Properties

Label 1152.4.d.o.577.4
Level $1152$
Weight $4$
Character 1152.577
Analytic conductor $67.970$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(577,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1152.577
Dual form 1152.4.d.o.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+18.4222i q^{5} +22.4222 q^{7} +O(q^{10})\) \(q+18.4222i q^{5} +22.4222 q^{7} -53.6888i q^{11} -7.15559i q^{13} -39.6888 q^{17} -125.689i q^{19} -99.1556 q^{23} -214.378 q^{25} +205.800i q^{29} -147.489 q^{31} +413.066i q^{35} +125.689i q^{37} -506.444 q^{41} -413.689i q^{43} -313.911 q^{47} +159.755 q^{49} +44.3331i q^{53} +989.066 q^{55} +324.000i q^{59} -324.000i q^{61} +131.822 q^{65} +464.266i q^{67} -1052.84 q^{71} -1022.27 q^{73} -1203.82i q^{77} +602.910 q^{79} -15.8217i q^{83} -731.156i q^{85} +381.378 q^{89} -160.444i q^{91} +2315.47 q^{95} +659.154 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{7} + 72 q^{17} - 512 q^{23} - 396 q^{25} + 160 q^{31} - 872 q^{41} - 448 q^{47} - 284 q^{49} + 3264 q^{55} - 1088 q^{65} - 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 1064 q^{89} + 4416 q^{95} - 2440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 18.4222i 1.64773i 0.566785 + 0.823866i \(0.308187\pi\)
−0.566785 + 0.823866i \(0.691813\pi\)
\(6\) 0 0
\(7\) 22.4222 1.21069 0.605343 0.795965i \(-0.293036\pi\)
0.605343 + 0.795965i \(0.293036\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 53.6888i − 1.47162i −0.677190 0.735809i \(-0.736802\pi\)
0.677190 0.735809i \(-0.263198\pi\)
\(12\) 0 0
\(13\) − 7.15559i − 0.152662i −0.997083 0.0763309i \(-0.975679\pi\)
0.997083 0.0763309i \(-0.0243205\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −39.6888 −0.566233 −0.283116 0.959086i \(-0.591368\pi\)
−0.283116 + 0.959086i \(0.591368\pi\)
\(18\) 0 0
\(19\) − 125.689i − 1.51763i −0.651306 0.758816i \(-0.725778\pi\)
0.651306 0.758816i \(-0.274222\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(27\) 0 0
\(28\) 0 0
\(29\) 205.800i 1.31780i 0.752232 + 0.658898i \(0.228977\pi\)
−0.752232 + 0.658898i \(0.771023\pi\)
\(30\) 0 0
\(31\) −147.489 −0.854508 −0.427254 0.904132i \(-0.640519\pi\)
−0.427254 + 0.904132i \(0.640519\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 413.066i 1.99489i
\(36\) 0 0
\(37\) 125.689i 0.558463i 0.960224 + 0.279231i \(0.0900796\pi\)
−0.960224 + 0.279231i \(0.909920\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −506.444 −1.92910 −0.964552 0.263892i \(-0.914994\pi\)
−0.964552 + 0.263892i \(0.914994\pi\)
\(42\) 0 0
\(43\) − 413.689i − 1.46714i −0.679615 0.733569i \(-0.737853\pi\)
0.679615 0.733569i \(-0.262147\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −313.911 −0.974226 −0.487113 0.873339i \(-0.661950\pi\)
−0.487113 + 0.873339i \(0.661950\pi\)
\(48\) 0 0
\(49\) 159.755 0.465759
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 44.3331i 0.114898i 0.998348 + 0.0574492i \(0.0182967\pi\)
−0.998348 + 0.0574492i \(0.981703\pi\)
\(54\) 0 0
\(55\) 989.066 2.42483
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 324.000i 0.714936i 0.933925 + 0.357468i \(0.116360\pi\)
−0.933925 + 0.357468i \(0.883640\pi\)
\(60\) 0 0
\(61\) − 324.000i − 0.680065i −0.940414 0.340032i \(-0.889562\pi\)
0.940414 0.340032i \(-0.110438\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 131.822 0.251546
\(66\) 0 0
\(67\) 464.266i 0.846554i 0.906000 + 0.423277i \(0.139120\pi\)
−0.906000 + 0.423277i \(0.860880\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 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 \(-0.805749\pi\)
−0.819501 + 0.573078i \(0.805749\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1203.82i − 1.78167i
\(78\) 0 0
\(79\) 602.910 0.858642 0.429321 0.903152i \(-0.358753\pi\)
0.429321 + 0.903152i \(0.358753\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 15.8217i − 0.0209236i −0.999945 0.0104618i \(-0.996670\pi\)
