Properties

Label 8048.2.a.s.1.20
Level $8048$
Weight $2$
Character 8048.1
Self dual yes
Analytic conductor $64.264$
Analytic rank $1$
Dimension $21$
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] = [8048,2,Mod(1,8048)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8048, 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("8048.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.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: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: no (minimal twist has level 2012)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 8048.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41348 q^{3} -0.469286 q^{5} +1.19654 q^{7} +2.82487 q^{9} +O(q^{10})\) \(q+2.41348 q^{3} -0.469286 q^{5} +1.19654 q^{7} +2.82487 q^{9} -4.91448 q^{11} -3.71686 q^{13} -1.13261 q^{15} +1.40011 q^{17} +8.52014 q^{19} +2.88782 q^{21} -7.20246 q^{23} -4.77977 q^{25} -0.422667 q^{27} +2.58126 q^{29} +6.69726 q^{31} -11.8610 q^{33} -0.561519 q^{35} -4.35249 q^{37} -8.97056 q^{39} -5.56215 q^{41} -8.72159 q^{43} -1.32567 q^{45} +3.39690 q^{47} -5.56830 q^{49} +3.37914 q^{51} +7.71607 q^{53} +2.30630 q^{55} +20.5632 q^{57} -9.21327 q^{59} -1.50326 q^{61} +3.38007 q^{63} +1.74427 q^{65} -7.93819 q^{67} -17.3830 q^{69} -1.98538 q^{71} +0.131679 q^{73} -11.5359 q^{75} -5.88036 q^{77} -4.67451 q^{79} -9.49471 q^{81} -5.94828 q^{83} -0.657054 q^{85} +6.22980 q^{87} +10.4631 q^{89} -4.44737 q^{91} +16.1637 q^{93} -3.99838 q^{95} -8.97655 q^{97} -13.8828 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q - 10 q^{3} + 3 q^{5} - 13 q^{7} + 21 q^{9} - 7 q^{11} + 12 q^{13} - 14 q^{15} + q^{17} - 14 q^{19} + 14 q^{21} - 26 q^{23} + 18 q^{25} - 37 q^{27} + 9 q^{29} - 28 q^{31} + 3 q^{33} - 20 q^{35} + 31 q^{37} - 29 q^{39} + 4 q^{41} - 38 q^{43} + 24 q^{45} - 9 q^{47} + 16 q^{49} - 15 q^{51} + 22 q^{53} - 35 q^{55} - q^{57} - 10 q^{59} + 22 q^{61} - 35 q^{63} - 14 q^{65} - 58 q^{67} + 15 q^{69} - 27 q^{71} - 6 q^{73} - 48 q^{75} + 16 q^{77} - 47 q^{79} + 29 q^{81} - 22 q^{83} + 14 q^{85} - 29 q^{87} + q^{89} - 51 q^{91} + 34 q^{93} - 23 q^{95} - 2 q^{97} - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.41348 1.39342 0.696711 0.717352i \(-0.254646\pi\)
0.696711 + 0.717352i \(0.254646\pi\)
\(4\) 0 0
\(5\) −0.469286 −0.209871 −0.104936 0.994479i \(-0.533464\pi\)
−0.104936 + 0.994479i \(0.533464\pi\)
\(6\) 0 0
\(7\) 1.19654 0.452249 0.226125 0.974098i \(-0.427394\pi\)
0.226125 + 0.974098i \(0.427394\pi\)
\(8\) 0 0
\(9\) 2.82487 0.941624
\(10\) 0 0
\(11\) −4.91448 −1.48177 −0.740886 0.671631i \(-0.765594\pi\)
−0.740886 + 0.671631i \(0.765594\pi\)
\(12\) 0 0
\(13\) −3.71686 −1.03087 −0.515436 0.856928i \(-0.672370\pi\)
−0.515436 + 0.856928i \(0.672370\pi\)
\(14\) 0 0
\(15\) −1.13261 −0.292439
\(16\) 0 0
\(17\) 1.40011 0.339578 0.169789 0.985480i \(-0.445691\pi\)
0.169789 + 0.985480i \(0.445691\pi\)
\(18\) 0 0
\(19\) 8.52014 1.95465 0.977327 0.211736i \(-0.0679115\pi\)
0.977327 + 0.211736i \(0.0679115\pi\)
\(20\) 0 0
\(21\) 2.88782 0.630174
\(22\) 0 0
\(23\) −7.20246 −1.50182 −0.750908 0.660407i \(-0.770384\pi\)
−0.750908 + 0.660407i \(0.770384\pi\)
\(24\) 0 0
\(25\) −4.77977 −0.955954
\(26\) 0 0
\(27\) −0.422667 −0.0813424
\(28\) 0 0
\(29\) 2.58126 0.479327 0.239664 0.970856i \(-0.422963\pi\)
0.239664 + 0.970856i \(0.422963\pi\)
\(30\) 0 0
\(31\) 6.69726 1.20286 0.601432 0.798924i \(-0.294597\pi\)
0.601432 + 0.798924i \(0.294597\pi\)
\(32\) 0 0
\(33\) −11.8610 −2.06473
\(34\) 0 0
\(35\) −0.561519 −0.0949140
\(36\) 0 0
\(37\) −4.35249 −0.715545 −0.357773 0.933809i \(-0.616464\pi\)
−0.357773 + 0.933809i \(0.616464\pi\)
\(38\) 0 0
\(39\) −8.97056 −1.43644
\(40\) 0 0
\(41\) −5.56215 −0.868661 −0.434331 0.900753i \(-0.643015\pi\)
−0.434331 + 0.900753i \(0.643015\pi\)
\(42\) 0 0
\(43\) −8.72159 −1.33003 −0.665015 0.746830i \(-0.731575\pi\)
−0.665015 + 0.746830i \(0.731575\pi\)
\(44\) 0 0
\(45\) −1.32567 −0.197620
\(46\) 0 0
\(47\) 3.39690 0.495488 0.247744 0.968825i \(-0.420311\pi\)
0.247744 + 0.968825i \(0.420311\pi\)
\(48\) 0 0
\(49\) −5.56830 −0.795471
\(50\) 0 0
\(51\) 3.37914 0.473175
\(52\) 0 0
\(53\) 7.71607 1.05988 0.529942 0.848034i \(-0.322214\pi\)
0.529942 + 0.848034i \(0.322214\pi\)
