Properties

Label 8512.2.a.bl.1.1
Level $8512$
Weight $2$
Character 8512.1
Self dual yes
Analytic conductor $67.969$
Analytic rank $1$
Dimension $3$
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] = [8512,2,Mod(1,8512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8512.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8512.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: \(67.9686622005\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.733.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 7x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 532)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.51820\) of defining polynomial
Character \(\chi\) \(=\) 8512.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.51820 q^{3} +2.85952 q^{5} +1.00000 q^{7} +3.34132 q^{9} +O(q^{10})\) \(q-2.51820 q^{3} +2.85952 q^{5} +1.00000 q^{7} +3.34132 q^{9} -2.34132 q^{11} +0.859523 q^{13} -7.20085 q^{15} +4.34132 q^{17} -1.00000 q^{19} -2.51820 q^{21} -4.85952 q^{23} +3.17687 q^{25} -0.859523 q^{27} -9.37772 q^{29} +7.37772 q^{31} +5.89592 q^{33} +2.85952 q^{35} -2.17687 q^{37} -2.16445 q^{39} +6.69507 q^{41} +0.353748 q^{43} +9.55460 q^{45} -5.54217 q^{47} +1.00000 q^{49} -10.9323 q^{51} -9.55460 q^{53} -6.69507 q^{55} +2.51820 q^{57} -14.5786 q^{59} +12.9323 q^{61} +3.34132 q^{63} +2.45783 q^{65} -4.34132 q^{67} +12.2372 q^{69} -7.89592 q^{71} +2.23725 q^{73} -8.00000 q^{75} -2.34132 q^{77} -7.36530 q^{79} -7.85952 q^{81} -1.83555 q^{83} +12.4141 q^{85} +23.6150 q^{87} -3.31735 q^{89} +0.859523 q^{91} -18.5786 q^{93} -2.85952 q^{95} +18.2248 q^{97} -7.82313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 2 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 2 q^{5} + 3 q^{7} + 6 q^{9} - 3 q^{11} - 8 q^{13} - 7 q^{15} + 9 q^{17} - 3 q^{19} - q^{21} - 4 q^{23} + 7 q^{25} + 8 q^{27} - 11 q^{29} + 5 q^{31} - 6 q^{33} - 2 q^{35} - 4 q^{37} - 5 q^{39} + 11 q^{41} - 4 q^{43} + 9 q^{45} + 2 q^{47} + 3 q^{49} + 4 q^{51} - 9 q^{53} - 11 q^{55} + q^{57} - 12 q^{59} + 2 q^{61} + 6 q^{63} + 26 q^{65} - 9 q^{67} + 9 q^{69} - 21 q^{73} - 24 q^{75} - 3 q^{77} - 6 q^{79} - 13 q^{81} - 7 q^{83} + 7 q^{85} + 26 q^{87} - 18 q^{89} - 8 q^{91} - 24 q^{93} + 2 q^{95} + 28 q^{97} - 26 q^{99}+O(q^{100}) \) 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
\(3\) −2.51820 −1.45388 −0.726941 0.686700i \(-0.759059\pi\)
−0.726941 + 0.686700i \(0.759059\pi\)
\(4\) 0 0
\(5\) 2.85952 1.27882 0.639409 0.768867i \(-0.279179\pi\)
0.639409 + 0.768867i \(0.279179\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 3.34132 1.11377
\(10\) 0 0
\(11\) −2.34132 −0.705936 −0.352968 0.935635i \(-0.614828\pi\)
−0.352968 + 0.935635i \(0.614828\pi\)
\(12\) 0 0
\(13\) 0.859523 0.238389 0.119194 0.992871i \(-0.461969\pi\)
0.119194 + 0.992871i \(0.461969\pi\)
\(14\) 0 0
\(15\) −7.20085 −1.85925
\(16\) 0 0
\(17\) 4.34132 1.05293 0.526463 0.850198i \(-0.323518\pi\)
0.526463 + 0.850198i \(0.323518\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.51820 −0.549516
\(22\) 0 0
\(23\) −4.85952 −1.01328 −0.506640 0.862158i \(-0.669113\pi\)
−0.506640 + 0.862158i \(0.669113\pi\)
\(24\) 0 0
\(25\) 3.17687 0.635375
\(26\) 0 0
\(27\) −0.859523 −0.165415
\(28\) 0 0
\(29\) −9.37772 −1.74140 −0.870700 0.491815i \(-0.836334\pi\)
−0.870700 + 0.491815i \(0.836334\pi\)
\(30\) 0 0
\(31\) 7.37772 1.32508 0.662539 0.749027i \(-0.269479\pi\)
0.662539 + 0.749027i \(0.269479\pi\)
\(32\) 0 0
\(33\) 5.89592 1.02635
\(34\) 0 0
\(35\) 2.85952 0.483348
\(36\) 0 0
\(37\) −2.17687 −0.357876 −0.178938 0.983860i \(-0.557266\pi\)
−0.178938 + 0.983860i \(0.557266\pi\)
\(38\) 0 0
\(39\) −2.16445 −0.346589
\(40\) 0 0
\(41\) 6.69507 1.04559 0.522797 0.852457i \(-0.324888\pi\)
0.522797 + 0.852457i \(0.324888\pi\)
\(42\) 0 0
\(43\) 0.353748 0.0539461 0.0269730 0.999636i \(-0.491413\pi\)
0.0269730 + 0.999636i \(0.491413\pi\)
\(44\) 0 0
\(45\) 9.55460 1.42432
\(46\) 0 0
\(47\) −5.54217 −0.808409 −0.404204 0.914669i \(-0.632452\pi\)
−0.404204 + 0.914669i \(0.632452\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −10.9323 −1.53083
\(52\) 0 0
\(53\) −9.55460 −1.31242 −0.656212 0.754576i \(-0.727842\pi\)
−0.656212 + 0.754576i \(0.727842\pi\)
\(54\) 0 0
\(55\) −6.69507 −0.902763
\(56\) 0 0
