Properties

Label 9680.2.a.dc.1.3
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.25903625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) 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 1.3
Root \(-0.220878\) 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)
 
\(f(q)\) \(=\) \(q-0.220878 q^{3} -1.00000 q^{5} +2.08772 q^{7} -2.95121 q^{9} +O(q^{10})\) \(q-0.220878 q^{3} -1.00000 q^{5} +2.08772 q^{7} -2.95121 q^{9} +1.87868 q^{13} +0.220878 q^{15} +5.28462 q^{17} +2.92388 q^{19} -0.461133 q^{21} -5.21521 q^{23} +1.00000 q^{25} +1.31449 q^{27} -3.81157 q^{29} -6.68385 q^{31} -2.08772 q^{35} -3.45754 q^{37} -0.414959 q^{39} +5.80287 q^{41} -0.884108 q^{43} +2.95121 q^{45} -7.69696 q^{47} -2.64141 q^{49} -1.16726 q^{51} -9.50720 q^{53} -0.645820 q^{57} +7.17454 q^{59} -7.88806 q^{61} -6.16132 q^{63} -1.87868 q^{65} +1.32232 q^{67} +1.15193 q^{69} +10.2325 q^{71} +8.17103 q^{73} -0.220878 q^{75} -5.23101 q^{79} +8.56330 q^{81} +9.63433 q^{83} -5.28462 q^{85} +0.841894 q^{87} -17.0929 q^{89} +3.92216 q^{91} +1.47632 q^{93} -2.92388 q^{95} -15.3296 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} - q^{13} - 3 q^{15} - 6 q^{17} - 7 q^{19} - 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} - 10 q^{29} - q^{31} + 7 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} - 9 q^{57} + 2 q^{59} - 6 q^{61} - 49 q^{63} + q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} - 10 q^{73} + 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} + 6 q^{85} - 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100}) \) 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
\(3\) −0.220878 −0.127524 −0.0637621 0.997965i \(-0.520310\pi\)
−0.0637621 + 0.997965i \(0.520310\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.08772 0.789085 0.394543 0.918878i \(-0.370903\pi\)
0.394543 + 0.918878i \(0.370903\pi\)
\(8\) 0 0
\(9\) −2.95121 −0.983738
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 1.87868 0.521052 0.260526 0.965467i \(-0.416104\pi\)
0.260526 + 0.965467i \(0.416104\pi\)
\(14\) 0 0
\(15\) 0.220878 0.0570305
\(16\) 0 0
\(17\) 5.28462 1.28171 0.640854 0.767663i \(-0.278580\pi\)
0.640854 + 0.767663i \(0.278580\pi\)
\(18\) 0 0
\(19\) 2.92388 0.670783 0.335391 0.942079i \(-0.391131\pi\)
0.335391 + 0.942079i \(0.391131\pi\)
\(20\) 0 0
\(21\) −0.461133 −0.100627
\(22\) 0 0
\(23\) −5.21521 −1.08745 −0.543724 0.839264i \(-0.682986\pi\)
−0.543724 + 0.839264i \(0.682986\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.31449 0.252974
\(28\) 0 0
\(29\) −3.81157 −0.707792 −0.353896 0.935285i \(-0.615143\pi\)
−0.353896 + 0.935285i \(0.615143\pi\)
\(30\) 0 0
\(31\) −6.68385 −1.20045 −0.600227 0.799829i \(-0.704923\pi\)
−0.600227 + 0.799829i \(0.704923\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.08772 −0.352890
\(36\) 0 0
\(37\) −3.45754 −0.568416 −0.284208 0.958763i \(-0.591731\pi\)
−0.284208 + 0.958763i \(0.591731\pi\)
\(38\) 0 0
\(39\) −0.414959 −0.0664467
\(40\) 0 0
\(41\) 5.80287 0.906256 0.453128 0.891446i \(-0.350308\pi\)
0.453128 + 0.891446i \(0.350308\pi\)
\(42\) 0 0
\(43\) −0.884108 −0.134825 −0.0674126 0.997725i \(-0.521474\pi\)
−0.0674126 + 0.997725i \(0.521474\pi\)
\(44\) 0 0
\(45\) 2.95121 0.439941
\(46\) 0 0
\(47\) −7.69696 −1.12272 −0.561359 0.827573i \(-0.689721\pi\)
−0.561359 + 0.827573i \(0.689721\pi\)
\(48\) 0 0
\(49\) −2.64141 −0.377345
\(50\) 0 0
\(51\) −1.16726 −0.163449
\(52\) 0 0
\(53\) −9.50720 −1.30591 −0.652957 0.757395i \(-0.726472\pi\)
−0.652957 + 0.757395i \(0.726472\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.645820 −0.0855410
\(58\) 0 0
\(59\) 7.17454 0.934045 0.467023 0.884245i \(-0.345327\pi\)
0.467023 + 0.884245i \(0.345327\pi\)
\(60\) 0 0
\(61\) −7.88806 −1.00996 −0.504981 0.863130i \(-0.668501\pi\)
−0.504981 + 0.863130i \(0.668501\pi\)
\(62\) 0 0
\(63\) −6.16132 −0.776253
\(64\) 0 0
\(65\) −1.87868 −0.233022
\(66\) 0 0
\(67\) 1.32232 0.161547 0.0807733 0.996732i \(-0.474261\pi\)
0.0807733 + 0.996732i \(0.474261\pi\)
\(68\) 0 0
\(69\) 1.15193 0.138676
\(70\) 0 0
\(71\) 10.2325 1.21437 0.607186 0.794560i \(-0.292298\pi\)
0.607186 + 0.794560i \(0.292298\pi\)
\(72\) 0 0
