Properties

Label 4900.2.e.s.2549.3
Level $4900$
Weight $2$
Character 4900.2549
Analytic conductor $39.127$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4900,2,Mod(2549,4900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4900.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.e (of order \(2\), degree \(1\), not 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: \(39.1266969904\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.4227136.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 9x^{4} + 22x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 700)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2549.3
Root \(-2.19869i\) of defining polynomial
Character \(\chi\) \(=\) 4900.2549
Dual form 4900.2.e.s.2549.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.364448i q^{3} +2.86718 q^{9} +O(q^{10})\) \(q-0.364448i q^{3} +2.86718 q^{9} +1.36445 q^{11} -2.63555i q^{13} -2.23163i q^{17} -6.23163 q^{19} -6.59607i q^{23} -2.13828i q^{27} -5.50273 q^{29} -4.50273 q^{31} -0.497270i q^{33} -2.09334i q^{37} -0.960522 q^{39} -9.32497 q^{41} +1.86718i q^{43} +6.86718i q^{47} -0.813312 q^{51} +10.1383i q^{53} +2.27110i q^{57} -1.63555 q^{59} -0.0394782 q^{61} -6.72890i q^{67} -2.40393 q^{69} -6.27110 q^{71} -4.00000i q^{73} -5.32497 q^{79} +7.82224 q^{81} -14.7738i q^{83} +2.00546i q^{87} +0.867178 q^{89} +1.64101i q^{93} +16.0988i q^{97} +3.91211 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{9} + 8 q^{11} - 4 q^{19} + 6 q^{31} + 28 q^{39} - 22 q^{41} - 6 q^{51} - 10 q^{59} - 34 q^{61} - 48 q^{69} - 38 q^{71} + 2 q^{79} + 46 q^{81} - 28 q^{89} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\) \(2451\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 0.364448i − 0.210414i −0.994450 0.105207i \(-0.966449\pi\)
0.994450 0.105207i \(-0.0335505\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.86718 0.955726
\(10\) 0 0
\(11\) 1.36445 0.411397 0.205698 0.978615i \(-0.434053\pi\)
0.205698 + 0.978615i \(0.434053\pi\)
\(12\) 0 0
\(13\) − 2.63555i − 0.730971i −0.930817 0.365485i \(-0.880903\pi\)
0.930817 0.365485i \(-0.119097\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.23163i − 0.541249i −0.962685 0.270624i \(-0.912770\pi\)
0.962685 0.270624i \(-0.0872301\pi\)
\(18\) 0 0
\(19\) −6.23163 −1.42963 −0.714816 0.699312i \(-0.753490\pi\)
−0.714816 + 0.699312i \(0.753490\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.59607i − 1.37538i −0.726006 0.687688i \(-0.758626\pi\)
0.726006 0.687688i \(-0.241374\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 2.13828i − 0.411512i
\(28\) 0 0
\(29\) −5.50273 −1.02183 −0.510916 0.859631i \(-0.670694\pi\)
−0.510916 + 0.859631i \(0.670694\pi\)
\(30\) 0 0
\(31\) −4.50273 −0.808714 −0.404357 0.914601i \(-0.632505\pi\)
−0.404357 + 0.914601i \(0.632505\pi\)
\(32\) 0 0
\(33\) − 0.497270i − 0.0865637i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.09334i − 0.344144i −0.985084 0.172072i \(-0.944954\pi\)
0.985084 0.172072i \(-0.0550461\pi\)
\(38\) 0 0
\(39\) −0.960522 −0.153807
\(40\) 0 0
\(41\) −9.32497 −1.45632 −0.728158 0.685410i \(-0.759623\pi\)
−0.728158 + 0.685410i \(0.759623\pi\)
\(42\) 0 0
\(43\) 1.86718i 0.284742i 0.989813 + 0.142371i \(0.0454726\pi\)
−0.989813 + 0.142371i \(0.954527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.86718i 1.00168i 0.865540 + 0.500840i \(0.166976\pi\)
−0.865540 + 0.500840i \(0.833024\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −0.813312 −0.113886
\(52\) 0 0
\(53\) 10.1383i 1.39260i 0.717751 + 0.696300i \(0.245172\pi\)
−0.717751 + 0.696300i \(0.754828\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.27110i 0.300815i
\(58\) 0 0
\(59\) −1.63555 −0.212931 −0.106465 0.994316i \(-0.533953\pi\)
−0.106465 + 0.994316i \(0.533953\pi\)
\(60\) 0 0
\(61\) −0.0394782 −0.00505467 −0.00252733 0.999997i \(-0.500804\pi\)
−0.00252733 + 0.999997i \(0.500804\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 6.72890i − 0.822065i −0.911621 0.411033i \(-0.865168\pi\)
0.911621 0.411033i \(-0.134832\pi\)
\(68\) 0 0
\(69\) −2.40393 −0.289399
