Properties

Label 1911.2.a.u.1.3
Level $1911$
Weight $2$
Character 1911.1
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $5$
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] = [1911,2,Mod(1,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, 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("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.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: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.375116.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + 7x^{2} + 2x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.562376\) of defining polynomial
Character \(\chi\) \(=\) 1911.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.0776754 q^{2} +1.00000 q^{3} -1.99397 q^{4} +2.20243 q^{5} +0.0776754 q^{6} -0.310233 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.0776754 q^{2} +1.00000 q^{3} -1.99397 q^{4} +2.20243 q^{5} +0.0776754 q^{6} -0.310233 q^{8} +1.00000 q^{9} +0.171074 q^{10} +2.91629 q^{11} -1.99397 q^{12} +1.00000 q^{13} +2.20243 q^{15} +3.96384 q^{16} +0.393942 q^{17} +0.0776754 q^{18} -0.124751 q^{19} -4.39156 q^{20} +0.226524 q^{22} +3.38187 q^{23} -0.310233 q^{24} -0.149317 q^{25} +0.0776754 q^{26} +1.00000 q^{27} -0.642223 q^{29} +0.171074 q^{30} -5.95780 q^{31} +0.928358 q^{32} +2.91629 q^{33} +0.0305996 q^{34} -1.99397 q^{36} +4.18670 q^{37} -0.00969009 q^{38} +1.00000 q^{39} -0.683265 q^{40} -0.434984 q^{41} -6.74934 q^{43} -5.81499 q^{44} +2.20243 q^{45} +0.262688 q^{46} +9.46321 q^{47} +3.96384 q^{48} -0.0115983 q^{50} +0.393942 q^{51} -1.99397 q^{52} +12.4338 q^{53} +0.0776754 q^{54} +6.42292 q^{55} -0.124751 q^{57} -0.0498849 q^{58} +7.33555 q^{59} -4.39156 q^{60} +2.39394 q^{61} -0.462775 q^{62} -7.85556 q^{64} +2.20243 q^{65} +0.226524 q^{66} +6.14047 q^{67} -0.785507 q^{68} +3.38187 q^{69} -14.0826 q^{71} -0.310233 q^{72} +7.55295 q^{73} +0.325204 q^{74} -0.149317 q^{75} +0.248750 q^{76} +0.0776754 q^{78} -2.37472 q^{79} +8.73006 q^{80} +1.00000 q^{81} -0.0337875 q^{82} +9.78804 q^{83} +0.867628 q^{85} -0.524258 q^{86} -0.642223 q^{87} -0.904729 q^{88} +6.40363 q^{89} +0.171074 q^{90} -6.74335 q^{92} -5.95780 q^{93} +0.735058 q^{94} -0.274755 q^{95} +0.928358 q^{96} -14.9096 q^{97} +2.91629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 5 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 5 q^{3} + 6 q^{4} + 3 q^{5} + 3 q^{8} + 5 q^{9} + 2 q^{10} - q^{11} + 6 q^{12} + 5 q^{13} + 3 q^{15} + 13 q^{17} + 7 q^{19} + 13 q^{20} - 19 q^{22} - 4 q^{23} + 3 q^{24} + 16 q^{25} + 5 q^{27} - 12 q^{29} + 2 q^{30} + 6 q^{31} + 21 q^{32} - q^{33} + 7 q^{34} + 6 q^{36} + 11 q^{37} + 14 q^{38} + 5 q^{39} + 11 q^{40} + 10 q^{41} + 10 q^{43} - 29 q^{44} + 3 q^{45} + q^{46} - 4 q^{47} - 29 q^{50} + 13 q^{51} + 6 q^{52} - 9 q^{53} - 12 q^{55} + 7 q^{57} + 34 q^{58} + 7 q^{59} + 13 q^{60} + 23 q^{61} - 24 q^{62} - 13 q^{64} + 3 q^{65} - 19 q^{66} + 25 q^{67} + 20 q^{68} - 4 q^{69} - 27 q^{71} + 3 q^{72} + 18 q^{73} + 15 q^{74} + 16 q^{75} + 2 q^{76} + 8 q^{79} + 41 q^{80} + 5 q^{81} + 26 q^{82} + 12 q^{83} + 10 q^{85} - 19 q^{86} - 12 q^{87} - 36 q^{88} + 29 q^{89} + 2 q^{90} - 50 q^{92} + 6 q^{93} - 2 q^{94} - 33 q^{95} + 21 q^{96} + 13 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0776754 0.0549248 0.0274624 0.999623i \(-0.491257\pi\)
0.0274624 + 0.999623i \(0.491257\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.99397 −0.996983
\(5\) 2.20243 0.984955 0.492478 0.870325i \(-0.336091\pi\)
0.492478 + 0.870325i \(0.336091\pi\)
\(6\) 0.0776754 0.0317108
\(7\) 0 0
\(8\) −0.310233 −0.109684
\(9\) 1.00000 0.333333
\(10\) 0.171074 0.0540984
\(11\) 2.91629 0.879295 0.439647 0.898170i \(-0.355103\pi\)
0.439647 + 0.898170i \(0.355103\pi\)
\(12\) −1.99397 −0.575609
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 2.20243 0.568664
\(16\) 3.96384 0.990959
\(17\) 0.393942 0.0955449 0.0477724 0.998858i \(-0.484788\pi\)
0.0477724 + 0.998858i \(0.484788\pi\)
\(18\) 0.0776754 0.0183083
\(19\) −0.124751 −0.0286199 −0.0143099 0.999898i \(-0.504555\pi\)
−0.0143099 + 0.999898i \(0.504555\pi\)
\(20\) −4.39156 −0.981984
\(21\) 0 0
\(22\) 0.226524 0.0482951
\(23\) 3.38187 0.705170 0.352585 0.935780i \(-0.385303\pi\)
0.352585 + 0.935780i \(0.385303\pi\)
\(24\) −0.310233 −0.0633260
\(25\) −0.149317 −0.0298635
\(26\) 0.0776754 0.0152334
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.642223 −0.119258 −0.0596289 0.998221i \(-0.518992\pi\)
−0.0596289 + 0.998221i \(0.518992\pi\)
\(30\) 0.171074 0.0312338
\(31\) −5.95780 −1.07005 −0.535026 0.844835i \(-0.679698\pi\)
−0.535026 + 0.844835i \(0.679698\pi\)
\(32\) 0.928358 0.164112
\(33\) 2.91629 0.507661
\(34\) 0.0305996 0.00524778
\(35\) 0 0
\(36\) −1.99397 −0.332328
\(37\) 4.18670 0.688290 0.344145 0.938917i \(-0.388169\pi\)
0.344145 + 0.938917i \(0.388169\pi\)
\(38\) −0.00969009 −0.00157194
\(39\) 1.00000 0.160128
\(40\) −0.683265 −0.108034
\(41\) −0.434984 −0.0679331 −0.0339665 0.999423i \(-0.510814\pi\)
−0.0339665 + 0.999423i \(0.510814\pi\)
\(42\) 0 0
\(43\) −6.74934 −1.02927 −0.514633 0.857411i \(-0.672072\pi\)
−0.514633 + 0.857411i \(0.672072\pi\)