0.999945 0.0104618i \(-0.00333016\pi\)
\(84\) 0 0
\(85\) − 731.156i − 0.933000i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 381.378 0.454224 0.227112 0.973869i \(-0.427072\pi\)
0.227112 + 0.973869i \(0.427072\pi\)
\(90\) 0 0
\(91\) − 160.444i − 0.184825i
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(99\) 0 0
\(100\) 0 0
\(101\) − 498.510i − 0.491125i −0.969381 0.245562i \(-0.921027\pi\)
0.969381 0.245562i \(-0.0789726\pi\)
\(102\) 0 0
\(103\) −196.821 −0.188285 −0.0941425 0.995559i \(-0.530011\pi\)
−0.0941425 + 0.995559i \(0.530011\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 359.378i 0.324695i 0.986734 + 0.162347i \(0.0519065\pi\)
−0.986734 + 0.162347i \(0.948093\pi\)
\(108\) 0 0
\(109\) − 1969.73i − 1.73088i −0.501011 0.865441i \(-0.667038\pi\)
0.501011 0.865441i \(-0.332962\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 693.643 0.577456 0.288728 0.957411i \(-0.406768\pi\)
0.288728 + 0.957411i \(0.406768\pi\)
\(114\) 0 0
\(115\) − 1826.66i − 1.48119i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −889.911 −0.685529
\(120\) 0 0
\(121\) −1551.49 −1.16566
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1646.53i − 1.17816i
\(126\) 0 0
\(127\) −2656.78 −1.85631 −0.928153 0.372199i \(-0.878604\pi\)
−0.928153 + 0.372199i \(0.878604\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 615.734i − 0.410664i −0.978692 0.205332i \(-0.934173\pi\)
0.978692 0.205332i \(-0.0658274\pi\)
\(132\) 0 0
\(133\) − 2818.22i − 1.83737i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −613.290 −0.382459 −0.191230 0.981545i \(-0.561247\pi\)
−0.191230 + 0.981545i \(0.561247\pi\)
\(138\) 0 0
\(139\) 1899.29i 1.15896i 0.814987 + 0.579480i \(0.196744\pi\)
−0.814987 + 0.579480i \(0.803256\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −384.175 −0.224660
\(144\) 0 0
\(145\) −3791.29 −2.17137
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 976.377i 0.536832i 0.963303 + 0.268416i \(0.0865001\pi\)
−0.963303 + 0.268416i \(0.913500\pi\)
\(150\) 0 0
\(151\) 683.132 0.368162 0.184081 0.982911i \(-0.441069\pi\)
0.184081 + 0.982911i \(0.441069\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 2717.07i − 1.40800i
\(156\) 0 0
\(157\) 511.109i 0.259815i 0.991526 + 0.129907i \(0.0414680\pi\)
−0.991526 + 0.129907i \(0.958532\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2223.29 −1.08832
\(162\) 0 0
\(163\) 2425.95i 1.16574i 0.812566 + 0.582869i \(0.198070\pi\)
−0.812566 + 0.582869i \(0.801930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 0 0
\(173\) − 2648.29i − 1.16385i −0.813243 0.581924i \(-0.802300\pi\)
0.813243 0.581924i \(-0.197700\pi\)
\(174\) 0 0
\(175\) −4806.82 −2.07635
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 2907.29i − 1.21397i −0.794713 0.606986i \(-0.792379\pi\)
0.794713 0.606986i \(-0.207621\pi\)
\(180\) 0 0
\(181\) 3682.80i 1.51238i 0.654354 + 0.756188i \(0.272940\pi\)
−0.654354 + 0.756188i \(0.727060\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2315.47 −0.920197
\(186\) 0 0
\(187\) 2130.85i 0.833277i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1279.56 0.484740 0.242370 0.970184i \(-0.422075\pi\)
0.242370 + 0.970184i \(0.422075\pi\)
\(192\) 0 0
\(193\) −4836.84 −1.80396 −0.901978 0.431783i \(-0.857885\pi\)
−0.901978 + 0.431783i \(0.857885\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2869.31i 1.03771i 0.854861 + 0.518857i \(0.173642\pi\)
−0.854861 + 0.518857i \(0.826358\pi\)
\(198\) 0 0
\(199\) 652.242 0.232343 0.116171 0.993229i \(-0.462938\pi\)
0.116171 + 0.993229i \(0.462938\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4614.49i 1.59544i
\(204\) 0 0
\(205\) − 9329.82i − 3.17865i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6748.08 −2.23337
