885 −0.220297
\(775\) −27.8060 −0.998820
\(776\) 10.4001 0.373342
\(777\) 19.3848 0.695426
\(778\) −16.0164 −0.574216
\(779\) 18.5737 0.665473
\(780\) 0.113923 0.00407909
\(781\) −43.3088 −1.54971
\(782\) −17.2315 −0.616197
\(783\) −0.991065 −0.0354178
\(784\) 0.0295321 0.00105472
\(785\) −0.427881 −0.0152717
\(786\) −8.02984 −0.286415
\(787\) −41.6570 −1.48491 −0.742456 0.669895i \(-0.766339\pi\)
−0.742456 + 0.669895i \(0.766339\pi\)
\(788\) 3.41934 0.121809
\(789\) −16.5275 −0.588396
\(790\) 0.670136 0.0238424
\(791\) 46.3077 1.64651
\(792\) −3.17383 −0.112777
\(793\) 1.34766 0.0478568
\(794\) −7.31848 −0.259723
\(795\) −0.820854 −0.0291127
\(796\) 12.7201 0.450851
\(797\) 47.6889 1.68923 0.844614 0.535376i \(-0.179830\pi\)
0.844614 + 0.535376i \(0.179830\pi\)
\(798\) 5.98565 0.211890
\(799\) −2.49758 −0.0883582
\(800\) −4.98702 −0.176318
\(801\) −8.95858 −0.316536
\(802\) 0.477356 0.0168560
\(803\) 5.04702 0.178106
\(804\) 10.6030 0.373938
\(805\) −1.48249 −0.0522507
\(806\) −5.57567 −0.196395
\(807\) −1.06645 −0.0375409
\(808\) −15.9846 −0.562336
\(809\) −12.8475 −0.451692 −0.225846 0.974163i \(-0.572515\pi\)
−0.225846 + 0.974163i \(0.572515\pi\)
\(810\) 0.113923 0.00400284
\(811\) −2.80173 −0.0983821 −0.0491911 0.998789i \(-0.515664\pi\)
−0.0491911 + 0.998789i \(0.515664\pi\)
\(812\) 2.62764 0.0922120
\(813\) 28.1094 0.985839
\(814\) 23.2050 0.813336
\(815\) −1.25823 −0.0440737
\(816\) −3.51081 −0.122903
\(817\) 13.8365 0.484079
\(818\) −14.8337 −0.518649
\(819\) −2.65133 −0.0926448
\(820\) −0.937263 −0.0327307
\(821\) 6.26417 0.218621 0.109310 0.994008i \(-0.465136\pi\)
0.109310 + 0.994008i \(0.465136\pi\)
\(822\) 0.471978 0.0164621
\(823\) 4.76304 0.166029 0.0830145 0.996548i \(-0.473545\pi\)
0.0830145 + 0.996548i \(0.473545\pi\)
\(824\) 1.00000 0.0348367
\(825\) −15.8280 −0.551059
\(826\) −22.2596 −0.774512
\(827\) −1.23342 −0.0428901 −0.0214451 0.999770i \(-0.506827\pi\)
−0.0214451 + 0.999770i \(0.506827\pi\)
\(828\) −4.90813 −0.170569
\(829\) 17.6034 0.611391 0.305696 0.952129i \(-0.401111\pi\)
0.305696 + 0.952129i \(0.401111\pi\)
\(830\) −1.38339 −0.0480181
\(831\) 8.95306 0.310578
\(832\) −1.00000 −0.0346688
\(833\) 0.103681 0.00359234
\(834\) −2.51841 −0.0872055
\(835\) 0.116207 0.00402152
\(836\) 7.16527 0.247816
\(837\) −5.57567 −0.192723
\(838\) 6.72517 0.232317
\(839\) −22.2105 −0.766792 −0.383396 0.923584i \(-0.625246\pi\)
−0.383396 + 0.923584i \(0.625246\pi\)
\(840\) −0.302047 −0.0104216
\(841\) −28.0178 −0.966131
\(842\) 23.3325 0.804092
\(843\) −23.4140 −0.806421
\(844\) −8.15531 −0.280717
\(845\) 0.113923 0.00391907
\(846\) −0.711399 −0.0244584
\(847\) −2.45721 −0.0844309
\(848\) 7.20535 0.247433
\(849\) 9.20239 0.315825
\(850\) −17.5085 −0.600535
\(851\) 35.8851 1.23013
\(852\) −13.6456 −0.467490
\(853\) 10.9095 0.373536 0.186768 0.982404i \(-0.440199\pi\)
