Properties

Label 2880.3.e.e.2431.1
Level $2880$
Weight $3$
Character 2880.2431
Analytic conductor $78.474$
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] = [2880,3,Mod(2431,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2431");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2880.e (of order \(2\), degree \(1\), not 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: \(78.4743161358\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.1
Root \(-0.309017 + 0.951057i\) of defining polynomial
Character \(\chi\) \(=\) 2880.2431
Dual form 2880.3.e.e.2431.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607 q^{5} -5.25731i q^{7} +O(q^{10})\) \(q-2.23607 q^{5} -5.25731i q^{7} +19.9192i q^{11} +8.47214 q^{13} -11.8885 q^{17} +15.2169i q^{19} +0.555029i q^{23} +5.00000 q^{25} -10.9443 q^{29} -8.29451i q^{31} +11.7557i q^{35} +18.3607 q^{37} +14.5836 q^{41} -22.2703i q^{43} -53.3902i q^{47} +21.3607 q^{49} -66.3607 q^{53} -44.5407i q^{55} +17.4370i q^{59} -90.1378 q^{61} -18.9443 q^{65} -50.2220i q^{67} +80.7868i q^{71} -5.55418 q^{73} +104.721 q^{77} -13.8448i q^{79} -76.2155i q^{83} +26.5836 q^{85} +111.443 q^{89} -44.5407i q^{91} -34.0260i q^{95} -92.8328 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{13} + 24 q^{17} + 20 q^{25} - 8 q^{29} - 16 q^{37} + 112 q^{41} - 4 q^{49} - 176 q^{53} - 128 q^{61} - 40 q^{65} + 264 q^{73} + 240 q^{77} + 160 q^{85} + 88 q^{89} - 264 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(5\) −2.23607 −0.447214
\(6\) 0 0
\(7\) − 5.25731i − 0.751044i −0.926813 0.375522i \(-0.877463\pi\)
0.926813 0.375522i \(-0.122537\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 19.9192i 1.81084i 0.424522 + 0.905418i \(0.360442\pi\)
−0.424522 + 0.905418i \(0.639558\pi\)
\(12\) 0 0
\(13\) 8.47214 0.651703 0.325851 0.945421i \(-0.394349\pi\)
0.325851 + 0.945421i \(0.394349\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.8885 −0.699326 −0.349663 0.936876i \(-0.613704\pi\)
−0.349663 + 0.936876i \(0.613704\pi\)
\(18\) 0 0
\(19\) 15.2169i 0.800890i 0.916321 + 0.400445i \(0.131144\pi\)
−0.916321 + 0.400445i \(0.868856\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.555029i 0.0241317i 0.999927 + 0.0120659i \(0.00384077\pi\)
−0.999927 + 0.0120659i \(0.996159\pi\)
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.9443 −0.377389 −0.188694 0.982036i \(-0.560426\pi\)
−0.188694 + 0.982036i \(0.560426\pi\)
\(30\) 0 0
\(31\) − 8.29451i − 0.267565i −0.991011 0.133782i \(-0.957288\pi\)
0.991011 0.133782i \(-0.0427123\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.7557i 0.335877i
\(36\) 0 0
\(37\) 18.3607 0.496235 0.248117 0.968730i \(-0.420188\pi\)
0.248117 + 0.968730i \(0.420188\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 14.5836 0.355697 0.177849 0.984058i \(-0.443086\pi\)
0.177849 + 0.984058i \(0.443086\pi\)
\(42\) 0 0
\(43\) − 22.2703i − 0.517915i −0.965889 0.258957i \(-0.916621\pi\)
0.965889 0.258957i \(-0.0833789\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 53.3902i − 1.13596i −0.823042 0.567981i \(-0.807725\pi\)
0.823042 0.567981i \(-0.192275\pi\)
\(48\) 0 0
\(49\) 21.3607 0.435932
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −66.3607 −1.25209 −0.626044 0.779788i \(-0.715327\pi\)
−0.626044 + 0.779788i \(0.715327\pi\)
\(54\) 0 0
\(55\) − 44.5407i − 0.809830i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 17.4370i 0.295543i 0.989022 + 0.147771i \(0.0472100\pi\)
−0.989022 + 0.147771i \(0.952790\pi\)
\(60\) 0 0
\(61\) −90.1378 −1.47767 −0.738834 0.673887i \(-0.764623\pi\)
−0.738834 + 0.673887i \(0.764623\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −18.9443 −0.291450
\(66\) 0 0
\(67\) − 50.2220i − 0.749582i −0.927109 0.374791i \(-0.877715\pi\)
0.927109 0.374791i \(-0.122285\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 80.7868i 1.13784i 0.822392 + 0.568921i \(0.192639\pi\)
−0.822392 + 0.568921i \(0.807361\pi\)
\(72\) 0 0
\(73\) −5.55418 −0.0760846 −0.0380423 0.999276i \(-0.512112\pi\)
−0.0380423 + 0.999276i \(0.512112\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 104.721 1.36002
\(78\) 0 0
\(79\) − 13.8448i − 0.175251i −0.996154 0.0876253i \(-0.972072\pi\)
