Properties

Label 2240.2.n.i.1119.9
Level $2240$
Weight $2$
Character 2240.1119
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
CM discriminant -56
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1119,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1119");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.n (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.101415451701035401216.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1119.9
Root \(-1.81768 - 0.332046i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1119
Dual form 2240.2.n.i.1119.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.78089 q^{3} +(-0.615370 - 2.14973i) q^{5} -2.64575 q^{7} +0.171573 q^{9} +O(q^{10})\) \(q+1.78089 q^{3} +(-0.615370 - 2.14973i) q^{5} -2.64575 q^{7} +0.171573 q^{9} -0.737669i q^{13} +(-1.09591 - 3.82843i) q^{15} +5.43275i q^{19} -4.71179 q^{21} -7.48331 q^{23} +(-4.24264 + 2.64575i) q^{25} -5.03712 q^{27} +(1.62811 + 5.68764i) q^{35} -1.31371i q^{39} +(-0.105581 - 0.368835i) q^{45} +7.00000 q^{49} +9.67513i q^{57} +6.45232i q^{59} -10.6543 q^{61} -0.453939 q^{63} +(-1.58579 + 0.453939i) q^{65} -13.3270 q^{69} -5.65685i q^{71} +(-7.55568 + 4.71179i) q^{75} +16.9706i q^{79} -9.48528 q^{81} -11.8551 q^{83} +1.95169i q^{91} +(11.6789 - 3.34315i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{9} + 112 q^{49} - 48 q^{65} - 16 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.78089 1.02820 0.514099 0.857731i \(-0.328126\pi\)
0.514099 + 0.857731i \(0.328126\pi\)
\(4\) 0 0
\(5\) −0.615370 2.14973i −0.275202 0.961387i
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 0 0
\(9\) 0.171573 0.0571910
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0.737669i 0.204593i −0.994754 0.102296i \(-0.967381\pi\)
0.994754 0.102296i \(-0.0326190\pi\)
\(14\) 0 0
\(15\) −1.09591 3.82843i −0.282962 0.988496i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.43275i 1.24636i 0.782080 + 0.623179i \(0.214159\pi\)
−0.782080 + 0.623179i \(0.785841\pi\)
\(20\) 0 0
\(21\) −4.71179 −1.02820
\(22\) 0 0
\(23\) −7.48331 −1.56038 −0.780189 0.625543i \(-0.784877\pi\)
−0.780189 + 0.625543i \(0.784877\pi\)
\(24\) 0 0
\(25\) −4.24264 + 2.64575i −0.848528 + 0.529150i
\(26\) 0 0
\(27\) −5.03712 −0.969394
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.62811 + 5.68764i 0.275202 + 0.961387i
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 1.31371i 0.210362i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −0.105581 0.368835i −0.0157390 0.0549826i
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.67513i 1.28150i
\(58\) 0 0
\(59\) 6.45232i 0.840021i 0.907519 + 0.420010i \(0.137974\pi\)
−0.907519 + 0.420010i \(0.862026\pi\)
\(60\) 0 0
\(61\) −10.6543 −1.36415 −0.682073 0.731284i \(-0.738922\pi\)
−0.682073 + 0.731284i \(0.738922\pi\)
\(62\) 0 0
\(63\) −0.453939 −0.0571910
\(64\) 0 0
\(65\) −1.58579 + 0.453939i −0.196693 + 0.0563042i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −13.3270 −1.60438
\(70\) 0 0
\(71\) 5.65685i 0.671345i −0.941979 0.335673i \(-0.891036\pi\)
0.941979 0.335673i \(-0.108964\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) −7.55568 + 4.71179i −0.872455 + 0.544071i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.9706i 1.90934i 0.297670 + 0.954669i \(0.403790\pi\)
−0.297670 + 0.954669i \(0.596210\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) −11.8551 −1.30127 −0.650635 0.759391i \(-0.725497\pi\)
−0.650635 + 0.759391i \(0.725497\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.95169i 0.204593i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.6789 3.34315i 1.19823 0.343000i
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −19.0583 −1.89638 −0.948188 0.317710i \(-0.897086\pi\)