\(44\) −5.81499 −0.876642
\(45\) 2.20243 0.328318
\(46\) 0.262688 0.0387313
\(47\) 9.46321 1.38035 0.690175 0.723642i \(-0.257533\pi\)
0.690175 + 0.723642i \(0.257533\pi\)
\(48\) 3.96384 0.572130
\(49\) 0 0
\(50\) −0.0115983 −0.00164024
\(51\) 0.393942 0.0551629
\(52\) −1.99397 −0.276513
\(53\) 12.4338 1.70792 0.853959 0.520341i \(-0.174195\pi\)
0.853959 + 0.520341i \(0.174195\pi\)
\(54\) 0.0776754 0.0105703
\(55\) 6.42292 0.866066
\(56\) 0 0
\(57\) −0.124751 −0.0165237
\(58\) −0.0498849 −0.00655021
\(59\) 7.33555 0.955007 0.477504 0.878630i \(-0.341542\pi\)
0.477504 + 0.878630i \(0.341542\pi\)
\(60\) −4.39156 −0.566949
\(61\) 2.39394 0.306513 0.153256 0.988186i \(-0.451024\pi\)
0.153256 + 0.988186i \(0.451024\pi\)
\(62\) −0.462775 −0.0587724
\(63\) 0 0
\(64\) −7.85556 −0.981945
\(65\) 2.20243 0.273177
\(66\) 0.226524 0.0278832
\(67\) 6.14047 0.750178 0.375089 0.926989i \(-0.377612\pi\)
0.375089 + 0.926989i \(0.377612\pi\)
\(68\) −0.785507 −0.0952567
\(69\) 3.38187 0.407130
\(70\) 0 0
\(71\) −14.0826 −1.67129 −0.835646 0.549269i \(-0.814906\pi\)
−0.835646 + 0.549269i \(0.814906\pi\)
\(72\) −0.310233 −0.0365613
\(73\) 7.55295 0.884006 0.442003 0.897014i \(-0.354268\pi\)
0.442003 + 0.897014i \(0.354268\pi\)
\(74\) 0.325204 0.0378042
\(75\) −0.149317 −0.0172417
\(76\) 0.248750 0.0285335
\(77\) 0 0
\(78\) 0.0776754 0.00879500
\(79\) −2.37472 −0.267177 −0.133589 0.991037i \(-0.542650\pi\)
−0.133589 + 0.991037i \(0.542650\pi\)
\(80\) 8.73006 0.976050
\(81\) 1.00000 0.111111
\(82\) −0.0337875 −0.00373121
\(83\) 9.78804 1.07438 0.537189 0.843462i \(-0.319486\pi\)
0.537189 + 0.843462i \(0.319486\pi\)
\(84\) 0 0
\(85\) 0.867628 0.0941074
\(86\) −0.524258 −0.0565322
\(87\) −0.642223 −0.0688535
\(88\) −0.904729 −0.0964445
\(89\) 6.40363 0.678784 0.339392 0.940645i \(-0.389779\pi\)
0.339392 + 0.940645i \(0.389779\pi\)
\(90\) 0.171074 0.0180328
\(91\) 0 0
\(92\) −6.74335 −0.703042
\(93\) −5.95780 −0.617795
\(94\) 0.735058 0.0758155
\(95\) −0.274755 −0.0281893
\(96\) 0.928358 0.0947502
\(97\) −14.9096 −1.51384 −0.756919 0.653509i \(-0.773296\pi\)
−0.756919 + 0.653509i \(0.773296\pi\)
\(98\) 0 0
\(99\) 2.91629 0.293098
\(100\) 0.297734 0.0297734
\(101\) 19.9821 1.98829 0.994145 0.108057i \(-0.0344630\pi\)
0.994145 + 0.108057i \(0.0344630\pi\)
\(102\) 0.0305996 0.00302981
\(103\) 14.6745 1.44592 0.722960 0.690890i \(-0.242781\pi\)
0.722960 + 0.690890i \(0.242781\pi\)
\(104\) −0.310233 −0.0304208
\(105\) 0 0
\(106\) 0.965802 0.0938070
\(107\) −5.03973 −0.487209 −0.243604 0.969875i \(-0.578330\pi\)
−0.243604 + 0.969875i \(0.578330\pi\)
\(108\) −1.99397 −0.191870
\(109\) −0.122682 −0.0117508 −0.00587542 0.999983i \(-0.501870\pi\)
−0.00587542 + 0.999983i \(0.501870\pi\)
\(110\) 0.498902 0.0475685
\(111\) 4.18670 0.397384
\(112\) 0 0
\(113\) −4.16138 −0.391470 −0.195735 0.980657i \(-0.562709\pi\)
−0.195735 + 0.980657i \(0.562709\pi\)
\(114\) −0.00969009 −0.000907560 0
\(115\) 7.44833 0.694560
\(116\) 1.28057 0.118898
\(117\) 1.00000 0.0924500
\(118\) 0.569792 0.0524536
\(119\) 0 0
\(120\) −0.683265 −0.0623733
\(121\) −2.49525 −0.226841
\(122\) 0.185950 0.0168352
\(123\) −0.434984 −0.0392212
\(124\) 11.8797 1.06682
\(125\) −11.3410 −1.01437
\(126\) 0 0
\(127\) 14.9711 1.32847 0.664237 0.747522i \(-0.268756\pi\)
0.664237 + 0.747522i \(0.268756\pi\)
\(128\) −2.46690 −0.218045
\(129\) −6.74934 −0.594246
\(130\) 0.171074 0.0150042
\(131\) −18.7422 −1.63751 −0.818755 0.574143i \(-0.805335\pi\)
−0.818755 + 0.574143i \(0.805335\pi\)
\(132\) −5.81499 −0.506130
\(133\) 0 0
\(134\) 0.476964 0.0412034
\(135\) 2.20243 0.189555
\(136\) −0.122214 −0.0104797
\(137\) −13.5905 −1.16111 −0.580557 0.814219i \(-0.697165\pi\)
−0.580557 + 0.814219i \(0.697165\pi\)
\(138\) 0.262688 0.0223615
\(139\) 15.7292 1.33413 0.667066 0.744999i \(-0.267550\pi\)
0.667066 + 0.744999i \(0.267550\pi\)
\(140\) 0 0
\(141\) 9.46321 0.796946
\(142\) −1.09387 −0.0917953
\(143\) 2.91629 0.243873
\(144\) 3.96384 0.330320
\(145\) −1.41445 −0.117464
\(146\) 0.586678 0.0485538
\(147\) 0 0
\(148\) −8.34815 −0.686213
\(149\) 3.34809 0.274286 0.137143 0.990551i \(-0.456208\pi\)
0.137143 + 0.990551i \(0.456208\pi\)
\(150\) −0.0115983 −0.000946995 0
\(151\) −20.4476 −1.66401 −0.832003 0.554771i \(-0.812806\pi\)
−0.832003 + 0.554771i \(0.812806\pi\)
\(152\) 0.0387019 0.00313914
\(153\) 0.393942 0.0318483
\(154\) 0 0
\(155\) −13.1216 −1.05395
\(156\) −1.99397 −0.159645
\(157\) −10.2014 −0.814158 −0.407079 0.913393i \(-0.633453\pi\)
−0.407079 + 0.913393i \(0.633453\pi\)
\(158\) −0.184457 −0.0146746
\(159\) 12.4338 0.986067
\(160\) 2.04464 0.161643
\(161\) 0 0
\(162\) 0.0776754 0.00610275
\(163\) 15.5835 1.22060 0.610299 0.792171i \(-0.291049\pi\)
0.610299 + 0.792171i \(0.291049\pi\)
\(164\) 0.867344 0.0677282
\(165\) 6.42292 0.500023
\(166\) 0.760290 0.0590099
\(167\) −1.53310 −0.118635 −0.0593174 0.998239i \(-0.518892\pi\)
−0.0593174 + 0.998239i \(0.518892\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0.0673933 0.00516883
\(171\) −0.124751 −0.00953996
\(172\) 13.4580 1.02616
\(173\) 13.9844 1.06322 0.531609 0.846990i \(-0.321588\pi\)
0.531609 + 0.846990i \(0.321588\pi\)
