Properties

Label 768.4.a.p.1.2
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $1$
Dimension $2$
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] = [768,4,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(45.3134668844\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +18.4222 q^{5} -22.4222 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +18.4222 q^{5} -22.4222 q^{7} +9.00000 q^{9} -53.6888 q^{11} -7.15559 q^{13} +55.2666 q^{15} +39.6888 q^{17} -125.689 q^{19} -67.2666 q^{21} -99.1556 q^{23} +214.378 q^{25} +27.0000 q^{27} -205.800 q^{29} -147.489 q^{31} -161.066 q^{33} -413.066 q^{35} -125.689 q^{37} -21.4668 q^{39} -506.444 q^{41} +413.689 q^{43} +165.800 q^{45} +313.911 q^{47} +159.755 q^{49} +119.066 q^{51} +44.3331 q^{53} -989.066 q^{55} -377.066 q^{57} +324.000 q^{59} -324.000 q^{61} -201.800 q^{63} -131.822 q^{65} +464.266 q^{67} -297.467 q^{69} -1052.84 q^{71} +1022.27 q^{73} +643.133 q^{75} +1203.82 q^{77} +602.910 q^{79} +81.0000 q^{81} +15.8217 q^{83} +731.156 q^{85} -617.400 q^{87} +381.378 q^{89} +160.444 q^{91} -442.466 q^{93} -2315.47 q^{95} +659.154 q^{97} -483.199 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 8 q^{5} - 16 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 8 q^{5} - 16 q^{7} + 18 q^{9} + 8 q^{11} - 72 q^{13} + 24 q^{15} - 36 q^{17} - 136 q^{19} - 48 q^{21} - 256 q^{23} + 198 q^{25} + 54 q^{27} - 152 q^{29} + 80 q^{31} + 24 q^{33} - 480 q^{35} - 136 q^{37} - 216 q^{39} - 436 q^{41} + 712 q^{43} + 72 q^{45} + 224 q^{47} - 142 q^{49} - 108 q^{51} - 344 q^{53} - 1632 q^{55} - 408 q^{57} + 648 q^{59} - 648 q^{61} - 144 q^{63} + 544 q^{65} - 456 q^{67} - 768 q^{69} - 2048 q^{71} + 660 q^{73} + 594 q^{75} + 1600 q^{77} - 496 q^{79} + 162 q^{81} - 776 q^{83} + 1520 q^{85} - 456 q^{87} + 532 q^{89} - 256 q^{91} + 240 q^{93} - 2208 q^{95} - 1220 q^{97} + 72 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 18.4222 1.64773 0.823866 0.566785i \(-0.191813\pi\)
0.823866 + 0.566785i \(0.191813\pi\)
\(6\) 0 0
\(7\) −22.4222 −1.21069 −0.605343 0.795965i \(-0.706964\pi\)
−0.605343 + 0.795965i \(0.706964\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −53.6888 −1.47162 −0.735809 0.677190i \(-0.763198\pi\)
−0.735809 + 0.677190i \(0.763198\pi\)
\(12\) 0 0
\(13\) −7.15559 −0.152662 −0.0763309 0.997083i \(-0.524321\pi\)
−0.0763309 + 0.997083i \(0.524321\pi\)
\(14\) 0 0
\(15\) 55.2666 0.951319
\(16\) 0 0
\(17\) 39.6888 0.566233 0.283116 0.959086i \(-0.408632\pi\)
0.283116 + 0.959086i \(0.408632\pi\)
\(18\) 0 0
\(19\) −125.689 −1.51763 −0.758816 0.651306i \(-0.774222\pi\)
−0.758816 + 0.651306i \(0.774222\pi\)
\(20\) 0 0
\(21\) −67.2666 −0.698989
\(22\) 0 0
\(23\) −99.1556 −0.898929 −0.449465 0.893298i \(-0.648385\pi\)
−0.449465 + 0.893298i \(0.648385\pi\)
\(24\) 0 0
\(25\) 214.378 1.71502
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −205.800 −1.31780 −0.658898 0.752232i \(-0.728977\pi\)
−0.658898 + 0.752232i \(0.728977\pi\)
\(30\) 0 0
\(31\) −147.489 −0.854508 −0.427254 0.904132i \(-0.640519\pi\)
−0.427254 + 0.904132i \(0.640519\pi\)
\(32\) 0 0
\(33\) −161.066 −0.849639
\(34\) 0 0
\(35\) −413.066 −1.99489
\(36\) 0 0
\(37\) −125.689 −0.558463 −0.279231 0.960224i \(-0.590080\pi\)
−0.279231 + 0.960224i \(0.590080\pi\)
\(38\) 0 0
\(39\) −21.4668 −0.0881393
\(40\) 0 0
\(41\) −506.444 −1.92910 −0.964552 0.263892i \(-0.914994\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(42\) 0 0
\(43\) 413.689 1.46714 0.733569 0.679615i \(-0.237853\pi\)
0.733569 + 0.679615i \(0.237853\pi\)
\(44\) 0 0
\(45\) 165.800 0.549244
\(46\) 0 0
\(47\) 313.911 0.974226 0.487113 0.873339i \(-0.338050\pi\)
0.487113 + 0.873339i \(0.338050\pi\)
\(48\) 0 0
\(49\) 159.755 0.465759
\(50\) 0 0
\(51\) 119.066 0.326914
\(52\) 0 0
\(53\) 44.3331 0.114898 0.0574492 0.998348i \(-0.481703\pi\)
0.0574492 + 0.998348i \(0.481703\pi\)
\(54\) 0 0
\(55\) −989.066 −2.42483
\(56\) 0 0
\(57\) −377.066 −0.876205
\(58\) 0 0
\(59\) 324.000 0.714936 0.357468 0.933925i \(-0.383640\pi\)
0.357468 + 0.933925i \(0.383640\pi\)
\(60\) 0 0
\(61\) −324.000 −0.680065 −0.340032 0.940414i \(-0.610438\pi\)
−0.340032 + 0.940414i \(0.610438\pi\)
\(62\) 0 0
\(63\) −201.800 −0.403562
\(64\) 0 0
\(65\) −131.822 −0.251546
\(66\) 0 0
\(67\) 464.266 0.846554 0.423277 0.906000i \(-0.360880\pi\)
