Properties

Label 6384.2.a.bj.1.1
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6384,2,Mod(1,6384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.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: \(50.9764966504\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1596)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.46410 q^{11} -4.00000 q^{13} +2.73205 q^{15} +4.19615 q^{17} +1.00000 q^{19} -1.00000 q^{21} +7.46410 q^{23} +2.46410 q^{25} -1.00000 q^{27} -6.19615 q^{29} -4.92820 q^{31} +3.46410 q^{33} -2.73205 q^{35} -7.46410 q^{37} +4.00000 q^{39} -9.46410 q^{41} +3.46410 q^{43} -2.73205 q^{45} -2.73205 q^{47} +1.00000 q^{49} -4.19615 q^{51} +8.73205 q^{53} +9.46410 q^{55} -1.00000 q^{57} +10.9282 q^{59} -14.3923 q^{61} +1.00000 q^{63} +10.9282 q^{65} -1.46410 q^{67} -7.46410 q^{69} -10.1962 q^{71} -15.8564 q^{73} -2.46410 q^{75} -3.46410 q^{77} +13.4641 q^{79} +1.00000 q^{81} +1.26795 q^{83} -11.4641 q^{85} +6.19615 q^{87} +9.46410 q^{89} -4.00000 q^{91} +4.92820 q^{93} -2.73205 q^{95} -6.00000 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{13} + 2 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{21} + 8 q^{23} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} - 2 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} - 2 q^{45} - 2 q^{47} + 2 q^{49} + 2 q^{51} + 14 q^{53} + 12 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} + 2 q^{63} + 8 q^{65} + 4 q^{67} - 8 q^{69} - 10 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{79} + 2 q^{81} + 6 q^{83} - 16 q^{85} + 2 q^{87} + 12 q^{89} - 8 q^{91} - 4 q^{93} - 2 q^{95} - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.73205 −1.22181 −0.610905 0.791704i \(-0.709194\pi\)
−0.610905 + 0.791704i \(0.709194\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 2.73205 0.705412
\(16\) 0 0
\(17\) 4.19615 1.01772 0.508858 0.860850i \(-0.330068\pi\)
0.508858 + 0.860850i \(0.330068\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 7.46410 1.55637 0.778186 0.628033i \(-0.216140\pi\)
0.778186 + 0.628033i \(0.216140\pi\)
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.19615 −1.15060 −0.575298 0.817944i \(-0.695114\pi\)
−0.575298 + 0.817944i \(0.695114\pi\)
\(30\) 0 0
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) 0 0
\(33\) 3.46410 0.603023
\(34\) 0 0
\(35\) −2.73205 −0.461801
\(36\) 0 0
\(37\) −7.46410 −1.22709 −0.613545 0.789659i \(-0.710257\pi\)
−0.613545 + 0.789659i \(0.710257\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −9.46410 −1.47804 −0.739022 0.673681i \(-0.764712\pi\)
−0.739022 + 0.673681i \(0.764712\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) −2.73205 −0.407270
\(46\) 0 0
\(47\) −2.73205 −0.398511 −0.199255 0.979948i \(-0.563852\pi\)
−0.199255 + 0.979948i \(0.563852\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.19615 −0.587579
\(52\) 0 0
\(53\) 8.73205 1.19944 0.599720 0.800210i \(-0.295279\pi\)
0.599720 + 0.800210i \(0.295279\pi\)
\(54\) 0 0
\(55\) 9.46410 1.27614
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 10.9282 1.42273 0.711365 0.702822i \(-0.248077\pi\)
0.711365 + 0.702822i \(0.248077\pi\)
\(60\) 0 0
\(61\) −14.3923 −1.84275 −0.921373 0.388680i \(-0.872931\pi\)
−0.921373 + 0.388680i \(0.872931\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 10.9282 1.35548
\(66\) 0 0
\(67\) −1.46410 −0.178868 −0.0894342 0.995993i \(-0.528506\pi\)
−0.0894342 + 0.995993i \(0.528506\pi\)
\(68\) 0 0
\(69\) −7.46410 −0.898572
\(70\) 0 0
\(71\) −10.1962 −1.21006 −0.605030 0.796202i \(-0.706839\pi\)
−0.605030 + 0.796202i \(0.706839\pi\)
\(72\) 0 0
\(73\) −15.8564 −1.85585 −0.927926 0.372764i \(-0.878410\pi\)
−0.927926 + 0.372764i \(0.878410\pi\)
\(74\) 0 0
\(75\) −2.46410 −0.284530
