Properties

Label 1728.4.d.i.865.7
Level $1728$
Weight $4$
Character 1728.865
Analytic conductor $101.955$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,4,Mod(865,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, 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("1728.865");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 170x^{12} + 7609x^{8} + 59868x^{4} + 104976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 865.7
Root \(0.890014 + 0.890014i\) of defining polynomial
Character \(\chi\) \(=\) 1728.865
Dual form 1728.4.d.i.865.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.31976i q^{5} -2.47210 q^{7} +O(q^{10})\) \(q-1.31976i q^{5} -2.47210 q^{7} -35.9931i q^{11} -65.5540i q^{13} +87.1175 q^{17} -9.13947i q^{19} +49.3929 q^{23} +123.258 q^{25} +122.355i q^{29} +138.042 q^{31} +3.26258i q^{35} +90.2859i q^{37} -93.1282 q^{41} +191.655i q^{43} -275.910 q^{47} -336.889 q^{49} -648.331i q^{53} -47.5023 q^{55} +173.633i q^{59} +297.865i q^{61} -86.5156 q^{65} -658.458i q^{67} -558.113 q^{71} +275.325 q^{73} +88.9784i q^{77} -625.878 q^{79} -119.135i q^{83} -114.974i q^{85} +551.721 q^{89} +162.056i q^{91} -12.0619 q^{95} +74.6796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 1.31976i − 0.118043i −0.998257 0.0590215i \(-0.981202\pi\)
0.998257 0.0590215i \(-0.0187980\pi\)
\(6\) 0 0
\(7\) −2.47210 −0.133481 −0.0667404 0.997770i \(-0.521260\pi\)
−0.0667404 + 0.997770i \(0.521260\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 35.9931i − 0.986575i −0.869866 0.493288i \(-0.835795\pi\)
0.869866 0.493288i \(-0.164205\pi\)
\(12\) 0 0
\(13\) − 65.5540i − 1.39857i −0.714843 0.699285i \(-0.753502\pi\)
0.714843 0.699285i \(-0.246498\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 87.1175 1.24289 0.621444 0.783459i \(-0.286546\pi\)
0.621444 + 0.783459i \(0.286546\pi\)
\(18\) 0 0
\(19\) − 9.13947i − 0.110355i −0.998477 0.0551773i \(-0.982428\pi\)
0.998477 0.0551773i \(-0.0175724\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 49.3929 0.447789 0.223894 0.974613i \(-0.428123\pi\)
0.223894 + 0.974613i \(0.428123\pi\)
\(24\) 0 0
\(25\) 123.258 0.986066
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 122.355i 0.783471i 0.920078 + 0.391736i \(0.128125\pi\)
−0.920078 + 0.391736i \(0.871875\pi\)
\(30\) 0 0
\(31\) 138.042 0.799775 0.399887 0.916564i \(-0.369049\pi\)
0.399887 + 0.916564i \(0.369049\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.26258i 0.0157565i
\(36\) 0 0
\(37\) 90.2859i 0.401160i 0.979677 + 0.200580i \(0.0642826\pi\)
−0.979677 + 0.200580i \(0.935717\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −93.1282 −0.354736 −0.177368 0.984145i \(-0.556758\pi\)
−0.177368 + 0.984145i \(0.556758\pi\)
\(42\) 0 0
\(43\) 191.655i 0.679701i 0.940480 + 0.339850i \(0.110376\pi\)
−0.940480 + 0.339850i \(0.889624\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −275.910 −0.856289 −0.428145 0.903710i \(-0.640833\pi\)
−0.428145 + 0.903710i \(0.640833\pi\)
\(48\) 0 0
\(49\) −336.889 −0.982183
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 648.331i − 1.68029i −0.542366 0.840143i \(-0.682471\pi\)
0.542366 0.840143i \(-0.317529\pi\)
\(54\) 0 0
\(55\) −47.5023 −0.116458
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 173.633i 0.383137i 0.981479 + 0.191569i \(0.0613575\pi\)
−0.981479 + 0.191569i \(0.938643\pi\)
\(60\) 0 0
\(61\) 297.865i 0.625208i 0.949884 + 0.312604i \(0.101201\pi\)
−0.949884 + 0.312604i \(0.898799\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −86.5156 −0.165091
\(66\) 0 0
\(67\) − 658.458i − 1.20065i −0.799757 0.600324i \(-0.795038\pi\)
0.799757 0.600324i \(-0.204962\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −558.113 −0.932899 −0.466450 0.884548i \(-0.654467\pi\)
−0.466450 + 0.884548i \(0.654467\pi\)
\(72\) 0 0
\(73\) 275.325 0.441429 0.220715 0.975338i \(-0.429161\pi\)
0.220715 + 0.975338i \(0.429161\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 88.9784i 0.131689i
\(78\) 0 0
\(79\) −625.878 −0.891351 −0.445676 0.895195i \(-0.647036\pi\)
