Properties

Label 2200.2.a.x.1.3
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.5670884447\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.54764.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 3x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.339102\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.339102 q^{3} +4.05237 q^{7} -2.88501 q^{9} +O(q^{10})\) \(q+0.339102 q^{3} +4.05237 q^{7} -2.88501 q^{9} +1.00000 q^{11} -4.00000 q^{13} -7.74350 q^{17} -7.06529 q^{19} +1.37417 q^{21} +2.72619 q^{23} -1.99562 q^{27} -4.73057 q^{29} +0.219729 q^{31} +0.339102 q^{33} -1.32618 q^{37} -1.35641 q^{39} -7.79587 q^{41} -11.1177 q^{43} +3.01292 q^{47} +9.42170 q^{49} -2.62583 q^{51} -5.03945 q^{53} -2.39585 q^{57} +10.9503 q^{59} +12.7306 q^{61} -11.6911 q^{63} -4.70034 q^{67} +0.924456 q^{69} -2.52860 q^{71} +4.10474 q^{73} +4.05237 q^{77} +13.9006 q^{79} +7.97831 q^{81} -3.63067 q^{83} -1.60415 q^{87} +9.88501 q^{89} -16.2095 q^{91} +0.0745107 q^{93} -12.4962 q^{97} -2.88501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - q^{7} + 7 q^{9} + 4 q^{11} - 16 q^{13} - 7 q^{17} - 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} - 15 q^{31} - q^{33} - 5 q^{37} + 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49} - 23 q^{51} - 5 q^{53} + 15 q^{57} + 6 q^{59} + 29 q^{61} - 40 q^{63} - 6 q^{67} - 30 q^{69} - q^{71} - 18 q^{73} - q^{77} - 20 q^{79} + 44 q^{81} - 26 q^{83} - 31 q^{87} + 21 q^{89} + 4 q^{91} - 25 q^{93} + 4 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.339102 0.195781 0.0978903 0.995197i \(-0.468791\pi\)
0.0978903 + 0.995197i \(0.468791\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.05237 1.53165 0.765826 0.643048i \(-0.222330\pi\)
0.765826 + 0.643048i \(0.222330\pi\)
\(8\) 0 0
\(9\) −2.88501 −0.961670
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.74350 −1.87807 −0.939037 0.343816i \(-0.888280\pi\)
−0.939037 + 0.343816i \(0.888280\pi\)
\(18\) 0 0
\(19\) −7.06529 −1.62089 −0.810445 0.585815i \(-0.800774\pi\)
−0.810445 + 0.585815i \(0.800774\pi\)
\(20\) 0 0
\(21\) 1.37417 0.299868
\(22\) 0 0
\(23\) 2.72619 0.568450 0.284225 0.958758i \(-0.408264\pi\)
0.284225 + 0.958758i \(0.408264\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.99562 −0.384057
\(28\) 0 0
\(29\) −4.73057 −0.878445 −0.439223 0.898378i \(-0.644746\pi\)
−0.439223 + 0.898378i \(0.644746\pi\)
\(30\) 0 0
\(31\) 0.219729 0.0394646 0.0197323 0.999805i \(-0.493719\pi\)
0.0197323 + 0.999805i \(0.493719\pi\)
\(32\) 0 0
\(33\) 0.339102 0.0590301
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.32618 −0.218022 −0.109011 0.994041i \(-0.534768\pi\)
−0.109011 + 0.994041i \(0.534768\pi\)
\(38\) 0 0
\(39\) −1.35641 −0.217199
\(40\) 0 0
\(41\) −7.79587 −1.21751 −0.608755 0.793358i \(-0.708331\pi\)
−0.608755 + 0.793358i \(0.708331\pi\)
\(42\) 0 0
\(43\) −11.1177 −1.69543 −0.847714 0.530454i \(-0.822022\pi\)
−0.847714 + 0.530454i \(0.822022\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.01292 0.439480 0.219740 0.975558i \(-0.429479\pi\)
0.219740 + 0.975558i \(0.429479\pi\)
\(48\) 0 0
\(49\) 9.42170 1.34596
\(50\) 0 0
\(51\) −2.62583 −0.367690
\(52\) 0 0
\(53\) −5.03945 −0.692221 −0.346111 0.938194i \(-0.612498\pi\)
−0.346111 + 0.938194i \(0.612498\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.39585 −0.317339
\(58\) 0 0
\(59\) 10.9503 1.42561 0.712804 0.701363i \(-0.247425\pi\)
0.712804 + 0.701363i \(0.247425\pi\)
\(60\) 0 0
\(61\) 12.7306 1.62998 0.814991 0.579473i \(-0.196742\pi\)
0.814991 + 0.579473i \(0.196742\pi\)
\(62\) 0 0
\(63\) −11.6911 −1.47294
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.70034 −0.574238 −0.287119 0.957895i \(-0.592698\pi\)
−0.287119 + 0.957895i \(0.592698\pi\)
\(68\) 0 0
\(69\) 0.924456 0.111291
\(70\) 0 0
\(71\) −2.52860 −0.300090 −0.150045 0.988679i \(-0.547942\pi\)
−0.150045 + 0.988679i \(0.547942\pi\)
\(72\) 0 0
\(73\) 4.10474 0.480423 0.240212 0.970721i \(-0.422783\pi\)
0.240212 + 0.970721i \(0.422783\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.05237 0.461810
\(78\) 0 0