\(210\) 0 0
\(211\) 537.511i 0.175373i 0.996148 + 0.0876866i \(0.0279474\pi\)
−0.996148 + 0.0876866i \(0.972053\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7621.06 2.41745
\(216\) 0 0
\(217\) −3307.02 −1.03454
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 283.997i 0.0864421i
\(222\) 0 0
\(223\) 4041.98 1.21377 0.606885 0.794790i \(-0.292419\pi\)
0.606885 + 0.794790i \(0.292419\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3070.22i − 0.897699i −0.893607 0.448850i \(-0.851834\pi\)
0.893607 0.448850i \(-0.148166\pi\)
\(228\) 0 0
\(229\) − 205.110i − 0.0591881i −0.999562 0.0295940i \(-0.990579\pi\)
0.999562 0.0295940i \(-0.00942145\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.3776 0.00376137 0.00188068 0.999998i \(-0.499401\pi\)
0.00188068 + 0.999998i \(0.499401\pi\)
\(234\) 0 0
\(235\) − 5782.93i − 1.60526i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4327.99 1.17136 0.585679 0.810543i \(-0.300828\pi\)
0.585679 + 0.810543i \(0.300828\pi\)
\(240\) 0 0
\(241\) −1508.31 −0.403150 −0.201575 0.979473i \(-0.564606\pi\)
−0.201575 + 0.979473i \(0.564606\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2943.04i 0.767446i
\(246\) 0 0
\(247\) −899.378 −0.231684
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 4871.47i − 1.22504i −0.790456 0.612518i \(-0.790157\pi\)
0.790456 0.612518i \(-0.209843\pi\)
\(252\) 0 0
\(253\) 5323.55i 1.32288i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1665.55 0.404258 0.202129 0.979359i \(-0.435214\pi\)
0.202129 + 0.979359i \(0.435214\pi\)
\(258\) 0 0
\(259\) 2818.22i 0.676122i
\(260\) 0 0
\(261\) 0 0
\(262\) 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
\(267\) 0 0
\(268\) 0 0
\(269\) − 5453.84i − 1.23616i −0.786116 0.618079i \(-0.787911\pi\)
0.786116 0.618079i \(-0.212089\pi\)
\(270\) 0 0
\(271\) 5416.20 1.21406 0.607031 0.794678i \(-0.292360\pi\)
0.607031 + 0.794678i \(0.292360\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11509.7i 2.52385i
\(276\) 0 0
\(277\) − 2648.75i − 0.574542i −0.957849 0.287271i \(-0.907252\pi\)
0.957849 0.287271i \(-0.0927479\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6664.57 1.41486 0.707429 0.706784i \(-0.249855\pi\)
0.707429 + 0.706784i \(0.249855\pi\)
\(282\) 0 0
\(283\) 5630.84i 1.18275i 0.806396 + 0.591376i \(0.201415\pi\)
−0.806396 + 0.591376i \(0.798585\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11355.6 −2.33554
\(288\) 0 0
\(289\) −3337.80 −0.679381
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 908.374i − 0.181119i −0.995891 0.0905593i \(-0.971135\pi\)
0.995891 0.0905593i \(-0.0288655\pi\)
\(294\) 0 0
\(295\) −5968.79 −1.17802
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 709.517i 0.137232i
\(300\) 0 0
\(301\) − 9275.82i − 1.77624i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5968.79 1.12056
\(306\) 0 0
\(307\) − 414.671i − 0.0770896i −0.999257 0.0385448i \(-0.987728\pi\)
0.999257 0.0385448i \(-0.0122722\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 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 \(-0.222431\pi\)
0.765623 + 0.643289i \(0.222431\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6774.73i − 1.20034i −0.799873 0.600169i \(-0.795100\pi\)
0.799873 0.600169i \(-0.204900\pi\)
\(318\) 0 0
\(319\) 11049.2 1.93929
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4988.44i 0.859332i
\(324\) 0 0
\(325\) 1534.00i 0.261818i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7038.57 −1.17948
\(330\) 0 0
\(331\) − 9292.36i − 1.54306i −0.636191 0.771532i \(-0.719491\pi\)
0.636191 0.771532i \(-0.280509\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 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
\(339\) 0 0
\(340\) 0 0
\(341\) 7918.49i 1.25751i
\(342\) 0 0
\(343\) −4108.75 −0.646798
