Properties

Label 1287.2.a.f.1.2
Level $1287$
Weight $2$
Character 1287.1
Self dual yes
Analytic conductor $10.277$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,2,Mod(1,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, 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("1287.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.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: \(10.2767467401\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1287.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.00000 q^{4} -0.732051 q^{5} -2.00000 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+1.73205 q^{2} +1.00000 q^{4} -0.732051 q^{5} -2.00000 q^{7} -1.73205 q^{8} -1.26795 q^{10} +1.00000 q^{11} -1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -6.73205 q^{17} -0.535898 q^{19} -0.732051 q^{20} +1.73205 q^{22} +2.00000 q^{23} -4.46410 q^{25} -1.73205 q^{26} -2.00000 q^{28} +4.19615 q^{29} -8.19615 q^{31} -5.19615 q^{32} -11.6603 q^{34} +1.46410 q^{35} -2.00000 q^{37} -0.928203 q^{38} +1.26795 q^{40} +5.46410 q^{41} -6.19615 q^{43} +1.00000 q^{44} +3.46410 q^{46} +1.46410 q^{47} -3.00000 q^{49} -7.73205 q^{50} -1.00000 q^{52} -2.00000 q^{53} -0.732051 q^{55} +3.46410 q^{56} +7.26795 q^{58} +13.4641 q^{59} +4.92820 q^{61} -14.1962 q^{62} +1.00000 q^{64} +0.732051 q^{65} -12.1962 q^{67} -6.73205 q^{68} +2.53590 q^{70} -6.92820 q^{71} +2.92820 q^{73} -3.46410 q^{74} -0.535898 q^{76} -2.00000 q^{77} -3.66025 q^{79} +3.66025 q^{80} +9.46410 q^{82} -13.8564 q^{83} +4.92820 q^{85} -10.7321 q^{86} -1.73205 q^{88} +15.6603 q^{89} +2.00000 q^{91} +2.00000 q^{92} +2.53590 q^{94} +0.392305 q^{95} +8.92820 q^{97} -5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{10} + 2 q^{11} - 2 q^{13} - 10 q^{16} - 10 q^{17} - 8 q^{19} + 2 q^{20} + 4 q^{23} - 2 q^{25} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 6 q^{34} - 4 q^{35} - 4 q^{37} + 12 q^{38} + 6 q^{40} + 4 q^{41} - 2 q^{43} + 2 q^{44} - 4 q^{47} - 6 q^{49} - 12 q^{50} - 2 q^{52} - 4 q^{53} + 2 q^{55} + 18 q^{58} + 20 q^{59} - 4 q^{61} - 18 q^{62} + 2 q^{64} - 2 q^{65} - 14 q^{67} - 10 q^{68} + 12 q^{70} - 8 q^{73} - 8 q^{76} - 4 q^{77} + 10 q^{79} - 10 q^{80} + 12 q^{82} - 4 q^{85} - 18 q^{86} + 14 q^{89} + 4 q^{91} + 4 q^{92} + 12 q^{94} - 20 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.732051 −0.327383 −0.163692 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) −1.26795 −0.400961
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −6.73205 −1.63276 −0.816381 0.577514i \(-0.804023\pi\)
−0.816381 + 0.577514i \(0.804023\pi\)
\(18\) 0 0
\(19\) −0.535898 −0.122944 −0.0614718 0.998109i \(-0.519579\pi\)
−0.0614718 + 0.998109i \(0.519579\pi\)
\(20\) −0.732051 −0.163692
\(21\) 0 0
\(22\) 1.73205 0.369274
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −4.46410 −0.892820
\(26\) −1.73205 −0.339683
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 4.19615 0.779206 0.389603 0.920983i \(-0.372612\pi\)
0.389603 + 0.920983i \(0.372612\pi\)
\(30\) 0 0
\(31\) −8.19615 −1.47207 −0.736036 0.676942i \(-0.763305\pi\)
−0.736036 + 0.676942i \(0.763305\pi\)
\(32\) −5.19615 −0.918559
\(33\) 0 0
\(34\) −11.6603 −1.99972
\(35\) 1.46410 0.247478
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −0.928203 −0.150574
\(39\) 0 0
\(40\) 1.26795 0.200480
\(41\) 5.46410 0.853349 0.426675 0.904405i \(-0.359685\pi\)
0.426675 + 0.904405i \(0.359685\pi\)
\(42\) 0 0
\(43\) −6.19615 −0.944904 −0.472452 0.881356i \(-0.656631\pi\)
−0.472452 + 0.881356i \(0.656631\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 3.46410 0.510754
\(47\) 1.46410 0.213561 0.106781 0.994283i \(-0.465946\pi\)
0.106781 + 0.994283i \(0.465946\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −7.73205 −1.09348
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) −0.732051 −0.0987097
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) 7.26795 0.954328
\(59\) 13.4641 1.75288 0.876438 0.481514i \(-0.159913\pi\)
0.876438 + 0.481514i \(0.159913\pi\)
\(60\) 0 0
\(61\) 4.92820 0.630992 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(62\) −14.1962 −1.80291
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.732051 0.0907997
\(66\) 0 0
\(67\) −12.1962 −1.49000 −0.744999 0.667066i \(-0.767550\pi\)
−0.744999 + 0.667066i \(0.767550\pi\)
\(68\) −6.73205 −0.816381
\(69\) 0 0
\(70\) 2.53590 0.303098
