Properties

Label 2592.2.i.bc.865.2
Level $2592$
Weight $2$
Character 2592.865
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
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] = [2592,2,Mod(865,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), 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: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.2
Root \(-0.651388 - 1.12824i\) of defining polynomial
Character \(\chi\) \(=\) 2592.865
Dual form 2592.2.i.bc.1729.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.80278 + 3.12250i) q^{5} +(-1.80278 + 3.12250i) q^{7} +O(q^{10})\) \(q+(1.80278 + 3.12250i) q^{5} +(-1.80278 + 3.12250i) q^{7} +(0.500000 - 0.866025i) q^{11} +(-2.00000 - 3.46410i) q^{13} -7.21110 q^{19} +(-3.00000 - 5.19615i) q^{23} +(-4.00000 + 6.92820i) q^{25} +(-3.60555 + 6.24500i) q^{29} +(1.80278 + 3.12250i) q^{31} -13.0000 q^{35} +10.0000 q^{37} +(-3.60555 - 6.24500i) q^{41} +(-3.60555 + 6.24500i) q^{43} +(-5.00000 + 8.66025i) q^{47} +(-3.00000 - 5.19615i) q^{49} +3.60555 q^{53} +3.60555 q^{55} +(-2.00000 - 3.46410i) q^{59} +(7.21110 - 12.4900i) q^{65} +(-3.60555 - 6.24500i) q^{67} -8.00000 q^{71} -3.00000 q^{73} +(1.80278 + 3.12250i) q^{77} +(7.21110 - 12.4900i) q^{79} +(4.50000 - 7.79423i) q^{83} -7.21110 q^{89} +14.4222 q^{91} +(-13.0000 - 22.5167i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{11} - 8 q^{13} - 12 q^{23} - 16 q^{25} - 52 q^{35} + 40 q^{37} - 20 q^{47} - 12 q^{49} - 8 q^{59} - 32 q^{71} - 12 q^{73} + 18 q^{83} - 52 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(5\) 1.80278 + 3.12250i 0.806226 + 1.39642i 0.915460 + 0.402408i \(0.131827\pi\)
−0.109235 + 0.994016i \(0.534840\pi\)
\(6\) 0 0
\(7\) −1.80278 + 3.12250i −0.681385 + 1.18019i 0.293173 + 0.956059i \(0.405289\pi\)
−0.974558 + 0.224134i \(0.928045\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i −0.780750 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) 0 0
\(13\) −2.00000 3.46410i −0.554700 0.960769i −0.997927 0.0643593i \(-0.979500\pi\)
0.443227 0.896410i \(-0.353834\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.21110 −1.65434 −0.827170 0.561951i \(-0.810051\pi\)
−0.827170 + 0.561951i \(0.810051\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) −4.00000 + 6.92820i −0.800000 + 1.38564i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.60555 + 6.24500i −0.669534 + 1.15967i 0.308500 + 0.951224i \(0.400173\pi\)
−0.978035 + 0.208443i \(0.933160\pi\)
\(30\) 0 0
\(31\) 1.80278 + 3.12250i 0.323788 + 0.560817i 0.981266 0.192656i \(-0.0617102\pi\)
−0.657478 + 0.753474i \(0.728377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −13.0000 −2.19740
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.60555 6.24500i −0.563093 0.975305i −0.997224 0.0744555i \(-0.976278\pi\)
0.434132 0.900849i \(-0.357055\pi\)
\(42\) 0 0
\(43\) −3.60555 + 6.24500i −0.549841 + 0.952353i 0.448444 + 0.893811i \(0.351979\pi\)
−0.998285 + 0.0585421i \(0.981355\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.00000 + 8.66025i −0.729325 + 1.26323i 0.227844 + 0.973698i \(0.426832\pi\)
−0.957169 + 0.289530i \(0.906501\pi\)
\(48\) 0 0
\(49\) −3.00000 5.19615i −0.428571 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.60555 0.495261 0.247630 0.968855i \(-0.420348\pi\)
0.247630 + 0.968855i \(0.420348\pi\)
\(54\) 0 0
\(55\) 3.60555 0.486172
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 3.46410i −0.260378 0.450988i 0.705965 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 0 0
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.21110 12.4900i 0.894427 1.54919i
\(66\) 0 0
\(67\) −3.60555 6.24500i −0.440488 0.762948i 0.557237 0.830353i \(-0.311861\pi\)
−0.997726 + 0.0674052i \(0.978528\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −3.00000 −0.351123 −0.175562 0.984468i \(-0.556174\pi\)
−0.175562 + 0.984468i \(0.556174\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.80278 + 3.12250i 0.205445 + 0.355842i
\(78\) 0 0
\(79\) 7.21110 12.4900i 0.811312 1.40523i −0.100633 0.994924i \(-0.532087\pi\)
