Properties

Label 1792.2.e.e.895.6
Level $1792$
Weight $2$
Character 1792.895
Analytic conductor $14.309$
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] = [1792,2,Mod(895,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.895");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12745506816.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 23x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 896)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.6
Root \(0.323042 + 0.323042i\) of defining polynomial
Character \(\chi\) \(=\) 1792.895
Dual form 1792.2.e.e.895.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.646084i q^{3} +0.646084 q^{5} +(-2.44949 + 1.00000i) q^{7} +2.58258 q^{9} +O(q^{10})\) \(q+0.646084i q^{3} +0.646084 q^{5} +(-2.44949 + 1.00000i) q^{7} +2.58258 q^{9} -3.58258 q^{11} +5.54506 q^{13} +0.417424i q^{15} -6.19115i q^{17} -6.83723i q^{19} +(-0.646084 - 1.58258i) q^{21} -5.58258i q^{23} -4.58258 q^{25} +3.60681i q^{27} +6.00000i q^{29} +6.19115 q^{31} -2.31464i q^{33} +(-1.58258 + 0.646084i) q^{35} -2.00000i q^{37} +3.58258i q^{39} +1.29217i q^{41} +3.58258 q^{43} +1.66856 q^{45} +6.19115 q^{47} +(5.00000 - 4.89898i) q^{49} +4.00000 q^{51} +9.16515i q^{53} -2.31464 q^{55} +4.41742 q^{57} +5.54506i q^{59} +10.4440 q^{61} +(-6.32599 + 2.58258i) q^{63} +3.58258 q^{65} +4.41742 q^{67} +3.60681 q^{69} -9.16515i q^{71} -7.48331i q^{73} -2.96073i q^{75} +(8.77548 - 3.58258i) q^{77} -2.00000i q^{79} +5.41742 q^{81} +0.646084i q^{83} -4.00000i q^{85} -3.87650 q^{87} -7.48331i q^{89} +(-13.5826 + 5.54506i) q^{91} +4.00000i q^{93} -4.41742i q^{95} -18.5734i q^{97} -9.25227 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{9} + 8 q^{11} + 24 q^{35} - 8 q^{43} + 40 q^{49} + 32 q^{51} + 72 q^{57} - 8 q^{65} + 72 q^{67} + 80 q^{81} - 72 q^{91} - 184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 0.646084i 0.373017i 0.982453 + 0.186508i \(0.0597171\pi\)
−0.982453 + 0.186508i \(0.940283\pi\)
\(4\) 0 0
\(5\) 0.646084 0.288937 0.144469 0.989509i \(-0.453853\pi\)
0.144469 + 0.989509i \(0.453853\pi\)
\(6\) 0 0
\(7\) −2.44949 + 1.00000i −0.925820 + 0.377964i
\(8\) 0 0
\(9\) 2.58258 0.860859
\(10\) 0 0
\(11\) −3.58258 −1.08019 −0.540094 0.841605i \(-0.681611\pi\)
−0.540094 + 0.841605i \(0.681611\pi\)
\(12\) 0 0
\(13\) 5.54506 1.53792 0.768962 0.639295i \(-0.220774\pi\)
0.768962 + 0.639295i \(0.220774\pi\)
\(14\) 0 0
\(15\) 0.417424i 0.107778i
\(16\) 0 0
\(17\) 6.19115i 1.50157i −0.660545 0.750787i \(-0.729675\pi\)
0.660545 0.750787i \(-0.270325\pi\)
\(18\) 0 0
\(19\) 6.83723i 1.56857i −0.620402 0.784284i \(-0.713030\pi\)
0.620402 0.784284i \(-0.286970\pi\)
\(20\) 0 0
\(21\) −0.646084 1.58258i −0.140987 0.345346i
\(22\) 0 0
\(23\) 5.58258i 1.16405i −0.813172 0.582024i \(-0.802261\pi\)
0.813172 0.582024i \(-0.197739\pi\)
\(24\) 0 0
\(25\) −4.58258 −0.916515
\(26\) 0 0
\(27\) 3.60681i 0.694131i
\(28\) 0 0
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 6.19115 1.11196 0.555981 0.831195i \(-0.312343\pi\)
0.555981 + 0.831195i \(0.312343\pi\)
\(32\) 0 0
\(33\) 2.31464i 0.402928i
\(34\) 0 0
\(35\) −1.58258 + 0.646084i −0.267504 + 0.109208i
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 3.58258i 0.573671i
\(40\) 0 0
\(41\) 1.29217i 0.201803i 0.994896 + 0.100901i \(0.0321726\pi\)
−0.994896 + 0.100901i \(0.967827\pi\)
\(42\) 0 0
\(43\) 3.58258 0.546338 0.273169 0.961966i \(-0.411928\pi\)
0.273169 + 0.961966i \(0.411928\pi\)
\(44\) 0 0
\(45\) 1.66856 0.248734
\(46\) 0 0
\(47\) 6.19115 0.903072 0.451536 0.892253i \(-0.350876\pi\)
0.451536 + 0.892253i \(0.350876\pi\)
\(48\) 0 0
\(49\) 5.00000 4.89898i 0.714286 0.699854i
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 9.16515i 1.25893i 0.777029 + 0.629465i \(0.216726\pi\)
−0.777029 + 0.629465i \(0.783274\pi\)
\(54\) 0 0
\(55\) −2.31464 −0.312107
\(56\) 0 0
\(57\) 4.41742 0.585102
\(58\) 0 0
\(59\) 5.54506i 0.721906i 0.932584 + 0.360953i \(0.117548\pi\)
−0.932584 + 0.360953i \(0.882452\pi\)
\(60\) 0 0
\(61\) 10.4440 1.33722 0.668611 0.743612i \(-0.266889\pi\)
0.668611 + 0.743612i \(0.266889\pi\)
\(62\) 0 0
\(63\) −6.32599 + 2.58258i −0.797000 + 0.325374i
\(64\) 0 0
\(65\) 3.58258 0.444364
\(66\) 0 0
\(67\) 4.41742 0.539674 0.269837 0.962906i \(-0.413030\pi\)
0.269837 + 0.962906i \(0.413030\pi\)
\(68\) 0 0
\(69\) 3.60681 0.434209
\(70\) 0 0
\(71\) 9.16515i 1.08770i −0.839181 0.543852i \(-0.816965\pi\)
0.839181 0.543852i \(-0.183035\pi\)
\(72\) 0 0
\(73\) 7.48331i 0.875856i −0.899010 0.437928i \(-0.855713\pi\)
