Properties

Label 4400.2.a.cc.1.1
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
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] = [4400,2,Mod(1,4400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4400, 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("4400.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.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: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.52434\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52434 q^{3} -3.46410 q^{7} +3.37228 q^{9} +O(q^{10})\) \(q-2.52434 q^{3} -3.46410 q^{7} +3.37228 q^{9} +1.00000 q^{11} -5.04868 q^{17} -4.00000 q^{19} +8.74456 q^{21} +2.52434 q^{23} -0.939764 q^{27} -2.74456 q^{29} +2.37228 q^{31} -2.52434 q^{33} -11.0371 q^{37} -2.74456 q^{41} -3.46410 q^{43} -6.63325 q^{47} +5.00000 q^{49} +12.7446 q^{51} -3.16915 q^{53} +10.0974 q^{57} +1.62772 q^{59} +10.7446 q^{61} -11.6819 q^{63} +0.644810 q^{67} -6.37228 q^{69} -7.11684 q^{71} -6.92820 q^{73} -3.46410 q^{77} -12.7446 q^{79} -7.74456 q^{81} +6.63325 q^{83} +6.92820 q^{87} -4.37228 q^{89} -5.98844 q^{93} -4.10891 q^{97} +3.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 4 q^{11} - 16 q^{19} + 12 q^{21} + 12 q^{29} - 2 q^{31} + 12 q^{41} + 20 q^{49} + 28 q^{51} + 18 q^{59} + 20 q^{61} - 14 q^{69} + 6 q^{71} - 28 q^{79} - 8 q^{81} - 6 q^{89} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −2.52434 −1.45743 −0.728714 0.684819i \(-0.759881\pi\)
−0.728714 + 0.684819i \(0.759881\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 3.37228 1.12409
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.04868 −1.22448 −0.612242 0.790671i \(-0.709732\pi\)
−0.612242 + 0.790671i \(0.709732\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 8.74456 1.90822
\(22\) 0 0
\(23\) 2.52434 0.526361 0.263180 0.964747i \(-0.415229\pi\)
0.263180 + 0.964747i \(0.415229\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.939764 −0.180858
\(28\) 0 0
\(29\) −2.74456 −0.509652 −0.254826 0.966987i \(-0.582018\pi\)
−0.254826 + 0.966987i \(0.582018\pi\)
\(30\) 0 0
\(31\) 2.37228 0.426074 0.213037 0.977044i \(-0.431664\pi\)
0.213037 + 0.977044i \(0.431664\pi\)
\(32\) 0 0
\(33\) −2.52434 −0.439431
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.0371 −1.81449 −0.907245 0.420602i \(-0.861819\pi\)
−0.907245 + 0.420602i \(0.861819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.74456 −0.428629 −0.214314 0.976765i \(-0.568752\pi\)
−0.214314 + 0.976765i \(0.568752\pi\)
\(42\) 0 0
\(43\) −3.46410 −0.528271 −0.264135 0.964486i \(-0.585087\pi\)
−0.264135 + 0.964486i \(0.585087\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.63325 −0.967559 −0.483779 0.875190i \(-0.660736\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 12.7446 1.78460
\(52\) 0 0
\(53\) −3.16915 −0.435316 −0.217658 0.976025i \(-0.569842\pi\)
−0.217658 + 0.976025i \(0.569842\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.0974 1.33743
\(58\) 0 0
\(59\) 1.62772 0.211911 0.105955 0.994371i \(-0.466210\pi\)
0.105955 + 0.994371i \(0.466210\pi\)
\(60\) 0 0
\(61\) 10.7446 1.37570 0.687850 0.725853i \(-0.258555\pi\)
0.687850 + 0.725853i \(0.258555\pi\)
\(62\) 0 0
\(63\) −11.6819 −1.47178
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.644810 0.0787761 0.0393880 0.999224i \(-0.487459\pi\)
0.0393880 + 0.999224i \(0.487459\pi\)
\(68\) 0 0
\(69\) −6.37228 −0.767133
\(70\) 0 0
\(71\) −7.11684 −0.844614 −0.422307 0.906453i \(-0.638780\pi\)
−0.422307 + 0.906453i \(0.638780\pi\)
\(72\) 0 0
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −12.7446 −1.43388 −0.716938 0.697137i \(-0.754457\pi\)
−0.716938 + 0.697137i \(0.754457\pi\)
\(80\) 0 0
\(81\) −7.74456 −0.860507
\(82\) 0 0
\(83\) 6.63325 0.728094 0.364047 0.931381i \(-0.381395\pi\)
0.364047 + 0.931381i \(0.381395\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.92820 0.742781
\(88\) 0 0
\(89\) −4.37228 −0.463461 −0.231730 0.972780i \(-0.574439\pi\)
−0.231730 + 0.972780i \(0.574439\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.98844 −0.620972
