Properties

Label 9680.2.a.ct.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.2
Root \(-0.477260\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.294963 q^{3} +1.00000 q^{5} -4.39026 q^{7} -2.91300 q^{9} +O(q^{10})\) \(q-0.294963 q^{3} +1.00000 q^{5} -4.39026 q^{7} -2.91300 q^{9} -5.96281 q^{13} -0.294963 q^{15} +1.66785 q^{17} +7.69351 q^{19} +1.29496 q^{21} +0.904706 q^{23} +1.00000 q^{25} +1.74411 q^{27} +4.73899 q^{29} +5.12608 q^{31} -4.39026 q^{35} -0.184976 q^{37} +1.75881 q^{39} +2.62237 q^{41} +3.59822 q^{43} -2.91300 q^{45} +0.776180 q^{47} +12.2744 q^{49} -0.491953 q^{51} -9.59554 q^{53} -2.26930 q^{57} +11.2538 q^{59} -13.1898 q^{61} +12.7888 q^{63} -5.96281 q^{65} +7.79954 q^{67} -0.266855 q^{69} +6.97072 q^{71} -12.7541 q^{73} -0.294963 q^{75} -10.0313 q^{79} +8.22454 q^{81} +4.09529 q^{83} +1.66785 q^{85} -1.39783 q^{87} -0.466291 q^{89} +26.1783 q^{91} -1.51200 q^{93} +7.69351 q^{95} +7.09963 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} - 3 q^{13} + 2 q^{15} - 11 q^{17} + 2 q^{19} + 2 q^{21} + 11 q^{23} + 4 q^{25} - q^{27} - 4 q^{29} + 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} - 13 q^{41} - 7 q^{43} - 4 q^{45} + q^{47} - 5 q^{49} - q^{51} - 15 q^{53} - 17 q^{57} + 17 q^{59} - 4 q^{61} + 15 q^{63} - 3 q^{65} - 7 q^{67} + 4 q^{69} + 15 q^{71} - 7 q^{73} + 2 q^{75} - 12 q^{79} - 8 q^{81} + 9 q^{83} - 11 q^{85} - 23 q^{87} - 12 q^{89} + 24 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97}+O(q^{100}) \) 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
\(3\) −0.294963 −0.170297 −0.0851485 0.996368i \(-0.527136\pi\)
−0.0851485 + 0.996368i \(0.527136\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.39026 −1.65936 −0.829681 0.558238i \(-0.811477\pi\)
−0.829681 + 0.558238i \(0.811477\pi\)
\(8\) 0 0
\(9\) −2.91300 −0.970999
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.96281 −1.65379 −0.826893 0.562359i \(-0.809894\pi\)
−0.826893 + 0.562359i \(0.809894\pi\)
\(14\) 0 0
\(15\) −0.294963 −0.0761591
\(16\) 0 0
\(17\) 1.66785 0.404513 0.202256 0.979333i \(-0.435173\pi\)
0.202256 + 0.979333i \(0.435173\pi\)
\(18\) 0 0
\(19\) 7.69351 1.76501 0.882506 0.470301i \(-0.155855\pi\)
0.882506 + 0.470301i \(0.155855\pi\)
\(20\) 0 0
\(21\) 1.29496 0.282584
\(22\) 0 0
\(23\) 0.904706 0.188644 0.0943221 0.995542i \(-0.469932\pi\)
0.0943221 + 0.995542i \(0.469932\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.74411 0.335655
\(28\) 0 0
\(29\) 4.73899 0.880008 0.440004 0.897996i \(-0.354977\pi\)
0.440004 + 0.897996i \(0.354977\pi\)
\(30\) 0 0
\(31\) 5.12608 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.39026 −0.742089
\(36\) 0 0
\(37\) −0.184976 −0.0304098 −0.0152049 0.999884i \(-0.504840\pi\)
−0.0152049 + 0.999884i \(0.504840\pi\)
\(38\) 0 0
\(39\) 1.75881 0.281635
\(40\) 0 0
\(41\) 2.62237 0.409545 0.204773 0.978810i \(-0.434355\pi\)
0.204773 + 0.978810i \(0.434355\pi\)
\(42\) 0 0
\(43\) 3.59822 0.548723 0.274361 0.961627i \(-0.411534\pi\)
0.274361 + 0.961627i \(0.411534\pi\)
\(44\) 0 0
\(45\) −2.91300 −0.434244
\(46\) 0 0
\(47\) 0.776180 0.113217 0.0566087 0.998396i \(-0.481971\pi\)
0.0566087 + 0.998396i \(0.481971\pi\)
\(48\) 0 0
\(49\) 12.2744 1.75348
\(50\) 0 0
\(51\) −0.491953 −0.0688872
\(52\) 0 0
\(53\) −9.59554 −1.31805 −0.659024 0.752122i \(-0.729031\pi\)
−0.659024 + 0.752122i \(0.729031\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.26930 −0.300576
\(58\) 0 0
\(59\) 11.2538 1.46512 0.732561 0.680701i \(-0.238325\pi\)
0.732561 + 0.680701i \(0.238325\pi\)
\(60\) 0 0
\(61\) −13.1898 −1.68878 −0.844390 0.535729i \(-0.820037\pi\)
−0.844390 + 0.535729i \(0.820037\pi\)
\(62\) 0 0
\(63\) 12.7888 1.61124
\(64\) 0 0
\(65\) −5.96281 −0.739596
\(66\) 0 0
\(67\) 7.79954 0.952866 0.476433 0.879211i \(-0.341930\pi\)
0.476433 + 0.879211i \(0.341930\pi\)
\(68\) 0 0
\(69\) −0.266855 −0.0321255
\(70\) 0 0
\(71\) 6.97072 0.827273 0.413636 0.910442i \(-0.364258\pi\)
0.413636 + 0.910442i \(0.364258\pi\)
\(72\) 0 0
\(73\) −12.7541 −1.49275 −0.746375 0.665526i \(-0.768207\pi\)
−0.746375 + 0.665526i \(0.768207\pi\)
\(74\) 0 0
\(75\) −0.294963 −0.0340594
