Properties

Label 6039.2.a.d.1.2
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $11$
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] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, 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("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.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: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 4x^{10} - 6x^{9} + 37x^{8} - 2x^{7} - 109x^{6} + 55x^{5} + 115x^{4} - 76x^{3} - 29x^{2} + 14x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2013)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.57504\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.57504 q^{2} +0.480756 q^{4} -0.664958 q^{5} -4.15209 q^{7} +2.39287 q^{8} +O(q^{10})\) \(q-1.57504 q^{2} +0.480756 q^{4} -0.664958 q^{5} -4.15209 q^{7} +2.39287 q^{8} +1.04734 q^{10} +1.00000 q^{11} +2.20668 q^{13} +6.53972 q^{14} -4.73039 q^{16} +0.555008 q^{17} -6.78925 q^{19} -0.319682 q^{20} -1.57504 q^{22} +4.88924 q^{23} -4.55783 q^{25} -3.47561 q^{26} -1.99614 q^{28} -6.82104 q^{29} -4.34874 q^{31} +2.66481 q^{32} -0.874161 q^{34} +2.76097 q^{35} -5.04417 q^{37} +10.6933 q^{38} -1.59116 q^{40} +2.62576 q^{41} +7.39322 q^{43} +0.480756 q^{44} -7.70076 q^{46} -8.12666 q^{47} +10.2399 q^{49} +7.17877 q^{50} +1.06087 q^{52} +10.8520 q^{53} -0.664958 q^{55} -9.93543 q^{56} +10.7434 q^{58} +12.5212 q^{59} +1.00000 q^{61} +6.84944 q^{62} +5.26359 q^{64} -1.46735 q^{65} -0.106558 q^{67} +0.266823 q^{68} -4.34864 q^{70} -14.5818 q^{71} +3.84843 q^{73} +7.94478 q^{74} -3.26397 q^{76} -4.15209 q^{77} -11.5447 q^{79} +3.14551 q^{80} -4.13569 q^{82} +11.0120 q^{83} -0.369057 q^{85} -11.6446 q^{86} +2.39287 q^{88} -13.7280 q^{89} -9.16235 q^{91} +2.35053 q^{92} +12.7998 q^{94} +4.51456 q^{95} -8.42214 q^{97} -16.1282 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 4 q^{2} + 6 q^{4} + 13 q^{5} - 5 q^{7} + 9 q^{8} + 6 q^{10} + 11 q^{11} - 3 q^{13} + 9 q^{14} + 4 q^{16} + 7 q^{17} - 8 q^{19} + 25 q^{20} + 4 q^{22} + 15 q^{23} + 4 q^{25} + 2 q^{26} + 13 q^{28} + 8 q^{29} - 17 q^{31} + 27 q^{32} - 18 q^{34} + 2 q^{35} - 10 q^{37} + 30 q^{38} + 10 q^{40} + 25 q^{41} - 7 q^{43} + 6 q^{44} + 32 q^{46} + 30 q^{47} - 2 q^{49} - 11 q^{50} - 7 q^{52} + 18 q^{53} + 13 q^{55} + 20 q^{56} - 13 q^{58} + 43 q^{59} + 11 q^{61} - 7 q^{62} + 25 q^{64} + 27 q^{65} - 30 q^{67} - 10 q^{68} - 4 q^{70} + 7 q^{71} + 6 q^{73} + 44 q^{74} - 19 q^{76} - 5 q^{77} + 17 q^{79} + 22 q^{80} + 8 q^{82} + 34 q^{83} + 10 q^{85} - 2 q^{86} + 9 q^{88} + 41 q^{89} - 39 q^{91} + 32 q^{92} + 55 q^{94} + 9 q^{95} - 41 q^{97} + 29 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.57504 −1.11372 −0.556861 0.830606i \(-0.687994\pi\)
−0.556861 + 0.830606i \(0.687994\pi\)
\(3\) 0 0
\(4\) 0.480756 0.240378
\(5\) −0.664958 −0.297378 −0.148689 0.988884i \(-0.547505\pi\)
−0.148689 + 0.988884i \(0.547505\pi\)
\(6\) 0 0
\(7\) −4.15209 −1.56934 −0.784672 0.619911i \(-0.787169\pi\)
−0.784672 + 0.619911i \(0.787169\pi\)
\(8\) 2.39287 0.846008
\(9\) 0 0
\(10\) 1.04734 0.331197
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.20668 0.612023 0.306012 0.952028i \(-0.401005\pi\)
0.306012 + 0.952028i \(0.401005\pi\)
\(14\) 6.53972 1.74781
\(15\) 0 0
\(16\) −4.73039 −1.18260
\(17\) 0.555008 0.134609 0.0673046 0.997732i \(-0.478560\pi\)
0.0673046 + 0.997732i \(0.478560\pi\)
\(18\) 0 0
\(19\) −6.78925 −1.55756 −0.778780 0.627297i \(-0.784161\pi\)
−0.778780 + 0.627297i \(0.784161\pi\)
\(20\) −0.319682 −0.0714831
\(21\) 0 0
\(22\) −1.57504 −0.335800
\(23\) 4.88924 1.01948 0.509739 0.860329i \(-0.329742\pi\)
0.509739 + 0.860329i \(0.329742\pi\)
\(24\) 0 0
\(25\) −4.55783 −0.911566
\(26\) −3.47561 −0.681624
\(27\) 0 0
\(28\) −1.99614 −0.377236
\(29\) −6.82104 −1.26663 −0.633317 0.773892i \(-0.718307\pi\)
−0.633317 + 0.773892i \(0.718307\pi\)
\(30\) 0 0
\(31\) −4.34874 −0.781057 −0.390528 0.920591i \(-0.627708\pi\)
−0.390528 + 0.920591i \(0.627708\pi\)
\(32\) 2.66481 0.471076
\(33\) 0 0
\(34\) −0.874161 −0.149917
\(35\) 2.76097 0.466689
\(36\) 0 0
\(37\) −5.04417 −0.829256 −0.414628 0.909991i \(-0.636088\pi\)
−0.414628 + 0.909991i \(0.636088\pi\)
\(38\) 10.6933 1.73469
\(39\) 0 0
\(40\) −1.59116 −0.251584
\(41\) 2.62576 0.410076 0.205038 0.978754i \(-0.434268\pi\)
0.205038 + 0.978754i \(0.434268\pi\)
\(42\) 0 0
\(43\) 7.39322 1.12746 0.563728 0.825961i \(-0.309367\pi\)
0.563728 + 0.825961i \(0.309367\pi\)
\(44\) 0.480756 0.0724767
\(45\) 0 0
\(46\) −7.70076 −1.13541
\(47\) −8.12666 −1.18540 −0.592698 0.805425i \(-0.701937\pi\)
