Properties

Label 1792.2.f.j.1791.1
Level $1792$
Weight $2$
Character 1792.1791
Analytic conductor $14.309$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,2,Mod(1791,1792)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.1791");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.f (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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 448)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1791.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1791
Dual form 1792.2.f.j.1791.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} -1.41421i q^{5} +(-2.44949 + 1.00000i) q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} -1.41421i q^{5} +(-2.44949 + 1.00000i) q^{7} -1.00000 q^{9} +3.46410i q^{11} +4.24264i q^{13} +2.00000i q^{15} -4.89898i q^{17} -4.24264 q^{19} +(3.46410 - 1.41421i) q^{21} +3.00000 q^{25} +5.65685 q^{27} +4.89898 q^{31} -4.89898i q^{33} +(1.41421 + 3.46410i) q^{35} +6.92820 q^{37} -6.00000i q^{39} -9.79796i q^{41} +3.46410i q^{43} +1.41421i q^{45} -4.89898 q^{47} +(5.00000 - 4.89898i) q^{49} +6.92820i q^{51} -13.8564 q^{53} +4.89898 q^{55} +6.00000 q^{57} -7.07107 q^{59} -4.24264i q^{61} +(2.44949 - 1.00000i) q^{63} +6.00000 q^{65} -10.3923i q^{67} +6.00000i q^{71} -4.89898i q^{73} -4.24264 q^{75} +(-3.46410 - 8.48528i) q^{77} -14.0000i q^{79} -5.00000 q^{81} +9.89949 q^{83} -6.92820 q^{85} -14.6969i q^{89} +(-4.24264 - 10.3923i) q^{91} -6.92820 q^{93} +6.00000i q^{95} -4.89898i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 24 q^{25} + 40 q^{49} + 48 q^{57} + 48 q^{65} - 40 q^{81}+O(q^{100}) \) 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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 1.41421i 0.632456i −0.948683 0.316228i \(-0.897584\pi\)
0.948683 0.316228i \(-0.102416\pi\)
\(6\) 0 0
\(7\) −2.44949 + 1.00000i −0.925820 + 0.377964i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 4.24264i 1.17670i 0.808608 + 0.588348i \(0.200222\pi\)
−0.808608 + 0.588348i \(0.799778\pi\)
\(14\) 0 0
\(15\) 2.00000i 0.516398i
\(16\) 0 0
\(17\) 4.89898i 1.18818i −0.804400 0.594089i \(-0.797513\pi\)
0.804400 0.594089i \(-0.202487\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) 3.46410 1.41421i 0.755929 0.308607i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 4.89898 0.879883 0.439941 0.898027i \(-0.354999\pi\)
0.439941 + 0.898027i \(0.354999\pi\)
\(32\) 0 0
\(33\) 4.89898i 0.852803i
\(34\) 0 0
\(35\) 1.41421 + 3.46410i 0.239046 + 0.585540i
\(36\) 0 0
\(37\) 6.92820 1.13899 0.569495 0.821995i \(-0.307139\pi\)
0.569495 + 0.821995i \(0.307139\pi\)
\(38\) 0 0
\(39\) 6.00000i 0.960769i
\(40\) 0 0
\(41\) 9.79796i 1.53018i −0.643921 0.765092i \(-0.722693\pi\)
0.643921 0.765092i \(-0.277307\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 0 0
\(45\) 1.41421i 0.210819i
\(46\) 0 0
\(47\) −4.89898 −0.714590 −0.357295 0.933992i \(-0.616301\pi\)
−0.357295 + 0.933992i \(0.616301\pi\)
\(48\) 0 0
\(49\) 5.00000 4.89898i 0.714286 0.699854i
\(50\) 0 0
\(51\) 6.92820i 0.970143i
\(52\) 0 0
\(53\) −13.8564 −1.90332 −0.951662 0.307148i \(-0.900625\pi\)
−0.951662 + 0.307148i \(0.900625\pi\)
\(54\) 0 0
\(55\) 4.89898 0.660578
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) −7.07107 −0.920575 −0.460287 0.887770i \(-0.652254\pi\)
−0.460287 + 0.887770i \(0.652254\pi\)
\(60\) 0 0
\(61\) 4.24264i 0.543214i −0.962408 0.271607i \(-0.912445\pi\)
0.962408 0.271607i \(-0.0875552\pi\)
\(62\) 0 0
\(63\) 2.44949 1.00000i 0.308607 0.125988i
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 10.3923i 1.26962i −0.772667 0.634811i \(-0.781078\pi\)
0.772667 0.634811i \(-0.218922\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i −0.958023 0.286691i \(-0.907445\pi\)
0.958023 0.286691i \(-0.0925553\pi\)
\(74\) 0 0
\(75\) −4.24264 −0.489898
\(76\) 0 0
\(77\) −3.46410 8.48528i −0.394771 0.966988i
\(78\) 0 0
\(79\) 14.0000i 1.57512i −0.616236 0.787562i \(-0.711343\pi\)
0.616236 0.787562i \(-0.288657\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) −6.92820 −0.751469