\(76\) 0 0
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) 13.4641 1.51483 0.757415 0.652934i \(-0.226462\pi\)
0.757415 + 0.652934i \(0.226462\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.26795 0.139176 0.0695878 0.997576i \(-0.477832\pi\)
0.0695878 + 0.997576i \(0.477832\pi\)
\(84\) 0 0
\(85\) −11.4641 −1.24346
\(86\) 0 0
\(87\) 6.19615 0.664297
\(88\) 0 0
\(89\) 9.46410 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 4.92820 0.511031
\(94\) 0 0
\(95\) −2.73205 −0.280302
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 0 0
\(99\) −3.46410 −0.348155
\(100\) 0 0
\(101\) −16.1962 −1.61158 −0.805789 0.592203i \(-0.798258\pi\)
−0.805789 + 0.592203i \(0.798258\pi\)
\(102\) 0 0
\(103\) 10.9282 1.07679 0.538394 0.842693i \(-0.319031\pi\)
0.538394 + 0.842693i \(0.319031\pi\)
\(104\) 0 0
\(105\) 2.73205 0.266621
\(106\) 0 0
\(107\) −11.6603 −1.12724 −0.563620 0.826034i \(-0.690592\pi\)
−0.563620 + 0.826034i \(0.690592\pi\)
\(108\) 0 0
\(109\) 3.46410 0.331801 0.165900 0.986143i \(-0.446947\pi\)
0.165900 + 0.986143i \(0.446947\pi\)
\(110\) 0 0
\(111\) 7.46410 0.708461
\(112\) 0 0
\(113\) 16.7321 1.57402 0.787009 0.616941i \(-0.211628\pi\)
0.787009 + 0.616941i \(0.211628\pi\)
\(114\) 0 0
\(115\) −20.3923 −1.90159
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 4.19615 0.384661
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 9.46410 0.853349
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −8.39230 −0.744697 −0.372348 0.928093i \(-0.621447\pi\)
−0.372348 + 0.928093i \(0.621447\pi\)
\(128\) 0 0
\(129\) −3.46410 −0.304997
\(130\) 0 0
\(131\) −12.1962 −1.06558 −0.532791 0.846247i \(-0.678857\pi\)
−0.532791 + 0.846247i \(0.678857\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 2.73205 0.235137
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 22.9282 1.94474 0.972372 0.233435i \(-0.0749966\pi\)
0.972372 + 0.233435i \(0.0749966\pi\)
\(140\) 0 0
\(141\) 2.73205 0.230080
\(142\) 0 0
\(143\) 13.8564 1.15873
\(144\) 0 0
\(145\) 16.9282 1.40581
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −10.3923 −0.851371 −0.425685 0.904871i \(-0.639967\pi\)
−0.425685 + 0.904871i \(0.639967\pi\)
\(150\) 0 0
\(151\) 1.07180 0.0872216 0.0436108 0.999049i \(-0.486114\pi\)
0.0436108 + 0.999049i \(0.486114\pi\)
\(152\) 0 0
\(153\) 4.19615 0.339239
\(154\) 0 0
\(155\) 13.4641 1.08146
\(156\) 0 0
\(157\) 6.39230 0.510161 0.255081 0.966920i \(-0.417898\pi\)
0.255081 + 0.966920i \(0.417898\pi\)
\(158\) 0 0
\(159\) −8.73205 −0.692497
\(160\) 0 0
\(161\) 7.46410 0.588254
\(162\) 0 0
\(163\) −15.4641 −1.21124 −0.605621 0.795753i \(-0.707075\pi\)
−0.605621 + 0.795753i \(0.707075\pi\)
\(164\) 0 0
\(165\) −9.46410 −0.736779
\(166\) 0 0
\(167\) 8.39230 0.649416 0.324708 0.945814i \(-0.394734\pi\)
0.324708 + 0.945814i \(0.394734\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 20.3923 1.55040 0.775199 0.631717i \(-0.217650\pi\)
0.775199 + 0.631717i \(0.217650\pi\)
\(174\) 0 0
\(175\) 2.46410 0.186269
\(176\) 0 0
\(177\) −10.9282 −0.821414
\(178\) 0 0
\(179\) 19.6603 1.46948 0.734738 0.678351i \(-0.237305\pi\)
0.734738 + 0.678351i \(0.237305\pi\)
\(180\) 0 0
\(181\) 18.7846 1.39625 0.698125 0.715976i \(-0.254018\pi\)
0.698125 + 0.715976i \(0.254018\pi\)
\(182\) 0 0
\(183\) 14.3923 1.06391
\(184\) 0 0
\(185\) 20.3923 1.49927
\(186\) 0 0
\(187\) −14.5359 −1.06297
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −0.535898 −0.0387762 −0.0193881 0.999812i \(-0.506172\pi\)
−0.0193881 + 0.999812i \(0.506172\pi\)
\(192\) 0 0
\(193\) 6.39230 0.460128 0.230064 0.973175i \(-0.426106\pi\)
0.230064 + 0.973175i \(0.426106\pi\)
\(194\) 0 0
\(195\) −10.9282 −0.782585
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −19.3205 −1.36959 −0.684797 0.728734i \(-0.740109\pi\)