\(73\) 8.17103 0.956346 0.478173 0.878266i \(-0.341299\pi\)
0.478173 + 0.878266i \(0.341299\pi\)
\(74\) 0 0
\(75\) −0.220878 −0.0255048
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.23101 −0.588534 −0.294267 0.955723i \(-0.595076\pi\)
−0.294267 + 0.955723i \(0.595076\pi\)
\(80\) 0 0
\(81\) 8.56330 0.951477
\(82\) 0 0
\(83\) 9.63433 1.05750 0.528752 0.848776i \(-0.322660\pi\)
0.528752 + 0.848776i \(0.322660\pi\)
\(84\) 0 0
\(85\) −5.28462 −0.573197
\(86\) 0 0
\(87\) 0.841894 0.0902605
\(88\) 0 0
\(89\) −17.0929 −1.81185 −0.905923 0.423443i \(-0.860821\pi\)
−0.905923 + 0.423443i \(0.860821\pi\)
\(90\) 0 0
\(91\) 3.92216 0.411154
\(92\) 0 0
\(93\) 1.47632 0.153087
\(94\) 0 0
\(95\) −2.92388 −0.299983
\(96\) 0 0
\(97\) −15.3296 −1.55648 −0.778242 0.627964i \(-0.783888\pi\)
−0.778242 + 0.627964i \(0.783888\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.5788 1.55015 0.775075 0.631869i \(-0.217712\pi\)
0.775075 + 0.631869i \(0.217712\pi\)
\(102\) 0 0
\(103\) 14.5038 1.42910 0.714549 0.699586i \(-0.246632\pi\)
0.714549 + 0.699586i \(0.246632\pi\)
\(104\) 0 0
\(105\) 0.461133 0.0450019
\(106\) 0 0
\(107\) −16.8702 −1.63090 −0.815450 0.578828i \(-0.803510\pi\)
−0.815450 + 0.578828i \(0.803510\pi\)
\(108\) 0 0
\(109\) −9.97408 −0.955344 −0.477672 0.878538i \(-0.658519\pi\)
−0.477672 + 0.878538i \(0.658519\pi\)
\(110\) 0 0
\(111\) 0.763696 0.0724868
\(112\) 0 0
\(113\) 0.00824325 0.000775459 0 0.000387730 1.00000i \(-0.499877\pi\)
0.000387730 1.00000i \(0.499877\pi\)
\(114\) 0 0
\(115\) 5.21521 0.486321
\(116\) 0 0
\(117\) −5.54438 −0.512578
\(118\) 0 0
\(119\) 11.0328 1.01138
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −1.28173 −0.115569
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.0723 0.893769 0.446885 0.894592i \(-0.352533\pi\)
0.446885 + 0.894592i \(0.352533\pi\)
\(128\) 0 0
\(129\) 0.195280 0.0171935
\(130\) 0 0
\(131\) 18.0249 1.57484 0.787422 0.616415i \(-0.211415\pi\)
0.787422 + 0.616415i \(0.211415\pi\)
\(132\) 0 0
\(133\) 6.10424 0.529305
\(134\) 0 0
\(135\) −1.31449 −0.113134
\(136\) 0 0
\(137\) 5.15806 0.440683 0.220341 0.975423i \(-0.429283\pi\)
0.220341 + 0.975423i \(0.429283\pi\)
\(138\) 0 0
\(139\) 1.61222 0.136747 0.0683735 0.997660i \(-0.478219\pi\)
0.0683735 + 0.997660i \(0.478219\pi\)
\(140\) 0 0
\(141\) 1.70009 0.143174
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.81157 0.316534
\(146\) 0 0
\(147\) 0.583431 0.0481205
\(148\) 0 0
\(149\) 3.00244 0.245970 0.122985 0.992409i \(-0.460753\pi\)
0.122985 + 0.992409i \(0.460753\pi\)
\(150\) 0 0
\(151\) −0.831273 −0.0676480 −0.0338240 0.999428i \(-0.510769\pi\)
−0.0338240 + 0.999428i \(0.510769\pi\)
\(152\) 0 0
\(153\) −15.5960 −1.26086
\(154\) 0 0
\(155\) 6.68385 0.536860
\(156\) 0 0
\(157\) −13.3102 −1.06227 −0.531133 0.847288i \(-0.678234\pi\)
−0.531133 + 0.847288i \(0.678234\pi\)
\(158\) 0 0
\(159\) 2.09993 0.166536
\(160\) 0 0
\(161\) −10.8879 −0.858088
\(162\) 0 0
\(163\) 6.36304 0.498392 0.249196 0.968453i \(-0.419834\pi\)
0.249196 + 0.968453i \(0.419834\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.25595 0.174571 0.0872854 0.996183i \(-0.472181\pi\)
0.0872854 + 0.996183i \(0.472181\pi\)
\(168\) 0 0
\(169\) −9.47056 −0.728505
\(170\) 0 0
\(171\) −8.62898 −0.659874
\(172\) 0 0
\(173\) −10.8245 −0.822968 −0.411484 0.911417i \(-0.634989\pi\)
−0.411484 + 0.911417i \(0.634989\pi\)
\(174\) 0 0
\(175\) 2.08772 0.157817
\(176\) 0 0
\(177\) −1.58470 −0.119113
\(178\) 0 0
\(179\) 11.5996 0.866993 0.433496 0.901155i \(-0.357280\pi\)
0.433496 + 0.901155i \(0.357280\pi\)
\(180\) 0 0
\(181\) −3.77485 −0.280582 −0.140291 0.990110i \(-0.544804\pi\)
−0.140291 + 0.990110i \(0.544804\pi\)
\(182\) 0 0
\(183\) 1.74230 0.128795
\(184\) 0 0
\(185\) 3.45754 0.254203
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.74430 0.199618
\(190\) 0 0
\(191\) 21.5286 1.55775 0.778876 0.627178i \(-0.215790\pi\)
0.778876 + 0.627178i \(0.215790\pi\)
\(192\) 0 0
\(193\) −17.2964 −1.24503 −0.622513 0.782610i \(-0.713888\pi\)
−0.622513 + 0.782610i \(0.713888\pi\)
\(194\) 0 0
\(195\) 0.414959 0.0297159
\(196\) 0 0
\(197\) 8.27858 0.589824 0.294912 0.955524i \(-0.404710\pi\)