\(71\) −6.92820 −0.822226 −0.411113 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) 2.92820 0.342720 0.171360 0.985208i \(-0.445184\pi\)
0.171360 + 0.985208i \(0.445184\pi\)
\(74\) −3.46410 −0.402694
\(75\) 0 0
\(76\) −0.535898 −0.0614718
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −3.66025 −0.411811 −0.205905 0.978572i \(-0.566014\pi\)
−0.205905 + 0.978572i \(0.566014\pi\)
\(80\) 3.66025 0.409229
\(81\) 0 0
\(82\) 9.46410 1.04514
\(83\) −13.8564 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 0 0
\(85\) 4.92820 0.534539
\(86\) −10.7321 −1.15727
\(87\) 0 0
\(88\) −1.73205 −0.184637
\(89\) 15.6603 1.65998 0.829992 0.557776i \(-0.188345\pi\)
0.829992 + 0.557776i \(0.188345\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) 2.53590 0.261558
\(95\) 0.392305 0.0402496
\(96\) 0 0
\(97\) 8.92820 0.906522 0.453261 0.891378i \(-0.350261\pi\)
0.453261 + 0.891378i \(0.350261\pi\)
\(98\) −5.19615 −0.524891
\(99\) 0 0
\(100\) −4.46410 −0.446410
\(101\) −8.19615 −0.815548 −0.407774 0.913083i \(-0.633695\pi\)
−0.407774 + 0.913083i \(0.633695\pi\)
\(102\) 0 0
\(103\) 19.3205 1.90371 0.951853 0.306554i \(-0.0991762\pi\)
0.951853 + 0.306554i \(0.0991762\pi\)
\(104\) 1.73205 0.169842
\(105\) 0 0
\(106\) −3.46410 −0.336463
\(107\) 8.39230 0.811315 0.405657 0.914025i \(-0.367043\pi\)
0.405657 + 0.914025i \(0.367043\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) −1.26795 −0.120894
\(111\) 0 0
\(112\) 10.0000 0.944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −1.46410 −0.136528
\(116\) 4.19615 0.389603
\(117\) 0 0
\(118\) 23.3205 2.14683
\(119\) 13.4641 1.23425
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 8.53590 0.772804
\(123\) 0 0
\(124\) −8.19615 −0.736036
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 13.1244 1.16460 0.582299 0.812975i \(-0.302153\pi\)
0.582299 + 0.812975i \(0.302153\pi\)
\(128\) 12.1244 1.07165
\(129\) 0 0
\(130\) 1.26795 0.111207
\(131\) 5.07180 0.443125 0.221562 0.975146i \(-0.428884\pi\)
0.221562 + 0.975146i \(0.428884\pi\)
\(132\) 0 0
\(133\) 1.07180 0.0929366
\(134\) −21.1244 −1.82487
\(135\) 0 0
\(136\) 11.6603 0.999859
\(137\) 14.5885 1.24638 0.623188 0.782072i \(-0.285837\pi\)
0.623188 + 0.782072i \(0.285837\pi\)
\(138\) 0 0
\(139\) −9.80385 −0.831551 −0.415776 0.909467i \(-0.636490\pi\)
−0.415776 + 0.909467i \(0.636490\pi\)
\(140\) 1.46410 0.123739
\(141\) 0 0
\(142\) −12.0000 −1.00702
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −3.07180 −0.255099
\(146\) 5.07180 0.419745
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −8.00000 −0.655386 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 0 0
\(151\) −4.53590 −0.369126 −0.184563 0.982821i \(-0.559087\pi\)
−0.184563 + 0.982821i \(0.559087\pi\)
\(152\) 0.928203 0.0752872
\(153\) 0 0
\(154\) −3.46410 −0.279145
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −15.3205 −1.22271 −0.611355 0.791357i \(-0.709375\pi\)
−0.611355 + 0.791357i \(0.709375\pi\)
\(158\) −6.33975 −0.504363
\(159\) 0 0
\(160\) 3.80385 0.300721
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −17.2679 −1.35253 −0.676265 0.736658i \(-0.736403\pi\)
−0.676265 + 0.736658i \(0.736403\pi\)
\(164\) 5.46410 0.426675
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) 9.85641 0.762712 0.381356 0.924428i \(-0.375457\pi\)
0.381356 + 0.924428i \(0.375457\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 8.53590 0.654674
\(171\) 0 0
\(172\) −6.19615 −0.472452
\(173\) −19.1244 −1.45400 −0.726999 0.686639i \(-0.759085\pi\)
−0.726999 + 0.686639i \(0.759085\pi\)
\(174\) 0 0
\(175\) 8.92820 0.674909
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) 27.1244 2.03306
\(179\) −7.85641 −0.587215 −0.293608 0.955926i \(-0.594856\pi\)
−0.293608 + 0.955926i \(0.594856\pi\)
\(180\) 0 0
\(181\) 8.39230 0.623795 0.311898 0.950116i \(-0.399035\pi\)
0.311898 + 0.950116i \(0.399035\pi\)
\(182\) 3.46410 0.256776
\(183\) 0 0
\(184\) −3.46410 −0.255377
\(185\) 1.46410 0.107643
\(186\) 0 0
\(187\) −6.73205 −0.492296
\(188\) 1.46410 0.106781
\(189\) 0 0
\(190\) 0.679492 0.0492955
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) −6.92820 −0.498703 −0.249351 0.968413i \(-0.580217\pi\)
−0.249351 + 0.968413i \(0.580217\pi\)
\(194\) 15.4641 1.11026
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −8.00000 −0.569976 −0.284988 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) 0 0
\(199\) 10.5359 0.746870 0.373435 0.927656i \(-0.378180\pi\)
0.373435 + 0.927656i \(0.378180\pi\)
