Properties

Label 9680.2.a.cc.1.3
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.404.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x - 1 \) 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 4840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.210756\) 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)
 
\(f(q)\) \(=\) \(q+2.74483 q^{3} -1.00000 q^{5} -0.210756 q^{7} +4.53407 q^{9} +O(q^{10})\) \(q+2.74483 q^{3} -1.00000 q^{5} -0.210756 q^{7} +4.53407 q^{9} -2.11256 q^{13} -2.74483 q^{15} -2.42151 q^{17} -4.11256 q^{19} -0.578488 q^{21} +5.37709 q^{23} +1.00000 q^{25} +4.21076 q^{27} +1.48965 q^{29} -2.95558 q^{31} +0.210756 q^{35} +7.48965 q^{37} -5.79861 q^{39} -0.0444180 q^{41} +10.2108 q^{43} -4.53407 q^{45} +7.81297 q^{47} -6.95558 q^{49} -6.64663 q^{51} +12.0237 q^{53} -11.2883 q^{57} -9.60221 q^{59} +8.37709 q^{61} -0.955582 q^{63} +2.11256 q^{65} +10.8811 q^{67} +14.7592 q^{69} +4.53407 q^{71} +16.0474 q^{73} +2.74483 q^{75} -11.4008 q^{79} -2.04442 q^{81} -9.60221 q^{83} +2.42151 q^{85} +4.08884 q^{87} -5.42151 q^{89} +0.445234 q^{91} -8.11256 q^{93} +4.11256 q^{95} +5.88744 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} + q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} + q^{7} + 6 q^{9} - 2 q^{13} + q^{15} - 4 q^{17} - 8 q^{19} - 5 q^{21} + 2 q^{23} + 3 q^{25} + 11 q^{27} - 14 q^{29} + 2 q^{31} - q^{35} + 4 q^{37} - 11 q^{41} + 29 q^{43} - 6 q^{45} - q^{47} - 10 q^{49} - 8 q^{51} + 10 q^{53} + 2 q^{57} - 6 q^{59} + 11 q^{61} + 8 q^{63} + 2 q^{65} - 7 q^{67} + 28 q^{69} + 6 q^{71} - 4 q^{73} - q^{75} + 6 q^{79} - 17 q^{81} - 6 q^{83} + 4 q^{85} + 34 q^{87} - 13 q^{89} - 28 q^{91} - 20 q^{93} + 8 q^{95} + 22 q^{97}+O(q^{100}) \) 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
\(3\) 2.74483 1.58473 0.792363 0.610050i \(-0.208851\pi\)
0.792363 + 0.610050i \(0.208851\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.210756 −0.0796582 −0.0398291 0.999207i \(-0.512681\pi\)
−0.0398291 + 0.999207i \(0.512681\pi\)
\(8\) 0 0
\(9\) 4.53407 1.51136
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.11256 −0.585918 −0.292959 0.956125i \(-0.594640\pi\)
−0.292959 + 0.956125i \(0.594640\pi\)
\(14\) 0 0
\(15\) −2.74483 −0.708711
\(16\) 0 0
\(17\) −2.42151 −0.587303 −0.293651 0.955913i \(-0.594870\pi\)
−0.293651 + 0.955913i \(0.594870\pi\)
\(18\) 0 0
\(19\) −4.11256 −0.943486 −0.471743 0.881736i \(-0.656375\pi\)
−0.471743 + 0.881736i \(0.656375\pi\)
\(20\) 0 0
\(21\) −0.578488 −0.126236
\(22\) 0 0
\(23\) 5.37709 1.12120 0.560601 0.828086i \(-0.310570\pi\)
0.560601 + 0.828086i \(0.310570\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.21076 0.810360
\(28\) 0 0
\(29\) 1.48965 0.276621 0.138311 0.990389i \(-0.455833\pi\)
0.138311 + 0.990389i \(0.455833\pi\)
\(30\) 0 0
\(31\) −2.95558 −0.530838 −0.265419 0.964133i \(-0.585510\pi\)
−0.265419 + 0.964133i \(0.585510\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.210756 0.0356242
\(36\) 0 0
\(37\) 7.48965 1.23129 0.615646 0.788023i \(-0.288895\pi\)
0.615646 + 0.788023i \(0.288895\pi\)
\(38\) 0 0
\(39\) −5.79861 −0.928520
\(40\) 0 0
\(41\) −0.0444180 −0.00693693 −0.00346847 0.999994i \(-0.501104\pi\)
−0.00346847 + 0.999994i \(0.501104\pi\)
\(42\) 0 0
\(43\) 10.2108 1.55713 0.778563 0.627567i \(-0.215949\pi\)
0.778563 + 0.627567i \(0.215949\pi\)
\(44\) 0 0
\(45\) −4.53407 −0.675899
\(46\) 0 0
\(47\) 7.81297 1.13964 0.569819 0.821770i \(-0.307013\pi\)
0.569819 + 0.821770i \(0.307013\pi\)
\(48\) 0 0
\(49\) −6.95558 −0.993655
\(50\) 0 0
\(51\) −6.64663 −0.930714
\(52\) 0 0
\(53\) 12.0237 1.65159 0.825793 0.563974i \(-0.190728\pi\)
0.825793 + 0.563974i \(0.190728\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −11.2883 −1.49517
\(58\) 0 0
\(59\) −9.60221 −1.25010 −0.625051 0.780584i \(-0.714922\pi\)
−0.625051 + 0.780584i \(0.714922\pi\)
\(60\) 0 0
\(61\) 8.37709 1.07258 0.536288 0.844035i \(-0.319826\pi\)
0.536288 + 0.844035i \(0.319826\pi\)
\(62\) 0 0
\(63\) −0.955582 −0.120392
\(64\) 0 0
\(65\) 2.11256 0.262031
\(66\) 0 0
\(67\) 10.8811 1.32934 0.664669 0.747138i \(-0.268572\pi\)
0.664669 + 0.747138i \(0.268572\pi\)
\(68\) 0 0