−0.445676 + 0.895195i \(0.647036\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 119.135i − 0.157551i −0.996892 0.0787755i \(-0.974899\pi\)
0.996892 0.0787755i \(-0.0251010\pi\)
\(84\) 0 0
\(85\) − 114.974i − 0.146714i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 551.721 0.657104 0.328552 0.944486i \(-0.393439\pi\)
0.328552 + 0.944486i \(0.393439\pi\)
\(90\) 0 0
\(91\) 162.056i 0.186682i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.0619 −0.0130266
\(96\) 0 0
\(97\) 74.6796 0.0781708 0.0390854 0.999236i \(-0.487556\pi\)
0.0390854 + 0.999236i \(0.487556\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 855.709i − 0.843032i −0.906821 0.421516i \(-0.861498\pi\)
0.906821 0.421516i \(-0.138502\pi\)
\(102\) 0 0
\(103\) −183.007 −0.175070 −0.0875349 0.996161i \(-0.527899\pi\)
−0.0875349 + 0.996161i \(0.527899\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1395.85i − 1.26114i −0.776133 0.630569i \(-0.782821\pi\)
0.776133 0.630569i \(-0.217179\pi\)
\(108\) 0 0
\(109\) − 590.120i − 0.518562i −0.965802 0.259281i \(-0.916514\pi\)
0.965802 0.259281i \(-0.0834856\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 606.251 0.504702 0.252351 0.967636i \(-0.418796\pi\)
0.252351 + 0.967636i \(0.418796\pi\)
\(114\) 0 0
\(115\) − 65.1868i − 0.0528583i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −215.363 −0.165902
\(120\) 0 0
\(121\) 35.4971 0.0266695
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 327.641i − 0.234441i
\(126\) 0 0
\(127\) 1258.67 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1933.54i − 1.28958i −0.764361 0.644788i \(-0.776946\pi\)
0.764361 0.644788i \(-0.223054\pi\)
\(132\) 0 0
\(133\) 22.5937i 0.0147302i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2156.67 1.34494 0.672469 0.740125i \(-0.265234\pi\)
0.672469 + 0.740125i \(0.265234\pi\)
\(138\) 0 0
\(139\) 2153.75i 1.31424i 0.753787 + 0.657119i \(0.228225\pi\)
−0.753787 + 0.657119i \(0.771775\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2359.49 −1.37979
\(144\) 0 0
\(145\) 161.479 0.0924832
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2268.88i − 1.24748i −0.781633 0.623738i \(-0.785613\pi\)
0.781633 0.623738i \(-0.214387\pi\)
\(150\) 0 0
\(151\) 1519.19 0.818743 0.409372 0.912368i \(-0.365748\pi\)
0.409372 + 0.912368i \(0.365748\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 182.182i − 0.0944078i
\(156\) 0 0
\(157\) 66.6451i 0.0338781i 0.999857 + 0.0169390i \(0.00539212\pi\)
−0.999857 + 0.0169390i \(0.994608\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −122.104 −0.0597712
\(162\) 0 0
\(163\) − 220.564i − 0.105987i −0.998595 0.0529937i \(-0.983124\pi\)
0.998595 0.0529937i \(-0.0168763\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3322.97 −1.53976 −0.769878 0.638191i \(-0.779683\pi\)
−0.769878 + 0.638191i \(0.779683\pi\)
\(168\) 0 0
\(169\) −2100.33 −0.955998
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 3687.70i − 1.62064i −0.585988 0.810320i \(-0.699293\pi\)
0.585988 0.810320i \(-0.300707\pi\)
\(174\) 0 0
\(175\) −304.706 −0.131621
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1344.27i 0.561315i 0.959808 + 0.280658i \(0.0905525\pi\)
−0.959808 + 0.280658i \(0.909447\pi\)
\(180\) 0 0
\(181\) − 168.428i − 0.0691665i −0.999402 0.0345832i \(-0.988990\pi\)
0.999402 0.0345832i \(-0.0110104\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 119.156 0.0473541
\(186\) 0 0
\(187\) − 3135.63i − 1.22620i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2866.99 −1.08611 −0.543057 0.839696i \(-0.682733\pi\)
−0.543057 + 0.839696i \(0.682733\pi\)
\(192\) 0 0
\(193\) 3021.80 1.12701 0.563507 0.826111i \(-0.309452\pi\)
0.563507 + 0.826111i \(0.309452\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 782.353i − 0.282946i −0.989942 0.141473i \(-0.954816\pi\)
0.989942 0.141473i \(-0.0451838\pi\)
\(198\) 0 0
\(199\) −1925.31 −0.685836 −0.342918 0.939365i \(-0.611415\pi\)
−0.342918 + 0.939365i \(0.611415\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 302.472i − 0.104578i
\(204\) 0 0