\(70\) 0 0
\(71\) −6.27110 −0.744243 −0.372122 0.928184i \(-0.621370\pi\)
−0.372122 + 0.928184i \(0.621370\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.32497 −0.599106 −0.299553 0.954080i \(-0.596838\pi\)
−0.299553 + 0.954080i \(0.596838\pi\)
\(80\) 0 0
\(81\) 7.82224 0.869138
\(82\) 0 0
\(83\) − 14.7738i − 1.62164i −0.585296 0.810819i \(-0.699022\pi\)
0.585296 0.810819i \(-0.300978\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00546i 0.215008i
\(88\) 0 0
\(89\) 0.867178 0.0919206 0.0459603 0.998943i \(-0.485365\pi\)
0.0459603 + 0.998943i \(0.485365\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.64101i 0.170165i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0988i 1.63459i 0.576222 + 0.817293i \(0.304526\pi\)
−0.576222 + 0.817293i \(0.695474\pi\)
\(98\) 0 0
\(99\) 3.91211 0.393182
\(100\) 0 0
\(101\) 15.6949 1.56170 0.780849 0.624719i \(-0.214787\pi\)
0.780849 + 0.624719i \(0.214787\pi\)
\(102\) 0 0
\(103\) − 13.3699i − 1.31738i −0.752416 0.658688i \(-0.771112\pi\)
0.752416 0.658688i \(-0.228888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.40393i 0.522417i 0.965282 + 0.261209i \(0.0841211\pi\)
−0.965282 + 0.261209i \(0.915879\pi\)
\(108\) 0 0
\(109\) 15.8277 1.51602 0.758009 0.652244i \(-0.226172\pi\)
0.758009 + 0.652244i \(0.226172\pi\)
\(110\) 0 0
\(111\) −0.762915 −0.0724127
\(112\) 0 0
\(113\) − 18.4238i − 1.73316i −0.499036 0.866581i \(-0.666312\pi\)
0.499036 0.866581i \(-0.333688\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 7.55660i − 0.698608i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −9.13828 −0.830753
\(122\) 0 0
\(123\) 3.39847i 0.306429i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.32497i 0.472515i 0.971691 + 0.236257i \(0.0759208\pi\)
−0.971691 + 0.236257i \(0.924079\pi\)
\(128\) 0 0
\(129\) 0.680489 0.0599137
\(130\) 0 0
\(131\) −16.2371 −1.41864 −0.709320 0.704886i \(-0.750998\pi\)
−0.709320 + 0.704886i \(0.750998\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.40939i 0.376719i 0.982100 + 0.188360i \(0.0603170\pi\)
−0.982100 + 0.188360i \(0.939683\pi\)
\(138\) 0 0
\(139\) −22.3699 −1.89739 −0.948695 0.316192i \(-0.897596\pi\)
−0.948695 + 0.316192i \(0.897596\pi\)
\(140\) 0 0
\(141\) 2.50273 0.210768
\(142\) 0 0
\(143\) − 3.59607i − 0.300719i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −22.5082 −1.84394 −0.921971 0.387258i \(-0.873422\pi\)
−0.921971 + 0.387258i \(0.873422\pi\)
\(150\) 0 0
\(151\) 13.5027 1.09884 0.549418 0.835547i \(-0.314849\pi\)
0.549418 + 0.835547i \(0.314849\pi\)
\(152\) 0 0
\(153\) − 6.39847i − 0.517285i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.5566i 1.24155i 0.783988 + 0.620776i \(0.213182\pi\)
−0.783988 + 0.620776i \(0.786818\pi\)
\(158\) 0 0
\(159\) 3.69488 0.293023
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) − 4.40393i − 0.344942i −0.985015 0.172471i \(-0.944825\pi\)
0.985015 0.172471i \(-0.0551751\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.09880i 0.704087i 0.935984 + 0.352043i \(0.114513\pi\)
−0.935984 + 0.352043i \(0.885487\pi\)
\(168\) 0 0
\(169\) 6.05387 0.465682
\(170\) 0 0
\(171\) −17.8672 −1.36634
\(172\) 0 0
\(173\) − 19.9605i − 1.51757i −0.651341 0.758785i \(-0.725793\pi\)
0.651341 0.758785i \(-0.274207\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.596074i 0.0448036i
\(178\) 0 0
\(179\) 13.1921 0.986027 0.493014 0.870022i \(-0.335895\pi\)
0.493014 + 0.870022i \(0.335895\pi\)
\(180\) 0 0
\(181\) −10.2316 −0.760511 −0.380255 0.924882i \(-0.624164\pi\)
−0.380255 + 0.924882i \(0.624164\pi\)
\(182\) 0 0
\(183\) 0.0143878i 0.00106357i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 3.04494i − 0.222668i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.0055 1.01340 0.506700 0.862123i \(-0.330865\pi\)
0.506700 + 0.862123i \(0.330865\pi\)
\(192\) 0 0
\(193\) 14.4094i 1.03721i 0.855014 + 0.518605i \(0.173549\pi\)
−0.855014 + 0.518605i \(0.826451\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 11.5566i − 0.823373i −0.911325 0.411687i \(-0.864940\pi\)