0.911946 0.410311i \(-0.134580\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.50000 7.79423i 0.493939 0.855528i −0.506036 0.862512i \(-0.668890\pi\)
0.999976 + 0.00698436i \(0.00222321\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.21110 −0.764375 −0.382188 0.924085i \(-0.624829\pi\)
−0.382188 + 0.924085i \(0.624829\pi\)
\(90\) 0 0
\(91\) 14.4222 1.51186
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.0000 22.5167i −1.33377 2.31016i
\(96\) 0 0
\(97\) −3.50000 + 6.06218i −0.355371 + 0.615521i −0.987181 0.159602i \(-0.948979\pi\)
0.631810 + 0.775123i \(0.282312\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.40833 9.36750i 0.538149 0.932101i −0.460855 0.887475i \(-0.652457\pi\)
0.999004 0.0446254i \(-0.0142094\pi\)
\(102\) 0 0
\(103\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.0000 −1.64345 −0.821726 0.569883i \(-0.806989\pi\)
−0.821726 + 0.569883i \(0.806989\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.60555 6.24500i −0.339182 0.587480i 0.645097 0.764100i \(-0.276817\pi\)
−0.984279 + 0.176620i \(0.943483\pi\)
\(114\) 0 0
\(115\) 10.8167 18.7350i 1.00866 1.74705i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 + 8.66025i 0.454545 + 0.787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) −3.60555 −0.319941 −0.159970 0.987122i \(-0.551140\pi\)
−0.159970 + 0.987122i \(0.551140\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.500000 + 0.866025i 0.0436852 + 0.0756650i 0.887041 0.461690i \(-0.152757\pi\)
−0.843356 + 0.537355i \(0.819423\pi\)
\(132\) 0 0
\(133\) 13.0000 22.5167i 1.12724 1.95244i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.8167 + 18.7350i −0.924129 + 1.60064i −0.131173 + 0.991359i \(0.541874\pi\)
−0.792956 + 0.609279i \(0.791459\pi\)
\(138\) 0 0
\(139\) 7.21110 + 12.4900i 0.611638 + 1.05939i 0.990964 + 0.134125i \(0.0428223\pi\)
−0.379327 + 0.925263i \(0.623844\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −26.0000 −2.15918
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.40833 9.36750i −0.443067 0.767415i 0.554848 0.831952i \(-0.312776\pi\)
−0.997915 + 0.0645365i \(0.979443\pi\)
\(150\) 0 0
\(151\) −5.40833 + 9.36750i −0.440123 + 0.762316i −0.997698 0.0678106i \(-0.978399\pi\)
0.557575 + 0.830127i \(0.311732\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.50000 + 11.2583i −0.522093 + 0.904291i
\(156\) 0 0
\(157\) 2.00000 + 3.46410i 0.159617 + 0.276465i 0.934731 0.355357i \(-0.115641\pi\)
−0.775113 + 0.631822i \(0.782307\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 21.6333 1.70494
\(162\) 0 0
\(163\) 14.4222 1.12963 0.564817 0.825216i \(-0.308947\pi\)
0.564817 + 0.825216i \(0.308947\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 + 1.73205i 0.0773823 + 0.134030i 0.902120 0.431486i \(-0.142010\pi\)
−0.824737 + 0.565516i \(0.808677\pi\)
\(168\) 0 0
\(169\) −1.50000 + 2.59808i −0.115385 + 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.40833 + 9.36750i −0.411187 + 0.712198i −0.995020 0.0996766i \(-0.968219\pi\)
0.583832 + 0.811874i \(0.301553\pi\)
\(174\) 0 0
\(175\) −14.4222 24.9800i −1.09022 1.88831i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 18.0278 + 31.2250i 1.32543 + 2.29571i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 + 10.3923i −0.434145 + 0.751961i −0.997225 0.0744412i \(-0.976283\pi\)
0.563081 + 0.826402i \(0.309616\pi\)
\(192\) 0 0
\(193\) 7.50000 + 12.9904i 0.539862 + 0.935068i 0.998911 + 0.0466572i \(0.0148568\pi\)
−0.459049 + 0.888411i \(0.651810\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0278 1.28442 0.642212 0.766527i \(-0.278017\pi\)
0.642212 + 0.766527i \(0.278017\pi\)
\(198\) 0 0
\(199\) −3.60555 −0.255591 −0.127795 0.991801i \(-0.540790\pi\)
−0.127795 + 0.991801i \(0.540790\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −13.0000 22.5167i −0.912421 1.58036i
\(204\) 0 0
\(205\) 13.0000 22.5167i 0.907959 1.57263i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.60555 + 6.24500i −0.249401 + 0.431976i
