Properties

Label 5445.2.a.bp.1.2
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5445,2,Mod(1,5445)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5445, 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("5445.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.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: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.477260\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.477260 q^{2} -1.77222 q^{4} +1.00000 q^{5} +2.68522 q^{7} +1.80033 q^{8} +O(q^{10})\) \(q-0.477260 q^{2} -1.77222 q^{4} +1.00000 q^{5} +2.68522 q^{7} +1.80033 q^{8} -0.477260 q^{10} -4.66785 q^{13} -1.28155 q^{14} +2.68522 q^{16} +4.62632 q^{17} -4.34044 q^{19} -1.77222 q^{20} -2.77222 q^{23} +1.00000 q^{25} +2.22778 q^{26} -4.75881 q^{28} +3.01341 q^{29} +2.38630 q^{31} -4.88221 q^{32} -2.20796 q^{34} +2.68522 q^{35} -10.6429 q^{37} +2.07152 q^{38} +1.80033 q^{40} +2.21041 q^{41} -7.06719 q^{43} +1.32307 q^{46} -4.36215 q^{47} +0.210405 q^{49} -0.477260 q^{50} +8.27247 q^{52} +6.33404 q^{53} +4.83428 q^{56} -1.43818 q^{58} -11.7473 q^{59} -3.98263 q^{61} -1.13889 q^{62} -3.04036 q^{64} -4.66785 q^{65} +7.31984 q^{67} -8.19888 q^{68} -1.28155 q^{70} -1.19571 q^{71} +1.02171 q^{73} +5.07943 q^{74} +7.69223 q^{76} +3.50213 q^{79} +2.68522 q^{80} -1.05494 q^{82} -11.1158 q^{83} +4.62632 q^{85} +3.37289 q^{86} -2.76978 q^{89} -12.5342 q^{91} +4.91300 q^{92} +2.08188 q^{94} -4.34044 q^{95} +18.5342 q^{97} -0.100418 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} - q^{4} + 4 q^{5} - 3 q^{7} + 3 q^{8} + q^{10} - q^{13} - 2 q^{14} - 3 q^{16} - q^{17} - 20 q^{19} - q^{20} - 5 q^{23} + 4 q^{25} + 15 q^{26} - 13 q^{28} + 12 q^{29} - 5 q^{31} - 8 q^{32} + 2 q^{34} - 3 q^{35} + 7 q^{37} - 20 q^{38} + 3 q^{40} + 11 q^{41} - 19 q^{43} + 4 q^{46} - 5 q^{47} + 3 q^{49} + q^{50} + 11 q^{52} + 11 q^{53} - 11 q^{56} - 14 q^{58} - 9 q^{59} - 12 q^{61} - 35 q^{62} - 3 q^{64} - q^{65} - 19 q^{67} - 3 q^{68} - 2 q^{70} - 5 q^{71} - 11 q^{73} - 34 q^{79} - 3 q^{80} - 6 q^{82} - 11 q^{83} - q^{85} - q^{86} + 8 q^{89} - 8 q^{91} + 12 q^{92} + q^{94} - 20 q^{95} + 32 q^{97} - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.477260 −0.337474 −0.168737 0.985661i \(-0.553969\pi\)
−0.168737 + 0.985661i \(0.553969\pi\)
\(3\) 0 0
\(4\) −1.77222 −0.886111
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.68522 1.01492 0.507459 0.861676i \(-0.330585\pi\)
0.507459 + 0.861676i \(0.330585\pi\)
\(8\) 1.80033 0.636513
\(9\) 0 0
\(10\) −0.477260 −0.150923
\(11\) 0 0
\(12\) 0 0
\(13\) −4.66785 −1.29463 −0.647314 0.762223i \(-0.724108\pi\)
−0.647314 + 0.762223i \(0.724108\pi\)
\(14\) −1.28155 −0.342508
\(15\) 0 0
\(16\) 2.68522 0.671305
\(17\) 4.62632 1.12205 0.561024 0.827799i \(-0.310407\pi\)
0.561024 + 0.827799i \(0.310407\pi\)
\(18\) 0 0
\(19\) −4.34044 −0.995766 −0.497883 0.867244i \(-0.665889\pi\)
−0.497883 + 0.867244i \(0.665889\pi\)
\(20\) −1.77222 −0.396281
\(21\) 0 0
\(22\) 0 0
\(23\) −2.77222 −0.578048 −0.289024 0.957322i \(-0.593331\pi\)
−0.289024 + 0.957322i \(0.593331\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.22778 0.436903
\(27\) 0 0
\(28\) −4.75881 −0.899330
\(29\) 3.01341 0.559577 0.279789 0.960062i \(-0.409736\pi\)
0.279789 + 0.960062i \(0.409736\pi\)
\(30\) 0 0
\(31\) 2.38630 0.428592 0.214296 0.976769i \(-0.431254\pi\)
0.214296 + 0.976769i \(0.431254\pi\)
\(32\) −4.88221 −0.863061
\(33\) 0 0
\(34\) −2.20796 −0.378662
\(35\) 2.68522 0.453885
\(36\) 0 0
\(37\) −10.6429 −1.74968 −0.874842 0.484409i \(-0.839035\pi\)
−0.874842 + 0.484409i \(0.839035\pi\)
\(38\) 2.07152 0.336045
\(39\) 0 0
\(40\) 1.80033 0.284657
\(41\) 2.21041 0.345207 0.172604 0.984991i \(-0.444782\pi\)
0.172604 + 0.984991i \(0.444782\pi\)
\(42\) 0 0
\(43\) −7.06719 −1.07774 −0.538868 0.842390i \(-0.681148\pi\)
−0.538868 + 0.842390i \(0.681148\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.32307 0.195076
\(47\) −4.36215 −0.636285 −0.318142 0.948043i \(-0.603059\pi\)
−0.318142 + 0.948043i \(0.603059\pi\)
\(48\) 0 0
\(49\) 0.210405 0.0300579
\(50\) −0.477260 −0.0674948
\(51\) 0 0
\(52\) 8.27247 1.14718
\(53\) 6.33404 0.870047 0.435024 0.900419i \(-0.356740\pi\)
0.435024 + 0.900419i \(0.356740\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.83428 0.646008
\(57\) 0 0
\(58\) −1.43818 −0.188843
\(59\) −11.7473 −1.52937 −0.764683 0.644407i \(-0.777104\pi\)
−0.764683 + 0.644407i \(0.777104\pi\)
\(60\) 0 0
\(61\) −3.98263 −0.509923 −0.254962 0.966951i \(-0.582063\pi\)
−0.254962 + 0.966951i \(0.582063\pi\)
\(62\) −1.13889 −0.144639
\(63\) 0 0
\(64\) −3.04036 −0.380044
\(65\) −4.66785 −0.578975
\(66\) 0 0
