Properties

Label 2736.2.k.h.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: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.2
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2736.2431
Dual form 2736.2.k.h.2431.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} +2.82843i q^{7} +O(q^{10})\) \(q+2.00000 q^{5} +2.82843i q^{7} +1.41421i q^{11} -2.82843i q^{13} +2.00000 q^{17} +(-1.00000 + 4.24264i) q^{19} +1.41421i q^{23} -1.00000 q^{25} +7.07107i q^{29} +6.00000 q^{31} +5.65685i q^{35} -11.3137i q^{37} +4.24264i q^{41} +7.07107i q^{47} -1.00000 q^{49} +9.89949i q^{53} +2.82843i q^{55} +8.00000 q^{59} -8.00000 q^{61} -5.65685i q^{65} -2.00000 q^{67} +8.00000 q^{71} -4.00000 q^{77} -8.00000 q^{79} +12.7279i q^{83} +4.00000 q^{85} -9.89949i q^{89} +8.00000 q^{91} +(-2.00000 + 8.48528i) q^{95} +2.82843i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 4 q^{17} - 2 q^{19} - 2 q^{25} + 12 q^{31} - 2 q^{49} + 16 q^{59} - 16 q^{61} - 4 q^{67} + 16 q^{71} - 8 q^{77} - 16 q^{79} + 8 q^{85} + 16 q^{91} - 4 q^{95}+O(q^{100}) \) 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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −1.00000 + 4.24264i −0.229416 + 0.973329i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.41421i 0.294884i 0.989071 + 0.147442i \(0.0471040\pi\)
−0.989071 + 0.147442i \(0.952896\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107i 1.31306i 0.754298 + 0.656532i \(0.227977\pi\)
−0.754298 + 0.656532i \(0.772023\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.65685i 0.956183i
\(36\) 0 0
\(37\) 11.3137i 1.85996i −0.367607 0.929981i \(-0.619823\pi\)
0.367607 0.929981i \(-0.380177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.24264i 0.662589i 0.943527 + 0.331295i \(0.107485\pi\)
−0.943527 + 0.331295i \(0.892515\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.07107i 1.03142i 0.856763 + 0.515711i \(0.172472\pi\)
−0.856763 + 0.515711i \(0.827528\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.89949i 1.35980i 0.733305 + 0.679900i \(0.237977\pi\)
−0.733305 + 0.679900i \(0.762023\pi\)
\(54\) 0 0
\(55\) 2.82843i 0.381385i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 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
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.7279i 1.39707i 0.715575 + 0.698535i \(0.246165\pi\)
−0.715575 + 0.698535i \(0.753835\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.89949i 1.04934i −0.851304 0.524672i \(-0.824188\pi\)
0.851304 0.524672i \(-0.175812\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 + 8.48528i −0.205196 + 0.870572i
\(96\) 0 0
\(97\) 2.82843i 0.287183i 0.989637 + 0.143592i \(0.0458652\pi\)
−0.989637 + 0.143592i \(0.954135\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 0 0
\(109\) 8.48528i 0.812743i 0.913708 + 0.406371i \(0.133206\pi\)
−0.913708 + 0.406371i \(0.866794\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.41421i 0.133038i 0.997785 + 0.0665190i \(0.0211893\pi\)
−0.997785 + 0.0665190i \(0.978811\pi\)
\(114\) 0 0
\(115\) 2.82843i 0.263752i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.65685i 0.518563i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.24264i 0.370681i −0.982674 0.185341i \(-0.940661\pi\)
0.982674 0.185341i \(-0.0593388\pi\)
\(132\) 0 0
\(133\) −12.0000 2.82843i −1.04053 0.245256i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 2.82843i 0.239904i −0.992780 0.119952i \(-0.961726\pi\)
0.992780 0.119952i \(-0.0382741\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 14.1421i 1.17444i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 8.48528i 0.664619i 0.943170 + 0.332309i \(0.107828\pi\)
−0.943170 + 0.332309i \(0.892172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.07107i 0.537603i −0.963196 0.268802i \(-0.913372\pi\)
0.963196 0.268802i \(-0.0866276\pi\)
\(174\) 0 0
\(175\) 2.82843i 0.213809i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 8.48528i 0.630706i −0.948974 0.315353i \(-0.897877\pi\)
0.948974 0.315353i \(-0.102123\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.6274i 1.66360i
\(186\) 0 0
