Properties

Label 1088.2.m.g.225.2
Level $1088$
Weight $2$
Character 1088.225
Analytic conductor $8.688$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1088,2,Mod(225,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1088.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1088.m (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: \(8.68772373992\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 225.2
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1088.225
Dual form 1088.2.m.g.353.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.41421 + 2.41421i) q^{3} +8.65685i q^{9} +O(q^{10})\) \(q+(2.41421 + 2.41421i) q^{3} +8.65685i q^{9} +(-4.41421 + 4.41421i) q^{11} +(2.82843 - 3.00000i) q^{17} -2.00000 q^{19} -5.00000i q^{25} +(-13.6569 + 13.6569i) q^{27} -21.3137 q^{33} +(8.65685 + 8.65685i) q^{41} +8.48528 q^{43} -7.00000i q^{49} +(14.0711 - 0.414214i) q^{51} +(-4.82843 - 4.82843i) q^{57} +14.1421 q^{59} +8.48528i q^{67} +(-9.48528 + 9.48528i) q^{73} +(12.0711 - 12.0711i) q^{75} -39.9706 q^{81} -2.82843 q^{83} +5.65685 q^{89} +(13.4853 - 13.4853i) q^{97} +(-38.2132 - 38.2132i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 12 q^{11} - 8 q^{19} - 32 q^{27} - 40 q^{33} + 12 q^{41} + 28 q^{51} - 8 q^{57} - 4 q^{73} + 20 q^{75} - 92 q^{81} + 20 q^{97} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 2.41421 + 2.41421i 1.39385 + 1.39385i 0.816497 + 0.577350i \(0.195913\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0 0
\(9\) 8.65685i 2.88562i
\(10\) 0 0
\(11\) −4.41421 + 4.41421i −1.33094 + 1.33094i −0.426401 + 0.904534i \(0.640219\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 3.00000i 0.685994 0.727607i
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) −13.6569 + 13.6569i −2.62826 + 2.62826i
\(28\) 0 0
\(29\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(30\) 0 0
\(31\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(32\) 0 0
\(33\) −21.3137 −3.71024
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.65685 + 8.65685i 1.35197 + 1.35197i 0.883452 + 0.468521i \(0.155213\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 14.0711 0.414214i 1.97034 0.0580015i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.82843 4.82843i −0.639541 0.639541i
\(58\) 0 0
\(59\) 14.1421 1.84115 0.920575 0.390567i \(-0.127721\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 0 0
\(61\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.48528i 1.03664i 0.855186 + 0.518321i \(0.173443\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(72\) 0 0
\(73\) −9.48528 + 9.48528i −1.11017 + 1.11017i −0.117041 + 0.993127i \(0.537341\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 12.0711 12.0711i 1.39385 1.39385i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(80\) 0 0
\(81\) −39.9706 −4.44117
\(82\) 0 0
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.65685 0.599625 0.299813 0.953998i \(-0.403076\pi\)
0.299813 + 0.953998i \(0.403076\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.4853 13.4853i 1.36922 1.36922i 0.507673 0.861550i \(-0.330506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) −38.2132 38.2132i −3.84057 3.84057i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.8995 + 12.8995i 1.24704 + 1.24704i 0.957020 + 0.290021i \(0.0936623\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.6569 14.6569i −1.37880 1.37880i −0.846649 0.532152i \(-0.821383\pi\)
−0.532152 0.846649i \(-0.678617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 27.9706i 2.54278i
\(122\) 0 0
\(123\) 41.7990i 3.76889i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 20.4853 + 20.4853i 1.80363 + 1.80363i
\(130\) 0 0
\(131\) 1.92893 + 1.92893i 0.168532 + 0.168532i 0.786334 0.617802i \(-0.211977\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 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
\(139\) −15.2426 15.2426i −1.29286 1.29286i −0.933008 0.359856i \(-0.882826\pi\)
−0.359856 0.933008i \(-0.617174\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 16.8995 16.8995i 1.39385 1.39385i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 25.9706 + 24.4853i 2.09960 + 1.97952i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.7279 + 11.7279i −0.918602 + 0.918602i −0.996928 0.0783260i \(-0.975042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 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
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 17.3137i 1.32401i
\(172\) 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
\(176\) 0 0
\(177\) 34.1421 + 34.1421i 2.56628 + 2.56628i