\(200\) 7.73205 0.546739
\(201\) 0 0
\(202\) −14.1962 −0.998838
\(203\) −8.39230 −0.589024
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 33.4641 2.33155
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) −0.535898 −0.0370689
\(210\) 0 0
\(211\) 19.2679 1.32646 0.663230 0.748415i \(-0.269185\pi\)
0.663230 + 0.748415i \(0.269185\pi\)
\(212\) −2.00000 −0.137361
\(213\) 0 0
\(214\) 14.5359 0.993654
\(215\) 4.53590 0.309346
\(216\) 0 0
\(217\) 16.3923 1.11278
\(218\) −13.8564 −0.938474
\(219\) 0 0
\(220\) −0.732051 −0.0493549
\(221\) 6.73205 0.452847
\(222\) 0 0
\(223\) −23.5167 −1.57479 −0.787396 0.616447i \(-0.788571\pi\)
−0.787396 + 0.616447i \(0.788571\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) −17.3205 −1.15214
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 0 0
\(229\) −15.4641 −1.02190 −0.510948 0.859611i \(-0.670706\pi\)
−0.510948 + 0.859611i \(0.670706\pi\)
\(230\) −2.53590 −0.167212
\(231\) 0 0
\(232\) −7.26795 −0.477164
\(233\) −4.19615 −0.274899 −0.137450 0.990509i \(-0.543890\pi\)
−0.137450 + 0.990509i \(0.543890\pi\)
\(234\) 0 0
\(235\) −1.07180 −0.0699163
\(236\) 13.4641 0.876438
\(237\) 0 0
\(238\) 23.3205 1.51164
\(239\) −22.3923 −1.44844 −0.724219 0.689570i \(-0.757799\pi\)
−0.724219 + 0.689570i \(0.757799\pi\)
\(240\) 0 0
\(241\) −18.7846 −1.21002 −0.605012 0.796217i \(-0.706832\pi\)
−0.605012 + 0.796217i \(0.706832\pi\)
\(242\) 1.73205 0.111340
\(243\) 0 0
\(244\) 4.92820 0.315496
\(245\) 2.19615 0.140307
\(246\) 0 0
\(247\) 0.535898 0.0340984
\(248\) 14.1962 0.901457
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 17.0718 1.07756 0.538781 0.842446i \(-0.318885\pi\)
0.538781 + 0.842446i \(0.318885\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 22.7321 1.42634
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −17.3205 −1.08042 −0.540212 0.841529i \(-0.681656\pi\)
−0.540212 + 0.841529i \(0.681656\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0.732051 0.0453999
\(261\) 0 0
\(262\) 8.78461 0.542715
\(263\) −9.46410 −0.583582 −0.291791 0.956482i \(-0.594251\pi\)
−0.291791 + 0.956482i \(0.594251\pi\)
\(264\) 0 0
\(265\) 1.46410 0.0899390
\(266\) 1.85641 0.113824
\(267\) 0 0
\(268\) −12.1962 −0.744999
\(269\) 17.3205 1.05605 0.528025 0.849229i \(-0.322933\pi\)
0.528025 + 0.849229i \(0.322933\pi\)
\(270\) 0 0
\(271\) 9.32051 0.566181 0.283090 0.959093i \(-0.408640\pi\)
0.283090 + 0.959093i \(0.408640\pi\)
\(272\) 33.6603 2.04095
\(273\) 0 0
\(274\) 25.2679 1.52649
\(275\) −4.46410 −0.269195
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −16.9808 −1.01844
\(279\) 0 0
\(280\) −2.53590 −0.151549
\(281\) −23.3205 −1.39118 −0.695592 0.718437i \(-0.744858\pi\)
−0.695592 + 0.718437i \(0.744858\pi\)
\(282\) 0 0
\(283\) 9.80385 0.582778 0.291389 0.956605i \(-0.405883\pi\)
0.291389 + 0.956605i \(0.405883\pi\)
\(284\) −6.92820 −0.411113
\(285\) 0 0
\(286\) −1.73205 −0.102418
\(287\) −10.9282 −0.645071
\(288\) 0 0
\(289\) 28.3205 1.66591
\(290\) −5.32051 −0.312431
\(291\) 0 0
\(292\) 2.92820 0.171360
\(293\) −13.0718 −0.763663 −0.381831 0.924232i \(-0.624706\pi\)
−0.381831 + 0.924232i \(0.624706\pi\)
\(294\) 0 0
\(295\) −9.85641 −0.573862
\(296\) 3.46410 0.201347
\(297\) 0 0
\(298\) −13.8564 −0.802680
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 12.3923 0.714281
\(302\) −7.85641 −0.452085
\(303\) 0 0
\(304\) 2.67949 0.153679
\(305\) −3.60770 −0.206576
\(306\) 0 0
\(307\) −20.5359 −1.17205 −0.586023 0.810295i \(-0.699307\pi\)
−0.586023 + 0.810295i \(0.699307\pi\)
\(308\) −2.00000 −0.113961
\(309\) 0 0
\(310\) 10.3923 0.590243
\(311\) 3.07180 0.174186 0.0870928 0.996200i \(-0.472242\pi\)
0.0870928 + 0.996200i \(0.472242\pi\)
\(312\) 0 0
\(313\) −9.46410 −0.534943 −0.267471 0.963566i \(-0.586188\pi\)
−0.267471 + 0.963566i \(0.586188\pi\)
\(314\) −26.5359 −1.49751
\(315\) 0 0
\(316\) −3.66025 −0.205905
\(317\) −17.1244 −0.961800 −0.480900 0.876776i \(-0.659690\pi\)
−0.480900 + 0.876776i \(0.659690\pi\)
\(318\) 0 0
\(319\) 4.19615 0.234939
\(320\) −0.732051 −0.0409229
\(321\) 0 0
\(322\) −6.92820 −0.386094
\(323\) 3.60770 0.200738
\(324\) 0 0
\(325\) 4.46410 0.247624
\(326\) −29.9090 −1.65650
\(327\) 0 0
\(328\) −9.46410 −0.522568
\(329\) −2.92820 −0.161437
\(330\) 0 0
\(331\) 23.5167 1.29259 0.646296 0.763087i \(-0.276317\pi\)
0.646296 + 0.763087i \(0.276317\pi\)
\(332\) −13.8564 −0.760469
\(333\) 0 0
\(334\) 17.0718 0.934127