\(57\) 2.51820 0.333544
\(58\) 0 0
\(59\) −14.5786 −1.89797 −0.948984 0.315324i \(-0.897887\pi\)
−0.948984 + 0.315324i \(0.897887\pi\)
\(60\) 0 0
\(61\) 12.9323 1.65581 0.827907 0.560866i \(-0.189532\pi\)
0.827907 + 0.560866i \(0.189532\pi\)
\(62\) 0 0
\(63\) 3.34132 0.420967
\(64\) 0 0
\(65\) 2.45783 0.304856
\(66\) 0 0
\(67\) −4.34132 −0.530377 −0.265189 0.964197i \(-0.585434\pi\)
−0.265189 + 0.964197i \(0.585434\pi\)
\(68\) 0 0
\(69\) 12.2372 1.47319
\(70\) 0 0
\(71\) −7.89592 −0.937073 −0.468537 0.883444i \(-0.655219\pi\)
−0.468537 + 0.883444i \(0.655219\pi\)
\(72\) 0 0
\(73\) 2.23725 0.261850 0.130925 0.991392i \(-0.458205\pi\)
0.130925 + 0.991392i \(0.458205\pi\)
\(74\) 0 0
\(75\) −8.00000 −0.923760
\(76\) 0 0
\(77\) −2.34132 −0.266819
\(78\) 0 0
\(79\) −7.36530 −0.828661 −0.414330 0.910127i \(-0.635984\pi\)
−0.414330 + 0.910127i \(0.635984\pi\)
\(80\) 0 0
\(81\) −7.85952 −0.873280
\(82\) 0 0
\(83\) −1.83555 −0.201478 −0.100739 0.994913i \(-0.532121\pi\)
−0.100739 + 0.994913i \(0.532121\pi\)
\(84\) 0 0
\(85\) 12.4141 1.34650
\(86\) 0 0
\(87\) 23.6150 2.53179
\(88\) 0 0
\(89\) −3.31735 −0.351638 −0.175819 0.984422i \(-0.556257\pi\)
−0.175819 + 0.984422i \(0.556257\pi\)
\(90\) 0 0
\(91\) 0.859523 0.0901025
\(92\) 0 0
\(93\) −18.5786 −1.92651
\(94\) 0 0
\(95\) −2.85952 −0.293381
\(96\) 0 0
\(97\) 18.2248 1.85045 0.925225 0.379418i \(-0.123876\pi\)
0.925225 + 0.379418i \(0.123876\pi\)
\(98\) 0 0
\(99\) −7.82313 −0.786254
\(100\) 0 0
\(101\) −7.18842 −0.715275 −0.357637 0.933860i \(-0.616418\pi\)
−0.357637 + 0.933860i \(0.616418\pi\)
\(102\) 0 0
\(103\) −5.21327 −0.513679 −0.256839 0.966454i \(-0.582681\pi\)
−0.256839 + 0.966454i \(0.582681\pi\)
\(104\) 0 0
\(105\) −7.20085 −0.702731
\(106\) 0 0
\(107\) −3.89592 −0.376633 −0.188316 0.982108i \(-0.560303\pi\)
−0.188316 + 0.982108i \(0.560303\pi\)
\(108\) 0 0
\(109\) −0.578570 −0.0554170 −0.0277085 0.999616i \(-0.508821\pi\)
−0.0277085 + 0.999616i \(0.508821\pi\)
\(110\) 0 0
\(111\) 5.48180 0.520310
\(112\) 0 0
\(113\) 4.51820 0.425036 0.212518 0.977157i \(-0.431834\pi\)
0.212518 + 0.977157i \(0.431834\pi\)
\(114\) 0 0
\(115\) −13.8959 −1.29580
\(116\) 0 0
\(117\) 2.87195 0.265512
\(118\) 0 0
\(119\) 4.34132 0.397969
\(120\) 0 0
\(121\) −5.51820 −0.501654
\(122\) 0 0
\(123\) −16.8595 −1.52017
\(124\) 0 0
\(125\) −5.21327 −0.466289
\(126\) 0 0
\(127\) 12.4017 1.10047 0.550236 0.835009i \(-0.314538\pi\)
0.550236 + 0.835009i \(0.314538\pi\)
\(128\) 0 0
\(129\) −0.890808 −0.0784313
\(130\) 0 0
\(131\) −1.98758 −0.173655 −0.0868277 0.996223i \(-0.527673\pi\)
−0.0868277 + 0.996223i \(0.527673\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −2.45783 −0.211536
\(136\) 0 0
\(137\) −13.4381 −1.14809 −0.574047 0.818822i \(-0.694627\pi\)
−0.574047 + 0.818822i \(0.694627\pi\)
\(138\) 0 0
\(139\) 11.6150 0.985169 0.492584 0.870265i \(-0.336052\pi\)
0.492584 + 0.870265i \(0.336052\pi\)
\(140\) 0 0
\(141\) 13.9563 1.17533
\(142\) 0 0
\(143\) −2.01242 −0.168287
\(144\) 0 0
\(145\) −26.8158 −2.22693
\(146\) 0 0
\(147\) −2.51820 −0.207698
\(148\) 0 0
\(149\) −4.10408 −0.336219 −0.168110 0.985768i \(-0.553766\pi\)
−0.168110 + 0.985768i \(0.553766\pi\)
\(150\) 0 0
\(151\) −11.7794 −0.958595 −0.479298 0.877652i \(-0.659109\pi\)
−0.479298 + 0.877652i \(0.659109\pi\)
\(152\) 0 0
\(153\) 14.5058 1.17272
\(154\) 0 0
\(155\) 21.0968 1.69453
\(156\) 0 0
\(157\) −16.8158 −1.34205 −0.671024 0.741436i \(-0.734145\pi\)
−0.671024 + 0.741436i \(0.734145\pi\)
\(158\) 0 0
\(159\) 24.0604 1.90811
\(160\) 0 0
\(161\) −4.85952 −0.382984
\(162\) 0 0
\(163\) 13.2736 1.03967 0.519836 0.854266i \(-0.325993\pi\)
0.519836 + 0.854266i \(0.325993\pi\)
\(164\) 0 0
\(165\) 16.8595 1.31251
\(166\) 0 0
\(167\) −14.2497 −1.10267 −0.551336 0.834283i \(-0.685882\pi\)
−0.551336 + 0.834283i \(0.685882\pi\)
\(168\) 0 0
\(169\) −12.2612 −0.943171
\(170\) 0 0
\(171\) −3.34132 −0.255517
\(172\) 0 0
\(173\) −20.6826 −1.57247 −0.786236 0.617926i \(-0.787973\pi\)
−0.786236 + 0.617926i \(0.787973\pi\)
\(174\) 0 0
\(175\) 3.17687 0.240149
\(176\) 0 0
\(177\) 36.7117 2.75942
\(178\) 0 0
\(179\) 14.0604 1.05092 0.525461 0.850818i \(-0.323893\pi\)
0.525461 + 0.850818i \(0.323893\pi\)
\(180\) 0 0
\(181\) −7.02397 −0.522088 −0.261044 0.965327i \(-0.584067\pi\)