0.294912 + 0.955524i \(0.404710\pi\)
\(198\) 0 0
\(199\) −14.4762 −1.02619 −0.513096 0.858331i \(-0.671501\pi\)
−0.513096 + 0.858331i \(0.671501\pi\)
\(200\) 0 0
\(201\) −0.292071 −0.0206011
\(202\) 0 0
\(203\) −7.95751 −0.558508
\(204\) 0 0
\(205\) −5.80287 −0.405290
\(206\) 0 0
\(207\) 15.3912 1.06976
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.12519 −0.559361 −0.279681 0.960093i \(-0.590229\pi\)
−0.279681 + 0.960093i \(0.590229\pi\)
\(212\) 0 0
\(213\) −2.26013 −0.154862
\(214\) 0 0
\(215\) 0.884108 0.0602957
\(216\) 0 0
\(217\) −13.9540 −0.947261
\(218\) 0 0
\(219\) −1.80480 −0.121957
\(220\) 0 0
\(221\) 9.92810 0.667837
\(222\) 0 0
\(223\) −12.3446 −0.826658 −0.413329 0.910582i \(-0.635634\pi\)
−0.413329 + 0.910582i \(0.635634\pi\)
\(224\) 0 0
\(225\) −2.95121 −0.196748
\(226\) 0 0
\(227\) −19.0269 −1.26286 −0.631430 0.775432i \(-0.717532\pi\)
−0.631430 + 0.775432i \(0.717532\pi\)
\(228\) 0 0
\(229\) 3.14648 0.207926 0.103963 0.994581i \(-0.466848\pi\)
0.103963 + 0.994581i \(0.466848\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.8983 −1.04153 −0.520767 0.853699i \(-0.674354\pi\)
−0.520767 + 0.853699i \(0.674354\pi\)
\(234\) 0 0
\(235\) 7.69696 0.502095
\(236\) 0 0
\(237\) 1.15542 0.0750523
\(238\) 0 0
\(239\) −15.7958 −1.02174 −0.510872 0.859657i \(-0.670677\pi\)
−0.510872 + 0.859657i \(0.670677\pi\)
\(240\) 0 0
\(241\) −16.2927 −1.04950 −0.524752 0.851255i \(-0.675842\pi\)
−0.524752 + 0.851255i \(0.675842\pi\)
\(242\) 0 0
\(243\) −5.83493 −0.374311
\(244\) 0 0
\(245\) 2.64141 0.168754
\(246\) 0 0
\(247\) 5.49302 0.349513
\(248\) 0 0
\(249\) −2.12801 −0.134857
\(250\) 0 0
\(251\) −9.75176 −0.615525 −0.307763 0.951463i \(-0.599580\pi\)
−0.307763 + 0.951463i \(0.599580\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 1.16726 0.0730965
\(256\) 0 0
\(257\) −17.9647 −1.12061 −0.560303 0.828288i \(-0.689315\pi\)
−0.560303 + 0.828288i \(0.689315\pi\)
\(258\) 0 0
\(259\) −7.21839 −0.448529
\(260\) 0 0
\(261\) 11.2488 0.696281
\(262\) 0 0
\(263\) −11.6324 −0.717284 −0.358642 0.933475i \(-0.616760\pi\)
−0.358642 + 0.933475i \(0.616760\pi\)
\(264\) 0 0
\(265\) 9.50720 0.584023
\(266\) 0 0
\(267\) 3.77545 0.231054
\(268\) 0 0
\(269\) −11.8652 −0.723437 −0.361718 0.932287i \(-0.617810\pi\)
−0.361718 + 0.932287i \(0.617810\pi\)
\(270\) 0 0
\(271\) −12.5270 −0.760963 −0.380481 0.924789i \(-0.624242\pi\)
−0.380481 + 0.924789i \(0.624242\pi\)
\(272\) 0 0
\(273\) −0.866320 −0.0524321
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 27.4409 1.64876 0.824381 0.566035i \(-0.191523\pi\)
0.824381 + 0.566035i \(0.191523\pi\)
\(278\) 0 0
\(279\) 19.7255 1.18093
\(280\) 0 0
\(281\) 25.7658 1.53706 0.768530 0.639814i \(-0.220989\pi\)
0.768530 + 0.639814i \(0.220989\pi\)
\(282\) 0 0
\(283\) −10.2068 −0.606731 −0.303366 0.952874i \(-0.598110\pi\)
−0.303366 + 0.952874i \(0.598110\pi\)
\(284\) 0 0
\(285\) 0.645820 0.0382551
\(286\) 0 0
\(287\) 12.1148 0.715113
\(288\) 0 0
\(289\) 10.9272 0.642776
\(290\) 0 0
\(291\) 3.38597 0.198489
\(292\) 0 0
\(293\) −26.0333 −1.52088 −0.760441 0.649407i \(-0.775017\pi\)
−0.760441 + 0.649407i \(0.775017\pi\)
\(294\) 0 0
\(295\) −7.17454 −0.417718
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.79771 −0.566616
\(300\) 0 0
\(301\) −1.84577 −0.106389
\(302\) 0 0
\(303\) −3.44102 −0.197682
\(304\) 0 0
\(305\) 7.88806 0.451669
\(306\) 0 0
\(307\) 10.1591 0.579808 0.289904 0.957056i \(-0.406377\pi\)
0.289904 + 0.957056i \(0.406377\pi\)
\(308\) 0 0
\(309\) −3.20356 −0.182244
\(310\) 0 0
\(311\) −6.32891 −0.358879 −0.179440 0.983769i \(-0.557428\pi\)
−0.179440 + 0.983769i \(0.557428\pi\)
\(312\) 0 0
\(313\) −21.9366 −1.23993 −0.619964 0.784630i \(-0.712853\pi\)
−0.619964 + 0.784630i \(0.712853\pi\)
\(314\) 0 0
\(315\) 6.16132 0.347151
\(316\) 0 0
\(317\) −24.2225 −1.36047 −0.680237 0.732993i \(-0.738123\pi\)
−0.680237 + 0.732993i \(0.738123\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.72625 0.207979
\(322\) 0 0
\(323\) 15.4516 0.859748
\(324\) 0 0
\(325\) 1.87868 0.104210
\(326\) 0 0
\(327\) 2.20306 0.121829
\(328\) 0 0
\(329\) −16.0691 −0.885920
\(330\) 0 0
\(331\) −29.3673 −1.61417 −0.807086 0.590433i \(-0.798957\pi\)