\(54\) 0 0
\(55\) 2.30630 0.310981
\(56\) 0 0
\(57\) 20.5632 2.72366
\(58\) 0 0
\(59\) −9.21327 −1.19947 −0.599733 0.800200i \(-0.704727\pi\)
−0.599733 + 0.800200i \(0.704727\pi\)
\(60\) 0 0
\(61\) −1.50326 −0.192473 −0.0962365 0.995358i \(-0.530681\pi\)
−0.0962365 + 0.995358i \(0.530681\pi\)
\(62\) 0 0
\(63\) 3.38007 0.425849
\(64\) 0 0
\(65\) 1.74427 0.216350
\(66\) 0 0
\(67\) −7.93819 −0.969804 −0.484902 0.874569i \(-0.661145\pi\)
−0.484902 + 0.874569i \(0.661145\pi\)
\(68\) 0 0
\(69\) −17.3830 −2.09266
\(70\) 0 0
\(71\) −1.98538 −0.235621 −0.117811 0.993036i \(-0.537588\pi\)
−0.117811 + 0.993036i \(0.537588\pi\)
\(72\) 0 0
\(73\) 0.131679 0.0154119 0.00770594 0.999970i \(-0.497547\pi\)
0.00770594 + 0.999970i \(0.497547\pi\)
\(74\) 0 0
\(75\) −11.5359 −1.33205
\(76\) 0 0
\(77\) −5.88036 −0.670130
\(78\) 0 0
\(79\) −4.67451 −0.525923 −0.262962 0.964806i \(-0.584699\pi\)
−0.262962 + 0.964806i \(0.584699\pi\)
\(80\) 0 0
\(81\) −9.49471 −1.05497
\(82\) 0 0
\(83\) −5.94828 −0.652909 −0.326455 0.945213i \(-0.605854\pi\)
−0.326455 + 0.945213i \(0.605854\pi\)
\(84\) 0 0
\(85\) −0.657054 −0.0712675
\(86\) 0 0
\(87\) 6.22980 0.667905
\(88\) 0 0
\(89\) 10.4631 1.10908 0.554542 0.832156i \(-0.312894\pi\)
0.554542 + 0.832156i \(0.312894\pi\)
\(90\) 0 0
\(91\) −4.44737 −0.466211
\(92\) 0 0
\(93\) 16.1637 1.67610
\(94\) 0 0
\(95\) −3.99838 −0.410225
\(96\) 0 0
\(97\) −8.97655 −0.911430 −0.455715 0.890126i \(-0.650616\pi\)
−0.455715 + 0.890126i \(0.650616\pi\)
\(98\) 0 0
\(99\) −13.8828 −1.39527
\(100\) 0 0
\(101\) 11.2532 1.11973 0.559867 0.828582i \(-0.310852\pi\)
0.559867 + 0.828582i \(0.310852\pi\)
\(102\) 0 0
\(103\) −5.95816 −0.587075 −0.293538 0.955948i \(-0.594833\pi\)
−0.293538 + 0.955948i \(0.594833\pi\)
\(104\) 0 0
\(105\) −1.35521 −0.132255
\(106\) 0 0
\(107\) 2.78724 0.269452 0.134726 0.990883i \(-0.456985\pi\)
0.134726 + 0.990883i \(0.456985\pi\)
\(108\) 0 0
\(109\) 10.4059 0.996708 0.498354 0.866974i \(-0.333938\pi\)
0.498354 + 0.866974i \(0.333938\pi\)
\(110\) 0 0
\(111\) −10.5046 −0.997056
\(112\) 0 0
\(113\) 12.7264 1.19720 0.598602 0.801047i \(-0.295723\pi\)
0.598602 + 0.801047i \(0.295723\pi\)
\(114\) 0 0
\(115\) 3.38001 0.315188
\(116\) 0 0
\(117\) −10.4997 −0.970694
\(118\) 0 0
\(119\) 1.67529 0.153574
\(120\) 0 0
\(121\) 13.1521 1.19565
\(122\) 0 0
\(123\) −13.4241 −1.21041
\(124\) 0 0
\(125\) 4.58951 0.410498
\(126\) 0 0
\(127\) −14.9016 −1.32230 −0.661152 0.750252i \(-0.729932\pi\)
−0.661152 + 0.750252i \(0.729932\pi\)
\(128\) 0 0
\(129\) −21.0494 −1.85329
\(130\) 0 0
\(131\) −1.38001 −0.120572 −0.0602858 0.998181i \(-0.519201\pi\)
−0.0602858 + 0.998181i \(0.519201\pi\)
\(132\) 0 0
\(133\) 10.1947 0.883990
\(134\) 0 0
\(135\) 0.198352 0.0170714
\(136\) 0 0
\(137\) −4.76427 −0.407039 −0.203519 0.979071i \(-0.565238\pi\)
−0.203519 + 0.979071i \(0.565238\pi\)
\(138\) 0 0
\(139\) −12.3225 −1.04519 −0.522593 0.852583i \(-0.675035\pi\)
−0.522593 + 0.852583i \(0.675035\pi\)
\(140\) 0 0
\(141\) 8.19833 0.690424
\(142\) 0 0
\(143\) 18.2664 1.52752
\(144\) 0 0
\(145\) −1.21135 −0.100597
\(146\) 0 0
\(147\) −13.4390 −1.10843
\(148\) 0 0
\(149\) −4.14290 −0.339400 −0.169700 0.985496i \(-0.554280\pi\)
−0.169700 + 0.985496i \(0.554280\pi\)
\(150\) 0 0
\(151\) −13.5882 −1.10579 −0.552895 0.833251i \(-0.686477\pi\)
−0.552895 + 0.833251i \(0.686477\pi\)
\(152\) 0 0
\(153\) 3.95514 0.319754
\(154\) 0 0
\(155\) −3.14293 −0.252446
\(156\) 0 0
\(157\) −3.32518 −0.265378 −0.132689 0.991158i \(-0.542361\pi\)
−0.132689 + 0.991158i \(0.542361\pi\)
\(158\) 0 0
\(159\) 18.6226 1.47686
\(160\) 0 0
\(161\) −8.61802 −0.679195
\(162\) 0 0
\(163\) −14.5563 −1.14014 −0.570069 0.821597i \(-0.693084\pi\)
−0.570069 + 0.821597i \(0.693084\pi\)
\(164\) 0 0
\(165\) 5.56620 0.433328
\(166\) 0 0
\(167\) −20.1284 −1.55758 −0.778790 0.627284i \(-0.784166\pi\)
−0.778790 + 0.627284i \(0.784166\pi\)
\(168\) 0 0
\(169\) 0.815066 0.0626974
\(170\) 0 0
\(171\) 24.0683 1.84055
\(172\) 0 0
\(173\) 13.2292 1.00580 0.502898 0.864346i \(-0.332267\pi\)
0.502898 + 0.864346i \(0.332267\pi\)
\(174\) 0 0
\(175\) −5.71918 −0.432329
\(176\) 0 0
\(177\) −22.2360 −1.67136
\(178\) 0 0
\(179\) 21.9923 1.64378 0.821889 0.569647i \(-0.192920\pi\)
