Properties

Label 3528.2.s.e.361.1
Level $3528$
Weight $2$
Character 3528.361
Analytic conductor $28.171$
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] = [3528,2,Mod(361,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("3528.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.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: \(28.1712218331\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3528.361
Dual form 3528.2.s.e.3313.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{5} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{11} -2.00000 q^{13} +(3.00000 - 5.19615i) q^{17} +(4.00000 + 6.92820i) q^{19} +(0.500000 - 0.866025i) q^{25} -6.00000 q^{29} +(4.00000 - 6.92820i) q^{31} +(1.00000 + 1.73205i) q^{37} +2.00000 q^{41} -4.00000 q^{43} +(4.00000 + 6.92820i) q^{47} +(3.00000 - 5.19615i) q^{53} +8.00000 q^{55} +(-3.00000 - 5.19615i) q^{61} +(2.00000 + 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} +8.00000 q^{71} +(5.00000 - 8.66025i) q^{73} +(-8.00000 - 13.8564i) q^{79} +8.00000 q^{83} -12.0000 q^{85} +(3.00000 + 5.19615i) q^{89} +(8.00000 - 13.8564i) q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 4 q^{11} - 4 q^{13} + 6 q^{17} + 8 q^{19} + q^{25} - 12 q^{29} + 8 q^{31} + 2 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} + 6 q^{53} + 16 q^{55} - 6 q^{61} + 4 q^{65} + 4 q^{67} + 16 q^{71} + 10 q^{73} - 16 q^{79} + 16 q^{83} - 24 q^{85} + 6 q^{89} + 16 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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 0.550990 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 5.19615i 0.727607 1.26025i −0.230285 0.973123i \(-0.573966\pi\)
0.957892 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) 4.00000 + 6.92820i 0.917663 + 1.58944i 0.802955 + 0.596040i \(0.203260\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 6.92820i 0.718421 1.24434i −0.243204 0.969975i \(-0.578198\pi\)
0.961625 0.274367i \(-0.0884683\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00000 5.19615i 0.412082 0.713746i −0.583036 0.812447i \(-0.698135\pi\)
0.995117 + 0.0987002i \(0.0314685\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −3.00000 5.19615i −0.384111 0.665299i 0.607535 0.794293i \(-0.292159\pi\)
−0.991645 + 0.128994i \(0.958825\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 + 3.46410i 0.248069 + 0.429669i
\(66\) 0 0
\(67\) 2.00000 3.46410i 0.244339 0.423207i −0.717607 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 5.00000 8.66025i 0.585206 1.01361i −0.409644 0.912245i \(-0.634347\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 13.8564i −0.900070 1.55897i −0.827401 0.561611i \(-0.810182\pi\)
−0.0726692 0.997356i \(-0.523152\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 13.8564i 0.820783 1.42164i
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.00000 + 1.73205i −0.0995037 + 0.172345i −0.911479 0.411346i \(-0.865059\pi\)
0.811976 + 0.583691i \(0.198392\pi\)
\(102\) 0 0
\(103\) −8.00000 13.8564i −0.788263 1.36531i −0.927030 0.374987i \(-0.877647\pi\)
0.138767 0.990325i \(-0.455686\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 10.3923i −0.580042 1.00466i −0.995474 0.0950377i \(-0.969703\pi\)
0.415432 0.909624i \(-0.363630\pi\)
\(108\) 0 0
\(109\) 5.00000 8.66025i 0.478913 0.829502i −0.520794 0.853682i \(-0.674364\pi\)
0.999708 + 0.0241802i \(0.00769755\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 6.92820i −0.349482 0.605320i 0.636676 0.771132i \(-0.280309\pi\)
