Properties

Label 1728.4.d.e.865.4
Level $1728$
Weight $4$
Character 1728.865
Analytic conductor $101.955$
Analytic rank $0$
Dimension $8$
CM no
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: \(8\)
Coefficient field: 8.0.2261390379264.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 865.4
Root \(1.29746 + 1.88096i\) of defining polynomial
Character \(\chi\) \(=\) 1728.865
Dual form 1728.4.d.e.865.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.50098i q^{5} +16.0706 q^{7} +O(q^{10})\) \(q-6.50098i q^{5} +16.0706 q^{7} +27.9124i q^{11} -35.5735i q^{13} +18.9124 q^{17} +76.7372i q^{19} -190.142 q^{23} +82.7372 q^{25} +138.134i q^{29} -265.085 q^{31} -104.474i q^{35} +14.9791i q^{37} -202.562 q^{41} -133.474i q^{43} +130.020 q^{47} -84.7372 q^{49} +131.633i q^{53} +181.458 q^{55} +477.124i q^{59} -582.906i q^{61} -231.263 q^{65} -20.0000i q^{67} -60.1222 q^{71} -321.212 q^{73} +448.568i q^{77} -210.009 q^{79} +587.737i q^{83} -122.949i q^{85} -55.5767 q^{89} -571.686i q^{91} +498.867 q^{95} -75.9488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 168 q^{17} - 296 q^{25} - 24 q^{41} + 280 q^{49} - 2808 q^{65} + 304 q^{73} - 6192 q^{89} + 3224 q^{97}+O(q^{100}) \) 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
\(5\) − 6.50098i − 0.581466i −0.956804 0.290733i \(-0.906101\pi\)
0.956804 0.290733i \(-0.0938991\pi\)
\(6\) 0 0
\(7\) 16.0706 0.867728 0.433864 0.900978i \(-0.357150\pi\)
0.433864 + 0.900978i \(0.357150\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 27.9124i 0.765082i 0.923938 + 0.382541i \(0.124951\pi\)
−0.923938 + 0.382541i \(0.875049\pi\)
\(12\) 0 0
\(13\) − 35.5735i − 0.758947i −0.925203 0.379474i \(-0.876105\pi\)
0.925203 0.379474i \(-0.123895\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.9124 0.269820 0.134910 0.990858i \(-0.456926\pi\)
0.134910 + 0.990858i \(0.456926\pi\)
\(18\) 0 0
\(19\) 76.7372i 0.926564i 0.886211 + 0.463282i \(0.153328\pi\)
−0.886211 + 0.463282i \(0.846672\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −190.142 −1.72380 −0.861898 0.507081i \(-0.830724\pi\)
−0.861898 + 0.507081i \(0.830724\pi\)
\(24\) 0 0
\(25\) 82.7372 0.661898
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 138.134i 0.884512i 0.896889 + 0.442256i \(0.145822\pi\)
−0.896889 + 0.442256i \(0.854178\pi\)
\(30\) 0 0
\(31\) −265.085 −1.53583 −0.767914 0.640553i \(-0.778705\pi\)
−0.767914 + 0.640553i \(0.778705\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 104.474i − 0.504554i
\(36\) 0 0
\(37\) 14.9791i 0.0665556i 0.999446 + 0.0332778i \(0.0105946\pi\)
−0.999446 + 0.0332778i \(0.989405\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −202.562 −0.771582 −0.385791 0.922586i \(-0.626071\pi\)
−0.385791 + 0.922586i \(0.626071\pi\)
\(42\) 0 0
\(43\) − 133.474i − 0.473364i −0.971587 0.236682i \(-0.923940\pi\)
0.971587 0.236682i \(-0.0760600\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 130.020 0.403517 0.201759 0.979435i \(-0.435334\pi\)
0.201759 + 0.979435i \(0.435334\pi\)
\(48\) 0 0
\(49\) −84.7372 −0.247047
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 131.633i 0.341155i 0.985344 + 0.170577i \(0.0545632\pi\)
−0.985344 + 0.170577i \(0.945437\pi\)
\(54\) 0 0
\(55\) 181.458 0.444869
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 477.124i 1.05282i 0.850231 + 0.526409i \(0.176462\pi\)
−0.850231 + 0.526409i \(0.823538\pi\)
\(60\) 0 0
\(61\) − 582.906i − 1.22350i −0.791052 0.611749i \(-0.790466\pi\)
0.791052 0.611749i \(-0.209534\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −231.263 −0.441302
\(66\) 0 0
\(67\) − 20.0000i − 0.0364685i −0.999834 0.0182342i \(-0.994196\pi\)
0.999834 0.0182342i \(-0.00580446\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −60.1222 −0.100496 −0.0502479 0.998737i \(-0.516001\pi\)
−0.0502479 + 0.998737i \(0.516001\pi\)
\(72\) 0 0
\(73\) −321.212 −0.515000 −0.257500 0.966278i \(-0.582899\pi\)
−0.257500 + 0.966278i \(0.582899\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 448.568i 0.663884i
\(78\) 0 0
\(79\) −210.009 −0.299086 −0.149543 0.988755i \(-0.547780\pi\)