\(79\) 13.9006 1.56394 0.781970 0.623316i \(-0.214215\pi\)
0.781970 + 0.623316i \(0.214215\pi\)
\(80\) 0 0
\(81\) 7.97831 0.886479
\(82\) 0 0
\(83\) −3.63067 −0.398518 −0.199259 0.979947i \(-0.563853\pi\)
−0.199259 + 0.979947i \(0.563853\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.60415 −0.171983
\(88\) 0 0
\(89\) 9.88501 1.04781 0.523904 0.851777i \(-0.324475\pi\)
0.523904 + 0.851777i \(0.324475\pi\)
\(90\) 0 0
\(91\) −16.2095 −1.69922
\(92\) 0 0
\(93\) 0.0745107 0.00772640
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.4962 −1.26880 −0.634399 0.773006i \(-0.718752\pi\)
−0.634399 + 0.773006i \(0.718752\pi\)
\(98\) 0 0
\(99\) −2.88501 −0.289954
\(100\) 0 0
\(101\) −0.643593 −0.0640399 −0.0320199 0.999487i \(-0.510194\pi\)
−0.0320199 + 0.999487i \(0.510194\pi\)
\(102\) 0 0
\(103\) −4.36933 −0.430523 −0.215261 0.976556i \(-0.569060\pi\)
−0.215261 + 0.976556i \(0.569060\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.98708 −0.868814 −0.434407 0.900717i \(-0.643042\pi\)
−0.434407 + 0.900717i \(0.643042\pi\)
\(108\) 0 0
\(109\) 18.2095 1.74415 0.872076 0.489371i \(-0.162773\pi\)
0.872076 + 0.489371i \(0.162773\pi\)
\(110\) 0 0
\(111\) −0.449710 −0.0426845
\(112\) 0 0
\(113\) −10.4173 −0.979979 −0.489989 0.871728i \(-0.662999\pi\)
−0.489989 + 0.871728i \(0.662999\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 11.5400 1.06688
\(118\) 0 0
\(119\) −31.3795 −2.87656
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −2.64359 −0.238365
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.77002 0.866949 0.433475 0.901166i \(-0.357287\pi\)
0.433475 + 0.901166i \(0.357287\pi\)
\(128\) 0 0
\(129\) −3.77002 −0.331932
\(130\) 0 0
\(131\) −12.0870 −1.05604 −0.528022 0.849231i \(-0.677066\pi\)
−0.528022 + 0.849231i \(0.677066\pi\)
\(132\) 0 0
\(133\) −28.6312 −2.48264
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.49621 0.384137 0.192069 0.981381i \(-0.438480\pi\)
0.192069 + 0.981381i \(0.438480\pi\)
\(138\) 0 0
\(139\) −8.33472 −0.706942 −0.353471 0.935446i \(-0.614999\pi\)
−0.353471 + 0.935446i \(0.614999\pi\)
\(140\) 0 0
\(141\) 1.02169 0.0860416
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.19492 0.263512
\(148\) 0 0
\(149\) −11.8136 −0.967810 −0.483905 0.875121i \(-0.660782\pi\)
−0.483905 + 0.875121i \(0.660782\pi\)
\(150\) 0 0
\(151\) −15.7700 −1.28335 −0.641673 0.766978i \(-0.721759\pi\)
−0.641673 + 0.766978i \(0.721759\pi\)
\(152\) 0 0
\(153\) 22.3401 1.80609
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.0302288 0.00241252 0.00120626 0.999999i \(-0.499616\pi\)
0.00120626 + 0.999999i \(0.499616\pi\)
\(158\) 0 0
\(159\) −1.70889 −0.135523
\(160\) 0 0
\(161\) 11.0475 0.870668
\(162\) 0 0
\(163\) 14.0782 1.10269 0.551345 0.834277i \(-0.314115\pi\)
0.551345 + 0.834277i \(0.314115\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.7306 −0.985121 −0.492561 0.870278i \(-0.663939\pi\)
−0.492561 + 0.870278i \(0.663939\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 20.3834 1.55876
\(172\) 0 0
\(173\) −7.32180 −0.556666 −0.278333 0.960485i \(-0.589782\pi\)
−0.278333 + 0.960485i \(0.589782\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.71327 0.279106
\(178\) 0 0
\(179\) 6.56254 0.490507 0.245254 0.969459i \(-0.421129\pi\)
0.245254 + 0.969459i \(0.421129\pi\)
\(180\) 0 0
\(181\) −7.59390 −0.564450 −0.282225 0.959348i \(-0.591072\pi\)
−0.282225 + 0.959348i \(0.591072\pi\)
\(182\) 0 0
\(183\) 4.31696 0.319119
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.74350 −0.566261
\(188\) 0 0
\(189\) −8.08698 −0.588241
\(190\) 0 0
\(191\) −12.6414 −0.914702 −0.457351 0.889286i \(-0.651202\pi\)
−0.457351 + 0.889286i \(0.651202\pi\)
\(192\) 0 0
\(193\) 4.42170 0.318281 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.678204 −0.0483200 −0.0241600 0.999708i \(-0.507691\pi\)
−0.0241600 + 0.999708i \(0.507691\pi\)
\(198\) 0 0
\(199\) 4.42170 0.313446 0.156723 0.987643i \(-0.449907\pi\)
0.156723 + 0.987643i \(0.449907\pi\)
\(200\) 0 0
\(201\) −1.59390 −0.112425
\(202\) 0 0
\(203\) −19.1700 −1.34547
