Properties

Label 2736.2.k.e.2431.2
Level $2736$
Weight $2$
Character 2736.2431
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(2431,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.k (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2431
Dual form 2736.2.k.e.2431.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.46410i q^{7} +O(q^{10})\) \(q+3.46410i q^{7} -3.46410i q^{11} +3.46410i q^{13} +6.00000 q^{17} +(-4.00000 - 1.73205i) q^{19} -5.00000 q^{25} +6.92820i q^{29} +10.0000 q^{31} +3.46410i q^{37} -6.92820i q^{41} +10.3923i q^{43} -6.92820i q^{47} -5.00000 q^{49} +13.8564i q^{53} -12.0000 q^{59} +10.0000 q^{61} +4.00000 q^{67} -12.0000 q^{71} -2.00000 q^{73} +12.0000 q^{77} -10.0000 q^{79} +3.46410i q^{83} +6.92820i q^{89} -12.0000 q^{91} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{17} - 8 q^{19} - 10 q^{25} + 20 q^{31} - 10 q^{49} - 24 q^{59} + 20 q^{61} + 8 q^{67} - 24 q^{71} - 4 q^{73} + 24 q^{77} - 20 q^{79} - 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 1.73205i −0.917663 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.46410i 0.569495i 0.958603 + 0.284747i \(0.0919097\pi\)
−0.958603 + 0.284747i \(0.908090\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 10.3923i 1.58481i 0.609994 + 0.792406i \(0.291172\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820i 1.01058i −0.862949 0.505291i \(-0.831385\pi\)
0.862949 0.505291i \(-0.168615\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 13.8564i 1.90332i 0.307148 + 0.951662i \(0.400625\pi\)
−0.307148 + 0.951662i \(0.599375\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 10.3923i 0.995402i 0.867349 + 0.497701i \(0.165822\pi\)
−0.867349 + 0.497701i \(0.834178\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.92820i 0.651751i 0.945413 + 0.325875i \(0.105659\pi\)
−0.945413 + 0.325875i \(0.894341\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.7846i 1.90532i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) 6.00000 13.8564i 0.520266 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i 0.989150 + 0.146911i \(0.0469330\pi\)
−0.989150 + 0.146911i \(0.953067\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −24.0000 −1.96616 −0.983078 0.183186i \(-0.941359\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.46410i 0.271329i 0.990755 + 0.135665i \(0.0433170\pi\)
−0.990755 + 0.135665i \(0.956683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.8564i 1.05348i 0.850026 + 0.526742i \(0.176586\pi\)
−0.850026 + 0.526742i \(0.823414\pi\)
\(174\) 0 0
\(175\) 17.3205i 1.30931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 3.46410i 0.257485i −0.991678 0.128742i \(-0.958906\pi\)
0.991678 0.128742i \(-0.0410940\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564i 1.00261i −0.865269 0.501307i \(-0.832853\pi\)
0.865269 0.501307i \(-0.167147\pi\)
\(192\) 0 0
\(193\) 13.8564i 0.997406i 0.866773 + 0.498703i \(0.166190\pi\)
−0.866773 + 0.498703i \(0.833810\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 3.46410i 0.245564i −0.992434 0.122782i \(-0.960818\pi\)
0.992434 0.122782i \(-0.0391815\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −24.0000 −1.68447
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 + 13.8564i −0.415029 + 0.958468i
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 34.6410i 2.35159i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 20.7846i 1.39812i
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.92820i 0.448148i −0.974572 0.224074i \(-0.928064\pi\)
0.974572 0.224074i \(-0.0719358\pi\)
\(240\) 0 0
\(241\) 27.7128i 1.78514i −0.450910 0.892570i \(-0.648900\pi\)
0.450910 0.892570i \(-0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 13.8564i 0.381771 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 31.1769i 1.96787i 0.178529 + 0.983935i \(0.442866\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.7846i 1.29651i 0.761424 + 0.648254i \(0.224501\pi\)
−0.761424 + 0.648254i \(0.775499\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.92820i 0.427211i 0.976920 + 0.213606i \(0.0685208\pi\)