−0.149543 + 0.988755i \(0.547780\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 587.737i 0.777260i 0.921394 + 0.388630i \(0.127051\pi\)
−0.921394 + 0.388630i \(0.872949\pi\)
\(84\) 0 0
\(85\) − 122.949i − 0.156891i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −55.5767 −0.0661924 −0.0330962 0.999452i \(-0.510537\pi\)
−0.0330962 + 0.999452i \(0.510537\pi\)
\(90\) 0 0
\(91\) − 571.686i − 0.658560i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 498.867 0.538765
\(96\) 0 0
\(97\) −75.9488 −0.0794994 −0.0397497 0.999210i \(-0.512656\pi\)
−0.0397497 + 0.999210i \(0.512656\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1805.57i − 1.77882i −0.457114 0.889408i \(-0.651117\pi\)
0.457114 0.889408i \(-0.348883\pi\)
\(102\) 0 0
\(103\) −896.519 −0.857637 −0.428819 0.903391i \(-0.641070\pi\)
−0.428819 + 0.903391i \(0.641070\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 554.226i 0.500739i 0.968150 + 0.250370i \(0.0805521\pi\)
−0.968150 + 0.250370i \(0.919448\pi\)
\(108\) 0 0
\(109\) 1256.94i 1.10452i 0.833672 + 0.552260i \(0.186234\pi\)
−0.833672 + 0.552260i \(0.813766\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1615.69 −1.34505 −0.672526 0.740073i \(-0.734791\pi\)
−0.672526 + 0.740073i \(0.734791\pi\)
\(114\) 0 0
\(115\) 1236.11i 1.00233i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 303.933 0.234130
\(120\) 0 0
\(121\) 551.898 0.414649
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1350.50i − 0.966336i
\(126\) 0 0
\(127\) 1861.53 1.30066 0.650330 0.759652i \(-0.274631\pi\)
0.650330 + 0.759652i \(0.274631\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1335.53i 0.890728i 0.895350 + 0.445364i \(0.146926\pi\)
−0.895350 + 0.445364i \(0.853074\pi\)
\(132\) 0 0
\(133\) 1233.21i 0.804006i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2153.20 −1.34277 −0.671387 0.741107i \(-0.734301\pi\)
−0.671387 + 0.741107i \(0.734301\pi\)
\(138\) 0 0
\(139\) 97.4744i 0.0594797i 0.999558 + 0.0297398i \(0.00946788\pi\)
−0.999558 + 0.0297398i \(0.990532\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 992.942 0.580657
\(144\) 0 0
\(145\) 898.007 0.514313
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1392.82i 0.765803i 0.923789 + 0.382901i \(0.125075\pi\)
−0.923789 + 0.382901i \(0.874925\pi\)
\(150\) 0 0
\(151\) 838.627 0.451963 0.225982 0.974132i \(-0.427441\pi\)
0.225982 + 0.974132i \(0.427441\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1723.31i 0.893032i
\(156\) 0 0
\(157\) − 2738.54i − 1.39210i −0.717994 0.696049i \(-0.754939\pi\)
0.717994 0.696049i \(-0.245061\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3055.69 −1.49579
\(162\) 0 0
\(163\) − 421.788i − 0.202681i −0.994852 0.101341i \(-0.967687\pi\)
0.994852 0.101341i \(-0.0323132\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3364.09 −1.55881 −0.779405 0.626520i \(-0.784479\pi\)
−0.779405 + 0.626520i \(0.784479\pi\)
\(168\) 0 0
\(169\) 931.526 0.423999
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1550.32i 0.681321i 0.940186 + 0.340660i \(0.110651\pi\)
−0.940186 + 0.340660i \(0.889349\pi\)
\(174\) 0 0
\(175\) 1329.63 0.574347
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4634.17i 1.93505i 0.252775 + 0.967525i \(0.418657\pi\)
−0.252775 + 0.967525i \(0.581343\pi\)
\(180\) 0 0
\(181\) 1688.80i 0.693522i 0.937953 + 0.346761i \(0.112719\pi\)
−0.937953 + 0.346761i \(0.887281\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 97.3792 0.0386998
\(186\) 0 0
\(187\) 527.891i 0.206434i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3125.26 1.18396 0.591980 0.805953i \(-0.298347\pi\)
0.591980 + 0.805953i \(0.298347\pi\)
\(192\) 0 0
\(193\) −2085.96 −0.777981 −0.388991 0.921242i \(-0.627176\pi\)
−0.388991 + 0.921242i \(0.627176\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2356.44i 0.852231i 0.904669 + 0.426115i \(0.140118\pi\)
−0.904669 + 0.426115i \(0.859882\pi\)
\(198\) 0 0
\(199\) −970.940 −0.345870 −0.172935 0.984933i \(-0.555325\pi\)
−0.172935 + 0.984933i \(0.555325\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2219.89i 0.767516i
