Properties

Label 2736.2.s.p.577.1
Level $2736$
Weight $2$
Character 2736.577
Analytic conductor $21.847$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(577,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.577
Dual form 2736.2.s.p.1873.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{5} +3.00000 q^{7} +O(q^{10})\) \(q+(1.00000 + 1.73205i) q^{5} +3.00000 q^{7} +(-2.50000 + 4.33013i) q^{13} +(-2.00000 - 3.46410i) q^{17} +(4.00000 + 1.73205i) q^{19} +(-3.00000 + 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-2.00000 + 3.46410i) q^{29} +7.00000 q^{31} +(3.00000 + 5.19615i) q^{35} -1.00000 q^{37} +(5.50000 + 9.52628i) q^{43} +(-3.00000 + 5.19615i) q^{47} +2.00000 q^{49} +(-1.00000 + 1.73205i) q^{53} +(-4.00000 - 6.92820i) q^{59} +(-3.50000 + 6.06218i) q^{61} -10.0000 q^{65} +(1.50000 - 2.59808i) q^{67} +(-3.00000 - 5.19615i) q^{71} +(-4.50000 - 7.79423i) q^{73} +(2.50000 + 4.33013i) q^{79} +16.0000 q^{83} +(4.00000 - 6.92820i) q^{85} +(6.00000 - 10.3923i) q^{89} +(-7.50000 + 12.9904i) q^{91} +(1.00000 + 8.66025i) q^{95} +(7.00000 + 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 6 q^{7} - 5 q^{13} - 4 q^{17} + 8 q^{19} - 6 q^{23} + q^{25} - 4 q^{29} + 14 q^{31} + 6 q^{35} - 2 q^{37} + 11 q^{43} - 6 q^{47} + 4 q^{49} - 2 q^{53} - 8 q^{59} - 7 q^{61} - 20 q^{65} + 3 q^{67} - 6 q^{71} - 9 q^{73} + 5 q^{79} + 32 q^{83} + 8 q^{85} + 12 q^{89} - 15 q^{91} + 2 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 3.46410i −0.485071 0.840168i 0.514782 0.857321i \(-0.327873\pi\)
−0.999853 + 0.0171533i \(0.994540\pi\)
\(18\) 0 0
\(19\) 4.00000 + 1.73205i 0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 + 3.46410i −0.371391 + 0.643268i −0.989780 0.142605i \(-0.954452\pi\)
0.618389 + 0.785872i \(0.287786\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 + 5.19615i 0.507093 + 0.878310i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 5.50000 + 9.52628i 0.838742 + 1.45274i 0.890947 + 0.454108i \(0.150042\pi\)
−0.0522047 + 0.998636i \(0.516625\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 + 5.19615i −0.437595 + 0.757937i −0.997503 0.0706177i \(-0.977503\pi\)
0.559908 + 0.828554i \(0.310836\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 + 1.73205i −0.137361 + 0.237915i −0.926497 0.376303i \(-0.877195\pi\)
0.789136 + 0.614218i \(0.210529\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 6.92820i −0.520756 0.901975i −0.999709 0.0241347i \(-0.992317\pi\)
0.478953 0.877841i \(-0.341016\pi\)
\(60\) 0 0
\(61\) −3.50000 + 6.06218i −0.448129 + 0.776182i −0.998264 0.0588933i \(-0.981243\pi\)
0.550135 + 0.835076i \(0.314576\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −10.0000 −1.24035
\(66\) 0 0
\(67\) 1.50000 2.59808i 0.183254 0.317406i −0.759733 0.650236i \(-0.774670\pi\)
0.942987 + 0.332830i \(0.108004\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.00000 5.19615i −0.356034 0.616670i 0.631260 0.775571i \(-0.282538\pi\)
−0.987294 + 0.158901i \(0.949205\pi\)
\(72\) 0 0
\(73\) −4.50000 7.79423i −0.526685 0.912245i −0.999517 0.0310925i \(-0.990101\pi\)
0.472831 0.881153i \(-0.343232\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.50000 + 4.33013i 0.281272 + 0.487177i 0.971698 0.236225i \(-0.0759104\pi\)
−0.690426 + 0.723403i \(0.742577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 4.00000 6.92820i 0.433861 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 10.3923i 0.635999 1.10158i −0.350304 0.936636i \(-0.613922\pi\)