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.6969i 1.55787i −0.627103 0.778936i \(-0.715760\pi\)
0.627103 0.778936i \(-0.284240\pi\)
\(90\) 0 0
\(91\) −4.24264 10.3923i −0.444750 1.08941i
\(92\) 0 0
\(93\) −6.92820 −0.718421
\(94\) 0 0
\(95\) 6.00000i 0.615587i
\(96\) 0 0
\(97\) 4.89898i 0.497416i −0.968579 0.248708i \(-0.919994\pi\)
0.968579 0.248708i \(-0.0800060\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 18.3848i 1.82935i 0.404186 + 0.914677i \(0.367555\pi\)
−0.404186 + 0.914677i \(0.632445\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −2.00000 4.89898i −0.195180 0.478091i
\(106\) 0 0
\(107\) 17.3205i 1.67444i −0.546869 0.837218i \(-0.684180\pi\)
0.546869 0.837218i \(-0.315820\pi\)
\(108\) 0 0
\(109\) 13.8564 1.32720 0.663602 0.748086i \(-0.269027\pi\)
0.663602 + 0.748086i \(0.269027\pi\)
\(110\) 0 0
\(111\) −9.79796 −0.929981
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.24264i 0.392232i
\(118\) 0 0
\(119\) 4.89898 + 12.0000i 0.449089 + 1.10004i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 13.8564i 1.24939i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) 4.89898i 0.431331i
\(130\) 0 0
\(131\) 15.5563 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(132\) 0 0
\(133\) 10.3923 4.24264i 0.901127 0.367884i
\(134\) 0 0
\(135\) 8.00000i 0.688530i
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 4.24264 0.359856 0.179928 0.983680i \(-0.442414\pi\)
0.179928 + 0.983680i \(0.442414\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 0 0
\(143\) −14.6969 −1.22902
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −7.07107 + 6.92820i −0.583212 + 0.571429i
\(148\) 0 0
\(149\) 6.92820 0.567581 0.283790 0.958886i \(-0.408408\pi\)
0.283790 + 0.958886i \(0.408408\pi\)
\(150\) 0 0
\(151\) 8.00000i 0.651031i −0.945537 0.325515i \(-0.894462\pi\)
0.945537 0.325515i \(-0.105538\pi\)
\(152\) 0 0
\(153\) 4.89898i 0.396059i
\(154\) 0 0
\(155\) 6.92820i 0.556487i
\(156\) 0 0
\(157\) 21.2132i 1.69300i 0.532390 + 0.846499i \(0.321294\pi\)
−0.532390 + 0.846499i \(0.678706\pi\)
\(158\) 0 0
\(159\) 19.5959 1.55406
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.46410i 0.271329i 0.990755 + 0.135665i \(0.0433170\pi\)
−0.990755 + 0.135665i \(0.956683\pi\)
\(164\) 0 0
\(165\) −6.92820 −0.539360
\(166\) 0 0
\(167\) −9.79796 −0.758189 −0.379094 0.925358i \(-0.623764\pi\)
−0.379094 + 0.925358i \(0.623764\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 4.24264 0.324443
\(172\) 0 0
\(173\) 7.07107i 0.537603i −0.963196 0.268802i \(-0.913372\pi\)
0.963196 0.268802i \(-0.0866276\pi\)
\(174\) 0 0
\(175\) −7.34847 + 3.00000i −0.555492 + 0.226779i
\(176\) 0 0
\(177\) 10.0000 0.751646
\(178\) 0 0
\(179\) 17.3205i 1.29460i −0.762237 0.647298i \(-0.775899\pi\)
0.762237 0.647298i \(-0.224101\pi\)
\(180\) 0 0
\(181\) 4.24264i 0.315353i 0.987491 + 0.157676i \(0.0504003\pi\)
−0.987491 + 0.157676i \(0.949600\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 9.79796i 0.720360i
\(186\) 0 0
\(187\) 16.9706 1.24101
\(188\) 0 0
\(189\) −13.8564 + 5.65685i −1.00791 + 0.411476i
\(190\) 0 0
\(191\) 18.0000i 1.30243i −0.758891 0.651217i \(-0.774259\pi\)
0.758891 0.651217i \(-0.225741\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −8.48528 −0.607644
\(196\) 0 0
\(197\) 6.92820 0.493614 0.246807 0.969065i \(-0.420619\pi\)
0.246807 + 0.969065i \(0.420619\pi\)
\(198\) 0 0
\(199\) −9.79796 −0.694559 −0.347279 0.937762i \(-0.612894\pi\)
−0.347279 + 0.937762i \(0.612894\pi\)
\(200\) 0 0
\(201\) 14.6969i 1.03664i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −13.8564 −0.967773
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.6969i 1.01661i
\(210\) 0 0
\(211\) 10.3923i 0.715436i −0.933830 0.357718i \(-0.883555\pi\)
0.933830 0.357718i \(-0.116445\pi\)
\(212\) 0 0
\(213\) 8.48528i 0.581402i
\(214\) 0 0
\(215\) 4.89898 0.334108
\(216\) 0 0
\(217\) −12.0000 + 4.89898i −0.814613 + 0.332564i
\(218\) 0 0
\(219\) 6.92820i 0.468165i
\(220\) 0 0
\(221\) 20.7846 1.39812
\(222\) 0 0
\(223\) 29.3939 1.96836 0.984180 0.177173i \(-0.0566951\pi\)
0.984180 + 0.177173i \(0.0566951\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) −9.89949 −0.657053 −0.328526 0.944495i \(-0.606552\pi\)