\(335\) 8.92820 0.487800
\(336\) 0 0
\(337\) 9.32051 0.507720 0.253860 0.967241i \(-0.418300\pi\)
0.253860 + 0.967241i \(0.418300\pi\)
\(338\) 1.73205 0.0942111
\(339\) 0 0
\(340\) 4.92820 0.267269
\(341\) −8.19615 −0.443847
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 10.7321 0.578633
\(345\) 0 0
\(346\) −33.1244 −1.78078
\(347\) 4.39230 0.235791 0.117896 0.993026i \(-0.462385\pi\)
0.117896 + 0.993026i \(0.462385\pi\)
\(348\) 0 0
\(349\) −3.85641 −0.206429 −0.103214 0.994659i \(-0.532913\pi\)
−0.103214 + 0.994659i \(0.532913\pi\)
\(350\) 15.4641 0.826591
\(351\) 0 0
\(352\) −5.19615 −0.276956
\(353\) −22.9808 −1.22314 −0.611571 0.791189i \(-0.709462\pi\)
−0.611571 + 0.791189i \(0.709462\pi\)
\(354\) 0 0
\(355\) 5.07180 0.269183
\(356\) 15.6603 0.829992
\(357\) 0 0
\(358\) −13.6077 −0.719189
\(359\) 16.5359 0.872731 0.436366 0.899769i \(-0.356265\pi\)
0.436366 + 0.899769i \(0.356265\pi\)
\(360\) 0 0
\(361\) −18.7128 −0.984885
\(362\) 14.5359 0.763990
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −2.14359 −0.112201
\(366\) 0 0
\(367\) 30.9282 1.61444 0.807220 0.590251i \(-0.200971\pi\)
0.807220 + 0.590251i \(0.200971\pi\)
\(368\) −10.0000 −0.521286
\(369\) 0 0
\(370\) 2.53590 0.131835
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) −16.2487 −0.841326 −0.420663 0.907217i \(-0.638203\pi\)
−0.420663 + 0.907217i \(0.638203\pi\)
\(374\) −11.6603 −0.602937
\(375\) 0 0
\(376\) −2.53590 −0.130779
\(377\) −4.19615 −0.216113
\(378\) 0 0
\(379\) 12.1962 0.626474 0.313237 0.949675i \(-0.398587\pi\)
0.313237 + 0.949675i \(0.398587\pi\)
\(380\) 0.392305 0.0201248
\(381\) 0 0
\(382\) 32.7846 1.67741
\(383\) −24.3923 −1.24639 −0.623194 0.782067i \(-0.714165\pi\)
−0.623194 + 0.782067i \(0.714165\pi\)
\(384\) 0 0
\(385\) 1.46410 0.0746175
\(386\) −12.0000 −0.610784
\(387\) 0 0
\(388\) 8.92820 0.453261
\(389\) −8.53590 −0.432787 −0.216394 0.976306i \(-0.569429\pi\)
−0.216394 + 0.976306i \(0.569429\pi\)
\(390\) 0 0
\(391\) −13.4641 −0.680909
\(392\) 5.19615 0.262445
\(393\) 0 0
\(394\) −13.8564 −0.698076
\(395\) 2.67949 0.134820
\(396\) 0 0
\(397\) 24.9282 1.25111 0.625555 0.780180i \(-0.284872\pi\)
0.625555 + 0.780180i \(0.284872\pi\)
\(398\) 18.2487 0.914725
\(399\) 0 0
\(400\) 22.3205 1.11603
\(401\) −26.1962 −1.30817 −0.654087 0.756420i \(-0.726947\pi\)
−0.654087 + 0.756420i \(0.726947\pi\)
\(402\) 0 0
\(403\) 8.19615 0.408279
\(404\) −8.19615 −0.407774
\(405\) 0 0
\(406\) −14.5359 −0.721405
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) −27.7128 −1.37031 −0.685155 0.728397i \(-0.740266\pi\)
−0.685155 + 0.728397i \(0.740266\pi\)
\(410\) −6.92820 −0.342160
\(411\) 0 0
\(412\) 19.3205 0.951853
\(413\) −26.9282 −1.32505
\(414\) 0 0
\(415\) 10.1436 0.497929
\(416\) 5.19615 0.254762
\(417\) 0 0
\(418\) −0.928203 −0.0453999
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 8.92820 0.435134 0.217567 0.976045i \(-0.430188\pi\)
0.217567 + 0.976045i \(0.430188\pi\)
\(422\) 33.3731 1.62458
\(423\) 0 0
\(424\) 3.46410 0.168232
\(425\) 30.0526 1.45776
\(426\) 0 0
\(427\) −9.85641 −0.476985
\(428\) 8.39230 0.405657
\(429\) 0 0
\(430\) 7.85641 0.378870
\(431\) 12.7846 0.615813 0.307906 0.951417i \(-0.400372\pi\)
0.307906 + 0.951417i \(0.400372\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 28.3923 1.36287
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −1.07180 −0.0512710
\(438\) 0 0
\(439\) −29.9090 −1.42748 −0.713739 0.700412i \(-0.753000\pi\)
−0.713739 + 0.700412i \(0.753000\pi\)
\(440\) 1.26795 0.0604471
\(441\) 0 0
\(442\) 11.6603 0.554622
\(443\) 4.78461 0.227324 0.113662 0.993519i \(-0.463742\pi\)
0.113662 + 0.993519i \(0.463742\pi\)
\(444\) 0 0
\(445\) −11.4641 −0.543451
\(446\) −40.7321 −1.92872
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 2.19615 0.103643 0.0518214 0.998656i \(-0.483497\pi\)
0.0518214 + 0.998656i \(0.483497\pi\)
\(450\) 0 0
\(451\) 5.46410 0.257294
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) 18.0000 0.844782
\(455\) −1.46410 −0.0686381
\(456\) 0 0
\(457\) −30.7846 −1.44004 −0.720022 0.693952i \(-0.755868\pi\)
−0.720022 + 0.693952i \(0.755868\pi\)
\(458\) −26.7846 −1.25156
\(459\) 0 0
\(460\) −1.46410 −0.0682641
\(461\) 17.4641 0.813384 0.406692 0.913565i \(-0.366682\pi\)
0.406692 + 0.913565i \(0.366682\pi\)
\(462\) 0 0
\(463\) 19.5167 0.907016 0.453508 0.891252i \(-0.350172\pi\)
0.453508 + 0.891252i \(0.350172\pi\)
\(464\) −20.9808 −0.974007