0.911325 0.411687i \(-0.135060\pi\)
\(198\) 0 0
\(199\) −5.40939 −0.383461 −0.191731 0.981448i \(-0.561410\pi\)
−0.191731 + 0.981448i \(0.561410\pi\)
\(200\) 0 0
\(201\) −2.45233 −0.172974
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 18.9121i − 1.31448i
\(208\) 0 0
\(209\) −8.50273 −0.588146
\(210\) 0 0
\(211\) −10.9660 −0.754929 −0.377465 0.926024i \(-0.623204\pi\)
−0.377465 + 0.926024i \(0.623204\pi\)
\(212\) 0 0
\(213\) 2.28549i 0.156599i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.45779 −0.0985085
\(220\) 0 0
\(221\) −5.88157 −0.395637
\(222\) 0 0
\(223\) − 21.5566i − 1.44354i −0.692135 0.721768i \(-0.743330\pi\)
0.692135 0.721768i \(-0.256670\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.4238i − 0.824595i −0.911049 0.412297i \(-0.864726\pi\)
0.911049 0.412297i \(-0.135274\pi\)
\(228\) 0 0
\(229\) −8.36445 −0.552738 −0.276369 0.961052i \(-0.589131\pi\)
−0.276369 + 0.961052i \(0.589131\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 14.0055i − 0.917528i −0.888558 0.458764i \(-0.848292\pi\)
0.888558 0.458764i \(-0.151708\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.94067i 0.126060i
\(238\) 0 0
\(239\) −16.9605 −1.09708 −0.548542 0.836123i \(-0.684817\pi\)
−0.548542 + 0.836123i \(0.684817\pi\)
\(240\) 0 0
\(241\) −10.3304 −0.665441 −0.332721 0.943025i \(-0.607967\pi\)
−0.332721 + 0.943025i \(0.607967\pi\)
\(242\) 0 0
\(243\) − 9.26564i − 0.594391i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.4238i 1.04502i
\(248\) 0 0
\(249\) −5.38429 −0.341216
\(250\) 0 0
\(251\) −19.5082 −1.23135 −0.615673 0.788002i \(-0.711116\pi\)
−0.615673 + 0.788002i \(0.711116\pi\)
\(252\) 0 0
\(253\) − 9.00000i − 0.565825i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.4633i 1.77549i 0.460337 + 0.887744i \(0.347729\pi\)
−0.460337 + 0.887744i \(0.652271\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −15.7773 −0.976591
\(262\) 0 0
\(263\) 28.5621i 1.76121i 0.473849 + 0.880606i \(0.342864\pi\)
−0.473849 + 0.880606i \(0.657136\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 0.316041i − 0.0193414i
\(268\) 0 0
\(269\) −9.19761 −0.560788 −0.280394 0.959885i \(-0.590465\pi\)
−0.280394 + 0.959885i \(0.590465\pi\)
\(270\) 0 0
\(271\) −3.86172 −0.234583 −0.117291 0.993098i \(-0.537421\pi\)
−0.117291 + 0.993098i \(0.537421\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.6410i − 0.639356i −0.947526 0.319678i \(-0.896425\pi\)
0.947526 0.319678i \(-0.103575\pi\)
\(278\) 0 0
\(279\) −12.9101 −0.772909
\(280\) 0 0
\(281\) 1.73436 0.103463 0.0517315 0.998661i \(-0.483526\pi\)
0.0517315 + 0.998661i \(0.483526\pi\)
\(282\) 0 0
\(283\) 5.91211i 0.351439i 0.984440 + 0.175719i \(0.0562251\pi\)
−0.984440 + 0.175719i \(0.943775\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.0198 0.707050
\(290\) 0 0
\(291\) 5.86718 0.343940
\(292\) 0 0
\(293\) − 24.3250i − 1.42108i −0.703657 0.710540i \(-0.748451\pi\)
0.703657 0.710540i \(-0.251549\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.91757i − 0.169295i
\(298\) 0 0
\(299\) −17.3843 −1.00536
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 5.71997i − 0.328604i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 23.5226i − 1.34250i −0.741229 0.671252i \(-0.765757\pi\)
0.741229 0.671252i \(-0.234243\pi\)
\(308\) 0 0
\(309\) −4.87264 −0.277195
\(310\) 0 0
\(311\) 8.81877 0.500067 0.250033 0.968237i \(-0.419558\pi\)
0.250033 + 0.968237i \(0.419558\pi\)
\(312\) 0 0
\(313\) 0.0933442i 0.00527612i 0.999997 + 0.00263806i \(0.000839722\pi\)
−0.999997 + 0.00263806i \(0.999160\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3.54221i − 0.198950i −0.995040 0.0994751i \(-0.968284\pi\)
0.995040 0.0994751i \(-0.0317164\pi\)
\(318\) 0 0
\(319\) −7.50819 −0.420378
\(320\) 0 0
\(321\) 1.96945 0.109924
\(322\) 0 0
\(323\) 13.9067i 0.773787i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 5.76837i − 0.318992i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −19.6499 −1.08006 −0.540029 0.841646i \(-0.681587\pi\)
−0.540029 + 0.841646i \(0.681587\pi\)
\(332\) 0 0