\(187\) 2.82843i 0.206835i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.7279i 0.920960i 0.887670 + 0.460480i \(0.152323\pi\)
−0.887670 + 0.460480i \(0.847677\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) 11.3137i 0.802008i 0.916076 + 0.401004i \(0.131339\pi\)
−0.916076 + 0.401004i \(0.868661\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −20.0000 −1.40372
\(204\) 0 0
\(205\) 8.48528i 0.592638i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 1.41421i −0.415029 0.0978232i
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.9706i 1.15204i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685i 0.380521i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 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
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 14.1421i 0.922531i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.8701i 1.73808i 0.494742 + 0.869040i \(0.335262\pi\)
−0.494742 + 0.869040i \(0.664738\pi\)
\(240\) 0 0
\(241\) 2.82843i 0.182195i 0.995842 + 0.0910975i \(0.0290375\pi\)
−0.995842 + 0.0910975i \(0.970963\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 12.0000 + 2.82843i 0.763542 + 0.179969i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0416i 1.51749i −0.651385 0.758747i \(-0.725812\pi\)
0.651385 0.758747i \(-0.274188\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.89949i 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) 0 0
\(259\) 32.0000 1.98838
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.0416i 1.48247i 0.671245 + 0.741235i \(0.265760\pi\)
−0.671245 + 0.741235i \(0.734240\pi\)
\(264\) 0 0
\(265\) 19.7990i 1.21624i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.0416i 1.46584i −0.680313 0.732922i \(-0.738156\pi\)
0.680313 0.732922i \(-0.261844\pi\)
\(270\) 0 0
\(271\) 11.3137i 0.687259i 0.939105 + 0.343629i \(0.111656\pi\)
−0.939105 + 0.343629i \(0.888344\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.41421i 0.0852803i
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.41421i 0.0843649i −0.999110 0.0421825i \(-0.986569\pi\)
0.999110 0.0421825i \(-0.0134311\pi\)
\(282\) 0 0
\(283\) 28.2843i 1.68133i −0.541559 0.840663i \(-0.682166\pi\)
0.541559 0.840663i \(-0.317834\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 29.6985i 1.73500i 0.497434 + 0.867502i \(0.334276\pi\)
−0.497434 + 0.867502i \(0.665724\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −16.0000 −0.916157
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.89949i 0.561349i 0.959803 + 0.280674i \(0.0905581\pi\)
−0.959803 + 0.280674i \(0.909442\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.07107i 0.397151i 0.980086 + 0.198575i \(0.0636315\pi\)
−0.980086 + 0.198575i \(0.936369\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.00000 + 8.48528i −0.111283 + 0.472134i
\(324\) 0 0
\(325\) 2.82843i 0.156893i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.0000 −1.10264
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 25.4558i 1.38667i −0.720616 0.693334i \(-0.756141\pi\)
0.720616 0.693334i \(-0.243859\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.48528i 0.459504i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.5269i 1.74614i −0.487598 0.873068i \(-0.662127\pi\)
0.487598 0.873068i \(-0.337873\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.3848i 0.970311i −0.874428 0.485156i \(-0.838763\pi\)
0.874428 0.485156i \(-0.161237\pi\)
\(360\) 0 0
\(361\) −17.0000 8.48528i −0.894737 0.446594i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 16.9706i 0.885856i −0.896557 0.442928i \(-0.853940\pi\)
0.896557 0.442928i \(-0.146060\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −28.0000 −1.45369
\(372\) 0 0
\(373\) 22.6274i 1.17160i −0.810454 0.585802i \(-0.800780\pi\)
0.810454 0.585802i \(-0.199220\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 2.82843i 0.143040i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.8701i 1.34183i 0.741536 + 0.670913i \(0.234098\pi\)
−0.741536 + 0.670913i \(0.765902\pi\)
\(402\) 0 0
\(403\) 16.9706i 0.845364i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) 25.4558i 1.25871i 0.777118 + 0.629355i \(0.216681\pi\)
