Properties

Label 324.3.g.d.53.4
Level $324$
Weight $3$
Character 324.53
Analytic conductor $8.828$
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] = [324,3,Mod(53,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.g (of order \(6\), 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: \(8.82836056527\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 53.4
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 324.53
Dual form 324.3.g.d.269.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(5.01910 + 2.89778i) q^{5} +(3.09808 + 5.36603i) q^{7} +O(q^{10})\) \(q+(5.01910 + 2.89778i) q^{5} +(3.09808 + 5.36603i) q^{7} +(-0.984508 + 0.568406i) q^{11} +(5.69615 - 9.86603i) q^{13} +31.2514i q^{17} -32.9808 q^{19} +(29.1301 + 16.8183i) q^{23} +(4.29423 + 7.43782i) q^{25} +(22.1420 - 12.7837i) q^{29} +(11.5885 - 20.0718i) q^{31} +35.9101i q^{35} +8.80385 q^{37} +(61.2137 + 35.3417i) q^{41} +(29.8827 + 51.7583i) q^{43} +(-52.1600 + 30.1146i) q^{47} +(5.30385 - 9.18653i) q^{49} -19.7718i q^{53} -6.58846 q^{55} +(-72.2364 - 41.7057i) q^{59} +(-15.3827 - 26.6436i) q^{61} +(57.1791 - 33.0124i) q^{65} +(5.29423 - 9.16987i) q^{67} +3.63342i q^{71} -17.2346 q^{73} +(-6.10016 - 3.52193i) q^{77} +(-56.6865 - 98.1840i) q^{79} +(-10.2313 + 5.90705i) q^{83} +(-90.5596 + 156.854i) q^{85} -111.252i q^{89} +70.5885 q^{91} +(-165.534 - 95.5709i) q^{95} +(-37.1577 - 64.3590i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} + 4 q^{13} - 56 q^{19} - 28 q^{25} - 32 q^{31} + 112 q^{37} + 52 q^{43} + 84 q^{49} + 72 q^{55} + 64 q^{61} - 20 q^{67} - 512 q^{73} - 308 q^{79} - 288 q^{85} + 440 q^{91} + 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) 5.01910 + 2.89778i 1.00382 + 0.579555i 0.909376 0.415975i \(-0.136560\pi\)
0.0944434 + 0.995530i \(0.469893\pi\)
\(6\) 0 0
\(7\) 3.09808 + 5.36603i 0.442582 + 0.766575i 0.997880 0.0650764i \(-0.0207291\pi\)
−0.555298 + 0.831651i \(0.687396\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.984508 + 0.568406i −0.0895007 + 0.0516733i −0.544082 0.839032i \(-0.683122\pi\)
0.454582 + 0.890705i \(0.349789\pi\)
\(12\) 0 0
\(13\) 5.69615 9.86603i 0.438166 0.758925i −0.559382 0.828910i \(-0.688962\pi\)
0.997548 + 0.0699846i \(0.0222950\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 31.2514i 1.83832i 0.393887 + 0.919159i \(0.371130\pi\)
−0.393887 + 0.919159i \(0.628870\pi\)
\(18\) 0 0
\(19\) −32.9808 −1.73583 −0.867915 0.496713i \(-0.834540\pi\)
−0.867915 + 0.496713i \(0.834540\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 29.1301 + 16.8183i 1.26653 + 0.731229i 0.974329 0.225130i \(-0.0722806\pi\)
0.292196 + 0.956358i \(0.405614\pi\)
\(24\) 0 0
\(25\) 4.29423 + 7.43782i 0.171769 + 0.297513i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 22.1420 12.7837i 0.763516 0.440816i −0.0670407 0.997750i \(-0.521356\pi\)
0.830557 + 0.556934i \(0.188022\pi\)
\(30\) 0 0
\(31\) 11.5885 20.0718i 0.373821 0.647477i −0.616329 0.787489i \(-0.711381\pi\)
0.990150 + 0.140012i \(0.0447140\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 35.9101i 1.02600i
\(36\) 0 0
\(37\) 8.80385 0.237942 0.118971 0.992898i \(-0.462040\pi\)
0.118971 + 0.992898i \(0.462040\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 61.2137 + 35.3417i 1.49302 + 0.861994i 0.999968 0.00800815i \(-0.00254910\pi\)
0.493049 + 0.870002i \(0.335882\pi\)
\(42\) 0 0
\(43\) 29.8827 + 51.7583i 0.694946 + 1.20368i 0.970199 + 0.242310i \(0.0779052\pi\)
−0.275253 + 0.961372i \(0.588762\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −52.1600 + 30.1146i −1.10979 + 0.640736i −0.938774 0.344533i \(-0.888037\pi\)
−0.171013 + 0.985269i \(0.554704\pi\)
\(48\) 0 0
\(49\) 5.30385 9.18653i 0.108242 0.187480i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 19.7718i 0.373053i −0.982450 0.186526i \(-0.940277\pi\)
0.982450 0.186526i \(-0.0597229\pi\)
\(54\) 0 0
\(55\) −6.58846 −0.119790
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −72.2364 41.7057i −1.22435 0.706876i −0.258504 0.966010i \(-0.583230\pi\)
−0.965841 + 0.259134i \(0.916563\pi\)
\(60\) 0 0
\(61\) −15.3827 26.6436i −0.252175 0.436780i 0.711949 0.702231i \(-0.247813\pi\)
−0.964124 + 0.265451i \(0.914479\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 57.1791 33.0124i 0.879678 0.507883i
\(66\) 0 0
\(67\) 5.29423 9.16987i 0.0790183 0.136864i −0.823808 0.566868i \(-0.808155\pi\)
0.902827 + 0.430005i \(0.141488\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.63342i 0.0511750i 0.999673 + 0.0255875i \(0.00814564\pi\)
−0.999673 + 0.0255875i \(0.991854\pi\)
\(72\) 0 0
\(73\) −17.2346 −0.236091 −0.118045 0.993008i \(-0.537663\pi\)
−0.118045 + 0.993008i \(0.537663\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.10016 3.52193i −0.0792229 0.0457394i