0.986303 0.164946i \(-0.0527450\pi\)
\(90\) 0 0
\(91\) −7.50000 + 12.9904i −0.786214 + 1.36176i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 + 8.66025i 0.102598 + 0.888523i
\(96\) 0 0
\(97\) 7.00000 + 12.1244i 0.710742 + 1.23104i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.00000 + 15.5885i −0.895533 + 1.55111i −0.0623905 + 0.998052i \(0.519872\pi\)
−0.833143 + 0.553058i \(0.813461\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) −9.00000 15.5885i −0.862044 1.49310i −0.869953 0.493135i \(-0.835851\pi\)
0.00790932 0.999969i \(-0.497482\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 10.3923i −0.550019 0.952661i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 6.00000 10.3923i 0.532414 0.922168i −0.466870 0.884326i \(-0.654618\pi\)
0.999284 0.0378419i \(-0.0120483\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) 12.0000 + 5.19615i 1.04053 + 0.450564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 + 15.5885i −0.768922 + 1.33181i 0.169226 + 0.985577i \(0.445873\pi\)
−0.938148 + 0.346235i \(0.887460\pi\)
\(138\) 0 0
\(139\) 2.50000 4.33013i 0.212047 0.367277i −0.740308 0.672268i \(-0.765320\pi\)
0.952355 + 0.304991i \(0.0986536\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.00000 3.46410i −0.163846 0.283790i 0.772399 0.635138i \(-0.219057\pi\)
−0.936245 + 0.351348i \(0.885723\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.00000 + 12.1244i 0.562254 + 0.973852i
\(156\) 0 0
\(157\) 6.50000 + 11.2583i 0.518756 + 0.898513i 0.999762 + 0.0217953i \(0.00693820\pi\)
−0.481006 + 0.876717i \(0.659728\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.00000 + 15.5885i −0.709299 + 1.22854i
\(162\) 0 0
\(163\) −7.00000 −0.548282 −0.274141 0.961689i \(-0.588394\pi\)
−0.274141 + 0.961689i \(0.588394\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 + 13.8564i −0.619059 + 1.07224i 0.370599 + 0.928793i \(0.379152\pi\)
−0.989658 + 0.143448i \(0.954181\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 + 3.46410i 0.152057 + 0.263371i 0.931984 0.362500i \(-0.118077\pi\)
−0.779926 + 0.625871i \(0.784744\pi\)
\(174\) 0 0
\(175\) 1.50000 2.59808i 0.113389 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) −7.00000 + 12.1244i −0.520306 + 0.901196i 0.479415 + 0.877588i \(0.340849\pi\)
−0.999721 + 0.0236082i \(0.992485\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.00000 1.73205i −0.0735215 0.127343i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) −1.50000 2.59808i −0.107972 0.187014i 0.806976 0.590584i \(-0.201102\pi\)
−0.914949 + 0.403570i \(0.867769\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −1.50000 + 2.59808i −0.106332 + 0.184173i −0.914282 0.405079i \(-0.867244\pi\)
0.807950 + 0.589252i \(0.200577\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.00000 + 10.3923i −0.421117 + 0.729397i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.50000 2.59808i −0.103264 0.178859i 0.809763 0.586756i \(-0.199595\pi\)
−0.913028 + 0.407898i \(0.866262\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −11.0000 + 19.0526i −0.750194 + 1.29937i
\(216\) 0 0
\(217\) 21.0000 1.42557
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) 0.500000 + 0.866025i 0.0334825 + 0.0579934i 0.882281 0.470723i \(-0.156007\pi\)
−0.848799 + 0.528716i \(0.822674\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.0000 + 22.5167i 0.851658 + 1.47512i 0.879711 + 0.475509i \(0.157736\pi\)