\(205\) 122.907i 0.0418741i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −328.958 −0.108873
\(210\) 0 0
\(211\) 1361.40i 0.444183i 0.975026 + 0.222092i \(0.0712884\pi\)
−0.975026 + 0.222092i \(0.928712\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 252.939 0.0802339
\(216\) 0 0
\(217\) −341.252 −0.106755
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 5710.90i − 1.73827i
\(222\) 0 0
\(223\) −5314.09 −1.59577 −0.797887 0.602806i \(-0.794049\pi\)
−0.797887 + 0.602806i \(0.794049\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3437.08i − 1.00496i −0.864587 0.502482i \(-0.832420\pi\)
0.864587 0.502482i \(-0.167580\pi\)
\(228\) 0 0
\(229\) 880.344i 0.254038i 0.991900 + 0.127019i \(0.0405409\pi\)
−0.991900 + 0.127019i \(0.959459\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2858.44 0.803701 0.401850 0.915705i \(-0.368367\pi\)
0.401850 + 0.915705i \(0.368367\pi\)
\(234\) 0 0
\(235\) 364.135i 0.101079i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5350.87 −1.44820 −0.724098 0.689697i \(-0.757744\pi\)
−0.724098 + 0.689697i \(0.757744\pi\)
\(240\) 0 0
\(241\) −2790.27 −0.745798 −0.372899 0.927872i \(-0.621636\pi\)
−0.372899 + 0.927872i \(0.621636\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 444.612i 0.115940i
\(246\) 0 0
\(247\) −599.129 −0.154339
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 3752.93i − 0.943757i −0.881664 0.471878i \(-0.843576\pi\)
0.881664 0.471878i \(-0.156424\pi\)
\(252\) 0 0
\(253\) − 1777.80i − 0.441777i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6299.43 1.52898 0.764490 0.644636i \(-0.222991\pi\)
0.764490 + 0.644636i \(0.222991\pi\)
\(258\) 0 0
\(259\) − 223.196i − 0.0535471i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7614.33 −1.78525 −0.892623 0.450805i \(-0.851137\pi\)
−0.892623 + 0.450805i \(0.851137\pi\)
\(264\) 0 0
\(265\) −855.642 −0.198346
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 6081.62i − 1.37845i −0.724547 0.689225i \(-0.757951\pi\)
0.724547 0.689225i \(-0.242049\pi\)
\(270\) 0 0
\(271\) 7989.97 1.79098 0.895491 0.445080i \(-0.146825\pi\)
0.895491 + 0.445080i \(0.146825\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4436.45i − 0.972828i
\(276\) 0 0
\(277\) 6087.60i 1.32046i 0.751061 + 0.660232i \(0.229542\pi\)
−0.751061 + 0.660232i \(0.770458\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6106.28 1.29633 0.648167 0.761498i \(-0.275536\pi\)
0.648167 + 0.761498i \(0.275536\pi\)
\(282\) 0 0
\(283\) − 5424.68i − 1.13945i −0.821836 0.569724i \(-0.807050\pi\)
0.821836 0.569724i \(-0.192950\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 230.222 0.0473504
\(288\) 0 0
\(289\) 2676.45 0.544770
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 1513.45i − 0.301763i −0.988552 0.150882i \(-0.951789\pi\)
0.988552 0.150882i \(-0.0482113\pi\)
\(294\) 0 0
\(295\) 229.154 0.0452267
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 3237.90i − 0.626264i
\(300\) 0 0
\(301\) − 473.790i − 0.0907269i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 393.110 0.0738014
\(306\) 0 0
\(307\) 6543.54i 1.21648i 0.793753 + 0.608241i \(0.208124\pi\)
−0.793753 + 0.608241i \(0.791876\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1284.46 0.234196 0.117098 0.993120i \(-0.462641\pi\)
0.117098 + 0.993120i \(0.462641\pi\)
\(312\) 0 0
\(313\) 4013.86 0.724847 0.362423 0.932014i \(-0.381949\pi\)
0.362423 + 0.932014i \(0.381949\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2203.61i − 0.390433i −0.980760 0.195216i \(-0.937459\pi\)
0.980760 0.195216i \(-0.0625409\pi\)
\(318\) 0 0
\(319\) 4403.92 0.772953
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 796.208i − 0.137159i
\(324\) 0 0
\(325\) − 8080.07i − 1.37908i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 682.076 0.114298
\(330\) 0 0
\(331\) 2467.37i 0.409725i 0.978791 + 0.204862i \(0.0656747\pi\)
−0.978791 + 0.204862i \(0.934325\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −869.007 −0.141728
\(336\) 0 0
\(337\) −7160.82 −1.15749 −0.578746 0.815508i \(-0.696458\pi\)
