Properties

Label 495.2.k.a.307.1
Level $495$
Weight $2$
Character 495.307
Analytic conductor $3.953$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(208,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.k (of order \(4\), degree \(2\), 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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 307.1
Root \(1.65831 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 495.307
Dual form 495.2.k.a.208.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{4} +(-1.65831 + 1.50000i) q^{5} -3.31662 q^{11} -4.00000 q^{16} +(-3.00000 - 3.31662i) q^{20} +(-2.84169 + 2.84169i) q^{23} +(0.500000 - 4.97494i) q^{25} -9.94987 q^{31} +(1.47494 + 1.47494i) q^{37} -6.63325i q^{44} +(9.31662 + 9.31662i) q^{47} +7.00000i q^{49} +(-3.63325 + 3.63325i) q^{53} +(5.50000 - 4.97494i) q^{55} +3.31662i q^{59} -8.00000i q^{64} +(11.4749 + 11.4749i) q^{67} +3.00000 q^{71} +(6.63325 - 6.00000i) q^{80} -9.00000i q^{89} +(-5.68338 - 5.68338i) q^{92} +(-3.52506 - 3.52506i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{16} - 12 q^{20} - 18 q^{23} + 2 q^{25} - 14 q^{37} + 24 q^{47} + 12 q^{53} + 22 q^{55} + 26 q^{67} + 12 q^{71} - 36 q^{92} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000i 1.00000i
\(5\) −1.65831 + 1.50000i −0.741620 + 0.670820i
\(6\) 0 0
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.31662 −1.00000
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 3.31662i −0.670820 0.741620i
\(21\) 0 0
\(22\) 0 0
\(23\) −2.84169 + 2.84169i −0.592533 + 0.592533i −0.938315 0.345782i \(-0.887614\pi\)
0.345782 + 0.938315i \(0.387614\pi\)
\(24\) 0 0
\(25\) 0.500000 4.97494i 0.100000 0.994987i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −9.94987 −1.78705 −0.893525 0.449013i \(-0.851776\pi\)
−0.893525 + 0.449013i \(0.851776\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.47494 + 1.47494i 0.242478 + 0.242478i 0.817875 0.575396i \(-0.195152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 9.31662 + 9.31662i 1.35897 + 1.35897i 0.875190 + 0.483779i \(0.160736\pi\)
0.483779 + 0.875190i \(0.339264\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.63325 + 3.63325i −0.499065 + 0.499065i −0.911147 0.412082i \(-0.864802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 5.50000 4.97494i 0.741620 0.670820i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.31662i 0.431788i 0.976417 + 0.215894i \(0.0692665\pi\)
−0.976417 + 0.215894i \(0.930733\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 11.4749 + 11.4749i 1.40189 + 1.40189i 0.794101 + 0.607785i \(0.207942\pi\)
0.607785 + 0.794101i \(0.292058\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 6.63325 6.00000i 0.741620 0.670820i
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000i 0.953998i −0.878904 0.476999i \(-0.841725\pi\)
0.878904 0.476999i \(-0.158275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.68338 5.68338i −0.592533 0.592533i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.52506 3.52506i −0.357916 0.357916i 0.505128 0.863044i \(-0.331445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 9.94987 + 1.00000i 0.994987 + 0.100000i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −7.94987 + 7.94987i −0.783324 + 0.783324i −0.980390 0.197066i \(-0.936859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.1583 12.1583i 1.14376 1.14376i 0.156001 0.987757i \(-0.450140\pi\)
0.987757 0.156001i \(-0.0498603\pi\)
\(114\) 0 0
\(115\) 0.449874 8.97494i 0.0419510 0.836917i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 19.8997i 1.78705i
\(125\) 6.63325 + 9.00000i 0.593296 + 0.804984i
\(126\) 0 0
\(127\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.1082 + 10.1082i 0.863601 + 0.863601i 0.991754 0.128154i \(-0.0409051\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −2.94987 + 2.94987i −0.242478 + 0.242478i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 16.5000 14.9248i 1.32531 1.19879i
\(156\) 0 0
\(157\) 16.4749 + 16.4749i 1.31484 + 1.31484i 0.917800 + 0.397043i \(0.129964\pi\)
