Properties

Label 1620.2.a.i.1.2
Level $1620$
Weight $2$
Character 1620.1
Self dual yes
Analytic conductor $12.936$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(1,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.a (trivial)

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: yes
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 180)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.08613\) of defining polynomial
Character \(\chi\) \(=\) 1620.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.73419 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.73419 q^{7} +5.52420 q^{11} -3.52420 q^{13} -5.52420 q^{17} +7.52420 q^{19} +0.734191 q^{23} +1.00000 q^{25} -4.46838 q^{29} +7.52420 q^{31} -2.73419 q^{35} +6.05582 q^{37} -1.05582 q^{41} -3.52420 q^{43} +1.20999 q^{47} +0.475800 q^{49} +1.46838 q^{53} -5.52420 q^{55} -1.46838 q^{59} +9.05582 q^{61} +3.52420 q^{65} -4.25839 q^{67} +10.0558 q^{71} +8.00000 q^{73} +15.1042 q^{77} +2.00000 q^{79} -5.26581 q^{83} +5.52420 q^{85} -3.00000 q^{89} -9.63583 q^{91} -7.52420 q^{95} +9.46838 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 3 q^{7} + 6 q^{13} + 6 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{29} + 6 q^{31} - 3 q^{35} + 12 q^{37} + 3 q^{41} + 6 q^{43} + 15 q^{47} + 18 q^{49} - 6 q^{53} + 6 q^{59} + 21 q^{61} - 6 q^{65} + 9 q^{67} + 24 q^{71} + 24 q^{73} + 6 q^{77} + 6 q^{79} - 21 q^{83} - 9 q^{89} - 6 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −0.447214
\(6\) 0 0
\(7\) 2.73419 1.03343 0.516714 0.856158i \(-0.327155\pi\)
0.516714 + 0.856158i \(0.327155\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.52420 1.66561 0.832804 0.553567i \(-0.186734\pi\)
0.832804 + 0.553567i \(0.186734\pi\)
\(12\) 0 0
\(13\) −3.52420 −0.977437 −0.488719 0.872442i \(-0.662536\pi\)
−0.488719 + 0.872442i \(0.662536\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.52420 −1.33982 −0.669908 0.742444i \(-0.733666\pi\)
−0.669908 + 0.742444i \(0.733666\pi\)
\(18\) 0 0
\(19\) 7.52420 1.72617 0.863085 0.505059i \(-0.168529\pi\)
0.863085 + 0.505059i \(0.168529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.734191 0.153089 0.0765447 0.997066i \(-0.475611\pi\)
0.0765447 + 0.997066i \(0.475611\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.46838 −0.829758 −0.414879 0.909877i \(-0.636176\pi\)
−0.414879 + 0.909877i \(0.636176\pi\)
\(30\) 0 0
\(31\) 7.52420 1.35139 0.675693 0.737183i \(-0.263844\pi\)
0.675693 + 0.737183i \(0.263844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.73419 −0.462163
\(36\) 0 0
\(37\) 6.05582 0.995570 0.497785 0.867300i \(-0.334147\pi\)
0.497785 + 0.867300i \(0.334147\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.05582 −0.164891 −0.0824455 0.996596i \(-0.526273\pi\)
−0.0824455 + 0.996596i \(0.526273\pi\)
\(42\) 0 0
\(43\) −3.52420 −0.537435 −0.268718 0.963219i \(-0.586600\pi\)
−0.268718 + 0.963219i \(0.586600\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.20999 0.176495 0.0882477 0.996099i \(-0.471873\pi\)
0.0882477 + 0.996099i \(0.471873\pi\)
\(48\) 0 0
\(49\) 0.475800 0.0679715
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.46838 0.201698 0.100849 0.994902i \(-0.467844\pi\)
0.100849 + 0.994902i \(0.467844\pi\)
\(54\) 0 0
\(55\) −5.52420 −0.744883
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.46838 −0.191167 −0.0955835 0.995421i \(-0.530472\pi\)
−0.0955835 + 0.995421i \(0.530472\pi\)
\(60\) 0 0
\(61\) 9.05582 1.15948 0.579739 0.814802i \(-0.303154\pi\)
0.579739 + 0.814802i \(0.303154\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.52420 0.437123
\(66\) 0 0
\(67\) −4.25839 −0.520245 −0.260123 0.965576i \(-0.583763\pi\)
−0.260123 + 0.965576i \(0.583763\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.0558 1.19341 0.596703 0.802462i \(-0.296477\pi\)
0.596703 + 0.802462i \(0.296477\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.1042 1.72129
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.26581 −0.577998 −0.288999 0.957329i \(-0.593322\pi\)
−0.288999 + 0.957329i \(0.593322\pi\)
