Properties

Label 384.4.d.e.193.3
Level $384$
Weight $4$
Character 384.193
Analytic conductor $22.657$
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] = [384,4,Mod(193,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, 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("384.193");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(22.6567334422\)
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: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.3
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.4.d.e.193.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} -18.4222i q^{5} +22.4222 q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} -18.4222i q^{5} +22.4222 q^{7} -9.00000 q^{9} +53.6888i q^{11} -7.15559i q^{13} +55.2666 q^{15} +39.6888 q^{17} -125.689i q^{19} +67.2666i q^{21} +99.1556 q^{23} -214.378 q^{25} -27.0000i q^{27} -205.800i q^{29} -147.489 q^{31} -161.066 q^{33} -413.066i q^{35} +125.689i q^{37} +21.4668 q^{39} +506.444 q^{41} -413.689i q^{43} +165.800i q^{45} +313.911 q^{47} +159.755 q^{49} +119.066i q^{51} -44.3331i q^{53} +989.066 q^{55} +377.066 q^{57} -324.000i q^{59} -324.000i q^{61} -201.800 q^{63} -131.822 q^{65} +464.266i q^{67} +297.467i q^{69} +1052.84 q^{71} -1022.27 q^{73} -643.133i q^{75} +1203.82i q^{77} +602.910 q^{79} +81.0000 q^{81} +15.8217i q^{83} -731.156i q^{85} +617.400 q^{87} -381.378 q^{89} -160.444i q^{91} -442.466i q^{93} -2315.47 q^{95} +659.154 q^{97} -483.199i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{7} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{7} - 36 q^{9} + 48 q^{15} - 72 q^{17} + 512 q^{23} - 396 q^{25} + 160 q^{31} + 48 q^{33} + 432 q^{39} + 872 q^{41} + 448 q^{47} - 284 q^{49} + 3264 q^{55} + 816 q^{57} - 288 q^{63} + 1088 q^{65} + 4096 q^{71} - 1320 q^{73} - 992 q^{79} + 324 q^{81} + 912 q^{87} - 1064 q^{89} - 4416 q^{95} - 2440 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) − 18.4222i − 1.64773i −0.566785 0.823866i \(-0.691813\pi\)
0.566785 0.823866i \(-0.308187\pi\)
\(6\) 0 0
\(7\) 22.4222 1.21069 0.605343 0.795965i \(-0.293036\pi\)
0.605343 + 0.795965i \(0.293036\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 53.6888i 1.47162i 0.677190 + 0.735809i \(0.263198\pi\)
−0.677190 + 0.735809i \(0.736802\pi\)
\(12\) 0 0
\(13\) − 7.15559i − 0.152662i −0.997083 0.0763309i \(-0.975679\pi\)
0.997083 0.0763309i \(-0.0243205\pi\)
\(14\) 0 0
\(15\) 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
\(19\) − 125.689i − 1.51763i −0.651306 0.758816i \(-0.725778\pi\)
0.651306 0.758816i \(-0.274222\pi\)
\(20\) 0 0
\(21\) 67.2666i 0.698989i
\(22\) 0 0
\(23\) 99.1556 0.898929 0.449465 0.893298i \(-0.351615\pi\)
0.449465 + 0.893298i \(0.351615\pi\)
\(24\) 0 0
\(25\) −214.378 −1.71502
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) − 205.800i − 1.31780i −0.752232 0.658898i \(-0.771023\pi\)
0.752232 0.658898i \(-0.228977\pi\)
\(30\) 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
\(35\) − 413.066i − 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
\(39\) 21.4668 0.0881393
\(40\) 0 0
\(41\) 506.444 1.92910 0.964552 0.263892i \(-0.0850063\pi\)
0.964552 + 0.263892i \(0.0850063\pi\)
\(42\) 0 0
\(43\) − 413.689i − 1.46714i −0.679615 0.733569i \(-0.737853\pi\)
0.679615 0.733569i \(-0.262147\pi\)
\(44\) 0 0
\(45\) 165.800i 0.549244i
\(46\) 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
\(51\) 119.066i 0.326914i
\(52\) 0 0
\(53\) − 44.3331i − 0.114898i −0.998348 0.0574492i \(-0.981703\pi\)