−0.777118 + 0.629355i \(0.783319\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 22.6274i 1.11342i
\(414\) 0 0
\(415\) 25.4558i 1.24958i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.3848i 0.898155i −0.893493 0.449078i \(-0.851753\pi\)
0.893493 0.449078i \(-0.148247\pi\)
\(420\) 0 0
\(421\) 14.1421i 0.689246i −0.938741 0.344623i \(-0.888007\pi\)
0.938741 0.344623i \(-0.111993\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 22.6274i 1.09502i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 33.9411i 1.63111i −0.578682 0.815553i \(-0.696433\pi\)
0.578682 0.815553i \(-0.303567\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 1.41421i −0.287019 0.0676510i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.8701i 1.27663i −0.769773 0.638317i \(-0.779631\pi\)
0.769773 0.638317i \(-0.220369\pi\)
\(444\) 0 0
\(445\) 19.7990i 0.938562i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.5563i 0.734150i 0.930191 + 0.367075i \(0.119641\pi\)
−0.930191 + 0.367075i \(0.880359\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) 16.9706i 0.788689i −0.918963 0.394344i \(-0.870972\pi\)
0.918963 0.394344i \(-0.129028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.0416i 1.11251i −0.831010 0.556257i \(-0.812237\pi\)
0.831010 0.556257i \(-0.187763\pi\)
\(468\) 0 0
\(469\) 5.65685i 0.261209i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 4.24264i 0.0458831 0.194666i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.07107i 0.323085i −0.986866 0.161543i \(-0.948353\pi\)
0.986866 0.161543i \(-0.0516469\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.65685i 0.256865i
\(486\) 0 0
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.0416i 1.08498i −0.840061 0.542492i \(-0.817481\pi\)
0.840061 0.542492i \(-0.182519\pi\)
\(492\) 0 0
\(493\) 14.1421i 0.636930i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.6274i 1.01498i
\(498\) 0 0
\(499\) 25.4558i 1.13956i −0.821797 0.569780i \(-0.807028\pi\)
0.821797 0.569780i \(-0.192972\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.7279i 0.567510i −0.958897 0.283755i \(-0.908420\pi\)
0.958897 0.283755i \(-0.0915802\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 24.0416i 1.06563i −0.846233 0.532813i \(-0.821135\pi\)
0.846233 0.532813i \(-0.178865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −10.0000 −0.439799
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.41421i 0.0619578i 0.999520 + 0.0309789i \(0.00986247\pi\)
−0.999520 + 0.0309789i \(0.990138\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) 21.0000 0.913043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) −40.0000 −1.72935
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.41421i 0.0609145i
\(540\) 0 0
\(541\) 12.0000 0.515920 0.257960 0.966156i \(-0.416950\pi\)
0.257960 + 0.966156i \(0.416950\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 16.9706i 0.726939i
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −30.0000 7.07107i −1.27804 0.301238i
\(552\) 0 0
\(553\) 22.6274i 0.962216i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) 2.82843i 0.118993i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.5269i 1.36360i −0.731539 0.681800i \(-0.761198\pi\)
0.731539 0.681800i \(-0.238802\pi\)
\(570\) 0 0
\(571\) 45.2548i 1.89386i −0.321446 0.946928i \(-0.604169\pi\)
0.321446 0.946928i \(-0.395831\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.41421i 0.0589768i
\(576\) 0 0
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) −14.0000 −0.579821
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.3553i 1.45927i −0.683836 0.729636i \(-0.739690\pi\)
0.683836 0.729636i \(-0.260310\pi\)
\(588\) 0 0
\(589\) −6.00000 + 25.4558i −0.247226 + 1.04889i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 11.3137i 0.463817i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 28.2843i 1.15374i −0.816836 0.576870i \(-0.804274\pi\)
0.816836 0.576870i \(-0.195726\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.0000 0.731804
\(606\) 0 0
\(607\) −34.0000 −1.38002 −0.690009 0.723801i \(-0.742393\pi\)
