Properties

Label 1620.3.b.a.809.17
Level $1620$
Weight $3$
Character 1620.809
Analytic conductor $44.142$
Analytic rank $0$
Dimension $24$
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] = [1620,3,Mod(809,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.809");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.b (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: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 809.17
Character \(\chi\) \(=\) 1620.809
Dual form 1620.3.b.a.809.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.82619 - 4.12464i) q^{5} +7.52777i q^{7} +O(q^{10})\) \(q+(2.82619 - 4.12464i) q^{5} +7.52777i q^{7} -6.67363i q^{11} +15.5692i q^{13} +18.9462 q^{17} -27.1064 q^{19} +19.6113 q^{23} +(-9.02534 - 23.3140i) q^{25} +6.97175i q^{29} +29.5191 q^{31} +(31.0494 + 21.2749i) q^{35} +52.2126i q^{37} +60.8539i q^{41} +10.6611i q^{43} -75.8093 q^{47} -7.66737 q^{49} -47.3234 q^{53} +(-27.5263 - 18.8609i) q^{55} -24.4204i q^{59} +40.7094 q^{61} +(64.2172 + 44.0013i) q^{65} -58.9100i q^{67} +33.9404i q^{71} +79.8213i q^{73} +50.2376 q^{77} +86.0506 q^{79} +153.076 q^{83} +(53.5456 - 78.1464i) q^{85} +83.3156i q^{89} -117.201 q^{91} +(-76.6077 + 111.804i) q^{95} +16.2175i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(5\) 2.82619 4.12464i 0.565237 0.824928i
\(6\) 0 0
\(7\) 7.52777i 1.07540i 0.843137 + 0.537698i \(0.180706\pi\)
−0.843137 + 0.537698i \(0.819294\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.67363i 0.606694i −0.952880 0.303347i \(-0.901896\pi\)
0.952880 0.303347i \(-0.0981041\pi\)
\(12\) 0 0
\(13\) 15.5692i 1.19763i 0.800888 + 0.598814i \(0.204361\pi\)
−0.800888 + 0.598814i \(0.795639\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.9462 1.11448 0.557242 0.830350i \(-0.311860\pi\)
0.557242 + 0.830350i \(0.311860\pi\)
\(18\) 0 0
\(19\) −27.1064 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 19.6113 0.852665 0.426333 0.904567i \(-0.359805\pi\)
0.426333 + 0.904567i \(0.359805\pi\)
\(24\) 0 0
\(25\) −9.02534 23.3140i −0.361014 0.932560i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.97175i 0.240405i 0.992749 + 0.120203i \(0.0383544\pi\)
−0.992749 + 0.120203i \(0.961646\pi\)
\(30\) 0 0
\(31\) 29.5191 0.952228 0.476114 0.879383i \(-0.342045\pi\)
0.476114 + 0.879383i \(0.342045\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 31.0494 + 21.2749i 0.887125 + 0.607854i
\(36\) 0 0
\(37\) 52.2126i 1.41115i 0.708634 + 0.705576i \(0.249312\pi\)
−0.708634 + 0.705576i \(0.750688\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 60.8539i 1.48424i 0.670266 + 0.742121i \(0.266180\pi\)
−0.670266 + 0.742121i \(0.733820\pi\)
\(42\) 0 0
\(43\) 10.6611i 0.247934i 0.992286 + 0.123967i \(0.0395616\pi\)
−0.992286 + 0.123967i \(0.960438\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −75.8093 −1.61296 −0.806482 0.591259i \(-0.798631\pi\)
−0.806482 + 0.591259i \(0.798631\pi\)
\(48\) 0 0
\(49\) −7.66737 −0.156477
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −47.3234 −0.892895 −0.446447 0.894810i \(-0.647311\pi\)
−0.446447 + 0.894810i \(0.647311\pi\)
\(54\) 0 0
\(55\) −27.5263 18.8609i −0.500479 0.342926i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 24.4204i 0.413905i −0.978351 0.206952i \(-0.933646\pi\)
0.978351 0.206952i \(-0.0663545\pi\)
\(60\) 0 0
\(61\) 40.7094 0.667367 0.333683 0.942685i \(-0.391708\pi\)
0.333683 + 0.942685i \(0.391708\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 64.2172 + 44.0013i 0.987957 + 0.676944i
\(66\) 0 0
\(67\) 58.9100i 0.879254i −0.898180 0.439627i \(-0.855111\pi\)
0.898180 0.439627i \(-0.144889\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 33.9404i 0.478034i 0.971015 + 0.239017i \(0.0768252\pi\)
−0.971015 + 0.239017i \(0.923175\pi\)
\(72\) 0 0
\(73\) 79.8213i 1.09344i 0.837315 + 0.546722i \(0.184124\pi\)
−0.837315 + 0.546722i \(0.815876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 50.2376 0.652436
\(78\) 0 0
\(79\) 86.0506 1.08925 0.544624 0.838680i \(-0.316672\pi\)
0.544624 + 0.838680i \(0.316672\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 153.076 1.84429 0.922143 0.386850i \(-0.126437\pi\)
0.922143 + 0.386850i \(0.126437\pi\)
\(84\) 0 0