\(67\) 7.31984 0.894260 0.447130 0.894469i \(-0.352446\pi\)
0.447130 + 0.894469i \(0.352446\pi\)
\(68\) −8.19888 −0.994260
\(69\) 0 0
\(70\) −1.28155 −0.153174
\(71\) −1.19571 −0.141905 −0.0709525 0.997480i \(-0.522604\pi\)
−0.0709525 + 0.997480i \(0.522604\pi\)
\(72\) 0 0
\(73\) 1.02171 0.119582 0.0597908 0.998211i \(-0.480957\pi\)
0.0597908 + 0.998211i \(0.480957\pi\)
\(74\) 5.07943 0.590472
\(75\) 0 0
\(76\) 7.69223 0.882360
\(77\) 0 0
\(78\) 0 0
\(79\) 3.50213 0.394021 0.197010 0.980401i \(-0.436877\pi\)
0.197010 + 0.980401i \(0.436877\pi\)
\(80\) 2.68522 0.300217
\(81\) 0 0
\(82\) −1.05494 −0.116498
\(83\) −11.1158 −1.22012 −0.610061 0.792355i \(-0.708855\pi\)
−0.610061 + 0.792355i \(0.708855\pi\)
\(84\) 0 0
\(85\) 4.62632 0.501795
\(86\) 3.37289 0.363708
\(87\) 0 0
\(88\) 0 0
\(89\) −2.76978 −0.293596 −0.146798 0.989167i \(-0.546897\pi\)
−0.146798 + 0.989167i \(0.546897\pi\)
\(90\) 0 0
\(91\) −12.5342 −1.31394
\(92\) 4.91300 0.512215
\(93\) 0 0
\(94\) 2.08188 0.214729
\(95\) −4.34044 −0.445320
\(96\) 0 0
\(97\) 18.5342 1.88186 0.940931 0.338597i \(-0.109952\pi\)
0.940931 + 0.338597i \(0.109952\pi\)
\(98\) −0.100418 −0.0101438
\(99\) 0 0
\(100\) −1.77222 −0.177222
\(101\) 7.11455 0.707925 0.353962 0.935260i \(-0.384834\pi\)
0.353962 + 0.935260i \(0.384834\pi\)
\(102\) 0 0
\(103\) 7.52835 0.741791 0.370895 0.928675i \(-0.379051\pi\)
0.370895 + 0.928675i \(0.379051\pi\)
\(104\) −8.40367 −0.824048
\(105\) 0 0
\(106\) −3.02298 −0.293618
\(107\) −18.0292 −1.74295 −0.871475 0.490441i \(-0.836836\pi\)
−0.871475 + 0.490441i \(0.836836\pi\)
\(108\) 0 0
\(109\) −16.3653 −1.56751 −0.783756 0.621068i \(-0.786699\pi\)
−0.783756 + 0.621068i \(0.786699\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.21041 0.681319
\(113\) 2.05377 0.193203 0.0966013 0.995323i \(-0.469203\pi\)
0.0966013 + 0.995323i \(0.469203\pi\)
\(114\) 0 0
\(115\) −2.77222 −0.258511
\(116\) −5.34044 −0.495848
\(117\) 0 0
\(118\) 5.60651 0.516121
\(119\) 12.4227 1.13879
\(120\) 0 0
\(121\) 0 0
\(122\) 1.90075 0.172086
\(123\) 0 0
\(124\) −4.22906 −0.379780
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.0762667 −0.00676758 −0.00338379 0.999994i \(-0.501077\pi\)
−0.00338379 + 0.999994i \(0.501077\pi\)
\(128\) 11.2155 0.991316
\(129\) 0 0
\(130\) 2.22778 0.195389
\(131\) 11.4831 1.00328 0.501642 0.865075i \(-0.332730\pi\)
0.501642 + 0.865075i \(0.332730\pi\)
\(132\) 0 0
\(133\) −11.6550 −1.01062
\(134\) −3.49346 −0.301789
\(135\) 0 0
\(136\) 8.32892 0.714199
\(137\) 18.3293 1.56598 0.782989 0.622036i \(-0.213694\pi\)
0.782989 + 0.622036i \(0.213694\pi\)
\(138\) 0 0
\(139\) −23.1874 −1.96673 −0.983363 0.181653i \(-0.941855\pi\)
−0.983363 + 0.181653i \(0.941855\pi\)
\(140\) −4.75881 −0.402193
\(141\) 0 0
\(142\) 0.570666 0.0478892
\(143\) 0 0
\(144\) 0 0
\(145\) 3.01341 0.250250
\(146\) −0.487619 −0.0403557
\(147\) 0 0
\(148\) 18.8616 1.55041
\(149\) −14.6646 −1.20137 −0.600686 0.799485i \(-0.705106\pi\)
−0.600686 + 0.799485i \(0.705106\pi\)
\(150\) 0 0
\(151\) −7.51902 −0.611889 −0.305944 0.952049i \(-0.598972\pi\)
−0.305944 + 0.952049i \(0.598972\pi\)
\(152\) −7.81423 −0.633818
\(153\) 0 0
\(154\) 0 0
\(155\) 2.38630 0.191672
\(156\) 0 0
\(157\) −13.3819 −1.06799 −0.533996 0.845487i \(-0.679310\pi\)
−0.533996 + 0.845487i \(0.679310\pi\)
\(158\) −1.67143 −0.132972
\(159\) 0 0
\(160\) −4.88221 −0.385973
\(161\) −7.44403 −0.586672
\(162\) 0 0
\(163\) −0.771990 −0.0604669 −0.0302335 0.999543i \(-0.509625\pi\)
−0.0302335 + 0.999543i \(0.509625\pi\)
\(164\) −3.91733 −0.305892
\(165\) 0 0
\(166\) 5.30514 0.411759
\(167\) 8.48232 0.656381 0.328191 0.944612i \(-0.393561\pi\)
0.328191 + 0.944612i \(0.393561\pi\)
\(168\) 0 0
\(169\) 8.78880 0.676062
\(170\) −2.20796 −0.169343
\(171\) 0 0
\(172\) 12.5246 0.954994
\(173\) −5.07871 −0.386127 −0.193064 0.981186i \(-0.561842\pi\)
−0.193064 + 0.981186i \(0.561842\pi\)
\(174\) 0 0
\(175\) 2.68522 0.202984
\(176\) 0 0
\(177\) 0 0
\(178\) 1.32190 0.0990809
\(179\) 11.3170 0.845876 0.422938 0.906159i \(-0.360999\pi\)
0.422938 + 0.906159i \(0.360999\pi\)
\(180\) 0 0
\(181\) 7.40006 0.550042 0.275021 0.961438i \(-0.411315\pi\)
0.275021 + 0.961438i \(0.411315\pi\)
\(182\) 5.98207 0.443421
\(183\) 0 0
\(184\) −4.99092 −0.367935
\(185\) −10.6429 −0.782482
\(186\) 0 0
\(187\) 0 0
\(188\) 7.73070 0.563819
\(189\) 0 0
\(190\) 2.07152 0.150284
\(191\) 5.15608 0.373081 0.186540 0.982447i \(-0.440272\pi\)
0.186540 + 0.982447i \(0.440272\pi\)
\(192\) 0 0
\(193\) −4.03230 −0.290251 −0.145126 0.989413i \(-0.546359\pi\)
−0.145126 + 0.989413i \(0.546359\pi\)
\(194\) −8.84563 −0.635079
\(195\) 0 0
\(196\) −0.372885 −0.0266346