−0.948188 + 0.317710i \(0.897086\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 2.89949 + 10.1291i 0.282962 + 0.988496i
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.8745i 1.49335i −0.665190 0.746674i \(-0.731650\pi\)
0.665190 0.746674i \(-0.268350\pi\)
\(114\) 0 0
\(115\) 4.60500 + 16.0871i 0.429419 + 1.50013i
\(116\) 0 0
\(117\) 0.126564i 0.0117008i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.29843 + 7.49240i 0.742234 + 0.670141i
\(126\) 0 0
\(127\) 22.4499 1.99211 0.996055 0.0887357i \(-0.0282826\pi\)
0.996055 + 0.0887357i \(0.0282826\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.99084i 0.348682i −0.984685 0.174341i \(-0.944221\pi\)
0.984685 0.174341i \(-0.0557795\pi\)
\(132\) 0 0
\(133\) 14.3737i 1.24636i
\(134\) 0 0
\(135\) 3.09969 + 10.8284i 0.266779 + 0.931963i
\(136\) 0 0
\(137\) 14.9666i 1.27869i −0.768922 0.639343i \(-0.779207\pi\)
0.768922 0.639343i \(-0.220793\pi\)
\(138\) 0 0
\(139\) 23.2603i 1.97292i 0.164012 + 0.986458i \(0.447557\pi\)
−0.164012 + 0.986458i \(0.552443\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 12.4662 1.02820
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i −0.913475 0.406894i \(-0.866612\pi\)
0.913475 0.406894i \(-0.133388\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 25.0590i 1.99993i −0.00842626 0.999964i \(-0.502682\pi\)
0.00842626 0.999964i \(-0.497318\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 19.7990 1.56038
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 12.4558 0.958142
\(170\) 0 0
\(171\) 0.932112i 0.0712804i
\(172\) 0 0
\(173\) 10.8119i 0.822014i 0.911632 + 0.411007i \(0.134823\pi\)
−0.911632 + 0.411007i \(0.865177\pi\)
\(174\) 0 0
\(175\) 11.2250 7.00000i 0.848528 0.529150i
\(176\) 0 0
\(177\) 11.4909i 0.863708i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −25.0009 −1.85830 −0.929150 0.369703i \(-0.879460\pi\)
−0.929150 + 0.369703i \(0.879460\pi\)
\(182\) 0 0
\(183\) −18.9742 −1.40261
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 13.3270 0.969394
\(190\) 0 0
\(191\) 22.6274i 1.63726i 0.574320 + 0.818631i \(0.305267\pi\)
−0.574320 + 0.818631i \(0.694733\pi\)
\(192\) 0 0
\(193\) 26.4575i 1.90445i −0.305392 0.952227i \(-0.598787\pi\)
0.305392 0.952227i \(-0.401213\pi\)
\(194\) 0 0
\(195\) −2.82411 + 0.808416i −0.202239 + 0.0578919i
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.28393 −0.0892396
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 10.0742i 0.690276i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −0.727922 + 0.453939i −0.0485281 + 0.0302626i
\(226\) 0 0
\(227\) −7.42912 −0.493088 −0.246544 0.969132i \(-0.579295\pi\)
−0.246544 + 0.969132i \(0.579295\pi\)
\(228\) 0 0
\(229\) −14.5577 −0.962000 −0.481000 0.876720i \(-0.659726\pi\)
−0.481000 + 0.876720i \(0.659726\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 29.9333i 1.96099i −0.196537 0.980497i \(-0.562969\pi\)
0.196537 0.980497i \(-0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 30.2227i 1.96318i
\(238\) 0 0
\(239\) 30.0000i 1.94054i −0.242028 0.970269i \(-0.577812\pi\)
0.242028 0.970269i \(-0.422188\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −1.78089 −0.114244
\(244\) 0 0
\(245\) −4.30759 15.0481i −0.275202 0.961387i
\(246\) 0 0
\(247\) 4.00757 0.254995
\(248\) 0 0
\(249\) −21.1127 −1.33796
\(250\) 0 0
\(251\) 31.6644i 1.99864i −0.0369181 0.999318i \(-0.511754\pi\)
0.0369181 0.999318i \(-0.488246\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.8745 −0.978864 −0.489432 0.872041i \(-0.662796\pi\)
−0.489432 + 0.872041i \(0.662796\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.4428 1.61224 0.806122 0.591749i \(-0.201562\pi\)
0.806122 + 0.591749i \(0.201562\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 3.47575i 0.210362i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 29.6640 1.76334 0.881672 0.471863i \(-0.156418\pi\)