\(465\) 0 0
\(466\) −7.26795 −0.336681
\(467\) 32.9282 1.52374 0.761868 0.647733i \(-0.224283\pi\)
0.761868 + 0.647733i \(0.224283\pi\)
\(468\) 0 0
\(469\) 24.3923 1.12633
\(470\) −1.85641 −0.0856296
\(471\) 0 0
\(472\) −23.3205 −1.07341
\(473\) −6.19615 −0.284899
\(474\) 0 0
\(475\) 2.39230 0.109766
\(476\) 13.4641 0.617126
\(477\) 0 0
\(478\) −38.7846 −1.77397
\(479\) −20.5359 −0.938309 −0.469155 0.883116i \(-0.655441\pi\)
−0.469155 + 0.883116i \(0.655441\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −32.5359 −1.48197
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −6.53590 −0.296780
\(486\) 0 0
\(487\) −25.6603 −1.16278 −0.581389 0.813626i \(-0.697490\pi\)
−0.581389 + 0.813626i \(0.697490\pi\)
\(488\) −8.53590 −0.386402
\(489\) 0 0
\(490\) 3.80385 0.171840
\(491\) −9.46410 −0.427109 −0.213554 0.976931i \(-0.568504\pi\)
−0.213554 + 0.976931i \(0.568504\pi\)
\(492\) 0 0
\(493\) −28.2487 −1.27226
\(494\) 0.928203 0.0417618
\(495\) 0 0
\(496\) 40.9808 1.84009
\(497\) 13.8564 0.621545
\(498\) 0 0
\(499\) −28.1962 −1.26223 −0.631117 0.775688i \(-0.717403\pi\)
−0.631117 + 0.775688i \(0.717403\pi\)
\(500\) 6.92820 0.309839
\(501\) 0 0
\(502\) 29.5692 1.31974
\(503\) 14.5359 0.648124 0.324062 0.946036i \(-0.394951\pi\)
0.324062 + 0.946036i \(0.394951\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 3.46410 0.153998
\(507\) 0 0
\(508\) 13.1244 0.582299
\(509\) 24.3397 1.07884 0.539420 0.842037i \(-0.318643\pi\)
0.539420 + 0.842037i \(0.318643\pi\)
\(510\) 0 0
\(511\) −5.85641 −0.259072
\(512\) 8.66025 0.382733
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) −14.1436 −0.623241
\(516\) 0 0
\(517\) 1.46410 0.0643911
\(518\) 6.92820 0.304408
\(519\) 0 0
\(520\) −1.26795 −0.0556033
\(521\) 27.0718 1.18604 0.593018 0.805189i \(-0.297936\pi\)
0.593018 + 0.805189i \(0.297936\pi\)
\(522\) 0 0
\(523\) −24.7321 −1.08146 −0.540729 0.841197i \(-0.681851\pi\)
−0.540729 + 0.841197i \(0.681851\pi\)
\(524\) 5.07180 0.221562
\(525\) 0 0
\(526\) −16.3923 −0.714738
\(527\) 55.1769 2.40354
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 2.53590 0.110152
\(531\) 0 0
\(532\) 1.07180 0.0464683
\(533\) −5.46410 −0.236677
\(534\) 0 0
\(535\) −6.14359 −0.265611
\(536\) 21.1244 0.912433
\(537\) 0 0
\(538\) 30.0000 1.29339
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 26.7846 1.15156 0.575780 0.817605i \(-0.304698\pi\)
0.575780 + 0.817605i \(0.304698\pi\)
\(542\) 16.1436 0.693427
\(543\) 0 0
\(544\) 34.9808 1.49979
\(545\) 5.85641 0.250861
\(546\) 0 0
\(547\) −38.9808 −1.66670 −0.833349 0.552748i \(-0.813579\pi\)
−0.833349 + 0.552748i \(0.813579\pi\)
\(548\) 14.5885 0.623188
\(549\) 0 0
\(550\) −7.73205 −0.329696
\(551\) −2.24871 −0.0957983
\(552\) 0 0
\(553\) 7.32051 0.311300
\(554\) −17.3205 −0.735878
\(555\) 0 0
\(556\) −9.80385 −0.415776
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 6.19615 0.262069
\(560\) −7.32051 −0.309348
\(561\) 0 0
\(562\) −40.3923 −1.70385
\(563\) −15.7128 −0.662216 −0.331108 0.943593i \(-0.607422\pi\)
−0.331108 + 0.943593i \(0.607422\pi\)
\(564\) 0 0
\(565\) 7.32051 0.307976
\(566\) 16.9808 0.713755
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 1.66025 0.0696015 0.0348007 0.999394i \(-0.488920\pi\)
0.0348007 + 0.999394i \(0.488920\pi\)
\(570\) 0 0
\(571\) −14.5885 −0.610508 −0.305254 0.952271i \(-0.598741\pi\)
−0.305254 + 0.952271i \(0.598741\pi\)
\(572\) −1.00000 −0.0418121
\(573\) 0 0
\(574\) −18.9282 −0.790048
\(575\) −8.92820 −0.372332
\(576\) 0 0
\(577\) 4.92820 0.205164 0.102582 0.994725i \(-0.467290\pi\)
0.102582 + 0.994725i \(0.467290\pi\)
\(578\) 49.0526 2.04032
\(579\) 0 0
\(580\) −3.07180 −0.127549
\(581\) 27.7128 1.14972
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) −5.07180 −0.209872
\(585\) 0 0
\(586\) −22.6410 −0.935292
\(587\) −9.46410 −0.390625 −0.195313 0.980741i \(-0.562572\pi\)
−0.195313 + 0.980741i \(0.562572\pi\)
\(588\) 0 0
\(589\) 4.39230 0.180982
\(590\) −17.0718 −0.702835
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −9.85641 −0.404073
\(596\) −8.00000 −0.327693
\(597\) 0 0
\(598\) −3.46410 −0.141658
\(599\) −27.7128 −1.13231 −0.566157 0.824297i \(-0.691571\pi\)
−0.566157 + 0.824297i \(0.691571\pi\)
\(600\) 0 0
\(601\) 27.8564 1.13629 0.568143 0.822930i \(-0.307662\pi\)
0.568143 + 0.822930i \(0.307662\pi\)
\(602\) 21.4641 0.874811
\(603\) 0 0
\(604\) −4.53590 −0.184563
\(605\) −0.732051 −0.0297621