0.397043 + 0.917800i \(0.370036\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −17.9499 + 17.9499i −1.40594 + 1.40594i −0.626608 + 0.779334i \(0.715557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 13.2665 1.00000
\(177\) 0 0
\(178\) 0 0
\(179\) 21.0000i 1.56961i 0.619740 + 0.784807i \(0.287238\pi\)
−0.619740 + 0.784807i \(0.712762\pi\)
\(180\) 0 0
\(181\) −9.94987 −0.739568 −0.369784 0.929118i \(-0.620568\pi\)
−0.369784 + 0.929118i \(0.620568\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.65831 0.233501i −0.342486 0.0171673i
\(186\) 0 0
\(187\) 0 0
\(188\) −18.6332 + 18.6332i −1.35897 + 1.35897i
\(189\) 0 0
\(190\) 0 0
\(191\) −23.2164 −1.67988 −0.839939 0.542681i \(-0.817409\pi\)
−0.839939 + 0.542681i \(0.817409\pi\)
\(192\) 0 0
\(193\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 19.8997i 1.41066i −0.708881 0.705328i \(-0.750800\pi\)
0.708881 0.705328i \(-0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −7.26650 7.26650i −0.499065 0.499065i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 9.94987 + 11.0000i 0.670820 + 0.741620i
\(221\) 0 0
\(222\) 0 0
\(223\) 14.4248 14.4248i 0.965957 0.965957i −0.0334825 0.999439i \(-0.510660\pi\)
0.999439 + 0.0334825i \(0.0106598\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(228\) 0 0
\(229\) 29.8496i 1.97252i 0.165205 + 0.986259i \(0.447172\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(234\) 0 0
\(235\) −29.4248 1.47494i −1.91946 0.0962143i
\(236\) −6.63325 −0.431788
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −10.5000 11.6082i −0.670820 0.741620i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) 9.42481 9.42481i 0.592533 0.592533i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −22.2665 22.2665i −1.38895 1.38895i −0.827541 0.561405i \(-0.810261\pi\)
−0.561405 0.827541i \(-0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(264\) 0 0
\(265\) 0.575188 11.4749i 0.0353335 0.704900i
\(266\) 0 0
\(267\) 0 0
\(268\) −22.9499 + 22.9499i −1.40189 + 1.40189i
\(269\) 13.2665i 0.808873i −0.914566 0.404436i \(-0.867468\pi\)
0.914566 0.404436i \(-0.132532\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.65831 + 16.5000i −0.100000 + 0.994987i
\(276\) 0 0
\(277\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 6.00000i 0.356034i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) 0 0
\(295\) −4.97494 5.50000i −0.289652 0.320222i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 24.4248 24.4248i 1.38057 1.38057i 0.536972 0.843600i \(-0.319568\pi\)
0.843600 0.536972i \(-0.180432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.1082 + 25.1082i 1.41022 + 1.41022i 0.758236 + 0.651981i \(0.226062\pi\)
0.651981 + 0.758236i \(0.273938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 12.0000 + 13.2665i 0.670820 + 0.741620i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −9.94987 −0.546895 −0.273447 0.961887i \(-0.588164\pi\)
−0.273447 + 0.961887i \(0.588164\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −36.2414 1.81662i −1.98008 0.0992528i
\(336\) 0 0
\(337\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.0000 1.78705
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.7414 13.7414i 0.731383 0.731383i −0.239511 0.970894i \(-0.576987\pi\)
0.970894 + 0.239511i \(0.0769871\pi\)
\(354\) 0 0
\(355\) −4.97494 + 4.50000i −0.264042 + 0.238835i
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −13.5251 13.5251i −0.706003 0.706003i 0.259690 0.965692i \(-0.416380\pi\)
−0.965692 + 0.259690i \(0.916380\pi\)
\(368\) 11.3668 11.3668i 0.592533 0.592533i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.8496i 1.53327i 0.642082 + 0.766636i \(0.278071\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.8417 + 17.8417i −0.911668 + 0.911668i −0.996403 0.0847358i \(-0.972995\pi\)
0.0847358 + 0.996403i \(0.472995\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 7.05013 7.05013i 0.357916 0.357916i
\(389\) 36.4829i 1.84976i 0.380265 + 0.924878i \(0.375833\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −20.8997 20.8997i −1.04893 1.04893i −0.998740 0.0501886i \(-0.984018\pi\)
−0.0501886 0.998740i \(-0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −2.00000 + 19.8997i −0.100000 + 0.994987i
\(401\) 26.5330 1.32499 0.662497 0.749064i \(-0.269497\pi\)