0.996154 0.0876253i \(-0.0279278\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 76.2155i − 0.918260i −0.888369 0.459130i \(-0.848161\pi\)
0.888369 0.459130i \(-0.151839\pi\)
\(84\) 0 0
\(85\) 26.5836 0.312748
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 111.443 1.25217 0.626083 0.779757i \(-0.284657\pi\)
0.626083 + 0.779757i \(0.284657\pi\)
\(90\) 0 0
\(91\) − 44.5407i − 0.489458i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 34.0260i − 0.358169i
\(96\) 0 0
\(97\) −92.8328 −0.957039 −0.478520 0.878077i \(-0.658826\pi\)
−0.478520 + 0.878077i \(0.658826\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 64.1115 0.634767 0.317383 0.948297i \(-0.397196\pi\)
0.317383 + 0.948297i \(0.397196\pi\)
\(102\) 0 0
\(103\) 137.769i 1.33757i 0.743458 + 0.668783i \(0.233184\pi\)
−0.743458 + 0.668783i \(0.766816\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 51.3320i − 0.479739i −0.970805 0.239869i \(-0.922895\pi\)
0.970805 0.239869i \(-0.0771046\pi\)
\(108\) 0 0
\(109\) −133.469 −1.22449 −0.612243 0.790669i \(-0.709733\pi\)
−0.612243 + 0.790669i \(0.709733\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −170.721 −1.51081 −0.755404 0.655259i \(-0.772559\pi\)
−0.755404 + 0.655259i \(0.772559\pi\)
\(114\) 0 0
\(115\) − 1.24108i − 0.0107920i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 62.5018i 0.525225i
\(120\) 0 0
\(121\) −275.774 −2.27912
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1803 −0.0894427
\(126\) 0 0
\(127\) − 198.637i − 1.56407i −0.623235 0.782035i \(-0.714182\pi\)
0.623235 0.782035i \(-0.285818\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.77041i 0.0593161i 0.999560 + 0.0296580i \(0.00944183\pi\)
−0.999560 + 0.0296580i \(0.990558\pi\)
\(132\) 0 0
\(133\) 80.0000 0.601504
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.832816 −0.00607895 −0.00303947 0.999995i \(-0.500967\pi\)
−0.00303947 + 0.999995i \(0.500967\pi\)
\(138\) 0 0
\(139\) 237.658i 1.70977i 0.518817 + 0.854885i \(0.326373\pi\)
−0.518817 + 0.854885i \(0.673627\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 168.758i 1.18013i
\(144\) 0 0
\(145\) 24.4721 0.168773
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −36.9706 −0.248125 −0.124062 0.992274i \(-0.539592\pi\)
−0.124062 + 0.992274i \(0.539592\pi\)
\(150\) 0 0
\(151\) 282.723i 1.87234i 0.351552 + 0.936168i \(0.385654\pi\)
−0.351552 + 0.936168i \(0.614346\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 18.5471i 0.119659i
\(156\) 0 0
\(157\) −204.748 −1.30413 −0.652063 0.758165i \(-0.726096\pi\)
−0.652063 + 0.758165i \(0.726096\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.91796 0.0181240
\(162\) 0 0
\(163\) − 107.235i − 0.657885i −0.944350 0.328943i \(-0.893308\pi\)
0.944350 0.328943i \(-0.106692\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 33.2090i − 0.198856i −0.995045 0.0994280i \(-0.968299\pi\)
0.995045 0.0994280i \(-0.0317013\pi\)
\(168\) 0 0
\(169\) −97.2229 −0.575284
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −226.361 −1.30844 −0.654222 0.756303i \(-0.727004\pi\)
−0.654222 + 0.756303i \(0.727004\pi\)
\(174\) 0 0
\(175\) − 26.2866i − 0.150209i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 224.337i 1.25328i 0.779308 + 0.626641i \(0.215571\pi\)
−0.779308 + 0.626641i \(0.784429\pi\)
\(180\) 0 0
\(181\) −86.2229 −0.476370 −0.238185 0.971220i \(-0.576552\pi\)
−0.238185 + 0.971220i \(0.576552\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −41.0557 −0.221923
\(186\) 0 0
\(187\) − 236.810i − 1.26636i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 31.0198i 0.162407i 0.996698 + 0.0812036i \(0.0258764\pi\)
−0.996698 + 0.0812036i \(0.974124\pi\)
\(192\) 0 0
\(193\) 110.223 0.571103 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −172.525 −0.875760 −0.437880 0.899033i \(-0.644271\pi\)
−0.437880 + 0.899033i \(0.644271\pi\)
\(198\) 0 0
\(199\) − 272.208i − 1.36788i −0.729538 0.683940i \(-0.760265\pi\)
0.729538 0.683940i \(-0.239735\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 57.5374i 0.283436i
\(204\) 0 0
\(205\) −32.6099 −0.159073
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −303.108 −1.45028
\(210\) 0 0
\(211\) 205.266i 0.972826i 0.873729 + 0.486413i \(0.161695\pi\)
−0.873729 + 0.486413i \(0.838305\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 49.7980i 0.231618i