−0.578746 + 0.815508i \(0.696458\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 4968.55i − 0.789038i
\(342\) 0 0
\(343\) 1680.75 0.264583
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1843.40i − 0.285184i −0.989782 0.142592i \(-0.954456\pi\)
0.989782 0.142592i \(-0.0455436\pi\)
\(348\) 0 0
\(349\) 5209.52i 0.799023i 0.916728 + 0.399512i \(0.130820\pi\)
−0.916728 + 0.399512i \(0.869180\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8650.90 −1.30437 −0.652183 0.758062i \(-0.726146\pi\)
−0.652183 + 0.758062i \(0.726146\pi\)
\(354\) 0 0
\(355\) 736.576i 0.110122i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12565.4 −1.84728 −0.923641 0.383260i \(-0.874801\pi\)
−0.923641 + 0.383260i \(0.874801\pi\)
\(360\) 0 0
\(361\) 6775.47 0.987822
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 363.363i − 0.0521076i
\(366\) 0 0
\(367\) −10796.9 −1.53567 −0.767837 0.640646i \(-0.778667\pi\)
−0.767837 + 0.640646i \(0.778667\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1602.74i 0.224286i
\(372\) 0 0
\(373\) − 8397.99i − 1.16577i −0.812555 0.582884i \(-0.801924\pi\)
0.812555 0.582884i \(-0.198076\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8020.83 1.09574
\(378\) 0 0
\(379\) − 242.519i − 0.0328690i −0.999865 0.0164345i \(-0.994768\pi\)
0.999865 0.0164345i \(-0.00523150\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3601.64 −0.480510 −0.240255 0.970710i \(-0.577231\pi\)
−0.240255 + 0.970710i \(0.577231\pi\)
\(384\) 0 0
\(385\) 117.430 0.0155449
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 13291.8i − 1.73245i −0.499656 0.866224i \(-0.666540\pi\)
0.499656 0.866224i \(-0.333460\pi\)
\(390\) 0 0
\(391\) 4302.99 0.556551
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 826.008i 0.105218i
\(396\) 0 0
\(397\) − 3255.27i − 0.411530i −0.978601 0.205765i \(-0.934032\pi\)
0.978601 0.205765i \(-0.0659682\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8209.13 −1.02231 −0.511153 0.859490i \(-0.670781\pi\)
−0.511153 + 0.859490i \(0.670781\pi\)
\(402\) 0 0
\(403\) − 9049.19i − 1.11854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3249.67 0.395774
\(408\) 0 0
\(409\) 4554.98 0.550683 0.275342 0.961346i \(-0.411209\pi\)
0.275342 + 0.961346i \(0.411209\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 429.238i − 0.0511414i
\(414\) 0 0
\(415\) −157.229 −0.0185978
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7550.04i 0.880295i 0.897925 + 0.440148i \(0.145074\pi\)
−0.897925 + 0.440148i \(0.854926\pi\)
\(420\) 0 0
\(421\) 11044.7i 1.27859i 0.768961 + 0.639296i \(0.220774\pi\)
−0.768961 + 0.639296i \(0.779226\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10737.9 1.22557
\(426\) 0 0
\(427\) − 736.351i − 0.0834532i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9156.47 −1.02332 −0.511661 0.859187i \(-0.670970\pi\)
−0.511661 + 0.859187i \(0.670970\pi\)
\(432\) 0 0
\(433\) −2250.45 −0.249768 −0.124884 0.992171i \(-0.539856\pi\)
−0.124884 + 0.992171i \(0.539856\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 451.425i − 0.0494156i
\(438\) 0 0
\(439\) −11848.2 −1.28812 −0.644061 0.764974i \(-0.722752\pi\)
−0.644061 + 0.764974i \(0.722752\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1274.32i 0.136670i 0.997662 + 0.0683348i \(0.0217686\pi\)
−0.997662 + 0.0683348i \(0.978231\pi\)
\(444\) 0 0
\(445\) − 728.139i − 0.0775665i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8201.95 −0.862080 −0.431040 0.902333i \(-0.641853\pi\)
−0.431040 + 0.902333i \(0.641853\pi\)
\(450\) 0 0
\(451\) 3351.97i 0.349974i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 213.875 0.0220365
\(456\) 0 0
\(457\) −16627.3 −1.70195 −0.850976 0.525205i \(-0.823989\pi\)
−0.850976 + 0.525205i \(0.823989\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8128.27i 0.821196i 0.911816 + 0.410598i \(0.134680\pi\)
−0.911816 + 0.410598i \(0.865320\pi\)
\(462\) 0 0
\(463\) −1415.26 −0.142057 −0.0710286 0.997474i \(-0.522628\pi\)
−0.0710286 + 0.997474i \(0.522628\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17479.2i − 1.73200i −0.500048 0.865998i \(-0.666684\pi\)