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3870.93i − 0.598855i −0.954119 0.299427i \(-0.903204\pi\)
0.954119 0.299427i \(-0.0967956\pi\)
\(348\) 0 0
\(349\) − 3474.57i − 0.532922i −0.963846 0.266461i \(-0.914146\pi\)
0.963846 0.266461i \(-0.0858543\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4308.58 0.649639 0.324820 0.945776i \(-0.394696\pi\)
0.324820 + 0.945776i \(0.394696\pi\)
\(354\) 0 0
\(355\) − 19395.7i − 2.89977i
\(356\) 0 0
\(357\) 0 0
\(358\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) − 18832.4i − 2.70064i
\(366\) 0 0
\(367\) −4427.66 −0.629760 −0.314880 0.949132i \(-0.601964\pi\)
−0.314880 + 0.949132i \(0.601964\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 994.045i 0.139106i
\(372\) 0 0
\(373\) 11278.5i 1.56562i 0.622258 + 0.782812i \(0.286215\pi\)
−0.622258 + 0.782812i \(0.713785\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1472.62 0.201177
\(378\) 0 0
\(379\) − 709.683i − 0.0961846i −0.998843 0.0480923i \(-0.984686\pi\)
0.998843 0.0480923i \(-0.0153142\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1233.69 −0.164592 −0.0822962 0.996608i \(-0.526225\pi\)
−0.0822962 + 0.996608i \(0.526225\pi\)
\(384\) 0 0
\(385\) 22177.1 2.93571
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6830.06i 0.890226i 0.895474 + 0.445113i \(0.146836\pi\)
−0.895474 + 0.445113i \(0.853164\pi\)
\(390\) 0 0
\(391\) 3935.37 0.509003
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11106.9i 1.41481i
\(396\) 0 0
\(397\) 11289.7i 1.42724i 0.700535 + 0.713618i \(0.252945\pi\)
−0.700535 + 0.713618i \(0.747055\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3055.59 0.380521 0.190261 0.981734i \(-0.439067\pi\)
0.190261 + 0.981734i \(0.439067\pi\)
\(402\) 0 0
\(403\) 1055.37i 0.130451i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6748.08 0.821843
\(408\) 0 0
\(409\) −4089.01 −0.494349 −0.247175 0.968971i \(-0.579502\pi\)
−0.247175 + 0.968971i \(0.579502\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7264.79i 0.865562i
\(414\) 0 0
\(415\) 291.471 0.0344765
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 15397.0i − 1.79520i −0.440806 0.897602i \(-0.645307\pi\)
0.440806 0.897602i \(-0.354693\pi\)
\(420\) 0 0
\(421\) 1034.45i 0.119753i 0.998206 + 0.0598766i \(0.0190707\pi\)
−0.998206 + 0.0598766i \(0.980929\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8508.40 0.971101
\(426\) 0 0
\(427\) − 7264.79i − 0.823344i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4943.86 −0.552523 −0.276261 0.961083i \(-0.589096\pi\)
−0.276261 + 0.961083i \(0.589096\pi\)
\(432\) 0 0
\(433\) 337.202 0.0374247 0.0187124 0.999825i \(-0.494043\pi\)
0.0187124 + 0.999825i \(0.494043\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12462.7i 1.36424i
\(438\) 0 0
\(439\) 4493.93 0.488573 0.244286 0.969703i \(-0.421446\pi\)
0.244286 + 0.969703i \(0.421446\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4292.26i 0.460341i 0.973150 + 0.230171i \(0.0739285\pi\)
−0.973150 + 0.230171i \(0.926072\pi\)
\(444\) 0 0
\(445\) 7025.82i 0.748440i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 4167.96 0.438081 0.219040 0.975716i \(-0.429707\pi\)
0.219040 + 0.975716i \(0.429707\pi\)
\(450\) 0 0
\(451\) 27190.4i 2.83890i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2955.73 0.304543
\(456\) 0 0
\(457\) −301.643 −0.0308759 −0.0154380 0.999881i \(-0.504914\pi\)
−0.0154380 + 0.999881i \(0.504914\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9611.88i 0.971085i 0.874213 + 0.485542i \(0.161378\pi\)
−0.874213 + 0.485542i \(0.838622\pi\)
\(462\) 0 0
\(463\) 13251.0 1.33008 0.665041 0.746807i \(-0.268414\pi\)
0.665041 + 0.746807i \(0.268414\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4432.00i − 0.439161i −0.975594 0.219581i \(-0.929531\pi\)
0.975594 0.219581i \(-0.0704689\pi\)
\(468\) 0 0