−0.807086 + 0.590433i \(0.798957\pi\)
\(332\) 0 0
\(333\) 10.2039 0.559172
\(334\) 0 0
\(335\) −1.32232 −0.0722458
\(336\) 0 0
\(337\) −5.77943 −0.314825 −0.157413 0.987533i \(-0.550315\pi\)
−0.157413 + 0.987533i \(0.550315\pi\)
\(338\) 0 0
\(339\) −0.00182075 −9.88898e−5 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1286 −1.08684
\(344\) 0 0
\(345\) −1.15193 −0.0620177
\(346\) 0 0
\(347\) 31.9559 1.71548 0.857742 0.514081i \(-0.171867\pi\)
0.857742 + 0.514081i \(0.171867\pi\)
\(348\) 0 0
\(349\) −10.9895 −0.588253 −0.294126 0.955767i \(-0.595029\pi\)
−0.294126 + 0.955767i \(0.595029\pi\)
\(350\) 0 0
\(351\) 2.46951 0.131813
\(352\) 0 0
\(353\) −19.8860 −1.05842 −0.529212 0.848489i \(-0.677513\pi\)
−0.529212 + 0.848489i \(0.677513\pi\)
\(354\) 0 0
\(355\) −10.2325 −0.543083
\(356\) 0 0
\(357\) −2.43691 −0.128975
\(358\) 0 0
\(359\) −33.1513 −1.74966 −0.874830 0.484429i \(-0.839027\pi\)
−0.874830 + 0.484429i \(0.839027\pi\)
\(360\) 0 0
\(361\) −10.4510 −0.550050
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.17103 −0.427691
\(366\) 0 0
\(367\) −20.2195 −1.05545 −0.527723 0.849416i \(-0.676954\pi\)
−0.527723 + 0.849416i \(0.676954\pi\)
\(368\) 0 0
\(369\) −17.1255 −0.891518
\(370\) 0 0
\(371\) −19.8484 −1.03048
\(372\) 0 0
\(373\) −11.6099 −0.601136 −0.300568 0.953760i \(-0.597176\pi\)
−0.300568 + 0.953760i \(0.597176\pi\)
\(374\) 0 0
\(375\) 0.220878 0.0114061
\(376\) 0 0
\(377\) −7.16073 −0.368796
\(378\) 0 0
\(379\) 2.61724 0.134438 0.0672192 0.997738i \(-0.478587\pi\)
0.0672192 + 0.997738i \(0.478587\pi\)
\(380\) 0 0
\(381\) −2.22475 −0.113977
\(382\) 0 0
\(383\) −7.75206 −0.396112 −0.198056 0.980191i \(-0.563463\pi\)
−0.198056 + 0.980191i \(0.563463\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.60919 0.132633
\(388\) 0 0
\(389\) 19.8235 1.00509 0.502545 0.864551i \(-0.332397\pi\)
0.502545 + 0.864551i \(0.332397\pi\)
\(390\) 0 0
\(391\) −27.5604 −1.39379
\(392\) 0 0
\(393\) −3.98131 −0.200831
\(394\) 0 0
\(395\) 5.23101 0.263201
\(396\) 0 0
\(397\) −17.8853 −0.897636 −0.448818 0.893623i \(-0.648155\pi\)
−0.448818 + 0.893623i \(0.648155\pi\)
\(398\) 0 0
\(399\) −1.34829 −0.0674991
\(400\) 0 0
\(401\) 34.2258 1.70916 0.854578 0.519323i \(-0.173816\pi\)
0.854578 + 0.519323i \(0.173816\pi\)
\(402\) 0 0
\(403\) −12.5568 −0.625499
\(404\) 0 0
\(405\) −8.56330 −0.425514
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −2.21701 −0.109624 −0.0548121 0.998497i \(-0.517456\pi\)
−0.0548121 + 0.998497i \(0.517456\pi\)
\(410\) 0 0
\(411\) −1.13930 −0.0561977
\(412\) 0 0
\(413\) 14.9784 0.737041
\(414\) 0 0
\(415\) −9.63433 −0.472930
\(416\) 0 0
\(417\) −0.356105 −0.0174385
\(418\) 0 0
\(419\) 17.7908 0.869137 0.434569 0.900639i \(-0.356901\pi\)
0.434569 + 0.900639i \(0.356901\pi\)
\(420\) 0 0
\(421\) 35.7958 1.74458 0.872290 0.488989i \(-0.162634\pi\)
0.872290 + 0.488989i \(0.162634\pi\)
\(422\) 0 0
\(423\) 22.7154 1.10446
\(424\) 0 0
\(425\) 5.28462 0.256342
\(426\) 0 0
\(427\) −16.4681 −0.796946
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.10027 0.149335 0.0746673 0.997209i \(-0.476211\pi\)
0.0746673 + 0.997209i \(0.476211\pi\)
\(432\) 0 0
\(433\) 19.3592 0.930345 0.465172 0.885220i \(-0.345992\pi\)
0.465172 + 0.885220i \(0.345992\pi\)
\(434\) 0 0
\(435\) −0.841894 −0.0403657
\(436\) 0 0
\(437\) −15.2486 −0.729441
\(438\) 0 0
\(439\) −40.6115 −1.93828 −0.969142 0.246504i \(-0.920718\pi\)
−0.969142 + 0.246504i \(0.920718\pi\)
\(440\) 0 0
\(441\) 7.79537 0.371208
\(442\) 0 0
\(443\) 21.5805 1.02532 0.512661 0.858591i \(-0.328660\pi\)
0.512661 + 0.858591i \(0.328660\pi\)
\(444\) 0 0
\(445\) 17.0929 0.810282
\(446\) 0 0
\(447\) −0.663174 −0.0313671
\(448\) 0 0
\(449\) −34.4259 −1.62466 −0.812330 0.583198i \(-0.801801\pi\)
−0.812330 + 0.583198i \(0.801801\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0.183610 0.00862676
\(454\) 0 0
\(455\) −3.92216 −0.183874
\(456\) 0 0
\(457\) −12.4701 −0.583327 −0.291664 0.956521i \(-0.594209\pi\)
−0.291664 + 0.956521i \(0.594209\pi\)
\(458\) 0 0
\(459\) 6.94660 0.324239
\(460\) 0 0
\(461\) −23.5302 −1.09591 −0.547956 0.836507i \(-0.684594\pi\)
−0.547956 + 0.836507i \(0.684594\pi\)