\(69\) 14.7592 1.77680
\(70\) 0 0
\(71\) 4.53407 0.538095 0.269048 0.963127i \(-0.413291\pi\)
0.269048 + 0.963127i \(0.413291\pi\)
\(72\) 0 0
\(73\) 16.0474 1.87821 0.939106 0.343628i \(-0.111656\pi\)
0.939106 + 0.343628i \(0.111656\pi\)
\(74\) 0 0
\(75\) 2.74483 0.316945
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −11.4008 −1.28269 −0.641346 0.767252i \(-0.721624\pi\)
−0.641346 + 0.767252i \(0.721624\pi\)
\(80\) 0 0
\(81\) −2.04442 −0.227158
\(82\) 0 0
\(83\) −9.60221 −1.05398 −0.526990 0.849872i \(-0.676679\pi\)
−0.526990 + 0.849872i \(0.676679\pi\)
\(84\) 0 0
\(85\) 2.42151 0.262650
\(86\) 0 0
\(87\) 4.08884 0.438369
\(88\) 0 0
\(89\) −5.42151 −0.574679 −0.287340 0.957829i \(-0.592771\pi\)
−0.287340 + 0.957829i \(0.592771\pi\)
\(90\) 0 0
\(91\) 0.445234 0.0466732
\(92\) 0 0
\(93\) −8.11256 −0.841233
\(94\) 0 0
\(95\) 4.11256 0.421940
\(96\) 0 0
\(97\) 5.88744 0.597779 0.298890 0.954288i \(-0.403384\pi\)
0.298890 + 0.954288i \(0.403384\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −4.48965 −0.446737 −0.223369 0.974734i \(-0.571705\pi\)
−0.223369 + 0.974734i \(0.571705\pi\)
\(102\) 0 0
\(103\) 12.7592 1.25720 0.628600 0.777729i \(-0.283628\pi\)
0.628600 + 0.777729i \(0.283628\pi\)
\(104\) 0 0
\(105\) 0.578488 0.0564547
\(106\) 0 0
\(107\) 14.4596 1.39786 0.698931 0.715189i \(-0.253659\pi\)
0.698931 + 0.715189i \(0.253659\pi\)
\(108\) 0 0
\(109\) 6.60221 0.632377 0.316189 0.948696i \(-0.397597\pi\)
0.316189 + 0.948696i \(0.397597\pi\)
\(110\) 0 0
\(111\) 20.5578 1.95126
\(112\) 0 0
\(113\) 6.84302 0.643738 0.321869 0.946784i \(-0.395689\pi\)
0.321869 + 0.946784i \(0.395689\pi\)
\(114\) 0 0
\(115\) −5.37709 −0.501417
\(116\) 0 0
\(117\) −9.57849 −0.885532
\(118\) 0 0
\(119\) 0.510348 0.0467835
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −0.121920 −0.0109931
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.32331 −0.738575 −0.369287 0.929315i \(-0.620398\pi\)
−0.369287 + 0.929315i \(0.620398\pi\)
\(128\) 0 0
\(129\) 28.0267 2.46762
\(130\) 0 0
\(131\) 3.04442 0.265992 0.132996 0.991117i \(-0.457540\pi\)
0.132996 + 0.991117i \(0.457540\pi\)
\(132\) 0 0
\(133\) 0.866746 0.0751564
\(134\) 0 0
\(135\) −4.21076 −0.362404
\(136\) 0 0
\(137\) 17.4245 1.48868 0.744339 0.667802i \(-0.232765\pi\)
0.744339 + 0.667802i \(0.232765\pi\)
\(138\) 0 0
\(139\) −8.02372 −0.680563 −0.340282 0.940324i \(-0.610522\pi\)
−0.340282 + 0.940324i \(0.610522\pi\)
\(140\) 0 0
\(141\) 21.4452 1.80601
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.48965 −0.123709
\(146\) 0 0
\(147\) −19.0919 −1.57467
\(148\) 0 0
\(149\) 3.33768 0.273433 0.136717 0.990610i \(-0.456345\pi\)
0.136717 + 0.990610i \(0.456345\pi\)
\(150\) 0 0
\(151\) 18.4452 1.50105 0.750526 0.660841i \(-0.229800\pi\)
0.750526 + 0.660841i \(0.229800\pi\)
\(152\) 0 0
\(153\) −10.9793 −0.887624
\(154\) 0 0
\(155\) 2.95558 0.237398
\(156\) 0 0
\(157\) 10.7542 0.858278 0.429139 0.903239i \(-0.358817\pi\)
0.429139 + 0.903239i \(0.358817\pi\)
\(158\) 0 0
\(159\) 33.0030 2.61731
\(160\) 0 0
\(161\) −1.13325 −0.0893129
\(162\) 0 0
\(163\) −12.3470 −0.967095 −0.483547 0.875318i \(-0.660652\pi\)
−0.483547 + 0.875318i \(0.660652\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 25.1901 1.94927 0.974633 0.223810i \(-0.0718497\pi\)
0.974633 + 0.223810i \(0.0718497\pi\)
\(168\) 0 0
\(169\) −8.53710 −0.656700
\(170\) 0 0
\(171\) −18.6466 −1.42594
\(172\) 0 0
\(173\) −20.8667 −1.58647 −0.793235 0.608916i \(-0.791605\pi\)
−0.793235 + 0.608916i \(0.791605\pi\)
\(174\) 0 0
\(175\) −0.210756 −0.0159316
\(176\) 0 0
\(177\) −26.3564 −1.98107
\(178\) 0 0
\(179\) −0.931860 −0.0696505 −0.0348252 0.999393i \(-0.511087\pi\)
−0.0348252 + 0.999393i \(0.511087\pi\)
\(180\) 0 0
\(181\) 11.8717 0.882420 0.441210 0.897404i \(-0.354549\pi\)
0.441210 + 0.897404i \(0.354549\pi\)
\(182\) 0 0
\(183\) 22.9937 1.69974
\(184\) 0 0
\(185\) −7.48965 −0.550650
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.887442 −0.0645519
\(190\) 0 0
\(191\) 17.2883 1.25093 0.625467 0.780250i \(-0.284908\pi\)
0.625467 + 0.780250i \(0.284908\pi\)
\(192\) 0 0
\(193\) −22.7829 −1.63995 −0.819975 0.572400i \(-0.806013\pi\)