0.423277 + 0.906000i \(0.360880\pi\)
\(68\) 0 0
\(69\) −297.467 −0.518997
\(70\) 0 0
\(71\) −1052.84 −1.75985 −0.879927 0.475109i \(-0.842409\pi\)
−0.879927 + 0.475109i \(0.842409\pi\)
\(72\) 0 0
\(73\) 1022.27 1.63900 0.819501 0.573078i \(-0.194251\pi\)
0.819501 + 0.573078i \(0.194251\pi\)
\(74\) 0 0
\(75\) 643.133 0.990168
\(76\) 0 0
\(77\) 1203.82 1.78167
\(78\) 0 0
\(79\) 602.910 0.858642 0.429321 0.903152i \(-0.358753\pi\)
0.429321 + 0.903152i \(0.358753\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 15.8217 0.0209236 0.0104618 0.999945i \(-0.496670\pi\)
0.0104618 + 0.999945i \(0.496670\pi\)
\(84\) 0 0
\(85\) 731.156 0.933000
\(86\) 0 0
\(87\) −617.400 −0.760830
\(88\) 0 0
\(89\) 381.378 0.454224 0.227112 0.973869i \(-0.427072\pi\)
0.227112 + 0.973869i \(0.427072\pi\)
\(90\) 0 0
\(91\) 160.444 0.184825
\(92\) 0 0
\(93\) −442.466 −0.493350
\(94\) 0 0
\(95\) −2315.47 −2.50065
\(96\) 0 0
\(97\) 659.154 0.689969 0.344984 0.938608i \(-0.387884\pi\)
0.344984 + 0.938608i \(0.387884\pi\)
\(98\) 0 0
\(99\) −483.199 −0.490539
\(100\) 0 0
\(101\) −498.510 −0.491125 −0.245562 0.969381i \(-0.578973\pi\)
−0.245562 + 0.969381i \(0.578973\pi\)
\(102\) 0 0
\(103\) 196.821 0.188285 0.0941425 0.995559i \(-0.469989\pi\)
0.0941425 + 0.995559i \(0.469989\pi\)
\(104\) 0 0
\(105\) −1239.20 −1.15175
\(106\) 0 0
\(107\) 359.378 0.324695 0.162347 0.986734i \(-0.448093\pi\)
0.162347 + 0.986734i \(0.448093\pi\)
\(108\) 0 0
\(109\) −1969.73 −1.73088 −0.865441 0.501011i \(-0.832962\pi\)
−0.865441 + 0.501011i \(0.832962\pi\)
\(110\) 0 0
\(111\) −377.066 −0.322429
\(112\) 0 0
\(113\) −693.643 −0.577456 −0.288728 0.957411i \(-0.593232\pi\)
−0.288728 + 0.957411i \(0.593232\pi\)
\(114\) 0 0
\(115\) −1826.66 −1.48119
\(116\) 0 0
\(117\) −64.4003 −0.0508873
\(118\) 0 0
\(119\) −889.911 −0.685529
\(120\) 0 0
\(121\) 1551.49 1.16566
\(122\) 0 0
\(123\) −1519.33 −1.11377
\(124\) 0 0
\(125\) 1646.53 1.17816
\(126\) 0 0
\(127\) −2656.78 −1.85631 −0.928153 0.372199i \(-0.878604\pi\)
−0.928153 + 0.372199i \(0.878604\pi\)
\(128\) 0 0
\(129\) 1241.07 0.847053
\(130\) 0 0
\(131\) 615.734 0.410664 0.205332 0.978692i \(-0.434173\pi\)
0.205332 + 0.978692i \(0.434173\pi\)
\(132\) 0 0
\(133\) 2818.22 1.83737
\(134\) 0 0
\(135\) 497.400 0.317106
\(136\) 0 0
\(137\) −613.290 −0.382459 −0.191230 0.981545i \(-0.561247\pi\)
−0.191230 + 0.981545i \(0.561247\pi\)
\(138\) 0 0
\(139\) −1899.29 −1.15896 −0.579480 0.814987i \(-0.696744\pi\)
−0.579480 + 0.814987i \(0.696744\pi\)
\(140\) 0 0
\(141\) 941.733 0.562469
\(142\) 0 0
\(143\) 384.175 0.224660
\(144\) 0 0
\(145\) −3791.29 −2.17137
\(146\) 0 0
\(147\) 479.266 0.268906
\(148\) 0 0
\(149\) 976.377 0.536832 0.268416 0.963303i \(-0.413500\pi\)
0.268416 + 0.963303i \(0.413500\pi\)
\(150\) 0 0
\(151\) −683.132 −0.368162 −0.184081 0.982911i \(-0.558931\pi\)
−0.184081 + 0.982911i \(0.558931\pi\)
\(152\) 0 0
\(153\) 357.199 0.188744
\(154\) 0 0
\(155\) −2717.07 −1.40800
\(156\) 0 0
\(157\) 511.109 0.259815 0.129907 0.991526i \(-0.458532\pi\)
0.129907 + 0.991526i \(0.458532\pi\)
\(158\) 0 0
\(159\) 132.999 0.0663366
\(160\) 0 0
\(161\) 2223.29 1.08832
\(162\) 0 0
\(163\) 2425.95 1.16574 0.582869 0.812566i \(-0.301930\pi\)
0.582869 + 0.812566i \(0.301930\pi\)
\(164\) 0 0
\(165\) −2967.20 −1.39998
\(166\) 0 0
\(167\) −337.332 −0.156309 −0.0781544 0.996941i \(-0.524903\pi\)
−0.0781544 + 0.996941i \(0.524903\pi\)
\(168\) 0 0
\(169\) −2145.80 −0.976694
\(170\) 0 0
\(171\) −1131.20 −0.505877
\(172\) 0 0
\(173\) 2648.29 1.16385 0.581924 0.813243i \(-0.302300\pi\)
0.581924 + 0.813243i \(0.302300\pi\)
\(174\) 0 0
\(175\) −4806.82 −2.07635
\(176\) 0 0
\(177\) 972.000 0.412768
\(178\) 0 0
\(179\) 2907.29 1.21397 0.606986 0.794713i \(-0.292379\pi\)
0.606986 + 0.794713i \(0.292379\pi\)
\(180\) 0 0
\(181\) −3682.80 −1.51238 −0.756188 0.654354i \(-0.772940\pi\)
−0.756188 + 0.654354i \(0.772940\pi\)
\(182\) 0 0
\(183\) −972.000 −0.392636
\(184\) 0 0
\(185\) −2315.47 −0.920197
\(186\) 0 0
\(187\) −2130.85 −0.833277
\(188\) 0 0
\(189\) −605.400 −0.232996
\(190\) 0 0
\(191\) −1279.56 −0.484740 −0.242370 0.970184i \(-0.577925\pi\)
−0.242370 + 0.970184i \(0.577925\pi\)
\(192\) 0 0
\(193\) −4836.84 −1.80396 −0.901978 0.431783i \(-0.857885\pi\)
−0.901978 + 0.431783i \(0.857885\pi\)
