Properties

Label 8034.2.a.p.1.5
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 12x^{6} + 12x^{5} + 43x^{4} - 38x^{3} - 49x^{2} + 23x + 20 \) 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 1.5
Root \(-2.01106\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.612396 q^{5} +1.00000 q^{6} -3.48353 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -0.612396 q^{5} +1.00000 q^{6} -3.48353 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.612396 q^{10} +5.85219 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.48353 q^{14} -0.612396 q^{15} +1.00000 q^{16} -4.41285 q^{17} +1.00000 q^{18} -6.11628 q^{19} -0.612396 q^{20} -3.48353 q^{21} +5.85219 q^{22} +5.11376 q^{23} +1.00000 q^{24} -4.62497 q^{25} +1.00000 q^{26} +1.00000 q^{27} -3.48353 q^{28} -7.19584 q^{29} -0.612396 q^{30} -4.63030 q^{31} +1.00000 q^{32} +5.85219 q^{33} -4.41285 q^{34} +2.13330 q^{35} +1.00000 q^{36} -4.90640 q^{37} -6.11628 q^{38} +1.00000 q^{39} -0.612396 q^{40} -2.73338 q^{41} -3.48353 q^{42} -6.64391 q^{43} +5.85219 q^{44} -0.612396 q^{45} +5.11376 q^{46} -4.75015 q^{47} +1.00000 q^{48} +5.13499 q^{49} -4.62497 q^{50} -4.41285 q^{51} +1.00000 q^{52} +2.85708 q^{53} +1.00000 q^{54} -3.58386 q^{55} -3.48353 q^{56} -6.11628 q^{57} -7.19584 q^{58} +10.0815 q^{59} -0.612396 q^{60} +14.1513 q^{61} -4.63030 q^{62} -3.48353 q^{63} +1.00000 q^{64} -0.612396 q^{65} +5.85219 q^{66} -4.74317 q^{67} -4.41285 q^{68} +5.11376 q^{69} +2.13330 q^{70} -5.14935 q^{71} +1.00000 q^{72} -2.75511 q^{73} -4.90640 q^{74} -4.62497 q^{75} -6.11628 q^{76} -20.3863 q^{77} +1.00000 q^{78} -14.8772 q^{79} -0.612396 q^{80} +1.00000 q^{81} -2.73338 q^{82} +17.5526 q^{83} -3.48353 q^{84} +2.70241 q^{85} -6.64391 q^{86} -7.19584 q^{87} +5.85219 q^{88} +8.15381 q^{89} -0.612396 q^{90} -3.48353 q^{91} +5.11376 q^{92} -4.63030 q^{93} -4.75015 q^{94} +3.74558 q^{95} +1.00000 q^{96} -16.6216 q^{97} +5.13499 q^{98} +5.85219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{5} + 8 q^{6} - 6 q^{7} + 8 q^{8} + 8 q^{9} - 8 q^{10} - 7 q^{11} + 8 q^{12} + 8 q^{13} - 6 q^{14} - 8 q^{15} + 8 q^{16} - 20 q^{17} + 8 q^{18} - 12 q^{19} - 8 q^{20} - 6 q^{21} - 7 q^{22} - 14 q^{23} + 8 q^{24} - 2 q^{25} + 8 q^{26} + 8 q^{27} - 6 q^{28} - 25 q^{29} - 8 q^{30} - 12 q^{31} + 8 q^{32} - 7 q^{33} - 20 q^{34} - 18 q^{35} + 8 q^{36} - 15 q^{37} - 12 q^{38} + 8 q^{39} - 8 q^{40} - 18 q^{41} - 6 q^{42} - 8 q^{43} - 7 q^{44} - 8 q^{45} - 14 q^{46} - 12 q^{47} + 8 q^{48} - 8 q^{49} - 2 q^{50} - 20 q^{51} + 8 q^{52} - 25 q^{53} + 8 q^{54} - 8 q^{55} - 6 q^{56} - 12 q^{57} - 25 q^{58} - 9 q^{59} - 8 q^{60} - 2 q^{61} - 12 q^{62} - 6 q^{63} + 8 q^{64} - 8 q^{65} - 7 q^{66} - 8 q^{67} - 20 q^{68} - 14 q^{69} - 18 q^{70} - 13 q^{71} + 8 q^{72} - 2 q^{73} - 15 q^{74} - 2 q^{75} - 12 q^{76} - 5 q^{77} + 8 q^{78} + q^{79} - 8 q^{80} + 8 q^{81} - 18 q^{82} - 6 q^{83} - 6 q^{84} + 5 q^{85} - 8 q^{86} - 25 q^{87} - 7 q^{88} - 17 q^{89} - 8 q^{90} - 6 q^{91} - 14 q^{92} - 12 q^{93} - 12 q^{94} + 10 q^{95} + 8 q^{96} + 19 q^{97} - 8 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −0.612396 −0.273872 −0.136936 0.990580i \(-0.543725\pi\)
−0.136936 + 0.990580i \(0.543725\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.48353 −1.31665 −0.658326 0.752733i \(-0.728735\pi\)
−0.658326 + 0.752733i \(0.728735\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.612396 −0.193656
\(11\) 5.85219 1.76450 0.882251 0.470780i \(-0.156027\pi\)
0.882251 + 0.470780i \(0.156027\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −3.48353 −0.931013
\(15\) −0.612396 −0.158120
\(16\) 1.00000 0.250000
\(17\) −4.41285 −1.07027 −0.535136 0.844766i \(-0.679740\pi\)
−0.535136 + 0.844766i \(0.679740\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.11628 −1.40317 −0.701585 0.712585i \(-0.747524\pi\)
−0.701585 + 0.712585i \(0.747524\pi\)
\(20\) −0.612396 −0.136936
\(21\) −3.48353 −0.760169
\(22\) 5.85219 1.24769
\(23\) 5.11376 1.06629 0.533146 0.846023i \(-0.321010\pi\)
0.533146 + 0.846023i \(0.321010\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.62497 −0.924994
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −3.48353 −0.658326
\(29\) −7.19584 −1.33623 −0.668117 0.744056i \(-0.732899\pi\)
−0.668117 + 0.744056i \(0.732899\pi\)
\(30\) −0.612396 −0.111808
\(31\) −4.63030 −0.831626 −0.415813 0.909450i \(-0.636503\pi\)
−0.415813 + 0.909450i \(0.636503\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.85219 1.01874
\(34\) −4.41285 −0.756797
\(35\) 2.13330 0.360593
\(36\) 1.00000 0.166667
\(37\) −4.90640 −0.806607 −0.403304 0.915066i \(-0.632138\pi\)
−0.403304 + 0.915066i \(0.632138\pi\)
\(38\) −6.11628 −0.992192
\(39\) 1.00000 0.160128
\(40\) −0.612396 −0.0968282
\(41\) −2.73338 −0.426883 −0.213441 0.976956i \(-0.568467\pi\)
−0.213441 + 0.976956i \(0.568467\pi\)
\(42\) −3.48353 −0.537521
\(43\) −6.64391 −1.01319 −0.506594 0.862185i \(-0.669096\pi\)
−0.506594 + 0.862185i \(0.669096\pi\)
\(44\) 5.85219 0.882251
\(45\) −0.612396 −0.0912905
\(46\) 5.11376 0.753982
\(47\) −4.75015 −0.692880 −0.346440 0.938072i \(-0.612610\pi\)
−0.346440 + 0.938072i \(0.612610\pi\)
\(48\) 1.00000 0.144338
\(49\) 5.13499 0.733570
\(50\) −4.62497 −0.654070
\(51\) −4.41285 −0.617922