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0313 −1.12861 −0.564303 0.825568i \(-0.690855\pi\)
−0.564303 + 0.825568i \(0.690855\pi\)
\(80\) 0 0
\(81\) 8.22454 0.913838
\(82\) 0 0
\(83\) 4.09529 0.449517 0.224758 0.974415i \(-0.427841\pi\)
0.224758 + 0.974415i \(0.427841\pi\)
\(84\) 0 0
\(85\) 1.66785 0.180904
\(86\) 0 0
\(87\) −1.39783 −0.149863
\(88\) 0 0
\(89\) −0.466291 −0.0494267 −0.0247134 0.999695i \(-0.507867\pi\)
−0.0247134 + 0.999695i \(0.507867\pi\)
\(90\) 0 0
\(91\) 26.1783 2.74423
\(92\) 0 0
\(93\) −1.51200 −0.156787
\(94\) 0 0
\(95\) 7.69351 0.789338
\(96\) 0 0
\(97\) 7.09963 0.720858 0.360429 0.932787i \(-0.382630\pi\)
0.360429 + 0.932787i \(0.382630\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.2851 1.62043 0.810214 0.586135i \(-0.199351\pi\)
0.810214 + 0.586135i \(0.199351\pi\)
\(102\) 0 0
\(103\) −3.19059 −0.314378 −0.157189 0.987569i \(-0.550243\pi\)
−0.157189 + 0.987569i \(0.550243\pi\)
\(104\) 0 0
\(105\) 1.29496 0.126375
\(106\) 0 0
\(107\) −13.1644 −1.27265 −0.636324 0.771422i \(-0.719546\pi\)
−0.636324 + 0.771422i \(0.719546\pi\)
\(108\) 0 0
\(109\) −1.33532 −0.127900 −0.0639502 0.997953i \(-0.520370\pi\)
−0.0639502 + 0.997953i \(0.520370\pi\)
\(110\) 0 0
\(111\) 0.0545610 0.00517870
\(112\) 0 0
\(113\) −1.67448 −0.157522 −0.0787611 0.996894i \(-0.525096\pi\)
−0.0787611 + 0.996894i \(0.525096\pi\)
\(114\) 0 0
\(115\) 0.904706 0.0843643
\(116\) 0 0
\(117\) 17.3696 1.60582
\(118\) 0 0
\(119\) −7.32228 −0.671232
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −0.773501 −0.0697443
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.901470 0.0799926 0.0399963 0.999200i \(-0.487265\pi\)
0.0399963 + 0.999200i \(0.487265\pi\)
\(128\) 0 0
\(129\) −1.06134 −0.0934458
\(130\) 0 0
\(131\) −14.5527 −1.27148 −0.635739 0.771904i \(-0.719305\pi\)
−0.635739 + 0.771904i \(0.719305\pi\)
\(132\) 0 0
\(133\) −33.7765 −2.92879
\(134\) 0 0
\(135\) 1.74411 0.150109
\(136\) 0 0
\(137\) −20.5895 −1.75908 −0.879540 0.475824i \(-0.842150\pi\)
−0.879540 + 0.475824i \(0.842150\pi\)
\(138\) 0 0
\(139\) −14.9841 −1.27094 −0.635469 0.772126i \(-0.719193\pi\)
−0.635469 + 0.772126i \(0.719193\pi\)
\(140\) 0 0
\(141\) −0.228944 −0.0192806
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.73899 0.393552
\(146\) 0 0
\(147\) −3.62048 −0.298612
\(148\) 0 0
\(149\) −10.9626 −0.898089 −0.449045 0.893509i \(-0.648236\pi\)
−0.449045 + 0.893509i \(0.648236\pi\)
\(150\) 0 0
\(151\) −5.91172 −0.481089 −0.240544 0.970638i \(-0.577326\pi\)
−0.240544 + 0.970638i \(0.577326\pi\)
\(152\) 0 0
\(153\) −4.85844 −0.392781
\(154\) 0 0
\(155\) 5.12608 0.411737
\(156\) 0 0
\(157\) −4.42628 −0.353256 −0.176628 0.984278i \(-0.556519\pi\)
−0.176628 + 0.984278i \(0.556519\pi\)
\(158\) 0 0
\(159\) 2.83033 0.224460
\(160\) 0 0
\(161\) −3.97189 −0.313029
\(162\) 0 0
\(163\) 7.38464 0.578410 0.289205 0.957267i \(-0.406609\pi\)
0.289205 + 0.957267i \(0.406609\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.4876 −1.50799 −0.753996 0.656879i \(-0.771876\pi\)
−0.753996 + 0.656879i \(0.771876\pi\)
\(168\) 0 0
\(169\) 22.5551 1.73501
\(170\) 0 0
\(171\) −22.4112 −1.71383
\(172\) 0 0
\(173\) −9.50805 −0.722883 −0.361442 0.932395i \(-0.617715\pi\)
−0.361442 + 0.932395i \(0.617715\pi\)
\(174\) 0 0
\(175\) −4.39026 −0.331872
\(176\) 0 0
\(177\) −3.31946 −0.249506
\(178\) 0 0
\(179\) −7.21253 −0.539090 −0.269545 0.962988i \(-0.586873\pi\)
−0.269545 + 0.962988i \(0.586873\pi\)
\(180\) 0 0
\(181\) −23.3895 −1.73853 −0.869263 0.494350i \(-0.835406\pi\)
−0.869263 + 0.494350i \(0.835406\pi\)
\(182\) 0 0
\(183\) 3.89050 0.287594
\(184\) 0 0
\(185\) −0.184976 −0.0135997
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −7.65711 −0.556973
\(190\) 0 0
\(191\) 4.16572 0.301421 0.150710 0.988578i \(-0.451844\pi\)
0.150710 + 0.988578i \(0.451844\pi\)
\(192\) 0 0
\(193\) 23.1660 1.66753 0.833763 0.552122i \(-0.186182\pi\)
0.833763 + 0.552122i \(0.186182\pi\)
\(194\) 0 0
\(195\) 1.75881 0.125951
\(196\) 0 0
\(197\) −10.1507 −0.723209 −0.361604 0.932332i \(-0.617771\pi\)
−0.361604 + 0.932332i \(0.617771\pi\)
\(198\) 0 0
\(199\) 17.0131 1.20603 0.603014 0.797731i \(-0.293966\pi\)