\(174\) −0.0498849 −0.00378176
\(175\) 0 0
\(176\) 11.5597 0.871345
\(177\) 7.33555 0.551374
\(178\) 0.497404 0.0372820
\(179\) −21.9304 −1.63915 −0.819577 0.572969i \(-0.805792\pi\)
−0.819577 + 0.572969i \(0.805792\pi\)
\(180\) −4.39156 −0.327328
\(181\) −0.955265 −0.0710043 −0.0355021 0.999370i \(-0.511303\pi\)
−0.0355021 + 0.999370i \(0.511303\pi\)
\(182\) 0 0
\(183\) 2.39394 0.176965
\(184\) −1.04917 −0.0773457
\(185\) 9.22091 0.677934
\(186\) −0.462775 −0.0339323
\(187\) 1.14885 0.0840121
\(188\) −18.8693 −1.37619
\(189\) 0 0
\(190\) −0.0213417 −0.00154829
\(191\) 19.6834 1.42424 0.712121 0.702056i \(-0.247735\pi\)
0.712121 + 0.702056i \(0.247735\pi\)
\(192\) −7.85556 −0.566926
\(193\) 15.0951 1.08657 0.543283 0.839550i \(-0.317181\pi\)
0.543283 + 0.839550i \(0.317181\pi\)
\(194\) −1.15811 −0.0831472
\(195\) 2.20243 0.157719
\(196\) 0 0
\(197\) −11.0323 −0.786015 −0.393008 0.919535i \(-0.628565\pi\)
−0.393008 + 0.919535i \(0.628565\pi\)
\(198\) 0.226524 0.0160984
\(199\) −7.17701 −0.508765 −0.254382 0.967104i \(-0.581872\pi\)
−0.254382 + 0.967104i \(0.581872\pi\)
\(200\) 0.0463231 0.00327554
\(201\) 6.14047 0.433116
\(202\) 1.55211 0.109206
\(203\) 0 0
\(204\) −0.785507 −0.0549965
\(205\) −0.958020 −0.0669110
\(206\) 1.13985 0.0794168
\(207\) 3.38187 0.235057
\(208\) 3.96384 0.274843
\(209\) −0.363811 −0.0251653
\(210\) 0 0
\(211\) 15.7119 1.08165 0.540825 0.841135i \(-0.318112\pi\)
0.540825 + 0.841135i \(0.318112\pi\)
\(212\) −24.7926 −1.70277
\(213\) −14.0826 −0.964921
\(214\) −0.391463 −0.0267598
\(215\) −14.8649 −1.01378
\(216\) −0.310233 −0.0211087
\(217\) 0 0
\(218\) −0.00952939 −0.000645412 0
\(219\) 7.55295 0.510381
\(220\) −12.8071 −0.863453
\(221\) 0.393942 0.0264994
\(222\) 0.325204 0.0218262
\(223\) −6.20889 −0.415778 −0.207889 0.978152i \(-0.566659\pi\)
−0.207889 + 0.978152i \(0.566659\pi\)
\(224\) 0 0
\(225\) −0.149317 −0.00995449
\(226\) −0.323237 −0.0215014
\(227\) −23.6879 −1.57222 −0.786111 0.618085i \(-0.787909\pi\)
−0.786111 + 0.618085i \(0.787909\pi\)
\(228\) 0.248750 0.0164738
\(229\) 9.04226 0.597530 0.298765 0.954327i \(-0.403425\pi\)
0.298765 + 0.954327i \(0.403425\pi\)
\(230\) 0.578552 0.0381486
\(231\) 0 0
\(232\) 0.199239 0.0130807
\(233\) 5.34449 0.350129 0.175065 0.984557i \(-0.443987\pi\)
0.175065 + 0.984557i \(0.443987\pi\)
\(234\) 0.0776754 0.00507780
\(235\) 20.8420 1.35958
\(236\) −14.6268 −0.952126
\(237\) −2.37472 −0.154255
\(238\) 0 0
\(239\) −27.2548 −1.76297 −0.881484 0.472215i \(-0.843455\pi\)
−0.881484 + 0.472215i \(0.843455\pi\)
\(240\) 8.73006 0.563523
\(241\) 15.6761 1.00979 0.504894 0.863182i \(-0.331532\pi\)
0.504894 + 0.863182i \(0.331532\pi\)
\(242\) −0.193819 −0.0124592
\(243\) 1.00000 0.0641500
\(244\) −4.77344 −0.305588
\(245\) 0 0
\(246\) −0.0337875 −0.00215422
\(247\) −0.124751 −0.00793773
\(248\) 1.84831 0.117368
\(249\) 9.78804 0.620292
\(250\) −0.880916 −0.0557140
\(251\) 18.2218 1.15015 0.575075 0.818101i \(-0.304973\pi\)
0.575075 + 0.818101i \(0.304973\pi\)
\(252\) 0 0
\(253\) 9.86253 0.620052
\(254\) 1.16289 0.0729662
\(255\) 0.867628 0.0543329
\(256\) 15.5195 0.969969
\(257\) −15.0614 −0.939502 −0.469751 0.882799i \(-0.655656\pi\)
−0.469751 + 0.882799i \(0.655656\pi\)
\(258\) −0.524258 −0.0326389
\(259\) 0 0
\(260\) −4.39156 −0.272353
\(261\) −0.642223 −0.0397526
\(262\) −1.45580 −0.0899399
\(263\) −27.7916 −1.71370 −0.856852 0.515563i \(-0.827583\pi\)
−0.856852 + 0.515563i \(0.827583\pi\)
\(264\) −0.904729 −0.0556822
\(265\) 27.3846 1.68222
\(266\) 0 0
\(267\) 6.40363 0.391896
\(268\) −12.2439 −0.747915
\(269\) 8.00809 0.488262 0.244131 0.969742i \(-0.421497\pi\)
0.244131 + 0.969742i \(0.421497\pi\)
\(270\) 0.171074 0.0104113
\(271\) −20.2645 −1.23098 −0.615490 0.788144i \(-0.711042\pi\)
−0.615490 + 0.788144i \(0.711042\pi\)
\(272\) 1.56152 0.0946811
\(273\) 0 0
\(274\) −1.05565 −0.0637740
\(275\) −0.435453 −0.0262588
\(276\) −6.74335 −0.405902
\(277\) −29.4339 −1.76851 −0.884256 0.467003i \(-0.845334\pi\)
−0.884256 + 0.467003i \(0.845334\pi\)
\(278\) 1.22177 0.0732769
\(279\) −5.95780 −0.356684
\(280\) 0 0
\(281\) −3.76735 −0.224741 −0.112371 0.993666i \(-0.535844\pi\)
−0.112371 + 0.993666i \(0.535844\pi\)
\(282\) 0.735058 0.0437721
\(283\) −22.4407 −1.33396 −0.666980 0.745075i \(-0.732414\pi\)
−0.666980 + 0.745075i \(0.732414\pi\)
\(284\) 28.0801 1.66625
\(285\) −0.274755 −0.0162751
\(286\) 0.226524 0.0133946
\(287\) 0 0
\(288\) 0.928358 0.0547040
\(289\) −16.8448 −0.990871
\(290\) −0.109868 −0.00645166
\(291\) −14.9096 −0.874015
\(292\) −15.0603 −0.881339
\(293\) −8.13635 −0.475331 −0.237665 0.971347i \(-0.576382\pi\)
−0.237665 + 0.971347i \(0.576382\pi\)
\(294\) 0 0
\(295\) 16.1560 0.940639
\(296\) −1.29885 −0.0754943
\(297\) 2.91629 0.169220
\(298\) 0.260064 0.0150651
\(299\) 3.38187 0.195579
\(300\) 0.297734 0.0171897
\(301\) 0 0
\(302\) −1.58828 −0.0913951
\(303\) 19.9821 1.14794
\(304\) −0.494493 −0.0283611
\(305\) 5.27248 0.301901
\(306\) 0.0305996 0.00174926
\(307\) −18.9247 −1.08009 −0.540046 0.841636i \(-0.681593\pi\)
−0.540046 + 0.841636i \(0.681593\pi\)
\(308\) 0 0
\(309\) 14.6745 0.834802