−0.0280525 + 0.999606i \(0.508931\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −10.5000 + 18.1865i −0.676364 + 1.17150i 0.299704 + 0.954032i \(0.403112\pi\)
−0.976068 + 0.217465i \(0.930221\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.00000 + 3.46410i 0.127775 + 0.221313i
\(246\) 0 0
\(247\) −17.5000 + 12.9904i −1.11350 + 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.00000 8.66025i 0.315597 0.546630i −0.663967 0.747762i \(-0.731128\pi\)
0.979564 + 0.201131i \(0.0644618\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 15.5885i 0.561405 0.972381i −0.435970 0.899961i \(-0.643595\pi\)
0.997374 0.0724199i \(-0.0230722\pi\)
\(258\) 0 0
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.00000 + 12.1244i 0.431638 + 0.747620i 0.997015 0.0772134i \(-0.0246023\pi\)
−0.565376 + 0.824833i \(0.691269\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.00000 + 3.46410i 0.121942 + 0.211210i 0.920534 0.390664i \(-0.127754\pi\)
−0.798591 + 0.601874i \(0.794421\pi\)
\(270\) 0 0
\(271\) −4.00000 6.92820i −0.242983 0.420858i 0.718580 0.695444i \(-0.244792\pi\)
−0.961563 + 0.274586i \(0.911459\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.0000 22.5167i 0.775515 1.34323i −0.158990 0.987280i \(-0.550824\pi\)
0.934505 0.355951i \(-0.115843\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.0294118 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 34.0000 1.98630 0.993151 0.116841i \(-0.0372769\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) 0 0
\(295\) 8.00000 13.8564i 0.465778 0.806751i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −15.0000 25.9808i −0.867472 1.50251i
\(300\) 0 0
\(301\) 16.5000 + 28.5788i 0.951044 + 1.64726i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 0 0
\(307\) −10.0000 17.3205i −0.570730 0.988534i −0.996491 0.0836980i \(-0.973327\pi\)
0.425761 0.904836i \(-0.360006\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) 11.0000 19.0526i 0.621757 1.07691i −0.367402 0.930062i \(-0.619753\pi\)
0.989158 0.146852i \(-0.0469141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 15.5885i 0.505490 0.875535i −0.494489 0.869184i \(-0.664645\pi\)
0.999980 0.00635137i \(-0.00202172\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.00000 17.3205i −0.111283 0.963739i
\(324\) 0 0
\(325\) 2.50000 + 4.33013i 0.138675 + 0.240192i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 + 15.5885i −0.496186 + 0.859419i
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −4.50000 7.79423i −0.245131 0.424579i 0.717038 0.697034i \(-0.245498\pi\)
−0.962168 + 0.272456i \(0.912164\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 15.5885i −0.483145 0.836832i 0.516667 0.856186i \(-0.327172\pi\)
−0.999813 + 0.0193540i \(0.993839\pi\)
\(348\) 0 0
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 6.00000 10.3923i 0.318447 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.00000 3.46410i −0.105556 0.182828i 0.808409 0.588621i \(-0.200329\pi\)
−0.913965 + 0.405793i \(0.866996\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.00000 15.5885i 0.471082 0.815937i
\(366\) 0 0
\(367\) 15.5000 26.8468i 0.809093 1.40139i −0.104399 0.994535i \(-0.533292\pi\)
0.913493 0.406855i \(-0.133375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.00000 + 5.19615i −0.155752 + 0.269771i
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.0000 17.3205i −0.515026 0.892052i
\(378\) 0 0