0.899010 0.437928i \(-0.144287\pi\)
\(74\) 0 0
\(75\) 2.96073i 0.341875i
\(76\) 0 0
\(77\) 8.77548 3.58258i 1.00006 0.408272i
\(78\) 0 0
\(79\) 2.00000i 0.225018i −0.993651 0.112509i \(-0.964111\pi\)
0.993651 0.112509i \(-0.0358886\pi\)
\(80\) 0 0
\(81\) 5.41742 0.601936
\(82\) 0 0
\(83\) 0.646084i 0.0709169i 0.999371 + 0.0354585i \(0.0112891\pi\)
−0.999371 + 0.0354585i \(0.988711\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 0 0
\(87\) −3.87650 −0.415605
\(88\) 0 0
\(89\) 7.48331i 0.793230i −0.917985 0.396615i \(-0.870185\pi\)
0.917985 0.396615i \(-0.129815\pi\)
\(90\) 0 0
\(91\) −13.5826 + 5.54506i −1.42384 + 0.581281i
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 4.41742i 0.453218i
\(96\) 0 0
\(97\) 18.5734i 1.88585i −0.333009 0.942924i \(-0.608064\pi\)
0.333009 0.942924i \(-0.391936\pi\)
\(98\) 0 0
\(99\) −9.25227 −0.929888
\(100\) 0 0
\(101\) 2.96073 0.294603 0.147302 0.989092i \(-0.452941\pi\)
0.147302 + 0.989092i \(0.452941\pi\)
\(102\) 0 0
\(103\) −1.29217 −0.127321 −0.0636605 0.997972i \(-0.520277\pi\)
−0.0636605 + 0.997972i \(0.520277\pi\)
\(104\) 0 0
\(105\) −0.417424 1.02248i −0.0407364 0.0997835i
\(106\) 0 0
\(107\) 15.5826 1.50642 0.753212 0.657778i \(-0.228503\pi\)
0.753212 + 0.657778i \(0.228503\pi\)
\(108\) 0 0
\(109\) 5.16515i 0.494732i 0.968922 + 0.247366i \(0.0795650\pi\)
−0.968922 + 0.247366i \(0.920435\pi\)
\(110\) 0 0
\(111\) 1.29217 0.122647
\(112\) 0 0
\(113\) −13.5826 −1.27774 −0.638871 0.769314i \(-0.720598\pi\)
−0.638871 + 0.769314i \(0.720598\pi\)
\(114\) 0 0
\(115\) 3.60681i 0.336337i
\(116\) 0 0
\(117\) 14.3205 1.32393
\(118\) 0 0
\(119\) 6.19115 + 15.1652i 0.567542 + 1.39019i
\(120\) 0 0
\(121\) 1.83485 0.166804
\(122\) 0 0
\(123\) −0.834849 −0.0752758
\(124\) 0 0
\(125\) −6.19115 −0.553753
\(126\) 0 0
\(127\) 17.5826i 1.56020i 0.625654 + 0.780101i \(0.284832\pi\)
−0.625654 + 0.780101i \(0.715168\pi\)
\(128\) 0 0
\(129\) 2.31464i 0.203793i
\(130\) 0 0
\(131\) 7.85971i 0.686706i 0.939206 + 0.343353i \(0.111563\pi\)
−0.939206 + 0.343353i \(0.888437\pi\)
\(132\) 0 0
\(133\) 6.83723 + 16.7477i 0.592863 + 1.45221i
\(134\) 0 0
\(135\) 2.33030i 0.200561i
\(136\) 0 0
\(137\) 1.16515 0.0995456 0.0497728 0.998761i \(-0.484150\pi\)
0.0497728 + 0.998761i \(0.484150\pi\)
\(138\) 0 0
\(139\) 19.2195i 1.63018i −0.579335 0.815089i \(-0.696688\pi\)
0.579335 0.815089i \(-0.303312\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) −19.8656 −1.66125
\(144\) 0 0
\(145\) 3.87650i 0.321926i
\(146\) 0 0
\(147\) 3.16515 + 3.23042i 0.261057 + 0.266440i
\(148\) 0 0
\(149\) 1.16515i 0.0954529i −0.998860 0.0477265i \(-0.984802\pi\)
0.998860 0.0477265i \(-0.0151976\pi\)
\(150\) 0 0
\(151\) 2.41742i 0.196727i 0.995151 + 0.0983636i \(0.0313608\pi\)
−0.995151 + 0.0983636i \(0.968639\pi\)
\(152\) 0 0
\(153\) 15.9891i 1.29264i
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 2.96073 0.236292 0.118146 0.992996i \(-0.462305\pi\)
0.118146 + 0.992996i \(0.462305\pi\)
\(158\) 0 0
\(159\) −5.92146 −0.469602
\(160\) 0 0
\(161\) 5.58258 + 13.6745i 0.439969 + 1.07770i
\(162\) 0 0
\(163\) 3.58258 0.280609 0.140304 0.990108i \(-0.455192\pi\)
0.140304 + 0.990108i \(0.455192\pi\)
\(164\) 0 0
\(165\) 1.49545i 0.116421i
\(166\) 0 0
\(167\) 11.0901 0.858180 0.429090 0.903262i \(-0.358834\pi\)
0.429090 + 0.903262i \(0.358834\pi\)
\(168\) 0 0
\(169\) 17.7477 1.36521
\(170\) 0 0
\(171\) 17.6577i 1.35032i
\(172\) 0 0
\(173\) −16.6352 −1.26475 −0.632375 0.774662i \(-0.717920\pi\)
−0.632375 + 0.774662i \(0.717920\pi\)
\(174\) 0 0
\(175\) 11.2250 4.58258i 0.848528 0.346410i
\(176\) 0 0
\(177\) −3.58258 −0.269283
\(178\) 0 0
\(179\) 7.58258 0.566748 0.283374 0.959009i \(-0.408546\pi\)
0.283374 + 0.959009i \(0.408546\pi\)
\(180\) 0 0
\(181\) −4.52259 −0.336161 −0.168081 0.985773i \(-0.553757\pi\)
−0.168081 + 0.985773i \(0.553757\pi\)
\(182\) 0 0
\(183\) 6.74773i 0.498806i
\(184\) 0 0
\(185\) 1.29217i 0.0950021i
\(186\) 0 0
\(187\) 22.1803i 1.62198i
\(188\) 0 0
\(189\) −3.60681 8.83485i −0.262357 0.642641i
\(190\) 0 0
\(191\) 21.1652i 1.53146i −0.643164 0.765728i \(-0.722379\pi\)
0.643164 0.765728i \(-0.277621\pi\)
\(192\) 0 0
\(193\) 9.58258 0.689769 0.344884 0.938645i \(-0.387918\pi\)
0.344884 + 0.938645i \(0.387918\pi\)
\(194\) 0 0
\(195\) 2.31464i 0.165755i
\(196\) 0 0
\(197\) 18.0000i 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 1.29217 0.0915993 0.0457997 0.998951i \(-0.485416\pi\)
0.0457997 + 0.998951i \(0.485416\pi\)
\(200\) 0 0