0.821889 + 0.569647i \(0.192920\pi\)
\(180\) 0 0
\(181\) −8.06078 −0.599153 −0.299576 0.954072i \(-0.596845\pi\)
−0.299576 + 0.954072i \(0.596845\pi\)
\(182\) 0 0
\(183\) −3.62809 −0.268196
\(184\) 0 0
\(185\) 2.04256 0.150172
\(186\) 0 0
\(187\) −6.88083 −0.503176
\(188\) 0 0
\(189\) −0.505738 −0.0367870
\(190\) 0 0
\(191\) 0.580381 0.0419949 0.0209975 0.999780i \(-0.493316\pi\)
0.0209975 + 0.999780i \(0.493316\pi\)
\(192\) 0 0
\(193\) −3.32214 −0.239133 −0.119567 0.992826i \(-0.538151\pi\)
−0.119567 + 0.992826i \(0.538151\pi\)
\(194\) 0 0
\(195\) 4.20976 0.301467
\(196\) 0 0
\(197\) −18.2811 −1.30247 −0.651236 0.758875i \(-0.725749\pi\)
−0.651236 + 0.758875i \(0.725749\pi\)
\(198\) 0 0
\(199\) 9.22831 0.654178 0.327089 0.944994i \(-0.393932\pi\)
0.327089 + 0.944994i \(0.393932\pi\)
\(200\) 0 0
\(201\) −19.1586 −1.35135
\(202\) 0 0
\(203\) 3.08857 0.216775
\(204\) 0 0
\(205\) 2.61024 0.182307
\(206\) 0 0
\(207\) −20.3460 −1.41415
\(208\) 0 0
\(209\) −41.8721 −2.89635
\(210\) 0 0
\(211\) −24.5409 −1.68947 −0.844734 0.535187i \(-0.820241\pi\)
−0.844734 + 0.535187i \(0.820241\pi\)
\(212\) 0 0
\(213\) −4.79167 −0.328320
\(214\) 0 0
\(215\) 4.09292 0.279135
\(216\) 0 0
\(217\) 8.01353 0.543994
\(218\) 0 0
\(219\) 0.317805 0.0214752
\(220\) 0 0
\(221\) −5.20403 −0.350061
\(222\) 0 0
\(223\) −20.6366 −1.38193 −0.690964 0.722889i \(-0.742814\pi\)
−0.690964 + 0.722889i \(0.742814\pi\)
\(224\) 0 0
\(225\) −13.5022 −0.900149
\(226\) 0 0
\(227\) −6.66140 −0.442132 −0.221066 0.975259i \(-0.570954\pi\)
−0.221066 + 0.975259i \(0.570954\pi\)
\(228\) 0 0
\(229\) 7.59570 0.501938 0.250969 0.967995i \(-0.419251\pi\)
0.250969 + 0.967995i \(0.419251\pi\)
\(230\) 0 0
\(231\) −14.1921 −0.933773
\(232\) 0 0
\(233\) 5.08213 0.332942 0.166471 0.986046i \(-0.446763\pi\)
0.166471 + 0.986046i \(0.446763\pi\)
\(234\) 0 0
\(235\) −1.59412 −0.103989
\(236\) 0 0
\(237\) −11.2818 −0.732833
\(238\) 0 0
\(239\) 18.2123 1.17805 0.589026 0.808114i \(-0.299511\pi\)
0.589026 + 0.808114i \(0.299511\pi\)
\(240\) 0 0
\(241\) −23.9353 −1.54181 −0.770904 0.636951i \(-0.780195\pi\)
−0.770904 + 0.636951i \(0.780195\pi\)
\(242\) 0 0
\(243\) −21.6473 −1.38867
\(244\) 0 0
\(245\) 2.61312 0.166946
\(246\) 0 0
\(247\) −31.6682 −2.01500
\(248\) 0 0
\(249\) −14.3560 −0.909778
\(250\) 0 0
\(251\) −0.886610 −0.0559623 −0.0279812 0.999608i \(-0.508908\pi\)
−0.0279812 + 0.999608i \(0.508908\pi\)
\(252\) 0 0
\(253\) 35.3963 2.22535
\(254\) 0 0
\(255\) −1.58579 −0.0993057
\(256\) 0 0
\(257\) 8.39773 0.523836 0.261918 0.965090i \(-0.415645\pi\)
0.261918 + 0.965090i \(0.415645\pi\)
\(258\) 0 0
\(259\) −5.20792 −0.323605
\(260\) 0 0
\(261\) 7.29172 0.451346
\(262\) 0 0
\(263\) −6.45866 −0.398258 −0.199129 0.979973i \(-0.563811\pi\)
−0.199129 + 0.979973i \(0.563811\pi\)
\(264\) 0 0
\(265\) −3.62104 −0.222439
\(266\) 0 0
\(267\) 25.2524 1.54542
\(268\) 0 0
\(269\) 27.7114 1.68960 0.844798 0.535085i \(-0.179721\pi\)
0.844798 + 0.535085i \(0.179721\pi\)
\(270\) 0 0
\(271\) 8.26899 0.502305 0.251153 0.967947i \(-0.419190\pi\)
0.251153 + 0.967947i \(0.419190\pi\)
\(272\) 0 0
\(273\) −10.7336 −0.649628
\(274\) 0 0
\(275\) 23.4901 1.41651
\(276\) 0 0
\(277\) 17.0883 1.02673 0.513367 0.858169i \(-0.328398\pi\)
0.513367 + 0.858169i \(0.328398\pi\)
\(278\) 0 0
\(279\) 18.9189 1.13265
\(280\) 0 0
\(281\) −5.50366 −0.328321 −0.164160 0.986434i \(-0.552491\pi\)
−0.164160 + 0.986434i \(0.552491\pi\)
\(282\) 0 0
\(283\) 12.1214 0.720541 0.360270 0.932848i \(-0.382684\pi\)
0.360270 + 0.932848i \(0.382684\pi\)
\(284\) 0 0
\(285\) −9.65001 −0.571617
\(286\) 0 0
\(287\) −6.65532 −0.392851
\(288\) 0 0
\(289\) −15.0397 −0.884687
\(290\) 0 0
\(291\) −21.6647 −1.27001
\(292\) 0 0
\(293\) 32.1420 1.87775 0.938877 0.344252i \(-0.111868\pi\)
0.938877 + 0.344252i \(0.111868\pi\)
\(294\) 0 0
\(295\) 4.32366 0.251733
\(296\) 0 0
\(297\) 2.07719 0.120531
\(298\) 0 0
\(299\) 26.7705 1.54818
\(300\) 0 0
\(301\) −10.4357 −0.601505
\(302\) 0 0
\(303\) 27.1593 1.56026
\(304\) 0 0
\(305\) 0.705460 0.0403945
\(306\) 0 0
\(307\) 16.8336 0.960746 0.480373 0.877064i \(-0.340501\pi\)
0.480373 + 0.877064i \(0.340501\pi\)
\(308\) 0 0
\(309\) −14.3799 −0.818044
\(310\) 0 0
\(311\) 22.1838 1.25793 0.628963 0.777435i \(-0.283480\pi\)