\(78\) 0 0
\(79\) −56.6865 98.1840i −0.717551 1.24283i −0.961967 0.273164i \(-0.911930\pi\)
0.244416 0.969670i \(-0.421404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.2313 + 5.90705i −0.123269 + 0.0711693i −0.560367 0.828245i \(-0.689340\pi\)
0.437098 + 0.899414i \(0.356006\pi\)
\(84\) 0 0
\(85\) −90.5596 + 156.854i −1.06541 + 1.84534i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 111.252i 1.25003i −0.780614 0.625013i \(-0.785094\pi\)
0.780614 0.625013i \(-0.214906\pi\)
\(90\) 0 0
\(91\) 70.5885 0.775697
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −165.534 95.5709i −1.74246 1.00601i
\(96\) 0 0
\(97\) −37.1577 64.3590i −0.383069 0.663495i 0.608430 0.793607i \(-0.291799\pi\)
−0.991499 + 0.130113i \(0.958466\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.9607 8.63759i 0.148126 0.0855206i −0.424105 0.905613i \(-0.639411\pi\)
0.572231 + 0.820092i \(0.306078\pi\)
\(102\) 0 0
\(103\) 56.0000 96.9948i 0.543689 0.941698i −0.454999 0.890492i \(-0.650360\pi\)
0.998688 0.0512055i \(-0.0163063\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 141.144i 1.31910i −0.751660 0.659551i \(-0.770746\pi\)
0.751660 0.659551i \(-0.229254\pi\)
\(108\) 0 0
\(109\) 171.708 1.57530 0.787650 0.616123i \(-0.211298\pi\)
0.787650 + 0.616123i \(0.211298\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.82598 + 2.78628i 0.0427078 + 0.0246574i 0.521202 0.853433i \(-0.325484\pi\)
−0.478494 + 0.878091i \(0.658817\pi\)
\(114\) 0 0
\(115\) 97.4711 + 168.825i 0.847575 + 1.46804i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −167.696 + 96.8192i −1.40921 + 0.813607i
\(120\) 0 0
\(121\) −59.8538 + 103.670i −0.494660 + 0.856776i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 95.1140i 0.760912i
\(126\) 0 0
\(127\) −68.2346 −0.537281 −0.268640 0.963241i \(-0.586574\pi\)
−0.268640 + 0.963241i \(0.586574\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −133.450 77.0474i −1.01870 0.588148i −0.104975 0.994475i \(-0.533476\pi\)
−0.913728 + 0.406327i \(0.866810\pi\)
\(132\) 0 0
\(133\) −102.177 176.976i −0.768247 1.33064i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 66.2328 38.2395i 0.483451 0.279121i −0.238403 0.971166i \(-0.576624\pi\)
0.721854 + 0.692046i \(0.243290\pi\)
\(138\) 0 0
\(139\) −47.0192 + 81.4397i −0.338268 + 0.585897i −0.984107 0.177576i \(-0.943174\pi\)
0.645839 + 0.763474i \(0.276508\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.9509i 0.0905658i
\(144\) 0 0
\(145\) 148.177 1.02191
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 58.1636 + 33.5808i 0.390360 + 0.225374i 0.682316 0.731057i \(-0.260973\pi\)
−0.291956 + 0.956432i \(0.594306\pi\)
\(150\) 0 0
\(151\) −23.1769 40.1436i −0.153490 0.265852i 0.779018 0.627001i \(-0.215718\pi\)
−0.932508 + 0.361149i \(0.882384\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 116.327 67.1615i 0.750498 0.433300i
\(156\) 0 0
\(157\) 84.7942 146.868i 0.540091 0.935464i −0.458808 0.888536i \(-0.651723\pi\)
0.998898 0.0469288i \(-0.0149434\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 208.417i 1.29452i
\(162\) 0 0
\(163\) −243.023 −1.49094 −0.745469 0.666540i \(-0.767775\pi\)
−0.745469 + 0.666540i \(0.767775\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 97.6215 + 56.3618i 0.584560 + 0.337496i 0.762944 0.646465i \(-0.223753\pi\)
−0.178383 + 0.983961i \(0.557087\pi\)
\(168\) 0 0
\(169\) 19.6077 + 33.9615i 0.116022 + 0.200956i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 162.677 93.9214i 0.940328 0.542898i 0.0502647 0.998736i \(-0.483993\pi\)
0.890063 + 0.455837i \(0.150660\pi\)
\(174\) 0 0
\(175\) −26.6077 + 46.0859i −0.152044 + 0.263348i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.3613i 0.0914042i −0.998955 0.0457021i \(-0.985448\pi\)
0.998955 0.0457021i \(-0.0145525\pi\)
\(180\) 0 0
\(181\) −227.608 −1.25750 −0.628751 0.777607i \(-0.716433\pi\)
−0.628751 + 0.777607i \(0.716433\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 44.1874 + 25.5116i 0.238851 + 0.137900i
\(186\) 0 0
\(187\) −17.7635 30.7673i −0.0949919 0.164531i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −315.720 + 182.281i −1.65299 + 0.954352i −0.677151 + 0.735844i \(0.736786\pi\)
−0.975835 + 0.218508i \(0.929881\pi\)
\(192\) 0 0
\(193\) 12.4423 21.5507i 0.0644678 0.111662i −0.831990 0.554791i \(-0.812798\pi\)
0.896458 + 0.443129i \(0.146132\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 171.259i 0.869333i 0.900592 + 0.434666i \(0.143134\pi\)
−0.900592 + 0.434666i \(0.856866\pi\)
\(198\) 0 0
\(199\) 122.431 0.615230 0.307615 0.951511i \(-0.400469\pi\)