−0.261044 + 0.965327i \(0.584067\pi\)
\(182\) 0 0
\(183\) −32.5661 −2.40736
\(184\) 0 0
\(185\) −6.22482 −0.457658
\(186\) 0 0
\(187\) −10.1645 −0.743298
\(188\) 0 0
\(189\) −0.859523 −0.0625211
\(190\) 0 0
\(191\) 15.6753 1.13423 0.567114 0.823639i \(-0.308060\pi\)
0.567114 + 0.823639i \(0.308060\pi\)
\(192\) 0 0
\(193\) −0.670226 −0.0482439 −0.0241220 0.999709i \(-0.507679\pi\)
−0.0241220 + 0.999709i \(0.507679\pi\)
\(194\) 0 0
\(195\) −6.18930 −0.443225
\(196\) 0 0
\(197\) 23.9811 1.70859 0.854293 0.519792i \(-0.173991\pi\)
0.854293 + 0.519792i \(0.173991\pi\)
\(198\) 0 0
\(199\) −16.9803 −1.20370 −0.601850 0.798609i \(-0.705569\pi\)
−0.601850 + 0.798609i \(0.705569\pi\)
\(200\) 0 0
\(201\) 10.9323 0.771106
\(202\) 0 0
\(203\) −9.37772 −0.658187
\(204\) 0 0
\(205\) 19.1447 1.33713
\(206\) 0 0
\(207\) −16.2372 −1.12857
\(208\) 0 0
\(209\) 2.34132 0.161953
\(210\) 0 0
\(211\) 12.8471 0.884431 0.442215 0.896909i \(-0.354193\pi\)
0.442215 + 0.896909i \(0.354193\pi\)
\(212\) 0 0
\(213\) 19.8835 1.36239
\(214\) 0 0
\(215\) 1.01155 0.0689872
\(216\) 0 0
\(217\) 7.37772 0.500832
\(218\) 0 0
\(219\) −5.63383 −0.380699
\(220\) 0 0
\(221\) 3.73147 0.251006
\(222\) 0 0
\(223\) −8.53062 −0.571253 −0.285626 0.958341i \(-0.592202\pi\)
−0.285626 + 0.958341i \(0.592202\pi\)
\(224\) 0 0
\(225\) 10.6150 0.707665
\(226\) 0 0
\(227\) 27.4985 1.82514 0.912569 0.408924i \(-0.134096\pi\)
0.912569 + 0.408924i \(0.134096\pi\)
\(228\) 0 0
\(229\) 25.4381 1.68100 0.840498 0.541814i \(-0.182262\pi\)
0.840498 + 0.541814i \(0.182262\pi\)
\(230\) 0 0
\(231\) 5.89592 0.387923
\(232\) 0 0
\(233\) 15.6587 1.02583 0.512917 0.858438i \(-0.328565\pi\)
0.512917 + 0.858438i \(0.328565\pi\)
\(234\) 0 0
\(235\) −15.8480 −1.03381
\(236\) 0 0
\(237\) 18.5473 1.20478
\(238\) 0 0
\(239\) 17.0051 1.09997 0.549985 0.835175i \(-0.314634\pi\)
0.549985 + 0.835175i \(0.314634\pi\)
\(240\) 0 0
\(241\) 2.40170 0.154707 0.0773534 0.997004i \(-0.475353\pi\)
0.0773534 + 0.997004i \(0.475353\pi\)
\(242\) 0 0
\(243\) 22.3704 1.43506
\(244\) 0 0
\(245\) 2.85952 0.182688
\(246\) 0 0
\(247\) −0.859523 −0.0546902
\(248\) 0 0
\(249\) 4.62228 0.292925
\(250\) 0 0
\(251\) 5.77942 0.364794 0.182397 0.983225i \(-0.441614\pi\)
0.182397 + 0.983225i \(0.441614\pi\)
\(252\) 0 0
\(253\) 11.3777 0.715311
\(254\) 0 0
\(255\) −31.2612 −1.95765
\(256\) 0 0
\(257\) 2.49335 0.155531 0.0777655 0.996972i \(-0.475221\pi\)
0.0777655 + 0.996972i \(0.475221\pi\)
\(258\) 0 0
\(259\) −2.17687 −0.135264
\(260\) 0 0
\(261\) −31.3340 −1.93953
\(262\) 0 0
\(263\) −6.72636 −0.414765 −0.207382 0.978260i \(-0.566494\pi\)
−0.207382 + 0.978260i \(0.566494\pi\)
\(264\) 0 0
\(265\) −27.3216 −1.67835
\(266\) 0 0
\(267\) 8.35375 0.511241
\(268\) 0 0
\(269\) −10.1893 −0.621252 −0.310626 0.950532i \(-0.600539\pi\)
−0.310626 + 0.950532i \(0.600539\pi\)
\(270\) 0 0
\(271\) −22.1083 −1.34298 −0.671492 0.741011i \(-0.734346\pi\)
−0.671492 + 0.741011i \(0.734346\pi\)
\(272\) 0 0
\(273\) −2.16445 −0.130999
\(274\) 0 0
\(275\) −7.43809 −0.448534
\(276\) 0 0
\(277\) 9.31735 0.559825 0.279913 0.960025i \(-0.409694\pi\)
0.279913 + 0.960025i \(0.409694\pi\)
\(278\) 0 0
\(279\) 24.6514 1.47584
\(280\) 0 0
\(281\) −12.9323 −0.771477 −0.385739 0.922608i \(-0.626053\pi\)
−0.385739 + 0.922608i \(0.626053\pi\)
\(282\) 0 0
\(283\) 19.9083 1.18343 0.591714 0.806148i \(-0.298451\pi\)
0.591714 + 0.806148i \(0.298451\pi\)
\(284\) 0 0
\(285\) 7.20085 0.426541
\(286\) 0 0
\(287\) 6.69507 0.395198
\(288\) 0 0
\(289\) 1.84710 0.108653
\(290\) 0 0
\(291\) −45.8937 −2.69034
\(292\) 0 0
\(293\) −20.1769 −1.17875 −0.589373 0.807861i \(-0.700625\pi\)
−0.589373 + 0.807861i \(0.700625\pi\)
\(294\) 0 0
\(295\) −41.6878 −2.42716
\(296\) 0 0
\(297\) 2.01242 0.116773
\(298\) 0 0
\(299\) −4.17687 −0.241555
\(300\) 0 0
\(301\) 0.353748 0.0203897
\(302\) 0 0
\(303\) 18.1019 1.03993
\(304\) 0 0
\(305\) 36.9803 2.11748
\(306\) 0 0
\(307\) −29.5713 −1.68772 −0.843860 0.536563i \(-0.819722\pi\)
−0.843860 + 0.536563i \(0.819722\pi\)
\(308\) 0 0
\(309\) 13.1281 0.746829
\(310\) 0 0
\(311\) 17.7794 1.00818 0.504089 0.863652i \(-0.331828\pi\)
0.504089 + 0.863652i \(0.331828\pi\)
\(312\) 0 0
\(313\) 24.0479 1.35927 0.679635 0.733550i \(-0.262138\pi\)
0.679635 + 0.733550i \(0.262138\pi\)