0.881672 + 0.471863i \(0.156418\pi\)
\(284\) 0 0
\(285\) 20.7989 5.95378i 1.23202 0.352671i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 32.1826i 1.88013i 0.340998 + 0.940064i \(0.389235\pi\)
−0.340998 + 0.940064i \(0.610765\pi\)
\(294\) 0 0
\(295\) 13.8707 3.97056i 0.807585 0.231175i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.52021i 0.319242i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −33.9408 −1.94985
\(304\) 0 0
\(305\) 6.55635 + 22.9039i 0.375415 + 1.31147i
\(306\) 0 0
\(307\) −32.6147 −1.86142 −0.930710 0.365758i \(-0.880810\pi\)
−0.930710 + 0.365758i \(0.880810\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0.279340 + 0.975845i 0.0157390 + 0.0549826i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.95169 + 3.12967i 0.108260 + 0.173603i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.4575i 1.44123i 0.693334 + 0.720616i \(0.256141\pi\)
−0.693334 + 0.720616i \(0.743859\pi\)
\(338\) 0 0
\(339\) 28.2708i 1.53546i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 8.20101 + 28.6493i 0.441528 + 1.54243i
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 3.26989 0.175033 0.0875167 0.996163i \(-0.472107\pi\)
0.0875167 + 0.996163i \(0.472107\pi\)
\(350\) 0 0
\(351\) 3.71573i 0.198331i
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −12.1607 + 3.48106i −0.645422 + 0.184755i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000i 0.316668i 0.987386 + 0.158334i \(0.0506123\pi\)
−0.987386 + 0.158334i \(0.949388\pi\)
\(360\) 0 0
\(361\) −10.5147 −0.553406
\(362\) 0 0
\(363\) −19.5898 −1.02820
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 14.7786 + 13.3431i 0.763164 + 0.689037i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 39.9809 2.04828
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 7.10726i 0.358514i
\(394\) 0 0
\(395\) 36.4821 10.4432i 1.83561 0.525453i
\(396\) 0 0
\(397\) 35.7444i 1.79396i 0.442072 + 0.896980i \(0.354244\pi\)
−0.442072 + 0.896980i \(0.645756\pi\)
\(398\) 0 0
\(399\) 25.5980i 1.28150i
\(400\) 0 0
\(401\) −36.7696 −1.83618 −0.918092 0.396368i \(-0.870271\pi\)
−0.918092 + 0.396368i \(0.870271\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 5.83695 + 20.3908i 0.290041 + 1.01322i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 26.6539i 1.31474i
\(412\) 0 0
\(413\) 17.0712i 0.840021i
\(414\) 0 0
\(415\) 7.29529 + 25.4853i 0.358112 + 1.25102i
\(416\) 0 0
\(417\) 41.4241i 2.02855i
\(418\) 0 0
\(419\) 36.5873i 1.78741i −0.448658 0.893704i \(-0.648098\pi\)
0.448658 0.893704i \(-0.351902\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.1887 1.36415
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000i 0.867029i 0.901146 + 0.433515i \(0.142727\pi\)
−0.901146 + 0.433515i \(0.857273\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 40.6549i 1.94479i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 1.20101 0.0571910
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 17.8089i 0.836736i
\(454\) 0 0
\(455\) 4.19560 1.20101i 0.196693 0.0563042i
\(456\) 0 0
\(457\) 5.29150i 0.247526i −0.992312 0.123763i \(-0.960504\pi\)
0.992312 0.123763i \(-0.0394963\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 34.4245 1.60331 0.801654 0.597789i \(-0.203954\pi\)
0.801654 + 0.597789i \(0.203954\pi\)
\(462\) 0 0
\(463\) −26.4575 −1.22958 −0.614792 0.788689i \(-0.710760\pi\)
−0.614792 + 0.788689i \(0.710760\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 37.6518 1.74232 0.871160 0.491000i \(-0.163368\pi\)
0.871160 + 0.491000i \(0.163368\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 44.6274i 2.05632i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −14.3737 23.0492i −0.659510 1.05757i
\(476\) 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