0.998348 0.0574492i \(-0.0182967\pi\)
\(54\) 0 0
\(55\) 989.066 2.42483
\(56\) 0 0
\(57\) 377.066 0.876205
\(58\) 0 0
\(59\) − 324.000i − 0.714936i −0.933925 0.357468i \(-0.883640\pi\)
0.933925 0.357468i \(-0.116360\pi\)
\(60\) 0 0
\(61\) − 324.000i − 0.680065i −0.940414 0.340032i \(-0.889562\pi\)
0.940414 0.340032i \(-0.110438\pi\)
\(62\) 0 0
\(63\) −201.800 −0.403562
\(64\) 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
\(69\) 297.467i 0.518997i
\(70\) 0 0
\(71\) 1052.84 1.75985 0.879927 0.475109i \(-0.157591\pi\)
0.879927 + 0.475109i \(0.157591\pi\)
\(72\) 0 0
\(73\) −1022.27 −1.63900 −0.819501 0.573078i \(-0.805749\pi\)
−0.819501 + 0.573078i \(0.805749\pi\)
\(74\) 0 0
\(75\) − 643.133i − 0.990168i
\(76\) 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
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 15.8217i 0.0209236i 0.999945 + 0.0104618i \(0.00333016\pi\)
−0.999945 + 0.0104618i \(0.996670\pi\)
\(84\) 0 0
\(85\) − 731.156i − 0.933000i
\(86\) 0 0
\(87\) 617.400 0.760830
\(88\) 0 0
\(89\) −381.378 −0.454224 −0.227112 0.973869i \(-0.572928\pi\)
−0.227112 + 0.973869i \(0.572928\pi\)
\(90\) 0 0
\(91\) − 160.444i − 0.184825i
\(92\) 0 0
\(93\) − 442.466i − 0.493350i
\(94\) 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
\(99\) − 483.199i − 0.490539i
\(100\) 0 0
\(101\) 498.510i 0.491125i 0.969381 + 0.245562i \(0.0789726\pi\)
−0.969381 + 0.245562i \(0.921027\pi\)
\(102\) 0 0
\(103\) −196.821 −0.188285 −0.0941425 0.995559i \(-0.530011\pi\)
−0.0941425 + 0.995559i \(0.530011\pi\)
\(104\) 0 0
\(105\) 1239.20 1.15175
\(106\) 0 0
\(107\) − 359.378i − 0.324695i −0.986734 0.162347i \(-0.948093\pi\)
0.986734 0.162347i \(-0.0519065\pi\)
\(108\) 0 0
\(109\) − 1969.73i − 1.73088i −0.501011 0.865441i \(-0.667038\pi\)
0.501011 0.865441i \(-0.332962\pi\)
\(110\) 0 0
\(111\) −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
\(115\) − 1826.66i − 1.48119i
\(116\) 0 0
\(117\) 64.4003i 0.0508873i
\(118\) 0 0
\(119\) 889.911 0.685529
\(120\) 0 0
\(121\) −1551.49 −1.16566
\(122\) 0 0
\(123\) 1519.33i 1.11377i
\(124\) 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
\(129\) 1241.07 0.847053
\(130\) 0 0
\(131\) 615.734i 0.410664i 0.978692 + 0.205332i \(0.0658274\pi\)
−0.978692 + 0.205332i \(0.934173\pi\)
\(132\) 0 0
\(133\) − 2818.22i − 1.83737i
\(134\) 0 0
\(135\) −497.400 −0.317106
\(136\) 0 0
\(137\) 613.290 0.382459 0.191230 0.981545i \(-0.438753\pi\)
0.191230 + 0.981545i \(0.438753\pi\)
\(138\) 0 0
\(139\) 1899.29i 1.15896i 0.814987 + 0.579480i \(0.196744\pi\)
−0.814987 + 0.579480i \(0.803256\pi\)
\(140\) 0 0
\(141\) 941.733i 0.562469i
\(142\) 0 0
\(143\) 384.175 0.224660
\(144\) 0 0
\(145\) −3791.29 −2.17137
\(146\) 0 0
\(147\) 479.266i 0.268906i
\(148\) 0 0
\(149\) − 976.377i − 0.536832i −0.963303 0.268416i \(-0.913500\pi\)
0.963303 0.268416i \(-0.0865001\pi\)
\(150\) 0 0
\(151\) 683.132 0.368162 0.184081 0.982911i \(-0.441069\pi\)
0.184081 + 0.982911i \(0.441069\pi\)
\(152\) 0 0
\(153\) −357.199 −0.188744
\(154\) 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
\(159\) 132.999 0.0663366
\(160\) 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
\(165\) 2967.20i 1.39998i
\(166\) 0 0
\(167\) 337.332 0.156309 0.0781544 0.996941i \(-0.475097\pi\)
0.0781544 + 0.996941i \(0.475097\pi\)
\(168\) 0 0
\(169\) 2145.80 0.976694
\(170\) 0 0
\(171\) 1131.20i 0.505877i
\(172\) 0 0
\(173\) 2648.29i 1.16385i 0.813243 + 0.581924i \(0.197700\pi\)
−0.813243 + 0.581924i \(0.802300\pi\)