\(469\) 10409.9i 1.02491i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −22210.5 −2.15907
\(474\) 0 0
\(475\) 26944.9i 2.60277i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9076.49 0.865794 0.432897 0.901443i \(-0.357491\pi\)
0.432897 + 0.901443i \(0.357491\pi\)
\(480\) 0 0
\(481\) 899.378 0.0852559
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12143.1i 1.13688i
\(486\) 0 0
\(487\) −3343.89 −0.311142 −0.155571 0.987825i \(-0.549722\pi\)
−0.155571 + 0.987825i \(0.549722\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2423.73i 0.222773i 0.993777 + 0.111386i \(0.0355291\pi\)
−0.993777 + 0.111386i \(0.964471\pi\)
\(492\) 0 0
\(493\) − 8167.95i − 0.746179i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −23607.1 −2.13063
\(498\) 0 0
\(499\) − 811.819i − 0.0728296i −0.999337 0.0364148i \(-0.988406\pi\)
0.999337 0.0364148i \(-0.0115938\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 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
\(507\) 0 0
\(508\) 0 0
\(509\) 5645.44i 0.491610i 0.969319 + 0.245805i \(0.0790523\pi\)
−0.969319 + 0.245805i \(0.920948\pi\)
\(510\) 0 0
\(511\) −22921.5 −1.98432
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 3625.88i − 0.310243i
\(516\) 0 0
\(517\) 16853.5i 1.43369i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12338.7 1.03756 0.518780 0.854908i \(-0.326386\pi\)
0.518780 + 0.854908i \(0.326386\pi\)
\(522\) 0 0
\(523\) − 10609.8i − 0.887062i −0.896259 0.443531i \(-0.853726\pi\)
0.896259 0.443531i \(-0.146274\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5853.65 0.483850
\(528\) 0 0
\(529\) −2335.17 −0.191926
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3623.91i 0.294501i
\(534\) 0 0
\(535\) −6620.53 −0.535010
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 8577.07i − 0.685419i
\(540\) 0 0
\(541\) 4035.42i 0.320696i 0.987061 + 0.160348i \(0.0512616\pi\)
−0.987061 + 0.160348i \(0.948738\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36286.8 2.85203
\(546\) 0 0
\(547\) − 7407.45i − 0.579012i −0.957176 0.289506i \(-0.906509\pi\)
0.957176 0.289506i \(-0.0934911\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 25866.7 1.99993
\(552\) 0 0
\(553\) 13518.6 1.03954
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9500.77i 0.722730i 0.932424 + 0.361365i \(0.117689\pi\)
−0.932424 + 0.361365i \(0.882311\pi\)
\(558\) 0 0
\(559\) −2960.19 −0.223976
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2700.26i 0.202136i 0.994880 + 0.101068i \(0.0322259\pi\)
−0.994880 + 0.101068i \(0.967774\pi\)
\(564\) 0 0
\(565\) 12778.4i 0.951492i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15904.9 1.17183 0.585913 0.810374i \(-0.300736\pi\)
0.585913 + 0.810374i \(0.300736\pi\)
\(570\) 0 0
\(571\) 18234.0i 1.33638i 0.743992 + 0.668188i \(0.232930\pi\)
−0.743992 + 0.668188i \(0.767070\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 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
\(579\) 0 0
\(580\) 0 0
\(581\) − 354.758i − 0.0253319i
\(582\) 0 0
\(583\) 2380.19 0.169086
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1616.99i 0.113697i 0.998383 + 0.0568485i \(0.0181052\pi\)
−0.998383 + 0.0568485i \(0.981895\pi\)
\(588\) 0 0
\(589\) 18537.7i 1.29683i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −8117.01 −0.562101 −0.281050 0.959693i \(-0.590683\pi\)
−0.281050 + 0.959693i \(0.590683\pi\)
\(594\) 0 0
\(595\) − 16394.1i − 1.12957i
\(596\) 0 0
\(597\) 0 0
\(598\) 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 \(-0.314101\pi\)
0.551381 + 0.834253i \(0.314101\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 28581.9i − 1.92069i
\(606\) 0 0
\(607\) −27725.7 −1.85396 −0.926980 0.375111i \(-0.877605\pi\)
−0.926980 + 0.375111i \(0.877605\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2246.22i 0.148727i
\(612\) 0 0
