Properties

Label 6422.2.a.bl.1.2
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $9$
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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 12x^{7} + 14x^{6} + 35x^{5} - 35x^{4} - 28x^{3} + 10x^{2} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.380781\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.66099 q^{3} +1.00000 q^{4} -1.69394 q^{5} -1.66099 q^{6} +0.961642 q^{7} +1.00000 q^{8} -0.241101 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.66099 q^{3} +1.00000 q^{4} -1.69394 q^{5} -1.66099 q^{6} +0.961642 q^{7} +1.00000 q^{8} -0.241101 q^{9} -1.69394 q^{10} -4.42494 q^{11} -1.66099 q^{12} +0.961642 q^{14} +2.81362 q^{15} +1.00000 q^{16} +1.01199 q^{17} -0.241101 q^{18} -1.00000 q^{19} -1.69394 q^{20} -1.59728 q^{21} -4.42494 q^{22} +6.07002 q^{23} -1.66099 q^{24} -2.13058 q^{25} +5.38345 q^{27} +0.961642 q^{28} +6.95493 q^{29} +2.81362 q^{30} +9.73717 q^{31} +1.00000 q^{32} +7.34980 q^{33} +1.01199 q^{34} -1.62896 q^{35} -0.241101 q^{36} -0.678162 q^{37} -1.00000 q^{38} -1.69394 q^{40} -11.2026 q^{41} -1.59728 q^{42} +5.44576 q^{43} -4.42494 q^{44} +0.408410 q^{45} +6.07002 q^{46} +3.91685 q^{47} -1.66099 q^{48} -6.07525 q^{49} -2.13058 q^{50} -1.68091 q^{51} +6.18790 q^{53} +5.38345 q^{54} +7.49557 q^{55} +0.961642 q^{56} +1.66099 q^{57} +6.95493 q^{58} +2.06496 q^{59} +2.81362 q^{60} -10.2252 q^{61} +9.73717 q^{62} -0.231853 q^{63} +1.00000 q^{64} +7.34980 q^{66} -5.84309 q^{67} +1.01199 q^{68} -10.0823 q^{69} -1.62896 q^{70} -14.6895 q^{71} -0.241101 q^{72} +2.12183 q^{73} -0.678162 q^{74} +3.53888 q^{75} -1.00000 q^{76} -4.25521 q^{77} -7.59795 q^{79} -1.69394 q^{80} -8.21857 q^{81} -11.2026 q^{82} +4.74343 q^{83} -1.59728 q^{84} -1.71425 q^{85} +5.44576 q^{86} -11.5521 q^{87} -4.42494 q^{88} -6.32590 q^{89} +0.408410 q^{90} +6.07002 q^{92} -16.1734 q^{93} +3.91685 q^{94} +1.69394 q^{95} -1.66099 q^{96} +4.22409 q^{97} -6.07525 q^{98} +1.06686 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + q^{3} + 9 q^{4} + q^{5} + q^{6} - 13 q^{7} + 9 q^{8} - 2 q^{9} + q^{10} - 3 q^{11} + q^{12} - 13 q^{14} - 3 q^{15} + 9 q^{16} - 8 q^{17} - 2 q^{18} - 9 q^{19} + q^{20} - 24 q^{21} - 3 q^{22} - 10 q^{23} + q^{24} + 8 q^{25} + 10 q^{27} - 13 q^{28} - 20 q^{29} - 3 q^{30} - q^{31} + 9 q^{32} + 2 q^{33} - 8 q^{34} + 4 q^{35} - 2 q^{36} - 15 q^{37} - 9 q^{38} + q^{40} - 19 q^{41} - 24 q^{42} - 16 q^{43} - 3 q^{44} - 15 q^{45} - 10 q^{46} - 18 q^{47} + q^{48} + 18 q^{49} + 8 q^{50} - 11 q^{51} + 17 q^{53} + 10 q^{54} - 26 q^{55} - 13 q^{56} - q^{57} - 20 q^{58} - 24 q^{59} - 3 q^{60} + 6 q^{61} - q^{62} - q^{63} + 9 q^{64} + 2 q^{66} - 29 q^{67} - 8 q^{68} - 12 q^{69} + 4 q^{70} - 23 q^{71} - 2 q^{72} - 38 q^{73} - 15 q^{74} + 11 q^{75} - 9 q^{76} - 40 q^{77} - 20 q^{79} + q^{80} - 31 q^{81} - 19 q^{82} - 20 q^{83} - 24 q^{84} - 39 q^{85} - 16 q^{86} - 10 q^{87} - 3 q^{88} + 7 q^{89} - 15 q^{90} - 10 q^{92} - 11 q^{93} - 18 q^{94} - q^{95} + q^{96} - 28 q^{97} + 18 q^{98} - 25 q^{99}+O(q^{100}) \) 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
\(3\) −1.66099 −0.958975 −0.479487 0.877549i \(-0.659177\pi\)
−0.479487 + 0.877549i \(0.659177\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.69394 −0.757551 −0.378775 0.925489i \(-0.623655\pi\)
−0.378775 + 0.925489i \(0.623655\pi\)
\(6\) −1.66099 −0.678098
\(7\) 0.961642 0.363466 0.181733 0.983348i \(-0.441829\pi\)
0.181733 + 0.983348i \(0.441829\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.241101 −0.0803671
\(10\) −1.69394 −0.535669
\(11\) −4.42494 −1.33417 −0.667085 0.744982i \(-0.732458\pi\)
−0.667085 + 0.744982i \(0.732458\pi\)
\(12\) −1.66099 −0.479487
\(13\) 0 0
\(14\) 0.961642 0.257010
\(15\) 2.81362 0.726472
\(16\) 1.00000 0.250000
\(17\) 1.01199 0.245444 0.122722 0.992441i \(-0.460838\pi\)
0.122722 + 0.992441i \(0.460838\pi\)
\(18\) −0.241101 −0.0568282
\(19\) −1.00000 −0.229416
\(20\) −1.69394 −0.378775
\(21\) −1.59728 −0.348555
\(22\) −4.42494 −0.943401
\(23\) 6.07002 1.26569 0.632844 0.774280i \(-0.281888\pi\)
0.632844 + 0.774280i \(0.281888\pi\)
\(24\) −1.66099 −0.339049
\(25\) −2.13058 −0.426117
\(26\) 0 0
\(27\) 5.38345 1.03604
\(28\) 0.961642 0.181733
\(29\) 6.95493 1.29150 0.645749 0.763550i \(-0.276545\pi\)
0.645749 + 0.763550i \(0.276545\pi\)
\(30\) 2.81362 0.513694
\(31\) 9.73717 1.74885 0.874424 0.485163i \(-0.161240\pi\)
0.874424 + 0.485163i \(0.161240\pi\)
\(32\) 1.00000 0.176777
\(33\) 7.34980 1.27944
\(34\) 1.01199 0.173555
\(35\) −1.62896 −0.275344
\(36\) −0.241101 −0.0401836
\(37\) −0.678162 −0.111489 −0.0557446 0.998445i \(-0.517753\pi\)
−0.0557446 + 0.998445i \(0.517753\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −1.69394 −0.267835
\(41\) −11.2026 −1.74955 −0.874777 0.484526i \(-0.838992\pi\)
−0.874777 + 0.484526i \(0.838992\pi\)
\(42\) −1.59728 −0.246466
\(43\) 5.44576 0.830470 0.415235 0.909714i \(-0.363699\pi\)
0.415235 + 0.909714i \(0.363699\pi\)
\(44\) −4.42494 −0.667085
\(45\) 0.408410 0.0608822
\(46\) 6.07002 0.894976
\(47\) 3.91685 0.571331 0.285666 0.958329i \(-0.407785\pi\)