−0.976920 + 0.213606i \(0.931479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i 0.676435 + 0.736502i \(0.263524\pi\)
−0.676435 + 0.736502i \(0.736476\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.3205i 1.04447i
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.7846i 1.23991i −0.784639 0.619953i \(-0.787152\pi\)
0.784639 0.619953i \(-0.212848\pi\)
\(282\) 0 0
\(283\) 10.3923i 0.617758i −0.951101 0.308879i \(-0.900046\pi\)
0.951101 0.308879i \(-0.0999539\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 13.8564i 0.809500i −0.914427 0.404750i \(-0.867359\pi\)
0.914427 0.404750i \(-0.132641\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −36.0000 −2.07501
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.8564i 0.785725i −0.919597 0.392862i \(-0.871485\pi\)
0.919597 0.392862i \(-0.128515\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −24.0000 10.3923i −1.33540 0.578243i
\(324\) 0 0
\(325\) 17.3205i 0.960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.92820i 0.377403i −0.982034 0.188702i \(-0.939572\pi\)
0.982034 0.188702i \(-0.0604279\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 34.6410i 1.87592i
\(342\) 0 0
\(343\) 6.92820i 0.374088i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.3923i 0.557888i 0.960307 + 0.278944i \(0.0899844\pi\)
−0.960307 + 0.278944i \(0.910016\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7846i 1.09697i −0.836160 0.548485i \(-0.815205\pi\)
0.836160 0.548485i \(-0.184795\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.46410i 0.180825i −0.995904 0.0904123i \(-0.971182\pi\)
0.995904 0.0904123i \(-0.0288185\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −48.0000 −2.49204
\(372\) 0 0
\(373\) 38.1051i 1.97301i −0.163737 0.986504i \(-0.552355\pi\)
0.163737 0.986504i \(-0.447645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 20.7846i 1.03793i −0.854794 0.518967i \(-0.826317\pi\)
0.854794 0.518967i \(-0.173683\pi\)
\(402\) 0 0
\(403\) 34.6410i 1.72559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 34.6410i 1.71289i −0.516240 0.856444i \(-0.672669\pi\)
0.516240 0.856444i \(-0.327331\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 41.5692i 2.04549i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.3205i 0.846162i −0.906092 0.423081i \(-0.860949\pi\)
0.906092 0.423081i \(-0.139051\pi\)
\(420\) 0 0
\(421\) 24.2487i 1.18181i 0.806741 + 0.590905i \(0.201229\pi\)
−0.806741 + 0.590905i \(0.798771\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) 34.6410i 1.67640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 20.7846i 0.998845i 0.866359 + 0.499422i \(0.166454\pi\)
−0.866359 + 0.499422i \(0.833546\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 2.00000 0.0954548 0.0477274 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.3205i 0.822922i −0.911427 0.411461i \(-0.865019\pi\)
0.911427 0.411461i \(-0.134981\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.6410i 1.63481i 0.576063 + 0.817405i \(0.304588\pi\)
−0.576063 + 0.817405i \(0.695412\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 3.46410i 0.160990i −0.996755 0.0804952i \(-0.974350\pi\)
0.996755 0.0804952i \(-0.0256502\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.1051i 1.76329i −0.471909 0.881647i \(-0.656435\pi\)
0.471909 0.881647i \(-0.343565\pi\)
\(468\) 0 0
\(469\) 13.8564i 0.639829i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 36.0000 1.65528
\(474\) 0 0
\(475\) 20.0000 + 8.66025i 0.917663 + 0.397360i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 20.7846i 0.949673i 0.880074 + 0.474837i \(0.157493\pi\)
−0.880074 + 0.474837i \(0.842507\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.3205i 0.781664i −0.920462 0.390832i \(-0.872187\pi\)
0.920462 0.390832i \(-0.127813\pi\)
\(492\) 0 0
\(493\) 41.5692i 1.87218i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 41.5692i 1.86463i
\(498\) 0 0
\(499\) 31.1769i 1.39567i −0.716258 0.697835i \(-0.754147\pi\)