\(210\) 0 0
\(211\) 3.60555 + 6.24500i 0.248216 + 0.429923i 0.963031 0.269391i \(-0.0868223\pi\)
−0.714815 + 0.699314i \(0.753489\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −26.0000 −1.77319
\(216\) 0 0
\(217\) −13.0000 −0.882498
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.21110 + 12.4900i −0.482891 + 0.836392i −0.999807 0.0196442i \(-0.993747\pi\)
0.516916 + 0.856036i \(0.327080\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.00000 10.3923i 0.398234 0.689761i −0.595274 0.803523i \(-0.702957\pi\)
0.993508 + 0.113761i \(0.0362899\pi\)
\(228\) 0 0
\(229\) 5.00000 + 8.66025i 0.330409 + 0.572286i 0.982592 0.185776i \(-0.0594799\pi\)
−0.652183 + 0.758062i \(0.726147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.6333 −1.41725 −0.708623 0.705588i \(-0.750683\pi\)
−0.708623 + 0.705588i \(0.750683\pi\)
\(234\) 0 0
\(235\) −36.0555 −2.35200
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.00000 + 5.19615i 0.194054 + 0.336111i 0.946590 0.322440i \(-0.104503\pi\)
−0.752536 + 0.658551i \(0.771170\pi\)
\(240\) 0 0
\(241\) −11.0000 + 19.0526i −0.708572 + 1.22728i 0.256814 + 0.966461i \(0.417327\pi\)
−0.965387 + 0.260822i \(0.916006\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 10.8167 18.7350i 0.691051 1.19693i
\(246\) 0 0
\(247\) 14.4222 + 24.9800i 0.917663 + 1.58944i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.21110 + 12.4900i 0.449816 + 0.779105i 0.998374 0.0570083i \(-0.0181561\pi\)
−0.548557 + 0.836113i \(0.684823\pi\)
\(258\) 0 0
\(259\) −18.0278 + 31.2250i −1.12019 + 1.94023i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.00000 15.5885i 0.554964 0.961225i −0.442943 0.896550i \(-0.646065\pi\)
0.997906 0.0646755i \(-0.0206012\pi\)
\(264\) 0 0
\(265\) 6.50000 + 11.2583i 0.399292 + 0.691594i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.6333 −1.31901 −0.659503 0.751702i \(-0.729233\pi\)
−0.659503 + 0.751702i \(0.729233\pi\)
\(270\) 0 0
\(271\) −10.8167 −0.657065 −0.328532 0.944493i \(-0.606554\pi\)
−0.328532 + 0.944493i \(0.606554\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 + 6.92820i 0.241209 + 0.417786i
\(276\) 0 0
\(277\) 4.00000 6.92820i 0.240337 0.416275i −0.720473 0.693482i \(-0.756075\pi\)
0.960810 + 0.277207i \(0.0894088\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(282\) 0 0
\(283\) −3.60555 6.24500i −0.214328 0.371227i 0.738737 0.673994i \(-0.235423\pi\)
−0.953064 + 0.302768i \(0.902089\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 26.0000 1.53473
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.8167 18.7350i −0.631916 1.09451i −0.987160 0.159736i \(-0.948936\pi\)
0.355244 0.934774i \(-0.384398\pi\)
\(294\) 0 0
\(295\) 7.21110 12.4900i 0.419847 0.727196i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 + 20.7846i −0.693978 + 1.20201i
\(300\) 0 0
\(301\) −13.0000 22.5167i −0.749308 1.29784i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.00000 + 15.5885i 0.510343 + 0.883940i 0.999928 + 0.0119847i \(0.00381495\pi\)
−0.489585 + 0.871956i \(0.662852\pi\)
\(312\) 0 0
\(313\) −0.500000 + 0.866025i −0.0282617 + 0.0489506i −0.879810 0.475325i \(-0.842331\pi\)
0.851549 + 0.524276i \(0.175664\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.40833 9.36750i 0.303762 0.526131i −0.673223 0.739440i \(-0.735091\pi\)
0.976985 + 0.213308i \(0.0684239\pi\)
\(318\) 0 0
\(319\) 3.60555 + 6.24500i 0.201872 + 0.349653i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 32.0000 1.77504
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −18.0278 31.2250i −0.993902 1.72149i
\(330\) 0 0
\(331\) −3.60555 + 6.24500i −0.198179 + 0.343256i −0.947938 0.318455i \(-0.896836\pi\)
0.749759 + 0.661711i \(0.230169\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 13.0000 22.5167i 0.710266 1.23022i
\(336\) 0 0
\(337\) −13.0000 22.5167i −0.708155 1.22656i −0.965541 0.260252i \(-0.916194\pi\)
0.257386 0.966309i \(-0.417139\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.60555 0.195252
\(342\) 0 0
\(343\) −3.60555 −0.194681