0.500048 0.865998i \(-0.333316\pi\)
\(468\) 0 0
\(469\) 1627.77i 0.160263i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6898.26 0.670576
\(474\) 0 0
\(475\) − 1126.52i − 0.108817i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6370.47 0.607670 0.303835 0.952725i \(-0.401733\pi\)
0.303835 + 0.952725i \(0.401733\pi\)
\(480\) 0 0
\(481\) 5918.60 0.561050
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 98.5592i − 0.00922751i
\(486\) 0 0
\(487\) 13658.3 1.27088 0.635440 0.772150i \(-0.280819\pi\)
0.635440 + 0.772150i \(0.280819\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 3242.02i − 0.297984i −0.988838 0.148992i \(-0.952397\pi\)
0.988838 0.148992i \(-0.0476029\pi\)
\(492\) 0 0
\(493\) 10659.2i 0.973767i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1379.71 0.124524
\(498\) 0 0
\(499\) − 388.342i − 0.0348388i −0.999848 0.0174194i \(-0.994455\pi\)
0.999848 0.0174194i \(-0.00554506\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12450.6 1.10367 0.551834 0.833954i \(-0.313928\pi\)
0.551834 + 0.833954i \(0.313928\pi\)
\(504\) 0 0
\(505\) −1129.33 −0.0995140
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12529.1i 1.09104i 0.838097 + 0.545521i \(0.183668\pi\)
−0.838097 + 0.545521i \(0.816332\pi\)
\(510\) 0 0
\(511\) −680.630 −0.0589223
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 241.525i 0.0206658i
\(516\) 0 0
\(517\) 9930.85i 0.844794i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5185.14 0.436017 0.218009 0.975947i \(-0.430044\pi\)
0.218009 + 0.975947i \(0.430044\pi\)
\(522\) 0 0
\(523\) 10344.4i 0.864877i 0.901663 + 0.432438i \(0.142347\pi\)
−0.901663 + 0.432438i \(0.857653\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12025.8 0.994030
\(528\) 0 0
\(529\) −9727.34 −0.799485
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6104.93i 0.496123i
\(534\) 0 0
\(535\) −1842.19 −0.148869
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12125.7i 0.968997i
\(540\) 0 0
\(541\) − 21105.5i − 1.67726i −0.544705 0.838628i \(-0.683358\pi\)
0.544705 0.838628i \(-0.316642\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −778.817 −0.0612126
\(546\) 0 0
\(547\) − 12491.4i − 0.976409i −0.872729 0.488204i \(-0.837652\pi\)
0.872729 0.488204i \(-0.162348\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1118.26 0.0864597
\(552\) 0 0
\(553\) 1547.23 0.118978
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8317.90i 0.632748i 0.948635 + 0.316374i \(0.102465\pi\)
−0.948635 + 0.316374i \(0.897535\pi\)
\(558\) 0 0
\(559\) 12563.8 0.950609
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8298.80i 0.621231i 0.950536 + 0.310615i \(0.100535\pi\)
−0.950536 + 0.310615i \(0.899465\pi\)
\(564\) 0 0
\(565\) − 800.106i − 0.0595765i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6133.13 −0.451870 −0.225935 0.974142i \(-0.572544\pi\)
−0.225935 + 0.974142i \(0.572544\pi\)
\(570\) 0 0
\(571\) 26180.9i 1.91880i 0.282043 + 0.959402i \(0.408988\pi\)
−0.282043 + 0.959402i \(0.591012\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6088.09 0.441549
\(576\) 0 0
\(577\) 22659.7 1.63489 0.817447 0.576004i \(-0.195389\pi\)
0.817447 + 0.576004i \(0.195389\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 294.513i 0.0210300i
\(582\) 0 0
\(583\) −23335.4 −1.65773
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 4163.23i − 0.292734i −0.989230 0.146367i \(-0.953242\pi\)
0.989230 0.146367i \(-0.0467580\pi\)
\(588\) 0 0
\(589\) − 1261.63i − 0.0882589i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −11485.2 −0.795350 −0.397675 0.917526i \(-0.630183\pi\)
−0.397675 + 0.917526i \(0.630183\pi\)
\(594\) 0 0
\(595\) 284.227i 0.0195835i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −10138.6 −0.691571 −0.345785 0.938314i \(-0.612387\pi\)
−0.345785 + 0.938314i \(0.612387\pi\)
\(600\) 0 0
\(601\) −5983.08 −0.406082 −0.203041 0.979170i \(-0.565082\pi\)
−0.203041 + 0.979170i \(0.565082\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 46.8477i − 0.00314815i
\(606\) 0 0