\(52\) 1.00000 0.138675
\(53\) 2.85708 0.392450 0.196225 0.980559i \(-0.437132\pi\)
0.196225 + 0.980559i \(0.437132\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.58386 −0.483247
\(56\) −3.48353 −0.465506
\(57\) −6.11628 −0.810121
\(58\) −7.19584 −0.944860
\(59\) 10.0815 1.31250 0.656251 0.754543i \(-0.272141\pi\)
0.656251 + 0.754543i \(0.272141\pi\)
\(60\) −0.612396 −0.0790599
\(61\) 14.1513 1.81188 0.905942 0.423402i \(-0.139164\pi\)
0.905942 + 0.423402i \(0.139164\pi\)
\(62\) −4.63030 −0.588049
\(63\) −3.48353 −0.438884
\(64\) 1.00000 0.125000
\(65\) −0.612396 −0.0759583
\(66\) 5.85219 0.720355
\(67\) −4.74317 −0.579471 −0.289735 0.957107i \(-0.593567\pi\)
−0.289735 + 0.957107i \(0.593567\pi\)
\(68\) −4.41285 −0.535136
\(69\) 5.11376 0.615624
\(70\) 2.13330 0.254978
\(71\) −5.14935 −0.611116 −0.305558 0.952174i \(-0.598843\pi\)
−0.305558 + 0.952174i \(0.598843\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.75511 −0.322461 −0.161231 0.986917i \(-0.551546\pi\)
−0.161231 + 0.986917i \(0.551546\pi\)
\(74\) −4.90640 −0.570358
\(75\) −4.62497 −0.534046
\(76\) −6.11628 −0.701585
\(77\) −20.3863 −2.32323
\(78\) 1.00000 0.113228
\(79\) −14.8772 −1.67381 −0.836905 0.547348i \(-0.815637\pi\)
−0.836905 + 0.547348i \(0.815637\pi\)
\(80\) −0.612396 −0.0684679
\(81\) 1.00000 0.111111
\(82\) −2.73338 −0.301852
\(83\) 17.5526 1.92665 0.963325 0.268338i \(-0.0864745\pi\)
0.963325 + 0.268338i \(0.0864745\pi\)
\(84\) −3.48353 −0.380084
\(85\) 2.70241 0.293117
\(86\) −6.64391 −0.716432
\(87\) −7.19584 −0.771475
\(88\) 5.85219 0.623846
\(89\) 8.15381 0.864302 0.432151 0.901801i \(-0.357755\pi\)
0.432151 + 0.901801i \(0.357755\pi\)
\(90\) −0.612396 −0.0645522
\(91\) −3.48353 −0.365173
\(92\) 5.11376 0.533146
\(93\) −4.63030 −0.480140
\(94\) −4.75015 −0.489940
\(95\) 3.74558 0.384289
\(96\) 1.00000 0.102062
\(97\) −16.6216 −1.68767 −0.843834 0.536604i \(-0.819707\pi\)
−0.843834 + 0.536604i \(0.819707\pi\)
\(98\) 5.13499 0.518713
\(99\) 5.85219 0.588167
\(100\) −4.62497 −0.462497
\(101\) 12.0412 1.19815 0.599073 0.800695i \(-0.295536\pi\)
0.599073 + 0.800695i \(0.295536\pi\)
\(102\) −4.41285 −0.436937
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 2.13330 0.208189
\(106\) 2.85708 0.277504
\(107\) −11.1516 −1.07807 −0.539034 0.842284i \(-0.681211\pi\)
−0.539034 + 0.842284i \(0.681211\pi\)
\(108\) 1.00000 0.0962250
\(109\) −17.3067 −1.65768 −0.828842 0.559483i \(-0.811000\pi\)
−0.828842 + 0.559483i \(0.811000\pi\)
\(110\) −3.58386 −0.341707
\(111\) −4.90640 −0.465695
\(112\) −3.48353 −0.329163
\(113\) −17.6891 −1.66405 −0.832026 0.554737i \(-0.812819\pi\)
−0.832026 + 0.554737i \(0.812819\pi\)
\(114\) −6.11628 −0.572842
\(115\) −3.13164 −0.292027
\(116\) −7.19584 −0.668117
\(117\) 1.00000 0.0924500
\(118\) 10.0815 0.928079
\(119\) 15.3723 1.40918
\(120\) −0.612396 −0.0559038
\(121\) 23.2481 2.11347
\(122\) 14.1513 1.28120
\(123\) −2.73338 −0.246461
\(124\) −4.63030 −0.415813
\(125\) 5.89429 0.527201
\(126\) −3.48353 −0.310338
\(127\) 5.50207 0.488230 0.244115 0.969746i \(-0.421503\pi\)
0.244115 + 0.969746i \(0.421503\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.64391 −0.584964
\(130\) −0.612396 −0.0537106
\(131\) −11.9690 −1.04574 −0.522870 0.852413i \(-0.675139\pi\)
−0.522870 + 0.852413i \(0.675139\pi\)
\(132\) 5.85219 0.509368
\(133\) 21.3063 1.84749
\(134\) −4.74317 −0.409748
\(135\) −0.612396 −0.0527066
\(136\) −4.41285 −0.378399
\(137\) 21.7646 1.85947 0.929736 0.368228i \(-0.120035\pi\)
0.929736 + 0.368228i \(0.120035\pi\)
\(138\) 5.11376 0.435312
\(139\) −20.0706 −1.70236 −0.851182 0.524871i \(-0.824114\pi\)
−0.851182 + 0.524871i \(0.824114\pi\)
\(140\) 2.13330 0.180297
\(141\) −4.75015 −0.400035
\(142\) −5.14935 −0.432124
\(143\) 5.85219 0.489385
\(144\) 1.00000 0.0833333
\(145\) 4.40670 0.365956
\(146\) −2.75511 −0.228015
\(147\) 5.13499 0.423527
\(148\) −4.90640 −0.403304
\(149\) 7.12490 0.583695 0.291847 0.956465i \(-0.405730\pi\)
0.291847 + 0.956465i \(0.405730\pi\)
\(150\) −4.62497 −0.377627
\(151\) −7.40255 −0.602411 −0.301206 0.953559i \(-0.597389\pi\)
−0.301206 + 0.953559i \(0.597389\pi\)
\(152\) −6.11628 −0.496096
\(153\) −4.41285 −0.356758
\(154\) −20.3863 −1.64277
\(155\) 2.83557 0.227759
\(156\) 1.00000 0.0800641
\(157\) 2.06155 0.164530 0.0822649 0.996610i \(-0.473785\pi\)
0.0822649 + 0.996610i \(0.473785\pi\)
\(158\) −14.8772 −1.18356
\(159\) 2.85708 0.226581
\(160\) −0.612396 −0.0484141
\(161\) −17.8139 −1.40393
\(162\) 1.00000 0.0785674
\(163\) 7.03191 0.550782 0.275391 0.961332i \(-0.411193\pi\)
0.275391 + 0.961332i \(0.411193\pi\)
\(164\) −2.73338 −0.213441
\(165\) −3.58386 −0.279003
\(166\) 17.5526 1.36235
\(167\) −5.22098 −0.404011 −0.202006 0.979384i \(-0.564746\pi\)
−0.202006 + 0.979384i \(0.564746\pi\)
\(168\) −3.48353 −0.268760
\(169\) 1.00000 0.0769231
\(170\) 2.70241 0.207265
\(171\) −6.11628 −0.467724
\(172\) −6.64391 −0.506594
\(173\) −0.772617 −0.0587410 −0.0293705 0.999569i \(-0.509350\pi\)
−0.0293705 + 0.999569i \(0.509350\pi\)
\(174\) −7.19584 −0.545515
\(175\) 16.1112 1.21789
\(176\) 5.85219 0.441125
\(177\) 10.0815 0.757773
\(178\) 8.15381 0.611154
\(179\) −10.5628 −0.789499 −0.394750 0.918789i \(-0.629169\pi\)