\(462\) 0 0
\(463\) 29.5796 1.37468 0.687340 0.726336i \(-0.258778\pi\)
0.687340 + 0.726336i \(0.258778\pi\)
\(464\) 0 0
\(465\) −1.47632 −0.0684626
\(466\) 0 0
\(467\) 21.8760 1.01230 0.506150 0.862446i \(-0.331068\pi\)
0.506150 + 0.862446i \(0.331068\pi\)
\(468\) 0 0
\(469\) 2.76063 0.127474
\(470\) 0 0
\(471\) 2.93992 0.135465
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.92388 0.134157
\(476\) 0 0
\(477\) 28.0578 1.28468
\(478\) 0 0
\(479\) 30.9759 1.41533 0.707664 0.706550i \(-0.249749\pi\)
0.707664 + 0.706550i \(0.249749\pi\)
\(480\) 0 0
\(481\) −6.49561 −0.296174
\(482\) 0 0
\(483\) 2.40491 0.109427
\(484\) 0 0
\(485\) 15.3296 0.696081
\(486\) 0 0
\(487\) 9.02364 0.408900 0.204450 0.978877i \(-0.434459\pi\)
0.204450 + 0.978877i \(0.434459\pi\)
\(488\) 0 0
\(489\) −1.40546 −0.0635570
\(490\) 0 0
\(491\) 29.4524 1.32917 0.664583 0.747214i \(-0.268609\pi\)
0.664583 + 0.747214i \(0.268609\pi\)
\(492\) 0 0
\(493\) −20.1427 −0.907182
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21.3626 0.958243
\(498\) 0 0
\(499\) −13.2183 −0.591732 −0.295866 0.955229i \(-0.595608\pi\)
−0.295866 + 0.955229i \(0.595608\pi\)
\(500\) 0 0
\(501\) −0.498291 −0.0222620
\(502\) 0 0
\(503\) 8.17945 0.364704 0.182352 0.983233i \(-0.441629\pi\)
0.182352 + 0.983233i \(0.441629\pi\)
\(504\) 0 0
\(505\) −15.5788 −0.693248
\(506\) 0 0
\(507\) 2.09184 0.0929019
\(508\) 0 0
\(509\) −7.71064 −0.341768 −0.170884 0.985291i \(-0.554662\pi\)
−0.170884 + 0.985291i \(0.554662\pi\)
\(510\) 0 0
\(511\) 17.0588 0.754639
\(512\) 0 0
\(513\) 3.84342 0.169691
\(514\) 0 0
\(515\) −14.5038 −0.639112
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 2.39089 0.104948
\(520\) 0 0
\(521\) −22.4464 −0.983396 −0.491698 0.870766i \(-0.663624\pi\)
−0.491698 + 0.870766i \(0.663624\pi\)
\(522\) 0 0
\(523\) 4.50517 0.196997 0.0984987 0.995137i \(-0.468596\pi\)
0.0984987 + 0.995137i \(0.468596\pi\)
\(524\) 0 0
\(525\) −0.461133 −0.0201255
\(526\) 0 0
\(527\) −35.3216 −1.53863
\(528\) 0 0
\(529\) 4.19845 0.182541
\(530\) 0 0
\(531\) −21.1736 −0.918855
\(532\) 0 0
\(533\) 10.9017 0.472206
\(534\) 0 0
\(535\) 16.8702 0.729360
\(536\) 0 0
\(537\) −2.56209 −0.110562
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −24.9692 −1.07351 −0.536755 0.843738i \(-0.680350\pi\)
−0.536755 + 0.843738i \(0.680350\pi\)
\(542\) 0 0
\(543\) 0.833782 0.0357810
\(544\) 0 0
\(545\) 9.97408 0.427243
\(546\) 0 0
\(547\) −9.75488 −0.417089 −0.208544 0.978013i \(-0.566873\pi\)
−0.208544 + 0.978013i \(0.566873\pi\)
\(548\) 0 0
\(549\) 23.2793 0.993538
\(550\) 0 0
\(551\) −11.1446 −0.474775
\(552\) 0 0
\(553\) −10.9209 −0.464404
\(554\) 0 0
\(555\) −0.763696 −0.0324171
\(556\) 0 0
\(557\) 35.0288 1.48422 0.742110 0.670278i \(-0.233825\pi\)
0.742110 + 0.670278i \(0.233825\pi\)
\(558\) 0 0
\(559\) −1.66096 −0.0702509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.82310 −0.0768347 −0.0384173 0.999262i \(-0.512232\pi\)
−0.0384173 + 0.999262i \(0.512232\pi\)
\(564\) 0 0
\(565\) −0.00824325 −0.000346796 0
\(566\) 0 0
\(567\) 17.8778 0.750797
\(568\) 0 0
\(569\) −12.9416 −0.542542 −0.271271 0.962503i \(-0.587444\pi\)
−0.271271 + 0.962503i \(0.587444\pi\)
\(570\) 0 0
\(571\) 8.07735 0.338027 0.169013 0.985614i \(-0.445942\pi\)
0.169013 + 0.985614i \(0.445942\pi\)
\(572\) 0 0
\(573\) −4.75519 −0.198651
\(574\) 0 0
\(575\) −5.21521 −0.217489
\(576\) 0 0
\(577\) 11.8678 0.494065 0.247032 0.969007i \(-0.420545\pi\)
0.247032 + 0.969007i \(0.420545\pi\)
\(578\) 0 0
\(579\) 3.82041 0.158771
\(580\) 0 0
\(581\) 20.1138 0.834461
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 5.54438 0.229232
\(586\) 0 0
\(587\) 43.5701 1.79833 0.899165 0.437611i \(-0.144175\pi\)
0.899165 + 0.437611i \(0.144175\pi\)
\(588\) 0 0
\(589\) −19.5427 −0.805244
\(590\) 0 0
\(591\) −1.82856 −0.0752168
\(592\) 0 0
\(593\) 20.0344 0.822715 0.411358 0.911474i \(-0.365055\pi\)
0.411358 + 0.911474i \(0.365055\pi\)
\(594\) 0 0
\(595\) −11.0328 −0.452302
\(596\) 0 0
\(597\) 3.19748 0.130864
\(598\) 0 0
\(599\) −14.1406 −0.577768 −0.288884 0.957364i \(-0.593284\pi\)
−0.288884 + 0.957364i \(0.593284\pi\)
\(600\) 0 0
\(601\) −38.5427 −1.57219 −0.786094 0.618106i \(-0.787900\pi\)