0.285666 + 0.958329i \(0.407785\pi\)
\(48\) −1.66099 −0.239744
\(49\) −6.07525 −0.867892
\(50\) −2.13058 −0.301310
\(51\) −1.68091 −0.235374
\(52\) 0 0
\(53\) 6.18790 0.849973 0.424986 0.905200i \(-0.360279\pi\)
0.424986 + 0.905200i \(0.360279\pi\)
\(54\) 5.38345 0.732594
\(55\) 7.49557 1.01070
\(56\) 0.961642 0.128505
\(57\) 1.66099 0.220004
\(58\) 6.95493 0.913227
\(59\) 2.06496 0.268835 0.134418 0.990925i \(-0.457084\pi\)
0.134418 + 0.990925i \(0.457084\pi\)
\(60\) 2.81362 0.363236
\(61\) −10.2252 −1.30920 −0.654602 0.755974i \(-0.727164\pi\)
−0.654602 + 0.755974i \(0.727164\pi\)
\(62\) 9.73717 1.23662
\(63\) −0.231853 −0.0292108
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 7.34980 0.904698
\(67\) −5.84309 −0.713847 −0.356924 0.934134i \(-0.616174\pi\)
−0.356924 + 0.934134i \(0.616174\pi\)
\(68\) 1.01199 0.122722
\(69\) −10.0823 −1.21376
\(70\) −1.62896 −0.194698
\(71\) −14.6895 −1.74332 −0.871662 0.490108i \(-0.836957\pi\)
−0.871662 + 0.490108i \(0.836957\pi\)
\(72\) −0.241101 −0.0284141
\(73\) 2.12183 0.248341 0.124171 0.992261i \(-0.460373\pi\)
0.124171 + 0.992261i \(0.460373\pi\)
\(74\) −0.678162 −0.0788347
\(75\) 3.53888 0.408635
\(76\) −1.00000 −0.114708
\(77\) −4.25521 −0.484926
\(78\) 0 0
\(79\) −7.59795 −0.854837 −0.427418 0.904054i \(-0.640577\pi\)
−0.427418 + 0.904054i \(0.640577\pi\)
\(80\) −1.69394 −0.189388
\(81\) −8.21857 −0.913174
\(82\) −11.2026 −1.23712
\(83\) 4.74343 0.520659 0.260330 0.965520i \(-0.416169\pi\)
0.260330 + 0.965520i \(0.416169\pi\)
\(84\) −1.59728 −0.174278
\(85\) −1.71425 −0.185936
\(86\) 5.44576 0.587231
\(87\) −11.5521 −1.23851
\(88\) −4.42494 −0.471700
\(89\) −6.32590 −0.670544 −0.335272 0.942121i \(-0.608828\pi\)
−0.335272 + 0.942121i \(0.608828\pi\)
\(90\) 0.408410 0.0430502
\(91\) 0 0
\(92\) 6.07002 0.632844
\(93\) −16.1734 −1.67710
\(94\) 3.91685 0.403992
\(95\) 1.69394 0.173794
\(96\) −1.66099 −0.169524
\(97\) 4.22409 0.428892 0.214446 0.976736i \(-0.431205\pi\)
0.214446 + 0.976736i \(0.431205\pi\)
\(98\) −6.07525 −0.613692
\(99\) 1.06686 0.107223
\(100\) −2.13058 −0.213058
\(101\) −5.33989 −0.531339 −0.265669 0.964064i \(-0.585593\pi\)
−0.265669 + 0.964064i \(0.585593\pi\)
\(102\) −1.68091 −0.166435
\(103\) −5.98387 −0.589608 −0.294804 0.955558i \(-0.595254\pi\)
−0.294804 + 0.955558i \(0.595254\pi\)
\(104\) 0 0
\(105\) 2.70569 0.264048
\(106\) 6.18790 0.601022
\(107\) 0.456797 0.0441603 0.0220801 0.999756i \(-0.492971\pi\)
0.0220801 + 0.999756i \(0.492971\pi\)
\(108\) 5.38345 0.518022
\(109\) −13.2279 −1.26700 −0.633500 0.773743i \(-0.718382\pi\)
−0.633500 + 0.773743i \(0.718382\pi\)
\(110\) 7.49557 0.714674
\(111\) 1.12642 0.106915
\(112\) 0.961642 0.0908666
\(113\) 11.5596 1.08743 0.543716 0.839269i \(-0.317017\pi\)
0.543716 + 0.839269i \(0.317017\pi\)
\(114\) 1.66099 0.155566
\(115\) −10.2822 −0.958822
\(116\) 6.95493 0.645749
\(117\) 0 0
\(118\) 2.06496 0.190095
\(119\) 0.973172 0.0892105
\(120\) 2.81362 0.256847
\(121\) 8.58011 0.780010
\(122\) −10.2252 −0.925747
\(123\) 18.6075 1.67778
\(124\) 9.73717 0.874424
\(125\) 12.0787 1.08036
\(126\) −0.231853 −0.0206551
\(127\) −2.75202 −0.244202 −0.122101 0.992518i \(-0.538963\pi\)
−0.122101 + 0.992518i \(0.538963\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.04537 −0.796400
\(130\) 0 0
\(131\) −11.9707 −1.04589 −0.522943 0.852368i \(-0.675166\pi\)
−0.522943 + 0.852368i \(0.675166\pi\)
\(132\) 7.34980 0.639718
\(133\) −0.961642 −0.0833849
\(134\) −5.84309 −0.504766
\(135\) −9.11921 −0.784857
\(136\) 1.01199 0.0867774
\(137\) −21.3846 −1.82701 −0.913505 0.406827i \(-0.866635\pi\)
−0.913505 + 0.406827i \(0.866635\pi\)
\(138\) −10.0823 −0.858259
\(139\) −12.8882 −1.09316 −0.546580 0.837407i \(-0.684070\pi\)
−0.546580 + 0.837407i \(0.684070\pi\)
\(140\) −1.62896 −0.137672
\(141\) −6.50586 −0.547892
\(142\) −14.6895 −1.23272
\(143\) 0 0
\(144\) −0.241101 −0.0200918
\(145\) −11.7812 −0.978375
\(146\) 2.12183 0.175604
\(147\) 10.0909 0.832287
\(148\) −0.678162 −0.0557446
\(149\) 23.2522 1.90489 0.952445 0.304709i \(-0.0985593\pi\)
0.952445 + 0.304709i \(0.0985593\pi\)
\(150\) 3.53888 0.288949
\(151\) −24.5628 −1.99889 −0.999446 0.0332861i \(-0.989403\pi\)
−0.999446 + 0.0332861i \(0.989403\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −0.243992 −0.0197256
\(154\) −4.25521 −0.342894
\(155\) −16.4941 −1.32484
\(156\) 0 0
\(157\) 9.81548 0.783360 0.391680 0.920101i \(-0.371894\pi\)
0.391680 + 0.920101i \(0.371894\pi\)
\(158\) −7.59795 −0.604461
\(159\) −10.2781 −0.815103
\(160\) −1.69394 −0.133917
\(161\) 5.83719 0.460035
\(162\) −8.21857 −0.645712
\(163\) 9.15174 0.716820 0.358410 0.933564i \(-0.383319\pi\)
0.358410 + 0.933564i \(0.383319\pi\)
\(164\) −11.2026 −0.874777
\(165\) −12.4501 −0.969238
\(166\) 4.74343 0.368162
\(167\) −16.3959 −1.26875 −0.634376 0.773024i \(-0.718743\pi\)
−0.634376 + 0.773024i \(0.718743\pi\)
\(168\) −1.59728 −0.123233
\(169\) 0 0
\(170\) −1.71425 −0.131477
\(171\) 0.241101 0.0184375
\(172\) 5.44576 0.415235
\(173\) 7.00746 0.532768 0.266384 0.963867i \(-0.414171\pi\)
0.266384 + 0.963867i \(0.414171\pi\)
\(174\) −11.5521 −0.875761