\(606\) 0 0
\(607\) 34.1962 1.38798 0.693990 0.719985i \(-0.255851\pi\)
0.693990 + 0.719985i \(0.255851\pi\)
\(608\) 2.78461 0.112931
\(609\) 0 0
\(610\) −6.24871 −0.253003
\(611\) −1.46410 −0.0592312
\(612\) 0 0
\(613\) 17.8564 0.721213 0.360607 0.932718i \(-0.382570\pi\)
0.360607 + 0.932718i \(0.382570\pi\)
\(614\) −35.5692 −1.43546
\(615\) 0 0
\(616\) 3.46410 0.139573
\(617\) −23.6603 −0.952526 −0.476263 0.879303i \(-0.658009\pi\)
−0.476263 + 0.879303i \(0.658009\pi\)
\(618\) 0 0
\(619\) −28.5885 −1.14907 −0.574534 0.818481i \(-0.694817\pi\)
−0.574534 + 0.818481i \(0.694817\pi\)
\(620\) 6.00000 0.240966
\(621\) 0 0
\(622\) 5.32051 0.213333
\(623\) −31.3205 −1.25483
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) −16.3923 −0.655168
\(627\) 0 0
\(628\) −15.3205 −0.611355
\(629\) 13.4641 0.536849
\(630\) 0 0
\(631\) −6.33975 −0.252381 −0.126191 0.992006i \(-0.540275\pi\)
−0.126191 + 0.992006i \(0.540275\pi\)
\(632\) 6.33975 0.252182
\(633\) 0 0
\(634\) −29.6603 −1.17796
\(635\) −9.60770 −0.381270
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 7.26795 0.287741
\(639\) 0 0
\(640\) −8.87564 −0.350841
\(641\) −19.8564 −0.784281 −0.392140 0.919905i \(-0.628265\pi\)
−0.392140 + 0.919905i \(0.628265\pi\)
\(642\) 0 0
\(643\) 30.0526 1.18516 0.592579 0.805513i \(-0.298110\pi\)
0.592579 + 0.805513i \(0.298110\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 6.24871 0.245852
\(647\) −5.07180 −0.199393 −0.0996965 0.995018i \(-0.531787\pi\)
−0.0996965 + 0.995018i \(0.531787\pi\)
\(648\) 0 0
\(649\) 13.4641 0.528512
\(650\) 7.73205 0.303276
\(651\) 0 0
\(652\) −17.2679 −0.676265
\(653\) −37.3205 −1.46046 −0.730232 0.683199i \(-0.760588\pi\)
−0.730232 + 0.683199i \(0.760588\pi\)
\(654\) 0 0
\(655\) −3.71281 −0.145072
\(656\) −27.3205 −1.06669
\(657\) 0 0
\(658\) −5.07180 −0.197719
\(659\) 25.0718 0.976659 0.488329 0.872659i \(-0.337607\pi\)
0.488329 + 0.872659i \(0.337607\pi\)
\(660\) 0 0
\(661\) −4.53590 −0.176426 −0.0882130 0.996102i \(-0.528116\pi\)
−0.0882130 + 0.996102i \(0.528116\pi\)
\(662\) 40.7321 1.58310
\(663\) 0 0
\(664\) 24.0000 0.931381
\(665\) −0.784610 −0.0304259
\(666\) 0 0
\(667\) 8.39230 0.324951
\(668\) 9.85641 0.381356
\(669\) 0 0
\(670\) 15.4641 0.597430
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) −23.4641 −0.904475 −0.452237 0.891898i \(-0.649374\pi\)
−0.452237 + 0.891898i \(0.649374\pi\)
\(674\) 16.1436 0.621828
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 40.3013 1.54890 0.774452 0.632633i \(-0.218026\pi\)
0.774452 + 0.632633i \(0.218026\pi\)
\(678\) 0 0
\(679\) −17.8564 −0.685266
\(680\) −8.53590 −0.327337
\(681\) 0 0
\(682\) −14.1962 −0.543599
\(683\) 24.7846 0.948357 0.474178 0.880429i \(-0.342745\pi\)
0.474178 + 0.880429i \(0.342745\pi\)
\(684\) 0 0
\(685\) −10.6795 −0.408042
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) 30.9808 1.18113
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 26.4449 1.00601 0.503005 0.864283i \(-0.332228\pi\)
0.503005 + 0.864283i \(0.332228\pi\)
\(692\) −19.1244 −0.726999
\(693\) 0 0
\(694\) 7.60770 0.288784
\(695\) 7.17691 0.272236
\(696\) 0 0
\(697\) −36.7846 −1.39332
\(698\) −6.67949 −0.252822
\(699\) 0 0
\(700\) 8.92820 0.337454
\(701\) 5.94744 0.224632 0.112316 0.993673i \(-0.464173\pi\)
0.112316 + 0.993673i \(0.464173\pi\)
\(702\) 0 0
\(703\) 1.07180 0.0404236
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −39.8038 −1.49804
\(707\) 16.3923 0.616496
\(708\) 0 0
\(709\) −4.53590 −0.170349 −0.0851746 0.996366i \(-0.527145\pi\)
−0.0851746 + 0.996366i \(0.527145\pi\)
\(710\) 8.78461 0.329681
\(711\) 0 0
\(712\) −27.1244 −1.01653
\(713\) −16.3923 −0.613897
\(714\) 0 0
\(715\) 0.732051 0.0273771
\(716\) −7.85641 −0.293608
\(717\) 0 0
\(718\) 28.6410 1.06887
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −38.6410 −1.43907
\(722\) −32.4115 −1.20623
\(723\) 0 0
\(724\) 8.39230 0.311898
\(725\) −18.7321 −0.695691
\(726\) 0 0
\(727\) 39.3205 1.45832 0.729158 0.684345i \(-0.239912\pi\)
0.729158 + 0.684345i \(0.239912\pi\)
\(728\) −3.46410 −0.128388
\(729\) 0 0
\(730\) −3.71281 −0.137417
\(731\) 41.7128 1.54280
\(732\) 0 0
\(733\) −22.9282 −0.846873 −0.423436 0.905926i \(-0.639176\pi\)
−0.423436 + 0.905926i \(0.639176\pi\)
\(734\) 53.5692 1.97728
\(735\) 0 0
\(736\) −10.3923 −0.383065
\(737\) −12.1962 −0.449251
\(738\) 0 0
\(739\) −2.00000 −0.0735712 −0.0367856 0.999323i \(-0.511712\pi\)