−0.786094 + 0.618106i \(0.787900\pi\)
\(602\) 0 0
\(603\) −3.90244 −0.158919
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.8199 0.642110 0.321055 0.947061i \(-0.395963\pi\)
0.321055 + 0.947061i \(0.395963\pi\)
\(608\) 0 0
\(609\) 1.75764 0.0712232
\(610\) 0 0
\(611\) −14.4601 −0.584994
\(612\) 0 0
\(613\) 4.94110 0.199569 0.0997847 0.995009i \(-0.468185\pi\)
0.0997847 + 0.995009i \(0.468185\pi\)
\(614\) 0 0
\(615\) 1.28173 0.0516842
\(616\) 0 0
\(617\) 8.96192 0.360793 0.180397 0.983594i \(-0.442262\pi\)
0.180397 + 0.983594i \(0.442262\pi\)
\(618\) 0 0
\(619\) 22.7324 0.913693 0.456846 0.889546i \(-0.348979\pi\)
0.456846 + 0.889546i \(0.348979\pi\)
\(620\) 0 0
\(621\) −6.85536 −0.275096
\(622\) 0 0
\(623\) −35.6853 −1.42970
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.2718 −0.728544
\(630\) 0 0
\(631\) 12.7142 0.506144 0.253072 0.967447i \(-0.418559\pi\)
0.253072 + 0.967447i \(0.418559\pi\)
\(632\) 0 0
\(633\) 1.79468 0.0713320
\(634\) 0 0
\(635\) −10.0723 −0.399706
\(636\) 0 0
\(637\) −4.96237 −0.196616
\(638\) 0 0
\(639\) −30.1982 −1.19462
\(640\) 0 0
\(641\) −13.5734 −0.536118 −0.268059 0.963402i \(-0.586382\pi\)
−0.268059 + 0.963402i \(0.586382\pi\)
\(642\) 0 0
\(643\) −22.7780 −0.898275 −0.449138 0.893463i \(-0.648269\pi\)
−0.449138 + 0.893463i \(0.648269\pi\)
\(644\) 0 0
\(645\) −0.195280 −0.00768915
\(646\) 0 0
\(647\) −12.7235 −0.500212 −0.250106 0.968218i \(-0.580465\pi\)
−0.250106 + 0.968218i \(0.580465\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 3.08214 0.120799
\(652\) 0 0
\(653\) −15.3118 −0.599197 −0.299598 0.954065i \(-0.596853\pi\)
−0.299598 + 0.954065i \(0.596853\pi\)
\(654\) 0 0
\(655\) −18.0249 −0.704291
\(656\) 0 0
\(657\) −24.1144 −0.940794
\(658\) 0 0
\(659\) 41.8861 1.63165 0.815826 0.578297i \(-0.196282\pi\)
0.815826 + 0.578297i \(0.196282\pi\)
\(660\) 0 0
\(661\) 3.83538 0.149179 0.0745895 0.997214i \(-0.476235\pi\)
0.0745895 + 0.997214i \(0.476235\pi\)
\(662\) 0 0
\(663\) −2.19290 −0.0851653
\(664\) 0 0
\(665\) −6.10424 −0.236712
\(666\) 0 0
\(667\) 19.8782 0.769686
\(668\) 0 0
\(669\) 2.72666 0.105419
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 28.5708 1.10132 0.550662 0.834728i \(-0.314375\pi\)
0.550662 + 0.834728i \(0.314375\pi\)
\(674\) 0 0
\(675\) 1.31449 0.0505949
\(676\) 0 0
\(677\) −18.9267 −0.727413 −0.363706 0.931514i \(-0.618489\pi\)
−0.363706 + 0.931514i \(0.618489\pi\)
\(678\) 0 0
\(679\) −32.0039 −1.22820
\(680\) 0 0
\(681\) 4.20263 0.161045
\(682\) 0 0
\(683\) −34.3479 −1.31429 −0.657143 0.753766i \(-0.728235\pi\)
−0.657143 + 0.753766i \(0.728235\pi\)
\(684\) 0 0
\(685\) −5.15806 −0.197079
\(686\) 0 0
\(687\) −0.694990 −0.0265155
\(688\) 0 0
\(689\) −17.8610 −0.680449
\(690\) 0 0
\(691\) −48.9701 −1.86291 −0.931456 0.363854i \(-0.881461\pi\)
−0.931456 + 0.363854i \(0.881461\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.61222 −0.0611551
\(696\) 0 0
\(697\) 30.6659 1.16156
\(698\) 0 0
\(699\) 3.51159 0.132821
\(700\) 0 0
\(701\) −17.1191 −0.646581 −0.323291 0.946300i \(-0.604789\pi\)
−0.323291 + 0.946300i \(0.604789\pi\)
\(702\) 0 0
\(703\) −10.1094 −0.381284
\(704\) 0 0
\(705\) −1.70009 −0.0640292
\(706\) 0 0
\(707\) 32.5243 1.22320
\(708\) 0 0
\(709\) 40.2510 1.51166 0.755829 0.654769i \(-0.227234\pi\)
0.755829 + 0.654769i \(0.227234\pi\)
\(710\) 0 0
\(711\) 15.4378 0.578963
\(712\) 0 0
\(713\) 34.8577 1.30543
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.48894 0.130297
\(718\) 0 0
\(719\) −31.1437 −1.16146 −0.580732 0.814094i \(-0.697234\pi\)
−0.580732 + 0.814094i \(0.697234\pi\)
\(720\) 0 0
\(721\) 30.2798 1.12768
\(722\) 0 0
\(723\) 3.59870 0.133837
\(724\) 0 0
\(725\) −3.81157 −0.141558
\(726\) 0 0
\(727\) 29.3135 1.08718 0.543589 0.839351i \(-0.317065\pi\)
0.543589 + 0.839351i \(0.317065\pi\)
\(728\) 0 0
\(729\) −24.4011 −0.903744
\(730\) 0 0
\(731\) −4.67217 −0.172807
\(732\) 0 0
\(733\) −27.1541 −1.00296 −0.501479 0.865170i \(-0.667211\pi\)
−0.501479 + 0.865170i \(0.667211\pi\)
\(734\) 0 0
\(735\) −0.583431 −0.0215202
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 10.6933 0.393361 0.196680 0.980468i \(-0.436984\pi\)