−0.328526 + 0.944495i \(0.606552\pi\)
\(228\) 0 0
\(229\) 21.2132i 1.40181i −0.713256 0.700904i \(-0.752780\pi\)
0.713256 0.700904i \(-0.247220\pi\)
\(230\) 0 0
\(231\) 4.89898 + 12.0000i 0.322329 + 0.789542i
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 6.92820i 0.451946i
\(236\) 0 0
\(237\) 19.7990i 1.28608i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 14.6969i 0.946713i 0.880871 + 0.473357i \(0.156958\pi\)
−0.880871 + 0.473357i \(0.843042\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) −6.92820 7.07107i −0.442627 0.451754i
\(246\) 0 0
\(247\) 18.0000i 1.14531i
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) −15.5563 −0.981908 −0.490954 0.871185i \(-0.663352\pi\)
−0.490954 + 0.871185i \(0.663352\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 9.79796 0.613572
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −16.9706 + 6.92820i −1.05450 + 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000i 1.10993i −0.831875 0.554964i \(-0.812732\pi\)
0.831875 0.554964i \(-0.187268\pi\)
\(264\) 0 0
\(265\) 19.5959i 1.20377i
\(266\) 0 0
\(267\) 20.7846i 1.27200i
\(268\) 0 0
\(269\) 9.89949i 0.603583i 0.953374 + 0.301791i \(0.0975846\pi\)
−0.953374 + 0.301791i \(0.902415\pi\)
\(270\) 0 0
\(271\) −19.5959 −1.19037 −0.595184 0.803590i \(-0.702921\pi\)
−0.595184 + 0.803590i \(0.702921\pi\)
\(272\) 0 0
\(273\) 6.00000 + 14.6969i 0.363137 + 0.889499i
\(274\) 0 0
\(275\) 10.3923i 0.626680i
\(276\) 0 0
\(277\) 13.8564 0.832551 0.416275 0.909239i \(-0.363335\pi\)
0.416275 + 0.909239i \(0.363335\pi\)
\(278\) 0 0
\(279\) −4.89898 −0.293294
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 12.7279 0.756596 0.378298 0.925684i \(-0.376509\pi\)
0.378298 + 0.925684i \(0.376509\pi\)
\(284\) 0 0
\(285\) 8.48528i 0.502625i
\(286\) 0 0
\(287\) 9.79796 + 24.0000i 0.578355 + 1.41668i
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 6.92820i 0.406138i
\(292\) 0 0
\(293\) 7.07107i 0.413096i 0.978436 + 0.206548i \(0.0662230\pi\)
−0.978436 + 0.206548i \(0.933777\pi\)
\(294\) 0 0
\(295\) 10.0000i 0.582223i
\(296\) 0 0
\(297\) 19.5959i 1.13707i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −3.46410 8.48528i −0.199667 0.489083i
\(302\) 0 0
\(303\) 26.0000i 1.49366i
\(304\) 0 0
\(305\) −6.00000 −0.343559
\(306\) 0 0
\(307\) 4.24264 0.242140 0.121070 0.992644i \(-0.461367\pi\)
0.121070 + 0.992644i \(0.461367\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.4949 1.38898 0.694489 0.719503i \(-0.255630\pi\)
0.694489 + 0.719503i \(0.255630\pi\)
\(312\) 0 0
\(313\) 19.5959i 1.10763i 0.832641 + 0.553813i \(0.186828\pi\)
−0.832641 + 0.553813i \(0.813172\pi\)
\(314\) 0 0
\(315\) −1.41421 3.46410i −0.0796819 0.195180i
\(316\) 0 0
\(317\) −6.92820 −0.389127 −0.194563 0.980890i \(-0.562329\pi\)
−0.194563 + 0.980890i \(0.562329\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 24.4949i 1.36717i
\(322\) 0 0
\(323\) 20.7846i 1.15649i
\(324\) 0 0
\(325\) 12.7279i 0.706018i
\(326\) 0 0
\(327\) −19.5959 −1.08366
\(328\) 0 0
\(329\) 12.0000 4.89898i 0.661581 0.270089i
\(330\) 0 0
\(331\) 3.46410i 0.190404i −0.995458 0.0952021i \(-0.969650\pi\)
0.995458 0.0952021i \(-0.0303497\pi\)
\(332\) 0 0
\(333\) −6.92820 −0.379663
\(334\) 0 0
\(335\) −14.6969 −0.802980
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.9706i 0.919007i
\(342\) 0 0
\(343\) −7.34847 + 17.0000i −0.396780 + 0.917914i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.46410i 0.185963i 0.995668 + 0.0929814i \(0.0296397\pi\)
−0.995668 + 0.0929814i \(0.970360\pi\)
\(348\) 0 0
\(349\) 4.24264i 0.227103i −0.993532 0.113552i \(-0.963777\pi\)
0.993532 0.113552i \(-0.0362227\pi\)
\(350\) 0 0
\(351\) 24.0000i 1.28103i
\(352\) 0 0
\(353\) 9.79796i 0.521493i 0.965407 + 0.260746i \(0.0839686\pi\)
−0.965407 + 0.260746i \(0.916031\pi\)
\(354\) 0 0
\(355\) 8.48528 0.450352
\(356\) 0 0
\(357\) −6.92820 16.9706i −0.366679 0.898177i
\(358\) 0 0
\(359\) 24.0000i 1.26667i −0.773877 0.633336i \(-0.781685\pi\)
0.773877 0.633336i \(-0.218315\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 1.41421 0.0742270