\(314\) 0 0
\(315\) 9.55460 0.538341
\(316\) 0 0
\(317\) −15.5109 −0.871178 −0.435589 0.900146i \(-0.643460\pi\)
−0.435589 + 0.900146i \(0.643460\pi\)
\(318\) 0 0
\(319\) 21.9563 1.22932
\(320\) 0 0
\(321\) 9.81070 0.547580
\(322\) 0 0
\(323\) −4.34132 −0.241558
\(324\) 0 0
\(325\) 2.73060 0.151466
\(326\) 0 0
\(327\) 1.45695 0.0805698
\(328\) 0 0
\(329\) −5.54217 −0.305550
\(330\) 0 0
\(331\) 29.8835 1.64255 0.821273 0.570536i \(-0.193264\pi\)
0.821273 + 0.570536i \(0.193264\pi\)
\(332\) 0 0
\(333\) −7.27364 −0.398593
\(334\) 0 0
\(335\) −12.4141 −0.678256
\(336\) 0 0
\(337\) −33.8771 −1.84540 −0.922701 0.385518i \(-0.874023\pi\)
−0.922701 + 0.385518i \(0.874023\pi\)
\(338\) 0 0
\(339\) −11.3777 −0.617953
\(340\) 0 0
\(341\) −17.2736 −0.935420
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 34.9927 1.88394
\(346\) 0 0
\(347\) −20.2124 −1.08506 −0.542529 0.840037i \(-0.682533\pi\)
−0.542529 + 0.840037i \(0.682533\pi\)
\(348\) 0 0
\(349\) 2.77431 0.148505 0.0742526 0.997239i \(-0.476343\pi\)
0.0742526 + 0.997239i \(0.476343\pi\)
\(350\) 0 0
\(351\) −0.738780 −0.0394332
\(352\) 0 0
\(353\) −19.3049 −1.02750 −0.513749 0.857941i \(-0.671744\pi\)
−0.513749 + 0.857941i \(0.671744\pi\)
\(354\) 0 0
\(355\) −22.5786 −1.19835
\(356\) 0 0
\(357\) −10.9323 −0.578600
\(358\) 0 0
\(359\) −28.5349 −1.50601 −0.753006 0.658013i \(-0.771397\pi\)
−0.753006 + 0.658013i \(0.771397\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 13.8959 0.729347
\(364\) 0 0
\(365\) 6.39746 0.334858
\(366\) 0 0
\(367\) 3.94387 0.205868 0.102934 0.994688i \(-0.467177\pi\)
0.102934 + 0.994688i \(0.467177\pi\)
\(368\) 0 0
\(369\) 22.3704 1.16456
\(370\) 0 0
\(371\) −9.55460 −0.496050
\(372\) 0 0
\(373\) −17.3216 −0.896878 −0.448439 0.893813i \(-0.648020\pi\)
−0.448439 + 0.893813i \(0.648020\pi\)
\(374\) 0 0
\(375\) 13.1281 0.677930
\(376\) 0 0
\(377\) −8.06037 −0.415130
\(378\) 0 0
\(379\) −16.2248 −0.833413 −0.416707 0.909041i \(-0.636816\pi\)
−0.416707 + 0.909041i \(0.636816\pi\)
\(380\) 0 0
\(381\) −31.2299 −1.59996
\(382\) 0 0
\(383\) 4.65136 0.237674 0.118837 0.992914i \(-0.462083\pi\)
0.118837 + 0.992914i \(0.462083\pi\)
\(384\) 0 0
\(385\) −6.69507 −0.341213
\(386\) 0 0
\(387\) 1.18199 0.0600838
\(388\) 0 0
\(389\) −11.6587 −0.591118 −0.295559 0.955324i \(-0.595506\pi\)
−0.295559 + 0.955324i \(0.595506\pi\)
\(390\) 0 0
\(391\) −21.0968 −1.06691
\(392\) 0 0
\(393\) 5.00511 0.252475
\(394\) 0 0
\(395\) −21.0612 −1.05971
\(396\) 0 0
\(397\) 33.9126 1.70202 0.851012 0.525146i \(-0.175989\pi\)
0.851012 + 0.525146i \(0.175989\pi\)
\(398\) 0 0
\(399\) 2.51820 0.126068
\(400\) 0 0
\(401\) 4.69507 0.234461 0.117230 0.993105i \(-0.462598\pi\)
0.117230 + 0.993105i \(0.462598\pi\)
\(402\) 0 0
\(403\) 6.34132 0.315884
\(404\) 0 0
\(405\) −22.4745 −1.11677
\(406\) 0 0
\(407\) 5.09677 0.252637
\(408\) 0 0
\(409\) −11.7628 −0.581631 −0.290815 0.956779i \(-0.593927\pi\)
−0.290815 + 0.956779i \(0.593927\pi\)
\(410\) 0 0
\(411\) 33.8398 1.66919
\(412\) 0 0
\(413\) −14.5786 −0.717365
\(414\) 0 0
\(415\) −5.24880 −0.257653
\(416\) 0 0
\(417\) −29.2488 −1.43232
\(418\) 0 0
\(419\) −31.7606 −1.55160 −0.775802 0.630976i \(-0.782655\pi\)
−0.775802 + 0.630976i \(0.782655\pi\)
\(420\) 0 0
\(421\) −12.8116 −0.624398 −0.312199 0.950017i \(-0.601066\pi\)
−0.312199 + 0.950017i \(0.601066\pi\)
\(422\) 0 0
\(423\) −18.5182 −0.900386
\(424\) 0 0
\(425\) 13.7918 0.669003
\(426\) 0 0
\(427\) 12.9323 0.625839
\(428\) 0 0
\(429\) 5.06768 0.244670
\(430\) 0 0
\(431\) −21.4381 −1.03264 −0.516318 0.856397i \(-0.672698\pi\)
−0.516318 + 0.856397i \(0.672698\pi\)
\(432\) 0 0
\(433\) −0.176874 −0.00850002 −0.00425001 0.999991i \(-0.501353\pi\)
−0.00425001 + 0.999991i \(0.501353\pi\)
\(434\) 0 0
\(435\) 67.5276 3.23770
\(436\) 0 0
\(437\) 4.85952 0.232463
\(438\) 0 0
\(439\) 4.14559 0.197858 0.0989291 0.995094i \(-0.468458\pi\)
0.0989291 + 0.995094i \(0.468458\pi\)
\(440\) 0 0
\(441\) 3.34132 0.159111
\(442\) 0 0
\(443\) 15.6753 0.744758 0.372379 0.928081i \(-0.378542\pi\)
0.372379 + 0.928081i \(0.378542\pi\)
\(444\) 0 0
\(445\) −9.48604 −0.449682
\(446\) 0 0
\(447\) 10.3349 0.488823
\(448\) 0 0
\(449\) −3.00731 −0.141924 −0.0709619 0.997479i \(-0.522607\pi\)
−0.0709619 + 0.997479i \(0.522607\pi\)
