Properties

Label 1638.2.a.x.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.37228 q^{10} +2.37228 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.37228 q^{17} +1.62772 q^{19} -2.37228 q^{20} +2.37228 q^{22} +3.62772 q^{23} +0.627719 q^{25} -1.00000 q^{26} -1.00000 q^{28} +6.37228 q^{29} -4.74456 q^{31} +1.00000 q^{32} +4.37228 q^{34} +2.37228 q^{35} -4.37228 q^{37} +1.62772 q^{38} -2.37228 q^{40} +8.74456 q^{41} +11.1168 q^{43} +2.37228 q^{44} +3.62772 q^{46} +1.25544 q^{47} +1.00000 q^{49} +0.627719 q^{50} -1.00000 q^{52} +8.74456 q^{53} -5.62772 q^{55} -1.00000 q^{56} +6.37228 q^{58} +2.00000 q^{59} +5.11684 q^{61} -4.74456 q^{62} +1.00000 q^{64} +2.37228 q^{65} +9.48913 q^{67} +4.37228 q^{68} +2.37228 q^{70} -4.74456 q^{71} -8.37228 q^{73} -4.37228 q^{74} +1.62772 q^{76} -2.37228 q^{77} +4.74456 q^{79} -2.37228 q^{80} +8.74456 q^{82} -6.00000 q^{83} -10.3723 q^{85} +11.1168 q^{86} +2.37228 q^{88} -3.25544 q^{89} +1.00000 q^{91} +3.62772 q^{92} +1.25544 q^{94} -3.86141 q^{95} +7.48913 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} - q^{11} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 3 q^{17} + 9 q^{19} + q^{20} - q^{22} + 13 q^{23} + 7 q^{25} - 2 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 3 q^{34} - q^{35} - 3 q^{37} + 9 q^{38} + q^{40} + 6 q^{41} + 5 q^{43} - q^{44} + 13 q^{46} + 14 q^{47} + 2 q^{49} + 7 q^{50} - 2 q^{52} + 6 q^{53} - 17 q^{55} - 2 q^{56} + 7 q^{58} + 4 q^{59} - 7 q^{61} + 2 q^{62} + 2 q^{64} - q^{65} - 4 q^{67} + 3 q^{68} - q^{70} + 2 q^{71} - 11 q^{73} - 3 q^{74} + 9 q^{76} + q^{77} - 2 q^{79} + q^{80} + 6 q^{82} - 12 q^{83} - 15 q^{85} + 5 q^{86} - q^{88} - 18 q^{89} + 2 q^{91} + 13 q^{92} + 14 q^{94} + 21 q^{95} - 8 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −2.37228 −1.06092 −0.530458 0.847711i \(-0.677980\pi\)
−0.530458 + 0.847711i \(0.677980\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.37228 −0.750181
\(11\) 2.37228 0.715270 0.357635 0.933862i \(-0.383583\pi\)
0.357635 + 0.933862i \(0.383583\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.37228 1.06043 0.530217 0.847862i \(-0.322110\pi\)
0.530217 + 0.847862i \(0.322110\pi\)
\(18\) 0 0
\(19\) 1.62772 0.373424 0.186712 0.982415i \(-0.440217\pi\)
0.186712 + 0.982415i \(0.440217\pi\)
\(20\) −2.37228 −0.530458
\(21\) 0 0
\(22\) 2.37228 0.505772
\(23\) 3.62772 0.756432 0.378216 0.925717i \(-0.376538\pi\)
0.378216 + 0.925717i \(0.376538\pi\)
\(24\) 0 0
\(25\) 0.627719 0.125544
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 6.37228 1.18330 0.591651 0.806194i \(-0.298476\pi\)
0.591651 + 0.806194i \(0.298476\pi\)
\(30\) 0 0
\(31\) −4.74456 −0.852149 −0.426074 0.904688i \(-0.640104\pi\)
−0.426074 + 0.904688i \(0.640104\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.37228 0.749840
\(35\) 2.37228 0.400989
\(36\) 0 0
\(37\) −4.37228 −0.718799 −0.359399 0.933184i \(-0.617018\pi\)
−0.359399 + 0.933184i \(0.617018\pi\)
\(38\) 1.62772 0.264051
\(39\) 0 0
\(40\) −2.37228 −0.375091
\(41\) 8.74456 1.36567 0.682836 0.730572i \(-0.260747\pi\)
0.682836 + 0.730572i \(0.260747\pi\)
\(42\) 0 0
\(43\) 11.1168 1.69530 0.847651 0.530554i \(-0.178016\pi\)
0.847651 + 0.530554i \(0.178016\pi\)
\(44\) 2.37228 0.357635
\(45\) 0 0
\(46\) 3.62772 0.534878
\(47\) 1.25544 0.183124 0.0915622 0.995799i \(-0.470814\pi\)
0.0915622 + 0.995799i \(0.470814\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.627719 0.0887728
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) 8.74456 1.20116 0.600579 0.799565i \(-0.294937\pi\)
0.600579 + 0.799565i \(0.294937\pi\)
\(54\) 0 0
\(55\) −5.62772 −0.758841
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.37228 0.836722
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) 5.11684 0.655145 0.327572 0.944826i \(-0.393769\pi\)
0.327572 + 0.944826i \(0.393769\pi\)
\(62\) −4.74456 −0.602560
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 2.37228 0.294245
\(66\) 0 0
\(67\) 9.48913 1.15928 0.579641 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(68\) 4.37228 0.530217
\(69\) 0 0
\(70\) 2.37228 0.283542
\(71\) −4.74456 −0.563076 −0.281538 0.959550i \(-0.590845\pi\)
−0.281538 + 0.959550i \(0.590845\pi\)
\(72\) 0 0
\(73\) −8.37228 −0.979901 −0.489951 0.871750i \(-0.662985\pi\)
−0.489951 + 0.871750i \(0.662985\pi\)
\(74\) −4.37228 −0.508267