\(310\) −1.01923 −0.0578882
\(311\) −11.6189 −0.658849 −0.329425 0.944182i \(-0.606855\pi\)
−0.329425 + 0.944182i \(0.606855\pi\)
\(312\) −0.310233 −0.0175635
\(313\) 11.7023 0.661455 0.330727 0.943726i \(-0.392706\pi\)
0.330727 + 0.943726i \(0.392706\pi\)
\(314\) −0.792395 −0.0447174
\(315\) 0 0
\(316\) 4.73512 0.266371
\(317\) −33.0069 −1.85385 −0.926926 0.375244i \(-0.877559\pi\)
−0.926926 + 0.375244i \(0.877559\pi\)
\(318\) 0.965802 0.0541595
\(319\) −1.87291 −0.104863
\(320\) −17.3013 −0.967172
\(321\) −5.03973 −0.281290
\(322\) 0 0
\(323\) −0.0491447 −0.00273448
\(324\) −1.99397 −0.110776
\(325\) −0.149317 −0.00828263
\(326\) 1.21046 0.0670411
\(327\) −0.122682 −0.00678435
\(328\) 0.134946 0.00745116
\(329\) 0 0
\(330\) 0.498902 0.0274637
\(331\) 14.7746 0.812085 0.406043 0.913854i \(-0.366908\pi\)
0.406043 + 0.913854i \(0.366908\pi\)
\(332\) −19.5170 −1.07114
\(333\) 4.18670 0.229430
\(334\) −0.119084 −0.00651599
\(335\) 13.5239 0.738892
\(336\) 0 0
\(337\) −13.0713 −0.712041 −0.356020 0.934478i \(-0.615867\pi\)
−0.356020 + 0.934478i \(0.615867\pi\)
\(338\) 0.0776754 0.00422498
\(339\) −4.16138 −0.226015
\(340\) −1.73002 −0.0938235
\(341\) −17.3747 −0.940892
\(342\) −0.00969009 −0.000523980 0
\(343\) 0 0
\(344\) 2.09387 0.112894
\(345\) 7.44833 0.401005
\(346\) 1.08625 0.0583970
\(347\) 6.37950 0.342469 0.171235 0.985230i \(-0.445224\pi\)
0.171235 + 0.985230i \(0.445224\pi\)
\(348\) 1.28057 0.0686458
\(349\) 10.3287 0.552881 0.276441 0.961031i \(-0.410845\pi\)
0.276441 + 0.961031i \(0.410845\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 2.70736 0.144303
\(353\) 30.1517 1.60481 0.802407 0.596777i \(-0.203552\pi\)
0.802407 + 0.596777i \(0.203552\pi\)
\(354\) 0.569792 0.0302841
\(355\) −31.0158 −1.64615
\(356\) −12.7686 −0.676736
\(357\) 0 0
\(358\) −1.70345 −0.0900302
\(359\) −29.1771 −1.53991 −0.769955 0.638098i \(-0.779721\pi\)
−0.769955 + 0.638098i \(0.779721\pi\)
\(360\) −0.683265 −0.0360112
\(361\) −18.9844 −0.999181
\(362\) −0.0742005 −0.00389989
\(363\) −2.49525 −0.130966
\(364\) 0 0
\(365\) 16.6348 0.870706
\(366\) 0.185950 0.00971978
\(367\) 4.48431 0.234079 0.117040 0.993127i \(-0.462660\pi\)
0.117040 + 0.993127i \(0.462660\pi\)
\(368\) 13.4052 0.698794
\(369\) −0.434984 −0.0226444
\(370\) 0.716237 0.0372354
\(371\) 0 0
\(372\) 11.8797 0.615932
\(373\) 23.9891 1.24211 0.621055 0.783767i \(-0.286704\pi\)
0.621055 + 0.783767i \(0.286704\pi\)
\(374\) 0.0892372 0.00461435
\(375\) −11.3410 −0.585646
\(376\) −2.93580 −0.151402
\(377\) −0.642223 −0.0330762
\(378\) 0 0
\(379\) 20.2228 1.03878 0.519389 0.854538i \(-0.326160\pi\)
0.519389 + 0.854538i \(0.326160\pi\)
\(380\) 0.547853 0.0281043
\(381\) 14.9711 0.766995
\(382\) 1.52892 0.0782262
\(383\) 23.0819 1.17943 0.589714 0.807612i \(-0.299240\pi\)
0.589714 + 0.807612i \(0.299240\pi\)
\(384\) −2.46690 −0.125888
\(385\) 0 0
\(386\) 1.17251 0.0596794
\(387\) −6.74934 −0.343088
\(388\) 29.7292 1.50927
\(389\) −33.3369 −1.69025 −0.845124 0.534570i \(-0.820474\pi\)
−0.845124 + 0.534570i \(0.820474\pi\)
\(390\) 0.171074 0.00866268
\(391\) 1.33226 0.0673754
\(392\) 0 0
\(393\) −18.7422 −0.945417
\(394\) −0.856934 −0.0431717
\(395\) −5.23015 −0.263157
\(396\) −5.81499 −0.292214
\(397\) 20.1880 1.01321 0.506605 0.862179i \(-0.330900\pi\)
0.506605 + 0.862179i \(0.330900\pi\)
\(398\) −0.557477 −0.0279438
\(399\) 0 0
\(400\) −0.591869 −0.0295935
\(401\) −4.21487 −0.210481 −0.105240 0.994447i \(-0.533561\pi\)
−0.105240 + 0.994447i \(0.533561\pi\)
\(402\) 0.476964 0.0237888
\(403\) −5.95780 −0.296779
\(404\) −39.8436 −1.98229
\(405\) 2.20243 0.109439
\(406\) 0 0
\(407\) 12.2096 0.605210
\(408\) −0.122214 −0.00605048
\(409\) −7.44627 −0.368195 −0.184097 0.982908i \(-0.558936\pi\)
−0.184097 + 0.982908i \(0.558936\pi\)
\(410\) −0.0744146 −0.00367507
\(411\) −13.5905 −0.670370
\(412\) −29.2604 −1.44156
\(413\) 0 0
\(414\) 0.262688 0.0129104
\(415\) 21.5574 1.05821
\(416\) 0.928358 0.0455165
\(417\) 15.7292 0.770262
\(418\) −0.0282591 −0.00138220
\(419\) 12.2817 0.600000 0.300000 0.953939i \(-0.403013\pi\)
0.300000 + 0.953939i \(0.403013\pi\)
\(420\) 0 0
\(421\) 26.9884 1.31533 0.657667 0.753309i \(-0.271544\pi\)
0.657667 + 0.753309i \(0.271544\pi\)
\(422\) 1.22042 0.0594093
\(423\) 9.46321 0.460117
\(424\) −3.85738 −0.187331
\(425\) −0.0588223 −0.00285330
\(426\) −1.09387 −0.0529981
\(427\) 0 0
\(428\) 10.0490 0.485739
\(429\) 2.91629 0.140800
\(430\) −1.15464 −0.0556816
\(431\) −20.3167 −0.978621 −0.489310 0.872110i \(-0.662752\pi\)
−0.489310 + 0.872110i \(0.662752\pi\)
\(432\) 3.96384 0.190710
\(433\) 9.02891 0.433902 0.216951 0.976183i \(-0.430389\pi\)
0.216951 + 0.976183i \(0.430389\pi\)
\(434\) 0 0
\(435\) −1.41445 −0.0678176
\(436\) 0.244624 0.0117154
\(437\) −0.421893 −0.0201819
\(438\) 0.586678 0.0280326
\(439\) −10.0349 −0.478940 −0.239470 0.970904i \(-0.576974\pi\)
−0.239470 + 0.970904i \(0.576974\pi\)
\(440\) −1.99260 −0.0949935
\(441\) 0 0
\(442\) 0.0305996 0.00145547
\(443\) −33.6409 −1.59833 −0.799165 0.601112i \(-0.794725\pi\)
−0.799165 + 0.601112i \(0.794725\pi\)
\(444\) −8.34815 −0.396185
\(445\) 14.1035 0.668571