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.10891 −0.417197 −0.208598 0.978001i \(-0.566890\pi\)
−0.208598 + 0.978001i \(0.566890\pi\)
\(98\) 0 0
\(99\) 3.37228 0.338927
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 10.3923 1.02398 0.511992 0.858990i \(-0.328908\pi\)
0.511992 + 0.858990i \(0.328908\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.63325 −0.641260 −0.320630 0.947204i \(-0.603895\pi\)
−0.320630 + 0.947204i \(0.603895\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 27.8614 2.64449
\(112\) 0 0
\(113\) 16.0858 1.51322 0.756612 0.653864i \(-0.226853\pi\)
0.756612 + 0.653864i \(0.226853\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 17.4891 1.60323
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.92820 0.624695
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −11.6819 −1.03660 −0.518302 0.855198i \(-0.673436\pi\)
−0.518302 + 0.855198i \(0.673436\pi\)
\(128\) 0 0
\(129\) 8.74456 0.769916
\(130\) 0 0
\(131\) −8.74456 −0.764016 −0.382008 0.924159i \(-0.624767\pi\)
−0.382008 + 0.924159i \(0.624767\pi\)
\(132\) 0 0
\(133\) 13.8564 1.20150
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.22938 −0.190469 −0.0952346 0.995455i \(-0.530360\pi\)
−0.0952346 + 0.995455i \(0.530360\pi\)
\(138\) 0 0
\(139\) −18.2337 −1.54656 −0.773281 0.634064i \(-0.781386\pi\)
−0.773281 + 0.634064i \(0.781386\pi\)
\(140\) 0 0
\(141\) 16.7446 1.41015
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −12.6217 −1.04102
\(148\) 0 0
\(149\) 11.4891 0.941226 0.470613 0.882340i \(-0.344033\pi\)
0.470613 + 0.882340i \(0.344033\pi\)
\(150\) 0 0
\(151\) −22.2337 −1.80935 −0.904676 0.426100i \(-0.859887\pi\)
−0.904676 + 0.426100i \(0.859887\pi\)
\(152\) 0 0
\(153\) −17.0256 −1.37643
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.39853 0.430850 0.215425 0.976520i \(-0.430886\pi\)
0.215425 + 0.976520i \(0.430886\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −8.74456 −0.689168
\(162\) 0 0
\(163\) 3.46410 0.271329 0.135665 0.990755i \(-0.456683\pi\)
0.135665 + 0.990755i \(0.456683\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.3692 1.73098 0.865490 0.500927i \(-0.167007\pi\)
0.865490 + 0.500927i \(0.167007\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −13.4891 −1.03154
\(172\) 0 0
\(173\) −1.87953 −0.142898 −0.0714489 0.997444i \(-0.522762\pi\)
−0.0714489 + 0.997444i \(0.522762\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.10891 −0.308845
\(178\) 0 0
\(179\) 15.8614 1.18554 0.592769 0.805373i \(-0.298035\pi\)
0.592769 + 0.805373i \(0.298035\pi\)
\(180\) 0 0
\(181\) 6.88316 0.511621 0.255810 0.966727i \(-0.417658\pi\)
0.255810 + 0.966727i \(0.417658\pi\)
\(182\) 0 0
\(183\) −27.1229 −2.00498
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −5.04868 −0.369196
\(188\) 0 0
\(189\) 3.25544 0.236798
\(190\) 0 0
\(191\) 13.6277 0.986067 0.493034 0.870010i \(-0.335888\pi\)
0.493034 + 0.870010i \(0.335888\pi\)
\(192\) 0 0
\(193\) 23.3639 1.68177 0.840883 0.541216i \(-0.182036\pi\)
0.840883 + 0.541216i \(0.182036\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.87953 0.133911 0.0669554 0.997756i \(-0.478671\pi\)
0.0669554 + 0.997756i \(0.478671\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −1.62772 −0.114810
\(202\) 0 0
\(203\) 9.50744 0.667292
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.51278 0.591679
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 21.4891 1.47937 0.739686 0.672952i \(-0.234974\pi\)
0.739686 + 0.672952i \(0.234974\pi\)
\(212\) 0 0
\(213\) 17.9653 1.23096
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.21782 −0.557862
\(218\) 0 0
\(219\) 17.4891 1.18181
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.57301 −0.507126 −0.253563 0.967319i \(-0.581603\pi\)
−0.253563 + 0.967319i \(0.581603\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.80240 0.650608 0.325304 0.945609i \(-0.394533\pi\)
0.325304 + 0.945609i \(0.394533\pi\)
\(228\) 0 0
\(229\) 20.3723 1.34624 0.673119 0.739534i \(-0.264954\pi\)