\(84\) 0 0
\(85\) 5.52420 0.599184
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −9.63583 −1.01011
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.52420 −0.771967
\(96\) 0 0
\(97\) 9.46838 0.961369 0.480684 0.876894i \(-0.340388\pi\)
0.480684 + 0.876894i \(0.340388\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.4684 1.34015 0.670077 0.742292i \(-0.266261\pi\)
0.670077 + 0.742292i \(0.266261\pi\)
\(102\) 0 0
\(103\) 18.5726 1.83001 0.915006 0.403440i \(-0.132185\pi\)
0.915006 + 0.403440i \(0.132185\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.2510 1.86106 0.930531 0.366213i \(-0.119346\pi\)
0.930531 + 0.366213i \(0.119346\pi\)
\(108\) 0 0
\(109\) −0.524200 −0.0502092 −0.0251046 0.999685i \(-0.507992\pi\)
−0.0251046 + 0.999685i \(0.507992\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.46838 −0.702566 −0.351283 0.936269i \(-0.614255\pi\)
−0.351283 + 0.936269i \(0.614255\pi\)
\(114\) 0 0
\(115\) −0.734191 −0.0684637
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −15.1042 −1.38460
\(120\) 0 0
\(121\) 19.5168 1.77425
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.25839 0.732814 0.366407 0.930455i \(-0.380588\pi\)
0.366407 + 0.930455i \(0.380588\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0968 −1.40638 −0.703192 0.711000i \(-0.748243\pi\)
−0.703192 + 0.711000i \(0.748243\pi\)
\(132\) 0 0
\(133\) 20.5726 1.78387
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.5168 −1.58200 −0.790998 0.611819i \(-0.790438\pi\)
−0.790998 + 0.611819i \(0.790438\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −19.4684 −1.62803
\(144\) 0 0
\(145\) 4.46838 0.371079
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.5726 1.11191 0.555955 0.831212i \(-0.312353\pi\)
0.555955 + 0.831212i \(0.312353\pi\)
\(150\) 0 0
\(151\) −12.4610 −1.01406 −0.507029 0.861929i \(-0.669256\pi\)
−0.507029 + 0.861929i \(0.669256\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.52420 −0.604358
\(156\) 0 0
\(157\) −1.58002 −0.126099 −0.0630496 0.998010i \(-0.520083\pi\)
−0.0630496 + 0.998010i \(0.520083\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.00742 0.158207
\(162\) 0 0
\(163\) −4.47580 −0.350572 −0.175286 0.984518i \(-0.556085\pi\)
−0.175286 + 0.984518i \(0.556085\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.7900 1.29925 0.649625 0.760255i \(-0.274926\pi\)
0.649625 + 0.760255i \(0.274926\pi\)
\(168\) 0 0
\(169\) −0.580017 −0.0446167
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −23.0484 −1.75234 −0.876169 0.482005i \(-0.839909\pi\)
−0.876169 + 0.482005i \(0.839909\pi\)
\(174\) 0 0
\(175\) 2.73419 0.206685
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.9926 0.971111 0.485556 0.874206i \(-0.338617\pi\)
0.485556 + 0.874206i \(0.338617\pi\)
\(180\) 0 0
\(181\) −24.5652 −1.82592 −0.912958 0.408054i \(-0.866207\pi\)
−0.912958 + 0.408054i \(0.866207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.05582 −0.445233
\(186\) 0 0
\(187\) −30.5168 −2.23161
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 15.1042 1.09290 0.546451 0.837491i \(-0.315978\pi\)
0.546451 + 0.837491i \(0.315978\pi\)
\(192\) 0 0
\(193\) −14.5726 −1.04896 −0.524479 0.851423i \(-0.675740\pi\)
−0.524479 + 0.851423i \(0.675740\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.53162 −0.322864 −0.161432 0.986884i \(-0.551611\pi\)
−0.161432 + 0.986884i \(0.551611\pi\)
\(198\) 0 0
\(199\) −7.41256 −0.525463 −0.262731 0.964869i \(-0.584623\pi\)
−0.262731 + 0.964869i \(0.584623\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.2174 −0.857494
\(204\) 0 0
\(205\) 1.05582 0.0737415
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 41.5652 2.87512
\(210\) 0 0
\(211\) −13.4126 −0.923359 −0.461680 0.887047i \(-0.652753\pi\)
−0.461680 + 0.887047i \(0.652753\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.52420 0.240348
\(216\) 0 0
\(217\) 20.5726 1.39656
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 19.4684 1.30959
\(222\) 0 0
\(223\) −8.79001 −0.588623 −0.294311 0.955710i \(-0.595090\pi\)