\(194\) 0 0
\(195\) −395.465 −0.145230
\(196\) 0 0
\(197\) 2869.31 1.03771 0.518857 0.854861i \(-0.326358\pi\)
0.518857 + 0.854861i \(0.326358\pi\)
\(198\) 0 0
\(199\) −652.242 −0.232343 −0.116171 0.993229i \(-0.537062\pi\)
−0.116171 + 0.993229i \(0.537062\pi\)
\(200\) 0 0
\(201\) 1392.80 0.488758
\(202\) 0 0
\(203\) 4614.49 1.59544
\(204\) 0 0
\(205\) −9329.82 −3.17865
\(206\) 0 0
\(207\) −892.400 −0.299643
\(208\) 0 0
\(209\) 6748.08 2.23337
\(210\) 0 0
\(211\) 537.511 0.175373 0.0876866 0.996148i \(-0.472053\pi\)
0.0876866 + 0.996148i \(0.472053\pi\)
\(212\) 0 0
\(213\) −3158.53 −1.01605
\(214\) 0 0
\(215\) 7621.06 2.41745
\(216\) 0 0
\(217\) 3307.02 1.03454
\(218\) 0 0
\(219\) 3066.80 0.946278
\(220\) 0 0
\(221\) −283.997 −0.0864421
\(222\) 0 0
\(223\) 4041.98 1.21377 0.606885 0.794790i \(-0.292419\pi\)
0.606885 + 0.794790i \(0.292419\pi\)
\(224\) 0 0
\(225\) 1929.40 0.571674
\(226\) 0 0
\(227\) 3070.22 0.897699 0.448850 0.893607i \(-0.351834\pi\)
0.448850 + 0.893607i \(0.351834\pi\)
\(228\) 0 0
\(229\) 205.110 0.0591881 0.0295940 0.999562i \(-0.490579\pi\)
0.0295940 + 0.999562i \(0.490579\pi\)
\(230\) 0 0
\(231\) 3611.47 1.02864
\(232\) 0 0
\(233\) 13.3776 0.00376137 0.00188068 0.999998i \(-0.499401\pi\)
0.00188068 + 0.999998i \(0.499401\pi\)
\(234\) 0 0
\(235\) 5782.93 1.60526
\(236\) 0 0
\(237\) 1808.73 0.495737
\(238\) 0 0
\(239\) −4327.99 −1.17136 −0.585679 0.810543i \(-0.699172\pi\)
−0.585679 + 0.810543i \(0.699172\pi\)
\(240\) 0 0
\(241\) −1508.31 −0.403150 −0.201575 0.979473i \(-0.564606\pi\)
−0.201575 + 0.979473i \(0.564606\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 2943.04 0.767446
\(246\) 0 0
\(247\) 899.378 0.231684
\(248\) 0 0
\(249\) 47.4652 0.0120803
\(250\) 0 0
\(251\) −4871.47 −1.22504 −0.612518 0.790456i \(-0.709843\pi\)
−0.612518 + 0.790456i \(0.709843\pi\)
\(252\) 0 0
\(253\) 5323.55 1.32288
\(254\) 0 0
\(255\) 2193.47 0.538668
\(256\) 0 0
\(257\) −1665.55 −0.404258 −0.202129 0.979359i \(-0.564786\pi\)
−0.202129 + 0.979359i \(0.564786\pi\)
\(258\) 0 0
\(259\) 2818.22 0.676122
\(260\) 0 0
\(261\) −1852.20 −0.439265
\(262\) 0 0
\(263\) −7167.64 −1.68052 −0.840258 0.542188i \(-0.817596\pi\)
−0.840258 + 0.542188i \(0.817596\pi\)
\(264\) 0 0
\(265\) 816.713 0.189322
\(266\) 0 0
\(267\) 1144.13 0.262246
\(268\) 0 0
\(269\) 5453.84 1.23616 0.618079 0.786116i \(-0.287911\pi\)
0.618079 + 0.786116i \(0.287911\pi\)
\(270\) 0 0
\(271\) 5416.20 1.21406 0.607031 0.794678i \(-0.292360\pi\)
0.607031 + 0.794678i \(0.292360\pi\)
\(272\) 0 0
\(273\) 481.332 0.106709
\(274\) 0 0
\(275\) −11509.7 −2.52385
\(276\) 0 0
\(277\) 2648.75 0.574542 0.287271 0.957849i \(-0.407252\pi\)
0.287271 + 0.957849i \(0.407252\pi\)
\(278\) 0 0
\(279\) −1327.40 −0.284836
\(280\) 0 0
\(281\) 6664.57 1.41486 0.707429 0.706784i \(-0.249855\pi\)
0.707429 + 0.706784i \(0.249855\pi\)
\(282\) 0 0
\(283\) −5630.84 −1.18275 −0.591376 0.806396i \(-0.701415\pi\)
−0.591376 + 0.806396i \(0.701415\pi\)
\(284\) 0 0
\(285\) −6946.40 −1.44375
\(286\) 0 0
\(287\) 11355.6 2.33554
\(288\) 0 0
\(289\) −3337.80 −0.679381
\(290\) 0 0
\(291\) 1977.46 0.398354
\(292\) 0 0
\(293\) −908.374 −0.181119 −0.0905593 0.995891i \(-0.528865\pi\)
−0.0905593 + 0.995891i \(0.528865\pi\)
\(294\) 0 0
\(295\) 5968.79 1.17802
\(296\) 0 0
\(297\) −1449.60 −0.283213
\(298\) 0 0
\(299\) 709.517 0.137232
\(300\) 0 0
\(301\) −9275.82 −1.77624
\(302\) 0 0
\(303\) −1495.53 −0.283551
\(304\) 0 0
\(305\) −5968.79 −1.12056
\(306\) 0 0
\(307\) −414.671 −0.0770896 −0.0385448 0.999257i \(-0.512272\pi\)
−0.0385448 + 0.999257i \(0.512272\pi\)
\(308\) 0 0
\(309\) 590.463 0.108706
\(310\) 0 0
\(311\) −1615.91 −0.294629 −0.147315 0.989090i \(-0.547063\pi\)
−0.147315 + 0.989090i \(0.547063\pi\)
\(312\) 0 0
\(313\) −8479.33 −1.53125 −0.765623 0.643289i \(-0.777569\pi\)
−0.765623 + 0.643289i \(0.777569\pi\)
\(314\) 0 0
\(315\) −3717.60 −0.664962
\(316\) 0 0
\(317\) 6774.73 1.20034 0.600169 0.799873i \(-0.295100\pi\)
0.600169 + 0.799873i \(0.295100\pi\)
\(318\) 0 0
\(319\) 11049.2 1.93929
\(320\) 0 0
\(321\) 1078.13 0.187463
\(322\) 0 0
\(323\) −4988.44 −0.859332
\(324\) 0 0
\(325\) −1534.00 −0.261818
\(326\) 0 0
\(327\) −5909.20 −0.999325
\(328\) 0 0
\(329\) −7038.57 −1.17948
\(330\) 0 0
\(331\) 9292.36 1.54306 0.771532 0.636191i \(-0.219491\pi\)