−0.592698 + 0.805425i \(0.701937\pi\)
\(48\) 0 0
\(49\) 10.2399 1.46284
\(50\) 7.17877 1.01523
\(51\) 0 0
\(52\) 1.06087 0.147117
\(53\) 10.8520 1.49063 0.745317 0.666710i \(-0.232298\pi\)
0.745317 + 0.666710i \(0.232298\pi\)
\(54\) 0 0
\(55\) −0.664958 −0.0896629
\(56\) −9.93543 −1.32768
\(57\) 0 0
\(58\) 10.7434 1.41068
\(59\) 12.5212 1.63012 0.815058 0.579380i \(-0.196705\pi\)
0.815058 + 0.579380i \(0.196705\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037
\(62\) 6.84944 0.869880
\(63\) 0 0
\(64\) 5.26359 0.657948
\(65\) −1.46735 −0.182002
\(66\) 0 0
\(67\) −0.106558 −0.0130181 −0.00650905 0.999979i \(-0.502072\pi\)
−0.00650905 + 0.999979i \(0.502072\pi\)
\(68\) 0.266823 0.0323571
\(69\) 0 0
\(70\) −4.34864 −0.519762
\(71\) −14.5818 −1.73054 −0.865269 0.501308i \(-0.832852\pi\)
−0.865269 + 0.501308i \(0.832852\pi\)
\(72\) 0 0
\(73\) 3.84843 0.450425 0.225212 0.974310i \(-0.427692\pi\)
0.225212 + 0.974310i \(0.427692\pi\)
\(74\) 7.94478 0.923562
\(75\) 0 0
\(76\) −3.26397 −0.374403
\(77\) −4.15209 −0.473175
\(78\) 0 0
\(79\) −11.5447 −1.29888 −0.649439 0.760414i \(-0.724996\pi\)
−0.649439 + 0.760414i \(0.724996\pi\)
\(80\) 3.14551 0.351678
\(81\) 0 0
\(82\) −4.13569 −0.456710
\(83\) 11.0120 1.20873 0.604364 0.796709i \(-0.293427\pi\)
0.604364 + 0.796709i \(0.293427\pi\)
\(84\) 0 0
\(85\) −0.369057 −0.0400298
\(86\) −11.6446 −1.25567
\(87\) 0 0
\(88\) 2.39287 0.255081
\(89\) −13.7280 −1.45516 −0.727582 0.686021i \(-0.759356\pi\)
−0.727582 + 0.686021i \(0.759356\pi\)
\(90\) 0 0
\(91\) −9.16235 −0.960475
\(92\) 2.35053 0.245060
\(93\) 0 0
\(94\) 12.7998 1.32020
\(95\) 4.51456 0.463184
\(96\) 0 0
\(97\) −8.42214 −0.855139 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(98\) −16.1282 −1.62920
\(99\) 0 0
\(100\) −2.19120 −0.219120
\(101\) −4.30166 −0.428031 −0.214016 0.976830i \(-0.568654\pi\)
−0.214016 + 0.976830i \(0.568654\pi\)
\(102\) 0 0
\(103\) −19.9878 −1.96945 −0.984727 0.174105i \(-0.944297\pi\)
−0.984727 + 0.174105i \(0.944297\pi\)
\(104\) 5.28031 0.517777
\(105\) 0 0
\(106\) −17.0923 −1.66015
\(107\) 1.93095 0.186672 0.0933360 0.995635i \(-0.470247\pi\)
0.0933360 + 0.995635i \(0.470247\pi\)
\(108\) 0 0
\(109\) −2.72415 −0.260927 −0.130463 0.991453i \(-0.541646\pi\)
−0.130463 + 0.991453i \(0.541646\pi\)
\(110\) 1.04734 0.0998596
\(111\) 0 0
\(112\) 19.6410 1.85590
\(113\) −12.9048 −1.21399 −0.606993 0.794708i \(-0.707624\pi\)
−0.606993 + 0.794708i \(0.707624\pi\)
\(114\) 0 0
\(115\) −3.25114 −0.303170
\(116\) −3.27925 −0.304471
\(117\) 0 0
\(118\) −19.7213 −1.81550
\(119\) −2.30445 −0.211248
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −1.57504 −0.142598
\(123\) 0 0
\(124\) −2.09068 −0.187749
\(125\) 6.35555 0.568458
\(126\) 0 0
\(127\) −0.247494 −0.0219616 −0.0109808 0.999940i \(-0.503495\pi\)
−0.0109808 + 0.999940i \(0.503495\pi\)
\(128\) −13.6200 −1.20385
\(129\) 0 0
\(130\) 2.31114 0.202700
\(131\) −10.1462 −0.886476 −0.443238 0.896404i \(-0.646170\pi\)
−0.443238 + 0.896404i \(0.646170\pi\)
\(132\) 0 0
\(133\) 28.1896 2.44435
\(134\) 0.167833 0.0144985
\(135\) 0 0
\(136\) 1.32806 0.113881
\(137\) 7.01435 0.599276 0.299638 0.954053i \(-0.403134\pi\)
0.299638 + 0.954053i \(0.403134\pi\)
\(138\) 0 0
\(139\) 20.8476 1.76827 0.884133 0.467235i \(-0.154750\pi\)
0.884133 + 0.467235i \(0.154750\pi\)
\(140\) 1.32735 0.112182
\(141\) 0 0
\(142\) 22.9669 1.92734
\(143\) 2.20668 0.184532
\(144\) 0 0
\(145\) 4.53570 0.376670
\(146\) −6.06144 −0.501648
\(147\) 0 0
\(148\) −2.42501 −0.199335
\(149\) 7.76168 0.635861 0.317931 0.948114i \(-0.397012\pi\)
0.317931 + 0.948114i \(0.397012\pi\)
\(150\) 0 0
\(151\) −5.96491 −0.485418 −0.242709 0.970099i \(-0.578036\pi\)
−0.242709 + 0.970099i \(0.578036\pi\)
\(152\) −16.2458 −1.31771
\(153\) 0 0
\(154\) 6.53972 0.526986
\(155\) 2.89173 0.232269
\(156\) 0 0
\(157\) −7.17348 −0.572506 −0.286253 0.958154i \(-0.592410\pi\)
−0.286253 + 0.958154i \(0.592410\pi\)
\(158\) 18.1833 1.44659
\(159\) 0 0
\(160\) −1.77198 −0.140088
\(161\) −20.3006 −1.59991
\(162\) 0 0
\(163\) −0.662100 −0.0518597 −0.0259298 0.999664i \(-0.508255\pi\)
−0.0259298 + 0.999664i \(0.508255\pi\)
\(164\) 1.26235 0.0985731
\(165\) 0 0
\(166\) −17.3444 −1.34619
\(167\) −24.6628 −1.90846 −0.954232 0.299066i \(-0.903325\pi\)
−0.954232 + 0.299066i \(0.903325\pi\)
\(168\) 0 0
\(169\) −8.13056 −0.625428
\(170\) 0.581280 0.0445821
\(171\) 0 0
\(172\) 3.55433 0.271015
\(173\) −18.8914 −1.43629 −0.718143 0.695895i \(-0.755008\pi\)
−0.718143 + 0.695895i \(0.755008\pi\)
\(174\) 0 0
\(175\) 18.9245 1.43056
\(176\) −4.73039 −0.356566
\(177\) 0 0
