Properties

Label 6422.2.a.bg.1.2
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $8$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
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: \( x^{8} - 4x^{7} - 4x^{6} + 22x^{5} + 3x^{4} - 28x^{3} + 7x^{2} + 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.47489\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.47489 q^{3} +1.00000 q^{4} +2.86363 q^{5} +2.47489 q^{6} +1.77068 q^{7} -1.00000 q^{8} +3.12506 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.47489 q^{3} +1.00000 q^{4} +2.86363 q^{5} +2.47489 q^{6} +1.77068 q^{7} -1.00000 q^{8} +3.12506 q^{9} -2.86363 q^{10} +0.764511 q^{11} -2.47489 q^{12} -1.77068 q^{14} -7.08716 q^{15} +1.00000 q^{16} +0.468182 q^{17} -3.12506 q^{18} +1.00000 q^{19} +2.86363 q^{20} -4.38223 q^{21} -0.764511 q^{22} -3.33675 q^{23} +2.47489 q^{24} +3.20038 q^{25} -0.309515 q^{27} +1.77068 q^{28} +3.54830 q^{29} +7.08716 q^{30} -2.73715 q^{31} -1.00000 q^{32} -1.89208 q^{33} -0.468182 q^{34} +5.07057 q^{35} +3.12506 q^{36} -5.74198 q^{37} -1.00000 q^{38} -2.86363 q^{40} -9.46441 q^{41} +4.38223 q^{42} +6.78094 q^{43} +0.764511 q^{44} +8.94902 q^{45} +3.33675 q^{46} -1.37434 q^{47} -2.47489 q^{48} -3.86470 q^{49} -3.20038 q^{50} -1.15870 q^{51} -9.55237 q^{53} +0.309515 q^{54} +2.18928 q^{55} -1.77068 q^{56} -2.47489 q^{57} -3.54830 q^{58} +5.76584 q^{59} -7.08716 q^{60} -2.36516 q^{61} +2.73715 q^{62} +5.53348 q^{63} +1.00000 q^{64} +1.89208 q^{66} -1.49066 q^{67} +0.468182 q^{68} +8.25808 q^{69} -5.07057 q^{70} -2.41091 q^{71} -3.12506 q^{72} -12.1769 q^{73} +5.74198 q^{74} -7.92058 q^{75} +1.00000 q^{76} +1.35370 q^{77} -9.31247 q^{79} +2.86363 q^{80} -8.60917 q^{81} +9.46441 q^{82} -3.26881 q^{83} -4.38223 q^{84} +1.34070 q^{85} -6.78094 q^{86} -8.78165 q^{87} -0.764511 q^{88} -14.5129 q^{89} -8.94902 q^{90} -3.33675 q^{92} +6.77413 q^{93} +1.37434 q^{94} +2.86363 q^{95} +2.47489 q^{96} -2.74409 q^{97} +3.86470 q^{98} +2.38914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{7} - 8 q^{8} - 2 q^{10} + 10 q^{11} - 4 q^{12} - 2 q^{14} + 8 q^{16} + 2 q^{17} + 8 q^{19} + 2 q^{20} - 12 q^{21} - 10 q^{22} - 8 q^{23} + 4 q^{24} - 14 q^{25} - 22 q^{27} + 2 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{32} - 4 q^{33} - 2 q^{34} - 10 q^{35} - 8 q^{38} - 2 q^{40} - 2 q^{41} + 12 q^{42} - 16 q^{43} + 10 q^{44} + 12 q^{45} + 8 q^{46} - 12 q^{47} - 4 q^{48} - 14 q^{49} + 14 q^{50} - 22 q^{51} - 24 q^{53} + 22 q^{54} - 10 q^{55} - 2 q^{56} - 4 q^{57} - 8 q^{58} + 4 q^{59} - 26 q^{61} - 12 q^{62} + 12 q^{63} + 8 q^{64} + 4 q^{66} - 10 q^{67} + 2 q^{68} + 6 q^{69} + 10 q^{70} + 36 q^{71} - 20 q^{73} + 6 q^{75} + 8 q^{76} - 12 q^{77} - 22 q^{79} + 2 q^{80} + 36 q^{81} + 2 q^{82} - 18 q^{83} - 12 q^{84} - 26 q^{85} + 16 q^{86} - 38 q^{87} - 10 q^{88} + 18 q^{89} - 12 q^{90} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 2 q^{95} + 4 q^{96} + 8 q^{97} + 14 q^{98} + 8 q^{99}+O(q^{100}) \) 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
\(3\) −2.47489 −1.42888 −0.714438 0.699699i \(-0.753318\pi\)
−0.714438 + 0.699699i \(0.753318\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.86363 1.28065 0.640327 0.768102i \(-0.278799\pi\)
0.640327 + 0.768102i \(0.278799\pi\)
\(6\) 2.47489 1.01037
\(7\) 1.77068 0.669253 0.334627 0.942351i \(-0.391390\pi\)
0.334627 + 0.942351i \(0.391390\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.12506 1.04169
\(10\) −2.86363 −0.905560
\(11\) 0.764511 0.230509 0.115254 0.993336i \(-0.463232\pi\)
0.115254 + 0.993336i \(0.463232\pi\)
\(12\) −2.47489 −0.714438
\(13\) 0 0
\(14\) −1.77068 −0.473234
\(15\) −7.08716 −1.82990
\(16\) 1.00000 0.250000
\(17\) 0.468182 0.113551 0.0567754 0.998387i \(-0.481918\pi\)
0.0567754 + 0.998387i \(0.481918\pi\)
\(18\) −3.12506 −0.736584
\(19\) 1.00000 0.229416
\(20\) 2.86363 0.640327
\(21\) −4.38223 −0.956280
\(22\) −0.764511 −0.162994
\(23\) −3.33675 −0.695760 −0.347880 0.937539i \(-0.613098\pi\)
−0.347880 + 0.937539i \(0.613098\pi\)
\(24\) 2.47489 0.505184
\(25\) 3.20038 0.640076
\(26\) 0 0
\(27\) −0.309515 −0.0595661
\(28\) 1.77068 0.334627
\(29\) 3.54830 0.658904 0.329452 0.944172i \(-0.393136\pi\)
0.329452 + 0.944172i \(0.393136\pi\)
\(30\) 7.08716 1.29393
\(31\) −2.73715 −0.491607 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.89208 −0.329368
\(34\) −0.468182 −0.0802925
\(35\) 5.07057 0.857082
\(36\) 3.12506 0.520844
\(37\) −5.74198 −0.943976 −0.471988 0.881605i \(-0.656463\pi\)
−0.471988 + 0.881605i \(0.656463\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −2.86363 −0.452780
\(41\) −9.46441 −1.47809 −0.739046 0.673655i \(-0.764723\pi\)
−0.739046 + 0.673655i \(0.764723\pi\)
\(42\) 4.38223 0.676192
\(43\) 6.78094 1.03408 0.517042 0.855960i \(-0.327033\pi\)
0.517042 + 0.855960i \(0.327033\pi\)
\(44\) 0.764511 0.115254
\(45\) 8.94902 1.33404
\(46\) 3.33675 0.491977
\(47\) −1.37434 −0.200467 −0.100234 0.994964i \(-0.531959\pi\)
−0.100234 + 0.994964i \(0.531959\pi\)
\(48\) −2.47489 −0.357219
\(49\) −3.86470 −0.552100
\(50\) −3.20038 −0.452602
\(51\) −1.15870 −0.162250