\(364\) 0 0
\(365\) −6.92820 −0.362639
\(366\) 0 0
\(367\) −9.79796 −0.511449 −0.255725 0.966750i \(-0.582314\pi\)
−0.255725 + 0.966750i \(0.582314\pi\)
\(368\) 0 0
\(369\) 9.79796i 0.510061i
\(370\) 0 0
\(371\) 33.9411 13.8564i 1.76214 0.719389i
\(372\) 0 0
\(373\) −13.8564 −0.717458 −0.358729 0.933442i \(-0.616790\pi\)
−0.358729 + 0.933442i \(0.616790\pi\)
\(374\) 0 0
\(375\) 16.0000i 0.826236i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.46410i 0.177939i −0.996034 0.0889695i \(-0.971643\pi\)
0.996034 0.0889695i \(-0.0283574\pi\)
\(380\) 0 0
\(381\) 22.6274i 1.15924i
\(382\) 0 0
\(383\) −4.89898 −0.250326 −0.125163 0.992136i \(-0.539945\pi\)
−0.125163 + 0.992136i \(0.539945\pi\)
\(384\) 0 0
\(385\) −12.0000 + 4.89898i −0.611577 + 0.249675i
\(386\) 0 0
\(387\) 3.46410i 0.176090i
\(388\) 0 0
\(389\) −6.92820 −0.351274 −0.175637 0.984455i \(-0.556198\pi\)
−0.175637 + 0.984455i \(0.556198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −22.0000 −1.10975
\(394\) 0 0
\(395\) −19.7990 −0.996195
\(396\) 0 0
\(397\) 4.24264i 0.212932i −0.994316 0.106466i \(-0.966046\pi\)
0.994316 0.106466i \(-0.0339535\pi\)
\(398\) 0 0
\(399\) −14.6969 + 6.00000i −0.735767 + 0.300376i
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 20.7846i 1.03536i
\(404\) 0 0
\(405\) 7.07107i 0.351364i
\(406\) 0 0
\(407\) 24.0000i 1.18964i
\(408\) 0 0
\(409\) 19.5959i 0.968956i 0.874804 + 0.484478i \(0.160990\pi\)
−0.874804 + 0.484478i \(0.839010\pi\)
\(410\) 0 0
\(411\) −25.4558 −1.25564
\(412\) 0 0
\(413\) 17.3205 7.07107i 0.852286 0.347945i
\(414\) 0 0
\(415\) 14.0000i 0.687233i
\(416\) 0 0
\(417\) −6.00000 −0.293821
\(418\) 0 0
\(419\) −7.07107 −0.345444 −0.172722 0.984971i \(-0.555256\pi\)
−0.172722 + 0.984971i \(0.555256\pi\)
\(420\) 0 0
\(421\) −20.7846 −1.01298 −0.506490 0.862246i \(-0.669057\pi\)
−0.506490 + 0.862246i \(0.669057\pi\)
\(422\) 0 0
\(423\) 4.89898 0.238197
\(424\) 0 0
\(425\) 14.6969i 0.712906i
\(426\) 0 0
\(427\) 4.24264 + 10.3923i 0.205316 + 0.502919i
\(428\) 0 0
\(429\) 20.7846 1.00349
\(430\) 0 0
\(431\) 24.0000i 1.15604i −0.816023 0.578020i \(-0.803826\pi\)
0.816023 0.578020i \(-0.196174\pi\)
\(432\) 0 0
\(433\) 34.2929i 1.64801i 0.566583 + 0.824005i \(0.308265\pi\)
−0.566583 + 0.824005i \(0.691735\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 24.4949 1.16908 0.584539 0.811366i \(-0.301275\pi\)
0.584539 + 0.811366i \(0.301275\pi\)
\(440\) 0 0
\(441\) −5.00000 + 4.89898i −0.238095 + 0.233285i
\(442\) 0 0
\(443\) 10.3923i 0.493753i 0.969047 + 0.246877i \(0.0794043\pi\)
−0.969047 + 0.246877i \(0.920596\pi\)
\(444\) 0 0
\(445\) −20.7846 −0.985285
\(446\) 0 0
\(447\) −9.79796 −0.463428
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 33.9411 1.59823
\(452\) 0 0
\(453\) 11.3137i 0.531564i
\(454\) 0 0
\(455\) −14.6969 + 6.00000i −0.689003 + 0.281284i
\(456\) 0 0
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) 0 0
\(459\) 27.7128i 1.29352i
\(460\) 0 0
\(461\) 15.5563i 0.724531i −0.932075 0.362266i \(-0.882003\pi\)
0.932075 0.362266i \(-0.117997\pi\)
\(462\) 0 0
\(463\) 14.0000i 0.650635i 0.945605 + 0.325318i \(0.105471\pi\)
−0.945605 + 0.325318i \(0.894529\pi\)
\(464\) 0 0
\(465\) 9.79796i 0.454369i
\(466\) 0 0
\(467\) −26.8701 −1.24340 −0.621699 0.783256i \(-0.713557\pi\)
−0.621699 + 0.783256i \(0.713557\pi\)
\(468\) 0 0
\(469\) 10.3923 + 25.4558i 0.479872 + 1.17544i
\(470\) 0 0
\(471\) 30.0000i 1.38233i
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −12.7279 −0.583997
\(476\) 0 0
\(477\) 13.8564 0.634441
\(478\) 0 0
\(479\) 34.2929 1.56688 0.783440 0.621467i \(-0.213463\pi\)
0.783440 + 0.621467i \(0.213463\pi\)
\(480\) 0 0
\(481\) 29.3939i 1.34025i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.92820 −0.314594
\(486\) 0 0
\(487\) 16.0000i 0.725029i 0.931978 + 0.362515i \(0.118082\pi\)
−0.931978 + 0.362515i \(0.881918\pi\)
\(488\) 0 0
\(489\) 4.89898i 0.221540i
\(490\) 0 0
\(491\) 38.1051i 1.71966i −0.510581 0.859830i \(-0.670569\pi\)
0.510581 0.859830i \(-0.329431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.89898 −0.220193