0.771532 + 0.636191i \(0.219491\pi\)
\(332\) 0 0
\(333\) −1131.20 −0.186154
\(334\) 0 0
\(335\) 8552.80 1.39489
\(336\) 0 0
\(337\) 6563.78 1.06098 0.530492 0.847690i \(-0.322007\pi\)
0.530492 + 0.847690i \(0.322007\pi\)
\(338\) 0 0
\(339\) −2080.93 −0.333394
\(340\) 0 0
\(341\) 7918.49 1.25751
\(342\) 0 0
\(343\) 4108.75 0.646798
\(344\) 0 0
\(345\) −5479.99 −0.855168
\(346\) 0 0
\(347\) −3870.93 −0.598855 −0.299427 0.954119i \(-0.596796\pi\)
−0.299427 + 0.954119i \(0.596796\pi\)
\(348\) 0 0
\(349\) −3474.57 −0.532922 −0.266461 0.963846i \(-0.585854\pi\)
−0.266461 + 0.963846i \(0.585854\pi\)
\(350\) 0 0
\(351\) −193.201 −0.0293798
\(352\) 0 0
\(353\) −4308.58 −0.649639 −0.324820 0.945776i \(-0.605304\pi\)
−0.324820 + 0.945776i \(0.605304\pi\)
\(354\) 0 0
\(355\) −19395.7 −2.89977
\(356\) 0 0
\(357\) −2669.73 −0.395791
\(358\) 0 0
\(359\) 8161.19 1.19981 0.599904 0.800072i \(-0.295205\pi\)
0.599904 + 0.800072i \(0.295205\pi\)
\(360\) 0 0
\(361\) 8938.68 1.30320
\(362\) 0 0
\(363\) 4654.47 0.672992
\(364\) 0 0
\(365\) 18832.4 2.70064
\(366\) 0 0
\(367\) −4427.66 −0.629760 −0.314880 0.949132i \(-0.601964\pi\)
−0.314880 + 0.949132i \(0.601964\pi\)
\(368\) 0 0
\(369\) −4558.00 −0.643035
\(370\) 0 0
\(371\) −994.045 −0.139106
\(372\) 0 0
\(373\) −11278.5 −1.56562 −0.782812 0.622258i \(-0.786215\pi\)
−0.782812 + 0.622258i \(0.786215\pi\)
\(374\) 0 0
\(375\) 4939.60 0.680213
\(376\) 0 0
\(377\) 1472.62 0.201177
\(378\) 0 0
\(379\) 709.683 0.0961846 0.0480923 0.998843i \(-0.484686\pi\)
0.0480923 + 0.998843i \(0.484686\pi\)
\(380\) 0 0
\(381\) −7970.33 −1.07174
\(382\) 0 0
\(383\) 1233.69 0.164592 0.0822962 0.996608i \(-0.473775\pi\)
0.0822962 + 0.996608i \(0.473775\pi\)
\(384\) 0 0
\(385\) 22177.1 2.93571
\(386\) 0 0
\(387\) 3723.20 0.489046
\(388\) 0 0
\(389\) 6830.06 0.890226 0.445113 0.895474i \(-0.353164\pi\)
0.445113 + 0.895474i \(0.353164\pi\)
\(390\) 0 0
\(391\) −3935.37 −0.509003
\(392\) 0 0
\(393\) 1847.20 0.237097
\(394\) 0 0
\(395\) 11106.9 1.41481
\(396\) 0 0
\(397\) 11289.7 1.42724 0.713618 0.700535i \(-0.247055\pi\)
0.713618 + 0.700535i \(0.247055\pi\)
\(398\) 0 0
\(399\) 8454.66 1.06081
\(400\) 0 0
\(401\) −3055.59 −0.380521 −0.190261 0.981734i \(-0.560933\pi\)
−0.190261 + 0.981734i \(0.560933\pi\)
\(402\) 0 0
\(403\) 1055.37 0.130451
\(404\) 0 0
\(405\) 1492.20 0.183081
\(406\) 0 0
\(407\) 6748.08 0.821843
\(408\) 0 0
\(409\) 4089.01 0.494349 0.247175 0.968971i \(-0.420498\pi\)
0.247175 + 0.968971i \(0.420498\pi\)
\(410\) 0 0
\(411\) −1839.87 −0.220813
\(412\) 0 0
\(413\) −7264.79 −0.865562
\(414\) 0 0
\(415\) 291.471 0.0344765
\(416\) 0 0
\(417\) −5697.86 −0.669126
\(418\) 0 0
\(419\) 15397.0 1.79520 0.897602 0.440806i \(-0.145307\pi\)
0.897602 + 0.440806i \(0.145307\pi\)
\(420\) 0 0
\(421\) −1034.45 −0.119753 −0.0598766 0.998206i \(-0.519071\pi\)
−0.0598766 + 0.998206i \(0.519071\pi\)
\(422\) 0 0
\(423\) 2825.20 0.324742
\(424\) 0 0
\(425\) 8508.40 0.971101
\(426\) 0 0
\(427\) 7264.79 0.823344
\(428\) 0 0
\(429\) 1152.53 0.129707
\(430\) 0 0
\(431\) 4943.86 0.552523 0.276261 0.961083i \(-0.410904\pi\)
0.276261 + 0.961083i \(0.410904\pi\)
\(432\) 0 0
\(433\) 337.202 0.0374247 0.0187124 0.999825i \(-0.494043\pi\)
0.0187124 + 0.999825i \(0.494043\pi\)
\(434\) 0 0
\(435\) −11373.9 −1.25364
\(436\) 0 0
\(437\) 12462.7 1.36424
\(438\) 0 0
\(439\) −4493.93 −0.488573 −0.244286 0.969703i \(-0.578554\pi\)
−0.244286 + 0.969703i \(0.578554\pi\)
\(440\) 0 0
\(441\) 1437.80 0.155253
\(442\) 0 0
\(443\) 4292.26 0.460341 0.230171 0.973150i \(-0.426072\pi\)
0.230171 + 0.973150i \(0.426072\pi\)
\(444\) 0 0
\(445\) 7025.82 0.748440
\(446\) 0 0
\(447\) 2929.13 0.309940
\(448\) 0 0
\(449\) −4167.96 −0.438081 −0.219040 0.975716i \(-0.570293\pi\)
−0.219040 + 0.975716i \(0.570293\pi\)
\(450\) 0 0
\(451\) 27190.4 2.83890
\(452\) 0 0
\(453\) −2049.40 −0.212559
\(454\) 0 0
\(455\) 2955.73 0.304543
\(456\) 0 0
\(457\) 301.643 0.0308759 0.0154380 0.999881i \(-0.495086\pi\)
0.0154380 + 0.999881i \(0.495086\pi\)
\(458\) 0 0
\(459\) 1071.60 0.108971
\(460\) 0 0
\(461\) −9611.88 −0.971085 −0.485542 0.874213i \(-0.661378\pi\)
−0.485542 + 0.874213i \(0.661378\pi\)
\(462\) 0 0
\(463\) 13251.0 1.33008 0.665041 0.746807i \(-0.268414\pi\)
0.665041 + 0.746807i \(0.268414\pi\)