\(52\) 0 0
\(53\) −9.55237 −1.31212 −0.656060 0.754709i \(-0.727778\pi\)
−0.656060 + 0.754709i \(0.727778\pi\)
\(54\) 0.309515 0.0421196
\(55\) 2.18928 0.295202
\(56\) −1.77068 −0.236617
\(57\) −2.47489 −0.327807
\(58\) −3.54830 −0.465915
\(59\) 5.76584 0.750649 0.375324 0.926894i \(-0.377531\pi\)
0.375324 + 0.926894i \(0.377531\pi\)
\(60\) −7.08716 −0.914948
\(61\) −2.36516 −0.302828 −0.151414 0.988470i \(-0.548383\pi\)
−0.151414 + 0.988470i \(0.548383\pi\)
\(62\) 2.73715 0.347618
\(63\) 5.53348 0.697153
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.89208 0.232899
\(67\) −1.49066 −0.182112 −0.0910562 0.995846i \(-0.529024\pi\)
−0.0910562 + 0.995846i \(0.529024\pi\)
\(68\) 0.468182 0.0567754
\(69\) 8.25808 0.994156
\(70\) −5.07057 −0.606049
\(71\) −2.41091 −0.286122 −0.143061 0.989714i \(-0.545695\pi\)
−0.143061 + 0.989714i \(0.545695\pi\)
\(72\) −3.12506 −0.368292
\(73\) −12.1769 −1.42520 −0.712598 0.701573i \(-0.752482\pi\)
−0.712598 + 0.701573i \(0.752482\pi\)
\(74\) 5.74198 0.667492
\(75\) −7.92058 −0.914590
\(76\) 1.00000 0.114708
\(77\) 1.35370 0.154269
\(78\) 0 0
\(79\) −9.31247 −1.04773 −0.523867 0.851800i \(-0.675511\pi\)
−0.523867 + 0.851800i \(0.675511\pi\)
\(80\) 2.86363 0.320164
\(81\) −8.60917 −0.956575
\(82\) 9.46441 1.04517
\(83\) −3.26881 −0.358798 −0.179399 0.983776i \(-0.557415\pi\)
−0.179399 + 0.983776i \(0.557415\pi\)
\(84\) −4.38223 −0.478140
\(85\) 1.34070 0.145419
\(86\) −6.78094 −0.731208
\(87\) −8.78165 −0.941492
\(88\) −0.764511 −0.0814971
\(89\) −14.5129 −1.53837 −0.769184 0.639027i \(-0.779337\pi\)
−0.769184 + 0.639027i \(0.779337\pi\)
\(90\) −8.94902 −0.943310
\(91\) 0 0
\(92\) −3.33675 −0.347880
\(93\) 6.77413 0.702445
\(94\) 1.37434 0.141752
\(95\) 2.86363 0.293802
\(96\) 2.47489 0.252592
\(97\) −2.74409 −0.278620 −0.139310 0.990249i \(-0.544488\pi\)
−0.139310 + 0.990249i \(0.544488\pi\)
\(98\) 3.86470 0.390394
\(99\) 2.38914 0.240118
\(100\) 3.20038 0.320038
\(101\) 13.3665 1.33002 0.665009 0.746835i \(-0.268428\pi\)
0.665009 + 0.746835i \(0.268428\pi\)
\(102\) 1.15870 0.114728
\(103\) −7.96934 −0.785242 −0.392621 0.919700i \(-0.628432\pi\)
−0.392621 + 0.919700i \(0.628432\pi\)
\(104\) 0 0
\(105\) −12.5491 −1.22466
\(106\) 9.55237 0.927808
\(107\) 5.93502 0.573760 0.286880 0.957966i \(-0.407382\pi\)
0.286880 + 0.957966i \(0.407382\pi\)
\(108\) −0.309515 −0.0297831
\(109\) 2.85280 0.273249 0.136624 0.990623i \(-0.456375\pi\)
0.136624 + 0.990623i \(0.456375\pi\)
\(110\) −2.18928 −0.208739
\(111\) 14.2107 1.34882
\(112\) 1.77068 0.167313
\(113\) −18.2867 −1.72027 −0.860135 0.510066i \(-0.829621\pi\)
−0.860135 + 0.510066i \(0.829621\pi\)
\(114\) 2.47489 0.231794
\(115\) −9.55522 −0.891029
\(116\) 3.54830 0.329452
\(117\) 0 0
\(118\) −5.76584 −0.530789
\(119\) 0.828999 0.0759943
\(120\) 7.08716 0.646966
\(121\) −10.4155 −0.946866
\(122\) 2.36516 0.214132
\(123\) 23.4233 2.11201
\(124\) −2.73715 −0.245803
\(125\) −5.15345 −0.460938
\(126\) −5.53348 −0.492961
\(127\) −14.0132 −1.24347 −0.621734 0.783228i \(-0.713572\pi\)
−0.621734 + 0.783228i \(0.713572\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −16.7821 −1.47758
\(130\) 0 0
\(131\) 14.9283 1.30429 0.652146 0.758094i \(-0.273869\pi\)
0.652146 + 0.758094i \(0.273869\pi\)
\(132\) −1.89208 −0.164684
\(133\) 1.77068 0.153537
\(134\) 1.49066 0.128773
\(135\) −0.886336 −0.0762836
\(136\) −0.468182 −0.0401463
\(137\) 9.46492 0.808643 0.404321 0.914617i \(-0.367508\pi\)
0.404321 + 0.914617i \(0.367508\pi\)
\(138\) −8.25808 −0.702974
\(139\) −21.9657 −1.86311 −0.931554 0.363604i \(-0.881546\pi\)
−0.931554 + 0.363604i \(0.881546\pi\)
\(140\) 5.07057 0.428541
\(141\) 3.40132 0.286443
\(142\) 2.41091 0.202319
\(143\) 0 0
\(144\) 3.12506 0.260422
\(145\) 10.1610 0.843828
\(146\) 12.1769 1.00777
\(147\) 9.56469 0.788882
\(148\) −5.74198 −0.471988
\(149\) −5.84283 −0.478663 −0.239332 0.970938i \(-0.576928\pi\)
−0.239332 + 0.970938i \(0.576928\pi\)
\(150\) 7.92058 0.646712
\(151\) 5.37075 0.437065 0.218533 0.975830i \(-0.429873\pi\)
0.218533 + 0.975830i \(0.429873\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 1.46310 0.118284
\(154\) −1.35370 −0.109084
\(155\) −7.83819 −0.629578
\(156\) 0 0
\(157\) 12.3174 0.983039 0.491519 0.870867i \(-0.336442\pi\)
0.491519 + 0.870867i \(0.336442\pi\)
\(158\) 9.31247 0.740860
\(159\) 23.6410 1.87486
\(160\) −2.86363 −0.226390
\(161\) −5.90831 −0.465640
\(162\) 8.60917 0.676401
\(163\) 3.62592 0.284004 0.142002 0.989866i \(-0.454646\pi\)
0.142002 + 0.989866i \(0.454646\pi\)
\(164\) −9.46441 −0.739046
\(165\) −5.41821 −0.421807
\(166\) 3.26881 0.253709
\(167\) 0.0446192 0.00345274 0.00172637 0.999999i \(-0.499450\pi\)
0.00172637 + 0.999999i \(0.499450\pi\)
\(168\) 4.38223 0.338096
\(169\) 0 0
\(170\) −1.34070 −0.102827
\(171\) 3.12506 0.238979
\(172\) 6.78094 0.517042
\(173\) −14.5771 −1.10827 −0.554137 0.832425i \(-0.686952\pi\)
−0.554137 + 0.832425i \(0.686952\pi\)
\(174\) 8.78165 0.665735
\(175\) 5.66684 0.428373
\(176\) 0.764511 0.0576271
\(177\) −14.2698 −1.07258
\(178\) 14.5129 1.08779
\(179\) −11.7631 −0.879214 −0.439607 0.898190i \(-0.644882\pi\)