\(446\) −0.482278 −0.0228365
\(447\) 3.34809 0.158359
\(448\) 0 0
\(449\) 1.21577 0.0573759 0.0286879 0.999588i \(-0.490867\pi\)
0.0286879 + 0.999588i \(0.490867\pi\)
\(450\) −0.0115983 −0.000546748 0
\(451\) −1.26854 −0.0597332
\(452\) 8.29766 0.390289
\(453\) −20.4476 −0.960714
\(454\) −1.83997 −0.0863540
\(455\) 0 0
\(456\) 0.0387019 0.00181238
\(457\) −28.4334 −1.33006 −0.665029 0.746817i \(-0.731581\pi\)
−0.665029 + 0.746817i \(0.731581\pi\)
\(458\) 0.702361 0.0328192
\(459\) 0.393942 0.0183876
\(460\) −14.8517 −0.692465
\(461\) −30.9864 −1.44318 −0.721590 0.692320i \(-0.756588\pi\)
−0.721590 + 0.692320i \(0.756588\pi\)
\(462\) 0 0
\(463\) −14.7992 −0.687779 −0.343889 0.939010i \(-0.611745\pi\)
−0.343889 + 0.939010i \(0.611745\pi\)
\(464\) −2.54567 −0.118180
\(465\) −13.1216 −0.608501
\(466\) 0.415135 0.0192308
\(467\) 1.82996 0.0846804 0.0423402 0.999103i \(-0.486519\pi\)
0.0423402 + 0.999103i \(0.486519\pi\)
\(468\) −1.99397 −0.0921711
\(469\) 0 0
\(470\) 1.61891 0.0746748
\(471\) −10.2014 −0.470054
\(472\) −2.27573 −0.104749
\(473\) −19.6830 −0.905027
\(474\) −0.184457 −0.00847241
\(475\) 0.0186275 0.000854689 0
\(476\) 0 0
\(477\) 12.4338 0.569306
\(478\) −2.11703 −0.0968306
\(479\) −9.34659 −0.427057 −0.213528 0.976937i \(-0.568496\pi\)
−0.213528 + 0.976937i \(0.568496\pi\)
\(480\) 2.04464 0.0933246
\(481\) 4.18670 0.190897
\(482\) 1.21765 0.0554623
\(483\) 0 0
\(484\) 4.97544 0.226156
\(485\) −32.8372 −1.49106
\(486\) 0.0776754 0.00352343
\(487\) 23.3739 1.05917 0.529587 0.848256i \(-0.322347\pi\)
0.529587 + 0.848256i \(0.322347\pi\)
\(488\) −0.742679 −0.0336195
\(489\) 15.5835 0.704712
\(490\) 0 0
\(491\) −21.4726 −0.969044 −0.484522 0.874779i \(-0.661007\pi\)
−0.484522 + 0.874779i \(0.661007\pi\)
\(492\) 0.867344 0.0391029
\(493\) −0.252998 −0.0113945
\(494\) −0.00969009 −0.000435978 0
\(495\) 6.42292 0.288689
\(496\) −23.6157 −1.06038
\(497\) 0 0
\(498\) 0.760290 0.0340694
\(499\) −22.1915 −0.993427 −0.496714 0.867915i \(-0.665460\pi\)
−0.496714 + 0.867915i \(0.665460\pi\)
\(500\) 22.6136 1.01131
\(501\) −1.53310 −0.0684939
\(502\) 1.41539 0.0631718
\(503\) −17.3231 −0.772399 −0.386200 0.922415i \(-0.626212\pi\)
−0.386200 + 0.922415i \(0.626212\pi\)
\(504\) 0 0
\(505\) 44.0090 1.95838
\(506\) 0.766076 0.0340562
\(507\) 1.00000 0.0444116
\(508\) −29.8520 −1.32447
\(509\) 26.8613 1.19061 0.595303 0.803502i \(-0.297032\pi\)
0.595303 + 0.803502i \(0.297032\pi\)
\(510\) 0.0673933 0.00298423
\(511\) 0 0
\(512\) 6.13928 0.271321
\(513\) −0.124751 −0.00550790
\(514\) −1.16990 −0.0516019
\(515\) 32.3195 1.42417
\(516\) 13.4580 0.592454
\(517\) 27.5975 1.21374
\(518\) 0 0
\(519\) 13.9844 0.613849
\(520\) −0.683265 −0.0299632
\(521\) 10.2029 0.446998 0.223499 0.974704i \(-0.428252\pi\)
0.223499 + 0.974704i \(0.428252\pi\)
\(522\) −0.0498849 −0.00218340
\(523\) −11.9147 −0.520992 −0.260496 0.965475i \(-0.583886\pi\)
−0.260496 + 0.965475i \(0.583886\pi\)
\(524\) 37.3712 1.63257
\(525\) 0 0
\(526\) −2.15872 −0.0941248
\(527\) −2.34703 −0.102238
\(528\) 11.5597 0.503071
\(529\) −11.5629 −0.502736
\(530\) 2.12711 0.0923957
\(531\) 7.33555 0.318336
\(532\) 0 0
\(533\) −0.434984 −0.0188412
\(534\) 0.497404 0.0215248
\(535\) −11.0996 −0.479879
\(536\) −1.90498 −0.0822825
\(537\) −21.9304 −0.946366
\(538\) 0.622031 0.0268177
\(539\) 0 0
\(540\) −4.39156 −0.188983
\(541\) 19.9687 0.858520 0.429260 0.903181i \(-0.358774\pi\)
0.429260 + 0.903181i \(0.358774\pi\)
\(542\) −1.57405 −0.0676114
\(543\) −0.955265 −0.0409943
\(544\) 0.365719 0.0156801
\(545\) −0.270199 −0.0115740
\(546\) 0 0
\(547\) −16.7640 −0.716778 −0.358389 0.933572i \(-0.616674\pi\)
−0.358389 + 0.933572i \(0.616674\pi\)
\(548\) 27.0990 1.15761
\(549\) 2.39394 0.102171
\(550\) −0.0338239 −0.00144226
\(551\) 0.0801180 0.00341314
\(552\) −1.04917 −0.0446556
\(553\) 0 0
\(554\) −2.28629 −0.0971351
\(555\) 9.22091 0.391406
\(556\) −31.3635 −1.33011
\(557\) 23.3797 0.990628 0.495314 0.868714i \(-0.335053\pi\)
0.495314 + 0.868714i \(0.335053\pi\)
\(558\) −0.462775 −0.0195908
\(559\) −6.74934 −0.285467
\(560\) 0 0
\(561\) 1.14885 0.0485044
\(562\) −0.292630 −0.0123439
\(563\) −42.0038 −1.77025 −0.885124 0.465355i \(-0.845927\pi\)
−0.885124 + 0.465355i \(0.845927\pi\)
\(564\) −18.8693 −0.794542
\(565\) −9.16514 −0.385580
\(566\) −1.74309 −0.0732675
\(567\) 0 0
\(568\) 4.36887 0.183314
\(569\) 12.5796 0.527366 0.263683 0.964609i \(-0.415063\pi\)
0.263683 + 0.964609i \(0.415063\pi\)
\(570\) −0.0213417 −0.000893906 0
\(571\) −8.79876 −0.368217 −0.184108 0.982906i \(-0.558940\pi\)
−0.184108 + 0.982906i \(0.558940\pi\)
\(572\) −5.81499 −0.243137
\(573\) 19.6834 0.822287
\(574\) 0 0
\(575\) −0.504972 −0.0210588
\(576\) −7.85556 −0.327315
\(577\) −1.41370 −0.0588531 −0.0294266 0.999567i \(-0.509368\pi\)
−0.0294266 + 0.999567i \(0.509368\pi\)
\(578\) −1.30843 −0.0544234
\(579\) 15.0951 0.627329
\(580\) 2.82036 0.117109
\(581\) 0 0
\(582\) −1.15811 −0.0480051
\(583\) 36.2607 1.50176
\(584\) −2.34317 −0.0969612
\(585\) 2.20243 0.0910591
\(586\) −0.631994 −0.0261074
\(587\) 24.3046 1.00316 0.501580 0.865111i \(-0.332752\pi\)
0.501580 + 0.865111i \(0.332752\pi\)