0.673119 + 0.739534i \(0.264954\pi\)
\(230\) 0 0
\(231\) 8.74456 0.575350
\(232\) 0 0
\(233\) −17.0256 −1.11538 −0.557691 0.830049i \(-0.688312\pi\)
−0.557691 + 0.830049i \(0.688312\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 32.1716 2.08977
\(238\) 0 0
\(239\) 3.25544 0.210577 0.105288 0.994442i \(-0.466423\pi\)
0.105288 + 0.994442i \(0.466423\pi\)
\(240\) 0 0
\(241\) 5.25544 0.338532 0.169266 0.985570i \(-0.445860\pi\)
0.169266 + 0.985570i \(0.445860\pi\)
\(242\) 0 0
\(243\) 22.3692 1.43498
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.7446 −1.06114
\(250\) 0 0
\(251\) 4.88316 0.308222 0.154111 0.988054i \(-0.450749\pi\)
0.154111 + 0.988054i \(0.450749\pi\)
\(252\) 0 0
\(253\) 2.52434 0.158704
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.9538 1.49419 0.747097 0.664715i \(-0.231447\pi\)
0.747097 + 0.664715i \(0.231447\pi\)
\(258\) 0 0
\(259\) 38.2337 2.37573
\(260\) 0 0
\(261\) −9.25544 −0.572897
\(262\) 0 0
\(263\) −14.1514 −0.872610 −0.436305 0.899799i \(-0.643713\pi\)
−0.436305 + 0.899799i \(0.643713\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 11.0371 0.675460
\(268\) 0 0
\(269\) 11.4891 0.700504 0.350252 0.936655i \(-0.386096\pi\)
0.350252 + 0.936655i \(0.386096\pi\)
\(270\) 0 0
\(271\) 9.48913 0.576423 0.288212 0.957567i \(-0.406939\pi\)
0.288212 + 0.957567i \(0.406939\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.21782 0.493761 0.246881 0.969046i \(-0.420594\pi\)
0.246881 + 0.969046i \(0.420594\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −23.4891 −1.40124 −0.700622 0.713533i \(-0.747094\pi\)
−0.700622 + 0.713533i \(0.747094\pi\)
\(282\) 0 0
\(283\) 4.75372 0.282579 0.141290 0.989968i \(-0.454875\pi\)
0.141290 + 0.989968i \(0.454875\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.50744 0.561207
\(288\) 0 0
\(289\) 8.48913 0.499360
\(290\) 0 0
\(291\) 10.3723 0.608034
\(292\) 0 0
\(293\) −10.0974 −0.589894 −0.294947 0.955514i \(-0.595302\pi\)
−0.294947 + 0.955514i \(0.595302\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.939764 −0.0545306
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −15.1460 −0.870117
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.1176 1.60475 0.802377 0.596817i \(-0.203568\pi\)
0.802377 + 0.596817i \(0.203568\pi\)
\(308\) 0 0
\(309\) −26.2337 −1.49238
\(310\) 0 0
\(311\) 17.4891 0.991717 0.495859 0.868403i \(-0.334853\pi\)
0.495859 + 0.868403i \(0.334853\pi\)
\(312\) 0 0
\(313\) −31.8217 −1.79867 −0.899335 0.437260i \(-0.855949\pi\)
−0.899335 + 0.437260i \(0.855949\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.51900 −0.197647 −0.0988235 0.995105i \(-0.531508\pi\)
−0.0988235 + 0.995105i \(0.531508\pi\)
\(318\) 0 0
\(319\) −2.74456 −0.153666
\(320\) 0 0
\(321\) 16.7446 0.934590
\(322\) 0 0
\(323\) 20.1947 1.12366
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −25.2434 −1.39596
\(328\) 0 0
\(329\) 22.9783 1.26683
\(330\) 0 0
\(331\) −3.11684 −0.171317 −0.0856586 0.996325i \(-0.527299\pi\)
−0.0856586 + 0.996325i \(0.527299\pi\)
\(332\) 0 0
\(333\) −37.2203 −2.03966
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.5668 −0.684556 −0.342278 0.939599i \(-0.611199\pi\)
−0.342278 + 0.939599i \(0.611199\pi\)
\(338\) 0 0
\(339\) −40.6060 −2.20541
\(340\) 0 0
\(341\) 2.37228 0.128466
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.2974 1.57277 0.786383 0.617739i \(-0.211951\pi\)
0.786383 + 0.617739i \(0.211951\pi\)
\(348\) 0 0
\(349\) −7.48913 −0.400884 −0.200442 0.979706i \(-0.564238\pi\)
−0.200442 + 0.979706i \(0.564238\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.7244 1.15627 0.578136 0.815941i \(-0.303780\pi\)
0.578136 + 0.815941i \(0.303780\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −44.1485 −2.33658
\(358\) 0 0
\(359\) −6.51087 −0.343631 −0.171815 0.985129i \(-0.554963\pi\)
−0.171815 + 0.985129i \(0.554963\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −2.52434 −0.132493
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 24.0087 1.25324 0.626621 0.779324i \(-0.284437\pi\)