−0.819975 + 0.572400i \(0.806013\pi\)
\(194\) 0 0
\(195\) 5.79861 0.415247
\(196\) 0 0
\(197\) −6.51035 −0.463843 −0.231922 0.972734i \(-0.574501\pi\)
−0.231922 + 0.972734i \(0.574501\pi\)
\(198\) 0 0
\(199\) −10.0474 −0.712244 −0.356122 0.934439i \(-0.615901\pi\)
−0.356122 + 0.934439i \(0.615901\pi\)
\(200\) 0 0
\(201\) 29.8667 2.10664
\(202\) 0 0
\(203\) −0.313953 −0.0220352
\(204\) 0 0
\(205\) 0.0444180 0.00310229
\(206\) 0 0
\(207\) 24.3801 1.69454
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.66232 0.114439 0.0572196 0.998362i \(-0.481776\pi\)
0.0572196 + 0.998362i \(0.481776\pi\)
\(212\) 0 0
\(213\) 12.4452 0.852733
\(214\) 0 0
\(215\) −10.2108 −0.696368
\(216\) 0 0
\(217\) 0.622906 0.0422856
\(218\) 0 0
\(219\) 44.0474 2.97645
\(220\) 0 0
\(221\) 5.11559 0.344111
\(222\) 0 0
\(223\) 18.0331 1.20758 0.603792 0.797142i \(-0.293656\pi\)
0.603792 + 0.797142i \(0.293656\pi\)
\(224\) 0 0
\(225\) 4.53407 0.302271
\(226\) 0 0
\(227\) 17.1012 1.13505 0.567524 0.823357i \(-0.307901\pi\)
0.567524 + 0.823357i \(0.307901\pi\)
\(228\) 0 0
\(229\) 21.2438 1.40383 0.701916 0.712260i \(-0.252328\pi\)
0.701916 + 0.712260i \(0.252328\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.51337 0.492218 0.246109 0.969242i \(-0.420848\pi\)
0.246109 + 0.969242i \(0.420848\pi\)
\(234\) 0 0
\(235\) −7.81297 −0.509662
\(236\) 0 0
\(237\) −31.2933 −2.03272
\(238\) 0 0
\(239\) 0.112558 0.00728080 0.00364040 0.999993i \(-0.498841\pi\)
0.00364040 + 0.999993i \(0.498841\pi\)
\(240\) 0 0
\(241\) −25.1600 −1.62070 −0.810349 0.585947i \(-0.800723\pi\)
−0.810349 + 0.585947i \(0.800723\pi\)
\(242\) 0 0
\(243\) −18.2438 −1.17034
\(244\) 0 0
\(245\) 6.95558 0.444376
\(246\) 0 0
\(247\) 8.68802 0.552805
\(248\) 0 0
\(249\) −26.3564 −1.67027
\(250\) 0 0
\(251\) 13.3771 0.844355 0.422177 0.906513i \(-0.361266\pi\)
0.422177 + 0.906513i \(0.361266\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 6.64663 0.416228
\(256\) 0 0
\(257\) 2.20139 0.137319 0.0686596 0.997640i \(-0.478128\pi\)
0.0686596 + 0.997640i \(0.478128\pi\)
\(258\) 0 0
\(259\) −1.57849 −0.0980825
\(260\) 0 0
\(261\) 6.75419 0.418074
\(262\) 0 0
\(263\) 5.88744 0.363035 0.181518 0.983388i \(-0.441899\pi\)
0.181518 + 0.983388i \(0.441899\pi\)
\(264\) 0 0
\(265\) −12.0237 −0.738611
\(266\) 0 0
\(267\) −14.8811 −0.910709
\(268\) 0 0
\(269\) −27.4927 −1.67626 −0.838129 0.545472i \(-0.816350\pi\)
−0.838129 + 0.545472i \(0.816350\pi\)
\(270\) 0 0
\(271\) 10.4690 0.635944 0.317972 0.948100i \(-0.396998\pi\)
0.317972 + 0.948100i \(0.396998\pi\)
\(272\) 0 0
\(273\) 1.22209 0.0739643
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −21.7622 −1.30756 −0.653782 0.756683i \(-0.726819\pi\)
−0.653782 + 0.756683i \(0.726819\pi\)
\(278\) 0 0
\(279\) −13.4008 −0.802286
\(280\) 0 0
\(281\) −15.9586 −0.952011 −0.476005 0.879442i \(-0.657916\pi\)
−0.476005 + 0.879442i \(0.657916\pi\)
\(282\) 0 0
\(283\) −13.1663 −0.782658 −0.391329 0.920251i \(-0.627984\pi\)
−0.391329 + 0.920251i \(0.627984\pi\)
\(284\) 0 0
\(285\) 11.2883 0.668659
\(286\) 0 0
\(287\) 0.00936136 0.000552584 0
\(288\) 0 0
\(289\) −11.1363 −0.655075
\(290\) 0 0
\(291\) 16.1600 0.947316
\(292\) 0 0
\(293\) 0.445234 0.0260109 0.0130054 0.999915i \(-0.495860\pi\)
0.0130054 + 0.999915i \(0.495860\pi\)
\(294\) 0 0
\(295\) 9.60221 0.559062
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.3594 −0.656932
\(300\) 0 0
\(301\) −2.15198 −0.124038
\(302\) 0 0
\(303\) −12.3233 −0.707956
\(304\) 0 0
\(305\) −8.37709 −0.479671
\(306\) 0 0
\(307\) −5.60221 −0.319735 −0.159868 0.987138i \(-0.551107\pi\)
−0.159868 + 0.987138i \(0.551107\pi\)
\(308\) 0 0
\(309\) 35.0217 1.99232
\(310\) 0 0
\(311\) −3.26953 −0.185398 −0.0926992 0.995694i \(-0.529549\pi\)
−0.0926992 + 0.995694i \(0.529549\pi\)
\(312\) 0 0
\(313\) −15.9162 −0.899635 −0.449817 0.893121i \(-0.648511\pi\)
−0.449817 + 0.893121i \(0.648511\pi\)
\(314\) 0 0
\(315\) 0.955582 0.0538409
\(316\) 0 0
\(317\) −20.8016 −1.16834 −0.584168 0.811633i \(-0.698579\pi\)
−0.584168 + 0.811633i \(0.698579\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 39.6891 2.21523
\(322\) 0 0