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.757359 + 25.7279i 0.0553836 + 1.88141i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 2.51472 + 2.51472i 0.181013 + 0.181013i 0.791797 0.610784i \(-0.209146\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(198\) 0 0
\(199\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(200\) 0 0
\(201\) −20.4853 + 20.4853i −1.44492 + 1.44492i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.82843 8.82843i 0.610675 0.610675i
\(210\) 0 0
\(211\) −5.72792 + 5.72792i −0.394326 + 0.394326i −0.876226 0.481900i \(-0.839947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −45.7990 −3.09481
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 43.2843 2.88562
\(226\) 0 0
\(227\) 16.4142 16.4142i 1.08945 1.08945i 0.0938647 0.995585i \(-0.470078\pi\)
0.995585 0.0938647i \(-0.0299221\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.1716 + 12.1716i −0.797386 + 0.797386i −0.982683 0.185296i \(-0.940675\pi\)
0.185296 + 0.982683i \(0.440675\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 21.4853 21.4853i 1.38399 1.38399i 0.546585 0.837404i \(-0.315928\pi\)
0.837404 0.546585i \(-0.184072\pi\)
\(242\) 0 0
\(243\) −55.5269 55.5269i −3.56205 3.56205i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.82843 6.82843i −0.432734 0.432734i
\(250\) 0 0
\(251\) 31.1127i 1.96382i −0.189358 0.981908i \(-0.560641\pi\)
0.189358 0.981908i \(-0.439359\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.3137i 0.705730i −0.935674 0.352865i \(-0.885208\pi\)
0.935674 0.352865i \(-0.114792\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 13.6569 + 13.6569i 0.835786 + 0.835786i
\(268\) 0 0
\(269\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.0711 + 22.0711i 1.33094 + 1.33094i
\(276\) 0 0
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000i 1.07379i 0.843649 + 0.536895i \(0.180403\pi\)
−0.843649 + 0.536895i \(0.819597\pi\)
\(282\) 0 0
\(283\) −23.7279 + 23.7279i −1.41048 + 1.41048i −0.653882 + 0.756596i \(0.726861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 16.9706i −0.0588235 0.998268i
\(290\) 0 0
\(291\) 65.1127 3.81697
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 120.569i 6.99610i
\(298\) 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
\(307\) 34.0000i 1.94048i −0.242140 0.970241i \(-0.577849\pi\)
0.242140 0.970241i \(-0.422151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(312\) 0 0
\(313\) −11.9706 11.9706i −0.676616 0.676616i 0.282617 0.959233i \(-0.408798\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 62.2843i 3.47637i
\(322\) 0 0
\(323\) −5.65685 + 6.00000i −0.314756 + 0.333849i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 25.4558 1.39918 0.699590 0.714545i \(-0.253366\pi\)
0.699590 + 0.714545i \(0.253366\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −9.97056 + 9.97056i −0.543131 + 0.543131i −0.924445 0.381314i \(-0.875472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 70.7696i 3.84367i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.3848 + 15.3848i −0.825898 + 0.825898i −0.986947 0.161048i \(-0.948512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 22.6274 1.20434 0.602168 0.798369i \(-0.294304\pi\)
0.602168 + 0.798369i \(0.294304\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 67.5269 67.5269i 3.54424 3.54424i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(368\) 0 0
\(369\) −74.9411 + 74.9411i −3.90128 + 3.90128i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −14.7574 14.7574i −0.758035 0.758035i 0.217930 0.975964i \(-0.430070\pi\)
−0.975964 + 0.217930i \(0.930070\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 73.4558i 3.73397i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 9.31371i 0.469814i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 16.7990 + 16.7990i 0.838902 + 0.838902i 0.988714 0.149813i \(-0.0478671\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −33.9411 −1.67828 −0.839140 0.543915i \(-0.816941\pi\)
−0.839140 + 0.543915i \(0.816941\pi\)
\(410\) 0 0
\(411\) 14.4853 + 14.4853i 0.714506 + 0.714506i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 73.5980i 3.60411i
\(418\) 0 0
\(419\) 9.38478 9.38478i 0.458476 0.458476i −0.439679 0.898155i \(-0.644908\pi\)
0.898155 + 0.439679i \(0.144908\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −15.0000 14.1421i −0.727607 0.685994i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(432\) 0 0
\(433\) 38.0000i 1.82616i −0.407777 0.913082i \(-0.633696\pi\)
0.407777 0.913082i \(-0.366304\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(440\) 0 0
\(441\) 60.5980 2.88562
\(442\) 0 0
\(443\) 2.82843i 0.134383i −0.997740 0.0671913i \(-0.978596\pi\)
0.997740 0.0671913i \(-0.0214038\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.1716 + 18.1716i 0.857570 + 0.857570i 0.991051 0.133482i \(-0.0426157\pi\)