\(464\) 0 0
\(465\) −8151.20 −0.812909
\(466\) 0 0
\(467\) 4432.00 0.439161 0.219581 0.975594i \(-0.429531\pi\)
0.219581 + 0.975594i \(0.429531\pi\)
\(468\) 0 0
\(469\) −10409.9 −1.02491
\(470\) 0 0
\(471\) 1533.33 0.150004
\(472\) 0 0
\(473\) −22210.5 −2.15907
\(474\) 0 0
\(475\) −26944.9 −2.60277
\(476\) 0 0
\(477\) 398.998 0.0382995
\(478\) 0 0
\(479\) −9076.49 −0.865794 −0.432897 0.901443i \(-0.642509\pi\)
−0.432897 + 0.901443i \(0.642509\pi\)
\(480\) 0 0
\(481\) 899.378 0.0852559
\(482\) 0 0
\(483\) 6669.86 0.628342
\(484\) 0 0
\(485\) 12143.1 1.13688
\(486\) 0 0
\(487\) 3343.89 0.311142 0.155571 0.987825i \(-0.450278\pi\)
0.155571 + 0.987825i \(0.450278\pi\)
\(488\) 0 0
\(489\) 7277.86 0.673040
\(490\) 0 0
\(491\) 2423.73 0.222773 0.111386 0.993777i \(-0.464471\pi\)
0.111386 + 0.993777i \(0.464471\pi\)
\(492\) 0 0
\(493\) −8167.95 −0.746179
\(494\) 0 0
\(495\) −8901.60 −0.808277
\(496\) 0 0
\(497\) 23607.1 2.13063
\(498\) 0 0
\(499\) −811.819 −0.0728296 −0.0364148 0.999337i \(-0.511594\pi\)
−0.0364148 + 0.999337i \(0.511594\pi\)
\(500\) 0 0
\(501\) −1012.00 −0.0902449
\(502\) 0 0
\(503\) −18192.4 −1.61264 −0.806320 0.591479i \(-0.798544\pi\)
−0.806320 + 0.591479i \(0.798544\pi\)
\(504\) 0 0
\(505\) −9183.65 −0.809242
\(506\) 0 0
\(507\) −6437.39 −0.563895
\(508\) 0 0
\(509\) −5645.44 −0.491610 −0.245805 0.969319i \(-0.579052\pi\)
−0.245805 + 0.969319i \(0.579052\pi\)
\(510\) 0 0
\(511\) −22921.5 −1.98432
\(512\) 0 0
\(513\) −3393.60 −0.292068
\(514\) 0 0
\(515\) 3625.88 0.310243
\(516\) 0 0
\(517\) −16853.5 −1.43369
\(518\) 0 0
\(519\) 7944.87 0.671948
\(520\) 0 0
\(521\) 12338.7 1.03756 0.518780 0.854908i \(-0.326386\pi\)
0.518780 + 0.854908i \(0.326386\pi\)
\(522\) 0 0
\(523\) 10609.8 0.887062 0.443531 0.896259i \(-0.353726\pi\)
0.443531 + 0.896259i \(0.353726\pi\)
\(524\) 0 0
\(525\) −14420.5 −1.19878
\(526\) 0 0
\(527\) −5853.65 −0.483850
\(528\) 0 0
\(529\) −2335.17 −0.191926
\(530\) 0 0
\(531\) 2916.00 0.238312
\(532\) 0 0
\(533\) 3623.91 0.294501
\(534\) 0 0
\(535\) 6620.53 0.535010
\(536\) 0 0
\(537\) 8721.86 0.700887
\(538\) 0 0
\(539\) −8577.07 −0.685419
\(540\) 0 0
\(541\) 4035.42 0.320696 0.160348 0.987061i \(-0.448738\pi\)
0.160348 + 0.987061i \(0.448738\pi\)
\(542\) 0 0
\(543\) −11048.4 −0.873171
\(544\) 0 0
\(545\) −36286.8 −2.85203
\(546\) 0 0
\(547\) −7407.45 −0.579012 −0.289506 0.957176i \(-0.593491\pi\)
−0.289506 + 0.957176i \(0.593491\pi\)
\(548\) 0 0
\(549\) −2916.00 −0.226688
\(550\) 0 0
\(551\) 25866.7 1.99993
\(552\) 0 0
\(553\) −13518.6 −1.03954
\(554\) 0 0
\(555\) −6946.40 −0.531276
\(556\) 0 0
\(557\) −9500.77 −0.722730 −0.361365 0.932424i \(-0.617689\pi\)
−0.361365 + 0.932424i \(0.617689\pi\)
\(558\) 0 0
\(559\) −2960.19 −0.223976
\(560\) 0 0
\(561\) −6392.54 −0.481093
\(562\) 0 0
\(563\) −2700.26 −0.202136 −0.101068 0.994880i \(-0.532226\pi\)
−0.101068 + 0.994880i \(0.532226\pi\)
\(564\) 0 0
\(565\) −12778.4 −0.951492
\(566\) 0 0
\(567\) −1816.20 −0.134521
\(568\) 0 0
\(569\) 15904.9 1.17183 0.585913 0.810374i \(-0.300736\pi\)
0.585913 + 0.810374i \(0.300736\pi\)
\(570\) 0 0
\(571\) −18234.0 −1.33638 −0.668188 0.743992i \(-0.732930\pi\)
−0.668188 + 0.743992i \(0.732930\pi\)
\(572\) 0 0
\(573\) −3838.67 −0.279865
\(574\) 0 0
\(575\) −21256.7 −1.54168
\(576\) 0 0
\(577\) 4869.57 0.351339 0.175670 0.984449i \(-0.443791\pi\)
0.175670 + 0.984449i \(0.443791\pi\)
\(578\) 0 0
\(579\) −14510.5 −1.04151
\(580\) 0 0
\(581\) −354.758 −0.0253319
\(582\) 0 0
\(583\) −2380.19 −0.169086
\(584\) 0 0
\(585\) −1186.40 −0.0838486
\(586\) 0 0
\(587\) 1616.99 0.113697 0.0568485 0.998383i \(-0.481895\pi\)
0.0568485 + 0.998383i \(0.481895\pi\)
\(588\) 0 0
\(589\) 18537.7 1.29683
\(590\) 0 0
\(591\) 8607.93 0.599125
\(592\) 0 0
\(593\) 8117.01 0.562101 0.281050 0.959693i \(-0.409317\pi\)
0.281050 + 0.959693i \(0.409317\pi\)
\(594\) 0 0
\(595\) −16394.1 −1.12957
\(596\) 0 0
\(597\) −1956.73 −0.134143
\(598\) 0 0
\(599\) −9536.40 −0.650495 −0.325248 0.945629i \(-0.605448\pi\)
−0.325248 + 0.945629i \(0.605448\pi\)
\(600\) 0 0
\(601\) −16247.8 −1.10276 −0.551381 0.834253i \(-0.685899\pi\)
−0.551381 + 0.834253i \(0.685899\pi\)
\(602\) 0 0
\(603\) 4178.39 0.282185
\(604\) 0 0
\(605\) 28581.9 1.92069
\(606\) 0 0
\(607\) −27725.7 −1.85396 −0.926980 0.375111i \(-0.877605\pi\)