−0.986157 + 0.165812i \(0.946976\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 + 5.19615i −0.256307 + 0.443937i −0.965250 0.261329i \(-0.915839\pi\)
0.708942 + 0.705266i \(0.249173\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 6.92820i 0.334497 0.579365i
\(144\) 0 0
\(145\) 6.00000 + 10.3923i 0.498273 + 0.863034i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) 9.00000 15.5885i 0.718278 1.24409i −0.243403 0.969925i \(-0.578264\pi\)
0.961681 0.274169i \(-0.0884028\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.00000 + 10.3923i 0.469956 + 0.813988i 0.999410 0.0343508i \(-0.0109363\pi\)
−0.529454 + 0.848339i \(0.677603\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.00000 15.5885i −0.684257 1.18517i −0.973670 0.227964i \(-0.926793\pi\)
0.289412 0.957205i \(-0.406540\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00000 + 3.46410i −0.149487 + 0.258919i −0.931038 0.364922i \(-0.881096\pi\)
0.781551 + 0.623841i \(0.214429\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.00000 3.46410i 0.147043 0.254686i
\(186\) 0 0
\(187\) 12.0000 + 20.7846i 0.877527 + 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 13.8564i −0.578860 1.00261i −0.995610 0.0935936i \(-0.970165\pi\)
0.416751 0.909021i \(-0.363169\pi\)
\(192\) 0 0
\(193\) 7.00000 12.1244i 0.503871 0.872730i −0.496119 0.868255i \(-0.665242\pi\)
0.999990 0.00447566i \(-0.00142465\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 3.46410i −0.139686 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 + 6.92820i 0.272798 + 0.472500i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 + 10.3923i −0.403604 + 0.699062i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000 6.92820i 0.265489 0.459841i −0.702202 0.711977i \(-0.747800\pi\)
0.967692 + 0.252136i \(0.0811332\pi\)
\(228\) 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
\(233\) −11.0000 19.0526i −0.720634 1.24817i −0.960746 0.277429i \(-0.910518\pi\)
0.240112 0.970745i \(-0.422816\pi\)
\(234\) 0 0
\(235\) 8.00000 13.8564i 0.521862 0.903892i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 5.00000 8.66025i 0.322078 0.557856i −0.658838 0.752285i \(-0.728952\pi\)
0.980917 + 0.194429i \(0.0622852\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 13.8564i −0.509028 0.881662i
\(248\) 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
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.00000 1.73205i −0.0623783 0.108042i 0.833150 0.553047i \(-0.186535\pi\)
−0.895528 + 0.445005i \(0.853202\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 20.7846i 0.739952 1.28163i −0.212565 0.977147i \(-0.568182\pi\)
0.952517 0.304487i \(-0.0984850\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.00000 12.1244i 0.426798 0.739235i −0.569789 0.821791i \(-0.692975\pi\)
0.996586 + 0.0825561i \(0.0263084\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 + 3.46410i 0.120605 + 0.208893i
\(276\) 0 0
\(277\) −11.0000 + 19.0526i −0.660926 + 1.14476i 0.319447 + 0.947604i \(0.396503\pi\)
−0.980373 + 0.197153i \(0.936830\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 8.00000 13.8564i 0.475551 0.823678i −0.524057 0.851683i \(-0.675582\pi\)
0.999608 + 0.0280052i \(0.00891551\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.00000 + 10.3923i −0.343559 + 0.595062i
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.00000 + 6.92820i −0.226819 + 0.392862i −0.956864 0.290537i \(-0.906166\pi\)
0.730044 + 0.683400i \(0.239499\pi\)