\(333\) − 6.00199i − 0.328907i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0449i 1.25534i 0.778480 + 0.627669i \(0.215991\pi\)
−0.778480 + 0.627669i \(0.784009\pi\)
\(338\) 0 0
\(339\) −6.71451 −0.364682
\(340\) 0 0
\(341\) −6.14374 −0.332702
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.5621i 1.21119i 0.795771 + 0.605597i \(0.207066\pi\)
−0.795771 + 0.605597i \(0.792934\pi\)
\(348\) 0 0
\(349\) 14.3250 0.766798 0.383399 0.923583i \(-0.374753\pi\)
0.383399 + 0.923583i \(0.374753\pi\)
\(350\) 0 0
\(351\) −5.63555 −0.300804
\(352\) 0 0
\(353\) − 32.7003i − 1.74046i −0.492643 0.870232i \(-0.663969\pi\)
0.492643 0.870232i \(-0.336031\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.1581 1.38057 0.690287 0.723536i \(-0.257484\pi\)
0.690287 + 0.723536i \(0.257484\pi\)
\(360\) 0 0
\(361\) 19.8332 1.04385
\(362\) 0 0
\(363\) 3.33043i 0.174802i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 25.2316i − 1.31708i −0.752546 0.658540i \(-0.771174\pi\)
0.752546 0.658540i \(-0.228826\pi\)
\(368\) 0 0
\(369\) −26.7363 −1.39184
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 5.69488i − 0.294870i −0.989072 0.147435i \(-0.952898\pi\)
0.989072 0.147435i \(-0.0471017\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.5027i 0.746929i
\(378\) 0 0
\(379\) −14.9660 −0.768751 −0.384375 0.923177i \(-0.625583\pi\)
−0.384375 + 0.923177i \(0.625583\pi\)
\(380\) 0 0
\(381\) 1.94067 0.0994238
\(382\) 0 0
\(383\) 24.4776i 1.25075i 0.780325 + 0.625374i \(0.215054\pi\)
−0.780325 + 0.625374i \(0.784946\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.35353i 0.272135i
\(388\) 0 0
\(389\) 31.2460 1.58424 0.792118 0.610368i \(-0.208978\pi\)
0.792118 + 0.610368i \(0.208978\pi\)
\(390\) 0 0
\(391\) −14.7200 −0.744421
\(392\) 0 0
\(393\) 5.91757i 0.298502i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.13282i − 0.0568547i −0.999596 0.0284274i \(-0.990950\pi\)
0.999596 0.0284274i \(-0.00904993\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.82224 −0.190874 −0.0954368 0.995435i \(-0.530425\pi\)
−0.0954368 + 0.995435i \(0.530425\pi\)
\(402\) 0 0
\(403\) 11.8672i 0.591146i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.85626i − 0.141580i
\(408\) 0 0
\(409\) 33.6609 1.66442 0.832211 0.554459i \(-0.187075\pi\)
0.832211 + 0.554459i \(0.187075\pi\)
\(410\) 0 0
\(411\) 1.60699 0.0792671
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.15267i 0.399238i
\(418\) 0 0
\(419\) −0.497270 −0.0242933 −0.0121466 0.999926i \(-0.503866\pi\)
−0.0121466 + 0.999926i \(0.503866\pi\)
\(420\) 0 0
\(421\) −1.28003 −0.0623850 −0.0311925 0.999513i \(-0.509930\pi\)
−0.0311925 + 0.999513i \(0.509930\pi\)
\(422\) 0 0
\(423\) 19.6894i 0.957332i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.31058 −0.0632755
\(430\) 0 0
\(431\) −10.2910 −0.495698 −0.247849 0.968799i \(-0.579724\pi\)
−0.247849 + 0.968799i \(0.579724\pi\)
\(432\) 0 0
\(433\) − 29.8870i − 1.43628i −0.695899 0.718139i \(-0.744994\pi\)
0.695899 0.718139i \(-0.255006\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 41.1043i 1.96628i
\(438\) 0 0
\(439\) −5.04841 −0.240947 −0.120474 0.992717i \(-0.538441\pi\)
−0.120474 + 0.992717i \(0.538441\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 11.7738i − 0.559392i −0.960089 0.279696i \(-0.909766\pi\)
0.960089 0.279696i \(-0.0902336\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.20307i 0.387992i
\(448\) 0 0
\(449\) 11.3843 0.537258 0.268629 0.963244i \(-0.413429\pi\)
0.268629 + 0.963244i \(0.413429\pi\)
\(450\) 0 0
\(451\) −12.7234 −0.599123
\(452\) 0 0
\(453\) − 4.92104i − 0.231211i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 37.2515i − 1.74255i −0.490795 0.871275i \(-0.663294\pi\)
0.490795 0.871275i \(-0.336706\pi\)
\(458\) 0 0
\(459\) −4.77184 −0.222731
\(460\) 0 0
\(461\) 15.0449 0.700713 0.350356 0.936616i \(-0.386061\pi\)
0.350356 + 0.936616i \(0.386061\pi\)
\(462\) 0 0
\(463\) − 31.2031i − 1.45013i −0.688681 0.725065i \(-0.741810\pi\)
0.688681 0.725065i \(-0.258190\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.7684i 0.868497i 0.900793 + 0.434248i \(0.142986\pi\)