\(175\) −2.04886 −0.154879
\(176\) −4.42494 −0.333543
\(177\) −3.42989 −0.257806
\(178\) −6.32590 −0.474146
\(179\) 16.0201 1.19740 0.598698 0.800975i \(-0.295685\pi\)
0.598698 + 0.800975i \(0.295685\pi\)
\(180\) 0.408410 0.0304411
\(181\) 2.81392 0.209157 0.104578 0.994517i \(-0.466651\pi\)
0.104578 + 0.994517i \(0.466651\pi\)
\(182\) 0 0
\(183\) 16.9840 1.25549
\(184\) 6.07002 0.447488
\(185\) 1.14876 0.0844587
\(186\) −16.1734 −1.18589
\(187\) −4.47800 −0.327464
\(188\) 3.91685 0.285666
\(189\) 5.17695 0.376568
\(190\) 1.69394 0.122891
\(191\) −7.85491 −0.568361 −0.284181 0.958771i \(-0.591721\pi\)
−0.284181 + 0.958771i \(0.591721\pi\)
\(192\) −1.66099 −0.119872
\(193\) −8.38465 −0.603540 −0.301770 0.953381i \(-0.597578\pi\)
−0.301770 + 0.953381i \(0.597578\pi\)
\(194\) 4.22409 0.303272
\(195\) 0 0
\(196\) −6.07525 −0.433946
\(197\) −10.1478 −0.723001 −0.361501 0.932372i \(-0.617735\pi\)
−0.361501 + 0.932372i \(0.617735\pi\)
\(198\) 1.06686 0.0758184
\(199\) 1.80755 0.128134 0.0640670 0.997946i \(-0.479593\pi\)
0.0640670 + 0.997946i \(0.479593\pi\)
\(200\) −2.13058 −0.150655
\(201\) 9.70533 0.684561
\(202\) −5.33989 −0.375713
\(203\) 6.68815 0.469416
\(204\) −1.68091 −0.117687
\(205\) 18.9765 1.32538
\(206\) −5.98387 −0.416916
\(207\) −1.46349 −0.101720
\(208\) 0 0
\(209\) 4.42494 0.306080
\(210\) 2.70569 0.186710
\(211\) −27.7630 −1.91128 −0.955642 0.294532i \(-0.904836\pi\)
−0.955642 + 0.294532i \(0.904836\pi\)
\(212\) 6.18790 0.424986
\(213\) 24.3992 1.67180
\(214\) 0.456797 0.0312260
\(215\) −9.22476 −0.629123
\(216\) 5.38345 0.366297
\(217\) 9.36367 0.635647
\(218\) −13.2279 −0.895904
\(219\) −3.52435 −0.238153
\(220\) 7.49557 0.505351
\(221\) 0 0
\(222\) 1.12642 0.0756005
\(223\) 7.00880 0.469344 0.234672 0.972075i \(-0.424598\pi\)
0.234672 + 0.972075i \(0.424598\pi\)
\(224\) 0.961642 0.0642524
\(225\) 0.513687 0.0342458
\(226\) 11.5596 0.768930
\(227\) 20.8321 1.38267 0.691337 0.722532i \(-0.257022\pi\)
0.691337 + 0.722532i \(0.257022\pi\)
\(228\) 1.66099 0.110002
\(229\) 6.04193 0.399262 0.199631 0.979871i \(-0.436026\pi\)
0.199631 + 0.979871i \(0.436026\pi\)
\(230\) −10.2822 −0.677990
\(231\) 7.06787 0.465032
\(232\) 6.95493 0.456613
\(233\) −3.21677 −0.210737 −0.105369 0.994433i \(-0.533602\pi\)
−0.105369 + 0.994433i \(0.533602\pi\)
\(234\) 0 0
\(235\) −6.63489 −0.432813
\(236\) 2.06496 0.134418
\(237\) 12.6201 0.819767
\(238\) 0.973172 0.0630814
\(239\) 7.65203 0.494969 0.247484 0.968892i \(-0.420396\pi\)
0.247484 + 0.968892i \(0.420396\pi\)
\(240\) 2.81362 0.181618
\(241\) 9.27475 0.597439 0.298720 0.954341i \(-0.403440\pi\)
0.298720 + 0.954341i \(0.403440\pi\)
\(242\) 8.58011 0.551550
\(243\) −2.49936 −0.160334
\(244\) −10.2252 −0.654602
\(245\) 10.2911 0.657473
\(246\) 18.6075 1.18637
\(247\) 0 0
\(248\) 9.73717 0.618311
\(249\) −7.87881 −0.499299
\(250\) 12.0787 0.763927
\(251\) −1.09472 −0.0690984 −0.0345492 0.999403i \(-0.511000\pi\)
−0.0345492 + 0.999403i \(0.511000\pi\)
\(252\) −0.231853 −0.0146054
\(253\) −26.8595 −1.68864
\(254\) −2.75202 −0.172677
\(255\) 2.84735 0.178308
\(256\) 1.00000 0.0625000
\(257\) 1.33399 0.0832118 0.0416059 0.999134i \(-0.486753\pi\)
0.0416059 + 0.999134i \(0.486753\pi\)
\(258\) −9.04537 −0.563140
\(259\) −0.652149 −0.0405226
\(260\) 0 0
\(261\) −1.67684 −0.103794
\(262\) −11.9707 −0.739553
\(263\) −20.5528 −1.26734 −0.633669 0.773604i \(-0.718452\pi\)
−0.633669 + 0.773604i \(0.718452\pi\)
\(264\) 7.34980 0.452349
\(265\) −10.4819 −0.643898
\(266\) −0.961642 −0.0589620
\(267\) 10.5073 0.643035
\(268\) −5.84309 −0.356924
\(269\) −20.8405 −1.27067 −0.635333 0.772238i \(-0.719137\pi\)
−0.635333 + 0.772238i \(0.719137\pi\)
\(270\) −9.11921 −0.554978
\(271\) −5.43640 −0.330238 −0.165119 0.986274i \(-0.552801\pi\)
−0.165119 + 0.986274i \(0.552801\pi\)
\(272\) 1.01199 0.0613609
\(273\) 0 0
\(274\) −21.3846 −1.29189
\(275\) 9.42770 0.568512
\(276\) −10.0823 −0.606881
\(277\) 24.6539 1.48131 0.740653 0.671888i \(-0.234516\pi\)
0.740653 + 0.671888i \(0.234516\pi\)
\(278\) −12.8882 −0.772980
\(279\) −2.34765 −0.140550
\(280\) −1.62896 −0.0973489
\(281\) −25.5274 −1.52284 −0.761419 0.648260i \(-0.775497\pi\)
−0.761419 + 0.648260i \(0.775497\pi\)
\(282\) −6.50586 −0.387418
\(283\) −16.6966 −0.992510 −0.496255 0.868177i \(-0.665292\pi\)
−0.496255 + 0.868177i \(0.665292\pi\)
\(284\) −14.6895 −0.871662
\(285\) −2.81362 −0.166664
\(286\) 0 0
\(287\) −10.7729 −0.635904
\(288\) −0.241101 −0.0142070
\(289\) −15.9759 −0.939757
\(290\) −11.7812 −0.691816
\(291\) −7.01619 −0.411297
\(292\) 2.12183 0.124171
\(293\) −21.2827 −1.24335 −0.621676 0.783275i \(-0.713548\pi\)
−0.621676 + 0.783275i \(0.713548\pi\)
\(294\) 10.0909 0.588516
\(295\) −3.49791 −0.203656
\(296\) −0.678162 −0.0394174
\(297\) −23.8214 −1.38226
\(298\) 23.2522 1.34696
\(299\) 0 0
\(300\) 3.53888 0.204318
\(301\) 5.23687 0.301848
\(302\) −24.5628 −1.41343
\(303\) 8.86952 0.509541
\(304\) −1.00000 −0.0573539
\(305\) 17.3208 0.991789
\(306\) −0.243992 −0.0139481
\(307\) −4.70636 −0.268606 −0.134303 0.990940i \(-0.542880\pi\)
−0.134303 + 0.990940i \(0.542880\pi\)
\(308\) −4.25521 −0.242463