\(85\) 53.5456 78.1464i 0.629948 0.919370i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 83.3156i 0.936130i 0.883694 + 0.468065i \(0.155049\pi\)
−0.883694 + 0.468065i \(0.844951\pi\)
\(90\) 0 0
\(91\) −117.201 −1.28792
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −76.6077 + 111.804i −0.806397 + 1.17689i
\(96\) 0 0
\(97\) 16.2175i 0.167190i 0.996500 + 0.0835952i \(0.0266403\pi\)
−0.996500 + 0.0835952i \(0.973360\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 127.041i 1.25783i 0.777473 + 0.628916i \(0.216501\pi\)
−0.777473 + 0.628916i \(0.783499\pi\)
\(102\) 0 0
\(103\) 131.249i 1.27426i −0.770755 0.637132i \(-0.780121\pi\)
0.770755 0.637132i \(-0.219879\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 131.922 1.23292 0.616458 0.787388i \(-0.288567\pi\)
0.616458 + 0.787388i \(0.288567\pi\)
\(108\) 0 0
\(109\) −201.576 −1.84932 −0.924662 0.380790i \(-0.875652\pi\)
−0.924662 + 0.380790i \(0.875652\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 33.6972 0.298205 0.149102 0.988822i \(-0.452362\pi\)
0.149102 + 0.988822i \(0.452362\pi\)
\(114\) 0 0
\(115\) 55.4252 80.8896i 0.481958 0.703388i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 142.623i 1.19851i
\(120\) 0 0
\(121\) 76.4626 0.631923
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −121.669 28.6634i −0.973354 0.229307i
\(126\) 0 0
\(127\) 118.081i 0.929768i 0.885372 + 0.464884i \(0.153904\pi\)
−0.885372 + 0.464884i \(0.846096\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 28.0634i 0.214224i −0.994247 0.107112i \(-0.965840\pi\)
0.994247 0.107112i \(-0.0341604\pi\)
\(132\) 0 0
\(133\) 204.051i 1.53422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 107.772 0.786656 0.393328 0.919398i \(-0.371324\pi\)
0.393328 + 0.919398i \(0.371324\pi\)
\(138\) 0 0
\(139\) 170.944 1.22981 0.614907 0.788600i \(-0.289194\pi\)
0.614907 + 0.788600i \(0.289194\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 103.903 0.726593
\(144\) 0 0
\(145\) 28.7560 + 19.7035i 0.198317 + 0.135886i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 81.6965i 0.548299i 0.961687 + 0.274149i \(0.0883963\pi\)
−0.961687 + 0.274149i \(0.911604\pi\)
\(150\) 0 0
\(151\) 9.21306 0.0610136 0.0305068 0.999535i \(-0.490288\pi\)
0.0305068 + 0.999535i \(0.490288\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 83.4264 121.756i 0.538235 0.785520i
\(156\) 0 0
\(157\) 288.365i 1.83672i 0.395749 + 0.918359i \(0.370485\pi\)
−0.395749 + 0.918359i \(0.629515\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 147.629i 0.916953i
\(162\) 0 0
\(163\) 69.1070i 0.423970i 0.977273 + 0.211985i \(0.0679927\pi\)
−0.977273 + 0.211985i \(0.932007\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 195.225 1.16901 0.584507 0.811389i \(-0.301288\pi\)
0.584507 + 0.811389i \(0.301288\pi\)
\(168\) 0 0
\(169\) −73.3987 −0.434312
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −59.3786 −0.343229 −0.171614 0.985164i \(-0.554898\pi\)
−0.171614 + 0.985164i \(0.554898\pi\)
\(174\) 0 0
\(175\) 175.503 67.9407i 1.00287 0.388233i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 82.1619i 0.459005i 0.973308 + 0.229503i \(0.0737099\pi\)
−0.973308 + 0.229503i \(0.926290\pi\)
\(180\) 0 0
\(181\) −242.120 −1.33768 −0.668840 0.743407i \(-0.733209\pi\)
−0.668840 + 0.743407i \(0.733209\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 215.358 + 147.563i 1.16410 + 0.797636i
\(186\) 0 0
\(187\) 126.440i 0.676151i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 27.4015i 0.143463i −0.997424 0.0717317i \(-0.977147\pi\)
0.997424 0.0717317i \(-0.0228525\pi\)
\(192\) 0 0
\(193\) 139.702i 0.723844i −0.932208 0.361922i \(-0.882121\pi\)
0.932208 0.361922i \(-0.117879\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 125.291 0.635994 0.317997 0.948092i \(-0.396990\pi\)
0.317997 + 0.948092i \(0.396990\pi\)
\(198\) 0 0
\(199\) −242.646 −1.21933 −0.609664 0.792660i \(-0.708696\pi\)
−0.609664 + 0.792660i \(0.708696\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −52.4818 −0.258531
\(204\) 0 0
\(205\) 251.001 + 171.984i 1.22439 + 0.838949i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 180.898i 0.865541i
\(210\) 0 0
\(211\) 163.529 0.775018 0.387509 0.921866i \(-0.373336\pi\)