\(204\) 0 0
\(205\) 1316.85i 0.448649i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2141.92 −0.708898
\(210\) 0 0
\(211\) − 1621.58i − 0.529073i −0.964376 0.264537i \(-0.914781\pi\)
0.964376 0.264537i \(-0.0852190\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −867.715 −0.275245
\(216\) 0 0
\(217\) −4260.07 −1.33268
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 672.781i − 0.204779i
\(222\) 0 0
\(223\) −3391.03 −1.01830 −0.509148 0.860679i \(-0.670040\pi\)
−0.509148 + 0.860679i \(0.670040\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3111.93i 0.909894i 0.890519 + 0.454947i \(0.150342\pi\)
−0.890519 + 0.454947i \(0.849658\pi\)
\(228\) 0 0
\(229\) 4512.85i 1.30226i 0.758966 + 0.651130i \(0.225705\pi\)
−0.758966 + 0.651130i \(0.774295\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6022.61 −1.69337 −0.846683 0.532097i \(-0.821404\pi\)
−0.846683 + 0.532097i \(0.821404\pi\)
\(234\) 0 0
\(235\) − 845.256i − 0.234632i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2371.15 −0.641745 −0.320872 0.947122i \(-0.603976\pi\)
−0.320872 + 0.947122i \(0.603976\pi\)
\(240\) 0 0
\(241\) 4124.74 1.10248 0.551241 0.834346i \(-0.314155\pi\)
0.551241 + 0.834346i \(0.314155\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 550.875i 0.143649i
\(246\) 0 0
\(247\) 2729.81 0.703214
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4354.18i 1.09495i 0.836821 + 0.547476i \(0.184411\pi\)
−0.836821 + 0.547476i \(0.815589\pi\)
\(252\) 0 0
\(253\) − 5307.32i − 1.31885i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7727.55 −1.87561 −0.937805 0.347163i \(-0.887145\pi\)
−0.937805 + 0.347163i \(0.887145\pi\)
\(258\) 0 0
\(259\) 240.723i 0.0577522i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3944.29 −0.924774 −0.462387 0.886678i \(-0.653007\pi\)
−0.462387 + 0.886678i \(0.653007\pi\)
\(264\) 0 0
\(265\) 855.744 0.198370
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 290.836i 0.0659204i 0.999457 + 0.0329602i \(0.0104935\pi\)
−0.999457 + 0.0329602i \(0.989507\pi\)
\(270\) 0 0
\(271\) −2842.92 −0.637252 −0.318626 0.947881i \(-0.603221\pi\)
−0.318626 + 0.947881i \(0.603221\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2309.39i 0.506406i
\(276\) 0 0
\(277\) − 6030.03i − 1.30798i −0.756505 0.653988i \(-0.773094\pi\)
0.756505 0.653988i \(-0.226906\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3545.58 −0.752711 −0.376356 0.926475i \(-0.622823\pi\)
−0.376356 + 0.926475i \(0.622823\pi\)
\(282\) 0 0
\(283\) 1499.48i 0.314964i 0.987522 + 0.157482i \(0.0503377\pi\)
−0.987522 + 0.157482i \(0.949662\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3255.28 −0.669524
\(288\) 0 0
\(289\) −4555.32 −0.927197
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1174.97i 0.234275i 0.993116 + 0.117137i \(0.0373718\pi\)
−0.993116 + 0.117137i \(0.962628\pi\)
\(294\) 0 0
\(295\) 3101.78 0.612177
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6764.01i 1.30827i
\(300\) 0 0
\(301\) − 2145.01i − 0.410752i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3789.46 −0.711423
\(306\) 0 0
\(307\) − 5403.90i − 1.00462i −0.864689 0.502308i \(-0.832484\pi\)
0.864689 0.502308i \(-0.167516\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9941.05 1.81256 0.906278 0.422682i \(-0.138911\pi\)
0.906278 + 0.422682i \(0.138911\pi\)
\(312\) 0 0
\(313\) 5905.11 1.06638 0.533189 0.845996i \(-0.320993\pi\)
0.533189 + 0.845996i \(0.320993\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 7828.23i − 1.38699i −0.720460 0.693497i \(-0.756069\pi\)
0.720460 0.693497i \(-0.243931\pi\)
\(318\) 0 0
\(319\) −3855.65 −0.676724
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1451.29i 0.250005i
\(324\) 0 0
\(325\) − 2943.25i − 0.502346i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2089.49 0.350144
\(330\) 0 0
\(331\) 11665.1i 1.93707i 0.248874 + 0.968536i \(0.419939\pi\)
−0.248874 + 0.968536i \(0.580061\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −130.020 −0.0212052
\(336\) 0 0