\(309\) 9.93917 0.565420
\(310\) −16.4941 −0.936804
\(311\) 4.94400 0.280348 0.140174 0.990127i \(-0.455234\pi\)
0.140174 + 0.990127i \(0.455234\pi\)
\(312\) 0 0
\(313\) 12.7603 0.721253 0.360627 0.932710i \(-0.382563\pi\)
0.360627 + 0.932710i \(0.382563\pi\)
\(314\) 9.81548 0.553919
\(315\) 0.392744 0.0221286
\(316\) −7.59795 −0.427418
\(317\) 13.6955 0.769218 0.384609 0.923080i \(-0.374336\pi\)
0.384609 + 0.923080i \(0.374336\pi\)
\(318\) −10.2781 −0.576365
\(319\) −30.7751 −1.72308
\(320\) −1.69394 −0.0946939
\(321\) −0.758738 −0.0423486
\(322\) 5.83719 0.325294
\(323\) −1.01199 −0.0563086
\(324\) −8.21857 −0.456587
\(325\) 0 0
\(326\) 9.15174 0.506868
\(327\) 21.9714 1.21502
\(328\) −11.2026 −0.618561
\(329\) 3.76661 0.207660
\(330\) −12.4501 −0.685355
\(331\) 6.78053 0.372692 0.186346 0.982484i \(-0.440336\pi\)
0.186346 + 0.982484i \(0.440336\pi\)
\(332\) 4.74343 0.260330
\(333\) 0.163506 0.00896006
\(334\) −16.3959 −0.897143
\(335\) 9.89782 0.540775
\(336\) −1.59728 −0.0871388
\(337\) 18.0123 0.981195 0.490597 0.871386i \(-0.336779\pi\)
0.490597 + 0.871386i \(0.336779\pi\)
\(338\) 0 0
\(339\) −19.2003 −1.04282
\(340\) −1.71425 −0.0929680
\(341\) −43.0864 −2.33326
\(342\) 0.241101 0.0130373
\(343\) −12.5737 −0.678916
\(344\) 5.44576 0.293616
\(345\) 17.0787 0.919487
\(346\) 7.00746 0.376724
\(347\) 24.5390 1.31732 0.658661 0.752440i \(-0.271123\pi\)
0.658661 + 0.752440i \(0.271123\pi\)
\(348\) −11.5521 −0.619257
\(349\) −4.97210 −0.266151 −0.133075 0.991106i \(-0.542485\pi\)
−0.133075 + 0.991106i \(0.542485\pi\)
\(350\) −2.04886 −0.109516
\(351\) 0 0
\(352\) −4.42494 −0.235850
\(353\) −13.5038 −0.718734 −0.359367 0.933196i \(-0.617007\pi\)
−0.359367 + 0.933196i \(0.617007\pi\)
\(354\) −3.42989 −0.182296
\(355\) 24.8831 1.32066
\(356\) −6.32590 −0.335272
\(357\) −1.61643 −0.0855507
\(358\) 16.0201 0.846687
\(359\) −35.4154 −1.86915 −0.934576 0.355764i \(-0.884221\pi\)
−0.934576 + 0.355764i \(0.884221\pi\)
\(360\) 0.408410 0.0215251
\(361\) 1.00000 0.0526316
\(362\) 2.81392 0.147896
\(363\) −14.2515 −0.748010
\(364\) 0 0
\(365\) −3.59424 −0.188131
\(366\) 16.9840 0.887768
\(367\) −35.1014 −1.83228 −0.916140 0.400858i \(-0.868712\pi\)
−0.916140 + 0.400858i \(0.868712\pi\)
\(368\) 6.07002 0.316422
\(369\) 2.70096 0.140607
\(370\) 1.14876 0.0597213
\(371\) 5.95054 0.308937
\(372\) −16.1734 −0.838551
\(373\) −5.02537 −0.260204 −0.130102 0.991501i \(-0.541530\pi\)
−0.130102 + 0.991501i \(0.541530\pi\)
\(374\) −4.47800 −0.231552
\(375\) −20.0627 −1.03603
\(376\) 3.91685 0.201996
\(377\) 0 0
\(378\) 5.17695 0.266273
\(379\) −19.8743 −1.02088 −0.510438 0.859915i \(-0.670517\pi\)
−0.510438 + 0.859915i \(0.670517\pi\)
\(380\) 1.69394 0.0868971
\(381\) 4.57108 0.234184
\(382\) −7.85491 −0.401892
\(383\) −13.1631 −0.672602 −0.336301 0.941754i \(-0.609176\pi\)
−0.336301 + 0.941754i \(0.609176\pi\)
\(384\) −1.66099 −0.0847622
\(385\) 7.20805 0.367356
\(386\) −8.38465 −0.426767
\(387\) −1.31298 −0.0667425
\(388\) 4.22409 0.214446
\(389\) 17.9823 0.911739 0.455869 0.890047i \(-0.349328\pi\)
0.455869 + 0.890047i \(0.349328\pi\)
\(390\) 0 0
\(391\) 6.14280 0.310655
\(392\) −6.07525 −0.306846
\(393\) 19.8833 1.00298
\(394\) −10.1478 −0.511239
\(395\) 12.8704 0.647582
\(396\) 1.06686 0.0536117
\(397\) −10.7231 −0.538176 −0.269088 0.963116i \(-0.586722\pi\)
−0.269088 + 0.963116i \(0.586722\pi\)
\(398\) 1.80755 0.0906044
\(399\) 1.59728 0.0799640
\(400\) −2.13058 −0.106529
\(401\) 23.5965 1.17835 0.589176 0.808004i \(-0.299452\pi\)
0.589176 + 0.808004i \(0.299452\pi\)
\(402\) 9.70533 0.484058
\(403\) 0 0
\(404\) −5.33989 −0.265669
\(405\) 13.9217 0.691776
\(406\) 6.68815 0.331927
\(407\) 3.00083 0.148745
\(408\) −1.68091 −0.0832174
\(409\) −2.70488 −0.133748 −0.0668739 0.997761i \(-0.521303\pi\)
−0.0668739 + 0.997761i \(0.521303\pi\)
\(410\) 18.9765 0.937182
\(411\) 35.5197 1.75206
\(412\) −5.98387 −0.294804
\(413\) 1.98575 0.0977125
\(414\) −1.46349 −0.0719267
\(415\) −8.03507 −0.394426
\(416\) 0 0
\(417\) 21.4071 1.04831
\(418\) 4.42494 0.216431
\(419\) 16.1618 0.789554 0.394777 0.918777i \(-0.370822\pi\)
0.394777 + 0.918777i \(0.370822\pi\)
\(420\) 2.70569 0.132024
\(421\) −16.8012 −0.818839 −0.409419 0.912346i \(-0.634269\pi\)
−0.409419 + 0.912346i \(0.634269\pi\)
\(422\) −27.7630 −1.35148
\(423\) −0.944358 −0.0459163
\(424\) 6.18790 0.300511
\(425\) −2.15613 −0.104588
\(426\) 24.3992 1.18214
\(427\) −9.83299 −0.475852
\(428\) 0.456797 0.0220801
\(429\) 0 0
\(430\) −9.22476 −0.444857
\(431\) −33.9067 −1.63323 −0.816615 0.577183i \(-0.804152\pi\)
−0.816615 + 0.577183i \(0.804152\pi\)
\(432\) 5.38345 0.259011
\(433\) 20.3139 0.976223 0.488111 0.872781i \(-0.337686\pi\)
0.488111 + 0.872781i \(0.337686\pi\)
\(434\) 9.36367 0.449471
\(435\) 19.5685 0.938237
\(436\) −13.2279 −0.633500
\(437\) −6.07002 −0.290369
\(438\) −3.52435 −0.168400
\(439\) −23.5505 −1.12400 −0.562001 0.827136i \(-0.689968\pi\)
−0.562001 + 0.827136i \(0.689968\pi\)
\(440\) 7.49557 0.357337
\(441\) 1.46475 0.0697500
\(442\) 0 0
\(443\) 26.0674 1.23850 0.619250 0.785194i \(-0.287437\pi\)