0.603014 + 0.797731i \(0.293966\pi\)
\(200\) 0 0
\(201\) −2.30058 −0.162270
\(202\) 0 0
\(203\) −20.8054 −1.46025
\(204\) 0 0
\(205\) 2.62237 0.183154
\(206\) 0 0
\(207\) −2.63541 −0.183173
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −6.01620 −0.414173 −0.207086 0.978323i \(-0.566398\pi\)
−0.207086 + 0.978323i \(0.566398\pi\)
\(212\) 0 0
\(213\) −2.05611 −0.140882
\(214\) 0 0
\(215\) 3.59822 0.245396
\(216\) 0 0
\(217\) −22.5048 −1.52773
\(218\) 0 0
\(219\) 3.76197 0.254211
\(220\) 0 0
\(221\) −9.94506 −0.668977
\(222\) 0 0
\(223\) −2.49407 −0.167016 −0.0835078 0.996507i \(-0.526612\pi\)
−0.0835078 + 0.996507i \(0.526612\pi\)
\(224\) 0 0
\(225\) −2.91300 −0.194200
\(226\) 0 0
\(227\) −24.9953 −1.65899 −0.829497 0.558512i \(-0.811373\pi\)
−0.829497 + 0.558512i \(0.811373\pi\)
\(228\) 0 0
\(229\) −2.77082 −0.183101 −0.0915506 0.995800i \(-0.529182\pi\)
−0.0915506 + 0.995800i \(0.529182\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.60025 0.628933 0.314467 0.949269i \(-0.398174\pi\)
0.314467 + 0.949269i \(0.398174\pi\)
\(234\) 0 0
\(235\) 0.776180 0.0506324
\(236\) 0 0
\(237\) 2.95885 0.192198
\(238\) 0 0
\(239\) 14.7692 0.955341 0.477671 0.878539i \(-0.341481\pi\)
0.477671 + 0.878539i \(0.341481\pi\)
\(240\) 0 0
\(241\) 10.4338 0.672099 0.336049 0.941844i \(-0.390909\pi\)
0.336049 + 0.941844i \(0.390909\pi\)
\(242\) 0 0
\(243\) −7.65828 −0.491279
\(244\) 0 0
\(245\) 12.2744 0.784180
\(246\) 0 0
\(247\) −45.8749 −2.91895
\(248\) 0 0
\(249\) −1.20796 −0.0765513
\(250\) 0 0
\(251\) 6.73394 0.425042 0.212521 0.977156i \(-0.431833\pi\)
0.212521 + 0.977156i \(0.431833\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −0.491953 −0.0308073
\(256\) 0 0
\(257\) 7.74192 0.482928 0.241464 0.970410i \(-0.422372\pi\)
0.241464 + 0.970410i \(0.422372\pi\)
\(258\) 0 0
\(259\) 0.812091 0.0504609
\(260\) 0 0
\(261\) −13.8047 −0.854487
\(262\) 0 0
\(263\) 18.1618 1.11990 0.559952 0.828525i \(-0.310819\pi\)
0.559952 + 0.828525i \(0.310819\pi\)
\(264\) 0 0
\(265\) −9.59554 −0.589449
\(266\) 0 0
\(267\) 0.137538 0.00841721
\(268\) 0 0
\(269\) −10.7267 −0.654021 −0.327011 0.945021i \(-0.606041\pi\)
−0.327011 + 0.945021i \(0.606041\pi\)
\(270\) 0 0
\(271\) −18.7875 −1.14126 −0.570630 0.821207i \(-0.693301\pi\)
−0.570630 + 0.821207i \(0.693301\pi\)
\(272\) 0 0
\(273\) −7.72162 −0.467334
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −28.8467 −1.73323 −0.866614 0.498978i \(-0.833709\pi\)
−0.866614 + 0.498978i \(0.833709\pi\)
\(278\) 0 0
\(279\) −14.9323 −0.893971
\(280\) 0 0
\(281\) −27.0542 −1.61392 −0.806959 0.590608i \(-0.798888\pi\)
−0.806959 + 0.590608i \(0.798888\pi\)
\(282\) 0 0
\(283\) 12.8731 0.765228 0.382614 0.923908i \(-0.375024\pi\)
0.382614 + 0.923908i \(0.375024\pi\)
\(284\) 0 0
\(285\) −2.26930 −0.134422
\(286\) 0 0
\(287\) −11.5129 −0.679583
\(288\) 0 0
\(289\) −14.2183 −0.836370
\(290\) 0 0
\(291\) −2.09413 −0.122760
\(292\) 0 0
\(293\) −5.88873 −0.344024 −0.172012 0.985095i \(-0.555027\pi\)
−0.172012 + 0.985095i \(0.555027\pi\)
\(294\) 0 0
\(295\) 11.2538 0.655223
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.39459 −0.311977
\(300\) 0 0
\(301\) −15.7971 −0.910529
\(302\) 0 0
\(303\) −4.80350 −0.275954
\(304\) 0 0
\(305\) −13.1898 −0.755246
\(306\) 0 0
\(307\) 6.85844 0.391432 0.195716 0.980661i \(-0.437297\pi\)
0.195716 + 0.980661i \(0.437297\pi\)
\(308\) 0 0
\(309\) 0.941105 0.0535376
\(310\) 0 0
\(311\) 14.2693 0.809138 0.404569 0.914508i \(-0.367422\pi\)
0.404569 + 0.914508i \(0.367422\pi\)
\(312\) 0 0
\(313\) 1.59973 0.0904220 0.0452110 0.998977i \(-0.485604\pi\)
0.0452110 + 0.998977i \(0.485604\pi\)
\(314\) 0 0
\(315\) 12.7888 0.720568
\(316\) 0 0
\(317\) −30.8556 −1.73302 −0.866511 0.499158i \(-0.833643\pi\)
−0.866511 + 0.499158i \(0.833643\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.88300 0.216728
\(322\) 0 0
\(323\) 12.8316 0.713970
\(324\) 0 0
\(325\) −5.96281 −0.330757
\(326\) 0 0
\(327\) 0.393870 0.0217810
\(328\) 0 0
\(329\) −3.40763 −0.187869
\(330\) 0 0
\(331\) 1.16816 0.0642079 0.0321040 0.999485i \(-0.489779\pi\)
0.0321040 + 0.999485i \(0.489779\pi\)
\(332\) 0 0