\(613\) − 927.190i − 0.0610911i −0.999533 0.0305456i \(-0.990276\pi\)
0.999533 0.0305456i \(-0.00972447\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18727.8 −1.22196 −0.610982 0.791644i \(-0.709225\pi\)
−0.610982 + 0.791644i \(0.709225\pi\)
\(618\) 0 0
\(619\) 3210.22i 0.208449i 0.994554 + 0.104224i \(0.0332360\pi\)
−0.994554 + 0.104224i \(0.966764\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8551.33 0.549922
\(624\) 0 0
\(625\) 3535.57 0.226276
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 4988.44i − 0.316220i
\(630\) 0 0
\(631\) −11911.2 −0.751468 −0.375734 0.926728i \(-0.622609\pi\)
−0.375734 + 0.926728i \(0.622609\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 48943.7i − 3.05869i
\(636\) 0 0
\(637\) − 1143.14i − 0.0711036i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −17232.6 −1.06185 −0.530924 0.847419i \(-0.678155\pi\)
−0.530924 + 0.847419i \(0.678155\pi\)
\(642\) 0 0
\(643\) 12754.6i 0.782262i 0.920335 + 0.391131i \(0.127916\pi\)
−0.920335 + 0.391131i \(0.872084\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 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
\(651\) 0 0
\(652\) 0 0
\(653\) 5198.99i 0.311565i 0.987791 + 0.155783i \(0.0497900\pi\)
−0.987791 + 0.155783i \(0.950210\pi\)
\(654\) 0 0
\(655\) 11343.2 0.676664
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6508.01i 0.384698i 0.981327 + 0.192349i \(0.0616105\pi\)
−0.981327 + 0.192349i \(0.938389\pi\)
\(660\) 0 0
\(661\) − 25280.2i − 1.48757i −0.668419 0.743785i \(-0.733028\pi\)
0.668419 0.743785i \(-0.266972\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 51917.8 3.02750
\(666\) 0 0
\(667\) − 20406.2i − 1.18460i
\(668\) 0 0
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 0 0
\(677\) − 6293.21i − 0.357264i −0.983916 0.178632i \(-0.942833\pi\)
0.983916 0.178632i \(-0.0571671\pi\)
\(678\) 0 0
\(679\) 14779.7 0.835335
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5675.91i 0.317984i 0.987280 + 0.158992i \(0.0508243\pi\)
−0.987280 + 0.158992i \(0.949176\pi\)
\(684\) 0 0
\(685\) − 11298.2i − 0.630190i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 317.229 0.0175406
\(690\) 0 0
\(691\) 3617.79i 0.199171i 0.995029 + 0.0995854i \(0.0317517\pi\)
−0.995029 + 0.0995854i \(0.968248\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −34989.1 −1.90966
\(696\) 0 0
\(697\) 20100.2 1.09232
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 7938.43i 0.427718i 0.976865 + 0.213859i \(0.0686033\pi\)
−0.976865 + 0.213859i \(0.931397\pi\)
\(702\) 0 0
\(703\) 15797.7 0.847540
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 11177.7i − 0.594597i
\(708\) 0 0
\(709\) 25691.6i 1.36088i 0.732802 + 0.680442i \(0.238212\pi\)
−0.732802 + 0.680442i \(0.761788\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14624.3 0.768142
\(714\) 0 0
\(715\) − 7077.35i − 0.370179i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 36803.3 1.90895 0.954473 0.298298i \(-0.0964190\pi\)
0.954473 + 0.298298i \(0.0964190\pi\)
\(720\) 0 0
\(721\) −4413.16 −0.227954
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 44118.9i − 2.26005i
\(726\) 0 0
\(727\) 28333.2 1.44542 0.722709 0.691152i \(-0.242897\pi\)
0.722709 + 0.691152i \(0.242897\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16418.8i 0.830742i
\(732\) 0 0
\(733\) − 10767.9i − 0.542592i −0.962496 0.271296i \(-0.912548\pi\)
0.962496 0.271296i \(-0.0874522\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 24925.9 1.24580
\(738\) 0 0
\(739\) 17301.4i 0.861221i 0.902538 + 0.430610i \(0.141702\pi\)
−0.902538 + 0.430610i \(0.858298\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(747\) 0 0
\(748\) 0 0
\(749\) 8058.04i 0.393103i
\(750\) 0 0
\(751\) −30052.8 −1.46024 −0.730121 0.683318i \(-0.760536\pi\)