\(216\) 0 0
\(217\) −43.6068 −0.200953
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −100.721 −0.455753
\(222\) 0 0
\(223\) 235.731i 1.05709i 0.848905 + 0.528545i \(0.177262\pi\)
−0.848905 + 0.528545i \(0.822738\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 58.5165i − 0.257782i −0.991659 0.128891i \(-0.958858\pi\)
0.991659 0.128891i \(-0.0411417\pi\)
\(228\) 0 0
\(229\) −162.721 −0.710574 −0.355287 0.934757i \(-0.615617\pi\)
−0.355287 + 0.934757i \(0.615617\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −319.050 −1.36931 −0.684656 0.728867i \(-0.740047\pi\)
−0.684656 + 0.728867i \(0.740047\pi\)
\(234\) 0 0
\(235\) 119.384i 0.508017i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 236.810i − 0.990837i −0.868654 0.495419i \(-0.835015\pi\)
0.868654 0.495419i \(-0.164985\pi\)
\(240\) 0 0
\(241\) −0.917961 −0.00380897 −0.00190448 0.999998i \(-0.500606\pi\)
−0.00190448 + 0.999998i \(0.500606\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −47.7639 −0.194955
\(246\) 0 0
\(247\) 128.920i 0.521942i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 136.690i − 0.544582i −0.962215 0.272291i \(-0.912219\pi\)
0.962215 0.272291i \(-0.0877813\pi\)
\(252\) 0 0
\(253\) −11.0557 −0.0436985
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 274.944 1.06982 0.534911 0.844908i \(-0.320345\pi\)
0.534911 + 0.844908i \(0.320345\pi\)
\(258\) 0 0
\(259\) − 96.5278i − 0.372694i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 406.385i − 1.54519i −0.634899 0.772596i \(-0.718958\pi\)
0.634899 0.772596i \(-0.281042\pi\)
\(264\) 0 0
\(265\) 148.387 0.559951
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −348.525 −1.29563 −0.647816 0.761797i \(-0.724317\pi\)
−0.647816 + 0.761797i \(0.724317\pi\)
\(270\) 0 0
\(271\) − 247.849i − 0.914571i −0.889320 0.457286i \(-0.848822\pi\)
0.889320 0.457286i \(-0.151178\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 99.5959i 0.362167i
\(276\) 0 0
\(277\) 54.7539 0.197667 0.0988337 0.995104i \(-0.468489\pi\)
0.0988337 + 0.995104i \(0.468489\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 50.3607 0.179220 0.0896098 0.995977i \(-0.471438\pi\)
0.0896098 + 0.995977i \(0.471438\pi\)
\(282\) 0 0
\(283\) 147.336i 0.520621i 0.965525 + 0.260310i \(0.0838249\pi\)
−0.965525 + 0.260310i \(0.916175\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 76.6705i − 0.267145i
\(288\) 0 0
\(289\) −147.663 −0.510943
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 178.859 0.610441 0.305220 0.952282i \(-0.401270\pi\)
0.305220 + 0.952282i \(0.401270\pi\)
\(294\) 0 0
\(295\) − 38.9904i − 0.132171i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.70228i 0.0157267i
\(300\) 0 0
\(301\) −117.082 −0.388977
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 201.554 0.660833
\(306\) 0 0
\(307\) 284.550i 0.926873i 0.886130 + 0.463436i \(0.153384\pi\)
−0.886130 + 0.463436i \(0.846616\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 282.199i − 0.907392i −0.891157 0.453696i \(-0.850105\pi\)
0.891157 0.453696i \(-0.149895\pi\)
\(312\) 0 0
\(313\) −567.548 −1.81325 −0.906626 0.421935i \(-0.861351\pi\)
−0.906626 + 0.421935i \(0.861351\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 161.141 0.508331 0.254165 0.967161i \(-0.418199\pi\)
0.254165 + 0.967161i \(0.418199\pi\)
\(318\) 0 0
\(319\) − 218.001i − 0.683389i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 180.907i − 0.560083i
\(324\) 0 0
\(325\) 42.3607 0.130341
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −280.689 −0.853158
\(330\) 0 0
\(331\) − 331.966i − 1.00292i −0.865181 0.501459i \(-0.832797\pi\)
0.865181 0.501459i \(-0.167203\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 112.300i 0.335223i
\(336\) 0 0
\(337\) −269.108 −0.798541 −0.399271 0.916833i \(-0.630737\pi\)
−0.399271 + 0.916833i \(0.630737\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 165.220 0.484516
\(342\) 0 0
\(343\) − 369.908i − 1.07845i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 503.075i − 1.44978i −0.688863 0.724892i \(-0.741890\pi\)
0.688863 0.724892i \(-0.258110\pi\)
\(348\) 0 0
\(349\) 0.504658 0.00144601 0.000723006 1.00000i \(-0.499770\pi\)