−0.294311 + 0.955710i \(0.595090\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.05582 0.269194 0.134597 0.990900i \(-0.457026\pi\)
0.134597 + 0.990900i \(0.457026\pi\)
\(228\) 0 0
\(229\) 2.58002 0.170492 0.0852462 0.996360i \(-0.472832\pi\)
0.0852462 + 0.996360i \(0.472832\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.94418 −0.127368 −0.0636838 0.997970i \(-0.520285\pi\)
−0.0636838 + 0.997970i \(0.520285\pi\)
\(234\) 0 0
\(235\) −1.20999 −0.0789311
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.46838 0.483089 0.241545 0.970390i \(-0.422346\pi\)
0.241545 + 0.970390i \(0.422346\pi\)
\(240\) 0 0
\(241\) 2.41256 0.155407 0.0777035 0.996977i \(-0.475241\pi\)
0.0777035 + 0.996977i \(0.475241\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.475800 −0.0303978
\(246\) 0 0
\(247\) −26.5168 −1.68722
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.99258 −0.441368 −0.220684 0.975345i \(-0.570829\pi\)
−0.220684 + 0.975345i \(0.570829\pi\)
\(252\) 0 0
\(253\) 4.05582 0.254987
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −28.0968 −1.75263 −0.876315 0.481738i \(-0.840006\pi\)
−0.876315 + 0.481738i \(0.840006\pi\)
\(258\) 0 0
\(259\) 16.5578 1.02885
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.63583 0.470846 0.235423 0.971893i \(-0.424352\pi\)
0.235423 + 0.971893i \(0.424352\pi\)
\(264\) 0 0
\(265\) −1.46838 −0.0902020
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.6210 −1.13534 −0.567671 0.823255i \(-0.692155\pi\)
−0.567671 + 0.823255i \(0.692155\pi\)
\(270\) 0 0
\(271\) 6.57260 0.399257 0.199628 0.979872i \(-0.436026\pi\)
0.199628 + 0.979872i \(0.436026\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.52420 0.333122
\(276\) 0 0
\(277\) 1.52420 0.0915803 0.0457901 0.998951i \(-0.485419\pi\)
0.0457901 + 0.998951i \(0.485419\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.5242 0.866441 0.433221 0.901288i \(-0.357377\pi\)
0.433221 + 0.901288i \(0.357377\pi\)
\(282\) 0 0
\(283\) −16.7342 −0.994744 −0.497372 0.867537i \(-0.665702\pi\)
−0.497372 + 0.867537i \(0.665702\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.88681 −0.170403
\(288\) 0 0
\(289\) 13.5168 0.795105
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −20.4610 −1.19534 −0.597671 0.801741i \(-0.703907\pi\)
−0.597671 + 0.801741i \(0.703907\pi\)
\(294\) 0 0
\(295\) 1.46838 0.0854925
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.58744 −0.149635
\(300\) 0 0
\(301\) −9.63583 −0.555400
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −9.05582 −0.518535
\(306\) 0 0
\(307\) −18.2026 −1.03888 −0.519438 0.854508i \(-0.673859\pi\)
−0.519438 + 0.854508i \(0.673859\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.10422 0.176024 0.0880120 0.996119i \(-0.471949\pi\)
0.0880120 + 0.996119i \(0.471949\pi\)
\(312\) 0 0
\(313\) −19.1042 −1.07983 −0.539917 0.841718i \(-0.681544\pi\)
−0.539917 + 0.841718i \(0.681544\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.00742 0.281245 0.140622 0.990063i \(-0.455090\pi\)
0.140622 + 0.990063i \(0.455090\pi\)
\(318\) 0 0
\(319\) −24.6842 −1.38205
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −41.5652 −2.31275
\(324\) 0 0
\(325\) −3.52420 −0.195487
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.30835 0.182395
\(330\) 0 0
\(331\) −3.48322 −0.191455 −0.0957275 0.995408i \(-0.530518\pi\)
−0.0957275 + 0.995408i \(0.530518\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.25839 0.232661
\(336\) 0 0
\(337\) 11.5800 0.630804 0.315402 0.948958i \(-0.397861\pi\)
0.315402 + 0.948958i \(0.397861\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 41.5652 2.25088
\(342\) 0 0
\(343\) −17.8384 −0.963183
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.4610 1.42050 0.710249 0.703950i \(-0.248582\pi\)
0.710249 + 0.703950i \(0.248582\pi\)
\(348\) 0 0
\(349\) −26.0336 −1.39354 −0.696772 0.717292i \(-0.745381\pi\)
−0.696772 + 0.717292i \(0.745381\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) −10.0558 −0.533707
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.04098 0.318831 0.159415 0.987212i \(-0.449039\pi\)