\(75\) 0 0
\(76\) 1.62772 0.186712
\(77\) −2.37228 −0.270347
\(78\) 0 0
\(79\) 4.74456 0.533805 0.266903 0.963724i \(-0.414000\pi\)
0.266903 + 0.963724i \(0.414000\pi\)
\(80\) −2.37228 −0.265229
\(81\) 0 0
\(82\) 8.74456 0.965675
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −10.3723 −1.12503
\(86\) 11.1168 1.19876
\(87\) 0 0
\(88\) 2.37228 0.252886
\(89\) −3.25544 −0.345076 −0.172538 0.985003i \(-0.555197\pi\)
−0.172538 + 0.985003i \(0.555197\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 3.62772 0.378216
\(93\) 0 0
\(94\) 1.25544 0.129488
\(95\) −3.86141 −0.396172
\(96\) 0 0
\(97\) 7.48913 0.760405 0.380203 0.924903i \(-0.375854\pi\)
0.380203 + 0.924903i \(0.375854\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 0.627719 0.0627719
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −11.8614 −1.16874 −0.584370 0.811488i \(-0.698658\pi\)
−0.584370 + 0.811488i \(0.698658\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 8.74456 0.849347
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −3.62772 −0.347472 −0.173736 0.984792i \(-0.555584\pi\)
−0.173736 + 0.984792i \(0.555584\pi\)
\(110\) −5.62772 −0.536582
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 20.0000 1.88144 0.940721 0.339182i \(-0.110150\pi\)
0.940721 + 0.339182i \(0.110150\pi\)
\(114\) 0 0
\(115\) −8.60597 −0.802511
\(116\) 6.37228 0.591651
\(117\) 0 0
\(118\) 2.00000 0.184115
\(119\) −4.37228 −0.400806
\(120\) 0 0
\(121\) −5.37228 −0.488389
\(122\) 5.11684 0.463257
\(123\) 0 0
\(124\) −4.74456 −0.426074
\(125\) 10.3723 0.927725
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.37228 0.208063
\(131\) −11.1168 −0.971283 −0.485642 0.874158i \(-0.661414\pi\)
−0.485642 + 0.874158i \(0.661414\pi\)
\(132\) 0 0
\(133\) −1.62772 −0.141141
\(134\) 9.48913 0.819736
\(135\) 0 0
\(136\) 4.37228 0.374920
\(137\) 1.11684 0.0954184 0.0477092 0.998861i \(-0.484808\pi\)
0.0477092 + 0.998861i \(0.484808\pi\)
\(138\) 0 0
\(139\) 0.744563 0.0631530 0.0315765 0.999501i \(-0.489947\pi\)
0.0315765 + 0.999501i \(0.489947\pi\)
\(140\) 2.37228 0.200494
\(141\) 0 0
\(142\) −4.74456 −0.398155
\(143\) −2.37228 −0.198380
\(144\) 0 0
\(145\) −15.1168 −1.25539
\(146\) −8.37228 −0.692895
\(147\) 0 0
\(148\) −4.37228 −0.359399
\(149\) −20.2337 −1.65761 −0.828804 0.559539i \(-0.810978\pi\)
−0.828804 + 0.559539i \(0.810978\pi\)
\(150\) 0 0
\(151\) 19.1168 1.55571 0.777853 0.628446i \(-0.216309\pi\)
0.777853 + 0.628446i \(0.216309\pi\)
\(152\) 1.62772 0.132025
\(153\) 0 0
\(154\) −2.37228 −0.191164
\(155\) 11.2554 0.904058
\(156\) 0 0
\(157\) −20.3723 −1.62589 −0.812943 0.582344i \(-0.802136\pi\)
−0.812943 + 0.582344i \(0.802136\pi\)
\(158\) 4.74456 0.377457
\(159\) 0 0
\(160\) −2.37228 −0.187545
\(161\) −3.62772 −0.285904
\(162\) 0 0
\(163\) −18.2337 −1.42817 −0.714086 0.700058i \(-0.753158\pi\)
−0.714086 + 0.700058i \(0.753158\pi\)
\(164\) 8.74456 0.682836
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 13.1168 1.01501 0.507506 0.861648i \(-0.330568\pi\)
0.507506 + 0.861648i \(0.330568\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −10.3723 −0.795518
\(171\) 0 0
\(172\) 11.1168 0.847651
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) −0.627719 −0.0474511
\(176\) 2.37228 0.178817
\(177\) 0 0
\(178\) −3.25544 −0.244005
\(179\) −2.74456 −0.205138 −0.102569 0.994726i \(-0.532706\pi\)
−0.102569 + 0.994726i \(0.532706\pi\)
\(180\) 0 0
\(181\) −23.4891 −1.74593 −0.872966 0.487780i \(-0.837807\pi\)
−0.872966 + 0.487780i \(0.837807\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) 3.62772 0.267439
\(185\) 10.3723 0.762585
\(186\) 0 0
\(187\) 10.3723 0.758496
\(188\) 1.25544 0.0915622
\(189\) 0 0
\(190\) −3.86141 −0.280136
\(191\) 5.11684 0.370242 0.185121 0.982716i \(-0.440732\pi\)
0.185121 + 0.982716i \(0.440732\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 7.48913 0.537688
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 11.8614 0.840833 0.420416 0.907331i \(-0.361884\pi\)
0.420416 + 0.907331i \(0.361884\pi\)
\(200\) 0.627719 0.0443864
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −6.37228 −0.447246
\(204\) 0 0
\(205\) −20.7446 −1.44886
\(206\) −11.8614 −0.826423
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 3.86141 0.267099
\(210\) 0 0
\(211\) −17.6277 −1.21354 −0.606771 0.794877i \(-0.707536\pi\)