−0.439607 + 0.898190i \(0.644882\pi\)
\(180\) 8.94902 0.667021
\(181\) 16.7531 1.24525 0.622625 0.782520i \(-0.286066\pi\)
0.622625 + 0.782520i \(0.286066\pi\)
\(182\) 0 0
\(183\) 5.85350 0.432703
\(184\) 3.33675 0.245988
\(185\) −16.4429 −1.20891
\(186\) −6.77413 −0.496704
\(187\) 0.357930 0.0261744
\(188\) −1.37434 −0.100234
\(189\) −0.548051 −0.0398648
\(190\) −2.86363 −0.207750
\(191\) 23.9089 1.72999 0.864995 0.501780i \(-0.167321\pi\)
0.864995 + 0.501780i \(0.167321\pi\)
\(192\) −2.47489 −0.178610
\(193\) 21.8221 1.57079 0.785393 0.618997i \(-0.212461\pi\)
0.785393 + 0.618997i \(0.212461\pi\)
\(194\) 2.74409 0.197014
\(195\) 0 0
\(196\) −3.86470 −0.276050
\(197\) −3.14538 −0.224099 −0.112049 0.993703i \(-0.535742\pi\)
−0.112049 + 0.993703i \(0.535742\pi\)
\(198\) −2.38914 −0.169789
\(199\) 6.98844 0.495397 0.247699 0.968837i \(-0.420326\pi\)
0.247699 + 0.968837i \(0.420326\pi\)
\(200\) −3.20038 −0.226301
\(201\) 3.68920 0.260216
\(202\) −13.3665 −0.940465
\(203\) 6.28290 0.440973
\(204\) −1.15870 −0.0811250
\(205\) −27.1026 −1.89293
\(206\) 7.96934 0.555250
\(207\) −10.4276 −0.724765
\(208\) 0 0
\(209\) 0.764511 0.0528823
\(210\) 12.5491 0.865969
\(211\) 18.2813 1.25854 0.629268 0.777188i \(-0.283355\pi\)
0.629268 + 0.777188i \(0.283355\pi\)
\(212\) −9.55237 −0.656060
\(213\) 5.96673 0.408833
\(214\) −5.93502 −0.405710
\(215\) 19.4181 1.32430
\(216\) 0.309515 0.0210598
\(217\) −4.84661 −0.329009
\(218\) −2.85280 −0.193216
\(219\) 30.1364 2.03643
\(220\) 2.18928 0.147601
\(221\) 0 0
\(222\) −14.2107 −0.953763
\(223\) −12.1425 −0.813119 −0.406560 0.913624i \(-0.633272\pi\)
−0.406560 + 0.913624i \(0.633272\pi\)
\(224\) −1.77068 −0.118308
\(225\) 10.0014 0.666759
\(226\) 18.2867 1.21642
\(227\) 22.4289 1.48866 0.744328 0.667814i \(-0.232770\pi\)
0.744328 + 0.667814i \(0.232770\pi\)
\(228\) −2.47489 −0.163903
\(229\) −9.13113 −0.603402 −0.301701 0.953403i \(-0.597554\pi\)
−0.301701 + 0.953403i \(0.597554\pi\)
\(230\) 9.55522 0.630052
\(231\) −3.35026 −0.220431
\(232\) −3.54830 −0.232958
\(233\) 14.0387 0.919708 0.459854 0.887994i \(-0.347902\pi\)
0.459854 + 0.887994i \(0.347902\pi\)
\(234\) 0 0
\(235\) −3.93559 −0.256729
\(236\) 5.76584 0.375324
\(237\) 23.0473 1.49708
\(238\) −0.828999 −0.0537361
\(239\) 14.0823 0.910907 0.455453 0.890260i \(-0.349477\pi\)
0.455453 + 0.890260i \(0.349477\pi\)
\(240\) −7.08716 −0.457474
\(241\) 30.4199 1.95951 0.979757 0.200188i \(-0.0641554\pi\)
0.979757 + 0.200188i \(0.0641554\pi\)
\(242\) 10.4155 0.669535
\(243\) 22.2353 1.42639
\(244\) −2.36516 −0.151414
\(245\) −11.0671 −0.707049
\(246\) −23.4233 −1.49342
\(247\) 0 0
\(248\) 2.73715 0.173809
\(249\) 8.08992 0.512678
\(250\) 5.15345 0.325933
\(251\) −4.08765 −0.258010 −0.129005 0.991644i \(-0.541178\pi\)
−0.129005 + 0.991644i \(0.541178\pi\)
\(252\) 5.53348 0.348576
\(253\) −2.55098 −0.160379
\(254\) 14.0132 0.879265
\(255\) −3.31808 −0.207786
\(256\) 1.00000 0.0625000
\(257\) −4.37406 −0.272846 −0.136423 0.990651i \(-0.543561\pi\)
−0.136423 + 0.990651i \(0.543561\pi\)
\(258\) 16.7821 1.04481
\(259\) −10.1672 −0.631759
\(260\) 0 0
\(261\) 11.0887 0.686372
\(262\) −14.9283 −0.922273
\(263\) −17.3927 −1.07248 −0.536241 0.844065i \(-0.680156\pi\)
−0.536241 + 0.844065i \(0.680156\pi\)
\(264\) 1.89208 0.116449
\(265\) −27.3545 −1.68037
\(266\) −1.77068 −0.108567
\(267\) 35.9179 2.19814
\(268\) −1.49066 −0.0910562
\(269\) −16.5567 −1.00948 −0.504739 0.863272i \(-0.668411\pi\)
−0.504739 + 0.863272i \(0.668411\pi\)
\(270\) 0.886336 0.0539407
\(271\) 25.6049 1.55539 0.777694 0.628643i \(-0.216389\pi\)
0.777694 + 0.628643i \(0.216389\pi\)
\(272\) 0.468182 0.0283877
\(273\) 0 0
\(274\) −9.46492 −0.571797
\(275\) 2.44672 0.147543
\(276\) 8.25808 0.497078
\(277\) −32.6858 −1.96390 −0.981951 0.189136i \(-0.939431\pi\)
−0.981951 + 0.189136i \(0.939431\pi\)
\(278\) 21.9657 1.31742
\(279\) −8.55376 −0.512100
\(280\) −5.07057 −0.303024
\(281\) 16.1688 0.964551 0.482275 0.876020i \(-0.339810\pi\)
0.482275 + 0.876020i \(0.339810\pi\)
\(282\) −3.40132 −0.202546
\(283\) −25.2008 −1.49803 −0.749017 0.662551i \(-0.769474\pi\)
−0.749017 + 0.662551i \(0.769474\pi\)
\(284\) −2.41091 −0.143061
\(285\) −7.08716 −0.419807
\(286\) 0 0
\(287\) −16.7584 −0.989218
\(288\) −3.12506 −0.184146
\(289\) −16.7808 −0.987106
\(290\) −10.1610 −0.596676
\(291\) 6.79131 0.398114
\(292\) −12.1769 −0.712598
\(293\) −31.1586 −1.82030 −0.910151 0.414276i \(-0.864035\pi\)
−0.910151 + 0.414276i \(0.864035\pi\)
\(294\) −9.56469 −0.557824
\(295\) 16.5112 0.961322
\(296\) 5.74198 0.333746
\(297\) −0.236627 −0.0137305
\(298\) 5.84283 0.338466
\(299\) 0 0
\(300\) −7.92058 −0.457295
\(301\) 12.0069 0.692064
\(302\) −5.37075 −0.309052
\(303\) −33.0806 −1.90043
\(304\) 1.00000 0.0573539
\(305\) −6.77295 −0.387818
\(306\) −1.46310 −0.0836397
\(307\) 33.0592 1.88679 0.943395 0.331671i \(-0.107612\pi\)
0.943395 + 0.331671i \(0.107612\pi\)
\(308\) 1.35370 0.0771343
\(309\) 19.7232 1.12201
\(310\) 7.83819 0.445179
\(311\) −7.30371 −0.414155 −0.207078 0.978324i \(-0.566395\pi\)
−0.207078 + 0.978324i \(0.566395\pi\)