0.159415 + 0.987212i \(0.449039\pi\)
\(360\) 0 0
\(361\) 37.6136 1.97966
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 22.4610 1.17245 0.586226 0.810147i \(-0.300613\pi\)
0.586226 + 0.810147i \(0.300613\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.01484 0.208440
\(372\) 0 0
\(373\) −2.53162 −0.131082 −0.0655411 0.997850i \(-0.520877\pi\)
−0.0655411 + 0.997850i \(0.520877\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.7475 0.811036
\(378\) 0 0
\(379\) −21.0484 −1.08118 −0.540592 0.841285i \(-0.681800\pi\)
−0.540592 + 0.841285i \(0.681800\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.63583 −0.390173 −0.195086 0.980786i \(-0.562499\pi\)
−0.195086 + 0.980786i \(0.562499\pi\)
\(384\) 0 0
\(385\) −15.1042 −0.769782
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.9293 −0.959756 −0.479878 0.877335i \(-0.659319\pi\)
−0.479878 + 0.877335i \(0.659319\pi\)
\(390\) 0 0
\(391\) −4.05582 −0.205112
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) −29.1600 −1.46350 −0.731750 0.681573i \(-0.761296\pi\)
−0.731750 + 0.681573i \(0.761296\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.6284 1.62939 0.814693 0.579893i \(-0.196906\pi\)
0.814693 + 0.579893i \(0.196906\pi\)
\(402\) 0 0
\(403\) −26.5168 −1.32089
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.4535 1.65823
\(408\) 0 0
\(409\) 32.2084 1.59260 0.796302 0.604899i \(-0.206786\pi\)
0.796302 + 0.604899i \(0.206786\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.01484 −0.197557
\(414\) 0 0
\(415\) 5.26581 0.258488
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −21.4126 −1.04607 −0.523036 0.852311i \(-0.675201\pi\)
−0.523036 + 0.852311i \(0.675201\pi\)
\(420\) 0 0
\(421\) −1.88836 −0.0920333 −0.0460166 0.998941i \(-0.514653\pi\)
−0.0460166 + 0.998941i \(0.514653\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.52420 −0.267963
\(426\) 0 0
\(427\) 24.7603 1.19824
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.11164 −0.101714 −0.0508569 0.998706i \(-0.516195\pi\)
−0.0508569 + 0.998706i \(0.516195\pi\)
\(432\) 0 0
\(433\) −9.52420 −0.457704 −0.228852 0.973461i \(-0.573497\pi\)
−0.228852 + 0.973461i \(0.573497\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.52420 0.264258
\(438\) 0 0
\(439\) −14.4051 −0.687520 −0.343760 0.939058i \(-0.611701\pi\)
−0.343760 + 0.939058i \(0.611701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.2100 −0.912694 −0.456347 0.889802i \(-0.650842\pi\)
−0.456347 + 0.889802i \(0.650842\pi\)
\(444\) 0 0
\(445\) 3.00000 0.142214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.5800 1.58474 0.792369 0.610041i \(-0.208847\pi\)
0.792369 + 0.610041i \(0.208847\pi\)
\(450\) 0 0
\(451\) −5.83255 −0.274644
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 9.63583 0.451735
\(456\) 0 0
\(457\) 2.16745 0.101389 0.0506946 0.998714i \(-0.483856\pi\)
0.0506946 + 0.998714i \(0.483856\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.7401 −1.19883 −0.599417 0.800437i \(-0.704601\pi\)
−0.599417 + 0.800437i \(0.704601\pi\)
\(462\) 0 0
\(463\) 12.0558 0.560281 0.280141 0.959959i \(-0.409619\pi\)
0.280141 + 0.959959i \(0.409619\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −14.4610 −0.669174 −0.334587 0.942365i \(-0.608597\pi\)
−0.334587 + 0.942365i \(0.608597\pi\)
\(468\) 0 0
\(469\) −11.6433 −0.537635
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −19.4684 −0.895157
\(474\) 0 0
\(475\) 7.52420 0.345234
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.9852 −0.913145 −0.456573 0.889686i \(-0.650923\pi\)
−0.456573 + 0.889686i \(0.650923\pi\)
\(480\) 0 0
\(481\) −21.3419 −0.973107
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9.46838 −0.429937
\(486\) 0 0
\(487\) −18.1526 −0.822574 −0.411287 0.911506i \(-0.634921\pi\)
−0.411287 + 0.911506i \(0.634921\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 39.5800 1.78622 0.893111 0.449837i \(-0.148518\pi\)
0.893111 + 0.449837i \(0.148518\pi\)
\(492\) 0 0
\(493\) 24.6842 1.11172
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.4945 1.23330