−0.606771 + 0.794877i \(0.707536\pi\)
\(212\) 8.74456 0.600579
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) −26.3723 −1.79857
\(216\) 0 0
\(217\) 4.74456 0.322082
\(218\) −3.62772 −0.245700
\(219\) 0 0
\(220\) −5.62772 −0.379421
\(221\) −4.37228 −0.294111
\(222\) 0 0
\(223\) −17.4891 −1.17116 −0.585579 0.810615i \(-0.699133\pi\)
−0.585579 + 0.810615i \(0.699133\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) 11.4891 0.762560 0.381280 0.924460i \(-0.375483\pi\)
0.381280 + 0.924460i \(0.375483\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −8.60597 −0.567461
\(231\) 0 0
\(232\) 6.37228 0.418361
\(233\) −1.48913 −0.0975558 −0.0487779 0.998810i \(-0.515533\pi\)
−0.0487779 + 0.998810i \(0.515533\pi\)
\(234\) 0 0
\(235\) −2.97825 −0.194280
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) −4.37228 −0.283413
\(239\) −5.48913 −0.355062 −0.177531 0.984115i \(-0.556811\pi\)
−0.177531 + 0.984115i \(0.556811\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −5.37228 −0.345343
\(243\) 0 0
\(244\) 5.11684 0.327572
\(245\) −2.37228 −0.151559
\(246\) 0 0
\(247\) −1.62772 −0.103569
\(248\) −4.74456 −0.301280
\(249\) 0 0
\(250\) 10.3723 0.656001
\(251\) −13.6277 −0.860174 −0.430087 0.902787i \(-0.641517\pi\)
−0.430087 + 0.902787i \(0.641517\pi\)
\(252\) 0 0
\(253\) 8.60597 0.541053
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.4891 0.716672 0.358336 0.933593i \(-0.383344\pi\)
0.358336 + 0.933593i \(0.383344\pi\)
\(258\) 0 0
\(259\) 4.37228 0.271680
\(260\) 2.37228 0.147123
\(261\) 0 0
\(262\) −11.1168 −0.686801
\(263\) 9.25544 0.570715 0.285357 0.958421i \(-0.407888\pi\)
0.285357 + 0.958421i \(0.407888\pi\)
\(264\) 0 0
\(265\) −20.7446 −1.27433
\(266\) −1.62772 −0.0998018
\(267\) 0 0
\(268\) 9.48913 0.579641
\(269\) 0.510875 0.0311486 0.0155743 0.999879i \(-0.495042\pi\)
0.0155743 + 0.999879i \(0.495042\pi\)
\(270\) 0 0
\(271\) 30.2337 1.83657 0.918283 0.395925i \(-0.129576\pi\)
0.918283 + 0.395925i \(0.129576\pi\)
\(272\) 4.37228 0.265108
\(273\) 0 0
\(274\) 1.11684 0.0674710
\(275\) 1.48913 0.0897976
\(276\) 0 0
\(277\) 26.7446 1.60693 0.803463 0.595355i \(-0.202989\pi\)
0.803463 + 0.595355i \(0.202989\pi\)
\(278\) 0.744563 0.0446559
\(279\) 0 0
\(280\) 2.37228 0.141771
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 14.9783 0.890365 0.445182 0.895440i \(-0.353139\pi\)
0.445182 + 0.895440i \(0.353139\pi\)
\(284\) −4.74456 −0.281538
\(285\) 0 0
\(286\) −2.37228 −0.140276
\(287\) −8.74456 −0.516175
\(288\) 0 0
\(289\) 2.11684 0.124520
\(290\) −15.1168 −0.887692
\(291\) 0 0
\(292\) −8.37228 −0.489951
\(293\) −28.7446 −1.67928 −0.839638 0.543147i \(-0.817233\pi\)
−0.839638 + 0.543147i \(0.817233\pi\)
\(294\) 0 0
\(295\) −4.74456 −0.276239
\(296\) −4.37228 −0.254134
\(297\) 0 0
\(298\) −20.2337 −1.17211
\(299\) −3.62772 −0.209796
\(300\) 0 0
\(301\) −11.1168 −0.640764
\(302\) 19.1168 1.10005
\(303\) 0 0
\(304\) 1.62772 0.0933561
\(305\) −12.1386 −0.695054
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −2.37228 −0.135173
\(309\) 0 0
\(310\) 11.2554 0.639266
\(311\) 17.4891 0.991717 0.495859 0.868403i \(-0.334853\pi\)
0.495859 + 0.868403i \(0.334853\pi\)
\(312\) 0 0
\(313\) 12.9783 0.733574 0.366787 0.930305i \(-0.380458\pi\)
0.366787 + 0.930305i \(0.380458\pi\)
\(314\) −20.3723 −1.14967
\(315\) 0 0
\(316\) 4.74456 0.266903
\(317\) −28.9783 −1.62758 −0.813790 0.581159i \(-0.802600\pi\)
−0.813790 + 0.581159i \(0.802600\pi\)
\(318\) 0 0
\(319\) 15.1168 0.846381
\(320\) −2.37228 −0.132615
\(321\) 0 0
\(322\) −3.62772 −0.202165
\(323\) 7.11684 0.395992
\(324\) 0 0
\(325\) −0.627719 −0.0348196
\(326\) −18.2337 −1.00987
\(327\) 0 0
\(328\) 8.74456 0.482838
\(329\) −1.25544 −0.0692145
\(330\) 0 0
\(331\) 16.7446 0.920364 0.460182 0.887824i \(-0.347784\pi\)
0.460182 + 0.887824i \(0.347784\pi\)
\(332\) −6.00000 −0.329293
\(333\) 0 0
\(334\) 13.1168 0.717722
\(335\) −22.5109 −1.22990
\(336\) 0 0
\(337\) −1.86141 −0.101397 −0.0506986 0.998714i \(-0.516145\pi\)
−0.0506986 + 0.998714i \(0.516145\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) −10.3723 −0.562516
\(341\) −11.2554 −0.609516
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 11.1168 0.599380
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 25.7228 1.38087 0.690436 0.723393i \(-0.257418\pi\)