\(178\) 21.6222 1.62065
\(179\) 9.09802 0.680018 0.340009 0.940422i \(-0.389570\pi\)
0.340009 + 0.940422i \(0.389570\pi\)
\(180\) 0 0
\(181\) 13.4415 0.999096 0.499548 0.866286i \(-0.333499\pi\)
0.499548 + 0.866286i \(0.333499\pi\)
\(182\) 14.4311 1.06970
\(183\) 0 0
\(184\) 11.6993 0.862486
\(185\) 3.35416 0.246603
\(186\) 0 0
\(187\) 0.555008 0.0405862
\(188\) −3.90694 −0.284943
\(189\) 0 0
\(190\) −7.11062 −0.515859
\(191\) −4.02432 −0.291190 −0.145595 0.989344i \(-0.546510\pi\)
−0.145595 + 0.989344i \(0.546510\pi\)
\(192\) 0 0
\(193\) 19.8101 1.42596 0.712980 0.701184i \(-0.247345\pi\)
0.712980 + 0.701184i \(0.247345\pi\)
\(194\) 13.2652 0.952387
\(195\) 0 0
\(196\) 4.92288 0.351634
\(197\) −0.313023 −0.0223019 −0.0111510 0.999938i \(-0.503550\pi\)
−0.0111510 + 0.999938i \(0.503550\pi\)
\(198\) 0 0
\(199\) −24.4725 −1.73481 −0.867405 0.497603i \(-0.834214\pi\)
−0.867405 + 0.497603i \(0.834214\pi\)
\(200\) −10.9063 −0.771193
\(201\) 0 0
\(202\) 6.77529 0.476708
\(203\) 28.3216 1.98779
\(204\) 0 0
\(205\) −1.74602 −0.121948
\(206\) 31.4816 2.19343
\(207\) 0 0
\(208\) −10.4385 −0.723776
\(209\) −6.78925 −0.469622
\(210\) 0 0
\(211\) −16.3430 −1.12510 −0.562550 0.826763i \(-0.690180\pi\)
−0.562550 + 0.826763i \(0.690180\pi\)
\(212\) 5.21715 0.358316
\(213\) 0 0
\(214\) −3.04133 −0.207901
\(215\) −4.91618 −0.335281
\(216\) 0 0
\(217\) 18.0564 1.22575
\(218\) 4.29066 0.290600
\(219\) 0 0
\(220\) −0.319682 −0.0215530
\(221\) 1.22473 0.0823840
\(222\) 0 0
\(223\) 7.06727 0.473259 0.236630 0.971600i \(-0.423957\pi\)
0.236630 + 0.971600i \(0.423957\pi\)
\(224\) −11.0645 −0.739280
\(225\) 0 0
\(226\) 20.3257 1.35204
\(227\) 28.2150 1.87269 0.936347 0.351075i \(-0.114184\pi\)
0.936347 + 0.351075i \(0.114184\pi\)
\(228\) 0 0
\(229\) 27.6985 1.83037 0.915184 0.403036i \(-0.132045\pi\)
0.915184 + 0.403036i \(0.132045\pi\)
\(230\) 5.12068 0.337648
\(231\) 0 0
\(232\) −16.3219 −1.07158
\(233\) −20.8520 −1.36606 −0.683030 0.730391i \(-0.739338\pi\)
−0.683030 + 0.730391i \(0.739338\pi\)
\(234\) 0 0
\(235\) 5.40389 0.352511
\(236\) 6.01962 0.391844
\(237\) 0 0
\(238\) 3.62960 0.235272
\(239\) 28.1349 1.81989 0.909947 0.414725i \(-0.136122\pi\)
0.909947 + 0.414725i \(0.136122\pi\)
\(240\) 0 0
\(241\) −6.54194 −0.421403 −0.210702 0.977550i \(-0.567575\pi\)
−0.210702 + 0.977550i \(0.567575\pi\)
\(242\) −1.57504 −0.101248
\(243\) 0 0
\(244\) 0.480756 0.0307772
\(245\) −6.80909 −0.435017
\(246\) 0 0
\(247\) −14.9817 −0.953263
\(248\) −10.4060 −0.660780
\(249\) 0 0
\(250\) −10.0103 −0.633105
\(251\) −9.23622 −0.582985 −0.291493 0.956573i \(-0.594152\pi\)
−0.291493 + 0.956573i \(0.594152\pi\)
\(252\) 0 0
\(253\) 4.88924 0.307384
\(254\) 0.389814 0.0244591
\(255\) 0 0
\(256\) 10.9249 0.682804
\(257\) 22.9183 1.42960 0.714802 0.699327i \(-0.246517\pi\)
0.714802 + 0.699327i \(0.246517\pi\)
\(258\) 0 0
\(259\) 20.9439 1.30139
\(260\) −0.705437 −0.0437493
\(261\) 0 0
\(262\) 15.9806 0.987288
\(263\) 23.9588 1.47736 0.738682 0.674054i \(-0.235448\pi\)
0.738682 + 0.674054i \(0.235448\pi\)
\(264\) 0 0
\(265\) −7.21611 −0.443282
\(266\) −44.3998 −2.72232
\(267\) 0 0
\(268\) −0.0512282 −0.00312926
\(269\) 21.8444 1.33188 0.665940 0.746005i \(-0.268031\pi\)
0.665940 + 0.746005i \(0.268031\pi\)
\(270\) 0 0
\(271\) 22.2094 1.34913 0.674563 0.738218i \(-0.264332\pi\)
0.674563 + 0.738218i \(0.264332\pi\)
\(272\) −2.62540 −0.159188
\(273\) 0 0
\(274\) −11.0479 −0.667427
\(275\) −4.55783 −0.274848
\(276\) 0 0
\(277\) 28.0104 1.68298 0.841490 0.540272i \(-0.181679\pi\)
0.841490 + 0.540272i \(0.181679\pi\)
\(278\) −32.8358 −1.96936
\(279\) 0 0
\(280\) 6.60664 0.394822
\(281\) 22.6750 1.35268 0.676340 0.736590i \(-0.263565\pi\)
0.676340 + 0.736590i \(0.263565\pi\)
\(282\) 0 0
\(283\) −22.1051 −1.31401 −0.657005 0.753886i \(-0.728177\pi\)
−0.657005 + 0.753886i \(0.728177\pi\)
\(284\) −7.01027 −0.415983
\(285\) 0 0
\(286\) −3.47561 −0.205517
\(287\) −10.9024 −0.643550
\(288\) 0 0
\(289\) −16.6920 −0.981880
\(290\) −7.14392 −0.419505
\(291\) 0 0
\(292\) 1.85015 0.108272
\(293\) 10.2956 0.601477 0.300738 0.953707i \(-0.402767\pi\)
0.300738 + 0.953707i \(0.402767\pi\)
\(294\) 0 0
\(295\) −8.32604 −0.484761
\(296\) −12.0701 −0.701558
\(297\) 0 0
\(298\) −12.2250 −0.708173
\(299\) 10.7890 0.623944
\(300\) 0 0
\(301\) −30.6973 −1.76936
\(302\) 9.39498 0.540620
\(303\) 0 0
\(304\) 32.1158 1.84196
\(305\) −0.664958 −0.0380754
\(306\) 0 0
\(307\) 32.4405 1.85147 0.925737 0.378167i \(-0.123445\pi\)
0.925737 + 0.378167i \(0.123445\pi\)
\(308\) −1.99614 −0.113741
\(309\) 0 0
\(310\) −4.55459 −0.258683