\(483\) 35.2598 1.60438
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 22.4499 1.01730 0.508652 0.860972i \(-0.330144\pi\)
0.508652 + 0.860972i \(0.330144\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.9666i 0.671345i
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 11.7279 + 40.9702i 0.521886 + 1.82315i
\(506\) 0 0
\(507\) 22.1825 0.985159
\(508\) 0 0
\(509\) 44.2704 1.96225 0.981125 0.193375i \(-0.0619433\pi\)
0.981125 + 0.193375i \(0.0619433\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 27.3654i 1.20821i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 19.2548i 0.845193i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −44.7754 −1.95789 −0.978946 0.204120i \(-0.934567\pi\)
−0.978946 + 0.204120i \(0.934567\pi\)
\(524\) 0 0
\(525\) 19.9905 12.4662i 0.872455 0.544071i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 33.0000 1.43478
\(530\) 0 0
\(531\) 1.10704i 0.0480416i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −44.5238 −1.91070
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −1.82799 −0.0780169
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 44.8999i 1.90934i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 26.9665 1.13650 0.568251 0.822855i \(-0.307620\pi\)
0.568251 + 0.822855i \(0.307620\pi\)
\(564\) 0 0
\(565\) −34.1258 + 9.76869i −1.43568 + 0.410972i
\(566\) 0 0
\(567\) 25.0957 1.05392
\(568\) 0 0
\(569\) 2.82843 0.118574 0.0592869 0.998241i \(-0.481117\pi\)
0.0592869 + 0.998241i \(0.481117\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 40.2970i 1.68343i
\(574\) 0 0
\(575\) 31.7490 19.7990i 1.32403 0.825675i
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 47.1179i 1.95816i
\(580\) 0 0
\(581\) 31.3657 1.30127
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −0.272078 + 0.0778836i −0.0112490 + 0.00322009i
\(586\) 0 0
\(587\) −48.3372 −1.99509 −0.997545 0.0700342i \(-0.977689\pi\)
−0.997545 + 0.0700342i \(0.977689\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.3137i 0.462266i 0.972922 + 0.231133i \(0.0742432\pi\)
−0.972922 + 0.231133i \(0.925757\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.76906 + 23.6470i 0.275202 + 0.961387i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 47.6235i 1.91725i −0.284670 0.958625i \(-0.591884\pi\)
0.284670 0.958625i \(-0.408116\pi\)
\(618\) 0 0
\(619\) 48.4724i 1.94827i 0.225968 + 0.974135i \(0.427446\pi\)
−0.225968 + 0.974135i \(0.572554\pi\)
\(620\) 0 0
\(621\) 37.6944 1.51262
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 22.4499i 0.440000 0.897998i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 33.9411i 1.35117i 0.737280 + 0.675587i \(0.236110\pi\)
−0.737280 + 0.675587i \(0.763890\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.8150 48.2612i −0.548232 1.91519i
\(636\) 0 0
\(637\) 5.16368i 0.204593i
\(638\) 0 0
\(639\) 0.970563i 0.0383949i
\(640\) 0 0
\(641\) 48.0833 1.89917 0.949587 0.313503i \(-0.101502\pi\)
0.949587 + 0.313503i \(0.101502\pi\)
\(642\) 0 0
\(643\) −23.4047 −0.922992 −0.461496 0.887142i \(-0.652687\pi\)
−0.461496 + 0.887142i \(0.652687\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) −8.57922 + 2.45584i −0.335218 + 0.0959578i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −7.59560 −0.295434 −0.147717 0.989030i \(-0.547193\pi\)
−0.147717 + 0.989030i \(0.547193\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −30.8995 + 8.84513i −1.19823 + 0.343000i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 44.8999i 1.73076i −0.501113 0.865382i \(-0.667076\pi\)
0.501113 0.865382i \(-0.332924\pi\)
\(674\) 0 0
\(675\) 21.3707 13.3270i 0.822558 0.512955i
\(676\) 0 0
\(677\) 42.8679i 1.64755i 0.566918 + 0.823775i \(0.308136\pi\)
−0.566918 + 0.823775i \(0.691864\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −13.2304 −0.506992