−0.684797 + 0.728734i \(0.740109\pi\)
\(200\) 0 0
\(201\) 1.46410 0.103270
\(202\) 0 0
\(203\) −6.19615 −0.434885
\(204\) 0 0
\(205\) 25.8564 1.80589
\(206\) 0 0
\(207\) 7.46410 0.518791
\(208\) 0 0
\(209\) −3.46410 −0.239617
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 10.1962 0.698629
\(214\) 0 0
\(215\) −9.46410 −0.645446
\(216\) 0 0
\(217\) −4.92820 −0.334548
\(218\) 0 0
\(219\) 15.8564 1.07148
\(220\) 0 0
\(221\) −16.7846 −1.12906
\(222\) 0 0
\(223\) 4.92820 0.330017 0.165008 0.986292i \(-0.447235\pi\)
0.165008 + 0.986292i \(0.447235\pi\)
\(224\) 0 0
\(225\) 2.46410 0.164273
\(226\) 0 0
\(227\) 12.3923 0.822506 0.411253 0.911521i \(-0.365091\pi\)
0.411253 + 0.911521i \(0.365091\pi\)
\(228\) 0 0
\(229\) −18.3923 −1.21540 −0.607699 0.794168i \(-0.707907\pi\)
−0.607699 + 0.794168i \(0.707907\pi\)
\(230\) 0 0
\(231\) 3.46410 0.227921
\(232\) 0 0
\(233\) 17.3205 1.13470 0.567352 0.823475i \(-0.307968\pi\)
0.567352 + 0.823475i \(0.307968\pi\)
\(234\) 0 0
\(235\) 7.46410 0.486904
\(236\) 0 0
\(237\) −13.4641 −0.874587
\(238\) 0 0
\(239\) 3.07180 0.198698 0.0993490 0.995053i \(-0.468324\pi\)
0.0993490 + 0.995053i \(0.468324\pi\)
\(240\) 0 0
\(241\) −29.7128 −1.91397 −0.956985 0.290137i \(-0.906299\pi\)
−0.956985 + 0.290137i \(0.906299\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.73205 −0.174544
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −1.26795 −0.0803530
\(250\) 0 0
\(251\) 28.1962 1.77973 0.889863 0.456228i \(-0.150800\pi\)
0.889863 + 0.456228i \(0.150800\pi\)
\(252\) 0 0
\(253\) −25.8564 −1.62558
\(254\) 0 0
\(255\) 11.4641 0.717910
\(256\) 0 0
\(257\) 5.07180 0.316370 0.158185 0.987409i \(-0.449436\pi\)
0.158185 + 0.987409i \(0.449436\pi\)
\(258\) 0 0
\(259\) −7.46410 −0.463797
\(260\) 0 0
\(261\) −6.19615 −0.383532
\(262\) 0 0
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) −23.8564 −1.46549
\(266\) 0 0
\(267\) −9.46410 −0.579194
\(268\) 0 0
\(269\) 13.8564 0.844840 0.422420 0.906400i \(-0.361181\pi\)
0.422420 + 0.906400i \(0.361181\pi\)
\(270\) 0 0
\(271\) −19.3205 −1.17364 −0.586819 0.809718i \(-0.699620\pi\)
−0.586819 + 0.809718i \(0.699620\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −8.53590 −0.514734
\(276\) 0 0
\(277\) 12.3923 0.744581 0.372291 0.928116i \(-0.378572\pi\)
0.372291 + 0.928116i \(0.378572\pi\)
\(278\) 0 0
\(279\) −4.92820 −0.295044
\(280\) 0 0
\(281\) 4.33975 0.258888 0.129444 0.991587i \(-0.458681\pi\)
0.129444 + 0.991587i \(0.458681\pi\)
\(282\) 0 0
\(283\) −5.46410 −0.324807 −0.162404 0.986724i \(-0.551925\pi\)
−0.162404 + 0.986724i \(0.551925\pi\)
\(284\) 0 0
\(285\) 2.73205 0.161833
\(286\) 0 0
\(287\) −9.46410 −0.558648
\(288\) 0 0
\(289\) 0.607695 0.0357468
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 0 0
\(293\) −2.53590 −0.148149 −0.0740744 0.997253i \(-0.523600\pi\)
−0.0740744 + 0.997253i \(0.523600\pi\)
\(294\) 0 0
\(295\) −29.8564 −1.73831
\(296\) 0 0
\(297\) 3.46410 0.201008
\(298\) 0 0
\(299\) −29.8564 −1.72664
\(300\) 0 0
\(301\) 3.46410 0.199667
\(302\) 0 0
\(303\) 16.1962 0.930445
\(304\) 0 0
\(305\) 39.3205 2.25149
\(306\) 0 0
\(307\) 27.8564 1.58985 0.794925 0.606708i \(-0.207510\pi\)
0.794925 + 0.606708i \(0.207510\pi\)
\(308\) 0 0
\(309\) −10.9282 −0.621684
\(310\) 0 0
\(311\) 14.7321 0.835378 0.417689 0.908590i \(-0.362840\pi\)
0.417689 + 0.908590i \(0.362840\pi\)
\(312\) 0 0
\(313\) −30.3923 −1.71787 −0.858937 0.512081i \(-0.828875\pi\)
−0.858937 + 0.512081i \(0.828875\pi\)
\(314\) 0 0
\(315\) −2.73205 −0.153934
\(316\) 0 0
\(317\) 2.87564 0.161512 0.0807561 0.996734i \(-0.474267\pi\)
0.0807561 + 0.996734i \(0.474267\pi\)
\(318\) 0 0
\(319\) 21.4641 1.20176
\(320\) 0 0
\(321\) 11.6603 0.650812