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −0.500000 0.866025i −0.0268414 0.0464907i 0.852293 0.523065i \(-0.175212\pi\)
−0.879134 + 0.476575i \(0.841878\pi\)
\(348\) 0 0
\(349\) 1.00000 1.73205i 0.0535288 0.0927146i −0.838019 0.545640i \(-0.816286\pi\)
0.891548 + 0.452926i \(0.149620\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.60555 6.24500i 0.191904 0.332388i −0.753977 0.656901i \(-0.771867\pi\)
0.945881 + 0.324513i \(0.105200\pi\)
\(354\) 0 0
\(355\) −14.4222 24.9800i −0.765451 1.32580i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) 33.0000 1.73684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.40833 9.36750i −0.283085 0.490317i
\(366\) 0 0
\(367\) −5.40833 + 9.36750i −0.282312 + 0.488979i −0.971954 0.235172i \(-0.924435\pi\)
0.689642 + 0.724151i \(0.257768\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.50000 + 11.2583i −0.337463 + 0.584503i
\(372\) 0 0
\(373\) 8.00000 + 13.8564i 0.414224 + 0.717458i 0.995347 0.0963587i \(-0.0307196\pi\)
−0.581122 + 0.813816i \(0.697386\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 28.8444 1.48556
\(378\) 0 0
\(379\) 14.4222 0.740819 0.370409 0.928869i \(-0.379217\pi\)
0.370409 + 0.928869i \(0.379217\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.00000 + 6.92820i 0.204390 + 0.354015i 0.949938 0.312437i \(-0.101145\pi\)
−0.745548 + 0.666452i \(0.767812\pi\)
\(384\) 0 0
\(385\) −6.50000 + 11.2583i −0.331271 + 0.573778i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.40833 + 9.36750i −0.274213 + 0.474951i −0.969936 0.243359i \(-0.921751\pi\)
0.695723 + 0.718310i \(0.255084\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 52.0000 2.61640
\(396\) 0 0
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.21110 12.4900i −0.360105 0.623721i 0.627873 0.778316i \(-0.283926\pi\)
−0.987978 + 0.154596i \(0.950593\pi\)
\(402\) 0 0
\(403\) 7.21110 12.4900i 0.359211 0.622171i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.00000 8.66025i 0.247841 0.429273i
\(408\) 0 0
\(409\) −3.50000 6.06218i −0.173064 0.299755i 0.766426 0.642333i \(-0.222033\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.4222 0.709670
\(414\) 0 0
\(415\) 32.4500 1.59291
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 + 3.46410i 0.0977064 + 0.169232i 0.910735 0.412991i \(-0.135516\pi\)
−0.813029 + 0.582224i \(0.802183\pi\)
\(420\) 0 0
\(421\) −16.0000 + 27.7128i −0.779792 + 1.35064i 0.152269 + 0.988339i \(0.451342\pi\)
−0.932061 + 0.362301i \(0.881991\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −35.0000 −1.68199 −0.840996 0.541041i \(-0.818030\pi\)
−0.840996 + 0.541041i \(0.818030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.6333 + 37.4700i 1.03486 + 1.79243i
\(438\) 0 0
\(439\) 9.01388 15.6125i 0.430209 0.745144i −0.566682 0.823937i \(-0.691773\pi\)
0.996891 + 0.0787928i \(0.0251066\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00000 10.3923i 0.285069 0.493753i −0.687557 0.726130i \(-0.741317\pi\)
0.972626 + 0.232377i \(0.0746503\pi\)
\(444\) 0 0
\(445\) −13.0000 22.5167i −0.616259 1.06739i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.21110 −0.340313 −0.170156 0.985417i \(-0.554427\pi\)
−0.170156 + 0.985417i \(0.554427\pi\)
\(450\) 0 0
\(451\) −7.21110 −0.339558
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 26.0000 + 45.0333i 1.21890 + 2.11119i
\(456\) 0 0
\(457\) −11.5000 + 19.9186i −0.537947 + 0.931752i 0.461067 + 0.887365i \(0.347467\pi\)
−0.999014 + 0.0443868i \(0.985867\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.40833 + 9.36750i −0.251891 + 0.436288i −0.964046 0.265734i \(-0.914386\pi\)
0.712156 + 0.702022i \(0.247719\pi\)
\(462\) 0 0
\(463\) 12.6194 + 21.8575i 0.586475 + 1.01580i 0.994690 + 0.102918i \(0.0328180\pi\)
−0.408215 + 0.912886i \(0.633849\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.0000 1.06431 0.532157 0.846646i \(-0.321382\pi\)
0.532157 + 0.846646i \(0.321382\pi\)
\(468\) 0 0
\(469\) 26.0000 1.20057
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.60555 + 6.24500i 0.165783 + 0.287145i