\(197\) −11.4176 −0.813469 −0.406734 0.913547i \(-0.633333\pi\)
−0.406734 + 0.913547i \(0.633333\pi\)
\(198\) 0 0
\(199\) −7.16644 −0.508015 −0.254008 0.967202i \(-0.581749\pi\)
−0.254008 + 0.967202i \(0.581749\pi\)
\(200\) 1.80033 0.127303
\(201\) 0 0
\(202\) −3.39549 −0.238906
\(203\) 8.09168 0.567925
\(204\) 0 0
\(205\) 2.21041 0.154381
\(206\) −3.59298 −0.250335
\(207\) 0 0
\(208\) −12.5342 −0.869090
\(209\) 0 0
\(210\) 0 0
\(211\) 3.48359 0.239820 0.119910 0.992785i \(-0.461739\pi\)
0.119910 + 0.992785i \(0.461739\pi\)
\(212\) −11.2253 −0.770959
\(213\) 0 0
\(214\) 8.60462 0.588200
\(215\) −7.06719 −0.481978
\(216\) 0 0
\(217\) 6.40774 0.434986
\(218\) 7.81051 0.528995
\(219\) 0 0
\(220\) 0 0
\(221\) −21.5950 −1.45264
\(222\) 0 0
\(223\) 10.5449 0.706141 0.353071 0.935597i \(-0.385138\pi\)
0.353071 + 0.935597i \(0.385138\pi\)
\(224\) −13.1098 −0.875936
\(225\) 0 0
\(226\) −0.980183 −0.0652008
\(227\) −0.216018 −0.0143376 −0.00716880 0.999974i \(-0.502282\pi\)
−0.00716880 + 0.999974i \(0.502282\pi\)
\(228\) 0 0
\(229\) −0.0757097 −0.00500304 −0.00250152 0.999997i \(-0.500796\pi\)
−0.00250152 + 0.999997i \(0.500796\pi\)
\(230\) 1.32307 0.0872407
\(231\) 0 0
\(232\) 5.42514 0.356178
\(233\) −15.1201 −0.990548 −0.495274 0.868737i \(-0.664932\pi\)
−0.495274 + 0.868737i \(0.664932\pi\)
\(234\) 0 0
\(235\) −4.36215 −0.284555
\(236\) 20.8188 1.35519
\(237\) 0 0
\(238\) −5.92886 −0.384311
\(239\) 23.1646 1.49839 0.749197 0.662347i \(-0.230439\pi\)
0.749197 + 0.662347i \(0.230439\pi\)
\(240\) 0 0
\(241\) −21.3349 −1.37430 −0.687151 0.726515i \(-0.741139\pi\)
−0.687151 + 0.726515i \(0.741139\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 7.05810 0.451849
\(245\) 0.210405 0.0134423
\(246\) 0 0
\(247\) 20.2605 1.28915
\(248\) 4.29613 0.272805
\(249\) 0 0
\(250\) −0.477260 −0.0301846
\(251\) 6.22186 0.392721 0.196360 0.980532i \(-0.437088\pi\)
0.196360 + 0.980532i \(0.437088\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.0363991 0.00228388
\(255\) 0 0
\(256\) 0.728021 0.0455013
\(257\) 14.2903 0.891406 0.445703 0.895181i \(-0.352954\pi\)
0.445703 + 0.895181i \(0.352954\pi\)
\(258\) 0 0
\(259\) −28.5785 −1.77578
\(260\) 8.27247 0.513037
\(261\) 0 0
\(262\) −5.48043 −0.338582
\(263\) −4.13132 −0.254748 −0.127374 0.991855i \(-0.540655\pi\)
−0.127374 + 0.991855i \(0.540655\pi\)
\(264\) 0 0
\(265\) 6.33404 0.389097
\(266\) 5.56249 0.341058
\(267\) 0 0
\(268\) −12.9724 −0.792414
\(269\) 1.68394 0.102672 0.0513359 0.998681i \(-0.483652\pi\)
0.0513359 + 0.998681i \(0.483652\pi\)
\(270\) 0 0
\(271\) −18.4310 −1.11960 −0.559801 0.828627i \(-0.689123\pi\)
−0.559801 + 0.828627i \(0.689123\pi\)
\(272\) 12.4227 0.753237
\(273\) 0 0
\(274\) −8.74784 −0.528476
\(275\) 0 0
\(276\) 0 0
\(277\) −3.42745 −0.205935 −0.102968 0.994685i \(-0.532834\pi\)
−0.102968 + 0.994685i \(0.532834\pi\)
\(278\) 11.0664 0.663718
\(279\) 0 0
\(280\) 4.83428 0.288904
\(281\) 22.8217 1.36143 0.680715 0.732548i \(-0.261669\pi\)
0.680715 + 0.732548i \(0.261669\pi\)
\(282\) 0 0
\(283\) −29.0991 −1.72976 −0.864880 0.501978i \(-0.832606\pi\)
−0.864880 + 0.501978i \(0.832606\pi\)
\(284\) 2.11907 0.125744
\(285\) 0 0
\(286\) 0 0
\(287\) 5.93542 0.350357
\(288\) 0 0
\(289\) 4.40288 0.258993
\(290\) −1.43818 −0.0844530
\(291\) 0 0
\(292\) −1.81069 −0.105963
\(293\) −21.2136 −1.23931 −0.619655 0.784874i \(-0.712727\pi\)
−0.619655 + 0.784874i \(0.712727\pi\)
\(294\) 0 0
\(295\) −11.7473 −0.683953
\(296\) −19.1608 −1.11370
\(297\) 0 0
\(298\) 6.99883 0.405432
\(299\) 12.9403 0.748358
\(300\) 0 0
\(301\) −18.9769 −1.09381
\(302\) 3.58853 0.206496
\(303\) 0 0
\(304\) −11.6550 −0.668463
\(305\) −3.98263 −0.228045
\(306\) 0 0
\(307\) −6.87520 −0.392388 −0.196194 0.980565i \(-0.562858\pi\)
−0.196194 + 0.980565i \(0.562858\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.13889 −0.0646844
\(311\) −25.1577 −1.42656 −0.713280 0.700879i \(-0.752791\pi\)
−0.713280 + 0.700879i \(0.752791\pi\)
\(312\) 0 0
\(313\) −11.5793 −0.654503 −0.327251 0.944937i \(-0.606122\pi\)
−0.327251 + 0.944937i \(0.606122\pi\)
\(314\) 6.38664 0.360419
\(315\) 0 0
\(316\) −6.20656 −0.349146
\(317\) −20.7413 −1.16495 −0.582473 0.812850i \(-0.697915\pi\)
−0.582473 + 0.812850i \(0.697915\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.04036 −0.169961
\(321\) 0 0
\(322\) 3.55274 0.197986
\(323\) −20.0803 −1.11730
\(324\) 0 0
\(325\) −4.66785 −0.258926
\(326\) 0.368440 0.0204060
\(327\) 0 0
\(328\) 3.97946 0.219729
\(329\) −11.7133 −0.645777
\(330\) 0 0
\(331\) −32.1415 −1.76665 −0.883327 0.468757i \(-0.844702\pi\)
−0.883327 + 0.468757i \(0.844702\pi\)
\(332\) 19.6997 1.08116
\(333\) 0 0
\(334\) −4.04827 −0.221511