\(322\) 0 0
\(323\) 4.19615 0.233480
\(324\) 0 0
\(325\) −9.85641 −0.546735
\(326\) 0 0
\(327\) −3.46410 −0.191565
\(328\) 0 0
\(329\) −2.73205 −0.150623
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −7.46410 −0.409030
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 34.3923 1.87347 0.936734 0.350042i \(-0.113833\pi\)
0.936734 + 0.350042i \(0.113833\pi\)
\(338\) 0 0
\(339\) −16.7321 −0.908760
\(340\) 0 0
\(341\) 17.0718 0.924490
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 20.3923 1.09788
\(346\) 0 0
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 0 0
\(349\) 31.8564 1.70523 0.852617 0.522536i \(-0.175014\pi\)
0.852617 + 0.522536i \(0.175014\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −6.05256 −0.322145 −0.161073 0.986943i \(-0.551495\pi\)
−0.161073 + 0.986943i \(0.551495\pi\)
\(354\) 0 0
\(355\) 27.8564 1.47846
\(356\) 0 0
\(357\) −4.19615 −0.222084
\(358\) 0 0
\(359\) 2.39230 0.126261 0.0631305 0.998005i \(-0.479892\pi\)
0.0631305 + 0.998005i \(0.479892\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 43.3205 2.26750
\(366\) 0 0
\(367\) −34.2487 −1.78777 −0.893884 0.448298i \(-0.852030\pi\)
−0.893884 + 0.448298i \(0.852030\pi\)
\(368\) 0 0
\(369\) −9.46410 −0.492681
\(370\) 0 0
\(371\) 8.73205 0.453345
\(372\) 0 0
\(373\) 22.7846 1.17974 0.589871 0.807497i \(-0.299179\pi\)
0.589871 + 0.807497i \(0.299179\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 24.7846 1.27647
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) 8.39230 0.429951
\(382\) 0 0
\(383\) 7.60770 0.388735 0.194368 0.980929i \(-0.437735\pi\)
0.194368 + 0.980929i \(0.437735\pi\)
\(384\) 0 0
\(385\) 9.46410 0.482335
\(386\) 0 0
\(387\) 3.46410 0.176090
\(388\) 0 0
\(389\) −27.1769 −1.37792 −0.688962 0.724797i \(-0.741933\pi\)
−0.688962 + 0.724797i \(0.741933\pi\)
\(390\) 0 0
\(391\) 31.3205 1.58395
\(392\) 0 0
\(393\) 12.1962 0.615215
\(394\) 0 0
\(395\) −36.7846 −1.85083
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −4.05256 −0.202375 −0.101188 0.994867i \(-0.532264\pi\)
−0.101188 + 0.994867i \(0.532264\pi\)
\(402\) 0 0
\(403\) 19.7128 0.981965
\(404\) 0 0
\(405\) −2.73205 −0.135757
\(406\) 0 0
\(407\) 25.8564 1.28165
\(408\) 0 0
\(409\) 18.9282 0.935939 0.467970 0.883745i \(-0.344986\pi\)
0.467970 + 0.883745i \(0.344986\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 10.9282 0.537742
\(414\) 0 0
\(415\) −3.46410 −0.170046
\(416\) 0 0
\(417\) −22.9282 −1.12280
\(418\) 0 0
\(419\) −10.0526 −0.491100 −0.245550 0.969384i \(-0.578968\pi\)
−0.245550 + 0.969384i \(0.578968\pi\)
\(420\) 0 0
\(421\) 15.0718 0.734554 0.367277 0.930112i \(-0.380290\pi\)
0.367277 + 0.930112i \(0.380290\pi\)
\(422\) 0 0
\(423\) −2.73205 −0.132837
\(424\) 0 0
\(425\) 10.3397 0.501551
\(426\) 0 0
\(427\) −14.3923 −0.696492
\(428\) 0 0
\(429\) −13.8564 −0.668994
\(430\) 0 0
\(431\) −13.5167 −0.651075 −0.325537 0.945529i \(-0.605545\pi\)
−0.325537 + 0.945529i \(0.605545\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) −16.9282 −0.811645
\(436\) 0 0
\(437\) 7.46410 0.357056
\(438\) 0 0
\(439\) 4.92820 0.235210 0.117605 0.993060i \(-0.462478\pi\)
0.117605 + 0.993060i \(0.462478\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.9282 1.18437 0.592187 0.805800i \(-0.298265\pi\)
0.592187 + 0.805800i \(0.298265\pi\)
\(444\) 0 0
\(445\) −25.8564 −1.22571
\(446\) 0 0
\(447\) 10.3923 0.491539
\(448\) 0 0
\(449\) 21.8038 1.02899 0.514494 0.857494i \(-0.327980\pi\)
0.514494 + 0.857494i \(0.327980\pi\)
\(450\) 0 0
\(451\) 32.7846 1.54377
\(452\) 0 0
\(453\) −1.07180 −0.0503574
\(454\) 0 0
\(455\) 10.9282 0.512322
\(456\) 0 0
\(457\) 24.3923 1.14102 0.570512 0.821289i \(-0.306745\pi\)
0.570512 + 0.821289i \(0.306745\pi\)
\(458\) 0 0