\(474\) 0 0
\(475\) 28.8444 49.9600i 1.32347 2.29232i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −21.0000 + 36.3731i −0.959514 + 1.66193i −0.235833 + 0.971794i \(0.575782\pi\)
−0.723681 + 0.690134i \(0.757551\pi\)
\(480\) 0 0
\(481\) −20.0000 34.6410i −0.911922 1.57949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25.2389 −1.14604
\(486\) 0 0
\(487\) 14.4222 0.653532 0.326766 0.945105i \(-0.394041\pi\)
0.326766 + 0.945105i \(0.394041\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.50000 + 6.06218i 0.157953 + 0.273582i 0.934130 0.356932i \(-0.116177\pi\)
−0.776178 + 0.630514i \(0.782844\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.4222 24.9800i 0.646924 1.12051i
\(498\) 0 0
\(499\) 10.8167 + 18.7350i 0.484220 + 0.838694i 0.999836 0.0181264i \(-0.00577013\pi\)
−0.515616 + 0.856820i \(0.672437\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) 39.0000 1.73548
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.40833 + 9.36750i 0.239720 + 0.415207i 0.960634 0.277818i \(-0.0896111\pi\)
−0.720914 + 0.693025i \(0.756278\pi\)
\(510\) 0 0
\(511\) 5.40833 9.36750i 0.239250 0.414394i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5.00000 + 8.66025i 0.219900 + 0.380878i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −43.2666 −1.89554 −0.947772 0.318947i \(-0.896671\pi\)
−0.947772 + 0.318947i \(0.896671\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.4222 + 24.9800i −0.624695 + 1.08200i
\(534\) 0 0
\(535\) −30.6472 53.0825i −1.32499 2.29496i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.60555 + 6.24500i 0.154445 + 0.267506i
\(546\) 0 0
\(547\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 26.0000 45.0333i 1.10764 1.91848i
\(552\) 0 0
\(553\) 26.0000 + 45.0333i 1.10563 + 1.91501i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.6611 1.68049 0.840247 0.542204i \(-0.182410\pi\)
0.840247 + 0.542204i \(0.182410\pi\)
\(558\) 0 0
\(559\) 28.8444 1.21999
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.5000 + 33.7750i 0.821827 + 1.42345i 0.904320 + 0.426855i \(0.140378\pi\)
−0.0824933 + 0.996592i \(0.526288\pi\)
\(564\) 0 0
\(565\) 13.0000 22.5167i 0.546914 0.947283i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.21110 12.4900i 0.302305 0.523608i −0.674353 0.738410i \(-0.735577\pi\)
0.976658 + 0.214802i \(0.0689105\pi\)
\(570\) 0 0
\(571\) −7.21110 12.4900i −0.301775 0.522690i 0.674763 0.738035i \(-0.264246\pi\)
−0.976538 + 0.215345i \(0.930913\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 48.0000 2.00174
\(576\) 0 0
\(577\) 6.00000 0.249783 0.124892 0.992170i \(-0.460142\pi\)
0.124892 + 0.992170i \(0.460142\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 16.2250 + 28.1025i 0.673126 + 1.16589i
\(582\) 0 0
\(583\) 1.80278 3.12250i 0.0746633 0.129321i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.50000 + 2.59808i −0.0619116 + 0.107234i −0.895320 0.445424i \(-0.853053\pi\)
0.833408 + 0.552658i \(0.186386\pi\)
\(588\) 0 0
\(589\) −13.0000 22.5167i −0.535656 0.927783i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −21.6333 −0.888373 −0.444187 0.895934i \(-0.646507\pi\)
−0.444187 + 0.895934i \(0.646507\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.00000 + 12.1244i 0.286012 + 0.495388i 0.972854 0.231419i \(-0.0743369\pi\)
−0.686842 + 0.726807i \(0.741004\pi\)
\(600\) 0 0
\(601\) 8.50000 14.7224i 0.346722 0.600541i −0.638943 0.769254i \(-0.720628\pi\)
0.985665 + 0.168714i \(0.0539613\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −18.0278 + 31.2250i −0.732933 + 1.26948i
\(606\) 0 0
\(607\) 7.21110 + 12.4900i 0.292690 + 0.506953i 0.974445 0.224627i \(-0.0721164\pi\)
−0.681755 + 0.731580i \(0.738783\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.8167 18.7350i −0.435462 0.754242i 0.561871 0.827225i \(-0.310082\pi\)
−0.997333 + 0.0729823i \(0.976748\pi\)
\(618\) 0 0
\(619\) −21.6333 + 37.4700i −0.869516 + 1.50605i −0.00702402 + 0.999975i \(0.502236\pi\)