0.186768 + 0.982404i \(0.440199\pi\)
\(854\) −3.57309 −0.122268
\(855\) −0.257193 −0.00879582
\(856\) 14.4797 0.494905
\(857\) 13.6307 0.465616 0.232808 0.972523i \(-0.425209\pi\)
0.232808 + 0.972523i \(0.425209\pi\)
\(858\) −3.17383 −0.108353
\(859\) 30.4612 1.03932 0.519662 0.854372i \(-0.326058\pi\)
0.519662 + 0.854372i \(0.326058\pi\)
\(860\) −0.698216 −0.0238090
\(861\) 21.8129 0.743383
\(862\) 7.72227 0.263021
\(863\) −4.66223 −0.158704 −0.0793520 0.996847i \(-0.525285\pi\)
−0.0793520 + 0.996847i \(0.525285\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −0.316179 −0.0107504
\(866\) 2.83059 0.0961875
\(867\) 4.67425 0.158746
\(868\) 14.7829 0.501765
\(869\) −18.6696 −0.633324
\(870\) −0.112905 −0.00382784
\(871\) 10.6030 0.359268
\(872\) −17.6514 −0.597753
\(873\) 10.4001 0.351990
\(874\) 11.0806 0.374808
\(875\) −3.01655 −0.101978
\(876\) 1.59020 0.0537278
\(877\) 23.9947 0.810245 0.405122 0.914262i \(-0.367229\pi\)
0.405122 + 0.914262i \(0.367229\pi\)
\(878\) 12.0883 0.407960
\(879\) 12.8741 0.434231
\(880\) −0.361572 −0.0121886
\(881\) −17.6645 −0.595133 −0.297567 0.954701i \(-0.596175\pi\)
−0.297567 + 0.954701i \(0.596175\pi\)
\(882\) 0.0295321 0.000994396 0
\(883\) 29.8929 1.00598 0.502988 0.864294i \(-0.332234\pi\)
0.502988 + 0.864294i \(0.332234\pi\)
\(884\) −3.51081 −0.118081
\(885\) 0.956458 0.0321510
\(886\) 10.8630 0.364948
\(887\) 13.8077 0.463616 0.231808 0.972762i \(-0.425536\pi\)
0.231808 + 0.972762i \(0.425536\pi\)
\(888\) 7.31136 0.245353
\(889\) −42.4733 −1.42451
\(890\) −1.02059 −0.0342101
\(891\) −3.17383 −0.106327
\(892\) 1.92053 0.0643041
\(893\) 1.60606 0.0537447
\(894\) 12.0279 0.402275
\(895\) 1.30137 0.0434999
\(896\) 2.65133 0.0885746
\(897\) −4.90813 −0.163878
\(898\) −18.7203 −0.624705
\(899\) 5.52585 0.184297
\(900\) −4.98702 −0.166234
\(901\) 25.2966 0.842751
\(902\) 26.1117 0.869424
\(903\) 16.2496 0.540752
\(904\) 17.4659 0.580906
\(905\) −1.65695 −0.0550787
\(906\) −19.7757 −0.657004
\(907\) 35.2314 1.16984 0.584920 0.811091i \(-0.301126\pi\)
0.584920 + 0.811091i \(0.301126\pi\)
\(908\) 7.73629 0.256738
\(909\) −15.9846 −0.530176
\(910\) −0.302047 −0.0100128
\(911\) 45.9998 1.52404 0.762021 0.647552i \(-0.224207\pi\)
0.762021 + 0.647552i \(0.224207\pi\)
\(912\) 2.25761 0.0747568
\(913\) 38.5405 1.27550
\(914\) −1.82995 −0.0605293
\(915\) 0.153529 0.00507552
\(916\) 3.09858 0.102380
\(917\) 21.2897 0.703049
\(918\) −3.51081 −0.115874
\(919\) −4.21035 −0.138887 −0.0694434 0.997586i \(-0.522122\pi\)
−0.0694434 + 0.997586i \(0.522122\pi\)
\(920\) −0.559148 −0.0184346
\(921\) 7.06703 0.232867
\(922\) 5.02597 0.165522
\(923\) −13.6456 −0.449150
\(924\) 8.41487 0.276829
\(925\) 36.4619 1.19886
\(926\) −22.5592 −0.741342
\(927\) 1.00000 0.0328443
\(928\) 0.991065 0.0325333