0.387509 + 0.921866i \(0.373336\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 43.9734 + 30.1304i 0.204528 + 0.140141i
\(216\) 0 0
\(217\) 222.213i 1.02402i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 294.977i 1.33474i
\(222\) 0 0
\(223\) 201.029i 0.901477i 0.892656 + 0.450739i \(0.148839\pi\)
−0.892656 + 0.450739i \(0.851161\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −155.168 −0.683559 −0.341780 0.939780i \(-0.611030\pi\)
−0.341780 + 0.939780i \(0.611030\pi\)
\(228\) 0 0
\(229\) −310.150 −1.35437 −0.677183 0.735814i \(-0.736800\pi\)
−0.677183 + 0.735814i \(0.736800\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 375.988 1.61368 0.806842 0.590768i \(-0.201175\pi\)
0.806842 + 0.590768i \(0.201175\pi\)
\(234\) 0 0
\(235\) −214.251 + 312.686i −0.911707 + 1.33058i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 308.019i 1.28878i −0.764696 0.644391i \(-0.777111\pi\)
0.764696 0.644391i \(-0.222889\pi\)
\(240\) 0 0
\(241\) 300.232 1.24577 0.622887 0.782312i \(-0.285959\pi\)
0.622887 + 0.782312i \(0.285959\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −21.6694 + 31.6251i −0.0884466 + 0.129082i
\(246\) 0 0
\(247\) 422.024i 1.70860i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 153.779i 0.612666i −0.951924 0.306333i \(-0.900898\pi\)
0.951924 0.306333i \(-0.0991021\pi\)
\(252\) 0 0
\(253\) 130.879i 0.517307i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 29.5779 0.115089 0.0575446 0.998343i \(-0.481673\pi\)
0.0575446 + 0.998343i \(0.481673\pi\)
\(258\) 0 0
\(259\) −393.045 −1.51755
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −258.391 −0.982475 −0.491237 0.871026i \(-0.663455\pi\)
−0.491237 + 0.871026i \(0.663455\pi\)
\(264\) 0 0
\(265\) −133.745 + 195.192i −0.504697 + 0.736574i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 443.969i 1.65044i 0.564810 + 0.825221i \(0.308949\pi\)
−0.564810 + 0.825221i \(0.691051\pi\)
\(270\) 0 0
\(271\) 233.582 0.861928 0.430964 0.902369i \(-0.358174\pi\)
0.430964 + 0.902369i \(0.358174\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −155.589 + 60.2318i −0.565779 + 0.219025i
\(276\) 0 0
\(277\) 199.108i 0.718802i 0.933183 + 0.359401i \(0.117019\pi\)
−0.933183 + 0.359401i \(0.882981\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 173.925i 0.618949i −0.950908 0.309474i \(-0.899847\pi\)
0.950908 0.309474i \(-0.100153\pi\)
\(282\) 0 0
\(283\) 221.755i 0.783587i 0.920053 + 0.391793i \(0.128145\pi\)
−0.920053 + 0.391793i \(0.871855\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −458.094 −1.59615
\(288\) 0 0
\(289\) 69.9596 0.242075
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −100.496 −0.342990 −0.171495 0.985185i \(-0.554860\pi\)
−0.171495 + 0.985185i \(0.554860\pi\)
\(294\) 0 0
\(295\) −100.725 69.0165i −0.341442 0.233954i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 305.331i 1.02118i
\(300\) 0 0
\(301\) −80.2547 −0.266627
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 115.052 167.912i 0.377221 0.550530i
\(306\) 0 0
\(307\) 214.703i 0.699360i −0.936869 0.349680i \(-0.886290\pi\)
0.936869 0.349680i \(-0.113710\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 347.773i 1.11824i 0.829087 + 0.559120i \(0.188861\pi\)
−0.829087 + 0.559120i \(0.811139\pi\)
\(312\) 0 0
\(313\) 254.365i 0.812668i −0.913725 0.406334i \(-0.866807\pi\)
0.913725 0.406334i \(-0.133193\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 417.859 1.31817 0.659083 0.752070i \(-0.270944\pi\)
0.659083 + 0.752070i \(0.270944\pi\)
\(318\) 0 0
\(319\) 46.5269 0.145852
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −513.564 −1.58998
\(324\) 0 0
\(325\) 362.980 140.517i 1.11686 0.432360i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 570.675i 1.73457i
\(330\) 0 0
\(331\) 38.0345 0.114908 0.0574540 0.998348i \(-0.481702\pi\)
0.0574540 + 0.998348i \(0.481702\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −242.983 166.491i −0.725322 0.496987i
\(336\) 0 0
\(337\) 342.230i 1.01552i −0.861499 0.507760i \(-0.830474\pi\)
0.861499 0.507760i \(-0.169526\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 196.999i 0.577711i
\(342\) 0 0
\(343\) 311.143i 0.907122i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −590.327 −1.70123 −0.850615 0.525788i \(-0.823770\pi\)