−0.926980 + 0.375111i \(0.877605\pi\)
\(608\) 0 0
\(609\) 13843.5 0.921125
\(610\) 0 0
\(611\) −2246.22 −0.148727
\(612\) 0 0
\(613\) 927.190 0.0610911 0.0305456 0.999533i \(-0.490276\pi\)
0.0305456 + 0.999533i \(0.490276\pi\)
\(614\) 0 0
\(615\) −27989.5 −1.83519
\(616\) 0 0
\(617\) −18727.8 −1.22196 −0.610982 0.791644i \(-0.709225\pi\)
−0.610982 + 0.791644i \(0.709225\pi\)
\(618\) 0 0
\(619\) −3210.22 −0.208449 −0.104224 0.994554i \(-0.533236\pi\)
−0.104224 + 0.994554i \(0.533236\pi\)
\(620\) 0 0
\(621\) −2677.20 −0.172999
\(622\) 0 0
\(623\) −8551.33 −0.549922
\(624\) 0 0
\(625\) 3535.57 0.226276
\(626\) 0 0
\(627\) 20244.3 1.28944
\(628\) 0 0
\(629\) −4988.44 −0.316220
\(630\) 0 0
\(631\) 11911.2 0.751468 0.375734 0.926728i \(-0.377391\pi\)
0.375734 + 0.926728i \(0.377391\pi\)
\(632\) 0 0
\(633\) 1612.53 0.101252
\(634\) 0 0
\(635\) −48943.7 −3.05869
\(636\) 0 0
\(637\) −1143.14 −0.0711036
\(638\) 0 0
\(639\) −9475.60 −0.586618
\(640\) 0 0
\(641\) 17232.6 1.06185 0.530924 0.847419i \(-0.321845\pi\)
0.530924 + 0.847419i \(0.321845\pi\)
\(642\) 0 0
\(643\) 12754.6 0.782262 0.391131 0.920335i \(-0.372084\pi\)
0.391131 + 0.920335i \(0.372084\pi\)
\(644\) 0 0
\(645\) 22863.2 1.39572
\(646\) 0 0
\(647\) −9441.73 −0.573714 −0.286857 0.957973i \(-0.592610\pi\)
−0.286857 + 0.957973i \(0.592610\pi\)
\(648\) 0 0
\(649\) −17395.2 −1.05211
\(650\) 0 0
\(651\) 9921.06 0.597292
\(652\) 0 0
\(653\) −5198.99 −0.311565 −0.155783 0.987791i \(-0.549790\pi\)
−0.155783 + 0.987791i \(0.549790\pi\)
\(654\) 0 0
\(655\) 11343.2 0.676664
\(656\) 0 0
\(657\) 9200.39 0.546334
\(658\) 0 0
\(659\) −6508.01 −0.384698 −0.192349 0.981327i \(-0.561611\pi\)
−0.192349 + 0.981327i \(0.561611\pi\)
\(660\) 0 0
\(661\) 25280.2 1.48757 0.743785 0.668419i \(-0.233028\pi\)
0.743785 + 0.668419i \(0.233028\pi\)
\(662\) 0 0
\(663\) −851.991 −0.0499074
\(664\) 0 0
\(665\) 51917.8 3.02750
\(666\) 0 0
\(667\) 20406.2 1.18460
\(668\) 0 0
\(669\) 12125.9 0.700770
\(670\) 0 0
\(671\) 17395.2 1.00079
\(672\) 0 0
\(673\) −5525.89 −0.316504 −0.158252 0.987399i \(-0.550586\pi\)
−0.158252 + 0.987399i \(0.550586\pi\)
\(674\) 0 0
\(675\) 5788.20 0.330056
\(676\) 0 0
\(677\) −6293.21 −0.357264 −0.178632 0.983916i \(-0.557167\pi\)
−0.178632 + 0.983916i \(0.557167\pi\)
\(678\) 0 0
\(679\) −14779.7 −0.835335
\(680\) 0 0
\(681\) 9210.66 0.518287
\(682\) 0 0
\(683\) 5675.91 0.317984 0.158992 0.987280i \(-0.449176\pi\)
0.158992 + 0.987280i \(0.449176\pi\)
\(684\) 0 0
\(685\) −11298.2 −0.630190
\(686\) 0 0
\(687\) 615.331 0.0341722
\(688\) 0 0
\(689\) −317.229 −0.0175406
\(690\) 0 0
\(691\) 3617.79 0.199171 0.0995854 0.995029i \(-0.468248\pi\)
0.0995854 + 0.995029i \(0.468248\pi\)
\(692\) 0 0
\(693\) 10834.4 0.593888
\(694\) 0 0
\(695\) −34989.1 −1.90966
\(696\) 0 0
\(697\) −20100.2 −1.09232
\(698\) 0 0
\(699\) 40.1329 0.00217163
\(700\) 0 0
\(701\) −7938.43 −0.427718 −0.213859 0.976865i \(-0.568603\pi\)
−0.213859 + 0.976865i \(0.568603\pi\)
\(702\) 0 0
\(703\) 15797.7 0.847540
\(704\) 0 0
\(705\) 17348.8 0.926799
\(706\) 0 0
\(707\) 11177.7 0.594597
\(708\) 0 0
\(709\) −25691.6 −1.36088 −0.680442 0.732802i \(-0.738212\pi\)
−0.680442 + 0.732802i \(0.738212\pi\)
\(710\) 0 0
\(711\) 5426.19 0.286214
\(712\) 0 0
\(713\) 14624.3 0.768142
\(714\) 0 0
\(715\) 7077.35 0.370179
\(716\) 0 0
\(717\) −12984.0 −0.676284
\(718\) 0 0
\(719\) −36803.3 −1.90895 −0.954473 0.298298i \(-0.903581\pi\)
−0.954473 + 0.298298i \(0.903581\pi\)
\(720\) 0 0
\(721\) −4413.16 −0.227954
\(722\) 0 0
\(723\) −4524.94 −0.232759
\(724\) 0 0
\(725\) −44118.9 −2.26005
\(726\) 0 0
\(727\) −28333.2 −1.44542 −0.722709 0.691152i \(-0.757103\pi\)
−0.722709 + 0.691152i \(0.757103\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 16418.8 0.830742
\(732\) 0 0
\(733\) −10767.9 −0.542592 −0.271296 0.962496i \(-0.587452\pi\)
−0.271296 + 0.962496i \(0.587452\pi\)
\(734\) 0 0
\(735\) 8829.13 0.443085
\(736\) 0 0
\(737\) −24925.9 −1.24580
\(738\) 0 0
\(739\) 17301.4 0.861221 0.430610 0.902538i \(-0.358298\pi\)
0.430610 + 0.902538i \(0.358298\pi\)
\(740\) 0 0
\(741\) 2698.13 0.133763
\(742\) 0 0
\(743\) −24110.8 −1.19050 −0.595248 0.803542i \(-0.702946\pi\)
−0.595248 + 0.803542i \(0.702946\pi\)
\(744\) 0 0
\(745\) 17987.0 0.884555
\(746\) 0 0
\(747\) 142.396 0.00697455