\(333\) 0.538833 0.0295279
\(334\) 0 0
\(335\) 7.79954 0.426134
\(336\) 0 0
\(337\) −14.0291 −0.764212 −0.382106 0.924119i \(-0.624801\pi\)
−0.382106 + 0.924119i \(0.624801\pi\)
\(338\) 0 0
\(339\) 0.493910 0.0268255
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −23.1558 −1.25029
\(344\) 0 0
\(345\) −0.266855 −0.0143670
\(346\) 0 0
\(347\) 33.3514 1.79040 0.895198 0.445669i \(-0.147034\pi\)
0.895198 + 0.445669i \(0.147034\pi\)
\(348\) 0 0
\(349\) 35.3268 1.89100 0.945500 0.325623i \(-0.105574\pi\)
0.945500 + 0.325623i \(0.105574\pi\)
\(350\) 0 0
\(351\) −10.3998 −0.555102
\(352\) 0 0
\(353\) 11.7558 0.625698 0.312849 0.949803i \(-0.398717\pi\)
0.312849 + 0.949803i \(0.398717\pi\)
\(354\) 0 0
\(355\) 6.97072 0.369968
\(356\) 0 0
\(357\) 2.15980 0.114309
\(358\) 0 0
\(359\) −2.72101 −0.143609 −0.0718047 0.997419i \(-0.522876\pi\)
−0.0718047 + 0.997419i \(0.522876\pi\)
\(360\) 0 0
\(361\) 40.1901 2.11527
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.7541 −0.667578
\(366\) 0 0
\(367\) −3.10358 −0.162006 −0.0810029 0.996714i \(-0.525812\pi\)
−0.0810029 + 0.996714i \(0.525812\pi\)
\(368\) 0 0
\(369\) −7.63895 −0.397668
\(370\) 0 0
\(371\) 42.1269 2.18712
\(372\) 0 0
\(373\) 27.4604 1.42185 0.710924 0.703269i \(-0.248277\pi\)
0.710924 + 0.703269i \(0.248277\pi\)
\(374\) 0 0
\(375\) −0.294963 −0.0152318
\(376\) 0 0
\(377\) −28.2577 −1.45535
\(378\) 0 0
\(379\) −13.9435 −0.716229 −0.358114 0.933678i \(-0.616580\pi\)
−0.358114 + 0.933678i \(0.616580\pi\)
\(380\) 0 0
\(381\) −0.265900 −0.0136225
\(382\) 0 0
\(383\) −8.15608 −0.416756 −0.208378 0.978048i \(-0.566818\pi\)
−0.208378 + 0.978048i \(0.566818\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −10.4816 −0.532809
\(388\) 0 0
\(389\) −3.10121 −0.157237 −0.0786187 0.996905i \(-0.525051\pi\)
−0.0786187 + 0.996905i \(0.525051\pi\)
\(390\) 0 0
\(391\) 1.50891 0.0763090
\(392\) 0 0
\(393\) 4.29252 0.216529
\(394\) 0 0
\(395\) −10.0313 −0.504728
\(396\) 0 0
\(397\) 23.1546 1.16209 0.581047 0.813870i \(-0.302643\pi\)
0.581047 + 0.813870i \(0.302643\pi\)
\(398\) 0 0
\(399\) 9.96281 0.498764
\(400\) 0 0
\(401\) −8.24375 −0.411673 −0.205837 0.978586i \(-0.565992\pi\)
−0.205837 + 0.978586i \(0.565992\pi\)
\(402\) 0 0
\(403\) −30.5658 −1.52259
\(404\) 0 0
\(405\) 8.22454 0.408681
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2.18649 −0.108115 −0.0540574 0.998538i \(-0.517215\pi\)
−0.0540574 + 0.998538i \(0.517215\pi\)
\(410\) 0 0
\(411\) 6.07314 0.299566
\(412\) 0 0
\(413\) −49.4071 −2.43117
\(414\) 0 0
\(415\) 4.09529 0.201030
\(416\) 0 0
\(417\) 4.41977 0.216437
\(418\) 0 0
\(419\) −10.7348 −0.524429 −0.262215 0.965010i \(-0.584453\pi\)
−0.262215 + 0.965010i \(0.584453\pi\)
\(420\) 0 0
\(421\) 18.7885 0.915696 0.457848 0.889031i \(-0.348620\pi\)
0.457848 + 0.889031i \(0.348620\pi\)
\(422\) 0 0
\(423\) −2.26101 −0.109934
\(424\) 0 0
\(425\) 1.66785 0.0809025
\(426\) 0 0
\(427\) 57.9066 2.80230
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.97878 −0.432493 −0.216246 0.976339i \(-0.569381\pi\)
−0.216246 + 0.976339i \(0.569381\pi\)
\(432\) 0 0
\(433\) 31.5059 1.51408 0.757039 0.653370i \(-0.226645\pi\)
0.757039 + 0.653370i \(0.226645\pi\)
\(434\) 0 0
\(435\) −1.39783 −0.0670206
\(436\) 0 0
\(437\) 6.96037 0.332959
\(438\) 0 0
\(439\) 27.4750 1.31131 0.655655 0.755060i \(-0.272393\pi\)
0.655655 + 0.755060i \(0.272393\pi\)
\(440\) 0 0
\(441\) −35.7552 −1.70263
\(442\) 0 0
\(443\) 30.1960 1.43466 0.717328 0.696736i \(-0.245365\pi\)
0.717328 + 0.696736i \(0.245365\pi\)
\(444\) 0 0
\(445\) −0.466291 −0.0221043
\(446\) 0 0
\(447\) 3.23355 0.152942
\(448\) 0 0
\(449\) −27.3820 −1.29224 −0.646118 0.763237i \(-0.723609\pi\)
−0.646118 + 0.763237i \(0.723609\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.74374 0.0819279
\(454\) 0 0
\(455\) 26.1783 1.22726
\(456\) 0 0
\(457\) −14.2884 −0.668385 −0.334193 0.942505i \(-0.608464\pi\)
−0.334193 + 0.942505i \(0.608464\pi\)
\(458\) 0 0
\(459\) 2.90892 0.135777
\(460\) 0 0
\(461\) −8.78738 −0.409269 −0.204635 0.978838i \(-0.565601\pi\)
−0.204635 + 0.978838i \(0.565601\pi\)
\(462\) 0 0
\(463\) 13.8394 0.643172 0.321586 0.946880i \(-0.395784\pi\)