0.626621 + 0.779324i \(0.284437\pi\)
\(368\) 0 0
\(369\) −9.25544 −0.481819
\(370\) 0 0
\(371\) 10.9783 0.569962
\(372\) 0 0
\(373\) 8.21782 0.425503 0.212751 0.977106i \(-0.431758\pi\)
0.212751 + 0.977106i \(0.431758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.37228 0.327322 0.163661 0.986517i \(-0.447670\pi\)
0.163661 + 0.986517i \(0.447670\pi\)
\(380\) 0 0
\(381\) 29.4891 1.51077
\(382\) 0 0
\(383\) −5.69349 −0.290924 −0.145462 0.989364i \(-0.546467\pi\)
−0.145462 + 0.989364i \(0.546467\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.6819 −0.593826
\(388\) 0 0
\(389\) −9.86141 −0.499993 −0.249997 0.968247i \(-0.580429\pi\)
−0.249997 + 0.968247i \(0.580429\pi\)
\(390\) 0 0
\(391\) −12.7446 −0.644520
\(392\) 0 0
\(393\) 22.0742 1.11350
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 23.3639 1.17260 0.586299 0.810095i \(-0.300584\pi\)
0.586299 + 0.810095i \(0.300584\pi\)
\(398\) 0 0
\(399\) −34.9783 −1.75110
\(400\) 0 0
\(401\) 11.4891 0.573740 0.286870 0.957970i \(-0.407385\pi\)
0.286870 + 0.957970i \(0.407385\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.0371 −0.547089
\(408\) 0 0
\(409\) 4.51087 0.223048 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(410\) 0 0
\(411\) 5.62772 0.277595
\(412\) 0 0
\(413\) −5.63858 −0.277457
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 46.0280 2.25400
\(418\) 0 0
\(419\) −22.9783 −1.12256 −0.561280 0.827626i \(-0.689691\pi\)
−0.561280 + 0.827626i \(0.689691\pi\)
\(420\) 0 0
\(421\) 31.4891 1.53469 0.767343 0.641237i \(-0.221578\pi\)
0.767343 + 0.641237i \(0.221578\pi\)
\(422\) 0 0
\(423\) −22.3692 −1.08763
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −37.2203 −1.80121
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.7228 −1.52803 −0.764017 0.645196i \(-0.776776\pi\)
−0.764017 + 0.645196i \(0.776776\pi\)
\(432\) 0 0
\(433\) −20.5446 −0.987308 −0.493654 0.869658i \(-0.664339\pi\)
−0.493654 + 0.869658i \(0.664339\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.0974 −0.483022
\(438\) 0 0
\(439\) 1.48913 0.0710721 0.0355360 0.999368i \(-0.488686\pi\)
0.0355360 + 0.999368i \(0.488686\pi\)
\(440\) 0 0
\(441\) 16.8614 0.802924
\(442\) 0 0
\(443\) 15.0911 0.717001 0.358500 0.933530i \(-0.383288\pi\)
0.358500 + 0.933530i \(0.383288\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −29.0024 −1.37177
\(448\) 0 0
\(449\) −21.8614 −1.03170 −0.515852 0.856678i \(-0.672524\pi\)
−0.515852 + 0.856678i \(0.672524\pi\)
\(450\) 0 0
\(451\) −2.74456 −0.129236
\(452\) 0 0
\(453\) 56.1253 2.63700
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −20.7846 −0.972263 −0.486132 0.873886i \(-0.661592\pi\)
−0.486132 + 0.873886i \(0.661592\pi\)
\(458\) 0 0
\(459\) 4.74456 0.221457
\(460\) 0 0
\(461\) −32.2337 −1.50127 −0.750636 0.660716i \(-0.770253\pi\)
−0.750636 + 0.660716i \(0.770253\pi\)
\(462\) 0 0
\(463\) 20.1398 0.935976 0.467988 0.883735i \(-0.344979\pi\)
0.467988 + 0.883735i \(0.344979\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.40387 0.203787 0.101893 0.994795i \(-0.467510\pi\)
0.101893 + 0.994795i \(0.467510\pi\)
\(468\) 0 0
\(469\) −2.23369 −0.103142
\(470\) 0 0
\(471\) −13.6277 −0.627932
\(472\) 0 0
\(473\) −3.46410 −0.159280
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.6873 −0.489336
\(478\) 0 0
\(479\) −17.4891 −0.799099 −0.399549 0.916712i \(-0.630833\pi\)
−0.399549 + 0.916712i \(0.630833\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 22.0742 1.00441
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −22.7190 −1.02950 −0.514749 0.857341i \(-0.672115\pi\)
−0.514749 + 0.857341i \(0.672115\pi\)
\(488\) 0 0
\(489\) −8.74456 −0.395443
\(490\) 0 0
\(491\) −29.4891 −1.33083 −0.665413 0.746476i \(-0.731744\pi\)
−0.665413 + 0.746476i \(0.731744\pi\)
\(492\) 0 0
\(493\) 13.8564 0.624061
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.6535 1.10586
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −56.4674 −2.52278
\(502\) 0 0
\(503\) −0.294954 −0.0131513 −0.00657567 0.999978i \(-0.502093\pi\)