\(588\) 0 0
\(589\) 0.743243 0.0306248
\(590\) 1.25492 0.0516644
\(591\) −11.0323 −0.453806
\(592\) 16.5954 0.682067
\(593\) 26.7063 1.09670 0.548348 0.836250i \(-0.315257\pi\)
0.548348 + 0.836250i \(0.315257\pi\)
\(594\) 0.226524 0.00929439
\(595\) 0 0
\(596\) −6.67597 −0.273459
\(597\) −7.17701 −0.293736
\(598\) 0.262688 0.0107421
\(599\) −33.6537 −1.37505 −0.687526 0.726160i \(-0.741303\pi\)
−0.687526 + 0.726160i \(0.741303\pi\)
\(600\) 0.0463231 0.00189113
\(601\) 12.5737 0.512890 0.256445 0.966559i \(-0.417449\pi\)
0.256445 + 0.966559i \(0.417449\pi\)
\(602\) 0 0
\(603\) 6.14047 0.250059
\(604\) 40.7719 1.65899
\(605\) −5.49560 −0.223428
\(606\) 1.55211 0.0630503
\(607\) −21.4417 −0.870292 −0.435146 0.900360i \(-0.643303\pi\)
−0.435146 + 0.900360i \(0.643303\pi\)
\(608\) −0.115814 −0.00469687
\(609\) 0 0
\(610\) 0.409542 0.0165819
\(611\) 9.46321 0.382840
\(612\) −0.785507 −0.0317522
\(613\) 30.7261 1.24102 0.620509 0.784200i \(-0.286926\pi\)
0.620509 + 0.784200i \(0.286926\pi\)
\(614\) −1.46999 −0.0593238
\(615\) −0.958020 −0.0386311
\(616\) 0 0
\(617\) 0.216739 0.00872559 0.00436279 0.999990i \(-0.498611\pi\)
0.00436279 + 0.999990i \(0.498611\pi\)
\(618\) 1.13985 0.0458513
\(619\) −30.4980 −1.22582 −0.612909 0.790153i \(-0.710001\pi\)
−0.612909 + 0.790153i \(0.710001\pi\)
\(620\) 26.1641 1.05077
\(621\) 3.38187 0.135710
\(622\) −0.902505 −0.0361871
\(623\) 0 0
\(624\) 3.96384 0.158680
\(625\) −24.2311 −0.969245
\(626\) 0.908983 0.0363303
\(627\) −0.363811 −0.0145292
\(628\) 20.3412 0.811702
\(629\) 1.64932 0.0657626
\(630\) 0 0
\(631\) −14.0505 −0.559343 −0.279671 0.960096i \(-0.590226\pi\)
−0.279671 + 0.960096i \(0.590226\pi\)
\(632\) 0.736717 0.0293050
\(633\) 15.7119 0.624490
\(634\) −2.56382 −0.101822
\(635\) 32.9729 1.30849
\(636\) −24.7926 −0.983092
\(637\) 0 0
\(638\) −0.145479 −0.00575956
\(639\) −14.0826 −0.557097
\(640\) −5.43317 −0.214765
\(641\) −18.1731 −0.717796 −0.358898 0.933377i \(-0.616847\pi\)
−0.358898 + 0.933377i \(0.616847\pi\)
\(642\) −0.391463 −0.0154498
\(643\) 7.27895 0.287054 0.143527 0.989646i \(-0.454156\pi\)
0.143527 + 0.989646i \(0.454156\pi\)
\(644\) 0 0
\(645\) −14.8649 −0.585306
\(646\) −0.00381733 −0.000150191 0
\(647\) −17.7797 −0.698992 −0.349496 0.936938i \(-0.613647\pi\)
−0.349496 + 0.936938i \(0.613647\pi\)
\(648\) −0.310233 −0.0121871
\(649\) 21.3926 0.839733
\(650\) −0.0115983 −0.000454922 0
\(651\) 0 0
\(652\) −31.0731 −1.21692
\(653\) 8.13253 0.318251 0.159125 0.987258i \(-0.449133\pi\)
0.159125 + 0.987258i \(0.449133\pi\)
\(654\) −0.00952939 −0.000372629 0
\(655\) −41.2782 −1.61287
\(656\) −1.72421 −0.0673189
\(657\) 7.55295 0.294669
\(658\) 0 0
\(659\) 10.0958 0.393278 0.196639 0.980476i \(-0.436997\pi\)
0.196639 + 0.980476i \(0.436997\pi\)
\(660\) −12.8071 −0.498515
\(661\) −24.3455 −0.946928 −0.473464 0.880813i \(-0.656997\pi\)
−0.473464 + 0.880813i \(0.656997\pi\)
\(662\) 1.14762 0.0446036
\(663\) 0.393942 0.0152994
\(664\) −3.03657 −0.117842
\(665\) 0 0
\(666\) 0.325204 0.0126014
\(667\) −2.17192 −0.0840970
\(668\) 3.05695 0.118277
\(669\) −6.20889 −0.240050
\(670\) 1.05048 0.0405835
\(671\) 6.98143 0.269515
\(672\) 0 0
\(673\) −39.4811 −1.52189 −0.760943 0.648819i \(-0.775263\pi\)
−0.760943 + 0.648819i \(0.775263\pi\)
\(674\) −1.01532 −0.0391087
\(675\) −0.149317 −0.00574722
\(676\) −1.99397 −0.0766910
\(677\) −17.9830 −0.691144 −0.345572 0.938392i \(-0.612315\pi\)
−0.345572 + 0.938392i \(0.612315\pi\)
\(678\) −0.323237 −0.0124138
\(679\) 0 0
\(680\) −0.269167 −0.0103221
\(681\) −23.6879 −0.907723
\(682\) −1.34959 −0.0516783
\(683\) 15.3265 0.586452 0.293226 0.956043i \(-0.405271\pi\)
0.293226 + 0.956043i \(0.405271\pi\)
\(684\) 0.248750 0.00951118
\(685\) −29.9321 −1.14365
\(686\) 0 0
\(687\) 9.04226 0.344984
\(688\) −26.7533 −1.01996
\(689\) 12.4338 0.473691
\(690\) 0.578552 0.0220251
\(691\) 25.2599 0.960931 0.480466 0.877014i \(-0.340468\pi\)
0.480466 + 0.877014i \(0.340468\pi\)
\(692\) −27.8845 −1.06001
\(693\) 0 0
\(694\) 0.495530 0.0188101
\(695\) 34.6424 1.31406
\(696\) 0.199239 0.00755212
\(697\) −0.171358 −0.00649066
\(698\) 0.802283 0.0303669
\(699\) 5.34449 0.202147
\(700\) 0 0
\(701\) 37.6169 1.42077 0.710385 0.703813i \(-0.248521\pi\)
0.710385 + 0.703813i \(0.248521\pi\)
\(702\) 0.0776754 0.00293167
\(703\) −0.522296 −0.0196988
\(704\) −22.9091 −0.863419
\(705\) 20.8420 0.784956
\(706\) 2.34205 0.0881441
\(707\) 0 0
\(708\) −14.6268 −0.549710
\(709\) 10.7266 0.402847 0.201423 0.979504i \(-0.435443\pi\)
0.201423 + 0.979504i \(0.435443\pi\)
\(710\) −2.40916 −0.0904143
\(711\) −2.37472 −0.0890590
\(712\) −1.98662 −0.0744516
\(713\) −20.1485 −0.754569
\(714\) 0 0
\(715\) 6.42292 0.240203
\(716\) 43.7285 1.63421
\(717\) −27.2548 −1.01785
\(718\) −2.26635 −0.0845792
\(719\) −12.3407 −0.460230 −0.230115 0.973163i \(-0.573910\pi\)
−0.230115 + 0.973163i \(0.573910\pi\)
\(720\) 8.73006 0.325350
\(721\) 0 0
\(722\) −1.47462 −0.0548798
\(723\) 15.6761 0.583001
\(724\) 1.90477 0.0707901
\(725\) 0.0958949 0.00356145
\(726\) −0.193819 −0.00719330
\(727\) −46.5464 −1.72631 −0.863155 0.504939i \(-0.831515\pi\)