\(498\) 0 0
\(499\) −42.6284 −1.90831 −0.954155 0.299313i \(-0.903243\pi\)
−0.954155 + 0.299313i \(0.903243\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 21.1952 0.945045 0.472523 0.881319i \(-0.343343\pi\)
0.472523 + 0.881319i \(0.343343\pi\)
\(504\) 0 0
\(505\) −13.4684 −0.599335
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.0336 −0.976620 −0.488310 0.872670i \(-0.662387\pi\)
−0.488310 + 0.872670i \(0.662387\pi\)
\(510\) 0 0
\(511\) 21.8735 0.967628
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −18.5726 −0.818406
\(516\) 0 0
\(517\) 6.68423 0.293972
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.40515 0.0615606 0.0307803 0.999526i \(-0.490201\pi\)
0.0307803 + 0.999526i \(0.490201\pi\)
\(522\) 0 0
\(523\) −11.8532 −0.518306 −0.259153 0.965836i \(-0.583443\pi\)
−0.259153 + 0.965836i \(0.583443\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −41.5652 −1.81061
\(528\) 0 0
\(529\) −22.4610 −0.976564
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.72091 0.161171
\(534\) 0 0
\(535\) −19.2510 −0.832292
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.62842 0.113214
\(540\) 0 0
\(541\) −8.98516 −0.386302 −0.193151 0.981169i \(-0.561871\pi\)
−0.193151 + 0.981169i \(0.561871\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.524200 0.0224542
\(546\) 0 0
\(547\) −6.20257 −0.265203 −0.132601 0.991169i \(-0.542333\pi\)
−0.132601 + 0.991169i \(0.542333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −33.6210 −1.43230
\(552\) 0 0
\(553\) 5.46838 0.232539
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 12.4200 0.525309
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.27323 −0.180095 −0.0900475 0.995937i \(-0.528702\pi\)
−0.0900475 + 0.995937i \(0.528702\pi\)
\(564\) 0 0
\(565\) 7.46838 0.314197
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 16.0968 0.674813 0.337406 0.941359i \(-0.390450\pi\)
0.337406 + 0.941359i \(0.390450\pi\)
\(570\) 0 0
\(571\) 10.9368 0.457689 0.228845 0.973463i \(-0.426505\pi\)
0.228845 + 0.973463i \(0.426505\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.734191 0.0306179
\(576\) 0 0
\(577\) −37.6620 −1.56789 −0.783944 0.620831i \(-0.786795\pi\)
−0.783944 + 0.620831i \(0.786795\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.3977 −0.597318
\(582\) 0 0
\(583\) 8.11164 0.335950
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.7974 0.899676 0.449838 0.893110i \(-0.351482\pi\)
0.449838 + 0.893110i \(0.351482\pi\)
\(588\) 0 0
\(589\) 56.6136 2.33272
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.2643 1.16067 0.580337 0.814377i \(-0.302921\pi\)
0.580337 + 0.814377i \(0.302921\pi\)
\(594\) 0 0
\(595\) 15.1042 0.619213
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.1116 0.576586 0.288293 0.957542i \(-0.406912\pi\)
0.288293 + 0.957542i \(0.406912\pi\)
\(600\) 0 0
\(601\) −28.2084 −1.15065 −0.575323 0.817926i \(-0.695124\pi\)
−0.575323 + 0.817926i \(0.695124\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −19.5168 −0.793470
\(606\) 0 0
\(607\) 28.3700 1.15150 0.575752 0.817624i \(-0.304709\pi\)
0.575752 + 0.817624i \(0.304709\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.26425 −0.172513
\(612\) 0 0
\(613\) 24.5726 0.992478 0.496239 0.868186i \(-0.334714\pi\)
0.496239 + 0.868186i \(0.334714\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −26.6284 −1.07202 −0.536010 0.844212i \(-0.680069\pi\)
−0.536010 + 0.844212i \(0.680069\pi\)
\(618\) 0 0
\(619\) 30.5726 1.22882 0.614408 0.788988i \(-0.289395\pi\)
0.614408 + 0.788988i \(0.289395\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.20257 −0.328629
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −33.4535 −1.33388
\(630\) 0 0
\(631\) 27.4684 1.09350 0.546750 0.837296i \(-0.315865\pi\)
0.546750 + 0.837296i \(0.315865\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.25839 −0.327724
\(636\) 0 0
\(637\) −1.67682 −0.0664379
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.5390 0.495262 0.247631 0.968854i \(-0.420348\pi\)
0.247631 + 0.968854i \(0.420348\pi\)