\(174\) 0 0
\(175\) −4806.82 −2.07635
\(176\) 0 0
\(177\) 972.000 0.412768
\(178\) 0 0
\(179\) 2907.29i 1.21397i 0.794713 + 0.606986i \(0.207621\pi\)
−0.794713 + 0.606986i \(0.792379\pi\)
\(180\) 0 0
\(181\) 3682.80i 1.51238i 0.654354 + 0.756188i \(0.272940\pi\)
−0.654354 + 0.756188i \(0.727060\pi\)
\(182\) 0 0
\(183\) 972.000 0.392636
\(184\) 0 0
\(185\) 2315.47 0.920197
\(186\) 0 0
\(187\) 2130.85i 0.833277i
\(188\) 0 0
\(189\) − 605.400i − 0.232996i
\(190\) 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
\(195\) − 395.465i − 0.145230i
\(196\) 0 0
\(197\) − 2869.31i − 1.03771i −0.854861 0.518857i \(-0.826358\pi\)
0.854861 0.518857i \(-0.173642\pi\)
\(198\) 0 0
\(199\) 652.242 0.232343 0.116171 0.993229i \(-0.462938\pi\)
0.116171 + 0.993229i \(0.462938\pi\)
\(200\) 0 0
\(201\) −1392.80 −0.488758
\(202\) 0 0
\(203\) − 4614.49i − 1.59544i
\(204\) 0 0
\(205\) − 9329.82i − 3.17865i
\(206\) 0 0
\(207\) −892.400 −0.299643
\(208\) 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
\(213\) 3158.53i 1.01605i
\(214\) 0 0
\(215\) −7621.06 −2.41745
\(216\) 0 0
\(217\) −3307.02 −1.03454
\(218\) 0 0
\(219\) − 3066.80i − 0.946278i
\(220\) 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
\(225\) 1929.40 0.571674
\(226\) 0 0
\(227\) 3070.22i 0.897699i 0.893607 + 0.448850i \(0.148166\pi\)
−0.893607 + 0.448850i \(0.851834\pi\)
\(228\) 0 0
\(229\) − 205.110i − 0.0591881i −0.999562 0.0295940i \(-0.990579\pi\)
0.999562 0.0295940i \(-0.00942145\pi\)
\(230\) 0 0
\(231\) −3611.47 −1.02864
\(232\) 0 0
\(233\) −13.3776 −0.00376137 −0.00188068 0.999998i \(-0.500599\pi\)
−0.00188068 + 0.999998i \(0.500599\pi\)
\(234\) 0 0
\(235\) − 5782.93i − 1.60526i
\(236\) 0 0
\(237\) 1808.73i 0.495737i
\(238\) 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
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) − 2943.04i − 0.767446i
\(246\) 0 0
\(247\) −899.378 −0.231684
\(248\) 0 0
\(249\) −47.4652 −0.0120803
\(250\) 0 0
\(251\) 4871.47i 1.22504i 0.790456 + 0.612518i \(0.209843\pi\)
−0.790456 + 0.612518i \(0.790157\pi\)
\(252\) 0 0
\(253\) 5323.55i 1.32288i
\(254\) 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
\(259\) 2818.22i 0.676122i
\(260\) 0 0
\(261\) 1852.20i 0.439265i
\(262\) 0 0
\(263\) 7167.64 1.68052 0.840258 0.542188i \(-0.182404\pi\)
0.840258 + 0.542188i \(0.182404\pi\)
\(264\) 0 0
\(265\) −816.713 −0.189322
\(266\) 0 0
\(267\) − 1144.13i − 0.262246i
\(268\) 0 0
\(269\) 5453.84i 1.23616i 0.786116 + 0.618079i \(0.212089\pi\)
−0.786116 + 0.618079i \(0.787911\pi\)
\(270\) 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
\(275\) − 11509.7i − 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
\(279\) 1327.40 0.284836
\(280\) 0 0
\(281\) −6664.57 −1.41486 −0.707429 0.706784i \(-0.750145\pi\)
−0.707429 + 0.706784i \(0.750145\pi\)
\(282\) 0 0
\(283\) 5630.84i 1.18275i 0.806396 + 0.591376i \(0.201415\pi\)
−0.806396 + 0.591376i \(0.798585\pi\)
\(284\) 0 0
\(285\) − 6946.40i − 1.44375i
\(286\) 0 0
\(287\) 11355.6 2.33554
\(288\) 0 0
\(289\) −3337.80 −0.679381
\(290\) 0 0
\(291\) 1977.46i 0.398354i
\(292\) 0 0
\(293\) 908.374i 0.181119i 0.995891 + 0.0905593i \(0.0288655\pi\)
−0.995891 + 0.0905593i \(0.971135\pi\)
\(294\) 0 0
\(295\) −5968.79 −1.17802
\(296\) 0 0
\(297\) 1449.60 0.283213
\(298\) 0 0
\(299\) − 709.517i − 0.137232i
\(300\) 0 0
\(301\) − 9275.82i − 1.77624i
\(302\) 0 0
\(303\) −1495.53 −0.283551
\(304\) 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
\(309\) − 590.463i − 0.108706i