−0.00657567 + 0.999978i \(0.502093\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 32.8164 1.45743
\(508\) 0 0
\(509\) 28.3723 1.25758 0.628790 0.777575i \(-0.283551\pi\)
0.628790 + 0.777575i \(0.283551\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) 3.75906 0.165966
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.63325 −0.291730
\(518\) 0 0
\(519\) 4.74456 0.208263
\(520\) 0 0
\(521\) 18.6060 0.815142 0.407571 0.913173i \(-0.366376\pi\)
0.407571 + 0.913173i \(0.366376\pi\)
\(522\) 0 0
\(523\) −9.10268 −0.398033 −0.199016 0.979996i \(-0.563775\pi\)
−0.199016 + 0.979996i \(0.563775\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11.9769 −0.521721
\(528\) 0 0
\(529\) −16.6277 −0.722944
\(530\) 0 0
\(531\) 5.48913 0.238208
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −40.0395 −1.72783
\(538\) 0 0
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) −0.233688 −0.0100470 −0.00502351 0.999987i \(-0.501599\pi\)
−0.00502351 + 0.999987i \(0.501599\pi\)
\(542\) 0 0
\(543\) −17.3754 −0.745650
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.10268 0.389203 0.194601 0.980882i \(-0.437659\pi\)
0.194601 + 0.980882i \(0.437659\pi\)
\(548\) 0 0
\(549\) 36.2337 1.54642
\(550\) 0 0
\(551\) 10.9783 0.467689
\(552\) 0 0
\(553\) 44.1485 1.87738
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.1716 1.36315 0.681577 0.731747i \(-0.261295\pi\)
0.681577 + 0.731747i \(0.261295\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 12.7446 0.538076
\(562\) 0 0
\(563\) 12.2718 0.517196 0.258598 0.965985i \(-0.416739\pi\)
0.258598 + 0.965985i \(0.416739\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 26.8280 1.12667
\(568\) 0 0
\(569\) −38.7446 −1.62426 −0.812128 0.583479i \(-0.801691\pi\)
−0.812128 + 0.583479i \(0.801691\pi\)
\(570\) 0 0
\(571\) 21.4891 0.899292 0.449646 0.893207i \(-0.351550\pi\)
0.449646 + 0.893207i \(0.351550\pi\)
\(572\) 0 0
\(573\) −34.4010 −1.43712
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 31.8217 1.32476 0.662378 0.749170i \(-0.269547\pi\)
0.662378 + 0.749170i \(0.269547\pi\)
\(578\) 0 0
\(579\) −58.9783 −2.45105
\(580\) 0 0
\(581\) −22.9783 −0.953298
\(582\) 0 0
\(583\) −3.16915 −0.131253
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0078 1.15600 0.578002 0.816035i \(-0.303833\pi\)
0.578002 + 0.816035i \(0.303833\pi\)
\(588\) 0 0
\(589\) −9.48913 −0.390993
\(590\) 0 0
\(591\) −4.74456 −0.195165
\(592\) 0 0
\(593\) 43.5586 1.78874 0.894368 0.447333i \(-0.147626\pi\)
0.894368 + 0.447333i \(0.147626\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −20.1947 −0.826514
\(598\) 0 0
\(599\) 34.9783 1.42917 0.714586 0.699547i \(-0.246615\pi\)
0.714586 + 0.699547i \(0.246615\pi\)
\(600\) 0 0
\(601\) 30.4674 1.24279 0.621395 0.783497i \(-0.286566\pi\)
0.621395 + 0.783497i \(0.286566\pi\)
\(602\) 0 0
\(603\) 2.17448 0.0885517
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.46410 −0.140604 −0.0703018 0.997526i \(-0.522396\pi\)
−0.0703018 + 0.997526i \(0.522396\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −44.1485 −1.78314 −0.891570 0.452883i \(-0.850395\pi\)
−0.891570 + 0.452883i \(0.850395\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.75906 −0.151334 −0.0756669 0.997133i \(-0.524109\pi\)
−0.0756669 + 0.997133i \(0.524109\pi\)
\(618\) 0 0
\(619\) 3.11684 0.125277 0.0626383 0.998036i \(-0.480049\pi\)
0.0626383 + 0.998036i \(0.480049\pi\)
\(620\) 0 0
\(621\) −2.37228 −0.0951964
\(622\) 0 0
\(623\) 15.1460 0.606813
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 10.0974 0.403249
\(628\) 0 0
\(629\) 55.7228 2.22181
\(630\) 0 0
\(631\) 16.6060 0.661073 0.330537 0.943793i \(-0.392770\pi\)
0.330537 + 0.943793i \(0.392770\pi\)
\(632\) 0 0
\(633\) −54.2458 −2.15608
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 0 0
\(641\) −19.6277 −0.775248 −0.387624 0.921818i \(-0.626704\pi\)
−0.387624 + 0.921818i \(0.626704\pi\)
\(642\) 0 0
\(643\) −39.1547 −1.54411 −0.772055 0.635556i \(-0.780771\pi\)