0.662497 + 0.749064i \(0.269497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.89181 4.89181i −0.242478 0.242478i
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −15.8997 15.8997i −0.783324 0.783324i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.0000i 1.17248i −0.810139 0.586238i \(-0.800608\pi\)
0.810139 0.586238i \(-0.199392\pi\)
\(420\) 0 0
\(421\) 39.7995 1.93971 0.969854 0.243685i \(-0.0783563\pi\)
0.969854 + 0.243685i \(0.0783563\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 29.4248 29.4248i 1.41407 1.41407i 0.696826 0.717241i \(-0.254595\pi\)
0.717241 0.696826i \(-0.245405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.7414 28.7414i 1.36555 1.36555i 0.498870 0.866677i \(-0.333748\pi\)
0.866677 0.498870i \(-0.166252\pi\)
\(444\) 0 0
\(445\) 13.5000 + 14.9248i 0.639961 + 0.707504i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.0000i 1.84052i −0.391303 0.920262i \(-0.627976\pi\)
0.391303 0.920262i \(-0.372024\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 24.3166 + 24.3166i 1.14376 + 1.14376i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 17.9499 + 0.899749i 0.836917 + 0.0419510i
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −0.575188 + 0.575188i −0.0267313 + 0.0267313i −0.720346 0.693615i \(-0.756017\pi\)
0.693615 + 0.720346i \(0.256017\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −23.0581 23.0581i −1.06700 1.06700i −0.997588 0.0694117i \(-0.977888\pi\)
−0.0694117 0.997588i \(-0.522112\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) 11.1332 + 0.558061i 0.505535 + 0.0253403i
\(486\) 0 0
\(487\) 26.4749 + 26.4749i 1.19969 + 1.19969i 0.974258 + 0.225436i \(0.0723806\pi\)
0.225436 + 0.974258i \(0.427619\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 39.7995 1.78705
\(497\) 0 0
\(498\) 0 0
\(499\) 19.8997i 0.890835i −0.895323 0.445418i \(-0.853055\pi\)
0.895323 0.445418i \(-0.146945\pi\)
\(500\) −18.0000 + 13.2665i −0.804984 + 0.593296i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.31662i 0.147007i 0.997295 + 0.0735034i \(0.0234180\pi\)
−0.997295 + 0.0735034i \(0.976582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.25856 25.1082i 0.0554589 1.10640i
\(516\) 0 0
\(517\) −30.8997 30.8997i −1.35897 1.35897i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 43.1161 1.88895 0.944476 0.328581i \(-0.106570\pi\)
0.944476 + 0.328581i \(0.106570\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.84962i 0.297810i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 23.2164i 1.00000i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) −20.2164 + 20.2164i −0.863601 + 0.863601i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(564\) 0 0
\(565\) −1.92481 + 38.3997i −0.0809774 + 1.61549i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.7164 + 15.5581i 0.530309 + 0.648816i
\(576\) 0 0
\(577\) −18.5251 18.5251i −0.771208 0.771208i 0.207109 0.978318i \(-0.433594\pi\)
−0.978318 + 0.207109i \(0.933594\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0501 12.0501i 0.499065 0.499065i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −20.6834 20.6834i −0.853694 0.853694i 0.136892 0.990586i \(-0.456289\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −5.89975 5.89975i −0.242478 0.242478i
\(593\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000i 1.47092i 0.677568 + 0.735460i \(0.263034\pi\)
−0.677568 + 0.735460i \(0.736966\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −18.2414 + 16.5000i −0.741620 + 0.670820i
\(606\) 0 0
\(607\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.73350 + 7.73350i 0.311339 + 0.311339i 0.845428 0.534089i \(-0.179345\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 0 0
\(619\) 1.00000i 0.0401934i −0.999798 0.0200967i \(-0.993603\pi\)
0.999798 0.0200967i \(-0.00639741\pi\)
\(620\) 29.8496 + 33.0000i 1.19879 + 1.32531i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −24.5000 4.97494i −0.980000 0.198997i
\(626\) 0 0
\(627\) 0 0
\(628\) −32.9499 + 32.9499i −1.31484 + 1.31484i
\(629\) 0 0
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −23.2164 −0.916992 −0.458496 0.888697i \(-0.651612\pi\)