\(312\) 0 0
\(313\) −7.00000 12.1244i −0.395663 0.685309i 0.597522 0.801852i \(-0.296152\pi\)
−0.993186 + 0.116543i \(0.962819\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 15.0000 + 25.9808i 0.842484 + 1.45922i 0.887788 + 0.460252i \(0.152241\pi\)
−0.0453045 + 0.998973i \(0.514426\pi\)
\(318\) 0 0
\(319\) 12.0000 20.7846i 0.671871 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −1.00000 + 1.73205i −0.0554700 + 0.0960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.00000 + 3.46410i 0.109930 + 0.190404i 0.915742 0.401768i \(-0.131604\pi\)
−0.805812 + 0.592172i \(0.798271\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 + 27.7128i 0.866449 + 1.50073i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 10.3923i 0.322097 0.557888i −0.658824 0.752297i \(-0.728946\pi\)
0.980921 + 0.194409i \(0.0622790\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.00000 12.1244i 0.372572 0.645314i −0.617388 0.786659i \(-0.711809\pi\)
0.989960 + 0.141344i \(0.0451425\pi\)
\(354\) 0 0
\(355\) −8.00000 13.8564i −0.424596 0.735422i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −22.5000 + 38.9711i −1.18421 + 2.05111i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) 0 0
\(367\) 8.00000 13.8564i 0.417597 0.723299i −0.578101 0.815966i \(-0.696206\pi\)
0.995697 + 0.0926670i \(0.0295392\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.0000 + 22.5167i 0.673114 + 1.16587i 0.977016 + 0.213165i \(0.0683772\pi\)
−0.303902 + 0.952703i \(0.598289\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 + 20.7846i 0.613171 + 1.06204i 0.990702 + 0.136047i \(0.0434398\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.00000 + 1.73205i −0.0507020 + 0.0878185i −0.890263 0.455448i \(-0.849479\pi\)
0.839561 + 0.543266i \(0.182813\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 + 27.7128i −0.805047 + 1.39438i
\(396\) 0 0
\(397\) −7.00000 12.1244i −0.351320 0.608504i 0.635161 0.772380i \(-0.280934\pi\)
−0.986481 + 0.163876i \(0.947600\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 + 15.5885i 0.449439 + 0.778450i 0.998350 0.0574304i \(-0.0182907\pi\)
−0.548911 + 0.835881i \(0.684957\pi\)
\(402\) 0 0
\(403\) −8.00000 + 13.8564i −0.398508 + 0.690237i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −7.00000 + 12.1244i −0.346128 + 0.599511i −0.985558 0.169338i \(-0.945837\pi\)
0.639430 + 0.768849i \(0.279170\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −8.00000 13.8564i −0.392705 0.680184i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 5.19615i −0.145521 0.252050i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.00000 + 6.92820i −0.192673 + 0.333720i −0.946135 0.323772i \(-0.895049\pi\)
0.753462 + 0.657491i \(0.228382\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 12.0000 + 20.7846i 0.572729 + 0.991995i 0.996284 + 0.0861252i \(0.0274485\pi\)
−0.423556 + 0.905870i \(0.639218\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.0000 + 31.1769i 0.855206 + 1.48126i 0.876454 + 0.481486i \(0.159903\pi\)
−0.0212481 + 0.999774i \(0.506764\pi\)
\(444\) 0 0
\(445\) 6.00000 10.3923i 0.284427 0.492642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) −4.00000 + 6.92820i −0.188353 + 0.326236i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.0000 + 32.9090i 0.888783 + 1.53942i 0.841316 + 0.540544i \(0.181781\pi\)
0.0474665 + 0.998873i \(0.484885\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.00000 + 6.92820i 0.185098 + 0.320599i 0.943610 0.331061i \(-0.107406\pi\)