\(201\) 2.85403i 0.201307i
\(202\) 0 0
\(203\) −6.00000 14.6969i −0.421117 1.03152i
\(204\) 0 0
\(205\) 0.834849i 0.0583084i
\(206\) 0 0
\(207\) 14.4174i 1.00208i
\(208\) 0 0
\(209\) 24.4949i 1.69435i
\(210\) 0 0
\(211\) −25.9129 −1.78392 −0.891958 0.452118i \(-0.850669\pi\)
−0.891958 + 0.452118i \(0.850669\pi\)
\(212\) 0 0
\(213\) 5.92146 0.405731
\(214\) 0 0
\(215\) 2.31464 0.157857
\(216\) 0 0
\(217\) −15.1652 + 6.19115i −1.02948 + 0.420282i
\(218\) 0 0
\(219\) 4.83485 0.326709
\(220\) 0 0
\(221\) 34.3303i 2.30931i
\(222\) 0 0
\(223\) −2.58434 −0.173060 −0.0865299 0.996249i \(-0.527578\pi\)
−0.0865299 + 0.996249i \(0.527578\pi\)
\(224\) 0 0
\(225\) −11.8348 −0.788990
\(226\) 0 0
\(227\) 12.7587i 0.846824i 0.905937 + 0.423412i \(0.139168\pi\)
−0.905937 + 0.423412i \(0.860832\pi\)
\(228\) 0 0
\(229\) −1.66856 −0.110262 −0.0551308 0.998479i \(-0.517558\pi\)
−0.0551308 + 0.998479i \(0.517558\pi\)
\(230\) 0 0
\(231\) 2.31464 + 5.66970i 0.152292 + 0.373039i
\(232\) 0 0
\(233\) −1.16515 −0.0763316 −0.0381658 0.999271i \(-0.512152\pi\)
−0.0381658 + 0.999271i \(0.512152\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 1.29217 0.0839353
\(238\) 0 0
\(239\) 25.5826i 1.65480i 0.561614 + 0.827400i \(0.310181\pi\)
−0.561614 + 0.827400i \(0.689819\pi\)
\(240\) 0 0
\(241\) 6.19115i 0.398807i −0.979917 0.199403i \(-0.936100\pi\)
0.979917 0.199403i \(-0.0639004\pi\)
\(242\) 0 0
\(243\) 14.3205i 0.918663i
\(244\) 0 0
\(245\) 3.23042 3.16515i 0.206384 0.202214i
\(246\) 0 0
\(247\) 37.9129i 2.41234i
\(248\) 0 0
\(249\) −0.417424 −0.0264532
\(250\) 0 0
\(251\) 10.4440i 0.659222i 0.944117 + 0.329611i \(0.106918\pi\)
−0.944117 + 0.329611i \(0.893082\pi\)
\(252\) 0 0
\(253\) 20.0000i 1.25739i
\(254\) 0 0
\(255\) 2.58434 0.161837
\(256\) 0 0
\(257\) 9.79796i 0.611180i 0.952163 + 0.305590i \(0.0988537\pi\)
−0.952163 + 0.305590i \(0.901146\pi\)
\(258\) 0 0
\(259\) 2.00000 + 4.89898i 0.124274 + 0.304408i
\(260\) 0 0
\(261\) 15.4955i 0.959145i
\(262\) 0 0
\(263\) 6.83485i 0.421455i 0.977545 + 0.210727i \(0.0675832\pi\)
−0.977545 + 0.210727i \(0.932417\pi\)
\(264\) 0 0
\(265\) 5.92146i 0.363752i
\(266\) 0 0
\(267\) 4.83485 0.295888
\(268\) 0 0
\(269\) −23.8488 −1.45409 −0.727044 0.686591i \(-0.759106\pi\)
−0.727044 + 0.686591i \(0.759106\pi\)
\(270\) 0 0
\(271\) −17.5510 −1.06615 −0.533073 0.846070i \(-0.678963\pi\)
−0.533073 + 0.846070i \(0.678963\pi\)
\(272\) 0 0
\(273\) −3.58258 8.77548i −0.216827 0.531116i
\(274\) 0 0
\(275\) 16.4174 0.990008
\(276\) 0 0
\(277\) 8.33030i 0.500519i 0.968179 + 0.250260i \(0.0805160\pi\)
−0.968179 + 0.250260i \(0.919484\pi\)
\(278\) 0 0
\(279\) 15.9891 0.957243
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 20.2420i 1.20326i 0.798774 + 0.601631i \(0.205482\pi\)
−0.798774 + 0.601631i \(0.794518\pi\)
\(284\) 0 0
\(285\) 2.85403 0.169058
\(286\) 0 0
\(287\) −1.29217 3.16515i −0.0762742 0.186833i
\(288\) 0 0
\(289\) −21.3303 −1.25472
\(290\) 0 0
\(291\) 12.0000 0.703452
\(292\) 0 0
\(293\) −1.66856 −0.0974783 −0.0487392 0.998812i \(-0.515520\pi\)
−0.0487392 + 0.998812i \(0.515520\pi\)
\(294\) 0 0
\(295\) 3.58258i 0.208586i
\(296\) 0 0
\(297\) 12.9217i 0.749792i
\(298\) 0 0
\(299\) 30.9557i 1.79022i
\(300\) 0 0
\(301\) −8.77548 + 3.58258i −0.505810 + 0.206496i
\(302\) 0 0
\(303\) 1.91288i 0.109892i
\(304\) 0 0
\(305\) 6.74773 0.386374
\(306\) 0 0
\(307\) 24.1185i 1.37652i −0.725466 0.688258i \(-0.758376\pi\)
0.725466 0.688258i \(-0.241624\pi\)
\(308\) 0 0
\(309\) 0.834849i 0.0474929i
\(310\) 0 0
\(311\) 0.269691 0.0152928 0.00764639 0.999971i \(-0.497566\pi\)
0.00764639 + 0.999971i \(0.497566\pi\)
\(312\) 0 0
\(313\) 11.0901i 0.626851i 0.949613 + 0.313426i \(0.101477\pi\)
−0.949613 + 0.313426i \(0.898523\pi\)
\(314\) 0 0
\(315\) −4.08712 + 1.66856i −0.230283 + 0.0940127i
\(316\) 0 0
\(317\) 8.33030i 0.467876i 0.972251 + 0.233938i \(0.0751613\pi\)
−0.972251 + 0.233938i \(0.924839\pi\)
\(318\) 0 0
\(319\) 21.4955i 1.20351i
\(320\) 0 0
\(321\) 10.0677i 0.561921i
\(322\) 0 0
\(323\) −42.3303 −2.35532
\(324\) 0 0
\(325\) −25.4107 −1.40953
\(326\) 0 0
\(327\) −3.33712 −0.184543
\(328\) 0 0
\(329\) −15.1652 + 6.19115i −0.836082 + 0.341329i
\(330\) 0 0
\(331\) 24.4174 1.34210 0.671052 0.741411i \(-0.265843\pi\)
0.671052 + 0.741411i \(0.265843\pi\)
\(332\) 0 0
\(333\) 5.16515i 0.283049i
\(334\) 0 0
\(335\) 2.85403 0.155932
\(336\) 0 0
\(337\) 8.74773 0.476519 0.238260 0.971202i \(-0.423423\pi\)
0.238260 + 0.971202i \(0.423423\pi\)
\(338\) 0 0
\(339\) 8.77548i 0.476619i