−0.690009 + 0.723801i \(0.742393\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.0000 0.809113
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 25.4558i 1.02316i −0.859237 0.511578i \(-0.829061\pi\)
0.859237 0.511578i \(-0.170939\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 28.0000 1.12180
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.6274i 0.902214i
\(630\) 0 0
\(631\) 14.1421i 0.562990i 0.959563 + 0.281495i \(0.0908302\pi\)
−0.959563 + 0.281495i \(0.909170\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 28.0000 1.11115
\(636\) 0 0
\(637\) 2.82843i 0.112066i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 41.0122i 1.61988i 0.586510 + 0.809942i \(0.300502\pi\)
−0.586510 + 0.809942i \(0.699498\pi\)
\(642\) 0 0
\(643\) 39.5980i 1.56159i −0.624786 0.780796i \(-0.714814\pi\)
0.624786 0.780796i \(-0.285186\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.07107i 0.277992i −0.990293 0.138996i \(-0.955612\pi\)
0.990293 0.138996i \(-0.0443876\pi\)
\(648\) 0 0
\(649\) 11.3137i 0.444102i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) 8.48528i 0.331547i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 39.5980i 1.54018i 0.637934 + 0.770091i \(0.279789\pi\)
−0.637934 + 0.770091i \(0.720211\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0000 5.65685i −0.930680 0.219363i
\(666\) 0 0
\(667\) −10.0000 −0.387202
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.3137i 0.436761i
\(672\) 0 0
\(673\) 28.2843i 1.09028i −0.838346 0.545139i \(-0.816477\pi\)
0.838346 0.545139i \(-0.183523\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.89949i 0.380468i 0.981739 + 0.190234i \(0.0609248\pi\)
−0.981739 + 0.190234i \(0.939075\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.0000 1.22445 0.612223 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 28.0000 1.06672
\(690\) 0 0
\(691\) 16.9706i 0.645591i 0.946469 + 0.322795i \(0.104623\pi\)
−0.946469 + 0.322795i \(0.895377\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.65685i 0.214577i
\(696\) 0 0
\(697\) 8.48528i 0.321403i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 48.0000 + 11.3137i 1.81035 + 0.426705i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.9706i 0.638244i
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.48528i 0.317776i
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.5269i 1.21305i 0.795065 + 0.606525i \(0.207437\pi\)
−0.795065 + 0.606525i \(0.792563\pi\)
\(720\) 0 0
\(721\) 22.6274i 0.842689i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.07107i 0.262613i
\(726\) 0 0
\(727\) 28.2843i 1.04901i −0.851409 0.524503i \(-0.824251\pi\)
0.851409 0.524503i \(-0.175749\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −52.0000 −1.92066 −0.960332 0.278859i \(-0.910044\pi\)
−0.960332 + 0.278859i \(0.910044\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.82843i 0.104186i
\(738\) 0 0
\(739\) 36.7696i 1.35259i 0.736631 + 0.676295i \(0.236415\pi\)
−0.736631 + 0.676295i \(0.763585\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 28.0000 1.02584
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 56.5685i 2.06697i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.6274i 0.817029i
\(768\) 0 0
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.2132i 0.762986i −0.924372 0.381493i \(-0.875410\pi\)
0.924372 0.381493i \(-0.124590\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.0000 4.24264i −0.644917 0.152008i
\(780\) 0 0
\(781\) 11.3137i 0.404836i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 16.0000 0.571064
\(786\) 0 0
\(787\) −42.0000 −1.49714 −0.748569 0.663057i \(-0.769259\pi\)
−0.748569 + 0.663057i \(0.769259\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 22.6274i 0.803523i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.07107i 0.250470i 0.992127 + 0.125235i \(0.0399685\pi\)
−0.992127 + 0.125235i \(0.960032\pi\)
\(798\) 0 0
\(799\) 14.1421i 0.500313i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 16.9706i 0.594453i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 8.48528i 0.295778i 0.989004 + 0.147889i \(0.0472479\pi\)
−0.989004 + 0.147889i \(0.952752\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 0 0