\(335\) 7.31984 0.399925
\(336\) 0 0
\(337\) −17.9964 −0.980326 −0.490163 0.871631i \(-0.663063\pi\)
−0.490163 + 0.871631i \(0.663063\pi\)
\(338\) −4.19454 −0.228153
\(339\) 0 0
\(340\) −8.19888 −0.444647
\(341\) 0 0
\(342\) 0 0
\(343\) −18.2316 −0.984411
\(344\) −12.7233 −0.685993
\(345\) 0 0
\(346\) 2.42387 0.130308
\(347\) 8.04981 0.432137 0.216068 0.976378i \(-0.430677\pi\)
0.216068 + 0.976378i \(0.430677\pi\)
\(348\) 0 0
\(349\) 19.2294 1.02932 0.514662 0.857393i \(-0.327917\pi\)
0.514662 + 0.857393i \(0.327917\pi\)
\(350\) −1.28155 −0.0685016
\(351\) 0 0
\(352\) 0 0
\(353\) −14.8497 −0.790371 −0.395186 0.918601i \(-0.629320\pi\)
−0.395186 + 0.918601i \(0.629320\pi\)
\(354\) 0 0
\(355\) −1.19571 −0.0634618
\(356\) 4.90866 0.260159
\(357\) 0 0
\(358\) −5.40117 −0.285461
\(359\) −10.2233 −0.539563 −0.269782 0.962922i \(-0.586952\pi\)
−0.269782 + 0.962922i \(0.586952\pi\)
\(360\) 0 0
\(361\) −0.160555 −0.00845028
\(362\) −3.53175 −0.185625
\(363\) 0 0
\(364\) 22.2134 1.16430
\(365\) 1.02171 0.0534785
\(366\) 0 0
\(367\) 14.1434 0.738279 0.369139 0.929374i \(-0.379653\pi\)
0.369139 + 0.929374i \(0.379653\pi\)
\(368\) −7.44403 −0.388047
\(369\) 0 0
\(370\) 5.07943 0.264067
\(371\) 17.0083 0.883026
\(372\) 0 0
\(373\) 12.4600 0.645154 0.322577 0.946543i \(-0.395451\pi\)
0.322577 + 0.946543i \(0.395451\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −7.85331 −0.405004
\(377\) −14.0662 −0.724444
\(378\) 0 0
\(379\) 16.3370 0.839174 0.419587 0.907715i \(-0.362175\pi\)
0.419587 + 0.907715i \(0.362175\pi\)
\(380\) 7.69223 0.394603
\(381\) 0 0
\(382\) −2.46079 −0.125905
\(383\) 0.812648 0.0415244 0.0207622 0.999784i \(-0.493391\pi\)
0.0207622 + 0.999784i \(0.493391\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.92445 0.0979522
\(387\) 0 0
\(388\) −32.8467 −1.66754
\(389\) 30.3941 1.54104 0.770521 0.637414i \(-0.219996\pi\)
0.770521 + 0.637414i \(0.219996\pi\)
\(390\) 0 0
\(391\) −12.8252 −0.648598
\(392\) 0.378799 0.0191322
\(393\) 0 0
\(394\) 5.44915 0.274524
\(395\) 3.50213 0.176211
\(396\) 0 0
\(397\) −14.8996 −0.747789 −0.373894 0.927471i \(-0.621978\pi\)
−0.373894 + 0.927471i \(0.621978\pi\)
\(398\) 3.42025 0.171442
\(399\) 0 0
\(400\) 2.68522 0.134261
\(401\) −12.1692 −0.607700 −0.303850 0.952720i \(-0.598272\pi\)
−0.303850 + 0.952720i \(0.598272\pi\)
\(402\) 0 0
\(403\) −11.1389 −0.554867
\(404\) −12.6086 −0.627300
\(405\) 0 0
\(406\) −3.86184 −0.191660
\(407\) 0 0
\(408\) 0 0
\(409\) −0.262108 −0.0129604 −0.00648020 0.999979i \(-0.502063\pi\)
−0.00648020 + 0.999979i \(0.502063\pi\)
\(410\) −1.05494 −0.0520997
\(411\) 0 0
\(412\) −13.3419 −0.657309
\(413\) −31.5440 −1.55218
\(414\) 0 0
\(415\) −11.1158 −0.545655
\(416\) 22.7894 1.11734
\(417\) 0 0
\(418\) 0 0
\(419\) 1.26916 0.0620023 0.0310012 0.999519i \(-0.490130\pi\)
0.0310012 + 0.999519i \(0.490130\pi\)
\(420\) 0 0
\(421\) −29.6655 −1.44581 −0.722903 0.690949i \(-0.757193\pi\)
−0.722903 + 0.690949i \(0.757193\pi\)
\(422\) −1.66258 −0.0809331
\(423\) 0 0
\(424\) 11.4034 0.553797
\(425\) 4.62632 0.224410
\(426\) 0 0
\(427\) −10.6942 −0.517530
\(428\) 31.9518 1.54445
\(429\) 0 0
\(430\) 3.37289 0.162655
\(431\) −31.3549 −1.51031 −0.755156 0.655545i \(-0.772439\pi\)
−0.755156 + 0.655545i \(0.772439\pi\)
\(432\) 0 0
\(433\) −26.0325 −1.25104 −0.625521 0.780207i \(-0.715114\pi\)
−0.625521 + 0.780207i \(0.715114\pi\)
\(434\) −3.05816 −0.146796
\(435\) 0 0
\(436\) 29.0030 1.38899
\(437\) 12.0327 0.575601
\(438\) 0 0
\(439\) 14.4191 0.688185 0.344093 0.938936i \(-0.388187\pi\)
0.344093 + 0.938936i \(0.388187\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.3064 0.490226
\(443\) 0.330608 0.0157077 0.00785383 0.999969i \(-0.497500\pi\)
0.00785383 + 0.999969i \(0.497500\pi\)
\(444\) 0 0
\(445\) −2.76978 −0.131300
\(446\) −5.03268 −0.238304
\(447\) 0 0
\(448\) −8.16402 −0.385714
\(449\) −8.50828 −0.401531 −0.200765 0.979639i \(-0.564343\pi\)
−0.200765 + 0.979639i \(0.564343\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −3.63974 −0.171199
\(453\) 0 0
\(454\) 0.103097 0.00483856
\(455\) −12.5342 −0.587612
\(456\) 0 0
\(457\) 1.13847 0.0532555 0.0266277 0.999645i \(-0.491523\pi\)
0.0266277 + 0.999645i \(0.491523\pi\)
\(458\) 0.0361332 0.00168839
\(459\) 0 0
\(460\) 4.91300 0.229070
\(461\) −14.5073 −0.675670 −0.337835 0.941205i \(-0.609695\pi\)
−0.337835 + 0.941205i \(0.609695\pi\)
\(462\) 0 0
\(463\) −4.89739 −0.227601 −0.113801 0.993504i \(-0.536302\pi\)
−0.113801 + 0.993504i \(0.536302\pi\)
\(464\) 8.09168 0.375647
\(465\) 0 0
\(466\) 7.21620 0.334284
\(467\) −32.4230 −1.50036 −0.750180 0.661234i \(-0.770033\pi\)
−0.750180 + 0.661234i \(0.770033\pi\)
\(468\) 0 0