−0.394750 + 0.918789i \(0.629169\pi\)
\(180\) −0.612396 −0.0456453
\(181\) 10.7815 0.801381 0.400690 0.916214i \(-0.368770\pi\)
0.400690 + 0.916214i \(0.368770\pi\)
\(182\) −3.48353 −0.258217
\(183\) 14.1513 1.04609
\(184\) 5.11376 0.376991
\(185\) 3.00466 0.220907
\(186\) −4.63030 −0.339510
\(187\) −25.8248 −1.88850
\(188\) −4.75015 −0.346440
\(189\) −3.48353 −0.253390
\(190\) 3.74558 0.271733
\(191\) 14.9225 1.07975 0.539875 0.841745i \(-0.318471\pi\)
0.539875 + 0.841745i \(0.318471\pi\)
\(192\) 1.00000 0.0721688
\(193\) −5.95264 −0.428480 −0.214240 0.976781i \(-0.568728\pi\)
−0.214240 + 0.976781i \(0.568728\pi\)
\(194\) −16.6216 −1.19336
\(195\) −0.612396 −0.0438546
\(196\) 5.13499 0.366785
\(197\) −0.0499995 −0.00356232 −0.00178116 0.999998i \(-0.500567\pi\)
−0.00178116 + 0.999998i \(0.500567\pi\)
\(198\) 5.85219 0.415897
\(199\) −21.6481 −1.53459 −0.767296 0.641293i \(-0.778398\pi\)
−0.767296 + 0.641293i \(0.778398\pi\)
\(200\) −4.62497 −0.327035
\(201\) −4.74317 −0.334558
\(202\) 12.0412 0.847217
\(203\) 25.0669 1.75935
\(204\) −4.41285 −0.308961
\(205\) 1.67391 0.116911
\(206\) 1.00000 0.0696733
\(207\) 5.11376 0.355431
\(208\) 1.00000 0.0693375
\(209\) −35.7936 −2.47590
\(210\) 2.13330 0.147212
\(211\) −22.1750 −1.52659 −0.763294 0.646051i \(-0.776419\pi\)
−0.763294 + 0.646051i \(0.776419\pi\)
\(212\) 2.85708 0.196225
\(213\) −5.14935 −0.352828
\(214\) −11.1516 −0.762310
\(215\) 4.06870 0.277483
\(216\) 1.00000 0.0680414
\(217\) 16.1298 1.09496
\(218\) −17.3067 −1.17216
\(219\) −2.75511 −0.186173
\(220\) −3.58386 −0.241623
\(221\) −4.41285 −0.296840
\(222\) −4.90640 −0.329296
\(223\) 4.05280 0.271395 0.135698 0.990750i \(-0.456672\pi\)
0.135698 + 0.990750i \(0.456672\pi\)
\(224\) −3.48353 −0.232753
\(225\) −4.62497 −0.308331
\(226\) −17.6891 −1.17666
\(227\) −7.40828 −0.491705 −0.245852 0.969307i \(-0.579068\pi\)
−0.245852 + 0.969307i \(0.579068\pi\)
\(228\) −6.11628 −0.405061
\(229\) −1.36046 −0.0899019 −0.0449509 0.998989i \(-0.514313\pi\)
−0.0449509 + 0.998989i \(0.514313\pi\)
\(230\) −3.13164 −0.206494
\(231\) −20.3863 −1.34132
\(232\) −7.19584 −0.472430
\(233\) 2.92599 0.191688 0.0958441 0.995396i \(-0.469445\pi\)
0.0958441 + 0.995396i \(0.469445\pi\)
\(234\) 1.00000 0.0653720
\(235\) 2.90897 0.189760
\(236\) 10.0815 0.656251
\(237\) −14.8772 −0.966375
\(238\) 15.3723 0.996438
\(239\) −9.80743 −0.634390 −0.317195 0.948360i \(-0.602741\pi\)
−0.317195 + 0.948360i \(0.602741\pi\)
\(240\) −0.612396 −0.0395300
\(241\) −3.18594 −0.205224 −0.102612 0.994721i \(-0.532720\pi\)
−0.102612 + 0.994721i \(0.532720\pi\)
\(242\) 23.2481 1.49445
\(243\) 1.00000 0.0641500
\(244\) 14.1513 0.905942
\(245\) −3.14465 −0.200904
\(246\) −2.73338 −0.174274
\(247\) −6.11628 −0.389170
\(248\) −4.63030 −0.294024
\(249\) 17.5526 1.11235
\(250\) 5.89429 0.372788
\(251\) −17.9970 −1.13596 −0.567981 0.823042i \(-0.692275\pi\)
−0.567981 + 0.823042i \(0.692275\pi\)
\(252\) −3.48353 −0.219442
\(253\) 29.9267 1.88147
\(254\) 5.50207 0.345231
\(255\) 2.70241 0.169231
\(256\) 1.00000 0.0625000
\(257\) −2.17579 −0.135722 −0.0678610 0.997695i \(-0.521617\pi\)
−0.0678610 + 0.997695i \(0.521617\pi\)
\(258\) −6.64391 −0.413632
\(259\) 17.0916 1.06202
\(260\) −0.612396 −0.0379792
\(261\) −7.19584 −0.445411
\(262\) −11.9690 −0.739449
\(263\) 18.6103 1.14756 0.573779 0.819010i \(-0.305477\pi\)
0.573779 + 0.819010i \(0.305477\pi\)
\(264\) 5.85219 0.360177
\(265\) −1.74966 −0.107481
\(266\) 21.3063 1.30637
\(267\) 8.15381 0.499005
\(268\) −4.74317 −0.289735
\(269\) 10.2151 0.622827 0.311413 0.950275i \(-0.399198\pi\)
0.311413 + 0.950275i \(0.399198\pi\)
\(270\) −0.612396 −0.0372692
\(271\) −9.90095 −0.601440 −0.300720 0.953713i \(-0.597227\pi\)
−0.300720 + 0.953713i \(0.597227\pi\)
\(272\) −4.41285 −0.267568
\(273\) −3.48353 −0.210833
\(274\) 21.7646 1.31484
\(275\) −27.0662 −1.63215
\(276\) 5.11376 0.307812
\(277\) −15.2883 −0.918585 −0.459292 0.888285i \(-0.651897\pi\)
−0.459292 + 0.888285i \(0.651897\pi\)
\(278\) −20.0706 −1.20375
\(279\) −4.63030 −0.277209
\(280\) 2.13330 0.127489
\(281\) −29.0625 −1.73372 −0.866860 0.498551i \(-0.833866\pi\)
−0.866860 + 0.498551i \(0.833866\pi\)
\(282\) −4.75015 −0.282867
\(283\) −8.88160 −0.527956 −0.263978 0.964529i \(-0.585035\pi\)
−0.263978 + 0.964529i \(0.585035\pi\)
\(284\) −5.14935 −0.305558
\(285\) 3.74558 0.221869
\(286\) 5.85219 0.346047
\(287\) 9.52182 0.562056
\(288\) 1.00000 0.0589256
\(289\) 2.47323 0.145484
\(290\) 4.40670 0.258770
\(291\) −16.6216 −0.974376
\(292\) −2.75511 −0.161231
\(293\) 20.6747 1.20783 0.603914 0.797050i \(-0.293607\pi\)
0.603914 + 0.797050i \(0.293607\pi\)
\(294\) 5.13499 0.299479
\(295\) −6.17388 −0.359457
\(296\) −4.90640 −0.285179
\(297\) 5.85219 0.339579
\(298\) 7.12490 0.412734
\(299\) 5.11376 0.295736
\(300\) −4.62497 −0.267023
\(301\) 23.1443 1.33401
\(302\) −7.40255 −0.425969
\(303\) 12.0412 0.691750
\(304\) −6.11628 −0.350793
\(305\) −8.66617 −0.496224
\(306\) −4.41285 −0.252266
\(307\) 28.7899 1.64313 0.821563 0.570117i \(-0.193102\pi\)
0.821563 + 0.570117i \(0.193102\pi\)
\(308\) −20.3863 −1.16162
\(309\) 1.00000 0.0568880
\(310\) 2.83557 0.161050
\(311\) −14.2384 −0.807386 −0.403693 0.914894i \(-0.632274\pi\)