0.307615 + 0.951511i \(0.400469\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 137.195 + 79.2096i 0.675837 + 0.390195i
\(204\) 0 0
\(205\) 204.825 + 354.767i 0.999146 + 1.73057i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 32.4698 18.7465i 0.155358 0.0896960i
\(210\) 0 0
\(211\) 186.256 322.604i 0.882729 1.52893i 0.0344337 0.999407i \(-0.489037\pi\)
0.848295 0.529524i \(-0.177629\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 346.373i 1.61104i
\(216\) 0 0
\(217\) 143.608 0.661787
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 308.327 + 178.013i 1.39515 + 0.805487i
\(222\) 0 0
\(223\) 160.060 + 277.231i 0.717756 + 1.24319i 0.961887 + 0.273448i \(0.0881640\pi\)
−0.244131 + 0.969742i \(0.578503\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.1848 7.61227i 0.0580830 0.0335342i −0.470677 0.882305i \(-0.655990\pi\)
0.528760 + 0.848771i \(0.322657\pi\)
\(228\) 0 0
\(229\) 89.8538 155.631i 0.392375 0.679613i −0.600387 0.799709i \(-0.704987\pi\)
0.992762 + 0.120096i \(0.0383203\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 150.796i 0.647193i −0.946195 0.323596i \(-0.895108\pi\)
0.946195 0.323596i \(-0.104892\pi\)
\(234\) 0 0
\(235\) −349.061 −1.48537
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 229.103 + 132.272i 0.958588 + 0.553441i 0.895738 0.444582i \(-0.146648\pi\)
0.0628500 + 0.998023i \(0.479981\pi\)
\(240\) 0 0
\(241\) −114.225 197.844i −0.473963 0.820927i 0.525593 0.850736i \(-0.323844\pi\)
−0.999556 + 0.0298087i \(0.990510\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 53.2411 30.7387i 0.217310 0.125464i
\(246\) 0 0
\(247\) −187.863 + 325.389i −0.760581 + 1.31736i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 47.2783i 0.188360i 0.995555 + 0.0941798i \(0.0300229\pi\)
−0.995555 + 0.0941798i \(0.969977\pi\)
\(252\) 0 0
\(253\) −38.2384 −0.151140
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.3351 + 12.8952i 0.0869069 + 0.0501757i 0.542824 0.839847i \(-0.317355\pi\)
−0.455917 + 0.890022i \(0.650689\pi\)
\(258\) 0 0
\(259\) 27.2750 + 47.2417i 0.105309 + 0.182400i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 247.017 142.615i 0.939228 0.542263i 0.0495095 0.998774i \(-0.484234\pi\)
0.889718 + 0.456510i \(0.150901\pi\)
\(264\) 0 0
\(265\) 57.2942 99.2365i 0.216205 0.374477i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 205.341i 0.763350i 0.924297 + 0.381675i \(0.124653\pi\)
−0.924297 + 0.381675i \(0.875347\pi\)
\(270\) 0 0
\(271\) 282.004 1.04060 0.520302 0.853982i \(-0.325819\pi\)
0.520302 + 0.853982i \(0.325819\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.45541 4.88173i −0.0307469 0.0177518i
\(276\) 0 0
\(277\) −24.3538 42.1821i −0.0879200 0.152282i 0.818712 0.574205i \(-0.194689\pi\)
−0.906632 + 0.421923i \(0.861355\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −213.659 + 123.356i −0.760353 + 0.438990i −0.829422 0.558622i \(-0.811330\pi\)
0.0690697 + 0.997612i \(0.477997\pi\)
\(282\) 0 0
\(283\) 176.473 305.660i 0.623580 1.08007i −0.365234 0.930916i \(-0.619011\pi\)
0.988814 0.149156i \(-0.0476556\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 437.966i 1.52601i
\(288\) 0 0
\(289\) −687.650 −2.37941
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −242.191 139.829i −0.826590 0.477232i 0.0260937 0.999660i \(-0.491693\pi\)
−0.852684 + 0.522428i \(0.825027\pi\)
\(294\) 0 0
\(295\) −241.708 418.650i −0.819348 1.41915i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 331.859 191.599i 1.10990 0.640798i
\(300\) 0 0
\(301\) −185.158 + 320.703i −0.615142 + 1.06546i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 178.302i 0.584598i
\(306\) 0 0
\(307\) 234.708 0.764520 0.382260 0.924055i \(-0.375146\pi\)
0.382260 + 0.924055i \(0.375146\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −204.123 117.850i −0.656343 0.378940i 0.134539 0.990908i \(-0.457045\pi\)
−0.790882 + 0.611969i \(0.790378\pi\)
\(312\) 0 0
\(313\) 178.835 + 309.751i 0.571357 + 0.989619i 0.996427 + 0.0844583i \(0.0269160\pi\)
−0.425070 + 0.905160i \(0.639751\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −523.067 + 301.993i −1.65005 + 0.952659i −0.673007 + 0.739636i \(0.734998\pi\)
−0.977047 + 0.213024i \(0.931669\pi\)
\(318\) 0 0
\(319\) −14.5326 + 25.1713i −0.0455568 + 0.0789068i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1030.69i 3.19101i
\(324\) 0 0
\(325\) 97.8423 0.301053
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −323.191 186.595i −0.982344 0.567157i
\(330\) 0 0
\(331\) 18.3712 + 31.8198i 0.0555021 + 0.0961324i 0.892442 0.451163i \(-0.148991\pi\)
−0.836939 + 0.547295i \(0.815657\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 53.1445 30.6830i 0.158640 0.0915910i
\(336\) 0 0