\(642\) 0 0
\(643\) −10.2584 −0.404551 −0.202276 0.979329i \(-0.564834\pi\)
−0.202276 + 0.979329i \(0.564834\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.7900 0.660083 0.330042 0.943966i \(-0.392937\pi\)
0.330042 + 0.943966i \(0.392937\pi\)
\(648\) 0 0
\(649\) −8.11164 −0.318410
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 50.0261 1.95767 0.978837 0.204641i \(-0.0656026\pi\)
0.978837 + 0.204641i \(0.0656026\pi\)
\(654\) 0 0
\(655\) 16.0968 0.628954
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.56518 0.216789 0.108394 0.994108i \(-0.465429\pi\)
0.108394 + 0.994108i \(0.465429\pi\)
\(660\) 0 0
\(661\) −39.2568 −1.52691 −0.763457 0.645859i \(-0.776499\pi\)
−0.763457 + 0.645859i \(0.776499\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −20.5726 −0.797771
\(666\) 0 0
\(667\) −3.28065 −0.127027
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 50.0261 1.93124
\(672\) 0 0
\(673\) 15.9442 0.614603 0.307302 0.951612i \(-0.400574\pi\)
0.307302 + 0.951612i \(0.400574\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −20.8958 −0.803090 −0.401545 0.915839i \(-0.631527\pi\)
−0.401545 + 0.915839i \(0.631527\pi\)
\(678\) 0 0
\(679\) 25.8884 0.993504
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −19.6358 −0.751344 −0.375672 0.926753i \(-0.622588\pi\)
−0.375672 + 0.926753i \(0.622588\pi\)
\(684\) 0 0
\(685\) 18.5168 0.707490
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.17487 −0.197147
\(690\) 0 0
\(691\) −43.1452 −1.64132 −0.820660 0.571416i \(-0.806394\pi\)
−0.820660 + 0.571416i \(0.806394\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 5.83255 0.220923
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 27.0410 1.02132 0.510662 0.859782i \(-0.329400\pi\)
0.510662 + 0.859782i \(0.329400\pi\)
\(702\) 0 0
\(703\) 45.5652 1.71852
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.8251 1.38495
\(708\) 0 0
\(709\) 43.5019 1.63375 0.816875 0.576815i \(-0.195705\pi\)
0.816875 + 0.576815i \(0.195705\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.52420 0.206883
\(714\) 0 0
\(715\) 19.4684 0.728076
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.40515 −0.164284 −0.0821421 0.996621i \(-0.526176\pi\)
−0.0821421 + 0.996621i \(0.526176\pi\)
\(720\) 0 0
\(721\) 50.7810 1.89118
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.46838 −0.165952
\(726\) 0 0
\(727\) 12.4817 0.462919 0.231460 0.972845i \(-0.425650\pi\)
0.231460 + 0.972845i \(0.425650\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19.4684 0.720064
\(732\) 0 0
\(733\) 39.0336 1.44174 0.720869 0.693072i \(-0.243743\pi\)
0.720869 + 0.693072i \(0.243743\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.5242 −0.866525
\(738\) 0 0
\(739\) −12.2935 −0.452224 −0.226112 0.974101i \(-0.572602\pi\)
−0.226112 + 0.974101i \(0.572602\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.8310 −1.49794 −0.748972 0.662602i \(-0.769452\pi\)
−0.748972 + 0.662602i \(0.769452\pi\)
\(744\) 0 0
\(745\) −13.5726 −0.497262
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 52.6358 1.92327
\(750\) 0 0
\(751\) −15.0894 −0.550619 −0.275310 0.961356i \(-0.588780\pi\)
−0.275310 + 0.961356i \(0.588780\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.4610 0.453501
\(756\) 0 0
\(757\) 34.4610 1.25251 0.626253 0.779620i \(-0.284588\pi\)
0.626253 + 0.779620i \(0.284588\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.35675 −0.302932 −0.151466 0.988462i \(-0.548399\pi\)
−0.151466 + 0.988462i \(0.548399\pi\)
\(762\) 0 0
\(763\) −1.43326 −0.0518876
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.17487 0.186854
\(768\) 0 0
\(769\) −18.3567 −0.661961 −0.330981 0.943638i \(-0.607379\pi\)
−0.330981 + 0.943638i \(0.607379\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −10.0968 −0.363157 −0.181578 0.983376i \(-0.558121\pi\)
−0.181578 + 0.983376i \(0.558121\pi\)
\(774\) 0 0
\(775\) 7.52420 0.270277
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.94418 −0.284630
\(780\) 0 0
\(781\) 55.5503 1.98775
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.58002 0.0563932