\(450\) 0 0
\(451\) −15.6753 −0.738123
\(452\) 0 0
\(453\) 29.6629 1.39369
\(454\) 0 0
\(455\) 2.45783 0.115225
\(456\) 0 0
\(457\) −22.7117 −1.06241 −0.531205 0.847243i \(-0.678261\pi\)
−0.531205 + 0.847243i \(0.678261\pi\)
\(458\) 0 0
\(459\) −3.73147 −0.174170
\(460\) 0 0
\(461\) 39.0406 1.81830 0.909152 0.416465i \(-0.136731\pi\)
0.909152 + 0.416465i \(0.136731\pi\)
\(462\) 0 0
\(463\) 26.2497 1.21993 0.609963 0.792430i \(-0.291184\pi\)
0.609963 + 0.792430i \(0.291184\pi\)
\(464\) 0 0
\(465\) −53.1259 −2.46365
\(466\) 0 0
\(467\) 6.63894 0.307214 0.153607 0.988132i \(-0.450911\pi\)
0.153607 + 0.988132i \(0.450911\pi\)
\(468\) 0 0
\(469\) −4.34132 −0.200464
\(470\) 0 0
\(471\) 42.3456 1.95118
\(472\) 0 0
\(473\) −0.828239 −0.0380825
\(474\) 0 0
\(475\) −3.17687 −0.145765
\(476\) 0 0
\(477\) −31.9250 −1.46175
\(478\) 0 0
\(479\) −0.0603715 −0.00275844 −0.00137922 0.999999i \(-0.500439\pi\)
−0.00137922 + 0.999999i \(0.500439\pi\)
\(480\) 0 0
\(481\) −1.87107 −0.0853136
\(482\) 0 0
\(483\) 12.2372 0.556814
\(484\) 0 0
\(485\) 52.1143 2.36639
\(486\) 0 0
\(487\) −28.6034 −1.29614 −0.648072 0.761579i \(-0.724425\pi\)
−0.648072 + 0.761579i \(0.724425\pi\)
\(488\) 0 0
\(489\) −33.4257 −1.51156
\(490\) 0 0
\(491\) −6.73060 −0.303748 −0.151874 0.988400i \(-0.548531\pi\)
−0.151874 + 0.988400i \(0.548531\pi\)
\(492\) 0 0
\(493\) −40.7117 −1.83356
\(494\) 0 0
\(495\) −22.3704 −1.00548
\(496\) 0 0
\(497\) −7.89592 −0.354180
\(498\) 0 0
\(499\) −39.5546 −1.77071 −0.885353 0.464919i \(-0.846084\pi\)
−0.885353 + 0.464919i \(0.846084\pi\)
\(500\) 0 0
\(501\) 35.8835 1.60316
\(502\) 0 0
\(503\) −28.8531 −1.28650 −0.643248 0.765658i \(-0.722413\pi\)
−0.643248 + 0.765658i \(0.722413\pi\)
\(504\) 0 0
\(505\) −20.5555 −0.914706
\(506\) 0 0
\(507\) 30.8762 1.37126
\(508\) 0 0
\(509\) −32.4496 −1.43831 −0.719153 0.694852i \(-0.755470\pi\)
−0.719153 + 0.694852i \(0.755470\pi\)
\(510\) 0 0
\(511\) 2.23725 0.0989699
\(512\) 0 0
\(513\) 0.859523 0.0379489
\(514\) 0 0
\(515\) −14.9075 −0.656902
\(516\) 0 0
\(517\) 12.9760 0.570685
\(518\) 0 0
\(519\) 52.0830 2.28619
\(520\) 0 0
\(521\) −8.88437 −0.389231 −0.194616 0.980880i \(-0.562346\pi\)
−0.194616 + 0.980880i \(0.562346\pi\)
\(522\) 0 0
\(523\) −31.0531 −1.35786 −0.678928 0.734205i \(-0.737555\pi\)
−0.678928 + 0.734205i \(0.737555\pi\)
\(524\) 0 0
\(525\) −8.00000 −0.349149
\(526\) 0 0
\(527\) 32.0291 1.39521
\(528\) 0 0
\(529\) 0.614968 0.0267377
\(530\) 0 0
\(531\) −48.7117 −2.11391
\(532\) 0 0
\(533\) 5.75457 0.249258
\(534\) 0 0
\(535\) −11.1405 −0.481645
\(536\) 0 0
\(537\) −35.4068 −1.52792
\(538\) 0 0
\(539\) −2.34132 −0.100848
\(540\) 0 0
\(541\) 40.1687 1.72699 0.863493 0.504360i \(-0.168272\pi\)
0.863493 + 0.504360i \(0.168272\pi\)
\(542\) 0 0
\(543\) 17.6878 0.759055
\(544\) 0 0
\(545\) −1.65443 −0.0708682
\(546\) 0 0
\(547\) −12.6638 −0.541464 −0.270732 0.962655i \(-0.587266\pi\)
−0.270732 + 0.962655i \(0.587266\pi\)
\(548\) 0 0
\(549\) 43.2111 1.84420
\(550\) 0 0
\(551\) 9.37772 0.399504
\(552\) 0 0
\(553\) −7.36530 −0.313204
\(554\) 0 0
\(555\) 15.6753 0.665381
\(556\) 0 0
\(557\) −30.7679 −1.30368 −0.651838 0.758358i \(-0.726002\pi\)
−0.651838 + 0.758358i \(0.726002\pi\)
\(558\) 0 0
\(559\) 0.304055 0.0128601
\(560\) 0 0
\(561\) 25.5961 1.08067
\(562\) 0 0
\(563\) −32.2976 −1.36118 −0.680591 0.732663i \(-0.738277\pi\)
−0.680591 + 0.732663i \(0.738277\pi\)
\(564\) 0 0
\(565\) 12.9199 0.543544
\(566\) 0 0
\(567\) −7.85952 −0.330069
\(568\) 0 0
\(569\) −32.4912 −1.36210 −0.681050 0.732237i \(-0.738476\pi\)
−0.681050 + 0.732237i \(0.738476\pi\)
\(570\) 0 0
\(571\) 23.9687 1.00306 0.501530 0.865140i \(-0.332771\pi\)
0.501530 + 0.865140i \(0.332771\pi\)
\(572\) 0 0
\(573\) −39.4736 −1.64903
\(574\) 0 0
\(575\) −15.4381 −0.643813
\(576\) 0 0
\(577\) −27.0655 −1.12675 −0.563375 0.826201i \(-0.690498\pi\)
−0.563375 + 0.826201i \(0.690498\pi\)
\(578\) 0 0
\(579\) 1.68776 0.0701410
\(580\) 0 0
\(581\) −1.83555 −0.0761514
\(582\) 0 0
\(583\) 22.3704 0.926488
\(584\) 0 0
\(585\) 8.21240 0.339541
\(586\) 0 0
\(587\) 9.15714 0.377956 0.188978 0.981981i \(-0.439483\pi\)
0.188978 + 0.981981i \(0.439483\pi\)
\(588\) 0 0
\(589\) −7.37772 −0.303994
\(590\) 0 0
\(591\) −60.3893 −2.48408
\(592\) 0 0