\(337\) −84.1384 + 145.732i −0.249669 + 0.432439i −0.963434 0.267946i \(-0.913655\pi\)
0.713765 + 0.700385i \(0.246988\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 26.3478i 0.0772663i
\(342\) 0 0
\(343\) 369.338 1.07679
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 152.947 + 88.3041i 0.440770 + 0.254479i 0.703924 0.710275i \(-0.251430\pi\)
−0.263154 + 0.964754i \(0.584763\pi\)
\(348\) 0 0
\(349\) −65.6077 113.636i −0.187988 0.325604i 0.756592 0.653888i \(-0.226863\pi\)
−0.944579 + 0.328284i \(0.893530\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −11.7952 + 6.80994i −0.0334140 + 0.0192916i −0.516614 0.856218i \(-0.672808\pi\)
0.483200 + 0.875510i \(0.339474\pi\)
\(354\) 0 0
\(355\) −10.5289 + 18.2365i −0.0296588 + 0.0513705i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 96.5853i 0.269040i 0.990911 + 0.134520i \(0.0429492\pi\)
−0.990911 + 0.134520i \(0.957051\pi\)
\(360\) 0 0
\(361\) 726.731 2.01310
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −86.5023 49.9421i −0.236993 0.136828i
\(366\) 0 0
\(367\) −127.473 220.790i −0.347338 0.601607i 0.638438 0.769674i \(-0.279581\pi\)
−0.985776 + 0.168067i \(0.946248\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 106.096 61.2545i 0.285973 0.165106i
\(372\) 0 0
\(373\) 45.5500 78.8949i 0.122118 0.211514i −0.798485 0.602015i \(-0.794365\pi\)
0.920603 + 0.390501i \(0.127698\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 291.271i 0.772602i
\(378\) 0 0
\(379\) 193.454 0.510432 0.255216 0.966884i \(-0.417853\pi\)
0.255216 + 0.966884i \(0.417853\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −288.347 166.477i −0.752865 0.434667i 0.0738632 0.997268i \(-0.476467\pi\)
−0.826728 + 0.562602i \(0.809801\pi\)
\(384\) 0 0
\(385\) −20.4115 35.3538i −0.0530170 0.0918281i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −35.6165 + 20.5632i −0.0915590 + 0.0528616i −0.545080 0.838384i \(-0.683501\pi\)
0.453521 + 0.891245i \(0.350168\pi\)
\(390\) 0 0
\(391\) −525.594 + 910.356i −1.34423 + 2.32828i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 657.060i 1.66344i
\(396\) 0 0
\(397\) −176.881 −0.445544 −0.222772 0.974871i \(-0.571510\pi\)
−0.222772 + 0.974871i \(0.571510\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −378.613 218.593i −0.944173 0.545119i −0.0529071 0.998599i \(-0.516849\pi\)
−0.891266 + 0.453481i \(0.850182\pi\)
\(402\) 0 0
\(403\) −132.019 228.664i −0.327591 0.567405i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.66746 + 5.00416i −0.0212960 + 0.0122952i
\(408\) 0 0
\(409\) 72.9519 126.356i 0.178367 0.308940i −0.762955 0.646452i \(-0.776252\pi\)
0.941321 + 0.337512i \(0.109585\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 516.830i 1.25140i
\(414\) 0 0
\(415\) −68.4693 −0.164986
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 115.555 + 66.7156i 0.275787 + 0.159226i 0.631515 0.775364i \(-0.282434\pi\)
−0.355728 + 0.934590i \(0.615767\pi\)
\(420\) 0 0
\(421\) −144.346 250.015i −0.342865 0.593859i 0.642099 0.766622i \(-0.278064\pi\)
−0.984963 + 0.172763i \(0.944731\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −232.442 + 134.201i −0.546923 + 0.315766i
\(426\) 0 0
\(427\) 95.3135 165.088i 0.223217 0.386622i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 669.565i 1.55351i 0.629800 + 0.776757i \(0.283137\pi\)
−0.629800 + 0.776757i \(0.716863\pi\)
\(432\) 0 0
\(433\) −510.654 −1.17934 −0.589669 0.807645i \(-0.700742\pi\)
−0.589669 + 0.807645i \(0.700742\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −960.732 554.679i −2.19847 1.26929i
\(438\) 0 0
\(439\) −38.6499 66.9437i −0.0880409 0.152491i 0.818642 0.574304i \(-0.194727\pi\)
−0.906683 + 0.421813i \(0.861394\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 657.617 379.676i 1.48446 0.857055i 0.484620 0.874725i \(-0.338958\pi\)
0.999844 + 0.0176696i \(0.00562472\pi\)
\(444\) 0 0
\(445\) 322.385 558.386i 0.724460 1.25480i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 541.149i 1.20523i 0.798032 + 0.602616i \(0.205875\pi\)
−0.798032 + 0.602616i \(0.794125\pi\)
\(450\) 0 0
\(451\) −80.3538 −0.178168
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 354.290 + 204.550i 0.778660 + 0.449560i
\(456\) 0 0
\(457\) 27.7942 + 48.1410i 0.0608189 + 0.105341i 0.894832 0.446404i \(-0.147295\pi\)
−0.834013 + 0.551745i \(0.813962\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −217.114 + 125.351i −0.470964 + 0.271911i −0.716643 0.697440i \(-0.754322\pi\)
0.245679 + 0.969351i \(0.420989\pi\)
\(462\) 0 0
\(463\) −285.512 + 494.520i −0.616656 + 1.06808i 0.373436 + 0.927656i \(0.378180\pi\)
−0.990092 + 0.140423i \(0.955154\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 72.2663i 0.154746i −0.997002 0.0773729i \(-0.975347\pi\)