\(337\) 1705.59 0.275696 0.137848 0.990453i \(-0.455982\pi\)
0.137848 + 0.990453i \(0.455982\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 7399.16i − 1.17504i
\(342\) 0 0
\(343\) −6873.98 −1.08210
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7701.30i 1.19143i 0.803195 + 0.595717i \(0.203132\pi\)
−0.803195 + 0.595717i \(0.796868\pi\)
\(348\) 0 0
\(349\) 6875.08i 1.05448i 0.849715 + 0.527242i \(0.176774\pi\)
−0.849715 + 0.527242i \(0.823226\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7008.39 −1.05671 −0.528355 0.849023i \(-0.677191\pi\)
−0.528355 + 0.849023i \(0.677191\pi\)
\(354\) 0 0
\(355\) 390.854i 0.0584348i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8996.79 1.32265 0.661327 0.750098i \(-0.269994\pi\)
0.661327 + 0.750098i \(0.269994\pi\)
\(360\) 0 0
\(361\) 970.400 0.141478
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2088.19i 0.299455i
\(366\) 0 0
\(367\) 4690.25 0.667109 0.333554 0.942731i \(-0.391752\pi\)
0.333554 + 0.942731i \(0.391752\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2115.42i 0.296029i
\(372\) 0 0
\(373\) 8072.72i 1.12062i 0.828285 + 0.560308i \(0.189317\pi\)
−0.828285 + 0.560308i \(0.810683\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4913.91 0.671298
\(378\) 0 0
\(379\) − 8379.80i − 1.13573i −0.823122 0.567864i \(-0.807770\pi\)
0.823122 0.567864i \(-0.192230\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 869.756 0.116038 0.0580189 0.998315i \(-0.481522\pi\)
0.0580189 + 0.998315i \(0.481522\pi\)
\(384\) 0 0
\(385\) 2916.13 0.386026
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8524.17i 1.11103i 0.831505 + 0.555517i \(0.187480\pi\)
−0.831505 + 0.555517i \(0.812520\pi\)
\(390\) 0 0
\(391\) −3596.04 −0.465114
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1365.26i 0.173908i
\(396\) 0 0
\(397\) − 3385.75i − 0.428025i −0.976831 0.214012i \(-0.931347\pi\)
0.976831 0.214012i \(-0.0686533\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −13391.3 −1.66766 −0.833829 0.552023i \(-0.813856\pi\)
−0.833829 + 0.552023i \(0.813856\pi\)
\(402\) 0 0
\(403\) 9430.01i 1.16561i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −418.104 −0.0509205
\(408\) 0 0
\(409\) −7965.98 −0.963062 −0.481531 0.876429i \(-0.659919\pi\)
−0.481531 + 0.876429i \(0.659919\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7667.65i 0.913560i
\(414\) 0 0
\(415\) 3820.87 0.451950
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6356.23i 0.741103i 0.928812 + 0.370552i \(0.120831\pi\)
−0.928812 + 0.370552i \(0.879169\pi\)
\(420\) 0 0
\(421\) − 6365.81i − 0.736937i −0.929640 0.368468i \(-0.879882\pi\)
0.929640 0.368468i \(-0.120118\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1564.76 0.178593
\(426\) 0 0
\(427\) − 9367.62i − 1.06166i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4612.28 −0.515466 −0.257733 0.966216i \(-0.582975\pi\)
−0.257733 + 0.966216i \(0.582975\pi\)
\(432\) 0 0
\(433\) −10400.1 −1.15426 −0.577131 0.816652i \(-0.695828\pi\)
−0.577131 + 0.816652i \(0.695828\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 14591.0i − 1.59721i
\(438\) 0 0
\(439\) −1480.23 −0.160928 −0.0804642 0.996757i \(-0.525640\pi\)
−0.0804642 + 0.996757i \(0.525640\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 5799.47i − 0.621989i −0.950412 0.310995i \(-0.899338\pi\)
0.950412 0.310995i \(-0.100662\pi\)
\(444\) 0 0
\(445\) 361.303i 0.0384886i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6011.34 −0.631832 −0.315916 0.948787i \(-0.602312\pi\)
−0.315916 + 0.948787i \(0.602312\pi\)
\(450\) 0 0
\(451\) − 5653.99i − 0.590324i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3716.52 −0.382930
\(456\) 0 0
\(457\) 7891.93 0.807810 0.403905 0.914801i \(-0.367653\pi\)
0.403905 + 0.914801i \(0.367653\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8338.72i 0.842458i 0.906954 + 0.421229i \(0.138401\pi\)
−0.906954 + 0.421229i \(0.861599\pi\)
\(462\) 0 0
\(463\) 16445.4 1.65072 0.825360 0.564607i \(-0.190972\pi\)
0.825360 + 0.564607i \(0.190972\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4975.50i − 0.493017i −0.969141 0.246508i \(-0.920717\pi\)