−0.863155 + 0.504939i \(0.831515\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.29212 0.0478233
\(731\) −2.65885 −0.0983410
\(732\) −4.77344 −0.176431
\(733\) −37.8843 −1.39929 −0.699644 0.714492i \(-0.746658\pi\)
−0.699644 + 0.714492i \(0.746658\pi\)
\(734\) 0.348321 0.0128568
\(735\) 0 0
\(736\) 3.13959 0.115727
\(737\) 17.9074 0.659628
\(738\) −0.0337875 −0.00124374
\(739\) 18.9782 0.698126 0.349063 0.937099i \(-0.386500\pi\)
0.349063 + 0.937099i \(0.386500\pi\)
\(740\) −18.3862 −0.675889
\(741\) −0.124751 −0.00458285
\(742\) 0 0
\(743\) 20.2127 0.741534 0.370767 0.928726i \(-0.379095\pi\)
0.370767 + 0.928726i \(0.379095\pi\)
\(744\) 1.84831 0.0677622
\(745\) 7.37392 0.270159
\(746\) 1.86336 0.0682226
\(747\) 9.78804 0.358126
\(748\) −2.29077 −0.0837587
\(749\) 0 0
\(750\) −0.880916 −0.0321665
\(751\) −47.1535 −1.72066 −0.860328 0.509742i \(-0.829741\pi\)
−0.860328 + 0.509742i \(0.829741\pi\)
\(752\) 37.5106 1.36787
\(753\) 18.2218 0.664040
\(754\) −0.0498849 −0.00181670
\(755\) −45.0344 −1.63897
\(756\) 0 0
\(757\) −14.7288 −0.535327 −0.267664 0.963512i \(-0.586252\pi\)
−0.267664 + 0.963512i \(0.586252\pi\)
\(758\) 1.57082 0.0570546
\(759\) 9.86253 0.357987
\(760\) 0.0852381 0.00309191
\(761\) −41.7494 −1.51341 −0.756706 0.653755i \(-0.773193\pi\)
−0.756706 + 0.653755i \(0.773193\pi\)
\(762\) 1.16289 0.0421270
\(763\) 0 0
\(764\) −39.2481 −1.41995
\(765\) 0.867628 0.0313691
\(766\) 1.79289 0.0647798
\(767\) 7.33555 0.264871
\(768\) 15.5195 0.560012
\(769\) 34.0121 1.22651 0.613254 0.789886i \(-0.289860\pi\)
0.613254 + 0.789886i \(0.289860\pi\)
\(770\) 0 0
\(771\) −15.0614 −0.542422
\(772\) −30.0990 −1.08329
\(773\) 17.0025 0.611537 0.305769 0.952106i \(-0.401087\pi\)
0.305769 + 0.952106i \(0.401087\pi\)
\(774\) −0.524258 −0.0188441
\(775\) 0.889603 0.0319555
\(776\) 4.62544 0.166044
\(777\) 0 0
\(778\) −2.58946 −0.0928366
\(779\) 0.0542648 0.00194424
\(780\) −4.39156 −0.157243
\(781\) −41.0688 −1.46956
\(782\) 0.103484 0.00370058
\(783\) −0.642223 −0.0229512
\(784\) 0 0
\(785\) −22.4678 −0.801909
\(786\) −1.45580 −0.0519268
\(787\) 31.1294 1.10964 0.554822 0.831969i \(-0.312786\pi\)
0.554822 + 0.831969i \(0.312786\pi\)
\(788\) 21.9979 0.783644
\(789\) −27.7916 −0.989407
\(790\) −0.406254 −0.0144539
\(791\) 0 0
\(792\) −0.904729 −0.0321482
\(793\) 2.39394 0.0850114
\(794\) 1.56811 0.0556503
\(795\) 27.3846 0.971231
\(796\) 14.3107 0.507230
\(797\) −10.5797 −0.374752 −0.187376 0.982288i \(-0.559998\pi\)
−0.187376 + 0.982288i \(0.559998\pi\)
\(798\) 0 0
\(799\) 3.72795 0.131885
\(800\) −0.138620 −0.00490095
\(801\) 6.40363 0.226261
\(802\) −0.327392 −0.0115606
\(803\) 22.0266 0.777302
\(804\) −12.2439 −0.431809
\(805\) 0 0
\(806\) −0.462775 −0.0163005
\(807\) 8.00809 0.281898
\(808\) −6.19909 −0.218083
\(809\) −19.0261 −0.668924 −0.334462 0.942409i \(-0.608555\pi\)
−0.334462 + 0.942409i \(0.608555\pi\)
\(810\) 0.171074 0.00601094
\(811\) 24.9894 0.877497 0.438748 0.898610i \(-0.355422\pi\)
0.438748 + 0.898610i \(0.355422\pi\)
\(812\) 0 0
\(813\) −20.2645 −0.710707
\(814\) 0.948389 0.0332410
\(815\) 34.3216 1.20223
\(816\) 1.56152 0.0546641
\(817\) 0.841988 0.0294574
\(818\) −0.578392 −0.0202230
\(819\) 0 0
\(820\) 1.91026 0.0667092
\(821\) 45.7000 1.59494 0.797471 0.603358i \(-0.206171\pi\)
0.797471 + 0.603358i \(0.206171\pi\)
\(822\) −1.05565 −0.0368199
\(823\) −10.3799 −0.361822 −0.180911 0.983499i \(-0.557905\pi\)
−0.180911 + 0.983499i \(0.557905\pi\)
\(824\) −4.55250 −0.158594
\(825\) −0.435453 −0.0151605
\(826\) 0 0
\(827\) −9.56239 −0.332517 −0.166258 0.986082i \(-0.553169\pi\)
−0.166258 + 0.986082i \(0.553169\pi\)
\(828\) −6.74335 −0.234347
\(829\) 17.8736 0.620777 0.310388 0.950610i \(-0.399541\pi\)
0.310388 + 0.950610i \(0.399541\pi\)
\(830\) 1.67448 0.0581221
\(831\) −29.4339 −1.02105
\(832\) −7.85556 −0.272343
\(833\) 0 0
\(834\) 1.22177 0.0423064
\(835\) −3.37654 −0.116850
\(836\) 0.725426 0.0250894
\(837\) −5.95780 −0.205932
\(838\) 0.953985 0.0329549
\(839\) −33.6620 −1.16214 −0.581071 0.813853i \(-0.697366\pi\)
−0.581071 + 0.813853i \(0.697366\pi\)
\(840\) 0 0
\(841\) −28.5876 −0.985778
\(842\) 2.09633 0.0722444
\(843\) −3.76735 −0.129754
\(844\) −31.3289 −1.07839
\(845\) 2.20243 0.0757658
\(846\) 0.735058 0.0252718
\(847\) 0 0
\(848\) 49.2857 1.69248
\(849\) −22.4407 −0.770163
\(850\) −0.00456904 −0.000156717 0
\(851\) 14.1589 0.485361
\(852\) 28.0801 0.962010
\(853\) 45.2796 1.55035 0.775173 0.631749i \(-0.217663\pi\)
0.775173 + 0.631749i \(0.217663\pi\)
\(854\) 0 0
\(855\) −0.274755 −0.00939643
\(856\) 1.56349 0.0534389
\(857\) 44.5722 1.52256 0.761278 0.648425i \(-0.224572\pi\)
0.761278 + 0.648425i \(0.224572\pi\)
\(858\) 0.226524 0.00773340
\(859\) 39.9009 1.36140 0.680701 0.732561i \(-0.261675\pi\)
0.680701 + 0.732561i \(0.261675\pi\)
\(860\) 29.6402 1.01072
\(861\) 0 0
\(862\) −1.57811 −0.0537505
\(863\) −3.11045 −0.105881 −0.0529406 0.998598i \(-0.516859\pi\)
−0.0529406 + 0.998598i \(0.516859\pi\)
\(864\) 0.928358 0.0315834
\(865\) 30.7997 1.04722
\(866\) 0.701324 0.0238320
\(867\) −16.8448 −0.572080
\(868\) 0 0
\(869\) −6.92538 −0.234927
\(870\) −0.109868 −0.00372487