0.997002 0.0773729i \(-0.0246532\pi\)
\(468\) 0 0
\(469\) 65.6077 0.139888
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −58.8395 33.9710i −0.124396 0.0718203i
\(474\) 0 0
\(475\) −141.627 245.305i −0.298162 0.516432i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −612.735 + 353.763i −1.27920 + 0.738544i −0.976701 0.214606i \(-0.931153\pi\)
−0.302496 + 0.953151i \(0.597820\pi\)
\(480\) 0 0
\(481\) 50.1481 86.8590i 0.104258 0.180580i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 430.699i 0.888039i
\(486\) 0 0
\(487\) 360.908 0.741083 0.370542 0.928816i \(-0.379172\pi\)
0.370542 + 0.928816i \(0.379172\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 588.914 + 340.010i 1.19942 + 0.692484i 0.960425 0.278538i \(-0.0898498\pi\)
0.238992 + 0.971022i \(0.423183\pi\)
\(492\) 0 0
\(493\) 399.508 + 691.967i 0.810360 + 1.40358i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.4971 + 11.2566i −0.0392295 + 0.0226491i
\(498\) 0 0
\(499\) −118.352 + 204.992i −0.237178 + 0.410805i −0.959903 0.280331i \(-0.909556\pi\)
0.722725 + 0.691135i \(0.242889\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 95.6715i 0.190202i −0.995468 0.0951009i \(-0.969683\pi\)
0.995468 0.0951009i \(-0.0303174\pi\)
\(504\) 0 0
\(505\) 100.119 0.198256
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 376.915 + 217.612i 0.740501 + 0.427529i 0.822252 0.569124i \(-0.192718\pi\)
−0.0817503 + 0.996653i \(0.526051\pi\)
\(510\) 0 0
\(511\) −53.3942 92.4815i −0.104490 0.180981i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 562.139 324.551i 1.09153 0.630196i
\(516\) 0 0
\(517\) 34.2346 59.2961i 0.0662178 0.114693i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 794.771i 1.52547i 0.646709 + 0.762737i \(0.276145\pi\)
−0.646709 + 0.762737i \(0.723855\pi\)
\(522\) 0 0
\(523\) −594.481 −1.13667 −0.568337 0.822796i \(-0.692413\pi\)
−0.568337 + 0.822796i \(0.692413\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 627.272 + 362.156i 1.19027 + 0.687202i
\(528\) 0 0
\(529\) 301.208 + 521.707i 0.569391 + 0.986214i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 697.365 402.624i 1.30838 0.755392i
\(534\) 0 0
\(535\) 409.004 708.415i 0.764493 1.32414i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0590i 0.0223728i
\(540\) 0 0
\(541\) 219.508 0.405744 0.202872 0.979205i \(-0.434972\pi\)
0.202872 + 0.979205i \(0.434972\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 861.818 + 497.571i 1.58132 + 0.912974i
\(546\) 0 0
\(547\) 292.296 + 506.272i 0.534362 + 0.925542i 0.999194 + 0.0401434i \(0.0127815\pi\)
−0.464832 + 0.885399i \(0.653885\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −730.259 + 421.615i −1.32533 + 0.765182i
\(552\) 0 0
\(553\) 351.238 608.363i 0.635151 1.10011i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 597.187i 1.07215i 0.844171 + 0.536075i \(0.180093\pi\)
−0.844171 + 0.536075i \(0.819907\pi\)
\(558\) 0 0
\(559\) 680.865 1.21801
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −311.957 180.108i −0.554097 0.319908i 0.196676 0.980469i \(-0.436985\pi\)
−0.750773 + 0.660561i \(0.770319\pi\)
\(564\) 0 0
\(565\) 16.1481 + 27.9693i 0.0285806 + 0.0495031i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −486.466 + 280.861i −0.854949 + 0.493605i −0.862318 0.506368i \(-0.830988\pi\)
0.00736827 + 0.999973i \(0.497655\pi\)
\(570\) 0 0
\(571\) 19.0385 32.9756i 0.0333423 0.0577506i −0.848873 0.528597i \(-0.822718\pi\)
0.882215 + 0.470847i \(0.156052\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 288.886i 0.502410i
\(576\) 0 0
\(577\) −747.008 −1.29464 −0.647320 0.762218i \(-0.724110\pi\)
−0.647320 + 0.762218i \(0.724110\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −63.3948 36.6010i −0.109113 0.0629965i
\(582\) 0 0
\(583\) 11.2384 + 19.4655i 0.0192768 + 0.0333885i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 519.419 299.887i 0.884870 0.510880i 0.0126092 0.999921i \(-0.495986\pi\)
0.872261 + 0.489040i \(0.162653\pi\)
\(588\) 0 0
\(589\) −382.196 + 661.983i −0.648890 + 1.12391i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 212.184i 0.357814i 0.983866 + 0.178907i \(0.0572561\pi\)
−0.983866 + 0.178907i \(0.942744\pi\)
\(594\) 0 0
\(595\) −1122.24 −1.88612
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 281.649 + 162.610i 0.470198 + 0.271469i 0.716323 0.697769i \(-0.245824\pi\)
−0.246124 + 0.969238i \(0.579157\pi\)
\(600\) 0 0
\(601\) −373.738 647.334i −0.621861 1.07709i −0.989139 0.146983i \(-0.953044\pi\)
0.367278 0.930111i \(-0.380290\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −600.824 + 346.886i −0.993098 + 0.573366i
\(606\) 0 0
\(607\) 61.8634 107.151i 0.101917 0.176525i −0.810558 0.585659i \(-0.800836\pi\)