0.969141 0.246508i \(-0.0792833\pi\)
\(468\) 0 0
\(469\) − 321.411i − 0.0316447i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3725.59 0.362163
\(474\) 0 0
\(475\) 6349.02i 0.613291i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −617.261 −0.0588797 −0.0294398 0.999567i \(-0.509372\pi\)
−0.0294398 + 0.999567i \(0.509372\pi\)
\(480\) 0 0
\(481\) 532.861 0.0505122
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 493.742i 0.0462261i
\(486\) 0 0
\(487\) 4775.91 0.444389 0.222194 0.975002i \(-0.428678\pi\)
0.222194 + 0.975002i \(0.428678\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 8682.25i − 0.798014i −0.916948 0.399007i \(-0.869355\pi\)
0.916948 0.399007i \(-0.130645\pi\)
\(492\) 0 0
\(493\) 2612.45i 0.238659i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −966.198 −0.0872030
\(498\) 0 0
\(499\) − 3011.15i − 0.270136i −0.990836 0.135068i \(-0.956875\pi\)
0.990836 0.135068i \(-0.0431252\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4948.29 −0.438635 −0.219318 0.975654i \(-0.570383\pi\)
−0.219318 + 0.975654i \(0.570383\pi\)
\(504\) 0 0
\(505\) −11737.9 −1.03432
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 8060.89i − 0.701950i −0.936385 0.350975i \(-0.885850\pi\)
0.936385 0.350975i \(-0.114150\pi\)
\(510\) 0 0
\(511\) −5162.05 −0.446880
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5828.25i 0.498687i
\(516\) 0 0
\(517\) 3629.16i 0.308724i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18392.8 1.54665 0.773325 0.634010i \(-0.218592\pi\)
0.773325 + 0.634010i \(0.218592\pi\)
\(522\) 0 0
\(523\) − 15531.7i − 1.29857i −0.760544 0.649286i \(-0.775068\pi\)
0.760544 0.649286i \(-0.224932\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5013.40 −0.414397
\(528\) 0 0
\(529\) 23986.9 1.97148
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7205.84i 0.585590i
\(534\) 0 0
\(535\) 3603.02 0.291163
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2365.22i − 0.189012i
\(540\) 0 0
\(541\) 24594.4i 1.95452i 0.212035 + 0.977262i \(0.431991\pi\)
−0.212035 + 0.977262i \(0.568009\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8171.32 0.642240
\(546\) 0 0
\(547\) 15437.8i 1.20671i 0.797471 + 0.603357i \(0.206170\pi\)
−0.797471 + 0.603357i \(0.793830\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10600.0 −0.819557
\(552\) 0 0
\(553\) −3374.96 −0.259526
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7720.89i 0.587333i 0.955908 + 0.293667i \(0.0948756\pi\)
−0.955908 + 0.293667i \(0.905124\pi\)
\(558\) 0 0
\(559\) −4748.15 −0.359258
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 4965.48i − 0.371705i −0.982578 0.185852i \(-0.940495\pi\)
0.982578 0.185852i \(-0.0595046\pi\)
\(564\) 0 0
\(565\) 10503.5i 0.782102i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 14638.2 1.07850 0.539250 0.842146i \(-0.318708\pi\)
0.539250 + 0.842146i \(0.318708\pi\)
\(570\) 0 0
\(571\) 10966.5i 0.803735i 0.915698 + 0.401867i \(0.131639\pi\)
−0.915698 + 0.401867i \(0.868361\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −15731.8 −1.14098
\(576\) 0 0
\(577\) −21576.8 −1.55676 −0.778382 0.627791i \(-0.783959\pi\)
−0.778382 + 0.627791i \(0.783959\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9445.26i 0.674450i
\(582\) 0 0
\(583\) −3674.19 −0.261011
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7747.02i − 0.544725i −0.962195 0.272362i \(-0.912195\pi\)
0.962195 0.272362i \(-0.0878050\pi\)
\(588\) 0 0
\(589\) − 20341.9i − 1.42304i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −9423.20 −0.652554 −0.326277 0.945274i \(-0.605794\pi\)
−0.326277 + 0.945274i \(0.605794\pi\)
\(594\) 0 0
\(595\) − 1975.86i − 0.136139i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16746.0 1.14227 0.571137 0.820855i \(-0.306503\pi\)
0.571137 + 0.820855i \(0.306503\pi\)
\(600\) 0 0
\(601\) 24440.4 1.65881 0.829405 0.558648i \(-0.188680\pi\)
0.829405 + 0.558648i \(0.188680\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3587.88i − 0.241104i
\(606\) 0 0