\(312\) 0 0
\(313\) 13.6996 0.774347 0.387173 0.922007i \(-0.373451\pi\)
0.387173 + 0.922007i \(0.373451\pi\)
\(314\) −12.3174 −0.695113
\(315\) 15.8458 0.892812
\(316\) −9.31247 −0.523867
\(317\) −9.54433 −0.536063 −0.268031 0.963410i \(-0.586373\pi\)
−0.268031 + 0.963410i \(0.586373\pi\)
\(318\) −23.6410 −1.32572
\(319\) 2.71272 0.151883
\(320\) 2.86363 0.160082
\(321\) −14.6885 −0.819833
\(322\) 5.90831 0.329257
\(323\) 0.468182 0.0260503
\(324\) −8.60917 −0.478287
\(325\) 0 0
\(326\) −3.62592 −0.200821
\(327\) −7.06035 −0.390438
\(328\) 9.46441 0.522584
\(329\) −2.43350 −0.134163
\(330\) 5.41821 0.298263
\(331\) 7.65204 0.420594 0.210297 0.977638i \(-0.432557\pi\)
0.210297 + 0.977638i \(0.432557\pi\)
\(332\) −3.26881 −0.179399
\(333\) −17.9440 −0.983328
\(334\) −0.0446192 −0.00244146
\(335\) −4.26869 −0.233223
\(336\) −4.38223 −0.239070
\(337\) −20.7810 −1.13201 −0.566007 0.824400i \(-0.691513\pi\)
−0.566007 + 0.824400i \(0.691513\pi\)
\(338\) 0 0
\(339\) 45.2576 2.45805
\(340\) 1.34070 0.0727097
\(341\) −2.09258 −0.113320
\(342\) −3.12506 −0.168984
\(343\) −19.2379 −1.03875
\(344\) −6.78094 −0.365604
\(345\) 23.6481 1.27317
\(346\) 14.5771 0.783668
\(347\) −29.8422 −1.60202 −0.801008 0.598654i \(-0.795702\pi\)
−0.801008 + 0.598654i \(0.795702\pi\)
\(348\) −8.78165 −0.470746
\(349\) −8.47281 −0.453539 −0.226770 0.973948i \(-0.572816\pi\)
−0.226770 + 0.973948i \(0.572816\pi\)
\(350\) −5.66684 −0.302906
\(351\) 0 0
\(352\) −0.764511 −0.0407485
\(353\) −10.1193 −0.538595 −0.269298 0.963057i \(-0.586792\pi\)
−0.269298 + 0.963057i \(0.586792\pi\)
\(354\) 14.2698 0.758432
\(355\) −6.90395 −0.366424
\(356\) −14.5129 −0.769184
\(357\) −2.05168 −0.108586
\(358\) 11.7631 0.621698
\(359\) 35.0123 1.84788 0.923939 0.382539i \(-0.124950\pi\)
0.923939 + 0.382539i \(0.124950\pi\)
\(360\) −8.94902 −0.471655
\(361\) 1.00000 0.0526316
\(362\) −16.7531 −0.880525
\(363\) 25.7772 1.35295
\(364\) 0 0
\(365\) −34.8701 −1.82518
\(366\) −5.85350 −0.305968
\(367\) 6.95348 0.362969 0.181484 0.983394i \(-0.441910\pi\)
0.181484 + 0.983394i \(0.441910\pi\)
\(368\) −3.33675 −0.173940
\(369\) −29.5769 −1.53971
\(370\) 16.4429 0.854826
\(371\) −16.9142 −0.878140
\(372\) 6.77413 0.351222
\(373\) 1.25498 0.0649805 0.0324903 0.999472i \(-0.489656\pi\)
0.0324903 + 0.999472i \(0.489656\pi\)
\(374\) −0.357930 −0.0185081
\(375\) 12.7542 0.658624
\(376\) 1.37434 0.0708759
\(377\) 0 0
\(378\) 0.548051 0.0281887
\(379\) −12.7877 −0.656862 −0.328431 0.944528i \(-0.606520\pi\)
−0.328431 + 0.944528i \(0.606520\pi\)
\(380\) 2.86363 0.146901
\(381\) 34.6810 1.77676
\(382\) −23.9089 −1.22329
\(383\) 3.94739 0.201702 0.100851 0.994902i \(-0.467843\pi\)
0.100851 + 0.994902i \(0.467843\pi\)
\(384\) 2.47489 0.126296
\(385\) 3.87650 0.197565
\(386\) −21.8221 −1.11071
\(387\) 21.1909 1.07719
\(388\) −2.74409 −0.139310
\(389\) 14.1388 0.716864 0.358432 0.933556i \(-0.383311\pi\)
0.358432 + 0.933556i \(0.383311\pi\)
\(390\) 0 0
\(391\) −1.56221 −0.0790042
\(392\) 3.86470 0.195197
\(393\) −36.9458 −1.86367
\(394\) 3.14538 0.158462
\(395\) −26.6675 −1.34179
\(396\) 2.38914 0.120059
\(397\) −12.9324 −0.649061 −0.324530 0.945875i \(-0.605206\pi\)
−0.324530 + 0.945875i \(0.605206\pi\)
\(398\) −6.98844 −0.350299
\(399\) −4.38223 −0.219386
\(400\) 3.20038 0.160019
\(401\) 1.92157 0.0959584 0.0479792 0.998848i \(-0.484722\pi\)
0.0479792 + 0.998848i \(0.484722\pi\)
\(402\) −3.68920 −0.184001
\(403\) 0 0
\(404\) 13.3665 0.665009
\(405\) −24.6535 −1.22504
\(406\) −6.28290 −0.311815
\(407\) −4.38980 −0.217595
\(408\) 1.15870 0.0573641
\(409\) 6.18073 0.305617 0.152809 0.988256i \(-0.451168\pi\)
0.152809 + 0.988256i \(0.451168\pi\)
\(410\) 27.1026 1.33850
\(411\) −23.4246 −1.15545
\(412\) −7.96934 −0.392621
\(413\) 10.2095 0.502374
\(414\) 10.4276 0.512486
\(415\) −9.36065 −0.459496
\(416\) 0 0
\(417\) 54.3626 2.66215
\(418\) −0.764511 −0.0373934
\(419\) 8.84081 0.431902 0.215951 0.976404i \(-0.430715\pi\)
0.215951 + 0.976404i \(0.430715\pi\)
\(420\) −12.5491 −0.612332
\(421\) −27.8790 −1.35874 −0.679369 0.733797i \(-0.737746\pi\)
−0.679369 + 0.733797i \(0.737746\pi\)
\(422\) −18.2813 −0.889919
\(423\) −4.29488 −0.208824
\(424\) 9.55237 0.463904
\(425\) 1.49836 0.0726812
\(426\) −5.96673 −0.289089
\(427\) −4.18794 −0.202669
\(428\) 5.93502 0.286880
\(429\) 0 0
\(430\) −19.4181 −0.936425
\(431\) −22.1378 −1.06634 −0.533169 0.846009i \(-0.678999\pi\)
−0.533169 + 0.846009i \(0.678999\pi\)
\(432\) −0.309515 −0.0148915
\(433\) −34.0289 −1.63532 −0.817662 0.575699i \(-0.804730\pi\)
−0.817662 + 0.575699i \(0.804730\pi\)
\(434\) 4.84661 0.232645
\(435\) −25.1474 −1.20573
\(436\) 2.85280 0.136624
\(437\) −3.33675 −0.159618
\(438\) −30.1364 −1.43997
\(439\) −16.2942 −0.777680 −0.388840 0.921305i \(-0.627124\pi\)
−0.388840 + 0.921305i \(0.627124\pi\)
\(440\) −2.18928 −0.104370
\(441\) −12.0774 −0.575116
\(442\) 0 0
\(443\) 19.2769 0.915873 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(444\) 14.2107 0.674412
\(445\) −41.5597 −1.97012
\(446\) 12.1425 0.574962
\(447\) 14.4603 0.683950
\(448\) 1.77068 0.0836567