−0.772055 + 0.635556i \(0.780771\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.0342 −1.61322 −0.806611 0.591083i \(-0.798701\pi\)
−0.806611 + 0.591083i \(0.798701\pi\)
\(648\) 0 0
\(649\) 1.62772 0.0638935
\(650\) 0 0
\(651\) 20.7446 0.813044
\(652\) 0 0
\(653\) 25.5932 1.00154 0.500770 0.865580i \(-0.333050\pi\)
0.500770 + 0.865580i \(0.333050\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −23.3639 −0.911511
\(658\) 0 0
\(659\) −32.7446 −1.27555 −0.637774 0.770224i \(-0.720144\pi\)
−0.637774 + 0.770224i \(0.720144\pi\)
\(660\) 0 0
\(661\) 35.3505 1.37498 0.687488 0.726196i \(-0.258713\pi\)
0.687488 + 0.726196i \(0.258713\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −6.92820 −0.268261
\(668\) 0 0
\(669\) 19.1168 0.739100
\(670\) 0 0
\(671\) 10.7446 0.414789
\(672\) 0 0
\(673\) −1.28962 −0.0497112 −0.0248556 0.999691i \(-0.507913\pi\)
−0.0248556 + 0.999691i \(0.507913\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 43.4487 1.66987 0.834935 0.550348i \(-0.185505\pi\)
0.834935 + 0.550348i \(0.185505\pi\)
\(678\) 0 0
\(679\) 14.2337 0.546239
\(680\) 0 0
\(681\) −24.7446 −0.948214
\(682\) 0 0
\(683\) −44.4434 −1.70058 −0.850290 0.526314i \(-0.823573\pi\)
−0.850290 + 0.526314i \(0.823573\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −51.4265 −1.96204
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −16.1386 −0.613941 −0.306971 0.951719i \(-0.599315\pi\)
−0.306971 + 0.951719i \(0.599315\pi\)
\(692\) 0 0
\(693\) −11.6819 −0.443760
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.8564 0.524849
\(698\) 0 0
\(699\) 42.9783 1.62559
\(700\) 0 0
\(701\) −35.4891 −1.34041 −0.670203 0.742178i \(-0.733793\pi\)
−0.670203 + 0.742178i \(0.733793\pi\)
\(702\) 0 0
\(703\) 44.1485 1.66509
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.7846 −0.781686
\(708\) 0 0
\(709\) 41.1168 1.54418 0.772088 0.635516i \(-0.219213\pi\)
0.772088 + 0.635516i \(0.219213\pi\)
\(710\) 0 0
\(711\) −42.9783 −1.61181
\(712\) 0 0
\(713\) 5.98844 0.224269
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.21782 −0.306900
\(718\) 0 0
\(719\) −21.3505 −0.796240 −0.398120 0.917333i \(-0.630337\pi\)
−0.398120 + 0.917333i \(0.630337\pi\)
\(720\) 0 0
\(721\) −36.0000 −1.34071
\(722\) 0 0
\(723\) −13.2665 −0.493386
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 15.7908 0.585650 0.292825 0.956166i \(-0.405405\pi\)
0.292825 + 0.956166i \(0.405405\pi\)
\(728\) 0 0
\(729\) −33.2337 −1.23088
\(730\) 0 0
\(731\) 17.4891 0.646859
\(732\) 0 0
\(733\) 30.2921 1.11886 0.559431 0.828877i \(-0.311020\pi\)
0.559431 + 0.828877i \(0.311020\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.644810 0.0237519
\(738\) 0 0
\(739\) −0.744563 −0.0273892 −0.0136946 0.999906i \(-0.504359\pi\)
−0.0136946 + 0.999906i \(0.504359\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.7793 0.799004 0.399502 0.916732i \(-0.369183\pi\)
0.399502 + 0.916732i \(0.369183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 22.3692 0.818446
\(748\) 0 0
\(749\) 22.9783 0.839607
\(750\) 0 0
\(751\) −21.6277 −0.789207 −0.394603 0.918852i \(-0.629118\pi\)
−0.394603 + 0.918852i \(0.629118\pi\)
\(752\) 0 0
\(753\) −12.3267 −0.449211
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −39.7995 −1.44654 −0.723269 0.690567i \(-0.757361\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) −6.37228 −0.231299
\(760\) 0 0
\(761\) 21.2554 0.770509 0.385255 0.922810i \(-0.374114\pi\)
0.385255 + 0.922810i \(0.374114\pi\)
\(762\) 0 0
\(763\) −34.6410 −1.25409
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −51.2119 −1.84675 −0.923375 0.383900i \(-0.874581\pi\)
−0.923375 + 0.383900i \(0.874581\pi\)
\(770\) 0 0
\(771\) −60.4674 −2.17768
\(772\) 0 0
\(773\) −30.8820 −1.11075 −0.555373 0.831601i \(-0.687425\pi\)
−0.555373 + 0.831601i \(0.687425\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −96.5147 −3.46245
\(778\) 0 0
\(779\) 10.9783 0.393337
\(780\) 0 0
\(781\) −7.11684 −0.254661
\(782\) 0 0
\(783\) 2.57924 0.0921745