0.000723006 1.00000i \(0.499770\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 335.994 0.951824 0.475912 0.879493i \(-0.342118\pi\)
0.475912 + 0.879493i \(0.342118\pi\)
\(354\) 0 0
\(355\) − 180.645i − 0.508859i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 98.4859i − 0.274334i −0.990548 0.137167i \(-0.956200\pi\)
0.990548 0.137167i \(-0.0437997\pi\)
\(360\) 0 0
\(361\) 129.446 0.358576
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.4195 0.0340261
\(366\) 0 0
\(367\) 498.473i 1.35824i 0.734029 + 0.679118i \(0.237638\pi\)
−0.734029 + 0.679118i \(0.762362\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 348.879i 0.940374i
\(372\) 0 0
\(373\) −600.354 −1.60953 −0.804765 0.593594i \(-0.797709\pi\)
−0.804765 + 0.593594i \(0.797709\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −92.7214 −0.245945
\(378\) 0 0
\(379\) − 303.490i − 0.800765i −0.916348 0.400383i \(-0.868877\pi\)
0.916348 0.400383i \(-0.131123\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 332.583i 0.868362i 0.900826 + 0.434181i \(0.142962\pi\)
−0.900826 + 0.434181i \(0.857038\pi\)
\(384\) 0 0
\(385\) −234.164 −0.608218
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 392.354 1.00862 0.504312 0.863522i \(-0.331746\pi\)
0.504312 + 0.863522i \(0.331746\pi\)
\(390\) 0 0
\(391\) − 6.59849i − 0.0168759i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 30.9579i 0.0783744i
\(396\) 0 0
\(397\) −334.190 −0.841789 −0.420895 0.907110i \(-0.638284\pi\)
−0.420895 + 0.907110i \(0.638284\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −121.003 −0.301753 −0.150877 0.988553i \(-0.548210\pi\)
−0.150877 + 0.988553i \(0.548210\pi\)
\(402\) 0 0
\(403\) − 70.2722i − 0.174373i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 365.730i 0.898599i
\(408\) 0 0
\(409\) −607.410 −1.48511 −0.742555 0.669785i \(-0.766386\pi\)
−0.742555 + 0.669785i \(0.766386\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 91.6718 0.221966
\(414\) 0 0
\(415\) 170.423i 0.410658i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 466.760i − 1.11398i −0.830518 0.556992i \(-0.811955\pi\)
0.830518 0.556992i \(-0.188045\pi\)
\(420\) 0 0
\(421\) 73.0883 0.173606 0.0868031 0.996225i \(-0.472335\pi\)
0.0868031 + 0.996225i \(0.472335\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −59.4427 −0.139865
\(426\) 0 0
\(427\) 473.882i 1.10979i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 463.630i 1.07571i 0.843038 + 0.537853i \(0.180765\pi\)
−0.843038 + 0.537853i \(0.819235\pi\)
\(432\) 0 0
\(433\) −99.8359 −0.230568 −0.115284 0.993333i \(-0.536778\pi\)
−0.115284 + 0.993333i \(0.536778\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.44582 −0.0193268
\(438\) 0 0
\(439\) 374.086i 0.852133i 0.904692 + 0.426066i \(0.140101\pi\)
−0.904692 + 0.426066i \(0.859899\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 290.100i − 0.654854i −0.944877 0.327427i \(-0.893818\pi\)
0.944877 0.327427i \(-0.106182\pi\)
\(444\) 0 0
\(445\) −249.193 −0.559985
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −299.921 −0.667976 −0.333988 0.942577i \(-0.608394\pi\)
−0.333988 + 0.942577i \(0.608394\pi\)
\(450\) 0 0
\(451\) 290.493i 0.644109i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 99.5959i 0.218892i
\(456\) 0 0
\(457\) 822.328 1.79941 0.899703 0.436503i \(-0.143783\pi\)
0.899703 + 0.436503i \(0.143783\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −456.885 −0.991075 −0.495537 0.868587i \(-0.665029\pi\)
−0.495537 + 0.868587i \(0.665029\pi\)
\(462\) 0 0
\(463\) 400.249i 0.864469i 0.901761 + 0.432234i \(0.142275\pi\)
−0.901761 + 0.432234i \(0.857725\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 913.145i 1.95534i 0.210139 + 0.977672i \(0.432608\pi\)
−0.210139 + 0.977672i \(0.567392\pi\)
\(468\) 0 0
\(469\) −264.033 −0.562969
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 443.607 0.937858
\(474\) 0 0
\(475\) 76.0845i 0.160178i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 526.131i 1.09840i 0.835692 + 0.549198i \(0.185067\pi\)
−0.835692 + 0.549198i \(0.814933\pi\)
\(480\) 0 0
\(481\) 155.554 0.323397
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 207.580 0.428001
\(486\) 0 0
\(487\) 443.541i 0.910762i 0.890297 + 0.455381i \(0.150497\pi\)