−0.133482 + 0.991051i \(0.542616\pi\)
\(450\) 0 0
\(451\) −76.4264 −3.59878
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000i 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) 0 0
\(459\) 2.34315 + 79.5980i 0.109369 + 3.71532i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −37.4558 + 37.4558i −1.72222 + 1.72222i
\(474\) 0 0
\(475\) 10.0000i 0.458831i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(488\) 0 0
\(489\) −56.6274 −2.56078
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −28.2132 + 28.2132i −1.26300 + 1.26300i −0.313363 + 0.949633i \(0.601456\pi\)
−0.949633 + 0.313363i \(0.898544\pi\)
\(500\) 0 0
\(501\) 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
\(507\) 31.3848 + 31.3848i 1.39385 + 1.39385i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 27.3137 27.3137i 1.20593 1.20593i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 19.6274 + 19.6274i 0.859893 + 0.859893i 0.991325 0.131432i \(-0.0419576\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 25.4558i 1.11311i −0.830812 0.556553i \(-0.812124\pi\)
0.830812 0.556553i \(-0.187876\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) 122.426i 5.31285i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 43.4558 + 43.4558i 1.87526 + 1.87526i
\(538\) 0 0
\(539\) 30.8995 + 30.8995i 1.33094 + 1.33094i
\(540\) 0 0
\(541\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.7574 18.7574i −0.802007 0.802007i 0.181402 0.983409i \(-0.441936\pi\)
−0.983409 + 0.181402i \(0.941936\pi\)
\(548\) 0 0
\(549\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −60.2843 + 63.9411i −2.54520 + 2.69960i
\(562\) 0 0
\(563\) 36.7696 1.54965 0.774826 0.632175i \(-0.217837\pi\)
0.774826 + 0.632175i \(0.217837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 22.6274i 0.948591i 0.880366 + 0.474295i \(0.157297\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) 32.2132 32.2132i 1.34808 1.34808i 0.460336 0.887745i \(-0.347729\pi\)
0.887745 0.460336i \(-0.152271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 12.1421i 0.504610i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 45.2548i 1.85839i 0.369586 + 0.929197i \(0.379500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 31.4853 + 31.4853i 1.28431 + 1.28431i 0.938190 + 0.346122i \(0.112502\pi\)
0.346122 + 0.938190i \(0.387498\pi\)
\(602\) 0 0
\(603\) −73.4558 −2.99135
\(604\) 0 0
\(605\) 0 0
\(606\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.79899 + 4.79899i −0.193200 + 0.193200i −0.797077 0.603877i \(-0.793622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) −8.21320 + 8.21320i −0.330117 + 0.330117i −0.852631 0.522514i \(-0.824994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 42.6274 1.70238
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −27.6569 −1.09926
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6.85786 + 6.85786i −0.270869 + 0.270869i −0.829450 0.558581i \(-0.811346\pi\)
0.558581 + 0.829450i \(0.311346\pi\)
\(642\) 0 0
\(643\) −20.7574 20.7574i −0.818590 0.818590i 0.167313 0.985904i \(-0.446491\pi\)
−0.985904 + 0.167313i \(0.946491\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −62.4264 + 62.4264i −2.45045 + 2.45045i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −82.1127 82.1127i −3.20352 3.20352i
\(658\) 0 0
\(659\) 18.0000i 0.701180i 0.936529 + 0.350590i \(0.114019\pi\)
−0.936529 + 0.350590i \(0.885981\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 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
\(673\) −30.4558 30.4558i −1.17399 1.17399i −0.981251 0.192736i \(-0.938264\pi\)
−0.192736 0.981251i \(-0.561736\pi\)
\(674\) 0 0
\(675\) 68.2843 + 68.2843i 2.62826 + 2.62826i
\(676\) 0 0
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 79.2548 3.03705
\(682\) 0 0
\(683\) −36.5563 36.5563i −1.39879 1.39879i −0.803543 0.595247i \(-0.797054\pi\)
−0.595247 0.803543i \(-0.702946\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −10.2721 + 10.2721i −0.390768 + 0.390768i −0.874961 0.484193i \(-0.839113\pi\)
0.484193 + 0.874961i \(0.339113\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 50.4558 1.48528i 1.91115 0.0562590i
\(698\) 0 0
\(699\) −58.7696 −2.22287
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 103.740 3.85814
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 148.196i 5.48874i
\(730\) 0 0
\(731\) 24.0000 25.4558i 0.887672 0.941518i
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −37.4558 37.4558i −1.37970 1.37970i
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 24.4853i 0.895870i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(752\) 0 0