−0.0367856 + 0.999323i \(0.511712\pi\)
\(740\) 1.46410 0.0538214
\(741\) 0 0
\(742\) 6.92820 0.254342
\(743\) −37.8564 −1.38882 −0.694408 0.719581i \(-0.744334\pi\)
−0.694408 + 0.719581i \(0.744334\pi\)
\(744\) 0 0
\(745\) 5.85641 0.214562
\(746\) −28.1436 −1.03041
\(747\) 0 0
\(748\) −6.73205 −0.246148
\(749\) −16.7846 −0.613296
\(750\) 0 0
\(751\) −3.21539 −0.117331 −0.0586656 0.998278i \(-0.518685\pi\)
−0.0586656 + 0.998278i \(0.518685\pi\)
\(752\) −7.32051 −0.266951
\(753\) 0 0
\(754\) −7.26795 −0.264683
\(755\) 3.32051 0.120846
\(756\) 0 0
\(757\) −9.46410 −0.343979 −0.171989 0.985099i \(-0.555019\pi\)
−0.171989 + 0.985099i \(0.555019\pi\)
\(758\) 21.1244 0.767271
\(759\) 0 0
\(760\) −0.679492 −0.0246478
\(761\) −54.6410 −1.98074 −0.990368 0.138463i \(-0.955784\pi\)
−0.990368 + 0.138463i \(0.955784\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 18.9282 0.684798
\(765\) 0 0
\(766\) −42.2487 −1.52651
\(767\) −13.4641 −0.486161
\(768\) 0 0
\(769\) 16.1436 0.582153 0.291076 0.956700i \(-0.405987\pi\)
0.291076 + 0.956700i \(0.405987\pi\)
\(770\) 2.53590 0.0913874
\(771\) 0 0
\(772\) −6.92820 −0.249351
\(773\) 49.5167 1.78099 0.890495 0.454993i \(-0.150358\pi\)
0.890495 + 0.454993i \(0.150358\pi\)
\(774\) 0 0
\(775\) 36.5885 1.31430
\(776\) −15.4641 −0.555129
\(777\) 0 0
\(778\) −14.7846 −0.530054
\(779\) −2.92820 −0.104914
\(780\) 0 0
\(781\) −6.92820 −0.247911
\(782\) −23.3205 −0.833940
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) 11.2154 0.400294
\(786\) 0 0
\(787\) −36.9282 −1.31635 −0.658174 0.752866i \(-0.728671\pi\)
−0.658174 + 0.752866i \(0.728671\pi\)
\(788\) −8.00000 −0.284988
\(789\) 0 0
\(790\) 4.64102 0.165120
\(791\) 20.0000 0.711118
\(792\) 0 0
\(793\) −4.92820 −0.175006
\(794\) 43.1769 1.53229
\(795\) 0 0
\(796\) 10.5359 0.373435
\(797\) 50.3923 1.78499 0.892494 0.451060i \(-0.148954\pi\)
0.892494 + 0.451060i \(0.148954\pi\)
\(798\) 0 0
\(799\) −9.85641 −0.348695
\(800\) 23.1962 0.820108
\(801\) 0 0
\(802\) −45.3731 −1.60218
\(803\) 2.92820 0.103334
\(804\) 0 0
\(805\) 2.92820 0.103206
\(806\) 14.1962 0.500038
\(807\) 0 0
\(808\) 14.1962 0.499419
\(809\) −42.8372 −1.50607 −0.753037 0.657978i \(-0.771412\pi\)
−0.753037 + 0.657978i \(0.771412\pi\)
\(810\) 0 0
\(811\) 22.3923 0.786300 0.393150 0.919474i \(-0.371385\pi\)
0.393150 + 0.919474i \(0.371385\pi\)
\(812\) −8.39230 −0.294512
\(813\) 0 0
\(814\) −3.46410 −0.121417
\(815\) 12.6410 0.442795
\(816\) 0 0
\(817\) 3.32051 0.116170
\(818\) −48.0000 −1.67828
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 12.6795 0.442517 0.221259 0.975215i \(-0.428983\pi\)
0.221259 + 0.975215i \(0.428983\pi\)
\(822\) 0 0
\(823\) 11.6077 0.404619 0.202309 0.979322i \(-0.435155\pi\)
0.202309 + 0.979322i \(0.435155\pi\)
\(824\) −33.4641 −1.16578
\(825\) 0 0
\(826\) −46.6410 −1.62285
\(827\) −20.2487 −0.704117 −0.352058 0.935978i \(-0.614518\pi\)
−0.352058 + 0.935978i \(0.614518\pi\)
\(828\) 0 0
\(829\) −47.8564 −1.66212 −0.831061 0.556182i \(-0.812266\pi\)
−0.831061 + 0.556182i \(0.812266\pi\)
\(830\) 17.5692 0.609837
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 20.1962 0.699755
\(834\) 0 0
\(835\) −7.21539 −0.249699
\(836\) −0.535898 −0.0185344
\(837\) 0 0
\(838\) −45.0333 −1.55565
\(839\) −23.7128 −0.818657 −0.409329 0.912387i \(-0.634237\pi\)
−0.409329 + 0.912387i \(0.634237\pi\)
\(840\) 0 0
\(841\) −11.3923 −0.392838
\(842\) 15.4641 0.532928
\(843\) 0 0
\(844\) 19.2679 0.663230
\(845\) −0.732051 −0.0251833
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 10.0000 0.343401
\(849\) 0 0
\(850\) 52.0526 1.78539
\(851\) −4.00000 −0.137118
\(852\) 0 0
\(853\) −12.1436 −0.415789 −0.207894 0.978151i \(-0.566661\pi\)
−0.207894 + 0.978151i \(0.566661\pi\)
\(854\) −17.0718 −0.584185
\(855\) 0 0
\(856\) −14.5359 −0.496827
\(857\) −31.8038 −1.08640 −0.543199 0.839604i \(-0.682787\pi\)
−0.543199 + 0.839604i \(0.682787\pi\)
\(858\) 0 0
\(859\) 49.9615 1.70467 0.852333 0.523000i \(-0.175187\pi\)
0.852333 + 0.523000i \(0.175187\pi\)
\(860\) 4.53590 0.154673
\(861\) 0 0
\(862\) 22.1436 0.754214
\(863\) −11.3205 −0.385355 −0.192677 0.981262i \(-0.561717\pi\)
−0.192677 + 0.981262i \(0.561717\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) −10.3923 −0.353145
\(867\) 0 0
\(868\) 16.3923 0.556391
\(869\) −3.66025 −0.124166
\(870\) 0 0
\(871\) 12.1962 0.413251
\(872\) 13.8564 0.469237
\(873\) 0 0
\(874\) −1.85641 −0.0627939