−0.758512 + 0.651660i \(0.774073\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 13.8564i 0.367840 0.637118i
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 20.7846i 0.548294 0.949673i −0.450098 0.892979i \(-0.648611\pi\)
0.998392 0.0566937i \(-0.0180558\pi\)
\(480\) 0 0
\(481\) −2.00000 3.46410i −0.0911922 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.00000 10.3923i −0.272446 0.471890i
\(486\) 0 0
\(487\) 8.00000 13.8564i 0.362515 0.627894i −0.625859 0.779936i \(-0.715252\pi\)
0.988374 + 0.152042i \(0.0485850\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −18.0000 + 31.1769i −0.810679 + 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.00000 + 3.46410i 0.0895323 + 0.155074i 0.907314 0.420455i \(-0.138129\pi\)
−0.817781 + 0.575529i \(0.804796\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17.0000 29.4449i −0.753512 1.30512i −0.946111 0.323843i \(-0.895025\pi\)
0.192599 0.981278i \(-0.438308\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.0000 + 27.7128i −0.705044 + 1.22117i
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.00000 + 15.5885i −0.394297 + 0.682943i −0.993011 0.118020i \(-0.962345\pi\)
0.598714 + 0.800963i \(0.295679\pi\)
\(522\) 0 0
\(523\) −16.0000 27.7128i −0.699631 1.21180i −0.968594 0.248646i \(-0.920014\pi\)
0.268963 0.963150i \(-0.413319\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 41.5692i −1.04546 1.81078i
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 0 0
\(535\) −12.0000 + 20.7846i −0.518805 + 0.898597i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 12.1244i −0.300954 0.521267i 0.675399 0.737453i \(-0.263972\pi\)
−0.976352 + 0.216186i \(0.930638\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 41.5692i −1.02243 1.77091i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.00000 12.1244i 0.296600 0.513725i −0.678756 0.734364i \(-0.737481\pi\)
0.975356 + 0.220638i \(0.0708140\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −16.0000 + 27.7128i −0.674320 + 1.16796i 0.302348 + 0.953198i \(0.402230\pi\)
−0.976667 + 0.214758i \(0.931104\pi\)
\(564\) 0 0
\(565\) 2.00000 + 3.46410i 0.0841406 + 0.145736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.0000 + 22.5167i 0.544988 + 0.943948i 0.998608 + 0.0527519i \(0.0167993\pi\)
−0.453619 + 0.891196i \(0.649867\pi\)
\(570\) 0 0
\(571\) −14.0000 + 24.2487i −0.585882 + 1.01478i 0.408883 + 0.912587i \(0.365918\pi\)
−0.994765 + 0.102190i \(0.967415\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.00000 + 12.1244i −0.291414 + 0.504744i −0.974144 0.225927i \(-0.927459\pi\)
0.682730 + 0.730670i \(0.260792\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 + 20.7846i 0.496989 + 0.860811i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 64.0000 2.63707
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −17.0000 29.4449i −0.698106 1.20916i −0.969122 0.246581i \(-0.920693\pi\)
0.271016 0.962575i \(-0.412640\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 + 20.7846i −0.490307 + 0.849236i −0.999938 0.0111569i \(-0.996449\pi\)
0.509631 + 0.860393i \(0.329782\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.00000 + 8.66025i −0.203279 + 0.352089i
\(606\) 0 0
\(607\) 16.0000 + 27.7128i 0.649420 + 1.12483i 0.983262 + 0.182199i \(0.0583216\pi\)
−0.333842 + 0.942629i \(0.608345\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 13.8564i −0.323645 0.560570i
\(612\) 0 0