\(310\) 0 0
\(311\) 1615.91 0.294629 0.147315 0.989090i \(-0.452937\pi\)
0.147315 + 0.989090i \(0.452937\pi\)
\(312\) 0 0
\(313\) 8479.33 1.53125 0.765623 0.643289i \(-0.222431\pi\)
0.765623 + 0.643289i \(0.222431\pi\)
\(314\) 0 0
\(315\) 3717.60i 0.664962i
\(316\) 0 0
\(317\) 6774.73i 1.20034i 0.799873 + 0.600169i \(0.204900\pi\)
−0.799873 + 0.600169i \(0.795100\pi\)
\(318\) 0 0
\(319\) 11049.2 1.93929
\(320\) 0 0
\(321\) 1078.13 0.187463
\(322\) 0 0
\(323\) − 4988.44i − 0.859332i
\(324\) 0 0
\(325\) 1534.00i 0.261818i
\(326\) 0 0
\(327\) 5909.20 0.999325
\(328\) 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
\(333\) − 1131.20i − 0.186154i
\(334\) 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
\(339\) − 2080.93i − 0.333394i
\(340\) 0 0
\(341\) − 7918.49i − 1.25751i
\(342\) 0 0
\(343\) −4108.75 −0.646798
\(344\) 0 0
\(345\) 5479.99 0.855168
\(346\) 0 0
\(347\) 3870.93i 0.598855i 0.954119 + 0.299427i \(0.0967956\pi\)
−0.954119 + 0.299427i \(0.903204\pi\)
\(348\) 0 0
\(349\) − 3474.57i − 0.532922i −0.963846 0.266461i \(-0.914146\pi\)
0.963846 0.266461i \(-0.0858543\pi\)
\(350\) 0 0
\(351\) −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
\(355\) − 19395.7i − 2.89977i
\(356\) 0 0
\(357\) 2669.73i 0.395791i
\(358\) 0 0
\(359\) −8161.19 −1.19981 −0.599904 0.800072i \(-0.704795\pi\)
−0.599904 + 0.800072i \(0.704795\pi\)
\(360\) 0 0
\(361\) −8938.68 −1.30320
\(362\) 0 0
\(363\) − 4654.47i − 0.672992i
\(364\) 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
\(369\) −4558.00 −0.643035
\(370\) 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
\(375\) −4939.60 −0.680213
\(376\) 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
\(381\) − 7970.33i − 1.07174i
\(382\) 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
\(387\) 3723.20i 0.489046i
\(388\) 0 0
\(389\) − 6830.06i − 0.890226i −0.895474 0.445113i \(-0.853164\pi\)
0.895474 0.445113i \(-0.146836\pi\)
\(390\) 0 0
\(391\) 3935.37 0.509003
\(392\) 0 0
\(393\) −1847.20 −0.237097
\(394\) 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
\(399\) 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
\(403\) 1055.37i 0.130451i
\(404\) 0 0
\(405\) − 1492.20i − 0.183081i
\(406\) 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
\(411\) 1839.87i 0.220813i
\(412\) 0 0
\(413\) − 7264.79i − 0.865562i
\(414\) 0 0
\(415\) 291.471 0.0344765
\(416\) 0 0
\(417\) −5697.86 −0.669126
\(418\) 0 0
\(419\) 15397.0i 1.79520i 0.440806 + 0.897602i \(0.354693\pi\)
−0.440806 + 0.897602i \(0.645307\pi\)
\(420\) 0 0
\(421\) 1034.45i 0.119753i 0.998206 + 0.0598766i \(0.0190707\pi\)
−0.998206 + 0.0598766i \(0.980929\pi\)
\(422\) 0 0
\(423\) −2825.20 −0.324742
\(424\) 0 0
\(425\) −8508.40 −0.971101
\(426\) 0 0
\(427\) − 7264.79i − 0.823344i
\(428\) 0 0
\(429\) 1152.53i 0.129707i
\(430\) 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
\(435\) − 11373.9i − 1.25364i
\(436\) 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
\(441\) −1437.80 −0.155253
\(442\) 0 0
\(443\) − 4292.26i − 0.460341i −0.973150 0.230171i \(-0.926072\pi\)
0.973150 0.230171i \(-0.0739285\pi\)
\(444\) 0 0
\(445\) 7025.82i 0.748440i
\(446\) 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
\(451\) 27190.4i 2.83890i
\(452\) 0 0
\(453\) 2049.40i 0.212559i
\(454\) 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
\(459\) − 1071.60i − 0.108971i
\(460\) 0 0
\(461\) − 9611.88i − 0.971085i −0.874213 0.485542i \(-0.838622\pi\)
0.874213 0.485542i \(-0.161378\pi\)