0.196680 + 0.980468i \(0.436984\pi\)
\(740\) 0 0
\(741\) −1.21329 −0.0445713
\(742\) 0 0
\(743\) −53.4828 −1.96209 −0.981046 0.193775i \(-0.937927\pi\)
−0.981046 + 0.193775i \(0.937927\pi\)
\(744\) 0 0
\(745\) −3.00244 −0.110001
\(746\) 0 0
\(747\) −28.4329 −1.04031
\(748\) 0 0
\(749\) −35.2202 −1.28692
\(750\) 0 0
\(751\) −49.3066 −1.79922 −0.899612 0.436690i \(-0.856151\pi\)
−0.899612 + 0.436690i \(0.856151\pi\)
\(752\) 0 0
\(753\) 2.15395 0.0784943
\(754\) 0 0
\(755\) 0.831273 0.0302531
\(756\) 0 0
\(757\) −40.7211 −1.48003 −0.740016 0.672589i \(-0.765182\pi\)
−0.740016 + 0.672589i \(0.765182\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −33.3345 −1.20838 −0.604188 0.796842i \(-0.706502\pi\)
−0.604188 + 0.796842i \(0.706502\pi\)
\(762\) 0 0
\(763\) −20.8231 −0.753848
\(764\) 0 0
\(765\) 15.5960 0.563876
\(766\) 0 0
\(767\) 13.4787 0.486686
\(768\) 0 0
\(769\) 43.5317 1.56979 0.784896 0.619627i \(-0.212716\pi\)
0.784896 + 0.619627i \(0.212716\pi\)
\(770\) 0 0
\(771\) 3.96801 0.142904
\(772\) 0 0
\(773\) −36.7061 −1.32023 −0.660113 0.751166i \(-0.729492\pi\)
−0.660113 + 0.751166i \(0.729492\pi\)
\(774\) 0 0
\(775\) −6.68385 −0.240091
\(776\) 0 0
\(777\) 1.59439 0.0571982
\(778\) 0 0
\(779\) 16.9669 0.607901
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −5.01029 −0.179053
\(784\) 0 0
\(785\) 13.3102 0.475060
\(786\) 0 0
\(787\) 26.2255 0.934837 0.467418 0.884036i \(-0.345184\pi\)
0.467418 + 0.884036i \(0.345184\pi\)
\(788\) 0 0
\(789\) 2.56934 0.0914710
\(790\) 0 0
\(791\) 0.0172096 0.000611903 0
\(792\) 0 0
\(793\) −14.8191 −0.526243
\(794\) 0 0
\(795\) −2.09993 −0.0744770
\(796\) 0 0
\(797\) −7.79065 −0.275959 −0.137980 0.990435i \(-0.544061\pi\)
−0.137980 + 0.990435i \(0.544061\pi\)
\(798\) 0 0
\(799\) −40.6755 −1.43900
\(800\) 0 0
\(801\) 50.4448 1.78238
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 10.8879 0.383749
\(806\) 0 0
\(807\) 2.62077 0.0922556
\(808\) 0 0
\(809\) −23.6845 −0.832704 −0.416352 0.909203i \(-0.636692\pi\)
−0.416352 + 0.909203i \(0.636692\pi\)
\(810\) 0 0
\(811\) −4.53483 −0.159239 −0.0796197 0.996825i \(-0.525371\pi\)
−0.0796197 + 0.996825i \(0.525371\pi\)
\(812\) 0 0
\(813\) 2.76695 0.0970411
\(814\) 0 0
\(815\) −6.36304 −0.222888
\(816\) 0 0
\(817\) −2.58502 −0.0904385
\(818\) 0 0
\(819\) −11.5751 −0.404468
\(820\) 0 0
\(821\) 20.1204 0.702207 0.351104 0.936337i \(-0.385806\pi\)
0.351104 + 0.936337i \(0.385806\pi\)
\(822\) 0 0
\(823\) 10.7823 0.375849 0.187924 0.982184i \(-0.439824\pi\)
0.187924 + 0.982184i \(0.439824\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 49.7772 1.73092 0.865462 0.500975i \(-0.167025\pi\)
0.865462 + 0.500975i \(0.167025\pi\)
\(828\) 0 0
\(829\) 15.8842 0.551682 0.275841 0.961203i \(-0.411044\pi\)
0.275841 + 0.961203i \(0.411044\pi\)
\(830\) 0 0
\(831\) −6.06109 −0.210257
\(832\) 0 0
\(833\) −13.9589 −0.483646
\(834\) 0 0
\(835\) −2.25595 −0.0780704
\(836\) 0 0
\(837\) −8.78588 −0.303684
\(838\) 0 0
\(839\) −5.99554 −0.206989 −0.103494 0.994630i \(-0.533002\pi\)
−0.103494 + 0.994630i \(0.533002\pi\)
\(840\) 0 0
\(841\) −14.4719 −0.499031
\(842\) 0 0
\(843\) −5.69111 −0.196012
\(844\) 0 0
\(845\) 9.47056 0.325797
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 2.25446 0.0773729
\(850\) 0 0
\(851\) 18.0318 0.618123
\(852\) 0 0
\(853\) 18.9322 0.648226 0.324113 0.946018i \(-0.394934\pi\)
0.324113 + 0.946018i \(0.394934\pi\)
\(854\) 0 0
\(855\) 8.62898 0.295105
\(856\) 0 0
\(857\) 33.4506 1.14265 0.571325 0.820724i \(-0.306429\pi\)
0.571325 + 0.820724i \(0.306429\pi\)
\(858\) 0 0
\(859\) 0.368811 0.0125837 0.00629184 0.999980i \(-0.497997\pi\)
0.00629184 + 0.999980i \(0.497997\pi\)
\(860\) 0 0
\(861\) −2.67589 −0.0911941
\(862\) 0 0
\(863\) 5.94784 0.202467 0.101233 0.994863i \(-0.467721\pi\)
0.101233 + 0.994863i \(0.467721\pi\)
\(864\) 0 0
\(865\) 10.8245 0.368042
\(866\) 0 0
\(867\) −2.41358 −0.0819695
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.48421 0.0841742
\(872\) 0 0
\(873\) 45.2409 1.53117
\(874\) 0 0
\(875\) −2.08772 −0.0705779
\(876\) 0 0
\(877\) −5.80988 −0.196186 −0.0980929 0.995177i \(-0.531274\pi\)
−0.0980929 + 0.995177i \(0.531274\pi\)