\(459\) −4.19615 −0.195860
\(460\) 0 0
\(461\) −18.3397 −0.854167 −0.427084 0.904212i \(-0.640459\pi\)
−0.427084 + 0.904212i \(0.640459\pi\)
\(462\) 0 0
\(463\) 1.85641 0.0862745 0.0431373 0.999069i \(-0.486265\pi\)
0.0431373 + 0.999069i \(0.486265\pi\)
\(464\) 0 0
\(465\) −13.4641 −0.624383
\(466\) 0 0
\(467\) 22.4449 1.03862 0.519312 0.854585i \(-0.326188\pi\)
0.519312 + 0.854585i \(0.326188\pi\)
\(468\) 0 0
\(469\) −1.46410 −0.0676059
\(470\) 0 0
\(471\) −6.39230 −0.294542
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 2.46410 0.113061
\(476\) 0 0
\(477\) 8.73205 0.399813
\(478\) 0 0
\(479\) −32.9808 −1.50693 −0.753465 0.657488i \(-0.771619\pi\)
−0.753465 + 0.657488i \(0.771619\pi\)
\(480\) 0 0
\(481\) 29.8564 1.36133
\(482\) 0 0
\(483\) −7.46410 −0.339628
\(484\) 0 0
\(485\) 16.3923 0.744336
\(486\) 0 0
\(487\) 10.9282 0.495204 0.247602 0.968862i \(-0.420357\pi\)
0.247602 + 0.968862i \(0.420357\pi\)
\(488\) 0 0
\(489\) 15.4641 0.699311
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) −26.0000 −1.17098
\(494\) 0 0
\(495\) 9.46410 0.425380
\(496\) 0 0
\(497\) −10.1962 −0.457360
\(498\) 0 0
\(499\) 6.92820 0.310149 0.155074 0.987903i \(-0.450438\pi\)
0.155074 + 0.987903i \(0.450438\pi\)
\(500\) 0 0
\(501\) −8.39230 −0.374941
\(502\) 0 0
\(503\) 18.7321 0.835221 0.417610 0.908626i \(-0.362868\pi\)
0.417610 + 0.908626i \(0.362868\pi\)
\(504\) 0 0
\(505\) 44.2487 1.96904
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) 32.7846 1.45315 0.726576 0.687086i \(-0.241110\pi\)
0.726576 + 0.687086i \(0.241110\pi\)
\(510\) 0 0
\(511\) −15.8564 −0.701446
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −29.8564 −1.31563
\(516\) 0 0
\(517\) 9.46410 0.416231
\(518\) 0 0
\(519\) −20.3923 −0.895123
\(520\) 0 0
\(521\) −16.0000 −0.700973 −0.350486 0.936568i \(-0.613984\pi\)
−0.350486 + 0.936568i \(0.613984\pi\)
\(522\) 0 0
\(523\) 22.7846 0.996301 0.498151 0.867090i \(-0.334013\pi\)
0.498151 + 0.867090i \(0.334013\pi\)
\(524\) 0 0
\(525\) −2.46410 −0.107542
\(526\) 0 0
\(527\) −20.6795 −0.900813
\(528\) 0 0
\(529\) 32.7128 1.42230
\(530\) 0 0
\(531\) 10.9282 0.474244
\(532\) 0 0
\(533\) 37.8564 1.63974
\(534\) 0 0
\(535\) 31.8564 1.37727
\(536\) 0 0
\(537\) −19.6603 −0.848403
\(538\) 0 0
\(539\) −3.46410 −0.149209
\(540\) 0 0
\(541\) −15.8564 −0.681720 −0.340860 0.940114i \(-0.610718\pi\)
−0.340860 + 0.940114i \(0.610718\pi\)
\(542\) 0 0
\(543\) −18.7846 −0.806125
\(544\) 0 0
\(545\) −9.46410 −0.405398
\(546\) 0 0
\(547\) 31.3205 1.33917 0.669584 0.742736i \(-0.266472\pi\)
0.669584 + 0.742736i \(0.266472\pi\)
\(548\) 0 0
\(549\) −14.3923 −0.614249
\(550\) 0 0
\(551\) −6.19615 −0.263965
\(552\) 0 0
\(553\) 13.4641 0.572552
\(554\) 0 0
\(555\) −20.3923 −0.865605
\(556\) 0 0
\(557\) 6.39230 0.270851 0.135425 0.990788i \(-0.456760\pi\)
0.135425 + 0.990788i \(0.456760\pi\)
\(558\) 0 0
\(559\) −13.8564 −0.586064
\(560\) 0 0
\(561\) 14.5359 0.613706
\(562\) 0 0
\(563\) −7.32051 −0.308523 −0.154261 0.988030i \(-0.549300\pi\)
−0.154261 + 0.988030i \(0.549300\pi\)
\(564\) 0 0
\(565\) −45.7128 −1.92315
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 37.9090 1.58923 0.794613 0.607116i \(-0.207674\pi\)
0.794613 + 0.607116i \(0.207674\pi\)
\(570\) 0 0
\(571\) 29.8564 1.24945 0.624726 0.780844i \(-0.285211\pi\)
0.624726 + 0.780844i \(0.285211\pi\)
\(572\) 0 0
\(573\) 0.535898 0.0223875
\(574\) 0 0
\(575\) 18.3923 0.767012
\(576\) 0 0
\(577\) 35.1769 1.46443 0.732217 0.681071i \(-0.238486\pi\)
0.732217 + 0.681071i \(0.238486\pi\)
\(578\) 0 0
\(579\) −6.39230 −0.265655
\(580\) 0 0
\(581\) 1.26795 0.0526034
\(582\) 0 0
\(583\) −30.2487 −1.25277
\(584\) 0 0
\(585\) 10.9282 0.451826
\(586\) 0 0
\(587\) −7.51666 −0.310246 −0.155123 0.987895i \(-0.549577\pi\)