\(311\) 4.29920 0.243785 0.121893 0.992543i \(-0.461104\pi\)
0.121893 + 0.992543i \(0.461104\pi\)
\(312\) 0 0
\(313\) 6.08806 0.344117 0.172059 0.985087i \(-0.444958\pi\)
0.172059 + 0.985087i \(0.444958\pi\)
\(314\) 11.2985 0.637613
\(315\) 0 0
\(316\) −5.55017 −0.312221
\(317\) 28.3357 1.59149 0.795745 0.605632i \(-0.207080\pi\)
0.795745 + 0.605632i \(0.207080\pi\)
\(318\) 0 0
\(319\) −6.82104 −0.381905
\(320\) −3.50006 −0.195660
\(321\) 0 0
\(322\) 31.9743 1.78186
\(323\) −3.76809 −0.209662
\(324\) 0 0
\(325\) −10.0577 −0.557900
\(326\) 1.04283 0.0577573
\(327\) 0 0
\(328\) 6.28312 0.346927
\(329\) 33.7427 1.86029
\(330\) 0 0
\(331\) 21.7095 1.19326 0.596630 0.802516i \(-0.296506\pi\)
0.596630 + 0.802516i \(0.296506\pi\)
\(332\) 5.29410 0.290551
\(333\) 0 0
\(334\) 38.8449 2.12550
\(335\) 0.0708564 0.00387130
\(336\) 0 0
\(337\) −9.98451 −0.543891 −0.271946 0.962313i \(-0.587667\pi\)
−0.271946 + 0.962313i \(0.587667\pi\)
\(338\) 12.8060 0.696553
\(339\) 0 0
\(340\) −0.177426 −0.00962229
\(341\) −4.34874 −0.235497
\(342\) 0 0
\(343\) −13.4523 −0.726355
\(344\) 17.6910 0.953836
\(345\) 0 0
\(346\) 29.7547 1.59962
\(347\) −6.67445 −0.358303 −0.179152 0.983821i \(-0.557335\pi\)
−0.179152 + 0.983821i \(0.557335\pi\)
\(348\) 0 0
\(349\) −26.9515 −1.44268 −0.721341 0.692580i \(-0.756474\pi\)
−0.721341 + 0.692580i \(0.756474\pi\)
\(350\) −29.8069 −1.59325
\(351\) 0 0
\(352\) 2.66481 0.142035
\(353\) 2.87667 0.153110 0.0765548 0.997065i \(-0.475608\pi\)
0.0765548 + 0.997065i \(0.475608\pi\)
\(354\) 0 0
\(355\) 9.69626 0.514624
\(356\) −6.59981 −0.349789
\(357\) 0 0
\(358\) −14.3298 −0.757351
\(359\) −11.6302 −0.613820 −0.306910 0.951738i \(-0.599295\pi\)
−0.306910 + 0.951738i \(0.599295\pi\)
\(360\) 0 0
\(361\) 27.0939 1.42599
\(362\) −21.1709 −1.11272
\(363\) 0 0
\(364\) −4.40485 −0.230877
\(365\) −2.55904 −0.133946
\(366\) 0 0
\(367\) −23.7261 −1.23849 −0.619245 0.785198i \(-0.712561\pi\)
−0.619245 + 0.785198i \(0.712561\pi\)
\(368\) −23.1280 −1.20563
\(369\) 0 0
\(370\) −5.28294 −0.274647
\(371\) −45.0585 −2.33932
\(372\) 0 0
\(373\) 20.6466 1.06904 0.534521 0.845155i \(-0.320492\pi\)
0.534521 + 0.845155i \(0.320492\pi\)
\(374\) −0.874161 −0.0452018
\(375\) 0 0
\(376\) −19.4461 −1.00285
\(377\) −15.0519 −0.775210
\(378\) 0 0
\(379\) −27.3322 −1.40396 −0.701980 0.712196i \(-0.747701\pi\)
−0.701980 + 0.712196i \(0.747701\pi\)
\(380\) 2.17040 0.111339
\(381\) 0 0
\(382\) 6.33848 0.324305
\(383\) 13.2603 0.677570 0.338785 0.940864i \(-0.389984\pi\)
0.338785 + 0.940864i \(0.389984\pi\)
\(384\) 0 0
\(385\) 2.76097 0.140712
\(386\) −31.2017 −1.58812
\(387\) 0 0
\(388\) −4.04899 −0.205556
\(389\) −21.9449 −1.11265 −0.556326 0.830964i \(-0.687789\pi\)
−0.556326 + 0.830964i \(0.687789\pi\)
\(390\) 0 0
\(391\) 2.71357 0.137231
\(392\) 24.5027 1.23757
\(393\) 0 0
\(394\) 0.493024 0.0248382
\(395\) 7.67672 0.386258
\(396\) 0 0
\(397\) −10.9744 −0.550791 −0.275395 0.961331i \(-0.588809\pi\)
−0.275395 + 0.961331i \(0.588809\pi\)
\(398\) 38.5452 1.93210
\(399\) 0 0
\(400\) 21.5603 1.07801
\(401\) 14.1612 0.707177 0.353589 0.935401i \(-0.384961\pi\)
0.353589 + 0.935401i \(0.384961\pi\)
\(402\) 0 0
\(403\) −9.59628 −0.478025
\(404\) −2.06805 −0.102889
\(405\) 0 0
\(406\) −44.6077 −2.21384
\(407\) −5.04417 −0.250030
\(408\) 0 0
\(409\) 20.1898 0.998323 0.499162 0.866509i \(-0.333641\pi\)
0.499162 + 0.866509i \(0.333641\pi\)
\(410\) 2.75006 0.135816
\(411\) 0 0
\(412\) −9.60924 −0.473413
\(413\) −51.9890 −2.55821
\(414\) 0 0
\(415\) −7.32253 −0.359449
\(416\) 5.88038 0.288309
\(417\) 0 0
\(418\) 10.6933 0.523029
\(419\) −6.66184 −0.325452 −0.162726 0.986671i \(-0.552029\pi\)
−0.162726 + 0.986671i \(0.552029\pi\)
\(420\) 0 0
\(421\) 34.0791 1.66092 0.830458 0.557082i \(-0.188079\pi\)
0.830458 + 0.557082i \(0.188079\pi\)
\(422\) 25.7409 1.25305
\(423\) 0 0
\(424\) 25.9674 1.26109
\(425\) −2.52963 −0.122705
\(426\) 0 0
\(427\) −4.15209 −0.200934
\(428\) 0.928315 0.0448718
\(429\) 0 0
\(430\) 7.74318 0.373409
\(431\) 3.49979 0.168579 0.0842894 0.996441i \(-0.473138\pi\)
0.0842894 + 0.996441i \(0.473138\pi\)
\(432\) 0 0
\(433\) 5.96208 0.286519 0.143260 0.989685i \(-0.454242\pi\)
0.143260 + 0.989685i \(0.454242\pi\)
\(434\) −28.4395 −1.36514
\(435\) 0 0
\(436\) −1.30965 −0.0627210
\(437\) −33.1943 −1.58790
\(438\) 0 0
\(439\) 35.7745 1.70742 0.853711 0.520746i \(-0.174346\pi\)
0.853711 + 0.520746i \(0.174346\pi\)
\(440\) −1.59116 −0.0758555
\(441\) 0 0
\(442\) −1.92899 −0.0917529
\(443\) 12.4530 0.591659 0.295829 0.955241i \(-0.404404\pi\)
0.295829 + 0.955241i \(0.404404\pi\)
\(444\) 0 0