\(323\) 9.95861 0.554112
\(324\) 0 0
\(325\) −2.11256 −0.117184
\(326\) 0 0
\(327\) 18.1219 1.00214
\(328\) 0 0
\(329\) −1.64663 −0.0907816
\(330\) 0 0
\(331\) −3.82733 −0.210369 −0.105184 0.994453i \(-0.533543\pi\)
−0.105184 + 0.994453i \(0.533543\pi\)
\(332\) 0 0
\(333\) 33.9586 1.86092
\(334\) 0 0
\(335\) −10.8811 −0.594498
\(336\) 0 0
\(337\) −27.8461 −1.51687 −0.758436 0.651748i \(-0.774036\pi\)
−0.758436 + 0.651748i \(0.774036\pi\)
\(338\) 0 0
\(339\) 18.7829 1.02015
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.94122 0.158811
\(344\) 0 0
\(345\) −14.7592 −0.794608
\(346\) 0 0
\(347\) 27.6403 1.48381 0.741904 0.670506i \(-0.233923\pi\)
0.741904 + 0.670506i \(0.233923\pi\)
\(348\) 0 0
\(349\) −18.5578 −0.993376 −0.496688 0.867929i \(-0.665451\pi\)
−0.496688 + 0.867929i \(0.665451\pi\)
\(350\) 0 0
\(351\) −8.89547 −0.474805
\(352\) 0 0
\(353\) −1.46593 −0.0780236 −0.0390118 0.999239i \(-0.512421\pi\)
−0.0390118 + 0.999239i \(0.512421\pi\)
\(354\) 0 0
\(355\) −4.53407 −0.240643
\(356\) 0 0
\(357\) 1.40082 0.0741390
\(358\) 0 0
\(359\) −0.445234 −0.0234986 −0.0117493 0.999931i \(-0.503740\pi\)
−0.0117493 + 0.999931i \(0.503740\pi\)
\(360\) 0 0
\(361\) −2.08686 −0.109835
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.0474 −0.839962
\(366\) 0 0
\(367\) 1.61657 0.0843843 0.0421922 0.999110i \(-0.486566\pi\)
0.0421922 + 0.999110i \(0.486566\pi\)
\(368\) 0 0
\(369\) −0.201395 −0.0104842
\(370\) 0 0
\(371\) −2.53407 −0.131562
\(372\) 0 0
\(373\) −4.22012 −0.218509 −0.109255 0.994014i \(-0.534846\pi\)
−0.109255 + 0.994014i \(0.534846\pi\)
\(374\) 0 0
\(375\) −2.74483 −0.141742
\(376\) 0 0
\(377\) −3.14698 −0.162078
\(378\) 0 0
\(379\) −2.33268 −0.119822 −0.0599108 0.998204i \(-0.519082\pi\)
−0.0599108 + 0.998204i \(0.519082\pi\)
\(380\) 0 0
\(381\) −22.8461 −1.17044
\(382\) 0 0
\(383\) −11.8461 −0.605305 −0.302652 0.953101i \(-0.597872\pi\)
−0.302652 + 0.953101i \(0.597872\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 46.2963 2.35337
\(388\) 0 0
\(389\) −28.0505 −1.42222 −0.711108 0.703083i \(-0.751806\pi\)
−0.711108 + 0.703083i \(0.751806\pi\)
\(390\) 0 0
\(391\) −13.0207 −0.658485
\(392\) 0 0
\(393\) 8.35640 0.421525
\(394\) 0 0
\(395\) 11.4008 0.573637
\(396\) 0 0
\(397\) 1.06814 0.0536084 0.0268042 0.999641i \(-0.491467\pi\)
0.0268042 + 0.999641i \(0.491467\pi\)
\(398\) 0 0
\(399\) 2.37907 0.119102
\(400\) 0 0
\(401\) 24.6860 1.23276 0.616381 0.787448i \(-0.288598\pi\)
0.616381 + 0.787448i \(0.288598\pi\)
\(402\) 0 0
\(403\) 6.24384 0.311028
\(404\) 0 0
\(405\) 2.04442 0.101588
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 12.0919 0.597904 0.298952 0.954268i \(-0.403363\pi\)
0.298952 + 0.954268i \(0.403363\pi\)
\(410\) 0 0
\(411\) 47.8273 2.35915
\(412\) 0 0
\(413\) 2.02372 0.0995809
\(414\) 0 0
\(415\) 9.60221 0.471354
\(416\) 0 0
\(417\) −22.0237 −1.07851
\(418\) 0 0
\(419\) −16.4927 −0.805720 −0.402860 0.915262i \(-0.631984\pi\)
−0.402860 + 0.915262i \(0.631984\pi\)
\(420\) 0 0
\(421\) 18.8874 0.920518 0.460259 0.887785i \(-0.347757\pi\)
0.460259 + 0.887785i \(0.347757\pi\)
\(422\) 0 0
\(423\) 35.4245 1.72240
\(424\) 0 0
\(425\) −2.42151 −0.117461
\(426\) 0 0
\(427\) −1.76552 −0.0854396
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.12825 −0.150683 −0.0753414 0.997158i \(-0.524005\pi\)
−0.0753414 + 0.997158i \(0.524005\pi\)
\(432\) 0 0
\(433\) 12.1126 0.582092 0.291046 0.956709i \(-0.405997\pi\)
0.291046 + 0.956709i \(0.405997\pi\)
\(434\) 0 0
\(435\) −4.08884 −0.196045
\(436\) 0 0
\(437\) −22.1136 −1.05784
\(438\) 0 0
\(439\) 34.5164 1.64738 0.823689 0.567042i \(-0.191912\pi\)
0.823689 + 0.567042i \(0.191912\pi\)
\(440\) 0 0
\(441\) −31.5371 −1.50177
\(442\) 0 0
\(443\) −13.2502 −0.629535 −0.314767 0.949169i \(-0.601927\pi\)
−0.314767 + 0.949169i \(0.601927\pi\)
\(444\) 0 0
\(445\) 5.42151 0.257004
\(446\) 0 0
\(447\) 9.16134 0.433316
\(448\) 0 0
\(449\) 38.4245 1.81337 0.906683 0.421813i \(-0.138606\pi\)
0.906683 + 0.421813i \(0.138606\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 50.6290 2.37876
\(454\) 0 0
\(455\) −0.445234 −0.0208729
\(456\) 0 0
\(457\) −29.3958 −1.37508 −0.687539 0.726147i \(-0.741309\pi\)