\(613\) 9.00000 15.5885i 0.363507 0.629612i −0.625029 0.780602i \(-0.714913\pi\)
0.988535 + 0.150990i \(0.0482461\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) −16.0000 + 27.7128i −0.643094 + 1.11387i 0.341644 + 0.939829i \(0.389016\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 + 13.8564i 0.317470 + 0.549875i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.0000 + 25.9808i −0.592464 + 1.02618i 0.401435 + 0.915888i \(0.368512\pi\)
−0.993899 + 0.110291i \(0.964822\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 27.7128i 0.629025 1.08950i −0.358723 0.933444i \(-0.616788\pi\)
0.987748 0.156059i \(-0.0498790\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −13.0000 22.5167i −0.508729 0.881145i −0.999949 0.0101092i \(-0.996782\pi\)
0.491220 0.871036i \(-0.336551\pi\)
\(654\) 0 0
\(655\) −8.00000 + 13.8564i −0.312586 + 0.541415i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 1.00000 1.73205i 0.0388955 0.0673690i −0.845922 0.533306i \(-0.820949\pi\)
0.884818 + 0.465937i \(0.154283\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.00000 + 5.19615i 0.115299 + 0.199704i 0.917899 0.396813i \(-0.129884\pi\)
−0.802600 + 0.596518i \(0.796551\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0000 31.1769i 0.688751 1.19295i −0.283491 0.958975i \(-0.591493\pi\)
0.972242 0.233977i \(-0.0751739\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.00000 + 10.3923i −0.228582 + 0.395915i
\(690\) 0 0
\(691\) −4.00000 6.92820i −0.152167 0.263561i 0.779857 0.625958i \(-0.215292\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.00000 13.8564i −0.303457 0.525603i
\(696\) 0 0
\(697\) 6.00000 10.3923i 0.227266 0.393637i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −8.00000 + 13.8564i −0.301726 + 0.522604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −15.0000 25.9808i −0.563337 0.975728i −0.997202 0.0747503i \(-0.976184\pi\)
0.433865 0.900978i \(-0.357149\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.00000 6.92820i −0.149175 0.258378i 0.781748 0.623595i \(-0.214328\pi\)
−0.930923 + 0.365216i \(0.880995\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 + 5.19615i −0.111417 + 0.192980i
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 + 20.7846i −0.443836 + 0.768747i
\(732\) 0 0
\(733\) 13.0000 + 22.5167i 0.480166 + 0.831672i 0.999741 0.0227529i \(-0.00724310\pi\)
−0.519575 + 0.854425i \(0.673910\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 + 13.8564i 0.294684 + 0.510407i
\(738\) 0 0
\(739\) −26.0000 + 45.0333i −0.956425 + 1.65658i −0.225354 + 0.974277i \(0.572354\pi\)
−0.731072 + 0.682300i \(0.760980\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) 6.00000 10.3923i 0.219823 0.380745i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 + 6.92820i 0.145962 + 0.252814i 0.929731 0.368238i \(-0.120039\pi\)
−0.783769 + 0.621052i \(0.786706\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 + 25.9808i 0.543750 + 0.941802i 0.998684 + 0.0512772i \(0.0163292\pi\)
−0.454935 + 0.890525i \(0.650337\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25.0000 + 43.3013i −0.899188 + 1.55744i −0.0706526 + 0.997501i \(0.522508\pi\)
−0.828535 + 0.559937i \(0.810825\pi\)
\(774\) 0 0
\(775\) −4.00000 6.92820i −0.143684 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 8.00000 + 13.8564i 0.286630 + 0.496457i
\(780\) 0 0
\(781\) −16.0000 + 27.7128i −0.572525 + 0.991642i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) −20.0000 + 34.6410i −0.712923 + 1.23482i 0.250832 + 0.968031i \(0.419296\pi\)