0.690436 + 0.723393i \(0.257418\pi\)
\(348\) 0 0
\(349\) −22.7446 −1.21749 −0.608744 0.793367i \(-0.708326\pi\)
−0.608744 + 0.793367i \(0.708326\pi\)
\(350\) −0.627719 −0.0335530
\(351\) 0 0
\(352\) 2.37228 0.126443
\(353\) −4.74456 −0.252528 −0.126264 0.991997i \(-0.540299\pi\)
−0.126264 + 0.991997i \(0.540299\pi\)
\(354\) 0 0
\(355\) 11.2554 0.597377
\(356\) −3.25544 −0.172538
\(357\) 0 0
\(358\) −2.74456 −0.145055
\(359\) −3.25544 −0.171815 −0.0859077 0.996303i \(-0.527379\pi\)
−0.0859077 + 0.996303i \(0.527379\pi\)
\(360\) 0 0
\(361\) −16.3505 −0.860554
\(362\) −23.4891 −1.23456
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 19.8614 1.03959
\(366\) 0 0
\(367\) 2.51087 0.131067 0.0655333 0.997850i \(-0.479125\pi\)
0.0655333 + 0.997850i \(0.479125\pi\)
\(368\) 3.62772 0.189108
\(369\) 0 0
\(370\) 10.3723 0.539229
\(371\) −8.74456 −0.453995
\(372\) 0 0
\(373\) −15.4891 −0.801997 −0.400998 0.916079i \(-0.631337\pi\)
−0.400998 + 0.916079i \(0.631337\pi\)
\(374\) 10.3723 0.536338
\(375\) 0 0
\(376\) 1.25544 0.0647442
\(377\) −6.37228 −0.328189
\(378\) 0 0
\(379\) −28.7446 −1.47651 −0.738255 0.674522i \(-0.764350\pi\)
−0.738255 + 0.674522i \(0.764350\pi\)
\(380\) −3.86141 −0.198086
\(381\) 0 0
\(382\) 5.11684 0.261801
\(383\) 6.60597 0.337549 0.168775 0.985655i \(-0.446019\pi\)
0.168775 + 0.985655i \(0.446019\pi\)
\(384\) 0 0
\(385\) 5.62772 0.286815
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) 7.48913 0.380203
\(389\) 3.25544 0.165057 0.0825286 0.996589i \(-0.473700\pi\)
0.0825286 + 0.996589i \(0.473700\pi\)
\(390\) 0 0
\(391\) 15.8614 0.802146
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −11.2554 −0.566323
\(396\) 0 0
\(397\) 5.25544 0.263763 0.131881 0.991265i \(-0.457898\pi\)
0.131881 + 0.991265i \(0.457898\pi\)
\(398\) 11.8614 0.594559
\(399\) 0 0
\(400\) 0.627719 0.0313859
\(401\) −35.4891 −1.77224 −0.886121 0.463454i \(-0.846610\pi\)
−0.886121 + 0.463454i \(0.846610\pi\)
\(402\) 0 0
\(403\) 4.74456 0.236343
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) −6.37228 −0.316251
\(407\) −10.3723 −0.514135
\(408\) 0 0
\(409\) 25.8614 1.27876 0.639382 0.768889i \(-0.279190\pi\)
0.639382 + 0.768889i \(0.279190\pi\)
\(410\) −20.7446 −1.02450
\(411\) 0 0
\(412\) −11.8614 −0.584370
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) 14.2337 0.698704
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 3.86141 0.188868
\(419\) −3.86141 −0.188642 −0.0943210 0.995542i \(-0.530068\pi\)
−0.0943210 + 0.995542i \(0.530068\pi\)
\(420\) 0 0
\(421\) −7.48913 −0.364998 −0.182499 0.983206i \(-0.558419\pi\)
−0.182499 + 0.983206i \(0.558419\pi\)
\(422\) −17.6277 −0.858104
\(423\) 0 0
\(424\) 8.74456 0.424674
\(425\) 2.74456 0.133131
\(426\) 0 0
\(427\) −5.11684 −0.247621
\(428\) 2.00000 0.0966736
\(429\) 0 0
\(430\) −26.3723 −1.27178
\(431\) −32.7446 −1.57725 −0.788625 0.614874i \(-0.789207\pi\)
−0.788625 + 0.614874i \(0.789207\pi\)
\(432\) 0 0
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 4.74456 0.227746
\(435\) 0 0
\(436\) −3.62772 −0.173736
\(437\) 5.90491 0.282470
\(438\) 0 0
\(439\) 9.62772 0.459506 0.229753 0.973249i \(-0.426208\pi\)
0.229753 + 0.973249i \(0.426208\pi\)
\(440\) −5.62772 −0.268291
\(441\) 0 0
\(442\) −4.37228 −0.207968
\(443\) 16.9783 0.806661 0.403331 0.915054i \(-0.367852\pi\)
0.403331 + 0.915054i \(0.367852\pi\)
\(444\) 0 0
\(445\) 7.72281 0.366096
\(446\) −17.4891 −0.828134
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −0.372281 −0.0175690 −0.00878452 0.999961i \(-0.502796\pi\)
−0.00878452 + 0.999961i \(0.502796\pi\)
\(450\) 0 0
\(451\) 20.7446 0.976823
\(452\) 20.0000 0.940721
\(453\) 0 0
\(454\) 11.4891 0.539211
\(455\) −2.37228 −0.111214
\(456\) 0 0
\(457\) 12.2337 0.572268 0.286134 0.958190i \(-0.407630\pi\)
0.286134 + 0.958190i \(0.407630\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −8.60597 −0.401255
\(461\) −36.6060 −1.70491 −0.852455 0.522801i \(-0.824887\pi\)
−0.852455 + 0.522801i \(0.824887\pi\)
\(462\) 0 0
\(463\) −17.3505 −0.806348 −0.403174 0.915123i \(-0.632093\pi\)
−0.403174 + 0.915123i \(0.632093\pi\)
\(464\) 6.37228 0.295826
\(465\) 0 0
\(466\) −1.48913 −0.0689824
\(467\) −34.0951 −1.57773 −0.788866 0.614565i \(-0.789332\pi\)
−0.788866 + 0.614565i \(0.789332\pi\)
\(468\) 0 0
\(469\) −9.48913 −0.438167