\(875\) −13.8564 −0.468432
\(876\) 0 0
\(877\) 15.8564 0.535433 0.267716 0.963498i \(-0.413731\pi\)
0.267716 + 0.963498i \(0.413731\pi\)
\(878\) −51.8038 −1.74830
\(879\) 0 0
\(880\) 3.66025 0.123387
\(881\) 28.6410 0.964940 0.482470 0.875912i \(-0.339740\pi\)
0.482470 + 0.875912i \(0.339740\pi\)
\(882\) 0 0
\(883\) −2.24871 −0.0756752 −0.0378376 0.999284i \(-0.512047\pi\)
−0.0378376 + 0.999284i \(0.512047\pi\)
\(884\) 6.73205 0.226423
\(885\) 0 0
\(886\) 8.28719 0.278413
\(887\) 58.6410 1.96897 0.984486 0.175461i \(-0.0561417\pi\)
0.984486 + 0.175461i \(0.0561417\pi\)
\(888\) 0 0
\(889\) −26.2487 −0.880354
\(890\) −19.8564 −0.665588
\(891\) 0 0
\(892\) −23.5167 −0.787396
\(893\) −0.784610 −0.0262560
\(894\) 0 0
\(895\) 5.75129 0.192244
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 3.80385 0.126936
\(899\) −34.3923 −1.14705
\(900\) 0 0
\(901\) 13.4641 0.448554
\(902\) 9.46410 0.315120
\(903\) 0 0
\(904\) 17.3205 0.576072
\(905\) −6.14359 −0.204220
\(906\) 0 0
\(907\) −2.92820 −0.0972294 −0.0486147 0.998818i \(-0.515481\pi\)
−0.0486147 + 0.998818i \(0.515481\pi\)
\(908\) 10.3923 0.344881
\(909\) 0 0
\(910\) −2.53590 −0.0840642
\(911\) −29.8564 −0.989187 −0.494593 0.869124i \(-0.664683\pi\)
−0.494593 + 0.869124i \(0.664683\pi\)
\(912\) 0 0
\(913\) −13.8564 −0.458580
\(914\) −53.3205 −1.76369
\(915\) 0 0
\(916\) −15.4641 −0.510948
\(917\) −10.1436 −0.334971
\(918\) 0 0
\(919\) 25.1244 0.828776 0.414388 0.910100i \(-0.363996\pi\)
0.414388 + 0.910100i \(0.363996\pi\)
\(920\) 2.53590 0.0836061
\(921\) 0 0
\(922\) 30.2487 0.996188
\(923\) 6.92820 0.228045
\(924\) 0 0
\(925\) 8.92820 0.293558
\(926\) 33.8038 1.11086
\(927\) 0 0
\(928\) −21.8038 −0.715746
\(929\) 26.8756 0.881761 0.440881 0.897566i \(-0.354666\pi\)
0.440881 + 0.897566i \(0.354666\pi\)
\(930\) 0 0
\(931\) 1.60770 0.0526901
\(932\) −4.19615 −0.137450
\(933\) 0 0
\(934\) 57.0333 1.86619
\(935\) 4.92820 0.161169
\(936\) 0 0
\(937\) −57.0333 −1.86320 −0.931599 0.363488i \(-0.881586\pi\)
−0.931599 + 0.363488i \(0.881586\pi\)
\(938\) 42.2487 1.37947
\(939\) 0 0
\(940\) −1.07180 −0.0349582
\(941\) −4.39230 −0.143185 −0.0715925 0.997434i \(-0.522808\pi\)
−0.0715925 + 0.997434i \(0.522808\pi\)
\(942\) 0 0
\(943\) 10.9282 0.355871
\(944\) −67.3205 −2.19110
\(945\) 0 0
\(946\) −10.7321 −0.348929
\(947\) −35.3205 −1.14776 −0.573881 0.818939i \(-0.694563\pi\)
−0.573881 + 0.818939i \(0.694563\pi\)
\(948\) 0 0
\(949\) −2.92820 −0.0950535
\(950\) 4.14359 0.134436
\(951\) 0 0
\(952\) −23.3205 −0.755822
\(953\) −10.7321 −0.347645 −0.173823 0.984777i \(-0.555612\pi\)
−0.173823 + 0.984777i \(0.555612\pi\)
\(954\) 0 0
\(955\) −13.8564 −0.448383
\(956\) −22.3923 −0.724219
\(957\) 0 0
\(958\) −35.5692 −1.14919
\(959\) −29.1769 −0.942172
\(960\) 0 0
\(961\) 36.1769 1.16700
\(962\) 3.46410 0.111687
\(963\) 0 0
\(964\) −18.7846 −0.605012
\(965\) 5.07180 0.163267
\(966\) 0 0
\(967\) 40.9282 1.31616 0.658081 0.752947i \(-0.271368\pi\)
0.658081 + 0.752947i \(0.271368\pi\)
\(968\) −1.73205 −0.0556702
\(969\) 0 0
\(970\) −11.3205 −0.363480
\(971\) 23.8564 0.765589 0.382794 0.923834i \(-0.374962\pi\)
0.382794 + 0.923834i \(0.374962\pi\)
\(972\) 0 0
\(973\) 19.6077 0.628594
\(974\) −44.4449 −1.42411
\(975\) 0 0
\(976\) −24.6410 −0.788740
\(977\) 0.339746 0.0108694 0.00543472 0.999985i \(-0.498270\pi\)
0.00543472 + 0.999985i \(0.498270\pi\)
\(978\) 0 0
\(979\) 15.6603 0.500504
\(980\) 2.19615 0.0701535
\(981\) 0 0
\(982\) −16.3923 −0.523099
\(983\) 46.9282 1.49678 0.748389 0.663260i \(-0.230828\pi\)
0.748389 + 0.663260i \(0.230828\pi\)
\(984\) 0 0
\(985\) 5.85641 0.186601
\(986\) −48.9282 −1.55819
\(987\) 0 0
\(988\) 0.535898 0.0170492
\(989\) −12.3923 −0.394052
\(990\) 0 0
\(991\) 57.1769 1.81628 0.908142 0.418662i \(-0.137501\pi\)
0.908142 + 0.418662i \(0.137501\pi\)
\(992\) 42.5885 1.35218
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) −7.71281 −0.244513
\(996\) 0 0
\(997\) 6.67949 0.211542 0.105771 0.994391i \(-0.466269\pi\)
0.105771 + 0.994391i \(0.466269\pi\)
\(998\) −48.8372 −1.54591
\(999\) 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 1287.2.a.f.1.2 2
3.2 odd 2 429.2.a.d.1.1 2
12.11 even 2 6864.2.a.bk.1.2 2
33.32 even 2 4719.2.a.n.1.2 2
39.38 odd 2 5577.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.a.d.1.1 2 3.2 odd 2
1287.2.a.f.1.2 2 1.1 even 1 trivial
4719.2.a.n.1.2 2 33.32 even 2
5577.2.a.h.1.2 2 39.38 odd 2
6864.2.a.bk.1.2 2 12.11 even 2