−0.403693 + 0.914894i \(0.632274\pi\)
\(312\) 1.00000 0.0566139
\(313\) −28.8988 −1.63346 −0.816728 0.577023i \(-0.804214\pi\)
−0.816728 + 0.577023i \(0.804214\pi\)
\(314\) 2.06155 0.116340
\(315\) 2.13330 0.120198
\(316\) −14.8772 −0.836905
\(317\) 17.5058 0.983225 0.491613 0.870814i \(-0.336408\pi\)
0.491613 + 0.870814i \(0.336408\pi\)
\(318\) 2.85708 0.160217
\(319\) −42.1114 −2.35779
\(320\) −0.612396 −0.0342340
\(321\) −11.1516 −0.622423
\(322\) −17.8139 −0.992732
\(323\) 26.9902 1.50178
\(324\) 1.00000 0.0555556
\(325\) −4.62497 −0.256547
\(326\) 7.03191 0.389461
\(327\) −17.3067 −0.957064
\(328\) −2.73338 −0.150926
\(329\) 16.5473 0.912282
\(330\) −3.58386 −0.197285
\(331\) 13.6127 0.748222 0.374111 0.927384i \(-0.377948\pi\)
0.374111 + 0.927384i \(0.377948\pi\)
\(332\) 17.5526 0.963325
\(333\) −4.90640 −0.268869
\(334\) −5.22098 −0.285679
\(335\) 2.90470 0.158701
\(336\) −3.48353 −0.190042
\(337\) −18.2169 −0.992337 −0.496169 0.868226i \(-0.665260\pi\)
−0.496169 + 0.868226i \(0.665260\pi\)
\(338\) 1.00000 0.0543928
\(339\) −17.6891 −0.960741
\(340\) 2.70241 0.146559
\(341\) −27.0974 −1.46741
\(342\) −6.11628 −0.330731
\(343\) 6.49681 0.350795
\(344\) −6.64391 −0.358216
\(345\) −3.13164 −0.168602
\(346\) −0.772617 −0.0415362
\(347\) 2.22190 0.119278 0.0596389 0.998220i \(-0.481005\pi\)
0.0596389 + 0.998220i \(0.481005\pi\)
\(348\) −7.19584 −0.385737
\(349\) −10.3813 −0.555698 −0.277849 0.960625i \(-0.589621\pi\)
−0.277849 + 0.960625i \(0.589621\pi\)
\(350\) 16.1112 0.861182
\(351\) 1.00000 0.0533761
\(352\) 5.85219 0.311923
\(353\) −19.5439 −1.04022 −0.520110 0.854099i \(-0.674109\pi\)
−0.520110 + 0.854099i \(0.674109\pi\)
\(354\) 10.0815 0.535827
\(355\) 3.15344 0.167367
\(356\) 8.15381 0.432151
\(357\) 15.3723 0.813588
\(358\) −10.5628 −0.558260
\(359\) 16.4233 0.866791 0.433396 0.901204i \(-0.357315\pi\)
0.433396 + 0.901204i \(0.357315\pi\)
\(360\) −0.612396 −0.0322761
\(361\) 18.4089 0.968888
\(362\) 10.7815 0.566662
\(363\) 23.2481 1.22021
\(364\) −3.48353 −0.182587
\(365\) 1.68722 0.0883130
\(366\) 14.1513 0.739698
\(367\) 9.91003 0.517299 0.258650 0.965971i \(-0.416722\pi\)
0.258650 + 0.965971i \(0.416722\pi\)
\(368\) 5.11376 0.266573
\(369\) −2.73338 −0.142294
\(370\) 3.00466 0.156205
\(371\) −9.95273 −0.516720
\(372\) −4.63030 −0.240070
\(373\) −16.0590 −0.831502 −0.415751 0.909478i \(-0.636481\pi\)
−0.415751 + 0.909478i \(0.636481\pi\)
\(374\) −25.8248 −1.33537
\(375\) 5.89429 0.304380
\(376\) −4.75015 −0.244970
\(377\) −7.19584 −0.370604
\(378\) −3.48353 −0.179174
\(379\) −24.6722 −1.26733 −0.633663 0.773609i \(-0.718449\pi\)
−0.633663 + 0.773609i \(0.718449\pi\)
\(380\) 3.74558 0.192144
\(381\) 5.50207 0.281880
\(382\) 14.9225 0.763499
\(383\) 13.5806 0.693935 0.346968 0.937877i \(-0.387211\pi\)
0.346968 + 0.937877i \(0.387211\pi\)
\(384\) 1.00000 0.0510310
\(385\) 12.4845 0.636268
\(386\) −5.95264 −0.302981
\(387\) −6.64391 −0.337729
\(388\) −16.6216 −0.843834
\(389\) 35.1204 1.78068 0.890339 0.455299i \(-0.150468\pi\)
0.890339 + 0.455299i \(0.150468\pi\)
\(390\) −0.612396 −0.0310099
\(391\) −22.5662 −1.14122
\(392\) 5.13499 0.259356
\(393\) −11.9690 −0.603758
\(394\) −0.0499995 −0.00251894
\(395\) 9.11070 0.458409
\(396\) 5.85219 0.294084
\(397\) 26.1141 1.31063 0.655315 0.755356i \(-0.272536\pi\)
0.655315 + 0.755356i \(0.272536\pi\)
\(398\) −21.6481 −1.08512
\(399\) 21.3063 1.06665
\(400\) −4.62497 −0.231249
\(401\) −30.5163 −1.52391 −0.761955 0.647630i \(-0.775760\pi\)
−0.761955 + 0.647630i \(0.775760\pi\)
\(402\) −4.74317 −0.236568
\(403\) −4.63030 −0.230652
\(404\) 12.0412 0.599073
\(405\) −0.612396 −0.0304302
\(406\) 25.0669 1.24405
\(407\) −28.7132 −1.42326
\(408\) −4.41285 −0.218469
\(409\) 7.07618 0.349895 0.174947 0.984578i \(-0.444024\pi\)
0.174947 + 0.984578i \(0.444024\pi\)
\(410\) 1.67391 0.0826686
\(411\) 21.7646 1.07357
\(412\) 1.00000 0.0492665
\(413\) −35.1193 −1.72811
\(414\) 5.11376 0.251327
\(415\) −10.7491 −0.527655
\(416\) 1.00000 0.0490290
\(417\) −20.0706 −0.982860
\(418\) −35.7936 −1.75072
\(419\) −8.13256 −0.397301 −0.198651 0.980070i \(-0.563656\pi\)
−0.198651 + 0.980070i \(0.563656\pi\)
\(420\) 2.13330 0.104094
\(421\) 33.9300 1.65365 0.826823 0.562462i \(-0.190146\pi\)
0.826823 + 0.562462i \(0.190146\pi\)
\(422\) −22.1750 −1.07946
\(423\) −4.75015 −0.230960
\(424\) 2.85708 0.138752
\(425\) 20.4093 0.989996
\(426\) −5.14935 −0.249487
\(427\) −49.2964 −2.38562
\(428\) −11.1516 −0.539034
\(429\) 5.85219 0.282546
\(430\) 4.06870 0.196210
\(431\) 25.4352 1.22517 0.612585 0.790405i \(-0.290130\pi\)
0.612585 + 0.790405i \(0.290130\pi\)
\(432\) 1.00000 0.0481125
\(433\) 6.45505 0.310210 0.155105 0.987898i \(-0.450428\pi\)
0.155105 + 0.987898i \(0.450428\pi\)
\(434\) 16.1298 0.774255
\(435\) 4.40670 0.211285
\(436\) −17.3067 −0.828842
\(437\) −31.2772 −1.49619
\(438\) −2.75511 −0.131644
\(439\) −5.52037 −0.263473 −0.131736 0.991285i \(-0.542055\pi\)
−0.131736 + 0.991285i \(0.542055\pi\)
\(440\) −3.58386 −0.170854
\(441\) 5.13499 0.244523
\(442\) −4.41285 −0.209898
\(443\) −1.12187 −0.0533016 −0.0266508 0.999645i \(-0.508484\pi\)
−0.0266508 + 0.999645i \(0.508484\pi\)
\(444\) −4.90640 −0.232847
\(445\) −4.99336 −0.236708