\(748\) 0 0
\(749\) −8058.04 −0.393103
\(750\) 0 0
\(751\) −30052.8 −1.46024 −0.730121 0.683318i \(-0.760536\pi\)
−0.730121 + 0.683318i \(0.760536\pi\)
\(752\) 0 0
\(753\) −14614.4 −0.707275
\(754\) 0 0
\(755\) −12584.8 −0.606633
\(756\) 0 0
\(757\) 25599.4 1.22910 0.614549 0.788879i \(-0.289338\pi\)
0.614549 + 0.788879i \(0.289338\pi\)
\(758\) 0 0
\(759\) 15970.6 0.763765
\(760\) 0 0
\(761\) −28904.5 −1.37685 −0.688427 0.725306i \(-0.741698\pi\)
−0.688427 + 0.725306i \(0.741698\pi\)
\(762\) 0 0
\(763\) 44165.7 2.09555
\(764\) 0 0
\(765\) 6580.40 0.311000
\(766\) 0 0
\(767\) −2318.41 −0.109143
\(768\) 0 0
\(769\) −8756.13 −0.410603 −0.205302 0.978699i \(-0.565818\pi\)
−0.205302 + 0.978699i \(0.565818\pi\)
\(770\) 0 0
\(771\) −4996.66 −0.233399
\(772\) 0 0
\(773\) 28289.8 1.31632 0.658159 0.752879i \(-0.271335\pi\)
0.658159 + 0.752879i \(0.271335\pi\)
\(774\) 0 0
\(775\) −31618.3 −1.46550
\(776\) 0 0
\(777\) 8454.66 0.390359
\(778\) 0 0
\(779\) 63654.4 2.92767
\(780\) 0 0
\(781\) 56526.0 2.58983
\(782\) 0 0
\(783\) −5556.60 −0.253610
\(784\) 0 0
\(785\) 9415.75 0.428105
\(786\) 0 0
\(787\) −4859.74 −0.220116 −0.110058 0.993925i \(-0.535104\pi\)
−0.110058 + 0.993925i \(0.535104\pi\)
\(788\) 0 0
\(789\) −21502.9 −0.970246
\(790\) 0 0
\(791\) 15553.0 0.699117
\(792\) 0 0
\(793\) 2318.41 0.103820
\(794\) 0 0
\(795\) 2450.14 0.109305
\(796\) 0 0
\(797\) 17361.7 0.771623 0.385811 0.922578i \(-0.373922\pi\)
0.385811 + 0.922578i \(0.373922\pi\)
\(798\) 0 0
\(799\) 12458.8 0.551638
\(800\) 0 0
\(801\) 3432.40 0.151408
\(802\) 0 0
\(803\) −54884.2 −2.41198
\(804\) 0 0
\(805\) 40957.8 1.79326
\(806\) 0 0
\(807\) 16361.5 0.713696
\(808\) 0 0
\(809\) −24475.3 −1.06367 −0.531833 0.846849i \(-0.678497\pi\)
−0.531833 + 0.846849i \(0.678497\pi\)
\(810\) 0 0
\(811\) −19875.4 −0.860566 −0.430283 0.902694i \(-0.641586\pi\)
−0.430283 + 0.902694i \(0.641586\pi\)
\(812\) 0 0
\(813\) 16248.6 0.700939
\(814\) 0 0
\(815\) 44691.4 1.92083
\(816\) 0 0
\(817\) −51996.1 −2.22658
\(818\) 0 0
\(819\) 1444.00 0.0616085
\(820\) 0 0
\(821\) 21682.1 0.921693 0.460846 0.887480i \(-0.347546\pi\)
0.460846 + 0.887480i \(0.347546\pi\)
\(822\) 0 0
\(823\) −6698.17 −0.283698 −0.141849 0.989888i \(-0.545305\pi\)
−0.141849 + 0.989888i \(0.545305\pi\)
\(824\) 0 0
\(825\) −34529.0 −1.45715
\(826\) 0 0
\(827\) 3390.85 0.142577 0.0712886 0.997456i \(-0.477289\pi\)
0.0712886 + 0.997456i \(0.477289\pi\)
\(828\) 0 0
\(829\) −40093.4 −1.67974 −0.839869 0.542789i \(-0.817368\pi\)
−0.839869 + 0.542789i \(0.817368\pi\)
\(830\) 0 0
\(831\) 7946.25 0.331712
\(832\) 0 0
\(833\) 6340.50 0.263728
\(834\) 0 0
\(835\) −6214.40 −0.257555
\(836\) 0 0
\(837\) −3982.19 −0.164450
\(838\) 0 0
\(839\) 6172.12 0.253975 0.126988 0.991904i \(-0.459469\pi\)
0.126988 + 0.991904i \(0.459469\pi\)
\(840\) 0 0
\(841\) 17964.6 0.736585
\(842\) 0 0
\(843\) 19993.7 0.816869
\(844\) 0 0
\(845\) −39530.3 −1.60933
\(846\) 0 0
\(847\) −34787.8 −1.41124
\(848\) 0 0
\(849\) −16892.5 −0.682862
\(850\) 0 0
\(851\) 12462.7 0.502018
\(852\) 0 0
\(853\) −276.632 −0.0111040 −0.00555198 0.999985i \(-0.501767\pi\)
−0.00555198 + 0.999985i \(0.501767\pi\)
\(854\) 0 0
\(855\) −20839.2 −0.833550
\(856\) 0 0
\(857\) 3704.41 0.147655 0.0738274 0.997271i \(-0.476479\pi\)
0.0738274 + 0.997271i \(0.476479\pi\)
\(858\) 0 0
\(859\) −26915.5 −1.06909 −0.534544 0.845141i \(-0.679516\pi\)
−0.534544 + 0.845141i \(0.679516\pi\)
\(860\) 0 0
\(861\) 34066.8 1.34842
\(862\) 0 0
\(863\) 23623.9 0.931828 0.465914 0.884830i \(-0.345726\pi\)
0.465914 + 0.884830i \(0.345726\pi\)
\(864\) 0 0
\(865\) 48787.3 1.91771
\(866\) 0 0
\(867\) −10013.4 −0.392241
\(868\) 0 0
\(869\) −32369.5 −1.26359
\(870\) 0 0
\(871\) −3322.10 −0.129236
\(872\) 0 0
\(873\) 5932.39 0.229990
\(874\) 0 0
\(875\) −36918.9 −1.42638
\(876\) 0 0
\(877\) 14094.0 0.542667 0.271333 0.962485i \(-0.412535\pi\)
0.271333 + 0.962485i \(0.412535\pi\)
\(878\) 0 0
\(879\) −2725.12 −0.104569
\(880\) 0 0
\(881\) 18967.3 0.725341 0.362671 0.931917i \(-0.381865\pi\)
0.362671 + 0.931917i \(0.381865\pi\)
\(882\) 0 0
\(883\) −32886.0 −1.25334 −0.626672 0.779283i \(-0.715583\pi\)
−0.626672 + 0.779283i \(0.715583\pi\)
\(884\) 0 0
\(885\) 17906.4 0.680132
\(886\) 0 0
\(887\) 27945.6 1.05786 0.528929 0.848666i \(-0.322594\pi\)