−0.155123 + 0.987895i \(0.549577\pi\)
\(588\) 0 0
\(589\) −4.92820 −0.203063
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) −27.5167 −1.12997 −0.564987 0.825100i \(-0.691119\pi\)
−0.564987 + 0.825100i \(0.691119\pi\)
\(594\) 0 0
\(595\) −11.4641 −0.469982
\(596\) 0 0
\(597\) 19.3205 0.790736
\(598\) 0 0
\(599\) 8.73205 0.356782 0.178391 0.983960i \(-0.442911\pi\)
0.178391 + 0.983960i \(0.442911\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −1.46410 −0.0596228
\(604\) 0 0
\(605\) −2.73205 −0.111074
\(606\) 0 0
\(607\) −8.78461 −0.356556 −0.178278 0.983980i \(-0.557053\pi\)
−0.178278 + 0.983980i \(0.557053\pi\)
\(608\) 0 0
\(609\) 6.19615 0.251081
\(610\) 0 0
\(611\) 10.9282 0.442108
\(612\) 0 0
\(613\) 0.392305 0.0158450 0.00792252 0.999969i \(-0.497478\pi\)
0.00792252 + 0.999969i \(0.497478\pi\)
\(614\) 0 0
\(615\) −25.8564 −1.04263
\(616\) 0 0
\(617\) 25.7128 1.03516 0.517579 0.855635i \(-0.326833\pi\)
0.517579 + 0.855635i \(0.326833\pi\)
\(618\) 0 0
\(619\) −3.21539 −0.129237 −0.0646187 0.997910i \(-0.520583\pi\)
−0.0646187 + 0.997910i \(0.520583\pi\)
\(620\) 0 0
\(621\) −7.46410 −0.299524
\(622\) 0 0
\(623\) 9.46410 0.379171
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 0 0
\(627\) 3.46410 0.138343
\(628\) 0 0
\(629\) −31.3205 −1.24883
\(630\) 0 0
\(631\) 36.2487 1.44304 0.721519 0.692394i \(-0.243444\pi\)
0.721519 + 0.692394i \(0.243444\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) 22.9282 0.909878
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −10.1962 −0.403354
\(640\) 0 0
\(641\) −36.4449 −1.43949 −0.719743 0.694241i \(-0.755740\pi\)
−0.719743 + 0.694241i \(0.755740\pi\)
\(642\) 0 0
\(643\) −44.1051 −1.73934 −0.869668 0.493637i \(-0.835667\pi\)
−0.869668 + 0.493637i \(0.835667\pi\)
\(644\) 0 0
\(645\) 9.46410 0.372649
\(646\) 0 0
\(647\) 16.9808 0.667583 0.333791 0.942647i \(-0.391672\pi\)
0.333791 + 0.942647i \(0.391672\pi\)
\(648\) 0 0
\(649\) −37.8564 −1.48599
\(650\) 0 0
\(651\) 4.92820 0.193152
\(652\) 0 0
\(653\) −29.7128 −1.16275 −0.581376 0.813635i \(-0.697485\pi\)
−0.581376 + 0.813635i \(0.697485\pi\)
\(654\) 0 0
\(655\) 33.3205 1.30194
\(656\) 0 0
\(657\) −15.8564 −0.618617
\(658\) 0 0
\(659\) 40.0526 1.56023 0.780113 0.625639i \(-0.215161\pi\)
0.780113 + 0.625639i \(0.215161\pi\)
\(660\) 0 0
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) 0 0
\(663\) 16.7846 0.651860
\(664\) 0 0
\(665\) −2.73205 −0.105944
\(666\) 0 0
\(667\) −46.2487 −1.79076
\(668\) 0 0
\(669\) −4.92820 −0.190535
\(670\) 0 0
\(671\) 49.8564 1.92469
\(672\) 0 0
\(673\) −32.2487 −1.24310 −0.621548 0.783376i \(-0.713496\pi\)
−0.621548 + 0.783376i \(0.713496\pi\)
\(674\) 0 0
\(675\) −2.46410 −0.0948433
\(676\) 0 0
\(677\) 17.0718 0.656122 0.328061 0.944656i \(-0.393605\pi\)
0.328061 + 0.944656i \(0.393605\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −12.3923 −0.474874
\(682\) 0 0
\(683\) 22.5885 0.864323 0.432162 0.901796i \(-0.357751\pi\)
0.432162 + 0.901796i \(0.357751\pi\)
\(684\) 0 0
\(685\) −49.1769 −1.87895
\(686\) 0 0
\(687\) 18.3923 0.701710
\(688\) 0 0
\(689\) −34.9282 −1.33066
\(690\) 0 0
\(691\) −41.4641 −1.57737 −0.788684 0.614798i \(-0.789237\pi\)
−0.788684 + 0.614798i \(0.789237\pi\)
\(692\) 0 0
\(693\) −3.46410 −0.131590
\(694\) 0 0
\(695\) −62.6410 −2.37611
\(696\) 0 0
\(697\) −39.7128 −1.50423
\(698\) 0 0
\(699\) −17.3205 −0.655122
\(700\) 0 0
\(701\) −9.21539 −0.348060 −0.174030 0.984740i \(-0.555679\pi\)
−0.174030 + 0.984740i \(0.555679\pi\)
\(702\) 0 0
\(703\) −7.46410 −0.281514
\(704\) 0 0
\(705\) −7.46410 −0.281114
\(706\) 0 0
\(707\) −16.1962 −0.609119
\(708\) 0 0
\(709\) −8.67949 −0.325965 −0.162983 0.986629i \(-0.552111\pi\)
−0.162983 + 0.986629i \(0.552111\pi\)