\(449\) 7.67409 0.362163 0.181081 0.983468i \(-0.442040\pi\)
0.181081 + 0.983468i \(0.442040\pi\)
\(450\) −10.0014 −0.471470
\(451\) −7.23564 −0.340713
\(452\) −18.2867 −0.860135
\(453\) −13.2920 −0.624512
\(454\) −22.4289 −1.05264
\(455\) 0 0
\(456\) 2.47489 0.115897
\(457\) −6.41003 −0.299848 −0.149924 0.988697i \(-0.547903\pi\)
−0.149924 + 0.988697i \(0.547903\pi\)
\(458\) 9.13113 0.426670
\(459\) −0.144909 −0.00676378
\(460\) −9.55522 −0.445514
\(461\) 12.3507 0.575227 0.287614 0.957747i \(-0.407138\pi\)
0.287614 + 0.957747i \(0.407138\pi\)
\(462\) 3.35026 0.155868
\(463\) 3.56113 0.165500 0.0827498 0.996570i \(-0.473630\pi\)
0.0827498 + 0.996570i \(0.473630\pi\)
\(464\) 3.54830 0.164726
\(465\) 19.3986 0.899589
\(466\) −14.0387 −0.650332
\(467\) −14.1047 −0.652689 −0.326345 0.945251i \(-0.605817\pi\)
−0.326345 + 0.945251i \(0.605817\pi\)
\(468\) 0 0
\(469\) −2.63947 −0.121879
\(470\) 3.93559 0.181535
\(471\) −30.4843 −1.40464
\(472\) −5.76584 −0.265394
\(473\) 5.18410 0.238365
\(474\) −23.0473 −1.05860
\(475\) 3.20038 0.146844
\(476\) 0.828999 0.0379971
\(477\) −29.8518 −1.36682
\(478\) −14.0823 −0.644108
\(479\) −8.23436 −0.376238 −0.188119 0.982146i \(-0.560239\pi\)
−0.188119 + 0.982146i \(0.560239\pi\)
\(480\) 7.08716 0.323483
\(481\) 0 0
\(482\) −30.4199 −1.38559
\(483\) 14.6224 0.665342
\(484\) −10.4155 −0.473433
\(485\) −7.85806 −0.356816
\(486\) −22.2353 −1.00861
\(487\) −17.2786 −0.782970 −0.391485 0.920184i \(-0.628039\pi\)
−0.391485 + 0.920184i \(0.628039\pi\)
\(488\) 2.36516 0.107066
\(489\) −8.97374 −0.405806
\(490\) 11.0671 0.499959
\(491\) −7.80350 −0.352167 −0.176083 0.984375i \(-0.556343\pi\)
−0.176083 + 0.984375i \(0.556343\pi\)
\(492\) 23.4233 1.05601
\(493\) 1.66125 0.0748190
\(494\) 0 0
\(495\) 6.84162 0.307508
\(496\) −2.73715 −0.122902
\(497\) −4.26894 −0.191488
\(498\) −8.08992 −0.362518
\(499\) −7.14315 −0.319771 −0.159886 0.987136i \(-0.551113\pi\)
−0.159886 + 0.987136i \(0.551113\pi\)
\(500\) −5.15345 −0.230469
\(501\) −0.110428 −0.00493354
\(502\) 4.08765 0.182441
\(503\) 19.4259 0.866158 0.433079 0.901356i \(-0.357427\pi\)
0.433079 + 0.901356i \(0.357427\pi\)
\(504\) −5.53348 −0.246481
\(505\) 38.2768 1.70329
\(506\) 2.55098 0.113405
\(507\) 0 0
\(508\) −14.0132 −0.621734
\(509\) −21.1831 −0.938923 −0.469462 0.882953i \(-0.655552\pi\)
−0.469462 + 0.882953i \(0.655552\pi\)
\(510\) 3.31808 0.146927
\(511\) −21.5613 −0.953817
\(512\) −1.00000 −0.0441942
\(513\) −0.309515 −0.0136654
\(514\) 4.37406 0.192931
\(515\) −22.8212 −1.00562
\(516\) −16.7821 −0.738789
\(517\) −1.05069 −0.0462095
\(518\) 10.1672 0.446721
\(519\) 36.0766 1.58359
\(520\) 0 0
\(521\) 39.4233 1.72716 0.863582 0.504208i \(-0.168215\pi\)
0.863582 + 0.504208i \(0.168215\pi\)
\(522\) −11.0887 −0.485338
\(523\) −24.0725 −1.05261 −0.526307 0.850294i \(-0.676424\pi\)
−0.526307 + 0.850294i \(0.676424\pi\)
\(524\) 14.9283 0.652146
\(525\) −14.0248 −0.612092
\(526\) 17.3927 0.758359
\(527\) −1.28148 −0.0558223
\(528\) −1.89208 −0.0823421
\(529\) −11.8661 −0.515917
\(530\) 27.3545 1.18820
\(531\) 18.0186 0.781941
\(532\) 1.77068 0.0767686
\(533\) 0 0
\(534\) −35.9179 −1.55432
\(535\) 16.9957 0.734789
\(536\) 1.49066 0.0643865
\(537\) 29.1123 1.25629
\(538\) 16.5567 0.713809
\(539\) −2.95460 −0.127264
\(540\) −0.886336 −0.0381418
\(541\) −31.8834 −1.37078 −0.685388 0.728178i \(-0.740368\pi\)
−0.685388 + 0.728178i \(0.740368\pi\)
\(542\) −25.6049 −1.09983
\(543\) −41.4621 −1.77931
\(544\) −0.468182 −0.0200731
\(545\) 8.16936 0.349937
\(546\) 0 0
\(547\) 42.6236 1.82245 0.911227 0.411904i \(-0.135136\pi\)
0.911227 + 0.411904i \(0.135136\pi\)
\(548\) 9.46492 0.404321
\(549\) −7.39127 −0.315452
\(550\) −2.44672 −0.104329
\(551\) 3.54830 0.151163
\(552\) −8.25808 −0.351487
\(553\) −16.4894 −0.701200
\(554\) 32.6858 1.38869
\(555\) 40.6943 1.72738
\(556\) −21.9657 −0.931554
\(557\) −7.79893 −0.330451 −0.165226 0.986256i \(-0.552835\pi\)
−0.165226 + 0.986256i \(0.552835\pi\)
\(558\) 8.55376 0.362110
\(559\) 0 0
\(560\) 5.07057 0.214271
\(561\) −0.885836 −0.0374000
\(562\) −16.1688 −0.682040
\(563\) −22.6549 −0.954788 −0.477394 0.878689i \(-0.658419\pi\)
−0.477394 + 0.878689i \(0.658419\pi\)
\(564\) 3.40132 0.143222
\(565\) −52.3664 −2.20307
\(566\) 25.2008 1.05927
\(567\) −15.2441 −0.640191
\(568\) 2.41091 0.101159
\(569\) 29.6724 1.24393 0.621965 0.783045i \(-0.286335\pi\)
0.621965 + 0.783045i \(0.286335\pi\)
\(570\) 7.08716 0.296848
\(571\) −13.9581 −0.584128 −0.292064 0.956399i \(-0.594342\pi\)
−0.292064 + 0.956399i \(0.594342\pi\)
\(572\) 0 0
\(573\) −59.1719 −2.47194
\(574\) 16.7584 0.699483
\(575\) −10.6789 −0.445340
\(576\) 3.12506 0.130211
\(577\) −15.4839 −0.644603 −0.322301 0.946637i \(-0.604456\pi\)
−0.322301 + 0.946637i \(0.604456\pi\)
\(578\) 16.7808 0.697989
\(579\) −54.0071 −2.24446
\(580\) 10.1610 0.421914
\(581\) −5.78800 −0.240127
\(582\) −6.79131 −0.281509
\(583\) −7.30289 −0.302455
\(584\) 12.1769 0.503883
\(585\) 0 0
\(586\) 31.1586 1.28715
\(587\) 29.6279 1.22288 0.611438 0.791293i \(-0.290592\pi\)
0.611438 + 0.791293i \(0.290592\pi\)
\(588\) 9.56469 0.394441