\(379\) 21.0000 1.07870 0.539349 0.842082i \(-0.318670\pi\)
0.539349 + 0.842082i \(0.318670\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 13.8564i −0.408781 0.708029i 0.585973 0.810331i \(-0.300713\pi\)
−0.994753 + 0.102302i \(0.967379\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −19.0000 + 32.9090i −0.963338 + 1.66855i −0.249323 + 0.968420i \(0.580208\pi\)
−0.714015 + 0.700130i \(0.753125\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.00000 + 8.66025i −0.251577 + 0.435745i
\(396\) 0 0
\(397\) −3.50000 6.06218i −0.175660 0.304252i 0.764730 0.644351i \(-0.222873\pi\)
−0.940389 + 0.340099i \(0.889539\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 20.7846i −0.599251 1.03793i −0.992932 0.118686i \(-0.962132\pi\)
0.393680 0.919247i \(-0.371202\pi\)
\(402\) 0 0
\(403\) −17.5000 + 30.3109i −0.871737 + 1.50989i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 13.0000 22.5167i 0.642809 1.11338i −0.341994 0.939702i \(-0.611102\pi\)
0.984803 0.173675i \(-0.0555643\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −12.0000 20.7846i −0.590481 1.02274i
\(414\) 0 0
\(415\) 16.0000 + 27.7128i 0.785409 + 1.36037i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −19.0000 32.9090i −0.926003 1.60388i −0.789940 0.613185i \(-0.789888\pi\)
−0.136064 0.990700i \(-0.543445\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −10.5000 + 18.1865i −0.508131 + 0.880108i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 25.9808i 0.722525 1.25145i −0.237460 0.971397i \(-0.576315\pi\)
0.959985 0.280052i \(-0.0903517\pi\)
\(432\) 0 0
\(433\) −9.50000 + 16.4545i −0.456541 + 0.790752i −0.998775 0.0494752i \(-0.984245\pi\)
0.542234 + 0.840227i \(0.317578\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −21.0000 + 15.5885i −1.00457 + 0.745697i
\(438\) 0 0
\(439\) −2.50000 4.33013i −0.119318 0.206666i 0.800179 0.599761i \(-0.204738\pi\)
−0.919498 + 0.393095i \(0.871404\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0000 17.3205i 0.475114 0.822922i −0.524479 0.851423i \(-0.675740\pi\)
0.999594 + 0.0285009i \(0.00907336\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −30.0000 −1.40642
\(456\) 0 0
\(457\) −23.0000 −1.07589 −0.537947 0.842978i \(-0.680800\pi\)
−0.537947 + 0.842978i \(0.680800\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.0000 + 20.7846i 0.558896 + 0.968036i 0.997589 + 0.0693989i \(0.0221081\pi\)
−0.438693 + 0.898637i \(0.644559\pi\)
\(462\) 0 0
\(463\) 1.00000 0.0464739 0.0232370 0.999730i \(-0.492603\pi\)
0.0232370 + 0.999730i \(0.492603\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) 4.50000 7.79423i 0.207791 0.359904i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.50000 2.59808i 0.160591 0.119208i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.00000 8.66025i 0.228456 0.395697i −0.728895 0.684626i \(-0.759966\pi\)
0.957351 + 0.288929i \(0.0932990\pi\)
\(480\) 0 0
\(481\) 2.50000 4.33013i 0.113990 0.197437i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.0000 + 24.2487i −0.635707 + 1.10108i
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.0000 + 22.5167i 0.586682 + 1.01616i 0.994663 + 0.103173i \(0.0328994\pi\)
−0.407982 + 0.912990i \(0.633767\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 15.5885i −0.403705 0.699238i
\(498\) 0 0
\(499\) −5.50000 9.52628i −0.246214 0.426455i 0.716258 0.697835i \(-0.245853\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 10.3923i 0.267527 0.463370i −0.700696 0.713460i \(-0.747127\pi\)