−0.458496 + 0.888697i \(0.651612\pi\)
\(642\) 0 0
\(643\) −5.57519 + 5.57519i −0.219864 + 0.219864i −0.808441 0.588577i \(-0.799688\pi\)
0.588577 + 0.808441i \(0.299688\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.05806 8.05806i −0.316795 0.316795i 0.530740 0.847535i \(-0.321914\pi\)
−0.847535 + 0.530740i \(0.821914\pi\)
\(648\) 0 0
\(649\) 11.0000i 0.431788i
\(650\) 0 0
\(651\) 0 0
\(652\) −35.8997 35.8997i −1.40594 1.40594i
\(653\) 27.1583 27.1583i 1.06279 1.06279i 0.0648948 0.997892i \(-0.479329\pi\)
0.997892 0.0648948i \(-0.0206712\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 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
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 26.0000 1.00000
\(677\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −35.2164 + 35.2164i −1.34752 + 1.34752i −0.459167 + 0.888350i \(0.651852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) −31.9248 1.60025i −1.21978 0.0611425i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 26.5330i 1.00000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 19.0000i 0.713560i 0.934188 + 0.356780i \(0.116125\pi\)
−0.934188 + 0.356780i \(0.883875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.2744 28.2744i 1.05889 1.05889i
\(714\) 0 0
\(715\) 0 0
\(716\) −42.0000 −1.56961
\(717\) 0 0
\(718\) 0 0
\(719\) 51.0000i 1.90198i 0.309223 + 0.950990i \(0.399931\pi\)
−0.309223 + 0.950990i \(0.600069\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 19.8997i 0.739568i
\(725\) 0 0
\(726\) 0 0
\(727\) 31.4749 + 31.4749i 1.16734 + 1.16734i 0.982831 + 0.184510i \(0.0590699\pi\)
0.184510 + 0.982831i \(0.440930\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −38.0581 38.0581i −1.40189 1.40189i
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0.467002 9.31662i 0.0171673 0.342486i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) −37.2665 37.2665i −1.35897 1.35897i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.899749 0.899749i −0.0327019 0.0327019i 0.690567 0.723269i \(-0.257361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 46.4327i 1.67988i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −33.6332 + 33.6332i −1.20970 + 1.20970i −0.238581 + 0.971123i \(0.576682\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) −4.97494 + 49.5000i −0.178705 + 1.77809i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −9.94987 −0.356034
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000i 1.00000i
\(785\) −52.0330 2.60819i −1.85714 0.0930902i
\(786\) 0 0
\(787\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 39.7995 1.41066
\(797\) 26.6913 + 26.6913i 0.945455 + 0.945455i 0.998587 0.0531327i \(-0.0169206\pi\)
−0.0531327 + 0.998587i \(0.516921\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.84169 56.6913i 0.0995400 1.98581i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 39.4248 39.4248i 1.37426 1.37426i 0.520246 0.854016i \(-0.325840\pi\)
0.854016 0.520246i \(-0.174160\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) 29.0000i 1.00721i 0.863934 + 0.503606i \(0.167994\pi\)
−0.863934 + 0.503606i \(0.832006\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.4829i 1.25953i 0.776786 + 0.629764i \(0.216849\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 19.5000 + 21.5581i 0.670820 + 0.741620i
\(846\) 0 0
\(847\) 0 0
\(848\) 14.5330 14.5330i 0.499065 0.499065i
\(849\) 0 0
\(850\) 0 0
\(851\) −8.38262 −0.287353
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(858\) 0 0
\(859\) 31.0000i 1.05771i −0.848713 0.528853i \(-0.822622\pi\)
0.848713 0.528853i \(-0.177378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.21637 + 5.21637i −0.177567 + 0.177567i −0.790295 0.612727i \(-0.790072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −22.0000 + 19.8997i −0.741620 + 0.670820i
\(881\) −57.0000 −1.92038 −0.960189 0.279350i \(-0.909881\pi\)
−0.960189 + 0.279350i \(0.909881\pi\)
\(882\) 0 0
\(883\) −37.9499 + 37.9499i −1.27711 + 1.27711i −0.334840 + 0.942275i \(0.608682\pi\)
−0.942275 + 0.334840i \(0.891318\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 28.8496 + 28.8496i 0.965957 + 0.965957i