0.619250 + 0.785194i \(0.287437\pi\)
\(444\) 1.12642 0.0534576
\(445\) 10.7157 0.507971
\(446\) 7.00880 0.331876
\(447\) −38.6217 −1.82674
\(448\) 0.961642 0.0454333
\(449\) 6.84222 0.322904 0.161452 0.986881i \(-0.448382\pi\)
0.161452 + 0.986881i \(0.448382\pi\)
\(450\) 0.513687 0.0242154
\(451\) 49.5709 2.33420
\(452\) 11.5596 0.543716
\(453\) 40.7986 1.91689
\(454\) 20.8321 0.977698
\(455\) 0 0
\(456\) 1.66099 0.0777831
\(457\) −24.4501 −1.14373 −0.571865 0.820348i \(-0.693780\pi\)
−0.571865 + 0.820348i \(0.693780\pi\)
\(458\) 6.04193 0.282321
\(459\) 5.44800 0.254291
\(460\) −10.2822 −0.479411
\(461\) 1.80454 0.0840456 0.0420228 0.999117i \(-0.486620\pi\)
0.0420228 + 0.999117i \(0.486620\pi\)
\(462\) 7.06787 0.328827
\(463\) 4.62667 0.215020 0.107510 0.994204i \(-0.465712\pi\)
0.107510 + 0.994204i \(0.465712\pi\)
\(464\) 6.95493 0.322874
\(465\) 27.3967 1.27049
\(466\) −3.21677 −0.149014
\(467\) 10.7837 0.499009 0.249504 0.968374i \(-0.419732\pi\)
0.249504 + 0.968374i \(0.419732\pi\)
\(468\) 0 0
\(469\) −5.61896 −0.259459
\(470\) −6.63489 −0.306045
\(471\) −16.3034 −0.751223
\(472\) 2.06496 0.0950475
\(473\) −24.0972 −1.10799
\(474\) 12.6201 0.579663
\(475\) 2.13058 0.0977578
\(476\) 0.973172 0.0446053
\(477\) −1.49191 −0.0683099
\(478\) 7.65203 0.349996
\(479\) 0.0903531 0.00412834 0.00206417 0.999998i \(-0.499343\pi\)
0.00206417 + 0.999998i \(0.499343\pi\)
\(480\) 2.81362 0.128423
\(481\) 0 0
\(482\) 9.27475 0.422453
\(483\) −9.69553 −0.441162
\(484\) 8.58011 0.390005
\(485\) −7.15534 −0.324907
\(486\) −2.49936 −0.113373
\(487\) −16.1552 −0.732064 −0.366032 0.930602i \(-0.619284\pi\)
−0.366032 + 0.930602i \(0.619284\pi\)
\(488\) −10.2252 −0.462873
\(489\) −15.2010 −0.687412
\(490\) 10.2911 0.464903
\(491\) 4.49239 0.202739 0.101369 0.994849i \(-0.467678\pi\)
0.101369 + 0.994849i \(0.467678\pi\)
\(492\) 18.6075 0.838889
\(493\) 7.03832 0.316990
\(494\) 0 0
\(495\) −1.80719 −0.0812272
\(496\) 9.73717 0.437212
\(497\) −14.1260 −0.633640
\(498\) −7.87881 −0.353058
\(499\) −10.3831 −0.464810 −0.232405 0.972619i \(-0.574659\pi\)
−0.232405 + 0.972619i \(0.574659\pi\)
\(500\) 12.0787 0.540178
\(501\) 27.2335 1.21670
\(502\) −1.09472 −0.0488599
\(503\) 32.2709 1.43889 0.719444 0.694551i \(-0.244397\pi\)
0.719444 + 0.694551i \(0.244397\pi\)
\(504\) −0.231853 −0.0103276
\(505\) 9.04543 0.402516
\(506\) −26.8595 −1.19405
\(507\) 0 0
\(508\) −2.75202 −0.122101
\(509\) −2.97775 −0.131986 −0.0659931 0.997820i \(-0.521022\pi\)
−0.0659931 + 0.997820i \(0.521022\pi\)
\(510\) 2.84735 0.126083
\(511\) 2.04044 0.0902638
\(512\) 1.00000 0.0441942
\(513\) −5.38345 −0.237685
\(514\) 1.33399 0.0588397
\(515\) 10.1363 0.446658
\(516\) −9.04537 −0.398200
\(517\) −17.3318 −0.762253
\(518\) −0.652149 −0.0286538
\(519\) −11.6394 −0.510911
\(520\) 0 0
\(521\) 3.76919 0.165131 0.0825657 0.996586i \(-0.473689\pi\)
0.0825657 + 0.996586i \(0.473689\pi\)
\(522\) −1.67684 −0.0733934
\(523\) 4.65115 0.203381 0.101690 0.994816i \(-0.467575\pi\)
0.101690 + 0.994816i \(0.467575\pi\)
\(524\) −11.9707 −0.522943
\(525\) 3.40314 0.148525
\(526\) −20.5528 −0.896143
\(527\) 9.85392 0.429244
\(528\) 7.34980 0.319859
\(529\) 13.8452 0.601964
\(530\) −10.4819 −0.455305
\(531\) −0.497865 −0.0216055
\(532\) −0.961642 −0.0416925
\(533\) 0 0
\(534\) 10.5073 0.454694
\(535\) −0.773785 −0.0334537
\(536\) −5.84309 −0.252383
\(537\) −26.6092 −1.14827
\(538\) −20.8405 −0.898497
\(539\) 26.8826 1.15792
\(540\) −9.11921 −0.392428
\(541\) −26.4333 −1.13646 −0.568229 0.822871i \(-0.692371\pi\)
−0.568229 + 0.822871i \(0.692371\pi\)
\(542\) −5.43640 −0.233513
\(543\) −4.67390 −0.200576
\(544\) 1.01199 0.0433887
\(545\) 22.4071 0.959817
\(546\) 0 0
\(547\) −38.2590 −1.63584 −0.817919 0.575333i \(-0.804872\pi\)
−0.817919 + 0.575333i \(0.804872\pi\)
\(548\) −21.3846 −0.913505
\(549\) 2.46531 0.105217
\(550\) 9.42770 0.401999
\(551\) −6.95493 −0.296290
\(552\) −10.0823 −0.429130
\(553\) −7.30651 −0.310704
\(554\) 24.6539 1.04744
\(555\) −1.90809 −0.0809938
\(556\) −12.8882 −0.546580
\(557\) −23.5589 −0.998221 −0.499111 0.866538i \(-0.666340\pi\)
−0.499111 + 0.866538i \(0.666340\pi\)
\(558\) −2.34765 −0.0993838
\(559\) 0 0
\(560\) −1.62896 −0.0688361
\(561\) 7.43792 0.314029
\(562\) −25.5274 −1.07681
\(563\) −5.38431 −0.226922 −0.113461 0.993542i \(-0.536194\pi\)
−0.113461 + 0.993542i \(0.536194\pi\)
\(564\) −6.50586 −0.273946
\(565\) −19.5811 −0.823785
\(566\) −16.6966 −0.701811
\(567\) −7.90332 −0.331908
\(568\) −14.6895 −0.616358
\(569\) 15.1819 0.636457 0.318229 0.948014i \(-0.396912\pi\)
0.318229 + 0.948014i \(0.396912\pi\)
\(570\) −2.81362 −0.117849
\(571\) 1.16095 0.0485842 0.0242921 0.999705i \(-0.492267\pi\)
0.0242921 + 0.999705i \(0.492267\pi\)
\(572\) 0 0
\(573\) 13.0469 0.545044
\(574\) −10.7729 −0.449652
\(575\) −12.9327 −0.539330
\(576\) −0.241101 −0.0100459
\(577\) 5.68941 0.236853 0.118427 0.992963i \(-0.462215\pi\)
0.118427 + 0.992963i \(0.462215\pi\)
\(578\) −15.9759 −0.664509
\(579\) 13.9268 0.578780
\(580\) −11.7812 −0.489188
\(581\) 4.56148 0.189242
\(582\) −7.01619 −0.290831
\(583\) −27.3811 −1.13401
\(584\) 2.12183 0.0878020