0.968223 + 0.250090i \(0.0804603\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.00000 8.66025i 0.221621 0.383859i −0.733679 0.679496i \(-0.762199\pi\)
0.955300 + 0.295637i \(0.0955319\pi\)
\(510\) 0 0
\(511\) −13.5000 23.3827i −0.597205 1.03439i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −13.0000 22.5167i −0.572848 0.992203i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 20.5000 35.5070i 0.896402 1.55261i 0.0643431 0.997928i \(-0.479505\pi\)
0.832059 0.554687i \(-0.187162\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.0000 24.2487i −0.609850 1.05629i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.0000 + 17.3205i 0.432338 + 0.748831i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.500000 + 0.866025i −0.0214967 + 0.0372333i −0.876574 0.481268i \(-0.840176\pi\)
0.855077 + 0.518501i \(0.173510\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 18.0000 31.1769i 0.771035 1.33547i
\(546\) 0 0
\(547\) −8.50000 + 14.7224i −0.363434 + 0.629486i −0.988524 0.151067i \(-0.951729\pi\)
0.625090 + 0.780553i \(0.285062\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −14.0000 + 10.3923i −0.596420 + 0.442727i
\(552\) 0 0
\(553\) 7.50000 + 12.9904i 0.318932 + 0.552407i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.0000 32.9090i 0.805056 1.39440i −0.111198 0.993798i \(-0.535469\pi\)
0.916253 0.400599i \(-0.131198\pi\)
\(558\) 0 0
\(559\) −55.0000 −2.32625
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −34.0000 −1.43293 −0.716465 0.697623i \(-0.754241\pi\)
−0.716465 + 0.697623i \(0.754241\pi\)
\(564\) 0 0
\(565\) 4.00000 + 6.92820i 0.168281 + 0.291472i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) 43.0000 1.79949 0.899747 0.436412i \(-0.143751\pi\)
0.899747 + 0.436412i \(0.143751\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.00000 + 5.19615i 0.125109 + 0.216695i
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 48.0000 1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.0000 + 38.1051i 0.908037 + 1.57277i 0.816788 + 0.576938i \(0.195753\pi\)
0.0912496 + 0.995828i \(0.470914\pi\)
\(588\) 0 0
\(589\) 28.0000 + 12.1244i 1.15372 + 0.499575i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.0000 20.7846i 0.492781 0.853522i −0.507184 0.861838i \(-0.669314\pi\)
0.999965 + 0.00831589i \(0.00264706\pi\)
\(594\) 0 0
\(595\) 12.0000 20.7846i 0.491952 0.852086i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.0000 + 27.7128i −0.653742 + 1.13231i 0.328465 + 0.944516i \(0.393469\pi\)
−0.982208 + 0.187799i \(0.939865\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.0000 19.0526i −0.447214 0.774597i
\(606\) 0 0
\(607\) 27.0000 1.09590 0.547948 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15.0000 25.9808i −0.606835 1.05107i
\(612\) 0 0
\(613\) −13.0000 22.5167i −0.525065 0.909439i −0.999574 0.0291886i \(-0.990708\pi\)
0.474509 0.880251i \(-0.342626\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 31.1769i 0.724653 1.25514i −0.234464 0.972125i \(-0.575334\pi\)
0.959117 0.283011i \(-0.0913331\pi\)
\(618\) 0 0
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 18.0000 31.1769i 0.721155 1.24908i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.00000 + 3.46410i 0.0797452 + 0.138123i
\(630\) 0 0
\(631\) −5.50000 + 9.52628i −0.218952 + 0.379235i −0.954488 0.298250i \(-0.903597\pi\)