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.86509 −0.546661
\(208\) 0 0
\(209\) −7.06529 −0.488717
\(210\) 0 0
\(211\) 11.1700 0.768977 0.384488 0.923130i \(-0.374378\pi\)
0.384488 + 0.923130i \(0.374378\pi\)
\(212\) 0 0
\(213\) −0.857454 −0.0587518
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.890425 0.0604460
\(218\) 0 0
\(219\) 1.39192 0.0940576
\(220\) 0 0
\(221\) 30.9740 2.08354
\(222\) 0 0
\(223\) 19.4487 1.30238 0.651190 0.758915i \(-0.274270\pi\)
0.651190 + 0.758915i \(0.274270\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −22.6047 −1.50032 −0.750162 0.661254i \(-0.770024\pi\)
−0.750162 + 0.661254i \(0.770024\pi\)
\(228\) 0 0
\(229\) −17.0550 −1.12703 −0.563514 0.826106i \(-0.690551\pi\)
−0.563514 + 0.826106i \(0.690551\pi\)
\(230\) 0 0
\(231\) 1.37417 0.0904135
\(232\) 0 0
\(233\) −14.5006 −0.949965 −0.474983 0.879995i \(-0.657546\pi\)
−0.474983 + 0.879995i \(0.657546\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.71372 0.306189
\(238\) 0 0
\(239\) 13.9006 0.899155 0.449578 0.893241i \(-0.351574\pi\)
0.449578 + 0.893241i \(0.351574\pi\)
\(240\) 0 0
\(241\) 15.0572 0.969920 0.484960 0.874536i \(-0.338834\pi\)
0.484960 + 0.874536i \(0.338834\pi\)
\(242\) 0 0
\(243\) 8.69231 0.557612
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 28.2612 1.79822
\(248\) 0 0
\(249\) −1.23117 −0.0780220
\(250\) 0 0
\(251\) 2.51084 0.158483 0.0792415 0.996855i \(-0.474750\pi\)
0.0792415 + 0.996855i \(0.474750\pi\)
\(252\) 0 0
\(253\) 2.72619 0.171394
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.64359 −0.414416 −0.207208 0.978297i \(-0.566438\pi\)
−0.207208 + 0.978297i \(0.566438\pi\)
\(258\) 0 0
\(259\) −5.37417 −0.333934
\(260\) 0 0
\(261\) 13.6478 0.844775
\(262\) 0 0
\(263\) −13.4088 −0.826821 −0.413410 0.910545i \(-0.635662\pi\)
−0.413410 + 0.910545i \(0.635662\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3.35203 0.205141
\(268\) 0 0
\(269\) −19.4611 −1.18657 −0.593284 0.804994i \(-0.702169\pi\)
−0.593284 + 0.804994i \(0.702169\pi\)
\(270\) 0 0
\(271\) 0.0530465 0.00322235 0.00161117 0.999999i \(-0.499487\pi\)
0.00161117 + 0.999999i \(0.499487\pi\)
\(272\) 0 0
\(273\) −5.49666 −0.332673
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.09597 −0.246103 −0.123052 0.992400i \(-0.539268\pi\)
−0.123052 + 0.992400i \(0.539268\pi\)
\(278\) 0 0
\(279\) −0.633922 −0.0379519
\(280\) 0 0
\(281\) 17.5400 1.04635 0.523176 0.852225i \(-0.324747\pi\)
0.523176 + 0.852225i \(0.324747\pi\)
\(282\) 0 0
\(283\) −15.1264 −0.899173 −0.449586 0.893237i \(-0.648429\pi\)
−0.449586 + 0.893237i \(0.648429\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −31.5917 −1.86480
\(288\) 0 0
\(289\) 42.9617 2.52716
\(290\) 0 0
\(291\) −4.23749 −0.248406
\(292\) 0 0
\(293\) 13.3564 0.780290 0.390145 0.920754i \(-0.372425\pi\)
0.390145 + 0.920754i \(0.372425\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.99562 −0.115797
\(298\) 0 0
\(299\) −10.9048 −0.630639
\(300\) 0 0
\(301\) −45.0529 −2.59680
\(302\) 0 0
\(303\) −0.218243 −0.0125378
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 15.7959 0.901518 0.450759 0.892646i \(-0.351153\pi\)
0.450759 + 0.892646i \(0.351153\pi\)
\(308\) 0 0
\(309\) −1.48165 −0.0842880
\(310\) 0 0
\(311\) −0.829968 −0.0470631 −0.0235316 0.999723i \(-0.507491\pi\)
−0.0235316 + 0.999723i \(0.507491\pi\)
\(312\) 0 0
\(313\) 24.5050 1.38510 0.692552 0.721368i \(-0.256487\pi\)
0.692552 + 0.721368i \(0.256487\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.4655 0.643968 0.321984 0.946745i \(-0.395650\pi\)
0.321984 + 0.946745i \(0.395650\pi\)
\(318\) 0 0
\(319\) −4.73057 −0.264861
\(320\) 0 0
\(321\) −3.04753 −0.170097
\(322\) 0 0
\(323\) 54.7101 3.04415
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.17487 0.341471
\(328\) 0 0
\(329\) 12.2095 0.673130
\(330\) 0 0
\(331\) −24.4114 −1.34177 −0.670887 0.741559i \(-0.734087\pi\)
−0.670887 + 0.741559i \(0.734087\pi\)
\(332\) 0 0
\(333\) 3.82604 0.209666
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.3871 0.783715 0.391857 0.920026i \(-0.371833\pi\)
0.391857 + 0.920026i \(0.371833\pi\)