−0.850615 + 0.525788i \(0.823770\pi\)
\(348\) 0 0
\(349\) 45.5246 0.130443 0.0652215 0.997871i \(-0.479225\pi\)
0.0652215 + 0.997871i \(0.479225\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 215.450 0.610339 0.305170 0.952298i \(-0.401287\pi\)
0.305170 + 0.952298i \(0.401287\pi\)
\(354\) 0 0
\(355\) 139.992 + 95.9219i 0.394344 + 0.270203i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 69.0790i 0.192421i 0.995361 + 0.0962104i \(0.0306722\pi\)
−0.995361 + 0.0962104i \(0.969328\pi\)
\(360\) 0 0
\(361\) 373.756 1.03534
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 329.234 + 225.590i 0.902012 + 0.618055i
\(366\) 0 0
\(367\) 234.990i 0.640299i −0.947367 0.320150i \(-0.896267\pi\)
0.947367 0.320150i \(-0.103733\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 356.240i 0.960216i
\(372\) 0 0
\(373\) 138.314i 0.370814i 0.982662 + 0.185407i \(0.0593604\pi\)
−0.982662 + 0.185407i \(0.940640\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −108.544 −0.287916
\(378\) 0 0
\(379\) −412.330 −1.08794 −0.543971 0.839104i \(-0.683080\pi\)
−0.543971 + 0.839104i \(0.683080\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 448.998 1.17232 0.586159 0.810196i \(-0.300639\pi\)
0.586159 + 0.810196i \(0.300639\pi\)
\(384\) 0 0
\(385\) 141.981 207.212i 0.368781 0.538213i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 361.573i 0.929494i 0.885444 + 0.464747i \(0.153855\pi\)
−0.885444 + 0.464747i \(0.846145\pi\)
\(390\) 0 0
\(391\) 371.560 0.950282
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 243.195 354.928i 0.615684 0.898552i
\(396\) 0 0
\(397\) 673.227i 1.69579i −0.530168 0.847893i \(-0.677871\pi\)
0.530168 0.847893i \(-0.322129\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 138.781i 0.346086i −0.984914 0.173043i \(-0.944640\pi\)
0.984914 0.173043i \(-0.0553600\pi\)
\(402\) 0 0
\(403\) 459.587i 1.14041i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 348.448 0.856137
\(408\) 0 0
\(409\) −635.790 −1.55450 −0.777249 0.629193i \(-0.783386\pi\)
−0.777249 + 0.629193i \(0.783386\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 183.831 0.445111
\(414\) 0 0
\(415\) 432.620 631.382i 1.04246 1.52140i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 785.128i 1.87381i −0.349578 0.936907i \(-0.613675\pi\)
0.349578 0.936907i \(-0.386325\pi\)
\(420\) 0 0
\(421\) −116.223 −0.276064 −0.138032 0.990428i \(-0.544078\pi\)
−0.138032 + 0.990428i \(0.544078\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −170.996 441.713i −0.402344 1.03932i
\(426\) 0 0
\(427\) 306.451i 0.717684i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 606.430i 1.40703i −0.710680 0.703515i \(-0.751613\pi\)
0.710680 0.703515i \(-0.248387\pi\)
\(432\) 0 0
\(433\) 638.451i 1.47448i −0.675629 0.737242i \(-0.736128\pi\)
0.675629 0.737242i \(-0.263872\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −531.591 −1.21646
\(438\) 0 0
\(439\) −752.450 −1.71401 −0.857005 0.515309i \(-0.827677\pi\)
−0.857005 + 0.515309i \(0.827677\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 308.710 0.696862 0.348431 0.937334i \(-0.386715\pi\)
0.348431 + 0.937334i \(0.386715\pi\)
\(444\) 0 0
\(445\) 343.647 + 235.465i 0.772240 + 0.529136i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 516.232i 1.14974i −0.818246 0.574869i \(-0.805053\pi\)
0.818246 0.574869i \(-0.194947\pi\)
\(450\) 0 0
\(451\) 406.117 0.900480
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −331.232 + 483.413i −0.727983 + 1.06245i
\(456\) 0 0
\(457\) 481.476i 1.05356i −0.850002 0.526779i \(-0.823400\pi\)
0.850002 0.526779i \(-0.176600\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 68.6414i 0.148897i 0.997225 + 0.0744484i \(0.0237196\pi\)
−0.997225 + 0.0744484i \(0.976280\pi\)
\(462\) 0 0
\(463\) 219.016i 0.473037i 0.971627 + 0.236518i \(0.0760064\pi\)
−0.971627 + 0.236518i \(0.923994\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −87.1086 −0.186528 −0.0932640 0.995641i \(-0.529730\pi\)
−0.0932640 + 0.995641i \(0.529730\pi\)
\(468\) 0 0
\(469\) 443.461 0.945546
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 71.1486 0.150420
\(474\) 0 0