0.735536 + 0.677485i \(0.236930\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) −5.00000 + 8.66025i −0.198107 + 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.00000 15.5885i −0.355479 0.615707i 0.631721 0.775196i \(-0.282349\pi\)
−0.987200 + 0.159489i \(0.949015\pi\)
\(642\) 0 0
\(643\) 14.5000 + 25.1147i 0.571824 + 0.990429i 0.996379 + 0.0850262i \(0.0270974\pi\)
−0.424555 + 0.905402i \(0.639569\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) −6.00000 + 10.3923i −0.234439 + 0.406061i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) 23.0000 39.8372i 0.894596 1.54949i 0.0602929 0.998181i \(-0.480797\pi\)
0.834303 0.551306i \(-0.185870\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 + 25.9808i 0.116335 + 1.00749i
\(666\) 0 0
\(667\) −12.0000 20.7846i −0.464642 0.804783i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −21.0000 −0.809491 −0.404745 0.914429i \(-0.632640\pi\)
−0.404745 + 0.914429i \(0.632640\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 0 0
\(679\) 21.0000 + 36.3731i 0.805906 + 1.39587i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.0000 −0.841807 −0.420903 0.907106i \(-0.638287\pi\)
−0.420903 + 0.907106i \(0.638287\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.00000 8.66025i −0.190485 0.329929i
\(690\) 0 0
\(691\) 36.0000 1.36950 0.684752 0.728776i \(-0.259910\pi\)
0.684752 + 0.728776i \(0.259910\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.0000 0.379322
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3.00000 5.19615i −0.113308 0.196256i 0.803794 0.594908i \(-0.202811\pi\)
−0.917102 + 0.398652i \(0.869478\pi\)
\(702\) 0 0
\(703\) −4.00000 1.73205i −0.150863 0.0653255i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −27.0000 + 46.7654i −1.01544 + 1.75879i
\(708\) 0 0
\(709\) 1.50000 2.59808i 0.0563337 0.0975728i −0.836483 0.547992i \(-0.815392\pi\)
0.892817 + 0.450420i \(0.148726\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −21.0000 + 36.3731i −0.786456 + 1.36218i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.0000 22.5167i −0.484818 0.839730i 0.515030 0.857172i \(-0.327781\pi\)
−0.999848 + 0.0174426i \(0.994448\pi\)
\(720\) 0 0
\(721\) −39.0000 −1.45244
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000 + 3.46410i 0.0742781 + 0.128654i
\(726\) 0 0
\(727\) 6.50000 + 11.2583i 0.241072 + 0.417548i 0.961020 0.276479i \(-0.0891678\pi\)
−0.719948 + 0.694028i \(0.755834\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 22.0000 38.1051i 0.813699 1.40937i
\(732\) 0 0
\(733\) −50.0000 −1.84679 −0.923396 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −14.5000 25.1147i −0.533391 0.923861i −0.999239 0.0389959i \(-0.987584\pi\)
0.465848 0.884865i \(-0.345749\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.00000 + 10.3923i 0.220119 + 0.381257i 0.954844 0.297108i \(-0.0960222\pi\)
−0.734725 + 0.678365i \(0.762689\pi\)
\(744\) 0 0
\(745\) 4.00000 6.92820i 0.146549 0.253830i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) 14.5000 25.1147i 0.529113 0.916450i −0.470311 0.882501i \(-0.655858\pi\)
0.999424 0.0339490i \(-0.0108084\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 + 27.7128i 0.582300 + 1.00857i
\(756\) 0 0
\(757\) 3.50000 + 6.06218i 0.127210 + 0.220334i 0.922595 0.385771i \(-0.126065\pi\)
−0.795385 + 0.606105i \(0.792731\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.0000 −0.724999 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(762\) 0 0