\(585\) 0 0
\(586\) −21.2827 −0.879182
\(587\) 41.9965 1.73338 0.866691 0.498845i \(-0.166242\pi\)
0.866691 + 0.498845i \(0.166242\pi\)
\(588\) 10.0909 0.416143
\(589\) −9.73717 −0.401213
\(590\) −3.49791 −0.144007
\(591\) 16.8554 0.693340
\(592\) −0.678162 −0.0278723
\(593\) 9.17552 0.376793 0.188397 0.982093i \(-0.439671\pi\)
0.188397 + 0.982093i \(0.439671\pi\)
\(594\) −23.8214 −0.977406
\(595\) −1.64849 −0.0675815
\(596\) 23.2522 0.952445
\(597\) −3.00233 −0.122877
\(598\) 0 0
\(599\) −38.4504 −1.57104 −0.785520 0.618837i \(-0.787604\pi\)
−0.785520 + 0.618837i \(0.787604\pi\)
\(600\) 3.53888 0.144474
\(601\) 17.4977 0.713748 0.356874 0.934153i \(-0.383843\pi\)
0.356874 + 0.934153i \(0.383843\pi\)
\(602\) 5.23687 0.213439
\(603\) 1.40878 0.0573698
\(604\) −24.5628 −0.999446
\(605\) −14.5341 −0.590897
\(606\) 8.86952 0.360300
\(607\) −22.8352 −0.926851 −0.463426 0.886136i \(-0.653380\pi\)
−0.463426 + 0.886136i \(0.653380\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −11.1090 −0.450158
\(610\) 17.3208 0.701300
\(611\) 0 0
\(612\) −0.243992 −0.00986280
\(613\) −44.8074 −1.80975 −0.904877 0.425674i \(-0.860037\pi\)
−0.904877 + 0.425674i \(0.860037\pi\)
\(614\) −4.70636 −0.189933
\(615\) −31.5198 −1.27100
\(616\) −4.25521 −0.171447
\(617\) −12.3912 −0.498850 −0.249425 0.968394i \(-0.580242\pi\)
−0.249425 + 0.968394i \(0.580242\pi\)
\(618\) 9.93917 0.399812
\(619\) −44.6247 −1.79362 −0.896809 0.442418i \(-0.854121\pi\)
−0.896809 + 0.442418i \(0.854121\pi\)
\(620\) −16.4941 −0.662421
\(621\) 32.6776 1.31131
\(622\) 4.94400 0.198236
\(623\) −6.08325 −0.243720
\(624\) 0 0
\(625\) −9.80770 −0.392308
\(626\) 12.7603 0.510003
\(627\) −7.34980 −0.293523
\(628\) 9.81548 0.391680
\(629\) −0.686293 −0.0273643
\(630\) 0.392744 0.0156473
\(631\) 21.6787 0.863014 0.431507 0.902110i \(-0.357982\pi\)
0.431507 + 0.902110i \(0.357982\pi\)
\(632\) −7.59795 −0.302230
\(633\) 46.1142 1.83287
\(634\) 13.6955 0.543919
\(635\) 4.66174 0.184996
\(636\) −10.2781 −0.407551
\(637\) 0 0
\(638\) −30.7751 −1.21840
\(639\) 3.54166 0.140106
\(640\) −1.69394 −0.0669587
\(641\) 1.22156 0.0482485 0.0241243 0.999709i \(-0.492320\pi\)
0.0241243 + 0.999709i \(0.492320\pi\)
\(642\) −0.758738 −0.0299450
\(643\) 20.1091 0.793024 0.396512 0.918030i \(-0.370221\pi\)
0.396512 + 0.918030i \(0.370221\pi\)
\(644\) 5.83719 0.230017
\(645\) 15.3223 0.603314
\(646\) −1.01199 −0.0398162
\(647\) 33.4789 1.31619 0.658095 0.752935i \(-0.271362\pi\)
0.658095 + 0.752935i \(0.271362\pi\)
\(648\) −8.21857 −0.322856
\(649\) −9.13733 −0.358672
\(650\) 0 0
\(651\) −15.5530 −0.609570
\(652\) 9.15174 0.358410
\(653\) 17.1761 0.672153 0.336077 0.941835i \(-0.390900\pi\)
0.336077 + 0.941835i \(0.390900\pi\)
\(654\) 21.9714 0.859149
\(655\) 20.2776 0.792312
\(656\) −11.2026 −0.437388
\(657\) −0.511576 −0.0199585
\(658\) 3.76661 0.146838
\(659\) −27.2804 −1.06269 −0.531347 0.847155i \(-0.678314\pi\)
−0.531347 + 0.847155i \(0.678314\pi\)
\(660\) −12.4501 −0.484619
\(661\) −45.5635 −1.77221 −0.886107 0.463481i \(-0.846600\pi\)
−0.886107 + 0.463481i \(0.846600\pi\)
\(662\) 6.78053 0.263533
\(663\) 0 0
\(664\) 4.74343 0.184081
\(665\) 1.62896 0.0631683
\(666\) 0.163506 0.00633572
\(667\) 42.2166 1.63463
\(668\) −16.3959 −0.634376
\(669\) −11.6416 −0.450089
\(670\) 9.89782 0.382386
\(671\) 45.2459 1.74670
\(672\) −1.59728 −0.0616164
\(673\) 28.5965 1.10232 0.551158 0.834401i \(-0.314186\pi\)
0.551158 + 0.834401i \(0.314186\pi\)
\(674\) 18.0123 0.693809
\(675\) −11.4699 −0.441476
\(676\) 0 0
\(677\) 29.9014 1.14921 0.574603 0.818433i \(-0.305157\pi\)
0.574603 + 0.818433i \(0.305157\pi\)
\(678\) −19.2003 −0.737385
\(679\) 4.06207 0.155888
\(680\) −1.71425 −0.0657383
\(681\) −34.6020 −1.32595
\(682\) −43.0864 −1.64986
\(683\) −32.4539 −1.24181 −0.620907 0.783884i \(-0.713236\pi\)
−0.620907 + 0.783884i \(0.713236\pi\)
\(684\) 0.241101 0.00921874
\(685\) 36.2242 1.38405
\(686\) −12.5737 −0.480066
\(687\) −10.0356 −0.382883
\(688\) 5.44576 0.207618
\(689\) 0 0
\(690\) 17.0787 0.650175
\(691\) −15.0559 −0.572752 −0.286376 0.958117i \(-0.592451\pi\)
−0.286376 + 0.958117i \(0.592451\pi\)
\(692\) 7.00746 0.266384
\(693\) 1.02594 0.0389721
\(694\) 24.5390 0.931487
\(695\) 21.8317 0.828124
\(696\) −11.5521 −0.437881
\(697\) −11.3369 −0.429417
\(698\) −4.97210 −0.188197
\(699\) 5.34303 0.202092
\(700\) −2.04886 −0.0774395
\(701\) 32.2158 1.21677 0.608387 0.793641i \(-0.291817\pi\)
0.608387 + 0.793641i \(0.291817\pi\)
\(702\) 0 0
\(703\) 0.678162 0.0255774
\(704\) −4.42494 −0.166771
\(705\) 11.0205 0.415056
\(706\) −13.5038 −0.508222
\(707\) −5.13506 −0.193124
\(708\) −3.42989 −0.128903
\(709\) 27.6450 1.03823 0.519116 0.854704i \(-0.326261\pi\)
0.519116 + 0.854704i \(0.326261\pi\)
\(710\) 24.8831 0.933845
\(711\) 1.83188 0.0687008
\(712\) −6.32590 −0.237073
\(713\) 59.1048 2.21349
\(714\) −1.61643 −0.0604934
\(715\) 0 0
\(716\) 16.0201 0.598698
\(717\) −12.7100 −0.474663
\(718\) −35.4154 −1.32169
\(719\) 8.31650 0.310153 0.155077 0.987902i \(-0.450438\pi\)
0.155077 + 0.987902i \(0.450438\pi\)
\(720\) 0.408410 0.0152206
\(721\) −5.75434 −0.214303