−0.900793 + 0.434248i \(0.857014\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.66957 0.261240
\(472\) 0 0
\(473\) 2.54767i 0.117142i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 29.0683i 1.33094i
\(478\) 0 0
\(479\) −0.714508 −0.0326467 −0.0163234 0.999867i \(-0.505196\pi\)
−0.0163234 + 0.999867i \(0.505196\pi\)
\(480\) 0 0
\(481\) −5.51712 −0.251559
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.6949i − 0.756517i −0.925700 0.378259i \(-0.876523\pi\)
0.925700 0.378259i \(-0.123477\pi\)
\(488\) 0 0
\(489\) −1.60500 −0.0725807
\(490\) 0 0
\(491\) 6.19761 0.279694 0.139847 0.990173i \(-0.455339\pi\)
0.139847 + 0.990173i \(0.455339\pi\)
\(492\) 0 0
\(493\) 12.2800i 0.553065i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.4687 −0.647708 −0.323854 0.946107i \(-0.604979\pi\)
−0.323854 + 0.946107i \(0.604979\pi\)
\(500\) 0 0
\(501\) 3.31604 0.148150
\(502\) 0 0
\(503\) 30.6949i 1.36862i 0.729193 + 0.684308i \(0.239896\pi\)
−0.729193 + 0.684308i \(0.760104\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2.20632i − 0.0979861i
\(508\) 0 0
\(509\) 44.4742 1.97128 0.985641 0.168852i \(-0.0540059\pi\)
0.985641 + 0.168852i \(0.0540059\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 13.3250i 0.588312i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9.36991i 0.412088i
\(518\) 0 0
\(519\) −7.27457 −0.319318
\(520\) 0 0
\(521\) 12.6499 0.554204 0.277102 0.960841i \(-0.410626\pi\)
0.277102 + 0.960841i \(0.410626\pi\)
\(522\) 0 0
\(523\) − 10.2711i − 0.449124i −0.974460 0.224562i \(-0.927905\pi\)
0.974460 0.224562i \(-0.0720951\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.0484i 0.437715i
\(528\) 0 0
\(529\) −20.5082 −0.891660
\(530\) 0 0
\(531\) −4.68942 −0.203503
\(532\) 0 0
\(533\) 24.5764i 1.06452i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4.80785i − 0.207474i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.13828 −0.0919319 −0.0459660 0.998943i \(-0.514637\pi\)
−0.0459660 + 0.998943i \(0.514637\pi\)
\(542\) 0 0
\(543\) 3.72890i 0.160022i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 32.0198i − 1.36907i −0.728980 0.684535i \(-0.760005\pi\)
0.728980 0.684535i \(-0.239995\pi\)
\(548\) 0 0
\(549\) −0.113191 −0.00483088
\(550\) 0 0
\(551\) 34.2910 1.46084
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 8.33043i − 0.352972i −0.984303 0.176486i \(-0.943527\pi\)
0.984303 0.176486i \(-0.0564730\pi\)
\(558\) 0 0
\(559\) 4.92104 0.208138
\(560\) 0 0
\(561\) −1.10972 −0.0468525
\(562\) 0 0
\(563\) − 19.4633i − 0.820278i −0.912023 0.410139i \(-0.865480\pi\)
0.912023 0.410139i \(-0.134520\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.8925 −0.792014 −0.396007 0.918247i \(-0.629604\pi\)
−0.396007 + 0.918247i \(0.629604\pi\)
\(570\) 0 0
\(571\) −18.6301 −0.779645 −0.389823 0.920890i \(-0.627464\pi\)
−0.389823 + 0.920890i \(0.627464\pi\)
\(572\) 0 0
\(573\) − 5.10426i − 0.213234i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.5926i 0.774020i 0.922075 + 0.387010i \(0.126492\pi\)
−0.922075 + 0.387010i \(0.873508\pi\)
\(578\) 0 0
\(579\) 5.25147 0.218244
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 13.8332i 0.572911i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 10.1383i − 0.418452i −0.977867 0.209226i \(-0.932906\pi\)
0.977867 0.209226i \(-0.0670944\pi\)
\(588\) 0 0
\(589\) 28.0593 1.15616
\(590\) 0 0
\(591\) −4.21178 −0.173249
\(592\) 0 0
\(593\) − 17.0055i − 0.698331i −0.937061 0.349165i \(-0.886465\pi\)
0.937061 0.349165i \(-0.113535\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.97144i 0.0806857i
\(598\) 0 0
\(599\) 8.18669 0.334499 0.167250 0.985915i \(-0.446511\pi\)
0.167250 + 0.985915i \(0.446511\pi\)
\(600\) 0 0
\(601\) 35.3359 1.44138 0.720690 0.693257i \(-0.243825\pi\)
0.720690 + 0.693257i \(0.243825\pi\)
\(602\) 0 0
\(603\) − 19.2929i − 0.785669i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.2316i − 1.14589i −0.819595 0.572943i \(-0.805802\pi\)
0.819595 0.572943i \(-0.194198\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.0988 0.732199