\(470\) −2.97825 −0.137376
\(471\) 0 0
\(472\) 2.00000 0.0920575
\(473\) 26.3723 1.21260
\(474\) 0 0
\(475\) 1.02175 0.0468811
\(476\) −4.37228 −0.200403
\(477\) 0 0
\(478\) −5.48913 −0.251067
\(479\) 39.3505 1.79797 0.898986 0.437978i \(-0.144305\pi\)
0.898986 + 0.437978i \(0.144305\pi\)
\(480\) 0 0
\(481\) 4.37228 0.199359
\(482\) −6.00000 −0.273293
\(483\) 0 0
\(484\) −5.37228 −0.244195
\(485\) −17.7663 −0.806727
\(486\) 0 0
\(487\) −32.4674 −1.47124 −0.735619 0.677396i \(-0.763108\pi\)
−0.735619 + 0.677396i \(0.763108\pi\)
\(488\) 5.11684 0.231629
\(489\) 0 0
\(490\) −2.37228 −0.107169
\(491\) 25.2554 1.13976 0.569881 0.821727i \(-0.306989\pi\)
0.569881 + 0.821727i \(0.306989\pi\)
\(492\) 0 0
\(493\) 27.8614 1.25481
\(494\) −1.62772 −0.0732345
\(495\) 0 0
\(496\) −4.74456 −0.213037
\(497\) 4.74456 0.212823
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 10.3723 0.463863
\(501\) 0 0
\(502\) −13.6277 −0.608235
\(503\) 21.4891 0.958153 0.479076 0.877773i \(-0.340972\pi\)
0.479076 + 0.877773i \(0.340972\pi\)
\(504\) 0 0
\(505\) 4.74456 0.211130
\(506\) 8.60597 0.382582
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −26.0951 −1.15664 −0.578322 0.815808i \(-0.696292\pi\)
−0.578322 + 0.815808i \(0.696292\pi\)
\(510\) 0 0
\(511\) 8.37228 0.370368
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 11.4891 0.506764
\(515\) 28.1386 1.23993
\(516\) 0 0
\(517\) 2.97825 0.130983
\(518\) 4.37228 0.192107
\(519\) 0 0
\(520\) 2.37228 0.104031
\(521\) −28.3723 −1.24301 −0.621506 0.783409i \(-0.713479\pi\)
−0.621506 + 0.783409i \(0.713479\pi\)
\(522\) 0 0
\(523\) 2.23369 0.0976724 0.0488362 0.998807i \(-0.484449\pi\)
0.0488362 + 0.998807i \(0.484449\pi\)
\(524\) −11.1168 −0.485642
\(525\) 0 0
\(526\) 9.25544 0.403556
\(527\) −20.7446 −0.903647
\(528\) 0 0
\(529\) −9.83966 −0.427811
\(530\) −20.7446 −0.901086
\(531\) 0 0
\(532\) −1.62772 −0.0705706
\(533\) −8.74456 −0.378769
\(534\) 0 0
\(535\) −4.74456 −0.205125
\(536\) 9.48913 0.409868
\(537\) 0 0
\(538\) 0.510875 0.0220254
\(539\) 2.37228 0.102181
\(540\) 0 0
\(541\) 23.3505 1.00392 0.501959 0.864891i \(-0.332613\pi\)
0.501959 + 0.864891i \(0.332613\pi\)
\(542\) 30.2337 1.29865
\(543\) 0 0
\(544\) 4.37228 0.187460
\(545\) 8.60597 0.368639
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 1.11684 0.0477092
\(549\) 0 0
\(550\) 1.48913 0.0634965
\(551\) 10.3723 0.441874
\(552\) 0 0
\(553\) −4.74456 −0.201759
\(554\) 26.7446 1.13627
\(555\) 0 0
\(556\) 0.744563 0.0315765
\(557\) 11.4891 0.486810 0.243405 0.969925i \(-0.421736\pi\)
0.243405 + 0.969925i \(0.421736\pi\)
\(558\) 0 0
\(559\) −11.1168 −0.470192
\(560\) 2.37228 0.100247
\(561\) 0 0
\(562\) 10.0000 0.421825
\(563\) −11.1168 −0.468519 −0.234260 0.972174i \(-0.575267\pi\)
−0.234260 + 0.972174i \(0.575267\pi\)
\(564\) 0 0
\(565\) −47.4456 −1.99605
\(566\) 14.9783 0.629583
\(567\) 0 0
\(568\) −4.74456 −0.199077
\(569\) 12.7446 0.534280 0.267140 0.963658i \(-0.413921\pi\)
0.267140 + 0.963658i \(0.413921\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −2.37228 −0.0991901
\(573\) 0 0
\(574\) −8.74456 −0.364991
\(575\) 2.27719 0.0949653
\(576\) 0 0
\(577\) −0.510875 −0.0212680 −0.0106340 0.999943i \(-0.503385\pi\)
−0.0106340 + 0.999943i \(0.503385\pi\)
\(578\) 2.11684 0.0880491
\(579\) 0 0
\(580\) −15.1168 −0.627693
\(581\) 6.00000 0.248922
\(582\) 0 0
\(583\) 20.7446 0.859152
\(584\) −8.37228 −0.346447
\(585\) 0 0
\(586\) −28.7446 −1.18743
\(587\) −27.4891 −1.13460 −0.567299 0.823512i \(-0.692012\pi\)
−0.567299 + 0.823512i \(0.692012\pi\)
\(588\) 0 0
\(589\) −7.72281 −0.318213
\(590\) −4.74456 −0.195331
\(591\) 0 0
\(592\) −4.37228 −0.179700
\(593\) −4.00000 −0.164260 −0.0821302 0.996622i \(-0.526172\pi\)
−0.0821302 + 0.996622i \(0.526172\pi\)
\(594\) 0 0
\(595\) 10.3723 0.425222
\(596\) −20.2337 −0.828804
\(597\) 0 0
\(598\) −3.62772 −0.148348
\(599\) −21.1168 −0.862811 −0.431405 0.902158i \(-0.641982\pi\)
−0.431405 + 0.902158i \(0.641982\pi\)
\(600\) 0 0
\(601\) −42.4674 −1.73228 −0.866140 0.499801i \(-0.833406\pi\)
−0.866140 + 0.499801i \(0.833406\pi\)
\(602\) −11.1168 −0.453089
\(603\) 0 0
\(604\) 19.1168 0.777853
\(605\) 12.7446 0.518140
\(606\) 0 0
\(607\) 19.1168 0.775929 0.387964 0.921674i \(-0.373178\pi\)
0.387964 + 0.921674i \(0.373178\pi\)
\(608\) 1.62772 0.0660127