\(722\) 1.00000 0.0372161
\(723\) −15.4053 −0.572929
\(724\) 2.81392 0.104578
\(725\) −14.8180 −0.550328
\(726\) −14.2515 −0.528923
\(727\) 20.4351 0.757897 0.378949 0.925418i \(-0.376286\pi\)
0.378949 + 0.925418i \(0.376286\pi\)
\(728\) 0 0
\(729\) 28.8071 1.06693
\(730\) −3.59424 −0.133029
\(731\) 5.51105 0.203834
\(732\) 16.9840 0.627747
\(733\) −42.1710 −1.55762 −0.778810 0.627260i \(-0.784176\pi\)
−0.778810 + 0.627260i \(0.784176\pi\)
\(734\) −35.1014 −1.29562
\(735\) −17.0934 −0.630500
\(736\) 6.07002 0.223744
\(737\) 25.8553 0.952393
\(738\) 2.70096 0.0994239
\(739\) 27.6948 1.01877 0.509384 0.860539i \(-0.329873\pi\)
0.509384 + 0.860539i \(0.329873\pi\)
\(740\) 1.14876 0.0422293
\(741\) 0 0
\(742\) 5.95054 0.218451
\(743\) −7.94485 −0.291468 −0.145734 0.989324i \(-0.546554\pi\)
−0.145734 + 0.989324i \(0.546554\pi\)
\(744\) −16.1734 −0.592945
\(745\) −39.3876 −1.44305
\(746\) −5.02537 −0.183992
\(747\) −1.14365 −0.0418439
\(748\) −4.47800 −0.163732
\(749\) 0.439276 0.0160508
\(750\) −20.0627 −0.732587
\(751\) 32.1477 1.17309 0.586543 0.809918i \(-0.300489\pi\)
0.586543 + 0.809918i \(0.300489\pi\)
\(752\) 3.91685 0.142833
\(753\) 1.81833 0.0662636
\(754\) 0 0
\(755\) 41.6078 1.51426
\(756\) 5.17695 0.188284
\(757\) −3.80527 −0.138305 −0.0691525 0.997606i \(-0.522030\pi\)
−0.0691525 + 0.997606i \(0.522030\pi\)
\(758\) −19.8743 −0.721868
\(759\) 44.6134 1.61937
\(760\) 1.69394 0.0614455
\(761\) −27.4468 −0.994944 −0.497472 0.867480i \(-0.665738\pi\)
−0.497472 + 0.867480i \(0.665738\pi\)
\(762\) 4.57108 0.165593
\(763\) −12.7205 −0.460512
\(764\) −7.85491 −0.284181
\(765\) 0.413307 0.0149431
\(766\) −13.1631 −0.475602
\(767\) 0 0
\(768\) −1.66099 −0.0599359
\(769\) 28.9834 1.04517 0.522584 0.852588i \(-0.324968\pi\)
0.522584 + 0.852588i \(0.324968\pi\)
\(770\) 7.20805 0.259760
\(771\) −2.21574 −0.0797981
\(772\) −8.38465 −0.301770
\(773\) 38.8125 1.39599 0.697995 0.716103i \(-0.254076\pi\)
0.697995 + 0.716103i \(0.254076\pi\)
\(774\) −1.31298 −0.0471941
\(775\) −20.7459 −0.745213
\(776\) 4.22409 0.151636
\(777\) 1.08321 0.0388601
\(778\) 17.9823 0.644697
\(779\) 11.2026 0.401375
\(780\) 0 0
\(781\) 65.0002 2.32589
\(782\) 6.14280 0.219666
\(783\) 37.4415 1.33805
\(784\) −6.07525 −0.216973
\(785\) −16.6268 −0.593435
\(786\) 19.8833 0.709213
\(787\) −41.3374 −1.47352 −0.736760 0.676154i \(-0.763645\pi\)
−0.736760 + 0.676154i \(0.763645\pi\)
\(788\) −10.1478 −0.361501
\(789\) 34.1380 1.21534
\(790\) 12.8704 0.457910
\(791\) 11.1162 0.395245
\(792\) 1.06686 0.0379092
\(793\) 0 0
\(794\) −10.7231 −0.380548
\(795\) 17.4104 0.617482
\(796\) 1.80755 0.0640670
\(797\) 41.7974 1.48054 0.740270 0.672309i \(-0.234698\pi\)
0.740270 + 0.672309i \(0.234698\pi\)
\(798\) 1.59728 0.0565431
\(799\) 3.96381 0.140230
\(800\) −2.13058 −0.0753275
\(801\) 1.52518 0.0538897
\(802\) 23.5965 0.833221
\(803\) −9.38898 −0.331330
\(804\) 9.70533 0.342281
\(805\) −9.88782 −0.348500
\(806\) 0 0
\(807\) 34.6159 1.21854
\(808\) −5.33989 −0.187857
\(809\) −15.4613 −0.543592 −0.271796 0.962355i \(-0.587618\pi\)
−0.271796 + 0.962355i \(0.587618\pi\)
\(810\) 13.9217 0.489159
\(811\) 40.5158 1.42270 0.711351 0.702837i \(-0.248084\pi\)
0.711351 + 0.702837i \(0.248084\pi\)
\(812\) 6.68815 0.234708
\(813\) 9.02982 0.316690
\(814\) 3.00083 0.105179
\(815\) −15.5025 −0.543027
\(816\) −1.68091 −0.0588436
\(817\) −5.44576 −0.190523
\(818\) −2.70488 −0.0945739
\(819\) 0 0
\(820\) 18.9765 0.662688
\(821\) −16.2439 −0.566915 −0.283457 0.958985i \(-0.591481\pi\)
−0.283457 + 0.958985i \(0.591481\pi\)
\(822\) 35.5197 1.23889
\(823\) −22.0517 −0.768674 −0.384337 0.923193i \(-0.625570\pi\)
−0.384337 + 0.923193i \(0.625570\pi\)
\(824\) −5.98387 −0.208458
\(825\) −15.6594 −0.545189
\(826\) 1.98575 0.0690932
\(827\) 20.1988 0.702380 0.351190 0.936304i \(-0.385777\pi\)
0.351190 + 0.936304i \(0.385777\pi\)
\(828\) −1.46349 −0.0508598
\(829\) −4.60677 −0.160000 −0.0799998 0.996795i \(-0.525492\pi\)
−0.0799998 + 0.996795i \(0.525492\pi\)
\(830\) −8.03507 −0.278901
\(831\) −40.9499 −1.42054
\(832\) 0 0
\(833\) −6.14809 −0.213019
\(834\) 21.4071 0.741269
\(835\) 27.7736 0.961145
\(836\) 4.42494 0.153040
\(837\) 52.4196 1.81188
\(838\) 16.1618 0.558299
\(839\) −17.0875 −0.589925 −0.294962 0.955509i \(-0.595307\pi\)
−0.294962 + 0.955509i \(0.595307\pi\)
\(840\) 2.70569 0.0933552
\(841\) 19.3710 0.667965
\(842\) −16.8012 −0.579006
\(843\) 42.4009 1.46036
\(844\) −27.7630 −0.955642
\(845\) 0 0
\(846\) −0.944358 −0.0324677
\(847\) 8.25099 0.283507
\(848\) 6.18790 0.212493
\(849\) 27.7329 0.951792
\(850\) −2.15613 −0.0739546
\(851\) −4.11646 −0.141110
\(852\) 24.3992 0.835902
\(853\) −42.6641 −1.46079 −0.730396 0.683024i \(-0.760665\pi\)
−0.730396 + 0.683024i \(0.760665\pi\)
\(854\) −9.83299 −0.336478
\(855\) −0.408410 −0.0139673
\(856\) 0.456797 0.0156130
\(857\) 38.0767 1.30068 0.650338 0.759645i \(-0.274627\pi\)
0.650338 + 0.759645i \(0.274627\pi\)
\(858\) 0 0
\(859\) −12.9531 −0.441955 −0.220978 0.975279i \(-0.570925\pi\)
−0.220978 + 0.975279i \(0.570925\pi\)
\(860\) −9.22476 −0.314562
\(861\) 17.8937 0.609816