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −32.1741 + 9.21001i −1.22931 + 0.351896i
\(686\) 0 0
\(687\) −25.9257 −0.989127
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 33.7035i 1.28214i 0.767482 + 0.641071i \(0.221510\pi\)
−0.767482 + 0.641071i \(0.778490\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 50.0034 14.3137i 1.89674 0.542950i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 53.3079i 2.01629i
\(700\) 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
\(707\) 50.4236 1.89638
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 2.91169i 0.109197i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 53.4267i 1.99526i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 25.2843 0.936454
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 6.99700i 0.258440i 0.991616 + 0.129220i \(0.0412474\pi\)
−0.991616 + 0.129220i \(0.958753\pi\)
\(734\) 0 0
\(735\) −7.67134 26.7990i −0.282962 0.988496i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 7.13704 0.262186
\(742\) 0 0
\(743\) 7.48331 0.274536 0.137268 0.990534i \(-0.456168\pi\)
0.137268 + 0.990534i \(0.456168\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.03402 −0.0744209
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 50.0000i 1.82453i −0.409605 0.912263i \(-0.634333\pi\)
0.409605 0.912263i \(-0.365667\pi\)
\(752\) 0 0
\(753\) 56.3908i 2.05499i
\(754\) 0 0
\(755\) −21.4973 + 6.15370i −0.782365 + 0.223956i
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.75968 0.171862
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.9233i 0.932395i −0.884681 0.466198i \(-0.845624\pi\)
0.884681 0.466198i \(-0.154376\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −53.8701 + 15.4206i −1.92270 + 0.550384i
\(786\) 0 0
\(787\) −55.4608 −1.97696 −0.988481 0.151344i \(-0.951640\pi\)
−0.988481 + 0.151344i \(0.951640\pi\)
\(788\) 0 0
\(789\) −28.2708 −1.00647
\(790\) 0 0
\(791\) 42.0000i 1.49335i
\(792\) 0 0
\(793\) 7.85937i 0.279094i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 45.2075i 1.60133i 0.599111 + 0.800666i \(0.295521\pi\)
−0.599111 + 0.800666i \(0.704479\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −12.1837 42.5624i −0.429419 1.50013i
\(806\) 0 0
\(807\) 47.0917 1.65771
\(808\) 0 0
\(809\) 31.1127 1.09386 0.546932 0.837177i \(-0.315796\pi\)
0.546932 + 0.837177i \(0.315796\pi\)
\(810\) 0 0
\(811\) 55.4345i 1.94657i 0.229604 + 0.973284i \(0.426257\pi\)
−0.229604 + 0.973284i \(0.573743\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0.334857i 0.0117008i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −26.4575 −0.922251 −0.461125 0.887335i \(-0.652554\pi\)
−0.461125 + 0.887335i \(0.652554\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −39.7697 −1.38126 −0.690630 0.723208i \(-0.742667\pi\)
−0.690630 + 0.723208i \(0.742667\pi\)
\(830\) 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
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 53.4267 1.84011
\(844\) 0 0
\(845\) −7.66495 26.7766i −0.263682 0.921145i
\(846\) 0 0
\(847\) 29.1033 1.00000
\(848\) 0 0
\(849\) 52.8284 1.81307
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.4297i 1.58972i 0.606790 + 0.794862i \(0.292457\pi\)
−0.606790 + 0.794862i \(0.707543\pi\)
\(854\) 0 0
\(855\) 2.00378 0.573593i 0.0685280 0.0196165i
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 22.8380i 0.779223i 0.920979 + 0.389612i \(0.127391\pi\)
−0.920979 + 0.389612i \(0.872609\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15.8745 −0.540375 −0.270187 0.962808i \(-0.587086\pi\)
−0.270187 + 0.962808i \(0.587086\pi\)
\(864\) 0 0
\(865\) 23.2426 6.65332i 0.790273 0.226220i
\(866\) 0 0
\(867\) −30.2751 −1.02820
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −21.9556 19.8230i −0.742234 0.670141i