\(893\) 0 0
\(894\) 0 0
\(895\) −31.5000 34.8246i −1.05293 1.16406i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.5000 14.9248i 0.548479 0.496118i
\(906\) 0 0
\(907\) 33.8496 + 33.8496i 1.12396 + 1.12396i 0.991140 + 0.132818i \(0.0424025\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6.63325 −0.219769 −0.109885 0.993944i \(-0.535048\pi\)
−0.109885 + 0.993944i \(0.535048\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −59.6992 −1.97252
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.07519 6.60025i 0.265511 0.217015i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 53.0660i 1.74104i 0.492134 + 0.870519i \(0.336217\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.94987 58.8496i 0.0962143 1.91946i
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 13.2665i 0.431788i
\(945\) 0 0
\(946\) 0 0
\(947\) 40.1082 + 40.1082i 1.30334 + 1.30334i 0.926126 + 0.377215i \(0.123118\pi\)
0.377215 + 0.926126i \(0.376882\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(954\) 0 0
\(955\) 38.5000 34.8246i 1.24583 1.12690i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 68.0000 2.19355
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 43.1161 1.38366 0.691831 0.722059i \(-0.256804\pi\)
0.691831 + 0.722059i \(0.256804\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41.6913 + 41.6913i 1.33382 + 1.33382i 0.901920 + 0.431903i \(0.142158\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) 29.8496i 0.953998i
\(980\) 23.2164 21.0000i 0.741620 0.670820i
\(981\) 0 0
\(982\) 0 0
\(983\) 43.7414 43.7414i 1.39514 1.39514i 0.581811 0.813324i \(-0.302344\pi\)
0.813324 0.581811i \(-0.197656\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −59.6992 −1.89641 −0.948205 0.317660i \(-0.897103\pi\)
−0.948205 + 0.317660i \(0.897103\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 29.8496 + 33.0000i 0.946297 + 1.04617i
\(996\) 0 0
\(997\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(998\) 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 495.2.k.a.307.1 4
3.2 odd 2 55.2.e.b.32.1 4
5.3 odd 4 inner 495.2.k.a.208.1 4
11.10 odd 2 CM 495.2.k.a.307.1 4
12.11 even 2 880.2.bd.d.417.2 4
15.2 even 4 275.2.e.a.43.2 4
15.8 even 4 55.2.e.b.43.1 yes 4
15.14 odd 2 275.2.e.a.32.2 4
33.2 even 10 605.2.m.a.282.2 16
33.5 odd 10 605.2.m.a.602.1 16
33.8 even 10 605.2.m.a.112.2 16
33.14 odd 10 605.2.m.a.112.2 16
33.17 even 10 605.2.m.a.602.1 16
33.20 odd 10 605.2.m.a.282.2 16
33.26 odd 10 605.2.m.a.457.2 16
33.29 even 10 605.2.m.a.457.2 16
33.32 even 2 55.2.e.b.32.1 4
55.43 even 4 inner 495.2.k.a.208.1 4
60.23 odd 4 880.2.bd.d.593.2 4
132.131 odd 2 880.2.bd.d.417.2 4
165.8 odd 20 605.2.m.a.233.2 16
165.32 odd 4 275.2.e.a.43.2 4
165.38 even 20 605.2.m.a.118.2 16
165.53 even 20 605.2.m.a.403.1 16
165.68 odd 20 605.2.m.a.403.1 16
165.83 odd 20 605.2.m.a.118.2 16
165.98 odd 4 55.2.e.b.43.1 yes 4
165.113 even 20 605.2.m.a.233.2 16
165.128 odd 20 605.2.m.a.578.2 16
165.158 even 20 605.2.m.a.578.2 16
165.164 even 2 275.2.e.a.32.2 4
660.263 even 4 880.2.bd.d.593.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.e.b.32.1 4 3.2 odd 2
55.2.e.b.32.1 4 33.32 even 2
55.2.e.b.43.1 yes 4 15.8 even 4
55.2.e.b.43.1 yes 4 165.98 odd 4
275.2.e.a.32.2 4 15.14 odd 2
275.2.e.a.32.2 4 165.164 even 2
275.2.e.a.43.2 4 15.2 even 4
275.2.e.a.43.2 4 165.32 odd 4
495.2.k.a.208.1 4 5.3 odd 4 inner
495.2.k.a.208.1 4 55.43 even 4 inner
495.2.k.a.307.1 4 1.1 even 1 trivial
495.2.k.a.307.1 4 11.10 odd 2 CM
605.2.m.a.112.2 16 33.8 even 10
605.2.m.a.112.2 16 33.14 odd 10
605.2.m.a.118.2 16 165.38 even 20
605.2.m.a.118.2 16 165.83 odd 20
605.2.m.a.233.2 16 165.8 odd 20
605.2.m.a.233.2 16 165.113 even 20
605.2.m.a.282.2 16 33.2 even 10
605.2.m.a.282.2 16 33.20 odd 10
605.2.m.a.403.1 16 165.53 even 20
605.2.m.a.403.1 16 165.68 odd 20
605.2.m.a.457.2 16 33.26 odd 10
605.2.m.a.457.2 16 33.29 even 10
605.2.m.a.578.2 16 165.128 odd 20
605.2.m.a.578.2 16 165.158 even 20
605.2.m.a.602.1 16 33.5 odd 10
605.2.m.a.602.1 16 33.17 even 10
880.2.bd.d.417.2 4 12.11 even 2
880.2.bd.d.417.2 4 132.131 odd 2
880.2.bd.d.593.2 4 60.23 odd 4
880.2.bd.d.593.2 4 660.263 even 4