\(612\) 0 0
\(613\) − 31.1581i − 1.25846i −0.777217 0.629232i \(-0.783369\pi\)
0.777217 0.629232i \(-0.216631\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 39.1527i − 1.57623i −0.615530 0.788114i \(-0.711058\pi\)
0.615530 0.788114i \(-0.288942\pi\)
\(618\) 0 0
\(619\) −39.3843 −1.58299 −0.791494 0.611177i \(-0.790696\pi\)
−0.791494 + 0.611177i \(0.790696\pi\)
\(620\) 0 0
\(621\) −14.1043 −0.565985
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 3.09880i 0.123754i
\(628\) 0 0
\(629\) −4.67156 −0.186267
\(630\) 0 0
\(631\) 22.7003 0.903686 0.451843 0.892097i \(-0.350767\pi\)
0.451843 + 0.892097i \(0.350767\pi\)
\(632\) 0 0
\(633\) 3.99653i 0.158848i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −17.9804 −0.711292
\(640\) 0 0
\(641\) 8.41285 0.332288 0.166144 0.986102i \(-0.446868\pi\)
0.166144 + 0.986102i \(0.446868\pi\)
\(642\) 0 0
\(643\) 29.7398i 1.17282i 0.810013 + 0.586412i \(0.199460\pi\)
−0.810013 + 0.586412i \(0.800540\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 21.6859i − 0.852563i −0.904591 0.426281i \(-0.859823\pi\)
0.904591 0.426281i \(-0.140177\pi\)
\(648\) 0 0
\(649\) −2.23163 −0.0875990
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.0702i 1.17674i 0.808592 + 0.588370i \(0.200230\pi\)
−0.808592 + 0.588370i \(0.799770\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 11.4687i − 0.447437i
\(658\) 0 0
\(659\) −2.37537 −0.0925311 −0.0462656 0.998929i \(-0.514732\pi\)
−0.0462656 + 0.998929i \(0.514732\pi\)
\(660\) 0 0
\(661\) 14.4633 0.562555 0.281278 0.959626i \(-0.409242\pi\)
0.281278 + 0.959626i \(0.409242\pi\)
\(662\) 0 0
\(663\) 2.14353i 0.0832476i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.2964i 1.40540i
\(668\) 0 0
\(669\) −7.85626 −0.303741
\(670\) 0 0
\(671\) −0.0538660 −0.00207947
\(672\) 0 0
\(673\) 8.99653i 0.346791i 0.984852 + 0.173395i \(0.0554738\pi\)
−0.984852 + 0.173395i \(0.944526\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.49181i 0.287934i 0.989583 + 0.143967i \(0.0459859\pi\)
−0.989583 + 0.143967i \(0.954014\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −4.52782 −0.173506
\(682\) 0 0
\(683\) − 42.3250i − 1.61952i −0.586761 0.809760i \(-0.699597\pi\)
0.586761 0.809760i \(-0.300403\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.04841i 0.116304i
\(688\) 0 0
\(689\) 26.7200 1.01795
\(690\) 0 0
\(691\) −28.6465 −1.08976 −0.544882 0.838513i \(-0.683425\pi\)
−0.544882 + 0.838513i \(0.683425\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.8098i 0.788229i
\(698\) 0 0
\(699\) −5.10426 −0.193061
\(700\) 0 0
\(701\) −20.4292 −0.771601 −0.385801 0.922582i \(-0.626075\pi\)
−0.385801 + 0.922582i \(0.626075\pi\)
\(702\) 0 0
\(703\) 13.0449i 0.491999i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −28.8277 −1.08265 −0.541323 0.840814i \(-0.682077\pi\)
−0.541323 + 0.840814i \(0.682077\pi\)
\(710\) 0 0
\(711\) −15.2676 −0.572581
\(712\) 0 0
\(713\) 29.7003i 1.11229i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.18123i 0.230842i
\(718\) 0 0
\(719\) 31.3106 1.16769 0.583844 0.811866i \(-0.301548\pi\)
0.583844 + 0.811866i \(0.301548\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.76490i 0.140018i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 8.17230i − 0.303094i −0.988450 0.151547i \(-0.951575\pi\)
0.988450 0.151547i \(-0.0484255\pi\)
\(728\) 0 0
\(729\) 20.0899 0.744069
\(730\) 0 0
\(731\) 4.16684 0.154116
\(732\) 0 0
\(733\) 21.4183i 0.791103i 0.918444 + 0.395552i \(0.129447\pi\)
−0.918444 + 0.395552i \(0.870553\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 9.18123i − 0.338195i
\(738\) 0 0
\(739\) −4.35006 −0.160020 −0.0800098 0.996794i \(-0.525495\pi\)
−0.0800098 + 0.996794i \(0.525495\pi\)
\(740\) 0 0
\(741\) 5.98561 0.219887
\(742\) 0 0
\(743\) 6.53328i 0.239683i 0.992793 + 0.119841i \(0.0382386\pi\)
−0.992793 + 0.119841i \(0.961761\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 42.3592i − 1.54984i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 34.4831 1.25831 0.629153 0.777281i \(-0.283402\pi\)