\(607\) −24645.3 −1.64798 −0.823988 0.566607i \(-0.808256\pi\)
−0.823988 + 0.566607i \(0.808256\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 4625.26i − 0.306248i
\(612\) 0 0
\(613\) − 13945.4i − 0.918842i −0.888219 0.459421i \(-0.848057\pi\)
0.888219 0.459421i \(-0.151943\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20120.0 1.31281 0.656403 0.754410i \(-0.272077\pi\)
0.656403 + 0.754410i \(0.272077\pi\)
\(618\) 0 0
\(619\) 15834.6i 1.02819i 0.857734 + 0.514093i \(0.171871\pi\)
−0.857734 + 0.514093i \(0.828129\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −893.149 −0.0574370
\(624\) 0 0
\(625\) 1562.60 0.100006
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 283.292i 0.0179580i
\(630\) 0 0
\(631\) 25542.9 1.61149 0.805744 0.592264i \(-0.201766\pi\)
0.805744 + 0.592264i \(0.201766\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 12101.8i − 0.756289i
\(636\) 0 0
\(637\) 3014.40i 0.187496i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4368.61 −0.269188 −0.134594 0.990901i \(-0.542973\pi\)
−0.134594 + 0.990901i \(0.542973\pi\)
\(642\) 0 0
\(643\) − 6253.46i − 0.383534i −0.981440 0.191767i \(-0.938578\pi\)
0.981440 0.191767i \(-0.0614218\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27111.2 1.64737 0.823687 0.567045i \(-0.191913\pi\)
0.823687 + 0.567045i \(0.191913\pi\)
\(648\) 0 0
\(649\) −13317.7 −0.805492
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11206.8i 0.671601i 0.941933 + 0.335801i \(0.109007\pi\)
−0.941933 + 0.335801i \(0.890993\pi\)
\(654\) 0 0
\(655\) 8682.23 0.517928
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 28324.2i − 1.67429i −0.546983 0.837143i \(-0.684224\pi\)
0.546983 0.837143i \(-0.315776\pi\)
\(660\) 0 0
\(661\) − 27069.6i − 1.59287i −0.604727 0.796433i \(-0.706718\pi\)
0.604727 0.796433i \(-0.293282\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8017.08 0.467502
\(666\) 0 0
\(667\) − 26265.1i − 1.52472i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 16270.3 0.936077
\(672\) 0 0
\(673\) 18620.6 1.06652 0.533262 0.845950i \(-0.320966\pi\)
0.533262 + 0.845950i \(0.320966\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 25175.3i − 1.42919i −0.699536 0.714597i \(-0.746610\pi\)
0.699536 0.714597i \(-0.253390\pi\)
\(678\) 0 0
\(679\) −1220.54 −0.0689839
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 13438.8i − 0.752886i −0.926440 0.376443i \(-0.877147\pi\)
0.926440 0.376443i \(-0.122853\pi\)
\(684\) 0 0
\(685\) 13997.9i 0.780777i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4682.65 0.258918
\(690\) 0 0
\(691\) 28254.8i 1.55552i 0.628563 + 0.777758i \(0.283643\pi\)
−0.628563 + 0.777758i \(0.716357\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 633.680 0.0345854
\(696\) 0 0
\(697\) −3830.93 −0.208188
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 32350.0i − 1.74300i −0.490397 0.871499i \(-0.663148\pi\)
0.490397 0.871499i \(-0.336852\pi\)
\(702\) 0 0
\(703\) −1149.46 −0.0616680
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 29016.4i − 1.54353i
\(708\) 0 0
\(709\) − 20041.0i − 1.06157i −0.847505 0.530787i \(-0.821897\pi\)
0.847505 0.530787i \(-0.178103\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 50403.8 2.64746
\(714\) 0 0
\(715\) − 6455.10i − 0.337632i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 11518.4 0.597447 0.298723 0.954340i \(-0.403439\pi\)
0.298723 + 0.954340i \(0.403439\pi\)
\(720\) 0 0
\(721\) −14407.6 −0.744196
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 11428.8i 0.585456i
\(726\) 0 0
\(727\) −16989.3 −0.866710 −0.433355 0.901223i \(-0.642670\pi\)
−0.433355 + 0.901223i \(0.642670\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 2524.32i − 0.127723i
\(732\) 0 0
\(733\) 11934.0i 0.601355i 0.953726 + 0.300677i \(0.0972127\pi\)
−0.953726 + 0.300677i \(0.902787\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 558.248 0.0279014
\(738\) 0 0
\(739\) − 19610.5i − 0.976160i −0.872799 0.488080i \(-0.837697\pi\)
0.872799 0.488080i \(-0.162303\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25308.6 −1.24964 −0.624820 0.780769i \(-0.714828\pi\)