\(469\) 19.6554 0.907601
\(470\) 2.08188 0.0960299
\(471\) 0 0
\(472\) −21.1490 −0.973461
\(473\) 0 0
\(474\) 0 0
\(475\) −4.34044 −0.199153
\(476\) −22.0158 −1.00909
\(477\) 0 0
\(478\) −11.0555 −0.505669
\(479\) −17.7354 −0.810352 −0.405176 0.914239i \(-0.632790\pi\)
−0.405176 + 0.914239i \(0.632790\pi\)
\(480\) 0 0
\(481\) 49.6795 2.26519
\(482\) 10.1823 0.463791
\(483\) 0 0
\(484\) 0 0
\(485\) 18.5342 0.841595
\(486\) 0 0
\(487\) −18.4603 −0.836514 −0.418257 0.908329i \(-0.637359\pi\)
−0.418257 + 0.908329i \(0.637359\pi\)
\(488\) −7.17005 −0.324573
\(489\) 0 0
\(490\) −0.100418 −0.00453642
\(491\) 11.4338 0.516002 0.258001 0.966145i \(-0.416936\pi\)
0.258001 + 0.966145i \(0.416936\pi\)
\(492\) 0 0
\(493\) 13.9410 0.627873
\(494\) −9.66954 −0.435053
\(495\) 0 0
\(496\) 6.40774 0.287716
\(497\) −3.21075 −0.144022
\(498\) 0 0
\(499\) 10.8815 0.487125 0.243562 0.969885i \(-0.421684\pi\)
0.243562 + 0.969885i \(0.421684\pi\)
\(500\) −1.77222 −0.0792562
\(501\) 0 0
\(502\) −2.96945 −0.132533
\(503\) −44.7247 −1.99417 −0.997087 0.0762683i \(-0.975699\pi\)
−0.997087 + 0.0762683i \(0.975699\pi\)
\(504\) 0 0
\(505\) 7.11455 0.316594
\(506\) 0 0
\(507\) 0 0
\(508\) 0.135162 0.00599683
\(509\) 13.8093 0.612088 0.306044 0.952017i \(-0.400994\pi\)
0.306044 + 0.952017i \(0.400994\pi\)
\(510\) 0 0
\(511\) 2.74350 0.121365
\(512\) −22.7784 −1.00667
\(513\) 0 0
\(514\) −6.82020 −0.300826
\(515\) 7.52835 0.331739
\(516\) 0 0
\(517\) 0 0
\(518\) 13.6394 0.599281
\(519\) 0 0
\(520\) −8.40367 −0.368525
\(521\) 3.64972 0.159897 0.0799486 0.996799i \(-0.474524\pi\)
0.0799486 + 0.996799i \(0.474524\pi\)
\(522\) 0 0
\(523\) −4.98707 −0.218069 −0.109035 0.994038i \(-0.534776\pi\)
−0.109035 + 0.994038i \(0.534776\pi\)
\(524\) −20.3506 −0.889021
\(525\) 0 0
\(526\) 1.97171 0.0859707
\(527\) 11.0398 0.480901
\(528\) 0 0
\(529\) −15.3148 −0.665860
\(530\) −3.02298 −0.131310
\(531\) 0 0
\(532\) 20.6553 0.895522
\(533\) −10.3178 −0.446915
\(534\) 0 0
\(535\) −18.0292 −0.779471
\(536\) 13.1781 0.569208
\(537\) 0 0
\(538\) −0.803678 −0.0346490
\(539\) 0 0
\(540\) 0 0
\(541\) 0.247594 0.0106449 0.00532246 0.999986i \(-0.498306\pi\)
0.00532246 + 0.999986i \(0.498306\pi\)
\(542\) 8.79637 0.377837
\(543\) 0 0
\(544\) −22.5867 −0.968396
\(545\) −16.3653 −0.701013
\(546\) 0 0
\(547\) −25.3120 −1.08226 −0.541132 0.840938i \(-0.682004\pi\)
−0.541132 + 0.840938i \(0.682004\pi\)
\(548\) −32.4836 −1.38763
\(549\) 0 0
\(550\) 0 0
\(551\) −13.0796 −0.557208
\(552\) 0 0
\(553\) 9.40400 0.399899
\(554\) 1.63578 0.0694978
\(555\) 0 0
\(556\) 41.0932 1.74274
\(557\) 38.6671 1.63838 0.819190 0.573523i \(-0.194424\pi\)
0.819190 + 0.573523i \(0.194424\pi\)
\(558\) 0 0
\(559\) 32.9885 1.39527
\(560\) 7.21041 0.304695
\(561\) 0 0
\(562\) −10.8919 −0.459447
\(563\) −13.9362 −0.587341 −0.293671 0.955907i \(-0.594877\pi\)
−0.293671 + 0.955907i \(0.594877\pi\)
\(564\) 0 0
\(565\) 2.05377 0.0864028
\(566\) 13.8878 0.583749
\(567\) 0 0
\(568\) −2.15268 −0.0903243
\(569\) −27.9125 −1.17015 −0.585076 0.810979i \(-0.698935\pi\)
−0.585076 + 0.810979i \(0.698935\pi\)
\(570\) 0 0
\(571\) −31.4113 −1.31452 −0.657261 0.753663i \(-0.728285\pi\)
−0.657261 + 0.753663i \(0.728285\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −2.83274 −0.118236
\(575\) −2.77222 −0.115610
\(576\) 0 0
\(577\) 20.7349 0.863206 0.431603 0.902064i \(-0.357948\pi\)
0.431603 + 0.902064i \(0.357948\pi\)
\(578\) −2.10132 −0.0874034
\(579\) 0 0
\(580\) −5.34044 −0.221750
\(581\) −29.8485 −1.23832
\(582\) 0 0
\(583\) 0 0
\(584\) 1.83941 0.0761153
\(585\) 0 0
\(586\) 10.1244 0.418235
\(587\) −15.2367 −0.628886 −0.314443 0.949276i \(-0.601818\pi\)
−0.314443 + 0.949276i \(0.601818\pi\)
\(588\) 0 0
\(589\) −10.3576 −0.426777
\(590\) 5.60651 0.230816
\(591\) 0 0
\(592\) −28.5785 −1.17457
\(593\) −27.5413 −1.13098 −0.565492 0.824754i \(-0.691314\pi\)
−0.565492 + 0.824754i \(0.691314\pi\)
\(594\) 0 0
\(595\) 12.4227 0.509281
\(596\) 25.9890 1.06455
\(597\) 0 0
\(598\) −6.17589 −0.252551
\(599\) 26.7331 1.09228 0.546142 0.837693i \(-0.316096\pi\)
0.546142 + 0.837693i \(0.316096\pi\)
\(600\) 0 0
\(601\) 2.40680 0.0981753 0.0490876 0.998794i \(-0.484369\pi\)
0.0490876 + 0.998794i \(0.484369\pi\)
\(602\) 9.05694 0.369133
\(603\) 0 0
\(604\) 13.3254 0.542202
\(605\) 0 0
\(606\) 0 0
\(607\) 10.2929 0.417774 0.208887 0.977940i \(-0.433016\pi\)
0.208887 + 0.977940i \(0.433016\pi\)
\(608\) 21.1910 0.859407
\(609\) 0 0
\(610\) 1.90075 0.0769591
\(611\) 20.3618 0.823752
\(612\) 0 0
\(613\) 27.8756 1.12589 0.562943 0.826496i \(-0.309669\pi\)
0.562943 + 0.826496i \(0.309669\pi\)
\(614\) 3.28126 0.132421
\(615\) 0 0
\(616\) 0 0