0.629153 + 0.777281i \(0.283402\pi\)
\(752\) 0 0
\(753\) 7.10972i 0.259093i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 44.2425i 1.60802i 0.594614 + 0.804011i \(0.297305\pi\)
−0.594614 + 0.804011i \(0.702695\pi\)
\(758\) 0 0
\(759\) −3.28003 −0.119058
\(760\) 0 0
\(761\) 23.8188 0.863430 0.431715 0.902010i \(-0.357909\pi\)
0.431715 + 0.902010i \(0.357909\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.31058i 0.155646i
\(768\) 0 0
\(769\) −0.158128 −0.00570225 −0.00285113 0.999996i \(-0.500908\pi\)
−0.00285113 + 0.999996i \(0.500908\pi\)
\(770\) 0 0
\(771\) 10.3734 0.373588
\(772\) 0 0
\(773\) 17.7991i 0.640191i 0.947385 + 0.320095i \(0.103715\pi\)
−0.947385 + 0.320095i \(0.896285\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 58.1097 2.08200
\(780\) 0 0
\(781\) −8.55660 −0.306179
\(782\) 0 0
\(783\) 11.7664i 0.420496i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 41.5226i − 1.48012i −0.672541 0.740060i \(-0.734797\pi\)
0.672541 0.740060i \(-0.265203\pi\)
\(788\) 0 0
\(789\) 10.4094 0.370584
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0.104047i 0.00369481i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 25.0933i 0.888852i 0.895816 + 0.444426i \(0.146592\pi\)
−0.895816 + 0.444426i \(0.853408\pi\)
\(798\) 0 0
\(799\) 15.3250 0.542158
\(800\) 0 0
\(801\) 2.48635 0.0878509
\(802\) 0 0
\(803\) − 5.45779i − 0.192601i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.35205i 0.117998i
\(808\) 0 0
\(809\) −12.6949 −0.446328 −0.223164 0.974781i \(-0.571639\pi\)
−0.223164 + 0.974781i \(0.571639\pi\)
\(810\) 0 0
\(811\) −17.9749 −0.631184 −0.315592 0.948895i \(-0.602203\pi\)
−0.315592 + 0.948895i \(0.602203\pi\)
\(812\) 0 0
\(813\) 1.40740i 0.0493595i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 11.6356i − 0.407076i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 20.0683 0.700387 0.350193 0.936677i \(-0.386116\pi\)
0.350193 + 0.936677i \(0.386116\pi\)
\(822\) 0 0
\(823\) − 10.6105i − 0.369857i −0.982752 0.184929i \(-0.940795\pi\)
0.982752 0.184929i \(-0.0592054\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 34.7793i − 1.20939i −0.796455 0.604697i \(-0.793294\pi\)
0.796455 0.604697i \(-0.206706\pi\)
\(828\) 0 0
\(829\) −26.2282 −0.910942 −0.455471 0.890251i \(-0.650529\pi\)
−0.455471 + 0.890251i \(0.650529\pi\)
\(830\) 0 0
\(831\) −3.87810 −0.134530
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.62810i 0.332796i
\(838\) 0 0
\(839\) 33.9210 1.17108 0.585542 0.810642i \(-0.300882\pi\)
0.585542 + 0.810642i \(0.300882\pi\)
\(840\) 0 0
\(841\) 1.28003 0.0441391
\(842\) 0 0
\(843\) − 0.632082i − 0.0217701i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2.15466 0.0739477
\(850\) 0 0
\(851\) −13.8079 −0.473327
\(852\) 0 0
\(853\) 57.6698i 1.97458i 0.158942 + 0.987288i \(0.449192\pi\)
−0.158942 + 0.987288i \(0.550808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.4039i 1.20938i 0.796463 + 0.604688i \(0.206702\pi\)
−0.796463 + 0.604688i \(0.793298\pi\)
\(858\) 0 0
\(859\) −13.4434 −0.458683 −0.229342 0.973346i \(-0.573657\pi\)
−0.229342 + 0.973346i \(0.573657\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 7.06478i − 0.240488i −0.992744 0.120244i \(-0.961632\pi\)
0.992744 0.120244i \(-0.0383677\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 4.38061i − 0.148773i
\(868\) 0 0
\(869\) −7.26564 −0.246470
\(870\) 0 0
\(871\) −17.7344 −0.600906
\(872\) 0 0
\(873\) 46.1581i 1.56222i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 15.0055i − 0.506698i −0.967375 0.253349i \(-0.918468\pi\)
0.967375 0.253349i \(-0.0815322\pi\)
\(878\) 0 0
\(879\) −8.86519 −0.299015
\(880\) 0 0
\(881\) 39.3699 1.32641 0.663203 0.748440i \(-0.269197\pi\)
0.663203 + 0.748440i \(0.269197\pi\)
\(882\) 0 0
\(883\) 58.2229i 1.95936i 0.200574 + 0.979679i \(0.435719\pi\)
−0.200574 + 0.979679i \(0.564281\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.2460i 0.847678i 0.905738 + 0.423839i \(0.139318\pi\)
−0.905738 + 0.423839i \(0.860682\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 10.6730 0.357560