\(609\) 0 0
\(610\) −12.1386 −0.491477
\(611\) −1.25544 −0.0507896
\(612\) 0 0
\(613\) −6.13859 −0.247935 −0.123968 0.992286i \(-0.539562\pi\)
−0.123968 + 0.992286i \(0.539562\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −2.37228 −0.0955819
\(617\) 9.39403 0.378189 0.189095 0.981959i \(-0.439445\pi\)
0.189095 + 0.981959i \(0.439445\pi\)
\(618\) 0 0
\(619\) 11.1168 0.446824 0.223412 0.974724i \(-0.428281\pi\)
0.223412 + 0.974724i \(0.428281\pi\)
\(620\) 11.2554 0.452029
\(621\) 0 0
\(622\) 17.4891 0.701250
\(623\) 3.25544 0.130426
\(624\) 0 0
\(625\) −27.7446 −1.10978
\(626\) 12.9783 0.518715
\(627\) 0 0
\(628\) −20.3723 −0.812943
\(629\) −19.1168 −0.762238
\(630\) 0 0
\(631\) 16.6060 0.661073 0.330537 0.943793i \(-0.392770\pi\)
0.330537 + 0.943793i \(0.392770\pi\)
\(632\) 4.74456 0.188729
\(633\) 0 0
\(634\) −28.9783 −1.15087
\(635\) 18.9783 0.753129
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 15.1168 0.598482
\(639\) 0 0
\(640\) −2.37228 −0.0937727
\(641\) 18.2337 0.720187 0.360094 0.932916i \(-0.382745\pi\)
0.360094 + 0.932916i \(0.382745\pi\)
\(642\) 0 0
\(643\) −0.138593 −0.00546559 −0.00273279 0.999996i \(-0.500870\pi\)
−0.00273279 + 0.999996i \(0.500870\pi\)
\(644\) −3.62772 −0.142952
\(645\) 0 0
\(646\) 7.11684 0.280008
\(647\) 21.4891 0.844825 0.422412 0.906404i \(-0.361183\pi\)
0.422412 + 0.906404i \(0.361183\pi\)
\(648\) 0 0
\(649\) 4.74456 0.186240
\(650\) −0.627719 −0.0246212
\(651\) 0 0
\(652\) −18.2337 −0.714086
\(653\) 8.13859 0.318488 0.159244 0.987239i \(-0.449094\pi\)
0.159244 + 0.987239i \(0.449094\pi\)
\(654\) 0 0
\(655\) 26.3723 1.03045
\(656\) 8.74456 0.341418
\(657\) 0 0
\(658\) −1.25544 −0.0489420
\(659\) 9.25544 0.360541 0.180270 0.983617i \(-0.442303\pi\)
0.180270 + 0.983617i \(0.442303\pi\)
\(660\) 0 0
\(661\) 10.7446 0.417915 0.208958 0.977925i \(-0.432993\pi\)
0.208958 + 0.977925i \(0.432993\pi\)
\(662\) 16.7446 0.650796
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) 3.86141 0.149739
\(666\) 0 0
\(667\) 23.1168 0.895088
\(668\) 13.1168 0.507506
\(669\) 0 0
\(670\) −22.5109 −0.869671
\(671\) 12.1386 0.468605
\(672\) 0 0
\(673\) 16.0951 0.620420 0.310210 0.950668i \(-0.399601\pi\)
0.310210 + 0.950668i \(0.399601\pi\)
\(674\) −1.86141 −0.0716987
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 35.4891 1.36396 0.681979 0.731372i \(-0.261120\pi\)
0.681979 + 0.731372i \(0.261120\pi\)
\(678\) 0 0
\(679\) −7.48913 −0.287406
\(680\) −10.3723 −0.397759
\(681\) 0 0
\(682\) −11.2554 −0.430993
\(683\) 11.1168 0.425374 0.212687 0.977120i \(-0.431778\pi\)
0.212687 + 0.977120i \(0.431778\pi\)
\(684\) 0 0
\(685\) −2.64947 −0.101231
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 11.1168 0.423826
\(689\) −8.74456 −0.333141
\(690\) 0 0
\(691\) 45.4891 1.73049 0.865244 0.501351i \(-0.167163\pi\)
0.865244 + 0.501351i \(0.167163\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) 25.7228 0.976425
\(695\) −1.76631 −0.0670000
\(696\) 0 0
\(697\) 38.2337 1.44820
\(698\) −22.7446 −0.860894
\(699\) 0 0
\(700\) −0.627719 −0.0237255
\(701\) 31.7228 1.19815 0.599077 0.800691i \(-0.295534\pi\)
0.599077 + 0.800691i \(0.295534\pi\)
\(702\) 0 0
\(703\) −7.11684 −0.268417
\(704\) 2.37228 0.0894087
\(705\) 0 0
\(706\) −4.74456 −0.178564
\(707\) 2.00000 0.0752177
\(708\) 0 0
\(709\) −28.5109 −1.07075 −0.535374 0.844615i \(-0.679829\pi\)
−0.535374 + 0.844615i \(0.679829\pi\)
\(710\) 11.2554 0.422409
\(711\) 0 0
\(712\) −3.25544 −0.122003
\(713\) −17.2119 −0.644592
\(714\) 0 0
\(715\) 5.62772 0.210465
\(716\) −2.74456 −0.102569
\(717\) 0 0
\(718\) −3.25544 −0.121492
\(719\) −45.4891 −1.69646 −0.848229 0.529630i \(-0.822331\pi\)
−0.848229 + 0.529630i \(0.822331\pi\)
\(720\) 0 0
\(721\) 11.8614 0.441742
\(722\) −16.3505 −0.608504
\(723\) 0 0
\(724\) −23.4891 −0.872966
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) 46.0951 1.70957 0.854786 0.518980i \(-0.173688\pi\)
0.854786 + 0.518980i \(0.173688\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) 19.8614 0.735104
\(731\) 48.6060 1.79776
\(732\) 0 0
\(733\) −30.7446 −1.13558 −0.567788 0.823175i \(-0.692201\pi\)
−0.567788 + 0.823175i \(0.692201\pi\)
\(734\) 2.51087 0.0926781
\(735\) 0 0
\(736\) 3.62772 0.133719
\(737\) 22.5109 0.829199
\(738\) 0 0
\(739\) 31.7228 1.16694 0.583471 0.812134i \(-0.301694\pi\)