0.321586 + 0.946880i \(0.395784\pi\)
\(464\) 0 0
\(465\) −1.51200 −0.0701175
\(466\) 0 0
\(467\) −24.8893 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(468\) 0 0
\(469\) −34.2420 −1.58115
\(470\) 0 0
\(471\) 1.30559 0.0601583
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.69351 0.353002
\(476\) 0 0
\(477\) 27.9518 1.27982
\(478\) 0 0
\(479\) −28.5906 −1.30634 −0.653169 0.757212i \(-0.726561\pi\)
−0.653169 + 0.757212i \(0.726561\pi\)
\(480\) 0 0
\(481\) 1.10297 0.0502913
\(482\) 0 0
\(483\) 1.17156 0.0533079
\(484\) 0 0
\(485\) 7.09963 0.322377
\(486\) 0 0
\(487\) −4.65371 −0.210880 −0.105440 0.994426i \(-0.533625\pi\)
−0.105440 + 0.994426i \(0.533625\pi\)
\(488\) 0 0
\(489\) −2.17820 −0.0985014
\(490\) 0 0
\(491\) 24.9360 1.12535 0.562673 0.826680i \(-0.309773\pi\)
0.562673 + 0.826680i \(0.309773\pi\)
\(492\) 0 0
\(493\) 7.90392 0.355974
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −30.6033 −1.37274
\(498\) 0 0
\(499\) −15.3848 −0.688719 −0.344359 0.938838i \(-0.611904\pi\)
−0.344359 + 0.938838i \(0.611904\pi\)
\(500\) 0 0
\(501\) 5.74810 0.256806
\(502\) 0 0
\(503\) −32.7511 −1.46030 −0.730149 0.683288i \(-0.760549\pi\)
−0.730149 + 0.683288i \(0.760549\pi\)
\(504\) 0 0
\(505\) 16.2851 0.724677
\(506\) 0 0
\(507\) −6.65292 −0.295467
\(508\) 0 0
\(509\) 15.7017 0.695964 0.347982 0.937501i \(-0.386867\pi\)
0.347982 + 0.937501i \(0.386867\pi\)
\(510\) 0 0
\(511\) 55.9936 2.47701
\(512\) 0 0
\(513\) 13.4184 0.592435
\(514\) 0 0
\(515\) −3.19059 −0.140594
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.80452 0.123105
\(520\) 0 0
\(521\) −33.6243 −1.47311 −0.736553 0.676380i \(-0.763548\pi\)
−0.736553 + 0.676380i \(0.763548\pi\)
\(522\) 0 0
\(523\) −22.9574 −1.00386 −0.501928 0.864910i \(-0.667376\pi\)
−0.501928 + 0.864910i \(0.667376\pi\)
\(524\) 0 0
\(525\) 1.29496 0.0565168
\(526\) 0 0
\(527\) 8.54952 0.372423
\(528\) 0 0
\(529\) −22.1815 −0.964413
\(530\) 0 0
\(531\) −32.7823 −1.42263
\(532\) 0 0
\(533\) −15.6367 −0.677300
\(534\) 0 0
\(535\) −13.1644 −0.569145
\(536\) 0 0
\(537\) 2.12743 0.0918053
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.7777 0.807314 0.403657 0.914910i \(-0.367739\pi\)
0.403657 + 0.914910i \(0.367739\pi\)
\(542\) 0 0
\(543\) 6.89902 0.296065
\(544\) 0 0
\(545\) −1.33532 −0.0571988
\(546\) 0 0
\(547\) 19.2785 0.824291 0.412146 0.911118i \(-0.364780\pi\)
0.412146 + 0.911118i \(0.364780\pi\)
\(548\) 0 0
\(549\) 38.4218 1.63980
\(550\) 0 0
\(551\) 36.4595 1.55323
\(552\) 0 0
\(553\) 44.0399 1.87277
\(554\) 0 0
\(555\) 0.0545610 0.00231598
\(556\) 0 0
\(557\) 17.9380 0.760057 0.380028 0.924975i \(-0.375914\pi\)
0.380028 + 0.924975i \(0.375914\pi\)
\(558\) 0 0
\(559\) −21.4555 −0.907470
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.81288 −0.0764038 −0.0382019 0.999270i \(-0.512163\pi\)
−0.0382019 + 0.999270i \(0.512163\pi\)
\(564\) 0 0
\(565\) −1.67448 −0.0704460
\(566\) 0 0
\(567\) −36.1079 −1.51639
\(568\) 0 0
\(569\) −8.53549 −0.357826 −0.178913 0.983865i \(-0.557258\pi\)
−0.178913 + 0.983865i \(0.557258\pi\)
\(570\) 0 0
\(571\) −32.2491 −1.34958 −0.674792 0.738008i \(-0.735767\pi\)
−0.674792 + 0.738008i \(0.735767\pi\)
\(572\) 0 0
\(573\) −1.22873 −0.0513310
\(574\) 0 0
\(575\) 0.904706 0.0377288
\(576\) 0 0
\(577\) 29.9652 1.24747 0.623734 0.781637i \(-0.285615\pi\)
0.623734 + 0.781637i \(0.285615\pi\)
\(578\) 0 0
\(579\) −6.83312 −0.283975
\(580\) 0 0
\(581\) −17.9794 −0.745911
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 17.3696 0.718147
\(586\) 0 0
\(587\) −9.80339 −0.404629 −0.202315 0.979321i \(-0.564846\pi\)
−0.202315 + 0.979321i \(0.564846\pi\)
\(588\) 0 0
\(589\) 39.4376 1.62500
\(590\) 0 0
\(591\) 2.99409 0.123160
\(592\) 0 0
\(593\) −15.4666 −0.635136 −0.317568 0.948235i \(-0.602866\pi\)
−0.317568 + 0.948235i \(0.602866\pi\)
\(594\) 0 0
\(595\) −7.32228 −0.300184
\(596\) 0 0
\(597\) −5.01824 −0.205383
\(598\) 0 0
\(599\) 0.278191 0.0113666 0.00568328 0.999984i \(-0.498191\pi\)
0.00568328 + 0.999984i \(0.498191\pi\)
\(600\) 0 0
\(601\) −30.0055 −1.22395 −0.611974 0.790878i \(-0.709624\pi\)
−0.611974 + 0.790878i \(0.709624\pi\)