\(786\) 0 0
\(787\) −14.5726 −0.519457 −0.259729 0.965682i \(-0.583633\pi\)
−0.259729 + 0.965682i \(0.583633\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20.4200 −0.726051
\(792\) 0 0
\(793\) −31.9145 −1.13332
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.07065 −0.0733463 −0.0366732 0.999327i \(-0.511676\pi\)
−0.0366732 + 0.999327i \(0.511676\pi\)
\(798\) 0 0
\(799\) −6.68423 −0.236471
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 44.1936 1.55956
\(804\) 0 0
\(805\) −2.00742 −0.0707522
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.37158 0.118539 0.0592693 0.998242i \(-0.481123\pi\)
0.0592693 + 0.998242i \(0.481123\pi\)
\(810\) 0 0
\(811\) −12.9368 −0.454271 −0.227136 0.973863i \(-0.572936\pi\)
−0.227136 + 0.973863i \(0.572936\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.47580 0.156780
\(816\) 0 0
\(817\) −26.5168 −0.927705
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21.9926 −0.767546 −0.383773 0.923427i \(-0.625376\pi\)
−0.383773 + 0.923427i \(0.625376\pi\)
\(822\) 0 0
\(823\) 14.9016 0.519439 0.259719 0.965684i \(-0.416370\pi\)
0.259719 + 0.965684i \(0.416370\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 56.5785 1.96743 0.983713 0.179747i \(-0.0575279\pi\)
0.983713 + 0.179747i \(0.0575279\pi\)
\(828\) 0 0
\(829\) 27.0558 0.939687 0.469844 0.882750i \(-0.344310\pi\)
0.469844 + 0.882750i \(0.344310\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.62842 −0.0910692
\(834\) 0 0
\(835\) −16.7900 −0.581042
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.5726 −1.19358 −0.596789 0.802398i \(-0.703557\pi\)
−0.596789 + 0.802398i \(0.703557\pi\)
\(840\) 0 0
\(841\) −9.03356 −0.311502
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.580017 0.0199532
\(846\) 0 0
\(847\) 53.3626 1.83356
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.44613 0.152411
\(852\) 0 0
\(853\) 16.9777 0.581307 0.290653 0.956828i \(-0.406127\pi\)
0.290653 + 0.956828i \(0.406127\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.41256 0.116571 0.0582855 0.998300i \(-0.481437\pi\)
0.0582855 + 0.998300i \(0.481437\pi\)
\(858\) 0 0
\(859\) −9.83255 −0.335482 −0.167741 0.985831i \(-0.553647\pi\)
−0.167741 + 0.985831i \(0.553647\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −48.9836 −1.66742 −0.833711 0.552202i \(-0.813788\pi\)
−0.833711 + 0.552202i \(0.813788\pi\)
\(864\) 0 0
\(865\) 23.0484 0.783669
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.0484 0.374791
\(870\) 0 0
\(871\) 15.0074 0.508507
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.73419 −0.0924325
\(876\) 0 0
\(877\) −12.6694 −0.427815 −0.213908 0.976854i \(-0.568619\pi\)
−0.213908 + 0.976854i \(0.568619\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −26.0894 −0.878974 −0.439487 0.898249i \(-0.644840\pi\)
−0.439487 + 0.898249i \(0.644840\pi\)
\(882\) 0 0
\(883\) 2.69321 0.0906337 0.0453169 0.998973i \(-0.485570\pi\)
0.0453169 + 0.998973i \(0.485570\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24.5578 −0.824569 −0.412284 0.911055i \(-0.635269\pi\)
−0.412284 + 0.911055i \(0.635269\pi\)
\(888\) 0 0
\(889\) 22.5800 0.757309
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.10422 0.304661
\(894\) 0 0
\(895\) −12.9926 −0.434294
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −33.6210 −1.12132
\(900\) 0 0
\(901\) −8.11164 −0.270238
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.5652 0.816574
\(906\) 0 0
\(907\) 34.3700 1.14124 0.570619 0.821215i \(-0.306703\pi\)
0.570619 + 0.821215i \(0.306703\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 37.5503 1.24410 0.622049 0.782978i \(-0.286300\pi\)
0.622049 + 0.782978i \(0.286300\pi\)
\(912\) 0 0
\(913\) −29.0894 −0.962718
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −44.0117 −1.45340
\(918\) 0 0
\(919\) 10.1116 0.333552 0.166776 0.985995i \(-0.446664\pi\)
0.166776 + 0.985995i \(0.446664\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −35.4387 −1.16648
\(924\) 0 0
\(925\) 6.05582 0.199114