−0.963755 + 0.266788i \(0.914038\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.00000 + 10.3923i 0.213066 + 0.369042i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 20.0000 + 34.6410i 0.705785 + 1.22245i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13.0000 22.5167i 0.457056 0.791644i −0.541748 0.840541i \(-0.682237\pi\)
0.998804 + 0.0488972i \(0.0155707\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.0000 20.7846i 0.420342 0.728053i
\(816\) 0 0
\(817\) −16.0000 27.7128i −0.559769 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.00000 8.66025i −0.174501 0.302245i 0.765487 0.643451i \(-0.222498\pi\)
−0.939989 + 0.341206i \(0.889165\pi\)
\(822\) 0 0
\(823\) −12.0000 + 20.7846i −0.418294 + 0.724506i −0.995768 0.0919029i \(-0.970705\pi\)
0.577474 + 0.816409i \(0.304038\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 13.0000 22.5167i 0.451509 0.782036i −0.546971 0.837151i \(-0.684219\pi\)
0.998480 + 0.0551154i \(0.0175527\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −16.0000 27.7128i −0.553703 0.959041i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.00000 + 15.5885i 0.309609 + 0.536259i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.00000 + 15.5885i −0.307434 + 0.532492i −0.977800 0.209539i \(-0.932804\pi\)
0.670366 + 0.742030i \(0.266137\pi\)
\(858\) 0 0
\(859\) −20.0000 34.6410i −0.682391 1.18194i −0.974249 0.225475i \(-0.927607\pi\)
0.291858 0.956462i \(-0.405727\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.00000 + 13.8564i 0.272323 + 0.471678i 0.969456 0.245264i \(-0.0788747\pi\)
−0.697133 + 0.716942i \(0.745541\pi\)
\(864\) 0 0
\(865\) −18.0000 + 31.1769i −0.612018 + 1.06005i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 64.0000 2.17105
\(870\) 0 0
\(871\) −4.00000 + 6.92820i −0.135535 + 0.234753i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3.00000 5.19615i −0.101303 0.175462i 0.810919 0.585159i \(-0.198968\pi\)
−0.912222 + 0.409697i \(0.865634\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.00000 + 13.8564i 0.268614 + 0.465253i 0.968504 0.248998i \(-0.0801012\pi\)
−0.699890 + 0.714250i \(0.746768\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −32.0000 + 55.4256i −1.07084 + 1.85475i
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −24.0000 + 41.5692i −0.800445 + 1.38641i
\(900\) 0 0
\(901\) −18.0000 31.1769i −0.599667 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 10.0000 + 17.3205i 0.332411 + 0.575753i
\(906\) 0 0
\(907\) 14.0000 24.2487i 0.464862 0.805165i −0.534333 0.845274i \(-0.679437\pi\)
0.999195 + 0.0401089i \(0.0127705\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) −16.0000 + 27.7128i −0.529523 + 0.917160i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 20.0000 + 34.6410i 0.659739 + 1.14270i 0.980683 + 0.195603i \(0.0626666\pi\)
−0.320944 + 0.947098i \(0.604000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −5.00000 8.66025i −0.164045 0.284134i 0.772271 0.635293i \(-0.219121\pi\)
−0.936316 + 0.351160i \(0.885787\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.0000 41.5692i 0.784884 1.35946i
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.0000 46.7654i 0.880175 1.52451i 0.0290288 0.999579i \(-0.490759\pi\)
0.851146 0.524929i \(-0.175908\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.0000 + 45.0333i 0.844886 + 1.46339i 0.885720 + 0.464220i \(0.153665\pi\)
−0.0408333 + 0.999166i \(0.513001\pi\)
\(948\) 0 0