−0.862492 + 0.506071i \(0.831098\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.0000 22.5167i 0.520834 0.902111i
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −25.2389 −1.00474 −0.502372 0.864652i \(-0.667539\pi\)
−0.502372 + 0.864652i \(0.667539\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.50000 11.2583i −0.257945 0.446773i
\(636\) 0 0
\(637\) −12.0000 + 20.7846i −0.475457 + 0.823516i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0278 + 31.2250i −0.712054 + 1.23331i 0.252031 + 0.967719i \(0.418901\pi\)
−0.964085 + 0.265594i \(0.914432\pi\)
\(642\) 0 0
\(643\) 21.6333 + 37.4700i 0.853134 + 1.47767i 0.878365 + 0.477990i \(0.158635\pi\)
−0.0252307 + 0.999682i \(0.508032\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.01388 + 15.6125i 0.352740 + 0.610964i 0.986729 0.162379i \(-0.0519166\pi\)
−0.633988 + 0.773343i \(0.718583\pi\)
\(654\) 0 0
\(655\) −1.80278 + 3.12250i −0.0704403 + 0.122006i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 13.5000 23.3827i 0.525885 0.910860i −0.473660 0.880708i \(-0.657067\pi\)
0.999545 0.0301523i \(-0.00959924\pi\)
\(660\) 0 0
\(661\) −7.00000 12.1244i −0.272268 0.471583i 0.697174 0.716902i \(-0.254441\pi\)
−0.969442 + 0.245319i \(0.921107\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 93.7443 3.63525
\(666\) 0 0
\(667\) 43.2666 1.67529
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 9.50000 16.4545i 0.366198 0.634274i −0.622770 0.782405i \(-0.713993\pi\)
0.988968 + 0.148132i \(0.0473259\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.60555 6.24500i 0.138573 0.240015i −0.788384 0.615184i \(-0.789082\pi\)
0.926957 + 0.375169i \(0.122415\pi\)
\(678\) 0 0
\(679\) −12.6194 21.8575i −0.484289 0.838814i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −78.0000 −2.98023
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7.21110 12.4900i −0.274721 0.475831i
\(690\) 0 0
\(691\) −7.21110 + 12.4900i −0.274323 + 0.475142i −0.969964 0.243248i \(-0.921787\pi\)
0.695641 + 0.718390i \(0.255120\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −26.0000 + 45.0333i −0.986236 + 1.70821i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.8167 −0.408539 −0.204270 0.978915i \(-0.565482\pi\)
−0.204270 + 0.978915i \(0.565482\pi\)
\(702\) 0 0
\(703\) −72.1110 −2.71972
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 19.5000 + 33.7750i 0.733373 + 1.27024i
\(708\) 0 0
\(709\) −2.00000 + 3.46410i −0.0751116 + 0.130097i −0.901135 0.433539i \(-0.857265\pi\)
0.826023 + 0.563636i \(0.190598\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.8167 18.7350i 0.405087 0.701631i
\(714\) 0 0
\(715\) −7.21110 12.4900i −0.269680 0.467099i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −28.8444 49.9600i −1.07125 1.85547i
\(726\) 0 0
\(727\) −16.2250 + 28.1025i −0.601751 + 1.04226i 0.390805 + 0.920474i \(0.372197\pi\)
−0.992556 + 0.121790i \(0.961137\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 23.0000 + 39.8372i 0.849524 + 1.47142i 0.881633 + 0.471935i \(0.156444\pi\)
−0.0321090 + 0.999484i \(0.510222\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.21110 −0.265624
\(738\) 0 0
\(739\) −28.8444 −1.06106 −0.530529 0.847667i \(-0.678007\pi\)
−0.530529 + 0.847667i \(0.678007\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18.0000 31.1769i −0.660356 1.14377i −0.980522 0.196409i \(-0.937072\pi\)
0.320166 0.947361i \(-0.396261\pi\)
\(744\) 0 0
\(745\) 19.5000 33.7750i 0.714425 1.23742i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.6472 53.0825i 1.11982 1.93959i
\(750\) 0 0
\(751\) 23.4361 + 40.5925i 0.855195 + 1.48124i 0.876464 + 0.481467i \(0.159896\pi\)
−0.0212693 + 0.999774i \(0.506771\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −39.0000 −1.41936
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) −3.60555 + 6.24500i −0.130530 + 0.226084i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 + 13.8564i −0.288863 + 0.500326i
\(768\) 0 0
\(769\) 5.50000 + 9.52628i 0.198335 + 0.343526i 0.947989 0.318304i \(-0.103113\pi\)