0.528929 + 0.848666i \(0.322594\pi\)
\(888\) 0 0
\(889\) 59570.8 2.24740
\(890\) 0 0
\(891\) −4348.79 −0.163513
\(892\) 0 0
\(893\) −39455.1 −1.47852
\(894\) 0 0
\(895\) 53558.6 2.00030
\(896\) 0 0
\(897\) 2128.55 0.0792310
\(898\) 0 0
\(899\) 30353.1 1.12607
\(900\) 0 0
\(901\) 1759.53 0.0650592
\(902\) 0 0
\(903\) −27827.4 −1.02551
\(904\) 0 0
\(905\) −67845.2 −2.49199
\(906\) 0 0
\(907\) 5544.89 0.202993 0.101497 0.994836i \(-0.467637\pi\)
0.101497 + 0.994836i \(0.467637\pi\)
\(908\) 0 0
\(909\) −4486.59 −0.163708
\(910\) 0 0
\(911\) −19638.9 −0.714233 −0.357116 0.934060i \(-0.616240\pi\)
−0.357116 + 0.934060i \(0.616240\pi\)
\(912\) 0 0
\(913\) −849.451 −0.0307916
\(914\) 0 0
\(915\) −17906.4 −0.646958
\(916\) 0 0
\(917\) −13806.1 −0.497184
\(918\) 0 0
\(919\) −22128.7 −0.794297 −0.397149 0.917754i \(-0.630000\pi\)
−0.397149 + 0.917754i \(0.630000\pi\)
\(920\) 0 0
\(921\) −1244.01 −0.0445077
\(922\) 0 0
\(923\) 7533.72 0.268663
\(924\) 0 0
\(925\) −26944.9 −0.957775
\(926\) 0 0
\(927\) 1771.39 0.0627616
\(928\) 0 0
\(929\) 17599.3 0.621544 0.310772 0.950484i \(-0.399412\pi\)
0.310772 + 0.950484i \(0.399412\pi\)
\(930\) 0 0
\(931\) −20079.5 −0.706850
\(932\) 0 0
\(933\) −4847.72 −0.170104
\(934\) 0 0
\(935\) −39254.9 −1.37302
\(936\) 0 0
\(937\) 441.299 0.0153859 0.00769297 0.999970i \(-0.497551\pi\)
0.00769297 + 0.999970i \(0.497551\pi\)
\(938\) 0 0
\(939\) −25438.0 −0.884065
\(940\) 0 0
\(941\) 5408.01 0.187350 0.0936749 0.995603i \(-0.470139\pi\)
0.0936749 + 0.995603i \(0.470139\pi\)
\(942\) 0 0
\(943\) 50216.8 1.73413
\(944\) 0 0
\(945\) −11152.8 −0.383916
\(946\) 0 0
\(947\) 34237.1 1.17482 0.587410 0.809289i \(-0.300148\pi\)
0.587410 + 0.809289i \(0.300148\pi\)
\(948\) 0 0
\(949\) −7314.92 −0.250213
\(950\) 0 0
\(951\) 20324.2 0.693015
\(952\) 0 0
\(953\) 21410.7 0.727766 0.363883 0.931445i \(-0.381451\pi\)
0.363883 + 0.931445i \(0.381451\pi\)
\(954\) 0 0
\(955\) −23572.2 −0.798722
\(956\) 0 0
\(957\) 33147.5 1.11965
\(958\) 0 0
\(959\) 13751.3 0.463038
\(960\) 0 0
\(961\) −8038.09 −0.269816
\(962\) 0 0
\(963\) 3234.40 0.108232
\(964\) 0 0
\(965\) −89105.3 −2.97243
\(966\) 0 0
\(967\) 53874.6 1.79161 0.895806 0.444445i \(-0.146599\pi\)
0.895806 + 0.444445i \(0.146599\pi\)
\(968\) 0 0
\(969\) −14965.3 −0.496136
\(970\) 0 0
\(971\) −42901.5 −1.41789 −0.708947 0.705262i \(-0.750829\pi\)
−0.708947 + 0.705262i \(0.750829\pi\)
\(972\) 0 0
\(973\) 42586.2 1.40314
\(974\) 0 0
\(975\) −4602.00 −0.151161
\(976\) 0 0
\(977\) 58636.5 1.92011 0.960055 0.279812i \(-0.0902721\pi\)
0.960055 + 0.279812i \(0.0902721\pi\)
\(978\) 0 0
\(979\) −20475.7 −0.668444
\(980\) 0 0
\(981\) −17727.6 −0.576961
\(982\) 0 0
\(983\) 25296.7 0.820793 0.410396 0.911907i \(-0.365390\pi\)
0.410396 + 0.911907i \(0.365390\pi\)
\(984\) 0 0
\(985\) 52859.0 1.70988
\(986\) 0 0
\(987\) −21115.7 −0.680973
\(988\) 0 0
\(989\) −41019.6 −1.31885
\(990\) 0 0
\(991\) −10605.2 −0.339944 −0.169972 0.985449i \(-0.554368\pi\)
−0.169972 + 0.985449i \(0.554368\pi\)
\(992\) 0 0
\(993\) 27877.1 0.890888
\(994\) 0 0
\(995\) −12015.7 −0.382839
\(996\) 0 0
\(997\) −5770.19 −0.183294 −0.0916469 0.995792i \(-0.529213\pi\)
−0.0916469 + 0.995792i \(0.529213\pi\)
\(998\) 0 0
\(999\) −3393.60 −0.107476
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 768.4.a.p.1.2 2
3.2 odd 2 2304.4.a.s.1.1 2
4.3 odd 2 768.4.a.j.1.2 2
8.3 odd 2 768.4.a.k.1.1 2
8.5 even 2 768.4.a.e.1.1 2
12.11 even 2 2304.4.a.t.1.1 2
16.3 odd 4 384.4.d.c.193.1 4
16.5 even 4 384.4.d.e.193.2 yes 4
16.11 odd 4 384.4.d.c.193.4 yes 4
16.13 even 4 384.4.d.e.193.3 yes 4
24.5 odd 2 2304.4.a.bp.1.2 2
24.11 even 2 2304.4.a.bq.1.2 2
48.5 odd 4 1152.4.d.o.577.1 4
48.11 even 4 1152.4.d.i.577.1 4
48.29 odd 4 1152.4.d.o.577.4 4
48.35 even 4 1152.4.d.i.577.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.1 4 16.3 odd 4
384.4.d.c.193.4 yes 4 16.11 odd 4
384.4.d.e.193.2 yes 4 16.5 even 4
384.4.d.e.193.3 yes 4 16.13 even 4
768.4.a.e.1.1 2 8.5 even 2
768.4.a.j.1.2 2 4.3 odd 2
768.4.a.k.1.1 2 8.3 odd 2
768.4.a.p.1.2 2 1.1 even 1 trivial
1152.4.d.i.577.1 4 48.11 even 4
1152.4.d.i.577.4 4 48.35 even 4
1152.4.d.o.577.1 4 48.5 odd 4
1152.4.d.o.577.4 4 48.29 odd 4
2304.4.a.s.1.1 2 3.2 odd 2
2304.4.a.t.1.1 2 12.11 even 2
2304.4.a.bp.1.2 2 24.5 odd 2
2304.4.a.bq.1.2 2 24.11 even 2