\(617\) 28.7216 1.15629 0.578143 0.815935i \(-0.303778\pi\)
0.578143 + 0.815935i \(0.303778\pi\)
\(618\) 0 0
\(619\) 22.6968 0.912261 0.456131 0.889913i \(-0.349235\pi\)
0.456131 + 0.889913i \(0.349235\pi\)
\(620\) −4.22906 −0.169843
\(621\) 0 0
\(622\) 12.0067 0.481427
\(623\) −7.43746 −0.297976
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 5.52635 0.220877
\(627\) 0 0
\(628\) 23.7157 0.946360
\(629\) −49.2375 −1.96323
\(630\) 0 0
\(631\) 28.5364 1.13602 0.568008 0.823023i \(-0.307714\pi\)
0.568008 + 0.823023i \(0.307714\pi\)
\(632\) 6.30500 0.250799
\(633\) 0 0
\(634\) 9.89897 0.393138
\(635\) −0.0762667 −0.00302655
\(636\) 0 0
\(637\) −0.982140 −0.0389138
\(638\) 0 0
\(639\) 0 0
\(640\) 11.2155 0.443330
\(641\) −2.37375 −0.0937575 −0.0468788 0.998901i \(-0.514927\pi\)
−0.0468788 + 0.998901i \(0.514927\pi\)
\(642\) 0 0
\(643\) 11.3603 0.448007 0.224004 0.974588i \(-0.428087\pi\)
0.224004 + 0.974588i \(0.428087\pi\)
\(644\) 13.1925 0.519856
\(645\) 0 0
\(646\) 9.58352 0.377059
\(647\) 48.4220 1.90366 0.951832 0.306621i \(-0.0991984\pi\)
0.951832 + 0.306621i \(0.0991984\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.22778 0.0873806
\(651\) 0 0
\(652\) 1.36814 0.0535804
\(653\) −29.7548 −1.16440 −0.582198 0.813047i \(-0.697807\pi\)
−0.582198 + 0.813047i \(0.697807\pi\)
\(654\) 0 0
\(655\) 11.4831 0.448682
\(656\) 5.93542 0.231739
\(657\) 0 0
\(658\) 5.59030 0.217933
\(659\) 28.4931 1.10993 0.554966 0.831873i \(-0.312731\pi\)
0.554966 + 0.831873i \(0.312731\pi\)
\(660\) 0 0
\(661\) −1.02875 −0.0400139 −0.0200070 0.999800i \(-0.506369\pi\)
−0.0200070 + 0.999800i \(0.506369\pi\)
\(662\) 15.3398 0.596200
\(663\) 0 0
\(664\) −20.0122 −0.776623
\(665\) −11.6550 −0.451963
\(666\) 0 0
\(667\) −8.35386 −0.323463
\(668\) −15.0326 −0.581627
\(669\) 0 0
\(670\) −3.49346 −0.134964
\(671\) 0 0
\(672\) 0 0
\(673\) 13.0369 0.502536 0.251268 0.967918i \(-0.419152\pi\)
0.251268 + 0.967918i \(0.419152\pi\)
\(674\) 8.58896 0.330834
\(675\) 0 0
\(676\) −15.5757 −0.599066
\(677\) 37.1064 1.42612 0.713058 0.701105i \(-0.247310\pi\)
0.713058 + 0.701105i \(0.247310\pi\)
\(678\) 0 0
\(679\) 49.7684 1.90994
\(680\) 8.32892 0.319399
\(681\) 0 0
\(682\) 0 0
\(683\) 32.8992 1.25885 0.629426 0.777061i \(-0.283290\pi\)
0.629426 + 0.777061i \(0.283290\pi\)
\(684\) 0 0
\(685\) 18.3293 0.700326
\(686\) 8.70119 0.332213
\(687\) 0 0
\(688\) −18.9769 −0.723489
\(689\) −29.5663 −1.12639
\(690\) 0 0
\(691\) 36.6998 1.39613 0.698064 0.716035i \(-0.254045\pi\)
0.698064 + 0.716035i \(0.254045\pi\)
\(692\) 9.00061 0.342152
\(693\) 0 0
\(694\) −3.84185 −0.145835
\(695\) −23.1874 −0.879546
\(696\) 0 0
\(697\) 10.2261 0.387339
\(698\) −9.17741 −0.347370
\(699\) 0 0
\(700\) −4.75881 −0.179866
\(701\) 36.2497 1.36913 0.684566 0.728951i \(-0.259992\pi\)
0.684566 + 0.728951i \(0.259992\pi\)
\(702\) 0 0
\(703\) 46.1949 1.74227
\(704\) 0 0
\(705\) 0 0
\(706\) 7.08718 0.266730
\(707\) 19.1041 0.718485
\(708\) 0 0
\(709\) −18.6013 −0.698585 −0.349293 0.937014i \(-0.613578\pi\)
−0.349293 + 0.937014i \(0.613578\pi\)
\(710\) 0.570666 0.0214167
\(711\) 0 0
\(712\) −4.98652 −0.186878
\(713\) −6.61536 −0.247747
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0563 −0.749540
\(717\) 0 0
\(718\) 4.87915 0.182088
\(719\) 37.4280 1.39583 0.697914 0.716181i \(-0.254112\pi\)
0.697914 + 0.716181i \(0.254112\pi\)
\(720\) 0 0
\(721\) 20.2153 0.752856
\(722\) 0.0766267 0.00285175
\(723\) 0 0
\(724\) −13.1146 −0.487399
\(725\) 3.01341 0.111915
\(726\) 0 0
\(727\) 14.6011 0.541526 0.270763 0.962646i \(-0.412724\pi\)
0.270763 + 0.962646i \(0.412724\pi\)
\(728\) −22.5657 −0.836341
\(729\) 0 0
\(730\) −0.487619 −0.0180476
\(731\) −32.6951 −1.20927
\(732\) 0 0
\(733\) 41.8369 1.54528 0.772640 0.634844i \(-0.218936\pi\)
0.772640 + 0.634844i \(0.218936\pi\)
\(734\) −6.75007 −0.249150
\(735\) 0 0
\(736\) 13.5346 0.498891
\(737\) 0 0
\(738\) 0 0
\(739\) −11.9299 −0.438849 −0.219424 0.975630i \(-0.570418\pi\)
−0.219424 + 0.975630i \(0.570418\pi\)
\(740\) 18.8616 0.693366
\(741\) 0 0
\(742\) −8.11738 −0.297998
\(743\) −46.6803 −1.71253 −0.856267 0.516534i \(-0.827222\pi\)
−0.856267 + 0.516534i \(0.827222\pi\)
\(744\) 0 0
\(745\) −14.6646 −0.537270
\(746\) −5.94666 −0.217723
\(747\) 0 0
\(748\) 0 0
\(749\) −48.4124 −1.76895
\(750\) 0 0
\(751\) 14.4039 0.525607 0.262803 0.964849i \(-0.415353\pi\)
0.262803 + 0.964849i \(0.415353\pi\)
\(752\) −11.7133 −0.427141
\(753\) 0 0
\(754\) 6.71322 0.244481
\(755\) −7.51902 −0.273645
\(756\) 0 0
\(757\) −16.0616 −0.583768 −0.291884 0.956454i \(-0.594282\pi\)
−0.291884 + 0.956454i \(0.594282\pi\)
\(758\) −7.79698 −0.283199
\(759\) 0 0
\(760\) −7.81423 −0.283452