\(340\) 0 0
\(341\) −22.1803 −1.20113
\(342\) 0 0
\(343\) −7.34847 + 17.0000i −0.396780 + 0.917914i
\(344\) 0 0
\(345\) 2.33030 0.125459
\(346\) 0 0
\(347\) −26.7477 −1.43589 −0.717947 0.696098i \(-0.754918\pi\)
−0.717947 + 0.696098i \(0.754918\pi\)
\(348\) 0 0
\(349\) −1.93825 −0.103752 −0.0518761 0.998654i \(-0.516520\pi\)
−0.0518761 + 0.998654i \(0.516520\pi\)
\(350\) 0 0
\(351\) 20.0000i 1.06752i
\(352\) 0 0
\(353\) 17.0116i 0.905435i 0.891654 + 0.452718i \(0.149545\pi\)
−0.891654 + 0.452718i \(0.850455\pi\)
\(354\) 0 0
\(355\) 5.92146i 0.314278i
\(356\) 0 0
\(357\) −9.79796 + 4.00000i −0.518563 + 0.211702i
\(358\) 0 0
\(359\) 13.5826i 0.716861i −0.933556 0.358430i \(-0.883312\pi\)
0.933556 0.358430i \(-0.116688\pi\)
\(360\) 0 0
\(361\) −27.7477 −1.46041
\(362\) 0 0
\(363\) 1.18547i 0.0622208i
\(364\) 0 0
\(365\) 4.83485i 0.253068i
\(366\) 0 0
\(367\) 29.3939 1.53435 0.767174 0.641439i \(-0.221662\pi\)
0.767174 + 0.641439i \(0.221662\pi\)
\(368\) 0 0
\(369\) 3.33712i 0.173724i
\(370\) 0 0
\(371\) −9.16515 22.4499i −0.475831 1.16554i
\(372\) 0 0
\(373\) 28.3303i 1.46689i −0.679750 0.733444i \(-0.737912\pi\)
0.679750 0.733444i \(-0.262088\pi\)
\(374\) 0 0
\(375\) 4.00000i 0.206559i
\(376\) 0 0
\(377\) 33.2704i 1.71351i
\(378\) 0 0
\(379\) −29.9129 −1.53652 −0.768261 0.640137i \(-0.778878\pi\)
−0.768261 + 0.640137i \(0.778878\pi\)
\(380\) 0 0
\(381\) −11.3598 −0.581981
\(382\) 0 0
\(383\) 23.7421 1.21317 0.606583 0.795020i \(-0.292540\pi\)
0.606583 + 0.795020i \(0.292540\pi\)
\(384\) 0 0
\(385\) 5.66970 2.31464i 0.288955 0.117965i
\(386\) 0 0
\(387\) 9.25227 0.470319
\(388\) 0 0
\(389\) 2.00000i 0.101404i −0.998714 0.0507020i \(-0.983854\pi\)
0.998714 0.0507020i \(-0.0161459\pi\)
\(390\) 0 0
\(391\) −34.5625 −1.74790
\(392\) 0 0
\(393\) −5.07803 −0.256153
\(394\) 0 0
\(395\) 1.29217i 0.0650160i
\(396\) 0 0
\(397\) 22.8263 1.14562 0.572811 0.819688i \(-0.305853\pi\)
0.572811 + 0.819688i \(0.305853\pi\)
\(398\) 0 0
\(399\) −10.8204 + 4.41742i −0.541699 + 0.221148i
\(400\) 0 0
\(401\) −6.41742 −0.320471 −0.160235 0.987079i \(-0.551225\pi\)
−0.160235 + 0.987079i \(0.551225\pi\)
\(402\) 0 0
\(403\) 34.3303 1.71011
\(404\) 0 0
\(405\) 3.50011 0.173922
\(406\) 0 0
\(407\) 7.16515i 0.355163i
\(408\) 0 0
\(409\) 31.2254i 1.54400i 0.635624 + 0.771999i \(0.280743\pi\)
−0.635624 + 0.771999i \(0.719257\pi\)
\(410\) 0 0
\(411\) 0.752785i 0.0371322i
\(412\) 0 0
\(413\) −5.54506 13.5826i −0.272855 0.668355i
\(414\) 0 0
\(415\) 0.417424i 0.0204906i
\(416\) 0 0
\(417\) 12.4174 0.608084
\(418\) 0 0
\(419\) 27.9950i 1.36765i 0.729648 + 0.683823i \(0.239684\pi\)
−0.729648 + 0.683823i \(0.760316\pi\)
\(420\) 0 0
\(421\) 31.4955i 1.53499i −0.641052 0.767497i \(-0.721502\pi\)
0.641052 0.767497i \(-0.278498\pi\)
\(422\) 0 0
\(423\) 15.9891 0.777417
\(424\) 0 0
\(425\) 28.3714i 1.37622i
\(426\) 0 0
\(427\) −25.5826 + 10.4440i −1.23803 + 0.505423i
\(428\) 0 0
\(429\) 12.8348i 0.619672i
\(430\) 0 0
\(431\) 4.74773i 0.228690i −0.993441 0.114345i \(-0.963523\pi\)
0.993441 0.114345i \(-0.0364769\pi\)
\(432\) 0 0
\(433\) 10.8204i 0.519997i −0.965609 0.259998i \(-0.916278\pi\)
0.965609 0.259998i \(-0.0837221\pi\)
\(434\) 0 0
\(435\) −2.50455 −0.120084
\(436\) 0 0
\(437\) −38.1694 −1.82589
\(438\) 0 0
\(439\) −19.8656 −0.948134 −0.474067 0.880489i \(-0.657215\pi\)
−0.474067 + 0.880489i \(0.657215\pi\)
\(440\) 0 0
\(441\) 12.9129 12.6520i 0.614899 0.602475i
\(442\) 0 0
\(443\) 22.7477 1.08078 0.540389 0.841416i \(-0.318277\pi\)
0.540389 + 0.841416i \(0.318277\pi\)
\(444\) 0 0
\(445\) 4.83485i 0.229194i
\(446\) 0 0
\(447\) 0.752785 0.0356055
\(448\) 0 0
\(449\) −35.4955 −1.67513 −0.837567 0.546335i \(-0.816023\pi\)
−0.837567 + 0.546335i \(0.816023\pi\)
\(450\) 0 0
\(451\) 4.62929i 0.217985i
\(452\) 0 0
\(453\) −1.56186 −0.0733825
\(454\) 0 0
\(455\) −8.77548 + 3.58258i −0.411401 + 0.167954i
\(456\) 0 0
\(457\) 4.74773 0.222089 0.111045 0.993815i \(-0.464580\pi\)
0.111045 + 0.993815i \(0.464580\pi\)
\(458\) 0 0
\(459\) 22.3303 1.04229
\(460\) 0 0
\(461\) −24.1185 −1.12331 −0.561655 0.827371i \(-0.689835\pi\)
−0.561655 + 0.827371i \(0.689835\pi\)
\(462\) 0 0
\(463\) 17.1652i 0.797732i 0.917009 + 0.398866i \(0.130596\pi\)
−0.917009 + 0.398866i \(0.869404\pi\)
\(464\) 0 0
\(465\) 2.58434i 0.119846i
\(466\) 0 0
\(467\) 36.2311i 1.67658i −0.545228 0.838288i \(-0.683557\pi\)
0.545228 0.838288i \(-0.316443\pi\)
\(468\) 0 0
\(469\) −10.8204 + 4.41742i −0.499641 + 0.203978i
\(470\) 0 0
\(471\) 1.91288i 0.0881408i