\(338\) 0 0
\(339\) −3.53253 −0.191861
\(340\) 0 0
\(341\) 0.219729 0.0118990
\(342\) 0 0
\(343\) 9.81363 0.529886
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.66528 0.0893969 0.0446985 0.999001i \(-0.485767\pi\)
0.0446985 + 0.999001i \(0.485767\pi\)
\(348\) 0 0
\(349\) 6.73866 0.360712 0.180356 0.983601i \(-0.442275\pi\)
0.180356 + 0.983601i \(0.442275\pi\)
\(350\) 0 0
\(351\) 7.98247 0.426073
\(352\) 0 0
\(353\) −23.1486 −1.23207 −0.616037 0.787717i \(-0.711263\pi\)
−0.616037 + 0.787717i \(0.711263\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −10.6409 −0.563174
\(358\) 0 0
\(359\) 20.9481 1.10560 0.552800 0.833314i \(-0.313559\pi\)
0.552800 + 0.833314i \(0.313559\pi\)
\(360\) 0 0
\(361\) 30.9184 1.62728
\(362\) 0 0
\(363\) 0.339102 0.0177982
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.4833 0.703822 0.351911 0.936033i \(-0.385532\pi\)
0.351911 + 0.936033i \(0.385532\pi\)
\(368\) 0 0
\(369\) 22.4912 1.17084
\(370\) 0 0
\(371\) −20.4217 −1.06024
\(372\) 0 0
\(373\) −7.47823 −0.387208 −0.193604 0.981080i \(-0.562018\pi\)
−0.193604 + 0.981080i \(0.562018\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.9223 0.974548
\(378\) 0 0
\(379\) 7.33807 0.376931 0.188466 0.982080i \(-0.439649\pi\)
0.188466 + 0.982080i \(0.439649\pi\)
\(380\) 0 0
\(381\) 3.31303 0.169732
\(382\) 0 0
\(383\) 20.2316 1.03379 0.516894 0.856050i \(-0.327088\pi\)
0.516894 + 0.856050i \(0.327088\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 32.0746 1.63044
\(388\) 0 0
\(389\) −0.406104 −0.0205903 −0.0102952 0.999947i \(-0.503277\pi\)
−0.0102952 + 0.999947i \(0.503277\pi\)
\(390\) 0 0
\(391\) −21.1103 −1.06759
\(392\) 0 0
\(393\) −4.09872 −0.206753
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.74833 0.338689 0.169345 0.985557i \(-0.445835\pi\)
0.169345 + 0.985557i \(0.445835\pi\)
\(398\) 0 0
\(399\) −9.70889 −0.486052
\(400\) 0 0
\(401\) −21.1700 −1.05718 −0.528590 0.848877i \(-0.677279\pi\)
−0.528590 + 0.848877i \(0.677279\pi\)
\(402\) 0 0
\(403\) −0.878918 −0.0437820
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.32618 −0.0657362
\(408\) 0 0
\(409\) 3.25167 0.160785 0.0803923 0.996763i \(-0.474383\pi\)
0.0803923 + 0.996763i \(0.474383\pi\)
\(410\) 0 0
\(411\) 1.52467 0.0752066
\(412\) 0 0
\(413\) 44.3747 2.18354
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.82632 −0.138405
\(418\) 0 0
\(419\) 0.0516929 0.00252536 0.00126268 0.999999i \(-0.499598\pi\)
0.00126268 + 0.999999i \(0.499598\pi\)
\(420\) 0 0
\(421\) 3.25167 0.158477 0.0792383 0.996856i \(-0.474751\pi\)
0.0792383 + 0.996856i \(0.474751\pi\)
\(422\) 0 0
\(423\) −8.69231 −0.422635
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 51.5890 2.49657
\(428\) 0 0
\(429\) −1.35641 −0.0654880
\(430\) 0 0
\(431\) 8.28303 0.398979 0.199490 0.979900i \(-0.436072\pi\)
0.199490 + 0.979900i \(0.436072\pi\)
\(432\) 0 0
\(433\) 12.2263 0.587557 0.293779 0.955873i \(-0.405087\pi\)
0.293779 + 0.955873i \(0.405087\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −19.2613 −0.921395
\(438\) 0 0
\(439\) −13.6706 −0.652463 −0.326232 0.945290i \(-0.605779\pi\)
−0.326232 + 0.945290i \(0.605779\pi\)
\(440\) 0 0
\(441\) −27.1817 −1.29437
\(442\) 0 0
\(443\) 5.27381 0.250566 0.125283 0.992121i \(-0.460016\pi\)
0.125283 + 0.992121i \(0.460016\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4.00602 −0.189478
\(448\) 0 0
\(449\) 10.9245 0.515557 0.257778 0.966204i \(-0.417010\pi\)
0.257778 + 0.966204i \(0.417010\pi\)
\(450\) 0 0
\(451\) −7.79587 −0.367093
\(452\) 0 0
\(453\) −5.34764 −0.251254
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.4843 −1.65988 −0.829942 0.557850i \(-0.811626\pi\)
−0.829942 + 0.557850i \(0.811626\pi\)
\(458\) 0 0
\(459\) 15.4531 0.721287
\(460\) 0 0
\(461\) −37.1657 −1.73098 −0.865490 0.500927i \(-0.832993\pi\)
−0.865490 + 0.500927i \(0.832993\pi\)
\(462\) 0 0
\(463\) 0.0309055 0.00143630 0.000718151 1.00000i \(-0.499771\pi\)
0.000718151 1.00000i \(0.499771\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −35.3219 −1.63450 −0.817250 0.576283i \(-0.804503\pi\)