\(761\) 38.8840 1.40954 0.704772 0.709433i \(-0.251049\pi\)
0.704772 + 0.709433i \(0.251049\pi\)
\(762\) 0 0
\(763\) −43.9445 −1.59090
\(764\) −9.13772 −0.330591
\(765\) 0 0
\(766\) −0.387844 −0.0140134
\(767\) 54.8345 1.97996
\(768\) 0 0
\(769\) 43.0017 1.55068 0.775341 0.631543i \(-0.217578\pi\)
0.775341 + 0.631543i \(0.217578\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.14613 0.257195
\(773\) 7.85462 0.282511 0.141256 0.989973i \(-0.454886\pi\)
0.141256 + 0.989973i \(0.454886\pi\)
\(774\) 0 0
\(775\) 2.38630 0.0857184
\(776\) 33.3677 1.19783
\(777\) 0 0
\(778\) −14.5059 −0.520061
\(779\) −9.59414 −0.343746
\(780\) 0 0
\(781\) 0 0
\(782\) 6.12096 0.218885
\(783\) 0 0
\(784\) 0.564984 0.0201780
\(785\) −13.3819 −0.477620
\(786\) 0 0
\(787\) −11.4249 −0.407253 −0.203627 0.979049i \(-0.565273\pi\)
−0.203627 + 0.979049i \(0.565273\pi\)
\(788\) 20.2345 0.720824
\(789\) 0 0
\(790\) −1.67143 −0.0594667
\(791\) 5.51483 0.196085
\(792\) 0 0
\(793\) 18.5903 0.660161
\(794\) 7.11097 0.252359
\(795\) 0 0
\(796\) 12.7005 0.450158
\(797\) 4.28502 0.151783 0.0758915 0.997116i \(-0.475820\pi\)
0.0758915 + 0.997116i \(0.475820\pi\)
\(798\) 0 0
\(799\) −20.1807 −0.713942
\(800\) −4.88221 −0.172612
\(801\) 0 0
\(802\) 5.80787 0.205083
\(803\) 0 0
\(804\) 0 0
\(805\) −7.44403 −0.262368
\(806\) 5.31614 0.187253
\(807\) 0 0
\(808\) 12.8086 0.450603
\(809\) 19.8484 0.697833 0.348917 0.937154i \(-0.386550\pi\)
0.348917 + 0.937154i \(0.386550\pi\)
\(810\) 0 0
\(811\) −3.13836 −0.110203 −0.0551014 0.998481i \(-0.517548\pi\)
−0.0551014 + 0.998481i \(0.517548\pi\)
\(812\) −14.3403 −0.503245
\(813\) 0 0
\(814\) 0 0
\(815\) −0.771990 −0.0270416
\(816\) 0 0
\(817\) 30.6747 1.07317
\(818\) 0.125094 0.00437379
\(819\) 0 0
\(820\) −3.91733 −0.136799
\(821\) −23.4999 −0.820151 −0.410076 0.912052i \(-0.634498\pi\)
−0.410076 + 0.912052i \(0.634498\pi\)
\(822\) 0 0
\(823\) −12.6564 −0.441173 −0.220586 0.975367i \(-0.570797\pi\)
−0.220586 + 0.975367i \(0.570797\pi\)
\(824\) 13.5535 0.472159
\(825\) 0 0
\(826\) 15.0547 0.523820
\(827\) −30.2236 −1.05098 −0.525488 0.850801i \(-0.676117\pi\)
−0.525488 + 0.850801i \(0.676117\pi\)
\(828\) 0 0
\(829\) 2.00166 0.0695204 0.0347602 0.999396i \(-0.488933\pi\)
0.0347602 + 0.999396i \(0.488933\pi\)
\(830\) 5.30514 0.184144
\(831\) 0 0
\(832\) 14.1919 0.492016
\(833\) 0.973403 0.0337264
\(834\) 0 0
\(835\) 8.48232 0.293543
\(836\) 0 0
\(837\) 0 0
\(838\) −0.605717 −0.0209242
\(839\) −35.3744 −1.22126 −0.610629 0.791917i \(-0.709083\pi\)
−0.610629 + 0.791917i \(0.709083\pi\)
\(840\) 0 0
\(841\) −19.9193 −0.686873
\(842\) 14.1581 0.487922
\(843\) 0 0
\(844\) −6.17370 −0.212508
\(845\) 8.78880 0.302344
\(846\) 0 0
\(847\) 0 0
\(848\) 17.0083 0.584067
\(849\) 0 0
\(850\) −2.20796 −0.0757324
\(851\) 29.5045 1.01140
\(852\) 0 0
\(853\) −18.2034 −0.623272 −0.311636 0.950202i \(-0.600877\pi\)
−0.311636 + 0.950202i \(0.600877\pi\)
\(854\) 5.10393 0.174653
\(855\) 0 0
\(856\) −32.4585 −1.10941
\(857\) 29.2837 1.00031 0.500156 0.865935i \(-0.333276\pi\)
0.500156 + 0.865935i \(0.333276\pi\)
\(858\) 0 0
\(859\) 8.44030 0.287979 0.143990 0.989579i \(-0.454007\pi\)
0.143990 + 0.989579i \(0.454007\pi\)
\(860\) 12.5246 0.427086
\(861\) 0 0
\(862\) 14.9644 0.509691
\(863\) 19.3487 0.658636 0.329318 0.944219i \(-0.393181\pi\)
0.329318 + 0.944219i \(0.393181\pi\)
\(864\) 0 0
\(865\) −5.07871 −0.172681
\(866\) 12.4243 0.422194
\(867\) 0 0
\(868\) −11.3559 −0.385446
\(869\) 0 0
\(870\) 0 0
\(871\) −34.1679 −1.15773
\(872\) −29.4630 −0.997743
\(873\) 0 0
\(874\) −5.74271 −0.194250
\(875\) 2.68522 0.0907770
\(876\) 0 0
\(877\) 17.5140 0.591405 0.295702 0.955280i \(-0.404446\pi\)
0.295702 + 0.955280i \(0.404446\pi\)
\(878\) −6.88165 −0.232245
\(879\) 0 0
\(880\) 0 0
\(881\) 20.0575 0.675754 0.337877 0.941190i \(-0.390291\pi\)
0.337877 + 0.941190i \(0.390291\pi\)
\(882\) 0 0
\(883\) 26.8980 0.905189 0.452594 0.891717i \(-0.350499\pi\)
0.452594 + 0.891717i \(0.350499\pi\)
\(884\) 38.2711 1.28720
\(885\) 0 0
\(886\) −0.157786 −0.00530092
\(887\) −6.80568 −0.228512 −0.114256 0.993451i \(-0.536448\pi\)
−0.114256 + 0.993451i \(0.536448\pi\)
\(888\) 0 0
\(889\) −0.204793 −0.00686854
\(890\) 1.32190 0.0443103
\(891\) 0 0
\(892\) −18.6880 −0.625720
\(893\) 18.9337 0.633591
\(894\) 0 0
\(895\) 11.3170 0.378287
\(896\) 30.1160 1.00610
\(897\) 0 0
\(898\) 4.06066 0.135506
\(899\) 7.19091 0.239830
\(900\) 0 0
\(901\) 29.3033 0.976235
\(902\) 0 0
\(903\) 0 0
\(904\) 3.69747 0.122976
\(905\) 7.40006 0.245986
\(906\) 0 0
\(907\) −21.3313 −0.708295 −0.354148 0.935190i \(-0.615229\pi\)
−0.354148 + 0.935190i \(0.615229\pi\)
\(908\) 0.382831 0.0127047