−0.890297 + 0.455381i \(0.849503\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 287.163i − 0.584854i −0.956288 0.292427i \(-0.905537\pi\)
0.956288 0.292427i \(-0.0944628\pi\)
\(492\) 0 0
\(493\) 130.111 0.263918
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 424.721 0.854570
\(498\) 0 0
\(499\) − 810.936i − 1.62512i −0.582876 0.812561i \(-0.698073\pi\)
0.582876 0.812561i \(-0.301927\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 642.471i − 1.27728i −0.769506 0.638639i \(-0.779498\pi\)
0.769506 0.638639i \(-0.220502\pi\)
\(504\) 0 0
\(505\) −143.358 −0.283876
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −915.050 −1.79774 −0.898870 0.438216i \(-0.855611\pi\)
−0.898870 + 0.438216i \(0.855611\pi\)
\(510\) 0 0
\(511\) 29.2000i 0.0571429i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 308.061i − 0.598177i
\(516\) 0 0
\(517\) 1063.49 2.05704
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1006.98 −1.93279 −0.966396 0.257058i \(-0.917247\pi\)
−0.966396 + 0.257058i \(0.917247\pi\)
\(522\) 0 0
\(523\) 774.173i 1.48025i 0.672467 + 0.740127i \(0.265235\pi\)
−0.672467 + 0.740127i \(0.734765\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 98.6096i 0.187115i
\(528\) 0 0
\(529\) 528.692 0.999418
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 123.554 0.231809
\(534\) 0 0
\(535\) 114.782i 0.214546i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 425.487i 0.789401i
\(540\) 0 0
\(541\) 259.115 0.478955 0.239477 0.970902i \(-0.423024\pi\)
0.239477 + 0.970902i \(0.423024\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 298.446 0.547607
\(546\) 0 0
\(547\) 149.818i 0.273890i 0.990579 + 0.136945i \(0.0437284\pi\)
−0.990579 + 0.136945i \(0.956272\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 166.538i − 0.302247i
\(552\) 0 0
\(553\) −72.7864 −0.131621
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 511.698 0.918668 0.459334 0.888264i \(-0.348088\pi\)
0.459334 + 0.888264i \(0.348088\pi\)
\(558\) 0 0
\(559\) − 188.677i − 0.337526i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 490.726i 0.871627i 0.900037 + 0.435814i \(0.143539\pi\)
−0.900037 + 0.435814i \(0.856461\pi\)
\(564\) 0 0
\(565\) 381.745 0.675654
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 232.748 0.409047 0.204523 0.978862i \(-0.434436\pi\)
0.204523 + 0.978862i \(0.434436\pi\)
\(570\) 0 0
\(571\) 210.755i 0.369098i 0.982823 + 0.184549i \(0.0590824\pi\)
−0.982823 + 0.184549i \(0.940918\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.77515i 0.00482634i
\(576\) 0 0
\(577\) 341.712 0.592222 0.296111 0.955154i \(-0.404310\pi\)
0.296111 + 0.955154i \(0.404310\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −400.689 −0.689654
\(582\) 0 0
\(583\) − 1321.85i − 2.26733i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 618.412i 1.05351i 0.850016 + 0.526756i \(0.176592\pi\)
−0.850016 + 0.526756i \(0.823408\pi\)
\(588\) 0 0
\(589\) 126.217 0.214290
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −120.663 −0.203478 −0.101739 0.994811i \(-0.532441\pi\)
−0.101739 + 0.994811i \(0.532441\pi\)
\(594\) 0 0
\(595\) − 139.758i − 0.234888i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 849.927i 1.41891i 0.704751 + 0.709455i \(0.251059\pi\)
−0.704751 + 0.709455i \(0.748941\pi\)
\(600\) 0 0
\(601\) −11.3576 −0.0188978 −0.00944890 0.999955i \(-0.503008\pi\)
−0.00944890 + 0.999955i \(0.503008\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 616.649 1.01926
\(606\) 0 0
\(607\) − 1115.12i − 1.83710i −0.395305 0.918550i \(-0.629361\pi\)
0.395305 0.918550i \(-0.370639\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 452.329i − 0.740309i
\(612\) 0 0
\(613\) −499.475 −0.814805 −0.407402 0.913249i \(-0.633565\pi\)
−0.407402 + 0.913249i \(0.633565\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −545.935 −0.884822 −0.442411 0.896813i \(-0.645877\pi\)
−0.442411 + 0.896813i \(0.645877\pi\)
\(618\) 0 0
\(619\) − 455.011i − 0.735075i −0.930009 0.367537i \(-0.880201\pi\)
0.930009 0.367537i \(-0.119799\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 585.889i − 0.940432i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −218.282 −0.347030
\(630\) 0 0
\(631\) 267.706i 0.424257i 0.977242 + 0.212128i \(0.0680395\pi\)