\(602\) 0 0
\(603\) −22.7200 −0.925232
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.2971 −0.905011 −0.452505 0.891762i \(-0.649470\pi\)
−0.452505 + 0.891762i \(0.649470\pi\)
\(608\) 0 0
\(609\) 6.13682 0.248676
\(610\) 0 0
\(611\) −4.62821 −0.187237
\(612\) 0 0
\(613\) 28.3748 1.14605 0.573024 0.819539i \(-0.305770\pi\)
0.573024 + 0.819539i \(0.305770\pi\)
\(614\) 0 0
\(615\) −0.773501 −0.0311906
\(616\) 0 0
\(617\) −15.1603 −0.610332 −0.305166 0.952299i \(-0.598712\pi\)
−0.305166 + 0.952299i \(0.598712\pi\)
\(618\) 0 0
\(619\) 29.5238 1.18666 0.593331 0.804959i \(-0.297813\pi\)
0.593331 + 0.804959i \(0.297813\pi\)
\(620\) 0 0
\(621\) 1.57791 0.0633194
\(622\) 0 0
\(623\) 2.04714 0.0820167
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.308511 −0.0123011
\(630\) 0 0
\(631\) 45.6894 1.81887 0.909433 0.415849i \(-0.136516\pi\)
0.909433 + 0.415849i \(0.136516\pi\)
\(632\) 0 0
\(633\) 1.77456 0.0705323
\(634\) 0 0
\(635\) 0.901470 0.0357738
\(636\) 0 0
\(637\) −73.1897 −2.89988
\(638\) 0 0
\(639\) −20.3057 −0.803281
\(640\) 0 0
\(641\) 36.1151 1.42646 0.713231 0.700929i \(-0.247231\pi\)
0.713231 + 0.700929i \(0.247231\pi\)
\(642\) 0 0
\(643\) 33.4972 1.32100 0.660500 0.750826i \(-0.270344\pi\)
0.660500 + 0.750826i \(0.270344\pi\)
\(644\) 0 0
\(645\) −1.06134 −0.0417902
\(646\) 0 0
\(647\) −39.2066 −1.54137 −0.770686 0.637215i \(-0.780086\pi\)
−0.770686 + 0.637215i \(0.780086\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 6.63808 0.260167
\(652\) 0 0
\(653\) 24.7595 0.968913 0.484456 0.874815i \(-0.339017\pi\)
0.484456 + 0.874815i \(0.339017\pi\)
\(654\) 0 0
\(655\) −14.5527 −0.568622
\(656\) 0 0
\(657\) 37.1525 1.44946
\(658\) 0 0
\(659\) 24.8813 0.969238 0.484619 0.874725i \(-0.338958\pi\)
0.484619 + 0.874725i \(0.338958\pi\)
\(660\) 0 0
\(661\) −5.94396 −0.231193 −0.115597 0.993296i \(-0.536878\pi\)
−0.115597 + 0.993296i \(0.536878\pi\)
\(662\) 0 0
\(663\) 2.93342 0.113925
\(664\) 0 0
\(665\) −33.7765 −1.30980
\(666\) 0 0
\(667\) 4.28739 0.166009
\(668\) 0 0
\(669\) 0.735660 0.0284422
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −37.8500 −1.45901 −0.729505 0.683975i \(-0.760250\pi\)
−0.729505 + 0.683975i \(0.760250\pi\)
\(674\) 0 0
\(675\) 1.74411 0.0671310
\(676\) 0 0
\(677\) −0.190100 −0.00730612 −0.00365306 0.999993i \(-0.501163\pi\)
−0.00365306 + 0.999993i \(0.501163\pi\)
\(678\) 0 0
\(679\) −31.1692 −1.19616
\(680\) 0 0
\(681\) 7.37267 0.282521
\(682\) 0 0
\(683\) −23.0773 −0.883030 −0.441515 0.897254i \(-0.645559\pi\)
−0.441515 + 0.897254i \(0.645559\pi\)
\(684\) 0 0
\(685\) −20.5895 −0.786685
\(686\) 0 0
\(687\) 0.817290 0.0311816
\(688\) 0 0
\(689\) 57.2164 2.17977
\(690\) 0 0
\(691\) −46.7082 −1.77686 −0.888432 0.459008i \(-0.848205\pi\)
−0.888432 + 0.459008i \(0.848205\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14.9841 −0.568381
\(696\) 0 0
\(697\) 4.37371 0.165666
\(698\) 0 0
\(699\) −2.83172 −0.107105
\(700\) 0 0
\(701\) −4.17239 −0.157589 −0.0787946 0.996891i \(-0.525107\pi\)
−0.0787946 + 0.996891i \(0.525107\pi\)
\(702\) 0 0
\(703\) −1.42311 −0.0536737
\(704\) 0 0
\(705\) −0.228944 −0.00862254
\(706\) 0 0
\(707\) −71.4957 −2.68887
\(708\) 0 0
\(709\) −8.74277 −0.328342 −0.164171 0.986432i \(-0.552495\pi\)
−0.164171 + 0.986432i \(0.552495\pi\)
\(710\) 0 0
\(711\) 29.2211 1.09588
\(712\) 0 0
\(713\) 4.63760 0.173679
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.35637 −0.162692
\(718\) 0 0
\(719\) 7.40160 0.276033 0.138017 0.990430i \(-0.455927\pi\)
0.138017 + 0.990430i \(0.455927\pi\)
\(720\) 0 0
\(721\) 14.0075 0.521667
\(722\) 0 0
\(723\) −3.07758 −0.114456
\(724\) 0 0
\(725\) 4.73899 0.176002
\(726\) 0 0
\(727\) −18.0187 −0.668275 −0.334137 0.942524i \(-0.608445\pi\)
−0.334137 + 0.942524i \(0.608445\pi\)
\(728\) 0 0
\(729\) −22.4147 −0.830175
\(730\) 0 0
\(731\) 6.00128 0.221965
\(732\) 0 0
\(733\) 36.3939 1.34424 0.672120 0.740442i \(-0.265384\pi\)
0.672120 + 0.740442i \(0.265384\pi\)
\(734\) 0 0
\(735\) −3.62048 −0.133543
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −12.4319 −0.457313 −0.228657 0.973507i \(-0.573433\pi\)
−0.228657 + 0.973507i \(0.573433\pi\)