\(909\) 0 0
\(910\) 5.98207 0.198304
\(911\) 27.5888 0.914059 0.457029 0.889452i \(-0.348913\pi\)
0.457029 + 0.889452i \(0.348913\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.543347 −0.0179723
\(915\) 0 0
\(916\) 0.134174 0.00443325
\(917\) 30.8347 1.01825
\(918\) 0 0
\(919\) −6.00889 −0.198215 −0.0991075 0.995077i \(-0.531599\pi\)
−0.0991075 + 0.995077i \(0.531599\pi\)
\(920\) −4.99092 −0.164546
\(921\) 0 0
\(922\) 6.92373 0.228021
\(923\) 5.58140 0.183714
\(924\) 0 0
\(925\) −10.6429 −0.349937
\(926\) 2.33733 0.0768094
\(927\) 0 0
\(928\) −14.7121 −0.482949
\(929\) −9.59739 −0.314880 −0.157440 0.987529i \(-0.550324\pi\)
−0.157440 + 0.987529i \(0.550324\pi\)
\(930\) 0 0
\(931\) −0.913252 −0.0299306
\(932\) 26.7961 0.877736
\(933\) 0 0
\(934\) 15.4742 0.506332
\(935\) 0 0
\(936\) 0 0
\(937\) −38.5917 −1.26074 −0.630369 0.776296i \(-0.717096\pi\)
−0.630369 + 0.776296i \(0.717096\pi\)
\(938\) −9.38072 −0.306291
\(939\) 0 0
\(940\) 7.73070 0.252148
\(941\) 29.1846 0.951391 0.475695 0.879610i \(-0.342196\pi\)
0.475695 + 0.879610i \(0.342196\pi\)
\(942\) 0 0
\(943\) −6.12774 −0.199547
\(944\) −31.5440 −1.02667
\(945\) 0 0
\(946\) 0 0
\(947\) −46.7623 −1.51957 −0.759785 0.650174i \(-0.774696\pi\)
−0.759785 + 0.650174i \(0.774696\pi\)
\(948\) 0 0
\(949\) −4.76917 −0.154814
\(950\) 2.07152 0.0672090
\(951\) 0 0
\(952\) 22.3650 0.724853
\(953\) 6.15238 0.199295 0.0996475 0.995023i \(-0.468228\pi\)
0.0996475 + 0.995023i \(0.468228\pi\)
\(954\) 0 0
\(955\) 5.15608 0.166847
\(956\) −41.0529 −1.32774
\(957\) 0 0
\(958\) 8.46440 0.273472
\(959\) 49.2182 1.58934
\(960\) 0 0
\(961\) −25.3056 −0.816309
\(962\) −23.7100 −0.764442
\(963\) 0 0
\(964\) 37.8102 1.21778
\(965\) −4.03230 −0.129804
\(966\) 0 0
\(967\) 3.39625 0.109216 0.0546080 0.998508i \(-0.482609\pi\)
0.0546080 + 0.998508i \(0.482609\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −8.84563 −0.284016
\(971\) 9.39500 0.301500 0.150750 0.988572i \(-0.451831\pi\)
0.150750 + 0.988572i \(0.451831\pi\)
\(972\) 0 0
\(973\) −62.2631 −1.99606
\(974\) 8.81035 0.282302
\(975\) 0 0
\(976\) −10.6942 −0.342314
\(977\) 16.5552 0.529649 0.264824 0.964297i \(-0.414686\pi\)
0.264824 + 0.964297i \(0.414686\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.372885 −0.0119114
\(981\) 0 0
\(982\) −5.45692 −0.174137
\(983\) −50.8180 −1.62084 −0.810421 0.585848i \(-0.800762\pi\)
−0.810421 + 0.585848i \(0.800762\pi\)
\(984\) 0 0
\(985\) −11.4176 −0.363794
\(986\) −6.65350 −0.211891
\(987\) 0 0
\(988\) −35.9062 −1.14233
\(989\) 19.5918 0.622983
\(990\) 0 0
\(991\) 11.3642 0.360996 0.180498 0.983575i \(-0.442229\pi\)
0.180498 + 0.983575i \(0.442229\pi\)
\(992\) −11.6504 −0.369901
\(993\) 0 0
\(994\) 1.53236 0.0486036
\(995\) −7.16644 −0.227191
\(996\) 0 0
\(997\) −29.1644 −0.923647 −0.461824 0.886972i \(-0.652805\pi\)
−0.461824 + 0.886972i \(0.652805\pi\)
\(998\) −5.19332 −0.164392
\(999\) 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 5445.2.a.bp.1.2 4
3.2 odd 2 605.2.a.j.1.3 4
11.3 even 5 495.2.n.e.361.1 8
11.4 even 5 495.2.n.e.181.1 8
11.10 odd 2 5445.2.a.bi.1.3 4
12.11 even 2 9680.2.a.cn.1.3 4
15.14 odd 2 3025.2.a.bd.1.2 4
33.2 even 10 605.2.g.e.81.2 8
33.5 odd 10 605.2.g.m.366.1 8
33.8 even 10 605.2.g.k.251.1 8
33.14 odd 10 55.2.g.b.31.2 yes 8
33.17 even 10 605.2.g.e.366.2 8
33.20 odd 10 605.2.g.m.81.1 8
33.26 odd 10 55.2.g.b.16.2 8
33.29 even 10 605.2.g.k.511.1 8
33.32 even 2 605.2.a.k.1.2 4
132.47 even 10 880.2.bo.h.801.1 8
132.59 even 10 880.2.bo.h.401.1 8
132.131 odd 2 9680.2.a.cm.1.3 4
165.14 odd 10 275.2.h.a.251.1 8
165.47 even 20 275.2.z.a.174.3 16
165.59 odd 10 275.2.h.a.126.1 8
165.92 even 20 275.2.z.a.49.2 16
165.113 even 20 275.2.z.a.174.2 16
165.158 even 20 275.2.z.a.49.3 16
165.164 even 2 3025.2.a.w.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.g.b.16.2 8 33.26 odd 10
55.2.g.b.31.2 yes 8 33.14 odd 10
275.2.h.a.126.1 8 165.59 odd 10
275.2.h.a.251.1 8 165.14 odd 10
275.2.z.a.49.2 16 165.92 even 20
275.2.z.a.49.3 16 165.158 even 20
275.2.z.a.174.2 16 165.113 even 20
275.2.z.a.174.3 16 165.47 even 20
495.2.n.e.181.1 8 11.4 even 5
495.2.n.e.361.1 8 11.3 even 5
605.2.a.j.1.3 4 3.2 odd 2
605.2.a.k.1.2 4 33.32 even 2
605.2.g.e.81.2 8 33.2 even 10
605.2.g.e.366.2 8 33.17 even 10
605.2.g.k.251.1 8 33.8 even 10
605.2.g.k.511.1 8 33.29 even 10
605.2.g.m.81.1 8 33.20 odd 10
605.2.g.m.366.1 8 33.5 odd 10
880.2.bo.h.401.1 8 132.59 even 10
880.2.bo.h.801.1 8 132.47 even 10
3025.2.a.w.1.3 4 165.164 even 2
3025.2.a.bd.1.2 4 15.14 odd 2
5445.2.a.bi.1.3 4 11.10 odd 2
5445.2.a.bp.1.2 4 1.1 even 1 trivial
9680.2.a.cm.1.3 4 132.131 odd 2
9680.2.a.cn.1.3 4 12.11 even 2