\(892\) 0 0
\(893\) − 42.7937i − 1.43204i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.33567i 0.211542i
\(898\) 0 0
\(899\) 24.7773 0.826369
\(900\) 0 0
\(901\) 22.6248 0.753743
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 27.1867i − 0.902719i −0.892342 0.451360i \(-0.850939\pi\)
0.892342 0.451360i \(-0.149061\pi\)
\(908\) 0 0
\(909\) 45.0000 1.49256
\(910\) 0 0
\(911\) −40.7882 −1.35137 −0.675687 0.737189i \(-0.736153\pi\)
−0.675687 + 0.737189i \(0.736153\pi\)
\(912\) 0 0
\(913\) − 20.1581i − 0.667137i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 47.0737 1.55282 0.776409 0.630229i \(-0.217039\pi\)
0.776409 + 0.630229i \(0.217039\pi\)
\(920\) 0 0
\(921\) −8.57276 −0.282482
\(922\) 0 0
\(923\) 16.5278i 0.544020i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 38.3339i − 1.25905i
\(928\) 0 0
\(929\) 17.9911 0.590268 0.295134 0.955456i \(-0.404636\pi\)
0.295134 + 0.955456i \(0.404636\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 3.21398i − 0.105221i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 37.7003i − 1.23162i −0.787896 0.615808i \(-0.788830\pi\)
0.787896 0.615808i \(-0.211170\pi\)
\(938\) 0 0
\(939\) 0.0340191 0.00111017
\(940\) 0 0
\(941\) −4.83316 −0.157556 −0.0787782 0.996892i \(-0.525102\pi\)
−0.0787782 + 0.996892i \(0.525102\pi\)
\(942\) 0 0
\(943\) 61.5082i 2.00298i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2.83663i − 0.0921780i −0.998937 0.0460890i \(-0.985324\pi\)
0.998937 0.0460890i \(-0.0146758\pi\)
\(948\) 0 0
\(949\) −10.5422 −0.342215
\(950\) 0 0
\(951\) −1.29095 −0.0418619
\(952\) 0 0
\(953\) − 17.7991i − 0.576571i −0.957545 0.288285i \(-0.906915\pi\)
0.957545 0.288285i \(-0.0930852\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 2.73634i 0.0884535i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −10.7254 −0.345982
\(962\) 0 0
\(963\) 15.4940i 0.499288i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 15.1581i − 0.487453i −0.969844 0.243726i \(-0.921630\pi\)
0.969844 0.243726i \(-0.0783698\pi\)
\(968\) 0 0
\(969\) 5.06825 0.162816
\(970\) 0 0
\(971\) 47.3843 1.52063 0.760317 0.649552i \(-0.225044\pi\)
0.760317 + 0.649552i \(0.225044\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 23.7200i 0.758869i 0.925219 + 0.379434i \(0.123881\pi\)
−0.925219 + 0.379434i \(0.876119\pi\)
\(978\) 0 0
\(979\) 1.18322 0.0378158
\(980\) 0 0
\(981\) 45.3808 1.44890
\(982\) 0 0
\(983\) 34.4543i 1.09892i 0.835519 + 0.549461i \(0.185167\pi\)
−0.835519 + 0.549461i \(0.814833\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.3160 0.391627
\(990\) 0 0
\(991\) −41.6609 −1.32340 −0.661700 0.749768i \(-0.730165\pi\)
−0.661700 + 0.749768i \(0.730165\pi\)
\(992\) 0 0
\(993\) 7.16138i 0.227260i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 24.7882i − 0.785051i −0.919741 0.392525i \(-0.871601\pi\)
0.919741 0.392525i \(-0.128399\pi\)
\(998\) 0 0
\(999\) −4.47616 −0.141619
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 4900.2.e.s.2549.3 6
5.2 odd 4 4900.2.a.bb.1.2 3
5.3 odd 4 4900.2.a.bd.1.2 3
5.4 even 2 inner 4900.2.e.s.2549.4 6
7.3 odd 6 700.2.r.d.149.4 12
7.5 odd 6 700.2.r.d.249.3 12
7.6 odd 2 4900.2.e.t.2549.4 6
35.3 even 12 700.2.i.e.401.2 yes 6
35.12 even 12 700.2.i.d.501.2 yes 6
35.13 even 4 4900.2.a.ba.1.2 3
35.17 even 12 700.2.i.d.401.2 6
35.19 odd 6 700.2.r.d.249.4 12
35.24 odd 6 700.2.r.d.149.3 12
35.27 even 4 4900.2.a.bc.1.2 3
35.33 even 12 700.2.i.e.501.2 yes 6
35.34 odd 2 4900.2.e.t.2549.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.i.d.401.2 6 35.17 even 12
700.2.i.d.501.2 yes 6 35.12 even 12
700.2.i.e.401.2 yes 6 35.3 even 12
700.2.i.e.501.2 yes 6 35.33 even 12
700.2.r.d.149.3 12 35.24 odd 6
700.2.r.d.149.4 12 7.3 odd 6
700.2.r.d.249.3 12 7.5 odd 6
700.2.r.d.249.4 12 35.19 odd 6
4900.2.a.ba.1.2 3 35.13 even 4
4900.2.a.bb.1.2 3 5.2 odd 4
4900.2.a.bc.1.2 3 35.27 even 4
4900.2.a.bd.1.2 3 5.3 odd 4
4900.2.e.s.2549.3 6 1.1 even 1 trivial
4900.2.e.s.2549.4 6 5.4 even 2 inner
4900.2.e.t.2549.3 6 35.34 odd 2
4900.2.e.t.2549.4 6 7.6 odd 2