\(829\) 53.7401i 1.86647i 0.359265 + 0.933236i \(0.383028\pi\)
−0.359265 + 0.933236i \(0.616972\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 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
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10.0000 0.344010
\(846\) 0 0
\(847\) 25.4558i 0.874673i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0416i 0.821246i −0.911805 0.410623i \(-0.865311\pi\)
0.911805 0.410623i \(-0.134689\pi\)
\(858\) 0 0
\(859\) 19.7990i 0.675533i −0.941230 0.337766i \(-0.890329\pi\)
0.941230 0.337766i \(-0.109671\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 14.1421i 0.480847i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.3137i 0.383791i
\(870\) 0 0
\(871\) 5.65685i 0.191675i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 33.9411i 1.14742i
\(876\) 0 0
\(877\) 33.9411i 1.14611i 0.819517 + 0.573055i \(0.194242\pi\)
−0.819517 + 0.573055i \(0.805758\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) 36.7696i 1.23739i 0.785630 + 0.618697i \(0.212339\pi\)
−0.785630 + 0.618697i \(0.787661\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 39.5980i 1.32807i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30.0000 7.07107i −1.00391 0.236624i
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 42.4264i 1.41500i
\(900\) 0 0
\(901\) 19.7990i 0.659600i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.9706i 0.564121i
\(906\) 0 0
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 14.1421i 0.466506i 0.972416 + 0.233253i \(0.0749370\pi\)
−0.972416 + 0.233253i \(0.925063\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 22.6274i 0.744791i
\(924\) 0 0
\(925\) 11.3137i 0.371992i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −58.0000 −1.90292 −0.951459 0.307775i \(-0.900416\pi\)
−0.951459 + 0.307775i \(0.900416\pi\)
\(930\) 0 0
\(931\) 1.00000 4.24264i 0.0327737 0.139047i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.65685i 0.184999i
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 43.8406i 1.42916i −0.699552 0.714582i \(-0.746617\pi\)
0.699552 0.714582i \(-0.253383\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.1838i 1.24081i −0.784283 0.620403i \(-0.786969\pi\)
0.784283 0.620403i \(-0.213031\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 32.5269i 1.05365i 0.849974 + 0.526825i \(0.176618\pi\)
−0.849974 + 0.526825i \(0.823382\pi\)
\(954\) 0 0
\(955\) 25.4558i 0.823732i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.9706i 0.548008i
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 48.0833i 1.54625i −0.634252 0.773127i \(-0.718692\pi\)
0.634252 0.773127i \(-0.281308\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.5563i 0.497692i −0.968543 0.248846i \(-0.919949\pi\)
0.968543 0.248846i \(-0.0800513\pi\)
\(978\) 0 0
\(979\) 14.0000 0.447442
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 28.0000 0.892154
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −42.0000 −1.33417 −0.667087 0.744980i \(-0.732459\pi\)
−0.667087 + 0.744980i \(0.732459\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.6274i 0.717337i
\(996\) 0 0
\(997\) −8.00000 −0.253363 −0.126681 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) 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.h.2431.2 2
3.2 odd 2 912.2.k.a.607.2 yes 2
4.3 odd 2 2736.2.k.i.2431.1 2
12.11 even 2 912.2.k.d.607.1 yes 2
19.18 odd 2 2736.2.k.i.2431.2 2
24.5 odd 2 3648.2.k.f.2431.2 2
24.11 even 2 3648.2.k.c.2431.1 2
57.56 even 2 912.2.k.d.607.2 yes 2
76.75 even 2 inner 2736.2.k.h.2431.1 2
228.227 odd 2 912.2.k.a.607.1 2
456.227 odd 2 3648.2.k.f.2431.1 2
456.341 even 2 3648.2.k.c.2431.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.k.a.607.1 2 228.227 odd 2
912.2.k.a.607.2 yes 2 3.2 odd 2
912.2.k.d.607.1 yes 2 12.11 even 2
912.2.k.d.607.2 yes 2 57.56 even 2
2736.2.k.h.2431.1 2 76.75 even 2 inner
2736.2.k.h.2431.2 2 1.1 even 1 trivial
2736.2.k.i.2431.1 2 4.3 odd 2
2736.2.k.i.2431.2 2 19.18 odd 2
3648.2.k.c.2431.1 2 24.11 even 2
3648.2.k.c.2431.2 2 456.341 even 2
3648.2.k.f.2431.1 2 456.227 odd 2
3648.2.k.f.2431.2 2 24.5 odd 2