\(593\) −12.6034 −0.517560 −0.258780 0.965936i \(-0.583321\pi\)
−0.258780 + 0.965936i \(0.583321\pi\)
\(594\) 0 0
\(595\) 12.4141 0.508929
\(596\) 0 0
\(597\) 42.7597 1.75004
\(598\) 0 0
\(599\) −37.7002 −1.54039 −0.770194 0.637810i \(-0.779841\pi\)
−0.770194 + 0.637810i \(0.779841\pi\)
\(600\) 0 0
\(601\) 20.0291 0.817004 0.408502 0.912758i \(-0.366051\pi\)
0.408502 + 0.912758i \(0.366051\pi\)
\(602\) 0 0
\(603\) −14.5058 −0.590721
\(604\) 0 0
\(605\) −15.7794 −0.641525
\(606\) 0 0
\(607\) 19.8480 0.805604 0.402802 0.915287i \(-0.368036\pi\)
0.402802 + 0.915287i \(0.368036\pi\)
\(608\) 0 0
\(609\) 23.6150 0.956927
\(610\) 0 0
\(611\) −4.76363 −0.192716
\(612\) 0 0
\(613\) −30.1498 −1.21774 −0.608870 0.793270i \(-0.708377\pi\)
−0.608870 + 0.793270i \(0.708377\pi\)
\(614\) 0 0
\(615\) −48.2102 −1.94402
\(616\) 0 0
\(617\) 16.6702 0.671118 0.335559 0.942019i \(-0.391075\pi\)
0.335559 + 0.942019i \(0.391075\pi\)
\(618\) 0 0
\(619\) −0.823999 −0.0331193 −0.0165596 0.999863i \(-0.505271\pi\)
−0.0165596 + 0.999863i \(0.505271\pi\)
\(620\) 0 0
\(621\) 4.17687 0.167612
\(622\) 0 0
\(623\) −3.31735 −0.132907
\(624\) 0 0
\(625\) −30.7918 −1.23167
\(626\) 0 0
\(627\) −5.89592 −0.235460
\(628\) 0 0
\(629\) −9.45052 −0.376817
\(630\) 0 0
\(631\) −40.6762 −1.61929 −0.809647 0.586917i \(-0.800342\pi\)
−0.809647 + 0.586917i \(0.800342\pi\)
\(632\) 0 0
\(633\) −32.3516 −1.28586
\(634\) 0 0
\(635\) 35.4629 1.40730
\(636\) 0 0
\(637\) 0.859523 0.0340556
\(638\) 0 0
\(639\) −26.3828 −1.04369
\(640\) 0 0
\(641\) −12.8407 −0.507176 −0.253588 0.967312i \(-0.581611\pi\)
−0.253588 + 0.967312i \(0.581611\pi\)
\(642\) 0 0
\(643\) −43.3820 −1.71082 −0.855409 0.517953i \(-0.826694\pi\)
−0.855409 + 0.517953i \(0.826694\pi\)
\(644\) 0 0
\(645\) −2.54729 −0.100299
\(646\) 0 0
\(647\) −35.0531 −1.37808 −0.689039 0.724724i \(-0.741967\pi\)
−0.689039 + 0.724724i \(0.741967\pi\)
\(648\) 0 0
\(649\) 34.1332 1.33984
\(650\) 0 0
\(651\) −18.5786 −0.728152
\(652\) 0 0
\(653\) −47.8085 −1.87089 −0.935446 0.353470i \(-0.885001\pi\)
−0.935446 + 0.353470i \(0.885001\pi\)
\(654\) 0 0
\(655\) −5.68352 −0.222074
\(656\) 0 0
\(657\) 7.47536 0.291642
\(658\) 0 0
\(659\) 40.6325 1.58282 0.791409 0.611287i \(-0.209348\pi\)
0.791409 + 0.611287i \(0.209348\pi\)
\(660\) 0 0
\(661\) 26.4265 1.02787 0.513937 0.857828i \(-0.328187\pi\)
0.513937 + 0.857828i \(0.328187\pi\)
\(662\) 0 0
\(663\) −9.39658 −0.364933
\(664\) 0 0
\(665\) −2.85952 −0.110888
\(666\) 0 0
\(667\) 45.5713 1.76453
\(668\) 0 0
\(669\) 21.4818 0.830534
\(670\) 0 0
\(671\) −30.2788 −1.16890
\(672\) 0 0
\(673\) −38.4077 −1.48051 −0.740254 0.672328i \(-0.765295\pi\)
−0.740254 + 0.672328i \(0.765295\pi\)
\(674\) 0 0
\(675\) −2.73060 −0.105101
\(676\) 0 0
\(677\) 21.3944 0.822253 0.411127 0.911578i \(-0.365135\pi\)
0.411127 + 0.911578i \(0.365135\pi\)
\(678\) 0 0
\(679\) 18.2248 0.699404
\(680\) 0 0
\(681\) −69.2466 −2.65354
\(682\) 0 0
\(683\) 34.6265 1.32495 0.662473 0.749085i \(-0.269507\pi\)
0.662473 + 0.749085i \(0.269507\pi\)
\(684\) 0 0
\(685\) −38.4265 −1.46820
\(686\) 0 0
\(687\) −64.0582 −2.44397
\(688\) 0 0
\(689\) −8.21240 −0.312867
\(690\) 0 0
\(691\) 45.8895 1.74572 0.872859 0.487972i \(-0.162263\pi\)
0.872859 + 0.487972i \(0.162263\pi\)
\(692\) 0 0
\(693\) −7.82313 −0.297176
\(694\) 0 0
\(695\) 33.2133 1.25985
\(696\) 0 0
\(697\) 29.0655 1.10093
\(698\) 0 0
\(699\) −39.4317 −1.49144
\(700\) 0 0
\(701\) 13.0843 0.494189 0.247094 0.968991i \(-0.420524\pi\)
0.247094 + 0.968991i \(0.420524\pi\)
\(702\) 0 0
\(703\) 2.17687 0.0821024
\(704\) 0 0
\(705\) 39.9083 1.50304
\(706\) 0 0
\(707\) −7.18842 −0.270349
\(708\) 0 0
\(709\) −43.8085 −1.64526 −0.822631 0.568575i \(-0.807495\pi\)
−0.822631 + 0.568575i \(0.807495\pi\)
\(710\) 0 0
\(711\) −24.6099 −0.922942
\(712\) 0 0
\(713\) −35.8522 −1.34268
\(714\) 0 0
\(715\) −5.75457 −0.215209
\(716\) 0 0
\(717\) −42.8223 −1.59923
\(718\) 0 0
\(719\) 46.2184 1.72365 0.861827 0.507202i \(-0.169320\pi\)
0.861827 + 0.507202i \(0.169320\pi\)
\(720\) 0 0
\(721\) −5.21327 −0.194152
\(722\) 0 0
\(723\) −6.04795 −0.224926
\(724\) 0 0
\(725\) −29.7918 −1.10644
\(726\) 0 0
\(727\) −41.8085 −1.55059 −0.775296 0.631598i \(-0.782399\pi\)
−0.775296 + 0.631598i \(0.782399\pi\)
\(728\) 0 0
\(729\) −32.7546 −1.21313