−0.977242 + 0.212128i \(0.931960\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 444.165i 0.699473i
\(636\) 0 0
\(637\) 180.971 0.284098
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 418.571 0.652997 0.326499 0.945198i \(-0.394131\pi\)
0.326499 + 0.945198i \(0.394131\pi\)
\(642\) 0 0
\(643\) − 439.339i − 0.683265i −0.939834 0.341633i \(-0.889020\pi\)
0.939834 0.341633i \(-0.110980\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 419.644i − 0.648600i −0.945954 0.324300i \(-0.894871\pi\)
0.945954 0.324300i \(-0.105129\pi\)
\(648\) 0 0
\(649\) −347.331 −0.535179
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −370.085 −0.566746 −0.283373 0.959010i \(-0.591453\pi\)
−0.283373 + 0.959010i \(0.591453\pi\)
\(654\) 0 0
\(655\) − 17.3752i − 0.0265270i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 322.823i − 0.489868i −0.969540 0.244934i \(-0.921234\pi\)
0.969540 0.244934i \(-0.0787664\pi\)
\(660\) 0 0
\(661\) 812.735 1.22955 0.614777 0.788701i \(-0.289246\pi\)
0.614777 + 0.788701i \(0.289246\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −178.885 −0.269001
\(666\) 0 0
\(667\) − 6.07439i − 0.00910703i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 1795.47i − 2.67581i
\(672\) 0 0
\(673\) 467.378 0.694469 0.347235 0.937778i \(-0.387121\pi\)
0.347235 + 0.937778i \(0.387121\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 548.237 0.809803 0.404902 0.914360i \(-0.367306\pi\)
0.404902 + 0.914360i \(0.367306\pi\)
\(678\) 0 0
\(679\) 488.051i 0.718779i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.9663i 0.0350898i 0.999846 + 0.0175449i \(0.00558500\pi\)
−0.999846 + 0.0175449i \(0.994415\pi\)
\(684\) 0 0
\(685\) 1.86223 0.00271859
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −562.217 −0.815989
\(690\) 0 0
\(691\) 186.981i 0.270595i 0.990805 + 0.135298i \(0.0431990\pi\)
−0.990805 + 0.135298i \(0.956801\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 531.420i − 0.764633i
\(696\) 0 0
\(697\) −173.378 −0.248748
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −706.636 −1.00804 −0.504020 0.863692i \(-0.668146\pi\)
−0.504020 + 0.863692i \(0.668146\pi\)
\(702\) 0 0
\(703\) 279.393i 0.397429i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 337.054i − 0.476738i
\(708\) 0 0
\(709\) 188.597 0.266005 0.133002 0.991116i \(-0.457538\pi\)
0.133002 + 0.991116i \(0.457538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.60369 0.00645679
\(714\) 0 0
\(715\) − 377.354i − 0.527769i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 156.085i − 0.217086i −0.994092 0.108543i \(-0.965381\pi\)
0.994092 0.108543i \(-0.0346186\pi\)
\(720\) 0 0
\(721\) 724.296 1.00457
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −54.7214 −0.0754777
\(726\) 0 0
\(727\) − 715.164i − 0.983719i −0.870675 0.491859i \(-0.836317\pi\)
0.870675 0.491859i \(-0.163683\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 264.762i 0.362191i
\(732\) 0 0
\(733\) −1233.29 −1.68252 −0.841259 0.540632i \(-0.818185\pi\)
−0.841259 + 0.540632i \(0.818185\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1000.38 1.35737
\(738\) 0 0
\(739\) − 8.55656i − 0.0115786i −0.999983 0.00578928i \(-0.998157\pi\)
0.999983 0.00578928i \(-0.00184280\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1010.56i 1.36011i 0.733163 + 0.680053i \(0.238043\pi\)
−0.733163 + 0.680053i \(0.761957\pi\)
\(744\) 0 0
\(745\) 82.6687 0.110965
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −269.868 −0.360305
\(750\) 0 0
\(751\) − 1104.31i − 1.47046i −0.677820 0.735228i \(-0.737075\pi\)
0.677820 0.735228i \(-0.262925\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 632.188i − 0.837335i
\(756\) 0 0
\(757\) 875.633 1.15671 0.578357 0.815783i \(-0.303694\pi\)
0.578357 + 0.815783i \(0.303694\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 647.207 0.850470 0.425235 0.905083i \(-0.360192\pi\)
0.425235 + 0.905083i \(0.360192\pi\)
\(762\) 0 0
\(763\) 701.688i 0.919644i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 147.729i 0.192606i
\(768\) 0 0
\(769\) 631.430 0.821106 0.410553 0.911837i \(-0.365336\pi\)
0.410553 + 0.911837i \(0.365336\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −421.522 −0.545306 −0.272653 0.962112i \(-0.587901\pi\)