\(475\) 244.644 + 631.959i 0.515041 + 1.33044i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 702.080i 1.46572i −0.680379 0.732860i \(-0.738185\pi\)
0.680379 0.732860i \(-0.261815\pi\)
\(480\) 0 0
\(481\) −812.907 −1.69003
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 66.8912 + 45.8336i 0.137920 + 0.0945022i
\(486\) 0 0
\(487\) 272.680i 0.559918i −0.960012 0.279959i \(-0.909679\pi\)
0.960012 0.279959i \(-0.0903208\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 764.337i 1.55669i −0.627834 0.778347i \(-0.716058\pi\)
0.627834 0.778347i \(-0.283942\pi\)
\(492\) 0 0
\(493\) 132.088i 0.267928i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −255.496 −0.514076
\(498\) 0 0
\(499\) −873.142 −1.74978 −0.874891 0.484319i \(-0.839067\pi\)
−0.874891 + 0.484319i \(0.839067\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −966.017 −1.92051 −0.960255 0.279123i \(-0.909956\pi\)
−0.960255 + 0.279123i \(0.909956\pi\)
\(504\) 0 0
\(505\) 523.999 + 359.042i 1.03762 + 0.710974i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 987.379i 1.93984i −0.243424 0.969920i \(-0.578271\pi\)
0.243424 0.969920i \(-0.421729\pi\)
\(510\) 0 0
\(511\) −600.877 −1.17588
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −541.356 370.935i −1.05118 0.720261i
\(516\) 0 0
\(517\) 505.923i 0.978575i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 952.317i 1.82786i 0.405868 + 0.913932i \(0.366969\pi\)
−0.405868 + 0.913932i \(0.633031\pi\)
\(522\) 0 0
\(523\) 660.769i 1.26342i 0.775205 + 0.631710i \(0.217647\pi\)
−0.775205 + 0.631710i \(0.782353\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 559.275 1.06124
\(528\) 0 0
\(529\) −144.397 −0.272962
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −947.444 −1.77757
\(534\) 0 0
\(535\) 372.836 544.131i 0.696890 1.01707i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 51.1692i 0.0949336i
\(540\) 0 0
\(541\) 520.946 0.962931 0.481466 0.876465i \(-0.340105\pi\)
0.481466 + 0.876465i \(0.340105\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −569.692 + 831.430i −1.04531 + 1.52556i
\(546\) 0 0
\(547\) 788.066i 1.44071i 0.693608 + 0.720353i \(0.256020\pi\)
−0.693608 + 0.720353i \(0.743980\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 188.979i 0.342974i
\(552\) 0 0
\(553\) 647.769i 1.17137i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −142.413 −0.255678 −0.127839 0.991795i \(-0.540804\pi\)
−0.127839 + 0.991795i \(0.540804\pi\)
\(558\) 0 0
\(559\) −165.985 −0.296932
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −832.795 −1.47921 −0.739605 0.673041i \(-0.764988\pi\)
−0.739605 + 0.673041i \(0.764988\pi\)
\(564\) 0 0
\(565\) 95.2345 138.989i 0.168557 0.245998i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1017.03i 1.78740i 0.448666 + 0.893699i \(0.351899\pi\)
−0.448666 + 0.893699i \(0.648101\pi\)
\(570\) 0 0
\(571\) 356.274 0.623948 0.311974 0.950091i \(-0.399010\pi\)
0.311974 + 0.950091i \(0.399010\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −176.999 457.218i −0.307824 0.795162i
\(576\) 0 0
\(577\) 269.258i 0.466651i −0.972399 0.233326i \(-0.925039\pi\)
0.972399 0.233326i \(-0.0749608\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1152.32i 1.98334i
\(582\) 0 0
\(583\) 315.819i 0.541714i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −716.019 −1.21979 −0.609897 0.792481i \(-0.708789\pi\)
−0.609897 + 0.792481i \(0.708789\pi\)
\(588\) 0 0
\(589\) −800.155 −1.35850
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 680.538 1.14762 0.573809 0.818989i \(-0.305465\pi\)
0.573809 + 0.818989i \(0.305465\pi\)
\(594\) 0 0
\(595\) 588.268 + 403.079i 0.988687 + 0.677444i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1006.72i 1.68066i −0.542073 0.840332i \(-0.682360\pi\)
0.542073 0.840332i \(-0.317640\pi\)
\(600\) 0 0
\(601\) 165.880 0.276007 0.138004 0.990432i \(-0.455931\pi\)
0.138004 + 0.990432i \(0.455931\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 216.098 315.381i 0.357186 0.521291i
\(606\) 0 0
\(607\) 201.754i 0.332379i −0.986094 0.166190i \(-0.946854\pi\)
0.986094 0.166190i \(-0.0531464\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1180.29i 1.93173i
\(612\) 0 0
\(613\) 741.425i 1.20950i −0.796415 0.604751i \(-0.793273\pi\)