−0.687539 + 0.726147i \(0.741309\pi\)
\(458\) 0 0
\(459\) −10.1964 −0.475927
\(460\) 0 0
\(461\) 28.5785 1.33103 0.665516 0.746383i \(-0.268211\pi\)
0.665516 + 0.746383i \(0.268211\pi\)
\(462\) 0 0
\(463\) 12.5247 0.582073 0.291036 0.956712i \(-0.406000\pi\)
0.291036 + 0.956712i \(0.406000\pi\)
\(464\) 0 0
\(465\) 8.11256 0.376211
\(466\) 0 0
\(467\) −28.8210 −1.33368 −0.666838 0.745202i \(-0.732353\pi\)
−0.666838 + 0.745202i \(0.732353\pi\)
\(468\) 0 0
\(469\) −2.29326 −0.105893
\(470\) 0 0
\(471\) 29.5184 1.36013
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.11256 −0.188697
\(476\) 0 0
\(477\) 54.5164 2.49613
\(478\) 0 0
\(479\) 23.1757 1.05892 0.529462 0.848333i \(-0.322394\pi\)
0.529462 + 0.848333i \(0.322394\pi\)
\(480\) 0 0
\(481\) −15.8223 −0.721436
\(482\) 0 0
\(483\) −3.11059 −0.141537
\(484\) 0 0
\(485\) −5.88744 −0.267335
\(486\) 0 0
\(487\) 17.5548 0.795482 0.397741 0.917498i \(-0.369794\pi\)
0.397741 + 0.917498i \(0.369794\pi\)
\(488\) 0 0
\(489\) −33.8905 −1.53258
\(490\) 0 0
\(491\) −8.19639 −0.369898 −0.184949 0.982748i \(-0.559212\pi\)
−0.184949 + 0.982748i \(0.559212\pi\)
\(492\) 0 0
\(493\) −3.60721 −0.162461
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.955582 −0.0428637
\(498\) 0 0
\(499\) 41.8223 1.87222 0.936112 0.351701i \(-0.114397\pi\)
0.936112 + 0.351701i \(0.114397\pi\)
\(500\) 0 0
\(501\) 69.1423 3.08905
\(502\) 0 0
\(503\) −23.3313 −1.04029 −0.520147 0.854077i \(-0.674123\pi\)
−0.520147 + 0.854077i \(0.674123\pi\)
\(504\) 0 0
\(505\) 4.48965 0.199787
\(506\) 0 0
\(507\) −23.4328 −1.04069
\(508\) 0 0
\(509\) −10.8510 −0.480964 −0.240482 0.970654i \(-0.577306\pi\)
−0.240482 + 0.970654i \(0.577306\pi\)
\(510\) 0 0
\(511\) −3.38209 −0.149615
\(512\) 0 0
\(513\) −17.3170 −0.764563
\(514\) 0 0
\(515\) −12.7592 −0.562237
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −57.2756 −2.51412
\(520\) 0 0
\(521\) −1.96058 −0.0858946 −0.0429473 0.999077i \(-0.513675\pi\)
−0.0429473 + 0.999077i \(0.513675\pi\)
\(522\) 0 0
\(523\) −24.8667 −1.08735 −0.543673 0.839297i \(-0.682967\pi\)
−0.543673 + 0.839297i \(0.682967\pi\)
\(524\) 0 0
\(525\) −0.578488 −0.0252473
\(526\) 0 0
\(527\) 7.15698 0.311763
\(528\) 0 0
\(529\) 5.91314 0.257093
\(530\) 0 0
\(531\) −43.5371 −1.88935
\(532\) 0 0
\(533\) 0.0938357 0.00406448
\(534\) 0 0
\(535\) −14.4596 −0.625143
\(536\) 0 0
\(537\) −2.55779 −0.110377
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.3140 0.744385 0.372192 0.928156i \(-0.378606\pi\)
0.372192 + 0.928156i \(0.378606\pi\)
\(542\) 0 0
\(543\) 32.5859 1.39839
\(544\) 0 0
\(545\) −6.60221 −0.282808
\(546\) 0 0
\(547\) −16.5341 −0.706946 −0.353473 0.935445i \(-0.614999\pi\)
−0.353473 + 0.935445i \(0.614999\pi\)
\(548\) 0 0
\(549\) 37.9823 1.62105
\(550\) 0 0
\(551\) −6.12628 −0.260988
\(552\) 0 0
\(553\) 2.40279 0.102177
\(554\) 0 0
\(555\) −20.5578 −0.872630
\(556\) 0 0
\(557\) 4.33268 0.183581 0.0917907 0.995778i \(-0.470741\pi\)
0.0917907 + 0.995778i \(0.470741\pi\)
\(558\) 0 0
\(559\) −21.5708 −0.912348
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 41.2375 1.73795 0.868977 0.494853i \(-0.164778\pi\)
0.868977 + 0.494853i \(0.164778\pi\)
\(564\) 0 0
\(565\) −6.84302 −0.287888
\(566\) 0 0
\(567\) 0.430873 0.0180950
\(568\) 0 0
\(569\) −23.6229 −0.990324 −0.495162 0.868801i \(-0.664891\pi\)
−0.495162 + 0.868801i \(0.664891\pi\)
\(570\) 0 0
\(571\) 0.871746 0.0364814 0.0182407 0.999834i \(-0.494193\pi\)
0.0182407 + 0.999834i \(0.494193\pi\)
\(572\) 0 0
\(573\) 47.4533 1.98239
\(574\) 0 0
\(575\) 5.37709 0.224240
\(576\) 0 0
\(577\) 23.8748 0.993920 0.496960 0.867774i \(-0.334450\pi\)
0.496960 + 0.867774i \(0.334450\pi\)
\(578\) 0 0
\(579\) −62.5351 −2.59887
\(580\) 0 0
\(581\) 2.02372 0.0839582
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 9.57849 0.396022
\(586\) 0 0
\(587\) −22.9649 −0.947865 −0.473932 0.880561i \(-0.657166\pi\)
−0.473932 + 0.880561i \(0.657166\pi\)
\(588\) 0 0
\(589\) 12.1550 0.500838
\(590\) 0 0
\(591\) −17.8698 −0.735064
\(592\) 0 0
\(593\) −12.2963 −0.504948 −0.252474 0.967604i \(-0.581244\pi\)
−0.252474 + 0.967604i \(0.581244\pi\)