−0.817250 + 0.576283i \(0.804503\pi\)
\(468\) 0 0
\(469\) −19.0475 −0.879533
\(470\) 0 0
\(471\) 0.0102506 0.000472324 0
\(472\) 0 0
\(473\) −11.1177 −0.511191
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 14.5389 0.665688
\(478\) 0 0
\(479\) −35.0787 −1.60279 −0.801394 0.598137i \(-0.795908\pi\)
−0.801394 + 0.598137i \(0.795908\pi\)
\(480\) 0 0
\(481\) 5.30471 0.241874
\(482\) 0 0
\(483\) 3.74624 0.170460
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −17.0663 −0.773346 −0.386673 0.922217i \(-0.626376\pi\)
−0.386673 + 0.922217i \(0.626376\pi\)
\(488\) 0 0
\(489\) 4.77395 0.215885
\(490\) 0 0
\(491\) 15.0122 0.677493 0.338747 0.940878i \(-0.389997\pi\)
0.338747 + 0.940878i \(0.389997\pi\)
\(492\) 0 0
\(493\) 36.6312 1.64979
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.2468 −0.459633
\(498\) 0 0
\(499\) −25.4967 −1.14139 −0.570694 0.821163i \(-0.693326\pi\)
−0.570694 + 0.821163i \(0.693326\pi\)
\(500\) 0 0
\(501\) −4.31696 −0.192868
\(502\) 0 0
\(503\) −36.5788 −1.63097 −0.815484 0.578779i \(-0.803529\pi\)
−0.815484 + 0.578779i \(0.803529\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.01731 0.0451801
\(508\) 0 0
\(509\) −14.1327 −0.626423 −0.313212 0.949683i \(-0.601405\pi\)
−0.313212 + 0.949683i \(0.601405\pi\)
\(510\) 0 0
\(511\) 16.6339 0.735841
\(512\) 0 0
\(513\) 14.0996 0.622514
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.01292 0.132508
\(518\) 0 0
\(519\) −2.48283 −0.108984
\(520\) 0 0
\(521\) 19.8551 0.869866 0.434933 0.900463i \(-0.356772\pi\)
0.434933 + 0.900463i \(0.356772\pi\)
\(522\) 0 0
\(523\) −7.79587 −0.340889 −0.170445 0.985367i \(-0.554520\pi\)
−0.170445 + 0.985367i \(0.554520\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.70147 −0.0741174
\(528\) 0 0
\(529\) −15.5679 −0.676864
\(530\) 0 0
\(531\) −31.5917 −1.37096
\(532\) 0 0
\(533\) 31.1835 1.35071
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.22537 0.0960317
\(538\) 0 0
\(539\) 9.42170 0.405821
\(540\) 0 0
\(541\) 4.21757 0.181327 0.0906637 0.995882i \(-0.471101\pi\)
0.0906637 + 0.995882i \(0.471101\pi\)
\(542\) 0 0
\(543\) −2.57510 −0.110508
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.4136 0.872823 0.436412 0.899747i \(-0.356249\pi\)
0.436412 + 0.899747i \(0.356249\pi\)
\(548\) 0 0
\(549\) −36.7278 −1.56751
\(550\) 0 0
\(551\) 33.4229 1.42386
\(552\) 0 0
\(553\) 56.3304 2.39541
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.4782 0.655834 0.327917 0.944707i \(-0.393653\pi\)
0.327917 + 0.944707i \(0.393653\pi\)
\(558\) 0 0
\(559\) 44.4707 1.88091
\(560\) 0 0
\(561\) −2.62583 −0.110863
\(562\) 0 0
\(563\) 1.08305 0.0456452 0.0228226 0.999740i \(-0.492735\pi\)
0.0228226 + 0.999740i \(0.492735\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 32.3311 1.35778
\(568\) 0 0
\(569\) 19.0572 0.798920 0.399460 0.916751i \(-0.369198\pi\)
0.399460 + 0.916751i \(0.369198\pi\)
\(570\) 0 0
\(571\) 23.9617 1.00277 0.501384 0.865225i \(-0.332824\pi\)
0.501384 + 0.865225i \(0.332824\pi\)
\(572\) 0 0
\(573\) −4.28673 −0.179081
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.57392 0.0655230 0.0327615 0.999463i \(-0.489570\pi\)
0.0327615 + 0.999463i \(0.489570\pi\)
\(578\) 0 0
\(579\) 1.49941 0.0623132
\(580\) 0 0
\(581\) −14.7128 −0.610390
\(582\) 0 0
\(583\) −5.03945 −0.208713
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.7488 0.980220 0.490110 0.871661i \(-0.336957\pi\)
0.490110 + 0.871661i \(0.336957\pi\)
\(588\) 0 0
\(589\) −1.55245 −0.0639677
\(590\) 0 0
\(591\) −0.229980 −0.00946012
\(592\) 0 0
\(593\) −14.1306 −0.580274 −0.290137 0.956985i \(-0.593701\pi\)
−0.290137 + 0.956985i \(0.593701\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.49941 0.0613666
\(598\) 0 0
\(599\) 23.8395 0.974054 0.487027 0.873387i \(-0.338081\pi\)
0.487027 + 0.873387i \(0.338081\pi\)
\(600\) 0 0
\(601\) 28.2353 1.15174 0.575871 0.817540i \(-0.304663\pi\)
0.575871 + 0.817540i \(0.304663\pi\)
\(602\) 0 0
\(603\) 13.5605 0.552228
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −29.6699 −1.20427 −0.602133 0.798396i \(-0.705682\pi\)