0.628963 + 0.777435i \(0.283480\pi\)
\(312\) 0 0
\(313\) −21.2574 −1.20154 −0.600769 0.799423i \(-0.705139\pi\)
−0.600769 + 0.799423i \(0.705139\pi\)
\(314\) 0 0
\(315\) −1.58622 −0.0893733
\(316\) 0 0
\(317\) −2.77219 −0.155702 −0.0778509 0.996965i \(-0.524806\pi\)
−0.0778509 + 0.996965i \(0.524806\pi\)
\(318\) 0 0
\(319\) −12.6855 −0.710253
\(320\) 0 0
\(321\) 6.72694 0.375461
\(322\) 0 0
\(323\) 11.9292 0.663757
\(324\) 0 0
\(325\) 17.7657 0.985466
\(326\) 0 0
\(327\) 25.1145 1.38883
\(328\) 0 0
\(329\) 4.06452 0.224084
\(330\) 0 0
\(331\) 13.6587 0.750750 0.375375 0.926873i \(-0.377514\pi\)
0.375375 + 0.926873i \(0.377514\pi\)
\(332\) 0 0
\(333\) −12.2952 −0.673775
\(334\) 0 0
\(335\) 3.72528 0.203534
\(336\) 0 0
\(337\) −17.3896 −0.947274 −0.473637 0.880720i \(-0.657059\pi\)
−0.473637 + 0.880720i \(0.657059\pi\)
\(338\) 0 0
\(339\) 30.7150 1.66821
\(340\) 0 0
\(341\) −32.9136 −1.78237
\(342\) 0 0
\(343\) −15.0384 −0.812000
\(344\) 0 0
\(345\) 8.15758 0.439189
\(346\) 0 0
\(347\) −26.1589 −1.40428 −0.702141 0.712038i \(-0.747772\pi\)
−0.702141 + 0.712038i \(0.747772\pi\)
\(348\) 0 0
\(349\) 19.9003 1.06524 0.532619 0.846355i \(-0.321208\pi\)
0.532619 + 0.846355i \(0.321208\pi\)
\(350\) 0 0
\(351\) 1.57100 0.0838536
\(352\) 0 0
\(353\) 33.8376 1.80099 0.900496 0.434864i \(-0.143204\pi\)
0.900496 + 0.434864i \(0.143204\pi\)
\(354\) 0 0
\(355\) 0.931711 0.0494501
\(356\) 0 0
\(357\) 4.04328 0.213993
\(358\) 0 0
\(359\) −6.84125 −0.361067 −0.180534 0.983569i \(-0.557782\pi\)
−0.180534 + 0.983569i \(0.557782\pi\)
\(360\) 0 0
\(361\) 53.5928 2.82067
\(362\) 0 0
\(363\) 31.7423 1.66604
\(364\) 0 0
\(365\) −0.0617952 −0.00323451
\(366\) 0 0
\(367\) 3.87311 0.202174 0.101087 0.994878i \(-0.467768\pi\)
0.101087 + 0.994878i \(0.467768\pi\)
\(368\) 0 0
\(369\) −15.7124 −0.817952
\(370\) 0 0
\(371\) 9.23257 0.479331
\(372\) 0 0
\(373\) 2.75357 0.142575 0.0712873 0.997456i \(-0.477289\pi\)
0.0712873 + 0.997456i \(0.477289\pi\)
\(374\) 0 0
\(375\) 11.0767 0.571997
\(376\) 0 0
\(377\) −9.59417 −0.494125
\(378\) 0 0
\(379\) 5.57451 0.286344 0.143172 0.989698i \(-0.454270\pi\)
0.143172 + 0.989698i \(0.454270\pi\)
\(380\) 0 0
\(381\) −35.9647 −1.84253
\(382\) 0 0
\(383\) −29.2495 −1.49458 −0.747291 0.664497i \(-0.768646\pi\)
−0.747291 + 0.664497i \(0.768646\pi\)
\(384\) 0 0
\(385\) 2.75957 0.140641
\(386\) 0 0
\(387\) −24.6374 −1.25239
\(388\) 0 0
\(389\) 9.17508 0.465195 0.232598 0.972573i \(-0.425277\pi\)
0.232598 + 0.972573i \(0.425277\pi\)
\(390\) 0 0
\(391\) −10.0843 −0.509983
\(392\) 0 0
\(393\) −3.33061 −0.168007
\(394\) 0 0
\(395\) 2.19368 0.110376
\(396\) 0 0
\(397\) −33.2492 −1.66873 −0.834365 0.551212i \(-0.814166\pi\)
−0.834365 + 0.551212i \(0.814166\pi\)
\(398\) 0 0
\(399\) 24.6046 1.23177
\(400\) 0 0
\(401\) −28.6476 −1.43059 −0.715297 0.698820i \(-0.753709\pi\)
−0.715297 + 0.698820i \(0.753709\pi\)
\(402\) 0 0
\(403\) −24.8928 −1.24000
\(404\) 0 0
\(405\) 4.45574 0.221407
\(406\) 0 0
\(407\) 21.3902 1.06027
\(408\) 0 0
\(409\) 10.5040 0.519389 0.259695 0.965691i \(-0.416378\pi\)
0.259695 + 0.965691i \(0.416378\pi\)
\(410\) 0 0
\(411\) −11.4985 −0.567177
\(412\) 0 0
\(413\) −11.0240 −0.542457
\(414\) 0 0
\(415\) 2.79145 0.137027
\(416\) 0 0
\(417\) −29.7402 −1.45638
\(418\) 0 0
\(419\) 14.6684 0.716597 0.358299 0.933607i \(-0.383357\pi\)
0.358299 + 0.933607i \(0.383357\pi\)
\(420\) 0 0
\(421\) −26.4857 −1.29083 −0.645416 0.763831i \(-0.723316\pi\)
−0.645416 + 0.763831i \(0.723316\pi\)
\(422\) 0 0
\(423\) 9.59580 0.466564
\(424\) 0 0
\(425\) −6.69223 −0.324621
\(426\) 0 0
\(427\) −1.79871 −0.0870457
\(428\) 0 0
\(429\) 44.0857 2.12848
\(430\) 0 0
\(431\) 26.8093 1.29136 0.645679 0.763609i \(-0.276575\pi\)
0.645679 + 0.763609i \(0.276575\pi\)
\(432\) 0 0
\(433\) −16.2785 −0.782296 −0.391148 0.920328i \(-0.627922\pi\)
−0.391148 + 0.920328i \(0.627922\pi\)
\(434\) 0 0
\(435\) −2.92356 −0.140174
\(436\) 0 0
\(437\) −61.3659 −2.93553
\(438\) 0 0
\(439\) 1.15846 0.0552904 0.0276452 0.999618i \(-0.491199\pi\)
0.0276452 + 0.999618i \(0.491199\pi\)
\(440\) 0 0
\(441\) −15.7297 −0.749034
\(442\) 0 0
\(443\) −11.2823 −0.536038 −0.268019 0.963414i \(-0.586369\pi\)
−0.268019 + 0.963414i \(0.586369\pi\)