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.8251 0.814486 0.407243 0.913320i \(-0.366490\pi\)
0.407243 + 0.913320i \(0.366490\pi\)
\(930\) 0 0
\(931\) 3.58002 0.117330
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 30.5168 0.998005
\(936\) 0 0
\(937\) 1.56518 0.0511322 0.0255661 0.999673i \(-0.491861\pi\)
0.0255661 + 0.999673i \(0.491861\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −10.4684 −0.341259 −0.170630 0.985335i \(-0.554580\pi\)
−0.170630 + 0.985335i \(0.554580\pi\)
\(942\) 0 0
\(943\) −0.775172 −0.0252431
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.19515 −0.103829 −0.0519143 0.998652i \(-0.516532\pi\)
−0.0519143 + 0.998652i \(0.516532\pi\)
\(948\) 0 0
\(949\) −28.1936 −0.915203
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −33.9293 −1.09908 −0.549540 0.835468i \(-0.685197\pi\)
−0.549540 + 0.835468i \(0.685197\pi\)
\(954\) 0 0
\(955\) −15.1042 −0.488761
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −50.6284 −1.63488
\(960\) 0 0
\(961\) 25.6136 0.826245
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14.5726 0.469109
\(966\) 0 0
\(967\) 38.4258 1.23569 0.617846 0.786299i \(-0.288006\pi\)
0.617846 + 0.786299i \(0.288006\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29.7326 0.954166 0.477083 0.878858i \(-0.341694\pi\)
0.477083 + 0.878858i \(0.341694\pi\)
\(972\) 0 0
\(973\) −10.9368 −0.350617
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −32.9777 −1.05505 −0.527526 0.849539i \(-0.676880\pi\)
−0.527526 + 0.849539i \(0.676880\pi\)
\(978\) 0 0
\(979\) −16.5726 −0.529663
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.37744 −0.235304 −0.117652 0.993055i \(-0.537537\pi\)
−0.117652 + 0.993055i \(0.537537\pi\)
\(984\) 0 0
\(985\) 4.53162 0.144389
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.58744 −0.0822757
\(990\) 0 0
\(991\) −4.51678 −0.143480 −0.0717401 0.997423i \(-0.522855\pi\)
−0.0717401 + 0.997423i \(0.522855\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.41256 0.234994
\(996\) 0 0
\(997\) −48.6284 −1.54008 −0.770039 0.637997i \(-0.779763\pi\)
−0.770039 + 0.637997i \(0.779763\pi\)
\(998\) 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 1620.2.a.i.1.2 3
3.2 odd 2 1620.2.a.j.1.2 3
4.3 odd 2 6480.2.a.bt.1.2 3
5.2 odd 4 8100.2.d.p.649.4 6
5.3 odd 4 8100.2.d.p.649.3 6
5.4 even 2 8100.2.a.v.1.2 3
9.2 odd 6 540.2.i.b.361.2 6
9.4 even 3 180.2.i.b.61.1 6
9.5 odd 6 540.2.i.b.181.2 6
9.7 even 3 180.2.i.b.121.1 yes 6
12.11 even 2 6480.2.a.bw.1.2 3
15.2 even 4 8100.2.d.o.649.4 6
15.8 even 4 8100.2.d.o.649.3 6
15.14 odd 2 8100.2.a.u.1.2 3
36.7 odd 6 720.2.q.k.481.3 6
36.11 even 6 2160.2.q.i.1441.2 6
36.23 even 6 2160.2.q.i.721.2 6
36.31 odd 6 720.2.q.k.241.3 6
45.2 even 12 2700.2.s.c.1549.4 12
45.4 even 6 900.2.i.c.601.3 6
45.7 odd 12 900.2.s.c.49.3 12
45.13 odd 12 900.2.s.c.349.3 12
45.14 odd 6 2700.2.i.c.1801.2 6
45.22 odd 12 900.2.s.c.349.4 12
45.23 even 12 2700.2.s.c.2449.4 12
45.29 odd 6 2700.2.i.c.901.2 6
45.32 even 12 2700.2.s.c.2449.3 12
45.34 even 6 900.2.i.c.301.3 6
45.38 even 12 2700.2.s.c.1549.3 12
45.43 odd 12 900.2.s.c.49.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.i.b.61.1 6 9.4 even 3
180.2.i.b.121.1 yes 6 9.7 even 3
540.2.i.b.181.2 6 9.5 odd 6
540.2.i.b.361.2 6 9.2 odd 6
720.2.q.k.241.3 6 36.31 odd 6
720.2.q.k.481.3 6 36.7 odd 6
900.2.i.c.301.3 6 45.34 even 6
900.2.i.c.601.3 6 45.4 even 6
900.2.s.c.49.3 12 45.7 odd 12
900.2.s.c.49.4 12 45.43 odd 12
900.2.s.c.349.3 12 45.13 odd 12
900.2.s.c.349.4 12 45.22 odd 12
1620.2.a.i.1.2 3 1.1 even 1 trivial
1620.2.a.j.1.2 3 3.2 odd 2
2160.2.q.i.721.2 6 36.23 even 6
2160.2.q.i.1441.2 6 36.11 even 6
2700.2.i.c.901.2 6 45.29 odd 6
2700.2.i.c.1801.2 6 45.14 odd 6
2700.2.s.c.1549.3 12 45.38 even 12
2700.2.s.c.1549.4 12 45.2 even 12
2700.2.s.c.2449.3 12 45.32 even 12
2700.2.s.c.2449.4 12 45.23 even 12
6480.2.a.bt.1.2 3 4.3 odd 2
6480.2.a.bw.1.2 3 12.11 even 2
8100.2.a.u.1.2 3 15.14 odd 2
8100.2.a.v.1.2 3 5.4 even 2
8100.2.d.o.649.3 6 15.8 even 4
8100.2.d.o.649.4 6 15.2 even 4
8100.2.d.p.649.3 6 5.3 odd 4
8100.2.d.p.649.4 6 5.2 odd 4