0.796415 0.604751i \(-0.206727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 695.522 1.12726 0.563632 0.826026i \(-0.309404\pi\)
0.563632 + 0.826026i \(0.309404\pi\)
\(618\) 0 0
\(619\) −465.817 −0.752532 −0.376266 0.926512i \(-0.622792\pi\)
−0.376266 + 0.926512i \(0.622792\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −627.181 −1.00671
\(624\) 0 0
\(625\) −462.086 + 420.834i −0.739338 + 0.673334i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 989.233i 1.57271i
\(630\) 0 0
\(631\) 340.458 0.539554 0.269777 0.962923i \(-0.413050\pi\)
0.269777 + 0.962923i \(0.413050\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 487.040 + 333.718i 0.766992 + 0.525540i
\(636\) 0 0
\(637\) 119.374i 0.187401i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1060.95i 1.65515i −0.561359 0.827573i \(-0.689721\pi\)
0.561359 0.827573i \(-0.310279\pi\)
\(642\) 0 0
\(643\) 969.459i 1.50771i −0.657039 0.753856i \(-0.728191\pi\)
0.657039 0.753856i \(-0.271809\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 761.354 1.17674 0.588372 0.808590i \(-0.299769\pi\)
0.588372 + 0.808590i \(0.299769\pi\)
\(648\) 0 0
\(649\) −162.973 −0.251113
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −363.354 −0.556438 −0.278219 0.960518i \(-0.589744\pi\)
−0.278219 + 0.960518i \(0.589744\pi\)
\(654\) 0 0
\(655\) −115.751 79.3123i −0.176720 0.121088i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 283.185i 0.429720i −0.976645 0.214860i \(-0.931070\pi\)
0.976645 0.214860i \(-0.0689295\pi\)
\(660\) 0 0
\(661\) −703.455 −1.06423 −0.532115 0.846672i \(-0.678602\pi\)
−0.532115 + 0.846672i \(0.678602\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −841.636 576.685i −1.26562 0.867196i
\(666\) 0 0
\(667\) 136.725i 0.204985i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 271.679i 0.404887i
\(672\) 0 0
\(673\) 450.802i 0.669839i 0.942247 + 0.334920i \(0.108709\pi\)
−0.942247 + 0.334920i \(0.891291\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −292.845 −0.432563 −0.216281 0.976331i \(-0.569393\pi\)
−0.216281 + 0.976331i \(0.569393\pi\)
\(678\) 0 0
\(679\) −122.081 −0.179796
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −161.354 −0.236243 −0.118121 0.992999i \(-0.537687\pi\)
−0.118121 + 0.992999i \(0.537687\pi\)
\(684\) 0 0
\(685\) 304.583 444.520i 0.444647 0.648934i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 736.786i 1.06936i
\(690\) 0 0
\(691\) 611.834 0.885432 0.442716 0.896662i \(-0.354015\pi\)
0.442716 + 0.896662i \(0.354015\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 483.120 705.083i 0.695136 1.01451i
\(696\) 0 0
\(697\) 1152.95i 1.65416i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 702.946i 1.00278i −0.865222 0.501388i \(-0.832823\pi\)
0.865222 0.501388i \(-0.167177\pi\)
\(702\) 0 0
\(703\) 1415.30i 2.01322i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −956.336 −1.35267
\(708\) 0 0
\(709\) −361.678 −0.510124 −0.255062 0.966925i \(-0.582096\pi\)
−0.255062 + 0.966925i \(0.582096\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 578.907 0.811932
\(714\) 0 0
\(715\) 293.649 428.562i 0.410698 0.599387i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 271.814i 0.378044i 0.981973 + 0.189022i \(0.0605318\pi\)
−0.981973 + 0.189022i \(0.939468\pi\)
\(720\) 0 0
\(721\) 988.014 1.37034
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 162.539 62.9224i 0.224192 0.0867896i
\(726\) 0 0
\(727\) 754.003i 1.03714i 0.855034 + 0.518572i \(0.173536\pi\)
−0.855034 + 0.518572i \(0.826464\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 201.989i 0.276318i
\(732\) 0 0
\(733\) 16.9221i 0.0230860i −0.999933 0.0115430i \(-0.996326\pi\)
0.999933 0.0115430i \(-0.00367434\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −393.144 −0.533438
\(738\) 0 0
\(739\) −874.635 −1.18354 −0.591769 0.806107i \(-0.701570\pi\)
−0.591769 + 0.806107i \(0.701570\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 272.840 0.367213 0.183607 0.983000i \(-0.441223\pi\)
0.183607 + 0.983000i \(0.441223\pi\)
\(744\) 0 0
\(745\) 336.969 + 230.890i 0.452307 + 0.309919i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 993.079i 1.32587i
\(750\) 0 0
\(751\) 1277.28 1.70077 0.850386 0.526160i \(-0.176369\pi\)