\(594\) 0 0
\(595\) −0.510348 −0.0209222
\(596\) 0 0
\(597\) −27.5785 −1.12871
\(598\) 0 0
\(599\) −43.9823 −1.79707 −0.898535 0.438903i \(-0.855367\pi\)
−0.898535 + 0.438903i \(0.855367\pi\)
\(600\) 0 0
\(601\) 41.5371 1.69433 0.847167 0.531327i \(-0.178306\pi\)
0.847167 + 0.531327i \(0.178306\pi\)
\(602\) 0 0
\(603\) 49.3357 2.00911
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 38.6002 1.56674 0.783368 0.621559i \(-0.213500\pi\)
0.783368 + 0.621559i \(0.213500\pi\)
\(608\) 0 0
\(609\) −0.861746 −0.0349197
\(610\) 0 0
\(611\) −16.5053 −0.667735
\(612\) 0 0
\(613\) −22.8304 −0.922109 −0.461055 0.887372i \(-0.652529\pi\)
−0.461055 + 0.887372i \(0.652529\pi\)
\(614\) 0 0
\(615\) 0.121920 0.00491628
\(616\) 0 0
\(617\) −27.0317 −1.08826 −0.544129 0.839002i \(-0.683140\pi\)
−0.544129 + 0.839002i \(0.683140\pi\)
\(618\) 0 0
\(619\) 24.6179 0.989477 0.494739 0.869042i \(-0.335264\pi\)
0.494739 + 0.869042i \(0.335264\pi\)
\(620\) 0 0
\(621\) 22.6416 0.908577
\(622\) 0 0
\(623\) 1.14262 0.0457779
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.1363 −0.723141
\(630\) 0 0
\(631\) 5.46093 0.217396 0.108698 0.994075i \(-0.465332\pi\)
0.108698 + 0.994075i \(0.465332\pi\)
\(632\) 0 0
\(633\) 4.56279 0.181355
\(634\) 0 0
\(635\) 8.32331 0.330301
\(636\) 0 0
\(637\) 14.6941 0.582200
\(638\) 0 0
\(639\) 20.5578 0.813254
\(640\) 0 0
\(641\) −9.20442 −0.363553 −0.181776 0.983340i \(-0.558185\pi\)
−0.181776 + 0.983340i \(0.558185\pi\)
\(642\) 0 0
\(643\) 21.9542 0.865791 0.432895 0.901444i \(-0.357492\pi\)
0.432895 + 0.901444i \(0.357492\pi\)
\(644\) 0 0
\(645\) −28.0267 −1.10355
\(646\) 0 0
\(647\) 30.6323 1.20428 0.602139 0.798391i \(-0.294315\pi\)
0.602139 + 0.798391i \(0.294315\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 1.70977 0.0670111
\(652\) 0 0
\(653\) 37.3594 1.46199 0.730994 0.682384i \(-0.239057\pi\)
0.730994 + 0.682384i \(0.239057\pi\)
\(654\) 0 0
\(655\) −3.04442 −0.118955
\(656\) 0 0
\(657\) 72.7602 2.83865
\(658\) 0 0
\(659\) 39.2231 1.52792 0.763958 0.645266i \(-0.223253\pi\)
0.763958 + 0.645266i \(0.223253\pi\)
\(660\) 0 0
\(661\) 10.9349 0.425318 0.212659 0.977126i \(-0.431788\pi\)
0.212659 + 0.977126i \(0.431788\pi\)
\(662\) 0 0
\(663\) 14.0414 0.545322
\(664\) 0 0
\(665\) −0.866746 −0.0336110
\(666\) 0 0
\(667\) 8.01000 0.310148
\(668\) 0 0
\(669\) 49.4977 1.91369
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 40.8254 1.57370 0.786851 0.617143i \(-0.211710\pi\)
0.786851 + 0.617143i \(0.211710\pi\)
\(674\) 0 0
\(675\) 4.21076 0.162072
\(676\) 0 0
\(677\) −25.0207 −0.961623 −0.480812 0.876824i \(-0.659658\pi\)
−0.480812 + 0.876824i \(0.659658\pi\)
\(678\) 0 0
\(679\) −1.24081 −0.0476180
\(680\) 0 0
\(681\) 46.9399 1.79874
\(682\) 0 0
\(683\) 16.8761 0.645746 0.322873 0.946442i \(-0.395351\pi\)
0.322873 + 0.946442i \(0.395351\pi\)
\(684\) 0 0
\(685\) −17.4245 −0.665757
\(686\) 0 0
\(687\) 58.3106 2.22469
\(688\) 0 0
\(689\) −25.4008 −0.967694
\(690\) 0 0
\(691\) −36.0899 −1.37292 −0.686462 0.727166i \(-0.740837\pi\)
−0.686462 + 0.727166i \(0.740837\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.02372 0.304357
\(696\) 0 0
\(697\) 0.107559 0.00407408
\(698\) 0 0
\(699\) 20.6229 0.780030
\(700\) 0 0
\(701\) −32.2438 −1.21783 −0.608917 0.793234i \(-0.708396\pi\)
−0.608917 + 0.793234i \(0.708396\pi\)
\(702\) 0 0
\(703\) −30.8016 −1.16171
\(704\) 0 0
\(705\) −21.4452 −0.807674
\(706\) 0 0
\(707\) 0.946221 0.0355863
\(708\) 0 0
\(709\) −16.8698 −0.633558 −0.316779 0.948499i \(-0.602601\pi\)
−0.316779 + 0.948499i \(0.602601\pi\)
\(710\) 0 0
\(711\) −51.6921 −1.93861
\(712\) 0 0
\(713\) −15.8924 −0.595177
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.308953 0.0115381
\(718\) 0 0
\(719\) −40.0424 −1.49333 −0.746666 0.665200i \(-0.768346\pi\)
−0.746666 + 0.665200i \(0.768346\pi\)
\(720\) 0 0
\(721\) −2.68907 −0.100146
\(722\) 0 0
\(723\) −69.0598 −2.56836
\(724\) 0 0
\(725\) 1.48965 0.0553243
\(726\) 0 0
\(727\) −23.7892 −0.882294 −0.441147 0.897435i \(-0.645428\pi\)
−0.441147 + 0.897435i \(0.645428\pi\)
\(728\) 0 0
\(729\) −43.9429 −1.62752
\(730\) 0 0
\(731\) −24.7255 −0.914504
\(732\) 0 0