−0.730121 + 0.683318i \(0.760536\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12584.8i 0.606633i
\(756\) 0 0
\(757\) − 25599.4i − 1.22910i −0.788879 0.614549i \(-0.789338\pi\)
0.788879 0.614549i \(-0.210662\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −28904.5 −1.37685 −0.688427 0.725306i \(-0.741698\pi\)
−0.688427 + 0.725306i \(0.741698\pi\)
\(762\) 0 0
\(763\) − 44165.7i − 2.09555i
\(764\) 0 0
\(765\) 0 0
\(766\) 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
\(771\) 0 0
\(772\) 0 0
\(773\) 28289.8i 1.31632i 0.752879 + 0.658159i \(0.228665\pi\)
−0.752879 + 0.658159i \(0.771335\pi\)
\(774\) 0 0
\(775\) 31618.3 1.46550
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 63654.4i 2.92767i
\(780\) 0 0
\(781\) 56526.0i 2.58983i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9415.75 −0.428105
\(786\) 0 0
\(787\) − 4859.74i − 0.220116i −0.993925 0.110058i \(-0.964896\pi\)
0.993925 0.110058i \(-0.0351036\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 15553.0 0.699117
\(792\) 0 0
\(793\) −2318.41 −0.103820
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 17361.7i − 0.771623i −0.922578 0.385811i \(-0.873922\pi\)
0.922578 0.385811i \(-0.126078\pi\)
\(798\) 0 0
\(799\) 12458.8 0.551638
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 54884.2i 2.41198i
\(804\) 0 0
\(805\) − 40957.8i − 1.79326i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −24475.3 −1.06367 −0.531833 0.846849i \(-0.678497\pi\)
−0.531833 + 0.846849i \(0.678497\pi\)
\(810\) 0 0
\(811\) 19875.4i 0.860566i 0.902694 + 0.430283i \(0.141586\pi\)
−0.902694 + 0.430283i \(0.858414\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −44691.4 −1.92083
\(816\) 0 0
\(817\) −51996.1 −2.22658
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21682.1i 0.921693i 0.887480 + 0.460846i \(0.152454\pi\)
−0.887480 + 0.460846i \(0.847546\pi\)
\(822\) 0 0
\(823\) 6698.17 0.283698 0.141849 0.989888i \(-0.454695\pi\)
0.141849 + 0.989888i \(0.454695\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3390.85i 0.142577i 0.997456 + 0.0712886i \(0.0227111\pi\)
−0.997456 + 0.0712886i \(0.977289\pi\)
\(828\) 0 0
\(829\) − 40093.4i − 1.67974i −0.542789 0.839869i \(-0.682632\pi\)
0.542789 0.839869i \(-0.317368\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6340.50 −0.263728
\(834\) 0 0
\(835\) − 6214.40i − 0.257555i
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(843\) 0 0
\(844\) 0 0
\(845\) 39530.3i 1.60933i
\(846\) 0 0
\(847\) −34787.8 −1.41124
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 12462.7i − 0.502018i
\(852\) 0 0
\(853\) 276.632i 0.0111040i 0.999985 + 0.00555198i \(0.00176726\pi\)
−0.999985 + 0.00555198i \(0.998233\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3704.41 0.147655 0.0738274 0.997271i \(-0.476479\pi\)
0.0738274 + 0.997271i \(0.476479\pi\)
\(858\) 0 0
\(859\) 26915.5i 1.06909i 0.845141 + 0.534544i \(0.179516\pi\)
−0.845141 + 0.534544i \(0.820484\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −23623.9 −0.931828 −0.465914 0.884830i \(-0.654274\pi\)
−0.465914 + 0.884830i \(0.654274\pi\)
\(864\) 0 0
\(865\) 48787.3 1.91771
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 32369.5i − 1.26359i
\(870\) 0 0
\(871\) 3322.10 0.129236
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 36918.9i − 1.42638i
\(876\) 0 0
\(877\) 14094.0i 0.542667i 0.962485 + 0.271333i \(0.0874646\pi\)
−0.962485 + 0.271333i \(0.912535\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18967.3 −0.725341 −0.362671 0.931917i \(-0.618135\pi\)
−0.362671 + 0.931917i \(0.618135\pi\)
\(882\) 0 0
\(883\) − 32886.0i − 1.25334i −0.779283 0.626672i \(-0.784417\pi\)
0.779283 0.626672i \(-0.215583\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) 39455.1i 1.47852i
\(894\) 0 0
\(895\) 53558.6 2.00030
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 30353.1i − 1.12607i