0.583471 + 0.812134i \(0.301694\pi\)
\(740\) 10.3723 0.381293
\(741\) 0 0
\(742\) −8.74456 −0.321023
\(743\) 44.4674 1.63135 0.815675 0.578511i \(-0.196366\pi\)
0.815675 + 0.578511i \(0.196366\pi\)
\(744\) 0 0
\(745\) 48.0000 1.75858
\(746\) −15.4891 −0.567097
\(747\) 0 0
\(748\) 10.3723 0.379248
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) −10.9783 −0.400602 −0.200301 0.979734i \(-0.564192\pi\)
−0.200301 + 0.979734i \(0.564192\pi\)
\(752\) 1.25544 0.0457811
\(753\) 0 0
\(754\) −6.37228 −0.232065
\(755\) −45.3505 −1.65047
\(756\) 0 0
\(757\) −43.2119 −1.57056 −0.785282 0.619138i \(-0.787482\pi\)
−0.785282 + 0.619138i \(0.787482\pi\)
\(758\) −28.7446 −1.04405
\(759\) 0 0
\(760\) −3.86141 −0.140068
\(761\) −46.9783 −1.70296 −0.851480 0.524387i \(-0.824295\pi\)
−0.851480 + 0.524387i \(0.824295\pi\)
\(762\) 0 0
\(763\) 3.62772 0.131332
\(764\) 5.11684 0.185121
\(765\) 0 0
\(766\) 6.60597 0.238683
\(767\) −2.00000 −0.0722158
\(768\) 0 0
\(769\) −19.6277 −0.707794 −0.353897 0.935284i \(-0.615144\pi\)
−0.353897 + 0.935284i \(0.615144\pi\)
\(770\) 5.62772 0.202809
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) −34.0951 −1.22632 −0.613158 0.789961i \(-0.710101\pi\)
−0.613158 + 0.789961i \(0.710101\pi\)
\(774\) 0 0
\(775\) −2.97825 −0.106982
\(776\) 7.48913 0.268844
\(777\) 0 0
\(778\) 3.25544 0.116713
\(779\) 14.2337 0.509975
\(780\) 0 0
\(781\) −11.2554 −0.402751
\(782\) 15.8614 0.567203
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 48.3288 1.72493
\(786\) 0 0
\(787\) 27.1168 0.966611 0.483306 0.875452i \(-0.339436\pi\)
0.483306 + 0.875452i \(0.339436\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −11.2554 −0.400450
\(791\) −20.0000 −0.711118
\(792\) 0 0
\(793\) −5.11684 −0.181704
\(794\) 5.25544 0.186508
\(795\) 0 0
\(796\) 11.8614 0.420416
\(797\) 49.7228 1.76127 0.880636 0.473793i \(-0.157116\pi\)
0.880636 + 0.473793i \(0.157116\pi\)
\(798\) 0 0
\(799\) 5.48913 0.194191
\(800\) 0.627719 0.0221932
\(801\) 0 0
\(802\) −35.4891 −1.25316
\(803\) −19.8614 −0.700894
\(804\) 0 0
\(805\) 8.60597 0.303321
\(806\) 4.74456 0.167120
\(807\) 0 0
\(808\) −2.00000 −0.0703598
\(809\) 32.7446 1.15124 0.575619 0.817718i \(-0.304761\pi\)
0.575619 + 0.817718i \(0.304761\pi\)
\(810\) 0 0
\(811\) 3.11684 0.109447 0.0547236 0.998502i \(-0.482572\pi\)
0.0547236 + 0.998502i \(0.482572\pi\)
\(812\) −6.37228 −0.223623
\(813\) 0 0
\(814\) −10.3723 −0.363548
\(815\) 43.2554 1.51517
\(816\) 0 0
\(817\) 18.0951 0.633067
\(818\) 25.8614 0.904223
\(819\) 0 0
\(820\) −20.7446 −0.724432
\(821\) −53.7228 −1.87494 −0.937470 0.348067i \(-0.886838\pi\)
−0.937470 + 0.348067i \(0.886838\pi\)
\(822\) 0 0
\(823\) −31.7228 −1.10579 −0.552894 0.833252i \(-0.686477\pi\)
−0.552894 + 0.833252i \(0.686477\pi\)
\(824\) −11.8614 −0.413212
\(825\) 0 0
\(826\) −2.00000 −0.0695889
\(827\) 34.3723 1.19524 0.597621 0.801779i \(-0.296113\pi\)
0.597621 + 0.801779i \(0.296113\pi\)
\(828\) 0 0
\(829\) −24.3723 −0.846484 −0.423242 0.906017i \(-0.639108\pi\)
−0.423242 + 0.906017i \(0.639108\pi\)
\(830\) 14.2337 0.494059
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 4.37228 0.151491
\(834\) 0 0
\(835\) −31.1168 −1.07684
\(836\) 3.86141 0.133550
\(837\) 0 0
\(838\) −3.86141 −0.133390
\(839\) −1.25544 −0.0433425 −0.0216713 0.999765i \(-0.506899\pi\)
−0.0216713 + 0.999765i \(0.506899\pi\)
\(840\) 0 0
\(841\) 11.6060 0.400206
\(842\) −7.48913 −0.258092
\(843\) 0 0
\(844\) −17.6277 −0.606771
\(845\) −2.37228 −0.0816090
\(846\) 0 0
\(847\) 5.37228 0.184594
\(848\) 8.74456 0.300290
\(849\) 0 0
\(850\) 2.74456 0.0941377
\(851\) −15.8614 −0.543722
\(852\) 0 0
\(853\) −39.2119 −1.34259 −0.671296 0.741190i \(-0.734262\pi\)
−0.671296 + 0.741190i \(0.734262\pi\)
\(854\) −5.11684 −0.175095
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) −0.978251 −0.0334164 −0.0167082 0.999860i \(-0.505319\pi\)
−0.0167082 + 0.999860i \(0.505319\pi\)
\(858\) 0 0
\(859\) −0.744563 −0.0254041 −0.0127021 0.999919i \(-0.504043\pi\)
−0.0127021 + 0.999919i \(0.504043\pi\)
\(860\) −26.3723 −0.899287
\(861\) 0 0
\(862\) −32.7446 −1.11528
\(863\) −18.9783 −0.646027 −0.323014 0.946394i \(-0.604696\pi\)
−0.323014 + 0.946394i \(0.604696\pi\)
\(864\) 0 0
\(865\) 23.7228 0.806600
\(866\) 10.0000 0.339814
\(867\) 0 0
\(868\) 4.74456 0.161041
\(869\) 11.2554 0.381815