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 57.3137i 1.93314i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 24.7022 7.07114i 0.830357 0.237694i
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −59.3970 −1.99211
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.83089i 0.328244i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15.3848 + 53.7450i 0.511407 + 1.78654i
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −3.26989 −0.108456
\(910\) 0 0
\(911\) 30.0000i 0.993944i 0.867766 + 0.496972i \(0.165555\pi\)
−0.867766 + 0.496972i \(0.834445\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 11.6761 + 40.7893i 0.386001 + 1.34845i
\(916\) 0 0
\(917\) 10.5588i 0.348682i
\(918\) 0 0
\(919\) 16.9706i 0.559807i −0.960028 0.279904i \(-0.909697\pi\)
0.960028 0.279904i \(-0.0903025\pi\)
\(920\) 0 0
\(921\) −58.0833 −1.91391
\(922\) 0 0
\(923\) −4.17289 −0.137352
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 38.0292i 1.24636i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 54.7135 1.78361 0.891805 0.452419i \(-0.149439\pi\)
0.891805 + 0.452419i \(0.149439\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −8.20101 28.6493i −0.266779 0.931963i
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 29.9333i 0.969633i 0.874616 + 0.484817i \(0.161114\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 48.6427 13.9242i 1.57404 0.450577i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 39.5980i 1.27869i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −56.8764 + 16.2811i −1.83092 + 0.524109i
\(966\) 0 0
\(967\) −22.4499 −0.721942 −0.360971 0.932577i \(-0.617555\pi\)
−0.360971 + 0.932577i \(0.617555\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 43.9717i 1.41112i 0.708650 + 0.705560i \(0.249305\pi\)
−0.708650 + 0.705560i \(0.750695\pi\)
\(972\) 0 0
\(973\) 61.5411i 1.97292i
\(974\) 0 0
\(975\) 3.47575 + 5.57359i 0.111313 + 0.178498i
\(976\) 0 0
\(977\) 59.8665i 1.91530i 0.287936 + 0.957650i \(0.407031\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 50.9117i 1.61726i −0.588315 0.808632i \(-0.700209\pi\)
0.588315 0.808632i \(-0.299791\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 63.0164i 1.99575i −0.0651544 0.997875i \(-0.520754\pi\)
0.0651544 0.997875i \(-0.479246\pi\)
\(998\) 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 2240.2.n.i.1119.9 yes 16
4.3 odd 2 inner 2240.2.n.i.1119.5 16
5.4 even 2 inner 2240.2.n.i.1119.6 yes 16
7.6 odd 2 inner 2240.2.n.i.1119.8 yes 16
8.3 odd 2 inner 2240.2.n.i.1119.12 yes 16
8.5 even 2 inner 2240.2.n.i.1119.8 yes 16
20.19 odd 2 inner 2240.2.n.i.1119.10 yes 16
28.27 even 2 inner 2240.2.n.i.1119.12 yes 16
35.34 odd 2 inner 2240.2.n.i.1119.11 yes 16
40.19 odd 2 inner 2240.2.n.i.1119.7 yes 16
40.29 even 2 inner 2240.2.n.i.1119.11 yes 16
56.13 odd 2 CM 2240.2.n.i.1119.9 yes 16
56.27 even 2 inner 2240.2.n.i.1119.5 16
140.139 even 2 inner 2240.2.n.i.1119.7 yes 16
280.69 odd 2 inner 2240.2.n.i.1119.6 yes 16
280.139 even 2 inner 2240.2.n.i.1119.10 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.n.i.1119.5 16 4.3 odd 2 inner
2240.2.n.i.1119.5 16 56.27 even 2 inner
2240.2.n.i.1119.6 yes 16 5.4 even 2 inner
2240.2.n.i.1119.6 yes 16 280.69 odd 2 inner
2240.2.n.i.1119.7 yes 16 40.19 odd 2 inner
2240.2.n.i.1119.7 yes 16 140.139 even 2 inner
2240.2.n.i.1119.8 yes 16 7.6 odd 2 inner
2240.2.n.i.1119.8 yes 16 8.5 even 2 inner
2240.2.n.i.1119.9 yes 16 1.1 even 1 trivial
2240.2.n.i.1119.9 yes 16 56.13 odd 2 CM
2240.2.n.i.1119.10 yes 16 20.19 odd 2 inner
2240.2.n.i.1119.10 yes 16 280.139 even 2 inner
2240.2.n.i.1119.11 yes 16 35.34 odd 2 inner
2240.2.n.i.1119.11 yes 16 40.29 even 2 inner
2240.2.n.i.1119.12 yes 16 8.3 odd 2 inner
2240.2.n.i.1119.12 yes 16 28.27 even 2 inner