\(496\) 0 0
\(497\) −6.00000 14.6969i −0.269137 0.659248i
\(498\) 0 0
\(499\) 24.2487i 1.08552i −0.839887 0.542761i \(-0.817379\pi\)
0.839887 0.542761i \(-0.182621\pi\)
\(500\) 0 0
\(501\) 13.8564 0.619059
\(502\) 0 0
\(503\) 24.4949 1.09217 0.546087 0.837729i \(-0.316117\pi\)
0.546087 + 0.837729i \(0.316117\pi\)
\(504\) 0 0
\(505\) 26.0000 1.15698
\(506\) 0 0
\(507\) 7.07107 0.314037
\(508\) 0 0
\(509\) 24.0416i 1.06563i 0.846233 + 0.532813i \(0.178865\pi\)
−0.846233 + 0.532813i \(0.821135\pi\)
\(510\) 0 0
\(511\) 4.89898 + 12.0000i 0.216718 + 0.530849i
\(512\) 0 0
\(513\) −24.0000 −1.05963
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.9706i 0.746364i
\(518\) 0 0
\(519\) 10.0000i 0.438951i
\(520\) 0 0
\(521\) 9.79796i 0.429256i −0.976696 0.214628i \(-0.931146\pi\)
0.976696 0.214628i \(-0.0688540\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0 0
\(525\) 10.3923 4.24264i 0.453557 0.185164i
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 7.07107 0.306858
\(532\) 0 0
\(533\) 41.5692 1.80056
\(534\) 0 0
\(535\) −24.4949 −1.05901
\(536\) 0 0
\(537\) 24.4949i 1.05703i
\(538\) 0 0
\(539\) 16.9706 + 17.3205i 0.730974 + 0.746047i
\(540\) 0 0
\(541\) 34.6410 1.48933 0.744667 0.667436i \(-0.232608\pi\)
0.744667 + 0.667436i \(0.232608\pi\)
\(542\) 0 0
\(543\) 6.00000i 0.257485i
\(544\) 0 0
\(545\) 19.5959i 0.839397i
\(546\) 0 0
\(547\) 24.2487i 1.03680i −0.855138 0.518400i \(-0.826528\pi\)
0.855138 0.518400i \(-0.173472\pi\)
\(548\) 0 0
\(549\) 4.24264i 0.181071i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 14.0000 + 34.2929i 0.595341 + 1.45828i
\(554\) 0 0
\(555\) 13.8564i 0.588172i
\(556\) 0 0
\(557\) −27.7128 −1.17423 −0.587115 0.809504i \(-0.699736\pi\)
−0.587115 + 0.809504i \(0.699736\pi\)
\(558\) 0 0
\(559\) −14.6969 −0.621614
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) −15.5563 −0.655622 −0.327811 0.944743i \(-0.606311\pi\)
−0.327811 + 0.944743i \(0.606311\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.2474 5.00000i 0.514344 0.209980i
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 38.1051i 1.59465i −0.603550 0.797325i \(-0.706248\pi\)
0.603550 0.797325i \(-0.293752\pi\)
\(572\) 0 0
\(573\) 25.4558i 1.06343i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 5.65685 0.235091
\(580\) 0 0
\(581\) −24.2487 + 9.89949i −1.00601 + 0.410700i
\(582\) 0 0
\(583\) 48.0000i 1.98796i
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 32.5269 1.34253 0.671265 0.741218i \(-0.265751\pi\)
0.671265 + 0.741218i \(0.265751\pi\)
\(588\) 0 0
\(589\) −20.7846 −0.856415
\(590\) 0 0
\(591\) −9.79796 −0.403034
\(592\) 0 0
\(593\) 9.79796i 0.402354i −0.979555 0.201177i \(-0.935523\pi\)
0.979555 0.201177i \(-0.0644766\pi\)
\(594\) 0 0
\(595\) 16.9706 6.92820i 0.695725 0.284029i
\(596\) 0 0
\(597\) 13.8564 0.567105
\(598\) 0 0
\(599\) 30.0000i 1.22577i −0.790173 0.612883i \(-0.790010\pi\)
0.790173 0.612883i \(-0.209990\pi\)
\(600\) 0 0
\(601\) 4.89898i 0.199834i −0.994996 0.0999168i \(-0.968142\pi\)
0.994996 0.0999168i \(-0.0318577\pi\)
\(602\) 0 0
\(603\) 10.3923i 0.423207i
\(604\) 0 0
\(605\) 1.41421i 0.0574960i
\(606\) 0 0
\(607\) 29.3939 1.19306 0.596530 0.802591i \(-0.296546\pi\)
0.596530 + 0.802591i \(0.296546\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.7846i 0.840855i
\(612\) 0 0
\(613\) 20.7846 0.839482 0.419741 0.907644i \(-0.362121\pi\)
0.419741 + 0.907644i \(0.362121\pi\)
\(614\) 0 0
\(615\) 19.5959 0.790184
\(616\) 0 0
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) 12.7279 0.511578 0.255789 0.966733i \(-0.417665\pi\)
0.255789 + 0.966733i \(0.417665\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.6969 + 36.0000i 0.588820 + 1.44231i
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 20.7846i 0.830057i
\(628\) 0 0
\(629\) 33.9411i 1.35332i
\(630\) 0 0
\(631\) 34.0000i 1.35352i 0.736204 + 0.676759i \(0.236616\pi\)
−0.736204 + 0.676759i \(0.763384\pi\)
\(632\) 0 0
\(633\) 14.6969i 0.584151i
\(634\) 0 0
\(635\) 22.6274 0.897942
\(636\) 0 0
\(637\) 20.7846 + 21.2132i 0.823516 + 0.840498i
\(638\) 0 0
\(639\) 6.00000i 0.237356i