−0.749654 + 0.661830i \(0.769780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −21.6333 −0.778096 −0.389048 0.921217i \(-0.627196\pi\)
−0.389048 + 0.921217i \(0.627196\pi\)
\(774\) 0 0
\(775\) −28.8444 −1.03612
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 26.0000 + 45.0333i 0.931547 + 1.61349i
\(780\) 0 0
\(781\) −4.00000 + 6.92820i −0.143131 + 0.247911i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.21110 + 12.4900i −0.257375 + 0.445787i
\(786\) 0 0
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 26.0000 0.924454
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.40833 9.36750i −0.191573 0.331814i 0.754199 0.656646i \(-0.228025\pi\)
−0.945772 + 0.324832i \(0.894692\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.50000 + 2.59808i −0.0529339 + 0.0916841i
\(804\) 0 0
\(805\) 39.0000 + 67.5500i 1.37457 + 2.38082i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −50.4777 −1.77470 −0.887351 0.461095i \(-0.847457\pi\)
−0.887351 + 0.461095i \(0.847457\pi\)
\(810\) 0 0
\(811\) −7.21110 −0.253216 −0.126608 0.991953i \(-0.540409\pi\)
−0.126608 + 0.991953i \(0.540409\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 26.0000 + 45.0333i 0.910740 + 1.57745i
\(816\) 0 0
\(817\) 26.0000 45.0333i 0.909625 1.57552i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.60555 + 6.24500i −0.125835 + 0.217952i −0.922059 0.387050i \(-0.873494\pi\)
0.796224 + 0.605002i \(0.206828\pi\)
\(822\) 0 0
\(823\) 9.01388 + 15.6125i 0.314204 + 0.544217i 0.979268 0.202569i \(-0.0649291\pi\)
−0.665064 + 0.746786i \(0.731596\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16.0000 −0.556375 −0.278187 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) 0 0
\(829\) −46.0000 −1.59765 −0.798823 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −3.60555 + 6.24500i −0.124775 + 0.216117i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −20.0000 + 34.6410i −0.690477 + 1.19594i 0.281205 + 0.959648i \(0.409266\pi\)
−0.971682 + 0.236293i \(0.924067\pi\)
\(840\) 0 0
\(841\) −11.5000 19.9186i −0.396552 0.686848i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −10.8167 −0.372104
\(846\) 0 0
\(847\) −36.0555 −1.23888
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −30.0000 51.9615i −1.02839 1.78122i
\(852\) 0 0
\(853\) 21.0000 36.3731i 0.719026 1.24539i −0.242360 0.970186i \(-0.577921\pi\)
0.961386 0.275204i \(-0.0887453\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 14.4222 24.9800i 0.492653 0.853300i −0.507311 0.861763i \(-0.669360\pi\)
0.999964 + 0.00846275i \(0.00269381\pi\)
\(858\) 0 0
\(859\) −14.4222 24.9800i −0.492079 0.852306i 0.507879 0.861428i \(-0.330430\pi\)
−0.999958 + 0.00912203i \(0.997096\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 0 0
\(865\) −39.0000 −1.32604
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −7.21110 12.4900i −0.244620 0.423694i
\(870\) 0 0
\(871\) −14.4222 + 24.9800i −0.488678 + 0.846415i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 19.5000 33.7750i 0.659220 1.14180i
\(876\) 0 0
\(877\) −17.0000 29.4449i −0.574049 0.994282i −0.996144 0.0877308i \(-0.972038\pi\)
0.422095 0.906552i \(-0.361295\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −43.2666 −1.45769 −0.728845 0.684679i \(-0.759942\pi\)
−0.728845 + 0.684679i \(0.759942\pi\)
\(882\) 0 0
\(883\) 50.4777 1.69871 0.849355 0.527822i \(-0.176991\pi\)
0.849355 + 0.527822i \(0.176991\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0000 31.1769i −0.604381 1.04682i −0.992149 0.125061i \(-0.960087\pi\)
0.387768 0.921757i \(-0.373246\pi\)
\(888\) 0 0
\(889\) 6.50000 11.2583i 0.218003 0.377592i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 36.0555 62.4500i 1.20655 2.08981i
\(894\) 0 0
\(895\) −27.0416 46.8375i −0.903902 1.56560i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −26.0000 −0.867149
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.4222 + 24.9800i 0.479410 + 0.830363i
\(906\) 0 0
\(907\) 3.60555 6.24500i 0.119720 0.207362i −0.799936 0.600085i \(-0.795134\pi\)