\(445\) 9.12854 0.432734
\(446\) −11.1312 −0.527079
\(447\) 0 0
\(448\) −21.8549 −1.03255
\(449\) 21.5415 1.01661 0.508304 0.861178i \(-0.330273\pi\)
0.508304 + 0.861178i \(0.330273\pi\)
\(450\) 0 0
\(451\) 2.62576 0.123642
\(452\) −6.20408 −0.291815
\(453\) 0 0
\(454\) −44.4398 −2.08566
\(455\) 6.09257 0.285624
\(456\) 0 0
\(457\) 14.2977 0.668820 0.334410 0.942428i \(-0.391463\pi\)
0.334410 + 0.942428i \(0.391463\pi\)
\(458\) −43.6263 −2.03852
\(459\) 0 0
\(460\) −1.56300 −0.0728754
\(461\) 16.7508 0.780162 0.390081 0.920781i \(-0.372447\pi\)
0.390081 + 0.920781i \(0.372447\pi\)
\(462\) 0 0
\(463\) −4.95141 −0.230111 −0.115056 0.993359i \(-0.536705\pi\)
−0.115056 + 0.993359i \(0.536705\pi\)
\(464\) 32.2661 1.49792
\(465\) 0 0
\(466\) 32.8427 1.52141
\(467\) 13.6976 0.633851 0.316926 0.948450i \(-0.397349\pi\)
0.316926 + 0.948450i \(0.397349\pi\)
\(468\) 0 0
\(469\) 0.442438 0.0204299
\(470\) −8.51135 −0.392599
\(471\) 0 0
\(472\) 29.9615 1.37909
\(473\) 7.39322 0.339940
\(474\) 0 0
\(475\) 30.9442 1.41982
\(476\) −1.10788 −0.0507794
\(477\) 0 0
\(478\) −44.3136 −2.02686
\(479\) 0.678782 0.0310143 0.0155072 0.999880i \(-0.495064\pi\)
0.0155072 + 0.999880i \(0.495064\pi\)
\(480\) 0 0
\(481\) −11.1309 −0.507524
\(482\) 10.3038 0.469327
\(483\) 0 0
\(484\) 0.480756 0.0218525
\(485\) 5.60037 0.254300
\(486\) 0 0
\(487\) −34.9423 −1.58339 −0.791693 0.610919i \(-0.790800\pi\)
−0.791693 + 0.610919i \(0.790800\pi\)
\(488\) 2.39287 0.108320
\(489\) 0 0
\(490\) 10.7246 0.484488
\(491\) 3.04634 0.137479 0.0687397 0.997635i \(-0.478102\pi\)
0.0687397 + 0.997635i \(0.478102\pi\)
\(492\) 0 0
\(493\) −3.78573 −0.170501
\(494\) 23.5968 1.06167
\(495\) 0 0
\(496\) 20.5712 0.923675
\(497\) 60.5449 2.71581
\(498\) 0 0
\(499\) −16.9286 −0.757828 −0.378914 0.925432i \(-0.623702\pi\)
−0.378914 + 0.925432i \(0.623702\pi\)
\(500\) 3.05547 0.136645
\(501\) 0 0
\(502\) 14.5474 0.649284
\(503\) −31.0454 −1.38425 −0.692124 0.721779i \(-0.743325\pi\)
−0.692124 + 0.721779i \(0.743325\pi\)
\(504\) 0 0
\(505\) 2.86042 0.127287
\(506\) −7.70076 −0.342340
\(507\) 0 0
\(508\) −0.118984 −0.00527908
\(509\) 0.726829 0.0322161 0.0161081 0.999870i \(-0.494872\pi\)
0.0161081 + 0.999870i \(0.494872\pi\)
\(510\) 0 0
\(511\) −15.9790 −0.706871
\(512\) 10.0329 0.443394
\(513\) 0 0
\(514\) −36.0973 −1.59218
\(515\) 13.2910 0.585673
\(516\) 0 0
\(517\) −8.12666 −0.357410
\(518\) −32.9875 −1.44939
\(519\) 0 0
\(520\) −3.51118 −0.153975
\(521\) 33.5607 1.47032 0.735161 0.677893i \(-0.237107\pi\)
0.735161 + 0.677893i \(0.237107\pi\)
\(522\) 0 0
\(523\) 0.242645 0.0106101 0.00530507 0.999986i \(-0.498311\pi\)
0.00530507 + 0.999986i \(0.498311\pi\)
\(524\) −4.87783 −0.213089
\(525\) 0 0
\(526\) −37.7362 −1.64537
\(527\) −2.41359 −0.105137
\(528\) 0 0
\(529\) 0.904686 0.0393342
\(530\) 11.3657 0.493693
\(531\) 0 0
\(532\) 13.5523 0.587567
\(533\) 5.79422 0.250976
\(534\) 0 0
\(535\) −1.28400 −0.0555122
\(536\) −0.254979 −0.0110134
\(537\) 0 0
\(538\) −34.4059 −1.48334
\(539\) 10.2399 0.441063
\(540\) 0 0
\(541\) −23.9828 −1.03110 −0.515551 0.856859i \(-0.672413\pi\)
−0.515551 + 0.856859i \(0.672413\pi\)
\(542\) −34.9807 −1.50255
\(543\) 0 0
\(544\) 1.47899 0.0634112
\(545\) 1.81145 0.0775939
\(546\) 0 0
\(547\) −33.8037 −1.44534 −0.722671 0.691192i \(-0.757086\pi\)
−0.722671 + 0.691192i \(0.757086\pi\)
\(548\) 3.37219 0.144053
\(549\) 0 0
\(550\) 7.17877 0.306104
\(551\) 46.3097 1.97286
\(552\) 0 0
\(553\) 47.9346 2.03839
\(554\) −44.1175 −1.87437
\(555\) 0 0
\(556\) 10.0226 0.425052
\(557\) −3.91424 −0.165852 −0.0829258 0.996556i \(-0.526426\pi\)
−0.0829258 + 0.996556i \(0.526426\pi\)
\(558\) 0 0
\(559\) 16.3145 0.690029
\(560\) −13.0604 −0.551904
\(561\) 0 0
\(562\) −35.7141 −1.50651
\(563\) 9.83958 0.414689 0.207344 0.978268i \(-0.433518\pi\)
0.207344 + 0.978268i \(0.433518\pi\)
\(564\) 0 0
\(565\) 8.58117 0.361013
\(566\) 34.8164 1.46344
\(567\) 0 0
\(568\) −34.8923 −1.46405
\(569\) 7.46080 0.312773 0.156387 0.987696i \(-0.450015\pi\)
0.156387 + 0.987696i \(0.450015\pi\)
\(570\) 0 0
\(571\) 32.1776 1.34659 0.673295 0.739374i \(-0.264879\pi\)
0.673295 + 0.739374i \(0.264879\pi\)
\(572\) 1.06087 0.0443574
\(573\) 0 0
\(574\) 17.1718 0.716736
\(575\) −22.2843 −0.929321
\(576\) 0 0
\(577\) −36.2854 −1.51058 −0.755291 0.655389i \(-0.772505\pi\)
−0.755291 + 0.655389i \(0.772505\pi\)
\(578\) 26.2905 1.09354
\(579\) 0 0
\(580\) 2.18056 0.0905430
\(581\) −45.7230 −1.89691
\(582\) 0 0
\(583\) 10.8520 0.449443
\(584\) 9.20880 0.381063
\(585\) 0 0
\(586\) −16.2160 −0.669878