\(607\) −2579.69 −0.172498 −0.0862492 0.996274i \(-0.527488\pi\)
−0.0862492 + 0.996274i \(0.527488\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18087.0i 1.19758i
\(612\) 0 0
\(613\) − 28274.9i − 1.86299i −0.363759 0.931493i \(-0.618507\pi\)
0.363759 0.931493i \(-0.381493\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8007.60 0.522486 0.261243 0.965273i \(-0.415868\pi\)
0.261243 + 0.965273i \(0.415868\pi\)
\(618\) 0 0
\(619\) − 27366.1i − 1.77696i −0.458915 0.888480i \(-0.651762\pi\)
0.458915 0.888480i \(-0.348238\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1363.91 −0.0877107
\(624\) 0 0
\(625\) 14974.9 0.958392
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7865.48i 0.498596i
\(630\) 0 0
\(631\) 27939.7 1.76270 0.881349 0.472466i \(-0.156636\pi\)
0.881349 + 0.472466i \(0.156636\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 1661.14i − 0.103812i
\(636\) 0 0
\(637\) 22084.4i 1.37365i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28620.9 −1.76358 −0.881792 0.471639i \(-0.843663\pi\)
−0.881792 + 0.471639i \(0.843663\pi\)
\(642\) 0 0
\(643\) − 4650.43i − 0.285218i −0.989779 0.142609i \(-0.954451\pi\)
0.989779 0.142609i \(-0.0455491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22838.1 1.38772 0.693862 0.720108i \(-0.255908\pi\)
0.693862 + 0.720108i \(0.255908\pi\)
\(648\) 0 0
\(649\) 6249.59 0.377994
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 8551.87i − 0.512497i −0.966611 0.256248i \(-0.917513\pi\)
0.966611 0.256248i \(-0.0824865\pi\)
\(654\) 0 0
\(655\) −2551.81 −0.152225
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10145.1i 0.599689i 0.953988 + 0.299845i \(0.0969348\pi\)
−0.953988 + 0.299845i \(0.903065\pi\)
\(660\) 0 0
\(661\) − 17372.6i − 1.02226i −0.859503 0.511131i \(-0.829227\pi\)
0.859503 0.511131i \(-0.170773\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 29.8182 0.00173880
\(666\) 0 0
\(667\) 6043.45i 0.350829i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10721.1 0.616815
\(672\) 0 0
\(673\) −19271.9 −1.10383 −0.551914 0.833901i \(-0.686102\pi\)
−0.551914 + 0.833901i \(0.686102\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 30042.8i 1.70552i 0.522300 + 0.852762i \(0.325074\pi\)
−0.522300 + 0.852762i \(0.674926\pi\)
\(678\) 0 0
\(679\) −184.615 −0.0104343
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17811.9i 0.997880i 0.866636 + 0.498940i \(0.166277\pi\)
−0.866636 + 0.498940i \(0.833723\pi\)
\(684\) 0 0
\(685\) − 2846.28i − 0.158760i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −42500.7 −2.35000
\(690\) 0 0
\(691\) − 2461.63i − 0.135521i −0.997702 0.0677604i \(-0.978415\pi\)
0.997702 0.0677604i \(-0.0215854\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2842.44 0.155137
\(696\) 0 0
\(697\) −8113.10 −0.440897
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13292.5i 0.716190i 0.933685 + 0.358095i \(0.116574\pi\)
−0.933685 + 0.358095i \(0.883426\pi\)
\(702\) 0 0
\(703\) 825.165 0.0442699
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2115.40i 0.112529i
\(708\) 0 0
\(709\) 27124.4i 1.43678i 0.695640 + 0.718390i \(0.255121\pi\)
−0.695640 + 0.718390i \(0.744879\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6818.28 0.358130
\(714\) 0 0
\(715\) 3113.96i 0.162875i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30827.3 1.59898 0.799489 0.600680i \(-0.205103\pi\)
0.799489 + 0.600680i \(0.205103\pi\)
\(720\) 0 0
\(721\) 452.411 0.0233684
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15081.2i 0.772554i
\(726\) 0 0
\(727\) −29322.3 −1.49588 −0.747939 0.663767i \(-0.768957\pi\)
−0.747939 + 0.663767i \(0.768957\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16696.5i 0.844791i
\(732\) 0 0
\(733\) 5700.95i 0.287271i 0.989631 + 0.143635i \(0.0458792\pi\)
−0.989631 + 0.143635i \(0.954121\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23699.9 −1.18453
\(738\) 0 0
\(739\) 26824.2i 1.33524i 0.744502 + 0.667621i \(0.232687\pi\)
−0.744502 + 0.667621i \(0.767313\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25329.2 1.25066 0.625328 0.780362i \(-0.284965\pi\)