\(949\) −10.0000 + 17.3205i −0.324614 + 0.562247i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −16.0000 + 27.7128i −0.517748 + 0.896766i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 28.5788i −0.532258 0.921898i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 41.5692i −0.770197 1.33402i −0.937455 0.348107i \(-0.886825\pi\)
0.167258 0.985913i \(-0.446509\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.00000 1.73205i 0.0319928 0.0554132i −0.849586 0.527451i \(-0.823148\pi\)
0.881579 + 0.472037i \(0.156481\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 0 0
\(985\) 6.00000 + 10.3923i 0.191176 + 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 13.8564i 0.254128 0.440163i −0.710530 0.703667i \(-0.751545\pi\)
0.964658 + 0.263504i \(0.0848781\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7.00000 + 12.1244i −0.221692 + 0.383982i −0.955322 0.295567i \(-0.904491\pi\)
0.733630 + 0.679549i \(0.237825\pi\)
\(998\) 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 3528.2.s.e.361.1 2
3.2 odd 2 392.2.i.d.361.1 2
7.2 even 3 inner 3528.2.s.e.3313.1 2
7.3 odd 6 504.2.a.c.1.1 1
7.4 even 3 3528.2.a.x.1.1 1
7.5 odd 6 3528.2.s.t.3313.1 2
7.6 odd 2 3528.2.s.t.361.1 2
12.11 even 2 784.2.i.g.753.1 2
21.2 odd 6 392.2.i.d.177.1 2
21.5 even 6 392.2.i.c.177.1 2
21.11 odd 6 392.2.a.d.1.1 1
21.17 even 6 56.2.a.a.1.1 1
21.20 even 2 392.2.i.c.361.1 2
28.3 even 6 1008.2.a.d.1.1 1
28.11 odd 6 7056.2.a.bo.1.1 1
56.3 even 6 4032.2.a.bk.1.1 1
56.45 odd 6 4032.2.a.bb.1.1 1
84.11 even 6 784.2.a.e.1.1 1
84.23 even 6 784.2.i.g.177.1 2
84.47 odd 6 784.2.i.e.177.1 2
84.59 odd 6 112.2.a.b.1.1 1
84.83 odd 2 784.2.i.e.753.1 2
105.17 odd 12 1400.2.g.g.449.1 2
105.38 odd 12 1400.2.g.g.449.2 2
105.59 even 6 1400.2.a.g.1.1 1
105.74 odd 6 9800.2.a.u.1.1 1
168.11 even 6 3136.2.a.p.1.1 1
168.53 odd 6 3136.2.a.q.1.1 1
168.59 odd 6 448.2.a.e.1.1 1
168.101 even 6 448.2.a.d.1.1 1
231.164 odd 6 6776.2.a.g.1.1 1
273.38 even 6 9464.2.a.c.1.1 1
336.59 odd 12 1792.2.b.d.897.2 2
336.101 even 12 1792.2.b.i.897.2 2
336.227 odd 12 1792.2.b.d.897.1 2
336.269 even 12 1792.2.b.i.897.1 2
420.59 odd 6 2800.2.a.p.1.1 1
420.143 even 12 2800.2.g.p.449.1 2
420.227 even 12 2800.2.g.p.449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.a.a.1.1 1 21.17 even 6
112.2.a.b.1.1 1 84.59 odd 6
392.2.a.d.1.1 1 21.11 odd 6
392.2.i.c.177.1 2 21.5 even 6
392.2.i.c.361.1 2 21.20 even 2
392.2.i.d.177.1 2 21.2 odd 6
392.2.i.d.361.1 2 3.2 odd 2
448.2.a.d.1.1 1 168.101 even 6
448.2.a.e.1.1 1 168.59 odd 6
504.2.a.c.1.1 1 7.3 odd 6
784.2.a.e.1.1 1 84.11 even 6
784.2.i.e.177.1 2 84.47 odd 6
784.2.i.e.753.1 2 84.83 odd 2
784.2.i.g.177.1 2 84.23 even 6
784.2.i.g.753.1 2 12.11 even 2
1008.2.a.d.1.1 1 28.3 even 6
1400.2.a.g.1.1 1 105.59 even 6
1400.2.g.g.449.1 2 105.17 odd 12
1400.2.g.g.449.2 2 105.38 odd 12
1792.2.b.d.897.1 2 336.227 odd 12
1792.2.b.d.897.2 2 336.59 odd 12
1792.2.b.i.897.1 2 336.269 even 12
1792.2.b.i.897.2 2 336.101 even 12
2800.2.a.p.1.1 1 420.59 odd 6
2800.2.g.p.449.1 2 420.143 even 12
2800.2.g.p.449.2 2 420.227 even 12
3136.2.a.p.1.1 1 168.11 even 6
3136.2.a.q.1.1 1 168.53 odd 6
3528.2.a.x.1.1 1 7.4 even 3
3528.2.s.e.361.1 2 1.1 even 1 trivial
3528.2.s.e.3313.1 2 7.2 even 3 inner
3528.2.s.t.361.1 2 7.6 odd 2
3528.2.s.t.3313.1 2 7.5 odd 6
4032.2.a.bb.1.1 1 56.45 odd 6
4032.2.a.bk.1.1 1 56.3 even 6
6776.2.a.g.1.1 1 231.164 odd 6
7056.2.a.bo.1.1 1 28.11 odd 6
9464.2.a.c.1.1 1 273.38 even 6
9800.2.a.u.1.1 1 105.74 odd 6