\(900\) 0 0
\(901\) − 1759.53i − 0.0650592i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −67845.2 −2.49199
\(906\) 0 0
\(907\) − 5544.89i − 0.202993i −0.994836 0.101497i \(-0.967637\pi\)
0.994836 0.101497i \(-0.0323631\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 19638.9 0.714233 0.357116 0.934060i \(-0.383760\pi\)
0.357116 + 0.934060i \(0.383760\pi\)
\(912\) 0 0
\(913\) −849.451 −0.0307916
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 13806.1i − 0.497184i
\(918\) 0 0
\(919\) 22128.7 0.794297 0.397149 0.917754i \(-0.370000\pi\)
0.397149 + 0.917754i \(0.370000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7533.72i 0.268663i
\(924\) 0 0
\(925\) − 26944.9i − 0.957775i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17599.3 −0.621544 −0.310772 0.950484i \(-0.600588\pi\)
−0.310772 + 0.950484i \(0.600588\pi\)
\(930\) 0 0
\(931\) − 20079.5i − 0.706850i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −39254.9 −1.37302
\(936\) 0 0
\(937\) −441.299 −0.0153859 −0.00769297 0.999970i \(-0.502449\pi\)
−0.00769297 + 0.999970i \(0.502449\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 5408.01i − 0.187350i −0.995603 0.0936749i \(-0.970139\pi\)
0.995603 0.0936749i \(-0.0298614\pi\)
\(942\) 0 0
\(943\) 50216.8 1.73413
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 34237.1i − 1.17482i −0.809289 0.587410i \(-0.800148\pi\)
0.809289 0.587410i \(-0.199852\pi\)
\(948\) 0 0
\(949\) 7314.92i 0.250213i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 21410.7 0.727766 0.363883 0.931445i \(-0.381451\pi\)
0.363883 + 0.931445i \(0.381451\pi\)
\(954\) 0 0
\(955\) 23572.2i 0.798722i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13751.3 −0.463038
\(960\) 0 0
\(961\) −8038.09 −0.269816
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 89105.3i − 2.97243i
\(966\) 0 0
\(967\) −53874.6 −1.79161 −0.895806 0.444445i \(-0.853401\pi\)
−0.895806 + 0.444445i \(0.853401\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 42901.5i − 1.41789i −0.705262 0.708947i \(-0.749171\pi\)
0.705262 0.708947i \(-0.250829\pi\)
\(972\) 0 0
\(973\) 42586.2i 1.40314i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −58636.5 −1.92011 −0.960055 0.279812i \(-0.909728\pi\)
−0.960055 + 0.279812i \(0.909728\pi\)
\(978\) 0 0
\(979\) − 20475.7i − 0.668444i
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(987\) 0 0
\(988\) 0 0
\(989\) 41019.6i 1.31885i
\(990\) 0 0
\(991\) −10605.2 −0.339944 −0.169972 0.985449i \(-0.554368\pi\)
−0.169972 + 0.985449i \(0.554368\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 12015.7i 0.382839i
\(996\) 0 0
\(997\) 5770.19i 0.183294i 0.995792 + 0.0916469i \(0.0292131\pi\)
−0.995792 + 0.0916469i \(0.970787\pi\)
\(998\) 0 0
\(999\) 0 0
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 1152.4.d.o.577.4 4
3.2 odd 2 384.4.d.e.193.3 yes 4
4.3 odd 2 1152.4.d.i.577.4 4
8.3 odd 2 1152.4.d.i.577.1 4
8.5 even 2 inner 1152.4.d.o.577.1 4
12.11 even 2 384.4.d.c.193.1 4
16.3 odd 4 2304.4.a.bq.1.2 2
16.5 even 4 2304.4.a.s.1.1 2
16.11 odd 4 2304.4.a.t.1.1 2
16.13 even 4 2304.4.a.bp.1.2 2
24.5 odd 2 384.4.d.e.193.2 yes 4
24.11 even 2 384.4.d.c.193.4 yes 4
48.5 odd 4 768.4.a.p.1.2 2
48.11 even 4 768.4.a.j.1.2 2
48.29 odd 4 768.4.a.e.1.1 2
48.35 even 4 768.4.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.1 4 12.11 even 2
384.4.d.c.193.4 yes 4 24.11 even 2
384.4.d.e.193.2 yes 4 24.5 odd 2
384.4.d.e.193.3 yes 4 3.2 odd 2
768.4.a.e.1.1 2 48.29 odd 4
768.4.a.j.1.2 2 48.11 even 4
768.4.a.k.1.1 2 48.35 even 4
768.4.a.p.1.2 2 48.5 odd 4
1152.4.d.i.577.1 4 8.3 odd 2
1152.4.d.i.577.4 4 4.3 odd 2
1152.4.d.o.577.1 4 8.5 even 2 inner
1152.4.d.o.577.4 4 1.1 even 1 trivial
2304.4.a.s.1.1 2 16.5 even 4
2304.4.a.t.1.1 2 16.11 odd 4
2304.4.a.bp.1.2 2 16.13 even 4
2304.4.a.bq.1.2 2 16.3 odd 4