\(878\) 0 0
\(879\) 5.75019 0.193949
\(880\) 0 0
\(881\) 27.3096 0.920083 0.460042 0.887897i \(-0.347834\pi\)
0.460042 + 0.887897i \(0.347834\pi\)
\(882\) 0 0
\(883\) −14.9649 −0.503609 −0.251804 0.967778i \(-0.581024\pi\)
−0.251804 + 0.967778i \(0.581024\pi\)
\(884\) 0 0
\(885\) 1.58470 0.0532691
\(886\) 0 0
\(887\) 9.81551 0.329573 0.164786 0.986329i \(-0.447307\pi\)
0.164786 + 0.986329i \(0.447307\pi\)
\(888\) 0 0
\(889\) 21.0281 0.705260
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −22.5050 −0.753100
\(894\) 0 0
\(895\) −11.5996 −0.387731
\(896\) 0 0
\(897\) 2.16410 0.0722573
\(898\) 0 0
\(899\) 25.4760 0.849672
\(900\) 0 0
\(901\) −50.2419 −1.67380
\(902\) 0 0
\(903\) 0.407691 0.0135671
\(904\) 0 0
\(905\) 3.77485 0.125480
\(906\) 0 0
\(907\) −54.5458 −1.81116 −0.905582 0.424172i \(-0.860565\pi\)
−0.905582 + 0.424172i \(0.860565\pi\)
\(908\) 0 0
\(909\) −45.9764 −1.52494
\(910\) 0 0
\(911\) −38.0190 −1.25962 −0.629812 0.776747i \(-0.716868\pi\)
−0.629812 + 0.776747i \(0.716868\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −1.74230 −0.0575987
\(916\) 0 0
\(917\) 37.6310 1.24269
\(918\) 0 0
\(919\) −55.3414 −1.82555 −0.912773 0.408467i \(-0.866063\pi\)
−0.912773 + 0.408467i \(0.866063\pi\)
\(920\) 0 0
\(921\) −2.24392 −0.0739396
\(922\) 0 0
\(923\) 19.2235 0.632751
\(924\) 0 0
\(925\) −3.45754 −0.113683
\(926\) 0 0
\(927\) −42.8037 −1.40586
\(928\) 0 0
\(929\) −0.719940 −0.0236205 −0.0118102 0.999930i \(-0.503759\pi\)
−0.0118102 + 0.999930i \(0.503759\pi\)
\(930\) 0 0
\(931\) −7.72316 −0.253116
\(932\) 0 0
\(933\) 1.39792 0.0457658
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −36.9846 −1.20823 −0.604117 0.796896i \(-0.706474\pi\)
−0.604117 + 0.796896i \(0.706474\pi\)
\(938\) 0 0
\(939\) 4.84531 0.158121
\(940\) 0 0
\(941\) 11.6784 0.380704 0.190352 0.981716i \(-0.439037\pi\)
0.190352 + 0.981716i \(0.439037\pi\)
\(942\) 0 0
\(943\) −30.2632 −0.985505
\(944\) 0 0
\(945\) −2.74430 −0.0892720
\(946\) 0 0
\(947\) 60.3234 1.96025 0.980123 0.198390i \(-0.0635713\pi\)
0.980123 + 0.198390i \(0.0635713\pi\)
\(948\) 0 0
\(949\) 15.3507 0.498306
\(950\) 0 0
\(951\) 5.35023 0.173493
\(952\) 0 0
\(953\) 2.02655 0.0656464 0.0328232 0.999461i \(-0.489550\pi\)
0.0328232 + 0.999461i \(0.489550\pi\)
\(954\) 0 0
\(955\) −21.5286 −0.696648
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 10.7686 0.347736
\(960\) 0 0
\(961\) 13.6738 0.441091
\(962\) 0 0
\(963\) 49.7874 1.60438
\(964\) 0 0
\(965\) 17.2964 0.556792
\(966\) 0 0
\(967\) −47.8068 −1.53736 −0.768682 0.639631i \(-0.779087\pi\)
−0.768682 + 0.639631i \(0.779087\pi\)
\(968\) 0 0
\(969\) −3.41292 −0.109639
\(970\) 0 0
\(971\) 7.71139 0.247470 0.123735 0.992315i \(-0.460513\pi\)
0.123735 + 0.992315i \(0.460513\pi\)
\(972\) 0 0
\(973\) 3.36588 0.107905
\(974\) 0 0
\(975\) −0.414959 −0.0132893
\(976\) 0 0
\(977\) −51.2365 −1.63920 −0.819600 0.572936i \(-0.805804\pi\)
−0.819600 + 0.572936i \(0.805804\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 29.4356 0.939808
\(982\) 0 0
\(983\) −33.1192 −1.05634 −0.528170 0.849139i \(-0.677121\pi\)
−0.528170 + 0.849139i \(0.677121\pi\)
\(984\) 0 0
\(985\) −8.27858 −0.263778
\(986\) 0 0
\(987\) 3.54932 0.112976
\(988\) 0 0
\(989\) 4.61081 0.146615
\(990\) 0 0
\(991\) −36.3110 −1.15346 −0.576729 0.816936i \(-0.695671\pi\)
−0.576729 + 0.816936i \(0.695671\pi\)
\(992\) 0 0
\(993\) 6.48660 0.205846
\(994\) 0 0
\(995\) 14.4762 0.458927
\(996\) 0 0
\(997\) 57.1807 1.81093 0.905466 0.424419i \(-0.139522\pi\)
0.905466 + 0.424419i \(0.139522\pi\)
\(998\) 0 0
\(999\) −4.54492 −0.143795
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.dc.1.3 6
4.3 odd 2 4840.2.a.bb.1.4 6
11.3 even 5 880.2.bo.i.801.2 12
11.4 even 5 880.2.bo.i.401.2 12
11.10 odd 2 9680.2.a.dd.1.3 6
44.3 odd 10 440.2.y.c.361.2 12
44.15 odd 10 440.2.y.c.401.2 yes 12
44.43 even 2 4840.2.a.ba.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.361.2 12 44.3 odd 10
440.2.y.c.401.2 yes 12 44.15 odd 10
880.2.bo.i.401.2 12 11.4 even 5
880.2.bo.i.801.2 12 11.3 even 5
4840.2.a.ba.1.4 6 44.43 even 2
4840.2.a.bb.1.4 6 4.3 odd 2
9680.2.a.dc.1.3 6 1.1 even 1 trivial
9680.2.a.dd.1.3 6 11.10 odd 2