−0.624820 + 0.780769i \(0.714828\pi\)
\(744\) 0 0
\(745\) 9054.73 0.445288
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8906.73i 0.434506i
\(750\) 0 0
\(751\) −11835.9 −0.575097 −0.287548 0.957766i \(-0.592840\pi\)
−0.287548 + 0.957766i \(0.592840\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 5451.90i − 0.262801i
\(756\) 0 0
\(757\) 13648.5i 0.655299i 0.944799 + 0.327650i \(0.106257\pi\)
−0.944799 + 0.327650i \(0.893743\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 33364.9 1.58932 0.794662 0.607052i \(-0.207648\pi\)
0.794662 + 0.607052i \(0.207648\pi\)
\(762\) 0 0
\(763\) 20199.7i 0.958423i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 16973.0 0.799033
\(768\) 0 0
\(769\) −17035.1 −0.798830 −0.399415 0.916770i \(-0.630787\pi\)
−0.399415 + 0.916770i \(0.630787\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18934.3i 0.881010i 0.897750 + 0.440505i \(0.145201\pi\)
−0.897750 + 0.440505i \(0.854799\pi\)
\(774\) 0 0
\(775\) −21932.4 −1.01656
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 15544.0i − 0.714921i
\(780\) 0 0
\(781\) − 1678.16i − 0.0768875i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −17803.2 −0.809458
\(786\) 0 0
\(787\) − 31964.7i − 1.44780i −0.689904 0.723901i \(-0.742347\pi\)
0.689904 0.723901i \(-0.257653\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −25965.0 −1.16714
\(792\) 0 0
\(793\) −20736.0 −0.928571
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 27500.2i − 1.22222i −0.791546 0.611110i \(-0.790723\pi\)
0.791546 0.611110i \(-0.209277\pi\)
\(798\) 0 0
\(799\) 2458.98 0.108877
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 8965.79i − 0.394017i
\(804\) 0 0
\(805\) 19865.0i 0.869749i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 41290.0 1.79441 0.897206 0.441613i \(-0.145593\pi\)
0.897206 + 0.441613i \(0.145593\pi\)
\(810\) 0 0
\(811\) − 25905.2i − 1.12164i −0.827936 0.560822i \(-0.810485\pi\)
0.827936 0.560822i \(-0.189515\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2742.04 −0.117852
\(816\) 0 0
\(817\) 10242.5 0.438602
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 9011.17i − 0.383060i −0.981487 0.191530i \(-0.938655\pi\)
0.981487 0.191530i \(-0.0613449\pi\)
\(822\) 0 0
\(823\) −7843.81 −0.332221 −0.166111 0.986107i \(-0.553121\pi\)
−0.166111 + 0.986107i \(0.553121\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3843.55i − 0.161612i −0.996730 0.0808061i \(-0.974251\pi\)
0.996730 0.0808061i \(-0.0257495\pi\)
\(828\) 0 0
\(829\) − 36286.9i − 1.52026i −0.649770 0.760131i \(-0.725135\pi\)
0.649770 0.760131i \(-0.274865\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1602.58 −0.0666582
\(834\) 0 0
\(835\) 21869.9i 0.906395i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 434.523 0.0178801 0.00894004 0.999960i \(-0.497154\pi\)
0.00894004 + 0.999960i \(0.497154\pi\)
\(840\) 0 0
\(841\) 5307.99 0.217639
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 6055.83i − 0.246541i
\(846\) 0 0
\(847\) 8869.30 0.359803
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2848.16i − 0.114728i
\(852\) 0 0
\(853\) − 20871.7i − 0.837787i −0.908035 0.418893i \(-0.862418\pi\)
0.908035 0.418893i \(-0.137582\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 34692.7 1.38282 0.691412 0.722460i \(-0.256989\pi\)
0.691412 + 0.722460i \(0.256989\pi\)
\(858\) 0 0
\(859\) 22844.9i 0.907403i 0.891154 + 0.453702i \(0.149897\pi\)
−0.891154 + 0.453702i \(0.850103\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3998.34 0.157712 0.0788558 0.996886i \(-0.474873\pi\)
0.0788558 + 0.996886i \(0.474873\pi\)
\(864\) 0 0
\(865\) 10078.6 0.396165
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 5861.85i − 0.228826i
\(870\) 0 0
\(871\) −711.470 −0.0276777
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 21703.2i − 0.838518i
\(876\) 0 0
\(877\) 16596.6i 0.639028i 0.947581 + 0.319514i \(0.103520\pi\)
−0.947581 + 0.319514i \(0.896480\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 342.063 0.0130811 0.00654053 0.999979i \(-0.497918\pi\)