\(446\) 4.05280 0.191906
\(447\) 7.12490 0.336996
\(448\) −3.48353 −0.164581
\(449\) −21.3872 −1.00932 −0.504662 0.863317i \(-0.668383\pi\)
−0.504662 + 0.863317i \(0.668383\pi\)
\(450\) −4.62497 −0.218023
\(451\) −15.9963 −0.753235
\(452\) −17.6891 −0.832026
\(453\) −7.40255 −0.347802
\(454\) −7.40828 −0.347688
\(455\) 2.13330 0.100011
\(456\) −6.11628 −0.286421
\(457\) −35.8670 −1.67779 −0.838894 0.544295i \(-0.816797\pi\)
−0.838894 + 0.544295i \(0.816797\pi\)
\(458\) −1.36046 −0.0635702
\(459\) −4.41285 −0.205974
\(460\) −3.13164 −0.146014
\(461\) 18.2746 0.851135 0.425567 0.904927i \(-0.360075\pi\)
0.425567 + 0.904927i \(0.360075\pi\)
\(462\) −20.3863 −0.948456
\(463\) −12.7228 −0.591279 −0.295640 0.955300i \(-0.595533\pi\)
−0.295640 + 0.955300i \(0.595533\pi\)
\(464\) −7.19584 −0.334058
\(465\) 2.83557 0.131497
\(466\) 2.92599 0.135544
\(467\) 4.92734 0.228010 0.114005 0.993480i \(-0.463632\pi\)
0.114005 + 0.993480i \(0.463632\pi\)
\(468\) 1.00000 0.0462250
\(469\) 16.5230 0.762961
\(470\) 2.90897 0.134181
\(471\) 2.06155 0.0949913
\(472\) 10.0815 0.464039
\(473\) −38.8814 −1.78777
\(474\) −14.8772 −0.683330
\(475\) 28.2876 1.29793
\(476\) 15.3723 0.704588
\(477\) 2.85708 0.130817
\(478\) −9.80743 −0.448581
\(479\) −37.8264 −1.72833 −0.864166 0.503207i \(-0.832153\pi\)
−0.864166 + 0.503207i \(0.832153\pi\)
\(480\) −0.612396 −0.0279519
\(481\) −4.90640 −0.223713
\(482\) −3.18594 −0.145116
\(483\) −17.8139 −0.810562
\(484\) 23.2481 1.05673
\(485\) 10.1790 0.462204
\(486\) 1.00000 0.0453609
\(487\) 33.2028 1.50456 0.752282 0.658841i \(-0.228953\pi\)
0.752282 + 0.658841i \(0.228953\pi\)
\(488\) 14.1513 0.640598
\(489\) 7.03191 0.317994
\(490\) −3.14465 −0.142061
\(491\) −32.9200 −1.48566 −0.742828 0.669482i \(-0.766516\pi\)
−0.742828 + 0.669482i \(0.766516\pi\)
\(492\) −2.73338 −0.123230
\(493\) 31.7541 1.43013
\(494\) −6.11628 −0.275184
\(495\) −3.58386 −0.161082
\(496\) −4.63030 −0.207907
\(497\) 17.9379 0.804626
\(498\) 17.5526 0.786551
\(499\) 43.7955 1.96055 0.980277 0.197628i \(-0.0633236\pi\)
0.980277 + 0.197628i \(0.0633236\pi\)
\(500\) 5.89429 0.263601
\(501\) −5.22098 −0.233256
\(502\) −17.9970 −0.803247
\(503\) 26.4282 1.17838 0.589188 0.807996i \(-0.299448\pi\)
0.589188 + 0.807996i \(0.299448\pi\)
\(504\) −3.48353 −0.155169
\(505\) −7.37398 −0.328138
\(506\) 29.9267 1.33040
\(507\) 1.00000 0.0444116
\(508\) 5.50207 0.244115
\(509\) −40.7276 −1.80522 −0.902609 0.430462i \(-0.858351\pi\)
−0.902609 + 0.430462i \(0.858351\pi\)
\(510\) 2.70241 0.119665
\(511\) 9.59751 0.424569
\(512\) 1.00000 0.0441942
\(513\) −6.11628 −0.270040
\(514\) −2.17579 −0.0959700
\(515\) −0.612396 −0.0269854
\(516\) −6.64391 −0.292482
\(517\) −27.7988 −1.22259
\(518\) 17.0916 0.750962
\(519\) −0.772617 −0.0339141
\(520\) −0.612396 −0.0268553
\(521\) −0.959434 −0.0420336 −0.0210168 0.999779i \(-0.506690\pi\)
−0.0210168 + 0.999779i \(0.506690\pi\)
\(522\) −7.19584 −0.314953
\(523\) 41.6081 1.81939 0.909697 0.415273i \(-0.136314\pi\)
0.909697 + 0.415273i \(0.136314\pi\)
\(524\) −11.9690 −0.522870
\(525\) 16.1112 0.703152
\(526\) 18.6103 0.811446
\(527\) 20.4328 0.890067
\(528\) 5.85219 0.254684
\(529\) 3.15050 0.136978
\(530\) −1.74966 −0.0760005
\(531\) 10.0815 0.437501
\(532\) 21.3063 0.923743
\(533\) −2.73338 −0.118396
\(534\) 8.15381 0.352850
\(535\) 6.82921 0.295252
\(536\) −4.74317 −0.204874
\(537\) −10.5628 −0.455818
\(538\) 10.2151 0.440405
\(539\) 30.0510 1.29439
\(540\) −0.612396 −0.0263533
\(541\) −1.42268 −0.0611656 −0.0305828 0.999532i \(-0.509736\pi\)
−0.0305828 + 0.999532i \(0.509736\pi\)
\(542\) −9.90095 −0.425282
\(543\) 10.7815 0.462677
\(544\) −4.41285 −0.189199
\(545\) 10.5986 0.453993
\(546\) −3.48353 −0.149081
\(547\) −5.60579 −0.239686 −0.119843 0.992793i \(-0.538239\pi\)
−0.119843 + 0.992793i \(0.538239\pi\)
\(548\) 21.7646 0.929736
\(549\) 14.1513 0.603961
\(550\) −27.0662 −1.15411
\(551\) 44.0118 1.87496
\(552\) 5.11376 0.217656
\(553\) 51.8250 2.20382
\(554\) −15.2883 −0.649537
\(555\) 3.00466 0.127541
\(556\) −20.0706 −0.851182
\(557\) −23.0865 −0.978208 −0.489104 0.872225i \(-0.662676\pi\)
−0.489104 + 0.872225i \(0.662676\pi\)
\(558\) −4.63030 −0.196016
\(559\) −6.64391 −0.281008
\(560\) 2.13330 0.0901483
\(561\) −25.8248 −1.09033
\(562\) −29.0625 −1.22593
\(563\) 15.9022 0.670200 0.335100 0.942183i \(-0.391230\pi\)
0.335100 + 0.942183i \(0.391230\pi\)
\(564\) −4.75015 −0.200017
\(565\) 10.8327 0.455737
\(566\) −8.88160 −0.373322
\(567\) −3.48353 −0.146295
\(568\) −5.14935 −0.216062
\(569\) 40.4754 1.69682 0.848409 0.529342i \(-0.177561\pi\)
0.848409 + 0.529342i \(0.177561\pi\)
\(570\) 3.74558 0.156885
\(571\) 24.1215 1.00945 0.504727 0.863279i \(-0.331593\pi\)
0.504727 + 0.863279i \(0.331593\pi\)
\(572\) 5.85219 0.244692
\(573\) 14.9225 0.623394
\(574\) 9.52182 0.397433
\(575\) −23.6510 −0.986314
\(576\) 1.00000 0.0416667
\(577\) −1.40846 −0.0586349 −0.0293174 0.999570i \(-0.509333\pi\)
−0.0293174 + 0.999570i \(0.509333\pi\)
\(578\) 2.47323 0.102873
\(579\) −5.95264 −0.247383
\(580\) 4.40670 0.182978
\(581\) −61.1451 −2.53673
\(582\) −16.6216 −0.688988
\(583\) 16.7202 0.692479
\(584\) −2.75511 −0.114007
\(585\) −0.612396 −0.0253194
\(586\) 20.6747 0.854063