\(710\) 0 0
\(711\) 13.4641 0.504943
\(712\) 0 0
\(713\) −36.7846 −1.37759
\(714\) 0 0
\(715\) −37.8564 −1.41575
\(716\) 0 0
\(717\) −3.07180 −0.114718
\(718\) 0 0
\(719\) −1.94744 −0.0726273 −0.0363136 0.999340i \(-0.511562\pi\)
−0.0363136 + 0.999340i \(0.511562\pi\)
\(720\) 0 0
\(721\) 10.9282 0.406988
\(722\) 0 0
\(723\) 29.7128 1.10503
\(724\) 0 0
\(725\) −15.2679 −0.567037
\(726\) 0 0
\(727\) −6.53590 −0.242403 −0.121202 0.992628i \(-0.538675\pi\)
−0.121202 + 0.992628i \(0.538675\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 14.5359 0.537630
\(732\) 0 0
\(733\) 16.6410 0.614650 0.307325 0.951605i \(-0.400566\pi\)
0.307325 + 0.951605i \(0.400566\pi\)
\(734\) 0 0
\(735\) 2.73205 0.100773
\(736\) 0 0
\(737\) 5.07180 0.186822
\(738\) 0 0
\(739\) 16.5359 0.608283 0.304141 0.952627i \(-0.401630\pi\)
0.304141 + 0.952627i \(0.401630\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) −10.5885 −0.388453 −0.194226 0.980957i \(-0.562220\pi\)
−0.194226 + 0.980957i \(0.562220\pi\)
\(744\) 0 0
\(745\) 28.3923 1.04021
\(746\) 0 0
\(747\) 1.26795 0.0463918
\(748\) 0 0
\(749\) −11.6603 −0.426056
\(750\) 0 0
\(751\) 28.7846 1.05037 0.525183 0.850990i \(-0.323997\pi\)
0.525183 + 0.850990i \(0.323997\pi\)
\(752\) 0 0
\(753\) −28.1962 −1.02752
\(754\) 0 0
\(755\) −2.92820 −0.106568
\(756\) 0 0
\(757\) −8.92820 −0.324501 −0.162251 0.986750i \(-0.551875\pi\)
−0.162251 + 0.986750i \(0.551875\pi\)
\(758\) 0 0
\(759\) 25.8564 0.938528
\(760\) 0 0
\(761\) 12.1962 0.442110 0.221055 0.975261i \(-0.429050\pi\)
0.221055 + 0.975261i \(0.429050\pi\)
\(762\) 0 0
\(763\) 3.46410 0.125409
\(764\) 0 0
\(765\) −11.4641 −0.414485
\(766\) 0 0
\(767\) −43.7128 −1.57838
\(768\) 0 0
\(769\) −18.6795 −0.673600 −0.336800 0.941576i \(-0.609345\pi\)
−0.336800 + 0.941576i \(0.609345\pi\)
\(770\) 0 0
\(771\) −5.07180 −0.182656
\(772\) 0 0
\(773\) −38.5359 −1.38604 −0.693020 0.720918i \(-0.743720\pi\)
−0.693020 + 0.720918i \(0.743720\pi\)
\(774\) 0 0
\(775\) −12.1436 −0.436211
\(776\) 0 0
\(777\) 7.46410 0.267773
\(778\) 0 0
\(779\) −9.46410 −0.339087
\(780\) 0 0
\(781\) 35.3205 1.26387
\(782\) 0 0
\(783\) 6.19615 0.221432
\(784\) 0 0
\(785\) −17.4641 −0.623321
\(786\) 0 0
\(787\) 1.21539 0.0433240 0.0216620 0.999765i \(-0.493104\pi\)
0.0216620 + 0.999765i \(0.493104\pi\)
\(788\) 0 0
\(789\) −2.00000 −0.0712019
\(790\) 0 0
\(791\) 16.7321 0.594923
\(792\) 0 0
\(793\) 57.5692 2.04434
\(794\) 0 0
\(795\) 23.8564 0.846100
\(796\) 0 0
\(797\) −41.1769 −1.45856 −0.729281 0.684215i \(-0.760145\pi\)
−0.729281 + 0.684215i \(0.760145\pi\)
\(798\) 0 0
\(799\) −11.4641 −0.405571
\(800\) 0 0
\(801\) 9.46410 0.334398
\(802\) 0 0
\(803\) 54.9282 1.93837
\(804\) 0 0
\(805\) −20.3923 −0.718734
\(806\) 0 0
\(807\) −13.8564 −0.487769
\(808\) 0 0
\(809\) −14.0000 −0.492214 −0.246107 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 0 0
\(811\) 33.0718 1.16131 0.580654 0.814150i \(-0.302797\pi\)
0.580654 + 0.814150i \(0.302797\pi\)
\(812\) 0 0
\(813\) 19.3205 0.677600
\(814\) 0 0
\(815\) 42.2487 1.47991
\(816\) 0 0
\(817\) 3.46410 0.121194
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 33.0333 1.15287 0.576435 0.817143i \(-0.304443\pi\)
0.576435 + 0.817143i \(0.304443\pi\)
\(822\) 0 0
\(823\) 51.4641 1.79393 0.896963 0.442106i \(-0.145768\pi\)
0.896963 + 0.442106i \(0.145768\pi\)
\(824\) 0 0
\(825\) 8.53590 0.297182
\(826\) 0 0
\(827\) −12.0526 −0.419109 −0.209554 0.977797i \(-0.567201\pi\)
−0.209554 + 0.977797i \(0.567201\pi\)
\(828\) 0 0
\(829\) −20.9282 −0.726867 −0.363433 0.931620i \(-0.618396\pi\)
−0.363433 + 0.931620i \(0.618396\pi\)
\(830\) 0 0
\(831\) −12.3923 −0.429884
\(832\) 0 0
\(833\) 4.19615 0.145388
\(834\) 0 0
\(835\) −22.9282 −0.793463