0.912474 + 0.409134i \(0.134169\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 686.149i 1.12299i
\(612\) 0 0
\(613\) −954.008 −1.55629 −0.778146 0.628083i \(-0.783840\pi\)
−0.778146 + 0.628083i \(0.783840\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 216.613 + 125.061i 0.351074 + 0.202693i 0.665158 0.746702i \(-0.268364\pi\)
−0.314084 + 0.949395i \(0.601697\pi\)
\(618\) 0 0
\(619\) 508.769 + 881.214i 0.821921 + 1.42361i 0.904250 + 0.427004i \(0.140431\pi\)
−0.0823287 + 0.996605i \(0.526236\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 596.983 344.668i 0.958239 0.553240i
\(624\) 0 0
\(625\) 382.975 663.332i 0.612760 1.06133i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 275.133i 0.437413i
\(630\) 0 0
\(631\) −75.6001 −0.119810 −0.0599050 0.998204i \(-0.519080\pi\)
−0.0599050 + 0.998204i \(0.519080\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −342.476 197.729i −0.539333 0.311384i
\(636\) 0 0
\(637\) −60.4230 104.656i −0.0948556 0.164295i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 209.895 121.183i 0.327450 0.189053i −0.327259 0.944935i \(-0.606125\pi\)
0.654708 + 0.755882i \(0.272791\pi\)
\(642\) 0 0
\(643\) 366.592 634.956i 0.570128 0.987490i −0.426424 0.904523i \(-0.640227\pi\)
0.996552 0.0829673i \(-0.0264397\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1177.72i 1.82028i 0.414296 + 0.910142i \(0.364028\pi\)
−0.414296 + 0.910142i \(0.635972\pi\)
\(648\) 0 0
\(649\) 94.8231 0.146106
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 550.151 + 317.630i 0.842497 + 0.486416i 0.858112 0.513462i \(-0.171637\pi\)
−0.0156151 + 0.999878i \(0.504971\pi\)
\(654\) 0 0
\(655\) −446.533 773.417i −0.681729 1.18079i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 793.616 458.194i 1.20427 0.695287i 0.242770 0.970084i \(-0.421944\pi\)
0.961502 + 0.274797i \(0.0886105\pi\)
\(660\) 0 0
\(661\) −206.679 + 357.978i −0.312676 + 0.541571i −0.978941 0.204145i \(-0.934559\pi\)
0.666265 + 0.745715i \(0.267892\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1184.34i 1.78097i
\(666\) 0 0
\(667\) 859.996 1.28935
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 30.2888 + 17.4872i 0.0451397 + 0.0260614i
\(672\) 0 0
\(673\) 366.227 + 634.324i 0.544171 + 0.942531i 0.998659 + 0.0517784i \(0.0164890\pi\)
−0.454488 + 0.890753i \(0.650178\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −507.219 + 292.843i −0.749215 + 0.432559i −0.825410 0.564533i \(-0.809056\pi\)
0.0761952 + 0.997093i \(0.475723\pi\)
\(678\) 0 0
\(679\) 230.235 398.778i 0.339079 0.587302i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1082.97i 1.58560i −0.609480 0.792801i \(-0.708622\pi\)
0.609480 0.792801i \(-0.291378\pi\)
\(684\) 0 0
\(685\) 443.238 0.647063
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −195.069 112.623i −0.283119 0.163459i
\(690\) 0 0
\(691\) 89.1788 + 154.462i 0.129058 + 0.223534i 0.923312 0.384051i \(-0.125471\pi\)
−0.794254 + 0.607586i \(0.792138\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −471.988 + 272.503i −0.679120 + 0.392090i
\(696\) 0 0
\(697\) −1104.48 + 1913.01i −1.58462 + 2.74464i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 638.090i 0.910257i 0.890426 + 0.455129i \(0.150407\pi\)
−0.890426 + 0.455129i \(0.849593\pi\)
\(702\) 0 0
\(703\) −290.358 −0.413026
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 92.6990 + 53.5198i 0.131116 + 0.0756999i
\(708\) 0 0
\(709\) −412.423 714.338i −0.581697 1.00753i −0.995278 0.0970614i \(-0.969056\pi\)
0.413582 0.910467i \(-0.364278\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 675.145 389.795i 0.946908 0.546698i
\(714\) 0 0
\(715\) −37.5289 + 65.0019i −0.0524879 + 0.0909117i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 748.451i 1.04096i −0.853874 0.520480i \(-0.825753\pi\)
0.853874 0.520480i \(-0.174247\pi\)
\(720\) 0 0
\(721\) 693.969 0.962509
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 190.165 + 109.792i 0.262297 + 0.151437i
\(726\) 0 0
\(727\) −13.1788 22.8264i −0.0181276 0.0313980i 0.856819 0.515617i \(-0.172437\pi\)
−0.874947 + 0.484219i \(0.839104\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1617.52 + 933.876i −2.21275 + 1.27753i
\(732\) 0 0
\(733\) 536.473 929.199i 0.731887 1.26767i −0.224189 0.974546i \(-0.571973\pi\)
0.956076 0.293119i \(-0.0946934\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0371i 0.0163325i
\(738\) 0 0
\(739\) 158.831 0.214926 0.107463 0.994209i \(-0.465727\pi\)
0.107463 + 0.994209i \(0.465727\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −142.080 82.0297i −0.191224 0.110403i 0.401331 0.915933i \(-0.368548\pi\)
−0.592556 + 0.805530i \(0.701881\pi\)