\(587\) 2.68345 0.110758 0.0553789 0.998465i \(-0.482363\pi\)
0.0553789 + 0.998465i \(0.482363\pi\)
\(588\) 5.13499 0.211764
\(589\) 28.3202 1.16691
\(590\) −6.17388 −0.254174
\(591\) −0.0499995 −0.00205670
\(592\) −4.90640 −0.201652
\(593\) −3.13502 −0.128740 −0.0643698 0.997926i \(-0.520504\pi\)
−0.0643698 + 0.997926i \(0.520504\pi\)
\(594\) 5.85219 0.240118
\(595\) −9.41393 −0.385933
\(596\) 7.12490 0.291847
\(597\) −21.6481 −0.885997
\(598\) 5.11376 0.209117
\(599\) −17.6222 −0.720023 −0.360011 0.932948i \(-0.617227\pi\)
−0.360011 + 0.932948i \(0.617227\pi\)
\(600\) −4.62497 −0.188814
\(601\) 18.4794 0.753790 0.376895 0.926256i \(-0.376992\pi\)
0.376895 + 0.926256i \(0.376992\pi\)
\(602\) 23.1443 0.943291
\(603\) −4.74317 −0.193157
\(604\) −7.40255 −0.301206
\(605\) −14.2371 −0.578818
\(606\) 12.0412 0.489141
\(607\) 13.1121 0.532203 0.266101 0.963945i \(-0.414264\pi\)
0.266101 + 0.963945i \(0.414264\pi\)
\(608\) −6.11628 −0.248048
\(609\) 25.0669 1.01576
\(610\) −8.66617 −0.350883
\(611\) −4.75015 −0.192170
\(612\) −4.41285 −0.178379
\(613\) −4.83225 −0.195173 −0.0975864 0.995227i \(-0.531112\pi\)
−0.0975864 + 0.995227i \(0.531112\pi\)
\(614\) 28.7899 1.16187
\(615\) 1.67391 0.0674986
\(616\) −20.3863 −0.821387
\(617\) −22.7084 −0.914205 −0.457103 0.889414i \(-0.651113\pi\)
−0.457103 + 0.889414i \(0.651113\pi\)
\(618\) 1.00000 0.0402259
\(619\) −2.60889 −0.104860 −0.0524302 0.998625i \(-0.516697\pi\)
−0.0524302 + 0.998625i \(0.516697\pi\)
\(620\) 2.83557 0.113879
\(621\) 5.11376 0.205208
\(622\) −14.2384 −0.570908
\(623\) −28.4040 −1.13798
\(624\) 1.00000 0.0400320
\(625\) 19.5152 0.780609
\(626\) −28.8988 −1.15503
\(627\) −35.7936 −1.42946
\(628\) 2.06155 0.0822649
\(629\) 21.6512 0.863290
\(630\) 2.13330 0.0849927
\(631\) −4.76038 −0.189508 −0.0947539 0.995501i \(-0.530206\pi\)
−0.0947539 + 0.995501i \(0.530206\pi\)
\(632\) −14.8772 −0.591781
\(633\) −22.1750 −0.881376
\(634\) 17.5058 0.695245
\(635\) −3.36944 −0.133712
\(636\) 2.85708 0.113291
\(637\) 5.13499 0.203456
\(638\) −42.1114 −1.66721
\(639\) −5.14935 −0.203705
\(640\) −0.612396 −0.0242071
\(641\) 2.57178 0.101579 0.0507895 0.998709i \(-0.483826\pi\)
0.0507895 + 0.998709i \(0.483826\pi\)
\(642\) −11.1516 −0.440120
\(643\) 49.6390 1.95757 0.978785 0.204891i \(-0.0656838\pi\)
0.978785 + 0.204891i \(0.0656838\pi\)
\(644\) −17.8139 −0.701967
\(645\) 4.06870 0.160205
\(646\) 26.9902 1.06192
\(647\) 20.9211 0.822492 0.411246 0.911524i \(-0.365094\pi\)
0.411246 + 0.911524i \(0.365094\pi\)
\(648\) 1.00000 0.0392837
\(649\) 58.9989 2.31591
\(650\) −4.62497 −0.181406
\(651\) 16.1298 0.632176
\(652\) 7.03191 0.275391
\(653\) 11.0240 0.431402 0.215701 0.976460i \(-0.430796\pi\)
0.215701 + 0.976460i \(0.430796\pi\)
\(654\) −17.3067 −0.676747
\(655\) 7.32978 0.286398
\(656\) −2.73338 −0.106721
\(657\) −2.75511 −0.107487
\(658\) 16.5473 0.645081
\(659\) −1.31811 −0.0513464 −0.0256732 0.999670i \(-0.508173\pi\)
−0.0256732 + 0.999670i \(0.508173\pi\)
\(660\) −3.58386 −0.139501
\(661\) −10.1024 −0.392940 −0.196470 0.980510i \(-0.562948\pi\)
−0.196470 + 0.980510i \(0.562948\pi\)
\(662\) 13.6127 0.529073
\(663\) −4.41285 −0.171381
\(664\) 17.5526 0.681173
\(665\) −13.0479 −0.505974
\(666\) −4.90640 −0.190119
\(667\) −36.7978 −1.42481
\(668\) −5.22098 −0.202006
\(669\) 4.05280 0.156690
\(670\) 2.90470 0.112218
\(671\) 82.8159 3.19707
\(672\) −3.48353 −0.134380
\(673\) 12.3916 0.477662 0.238831 0.971061i \(-0.423236\pi\)
0.238831 + 0.971061i \(0.423236\pi\)
\(674\) −18.2169 −0.701688
\(675\) −4.62497 −0.178015
\(676\) 1.00000 0.0384615
\(677\) −38.3673 −1.47457 −0.737287 0.675580i \(-0.763893\pi\)
−0.737287 + 0.675580i \(0.763893\pi\)
\(678\) −17.6891 −0.679346
\(679\) 57.9019 2.22207
\(680\) 2.70241 0.103633
\(681\) −7.40828 −0.283886
\(682\) −27.0974 −1.03761
\(683\) 40.7863 1.56065 0.780323 0.625377i \(-0.215055\pi\)
0.780323 + 0.625377i \(0.215055\pi\)
\(684\) −6.11628 −0.233862
\(685\) −13.3285 −0.509256
\(686\) 6.49681 0.248049
\(687\) −1.36046 −0.0519049
\(688\) −6.64391 −0.253297
\(689\) 2.85708 0.108846
\(690\) −3.13164 −0.119220
\(691\) 28.9316 1.10061 0.550306 0.834963i \(-0.314511\pi\)
0.550306 + 0.834963i \(0.314511\pi\)
\(692\) −0.772617 −0.0293705
\(693\) −20.3863 −0.774411
\(694\) 2.22190 0.0843422
\(695\) 12.2911 0.466229
\(696\) −7.19584 −0.272758
\(697\) 12.0620 0.456881
\(698\) −10.3813 −0.392938
\(699\) 2.92599 0.110671
\(700\) 16.1112 0.608947
\(701\) −25.1536 −0.950038 −0.475019 0.879976i \(-0.657559\pi\)
−0.475019 + 0.879976i \(0.657559\pi\)
\(702\) 1.00000 0.0377426
\(703\) 30.0089 1.13181
\(704\) 5.85219 0.220563
\(705\) 2.90897 0.109558
\(706\) −19.5439 −0.735546
\(707\) −41.9459 −1.57754
\(708\) 10.0815 0.378887
\(709\) 7.09284 0.266377 0.133189 0.991091i \(-0.457478\pi\)
0.133189 + 0.991091i \(0.457478\pi\)
\(710\) 3.15344 0.118347
\(711\) −14.8772 −0.557937
\(712\) 8.15381 0.305577
\(713\) −23.6782 −0.886756
\(714\) 15.3723 0.575294
\(715\) −3.58386 −0.134029
\(716\) −10.5628 −0.394750
\(717\) −9.80743 −0.366265
\(718\) 16.4233 0.612914
\(719\) 52.2800 1.94971 0.974857 0.222833i \(-0.0715303\pi\)
0.974857 + 0.222833i \(0.0715303\pi\)
\(720\) −0.612396 −0.0228226
\(721\) −3.48353 −0.129733
\(722\) 18.4089 0.685108