0.00654053 + 0.999979i \(0.497918\pi\)
\(882\) 0 0
\(883\) − 14529.0i − 0.553725i −0.960910 0.276862i \(-0.910705\pi\)
0.960910 0.276862i \(-0.0892946\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33035.3 −1.25053 −0.625263 0.780414i \(-0.715008\pi\)
−0.625263 + 0.780414i \(0.715008\pi\)
\(888\) 0 0
\(889\) 29915.8 1.12862
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9977.35i 0.373885i
\(894\) 0 0
\(895\) 30126.6 1.12517
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 36617.3i − 1.35846i
\(900\) 0 0
\(901\) 2489.50i 0.0920501i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10978.9 0.403259
\(906\) 0 0
\(907\) 52453.7i 1.92028i 0.279515 + 0.960141i \(0.409826\pi\)
−0.279515 + 0.960141i \(0.590174\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48342.7 1.75814 0.879070 0.476692i \(-0.158164\pi\)
0.879070 + 0.476692i \(0.158164\pi\)
\(912\) 0 0
\(913\) −16405.2 −0.594668
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21462.6i 0.772910i
\(918\) 0 0
\(919\) 21911.9 0.786514 0.393257 0.919429i \(-0.371348\pi\)
0.393257 + 0.919429i \(0.371348\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2138.76i 0.0762710i
\(924\) 0 0
\(925\) 1239.33i 0.0440530i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25184.7 −0.889433 −0.444717 0.895671i \(-0.646696\pi\)
−0.444717 + 0.895671i \(0.646696\pi\)
\(930\) 0 0
\(931\) − 6502.50i − 0.228905i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3431.81 0.120034
\(936\) 0 0
\(937\) −37948.3 −1.32307 −0.661536 0.749914i \(-0.730095\pi\)
−0.661536 + 0.749914i \(0.730095\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 37907.9i 1.31325i 0.754219 + 0.656623i \(0.228016\pi\)
−0.754219 + 0.656623i \(0.771984\pi\)
\(942\) 0 0
\(943\) 38515.5 1.33005
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 20062.2i − 0.688419i −0.938893 0.344209i \(-0.888147\pi\)
0.938893 0.344209i \(-0.111853\pi\)
\(948\) 0 0
\(949\) 11426.6i 0.390858i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 43232.8 1.46952 0.734758 0.678329i \(-0.237296\pi\)
0.734758 + 0.678329i \(0.237296\pi\)
\(954\) 0 0
\(955\) − 20317.3i − 0.688432i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −34603.1 −1.16516
\(960\) 0 0
\(961\) 40479.1 1.35877
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 13560.8i 0.452369i
\(966\) 0 0
\(967\) −9292.66 −0.309030 −0.154515 0.987990i \(-0.549381\pi\)
−0.154515 + 0.987990i \(0.549381\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 38499.0i − 1.27239i −0.771528 0.636196i \(-0.780507\pi\)
0.771528 0.636196i \(-0.219493\pi\)
\(972\) 0 0
\(973\) 1566.47i 0.0516122i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9843.93 −0.322349 −0.161175 0.986926i \(-0.551528\pi\)
−0.161175 + 0.986926i \(0.551528\pi\)
\(978\) 0 0
\(979\) − 1551.28i − 0.0506426i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −43241.9 −1.40305 −0.701527 0.712643i \(-0.747498\pi\)
−0.701527 + 0.712643i \(0.747498\pi\)
\(984\) 0 0
\(985\) 15319.2 0.495543
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25379.1i 0.815984i
\(990\) 0 0
\(991\) −34891.3 −1.11842 −0.559212 0.829025i \(-0.688896\pi\)
−0.559212 + 0.829025i \(0.688896\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6312.07i 0.201111i
\(996\) 0 0
\(997\) 50166.9i 1.59358i 0.604256 + 0.796790i \(0.293471\pi\)
−0.604256 + 0.796790i \(0.706529\pi\)
\(998\) 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.e.865.4 yes 8
3.2 odd 2 1728.4.d.h.865.6 yes 8
4.3 odd 2 inner 1728.4.d.e.865.3 8
8.3 odd 2 inner 1728.4.d.e.865.5 yes 8
8.5 even 2 inner 1728.4.d.e.865.6 yes 8
12.11 even 2 1728.4.d.h.865.5 yes 8
24.5 odd 2 1728.4.d.h.865.4 yes 8
24.11 even 2 1728.4.d.h.865.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.4.d.e.865.3 8 4.3 odd 2 inner
1728.4.d.e.865.4 yes 8 1.1 even 1 trivial
1728.4.d.e.865.5 yes 8 8.3 odd 2 inner
1728.4.d.e.865.6 yes 8 8.5 even 2 inner
1728.4.d.h.865.3 yes 8 24.11 even 2
1728.4.d.h.865.4 yes 8 24.5 odd 2
1728.4.d.h.865.5 yes 8 12.11 even 2
1728.4.d.h.865.6 yes 8 3.2 odd 2