\(472\) 0 0
\(473\) −12.8348 −0.590147
\(474\) 0 0
\(475\) 31.3321i 1.43762i
\(476\) 0 0
\(477\) 23.6697i 1.08376i
\(478\) 0 0
\(479\) −15.9891 −0.730561 −0.365280 0.930898i \(-0.619027\pi\)
−0.365280 + 0.930898i \(0.619027\pi\)
\(480\) 0 0
\(481\) 11.0901i 0.505666i
\(482\) 0 0
\(483\) −8.83485 + 3.60681i −0.402000 + 0.164116i
\(484\) 0 0
\(485\) 12.0000i 0.544892i
\(486\) 0 0
\(487\) 7.25227i 0.328632i −0.986408 0.164316i \(-0.947458\pi\)
0.986408 0.164316i \(-0.0525417\pi\)
\(488\) 0 0
\(489\) 2.31464i 0.104672i
\(490\) 0 0
\(491\) 3.58258 0.161679 0.0808397 0.996727i \(-0.474240\pi\)
0.0808397 + 0.996727i \(0.474240\pi\)
\(492\) 0 0
\(493\) 37.1469 1.67301
\(494\) 0 0
\(495\) −5.97774 −0.268680
\(496\) 0 0
\(497\) 9.16515 + 22.4499i 0.411113 + 1.00702i
\(498\) 0 0
\(499\) 11.5826 0.518507 0.259254 0.965809i \(-0.416523\pi\)
0.259254 + 0.965809i \(0.416523\pi\)
\(500\) 0 0
\(501\) 7.16515i 0.320115i
\(502\) 0 0
\(503\) −25.0343 −1.11622 −0.558112 0.829766i \(-0.688474\pi\)
−0.558112 + 0.829766i \(0.688474\pi\)
\(504\) 0 0
\(505\) 1.91288 0.0851220
\(506\) 0 0
\(507\) 11.4665i 0.509246i
\(508\) 0 0
\(509\) −36.5008 −1.61787 −0.808935 0.587898i \(-0.799955\pi\)
−0.808935 + 0.587898i \(0.799955\pi\)
\(510\) 0 0
\(511\) 7.48331 + 18.3303i 0.331042 + 0.810885i
\(512\) 0 0
\(513\) 24.6606 1.08879
\(514\) 0 0
\(515\) −0.834849 −0.0367878
\(516\) 0 0
\(517\) −22.1803 −0.975486
\(518\) 0 0
\(519\) 10.7477i 0.471773i
\(520\) 0 0
\(521\) 20.8881i 0.915124i 0.889178 + 0.457562i \(0.151277\pi\)
−0.889178 + 0.457562i \(0.848723\pi\)
\(522\) 0 0
\(523\) 0.915775i 0.0400440i 0.999800 + 0.0200220i \(0.00637363\pi\)
−0.999800 + 0.0200220i \(0.993626\pi\)
\(524\) 0 0
\(525\) 2.96073 + 7.25227i 0.129217 + 0.316515i
\(526\) 0 0
\(527\) 38.3303i 1.66969i
\(528\) 0 0
\(529\) −8.16515 −0.355007
\(530\) 0 0
\(531\) 14.3205i 0.621459i
\(532\) 0 0
\(533\) 7.16515i 0.310357i
\(534\) 0 0
\(535\) 10.0677 0.435262
\(536\) 0 0
\(537\) 4.89898i 0.211407i
\(538\) 0 0
\(539\) −17.9129 + 17.5510i −0.771562 + 0.755974i
\(540\) 0 0
\(541\) 18.8348i 0.809773i 0.914367 + 0.404887i \(0.132689\pi\)
−0.914367 + 0.404887i \(0.867311\pi\)
\(542\) 0 0
\(543\) 2.92197i 0.125394i
\(544\) 0 0
\(545\) 3.33712i 0.142947i
\(546\) 0 0
\(547\) 11.5826 0.495235 0.247618 0.968858i \(-0.420352\pi\)
0.247618 + 0.968858i \(0.420352\pi\)
\(548\) 0 0
\(549\) 26.9725 1.15116
\(550\) 0 0
\(551\) 41.0234 1.74765
\(552\) 0 0
\(553\) 2.00000 + 4.89898i 0.0850487 + 0.208326i
\(554\) 0 0
\(555\) 0.834849 0.0354373
\(556\) 0 0
\(557\) 27.4955i 1.16502i 0.812824 + 0.582510i \(0.197929\pi\)
−0.812824 + 0.582510i \(0.802071\pi\)
\(558\) 0 0
\(559\) 19.8656 0.840226
\(560\) 0 0
\(561\) −14.3303 −0.605026
\(562\) 0 0
\(563\) 1.66856i 0.0703214i −0.999382 0.0351607i \(-0.988806\pi\)
0.999382 0.0351607i \(-0.0111943\pi\)
\(564\) 0 0
\(565\) −8.77548 −0.369187
\(566\) 0 0
\(567\) −13.2699 + 5.41742i −0.557284 + 0.227510i
\(568\) 0 0
\(569\) −17.5826 −0.737100 −0.368550 0.929608i \(-0.620146\pi\)
−0.368550 + 0.929608i \(0.620146\pi\)
\(570\) 0 0
\(571\) 3.58258 0.149926 0.0749631 0.997186i \(-0.476116\pi\)
0.0749631 + 0.997186i \(0.476116\pi\)
\(572\) 0 0
\(573\) 13.6745 0.571259
\(574\) 0 0
\(575\) 25.5826i 1.06687i
\(576\) 0 0
\(577\) 34.5625i 1.43886i −0.694566 0.719429i \(-0.744404\pi\)
0.694566 0.719429i \(-0.255596\pi\)
\(578\) 0 0
\(579\) 6.19115i 0.257295i
\(580\) 0 0
\(581\) −0.646084 1.58258i −0.0268041 0.0656563i
\(582\) 0 0
\(583\) 32.8348i 1.35988i
\(584\) 0 0
\(585\) 9.25227 0.382534
\(586\) 0 0
\(587\) 6.83723i 0.282203i −0.989995 0.141101i \(-0.954936\pi\)
0.989995 0.141101i \(-0.0450643\pi\)
\(588\) 0 0
\(589\) 42.3303i 1.74419i
\(590\) 0 0
\(591\) 11.6295 0.478374
\(592\) 0 0
\(593\) 5.16867i 0.212252i −0.994353 0.106126i \(-0.966155\pi\)
0.994353 0.106126i \(-0.0338447\pi\)
\(594\) 0 0
\(595\) 4.00000 + 9.79796i 0.163984 + 0.401677i
\(596\) 0 0
\(597\) 0.834849i 0.0341681i
\(598\) 0 0
\(599\) 10.0000i 0.408589i 0.978909 + 0.204294i \(0.0654900\pi\)
−0.978909 + 0.204294i \(0.934510\pi\)
\(600\) 0 0
\(601\) 19.8656i 0.810335i −0.914242 0.405168i \(-0.867213\pi\)
0.914242 0.405168i \(-0.132787\pi\)
\(602\) 0 0
\(603\) 11.4083 0.464583
\(604\) 0 0
\(605\) 1.18547 0.0481960
\(606\) 0 0
\(607\) 27.3489 1.11006 0.555029 0.831831i \(-0.312707\pi\)
0.555029 + 0.831831i \(0.312707\pi\)
\(608\) 0 0
\(609\) 9.49545 3.87650i 0.384775 0.157084i
\(610\) 0 0
\(611\) 34.3303 1.38886
\(612\) 0 0
\(613\) 21.1652i 0.854852i 0.904050 + 0.427426i \(0.140580\pi\)