−0.602133 + 0.798396i \(0.705682\pi\)
\(608\) 0 0
\(609\) −6.50059 −0.263417
\(610\) 0 0
\(611\) −12.0517 −0.487559
\(612\) 0 0
\(613\) −9.34023 −0.377248 −0.188624 0.982049i \(-0.560403\pi\)
−0.188624 + 0.982049i \(0.560403\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.7754 0.715609 0.357805 0.933796i \(-0.383525\pi\)
0.357805 + 0.933796i \(0.383525\pi\)
\(618\) 0 0
\(619\) −8.64143 −0.347328 −0.173664 0.984805i \(-0.555561\pi\)
−0.173664 + 0.984805i \(0.555561\pi\)
\(620\) 0 0
\(621\) −5.44044 −0.218317
\(622\) 0 0
\(623\) 40.0577 1.60488
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.39585 −0.0956812
\(628\) 0 0
\(629\) 10.2693 0.409462
\(630\) 0 0
\(631\) 20.0414 0.797837 0.398919 0.916986i \(-0.369386\pi\)
0.398919 + 0.916986i \(0.369386\pi\)
\(632\) 0 0
\(633\) 3.78778 0.150551
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −37.6868 −1.49321
\(638\) 0 0
\(639\) 7.29504 0.288587
\(640\) 0 0
\(641\) 48.0113 1.89633 0.948166 0.317777i \(-0.102936\pi\)
0.948166 + 0.317777i \(0.102936\pi\)
\(642\) 0 0
\(643\) −35.9396 −1.41732 −0.708660 0.705550i \(-0.750700\pi\)
−0.708660 + 0.705550i \(0.750700\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.9172 −1.29411 −0.647055 0.762443i \(-0.724000\pi\)
−0.647055 + 0.762443i \(0.724000\pi\)
\(648\) 0 0
\(649\) 10.9503 0.429837
\(650\) 0 0
\(651\) 0.301945 0.0118341
\(652\) 0 0
\(653\) 3.34461 0.130885 0.0654424 0.997856i \(-0.479154\pi\)
0.0654424 + 0.997856i \(0.479154\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −11.8422 −0.462009
\(658\) 0 0
\(659\) −8.65168 −0.337022 −0.168511 0.985700i \(-0.553896\pi\)
−0.168511 + 0.985700i \(0.553896\pi\)
\(660\) 0 0
\(661\) −29.0550 −1.13011 −0.565055 0.825053i \(-0.691145\pi\)
−0.565055 + 0.825053i \(0.691145\pi\)
\(662\) 0 0
\(663\) 10.5033 0.407916
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.8964 −0.499352
\(668\) 0 0
\(669\) 6.59508 0.254981
\(670\) 0 0
\(671\) 12.7306 0.491458
\(672\) 0 0
\(673\) −5.71765 −0.220399 −0.110200 0.993909i \(-0.535149\pi\)
−0.110200 + 0.993909i \(0.535149\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.0701 0.387026 0.193513 0.981098i \(-0.438012\pi\)
0.193513 + 0.981098i \(0.438012\pi\)
\(678\) 0 0
\(679\) −50.6393 −1.94336
\(680\) 0 0
\(681\) −7.66528 −0.293734
\(682\) 0 0
\(683\) −3.79120 −0.145066 −0.0725331 0.997366i \(-0.523108\pi\)
−0.0725331 + 0.997366i \(0.523108\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.78340 −0.220650
\(688\) 0 0
\(689\) 20.1578 0.767950
\(690\) 0 0
\(691\) 27.5637 1.04857 0.524287 0.851542i \(-0.324332\pi\)
0.524287 + 0.851542i \(0.324332\pi\)
\(692\) 0 0
\(693\) −11.6911 −0.444109
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 60.3673 2.28657
\(698\) 0 0
\(699\) −4.91718 −0.185985
\(700\) 0 0
\(701\) 13.6830 0.516801 0.258401 0.966038i \(-0.416805\pi\)
0.258401 + 0.966038i \(0.416805\pi\)
\(702\) 0 0
\(703\) 9.36984 0.353390
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.60808 −0.0980868
\(708\) 0 0
\(709\) 14.3584 0.539241 0.269621 0.962967i \(-0.413102\pi\)
0.269621 + 0.962967i \(0.413102\pi\)
\(710\) 0 0
\(711\) −40.1034 −1.50399
\(712\) 0 0
\(713\) 0.599025 0.0224336
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.71372 0.176037
\(718\) 0 0
\(719\) 48.8332 1.82117 0.910585 0.413323i \(-0.135632\pi\)
0.910585 + 0.413323i \(0.135632\pi\)
\(720\) 0 0
\(721\) −17.7061 −0.659411
\(722\) 0 0
\(723\) 5.10593 0.189891
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 32.2566 1.19633 0.598165 0.801373i \(-0.295897\pi\)
0.598165 + 0.801373i \(0.295897\pi\)
\(728\) 0 0
\(729\) −20.9874 −0.777310
\(730\) 0 0
\(731\) 86.0896 3.18414
\(732\) 0 0
\(733\) 18.8176 0.695042 0.347521 0.937672i \(-0.387024\pi\)
0.347521 + 0.937672i \(0.387024\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.70034 −0.173139
\(738\) 0 0
\(739\) −14.3089 −0.526360 −0.263180 0.964747i \(-0.584771\pi\)
−0.263180 + 0.964747i \(0.584771\pi\)
\(740\) 0 0
\(741\) 9.58342 0.352056
\(742\) 0 0
\(743\) −43.8448 −1.60851 −0.804255 0.594284i \(-0.797435\pi\)