\(589\) −2.73715 −0.112782
\(590\) −16.5112 −0.679757
\(591\) 7.78445 0.320210
\(592\) −5.74198 −0.235994
\(593\) 42.1120 1.72933 0.864666 0.502348i \(-0.167530\pi\)
0.864666 + 0.502348i \(0.167530\pi\)
\(594\) 0.236627 0.00970893
\(595\) 2.37395 0.0973224
\(596\) −5.84283 −0.239332
\(597\) −17.2956 −0.707862
\(598\) 0 0
\(599\) −20.9276 −0.855077 −0.427538 0.903997i \(-0.640619\pi\)
−0.427538 + 0.903997i \(0.640619\pi\)
\(600\) 7.92058 0.323356
\(601\) 25.8107 1.05284 0.526420 0.850224i \(-0.323534\pi\)
0.526420 + 0.850224i \(0.323534\pi\)
\(602\) −12.0069 −0.489363
\(603\) −4.65839 −0.189704
\(604\) 5.37075 0.218533
\(605\) −29.8262 −1.21261
\(606\) 33.0806 1.34381
\(607\) 4.65459 0.188924 0.0944620 0.995528i \(-0.469887\pi\)
0.0944620 + 0.995528i \(0.469887\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −15.5495 −0.630096
\(610\) 6.77295 0.274229
\(611\) 0 0
\(612\) 1.46310 0.0591422
\(613\) −10.0995 −0.407916 −0.203958 0.978980i \(-0.565381\pi\)
−0.203958 + 0.978980i \(0.565381\pi\)
\(614\) −33.0592 −1.33416
\(615\) 67.0758 2.70476
\(616\) −1.35370 −0.0545422
\(617\) 43.7568 1.76158 0.880791 0.473504i \(-0.157011\pi\)
0.880791 + 0.473504i \(0.157011\pi\)
\(618\) −19.7232 −0.793384
\(619\) −44.1459 −1.77437 −0.887186 0.461412i \(-0.847343\pi\)
−0.887186 + 0.461412i \(0.847343\pi\)
\(620\) −7.83819 −0.314789
\(621\) 1.03277 0.0414437
\(622\) 7.30371 0.292852
\(623\) −25.6978 −1.02956
\(624\) 0 0
\(625\) −30.7595 −1.23038
\(626\) −13.6996 −0.547546
\(627\) −1.89208 −0.0755623
\(628\) 12.3174 0.491519
\(629\) −2.68829 −0.107189
\(630\) −15.8458 −0.631313
\(631\) −0.00599428 −0.000238628 0 −0.000119314 1.00000i \(-0.500038\pi\)
−0.000119314 1.00000i \(0.500038\pi\)
\(632\) 9.31247 0.370430
\(633\) −45.2441 −1.79829
\(634\) 9.54433 0.379054
\(635\) −40.1286 −1.59245
\(636\) 23.6410 0.937428
\(637\) 0 0
\(638\) −2.71272 −0.107397
\(639\) −7.53424 −0.298050
\(640\) −2.86363 −0.113195
\(641\) −11.7773 −0.465176 −0.232588 0.972575i \(-0.574719\pi\)
−0.232588 + 0.972575i \(0.574719\pi\)
\(642\) 14.6885 0.579709
\(643\) −24.8665 −0.980640 −0.490320 0.871542i \(-0.663120\pi\)
−0.490320 + 0.871542i \(0.663120\pi\)
\(644\) −5.90831 −0.232820
\(645\) −48.0576 −1.89227
\(646\) −0.468182 −0.0184204
\(647\) −35.0553 −1.37816 −0.689082 0.724683i \(-0.741986\pi\)
−0.689082 + 0.724683i \(0.741986\pi\)
\(648\) 8.60917 0.338200
\(649\) 4.40805 0.173031
\(650\) 0 0
\(651\) 11.9948 0.470114
\(652\) 3.62592 0.142002
\(653\) 28.2767 1.10655 0.553277 0.832997i \(-0.313377\pi\)
0.553277 + 0.832997i \(0.313377\pi\)
\(654\) 7.06035 0.276082
\(655\) 42.7491 1.67035
\(656\) −9.46441 −0.369523
\(657\) −38.0535 −1.48461
\(658\) 2.43350 0.0948679
\(659\) −8.23201 −0.320673 −0.160337 0.987062i \(-0.551258\pi\)
−0.160337 + 0.987062i \(0.551258\pi\)
\(660\) −5.41821 −0.210903
\(661\) 17.1371 0.666556 0.333278 0.942829i \(-0.391845\pi\)
0.333278 + 0.942829i \(0.391845\pi\)
\(662\) −7.65204 −0.297405
\(663\) 0 0
\(664\) 3.26881 0.126854
\(665\) 5.07057 0.196628
\(666\) 17.9440 0.695318
\(667\) −11.8398 −0.458439
\(668\) 0.0446192 0.00172637
\(669\) 30.0512 1.16185
\(670\) 4.26869 0.164914
\(671\) −1.80819 −0.0698044
\(672\) 4.38223 0.169048
\(673\) 13.7359 0.529479 0.264739 0.964320i \(-0.414714\pi\)
0.264739 + 0.964320i \(0.414714\pi\)
\(674\) 20.7810 0.800455
\(675\) −0.990565 −0.0381268
\(676\) 0 0
\(677\) 3.07430 0.118155 0.0590774 0.998253i \(-0.481184\pi\)
0.0590774 + 0.998253i \(0.481184\pi\)
\(678\) −45.2576 −1.73811
\(679\) −4.85890 −0.186468
\(680\) −1.34070 −0.0514135
\(681\) −55.5089 −2.12710
\(682\) 2.09258 0.0801290
\(683\) −32.0415 −1.22603 −0.613016 0.790070i \(-0.710044\pi\)
−0.613016 + 0.790070i \(0.710044\pi\)
\(684\) 3.12506 0.119490
\(685\) 27.1040 1.03559
\(686\) 19.2379 0.734506
\(687\) 22.5985 0.862187
\(688\) 6.78094 0.258521
\(689\) 0 0
\(690\) −23.6481 −0.900267
\(691\) −3.10166 −0.117993 −0.0589964 0.998258i \(-0.518790\pi\)
−0.0589964 + 0.998258i \(0.518790\pi\)
\(692\) −14.5771 −0.554137
\(693\) 4.23040 0.160700
\(694\) 29.8422 1.13280
\(695\) −62.9017 −2.38600
\(696\) 8.78165 0.332868
\(697\) −4.43106 −0.167839
\(698\) 8.47281 0.320701
\(699\) −34.7443 −1.31415
\(700\) 5.66684 0.214187
\(701\) −3.93054 −0.148454 −0.0742272 0.997241i \(-0.523649\pi\)
−0.0742272 + 0.997241i \(0.523649\pi\)
\(702\) 0 0
\(703\) −5.74198 −0.216563
\(704\) 0.764511 0.0288136
\(705\) 9.74013 0.366835
\(706\) 10.1193 0.380844
\(707\) 23.6678 0.890119
\(708\) −14.2698 −0.536292
\(709\) 36.5498 1.37266 0.686328 0.727293i \(-0.259222\pi\)
0.686328 + 0.727293i \(0.259222\pi\)
\(710\) 6.90395 0.259101
\(711\) −29.1020 −1.09141
\(712\) 14.5129 0.543896
\(713\) 9.13318 0.342040
\(714\) 2.05168 0.0767822
\(715\) 0 0
\(716\) −11.7631 −0.439607
\(717\) −34.8520 −1.30157
\(718\) −35.0123 −1.30665
\(719\) 42.2264 1.57478 0.787389 0.616456i \(-0.211432\pi\)
0.787389 + 0.616456i \(0.211432\pi\)
\(720\) 8.94902 0.333510
\(721\) −14.1111 −0.525526
\(722\) −1.00000 −0.0372161
\(723\) −75.2857 −2.79990
\(724\) 16.7531 0.622625
\(725\) 11.3559 0.421748
\(726\) −25.7772 −0.956683