\(730\) 0 0
\(731\) 1.53574 0.0568012
\(732\) 0 0
\(733\) 2.30405 0.0851022 0.0425511 0.999094i \(-0.486451\pi\)
0.0425511 + 0.999094i \(0.486451\pi\)
\(734\) 0 0
\(735\) −7.20085 −0.265607
\(736\) 0 0
\(737\) 10.1645 0.374412
\(738\) 0 0
\(739\) 15.1405 0.556951 0.278476 0.960443i \(-0.410171\pi\)
0.278476 + 0.960443i \(0.410171\pi\)
\(740\) 0 0
\(741\) 2.16445 0.0795131
\(742\) 0 0
\(743\) 51.2714 1.88097 0.940483 0.339839i \(-0.110373\pi\)
0.940483 + 0.339839i \(0.110373\pi\)
\(744\) 0 0
\(745\) −11.7357 −0.429963
\(746\) 0 0
\(747\) −6.13317 −0.224401
\(748\) 0 0
\(749\) −3.89592 −0.142354
\(750\) 0 0
\(751\) −7.57770 −0.276514 −0.138257 0.990396i \(-0.544150\pi\)
−0.138257 + 0.990396i \(0.544150\pi\)
\(752\) 0 0
\(753\) −14.5537 −0.530367
\(754\) 0 0
\(755\) −33.6835 −1.22587
\(756\) 0 0
\(757\) −35.5109 −1.29067 −0.645333 0.763902i \(-0.723281\pi\)
−0.645333 + 0.763902i \(0.723281\pi\)
\(758\) 0 0
\(759\) −28.6514 −1.03998
\(760\) 0 0
\(761\) 33.2861 1.20662 0.603309 0.797507i \(-0.293848\pi\)
0.603309 + 0.797507i \(0.293848\pi\)
\(762\) 0 0
\(763\) −0.578570 −0.0209456
\(764\) 0 0
\(765\) 41.4796 1.49970
\(766\) 0 0
\(767\) −12.5306 −0.452455
\(768\) 0 0
\(769\) 52.6680 1.89926 0.949629 0.313377i \(-0.101460\pi\)
0.949629 + 0.313377i \(0.101460\pi\)
\(770\) 0 0
\(771\) −6.27876 −0.226124
\(772\) 0 0
\(773\) 3.62141 0.130253 0.0651264 0.997877i \(-0.479255\pi\)
0.0651264 + 0.997877i \(0.479255\pi\)
\(774\) 0 0
\(775\) 23.4381 0.841921
\(776\) 0 0
\(777\) 5.48180 0.196659
\(778\) 0 0
\(779\) −6.69507 −0.239876
\(780\) 0 0
\(781\) 18.4869 0.661514
\(782\) 0 0
\(783\) 8.06037 0.288054
\(784\) 0 0
\(785\) −48.0852 −1.71624
\(786\) 0 0
\(787\) 7.85441 0.279979 0.139990 0.990153i \(-0.455293\pi\)
0.139990 + 0.990153i \(0.455293\pi\)
\(788\) 0 0
\(789\) 16.9383 0.603020
\(790\) 0 0
\(791\) 4.51820 0.160649
\(792\) 0 0
\(793\) 11.1156 0.394728
\(794\) 0 0
\(795\) 68.8012 2.44013
\(796\) 0 0
\(797\) −20.0355 −0.709695 −0.354847 0.934924i \(-0.615467\pi\)
−0.354847 + 0.934924i \(0.615467\pi\)
\(798\) 0 0
\(799\) −24.0604 −0.851195
\(800\) 0 0
\(801\) −11.0843 −0.391646
\(802\) 0 0
\(803\) −5.23812 −0.184849
\(804\) 0 0
\(805\) −13.8959 −0.489767
\(806\) 0 0
\(807\) 25.6587 0.903228
\(808\) 0 0
\(809\) 21.4942 0.755697 0.377848 0.925867i \(-0.376664\pi\)
0.377848 + 0.925867i \(0.376664\pi\)
\(810\) 0 0
\(811\) −1.15714 −0.0406327 −0.0203163 0.999794i \(-0.506467\pi\)
−0.0203163 + 0.999794i \(0.506467\pi\)
\(812\) 0 0
\(813\) 55.6731 1.95254
\(814\) 0 0
\(815\) 37.9563 1.32955
\(816\) 0 0
\(817\) −0.353748 −0.0123761
\(818\) 0 0
\(819\) 2.87195 0.100354
\(820\) 0 0
\(821\) 25.6463 0.895060 0.447530 0.894269i \(-0.352304\pi\)
0.447530 + 0.894269i \(0.352304\pi\)
\(822\) 0 0
\(823\) −13.8398 −0.482425 −0.241212 0.970472i \(-0.577545\pi\)
−0.241212 + 0.970472i \(0.577545\pi\)
\(824\) 0 0
\(825\) 18.7306 0.652116
\(826\) 0 0
\(827\) 7.23812 0.251694 0.125847 0.992050i \(-0.459835\pi\)
0.125847 + 0.992050i \(0.459835\pi\)
\(828\) 0 0
\(829\) −56.2415 −1.95335 −0.976674 0.214729i \(-0.931113\pi\)
−0.976674 + 0.214729i \(0.931113\pi\)
\(830\) 0 0
\(831\) −23.4629 −0.813920
\(832\) 0 0
\(833\) 4.34132 0.150418
\(834\) 0 0
\(835\) −40.7473 −1.41012
\(836\) 0 0
\(837\) −6.34132 −0.219188
\(838\) 0 0
\(839\) 21.9272 0.757011 0.378506 0.925599i \(-0.376438\pi\)
0.378506 + 0.925599i \(0.376438\pi\)
\(840\) 0 0
\(841\) 58.9417 2.03247
\(842\) 0 0
\(843\) 32.5661 1.12164
\(844\) 0 0
\(845\) −35.0612 −1.20614
\(846\) 0 0
\(847\) −5.51820 −0.189608
\(848\) 0 0
\(849\) −50.1332 −1.72057
\(850\) 0 0
\(851\) 10.5786 0.362629
\(852\) 0 0
\(853\) −21.2008 −0.725903 −0.362952 0.931808i \(-0.618231\pi\)
−0.362952 + 0.931808i \(0.618231\pi\)
\(854\) 0 0
\(855\) −9.55460 −0.326760
\(856\) 0 0
\(857\) 35.0240 1.19640 0.598198 0.801348i \(-0.295884\pi\)
0.598198 + 0.801348i \(0.295884\pi\)
\(858\) 0 0
\(859\) −43.6441 −1.48912 −0.744558 0.667558i \(-0.767340\pi\)
−0.744558 + 0.667558i \(0.767340\pi\)
\(860\) 0 0
\(861\) −16.8595 −0.574571
\(862\) 0 0
\(863\) 5.32159 0.181149 0.0905745 0.995890i \(-0.471130\pi\)
0.0905745 + 0.995890i \(0.471130\pi\)
\(864\) 0 0
\(865\) −59.1425 −2.01091
\(866\) 0 0
\(867\) −4.65136 −0.157969
\(868\) 0 0
\(869\) 17.2446 0.584981