\(744\) 0 0
\(745\) 194.619 + 337.090i 0.261234 + 0.452470i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 757.382 437.275i 1.01119 0.583811i
\(750\) 0 0
\(751\) −116.198 + 201.261i −0.154724 + 0.267991i −0.932959 0.359984i \(-0.882782\pi\)
0.778234 + 0.627974i \(0.216116\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 268.646i 0.355823i
\(756\) 0 0
\(757\) 1012.72 1.33781 0.668905 0.743348i \(-0.266763\pi\)
0.668905 + 0.743348i \(0.266763\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −501.427 289.499i −0.658905 0.380419i 0.132954 0.991122i \(-0.457554\pi\)
−0.791860 + 0.610703i \(0.790887\pi\)
\(762\) 0 0
\(763\) 531.963 + 921.388i 0.697200 + 1.20759i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −822.939 + 475.124i −1.07293 + 0.619458i
\(768\) 0 0
\(769\) 133.635 231.462i 0.173777 0.300991i −0.765960 0.642888i \(-0.777736\pi\)
0.939737 + 0.341897i \(0.111069\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1356.98i 1.75548i −0.479140 0.877739i \(-0.659051\pi\)
0.479140 0.877739i \(-0.340949\pi\)
\(774\) 0 0
\(775\) 199.054 0.256844
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2018.87 1165.60i −2.59162 1.49627i
\(780\) 0 0
\(781\) −2.06526 3.57714i −0.00264438 0.00458020i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 851.181 491.430i 1.08431 0.626025i
\(786\) 0 0
\(787\) −28.9404 + 50.1262i −0.0367731 + 0.0636928i −0.883826 0.467815i \(-0.845041\pi\)
0.847053 + 0.531508i \(0.178375\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 34.5285i 0.0436517i
\(792\) 0 0
\(793\) −350.488 −0.441978
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −142.793 82.4418i −0.179164 0.103440i 0.407736 0.913100i \(-0.366318\pi\)
−0.586900 + 0.809660i \(0.699652\pi\)
\(798\) 0 0
\(799\) −941.123 1630.07i −1.17788 2.04014i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 16.9676 9.79627i 0.0211303 0.0121996i
\(804\) 0 0
\(805\) −603.946 + 1046.07i −0.750244 + 1.29946i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 981.255i 1.21292i −0.795113 0.606461i \(-0.792588\pi\)
0.795113 0.606461i \(-0.207412\pi\)
\(810\) 0 0
\(811\) −289.877 −0.357432 −0.178716 0.983901i \(-0.557194\pi\)
−0.178716 + 0.983901i \(0.557194\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1219.76 704.227i −1.49663 0.864082i
\(816\) 0 0
\(817\) −985.554 1707.03i −1.20631 2.08939i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −114.436 + 66.0695i −0.139386 + 0.0804745i −0.568071 0.822979i \(-0.692310\pi\)
0.428685 + 0.903454i \(0.358977\pi\)
\(822\) 0 0
\(823\) −243.096 + 421.055i −0.295378 + 0.511610i −0.975073 0.221885i \(-0.928779\pi\)
0.679695 + 0.733495i \(0.262112\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 258.192i 0.312203i 0.987741 + 0.156101i \(0.0498927\pi\)
−0.987741 + 0.156101i \(0.950107\pi\)
\(828\) 0 0
\(829\) −376.400 −0.454041 −0.227020 0.973890i \(-0.572898\pi\)
−0.227020 + 0.973890i \(0.572898\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 287.092 + 165.753i 0.344648 + 0.198983i
\(834\) 0 0
\(835\) 326.648 + 565.771i 0.391195 + 0.677570i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −252.519 + 145.792i −0.300976 + 0.173768i −0.642881 0.765966i \(-0.722261\pi\)
0.341905 + 0.939734i \(0.388928\pi\)
\(840\) 0 0
\(841\) −93.6556 + 162.216i −0.111362 + 0.192885i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 227.275i 0.268964i
\(846\) 0 0
\(847\) −741.727 −0.875711
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 256.457 + 148.065i 0.301359 + 0.173990i
\(852\) 0 0
\(853\) −96.7691 167.609i −0.113446 0.196494i 0.803712 0.595019i \(-0.202855\pi\)
−0.917157 + 0.398525i \(0.869522\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −165.998 + 95.8387i −0.193696 + 0.111830i −0.593712 0.804678i \(-0.702338\pi\)
0.400016 + 0.916508i \(0.369005\pi\)
\(858\) 0 0
\(859\) −598.258 + 1036.21i −0.696458 + 1.20630i 0.273228 + 0.961949i \(0.411909\pi\)
−0.969687 + 0.244352i \(0.921425\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 692.323i 0.802228i −0.916028 0.401114i \(-0.868623\pi\)
0.916028 0.401114i \(-0.131377\pi\)
\(864\) 0 0
\(865\) 1088.65 1.25856
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 111.617 + 64.4419i 0.128443 + 0.0741564i
\(870\) 0 0
\(871\) −60.3135 104.466i −0.0692462 0.119938i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 510.384 294.670i 0.583296 0.336766i
\(876\) 0 0
\(877\) 513.306 889.072i 0.585297 1.01376i −0.409541 0.912292i \(-0.634311\pi\)
0.994838 0.101473i \(-0.0323555\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 984.820i 1.11784i 0.829220 + 0.558922i \(0.188785\pi\)
−0.829220 + 0.558922i \(0.811215\pi\)
\(882\) 0 0
\(883\) 1038.19 1.17576 0.587878 0.808950i \(-0.299964\pi\)