0.850386 + 0.526160i \(0.176369\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 26.0378 38.0006i 0.0344872 0.0503319i
\(756\) 0 0
\(757\) 1087.23i 1.43624i 0.695921 + 0.718118i \(0.254996\pi\)
−0.695921 + 0.718118i \(0.745004\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1017.04i 1.33645i 0.743958 + 0.668226i \(0.232946\pi\)
−0.743958 + 0.668226i \(0.767054\pi\)
\(762\) 0 0
\(763\) 1517.42i 1.98876i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 380.205 0.495704
\(768\) 0 0
\(769\) −205.301 −0.266971 −0.133486 0.991051i \(-0.542617\pi\)
−0.133486 + 0.991051i \(0.542617\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 254.726 0.329530 0.164765 0.986333i \(-0.447313\pi\)
0.164765 + 0.986333i \(0.447313\pi\)
\(774\) 0 0
\(775\) −266.420 688.208i −0.343768 0.888010i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1649.53i 2.11750i
\(780\) 0 0
\(781\) 226.506 0.290020
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1189.40 + 814.972i 1.51516 + 1.03818i
\(786\) 0 0
\(787\) 66.9905i 0.0851214i 0.999094 + 0.0425607i \(0.0135516\pi\)
−0.999094 + 0.0425607i \(0.986448\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 253.665i 0.320688i
\(792\) 0 0
\(793\) 633.811i 0.799257i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 642.644 0.806329 0.403164 0.915128i \(-0.367910\pi\)
0.403164 + 0.915128i \(0.367910\pi\)
\(798\) 0 0
\(799\) −1436.30 −1.79762
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 532.698 0.663385
\(804\) 0 0
\(805\) 608.918 + 417.228i 0.756420 + 0.518296i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 479.966i 0.593283i 0.954989 + 0.296642i \(0.0958667\pi\)
−0.954989 + 0.296642i \(0.904133\pi\)
\(810\) 0 0
\(811\) −478.009 −0.589407 −0.294703 0.955589i \(-0.595221\pi\)
−0.294703 + 0.955589i \(0.595221\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 285.042 + 195.309i 0.349745 + 0.239643i
\(816\) 0 0
\(817\) 288.985i 0.353715i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 894.509i 1.08954i −0.838587 0.544768i \(-0.816618\pi\)
0.838587 0.544768i \(-0.183382\pi\)
\(822\) 0 0
\(823\) 376.036i 0.456909i 0.973555 + 0.228455i \(0.0733672\pi\)
−0.973555 + 0.228455i \(0.926633\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −363.894 −0.440017 −0.220008 0.975498i \(-0.570608\pi\)
−0.220008 + 0.975498i \(0.570608\pi\)
\(828\) 0 0
\(829\) −398.314 −0.480475 −0.240238 0.970714i \(-0.577225\pi\)
−0.240238 + 0.970714i \(0.577225\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −145.268 −0.174391
\(834\) 0 0
\(835\) 551.743 805.234i 0.660770 0.964352i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1499.60i 1.78736i 0.448702 + 0.893682i \(0.351887\pi\)
−0.448702 + 0.893682i \(0.648113\pi\)
\(840\) 0 0
\(841\) 792.395 0.942205
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −207.438 + 302.743i −0.245489 + 0.358276i
\(846\) 0 0
\(847\) 575.593i 0.679567i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1023.96i 1.20324i
\(852\) 0 0
\(853\) 1524.29i 1.78697i −0.449093 0.893485i \(-0.648253\pi\)
0.449093 0.893485i \(-0.351747\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −931.475 −1.08690 −0.543451 0.839441i \(-0.682883\pi\)
−0.543451 + 0.839441i \(0.682883\pi\)
\(858\) 0 0
\(859\) −102.105 −0.118865 −0.0594325 0.998232i \(-0.518929\pi\)
−0.0594325 + 0.998232i \(0.518929\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 971.646 1.12589 0.562947 0.826493i \(-0.309668\pi\)
0.562947 + 0.826493i \(0.309668\pi\)
\(864\) 0 0
\(865\) −167.815 + 244.915i −0.194006 + 0.283139i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 574.270i 0.660840i
\(870\) 0 0
\(871\) 917.179 1.05302
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 215.772 915.899i 0.246596 1.04674i
\(876\) 0 0
\(877\) 771.593i 0.879810i −0.898044 0.439905i \(-0.855012\pi\)
0.898044 0.439905i \(-0.144988\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 279.972i 0.317789i −0.987296 0.158895i \(-0.949207\pi\)
0.987296 0.158895i \(-0.0507930\pi\)
\(882\) 0 0
\(883\) 75.4516i 0.0854492i −0.999087 0.0427246i \(-0.986396\pi\)
0.999087 0.0427246i \(-0.0136038\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −669.288 −0.754553 −0.377276 0.926101i \(-0.623139\pi\)