\(733\) 34.9666 1.29152 0.645761 0.763540i \(-0.276540\pi\)
0.645761 + 0.763540i \(0.276540\pi\)
\(734\) 0 0
\(735\) 19.0919 0.704214
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 31.8510 1.17166 0.585830 0.810434i \(-0.300769\pi\)
0.585830 + 0.810434i \(0.300769\pi\)
\(740\) 0 0
\(741\) 23.8471 0.876045
\(742\) 0 0
\(743\) 26.7448 0.981173 0.490586 0.871393i \(-0.336783\pi\)
0.490586 + 0.871393i \(0.336783\pi\)
\(744\) 0 0
\(745\) −3.33768 −0.122283
\(746\) 0 0
\(747\) −43.5371 −1.59294
\(748\) 0 0
\(749\) −3.04744 −0.111351
\(750\) 0 0
\(751\) 6.45023 0.235372 0.117686 0.993051i \(-0.462452\pi\)
0.117686 + 0.993051i \(0.462452\pi\)
\(752\) 0 0
\(753\) 36.7178 1.33807
\(754\) 0 0
\(755\) −18.4452 −0.671291
\(756\) 0 0
\(757\) 13.3771 0.486199 0.243099 0.970001i \(-0.421836\pi\)
0.243099 + 0.970001i \(0.421836\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11.1156 0.402940 0.201470 0.979495i \(-0.435428\pi\)
0.201470 + 0.979495i \(0.435428\pi\)
\(762\) 0 0
\(763\) −1.39145 −0.0503740
\(764\) 0 0
\(765\) 10.9793 0.396958
\(766\) 0 0
\(767\) 20.2852 0.732457
\(768\) 0 0
\(769\) 24.8717 0.896898 0.448449 0.893808i \(-0.351977\pi\)
0.448449 + 0.893808i \(0.351977\pi\)
\(770\) 0 0
\(771\) 6.04245 0.217613
\(772\) 0 0
\(773\) −38.3564 −1.37958 −0.689792 0.724008i \(-0.742298\pi\)
−0.689792 + 0.724008i \(0.742298\pi\)
\(774\) 0 0
\(775\) −2.95558 −0.106168
\(776\) 0 0
\(777\) −4.33268 −0.155434
\(778\) 0 0
\(779\) 0.182672 0.00654490
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.27256 0.224163
\(784\) 0 0
\(785\) −10.7542 −0.383833
\(786\) 0 0
\(787\) 15.2552 0.543788 0.271894 0.962327i \(-0.412350\pi\)
0.271894 + 0.962327i \(0.412350\pi\)
\(788\) 0 0
\(789\) 16.1600 0.575311
\(790\) 0 0
\(791\) −1.44221 −0.0512790
\(792\) 0 0
\(793\) −17.6971 −0.628442
\(794\) 0 0
\(795\) −33.0030 −1.17050
\(796\) 0 0
\(797\) −47.9222 −1.69749 −0.848746 0.528801i \(-0.822642\pi\)
−0.848746 + 0.528801i \(0.822642\pi\)
\(798\) 0 0
\(799\) −18.9192 −0.669313
\(800\) 0 0
\(801\) −24.5815 −0.868545
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 1.13325 0.0399420
\(806\) 0 0
\(807\) −75.4626 −2.65641
\(808\) 0 0
\(809\) −39.2331 −1.37936 −0.689682 0.724112i \(-0.742250\pi\)
−0.689682 + 0.724112i \(0.742250\pi\)
\(810\) 0 0
\(811\) −44.4740 −1.56169 −0.780846 0.624724i \(-0.785212\pi\)
−0.780846 + 0.624724i \(0.785212\pi\)
\(812\) 0 0
\(813\) 28.7355 1.00780
\(814\) 0 0
\(815\) 12.3470 0.432498
\(816\) 0 0
\(817\) −41.9923 −1.46913
\(818\) 0 0
\(819\) 2.01872 0.0705399
\(820\) 0 0
\(821\) −5.41651 −0.189038 −0.0945188 0.995523i \(-0.530131\pi\)
−0.0945188 + 0.995523i \(0.530131\pi\)
\(822\) 0 0
\(823\) −34.6146 −1.20659 −0.603295 0.797518i \(-0.706146\pi\)
−0.603295 + 0.797518i \(0.706146\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 42.5197 1.47856 0.739278 0.673401i \(-0.235167\pi\)
0.739278 + 0.673401i \(0.235167\pi\)
\(828\) 0 0
\(829\) 15.5942 0.541608 0.270804 0.962634i \(-0.412710\pi\)
0.270804 + 0.962634i \(0.412710\pi\)
\(830\) 0 0
\(831\) −59.7335 −2.07213
\(832\) 0 0
\(833\) 16.8430 0.583576
\(834\) 0 0
\(835\) −25.1901 −0.871738
\(836\) 0 0
\(837\) −12.4452 −0.430170
\(838\) 0 0
\(839\) −10.4977 −0.362420 −0.181210 0.983444i \(-0.558001\pi\)
−0.181210 + 0.983444i \(0.558001\pi\)
\(840\) 0 0
\(841\) −26.7809 −0.923481
\(842\) 0 0
\(843\) −43.8036 −1.50868
\(844\) 0 0
\(845\) 8.53710 0.293685
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −36.1393 −1.24030
\(850\) 0 0
\(851\) 40.2726 1.38053
\(852\) 0 0
\(853\) 25.1156 0.859941 0.429971 0.902843i \(-0.358524\pi\)
0.429971 + 0.902843i \(0.358524\pi\)
\(854\) 0 0
\(855\) 18.6466 0.637701
\(856\) 0 0
\(857\) −4.46896 −0.152657 −0.0763283 0.997083i \(-0.524320\pi\)
−0.0763283 + 0.997083i \(0.524320\pi\)
\(858\) 0 0
\(859\) −44.0424 −1.50271 −0.751354 0.659899i \(-0.770599\pi\)
−0.751354 + 0.659899i \(0.770599\pi\)
\(860\) 0 0
\(861\) 0.0256953 0.000875694 0
\(862\) 0 0
\(863\) −20.9412 −0.712847 −0.356424 0.934324i \(-0.616004\pi\)
−0.356424 + 0.934324i \(0.616004\pi\)
\(864\) 0 0
\(865\) 20.8667 0.709491
\(866\) 0 0
\(867\) −30.5672 −1.03811