0.919657 + 0.392723i \(0.128467\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 15.0000 25.9808i 0.496972 0.860781i −0.503022 0.864274i \(-0.667778\pi\)
0.999994 + 0.00349271i \(0.00111177\pi\)
\(912\) 0 0
\(913\) −4.50000 7.79423i −0.148928 0.257951i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.60555 −0.119066
\(918\) 0 0
\(919\) 54.0833 1.78404 0.892021 0.451994i \(-0.149287\pi\)
0.892021 + 0.451994i \(0.149287\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.0000 + 27.7128i 0.526646 + 0.912178i
\(924\) 0 0
\(925\) −40.0000 + 69.2820i −1.31519 + 2.27798i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −21.6333 + 37.4700i −0.709766 + 1.22935i 0.255178 + 0.966894i \(0.417866\pi\)
−0.964944 + 0.262456i \(0.915467\pi\)
\(930\) 0 0
\(931\) 21.6333 + 37.4700i 0.709003 + 1.22803i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.6194 21.8575i −0.411382 0.712534i 0.583659 0.811999i \(-0.301620\pi\)
−0.995041 + 0.0994646i \(0.968287\pi\)
\(942\) 0 0
\(943\) −21.6333 + 37.4700i −0.704477 + 1.22019i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.5000 23.3827i 0.438691 0.759835i −0.558898 0.829237i \(-0.688776\pi\)
0.997589 + 0.0694014i \(0.0221089\pi\)
\(948\) 0 0
\(949\) 6.00000 + 10.3923i 0.194768 + 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 43.2666 1.40154 0.700772 0.713386i \(-0.252839\pi\)
0.700772 + 0.713386i \(0.252839\pi\)
\(954\) 0 0
\(955\) −43.2666 −1.40007
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −39.0000 67.5500i −1.25938 2.18130i
\(960\) 0 0
\(961\) 9.00000 15.5885i 0.290323 0.502853i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −27.0416 + 46.8375i −0.870501 + 1.50775i
\(966\) 0 0
\(967\) 5.40833 + 9.36750i 0.173920 + 0.301238i 0.939787 0.341761i \(-0.111023\pi\)
−0.765867 + 0.642999i \(0.777690\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.0000 −0.802288 −0.401144 0.916015i \(-0.631387\pi\)
−0.401144 + 0.916015i \(0.631387\pi\)
\(972\) 0 0
\(973\) −52.0000 −1.66704
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0278 + 31.2250i 0.576759 + 0.998976i 0.995848 + 0.0910308i \(0.0290162\pi\)
−0.419089 + 0.907945i \(0.637650\pi\)
\(978\) 0 0
\(979\) −3.60555 + 6.24500i −0.115234 + 0.199591i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.00000 5.19615i 0.0956851 0.165732i −0.814209 0.580572i \(-0.802829\pi\)
0.909894 + 0.414840i \(0.136162\pi\)
\(984\) 0 0
\(985\) 32.5000 + 56.2917i 1.03554 + 1.79360i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 43.2666 1.37580
\(990\) 0 0
\(991\) 61.2944 1.94708 0.973540 0.228517i \(-0.0733876\pi\)
0.973540 + 0.228517i \(0.0733876\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.50000 11.2583i −0.206064 0.356913i
\(996\) 0 0
\(997\) −18.0000 + 31.1769i −0.570066 + 0.987383i 0.426493 + 0.904491i \(0.359749\pi\)
−0.996559 + 0.0828918i \(0.973584\pi\)
\(998\) 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 2592.2.i.bc.865.2 4
3.2 odd 2 2592.2.i.bb.865.1 4
4.3 odd 2 2592.2.i.bb.865.2 4
9.2 odd 6 864.2.a.n.1.2 yes 2
9.4 even 3 inner 2592.2.i.bc.1729.2 4
9.5 odd 6 2592.2.i.bb.1729.1 4
9.7 even 3 864.2.a.m.1.1 2
12.11 even 2 inner 2592.2.i.bc.865.1 4
36.7 odd 6 864.2.a.n.1.1 yes 2
36.11 even 6 864.2.a.m.1.2 yes 2
36.23 even 6 inner 2592.2.i.bc.1729.1 4
36.31 odd 6 2592.2.i.bb.1729.2 4
72.11 even 6 1728.2.a.bd.1.1 2
72.29 odd 6 1728.2.a.bc.1.1 2
72.43 odd 6 1728.2.a.bc.1.2 2
72.61 even 6 1728.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.m.1.1 2 9.7 even 3
864.2.a.m.1.2 yes 2 36.11 even 6
864.2.a.n.1.1 yes 2 36.7 odd 6
864.2.a.n.1.2 yes 2 9.2 odd 6
1728.2.a.bc.1.1 2 72.29 odd 6
1728.2.a.bc.1.2 2 72.43 odd 6
1728.2.a.bd.1.1 2 72.11 even 6
1728.2.a.bd.1.2 2 72.61 even 6
2592.2.i.bb.865.1 4 3.2 odd 2
2592.2.i.bb.865.2 4 4.3 odd 2
2592.2.i.bb.1729.1 4 9.5 odd 6
2592.2.i.bb.1729.2 4 36.31 odd 6
2592.2.i.bc.865.1 4 12.11 even 2 inner
2592.2.i.bc.865.2 4 1.1 even 1 trivial
2592.2.i.bc.1729.1 4 36.23 even 6 inner
2592.2.i.bc.1729.2 4 9.4 even 3 inner