\(870\) 0 0
\(871\) −3.73147 −0.126436
\(872\) 0 0
\(873\) 60.8950 2.06099
\(874\) 0 0
\(875\) −5.21327 −0.176241
\(876\) 0 0
\(877\) 15.6942 0.529955 0.264978 0.964255i \(-0.414635\pi\)
0.264978 + 0.964255i \(0.414635\pi\)
\(878\) 0 0
\(879\) 50.8094 1.71376
\(880\) 0 0
\(881\) −22.7743 −0.767286 −0.383643 0.923482i \(-0.625331\pi\)
−0.383643 + 0.923482i \(0.625331\pi\)
\(882\) 0 0
\(883\) 41.4548 1.39506 0.697532 0.716554i \(-0.254282\pi\)
0.697532 + 0.716554i \(0.254282\pi\)
\(884\) 0 0
\(885\) 104.978 3.52880
\(886\) 0 0
\(887\) 30.0415 1.00870 0.504348 0.863501i \(-0.331733\pi\)
0.504348 + 0.863501i \(0.331733\pi\)
\(888\) 0 0
\(889\) 12.4017 0.415940
\(890\) 0 0
\(891\) 18.4017 0.616480
\(892\) 0 0
\(893\) 5.54217 0.185462
\(894\) 0 0
\(895\) 40.2060 1.34394
\(896\) 0 0
\(897\) 10.5182 0.351192
\(898\) 0 0
\(899\) −69.1862 −2.30749
\(900\) 0 0
\(901\) −41.4796 −1.38189
\(902\) 0 0
\(903\) −0.890808 −0.0296442
\(904\) 0 0
\(905\) −20.0852 −0.667655
\(906\) 0 0
\(907\) −24.9323 −0.827864 −0.413932 0.910308i \(-0.635845\pi\)
−0.413932 + 0.910308i \(0.635845\pi\)
\(908\) 0 0
\(909\) −24.0189 −0.796655
\(910\) 0 0
\(911\) 34.8034 1.15309 0.576544 0.817066i \(-0.304401\pi\)
0.576544 + 0.817066i \(0.304401\pi\)
\(912\) 0 0
\(913\) 4.29762 0.142230
\(914\) 0 0
\(915\) −93.1237 −3.07857
\(916\) 0 0
\(917\) −1.98758 −0.0656356
\(918\) 0 0
\(919\) 2.10408 0.0694072 0.0347036 0.999398i \(-0.488951\pi\)
0.0347036 + 0.999398i \(0.488951\pi\)
\(920\) 0 0
\(921\) 74.4663 2.45375
\(922\) 0 0
\(923\) −6.78673 −0.223388
\(924\) 0 0
\(925\) −6.91565 −0.227385
\(926\) 0 0
\(927\) −17.4192 −0.572123
\(928\) 0 0
\(929\) 15.2985 0.501927 0.250964 0.967997i \(-0.419253\pi\)
0.250964 + 0.967997i \(0.419253\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −44.7721 −1.46577
\(934\) 0 0
\(935\) −29.0655 −0.950543
\(936\) 0 0
\(937\) 41.8522 1.36725 0.683626 0.729832i \(-0.260402\pi\)
0.683626 + 0.729832i \(0.260402\pi\)
\(938\) 0 0
\(939\) −60.5575 −1.97622
\(940\) 0 0
\(941\) −15.7918 −0.514799 −0.257400 0.966305i \(-0.582866\pi\)
−0.257400 + 0.966305i \(0.582866\pi\)
\(942\) 0 0
\(943\) −32.5349 −1.05948
\(944\) 0 0
\(945\) −2.45783 −0.0799531
\(946\) 0 0
\(947\) 15.1383 0.491928 0.245964 0.969279i \(-0.420896\pi\)
0.245964 + 0.969279i \(0.420896\pi\)
\(948\) 0 0
\(949\) 1.92296 0.0624221
\(950\) 0 0
\(951\) 39.0595 1.26659
\(952\) 0 0
\(953\) 19.5961 0.634780 0.317390 0.948295i \(-0.397194\pi\)
0.317390 + 0.948295i \(0.397194\pi\)
\(954\) 0 0
\(955\) 44.8240 1.45047
\(956\) 0 0
\(957\) −55.2903 −1.78728
\(958\) 0 0
\(959\) −13.4381 −0.433939
\(960\) 0 0
\(961\) 23.4308 0.755832
\(962\) 0 0
\(963\) −13.0175 −0.419484
\(964\) 0 0
\(965\) −1.91653 −0.0616952
\(966\) 0 0
\(967\) 12.0852 0.388634 0.194317 0.980939i \(-0.437751\pi\)
0.194317 + 0.980939i \(0.437751\pi\)
\(968\) 0 0
\(969\) 10.9323 0.351197
\(970\) 0 0
\(971\) 17.3589 0.557072 0.278536 0.960426i \(-0.410151\pi\)
0.278536 + 0.960426i \(0.410151\pi\)
\(972\) 0 0
\(973\) 11.6150 0.372359
\(974\) 0 0
\(975\) −6.87619 −0.220214
\(976\) 0 0
\(977\) −40.9969 −1.31161 −0.655804 0.754931i \(-0.727670\pi\)
−0.655804 + 0.754931i \(0.727670\pi\)
\(978\) 0 0
\(979\) 7.76699 0.248234
\(980\) 0 0
\(981\) −1.93319 −0.0617220
\(982\) 0 0
\(983\) −54.1126 −1.72592 −0.862961 0.505270i \(-0.831393\pi\)
−0.862961 + 0.505270i \(0.831393\pi\)
\(984\) 0 0
\(985\) 68.5746 2.18497
\(986\) 0 0
\(987\) 13.9563 0.444234
\(988\) 0 0
\(989\) −1.71905 −0.0546625
\(990\) 0 0
\(991\) 26.9241 0.855273 0.427637 0.903951i \(-0.359346\pi\)
0.427637 + 0.903951i \(0.359346\pi\)
\(992\) 0 0
\(993\) −75.2526 −2.38807
\(994\) 0 0
\(995\) −48.5555 −1.53931
\(996\) 0 0
\(997\) −24.7845 −0.784934 −0.392467 0.919766i \(-0.628378\pi\)
−0.392467 + 0.919766i \(0.628378\pi\)
\(998\) 0 0
\(999\) 1.87107 0.0591982
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 8512.2.a.bl.1.1 3
4.3 odd 2 8512.2.a.bn.1.3 3
8.3 odd 2 532.2.a.e.1.1 3
8.5 even 2 2128.2.a.r.1.3 3
24.11 even 2 4788.2.a.o.1.3 3
56.27 even 2 3724.2.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
532.2.a.e.1.1 3 8.3 odd 2
2128.2.a.r.1.3 3 8.5 even 2
3724.2.a.i.1.3 3 56.27 even 2
4788.2.a.o.1.3 3 24.11 even 2
8512.2.a.bl.1.1 3 1.1 even 1 trivial
8512.2.a.bn.1.3 3 4.3 odd 2