0.625328 + 0.780362i \(0.284965\pi\)
\(744\) 0 0
\(745\) −2994.38 −0.147256
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3450.68i 0.168338i
\(750\) 0 0
\(751\) 9862.86 0.479229 0.239614 0.970868i \(-0.422979\pi\)
0.239614 + 0.970868i \(0.422979\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 2004.97i − 0.0966469i
\(756\) 0 0
\(757\) 12425.8i 0.596595i 0.954473 + 0.298298i \(0.0964188\pi\)
−0.954473 + 0.298298i \(0.903581\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38708.2 1.84385 0.921926 0.387365i \(-0.126615\pi\)
0.921926 + 0.387365i \(0.126615\pi\)
\(762\) 0 0
\(763\) 1458.83i 0.0692180i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11382.3 0.535844
\(768\) 0 0
\(769\) −17734.4 −0.831626 −0.415813 0.909450i \(-0.636503\pi\)
−0.415813 + 0.909450i \(0.636503\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 32273.9i 1.50170i 0.660475 + 0.750848i \(0.270355\pi\)
−0.660475 + 0.750848i \(0.729645\pi\)
\(774\) 0 0
\(775\) 17014.8 0.788631
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 851.143i 0.0391468i
\(780\) 0 0
\(781\) 20088.2i 0.920375i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 87.9556 0.00399907
\(786\) 0 0
\(787\) − 24819.5i − 1.12417i −0.827080 0.562084i \(-0.810000\pi\)
0.827080 0.562084i \(-0.190000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1498.71 −0.0673679
\(792\) 0 0
\(793\) 19526.2 0.874397
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15582.6i 0.692553i 0.938132 + 0.346277i \(0.112554\pi\)
−0.938132 + 0.346277i \(0.887446\pi\)
\(798\) 0 0
\(799\) −24036.6 −1.06427
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 9909.80i − 0.435503i
\(804\) 0 0
\(805\) 161.148i 0.00705556i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13815.9 0.600422 0.300211 0.953873i \(-0.402943\pi\)
0.300211 + 0.953873i \(0.402943\pi\)
\(810\) 0 0
\(811\) − 26934.2i − 1.16620i −0.812400 0.583100i \(-0.801839\pi\)
0.812400 0.583100i \(-0.198161\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −291.092 −0.0125111
\(816\) 0 0
\(817\) 1751.63 0.0750081
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8300.96i 0.352869i 0.984312 + 0.176434i \(0.0564564\pi\)
−0.984312 + 0.176434i \(0.943544\pi\)
\(822\) 0 0
\(823\) 27975.6 1.18489 0.592447 0.805609i \(-0.298162\pi\)
0.592447 + 0.805609i \(0.298162\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 11664.8i − 0.490476i −0.969463 0.245238i \(-0.921134\pi\)
0.969463 0.245238i \(-0.0788662\pi\)
\(828\) 0 0
\(829\) − 22133.6i − 0.927298i −0.886019 0.463649i \(-0.846540\pi\)
0.886019 0.463649i \(-0.153460\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −29348.9 −1.22074
\(834\) 0 0
\(835\) 4385.53i 0.181757i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 44278.9 1.82202 0.911012 0.412380i \(-0.135302\pi\)
0.911012 + 0.412380i \(0.135302\pi\)
\(840\) 0 0
\(841\) 9418.38 0.386173
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2771.93i 0.112849i
\(846\) 0 0
\(847\) −87.7524 −0.00355987
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4459.48i 0.179635i
\(852\) 0 0
\(853\) 32745.5i 1.31440i 0.753715 + 0.657202i \(0.228260\pi\)
−0.753715 + 0.657202i \(0.771740\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11799.2 0.470305 0.235153 0.971958i \(-0.424441\pi\)
0.235153 + 0.971958i \(0.424441\pi\)
\(858\) 0 0
\(859\) − 6975.49i − 0.277067i −0.990358 0.138534i \(-0.955761\pi\)
0.990358 0.138534i \(-0.0442389\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8615.25 −0.339822 −0.169911 0.985459i \(-0.554348\pi\)
−0.169911 + 0.985459i \(0.554348\pi\)
\(864\) 0 0
\(865\) −4866.88 −0.191305
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 22527.3i 0.879385i
\(870\) 0 0
\(871\) −43164.6 −1.67919
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 809.961i 0.0312934i
\(876\) 0 0
\(877\) − 32197.0i − 1.23970i −0.784720 0.619850i \(-0.787193\pi\)
0.784720 0.619850i \(-0.212807\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30615.2 −1.17077 −0.585387 0.810754i \(-0.699057\pi\)