\(444\) 0 0
\(445\) −4.91017 −0.232765
\(446\) 0 0
\(447\) −9.99880 −0.472927
\(448\) 0 0
\(449\) 5.72243 0.270058 0.135029 0.990842i \(-0.456887\pi\)
0.135029 + 0.990842i \(0.456887\pi\)
\(450\) 0 0
\(451\) 27.3351 1.28716
\(452\) 0 0
\(453\) −32.7947 −1.54083
\(454\) 0 0
\(455\) 2.08709 0.0978442
\(456\) 0 0
\(457\) −6.47866 −0.303059 −0.151529 0.988453i \(-0.548420\pi\)
−0.151529 + 0.988453i \(0.548420\pi\)
\(458\) 0 0
\(459\) −0.591783 −0.0276221
\(460\) 0 0
\(461\) 9.28966 0.432662 0.216331 0.976320i \(-0.430591\pi\)
0.216331 + 0.976320i \(0.430591\pi\)
\(462\) 0 0
\(463\) 22.4220 1.04204 0.521019 0.853545i \(-0.325552\pi\)
0.521019 + 0.853545i \(0.325552\pi\)
\(464\) 0 0
\(465\) −7.58539 −0.351764
\(466\) 0 0
\(467\) 5.00761 0.231724 0.115862 0.993265i \(-0.463037\pi\)
0.115862 + 0.993265i \(0.463037\pi\)
\(468\) 0 0
\(469\) −9.49835 −0.438593
\(470\) 0 0
\(471\) −8.02525 −0.369784
\(472\) 0 0
\(473\) 42.8621 1.97080
\(474\) 0 0
\(475\) −40.7243 −1.86856
\(476\) 0 0
\(477\) 21.7969 0.998012
\(478\) 0 0
\(479\) −19.9556 −0.911793 −0.455896 0.890033i \(-0.650681\pi\)
−0.455896 + 0.890033i \(0.650681\pi\)
\(480\) 0 0
\(481\) 16.1776 0.737636
\(482\) 0 0
\(483\) −20.7994 −0.946405
\(484\) 0 0
\(485\) 4.21257 0.191283
\(486\) 0 0
\(487\) 34.3497 1.55653 0.778267 0.627934i \(-0.216099\pi\)
0.778267 + 0.627934i \(0.216099\pi\)
\(488\) 0 0
\(489\) −35.1313 −1.58869
\(490\) 0 0
\(491\) −12.5459 −0.566187 −0.283094 0.959092i \(-0.591361\pi\)
−0.283094 + 0.959092i \(0.591361\pi\)
\(492\) 0 0
\(493\) 3.61405 0.162769
\(494\) 0 0
\(495\) 6.51499 0.292827
\(496\) 0 0
\(497\) −2.37558 −0.106559
\(498\) 0 0
\(499\) 9.00374 0.403063 0.201531 0.979482i \(-0.435408\pi\)
0.201531 + 0.979482i \(0.435408\pi\)
\(500\) 0 0
\(501\) −48.5794 −2.17037
\(502\) 0 0
\(503\) 1.00000 0.0445878
\(504\) 0 0
\(505\) −5.28097 −0.235000
\(506\) 0 0
\(507\) 1.96714 0.0873639
\(508\) 0 0
\(509\) 30.3723 1.34623 0.673114 0.739539i \(-0.264957\pi\)
0.673114 + 0.739539i \(0.264957\pi\)
\(510\) 0 0
\(511\) 0.157559 0.00697001
\(512\) 0 0
\(513\) −3.60119 −0.158996
\(514\) 0 0
\(515\) 2.79608 0.123210
\(516\) 0 0
\(517\) −16.6940 −0.734201
\(518\) 0 0
\(519\) 31.9284 1.40150
\(520\) 0 0
\(521\) 32.2835 1.41437 0.707183 0.707030i \(-0.249966\pi\)
0.707183 + 0.707030i \(0.249966\pi\)
\(522\) 0 0
\(523\) −6.59901 −0.288554 −0.144277 0.989537i \(-0.546086\pi\)
−0.144277 + 0.989537i \(0.546086\pi\)
\(524\) 0 0
\(525\) −13.8031 −0.602417
\(526\) 0 0
\(527\) 9.37693 0.408466
\(528\) 0 0
\(529\) 28.8754 1.25545
\(530\) 0 0
\(531\) −26.0263 −1.12945
\(532\) 0 0
\(533\) 20.6737 0.895479
\(534\) 0 0
\(535\) −1.30801 −0.0565503
\(536\) 0 0
\(537\) 53.0778 2.29048
\(538\) 0 0
\(539\) 27.3653 1.17871
\(540\) 0 0
\(541\) −27.7942 −1.19497 −0.597483 0.801882i \(-0.703832\pi\)
−0.597483 + 0.801882i \(0.703832\pi\)
\(542\) 0 0
\(543\) −19.4545 −0.834872
\(544\) 0 0
\(545\) −4.88336 −0.209180
\(546\) 0 0
\(547\) −27.1074 −1.15903 −0.579515 0.814961i \(-0.696758\pi\)
−0.579515 + 0.814961i \(0.696758\pi\)
\(548\) 0 0
\(549\) −4.24652 −0.181237
\(550\) 0 0
\(551\) 21.9927 0.936919
\(552\) 0 0
\(553\) −5.59323 −0.237848
\(554\) 0 0
\(555\) 4.92968 0.209253
\(556\) 0 0
\(557\) −18.6307 −0.789408 −0.394704 0.918808i \(-0.629153\pi\)
−0.394704 + 0.918808i \(0.629153\pi\)
\(558\) 0 0
\(559\) 32.4169 1.37109
\(560\) 0 0
\(561\) −16.6067 −0.701137
\(562\) 0 0
\(563\) −21.4672 −0.904734 −0.452367 0.891832i \(-0.649420\pi\)
−0.452367 + 0.891832i \(0.649420\pi\)
\(564\) 0 0
\(565\) −5.97234 −0.251258
\(566\) 0 0
\(567\) −11.3608 −0.477108
\(568\) 0 0
\(569\) 9.05449 0.379584 0.189792 0.981824i \(-0.439219\pi\)
0.189792 + 0.981824i \(0.439219\pi\)
\(570\) 0 0
\(571\) 42.2430 1.76781 0.883907 0.467663i \(-0.154904\pi\)
0.883907 + 0.467663i \(0.154904\pi\)
\(572\) 0 0
\(573\) 1.40074 0.0585166
\(574\) 0 0
\(575\) 34.4261 1.43567
\(576\) 0 0
\(577\) 32.2208 1.34137 0.670686 0.741742i \(-0.266000\pi\)
0.670686 + 0.741742i \(0.266000\pi\)
\(578\) 0 0
\(579\) −8.01792 −0.333213
\(580\) 0 0
\(581\) −7.11735 −0.295277
\(582\) 0 0
\(583\) −37.9205 −1.57051
\(584\) 0 0
\(585\) 4.92734 0.203721
\(586\) 0 0
\(587\) 41.8527 1.72745 0.863723 0.503967i \(-0.168127\pi\)
0.863723 + 0.503967i \(0.168127\pi\)