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) −38.1838 −1.50582 −0.752910 0.658123i \(-0.771351\pi\)
−0.752910 + 0.658123i \(0.771351\pi\)
\(644\) 0 0
\(645\) −6.92820 −0.272798
\(646\) 0 0
\(647\) −29.3939 −1.15559 −0.577796 0.816181i \(-0.696087\pi\)
−0.577796 + 0.816181i \(0.696087\pi\)
\(648\) 0 0
\(649\) 24.4949i 0.961509i
\(650\) 0 0
\(651\) 16.9706 6.92820i 0.665129 0.271538i
\(652\) 0 0
\(653\) 6.92820 0.271122 0.135561 0.990769i \(-0.456716\pi\)
0.135561 + 0.990769i \(0.456716\pi\)
\(654\) 0 0
\(655\) 22.0000i 0.859611i
\(656\) 0 0
\(657\) 4.89898i 0.191127i
\(658\) 0 0
\(659\) 24.2487i 0.944596i 0.881439 + 0.472298i \(0.156575\pi\)
−0.881439 + 0.472298i \(0.843425\pi\)
\(660\) 0 0
\(661\) 21.2132i 0.825098i −0.910935 0.412549i \(-0.864639\pi\)
0.910935 0.412549i \(-0.135361\pi\)
\(662\) 0 0
\(663\) −29.3939 −1.14156
\(664\) 0 0
\(665\) −6.00000 14.6969i −0.232670 0.569923i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −41.5692 −1.60716
\(670\) 0 0
\(671\) 14.6969 0.567369
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 16.9706 0.653197
\(676\) 0 0
\(677\) 41.0122i 1.57623i −0.615530 0.788113i \(-0.711058\pi\)
0.615530 0.788113i \(-0.288942\pi\)
\(678\) 0 0
\(679\) 4.89898 + 12.0000i 0.188006 + 0.460518i
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 10.3923i 0.397650i 0.980035 + 0.198825i \(0.0637126\pi\)
−0.980035 + 0.198825i \(0.936287\pi\)
\(684\) 0 0
\(685\) 25.4558i 0.972618i
\(686\) 0 0
\(687\) 30.0000i 1.14457i
\(688\) 0 0
\(689\) 58.7878i 2.23964i
\(690\) 0 0
\(691\) −12.7279 −0.484193 −0.242096 0.970252i \(-0.577835\pi\)
−0.242096 + 0.970252i \(0.577835\pi\)
\(692\) 0 0
\(693\) 3.46410 + 8.48528i 0.131590 + 0.322329i
\(694\) 0 0
\(695\) 6.00000i 0.227593i
\(696\) 0 0
\(697\) −48.0000 −1.81813
\(698\) 0 0
\(699\) 8.48528 0.320943
\(700\) 0 0
\(701\) −34.6410 −1.30837 −0.654187 0.756333i \(-0.726989\pi\)
−0.654187 + 0.756333i \(0.726989\pi\)
\(702\) 0 0
\(703\) −29.3939 −1.10861
\(704\) 0 0
\(705\) 9.79796i 0.369012i
\(706\) 0 0
\(707\) −18.3848 45.0333i −0.691431 1.69365i
\(708\) 0 0
\(709\) −27.7128 −1.04078 −0.520388 0.853930i \(-0.674213\pi\)
−0.520388 + 0.853930i \(0.674213\pi\)
\(710\) 0 0
\(711\) 14.0000i 0.525041i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 20.7846i 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.89898 0.182701 0.0913506 0.995819i \(-0.470882\pi\)
0.0913506 + 0.995819i \(0.470882\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 20.7846i 0.772988i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −19.5959 −0.726772 −0.363386 0.931639i \(-0.618379\pi\)
−0.363386 + 0.931639i \(0.618379\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) 12.7279i 0.470117i 0.971981 + 0.235058i \(0.0755281\pi\)
−0.971981 + 0.235058i \(0.924472\pi\)
\(734\) 0 0
\(735\) 9.79796 + 10.0000i 0.361403 + 0.368856i
\(736\) 0 0
\(737\) 36.0000 1.32608
\(738\) 0 0
\(739\) 31.1769i 1.14686i 0.819254 + 0.573431i \(0.194388\pi\)
−0.819254 + 0.573431i \(0.805612\pi\)
\(740\) 0 0
\(741\) 25.4558i 0.935144i
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) 9.79796i 0.358969i
\(746\) 0 0
\(747\) −9.89949 −0.362204
\(748\) 0 0
\(749\) 17.3205 + 42.4264i 0.632878 + 1.55023i
\(750\) 0 0
\(751\) 8.00000i 0.291924i −0.989290 0.145962i \(-0.953372\pi\)
0.989290 0.145962i \(-0.0466277\pi\)
\(752\) 0 0
\(753\) 22.0000 0.801725
\(754\) 0 0
\(755\) −11.3137 −0.411748
\(756\) 0 0
\(757\) −34.6410 −1.25905 −0.629525 0.776981i \(-0.716750\pi\)
−0.629525 + 0.776981i \(0.716750\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.5959i 0.710351i −0.934800 0.355176i \(-0.884421\pi\)
0.934800 0.355176i \(-0.115579\pi\)
\(762\) 0 0
\(763\) −33.9411 + 13.8564i −1.22875 + 0.501636i
\(764\) 0 0
\(765\) 6.92820 0.250490
\(766\) 0 0
\(767\) 30.0000i 1.08324i
\(768\) 0 0
\(769\) 24.4949i 0.883309i −0.897185 0.441654i \(-0.854392\pi\)
0.897185 0.441654i \(-0.145608\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 35.3553i 1.27164i −0.771836 0.635822i \(-0.780661\pi\)
0.771836 0.635822i \(-0.219339\pi\)
\(774\) 0 0
\(775\) 14.6969 0.527930
\(776\) 0 0