\(462\) 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
\(467\) 4432.00i 0.439161i 0.975594 + 0.219581i \(0.0704689\pi\)
−0.975594 + 0.219581i \(0.929531\pi\)
\(468\) 0 0
\(469\) 10409.9i 1.02491i
\(470\) 0 0
\(471\) −1533.33 −0.150004
\(472\) 0 0
\(473\) 22210.5 2.15907
\(474\) 0 0
\(475\) 26944.9i 2.60277i
\(476\) 0 0
\(477\) 398.998i 0.0382995i
\(478\) 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
\(483\) 6669.86i 0.628342i
\(484\) 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
\(489\) −7277.86 −0.673040
\(490\) 0 0
\(491\) − 2423.73i − 0.222773i −0.993777 0.111386i \(-0.964471\pi\)
0.993777 0.111386i \(-0.0355291\pi\)
\(492\) 0 0
\(493\) − 8167.95i − 0.746179i
\(494\) 0 0
\(495\) −8901.60 −0.808277
\(496\) 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
\(501\) 1012.00i 0.0902449i
\(502\) 0 0
\(503\) 18192.4 1.61264 0.806320 0.591479i \(-0.201456\pi\)
0.806320 + 0.591479i \(0.201456\pi\)
\(504\) 0 0
\(505\) 9183.65 0.809242
\(506\) 0 0
\(507\) 6437.39i 0.563895i
\(508\) 0 0
\(509\) − 5645.44i − 0.491610i −0.969319 0.245805i \(-0.920948\pi\)
0.969319 0.245805i \(-0.0790523\pi\)
\(510\) 0 0
\(511\) −22921.5 −1.98432
\(512\) 0 0
\(513\) −3393.60 −0.292068
\(514\) 0 0
\(515\) 3625.88i 0.310243i
\(516\) 0 0
\(517\) 16853.5i 1.43369i
\(518\) 0 0
\(519\) −7944.87 −0.671948
\(520\) 0 0
\(521\) −12338.7 −1.03756 −0.518780 0.854908i \(-0.673614\pi\)
−0.518780 + 0.854908i \(0.673614\pi\)
\(522\) 0 0
\(523\) − 10609.8i − 0.887062i −0.896259 0.443531i \(-0.853726\pi\)
0.896259 0.443531i \(-0.146274\pi\)
\(524\) 0 0
\(525\) − 14420.5i − 1.19878i
\(526\) 0 0
\(527\) −5853.65 −0.483850
\(528\) 0 0
\(529\) −2335.17 −0.191926
\(530\) 0 0
\(531\) 2916.00i 0.238312i
\(532\) 0 0
\(533\) − 3623.91i − 0.294501i
\(534\) 0 0
\(535\) −6620.53 −0.535010
\(536\) 0 0
\(537\) −8721.86 −0.700887
\(538\) 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
\(543\) −11048.4 −0.873171
\(544\) 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
\(549\) 2916.00i 0.226688i
\(550\) 0 0
\(551\) −25866.7 −1.99993
\(552\) 0 0
\(553\) 13518.6 1.03954
\(554\) 0 0
\(555\) 6946.40i 0.531276i
\(556\) 0 0
\(557\) − 9500.77i − 0.722730i −0.932424 0.361365i \(-0.882311\pi\)
0.932424 0.361365i \(-0.117689\pi\)
\(558\) 0 0
\(559\) −2960.19 −0.223976
\(560\) 0 0
\(561\) −6392.54 −0.481093
\(562\) 0 0
\(563\) − 2700.26i − 0.202136i −0.994880 0.101068i \(-0.967774\pi\)
0.994880 0.101068i \(-0.0322259\pi\)
\(564\) 0 0
\(565\) 12778.4i 0.951492i
\(566\) 0 0
\(567\) 1816.20 0.134521
\(568\) 0 0
\(569\) −15904.9 −1.17183 −0.585913 0.810374i \(-0.699264\pi\)
−0.585913 + 0.810374i \(0.699264\pi\)
\(570\) 0 0
\(571\) 18234.0i 1.33638i 0.743992 + 0.668188i \(0.232930\pi\)
−0.743992 + 0.668188i \(0.767070\pi\)
\(572\) 0 0
\(573\) − 3838.67i − 0.279865i
\(574\) 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
\(579\) − 14510.5i − 1.04151i
\(580\) 0 0
\(581\) 354.758i 0.0253319i
\(582\) 0 0
\(583\) 2380.19 0.169086
\(584\) 0 0
\(585\) 1186.40 0.0838486
\(586\) 0 0
\(587\) − 1616.99i − 0.113697i −0.998383 0.0568485i \(-0.981895\pi\)
0.998383 0.0568485i \(-0.0181052\pi\)
\(588\) 0 0
\(589\) 18537.7i 1.29683i
\(590\) 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
\(595\) − 16394.1i − 1.12957i
\(596\) 0 0
\(597\) 1956.73i 0.134143i
\(598\) 0 0
\(599\) 9536.40 0.650495 0.325248 0.945629i \(-0.394552\pi\)
0.325248 + 0.945629i \(0.394552\pi\)
\(600\) 0 0
\(601\) 16247.8 1.10276 0.551381 0.834253i \(-0.314101\pi\)