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −22.9870 −0.778884
\(872\) 0 0
\(873\) 26.6941 0.903457
\(874\) 0 0
\(875\) 0.210756 0.00712485
\(876\) 0 0
\(877\) −42.6764 −1.44108 −0.720540 0.693413i \(-0.756106\pi\)
−0.720540 + 0.693413i \(0.756106\pi\)
\(878\) 0 0
\(879\) 1.22209 0.0412201
\(880\) 0 0
\(881\) −48.5608 −1.63606 −0.818028 0.575179i \(-0.804933\pi\)
−0.818028 + 0.575179i \(0.804933\pi\)
\(882\) 0 0
\(883\) 22.0424 0.741787 0.370894 0.928675i \(-0.379051\pi\)
0.370894 + 0.928675i \(0.379051\pi\)
\(884\) 0 0
\(885\) 26.3564 0.885961
\(886\) 0 0
\(887\) −36.3056 −1.21902 −0.609512 0.792777i \(-0.708635\pi\)
−0.609512 + 0.792777i \(0.708635\pi\)
\(888\) 0 0
\(889\) 1.75419 0.0588336
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −32.1313 −1.07523
\(894\) 0 0
\(895\) 0.931860 0.0311486
\(896\) 0 0
\(897\) −31.1796 −1.04106
\(898\) 0 0
\(899\) −4.40279 −0.146841
\(900\) 0 0
\(901\) −29.1156 −0.969981
\(902\) 0 0
\(903\) −5.90680 −0.196566
\(904\) 0 0
\(905\) −11.8717 −0.394630
\(906\) 0 0
\(907\) −8.59088 −0.285255 −0.142628 0.989776i \(-0.545555\pi\)
−0.142628 + 0.989776i \(0.545555\pi\)
\(908\) 0 0
\(909\) −20.3564 −0.675179
\(910\) 0 0
\(911\) 3.30593 0.109530 0.0547651 0.998499i \(-0.482559\pi\)
0.0547651 + 0.998499i \(0.482559\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −22.9937 −0.760147
\(916\) 0 0
\(917\) −0.641629 −0.0211885
\(918\) 0 0
\(919\) 18.7592 0.618808 0.309404 0.950931i \(-0.399870\pi\)
0.309404 + 0.950931i \(0.399870\pi\)
\(920\) 0 0
\(921\) −15.3771 −0.506692
\(922\) 0 0
\(923\) −9.57849 −0.315280
\(924\) 0 0
\(925\) 7.48965 0.246258
\(926\) 0 0
\(927\) 57.8510 1.90008
\(928\) 0 0
\(929\) 9.77488 0.320704 0.160352 0.987060i \(-0.448737\pi\)
0.160352 + 0.987060i \(0.448737\pi\)
\(930\) 0 0
\(931\) 28.6052 0.937499
\(932\) 0 0
\(933\) −8.97430 −0.293806
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 51.6734 1.68810 0.844048 0.536268i \(-0.180166\pi\)
0.844048 + 0.536268i \(0.180166\pi\)
\(938\) 0 0
\(939\) −43.6871 −1.42567
\(940\) 0 0
\(941\) 1.93186 0.0629768 0.0314884 0.999504i \(-0.489975\pi\)
0.0314884 + 0.999504i \(0.489975\pi\)
\(942\) 0 0
\(943\) −0.238840 −0.00777770
\(944\) 0 0
\(945\) 0.887442 0.0288685
\(946\) 0 0
\(947\) −27.5735 −0.896018 −0.448009 0.894029i \(-0.647867\pi\)
−0.448009 + 0.894029i \(0.647867\pi\)
\(948\) 0 0
\(949\) −33.9012 −1.10048
\(950\) 0 0
\(951\) −57.0969 −1.85149
\(952\) 0 0
\(953\) 9.22314 0.298767 0.149383 0.988779i \(-0.452271\pi\)
0.149383 + 0.988779i \(0.452271\pi\)
\(954\) 0 0
\(955\) −17.2883 −0.559435
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.67232 −0.118586
\(960\) 0 0
\(961\) −22.2645 −0.718211
\(962\) 0 0
\(963\) 65.5608 2.11267
\(964\) 0 0
\(965\) 22.7829 0.733408
\(966\) 0 0
\(967\) 24.6290 0.792014 0.396007 0.918248i \(-0.370396\pi\)
0.396007 + 0.918248i \(0.370396\pi\)
\(968\) 0 0
\(969\) 27.3346 0.878115
\(970\) 0 0
\(971\) 6.69407 0.214823 0.107412 0.994215i \(-0.465744\pi\)
0.107412 + 0.994215i \(0.465744\pi\)
\(972\) 0 0
\(973\) 1.69105 0.0542125
\(974\) 0 0
\(975\) −5.79861 −0.185704
\(976\) 0 0
\(977\) −23.4422 −0.749983 −0.374991 0.927028i \(-0.622354\pi\)
−0.374991 + 0.927028i \(0.622354\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 29.9349 0.955747
\(982\) 0 0
\(983\) −12.7735 −0.407413 −0.203706 0.979032i \(-0.565299\pi\)
−0.203706 + 0.979032i \(0.565299\pi\)
\(984\) 0 0
\(985\) 6.51035 0.207437
\(986\) 0 0
\(987\) −4.51971 −0.143864
\(988\) 0 0
\(989\) 54.9042 1.74585
\(990\) 0 0
\(991\) 12.5053 0.397245 0.198623 0.980076i \(-0.436353\pi\)
0.198623 + 0.980076i \(0.436353\pi\)
\(992\) 0 0
\(993\) −10.5053 −0.333377
\(994\) 0 0
\(995\) 10.0474 0.318525
\(996\) 0 0
\(997\) 40.1787 1.27247 0.636237 0.771494i \(-0.280490\pi\)
0.636237 + 0.771494i \(0.280490\pi\)
\(998\) 0 0
\(999\) 31.5371 0.997790
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.cc.1.3 3
4.3 odd 2 4840.2.a.t.1.1 3
11.10 odd 2 9680.2.a.ca.1.3 3
44.43 even 2 4840.2.a.u.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.t.1.1 3 4.3 odd 2
4840.2.a.u.1.1 yes 3 44.43 even 2
9680.2.a.ca.1.3 3 11.10 odd 2
9680.2.a.cc.1.3 3 1.1 even 1 trivial