\(740\) 0 0
\(741\) 13.5314 0.497089
\(742\) 0 0
\(743\) 37.3696 1.37096 0.685479 0.728092i \(-0.259593\pi\)
0.685479 + 0.728092i \(0.259593\pi\)
\(744\) 0 0
\(745\) −10.9626 −0.401638
\(746\) 0 0
\(747\) −11.9296 −0.436480
\(748\) 0 0
\(749\) 57.7950 2.11178
\(750\) 0 0
\(751\) 7.89958 0.288260 0.144130 0.989559i \(-0.453962\pi\)
0.144130 + 0.989559i \(0.453962\pi\)
\(752\) 0 0
\(753\) −1.98626 −0.0723834
\(754\) 0 0
\(755\) −5.91172 −0.215149
\(756\) 0 0
\(757\) −3.17036 −0.115229 −0.0576143 0.998339i \(-0.518349\pi\)
−0.0576143 + 0.998339i \(0.518349\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −46.7689 −1.69537 −0.847685 0.530499i \(-0.822004\pi\)
−0.847685 + 0.530499i \(0.822004\pi\)
\(762\) 0 0
\(763\) 5.86239 0.212233
\(764\) 0 0
\(765\) −4.85844 −0.175657
\(766\) 0 0
\(767\) −67.1044 −2.42300
\(768\) 0 0
\(769\) 1.95610 0.0705388 0.0352694 0.999378i \(-0.488771\pi\)
0.0352694 + 0.999378i \(0.488771\pi\)
\(770\) 0 0
\(771\) −2.28358 −0.0822411
\(772\) 0 0
\(773\) −20.4428 −0.735277 −0.367639 0.929969i \(-0.619834\pi\)
−0.367639 + 0.929969i \(0.619834\pi\)
\(774\) 0 0
\(775\) 5.12608 0.184134
\(776\) 0 0
\(777\) −0.239537 −0.00859333
\(778\) 0 0
\(779\) 20.1752 0.722852
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 8.26534 0.295379
\(784\) 0 0
\(785\) −4.42628 −0.157981
\(786\) 0 0
\(787\) −2.59196 −0.0923933 −0.0461967 0.998932i \(-0.514710\pi\)
−0.0461967 + 0.998932i \(0.514710\pi\)
\(788\) 0 0
\(789\) −5.35706 −0.190716
\(790\) 0 0
\(791\) 7.35141 0.261386
\(792\) 0 0
\(793\) 78.6483 2.79288
\(794\) 0 0
\(795\) 2.83033 0.100381
\(796\) 0 0
\(797\) 22.3867 0.792977 0.396488 0.918040i \(-0.370229\pi\)
0.396488 + 0.918040i \(0.370229\pi\)
\(798\) 0 0
\(799\) 1.29455 0.0457979
\(800\) 0 0
\(801\) 1.35830 0.0479933
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −3.97189 −0.139991
\(806\) 0 0
\(807\) 3.16399 0.111378
\(808\) 0 0
\(809\) −11.9561 −0.420354 −0.210177 0.977663i \(-0.567404\pi\)
−0.210177 + 0.977663i \(0.567404\pi\)
\(810\) 0 0
\(811\) 18.4696 0.648554 0.324277 0.945962i \(-0.394879\pi\)
0.324277 + 0.945962i \(0.394879\pi\)
\(812\) 0 0
\(813\) 5.54162 0.194353
\(814\) 0 0
\(815\) 7.38464 0.258673
\(816\) 0 0
\(817\) 27.6829 0.968503
\(818\) 0 0
\(819\) −76.2572 −2.66464
\(820\) 0 0
\(821\) −55.4734 −1.93604 −0.968018 0.250881i \(-0.919280\pi\)
−0.968018 + 0.250881i \(0.919280\pi\)
\(822\) 0 0
\(823\) −43.7133 −1.52375 −0.761874 0.647725i \(-0.775721\pi\)
−0.761874 + 0.647725i \(0.775721\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.7396 1.10369 0.551846 0.833946i \(-0.313924\pi\)
0.551846 + 0.833946i \(0.313924\pi\)
\(828\) 0 0
\(829\) −35.0302 −1.21665 −0.608325 0.793688i \(-0.708158\pi\)
−0.608325 + 0.793688i \(0.708158\pi\)
\(830\) 0 0
\(831\) 8.50870 0.295164
\(832\) 0 0
\(833\) 20.4718 0.709304
\(834\) 0 0
\(835\) −19.4876 −0.674394
\(836\) 0 0
\(837\) 8.94047 0.309028
\(838\) 0 0
\(839\) 15.8542 0.547349 0.273674 0.961822i \(-0.411761\pi\)
0.273674 + 0.961822i \(0.411761\pi\)
\(840\) 0 0
\(841\) −6.54197 −0.225585
\(842\) 0 0
\(843\) 7.97998 0.274845
\(844\) 0 0
\(845\) 22.5551 0.775919
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3.79710 −0.130316
\(850\) 0 0
\(851\) −0.167349 −0.00573664
\(852\) 0 0
\(853\) 22.5865 0.773346 0.386673 0.922217i \(-0.373624\pi\)
0.386673 + 0.922217i \(0.373624\pi\)
\(854\) 0 0
\(855\) −22.4112 −0.766446
\(856\) 0 0
\(857\) −40.9108 −1.39749 −0.698744 0.715372i \(-0.746257\pi\)
−0.698744 + 0.715372i \(0.746257\pi\)
\(858\) 0 0
\(859\) −15.9523 −0.544284 −0.272142 0.962257i \(-0.587732\pi\)
−0.272142 + 0.962257i \(0.587732\pi\)
\(860\) 0 0
\(861\) 3.39587 0.115731
\(862\) 0 0
\(863\) −7.28894 −0.248118 −0.124059 0.992275i \(-0.539591\pi\)
−0.124059 + 0.992275i \(0.539591\pi\)
\(864\) 0 0
\(865\) −9.50805 −0.323283
\(866\) 0 0
\(867\) 4.19387 0.142431
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −46.5072 −1.57584
\(872\) 0 0
\(873\) −20.6812 −0.699952
\(874\) 0 0
\(875\) −4.39026 −0.148418
\(876\) 0 0
\(877\) −7.14718 −0.241343 −0.120671 0.992692i \(-0.538505\pi\)
−0.120671 + 0.992692i \(0.538505\pi\)
\(878\) 0 0