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −4.75372 −0.169452 −0.0847259 0.996404i \(-0.527001\pi\)
−0.0847259 + 0.996404i \(0.527001\pi\)
\(788\) 0 0
\(789\) 35.7228 1.27177
\(790\) 0 0
\(791\) −55.7228 −1.98128
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −31.2318 −1.10629 −0.553144 0.833086i \(-0.686572\pi\)
−0.553144 + 0.833086i \(0.686572\pi\)
\(798\) 0 0
\(799\) 33.4891 1.18476
\(800\) 0 0
\(801\) −14.7446 −0.520974
\(802\) 0 0
\(803\) −6.92820 −0.244491
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −29.0024 −1.02093
\(808\) 0 0
\(809\) −21.2554 −0.747301 −0.373651 0.927569i \(-0.621894\pi\)
−0.373651 + 0.927569i \(0.621894\pi\)
\(810\) 0 0
\(811\) −34.2337 −1.20211 −0.601054 0.799209i \(-0.705252\pi\)
−0.601054 + 0.799209i \(0.705252\pi\)
\(812\) 0 0
\(813\) −23.9538 −0.840095
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 13.8564 0.484774
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) 33.5161 1.16830 0.584149 0.811646i \(-0.301428\pi\)
0.584149 + 0.811646i \(0.301428\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.0202 0.626624 0.313312 0.949650i \(-0.398561\pi\)
0.313312 + 0.949650i \(0.398561\pi\)
\(828\) 0 0
\(829\) 31.3505 1.08885 0.544424 0.838810i \(-0.316748\pi\)
0.544424 + 0.838810i \(0.316748\pi\)
\(830\) 0 0
\(831\) −20.7446 −0.719621
\(832\) 0 0
\(833\) −25.2434 −0.874631
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.22938 −0.0770588
\(838\) 0 0
\(839\) −7.11684 −0.245701 −0.122850 0.992425i \(-0.539204\pi\)
−0.122850 + 0.992425i \(0.539204\pi\)
\(840\) 0 0
\(841\) −21.4674 −0.740254
\(842\) 0 0
\(843\) 59.2945 2.04221
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.46410 −0.119028
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −27.8614 −0.955077
\(852\) 0 0
\(853\) −24.6535 −0.844119 −0.422059 0.906568i \(-0.638693\pi\)
−0.422059 + 0.906568i \(0.638693\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.6873 −0.365070 −0.182535 0.983199i \(-0.558430\pi\)
−0.182535 + 0.983199i \(0.558430\pi\)
\(858\) 0 0
\(859\) −11.1168 −0.379302 −0.189651 0.981852i \(-0.560736\pi\)
−0.189651 + 0.981852i \(0.560736\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −23.6588 −0.805355 −0.402678 0.915342i \(-0.631920\pi\)
−0.402678 + 0.915342i \(0.631920\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −21.4294 −0.727781
\(868\) 0 0
\(869\) −12.7446 −0.432330
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −13.8564 −0.468968
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 25.4891 0.859727
\(880\) 0 0
\(881\) 21.8614 0.736530 0.368265 0.929721i \(-0.379952\pi\)
0.368265 + 0.929721i \(0.379952\pi\)
\(882\) 0 0
\(883\) −24.2487 −0.816034 −0.408017 0.912974i \(-0.633780\pi\)
−0.408017 + 0.912974i \(0.633780\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.1514 0.475156 0.237578 0.971368i \(-0.423646\pi\)
0.237578 + 0.971368i \(0.423646\pi\)
\(888\) 0 0
\(889\) 40.4674 1.35723
\(890\) 0 0
\(891\) −7.74456 −0.259453
\(892\) 0 0
\(893\) 26.5330 0.887893
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −6.51087 −0.217150
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 0 0
\(903\) −30.2921 −1.00806
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −19.8997 −0.660760 −0.330380 0.943848i \(-0.607177\pi\)
−0.330380 + 0.943848i \(0.607177\pi\)
\(908\) 0 0
\(909\) 20.2337 0.671109
\(910\) 0 0
\(911\) 30.5109 1.01087 0.505435 0.862865i \(-0.331332\pi\)
0.505435 + 0.862865i \(0.331332\pi\)
\(912\) 0 0
\(913\) 6.63325 0.219529
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 30.2921 1.00033
\(918\) 0 0
\(919\) −6.23369 −0.205630 −0.102815 0.994700i \(-0.532785\pi\)
−0.102815 + 0.994700i \(0.532785\pi\)
\(920\) 0 0
\(921\) −70.9783 −2.33881
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 35.0458 1.15105
\(928\) 0 0
\(929\) −52.9783 −1.73816 −0.869080 0.494672i \(-0.835288\pi\)
−0.869080 + 0.494672i \(0.835288\pi\)
\(930\) 0 0
\(931\) −20.0000 −0.655474
\(932\) 0 0
\(933\) −44.1485 −1.44536
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −25.9431 −0.847524 −0.423762 0.905774i \(-0.639291\pi\)