0.551381 + 0.834253i \(0.314101\pi\)
\(602\) 0 0
\(603\) − 4178.39i − 0.282185i
\(604\) 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
\(609\) 13843.5 0.921125
\(610\) 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
\(615\) 27989.5 1.83519
\(616\) 0 0
\(617\) 18727.8 1.22196 0.610982 0.791644i \(-0.290775\pi\)
0.610982 + 0.791644i \(0.290775\pi\)
\(618\) 0 0
\(619\) 3210.22i 0.208449i 0.994554 + 0.104224i \(0.0332360\pi\)
−0.994554 + 0.104224i \(0.966764\pi\)
\(620\) 0 0
\(621\) − 2677.20i − 0.172999i
\(622\) 0 0
\(623\) −8551.33 −0.549922
\(624\) 0 0
\(625\) 3535.57 0.226276
\(626\) 0 0
\(627\) 20244.3i 1.28944i
\(628\) 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
\(633\) −1612.53 −0.101252
\(634\) 0 0
\(635\) 48943.7i 3.05869i
\(636\) 0 0
\(637\) − 1143.14i − 0.0711036i
\(638\) 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
\(643\) 12754.6i 0.782262i 0.920335 + 0.391131i \(0.127916\pi\)
−0.920335 + 0.391131i \(0.872084\pi\)
\(644\) 0 0
\(645\) − 22863.2i − 1.39572i
\(646\) 0 0
\(647\) 9441.73 0.573714 0.286857 0.957973i \(-0.407390\pi\)
0.286857 + 0.957973i \(0.407390\pi\)
\(648\) 0 0
\(649\) 17395.2 1.05211
\(650\) 0 0
\(651\) − 9921.06i − 0.597292i
\(652\) 0 0
\(653\) − 5198.99i − 0.311565i −0.987791 0.155783i \(-0.950210\pi\)
0.987791 0.155783i \(-0.0497900\pi\)
\(654\) 0 0
\(655\) 11343.2 0.676664
\(656\) 0 0
\(657\) 9200.39 0.546334
\(658\) 0 0
\(659\) − 6508.01i − 0.384698i −0.981327 0.192349i \(-0.938389\pi\)
0.981327 0.192349i \(-0.0616105\pi\)
\(660\) 0 0
\(661\) − 25280.2i − 1.48757i −0.668419 0.743785i \(-0.733028\pi\)
0.668419 0.743785i \(-0.266972\pi\)
\(662\) 0 0
\(663\) 851.991 0.0499074
\(664\) 0 0
\(665\) −51917.8 −3.02750
\(666\) 0 0
\(667\) − 20406.2i − 1.18460i
\(668\) 0 0
\(669\) 12125.9i 0.700770i
\(670\) 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
\(675\) 5788.20i 0.330056i
\(676\) 0 0
\(677\) 6293.21i 0.357264i 0.983916 + 0.178632i \(0.0571671\pi\)
−0.983916 + 0.178632i \(0.942833\pi\)
\(678\) 0 0
\(679\) 14779.7 0.835335
\(680\) 0 0
\(681\) −9210.66 −0.518287
\(682\) 0 0
\(683\) − 5675.91i − 0.317984i −0.987280 0.158992i \(-0.949176\pi\)
0.987280 0.158992i \(-0.0508243\pi\)
\(684\) 0 0
\(685\) − 11298.2i − 0.630190i
\(686\) 0 0
\(687\) 615.331 0.0341722
\(688\) 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
\(693\) − 10834.4i − 0.593888i
\(694\) 0 0
\(695\) 34989.1 1.90966
\(696\) 0 0
\(697\) 20100.2 1.09232
\(698\) 0 0
\(699\) − 40.1329i − 0.00217163i
\(700\) 0 0
\(701\) − 7938.43i − 0.427718i −0.976865 0.213859i \(-0.931397\pi\)
0.976865 0.213859i \(-0.0686033\pi\)
\(702\) 0 0
\(703\) 15797.7 0.847540
\(704\) 0 0
\(705\) 17348.8 0.926799
\(706\) 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
\(711\) −5426.19 −0.286214
\(712\) 0 0
\(713\) −14624.3 −0.768142
\(714\) 0 0
\(715\) − 7077.35i − 0.370179i
\(716\) 0 0
\(717\) − 12984.0i − 0.676284i
\(718\) 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
\(723\) − 4524.94i − 0.232759i
\(724\) 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
\(729\) −729.000 −0.0370370
\(730\) 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
\(735\) 8829.13 0.443085
\(736\) 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
\(741\) − 2698.13i − 0.133763i
\(742\) 0 0
\(743\) 24110.8 1.19050 0.595248 0.803542i \(-0.297054\pi\)
0.595248 + 0.803542i \(0.297054\pi\)
\(744\) 0 0
\(745\) −17987.0 −0.884555
\(746\) 0 0
\(747\) − 142.396i − 0.00697455i