0.716258 0.697835i \(-0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13.8564i 0.617827i −0.951090 0.308913i \(-0.900035\pi\)
0.951090 0.308913i \(-0.0999653\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.92820i 0.307087i −0.988142 0.153544i \(-0.950931\pi\)
0.988142 0.153544i \(-0.0490686\pi\)
\(510\) 0 0
\(511\) 6.92820i 0.306486i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.7846i 0.910590i −0.890341 0.455295i \(-0.849534\pi\)
0.890341 0.455295i \(-0.150466\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 60.0000 2.61364
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 17.3205i 0.746047i
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 27.7128i 0.511217 1.18061i
\(552\) 0 0
\(553\) 34.6410i 1.47309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.7846i 0.871336i −0.900107 0.435668i \(-0.856512\pi\)
0.900107 0.435668i \(-0.143488\pi\)
\(570\) 0 0
\(571\) 24.2487i 1.01478i 0.861717 + 0.507388i \(0.169389\pi\)
−0.861717 + 0.507388i \(0.830611\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 48.0000 1.98796
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.3923i 0.428936i −0.976731 0.214468i \(-0.931198\pi\)
0.976731 0.214468i \(-0.0688018\pi\)
\(588\) 0 0
\(589\) −40.0000 17.3205i −1.64817 0.713679i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 20.7846i 0.847822i −0.905704 0.423911i \(-0.860657\pi\)
0.905704 0.423911i \(-0.139343\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) 45.0333i 1.81004i −0.425367 0.905021i \(-0.639855\pi\)
0.425367 0.905021i \(-0.360145\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.7846i 0.828737i
\(630\) 0 0
\(631\) 17.3205i 0.689519i 0.938691 + 0.344759i \(0.112039\pi\)
−0.938691 + 0.344759i \(0.887961\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.3205i 0.686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.92820i 0.273648i 0.990595 + 0.136824i \(0.0436894\pi\)
−0.990595 + 0.136824i \(0.956311\pi\)
\(642\) 0 0
\(643\) 31.1769i 1.22950i 0.788723 + 0.614749i \(0.210743\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.8564i 0.544752i 0.962191 + 0.272376i \(0.0878094\pi\)
−0.962191 + 0.272376i \(0.912191\pi\)
\(648\) 0 0
\(649\) 41.5692i 1.63173i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 24.2487i 0.943166i −0.881822 0.471583i \(-0.843683\pi\)
0.881822 0.471583i \(-0.156317\pi\)
\(662\) 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
\(671\) 34.6410i 1.33730i
\(672\) 0 0
\(673\) 34.6410i 1.33531i −0.744469 0.667657i \(-0.767297\pi\)
0.744469 0.667657i \(-0.232703\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) 31.1769i 1.18603i −0.805193 0.593013i \(-0.797938\pi\)
0.805193 0.593013i \(-0.202062\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 41.5692i 1.57455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 6.00000 13.8564i 0.226294 0.522604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 34.6410i 1.29189i 0.763383 + 0.645946i \(0.223537\pi\)
−0.763383 + 0.645946i \(0.776463\pi\)
\(720\) 0 0
\(721\) 6.92820i 0.258020i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 34.6410i 1.28654i
\(726\) 0 0
\(727\) 10.3923i 0.385429i 0.981255 + 0.192715i \(0.0617292\pi\)
−0.981255 + 0.192715i \(0.938271\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 62.3538i 2.30624i
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.8564i 0.510407i
\(738\) 0 0
\(739\) 10.3923i 0.382287i −0.981562 0.191144i \(-0.938780\pi\)
0.981562 0.191144i \(-0.0612196\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 41.5692i 1.51891i
\(750\) 0 0
\(751\) 38.0000 1.38664 0.693320 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −36.0000 −1.30329
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 41.5692i 1.50098i
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.6410i 1.24595i 0.782241 + 0.622975i \(0.214076\pi\)
−0.782241 + 0.622975i \(0.785924\pi\)
\(774\) 0 0
\(775\) −50.0000 −1.79605
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 + 27.7128i −0.429945 + 0.992915i