\(727\) −17.3320 −0.642809 −0.321405 0.946942i \(-0.604155\pi\)
−0.321405 + 0.946942i \(0.604155\pi\)
\(728\) 0 0
\(729\) −29.2022 −1.08156
\(730\) 34.8701 1.29060
\(731\) 3.17472 0.117421
\(732\) 5.85350 0.216352
\(733\) −2.27653 −0.0840856 −0.0420428 0.999116i \(-0.513387\pi\)
−0.0420428 + 0.999116i \(0.513387\pi\)
\(734\) −6.95348 −0.256658
\(735\) 27.3897 1.01029
\(736\) 3.33675 0.122994
\(737\) −1.13962 −0.0419785
\(738\) 29.5769 1.08874
\(739\) 30.6823 1.12867 0.564334 0.825547i \(-0.309133\pi\)
0.564334 + 0.825547i \(0.309133\pi\)
\(740\) −16.4429 −0.604453
\(741\) 0 0
\(742\) 16.9142 0.620939
\(743\) −3.84335 −0.140999 −0.0704995 0.997512i \(-0.522459\pi\)
−0.0704995 + 0.997512i \(0.522459\pi\)
\(744\) −6.77413 −0.248352
\(745\) −16.7317 −0.613002
\(746\) −1.25498 −0.0459482
\(747\) −10.2152 −0.373755
\(748\) 0.357930 0.0130872
\(749\) 10.5090 0.383991
\(750\) −12.7542 −0.465717
\(751\) 8.93762 0.326139 0.163069 0.986615i \(-0.447861\pi\)
0.163069 + 0.986615i \(0.447861\pi\)
\(752\) −1.37434 −0.0501168
\(753\) 10.1165 0.368665
\(754\) 0 0
\(755\) 15.3798 0.559730
\(756\) −0.548051 −0.0199324
\(757\) −10.4195 −0.378703 −0.189351 0.981909i \(-0.560638\pi\)
−0.189351 + 0.981909i \(0.560638\pi\)
\(758\) 12.7877 0.464472
\(759\) 6.31339 0.229161
\(760\) −2.86363 −0.103875
\(761\) −45.1809 −1.63781 −0.818903 0.573932i \(-0.805417\pi\)
−0.818903 + 0.573932i \(0.805417\pi\)
\(762\) −34.6810 −1.25636
\(763\) 5.05139 0.182873
\(764\) 23.9089 0.864995
\(765\) 4.18977 0.151482
\(766\) −3.94739 −0.142625
\(767\) 0 0
\(768\) −2.47489 −0.0893048
\(769\) −31.0575 −1.11996 −0.559981 0.828506i \(-0.689191\pi\)
−0.559981 + 0.828506i \(0.689191\pi\)
\(770\) −3.87650 −0.139699
\(771\) 10.8253 0.389864
\(772\) 21.8221 0.785393
\(773\) 10.6459 0.382906 0.191453 0.981502i \(-0.438680\pi\)
0.191453 + 0.981502i \(0.438680\pi\)
\(774\) −21.1909 −0.761690
\(775\) −8.75992 −0.314666
\(776\) 2.74409 0.0985071
\(777\) 25.1627 0.902705
\(778\) −14.1388 −0.506900
\(779\) −9.46441 −0.339098
\(780\) 0 0
\(781\) −1.84317 −0.0659536
\(782\) 1.56221 0.0558644
\(783\) −1.09825 −0.0392483
\(784\) −3.86470 −0.138025
\(785\) 35.2726 1.25893
\(786\) 36.9458 1.31781
\(787\) 13.9279 0.496477 0.248238 0.968699i \(-0.420148\pi\)
0.248238 + 0.968699i \(0.420148\pi\)
\(788\) −3.14538 −0.112049
\(789\) 43.0450 1.53244
\(790\) 26.6675 0.948786
\(791\) −32.3799 −1.15130
\(792\) −2.38914 −0.0848945
\(793\) 0 0
\(794\) 12.9324 0.458955
\(795\) 67.6992 2.40104
\(796\) 6.98844 0.247699
\(797\) −43.3987 −1.53726 −0.768631 0.639693i \(-0.779062\pi\)
−0.768631 + 0.639693i \(0.779062\pi\)
\(798\) 4.38223 0.155129
\(799\) −0.643439 −0.0227632
\(800\) −3.20038 −0.113151
\(801\) −45.3539 −1.60250
\(802\) −1.92157 −0.0678529
\(803\) −9.30935 −0.328520
\(804\) 3.68920 0.130108
\(805\) −16.9192 −0.596324
\(806\) 0 0
\(807\) 40.9759 1.44242
\(808\) −13.3665 −0.470233
\(809\) 21.9861 0.772991 0.386495 0.922291i \(-0.373685\pi\)
0.386495 + 0.922291i \(0.373685\pi\)
\(810\) 24.6535 0.866235
\(811\) −34.1994 −1.20090 −0.600452 0.799661i \(-0.705013\pi\)
−0.600452 + 0.799661i \(0.705013\pi\)
\(812\) 6.28290 0.220487
\(813\) −63.3693 −2.22246
\(814\) 4.38980 0.153863
\(815\) 10.3833 0.363711
\(816\) −1.15870 −0.0405625
\(817\) 6.78094 0.237235
\(818\) −6.18073 −0.216104
\(819\) 0 0
\(820\) −27.1026 −0.946463
\(821\) 21.6637 0.756067 0.378033 0.925792i \(-0.376600\pi\)
0.378033 + 0.925792i \(0.376600\pi\)
\(822\) 23.4246 0.817027
\(823\) −34.3177 −1.19624 −0.598121 0.801406i \(-0.704086\pi\)
−0.598121 + 0.801406i \(0.704086\pi\)
\(824\) 7.96934 0.277625
\(825\) −6.05537 −0.210821
\(826\) −10.2095 −0.355232
\(827\) 48.1657 1.67489 0.837444 0.546524i \(-0.184049\pi\)
0.837444 + 0.546524i \(0.184049\pi\)
\(828\) −10.4276 −0.362382
\(829\) 35.0387 1.21695 0.608473 0.793575i \(-0.291783\pi\)
0.608473 + 0.793575i \(0.291783\pi\)
\(830\) 9.36065 0.324913
\(831\) 80.8937 2.80617
\(832\) 0 0
\(833\) −1.80938 −0.0626914
\(834\) −54.3626 −1.88242
\(835\) 0.127773 0.00442177
\(836\) 0.764511 0.0264411
\(837\) 0.847188 0.0292831
\(838\) −8.84081 −0.305401
\(839\) 39.5247 1.36454 0.682272 0.731098i \(-0.260992\pi\)
0.682272 + 0.731098i \(0.260992\pi\)
\(840\) 12.5491 0.432984
\(841\) −16.4095 −0.565846
\(842\) 27.8790 0.960772
\(843\) −40.0160 −1.37822
\(844\) 18.2813 0.629268
\(845\) 0 0
\(846\) 4.29488 0.147661
\(847\) −18.4425 −0.633693
\(848\) −9.55237 −0.328030
\(849\) 62.3692 2.14051
\(850\) −1.49836 −0.0513933
\(851\) 19.1596 0.656781
\(852\) 5.96673 0.204417
\(853\) 29.7468 1.01851 0.509256 0.860615i \(-0.329921\pi\)
0.509256 + 0.860615i \(0.329921\pi\)
\(854\) 4.18794 0.143308
\(855\) 8.94902 0.306050
\(856\) −5.93502 −0.202855
\(857\) −52.6321 −1.79788 −0.898939 0.438074i \(-0.855661\pi\)
−0.898939 + 0.438074i \(0.855661\pi\)
\(858\) 0 0
\(859\) −22.3911 −0.763974 −0.381987 0.924168i \(-0.624760\pi\)
−0.381987 + 0.924168i \(0.624760\pi\)
\(860\) 19.4181 0.662152
\(861\) 41.4752 1.41347
\(862\) 22.1378 0.754015
\(863\) −16.1610 −0.550128 −0.275064 0.961426i \(-0.588699\pi\)
−0.275064 + 0.961426i \(0.588699\pi\)
\(864\) 0.309515 0.0105299