−0.272653 + 0.962112i \(0.587901\pi\)
\(774\) 0 0
\(775\) − 41.4725i − 0.0535129i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 221.917i 0.284874i
\(780\) 0 0
\(781\) −1609.21 −2.06044
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 457.830 0.583223
\(786\) 0 0
\(787\) − 838.633i − 1.06561i −0.846239 0.532804i \(-0.821138\pi\)
0.846239 0.532804i \(-0.178862\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 897.535i 1.13468i
\(792\) 0 0
\(793\) −763.659 −0.963001
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1213.57 1.52268 0.761339 0.648354i \(-0.224542\pi\)
0.761339 + 0.648354i \(0.224542\pi\)
\(798\) 0 0
\(799\) 634.732i 0.794408i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 110.635i − 0.137777i
\(804\) 0 0
\(805\) −6.52476 −0.00810529
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 229.214 0.283330 0.141665 0.989915i \(-0.454754\pi\)
0.141665 + 0.989915i \(0.454754\pi\)
\(810\) 0 0
\(811\) − 454.225i − 0.560080i −0.959988 0.280040i \(-0.909652\pi\)
0.959988 0.280040i \(-0.0903478\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 239.785i 0.294215i
\(816\) 0 0
\(817\) 338.885 0.414792
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1130.90 1.37747 0.688733 0.725015i \(-0.258167\pi\)
0.688733 + 0.725015i \(0.258167\pi\)
\(822\) 0 0
\(823\) 780.148i 0.947931i 0.880543 + 0.473966i \(0.157178\pi\)
−0.880543 + 0.473966i \(0.842822\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 209.175i 0.252932i 0.991971 + 0.126466i \(0.0403635\pi\)
−0.991971 + 0.126466i \(0.959636\pi\)
\(828\) 0 0
\(829\) 508.525 0.613419 0.306710 0.951803i \(-0.400772\pi\)
0.306710 + 0.951803i \(0.400772\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −253.947 −0.304859
\(834\) 0 0
\(835\) 74.2575i 0.0889311i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 274.028i 0.326613i 0.986575 + 0.163306i \(0.0522159\pi\)
−0.986575 + 0.163306i \(0.947784\pi\)
\(840\) 0 0
\(841\) −721.223 −0.857578
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 217.397 0.257275
\(846\) 0 0
\(847\) 1449.83i 1.71172i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.1907i 0.0119750i
\(852\) 0 0
\(853\) 1583.28 1.85613 0.928066 0.372416i \(-0.121471\pi\)
0.928066 + 0.372416i \(0.121471\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1007.38 1.17547 0.587735 0.809054i \(-0.300020\pi\)
0.587735 + 0.809054i \(0.300020\pi\)
\(858\) 0 0
\(859\) 76.6086i 0.0891835i 0.999005 + 0.0445917i \(0.0141987\pi\)
−0.999005 + 0.0445917i \(0.985801\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 255.450i 0.296002i 0.988987 + 0.148001i \(0.0472839\pi\)
−0.988987 + 0.148001i \(0.952716\pi\)
\(864\) 0 0
\(865\) 506.158 0.585154
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 275.777 0.317350
\(870\) 0 0
\(871\) − 425.487i − 0.488504i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 58.7785i 0.0671755i
\(876\) 0 0
\(877\) −601.522 −0.685886 −0.342943 0.939356i \(-0.611424\pi\)
−0.342943 + 0.939356i \(0.611424\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −237.850 −0.269977 −0.134989 0.990847i \(-0.543100\pi\)
−0.134989 + 0.990847i \(0.543100\pi\)
\(882\) 0 0
\(883\) 1.30294i 0.00147559i 1.00000 0.000737794i \(0.000234847\pi\)
−1.00000 0.000737794i \(0.999765\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 536.353i 0.604682i 0.953200 + 0.302341i \(0.0977682\pi\)
−0.953200 + 0.302341i \(0.902232\pi\)
\(888\) 0 0
\(889\) −1044.30 −1.17469
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 812.433 0.909780
\(894\) 0 0
\(895\) − 501.634i − 0.560485i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 90.7773i 0.100976i
\(900\) 0 0
\(901\) 788.932 0.875618
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 192.800 0.213039
\(906\) 0 0
\(907\) − 332.159i − 0.366217i −0.983093 0.183108i \(-0.941384\pi\)
0.983093 0.183108i \(-0.0586159\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1450.06i 1.59172i 0.605478 + 0.795862i \(0.292982\pi\)
−0.605478 + 0.795862i \(0.707018\pi\)
\(912\) 0 0
\(913\) 1518.15 1.66282
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 40.8514 0.0445490
\(918\) 0 0
\(919\) 814.405i 0.886186i 0.896476 + 0.443093i \(0.146119\pi\)