\(780\) 0 0
\(781\) 41.5692i 1.48746i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 34.6410i 1.23014i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.4974i 1.71787i −0.512087 0.858933i \(-0.671128\pi\)
0.512087 0.858933i \(-0.328872\pi\)
\(798\) 0 0
\(799\) 41.5692i 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.92820i 0.244491i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 18.0000 41.5692i 0.629740 1.45432i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.0000 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) 0 0
\(823\) 24.2487i 0.845257i −0.906303 0.422628i \(-0.861108\pi\)
0.906303 0.422628i \(-0.138892\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 24.2487i 0.842193i −0.907016 0.421096i \(-0.861645\pi\)
0.907016 0.421096i \(-0.138355\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −30.0000 −1.03944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.46410i 0.119028i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.92820i 0.236663i 0.992974 + 0.118331i \(0.0377545\pi\)
−0.992974 + 0.118331i \(0.962245\pi\)
\(858\) 0 0
\(859\) 17.3205i 0.590968i −0.955348 0.295484i \(-0.904519\pi\)
0.955348 0.295484i \(-0.0954809\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34.6410i 1.17512i
\(870\) 0 0
\(871\) 13.8564i 0.469506i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 24.2487i 0.818821i −0.912350 0.409410i \(-0.865734\pi\)
0.912350 0.409410i \(-0.134266\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 24.2487i 0.816034i 0.912974 + 0.408017i \(0.133780\pi\)
−0.912974 + 0.408017i \(0.866220\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 6.92820i 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.0000 + 27.7128i −0.401565 + 0.927374i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 69.2820i 2.31069i
\(900\) 0 0
\(901\) 83.1384i 2.76974i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −36.0000 −1.18882
\(918\) 0 0
\(919\) 10.3923i 0.342811i −0.985201 0.171405i \(-0.945169\pi\)
0.985201 0.171405i \(-0.0548307\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 41.5692i 1.36827i
\(924\) 0 0
\(925\) 17.3205i 0.569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 20.0000 + 8.66025i 0.655474 + 0.283828i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.7846i 0.677559i 0.940866 + 0.338779i \(0.110014\pi\)
−0.940866 + 0.338779i \(0.889986\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.1051i 1.23825i 0.785292 + 0.619125i \(0.212513\pi\)
−0.785292 + 0.619125i \(0.787487\pi\)
\(948\) 0 0
\(949\) 6.92820i 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.4974i 1.57099i −0.618871 0.785493i \(-0.712410\pi\)
0.618871 0.785493i \(-0.287590\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 62.3538i 2.01351i
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.2487i 0.779786i −0.920860 0.389893i \(-0.872512\pi\)
0.920860 0.389893i \(-0.127488\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 48.4974i 1.55157i −0.630997 0.775785i \(-0.717354\pi\)
0.630997 0.775785i \(-0.282646\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 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.k.e.2431.2 2
3.2 odd 2 912.2.k.e.607.2 yes 2
4.3 odd 2 2736.2.k.f.2431.1 2
12.11 even 2 912.2.k.b.607.1 2
19.18 odd 2 2736.2.k.f.2431.2 2
24.5 odd 2 3648.2.k.b.2431.2 2
24.11 even 2 3648.2.k.e.2431.1 2
57.56 even 2 912.2.k.b.607.2 yes 2
76.75 even 2 inner 2736.2.k.e.2431.1 2
228.227 odd 2 912.2.k.e.607.1 yes 2
456.227 odd 2 3648.2.k.b.2431.1 2
456.341 even 2 3648.2.k.e.2431.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.k.b.607.1 2 12.11 even 2
912.2.k.b.607.2 yes 2 57.56 even 2
912.2.k.e.607.1 yes 2 228.227 odd 2
912.2.k.e.607.2 yes 2 3.2 odd 2
2736.2.k.e.2431.1 2 76.75 even 2 inner
2736.2.k.e.2431.2 2 1.1 even 1 trivial
2736.2.k.f.2431.1 2 4.3 odd 2
2736.2.k.f.2431.2 2 19.18 odd 2
3648.2.k.b.2431.1 2 456.227 odd 2
3648.2.k.b.2431.2 2 24.5 odd 2
3648.2.k.e.2431.1 2 24.11 even 2
3648.2.k.e.2431.2 2 456.341 even 2