\(748\) 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
\(753\) −14614.4 −0.707275
\(754\) 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
\(759\) −15970.6 −0.763765
\(760\) 0 0
\(761\) 28904.5 1.37685 0.688427 0.725306i \(-0.258302\pi\)
0.688427 + 0.725306i \(0.258302\pi\)
\(762\) 0 0
\(763\) − 44165.7i − 2.09555i
\(764\) 0 0
\(765\) 6580.40i 0.311000i
\(766\) 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
\(771\) − 4996.66i − 0.233399i
\(772\) 0 0
\(773\) − 28289.8i − 1.31632i −0.752879 0.658159i \(-0.771335\pi\)
0.752879 0.658159i \(-0.228665\pi\)
\(774\) 0 0
\(775\) 31618.3 1.46550
\(776\) 0 0
\(777\) −8454.66 −0.390359
\(778\) 0 0
\(779\) − 63654.4i − 2.92767i
\(780\) 0 0
\(781\) 56526.0i 2.58983i
\(782\) 0 0
\(783\) −5556.60 −0.253610
\(784\) 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
\(789\) 21502.9i 0.970246i
\(790\) 0 0
\(791\) −15553.0 −0.699117
\(792\) 0 0
\(793\) −2318.41 −0.103820
\(794\) 0 0
\(795\) − 2450.14i − 0.109305i
\(796\) 0 0
\(797\) 17361.7i 0.771623i 0.922578 + 0.385811i \(0.126078\pi\)
−0.922578 + 0.385811i \(0.873922\pi\)
\(798\) 0 0
\(799\) 12458.8 0.551638
\(800\) 0 0
\(801\) 3432.40 0.151408
\(802\) 0 0
\(803\) − 54884.2i − 2.41198i
\(804\) 0 0
\(805\) − 40957.8i − 1.79326i
\(806\) 0 0
\(807\) −16361.5 −0.713696
\(808\) 0 0
\(809\) 24475.3 1.06367 0.531833 0.846849i \(-0.321503\pi\)
0.531833 + 0.846849i \(0.321503\pi\)
\(810\) 0 0
\(811\) 19875.4i 0.860566i 0.902694 + 0.430283i \(0.141586\pi\)
−0.902694 + 0.430283i \(0.858414\pi\)
\(812\) 0 0
\(813\) 16248.6i 0.700939i
\(814\) 0 0
\(815\) 44691.4 1.92083
\(816\) 0 0
\(817\) −51996.1 −2.22658
\(818\) 0 0
\(819\) 1444.00i 0.0616085i
\(820\) 0 0
\(821\) − 21682.1i − 0.921693i −0.887480 0.460846i \(-0.847546\pi\)
0.887480 0.460846i \(-0.152454\pi\)
\(822\) 0 0
\(823\) 6698.17 0.283698 0.141849 0.989888i \(-0.454695\pi\)
0.141849 + 0.989888i \(0.454695\pi\)
\(824\) 0 0
\(825\) 34529.0 1.45715
\(826\) 0 0
\(827\) − 3390.85i − 0.142577i −0.997456 0.0712886i \(-0.977289\pi\)
0.997456 0.0712886i \(-0.0227111\pi\)
\(828\) 0 0
\(829\) − 40093.4i − 1.67974i −0.542789 0.839869i \(-0.682632\pi\)
0.542789 0.839869i \(-0.317368\pi\)
\(830\) 0 0
\(831\) 7946.25 0.331712
\(832\) 0 0
\(833\) 6340.50 0.263728
\(834\) 0 0
\(835\) − 6214.40i − 0.257555i
\(836\) 0 0
\(837\) 3982.19i 0.164450i
\(838\) 0 0
\(839\) −6172.12 −0.253975 −0.126988 0.991904i \(-0.540531\pi\)
−0.126988 + 0.991904i \(0.540531\pi\)
\(840\) 0 0
\(841\) −17964.6 −0.736585
\(842\) 0 0
\(843\) − 19993.7i − 0.816869i
\(844\) 0 0
\(845\) − 39530.3i − 1.60933i
\(846\) 0 0
\(847\) −34787.8 −1.41124
\(848\) 0 0
\(849\) −16892.5 −0.682862
\(850\) 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
\(855\) 20839.2 0.833550
\(856\) 0 0
\(857\) −3704.41 −0.147655 −0.0738274 0.997271i \(-0.523521\pi\)
−0.0738274 + 0.997271i \(0.523521\pi\)
\(858\) 0 0
\(859\) 26915.5i 1.06909i 0.845141 + 0.534544i \(0.179516\pi\)
−0.845141 + 0.534544i \(0.820484\pi\)
\(860\) 0 0
\(861\) 34066.8i 1.34842i
\(862\) 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
\(867\) − 10013.4i − 0.392241i
\(868\) 0 0
\(869\) 32369.5i 1.26359i
\(870\) 0 0
\(871\) 3322.10 0.129236
\(872\) 0 0
\(873\) −5932.39 −0.229990
\(874\) 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
\(879\) −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
\(883\) − 32886.0i − 1.25334i −0.779283 0.626672i \(-0.784417\pi\)
0.779283 0.626672i \(-0.215583\pi\)
\(884\) 0 0
\(885\) − 17906.4i − 0.680132i
\(886\) 0 0