−0.896476 + 0.443093i \(0.853881\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 684.437i 0.741535i
\(924\) 0 0
\(925\) 91.8034 0.0992469
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −400.039 −0.430612 −0.215306 0.976547i \(-0.569075\pi\)
−0.215306 + 0.976547i \(0.569075\pi\)
\(930\) 0 0
\(931\) 325.043i 0.349134i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 529.524i 0.566335i
\(936\) 0 0
\(937\) 249.279 0.266039 0.133020 0.991113i \(-0.457533\pi\)
0.133020 + 0.991113i \(0.457533\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 724.229 0.769638 0.384819 0.922992i \(-0.374264\pi\)
0.384819 + 0.922992i \(0.374264\pi\)
\(942\) 0 0
\(943\) 8.09432i 0.00858358i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1141.54i − 1.20542i −0.797959 0.602712i \(-0.794087\pi\)
0.797959 0.602712i \(-0.205913\pi\)
\(948\) 0 0
\(949\) −47.0557 −0.0495845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1295.33 −1.35921 −0.679604 0.733579i \(-0.737848\pi\)
−0.679604 + 0.733579i \(0.737848\pi\)
\(954\) 0 0
\(955\) − 69.3623i − 0.0726307i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.37837i 0.00456556i
\(960\) 0 0
\(961\) 892.201 0.928409
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −246.466 −0.255405
\(966\) 0 0
\(967\) − 398.477i − 0.412075i −0.978544 0.206037i \(-0.933943\pi\)
0.978544 0.206037i \(-0.0660569\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 928.093i − 0.955811i −0.878411 0.477906i \(-0.841396\pi\)
0.878411 0.477906i \(-0.158604\pi\)
\(972\) 0 0
\(973\) 1249.44 1.28411
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1378.05 1.41049 0.705247 0.708962i \(-0.250836\pi\)
0.705247 + 0.708962i \(0.250836\pi\)
\(978\) 0 0
\(979\) 2219.85i 2.26747i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 311.291i − 0.316675i −0.987385 0.158337i \(-0.949387\pi\)
0.987385 0.158337i \(-0.0506134\pi\)
\(984\) 0 0
\(985\) 385.777 0.391652
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.3607 0.0124982
\(990\) 0 0
\(991\) 961.147i 0.969876i 0.874549 + 0.484938i \(0.161158\pi\)
−0.874549 + 0.484938i \(0.838842\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 608.676i 0.611735i
\(996\) 0 0
\(997\) 1089.68 1.09296 0.546479 0.837473i \(-0.315968\pi\)
0.546479 + 0.837473i \(0.315968\pi\)
\(998\) 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 2880.3.e.e.2431.1 4
3.2 odd 2 320.3.b.c.191.3 4
4.3 odd 2 inner 2880.3.e.e.2431.2 4
8.3 odd 2 180.3.c.a.91.2 4
8.5 even 2 180.3.c.a.91.1 4
12.11 even 2 320.3.b.c.191.2 4
15.2 even 4 1600.3.h.n.1599.5 8
15.8 even 4 1600.3.h.n.1599.3 8
15.14 odd 2 1600.3.b.s.1151.2 4
24.5 odd 2 20.3.b.a.11.4 yes 4
24.11 even 2 20.3.b.a.11.3 4
40.3 even 4 900.3.f.e.199.8 8
40.13 odd 4 900.3.f.e.199.2 8
40.19 odd 2 900.3.c.k.451.3 4
40.27 even 4 900.3.f.e.199.1 8
40.29 even 2 900.3.c.k.451.4 4
40.37 odd 4 900.3.f.e.199.7 8
48.5 odd 4 1280.3.g.e.1151.6 8
48.11 even 4 1280.3.g.e.1151.4 8
48.29 odd 4 1280.3.g.e.1151.3 8
48.35 even 4 1280.3.g.e.1151.5 8
60.23 odd 4 1600.3.h.n.1599.6 8
60.47 odd 4 1600.3.h.n.1599.4 8
60.59 even 2 1600.3.b.s.1151.3 4
120.29 odd 2 100.3.b.f.51.1 4
120.53 even 4 100.3.d.b.99.7 8
120.59 even 2 100.3.b.f.51.2 4
120.77 even 4 100.3.d.b.99.2 8
120.83 odd 4 100.3.d.b.99.1 8
120.107 odd 4 100.3.d.b.99.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.b.a.11.3 4 24.11 even 2
20.3.b.a.11.4 yes 4 24.5 odd 2
100.3.b.f.51.1 4 120.29 odd 2
100.3.b.f.51.2 4 120.59 even 2
100.3.d.b.99.1 8 120.83 odd 4
100.3.d.b.99.2 8 120.77 even 4
100.3.d.b.99.7 8 120.53 even 4
100.3.d.b.99.8 8 120.107 odd 4
180.3.c.a.91.1 4 8.5 even 2
180.3.c.a.91.2 4 8.3 odd 2
320.3.b.c.191.2 4 12.11 even 2
320.3.b.c.191.3 4 3.2 odd 2
900.3.c.k.451.3 4 40.19 odd 2
900.3.c.k.451.4 4 40.29 even 2
900.3.f.e.199.1 8 40.27 even 4
900.3.f.e.199.2 8 40.13 odd 4
900.3.f.e.199.7 8 40.37 odd 4
900.3.f.e.199.8 8 40.3 even 4
1280.3.g.e.1151.3 8 48.29 odd 4
1280.3.g.e.1151.4 8 48.11 even 4
1280.3.g.e.1151.5 8 48.35 even 4
1280.3.g.e.1151.6 8 48.5 odd 4
1600.3.b.s.1151.2 4 15.14 odd 2
1600.3.b.s.1151.3 4 60.59 even 2
1600.3.h.n.1599.3 8 15.8 even 4
1600.3.h.n.1599.4 8 60.47 odd 4
1600.3.h.n.1599.5 8 15.2 even 4
1600.3.h.n.1599.6 8 60.23 odd 4
2880.3.e.e.2431.1 4 1.1 even 1 trivial
2880.3.e.e.2431.2 4 4.3 odd 2 inner