\(588\) 0 0
\(589\) 57.0616 2.35118
\(590\) 0 0
\(591\) −44.1209 −1.81489
\(592\) 0 0
\(593\) 28.7022 1.17866 0.589329 0.807893i \(-0.299392\pi\)
0.589329 + 0.807893i \(0.299392\pi\)
\(594\) 0 0
\(595\) −0.786191 −0.0322307
\(596\) 0 0
\(597\) 22.2723 0.911546
\(598\) 0 0
\(599\) −0.109129 −0.00445889 −0.00222944 0.999998i \(-0.500710\pi\)
−0.00222944 + 0.999998i \(0.500710\pi\)
\(600\) 0 0
\(601\) −13.4723 −0.549546 −0.274773 0.961509i \(-0.588603\pi\)
−0.274773 + 0.961509i \(0.588603\pi\)
\(602\) 0 0
\(603\) −22.4244 −0.913191
\(604\) 0 0
\(605\) −6.17210 −0.250932
\(606\) 0 0
\(607\) −7.37863 −0.299489 −0.149745 0.988725i \(-0.547845\pi\)
−0.149745 + 0.988725i \(0.547845\pi\)
\(608\) 0 0
\(609\) 7.45420 0.302059
\(610\) 0 0
\(611\) −12.6258 −0.510785
\(612\) 0 0
\(613\) −28.2749 −1.14201 −0.571007 0.820946i \(-0.693447\pi\)
−0.571007 + 0.820946i \(0.693447\pi\)
\(614\) 0 0
\(615\) 6.29975 0.254030
\(616\) 0 0
\(617\) −24.9131 −1.00296 −0.501481 0.865168i \(-0.667211\pi\)
−0.501481 + 0.865168i \(0.667211\pi\)
\(618\) 0 0
\(619\) −46.2744 −1.85992 −0.929962 0.367655i \(-0.880161\pi\)
−0.929962 + 0.367655i \(0.880161\pi\)
\(620\) 0 0
\(621\) 3.04424 0.122161
\(622\) 0 0
\(623\) 12.5195 0.501582
\(624\) 0 0
\(625\) 21.7451 0.869802
\(626\) 0 0
\(627\) −101.057 −4.03584
\(628\) 0 0
\(629\) −6.09399 −0.242983
\(630\) 0 0
\(631\) 32.8627 1.30825 0.654123 0.756389i \(-0.273038\pi\)
0.654123 + 0.756389i \(0.273038\pi\)
\(632\) 0 0
\(633\) −59.2290 −2.35414
\(634\) 0 0
\(635\) 6.99311 0.277513
\(636\) 0 0
\(637\) 20.6966 0.820029
\(638\) 0 0
\(639\) −5.60844 −0.221867
\(640\) 0 0
\(641\) 21.8874 0.864501 0.432250 0.901754i \(-0.357720\pi\)
0.432250 + 0.901754i \(0.357720\pi\)
\(642\) 0 0
\(643\) −21.3591 −0.842322 −0.421161 0.906986i \(-0.638377\pi\)
−0.421161 + 0.906986i \(0.638377\pi\)
\(644\) 0 0
\(645\) 9.87817 0.388952
\(646\) 0 0
\(647\) −38.8909 −1.52896 −0.764480 0.644648i \(-0.777004\pi\)
−0.764480 + 0.644648i \(0.777004\pi\)
\(648\) 0 0
\(649\) 45.2785 1.77733
\(650\) 0 0
\(651\) 19.3405 0.758013
\(652\) 0 0
\(653\) −0.851542 −0.0333234 −0.0166617 0.999861i \(-0.505304\pi\)
−0.0166617 + 0.999861i \(0.505304\pi\)
\(654\) 0 0
\(655\) 0.647617 0.0253045
\(656\) 0 0
\(657\) 0.371977 0.0145122
\(658\) 0 0
\(659\) 23.5657 0.917991 0.458996 0.888439i \(-0.348209\pi\)
0.458996 + 0.888439i \(0.348209\pi\)
\(660\) 0 0
\(661\) 43.6262 1.69686 0.848431 0.529306i \(-0.177548\pi\)
0.848431 + 0.529306i \(0.177548\pi\)
\(662\) 0 0
\(663\) −12.5598 −0.487783
\(664\) 0 0
\(665\) −4.78422 −0.185524
\(666\) 0 0
\(667\) −18.5914 −0.719861
\(668\) 0 0
\(669\) −49.8060 −1.92561
\(670\) 0 0
\(671\) 7.38775 0.285201
\(672\) 0 0
\(673\) −13.0843 −0.504364 −0.252182 0.967680i \(-0.581148\pi\)
−0.252182 + 0.967680i \(0.581148\pi\)
\(674\) 0 0
\(675\) 2.02025 0.0777596
\(676\) 0 0
\(677\) −9.40790 −0.361575 −0.180788 0.983522i \(-0.557865\pi\)
−0.180788 + 0.983522i \(0.557865\pi\)
\(678\) 0 0
\(679\) −10.7408 −0.412194
\(680\) 0 0
\(681\) −16.0771 −0.616077
\(682\) 0 0
\(683\) −38.6814 −1.48010 −0.740051 0.672550i \(-0.765199\pi\)
−0.740051 + 0.672550i \(0.765199\pi\)
\(684\) 0 0
\(685\) 2.23581 0.0854257
\(686\) 0 0
\(687\) 18.3320 0.699411
\(688\) 0 0
\(689\) −28.6796 −1.09260
\(690\) 0 0
\(691\) 6.87534 0.261550 0.130775 0.991412i \(-0.458253\pi\)
0.130775 + 0.991412i \(0.458253\pi\)
\(692\) 0 0
\(693\) −16.6113 −0.631010
\(694\) 0 0
\(695\) 5.78280 0.219354
\(696\) 0 0
\(697\) −7.78764 −0.294978
\(698\) 0 0
\(699\) 12.2656 0.463928
\(700\) 0 0
\(701\) 42.7236 1.61365 0.806825 0.590791i \(-0.201184\pi\)
0.806825 + 0.590791i \(0.201184\pi\)
\(702\) 0 0
\(703\) −37.0838 −1.39864
\(704\) 0 0
\(705\) −3.84736 −0.144900
\(706\) 0 0
\(707\) 13.4649 0.506399
\(708\) 0 0
\(709\) 27.2041 1.02167 0.510836 0.859678i \(-0.329336\pi\)
0.510836 + 0.859678i \(0.329336\pi\)
\(710\) 0 0
\(711\) −13.2049 −0.495222
\(712\) 0 0
\(713\) −48.2367 −1.80648
\(714\) 0 0
\(715\) −8.57219 −0.320582
\(716\) 0 0
\(717\) 43.9549 1.64152
\(718\) 0 0
\(719\) −37.5483 −1.40031 −0.700157 0.713989i \(-0.746887\pi\)
−0.700157 + 0.713989i \(0.746887\pi\)
\(720\) 0 0
\(721\) −7.12917 −0.265504
\(722\) 0 0
\(723\) −57.7673 −2.14839
\(724\) 0 0