−0.423762 + 0.905774i \(0.639291\pi\)
\(938\) 0 0
\(939\) 80.3288 2.62143
\(940\) 0 0
\(941\) 10.4674 0.341227 0.170613 0.985338i \(-0.445425\pi\)
0.170613 + 0.985338i \(0.445425\pi\)
\(942\) 0 0
\(943\) −6.92820 −0.225613
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −56.1802 −1.82561 −0.912806 0.408393i \(-0.866089\pi\)
−0.912806 + 0.408393i \(0.866089\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 8.88316 0.288056
\(952\) 0 0
\(953\) −41.6790 −1.35012 −0.675058 0.737765i \(-0.735881\pi\)
−0.675058 + 0.737765i \(0.735881\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.92820 0.223957
\(958\) 0 0
\(959\) 7.72281 0.249383
\(960\) 0 0
\(961\) −25.3723 −0.818461
\(962\) 0 0
\(963\) −22.3692 −0.720837
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 46.3229 1.48965 0.744823 0.667262i \(-0.232534\pi\)
0.744823 + 0.667262i \(0.232534\pi\)
\(968\) 0 0
\(969\) −50.9783 −1.63766
\(970\) 0 0
\(971\) −54.0951 −1.73599 −0.867997 0.496569i \(-0.834593\pi\)
−0.867997 + 0.496569i \(0.834593\pi\)
\(972\) 0 0
\(973\) 63.1633 2.02492
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.3630 −0.875419 −0.437709 0.899117i \(-0.644210\pi\)
−0.437709 + 0.899117i \(0.644210\pi\)
\(978\) 0 0
\(979\) −4.37228 −0.139739
\(980\) 0 0
\(981\) 33.7228 1.07669
\(982\) 0 0
\(983\) 8.16292 0.260357 0.130178 0.991491i \(-0.458445\pi\)
0.130178 + 0.991491i \(0.458445\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −58.0049 −1.84632
\(988\) 0 0
\(989\) −8.74456 −0.278061
\(990\) 0 0
\(991\) 26.9783 0.856992 0.428496 0.903544i \(-0.359044\pi\)
0.428496 + 0.903544i \(0.359044\pi\)
\(992\) 0 0
\(993\) 7.86797 0.249682
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 22.0742 0.699098 0.349549 0.936918i \(-0.386335\pi\)
0.349549 + 0.936918i \(0.386335\pi\)
\(998\) 0 0
\(999\) 10.3723 0.328164
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 4400.2.a.cc.1.1 4
4.3 odd 2 275.2.a.h.1.3 4
5.2 odd 4 880.2.b.h.529.4 4
5.3 odd 4 880.2.b.h.529.1 4
5.4 even 2 inner 4400.2.a.cc.1.4 4
12.11 even 2 2475.2.a.bi.1.2 4
20.3 even 4 55.2.b.a.34.2 4
20.7 even 4 55.2.b.a.34.3 yes 4
20.19 odd 2 275.2.a.h.1.2 4
44.43 even 2 3025.2.a.ba.1.2 4
60.23 odd 4 495.2.c.a.199.3 4
60.47 odd 4 495.2.c.a.199.2 4
60.59 even 2 2475.2.a.bi.1.3 4
220.3 even 20 605.2.j.i.9.2 16
220.7 odd 20 605.2.j.j.269.3 16
220.27 even 20 605.2.j.i.124.2 16
220.43 odd 4 605.2.b.c.364.3 4
220.47 even 20 605.2.j.i.9.3 16
220.63 odd 20 605.2.j.j.9.3 16
220.83 odd 20 605.2.j.j.124.2 16
220.87 odd 4 605.2.b.c.364.2 4
220.103 even 20 605.2.j.i.269.3 16
220.107 odd 20 605.2.j.j.9.2 16
220.123 odd 20 605.2.j.j.444.3 16
220.127 odd 20 605.2.j.j.124.3 16
220.147 even 20 605.2.j.i.269.2 16
220.163 even 20 605.2.j.i.444.2 16
220.167 odd 20 605.2.j.j.444.2 16
220.183 odd 20 605.2.j.j.269.2 16
220.203 even 20 605.2.j.i.124.3 16
220.207 even 20 605.2.j.i.444.3 16
220.219 even 2 3025.2.a.ba.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.b.a.34.2 4 20.3 even 4
55.2.b.a.34.3 yes 4 20.7 even 4
275.2.a.h.1.2 4 20.19 odd 2
275.2.a.h.1.3 4 4.3 odd 2
495.2.c.a.199.2 4 60.47 odd 4
495.2.c.a.199.3 4 60.23 odd 4
605.2.b.c.364.2 4 220.87 odd 4
605.2.b.c.364.3 4 220.43 odd 4
605.2.j.i.9.2 16 220.3 even 20
605.2.j.i.9.3 16 220.47 even 20
605.2.j.i.124.2 16 220.27 even 20
605.2.j.i.124.3 16 220.203 even 20
605.2.j.i.269.2 16 220.147 even 20
605.2.j.i.269.3 16 220.103 even 20
605.2.j.i.444.2 16 220.163 even 20
605.2.j.i.444.3 16 220.207 even 20
605.2.j.j.9.2 16 220.107 odd 20
605.2.j.j.9.3 16 220.63 odd 20
605.2.j.j.124.2 16 220.83 odd 20
605.2.j.j.124.3 16 220.127 odd 20
605.2.j.j.269.2 16 220.183 odd 20
605.2.j.j.269.3 16 220.7 odd 20
605.2.j.j.444.2 16 220.167 odd 20
605.2.j.j.444.3 16 220.123 odd 20
880.2.b.h.529.1 4 5.3 odd 4
880.2.b.h.529.4 4 5.2 odd 4
2475.2.a.bi.1.2 4 12.11 even 2
2475.2.a.bi.1.3 4 60.59 even 2
3025.2.a.ba.1.2 4 44.43 even 2
3025.2.a.ba.1.3 4 220.219 even 2
4400.2.a.cc.1.1 4 1.1 even 1 trivial
4400.2.a.cc.1.4 4 5.4 even 2 inner