\(865\) −41.7434 −1.41932
\(866\) 34.0289 1.15635
\(867\) 41.5306 1.41045
\(868\) −4.84661 −0.164505
\(869\) −7.11948 −0.241512
\(870\) 25.1474 0.852577
\(871\) 0 0
\(872\) −2.85280 −0.0966080
\(873\) −8.57545 −0.290235
\(874\) 3.33675 0.112867
\(875\) −9.12509 −0.308484
\(876\) 30.1364 1.01821
\(877\) 7.90458 0.266919 0.133459 0.991054i \(-0.457391\pi\)
0.133459 + 0.991054i \(0.457391\pi\)
\(878\) 16.2942 0.549903
\(879\) 77.1139 2.60099
\(880\) 2.18928 0.0738005
\(881\) 48.9018 1.64754 0.823771 0.566922i \(-0.191866\pi\)
0.823771 + 0.566922i \(0.191866\pi\)
\(882\) 12.0774 0.406668
\(883\) 3.81146 0.128266 0.0641329 0.997941i \(-0.479572\pi\)
0.0641329 + 0.997941i \(0.479572\pi\)
\(884\) 0 0
\(885\) −40.8635 −1.37361
\(886\) −19.2769 −0.647620
\(887\) 30.7116 1.03119 0.515597 0.856831i \(-0.327570\pi\)
0.515597 + 0.856831i \(0.327570\pi\)
\(888\) −14.2107 −0.476881
\(889\) −24.8128 −0.832195
\(890\) 41.5597 1.39308
\(891\) −6.58180 −0.220499
\(892\) −12.1425 −0.406560
\(893\) −1.37434 −0.0459904
\(894\) −14.4603 −0.483626
\(895\) −33.6851 −1.12597
\(896\) −1.77068 −0.0591542
\(897\) 0 0
\(898\) −7.67409 −0.256088
\(899\) −9.71224 −0.323921
\(900\) 10.0014 0.333380
\(901\) −4.47225 −0.148992
\(902\) 7.23564 0.240920
\(903\) −29.7156 −0.988874
\(904\) 18.2867 0.608208
\(905\) 47.9748 1.59474
\(906\) 13.2920 0.441597
\(907\) −19.8204 −0.658125 −0.329063 0.944308i \(-0.606733\pi\)
−0.329063 + 0.944308i \(0.606733\pi\)
\(908\) 22.4289 0.744328
\(909\) 41.7712 1.38546
\(910\) 0 0
\(911\) 6.87384 0.227741 0.113870 0.993496i \(-0.463675\pi\)
0.113870 + 0.993496i \(0.463675\pi\)
\(912\) −2.47489 −0.0819517
\(913\) −2.49904 −0.0827060
\(914\) 6.41003 0.212025
\(915\) 16.7623 0.554144
\(916\) −9.13113 −0.301701
\(917\) 26.4332 0.872901
\(918\) 0.144909 0.00478272
\(919\) −25.0260 −0.825532 −0.412766 0.910837i \(-0.635437\pi\)
−0.412766 + 0.910837i \(0.635437\pi\)
\(920\) 9.55522 0.315026
\(921\) −81.8179 −2.69599
\(922\) −12.3507 −0.406747
\(923\) 0 0
\(924\) −3.35026 −0.110215
\(925\) −18.3765 −0.604216
\(926\) −3.56113 −0.117026
\(927\) −24.9047 −0.817977
\(928\) −3.54830 −0.116479
\(929\) −52.0702 −1.70837 −0.854184 0.519971i \(-0.825943\pi\)
−0.854184 + 0.519971i \(0.825943\pi\)
\(930\) −19.3986 −0.636106
\(931\) −3.86470 −0.126660
\(932\) 14.0387 0.459854
\(933\) 18.0759 0.591777
\(934\) 14.1047 0.461521
\(935\) 1.02498 0.0335204
\(936\) 0 0
\(937\) −21.3297 −0.696812 −0.348406 0.937344i \(-0.613277\pi\)
−0.348406 + 0.937344i \(0.613277\pi\)
\(938\) 2.63947 0.0861818
\(939\) −33.9049 −1.10645
\(940\) −3.93559 −0.128365
\(941\) −53.3220 −1.73825 −0.869124 0.494594i \(-0.835317\pi\)
−0.869124 + 0.494594i \(0.835317\pi\)
\(942\) 30.4843 0.993231
\(943\) 31.5804 1.02840
\(944\) 5.76584 0.187662
\(945\) −1.56941 −0.0510531
\(946\) −5.18410 −0.168550
\(947\) −26.0970 −0.848039 −0.424020 0.905653i \(-0.639381\pi\)
−0.424020 + 0.905653i \(0.639381\pi\)
\(948\) 23.0473 0.748542
\(949\) 0 0
\(950\) −3.20038 −0.103834
\(951\) 23.6211 0.765968
\(952\) −0.828999 −0.0268680
\(953\) 17.9036 0.579954 0.289977 0.957034i \(-0.406352\pi\)
0.289977 + 0.957034i \(0.406352\pi\)
\(954\) 29.8518 0.966486
\(955\) 68.4664 2.21552
\(956\) 14.0823 0.455453
\(957\) −6.71366 −0.217022
\(958\) 8.23436 0.266040
\(959\) 16.7593 0.541187
\(960\) −7.08716 −0.228737
\(961\) −23.5080 −0.758323
\(962\) 0 0
\(963\) 18.5473 0.597679
\(964\) 30.4199 0.979757
\(965\) 62.4903 2.01164
\(966\) −14.6224 −0.470468
\(967\) 52.6412 1.69283 0.846414 0.532525i \(-0.178757\pi\)
0.846414 + 0.532525i \(0.178757\pi\)
\(968\) 10.4155 0.334768
\(969\) −1.15870 −0.0372227
\(970\) 7.85806 0.252307
\(971\) 25.3264 0.812763 0.406382 0.913703i \(-0.366790\pi\)
0.406382 + 0.913703i \(0.366790\pi\)
\(972\) 22.2353 0.713197
\(973\) −38.8942 −1.24689
\(974\) 17.2786 0.553643
\(975\) 0 0
\(976\) −2.36516 −0.0757069
\(977\) −55.5077 −1.77585 −0.887924 0.459990i \(-0.847853\pi\)
−0.887924 + 0.459990i \(0.847853\pi\)
\(978\) 8.97374 0.286948
\(979\) −11.0953 −0.354607
\(980\) −11.0671 −0.353525
\(981\) 8.91517 0.284640
\(982\) 7.80350 0.249020
\(983\) 4.80383 0.153218 0.0766092 0.997061i \(-0.475591\pi\)
0.0766092 + 0.997061i \(0.475591\pi\)
\(984\) −23.4233 −0.746708
\(985\) −9.00720 −0.286993
\(986\) −1.66125 −0.0529050
\(987\) 6.02265 0.191703
\(988\) 0 0
\(989\) −22.6263 −0.719475
\(990\) −6.84162 −0.217441
\(991\) 0.0302006 0.000959354 0 0.000479677 1.00000i \(-0.499847\pi\)
0.000479677 1.00000i \(0.499847\pi\)
\(992\) 2.73715 0.0869046
\(993\) −18.9379 −0.600977
\(994\) 4.26894 0.135403
\(995\) 20.0123 0.634433
\(996\) 8.08992 0.256339
\(997\) 14.6989 0.465519 0.232760 0.972534i \(-0.425224\pi\)
0.232760 + 0.972534i \(0.425224\pi\)
\(998\) 7.14315 0.226113
\(999\) 1.77723 0.0562290
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 6422.2.a.bg.1.2 8
13.2 odd 12 494.2.m.a.381.4 yes 16
13.7 odd 12 494.2.m.a.153.4 16
13.12 even 2 6422.2.a.bh.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.m.a.153.4 16 13.7 odd 12
494.2.m.a.381.4 yes 16 13.2 odd 12
6422.2.a.bg.1.2 8 1.1 even 1 trivial
6422.2.a.bh.1.2 8 13.12 even 2