\(887\) −27945.6 −1.05786 −0.528929 0.848666i \(-0.677406\pi\)
−0.528929 + 0.848666i \(0.677406\pi\)
\(888\) 0 0
\(889\) −59570.8 −2.24740
\(890\) 0 0
\(891\) 4348.79i 0.163513i
\(892\) 0 0
\(893\) − 39455.1i − 1.47852i
\(894\) 0 0
\(895\) 53558.6 2.00030
\(896\) 0 0
\(897\) 2128.55 0.0792310
\(898\) 0 0
\(899\) 30353.1i 1.12607i
\(900\) 0 0
\(901\) − 1759.53i − 0.0650592i
\(902\) 0 0
\(903\) 27827.4 1.02551
\(904\) 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
\(909\) − 4486.59i − 0.163708i
\(910\) 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
\(915\) − 17906.4i − 0.646958i
\(916\) 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
\(921\) 1244.01 0.0445077
\(922\) 0 0
\(923\) − 7533.72i − 0.268663i
\(924\) 0 0
\(925\) − 26944.9i − 0.957775i
\(926\) 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
\(931\) − 20079.5i − 0.706850i
\(932\) 0 0
\(933\) 4847.72i 0.170104i
\(934\) 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
\(939\) 25438.0i 0.884065i
\(940\) 0 0
\(941\) 5408.01i 0.187350i 0.995603 + 0.0936749i \(0.0298614\pi\)
−0.995603 + 0.0936749i \(0.970139\pi\)
\(942\) 0 0
\(943\) 50216.8 1.73413
\(944\) 0 0
\(945\) −11152.8 −0.383916
\(946\) 0 0
\(947\) 34237.1i 1.17482i 0.809289 + 0.587410i \(0.199852\pi\)
−0.809289 + 0.587410i \(0.800148\pi\)
\(948\) 0 0
\(949\) 7314.92i 0.250213i
\(950\) 0 0
\(951\) −20324.2 −0.693015
\(952\) 0 0
\(953\) −21410.7 −0.727766 −0.363883 0.931445i \(-0.618549\pi\)
−0.363883 + 0.931445i \(0.618549\pi\)
\(954\) 0 0
\(955\) 23572.2i 0.798722i
\(956\) 0 0
\(957\) 33147.5i 1.11965i
\(958\) 0 0
\(959\) 13751.3 0.463038
\(960\) 0 0
\(961\) −8038.09 −0.269816
\(962\) 0 0
\(963\) 3234.40i 0.108232i
\(964\) 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
\(969\) 14965.3 0.496136
\(970\) 0 0
\(971\) 42901.5i 1.41789i 0.705262 + 0.708947i \(0.250829\pi\)
−0.705262 + 0.708947i \(0.749171\pi\)
\(972\) 0 0
\(973\) 42586.2i 1.40314i
\(974\) 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
\(979\) − 20475.7i − 0.668444i
\(980\) 0 0
\(981\) 17727.6i 0.576961i
\(982\) 0 0
\(983\) −25296.7 −0.820793 −0.410396 0.911907i \(-0.634610\pi\)
−0.410396 + 0.911907i \(0.634610\pi\)
\(984\) 0 0
\(985\) −52859.0 −1.70988
\(986\) 0 0
\(987\) 21115.7i 0.680973i
\(988\) 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
\(993\) 27877.1 0.890888
\(994\) 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
\(999\) 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 384.4.d.e.193.3 yes 4
3.2 odd 2 1152.4.d.o.577.4 4
4.3 odd 2 384.4.d.c.193.1 4
8.3 odd 2 384.4.d.c.193.4 yes 4
8.5 even 2 inner 384.4.d.e.193.2 yes 4
12.11 even 2 1152.4.d.i.577.4 4
16.3 odd 4 768.4.a.k.1.1 2
16.5 even 4 768.4.a.p.1.2 2
16.11 odd 4 768.4.a.j.1.2 2
16.13 even 4 768.4.a.e.1.1 2
24.5 odd 2 1152.4.d.o.577.1 4
24.11 even 2 1152.4.d.i.577.1 4
48.5 odd 4 2304.4.a.s.1.1 2
48.11 even 4 2304.4.a.t.1.1 2
48.29 odd 4 2304.4.a.bp.1.2 2
48.35 even 4 2304.4.a.bq.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.c.193.1 4 4.3 odd 2
384.4.d.c.193.4 yes 4 8.3 odd 2
384.4.d.e.193.2 yes 4 8.5 even 2 inner
384.4.d.e.193.3 yes 4 1.1 even 1 trivial
768.4.a.e.1.1 2 16.13 even 4
768.4.a.j.1.2 2 16.11 odd 4
768.4.a.k.1.1 2 16.3 odd 4
768.4.a.p.1.2 2 16.5 even 4
1152.4.d.i.577.1 4 24.11 even 2
1152.4.d.i.577.4 4 12.11 even 2
1152.4.d.o.577.1 4 24.5 odd 2
1152.4.d.o.577.4 4 3.2 odd 2
2304.4.a.s.1.1 2 48.5 odd 4
2304.4.a.t.1.1 2 48.11 even 4
2304.4.a.bp.1.2 2 48.29 odd 4
2304.4.a.bq.1.2 2 48.35 even 4