−0.904050 + 0.427426i \(0.859420\pi\)
\(614\) 0 0
\(615\) −0.539382 −0.0217500
\(616\) 0 0
\(617\) −2.41742 −0.0973218 −0.0486609 0.998815i \(-0.515495\pi\)
−0.0486609 + 0.998815i \(0.515495\pi\)
\(618\) 0 0
\(619\) 29.2872i 1.17715i −0.808442 0.588575i \(-0.799689\pi\)
0.808442 0.588575i \(-0.200311\pi\)
\(620\) 0 0
\(621\) 20.1353 0.808002
\(622\) 0 0
\(623\) 7.48331 + 18.3303i 0.299813 + 0.734388i
\(624\) 0 0
\(625\) 18.9129 0.756515
\(626\) 0 0
\(627\) −15.8258 −0.632020
\(628\) 0 0
\(629\) −12.3823 −0.493714
\(630\) 0 0
\(631\) 3.66970i 0.146088i 0.997329 + 0.0730442i \(0.0232714\pi\)
−0.997329 + 0.0730442i \(0.976729\pi\)
\(632\) 0 0
\(633\) 16.7419i 0.665430i
\(634\) 0 0
\(635\) 11.3598i 0.450801i
\(636\) 0 0
\(637\) 27.7253 27.1652i 1.09852 1.07632i
\(638\) 0 0
\(639\) 23.6697i 0.936359i
\(640\) 0 0
\(641\) −20.7477 −0.819486 −0.409743 0.912201i \(-0.634382\pi\)
−0.409743 + 0.912201i \(0.634382\pi\)
\(642\) 0 0
\(643\) 24.3882i 0.961777i −0.876782 0.480888i \(-0.840314\pi\)
0.876782 0.480888i \(-0.159686\pi\)
\(644\) 0 0
\(645\) 1.49545i 0.0588835i
\(646\) 0 0
\(647\) −43.0683 −1.69319 −0.846596 0.532237i \(-0.821352\pi\)
−0.846596 + 0.532237i \(0.821352\pi\)
\(648\) 0 0
\(649\) 19.8656i 0.779793i
\(650\) 0 0
\(651\) −4.00000 9.79796i −0.156772 0.384012i
\(652\) 0 0
\(653\) 27.4955i 1.07598i −0.842951 0.537990i \(-0.819184\pi\)
0.842951 0.537990i \(-0.180816\pi\)
\(654\) 0 0
\(655\) 5.07803i 0.198415i
\(656\) 0 0
\(657\) 19.3262i 0.753988i
\(658\) 0 0
\(659\) −17.2523 −0.672053 −0.336027 0.941853i \(-0.609083\pi\)
−0.336027 + 0.941853i \(0.609083\pi\)
\(660\) 0 0
\(661\) 47.8606 1.86156 0.930781 0.365577i \(-0.119128\pi\)
0.930781 + 0.365577i \(0.119128\pi\)
\(662\) 0 0
\(663\) 22.1803 0.861410
\(664\) 0 0
\(665\) 4.41742 + 10.8204i 0.171300 + 0.419598i
\(666\) 0 0
\(667\) 33.4955 1.29695
\(668\) 0 0
\(669\) 1.66970i 0.0645542i
\(670\) 0 0
\(671\) −37.4166 −1.44445
\(672\) 0 0
\(673\) 36.3303 1.40043 0.700215 0.713932i \(-0.253087\pi\)
0.700215 + 0.713932i \(0.253087\pi\)
\(674\) 0 0
\(675\) 16.5285i 0.636182i
\(676\) 0 0
\(677\) −11.7362 −0.451059 −0.225530 0.974236i \(-0.572411\pi\)
−0.225530 + 0.974236i \(0.572411\pi\)
\(678\) 0 0
\(679\) 18.5734 + 45.4955i 0.712783 + 1.74596i
\(680\) 0 0
\(681\) −8.24318 −0.315879
\(682\) 0 0
\(683\) −0.417424 −0.0159723 −0.00798615 0.999968i \(-0.502542\pi\)
−0.00798615 + 0.999968i \(0.502542\pi\)
\(684\) 0 0
\(685\) 0.752785 0.0287625
\(686\) 0 0
\(687\) 1.07803i 0.0411294i
\(688\) 0 0
\(689\) 50.8213i 1.93614i
\(690\) 0 0
\(691\) 25.4107i 0.966668i 0.875436 + 0.483334i \(0.160574\pi\)
−0.875436 + 0.483334i \(0.839426\pi\)
\(692\) 0 0
\(693\) 22.6633 9.25227i 0.860909 0.351465i
\(694\) 0 0
\(695\) 12.4174i 0.471020i
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) 0.752785i 0.0284730i
\(700\) 0 0
\(701\) 39.4955i 1.49172i 0.666101 + 0.745861i \(0.267962\pi\)
−0.666101 + 0.745861i \(0.732038\pi\)
\(702\) 0 0
\(703\) −13.6745 −0.515742
\(704\) 0 0
\(705\) 2.58434i 0.0973317i
\(706\) 0 0
\(707\) −7.25227 + 2.96073i −0.272750 + 0.111350i
\(708\) 0 0
\(709\) 5.16515i 0.193981i −0.995285 0.0969907i \(-0.969078\pi\)
0.995285 0.0969907i \(-0.0309217\pi\)
\(710\) 0 0
\(711\) 5.16515i 0.193708i
\(712\) 0 0
\(713\) 34.5625i 1.29438i
\(714\) 0 0
\(715\) −12.8348 −0.479996
\(716\) 0 0
\(717\) −16.5285 −0.617268
\(718\) 0 0
\(719\) −40.7537 −1.51986 −0.759928 0.650007i \(-0.774766\pi\)
−0.759928 + 0.650007i \(0.774766\pi\)
\(720\) 0 0
\(721\) 3.16515 1.29217i 0.117876 0.0481228i
\(722\) 0 0
\(723\) 4.00000 0.148762
\(724\) 0 0
\(725\) 27.4955i 1.02116i
\(726\) 0 0
\(727\) 31.2254 1.15809 0.579043 0.815297i \(-0.303426\pi\)
0.579043 + 0.815297i \(0.303426\pi\)
\(728\) 0 0
\(729\) 7.00000 0.259259
\(730\) 0 0
\(731\) 22.1803i 0.820366i
\(732\) 0 0
\(733\) −7.10692 −0.262500 −0.131250 0.991349i \(-0.541899\pi\)
−0.131250 + 0.991349i \(0.541899\pi\)
\(734\) 0 0
\(735\) 2.04495 + 2.08712i 0.0754292 + 0.0769846i
\(736\) 0 0
\(737\) −15.8258 −0.582949
\(738\) 0 0
\(739\) −26.7477 −0.983931 −0.491966 0.870615i \(-0.663721\pi\)
−0.491966 + 0.870615i \(0.663721\pi\)
\(740\) 0 0
\(741\) 24.4949 0.899843
\(742\) 0 0
\(743\) 40.7477i 1.49489i 0.664324 + 0.747445i \(0.268719\pi\)
−0.664324 + 0.747445i \(0.731281\pi\)
\(744\) 0 0
\(745\) 0.752785i 0.0275799i
\(746\) 0 0
\(747\) 1.66856i 0.0610494i
\(748\) 0 0
\(749\) −38.1694 + 15.5826i −1.39468 + 0.569375i
\(750\) 0 0
\(751\) 11.0780i 0.404243i −0.979360 0.202122i \(-0.935216\pi\)