\(929\) 32.5571 1.06816 0.534082 0.845433i \(-0.320657\pi\)
0.534082 + 0.845433i \(0.320657\pi\)
\(930\) −0.635196 −0.0208289
\(931\) −0.0666718 −0.00218508
\(932\) 3.97487 0.130201
\(933\) −8.92110 −0.292064
\(934\) 31.3275 1.02507
\(935\) −1.26941 −0.0415141
\(936\) −1.00000 −0.0326860
\(937\) −30.1835 −0.986053 −0.493027 0.870014i \(-0.664109\pi\)
−0.493027 + 0.870014i \(0.664109\pi\)
\(938\) −28.1119 −0.917887
\(939\) 13.7395 0.448373
\(940\) −0.0810446 −0.00264338
\(941\) −22.3403 −0.728273 −0.364136 0.931346i \(-0.618636\pi\)
−0.364136 + 0.931346i \(0.618636\pi\)
\(942\) 3.75589 0.122373
\(943\) 40.3801 1.31496
\(944\) −8.39566 −0.273255
\(945\) −0.302047 −0.00982558
\(946\) 19.4519 0.632437
\(947\) 52.4983 1.70596 0.852982 0.521940i \(-0.174792\pi\)
0.852982 + 0.521940i \(0.174792\pi\)
\(948\) −5.88236 −0.191050
\(949\) 1.59020 0.0516200
\(950\) 11.2587 0.365282
\(951\) −21.1401 −0.685513
\(952\) 9.30829 0.301683
\(953\) 31.1909 1.01037 0.505186 0.863010i \(-0.331424\pi\)
0.505186 + 0.863010i \(0.331424\pi\)
\(954\) 7.20535 0.233282
\(955\) −1.30655 −0.0422791
\(956\) −17.6397 −0.570509
\(957\) 3.14547 0.101679
\(958\) −9.71552 −0.313894
\(959\) −1.25137 −0.0404088
\(960\) −0.113923 −0.00367684
\(961\) 0.0880923 0.00284169
\(962\) 7.31136 0.235728
\(963\) 14.4797 0.466601
\(964\) 24.3235 0.783408
\(965\) −0.803258 −0.0258578
\(966\) 13.0131 0.418688
\(967\) 13.8566 0.445599 0.222800 0.974864i \(-0.428480\pi\)
0.222800 + 0.974864i \(0.428480\pi\)
\(968\) −0.926787 −0.0297880
\(969\) 7.92602 0.254620
\(970\) 1.18481 0.0380420
\(971\) 18.3937 0.590282 0.295141 0.955454i \(-0.404633\pi\)
0.295141 + 0.955454i \(0.404633\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.67714 0.214059
\(974\) −22.4933 −0.720731
\(975\) −4.98702 −0.159713
\(976\) −1.34766 −0.0431375
\(977\) 19.2772 0.616732 0.308366 0.951268i \(-0.400218\pi\)
0.308366 + 0.951268i \(0.400218\pi\)
\(978\) 11.0445 0.353166
\(979\) 28.4330 0.908723
\(980\) 0.00336438 0.000107471 0
\(981\) −17.6514 −0.563567
\(982\) −13.9549 −0.445320
\(983\) −56.7511 −1.81008 −0.905039 0.425329i \(-0.860158\pi\)
−0.905039 + 0.425329i \(0.860158\pi\)
\(984\) 8.22718 0.262273
\(985\) 0.389541 0.0124118
\(986\) 3.47944 0.110808
\(987\) 1.88615 0.0600369
\(988\) 2.25761 0.0718240
\(989\) 30.0812 0.956526
\(990\) −0.361572 −0.0114915
\(991\) 4.71272 0.149705 0.0748523 0.997195i \(-0.476151\pi\)
0.0748523 + 0.997195i \(0.476151\pi\)
\(992\) 5.57567 0.177028
\(993\) 19.2913 0.612190
\(994\) 36.1789 1.14752
\(995\) 1.44911 0.0459398
\(996\) 12.1432 0.384772
\(997\) 9.06102 0.286965 0.143483 0.989653i \(-0.454170\pi\)
0.143483 + 0.989653i \(0.454170\pi\)
\(998\) 22.1708 0.701805
\(999\) 7.31136 0.231321
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.u.1.6 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.u.1.6 11 1.1 even 1 trivial