\(777\) 24.0000 9.79796i 0.860995 0.351500i
\(778\) 0 0
\(779\) 41.5692i 1.48937i
\(780\) 0 0
\(781\) −20.7846 −0.743732
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −21.2132 −0.756169 −0.378085 0.925771i \(-0.623417\pi\)
−0.378085 + 0.925771i \(0.623417\pi\)
\(788\) 0 0
\(789\) 25.4558i 0.906252i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 18.0000 0.639199
\(794\) 0 0
\(795\) 27.7128i 0.982872i
\(796\) 0 0
\(797\) 15.5563i 0.551034i 0.961296 + 0.275517i \(0.0888491\pi\)
−0.961296 + 0.275517i \(0.911151\pi\)
\(798\) 0 0
\(799\) 24.0000i 0.849059i
\(800\) 0 0
\(801\) 14.6969i 0.519291i
\(802\) 0 0
\(803\) 16.9706 0.598878
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000i 0.492823i
\(808\) 0 0
\(809\) 48.0000 1.68759 0.843795 0.536666i \(-0.180316\pi\)
0.843795 + 0.536666i \(0.180316\pi\)
\(810\) 0 0
\(811\) −55.1543 −1.93673 −0.968365 0.249537i \(-0.919722\pi\)
−0.968365 + 0.249537i \(0.919722\pi\)
\(812\) 0 0
\(813\) 27.7128 0.971931
\(814\) 0 0
\(815\) 4.89898 0.171604
\(816\) 0 0
\(817\) 14.6969i 0.514181i
\(818\) 0 0
\(819\) 4.24264 + 10.3923i 0.148250 + 0.363137i
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 22.0000i 0.766872i 0.923567 + 0.383436i \(0.125259\pi\)
−0.923567 + 0.383436i \(0.874741\pi\)
\(824\) 0 0
\(825\) 14.6969i 0.511682i
\(826\) 0 0
\(827\) 10.3923i 0.361376i −0.983540 0.180688i \(-0.942168\pi\)
0.983540 0.180688i \(-0.0578324\pi\)
\(828\) 0 0
\(829\) 55.1543i 1.91559i −0.287454 0.957795i \(-0.592809\pi\)
0.287454 0.957795i \(-0.407191\pi\)
\(830\) 0 0
\(831\) −19.5959 −0.679775
\(832\) 0 0
\(833\) −24.0000 24.4949i −0.831551 0.848698i
\(834\) 0 0
\(835\) 13.8564i 0.479521i
\(836\) 0 0
\(837\) 27.7128 0.957895
\(838\) 0 0
\(839\) −48.9898 −1.69132 −0.845658 0.533726i \(-0.820792\pi\)
−0.845658 + 0.533726i \(0.820792\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −8.48528 −0.292249
\(844\) 0 0
\(845\) 7.07107i 0.243252i
\(846\) 0 0
\(847\) 2.44949 1.00000i 0.0841655 0.0343604i
\(848\) 0 0
\(849\) −18.0000 −0.617758
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 4.24264i 0.145265i −0.997359 0.0726326i \(-0.976860\pi\)
0.997359 0.0726326i \(-0.0231401\pi\)
\(854\) 0 0
\(855\) 6.00000i 0.205196i
\(856\) 0 0
\(857\) 39.1918i 1.33877i −0.742917 0.669384i \(-0.766558\pi\)
0.742917 0.669384i \(-0.233442\pi\)
\(858\) 0 0
\(859\) 29.6985 1.01330 0.506650 0.862152i \(-0.330884\pi\)
0.506650 + 0.862152i \(0.330884\pi\)
\(860\) 0 0
\(861\) −13.8564 33.9411i −0.472225 1.15671i
\(862\) 0 0
\(863\) 6.00000i 0.204242i 0.994772 + 0.102121i \(0.0325630\pi\)
−0.994772 + 0.102121i \(0.967437\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 9.89949 0.336204
\(868\) 0 0
\(869\) 48.4974 1.64516
\(870\) 0 0
\(871\) 44.0908 1.49396
\(872\) 0 0
\(873\) 4.89898i 0.165805i
\(874\) 0 0
\(875\) 11.3137 + 27.7128i 0.382473 + 0.936864i
\(876\) 0 0
\(877\) −6.92820 −0.233949 −0.116974 0.993135i \(-0.537320\pi\)
−0.116974 + 0.993135i \(0.537320\pi\)
\(878\) 0 0
\(879\) 10.0000i 0.337292i
\(880\) 0 0
\(881\) 29.3939i 0.990305i −0.868806 0.495152i \(-0.835112\pi\)
0.868806 0.495152i \(-0.164888\pi\)
\(882\) 0 0
\(883\) 24.2487i 0.816034i 0.912974 + 0.408017i \(0.133780\pi\)
−0.912974 + 0.408017i \(0.866220\pi\)
\(884\) 0 0
\(885\) 14.1421i 0.475383i
\(886\) 0 0
\(887\) 19.5959 0.657967 0.328983 0.944336i \(-0.393294\pi\)
0.328983 + 0.944336i \(0.393294\pi\)
\(888\) 0 0
\(889\) −16.0000 39.1918i −0.536623 1.31445i
\(890\) 0 0
\(891\) 17.3205i 0.580259i
\(892\) 0 0
\(893\) 20.7846 0.695530
\(894\) 0 0
\(895\) −24.4949 −0.818774
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 67.8823i 2.26149i
\(902\) 0 0
\(903\) 4.89898 + 12.0000i 0.163028 + 0.399335i
\(904\) 0 0
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 17.3205i 0.575118i −0.957763 0.287559i \(-0.907156\pi\)
0.957763 0.287559i \(-0.0928437\pi\)
\(908\) 0 0
\(909\) 18.3848i 0.609785i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 34.2929i 1.13493i
\(914\) 0 0
\(915\) 8.48528 0.280515
\(916\) 0 0
\(917\) −38.1051 + 15.5563i −1.25834 + 0.513716i
\(918\) 0 0
\(919\) 34.0000i 1.12156i −0.827966 0.560778i \(-0.810502\pi\)