\(725\) −12.3378 −0.458215
\(726\) 0 0
\(727\) −9.60879 −0.356370 −0.178185 0.983997i \(-0.557023\pi\)
−0.178185 + 0.983997i \(0.557023\pi\)
\(728\) 0 0
\(729\) −23.7611 −0.880039
\(730\) 0 0
\(731\) −12.2112 −0.451648
\(732\) 0 0
\(733\) 11.2399 0.415156 0.207578 0.978218i \(-0.433442\pi\)
0.207578 + 0.978218i \(0.433442\pi\)
\(734\) 0 0
\(735\) 6.30671 0.232627
\(736\) 0 0
\(737\) 39.0121 1.43703
\(738\) 0 0
\(739\) 37.3498 1.37393 0.686967 0.726688i \(-0.258942\pi\)
0.686967 + 0.726688i \(0.258942\pi\)
\(740\) 0 0
\(741\) −76.4304 −2.80774
\(742\) 0 0
\(743\) 9.14462 0.335483 0.167742 0.985831i \(-0.446353\pi\)
0.167742 + 0.985831i \(0.446353\pi\)
\(744\) 0 0
\(745\) 1.94421 0.0712302
\(746\) 0 0
\(747\) −16.8031 −0.614795
\(748\) 0 0
\(749\) 3.33504 0.121860
\(750\) 0 0
\(751\) −20.5312 −0.749195 −0.374598 0.927187i \(-0.622219\pi\)
−0.374598 + 0.927187i \(0.622219\pi\)
\(752\) 0 0
\(753\) −2.13981 −0.0779791
\(754\) 0 0
\(755\) 6.37674 0.232073
\(756\) 0 0
\(757\) 44.1360 1.60415 0.802075 0.597223i \(-0.203729\pi\)
0.802075 + 0.597223i \(0.203729\pi\)
\(758\) 0 0
\(759\) 85.4282 3.10085
\(760\) 0 0
\(761\) −36.4882 −1.32270 −0.661348 0.750079i \(-0.730015\pi\)
−0.661348 + 0.750079i \(0.730015\pi\)
\(762\) 0 0
\(763\) 12.4511 0.450760
\(764\) 0 0
\(765\) −1.85609 −0.0671072
\(766\) 0 0
\(767\) 34.2445 1.23650
\(768\) 0 0
\(769\) −30.0855 −1.08491 −0.542456 0.840084i \(-0.682505\pi\)
−0.542456 + 0.840084i \(0.682505\pi\)
\(770\) 0 0
\(771\) 20.2677 0.729925
\(772\) 0 0
\(773\) −38.0194 −1.36746 −0.683732 0.729734i \(-0.739644\pi\)
−0.683732 + 0.729734i \(0.739644\pi\)
\(774\) 0 0
\(775\) −32.0114 −1.14988
\(776\) 0 0
\(777\) −12.5692 −0.450918
\(778\) 0 0
\(779\) −47.3903 −1.69793
\(780\) 0 0
\(781\) 9.75711 0.349137
\(782\) 0 0
\(783\) −1.09101 −0.0389896
\(784\) 0 0
\(785\) 1.56046 0.0556953
\(786\) 0 0
\(787\) −10.8636 −0.387247 −0.193624 0.981076i \(-0.562024\pi\)
−0.193624 + 0.981076i \(0.562024\pi\)
\(788\) 0 0
\(789\) −15.5878 −0.554941
\(790\) 0 0
\(791\) 15.2277 0.541434
\(792\) 0 0
\(793\) 5.58742 0.198415
\(794\) 0 0
\(795\) −8.73931 −0.309951
\(796\) 0 0
\(797\) 37.2809 1.32056 0.660279 0.751020i \(-0.270438\pi\)
0.660279 + 0.751020i \(0.270438\pi\)
\(798\) 0 0
\(799\) 4.75604 0.168257
\(800\) 0 0
\(801\) 29.5568 1.04434
\(802\) 0 0
\(803\) −0.647134 −0.0228369
\(804\) 0 0
\(805\) 4.04431 0.142543
\(806\) 0 0
\(807\) 66.8809 2.35432
\(808\) 0 0
\(809\) 0.187390 0.00658828 0.00329414 0.999995i \(-0.498951\pi\)
0.00329414 + 0.999995i \(0.498951\pi\)
\(810\) 0 0
\(811\) 23.9796 0.842036 0.421018 0.907052i \(-0.361673\pi\)
0.421018 + 0.907052i \(0.361673\pi\)
\(812\) 0 0
\(813\) 19.9570 0.699923
\(814\) 0 0
\(815\) 6.83107 0.239282
\(816\) 0 0
\(817\) −74.3091 −2.59975
\(818\) 0 0
\(819\) −12.5632 −0.438995
\(820\) 0 0
\(821\) −38.9517 −1.35942 −0.679711 0.733480i \(-0.737895\pi\)
−0.679711 + 0.733480i \(0.737895\pi\)
\(822\) 0 0
\(823\) 29.6913 1.03497 0.517486 0.855691i \(-0.326868\pi\)
0.517486 + 0.855691i \(0.326868\pi\)
\(824\) 0 0
\(825\) 56.6928 1.97379
\(826\) 0 0
\(827\) −48.3192 −1.68022 −0.840111 0.542414i \(-0.817510\pi\)
−0.840111 + 0.542414i \(0.817510\pi\)
\(828\) 0 0
\(829\) 30.5168 1.05989 0.529946 0.848032i \(-0.322212\pi\)
0.529946 + 0.848032i \(0.322212\pi\)
\(830\) 0 0
\(831\) 41.2421 1.43067
\(832\) 0 0
\(833\) −7.79625 −0.270124
\(834\) 0 0
\(835\) 9.44597 0.326891
\(836\) 0 0
\(837\) −2.83071 −0.0978438
\(838\) 0 0
\(839\) −16.5394 −0.571004 −0.285502 0.958378i \(-0.592160\pi\)
−0.285502 + 0.958378i \(0.592160\pi\)
\(840\) 0 0
\(841\) −22.3371 −0.770245
\(842\) 0 0
\(843\) −13.2829 −0.457489
\(844\) 0 0
\(845\) −0.382499 −0.0131584
\(846\) 0 0
\(847\) 15.7370 0.540730
\(848\) 0 0
\(849\) 29.2546 1.00402
\(850\) 0 0
\(851\) 31.3486 1.07462
\(852\) 0 0
\(853\) 22.1321 0.757788 0.378894 0.925440i \(-0.376305\pi\)
0.378894 + 0.925440i \(0.376305\pi\)
\(854\) 0 0
\(855\) −11.2949 −0.386278
\(856\) 0 0
\(857\) 29.2714 0.999892 0.499946 0.866057i \(-0.333353\pi\)
0.499946 + 0.866057i \(0.333353\pi\)
\(858\) 0 0
\(859\) −18.2672 −0.623270 −0.311635 0.950202i \(-0.600877\pi\)
−0.311635 + 0.950202i \(0.600877\pi\)
\(860\) 0 0
\(861\) −16.0625 −0.547408
\(862\) 0 0
\(863\) 14.1450 0.481501 0.240751 0.970587i \(-0.422606\pi\)