\(587\) 32.1672 1.32768 0.663841 0.747874i \(-0.268925\pi\)
0.663841 + 0.747874i \(0.268925\pi\)
\(588\) 0 0
\(589\) 29.5247 1.21654
\(590\) 13.1139 0.539889
\(591\) 0 0
\(592\) 23.8609 0.980676
\(593\) 31.5275 1.29468 0.647340 0.762201i \(-0.275881\pi\)
0.647340 + 0.762201i \(0.275881\pi\)
\(594\) 0 0
\(595\) 1.53236 0.0628206
\(596\) 3.73147 0.152847
\(597\) 0 0
\(598\) −16.9931 −0.694900
\(599\) 9.91924 0.405289 0.202645 0.979252i \(-0.435046\pi\)
0.202645 + 0.979252i \(0.435046\pi\)
\(600\) 0 0
\(601\) −6.16233 −0.251367 −0.125683 0.992070i \(-0.540112\pi\)
−0.125683 + 0.992070i \(0.540112\pi\)
\(602\) 48.3496 1.97058
\(603\) 0 0
\(604\) −2.86766 −0.116684
\(605\) −0.664958 −0.0270344
\(606\) 0 0
\(607\) −14.6484 −0.594560 −0.297280 0.954790i \(-0.596079\pi\)
−0.297280 + 0.954790i \(0.596079\pi\)
\(608\) −18.0920 −0.733729
\(609\) 0 0
\(610\) 1.04734 0.0424054
\(611\) −17.9330 −0.725490
\(612\) 0 0
\(613\) 23.1981 0.936962 0.468481 0.883474i \(-0.344801\pi\)
0.468481 + 0.883474i \(0.344801\pi\)
\(614\) −51.0951 −2.06203
\(615\) 0 0
\(616\) −9.93543 −0.400310
\(617\) 0.149954 0.00603693 0.00301847 0.999995i \(-0.499039\pi\)
0.00301847 + 0.999995i \(0.499039\pi\)
\(618\) 0 0
\(619\) 27.4179 1.10202 0.551009 0.834499i \(-0.314243\pi\)
0.551009 + 0.834499i \(0.314243\pi\)
\(620\) 1.39021 0.0558324
\(621\) 0 0
\(622\) −6.77141 −0.271509
\(623\) 56.9999 2.28365
\(624\) 0 0
\(625\) 18.5630 0.742519
\(626\) −9.58894 −0.383251
\(627\) 0 0
\(628\) −3.44869 −0.137618
\(629\) −2.79955 −0.111626
\(630\) 0 0
\(631\) 4.66672 0.185779 0.0928896 0.995676i \(-0.470390\pi\)
0.0928896 + 0.995676i \(0.470390\pi\)
\(632\) −27.6249 −1.09886
\(633\) 0 0
\(634\) −44.6298 −1.77248
\(635\) 0.164573 0.00653089
\(636\) 0 0
\(637\) 22.5962 0.895292
\(638\) 10.7434 0.425336
\(639\) 0 0
\(640\) 9.05672 0.357998
\(641\) −15.1800 −0.599575 −0.299788 0.954006i \(-0.596916\pi\)
−0.299788 + 0.954006i \(0.596916\pi\)
\(642\) 0 0
\(643\) −26.8497 −1.05885 −0.529425 0.848357i \(-0.677592\pi\)
−0.529425 + 0.848357i \(0.677592\pi\)
\(644\) −9.75963 −0.384583
\(645\) 0 0
\(646\) 5.93489 0.233505
\(647\) −26.4109 −1.03832 −0.519159 0.854678i \(-0.673755\pi\)
−0.519159 + 0.854678i \(0.673755\pi\)
\(648\) 0 0
\(649\) 12.5212 0.491498
\(650\) 15.8413 0.621345
\(651\) 0 0
\(652\) −0.318308 −0.0124659
\(653\) −3.27170 −0.128032 −0.0640158 0.997949i \(-0.520391\pi\)
−0.0640158 + 0.997949i \(0.520391\pi\)
\(654\) 0 0
\(655\) 6.74678 0.263618
\(656\) −12.4209 −0.484954
\(657\) 0 0
\(658\) −53.1461 −2.07185
\(659\) −21.3103 −0.830133 −0.415067 0.909791i \(-0.636242\pi\)
−0.415067 + 0.909791i \(0.636242\pi\)
\(660\) 0 0
\(661\) 32.1532 1.25061 0.625306 0.780379i \(-0.284974\pi\)
0.625306 + 0.780379i \(0.284974\pi\)
\(662\) −34.1933 −1.32896
\(663\) 0 0
\(664\) 26.3504 1.02259
\(665\) −18.7449 −0.726896
\(666\) 0 0
\(667\) −33.3497 −1.29131
\(668\) −11.8568 −0.458753
\(669\) 0 0
\(670\) −0.111602 −0.00431155
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) −8.45786 −0.326027 −0.163013 0.986624i \(-0.552121\pi\)
−0.163013 + 0.986624i \(0.552121\pi\)
\(674\) 15.7260 0.605744
\(675\) 0 0
\(676\) −3.90881 −0.150339
\(677\) −7.96311 −0.306047 −0.153024 0.988223i \(-0.548901\pi\)
−0.153024 + 0.988223i \(0.548901\pi\)
\(678\) 0 0
\(679\) 34.9695 1.34201
\(680\) −0.883106 −0.0338656
\(681\) 0 0
\(682\) 6.84944 0.262279
\(683\) −17.0017 −0.650550 −0.325275 0.945619i \(-0.605457\pi\)
−0.325275 + 0.945619i \(0.605457\pi\)
\(684\) 0 0
\(685\) −4.66424 −0.178212
\(686\) 21.1879 0.808958
\(687\) 0 0
\(688\) −34.9728 −1.33332
\(689\) 23.9469 0.912303
\(690\) 0 0
\(691\) −17.2525 −0.656317 −0.328158 0.944623i \(-0.606428\pi\)
−0.328158 + 0.944623i \(0.606428\pi\)
\(692\) −9.08215 −0.345251
\(693\) 0 0
\(694\) 10.5125 0.399050
\(695\) −13.8627 −0.525844
\(696\) 0 0
\(697\) 1.45732 0.0552000
\(698\) 42.4498 1.60675
\(699\) 0 0
\(700\) 9.09808 0.343875
\(701\) 51.6390 1.95038 0.975188 0.221377i \(-0.0710551\pi\)
0.975188 + 0.221377i \(0.0710551\pi\)
\(702\) 0 0
\(703\) 34.2461 1.29162
\(704\) 5.26359 0.198379
\(705\) 0 0
\(706\) −4.53087 −0.170522
\(707\) 17.8609 0.671728
\(708\) 0 0
\(709\) 36.0695 1.35462 0.677310 0.735698i \(-0.263146\pi\)
0.677310 + 0.735698i \(0.263146\pi\)
\(710\) −15.2720 −0.573149
\(711\) 0 0
\(712\) −32.8493 −1.23108
\(713\) −21.2620 −0.796269
\(714\) 0 0
\(715\) −1.46735 −0.0548758
\(716\) 4.37392 0.163461
\(717\) 0 0
\(718\) 18.3181 0.683626
\(719\) −7.32908 −0.273329 −0.136664 0.990617i \(-0.543638\pi\)
−0.136664 + 0.990617i \(0.543638\pi\)
\(720\) 0 0
\(721\) 82.9911 3.09075
\(722\) −42.6740 −1.58816