0.979360 0.202122i \(-0.0647836\pi\)
\(752\) 0 0
\(753\) −6.74773 −0.245901
\(754\) 0 0
\(755\) 1.56186i 0.0568419i
\(756\) 0 0
\(757\) 30.8348i 1.12071i 0.828252 + 0.560356i \(0.189336\pi\)
−0.828252 + 0.560356i \(0.810664\pi\)
\(758\) 0 0
\(759\) −12.9217 −0.469027
\(760\) 0 0
\(761\) 28.6411i 1.03824i −0.854702 0.519119i \(-0.826260\pi\)
0.854702 0.519119i \(-0.173740\pi\)
\(762\) 0 0
\(763\) −5.16515 12.6520i −0.186991 0.458033i
\(764\) 0 0
\(765\) 10.3303i 0.373493i
\(766\) 0 0
\(767\) 30.7477i 1.11024i
\(768\) 0 0
\(769\) 35.5850i 1.28323i 0.767027 + 0.641614i \(0.221735\pi\)
−0.767027 + 0.641614i \(0.778265\pi\)
\(770\) 0 0
\(771\) −6.33030 −0.227980
\(772\) 0 0
\(773\) 10.4440 0.375646 0.187823 0.982203i \(-0.439857\pi\)
0.187823 + 0.982203i \(0.439857\pi\)
\(774\) 0 0
\(775\) −28.3714 −1.01913
\(776\) 0 0
\(777\) −3.16515 + 1.29217i −0.113549 + 0.0463563i
\(778\) 0 0
\(779\) 8.83485 0.316541
\(780\) 0 0
\(781\) 32.8348i 1.17492i
\(782\) 0 0
\(783\) −21.6409 −0.773382
\(784\) 0 0
\(785\) 1.91288 0.0682736
\(786\) 0 0
\(787\) 8.12940i 0.289782i 0.989448 + 0.144891i \(0.0462831\pi\)
−0.989448 + 0.144891i \(0.953717\pi\)
\(788\) 0 0
\(789\) −4.41589 −0.157210
\(790\) 0 0
\(791\) 33.2704 13.5826i 1.18296 0.482941i
\(792\) 0 0
\(793\) 57.9129 2.05655
\(794\) 0 0
\(795\) −3.82576 −0.135686
\(796\) 0 0
\(797\) 5.54506 0.196416 0.0982081 0.995166i \(-0.468689\pi\)
0.0982081 + 0.995166i \(0.468689\pi\)
\(798\) 0 0
\(799\) 38.3303i 1.35603i
\(800\) 0 0
\(801\) 19.3262i 0.682859i
\(802\) 0 0
\(803\) 26.8095i 0.946088i
\(804\) 0 0
\(805\) 3.60681 + 8.83485i 0.127123 + 0.311387i
\(806\) 0 0
\(807\) 15.4083i 0.542399i
\(808\) 0 0
\(809\) −1.58258 −0.0556404 −0.0278202 0.999613i \(-0.508857\pi\)
−0.0278202 + 0.999613i \(0.508857\pi\)
\(810\) 0 0
\(811\) 1.93825i 0.0680612i −0.999421 0.0340306i \(-0.989166\pi\)
0.999421 0.0340306i \(-0.0108344\pi\)
\(812\) 0 0
\(813\) 11.3394i 0.397690i
\(814\) 0 0
\(815\) 2.31464 0.0810784
\(816\) 0 0
\(817\) 24.4949i 0.856968i
\(818\) 0 0
\(819\) −35.0780 + 14.3205i −1.22573 + 0.500400i
\(820\) 0 0
\(821\) 54.6606i 1.90767i 0.300334 + 0.953834i \(0.402902\pi\)
−0.300334 + 0.953834i \(0.597098\pi\)
\(822\) 0 0
\(823\) 18.8348i 0.656542i −0.944584 0.328271i \(-0.893534\pi\)
0.944584 0.328271i \(-0.106466\pi\)
\(824\) 0 0
\(825\) 10.6070i 0.369289i
\(826\) 0 0
\(827\) −25.9129 −0.901079 −0.450540 0.892756i \(-0.648768\pi\)
−0.450540 + 0.892756i \(0.648768\pi\)
\(828\) 0 0
\(829\) −3.98320 −0.138342 −0.0691712 0.997605i \(-0.522035\pi\)
−0.0691712 + 0.997605i \(0.522035\pi\)
\(830\) 0 0
\(831\) −5.38207 −0.186702
\(832\) 0 0
\(833\) −30.3303 30.9557i −1.05088 1.07255i
\(834\) 0 0
\(835\) 7.16515 0.247960
\(836\) 0 0
\(837\) 22.3303i 0.771848i
\(838\) 0 0
\(839\) 20.8881 0.721137 0.360568 0.932733i \(-0.382583\pi\)
0.360568 + 0.932733i \(0.382583\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 6.46084i 0.222523i
\(844\) 0 0
\(845\) 11.4665 0.394460
\(846\) 0 0
\(847\) −4.49444 + 1.83485i −0.154431 + 0.0630461i
\(848\) 0 0
\(849\) −13.0780 −0.448837
\(850\) 0 0
\(851\) −11.1652 −0.382736
\(852\) 0 0
\(853\) 47.3212 1.62025 0.810124 0.586258i \(-0.199400\pi\)
0.810124 + 0.586258i \(0.199400\pi\)
\(854\) 0 0
\(855\) 11.4083i 0.390157i
\(856\) 0 0
\(857\) 3.33712i 0.113994i −0.998374 0.0569969i \(-0.981847\pi\)
0.998374 0.0569969i \(-0.0181525\pi\)
\(858\) 0 0
\(859\) 13.7812i 0.470207i −0.971970 0.235104i \(-0.924457\pi\)
0.971970 0.235104i \(-0.0755429\pi\)
\(860\) 0 0
\(861\) 2.04495 0.834849i 0.0696918 0.0284516i
\(862\) 0 0
\(863\) 14.8348i 0.504984i −0.967599 0.252492i \(-0.918750\pi\)
0.967599 0.252492i \(-0.0812502\pi\)
\(864\) 0 0
\(865\) −10.7477 −0.365434
\(866\) 0 0
\(867\) 13.7812i 0.468033i
\(868\) 0 0
\(869\) 7.16515i 0.243061i
\(870\) 0 0
\(871\) 24.4949 0.829978
\(872\) 0 0
\(873\) 47.9673i 1.62345i
\(874\) 0 0
\(875\) 15.1652 6.19115i 0.512676 0.209299i
\(876\) 0 0
\(877\) 32.3303i 1.09172i 0.837877 + 0.545858i \(0.183796\pi\)
−0.837877 + 0.545858i \(0.816204\pi\)
\(878\) 0 0
\(879\) 1.07803i 0.0363610i
\(880\) 0 0
\(881\) 35.1019i 1.18261i −0.806447 0.591307i \(-0.798612\pi\)
0.806447 0.591307i \(-0.201388\pi\)
\(882\) 0 0
\(883\) 13.0780 0.440111 0.220055 0.975487i \(-0.429376\pi\)
0.220055 + 0.975487i \(0.429376\pi\)
\(884\) 0 0
\(885\) −2.31464 −0.0778059
\(886\) 0 0
\(887\) −28.6411 −0.961674 −0.480837 0.876810i \(-0.659667\pi\)
−0.480837 + 0.876810i \(0.659667\pi\)
\(888\) 0 0
\(889\) −17.5826 43.0683i −0.589701 1.44447i