0.240751 + 0.970587i \(0.422606\pi\)
\(864\) 0 0
\(865\) −6.20828 −0.211088
\(866\) 0 0
\(867\) −36.2979 −1.23274
\(868\) 0 0
\(869\) 22.9728 0.779298
\(870\) 0 0
\(871\) 29.5051 0.999744
\(872\) 0 0
\(873\) −25.3576 −0.858225
\(874\) 0 0
\(875\) 5.49153 0.185647
\(876\) 0 0
\(877\) 25.2115 0.851333 0.425666 0.904880i \(-0.360040\pi\)
0.425666 + 0.904880i \(0.360040\pi\)
\(878\) 0 0
\(879\) 77.5739 2.61650
\(880\) 0 0
\(881\) 24.8475 0.837133 0.418567 0.908186i \(-0.362533\pi\)
0.418567 + 0.908186i \(0.362533\pi\)
\(882\) 0 0
\(883\) 51.9797 1.74926 0.874629 0.484793i \(-0.161105\pi\)
0.874629 + 0.484793i \(0.161105\pi\)
\(884\) 0 0
\(885\) 10.4351 0.350771
\(886\) 0 0
\(887\) 7.94768 0.266857 0.133428 0.991058i \(-0.457401\pi\)
0.133428 + 0.991058i \(0.457401\pi\)
\(888\) 0 0
\(889\) −17.8303 −0.598010
\(890\) 0 0
\(891\) 46.6616 1.56322
\(892\) 0 0
\(893\) 28.9420 0.968508
\(894\) 0 0
\(895\) −10.3207 −0.344982
\(896\) 0 0
\(897\) 64.6101 2.15727
\(898\) 0 0
\(899\) 17.2873 0.576565
\(900\) 0 0
\(901\) 10.8034 0.359913
\(902\) 0 0
\(903\) −25.1864 −0.838150
\(904\) 0 0
\(905\) 3.78281 0.125745
\(906\) 0 0
\(907\) −37.7947 −1.25495 −0.627476 0.778636i \(-0.715912\pi\)
−0.627476 + 0.778636i \(0.715912\pi\)
\(908\) 0 0
\(909\) 31.7888 1.05437
\(910\) 0 0
\(911\) −8.62265 −0.285681 −0.142841 0.989746i \(-0.545624\pi\)
−0.142841 + 0.989746i \(0.545624\pi\)
\(912\) 0 0
\(913\) 29.2327 0.967462
\(914\) 0 0
\(915\) 1.70261 0.0562866
\(916\) 0 0
\(917\) −1.65123 −0.0545284
\(918\) 0 0
\(919\) 13.6950 0.451756 0.225878 0.974156i \(-0.427475\pi\)
0.225878 + 0.974156i \(0.427475\pi\)
\(920\) 0 0
\(921\) 40.6276 1.33872
\(922\) 0 0
\(923\) 7.37938 0.242895
\(924\) 0 0
\(925\) 20.8039 0.684028
\(926\) 0 0
\(927\) −16.8311 −0.552804
\(928\) 0 0
\(929\) −31.6443 −1.03822 −0.519108 0.854709i \(-0.673736\pi\)
−0.519108 + 0.854709i \(0.673736\pi\)
\(930\) 0 0
\(931\) −47.4427 −1.55487
\(932\) 0 0
\(933\) 53.5400 1.75282
\(934\) 0 0
\(935\) 3.22908 0.105602
\(936\) 0 0
\(937\) −23.0094 −0.751684 −0.375842 0.926684i \(-0.622646\pi\)
−0.375842 + 0.926684i \(0.622646\pi\)
\(938\) 0 0
\(939\) −51.3042 −1.67425
\(940\) 0 0
\(941\) 31.9570 1.04177 0.520884 0.853628i \(-0.325602\pi\)
0.520884 + 0.853628i \(0.325602\pi\)
\(942\) 0 0
\(943\) 40.0611 1.30457
\(944\) 0 0
\(945\) 0.237336 0.00772053
\(946\) 0 0
\(947\) 7.64024 0.248274 0.124137 0.992265i \(-0.460384\pi\)
0.124137 + 0.992265i \(0.460384\pi\)
\(948\) 0 0
\(949\) −0.489433 −0.0158877
\(950\) 0 0
\(951\) −6.69062 −0.216958
\(952\) 0 0
\(953\) −0.845695 −0.0273947 −0.0136974 0.999906i \(-0.504360\pi\)
−0.0136974 + 0.999906i \(0.504360\pi\)
\(954\) 0 0
\(955\) −0.272365 −0.00881352
\(956\) 0 0
\(957\) −30.6162 −0.989682
\(958\) 0 0
\(959\) −5.70063 −0.184083
\(960\) 0 0
\(961\) 13.8533 0.446881
\(962\) 0 0
\(963\) 7.87359 0.253723
\(964\) 0 0
\(965\) 1.55904 0.0501872
\(966\) 0 0
\(967\) 56.5661 1.81904 0.909521 0.415657i \(-0.136448\pi\)
0.909521 + 0.415657i \(0.136448\pi\)
\(968\) 0 0
\(969\) 28.7908 0.924893
\(970\) 0 0
\(971\) −28.3347 −0.909304 −0.454652 0.890669i \(-0.650236\pi\)
−0.454652 + 0.890669i \(0.650236\pi\)
\(972\) 0 0
\(973\) −14.7444 −0.472684
\(974\) 0 0
\(975\) 42.8772 1.37317
\(976\) 0 0
\(977\) 2.15495 0.0689429 0.0344715 0.999406i \(-0.489025\pi\)
0.0344715 + 0.999406i \(0.489025\pi\)
\(978\) 0 0
\(979\) −51.4205 −1.64341
\(980\) 0 0
\(981\) 29.3954 0.938524
\(982\) 0 0
\(983\) −27.7182 −0.884074 −0.442037 0.896997i \(-0.645744\pi\)
−0.442037 + 0.896997i \(0.645744\pi\)
\(984\) 0 0
\(985\) 8.57905 0.273351
\(986\) 0 0
\(987\) 9.80962 0.312244
\(988\) 0 0
\(989\) 62.8168 1.99746
\(990\) 0 0
\(991\) −36.5255 −1.16027 −0.580135 0.814520i \(-0.697000\pi\)
−0.580135 + 0.814520i \(0.697000\pi\)
\(992\) 0 0
\(993\) 32.9650 1.04611
\(994\) 0 0
\(995\) −4.33072 −0.137293
\(996\) 0 0
\(997\) −29.0526 −0.920105 −0.460052 0.887892i \(-0.652169\pi\)
−0.460052 + 0.887892i \(0.652169\pi\)
\(998\) 0 0
\(999\) 1.83966 0.0582042
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 8048.2.a.s.1.20 21
4.3 odd 2 2012.2.a.b.1.2 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2012.2.a.b.1.2 21 4.3 odd 2
8048.2.a.s.1.20 21 1.1 even 1 trivial