\(723\) 0 0
\(724\) 6.46206 0.240161
\(725\) 31.0891 1.15462
\(726\) 0 0
\(727\) −29.5709 −1.09672 −0.548361 0.836242i \(-0.684748\pi\)
−0.548361 + 0.836242i \(0.684748\pi\)
\(728\) −21.9243 −0.812570
\(729\) 0 0
\(730\) 4.03060 0.149179
\(731\) 4.10330 0.151766
\(732\) 0 0
\(733\) 48.3129 1.78448 0.892239 0.451563i \(-0.149134\pi\)
0.892239 + 0.451563i \(0.149134\pi\)
\(734\) 37.3695 1.37933
\(735\) 0 0
\(736\) 13.0289 0.480251
\(737\) −0.106558 −0.00392510
\(738\) 0 0
\(739\) 24.3822 0.896915 0.448457 0.893804i \(-0.351974\pi\)
0.448457 + 0.893804i \(0.351974\pi\)
\(740\) 1.61253 0.0592778
\(741\) 0 0
\(742\) 70.9689 2.60535
\(743\) 26.4089 0.968849 0.484424 0.874833i \(-0.339029\pi\)
0.484424 + 0.874833i \(0.339029\pi\)
\(744\) 0 0
\(745\) −5.16119 −0.189091
\(746\) −32.5193 −1.19062
\(747\) 0 0
\(748\) 0.266823 0.00975603
\(749\) −8.01748 −0.292952
\(750\) 0 0
\(751\) −0.0998497 −0.00364357 −0.00182178 0.999998i \(-0.500580\pi\)
−0.00182178 + 0.999998i \(0.500580\pi\)
\(752\) 38.4423 1.40184
\(753\) 0 0
\(754\) 23.7073 0.863369
\(755\) 3.96641 0.144353
\(756\) 0 0
\(757\) 6.11193 0.222142 0.111071 0.993812i \(-0.464572\pi\)
0.111071 + 0.993812i \(0.464572\pi\)
\(758\) 43.0493 1.56362
\(759\) 0 0
\(760\) 10.8028 0.391858
\(761\) 45.5077 1.64965 0.824827 0.565385i \(-0.191273\pi\)
0.824827 + 0.565385i \(0.191273\pi\)
\(762\) 0 0
\(763\) 11.3109 0.409484
\(764\) −1.93472 −0.0699956
\(765\) 0 0
\(766\) −20.8855 −0.754625
\(767\) 27.6302 0.997668
\(768\) 0 0
\(769\) −12.4644 −0.449478 −0.224739 0.974419i \(-0.572153\pi\)
−0.224739 + 0.974419i \(0.572153\pi\)
\(770\) −4.34864 −0.156714
\(771\) 0 0
\(772\) 9.52381 0.342769
\(773\) 21.9139 0.788190 0.394095 0.919070i \(-0.371058\pi\)
0.394095 + 0.919070i \(0.371058\pi\)
\(774\) 0 0
\(775\) 19.8208 0.711985
\(776\) −20.1531 −0.723454
\(777\) 0 0
\(778\) 34.5642 1.23919
\(779\) −17.8270 −0.638717
\(780\) 0 0
\(781\) −14.5818 −0.521777
\(782\) −4.27398 −0.152837
\(783\) 0 0
\(784\) −48.4386 −1.72995
\(785\) 4.77006 0.170251
\(786\) 0 0
\(787\) −17.5182 −0.624456 −0.312228 0.950007i \(-0.601075\pi\)
−0.312228 + 0.950007i \(0.601075\pi\)
\(788\) −0.150487 −0.00536089
\(789\) 0 0
\(790\) −12.0912 −0.430184
\(791\) 53.5821 1.90516
\(792\) 0 0
\(793\) 2.20668 0.0783615
\(794\) 17.2852 0.613428
\(795\) 0 0
\(796\) −11.7653 −0.417010
\(797\) −40.6397 −1.43953 −0.719766 0.694217i \(-0.755751\pi\)
−0.719766 + 0.694217i \(0.755751\pi\)
\(798\) 0 0
\(799\) −4.51036 −0.159565
\(800\) −12.1457 −0.429417
\(801\) 0 0
\(802\) −22.3045 −0.787600
\(803\) 3.84843 0.135808
\(804\) 0 0
\(805\) 13.4990 0.475778
\(806\) 15.1145 0.532387
\(807\) 0 0
\(808\) −10.2933 −0.362118
\(809\) −7.30213 −0.256729 −0.128365 0.991727i \(-0.540973\pi\)
−0.128365 + 0.991727i \(0.540973\pi\)
\(810\) 0 0
\(811\) 5.58750 0.196204 0.0981019 0.995176i \(-0.468723\pi\)
0.0981019 + 0.995176i \(0.468723\pi\)
\(812\) 13.6158 0.477820
\(813\) 0 0
\(814\) 7.94478 0.278464
\(815\) 0.440268 0.0154219
\(816\) 0 0
\(817\) −50.1944 −1.75608
\(818\) −31.7998 −1.11186
\(819\) 0 0
\(820\) −0.839410 −0.0293135
\(821\) 48.2701 1.68464 0.842318 0.538980i \(-0.181190\pi\)
0.842318 + 0.538980i \(0.181190\pi\)
\(822\) 0 0
\(823\) −31.9312 −1.11305 −0.556526 0.830830i \(-0.687866\pi\)
−0.556526 + 0.830830i \(0.687866\pi\)
\(824\) −47.8282 −1.66617
\(825\) 0 0
\(826\) 81.8848 2.84914
\(827\) 45.8879 1.59568 0.797839 0.602871i \(-0.205977\pi\)
0.797839 + 0.602871i \(0.205977\pi\)
\(828\) 0 0
\(829\) −19.3612 −0.672441 −0.336221 0.941783i \(-0.609149\pi\)
−0.336221 + 0.941783i \(0.609149\pi\)
\(830\) 11.5333 0.400327
\(831\) 0 0
\(832\) 11.6151 0.402680
\(833\) 5.68322 0.196912
\(834\) 0 0
\(835\) 16.3997 0.567536
\(836\) −3.26397 −0.112887
\(837\) 0 0
\(838\) 10.4927 0.362463
\(839\) 1.48442 0.0512479 0.0256239 0.999672i \(-0.491843\pi\)
0.0256239 + 0.999672i \(0.491843\pi\)
\(840\) 0 0
\(841\) 17.5265 0.604363
\(842\) −53.6760 −1.84980
\(843\) 0 0
\(844\) −7.85700 −0.270449
\(845\) 5.40648 0.185989
\(846\) 0 0
\(847\) −4.15209 −0.142668
\(848\) −51.3341 −1.76282
\(849\) 0 0
\(850\) 3.98428 0.136660
\(851\) −24.6622 −0.845408
\(852\) 0 0
\(853\) −1.91635 −0.0656146 −0.0328073 0.999462i \(-0.510445\pi\)
−0.0328073 + 0.999462i \(0.510445\pi\)
\(854\) 6.53972 0.223785
\(855\) 0 0
\(856\) 4.62052 0.157926
\(857\) 37.2576 1.27270 0.636348 0.771402i \(-0.280444\pi\)
0.636348 + 0.771402i \(0.280444\pi\)
\(858\) 0 0
\(859\) −16.6063 −0.566600 −0.283300 0.959031i \(-0.591429\pi\)
−0.283300 + 0.959031i \(0.591429\pi\)
\(860\) −2.36348 −0.0805940
\(861\) 0 0
\(862\) −5.51231 −0.187750