0.587878 + 0.808950i \(0.299964\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −146.847 84.7821i −0.165555 0.0955830i 0.414933 0.909852i \(-0.363805\pi\)
−0.580488 + 0.814269i \(0.697138\pi\)
\(888\) 0 0
\(889\) −211.396 366.149i −0.237791 0.411866i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1720.28 993.202i 1.92640 1.11221i
\(894\) 0 0
\(895\) 47.4115 82.1192i 0.0529738 0.0917533i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 592.572i 0.659146i
\(900\) 0 0
\(901\) 617.896 0.685789
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1142.39 659.556i −1.26230 0.728792i
\(906\) 0 0
\(907\) −365.981 633.897i −0.403507 0.698894i 0.590640 0.806936i \(-0.298876\pi\)
−0.994146 + 0.108041i \(0.965542\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −853.652 + 492.856i −0.937049 + 0.541006i −0.889034 0.457841i \(-0.848623\pi\)
−0.0480151 + 0.998847i \(0.515290\pi\)
\(912\) 0 0
\(913\) 6.71521 11.6311i 0.00735510 0.0127394i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 954.795i 1.04122i
\(918\) 0 0
\(919\) −839.650 −0.913656 −0.456828 0.889555i \(-0.651015\pi\)
−0.456828 + 0.889555i \(0.651015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 35.8475 + 20.6965i 0.0388380 + 0.0224231i
\(924\) 0 0
\(925\) 37.8057 + 65.4815i 0.0408711 + 0.0707908i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −519.360 + 299.853i −0.559053 + 0.322769i −0.752765 0.658289i \(-0.771280\pi\)
0.193712 + 0.981058i \(0.437947\pi\)
\(930\) 0 0
\(931\) −174.925 + 302.979i −0.187889 + 0.325434i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 205.899i 0.220212i
\(936\) 0 0
\(937\) 1610.24 1.71850 0.859252 0.511553i \(-0.170930\pi\)
0.859252 + 0.511553i \(0.170930\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 762.189 + 440.050i 0.809978 + 0.467641i 0.846948 0.531675i \(-0.178437\pi\)
−0.0369704 + 0.999316i \(0.511771\pi\)
\(942\) 0 0
\(943\) 1188.77 + 2059.02i 1.26063 + 2.18347i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −853.072 + 492.522i −0.900816 + 0.520086i −0.877465 0.479641i \(-0.840767\pi\)
−0.0233509 + 0.999727i \(0.507434\pi\)
\(948\) 0 0
\(949\) −98.1711 + 170.037i −0.103447 + 0.179175i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 845.550i 0.887251i −0.896212 0.443625i \(-0.853692\pi\)
0.896212 0.443625i \(-0.146308\pi\)
\(954\) 0 0
\(955\) −2112.84 −2.21240
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 410.388 + 236.938i 0.427934 + 0.247068i
\(960\) 0 0
\(961\) 211.915 + 367.048i 0.220515 + 0.381944i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 124.898 72.1100i 0.129428 0.0747253i
\(966\) 0 0
\(967\) 513.471 889.358i 0.530994 0.919708i −0.468352 0.883542i \(-0.655152\pi\)
0.999346 0.0361664i \(-0.0115146\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 338.171i 0.348271i 0.984722 + 0.174135i \(0.0557130\pi\)
−0.984722 + 0.174135i \(0.944287\pi\)
\(972\) 0 0
\(973\) −582.677 −0.598846
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 789.485 + 455.809i 0.808070 + 0.466540i 0.846285 0.532730i \(-0.178834\pi\)
−0.0382151 + 0.999270i \(0.512167\pi\)
\(978\) 0 0
\(979\) 63.2365 + 109.529i 0.0645930 + 0.111878i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −829.618 + 478.980i −0.843966 + 0.487264i −0.858610 0.512629i \(-0.828672\pi\)
0.0146445 + 0.999893i \(0.495338\pi\)
\(984\) 0 0
\(985\) −496.269 + 859.563i −0.503827 + 0.872653i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2010.30i 2.03266i
\(990\) 0 0
\(991\) 1771.18 1.78727 0.893633 0.448798i \(-0.148148\pi\)
0.893633 + 0.448798i \(0.148148\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 614.492 + 354.777i 0.617580 + 0.356560i
\(996\) 0 0
\(997\) 544.325 + 942.799i 0.545963 + 0.945635i 0.998546 + 0.0539130i \(0.0171694\pi\)
−0.452583 + 0.891722i \(0.649497\pi\)
\(998\) 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 324.3.g.d.53.4 8
3.2 odd 2 inner 324.3.g.d.53.1 8
4.3 odd 2 1296.3.q.n.1025.4 8
9.2 odd 6 inner 324.3.g.d.269.4 8
9.4 even 3 324.3.c.a.161.1 4
9.5 odd 6 324.3.c.a.161.4 yes 4
9.7 even 3 inner 324.3.g.d.269.1 8
12.11 even 2 1296.3.q.n.1025.1 8
36.7 odd 6 1296.3.q.n.593.1 8
36.11 even 6 1296.3.q.n.593.4 8
36.23 even 6 1296.3.e.f.161.4 4
36.31 odd 6 1296.3.e.f.161.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.3.c.a.161.1 4 9.4 even 3
324.3.c.a.161.4 yes 4 9.5 odd 6
324.3.g.d.53.1 8 3.2 odd 2 inner
324.3.g.d.53.4 8 1.1 even 1 trivial
324.3.g.d.269.1 8 9.7 even 3 inner
324.3.g.d.269.4 8 9.2 odd 6 inner
1296.3.e.f.161.1 4 36.31 odd 6
1296.3.e.f.161.4 4 36.23 even 6
1296.3.q.n.593.1 8 36.7 odd 6
1296.3.q.n.593.4 8 36.11 even 6
1296.3.q.n.1025.1 8 12.11 even 2
1296.3.q.n.1025.4 8 4.3 odd 2