\(753\) 75.1127 75.1127i 2.73726 2.73726i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −54.0000 −1.95750 −0.978749 0.205061i \(-0.934261\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 27.3137 27.3137i 0.983679 0.983679i
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.3137 17.3137i −0.620328 0.620328i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −12.2721 12.2721i −0.437452 0.437452i 0.453701 0.891154i \(-0.350103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 48.9706i 1.73029i
\(802\) 0 0
\(803\) 83.7401i 2.95512i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31.2843 31.2843i −1.09990 1.09990i −0.994422 0.105474i \(-0.966364\pi\)
−0.105474 0.994422i \(-0.533636\pi\)
\(810\) 0 0
\(811\) −2.21320 2.21320i −0.0777161 0.0777161i 0.667180 0.744896i \(-0.267501\pi\)
−0.744896 + 0.667180i \(0.767501\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −16.9706 −0.593725
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(824\) 0 0
\(825\) 106.569i 3.71024i
\(826\) 0 0
\(827\) 17.1005 17.1005i 0.594643 0.594643i −0.344239 0.938882i \(-0.611863\pi\)
0.938882 + 0.344239i \(0.111863\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21.0000 19.7990i −0.727607 0.685994i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(840\) 0 0
\(841\) 29.0000i 1.00000i
\(842\) 0 0
\(843\) −43.4558 + 43.4558i −1.49670 + 1.49670i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −114.569 −3.93198
\(850\) 0 0
\(851\) 0 0
\(852\) 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
\(857\) 15.6863 + 15.6863i 0.535834 + 0.535834i 0.922302 0.386469i \(-0.126305\pi\)
−0.386469 + 0.922302i \(0.626305\pi\)
\(858\) 0 0
\(859\) −8.48528 −0.289514 −0.144757 0.989467i \(-0.546240\pi\)
−0.144757 + 0.989467i \(0.546240\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 38.5563 43.3848i 1.30944 1.47342i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 116.740 + 116.740i 3.95105 + 3.95105i
\(874\) 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
\(880\) 0 0
\(881\) 37.2843 37.2843i 1.25614 1.25614i 0.303218 0.952921i \(-0.401939\pi\)
0.952921 0.303218i \(-0.0980609\pi\)
\(882\) 0 0
\(883\) 59.3970i 1.99887i 0.0336527 + 0.999434i \(0.489286\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) 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
\(891\) 176.439 176.439i 5.91092 5.91092i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) −24.6985 + 24.6985i −0.820100 + 0.820100i −0.986122 0.166022i \(-0.946908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(912\) 0 0
\(913\) 12.4853 12.4853i 0.413203 0.413203i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 82.0833 82.0833i 2.70474 2.70474i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 41.1421 + 41.1421i 1.34983 + 1.34983i 0.885841 + 0.463988i \(0.153582\pi\)
0.463988 + 0.885841i \(0.346418\pi\)
\(930\) 0 0
\(931\) 14.0000i 0.458831i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34.0000i 1.11073i 0.831606 + 0.555366i \(0.187422\pi\)
−0.831606 + 0.555366i \(0.812578\pi\)
\(938\) 0 0
\(939\) 57.7990i 1.88620i
\(940\) 0 0
\(941\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.8701 11.8701i −0.385725 0.385725i 0.487435 0.873160i \(-0.337933\pi\)
−0.873160 + 0.487435i \(0.837933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −45.2548 −1.46595 −0.732974 0.680257i \(-0.761868\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000i 1.00000i
\(962\) 0 0
\(963\) −111.669 + 111.669i −3.59848 + 3.59848i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) −28.1421 + 0.828427i −0.904056 + 0.0266129i
\(970\) 0 0
\(971\) −31.1127 −0.998454 −0.499227 0.866471i \(-0.666383\pi\)
−0.499227 + 0.866471i \(0.666383\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 62.2254i 1.99077i 0.0959785 + 0.995383i \(0.469402\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 0 0
\(979\) −24.9706 + 24.9706i −0.798063 + 0.798063i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(992\) 0 0
\(993\) 61.4558 + 61.4558i 1.95024 + 1.95024i
\(994\) 0 0
\(995\) 0 0
\(996\) 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 1088.2.m.g.225.2 yes 4
4.3 odd 2 1088.2.m.f.225.1 4
8.3 odd 2 CM 1088.2.m.g.225.2 yes 4
8.5 even 2 1088.2.m.f.225.1 4
17.13 even 4 1088.2.m.f.353.1 yes 4
68.47 odd 4 inner 1088.2.m.g.353.2 yes 4
136.13 even 4 inner 1088.2.m.g.353.2 yes 4
136.115 odd 4 1088.2.m.f.353.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1088.2.m.f.225.1 4 4.3 odd 2
1088.2.m.f.225.1 4 8.5 even 2
1088.2.m.f.353.1 yes 4 17.13 even 4
1088.2.m.f.353.1 yes 4 136.115 odd 4
1088.2.m.g.225.2 yes 4 1.1 even 1 trivial
1088.2.m.g.225.2 yes 4 8.3 odd 2 CM
1088.2.m.g.353.2 yes 4 68.47 odd 4 inner
1088.2.m.g.353.2 yes 4 136.13 even 4 inner