\(723\) −3.18594 −0.118486
\(724\) 10.7815 0.400690
\(725\) 33.2805 1.23601
\(726\) 23.2481 0.862819
\(727\) 29.6708 1.10043 0.550214 0.835024i \(-0.314546\pi\)
0.550214 + 0.835024i \(0.314546\pi\)
\(728\) −3.48353 −0.129108
\(729\) 1.00000 0.0370370
\(730\) 1.68722 0.0624467
\(731\) 29.3186 1.08439
\(732\) 14.1513 0.523046
\(733\) 26.7669 0.988658 0.494329 0.869275i \(-0.335414\pi\)
0.494329 + 0.869275i \(0.335414\pi\)
\(734\) 9.91003 0.365786
\(735\) −3.14465 −0.115992
\(736\) 5.11376 0.188496
\(737\) −27.7580 −1.02248
\(738\) −2.73338 −0.100617
\(739\) −7.88082 −0.289901 −0.144950 0.989439i \(-0.546302\pi\)
−0.144950 + 0.989439i \(0.546302\pi\)
\(740\) 3.00466 0.110453
\(741\) −6.11628 −0.224687
\(742\) −9.95273 −0.365376
\(743\) 28.1717 1.03352 0.516759 0.856131i \(-0.327138\pi\)
0.516759 + 0.856131i \(0.327138\pi\)
\(744\) −4.63030 −0.169755
\(745\) −4.36326 −0.159857
\(746\) −16.0590 −0.587961
\(747\) 17.5526 0.642216
\(748\) −25.8248 −0.944249
\(749\) 38.8470 1.41944
\(750\) 5.89429 0.215229
\(751\) −4.01398 −0.146472 −0.0732360 0.997315i \(-0.523333\pi\)
−0.0732360 + 0.997315i \(0.523333\pi\)
\(752\) −4.75015 −0.173220
\(753\) −17.9970 −0.655848
\(754\) −7.19584 −0.262057
\(755\) 4.53329 0.164983
\(756\) −3.48353 −0.126695
\(757\) −33.4646 −1.21629 −0.608147 0.793825i \(-0.708087\pi\)
−0.608147 + 0.793825i \(0.708087\pi\)
\(758\) −24.6722 −0.896135
\(759\) 29.9267 1.08627
\(760\) 3.74558 0.135867
\(761\) 3.50752 0.127147 0.0635737 0.997977i \(-0.479750\pi\)
0.0635737 + 0.997977i \(0.479750\pi\)
\(762\) 5.50207 0.199319
\(763\) 60.2885 2.18259
\(764\) 14.9225 0.539875
\(765\) 2.70241 0.0977058
\(766\) 13.5806 0.490686
\(767\) 10.0815 0.364022
\(768\) 1.00000 0.0360844
\(769\) 8.88559 0.320423 0.160211 0.987083i \(-0.448782\pi\)
0.160211 + 0.987083i \(0.448782\pi\)
\(770\) 12.4845 0.449909
\(771\) −2.17579 −0.0783592
\(772\) −5.95264 −0.214240
\(773\) 30.8947 1.11120 0.555602 0.831448i \(-0.312488\pi\)
0.555602 + 0.831448i \(0.312488\pi\)
\(774\) −6.64391 −0.238811
\(775\) 21.4150 0.769250
\(776\) −16.6216 −0.596681
\(777\) 17.0916 0.613158
\(778\) 35.1204 1.25913
\(779\) 16.7181 0.598989
\(780\) −0.612396 −0.0219273
\(781\) −30.1350 −1.07831
\(782\) −22.5662 −0.806967
\(783\) −7.19584 −0.257158
\(784\) 5.13499 0.183393
\(785\) −1.26249 −0.0450600
\(786\) −11.9690 −0.426921
\(787\) 31.7040 1.13013 0.565063 0.825048i \(-0.308852\pi\)
0.565063 + 0.825048i \(0.308852\pi\)
\(788\) −0.0499995 −0.00178116
\(789\) 18.6103 0.662543
\(790\) 9.11070 0.324144
\(791\) 61.6206 2.19098
\(792\) 5.85219 0.207949
\(793\) 14.1513 0.502526
\(794\) 26.1141 0.926755
\(795\) −1.74966 −0.0620542
\(796\) −21.6481 −0.767296
\(797\) 48.6722 1.72406 0.862028 0.506861i \(-0.169194\pi\)
0.862028 + 0.506861i \(0.169194\pi\)
\(798\) 21.3063 0.754233
\(799\) 20.9617 0.741571
\(800\) −4.62497 −0.163517
\(801\) 8.15381 0.288101
\(802\) −30.5163 −1.07757
\(803\) −16.1234 −0.568983
\(804\) −4.74317 −0.167279
\(805\) 10.9092 0.384498
\(806\) −4.63030 −0.163095
\(807\) 10.2151 0.359589
\(808\) 12.0412 0.423608
\(809\) 6.54336 0.230052 0.115026 0.993362i \(-0.463305\pi\)
0.115026 + 0.993362i \(0.463305\pi\)
\(810\) −0.612396 −0.0215174
\(811\) −19.0964 −0.670565 −0.335282 0.942118i \(-0.608832\pi\)
−0.335282 + 0.942118i \(0.608832\pi\)
\(812\) 25.0669 0.879677
\(813\) −9.90095 −0.347241
\(814\) −28.7132 −1.00640
\(815\) −4.30631 −0.150843
\(816\) −4.41285 −0.154481
\(817\) 40.6360 1.42168
\(818\) 7.07618 0.247413
\(819\) −3.48353 −0.121724
\(820\) 1.67391 0.0584555
\(821\) 41.0461 1.43252 0.716259 0.697835i \(-0.245853\pi\)
0.716259 + 0.697835i \(0.245853\pi\)
\(822\) 21.7646 0.759126
\(823\) 12.4626 0.434420 0.217210 0.976125i \(-0.430304\pi\)
0.217210 + 0.976125i \(0.430304\pi\)
\(824\) 1.00000 0.0348367
\(825\) −27.0662 −0.942325
\(826\) −35.1193 −1.22196
\(827\) 13.2011 0.459048 0.229524 0.973303i \(-0.426283\pi\)
0.229524 + 0.973303i \(0.426283\pi\)
\(828\) 5.11376 0.177715
\(829\) −13.7036 −0.475945 −0.237972 0.971272i \(-0.576483\pi\)
−0.237972 + 0.971272i \(0.576483\pi\)
\(830\) −10.7491 −0.373108
\(831\) −15.2883 −0.530345
\(832\) 1.00000 0.0346688
\(833\) −22.6599 −0.785121
\(834\) −20.0706 −0.694987
\(835\) 3.19730 0.110647
\(836\) −35.7936 −1.23795
\(837\) −4.63030 −0.160047
\(838\) −8.13256 −0.280935
\(839\) 24.5301 0.846872 0.423436 0.905926i \(-0.360824\pi\)
0.423436 + 0.905926i \(0.360824\pi\)
\(840\) 2.13330 0.0736058
\(841\) 22.7801 0.785520
\(842\) 33.9300 1.16930
\(843\) −29.0625 −1.00096
\(844\) −22.1750 −0.763294
\(845\) −0.612396 −0.0210670
\(846\) −4.75015 −0.163313
\(847\) −80.9856 −2.78270
\(848\) 2.85708 0.0981126
\(849\) −8.88160 −0.304816
\(850\) 20.4093 0.700033
\(851\) −25.0901 −0.860079
\(852\) −5.14935 −0.176414
\(853\) 3.99574 0.136812 0.0684058 0.997658i \(-0.478209\pi\)
0.0684058 + 0.997658i \(0.478209\pi\)
\(854\) −49.2964 −1.68689
\(855\) 3.74558 0.128096
\(856\) −11.1516 −0.381155
\(857\) 11.1097 0.379500 0.189750 0.981832i \(-0.439232\pi\)
0.189750 + 0.981832i \(0.439232\pi\)
\(858\) 5.85219 0.199790
\(859\) 55.0781 1.87924 0.939620 0.342220i \(-0.111179\pi\)
0.939620 + 0.342220i \(0.111179\pi\)
\(860\) 4.06870 0.138742
\(861\) 9.52182 0.324503