0.827966 0.560778i \(-0.189498\pi\)
\(920\) 0 0
\(921\) −6.00000 −0.197707
\(922\) 0 0
\(923\) −25.4558 −0.837889
\(924\) 0 0
\(925\) 20.7846 0.683394
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 53.8888i 1.76803i −0.467455 0.884017i \(-0.654829\pi\)
0.467455 0.884017i \(-0.345171\pi\)
\(930\) 0 0
\(931\) −21.2132 + 20.7846i −0.695235 + 0.681188i
\(932\) 0 0
\(933\) −34.6410 −1.13410
\(934\) 0 0
\(935\) 24.0000i 0.784884i
\(936\) 0 0
\(937\) 4.89898i 0.160043i −0.996793 0.0800213i \(-0.974501\pi\)
0.996793 0.0800213i \(-0.0254988\pi\)
\(938\) 0 0
\(939\) 27.7128i 0.904373i
\(940\) 0 0
\(941\) 1.41421i 0.0461020i −0.999734 0.0230510i \(-0.992662\pi\)
0.999734 0.0230510i \(-0.00733802\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 8.00000 + 19.5959i 0.260240 + 0.637455i
\(946\) 0 0
\(947\) 24.2487i 0.787977i 0.919115 + 0.393989i \(0.128905\pi\)
−0.919115 + 0.393989i \(0.871095\pi\)
\(948\) 0 0
\(949\) 20.7846 0.674697
\(950\) 0 0
\(951\) 9.79796 0.317721
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) −25.4558 −0.823732
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −44.0908 + 18.0000i −1.42377 + 0.581250i
\(960\) 0 0
\(961\) −7.00000 −0.225806
\(962\) 0 0
\(963\) 17.3205i 0.558146i
\(964\) 0 0
\(965\) 5.65685i 0.182101i
\(966\) 0 0
\(967\) 8.00000i 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 0 0
\(969\) 29.3939i 0.944267i
\(970\) 0 0
\(971\) −7.07107 −0.226921 −0.113461 0.993542i \(-0.536194\pi\)
−0.113461 + 0.993542i \(0.536194\pi\)
\(972\) 0 0
\(973\) −10.3923 + 4.24264i −0.333162 + 0.136013i
\(974\) 0 0
\(975\) 18.0000i 0.576461i
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 50.9117 1.62714
\(980\) 0 0
\(981\) −13.8564 −0.442401
\(982\) 0 0
\(983\) 9.79796 0.312506 0.156253 0.987717i \(-0.450058\pi\)
0.156253 + 0.987717i \(0.450058\pi\)
\(984\) 0 0
\(985\) 9.79796i 0.312189i
\(986\) 0 0
\(987\) −16.9706 + 6.92820i −0.540179 + 0.220527i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 2.00000i 0.0635321i 0.999495 + 0.0317660i \(0.0101131\pi\)
−0.999495 + 0.0317660i \(0.989887\pi\)
\(992\) 0 0
\(993\) 4.89898i 0.155464i
\(994\) 0 0
\(995\) 13.8564i 0.439278i
\(996\) 0 0
\(997\) 4.24264i 0.134366i 0.997741 + 0.0671829i \(0.0214011\pi\)
−0.997741 + 0.0671829i \(0.978599\pi\)
\(998\) 0 0
\(999\) 39.1918 1.23997
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.f.j.1791.1 8
4.3 odd 2 inner 1792.2.f.j.1791.6 8
7.6 odd 2 inner 1792.2.f.j.1791.8 8
8.3 odd 2 inner 1792.2.f.j.1791.4 8
8.5 even 2 inner 1792.2.f.j.1791.7 8
16.3 odd 4 448.2.e.b.223.5 yes 8
16.5 even 4 448.2.e.b.223.8 yes 8
16.11 odd 4 448.2.e.b.223.3 yes 8
16.13 even 4 448.2.e.b.223.2 yes 8
28.27 even 2 inner 1792.2.f.j.1791.3 8
48.5 odd 4 4032.2.p.f.1567.3 8
48.11 even 4 4032.2.p.f.1567.2 8
48.29 odd 4 4032.2.p.f.1567.7 8
48.35 even 4 4032.2.p.f.1567.6 8
56.13 odd 2 inner 1792.2.f.j.1791.2 8
56.27 even 2 inner 1792.2.f.j.1791.5 8
112.13 odd 4 448.2.e.b.223.7 yes 8
112.27 even 4 448.2.e.b.223.6 yes 8
112.69 odd 4 448.2.e.b.223.1 8
112.83 even 4 448.2.e.b.223.4 yes 8
336.83 odd 4 4032.2.p.f.1567.4 8
336.125 even 4 4032.2.p.f.1567.1 8
336.251 odd 4 4032.2.p.f.1567.8 8
336.293 even 4 4032.2.p.f.1567.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
448.2.e.b.223.1 8 112.69 odd 4
448.2.e.b.223.2 yes 8 16.13 even 4
448.2.e.b.223.3 yes 8 16.11 odd 4
448.2.e.b.223.4 yes 8 112.83 even 4
448.2.e.b.223.5 yes 8 16.3 odd 4
448.2.e.b.223.6 yes 8 112.27 even 4
448.2.e.b.223.7 yes 8 112.13 odd 4
448.2.e.b.223.8 yes 8 16.5 even 4
1792.2.f.j.1791.1 8 1.1 even 1 trivial
1792.2.f.j.1791.2 8 56.13 odd 2 inner
1792.2.f.j.1791.3 8 28.27 even 2 inner
1792.2.f.j.1791.4 8 8.3 odd 2 inner
1792.2.f.j.1791.5 8 56.27 even 2 inner
1792.2.f.j.1791.6 8 4.3 odd 2 inner
1792.2.f.j.1791.7 8 8.5 even 2 inner
1792.2.f.j.1791.8 8 7.6 odd 2 inner
4032.2.p.f.1567.1 8 336.125 even 4
4032.2.p.f.1567.2 8 48.11 even 4
4032.2.p.f.1567.3 8 48.5 odd 4
4032.2.p.f.1567.4 8 336.83 odd 4
4032.2.p.f.1567.5 8 336.293 even 4
4032.2.p.f.1567.6 8 48.35 even 4
4032.2.p.f.1567.7 8 48.29 odd 4
4032.2.p.f.1567.8 8 336.251 odd 4