\(870\) 0 0
\(871\) −9.48913 −0.321527
\(872\) −3.62772 −0.122850
\(873\) 0 0
\(874\) 5.90491 0.199736
\(875\) −10.3723 −0.350647
\(876\) 0 0
\(877\) −52.9783 −1.78895 −0.894474 0.447120i \(-0.852450\pi\)
−0.894474 + 0.447120i \(0.852450\pi\)
\(878\) 9.62772 0.324920
\(879\) 0 0
\(880\) −5.62772 −0.189710
\(881\) −14.1386 −0.476341 −0.238171 0.971223i \(-0.576548\pi\)
−0.238171 + 0.971223i \(0.576548\pi\)
\(882\) 0 0
\(883\) −30.3723 −1.02211 −0.511054 0.859548i \(-0.670745\pi\)
−0.511054 + 0.859548i \(0.670745\pi\)
\(884\) −4.37228 −0.147056
\(885\) 0 0
\(886\) 16.9783 0.570395
\(887\) 53.2119 1.78668 0.893341 0.449379i \(-0.148355\pi\)
0.893341 + 0.449379i \(0.148355\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 7.72281 0.258869
\(891\) 0 0
\(892\) −17.4891 −0.585579
\(893\) 2.04350 0.0683831
\(894\) 0 0
\(895\) 6.51087 0.217635
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −0.372281 −0.0124232
\(899\) −30.2337 −1.00835
\(900\) 0 0
\(901\) 38.2337 1.27375
\(902\) 20.7446 0.690718
\(903\) 0 0
\(904\) 20.0000 0.665190
\(905\) 55.7228 1.85229
\(906\) 0 0
\(907\) −57.9565 −1.92441 −0.962207 0.272319i \(-0.912209\pi\)
−0.962207 + 0.272319i \(0.912209\pi\)
\(908\) 11.4891 0.381280
\(909\) 0 0
\(910\) −2.37228 −0.0786404
\(911\) 22.8832 0.758153 0.379076 0.925365i \(-0.376242\pi\)
0.379076 + 0.925365i \(0.376242\pi\)
\(912\) 0 0
\(913\) −14.2337 −0.471066
\(914\) 12.2337 0.404654
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 11.1168 0.367111
\(918\) 0 0
\(919\) −49.2119 −1.62335 −0.811676 0.584108i \(-0.801444\pi\)
−0.811676 + 0.584108i \(0.801444\pi\)
\(920\) −8.60597 −0.283730
\(921\) 0 0
\(922\) −36.6060 −1.20555
\(923\) 4.74456 0.156169
\(924\) 0 0
\(925\) −2.74456 −0.0902407
\(926\) −17.3505 −0.570174
\(927\) 0 0
\(928\) 6.37228 0.209180
\(929\) 15.2554 0.500515 0.250257 0.968179i \(-0.419485\pi\)
0.250257 + 0.968179i \(0.419485\pi\)
\(930\) 0 0
\(931\) 1.62772 0.0533463
\(932\) −1.48913 −0.0487779
\(933\) 0 0
\(934\) −34.0951 −1.11563
\(935\) −24.6060 −0.804701
\(936\) 0 0
\(937\) 24.5109 0.800735 0.400368 0.916355i \(-0.368882\pi\)
0.400368 + 0.916355i \(0.368882\pi\)
\(938\) −9.48913 −0.309831
\(939\) 0 0
\(940\) −2.97825 −0.0971398
\(941\) 24.7446 0.806650 0.403325 0.915057i \(-0.367854\pi\)
0.403325 + 0.915057i \(0.367854\pi\)
\(942\) 0 0
\(943\) 31.7228 1.03304
\(944\) 2.00000 0.0650945
\(945\) 0 0
\(946\) 26.3723 0.857437
\(947\) −8.13859 −0.264469 −0.132234 0.991218i \(-0.542215\pi\)
−0.132234 + 0.991218i \(0.542215\pi\)
\(948\) 0 0
\(949\) 8.37228 0.271776
\(950\) 1.02175 0.0331499
\(951\) 0 0
\(952\) −4.37228 −0.141706
\(953\) −45.4891 −1.47354 −0.736769 0.676145i \(-0.763649\pi\)
−0.736769 + 0.676145i \(0.763649\pi\)
\(954\) 0 0
\(955\) −12.1386 −0.392796
\(956\) −5.48913 −0.177531
\(957\) 0 0
\(958\) 39.3505 1.27136
\(959\) −1.11684 −0.0360648
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 4.37228 0.140968
\(963\) 0 0
\(964\) −6.00000 −0.193247
\(965\) −4.74456 −0.152733
\(966\) 0 0
\(967\) 24.8832 0.800188 0.400094 0.916474i \(-0.368977\pi\)
0.400094 + 0.916474i \(0.368977\pi\)
\(968\) −5.37228 −0.172672
\(969\) 0 0
\(970\) −17.7663 −0.570442
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −0.744563 −0.0238696
\(974\) −32.4674 −1.04032
\(975\) 0 0
\(976\) 5.11684 0.163786
\(977\) −25.1168 −0.803559 −0.401780 0.915736i \(-0.631608\pi\)
−0.401780 + 0.915736i \(0.631608\pi\)
\(978\) 0 0
\(979\) −7.72281 −0.246822
\(980\) −2.37228 −0.0757797
\(981\) 0 0
\(982\) 25.2554 0.805933
\(983\) 35.3505 1.12751 0.563753 0.825943i \(-0.309357\pi\)
0.563753 + 0.825943i \(0.309357\pi\)
\(984\) 0 0
\(985\) −14.2337 −0.453523
\(986\) 27.8614 0.887288
\(987\) 0 0
\(988\) −1.62772 −0.0517846
\(989\) 40.3288 1.28238
\(990\) 0 0
\(991\) 14.2337 0.452148 0.226074 0.974110i \(-0.427411\pi\)
0.226074 + 0.974110i \(0.427411\pi\)
\(992\) −4.74456 −0.150640
\(993\) 0 0
\(994\) 4.74456 0.150488
\(995\) −28.1386 −0.892053
\(996\) 0 0
\(997\) 20.5109 0.649586 0.324793 0.945785i \(-0.394705\pi\)
0.324793 + 0.945785i \(0.394705\pi\)
\(998\) −20.0000 −0.633089
\(999\) 0 0
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 1638.2.a.x.1.1 yes 2
3.2 odd 2 1638.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1638.2.a.v.1.2 2 3.2 odd 2
1638.2.a.x.1.1 yes 2 1.1 even 1 trivial