\(862\) 25.4352 0.866326
\(863\) −52.6209 −1.79124 −0.895618 0.444824i \(-0.853266\pi\)
−0.895618 + 0.444824i \(0.853266\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0.473147 0.0160875
\(866\) 6.45505 0.219352
\(867\) 2.47323 0.0839953
\(868\) 16.1298 0.547481
\(869\) −87.0639 −2.95344
\(870\) 4.40670 0.149401
\(871\) −4.74317 −0.160716
\(872\) −17.3067 −0.586080
\(873\) −16.6216 −0.562556
\(874\) −31.2772 −1.05797
\(875\) −20.5329 −0.694140
\(876\) −2.75511 −0.0930865
\(877\) −21.1462 −0.714058 −0.357029 0.934093i \(-0.616210\pi\)
−0.357029 + 0.934093i \(0.616210\pi\)
\(878\) −5.52037 −0.186303
\(879\) 20.6747 0.697339
\(880\) −3.58386 −0.120812
\(881\) 15.9866 0.538601 0.269301 0.963056i \(-0.413207\pi\)
0.269301 + 0.963056i \(0.413207\pi\)
\(882\) 5.13499 0.172904
\(883\) 19.3224 0.650252 0.325126 0.945671i \(-0.394593\pi\)
0.325126 + 0.945671i \(0.394593\pi\)
\(884\) −4.41285 −0.148420
\(885\) −6.17388 −0.207533
\(886\) −1.12187 −0.0376899
\(887\) −54.5289 −1.83090 −0.915451 0.402430i \(-0.868166\pi\)
−0.915451 + 0.402430i \(0.868166\pi\)
\(888\) −4.90640 −0.164648
\(889\) −19.1666 −0.642828
\(890\) −4.99336 −0.167378
\(891\) 5.85219 0.196056
\(892\) 4.05280 0.135698
\(893\) 29.0532 0.972229
\(894\) 7.12490 0.238292
\(895\) 6.46860 0.216221
\(896\) −3.48353 −0.116377
\(897\) 5.11376 0.170743
\(898\) −21.3872 −0.713700
\(899\) 33.3189 1.11125
\(900\) −4.62497 −0.154166
\(901\) −12.6079 −0.420029
\(902\) −15.9963 −0.532618
\(903\) 23.1443 0.770194
\(904\) −17.6891 −0.588331
\(905\) −6.60253 −0.219475
\(906\) −7.40255 −0.245933
\(907\) −10.9245 −0.362741 −0.181370 0.983415i \(-0.558053\pi\)
−0.181370 + 0.983415i \(0.558053\pi\)
\(908\) −7.40828 −0.245852
\(909\) 12.0412 0.399382
\(910\) 2.13330 0.0707182
\(911\) −25.7835 −0.854244 −0.427122 0.904194i \(-0.640473\pi\)
−0.427122 + 0.904194i \(0.640473\pi\)
\(912\) −6.11628 −0.202530
\(913\) 102.721 3.39958
\(914\) −35.8670 −1.18638
\(915\) −8.66617 −0.286495
\(916\) −1.36046 −0.0449509
\(917\) 41.6945 1.37687
\(918\) −4.41285 −0.145646
\(919\) −26.0283 −0.858595 −0.429298 0.903163i \(-0.641239\pi\)
−0.429298 + 0.903163i \(0.641239\pi\)
\(920\) −3.13164 −0.103247
\(921\) 28.7899 0.948660
\(922\) 18.2746 0.601843
\(923\) −5.14935 −0.169493
\(924\) −20.3863 −0.670660
\(925\) 22.6920 0.746107
\(926\) −12.7228 −0.418097
\(927\) 1.00000 0.0328443
\(928\) −7.19584 −0.236215
\(929\) −49.0183 −1.60824 −0.804119 0.594469i \(-0.797362\pi\)
−0.804119 + 0.594469i \(0.797362\pi\)
\(930\) 2.83557 0.0929822
\(931\) −31.4071 −1.02932
\(932\) 2.92599 0.0958441
\(933\) −14.2384 −0.466145
\(934\) 4.92734 0.161228
\(935\) 15.8150 0.517206
\(936\) 1.00000 0.0326860
\(937\) −6.72812 −0.219798 −0.109899 0.993943i \(-0.535053\pi\)
−0.109899 + 0.993943i \(0.535053\pi\)
\(938\) 16.5230 0.539495
\(939\) −28.8988 −0.943076
\(940\) 2.90897 0.0948801
\(941\) 31.5977 1.03006 0.515028 0.857173i \(-0.327781\pi\)
0.515028 + 0.857173i \(0.327781\pi\)
\(942\) 2.06155 0.0671690
\(943\) −13.9779 −0.455181
\(944\) 10.0815 0.328125
\(945\) 2.13330 0.0693962
\(946\) −38.8814 −1.26414
\(947\) −17.8014 −0.578467 −0.289234 0.957259i \(-0.593400\pi\)
−0.289234 + 0.957259i \(0.593400\pi\)
\(948\) −14.8772 −0.483187
\(949\) −2.75511 −0.0894347
\(950\) 28.2876 0.917772
\(951\) 17.5058 0.567665
\(952\) 15.3723 0.498219
\(953\) 2.08963 0.0676898 0.0338449 0.999427i \(-0.489225\pi\)
0.0338449 + 0.999427i \(0.489225\pi\)
\(954\) 2.85708 0.0925014
\(955\) −9.13844 −0.295713
\(956\) −9.80743 −0.317195
\(957\) −42.1114 −1.36127
\(958\) −37.8264 −1.22211
\(959\) −75.8175 −2.44828
\(960\) −0.612396 −0.0197650
\(961\) −9.56033 −0.308398
\(962\) −4.90640 −0.158189
\(963\) −11.1516 −0.359356
\(964\) −3.18594 −0.102612
\(965\) 3.64537 0.117349
\(966\) −17.8139 −0.573154
\(967\) 4.77847 0.153665 0.0768326 0.997044i \(-0.475519\pi\)
0.0768326 + 0.997044i \(0.475519\pi\)
\(968\) 23.2481 0.747223
\(969\) 26.9902 0.867051
\(970\) 10.1790 0.326828
\(971\) 50.8623 1.63225 0.816124 0.577877i \(-0.196119\pi\)
0.816124 + 0.577877i \(0.196119\pi\)
\(972\) 1.00000 0.0320750
\(973\) 69.9165 2.24142
\(974\) 33.2028 1.06389
\(975\) −4.62497 −0.148118
\(976\) 14.1513 0.452971
\(977\) 42.7067 1.36631 0.683155 0.730274i \(-0.260607\pi\)
0.683155 + 0.730274i \(0.260607\pi\)
\(978\) 7.03191 0.224856
\(979\) 47.7176 1.52506
\(980\) −3.14465 −0.100452
\(981\) −17.3067 −0.552561
\(982\) −32.9200 −1.05052
\(983\) −48.5646 −1.54897 −0.774485 0.632592i \(-0.781991\pi\)
−0.774485 + 0.632592i \(0.781991\pi\)
\(984\) −2.73338 −0.0871370
\(985\) 0.0306195 0.000975618 0
\(986\) 31.7541 1.01126
\(987\) 16.5473 0.526706
\(988\) −6.11628 −0.194585
\(989\) −33.9754 −1.08035
\(990\) −3.58386 −0.113902
\(991\) −12.7901 −0.406292 −0.203146 0.979148i \(-0.565117\pi\)
−0.203146 + 0.979148i \(0.565117\pi\)
\(992\) −4.63030 −0.147012
\(993\) 13.6127 0.431986
\(994\) 17.9379 0.568957
\(995\) 13.2572 0.420281
\(996\) 17.5526 0.556176
\(997\) −16.0228 −0.507447 −0.253723 0.967277i \(-0.581655\pi\)
−0.253723 + 0.967277i \(0.581655\pi\)
\(998\) 43.7955 1.38632
\(999\) −4.90640 −0.155232
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.p.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.p.1.5 8 1.1 even 1 trivial