\(890\) 0 0
\(891\) −19.4083 −0.650204
\(892\) 0 0
\(893\) 42.3303i 1.41653i
\(894\) 0 0
\(895\) 4.89898 0.163755
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) 37.1469i 1.23892i
\(900\) 0 0
\(901\) 56.7428 1.89038
\(902\) 0 0
\(903\) −2.31464 5.66970i −0.0770265 0.188676i
\(904\) 0 0
\(905\) −2.92197 −0.0971296
\(906\) 0 0
\(907\) −5.91288 −0.196334 −0.0981670 0.995170i \(-0.531298\pi\)
−0.0981670 + 0.995170i \(0.531298\pi\)
\(908\) 0 0
\(909\) 7.64630 0.253612
\(910\) 0 0
\(911\) 14.4174i 0.477671i −0.971060 0.238835i \(-0.923234\pi\)
0.971060 0.238835i \(-0.0767656\pi\)
\(912\) 0 0
\(913\) 2.31464i 0.0766035i
\(914\) 0 0
\(915\) 4.35960i 0.144124i
\(916\) 0 0
\(917\) −7.85971 19.2523i −0.259550 0.635766i
\(918\) 0 0
\(919\) 13.1652i 0.434278i 0.976141 + 0.217139i \(0.0696725\pi\)
−0.976141 + 0.217139i \(0.930327\pi\)
\(920\) 0 0
\(921\) 15.5826 0.513463
\(922\) 0 0
\(923\) 50.8213i 1.67280i
\(924\) 0 0
\(925\) 9.16515i 0.301348i
\(926\) 0 0
\(927\) −3.33712 −0.109605
\(928\) 0 0
\(929\) 33.0007i 1.08272i 0.840792 + 0.541359i \(0.182090\pi\)
−0.840792 + 0.541359i \(0.817910\pi\)
\(930\) 0 0
\(931\) −33.4955 34.1862i −1.09777 1.12041i
\(932\) 0 0
\(933\) 0.174243i 0.00570446i
\(934\) 0 0
\(935\) 14.3303i 0.468651i
\(936\) 0 0
\(937\) 6.94393i 0.226848i −0.993547 0.113424i \(-0.963818\pi\)
0.993547 0.113424i \(-0.0361819\pi\)
\(938\) 0 0
\(939\) −7.16515 −0.233826
\(940\) 0 0
\(941\) 25.6804 0.837156 0.418578 0.908181i \(-0.362529\pi\)
0.418578 + 0.908181i \(0.362529\pi\)
\(942\) 0 0
\(943\) 7.21362 0.234908
\(944\) 0 0
\(945\) −2.33030 5.70805i −0.0758048 0.185683i
\(946\) 0 0
\(947\) 46.7477 1.51910 0.759549 0.650451i \(-0.225420\pi\)
0.759549 + 0.650451i \(0.225420\pi\)
\(948\) 0 0
\(949\) 41.4955i 1.34700i
\(950\) 0 0
\(951\) −5.38207 −0.174526
\(952\) 0 0
\(953\) −8.33030 −0.269845 −0.134922 0.990856i \(-0.543079\pi\)
−0.134922 + 0.990856i \(0.543079\pi\)
\(954\) 0 0
\(955\) 13.6745i 0.442495i
\(956\) 0 0
\(957\) 13.8879 0.448931
\(958\) 0 0
\(959\) −2.85403 + 1.16515i −0.0921613 + 0.0376247i
\(960\) 0 0
\(961\) 7.33030 0.236461
\(962\) 0 0
\(963\) 40.2432 1.29682
\(964\) 0 0
\(965\) 6.19115 0.199300
\(966\) 0 0
\(967\) 47.0780i 1.51393i 0.653457 + 0.756964i \(0.273318\pi\)
−0.653457 + 0.756964i \(0.726682\pi\)
\(968\) 0 0
\(969\) 27.3489i 0.878574i
\(970\) 0 0
\(971\) 37.5233i 1.20418i 0.798429 + 0.602090i \(0.205665\pi\)
−0.798429 + 0.602090i \(0.794335\pi\)
\(972\) 0 0
\(973\) 19.2195 + 47.0780i 0.616150 + 1.50925i
\(974\) 0 0
\(975\) 16.4174i 0.525778i
\(976\) 0 0
\(977\) 31.4955 1.00763 0.503814 0.863812i \(-0.331930\pi\)
0.503814 + 0.863812i \(0.331930\pi\)
\(978\) 0 0
\(979\) 26.8095i 0.856837i
\(980\) 0 0
\(981\) 13.3394i 0.425894i
\(982\) 0 0
\(983\) 48.2370 1.53852 0.769261 0.638935i \(-0.220625\pi\)
0.769261 + 0.638935i \(0.220625\pi\)
\(984\) 0 0
\(985\) 11.6295i 0.370547i
\(986\) 0 0
\(987\) −4.00000 9.79796i −0.127321 0.311872i
\(988\) 0 0
\(989\) 20.0000i 0.635963i
\(990\) 0 0
\(991\) 16.3303i 0.518749i −0.965777 0.259375i \(-0.916484\pi\)
0.965777 0.259375i \(-0.0835164\pi\)
\(992\) 0 0
\(993\) 15.7757i 0.500627i
\(994\) 0 0
\(995\) 0.834849 0.0264665
\(996\) 0 0
\(997\) 7.85971 0.248919 0.124460 0.992225i \(-0.460280\pi\)
0.124460 + 0.992225i \(0.460280\pi\)
\(998\) 0 0
\(999\) 7.21362 0.228229
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 1792.2.e.e.895.6 8
4.3 odd 2 1792.2.e.d.895.4 8
7.6 odd 2 inner 1792.2.e.e.895.3 8
8.3 odd 2 inner 1792.2.e.e.895.5 8
8.5 even 2 1792.2.e.d.895.3 8
16.3 odd 4 896.2.f.d.895.5 yes 8
16.5 even 4 896.2.f.c.895.6 yes 8
16.11 odd 4 896.2.f.c.895.4 yes 8
16.13 even 4 896.2.f.d.895.3 yes 8
28.27 even 2 1792.2.e.d.895.5 8
56.13 odd 2 1792.2.e.d.895.6 8
56.27 even 2 inner 1792.2.e.e.895.4 8
112.13 odd 4 896.2.f.d.895.6 yes 8
112.27 even 4 896.2.f.c.895.5 yes 8
112.69 odd 4 896.2.f.c.895.3 8
112.83 even 4 896.2.f.d.895.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.f.c.895.3 8 112.69 odd 4
896.2.f.c.895.4 yes 8 16.11 odd 4
896.2.f.c.895.5 yes 8 112.27 even 4
896.2.f.c.895.6 yes 8 16.5 even 4
896.2.f.d.895.3 yes 8 16.13 even 4
896.2.f.d.895.4 yes 8 112.83 even 4
896.2.f.d.895.5 yes 8 16.3 odd 4
896.2.f.d.895.6 yes 8 112.13 odd 4
1792.2.e.d.895.3 8 8.5 even 2
1792.2.e.d.895.4 8 4.3 odd 2
1792.2.e.d.895.5 8 28.27 even 2
1792.2.e.d.895.6 8 56.13 odd 2
1792.2.e.e.895.3 8 7.6 odd 2 inner
1792.2.e.e.895.4 8 56.27 even 2 inner
1792.2.e.e.895.5 8 8.3 odd 2 inner
1792.2.e.e.895.6 8 1.1 even 1 trivial