\(871\) 6.14047 0.208062
\(872\) 0.0380601 0.00128888
\(873\) −14.9096 −0.504613
\(874\) −0.0327707 −0.00110848
\(875\) 0 0
\(876\) −15.0603 −0.508841
\(877\) −4.64274 −0.156774 −0.0783872 0.996923i \(-0.524977\pi\)
−0.0783872 + 0.996923i \(0.524977\pi\)
\(878\) −0.779466 −0.0263057
\(879\) −8.13635 −0.274432
\(880\) 25.4594 0.858236
\(881\) 42.8058 1.44216 0.721082 0.692850i \(-0.243645\pi\)
0.721082 + 0.692850i \(0.243645\pi\)
\(882\) 0 0
\(883\) −9.60368 −0.323190 −0.161595 0.986857i \(-0.551664\pi\)
−0.161595 + 0.986857i \(0.551664\pi\)
\(884\) −0.785507 −0.0264194
\(885\) 16.1560 0.543078
\(886\) −2.61307 −0.0877879
\(887\) 10.4937 0.352346 0.176173 0.984359i \(-0.443628\pi\)
0.176173 + 0.984359i \(0.443628\pi\)
\(888\) −1.29885 −0.0435866
\(889\) 0 0
\(890\) 1.09550 0.0367211
\(891\) 2.91629 0.0976994
\(892\) 12.3803 0.414524
\(893\) −1.18055 −0.0395055
\(894\) 0.260064 0.00869784
\(895\) −48.3001 −1.61449
\(896\) 0 0
\(897\) 3.38187 0.112918
\(898\) 0.0944356 0.00315136
\(899\) 3.82624 0.127612
\(900\) 0.297734 0.00992446
\(901\) 4.89820 0.163183
\(902\) −0.0985343 −0.00328083
\(903\) 0 0
\(904\) 1.29100 0.0429380
\(905\) −2.10390 −0.0699360
\(906\) −1.58828 −0.0527670
\(907\) 54.3480 1.80460 0.902298 0.431113i \(-0.141879\pi\)
0.902298 + 0.431113i \(0.141879\pi\)
\(908\) 47.2329 1.56748
\(909\) 19.9821 0.662763
\(910\) 0 0
\(911\) 29.9548 0.992445 0.496223 0.868195i \(-0.334720\pi\)
0.496223 + 0.868195i \(0.334720\pi\)
\(912\) −0.494493 −0.0163743
\(913\) 28.5448 0.944695
\(914\) −2.20858 −0.0730532
\(915\) 5.27248 0.174303
\(916\) −18.0300 −0.595727
\(917\) 0 0
\(918\) 0.0305996 0.00100994
\(919\) 30.8093 1.01631 0.508153 0.861267i \(-0.330329\pi\)
0.508153 + 0.861267i \(0.330329\pi\)
\(920\) −2.31072 −0.0761821
\(921\) −18.9247 −0.623591
\(922\) −2.40688 −0.0792664
\(923\) −14.0826 −0.463533
\(924\) 0 0
\(925\) −0.625147 −0.0205547
\(926\) −1.14954 −0.0377761
\(927\) 14.6745 0.481973
\(928\) −0.596213 −0.0195716
\(929\) 19.8880 0.652503 0.326252 0.945283i \(-0.394214\pi\)
0.326252 + 0.945283i \(0.394214\pi\)
\(930\) −1.01923 −0.0334218
\(931\) 0 0
\(932\) −10.6567 −0.349073
\(933\) −11.6189 −0.380387
\(934\) 0.142143 0.00465105
\(935\) 2.53025 0.0827482
\(936\) −0.310233 −0.0101403
\(937\) 50.6240 1.65382 0.826908 0.562338i \(-0.190098\pi\)
0.826908 + 0.562338i \(0.190098\pi\)
\(938\) 0 0
\(939\) 11.7023 0.381891
\(940\) −41.5583 −1.35548
\(941\) −21.1170 −0.688394 −0.344197 0.938897i \(-0.611849\pi\)
−0.344197 + 0.938897i \(0.611849\pi\)
\(942\) −0.792395 −0.0258176
\(943\) −1.47106 −0.0479044
\(944\) 29.0769 0.946373
\(945\) 0 0
\(946\) −1.52889 −0.0497084
\(947\) 33.0962 1.07548 0.537741 0.843110i \(-0.319278\pi\)
0.537741 + 0.843110i \(0.319278\pi\)
\(948\) 4.73512 0.153789
\(949\) 7.55295 0.245179
\(950\) 0.00144690 4.69436e−5 0
\(951\) −33.0069 −1.07032
\(952\) 0 0
\(953\) −3.06989 −0.0994436 −0.0497218 0.998763i \(-0.515833\pi\)
−0.0497218 + 0.998763i \(0.515833\pi\)
\(954\) 0.965802 0.0312690
\(955\) 43.3513 1.40282
\(956\) 54.3452 1.75765
\(957\) −1.87291 −0.0605425
\(958\) −0.726000 −0.0234560
\(959\) 0 0
\(960\) −17.3013 −0.558397
\(961\) 4.49541 0.145013
\(962\) 0.325204 0.0104850
\(963\) −5.03973 −0.162403
\(964\) −31.2576 −1.00674
\(965\) 33.2457 1.07022
\(966\) 0 0
\(967\) 10.0334 0.322651 0.161325 0.986901i \(-0.448423\pi\)
0.161325 + 0.986901i \(0.448423\pi\)
\(968\) 0.774107 0.0248807
\(969\) −0.0491447 −0.00157875
\(970\) −2.55064 −0.0818963
\(971\) 19.4697 0.624812 0.312406 0.949949i \(-0.398865\pi\)
0.312406 + 0.949949i \(0.398865\pi\)
\(972\) −1.99397 −0.0639565
\(973\) 0 0
\(974\) 1.81558 0.0581749
\(975\) −0.149317 −0.00478198
\(976\) 9.48919 0.303742
\(977\) 1.86853 0.0597795 0.0298898 0.999553i \(-0.490484\pi\)
0.0298898 + 0.999553i \(0.490484\pi\)
\(978\) 1.21046 0.0387062
\(979\) 18.6749 0.596851
\(980\) 0 0
\(981\) −0.122682 −0.00391694
\(982\) −1.66789 −0.0532246
\(983\) −32.2863 −1.02977 −0.514886 0.857258i \(-0.672166\pi\)
−0.514886 + 0.857258i \(0.672166\pi\)
\(984\) 0.134946 0.00430193
\(985\) −24.2977 −0.774190
\(986\) −0.0196517 −0.000625839 0
\(987\) 0 0
\(988\) 0.248750 0.00791378
\(989\) −22.8254 −0.725806
\(990\) 0.498902 0.0158562
\(991\) 56.8365 1.80547 0.902736 0.430195i \(-0.141555\pi\)
0.902736 + 0.430195i \(0.141555\pi\)
\(992\) −5.53097 −0.175609
\(993\) 14.7746 0.468858
\(994\) 0 0
\(995\) −15.8068 −0.501111
\(996\) −19.5170 −0.618421
\(997\) 24.1136 0.763685 0.381843 0.924227i \(-0.375290\pi\)
0.381843 + 0.924227i \(0.375290\pi\)
\(998\) −1.72373 −0.0545638
\(999\) 4.18670 0.132461
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 1911.2.a.u.1.3 5
3.2 odd 2 5733.2.a.bp.1.3 5
7.3 odd 6 273.2.i.e.79.3 10
7.5 odd 6 273.2.i.e.235.3 yes 10
7.6 odd 2 1911.2.a.t.1.3 5
21.5 even 6 819.2.j.g.235.3 10
21.17 even 6 819.2.j.g.352.3 10
21.20 even 2 5733.2.a.bq.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.i.e.79.3 10 7.3 odd 6
273.2.i.e.235.3 yes 10 7.5 odd 6
819.2.j.g.235.3 10 21.5 even 6
819.2.j.g.352.3 10 21.17 even 6
1911.2.a.t.1.3 5 7.6 odd 2
1911.2.a.u.1.3 5 1.1 even 1 trivial
5733.2.a.bp.1.3 5 3.2 odd 2
5733.2.a.bq.1.3 5 21.20 even 2