\(879\) 1.73696 0.0585861
\(880\) 0 0
\(881\) −17.5763 −0.592160 −0.296080 0.955163i \(-0.595680\pi\)
−0.296080 + 0.955163i \(0.595680\pi\)
\(882\) 0 0
\(883\) 6.26855 0.210954 0.105477 0.994422i \(-0.466363\pi\)
0.105477 + 0.994422i \(0.466363\pi\)
\(884\) 0 0
\(885\) −3.31946 −0.111582
\(886\) 0 0
\(887\) −24.6894 −0.828989 −0.414495 0.910052i \(-0.636042\pi\)
−0.414495 + 0.910052i \(0.636042\pi\)
\(888\) 0 0
\(889\) −3.95769 −0.132737
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.97155 0.199830
\(894\) 0 0
\(895\) −7.21253 −0.241088
\(896\) 0 0
\(897\) 1.59120 0.0531288
\(898\) 0 0
\(899\) 24.2924 0.810199
\(900\) 0 0
\(901\) −16.0039 −0.533167
\(902\) 0 0
\(903\) 4.65956 0.155060
\(904\) 0 0
\(905\) −23.3895 −0.777492
\(906\) 0 0
\(907\) −25.4380 −0.844655 −0.422327 0.906443i \(-0.638787\pi\)
−0.422327 + 0.906443i \(0.638787\pi\)
\(908\) 0 0
\(909\) −47.4384 −1.57343
\(910\) 0 0
\(911\) 24.5296 0.812701 0.406350 0.913717i \(-0.366801\pi\)
0.406350 + 0.913717i \(0.366801\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 3.89050 0.128616
\(916\) 0 0
\(917\) 63.8903 2.10984
\(918\) 0 0
\(919\) 21.2250 0.700148 0.350074 0.936722i \(-0.386156\pi\)
0.350074 + 0.936722i \(0.386156\pi\)
\(920\) 0 0
\(921\) −2.02298 −0.0666596
\(922\) 0 0
\(923\) −41.5651 −1.36813
\(924\) 0 0
\(925\) −0.184976 −0.00608196
\(926\) 0 0
\(927\) 9.29417 0.305261
\(928\) 0 0
\(929\) 34.2535 1.12382 0.561910 0.827198i \(-0.310067\pi\)
0.561910 + 0.827198i \(0.310067\pi\)
\(930\) 0 0
\(931\) 94.4329 3.09491
\(932\) 0 0
\(933\) −4.20891 −0.137794
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −34.9322 −1.14119 −0.570593 0.821233i \(-0.693287\pi\)
−0.570593 + 0.821233i \(0.693287\pi\)
\(938\) 0 0
\(939\) −0.471860 −0.0153986
\(940\) 0 0
\(941\) 18.7014 0.609649 0.304825 0.952409i \(-0.401402\pi\)
0.304825 + 0.952409i \(0.401402\pi\)
\(942\) 0 0
\(943\) 2.37247 0.0772583
\(944\) 0 0
\(945\) −7.65711 −0.249086
\(946\) 0 0
\(947\) −12.7418 −0.414052 −0.207026 0.978335i \(-0.566379\pi\)
−0.207026 + 0.978335i \(0.566379\pi\)
\(948\) 0 0
\(949\) 76.0501 2.46869
\(950\) 0 0
\(951\) 9.10125 0.295128
\(952\) 0 0
\(953\) −5.77686 −0.187131 −0.0935654 0.995613i \(-0.529826\pi\)
−0.0935654 + 0.995613i \(0.529826\pi\)
\(954\) 0 0
\(955\) 4.16572 0.134799
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 90.3933 2.91895
\(960\) 0 0
\(961\) −4.72330 −0.152364
\(962\) 0 0
\(963\) 38.3478 1.23574
\(964\) 0 0
\(965\) 23.1660 0.745741
\(966\) 0 0
\(967\) −36.2900 −1.16701 −0.583504 0.812110i \(-0.698319\pi\)
−0.583504 + 0.812110i \(0.698319\pi\)
\(968\) 0 0
\(969\) −3.78485 −0.121587
\(970\) 0 0
\(971\) 6.98883 0.224282 0.112141 0.993692i \(-0.464229\pi\)
0.112141 + 0.993692i \(0.464229\pi\)
\(972\) 0 0
\(973\) 65.7842 2.10895
\(974\) 0 0
\(975\) 1.75881 0.0563269
\(976\) 0 0
\(977\) 25.9994 0.831795 0.415897 0.909412i \(-0.363468\pi\)
0.415897 + 0.909412i \(0.363468\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.88978 0.124191
\(982\) 0 0
\(983\) −28.6461 −0.913669 −0.456834 0.889552i \(-0.651017\pi\)
−0.456834 + 0.889552i \(0.651017\pi\)
\(984\) 0 0
\(985\) −10.1507 −0.323429
\(986\) 0 0
\(987\) 1.00512 0.0319935
\(988\) 0 0
\(989\) 3.25533 0.103513
\(990\) 0 0
\(991\) −27.9673 −0.888409 −0.444205 0.895925i \(-0.646514\pi\)
−0.444205 + 0.895925i \(0.646514\pi\)
\(992\) 0 0
\(993\) −0.344564 −0.0109344
\(994\) 0 0
\(995\) 17.0131 0.539352
\(996\) 0 0
\(997\) −13.0579 −0.413547 −0.206773 0.978389i \(-0.566296\pi\)
−0.206773 + 0.978389i \(0.566296\pi\)
\(998\) 0 0
\(999\) −0.322619 −0.0102072
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 9680.2.a.ct.1.2 4
4.3 odd 2 4840.2.a.z.1.3 4
11.2 odd 10 880.2.bo.d.81.1 8
11.6 odd 10 880.2.bo.d.641.1 8
11.10 odd 2 9680.2.a.cu.1.2 4
44.35 even 10 440.2.y.a.81.2 8
44.39 even 10 440.2.y.a.201.2 yes 8
44.43 even 2 4840.2.a.y.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.2 8 44.35 even 10
440.2.y.a.201.2 yes 8 44.39 even 10
880.2.bo.d.81.1 8 11.2 odd 10
880.2.bo.d.641.1 8 11.6 odd 10
4840.2.a.y.1.3 4 44.43 even 2
4840.2.a.z.1.3 4 4.3 odd 2
9680.2.a.ct.1.2 4 1.1 even 1 trivial
9680.2.a.cu.1.2 4 11.10 odd 2