\(763\) −27.0000 46.7654i −0.977466 1.69302i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 18.5000 32.0429i 0.667127 1.15550i −0.311577 0.950221i \(-0.600857\pi\)
0.978704 0.205277i \(-0.0658095\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17.0000 29.4449i 0.611448 1.05906i −0.379549 0.925172i \(-0.623921\pi\)
0.990997 0.133887i \(-0.0427458\pi\)
\(774\) 0 0
\(775\) 3.50000 6.06218i 0.125724 0.217760i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −13.0000 + 22.5167i −0.463990 + 0.803654i
\(786\) 0 0
\(787\) 43.0000 1.53278 0.766392 0.642373i \(-0.222050\pi\)
0.766392 + 0.642373i \(0.222050\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −17.5000 30.3109i −0.621443 1.07637i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −36.0000 −1.26883
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.00000 12.1244i −0.245199 0.424698i
\(816\) 0 0
\(817\) 5.50000 + 47.6314i 0.192421 + 1.66641i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.00000 3.46410i 0.0698005 0.120898i −0.829013 0.559229i \(-0.811097\pi\)
0.898813 + 0.438331i \(0.144430\pi\)
\(822\) 0 0
\(823\) −16.0000 + 27.7128i −0.557725 + 0.966008i 0.439961 + 0.898017i \(0.354992\pi\)
−0.997686 + 0.0679910i \(0.978341\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −16.0000 + 27.7128i −0.556375 + 0.963669i 0.441421 + 0.897300i \(0.354475\pi\)
−0.997795 + 0.0663686i \(0.978859\pi\)
\(828\) 0 0
\(829\) −27.0000 −0.937749 −0.468874 0.883265i \(-0.655340\pi\)
−0.468874 + 0.883265i \(0.655340\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.00000 6.92820i −0.138592 0.240048i
\(834\) 0 0
\(835\) −32.0000 −1.10741
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.0000 36.3731i −0.725001 1.25574i −0.958974 0.283495i \(-0.908506\pi\)
0.233973 0.972243i \(-0.424827\pi\)
\(840\) 0 0
\(841\) 6.50000 + 11.2583i 0.224138 + 0.388218i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.0000 20.7846i 0.412813 0.715012i
\(846\) 0 0
\(847\) −33.0000 −1.13389
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.00000 5.19615i 0.102839 0.178122i
\(852\) 0 0
\(853\) −7.50000 12.9904i −0.256795 0.444782i 0.708586 0.705624i \(-0.249333\pi\)
−0.965382 + 0.260842i \(0.916000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19.0000 32.9090i −0.649028 1.12415i −0.983355 0.181692i \(-0.941843\pi\)
0.334328 0.942457i \(-0.391491\pi\)
\(858\) 0 0
\(859\) −0.500000 + 0.866025i −0.0170598 + 0.0295484i −0.874429 0.485153i \(-0.838764\pi\)
0.857369 + 0.514701i \(0.172097\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −4.00000 + 6.92820i −0.136004 + 0.235566i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 7.50000 + 12.9904i 0.254128 + 0.440162i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 36.0000 1.21702
\(876\) 0 0
\(877\) 2.50000 + 4.33013i 0.0844190 + 0.146218i 0.905143 0.425106i \(-0.139763\pi\)
−0.820724 + 0.571324i \(0.806430\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 0 0
\(883\) 5.50000 9.52628i 0.185090 0.320585i −0.758517 0.651653i \(-0.774076\pi\)
0.943607 + 0.331068i \(0.107409\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.0000 45.0333i 0.872995 1.51207i 0.0141108 0.999900i \(-0.495508\pi\)
0.858884 0.512170i \(-0.171158\pi\)
\(888\) 0 0
\(889\) 18.0000 31.1769i 0.603701 1.04564i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −21.0000 + 15.5885i −0.702738 + 0.521648i
\(894\) 0 0
\(895\) 6.00000 + 10.3923i 0.200558 + 0.347376i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −14.0000 + 24.2487i −0.466926 + 0.808740i