−0.377276 + 0.926101i \(0.623139\pi\)
\(888\) 0 0
\(889\) −888.884 −0.999869
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2054.92 2.30114
\(894\) 0 0
\(895\) 338.888 + 232.205i 0.378646 + 0.259447i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 205.800i 0.228921i
\(900\) 0 0
\(901\) −896.601 −0.995117
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −684.276 + 998.658i −0.756106 + 1.10349i
\(906\) 0 0
\(907\) 459.134i 0.506212i 0.967439 + 0.253106i \(0.0814521\pi\)
−0.967439 + 0.253106i \(0.918548\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 670.249i 0.735729i 0.929879 + 0.367864i \(0.119911\pi\)
−0.929879 + 0.367864i \(0.880089\pi\)
\(912\) 0 0
\(913\) 1021.57i 1.11892i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 211.255 0.230376
\(918\) 0 0
\(919\) −22.6103 −0.0246032 −0.0123016 0.999924i \(-0.503916\pi\)
−0.0123016 + 0.999924i \(0.503916\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −528.424 −0.572507
\(924\) 0 0
\(925\) 1217.29 471.237i 1.31598 0.509445i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 944.272i 1.01644i 0.861228 + 0.508220i \(0.169696\pi\)
−0.861228 + 0.508220i \(0.830304\pi\)
\(930\) 0 0
\(931\) 207.835 0.223238
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −521.520 357.343i −0.557776 0.382185i
\(936\) 0 0
\(937\) 1037.45i 1.10721i −0.832781 0.553603i \(-0.813253\pi\)
0.832781 0.553603i \(-0.186747\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 410.283i 0.436007i −0.975948 0.218004i \(-0.930046\pi\)
0.975948 0.218004i \(-0.0699544\pi\)
\(942\) 0 0
\(943\) 1193.42i 1.26556i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −753.302 −0.795462 −0.397731 0.917502i \(-0.630202\pi\)
−0.397731 + 0.917502i \(0.630202\pi\)
\(948\) 0 0
\(949\) −1242.75 −1.30954
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −661.356 −0.693973 −0.346986 0.937870i \(-0.612795\pi\)
−0.346986 + 0.937870i \(0.612795\pi\)
\(954\) 0 0
\(955\) −113.021 77.4418i −0.118347 0.0810909i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 811.282i 0.845966i
\(960\) 0 0
\(961\) −89.6241 −0.0932613
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −576.221 394.824i −0.597120 0.409144i
\(966\) 0 0
\(967\) 1584.38i 1.63845i −0.573470 0.819227i \(-0.694403\pi\)
0.573470 0.819227i \(-0.305597\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 982.405i 1.01175i 0.862608 + 0.505873i \(0.168829\pi\)
−0.862608 + 0.505873i \(0.831171\pi\)
\(972\) 0 0
\(973\) 1286.83i 1.32254i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1891.35 1.93588 0.967939 0.251185i \(-0.0808201\pi\)
0.967939 + 0.251185i \(0.0808201\pi\)
\(978\) 0 0
\(979\) 556.017 0.567944
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1431.06 −1.45581 −0.727906 0.685677i \(-0.759506\pi\)
−0.727906 + 0.685677i \(0.759506\pi\)
\(984\) 0 0
\(985\) 354.095 516.780i 0.359488 0.524650i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 209.079i 0.211404i
\(990\) 0 0
\(991\) 159.789 0.161240 0.0806200 0.996745i \(-0.474310\pi\)
0.0806200 + 0.996745i \(0.474310\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −685.764 + 1000.83i −0.689210 + 1.00586i
\(996\) 0 0
\(997\) 651.900i 0.653861i 0.945048 + 0.326931i \(0.106014\pi\)
−0.945048 + 0.326931i \(0.893986\pi\)
\(998\) 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 1620.3.b.a.809.17 yes 24
3.2 odd 2 inner 1620.3.b.a.809.8 yes 24
5.4 even 2 inner 1620.3.b.a.809.7 24
9.2 odd 6 1620.3.t.f.1349.10 48
9.4 even 3 1620.3.t.f.269.2 48
9.5 odd 6 1620.3.t.f.269.23 48
9.7 even 3 1620.3.t.f.1349.15 48
15.14 odd 2 inner 1620.3.b.a.809.18 yes 24
45.4 even 6 1620.3.t.f.269.10 48
45.14 odd 6 1620.3.t.f.269.15 48
45.29 odd 6 1620.3.t.f.1349.2 48
45.34 even 6 1620.3.t.f.1349.23 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.3.b.a.809.7 24 5.4 even 2 inner
1620.3.b.a.809.8 yes 24 3.2 odd 2 inner
1620.3.b.a.809.17 yes 24 1.1 even 1 trivial
1620.3.b.a.809.18 yes 24 15.14 odd 2 inner
1620.3.t.f.269.2 48 9.4 even 3
1620.3.t.f.269.10 48 45.4 even 6
1620.3.t.f.269.15 48 45.14 odd 6
1620.3.t.f.269.23 48 9.5 odd 6
1620.3.t.f.1349.2 48 45.29 odd 6
1620.3.t.f.1349.10 48 9.2 odd 6
1620.3.t.f.1349.15 48 9.7 even 3
1620.3.t.f.1349.23 48 45.34 even 6