−0.585387 + 0.810754i \(0.699057\pi\)
\(882\) 0 0
\(883\) 37640.3i 1.43454i 0.696796 + 0.717269i \(0.254608\pi\)
−0.696796 + 0.717269i \(0.745392\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 21465.3 0.812553 0.406276 0.913750i \(-0.366827\pi\)
0.406276 + 0.913750i \(0.366827\pi\)
\(888\) 0 0
\(889\) −3111.55 −0.117388
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2521.67i 0.0944955i
\(894\) 0 0
\(895\) 1774.11 0.0662593
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 16890.0i 0.626600i
\(900\) 0 0
\(901\) − 56481.0i − 2.08841i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −222.284 −0.00816462
\(906\) 0 0
\(907\) − 8317.31i − 0.304489i −0.988343 0.152245i \(-0.951350\pi\)
0.988343 0.152245i \(-0.0486502\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28334.9 1.03049 0.515246 0.857043i \(-0.327701\pi\)
0.515246 + 0.857043i \(0.327701\pi\)
\(912\) 0 0
\(913\) −4288.03 −0.155436
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4779.91i 0.172134i
\(918\) 0 0
\(919\) 18433.3 0.661652 0.330826 0.943692i \(-0.392673\pi\)
0.330826 + 0.943692i \(0.392673\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 36586.6i 1.30473i
\(924\) 0 0
\(925\) 11128.5i 0.395570i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −20036.0 −0.707600 −0.353800 0.935321i \(-0.615111\pi\)
−0.353800 + 0.935321i \(0.615111\pi\)
\(930\) 0 0
\(931\) 3078.99i 0.108388i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4138.28 −0.144745
\(936\) 0 0
\(937\) 24844.2 0.866196 0.433098 0.901347i \(-0.357420\pi\)
0.433098 + 0.901347i \(0.357420\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12898.8i 0.446853i 0.974721 + 0.223426i \(0.0717243\pi\)
−0.974721 + 0.223426i \(0.928276\pi\)
\(942\) 0 0
\(943\) −4599.88 −0.158847
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45388.5i 1.55747i 0.627350 + 0.778737i \(0.284139\pi\)
−0.627350 + 0.778737i \(0.715861\pi\)
\(948\) 0 0
\(949\) − 18048.7i − 0.617370i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13191.2 0.448378 0.224189 0.974546i \(-0.428027\pi\)
0.224189 + 0.974546i \(0.428027\pi\)
\(954\) 0 0
\(955\) 3783.74i 0.128208i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5331.49 −0.179523
\(960\) 0 0
\(961\) −10735.5 −0.360360
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 3988.05i − 0.133036i
\(966\) 0 0
\(967\) 3495.31 0.116237 0.0581186 0.998310i \(-0.481490\pi\)
0.0581186 + 0.998310i \(0.481490\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 21176.9i − 0.699896i −0.936769 0.349948i \(-0.886199\pi\)
0.936769 0.349948i \(-0.113801\pi\)
\(972\) 0 0
\(973\) − 5324.29i − 0.175425i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −58221.0 −1.90650 −0.953252 0.302178i \(-0.902286\pi\)
−0.953252 + 0.302178i \(0.902286\pi\)
\(978\) 0 0
\(979\) − 19858.1i − 0.648283i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −32317.8 −1.04860 −0.524302 0.851533i \(-0.675674\pi\)
−0.524302 + 0.851533i \(0.675674\pi\)
\(984\) 0 0
\(985\) −1032.52 −0.0333998
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9466.41i 0.304362i
\(990\) 0 0
\(991\) 43887.8 1.40680 0.703401 0.710793i \(-0.251664\pi\)
0.703401 + 0.710793i \(0.251664\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2540.94i 0.0809581i
\(996\) 0 0
\(997\) 19646.1i 0.624071i 0.950070 + 0.312036i \(0.101011\pi\)
−0.950070 + 0.312036i \(0.898989\pi\)
\(998\) 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 1728.4.d.i.865.7 16
3.2 odd 2 1728.4.d.k.865.9 yes 16
4.3 odd 2 inner 1728.4.d.i.865.8 yes 16
8.3 odd 2 inner 1728.4.d.i.865.10 yes 16
8.5 even 2 inner 1728.4.d.i.865.9 yes 16
12.11 even 2 1728.4.d.k.865.10 yes 16
24.5 odd 2 1728.4.d.k.865.7 yes 16
24.11 even 2 1728.4.d.k.865.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.4.d.i.865.7 16 1.1 even 1 trivial
1728.4.d.i.865.8 yes 16 4.3 odd 2 inner
1728.4.d.i.865.9 yes 16 8.5 even 2 inner
1728.4.d.i.865.10 yes 16 8.3 odd 2 inner
1728.4.d.k.865.7 yes 16 24.5 odd 2
1728.4.d.k.865.8 yes 16 24.11 even 2
1728.4.d.k.865.9 yes 16 3.2 odd 2
1728.4.d.k.865.10 yes 16 12.11 even 2