−0.804255 + 0.594284i \(0.797435\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 10.4745 0.383243
\(748\) 0 0
\(749\) −36.4190 −1.33072
\(750\) 0 0
\(751\) 23.0903 0.842578 0.421289 0.906926i \(-0.361578\pi\)
0.421289 + 0.906926i \(0.361578\pi\)
\(752\) 0 0
\(753\) 0.851432 0.0310279
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19.0270 0.691549 0.345775 0.938318i \(-0.387616\pi\)
0.345775 + 0.938318i \(0.387616\pi\)
\(758\) 0 0
\(759\) 0.924456 0.0335556
\(760\) 0 0
\(761\) −35.9537 −1.30332 −0.651659 0.758512i \(-0.725927\pi\)
−0.651659 + 0.758512i \(0.725927\pi\)
\(762\) 0 0
\(763\) 73.7915 2.67143
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −43.8012 −1.58157
\(768\) 0 0
\(769\) −17.4665 −0.629858 −0.314929 0.949115i \(-0.601981\pi\)
−0.314929 + 0.949115i \(0.601981\pi\)
\(770\) 0 0
\(771\) −2.25285 −0.0811346
\(772\) 0 0
\(773\) −10.5361 −0.378958 −0.189479 0.981885i \(-0.560680\pi\)
−0.189479 + 0.981885i \(0.560680\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.82239 −0.0653779
\(778\) 0 0
\(779\) 55.0801 1.97345
\(780\) 0 0
\(781\) −2.52860 −0.0904805
\(782\) 0 0
\(783\) 9.44042 0.337373
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −37.8748 −1.35009 −0.675045 0.737777i \(-0.735876\pi\)
−0.675045 + 0.737777i \(0.735876\pi\)
\(788\) 0 0
\(789\) −4.54694 −0.161875
\(790\) 0 0
\(791\) −42.2148 −1.50099
\(792\) 0 0
\(793\) −50.9223 −1.80830
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.9574 −1.06114 −0.530572 0.847640i \(-0.678023\pi\)
−0.530572 + 0.847640i \(0.678023\pi\)
\(798\) 0 0
\(799\) −23.3306 −0.825376
\(800\) 0 0
\(801\) −28.5184 −1.00765
\(802\) 0 0
\(803\) 4.10474 0.144853
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.59931 −0.232307
\(808\) 0 0
\(809\) −2.59589 −0.0912667 −0.0456333 0.998958i \(-0.514531\pi\)
−0.0456333 + 0.998958i \(0.514531\pi\)
\(810\) 0 0
\(811\) 39.2012 1.37654 0.688271 0.725454i \(-0.258370\pi\)
0.688271 + 0.725454i \(0.258370\pi\)
\(812\) 0 0
\(813\) 0.0179882 0.000630873 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 78.5495 2.74810
\(818\) 0 0
\(819\) 46.7645 1.63408
\(820\) 0 0
\(821\) −35.7578 −1.24796 −0.623979 0.781441i \(-0.714485\pi\)
−0.623979 + 0.781441i \(0.714485\pi\)
\(822\) 0 0
\(823\) 17.3093 0.603365 0.301683 0.953408i \(-0.402452\pi\)
0.301683 + 0.953408i \(0.402452\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.9957 0.903958 0.451979 0.892029i \(-0.350718\pi\)
0.451979 + 0.892029i \(0.350718\pi\)
\(828\) 0 0
\(829\) −46.6902 −1.62162 −0.810808 0.585312i \(-0.800972\pi\)
−0.810808 + 0.585312i \(0.800972\pi\)
\(830\) 0 0
\(831\) −1.38895 −0.0481822
\(832\) 0 0
\(833\) −72.9569 −2.52781
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.438496 −0.0151566
\(838\) 0 0
\(839\) −9.45364 −0.326376 −0.163188 0.986595i \(-0.552178\pi\)
−0.163188 + 0.986595i \(0.552178\pi\)
\(840\) 0 0
\(841\) −6.62168 −0.228334
\(842\) 0 0
\(843\) 5.94786 0.204855
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.05237 0.139241
\(848\) 0 0
\(849\) −5.12940 −0.176041
\(850\) 0 0
\(851\) −3.61542 −0.123935
\(852\) 0 0
\(853\) −21.5733 −0.738656 −0.369328 0.929299i \(-0.620412\pi\)
−0.369328 + 0.929299i \(0.620412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −19.5783 −0.668782 −0.334391 0.942434i \(-0.608531\pi\)
−0.334391 + 0.942434i \(0.608531\pi\)
\(858\) 0 0
\(859\) −28.3598 −0.967622 −0.483811 0.875172i \(-0.660748\pi\)
−0.483811 + 0.875172i \(0.660748\pi\)
\(860\) 0 0
\(861\) −10.7128 −0.365092
\(862\) 0 0
\(863\) 9.05630 0.308280 0.154140 0.988049i \(-0.450739\pi\)
0.154140 + 0.988049i \(0.450739\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.5684 0.494769
\(868\) 0 0
\(869\) 13.9006 0.471546
\(870\) 0 0
\(871\) 18.8014 0.637060
\(872\) 0 0
\(873\) 36.0517 1.22016
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −53.2624 −1.79854 −0.899271 0.437392i \(-0.855902\pi\)
−0.899271 + 0.437392i \(0.855902\pi\)
\(878\) 0 0
\(879\) 4.52918 0.152766
\(880\) 0 0
\(881\) 33.7557 1.13726 0.568629 0.822594i \(-0.307474\pi\)
0.568629 + 0.822594i \(0.307474\pi\)