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −28.0000 −0.930751
\(906\) 0 0
\(907\) 14.0000 + 24.2487i 0.464862 + 0.805165i 0.999195 0.0401089i \(-0.0127705\pi\)
−0.534333 + 0.845274i \(0.679437\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.00000 + 15.5885i 0.297206 + 0.514776i
\(918\) 0 0
\(919\) −31.0000 −1.02260 −0.511298 0.859404i \(-0.670835\pi\)
−0.511298 + 0.859404i \(0.670835\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.0000 0.987462
\(924\) 0 0
\(925\) −0.500000 + 0.866025i −0.0164399 + 0.0284747i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.00000 + 15.5885i 0.295280 + 0.511441i 0.975050 0.221985i \(-0.0712536\pi\)
−0.679770 + 0.733426i \(0.737920\pi\)
\(930\) 0 0
\(931\) 8.00000 + 3.46410i 0.262189 + 0.113531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 24.5000 42.4352i 0.800380 1.38630i −0.118986 0.992896i \(-0.537964\pi\)
0.919366 0.393403i \(-0.128702\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.0000 + 34.6410i −0.651981 + 1.12926i 0.330660 + 0.943750i \(0.392729\pi\)
−0.982641 + 0.185515i \(0.940605\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.0000 + 19.0526i 0.357452 + 0.619125i 0.987534 0.157403i \(-0.0503122\pi\)
−0.630082 + 0.776528i \(0.716979\pi\)
\(948\) 0 0
\(949\) 45.0000 1.46076
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20.0000 + 34.6410i 0.647864 + 1.12213i 0.983632 + 0.180188i \(0.0576706\pi\)
−0.335769 + 0.941944i \(0.608996\pi\)
\(954\) 0 0
\(955\) 6.00000 + 10.3923i 0.194155 + 0.336287i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −27.0000 + 46.7654i −0.871875 + 1.51013i
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.00000 5.19615i 0.0965734 0.167270i
\(966\) 0 0
\(967\) −28.5000 49.3634i −0.916498 1.58742i −0.804693 0.593691i \(-0.797670\pi\)
−0.111805 0.993730i \(-0.535663\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.0000 43.3013i −0.802288 1.38960i −0.918107 0.396333i \(-0.870282\pi\)
0.115818 0.993270i \(-0.463051\pi\)
\(972\) 0 0
\(973\) 7.50000 12.9904i 0.240439 0.416452i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10.0000 17.3205i −0.318950 0.552438i 0.661319 0.750105i \(-0.269997\pi\)
−0.980269 + 0.197666i \(0.936664\pi\)
\(984\) 0 0
\(985\) 12.0000 + 20.7846i 0.382352 + 0.662253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −66.0000 −2.09868
\(990\) 0 0
\(991\) −12.5000 21.6506i −0.397076 0.687755i 0.596288 0.802771i \(-0.296642\pi\)
−0.993364 + 0.115015i \(0.963308\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) 8.50000 14.7224i 0.269198 0.466264i −0.699457 0.714675i \(-0.746575\pi\)
0.968655 + 0.248410i \(0.0799082\pi\)
\(998\) 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 2736.2.s.p.577.1 2
3.2 odd 2 912.2.q.e.577.1 2
4.3 odd 2 1368.2.s.e.577.1 2
12.11 even 2 456.2.q.a.121.1 yes 2
19.11 even 3 inner 2736.2.s.p.1873.1 2
57.11 odd 6 912.2.q.e.49.1 2
76.11 odd 6 1368.2.s.e.505.1 2
228.11 even 6 456.2.q.a.49.1 2
228.83 even 6 8664.2.a.m.1.1 1
228.107 odd 6 8664.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.q.a.49.1 2 228.11 even 6
456.2.q.a.121.1 yes 2 12.11 even 2
912.2.q.e.49.1 2 57.11 odd 6
912.2.q.e.577.1 2 3.2 odd 2
1368.2.s.e.505.1 2 76.11 odd 6
1368.2.s.e.577.1 2 4.3 odd 2
2736.2.s.p.577.1 2 1.1 even 1 trivial
2736.2.s.p.1873.1 2 19.11 even 3 inner
8664.2.a.e.1.1 1 228.107 odd 6
8664.2.a.m.1.1 1 228.83 even 6