\(863\) 8.66162 0.294845 0.147422 0.989074i \(-0.452902\pi\)
0.147422 + 0.989074i \(0.452902\pi\)
\(864\) 0 0
\(865\) 12.5620 0.427120
\(866\) −9.39052 −0.319103
\(867\) 0 0
\(868\) 8.68070 0.294642
\(869\) −11.5447 −0.391626
\(870\) 0 0
\(871\) −0.235139 −0.00796738
\(872\) −6.51856 −0.220746
\(873\) 0 0
\(874\) 52.2824 1.76848
\(875\) −26.3889 −0.892106
\(876\) 0 0
\(877\) −23.4656 −0.792378 −0.396189 0.918169i \(-0.629667\pi\)
−0.396189 + 0.918169i \(0.629667\pi\)
\(878\) −56.3463 −1.90160
\(879\) 0 0
\(880\) 3.14551 0.106035
\(881\) 5.76381 0.194188 0.0970938 0.995275i \(-0.469045\pi\)
0.0970938 + 0.995275i \(0.469045\pi\)
\(882\) 0 0
\(883\) 29.0682 0.978224 0.489112 0.872221i \(-0.337321\pi\)
0.489112 + 0.872221i \(0.337321\pi\)
\(884\) 0.588794 0.0198033
\(885\) 0 0
\(886\) −19.6139 −0.658943
\(887\) −35.6351 −1.19651 −0.598255 0.801306i \(-0.704139\pi\)
−0.598255 + 0.801306i \(0.704139\pi\)
\(888\) 0 0
\(889\) 1.02762 0.0344653
\(890\) −14.3778 −0.481946
\(891\) 0 0
\(892\) 3.39763 0.113761
\(893\) 55.1739 1.84633
\(894\) 0 0
\(895\) −6.04980 −0.202222
\(896\) 56.5515 1.88925
\(897\) 0 0
\(898\) −33.9288 −1.13222
\(899\) 29.6629 0.989313
\(900\) 0 0
\(901\) 6.02294 0.200653
\(902\) −4.13569 −0.137703
\(903\) 0 0
\(904\) −30.8796 −1.02704
\(905\) −8.93801 −0.297109
\(906\) 0 0
\(907\) 5.48853 0.182244 0.0911218 0.995840i \(-0.470955\pi\)
0.0911218 + 0.995840i \(0.470955\pi\)
\(908\) 13.5645 0.450154
\(909\) 0 0
\(910\) −9.59606 −0.318106
\(911\) −12.8974 −0.427310 −0.213655 0.976909i \(-0.568537\pi\)
−0.213655 + 0.976909i \(0.568537\pi\)
\(912\) 0 0
\(913\) 11.0120 0.364445
\(914\) −22.5195 −0.744880
\(915\) 0 0
\(916\) 13.3162 0.439980
\(917\) 42.1279 1.39119
\(918\) 0 0
\(919\) 49.0449 1.61784 0.808921 0.587918i \(-0.200052\pi\)
0.808921 + 0.587918i \(0.200052\pi\)
\(920\) −7.77956 −0.256485
\(921\) 0 0
\(922\) −26.3832 −0.868884
\(923\) −32.1773 −1.05913
\(924\) 0 0
\(925\) 22.9905 0.755922
\(926\) 7.79867 0.256280
\(927\) 0 0
\(928\) −18.1768 −0.596681
\(929\) 28.3566 0.930351 0.465176 0.885218i \(-0.345991\pi\)
0.465176 + 0.885218i \(0.345991\pi\)
\(930\) 0 0
\(931\) −69.5211 −2.27846
\(932\) −10.0247 −0.328370
\(933\) 0 0
\(934\) −21.5743 −0.705934
\(935\) −0.369057 −0.0120695
\(936\) 0 0
\(937\) 9.87929 0.322742 0.161371 0.986894i \(-0.448408\pi\)
0.161371 + 0.986894i \(0.448408\pi\)
\(938\) −0.696858 −0.0227532
\(939\) 0 0
\(940\) 2.59795 0.0847358
\(941\) 22.6523 0.738444 0.369222 0.929341i \(-0.379624\pi\)
0.369222 + 0.929341i \(0.379624\pi\)
\(942\) 0 0
\(943\) 12.8380 0.418063
\(944\) −59.2299 −1.92777
\(945\) 0 0
\(946\) −11.6446 −0.378599
\(947\) 31.5071 1.02384 0.511922 0.859032i \(-0.328934\pi\)
0.511922 + 0.859032i \(0.328934\pi\)
\(948\) 0 0
\(949\) 8.49226 0.275670
\(950\) −48.7385 −1.58128
\(951\) 0 0
\(952\) −5.51424 −0.178718
\(953\) −39.0222 −1.26405 −0.632027 0.774946i \(-0.717777\pi\)
−0.632027 + 0.774946i \(0.717777\pi\)
\(954\) 0 0
\(955\) 2.67601 0.0865935
\(956\) 13.5260 0.437462
\(957\) 0 0
\(958\) −1.06911 −0.0345414
\(959\) −29.1242 −0.940470
\(960\) 0 0
\(961\) −12.0885 −0.389951
\(962\) 17.5316 0.565241
\(963\) 0 0
\(964\) −3.14508 −0.101296
\(965\) −13.1729 −0.424049
\(966\) 0 0
\(967\) 22.1623 0.712693 0.356346 0.934354i \(-0.384022\pi\)
0.356346 + 0.934354i \(0.384022\pi\)
\(968\) 2.39287 0.0769098
\(969\) 0 0
\(970\) −8.82081 −0.283219
\(971\) −37.9123 −1.21666 −0.608331 0.793683i \(-0.708161\pi\)
−0.608331 + 0.793683i \(0.708161\pi\)
\(972\) 0 0
\(973\) −86.5610 −2.77502
\(974\) 55.0356 1.76345
\(975\) 0 0
\(976\) −4.73039 −0.151416
\(977\) 3.54195 0.113317 0.0566584 0.998394i \(-0.481955\pi\)
0.0566584 + 0.998394i \(0.481955\pi\)
\(978\) 0 0
\(979\) −13.7280 −0.438749
\(980\) −3.27351 −0.104568
\(981\) 0 0
\(982\) −4.79811 −0.153114
\(983\) −47.1388 −1.50349 −0.751747 0.659452i \(-0.770789\pi\)
−0.751747 + 0.659452i \(0.770789\pi\)
\(984\) 0 0
\(985\) 0.208147 0.00663211
\(986\) 5.96268 0.189890
\(987\) 0 0
\(988\) −7.20254 −0.229143
\(989\) 36.1472 1.14941
\(990\) 0 0
\(991\) 36.7061 1.16601 0.583004 0.812469i \(-0.301877\pi\)
0.583004 + 0.812469i \(0.301877\pi\)
\(992\) −11.5886 −0.367937
\(993\) 0 0
\(994\) −95.3607 −3.02466
\(995\) 16.2732 0.515895
\(996\) 0 0
\(997\) −28.5336 −0.903670 −0.451835 0.892102i \(-0.649230\pi\)
−0.451835 + 0.892102i \(0.649230\pi\)
\(998\) 26.6632 0.844010
\(999\) 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 6039.2.a.d.1.2 11
3.2 odd 2 2013.2.a.a.1.10 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.a.1.10 11 3.2 odd 2
6039.2.a.d.1.2 11 1.1 even 1 trivial