\(882\) 0 0
\(883\) 0.157109 0.00528714 0.00264357 0.999997i \(-0.499159\pi\)
0.00264357 + 0.999997i \(0.499159\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.63050 0.289784 0.144892 0.989447i \(-0.453717\pi\)
0.144892 + 0.989447i \(0.453717\pi\)
\(888\) 0 0
\(889\) 39.5917 1.32786
\(890\) 0 0
\(891\) 7.97831 0.267284
\(892\) 0 0
\(893\) −21.2872 −0.712348
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −3.69783 −0.123467
\(898\) 0 0
\(899\) −1.03945 −0.0346675
\(900\) 0 0
\(901\) 39.0229 1.30004
\(902\) 0 0
\(903\) −15.2775 −0.508404
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.69596 0.222336 0.111168 0.993802i \(-0.464541\pi\)
0.111168 + 0.993802i \(0.464541\pi\)
\(908\) 0 0
\(909\) 1.85677 0.0615852
\(910\) 0 0
\(911\) −29.9345 −0.991776 −0.495888 0.868387i \(-0.665157\pi\)
−0.495888 + 0.868387i \(0.665157\pi\)
\(912\) 0 0
\(913\) −3.63067 −0.120158
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −48.9809 −1.61749
\(918\) 0 0
\(919\) 35.3618 1.16648 0.583238 0.812301i \(-0.301785\pi\)
0.583238 + 0.812301i \(0.301785\pi\)
\(920\) 0 0
\(921\) 5.35641 0.176500
\(922\) 0 0
\(923\) 10.1144 0.332920
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.6056 0.414021
\(928\) 0 0
\(929\) −27.2134 −0.892843 −0.446421 0.894823i \(-0.647302\pi\)
−0.446421 + 0.894823i \(0.647302\pi\)
\(930\) 0 0
\(931\) −66.5671 −2.18165
\(932\) 0 0
\(933\) −0.281444 −0.00921405
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.3659 0.534651 0.267326 0.963606i \(-0.413860\pi\)
0.267326 + 0.963606i \(0.413860\pi\)
\(938\) 0 0
\(939\) 8.30968 0.271176
\(940\) 0 0
\(941\) −18.0134 −0.587221 −0.293611 0.955925i \(-0.594857\pi\)
−0.293611 + 0.955925i \(0.594857\pi\)
\(942\) 0 0
\(943\) −21.2530 −0.692094
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −47.9751 −1.55898 −0.779491 0.626414i \(-0.784522\pi\)
−0.779491 + 0.626414i \(0.784522\pi\)
\(948\) 0 0
\(949\) −16.4190 −0.532982
\(950\) 0 0
\(951\) 3.88798 0.126076
\(952\) 0 0
\(953\) −40.7788 −1.32096 −0.660478 0.750845i \(-0.729646\pi\)
−0.660478 + 0.750845i \(0.729646\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.60415 −0.0518547
\(958\) 0 0
\(959\) 18.2203 0.588364
\(960\) 0 0
\(961\) −30.9517 −0.998443
\(962\) 0 0
\(963\) 25.9278 0.835512
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −0.835313 −0.0268618 −0.0134309 0.999910i \(-0.504275\pi\)
−0.0134309 + 0.999910i \(0.504275\pi\)
\(968\) 0 0
\(969\) 18.5523 0.595985
\(970\) 0 0
\(971\) 27.3542 0.877839 0.438920 0.898526i \(-0.355361\pi\)
0.438920 + 0.898526i \(0.355361\pi\)
\(972\) 0 0
\(973\) −33.7754 −1.08279
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.40765 0.141013 0.0705066 0.997511i \(-0.477538\pi\)
0.0705066 + 0.997511i \(0.477538\pi\)
\(978\) 0 0
\(979\) 9.88501 0.315926
\(980\) 0 0
\(981\) −52.5345 −1.67730
\(982\) 0 0
\(983\) −28.3968 −0.905718 −0.452859 0.891582i \(-0.649596\pi\)
−0.452859 + 0.891582i \(0.649596\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.14026 0.131786
\(988\) 0 0
\(989\) −30.3089 −0.963766
\(990\) 0 0
\(991\) −6.76451 −0.214882 −0.107441 0.994211i \(-0.534266\pi\)
−0.107441 + 0.994211i \(0.534266\pi\)
\(992\) 0 0
\(993\) −8.27797 −0.262693
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33.5747 1.06332 0.531660 0.846958i \(-0.321568\pi\)
0.531660 + 0.846958i \(0.321568\pi\)
\(998\) 0 0
\(999\) 2.64655 0.0837330
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 2200.2.a.x.1.3 4
4.3 odd 2 4400.2.a.ce.1.2 4
5.2 odd 4 440.2.b.d.89.4 8
5.3 odd 4 440.2.b.d.89.5 yes 8
5.4 even 2 2200.2.a.y.1.2 4
15.2 even 4 3960.2.d.f.3169.4 8
15.8 even 4 3960.2.d.f.3169.3 8
20.3 even 4 880.2.b.j.529.4 8
20.7 even 4 880.2.b.j.529.5 8
20.19 odd 2 4400.2.a.cb.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.b.d.89.4 8 5.2 odd 4
440.2.b.d.89.5 yes 8 5.3 odd 4
880.2.b.j.529.4 8 20.3 even 4
880.2.b.j.529.5 8 20.7 even 4
2200.2.a.x.1.3 4 1.1 even 1 trivial
2200.2.a.y.1.2 4 5.4 even 2
3960.2.d.f.3169.3 8 15.8 even 4
3960.2.d.f.3169.4 8 15.2 even 4
4400.2.a.cb.1.3 4 20.19 odd 2
4400.2.a.ce.1.2 4 4.3 odd 2