\(862\) −33.9067 −1.15487
\(863\) 35.8320 1.21974 0.609868 0.792503i \(-0.291222\pi\)
0.609868 + 0.792503i \(0.291222\pi\)
\(864\) 5.38345 0.183149
\(865\) −11.8702 −0.403599
\(866\) 20.3139 0.690294
\(867\) 26.5358 0.901204
\(868\) 9.36367 0.317824
\(869\) 33.6205 1.14050
\(870\) 19.5685 0.663434
\(871\) 0 0
\(872\) −13.2279 −0.447952
\(873\) −1.01844 −0.0344688
\(874\) −6.07002 −0.205322
\(875\) 11.6154 0.392673
\(876\) −3.52435 −0.119077
\(877\) 20.1701 0.681095 0.340548 0.940227i \(-0.389388\pi\)
0.340548 + 0.940227i \(0.389388\pi\)
\(878\) −23.5505 −0.794790
\(879\) 35.3505 1.19234
\(880\) 7.49557 0.252675
\(881\) 13.5626 0.456934 0.228467 0.973552i \(-0.426629\pi\)
0.228467 + 0.973552i \(0.426629\pi\)
\(882\) 1.46475 0.0493207
\(883\) −24.7191 −0.831863 −0.415932 0.909396i \(-0.636544\pi\)
−0.415932 + 0.909396i \(0.636544\pi\)
\(884\) 0 0
\(885\) 5.81001 0.195301
\(886\) 26.0674 0.875752
\(887\) 19.4900 0.654409 0.327204 0.944954i \(-0.393893\pi\)
0.327204 + 0.944954i \(0.393893\pi\)
\(888\) 1.12642 0.0378003
\(889\) −2.64646 −0.0887593
\(890\) 10.7157 0.359190
\(891\) 36.3667 1.21833
\(892\) 7.00880 0.234672
\(893\) −3.91685 −0.131072
\(894\) −38.6217 −1.29170
\(895\) −27.1370 −0.907089
\(896\) 0.961642 0.0321262
\(897\) 0 0
\(898\) 6.84222 0.228328
\(899\) 67.7213 2.25863
\(900\) 0.513687 0.0171229
\(901\) 6.26209 0.208620
\(902\) 49.5709 1.65053
\(903\) −8.69840 −0.289465
\(904\) 11.5596 0.384465
\(905\) −4.76660 −0.158447
\(906\) 40.7986 1.35544
\(907\) 31.2071 1.03621 0.518107 0.855316i \(-0.326637\pi\)
0.518107 + 0.855316i \(0.326637\pi\)
\(908\) 20.8321 0.691337
\(909\) 1.28746 0.0427022
\(910\) 0 0
\(911\) 13.0732 0.433135 0.216567 0.976268i \(-0.430514\pi\)
0.216567 + 0.976268i \(0.430514\pi\)
\(912\) 1.66099 0.0550010
\(913\) −20.9894 −0.694648
\(914\) −24.4501 −0.808739
\(915\) −28.7698 −0.951100
\(916\) 6.04193 0.199631
\(917\) −11.5115 −0.380144
\(918\) 5.44800 0.179811
\(919\) −56.2420 −1.85525 −0.927627 0.373508i \(-0.878155\pi\)
−0.927627 + 0.373508i \(0.878155\pi\)
\(920\) −10.2822 −0.338995
\(921\) 7.81722 0.257586
\(922\) 1.80454 0.0594292
\(923\) 0 0
\(924\) 7.06787 0.232516
\(925\) 1.44488 0.0475074
\(926\) 4.62667 0.152042
\(927\) 1.44272 0.0473851
\(928\) 6.95493 0.228307
\(929\) −29.1740 −0.957169 −0.478585 0.878041i \(-0.658850\pi\)
−0.478585 + 0.878041i \(0.658850\pi\)
\(930\) 27.3967 0.898372
\(931\) 6.07525 0.199108
\(932\) −3.21677 −0.105369
\(933\) −8.21195 −0.268847
\(934\) 10.7837 0.352853
\(935\) 7.58544 0.248070
\(936\) 0 0
\(937\) −41.5894 −1.35867 −0.679333 0.733830i \(-0.737731\pi\)
−0.679333 + 0.733830i \(0.737731\pi\)
\(938\) −5.61896 −0.183466
\(939\) −21.1947 −0.691664
\(940\) −6.63489 −0.216406
\(941\) 28.2701 0.921578 0.460789 0.887510i \(-0.347567\pi\)
0.460789 + 0.887510i \(0.347567\pi\)
\(942\) −16.3034 −0.531195
\(943\) −68.0001 −2.21439
\(944\) 2.06496 0.0672088
\(945\) −8.76942 −0.285269
\(946\) −24.0972 −0.783466
\(947\) −44.0304 −1.43079 −0.715397 0.698718i \(-0.753754\pi\)
−0.715397 + 0.698718i \(0.753754\pi\)
\(948\) 12.6201 0.409883
\(949\) 0 0
\(950\) 2.13058 0.0691252
\(951\) −22.7482 −0.737660
\(952\) 0.973172 0.0315407
\(953\) −52.4548 −1.69918 −0.849589 0.527445i \(-0.823150\pi\)
−0.849589 + 0.527445i \(0.823150\pi\)
\(954\) −1.49191 −0.0483024
\(955\) 13.3057 0.430562
\(956\) 7.65203 0.247484
\(957\) 51.1173 1.65239
\(958\) 0.0903531 0.00291918
\(959\) −20.5643 −0.664057
\(960\) 2.81362 0.0908090
\(961\) 63.8125 2.05847
\(962\) 0 0
\(963\) −0.110135 −0.00354904
\(964\) 9.27475 0.298720
\(965\) 14.2031 0.457213
\(966\) −9.69553 −0.311948
\(967\) 8.38307 0.269581 0.134791 0.990874i \(-0.456964\pi\)
0.134791 + 0.990874i \(0.456964\pi\)
\(968\) 8.58011 0.275775
\(969\) 1.68091 0.0539986
\(970\) −7.15534 −0.229744
\(971\) −20.7629 −0.666312 −0.333156 0.942872i \(-0.608114\pi\)
−0.333156 + 0.942872i \(0.608114\pi\)
\(972\) −2.49936 −0.0801670
\(973\) −12.3938 −0.397327
\(974\) −16.1552 −0.517647
\(975\) 0 0
\(976\) −10.2252 −0.327301
\(977\) −58.2524 −1.86366 −0.931830 0.362895i \(-0.881788\pi\)
−0.931830 + 0.362895i \(0.881788\pi\)
\(978\) −15.2010 −0.486074
\(979\) 27.9917 0.894620
\(980\) 10.2911 0.328736
\(981\) 3.18926 0.101825
\(982\) 4.49239 0.143358
\(983\) −2.89705 −0.0924015 −0.0462007 0.998932i \(-0.514711\pi\)
−0.0462007 + 0.998932i \(0.514711\pi\)
\(984\) 18.6075 0.593184
\(985\) 17.1897 0.547710
\(986\) 7.03832 0.224146
\(987\) −6.25631 −0.199141
\(988\) 0 0
\(989\) 33.0559 1.05112
\(990\) −1.80719 −0.0574363
\(991\) 8.67737 0.275646 0.137823 0.990457i \(-0.455990\pi\)
0.137823 + 0.990457i \(0.455990\pi\)
\(992\) 9.73717 0.309156
\(993\) −11.2624 −0.357402
\(994\) −14.1260 −0.448051
\(995\) −3.06188 −0.0970680
\(996\) −7.87881 −0.249650
\(997\) 37.0456 1.17324 0.586622 0.809861i \(-0.300457\pi\)
0.586622 + 0.809861i \(0.300457\pi\)
\(998\) −10.3831 −0.328670
\(999\) −3.65085 −0.115508
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.bl.1.2 yes 9
13.12 even 2 6422.2.a.bj.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6422.2.a.bj.1.2 9 13.12 even 2
6422.2.a.bl.1.2 yes 9 1.1 even 1 trivial