\(836\) 0 0
\(837\) 4.92820 0.170344
\(838\) 0 0
\(839\) 17.0718 0.589384 0.294692 0.955592i \(-0.404783\pi\)
0.294692 + 0.955592i \(0.404783\pi\)
\(840\) 0 0
\(841\) 9.39230 0.323873
\(842\) 0 0
\(843\) −4.33975 −0.149469
\(844\) 0 0
\(845\) −8.19615 −0.281956
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 5.46410 0.187527
\(850\) 0 0
\(851\) −55.7128 −1.90981
\(852\) 0 0
\(853\) −27.8564 −0.953785 −0.476893 0.878962i \(-0.658237\pi\)
−0.476893 + 0.878962i \(0.658237\pi\)
\(854\) 0 0
\(855\) −2.73205 −0.0934342
\(856\) 0 0
\(857\) −35.3205 −1.20653 −0.603263 0.797542i \(-0.706133\pi\)
−0.603263 + 0.797542i \(0.706133\pi\)
\(858\) 0 0
\(859\) −33.5692 −1.14537 −0.572683 0.819777i \(-0.694098\pi\)
−0.572683 + 0.819777i \(0.694098\pi\)
\(860\) 0 0
\(861\) 9.46410 0.322536
\(862\) 0 0
\(863\) −19.2679 −0.655889 −0.327944 0.944697i \(-0.606356\pi\)
−0.327944 + 0.944697i \(0.606356\pi\)
\(864\) 0 0
\(865\) −55.7128 −1.89429
\(866\) 0 0
\(867\) −0.607695 −0.0206384
\(868\) 0 0
\(869\) −46.6410 −1.58219
\(870\) 0 0
\(871\) 5.85641 0.198437
\(872\) 0 0
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) −40.6410 −1.37235 −0.686175 0.727437i \(-0.740711\pi\)
−0.686175 + 0.727437i \(0.740711\pi\)
\(878\) 0 0
\(879\) 2.53590 0.0855337
\(880\) 0 0
\(881\) 34.7321 1.17015 0.585076 0.810978i \(-0.301065\pi\)
0.585076 + 0.810978i \(0.301065\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 29.8564 1.00361
\(886\) 0 0
\(887\) −30.2487 −1.01565 −0.507826 0.861460i \(-0.669551\pi\)
−0.507826 + 0.861460i \(0.669551\pi\)
\(888\) 0 0
\(889\) −8.39230 −0.281469
\(890\) 0 0
\(891\) −3.46410 −0.116052
\(892\) 0 0
\(893\) −2.73205 −0.0914246
\(894\) 0 0
\(895\) −53.7128 −1.79542
\(896\) 0 0
\(897\) 29.8564 0.996876
\(898\) 0 0
\(899\) 30.5359 1.01843
\(900\) 0 0
\(901\) 36.6410 1.22069
\(902\) 0 0
\(903\) −3.46410 −0.115278
\(904\) 0 0
\(905\) −51.3205 −1.70595
\(906\) 0 0
\(907\) −33.0718 −1.09813 −0.549065 0.835779i \(-0.685016\pi\)
−0.549065 + 0.835779i \(0.685016\pi\)
\(908\) 0 0
\(909\) −16.1962 −0.537192
\(910\) 0 0
\(911\) −43.3731 −1.43701 −0.718507 0.695520i \(-0.755174\pi\)
−0.718507 + 0.695520i \(0.755174\pi\)
\(912\) 0 0
\(913\) −4.39230 −0.145364
\(914\) 0 0
\(915\) −39.3205 −1.29990
\(916\) 0 0
\(917\) −12.1962 −0.402752
\(918\) 0 0
\(919\) 3.71281 0.122474 0.0612372 0.998123i \(-0.480495\pi\)
0.0612372 + 0.998123i \(0.480495\pi\)
\(920\) 0 0
\(921\) −27.8564 −0.917900
\(922\) 0 0
\(923\) 40.7846 1.34244
\(924\) 0 0
\(925\) −18.3923 −0.604735
\(926\) 0 0
\(927\) 10.9282 0.358929
\(928\) 0 0
\(929\) −21.6603 −0.710650 −0.355325 0.934743i \(-0.615630\pi\)
−0.355325 + 0.934743i \(0.615630\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −14.7321 −0.482306
\(934\) 0 0
\(935\) 39.7128 1.29875
\(936\) 0 0
\(937\) 7.07180 0.231026 0.115513 0.993306i \(-0.463149\pi\)
0.115513 + 0.993306i \(0.463149\pi\)
\(938\) 0 0
\(939\) 30.3923 0.991815
\(940\) 0 0
\(941\) 18.6410 0.607680 0.303840 0.952723i \(-0.401731\pi\)
0.303840 + 0.952723i \(0.401731\pi\)
\(942\) 0 0
\(943\) −70.6410 −2.30039
\(944\) 0 0
\(945\) 2.73205 0.0888736
\(946\) 0 0
\(947\) 7.17691 0.233218 0.116609 0.993178i \(-0.462798\pi\)
0.116609 + 0.993178i \(0.462798\pi\)
\(948\) 0 0
\(949\) 63.4256 2.05888
\(950\) 0 0
\(951\) −2.87564 −0.0932492
\(952\) 0 0
\(953\) 0.732051 0.0237135 0.0118567 0.999930i \(-0.496226\pi\)
0.0118567 + 0.999930i \(0.496226\pi\)
\(954\) 0 0
\(955\) 1.46410 0.0473772
\(956\) 0 0
\(957\) −21.4641 −0.693836
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) 0 0
\(963\) −11.6603 −0.375746
\(964\) 0 0
\(965\) −17.4641 −0.562189
\(966\) 0 0
\(967\) −14.3923 −0.462825 −0.231413 0.972856i \(-0.574335\pi\)
−0.231413 + 0.972856i \(0.574335\pi\)
\(968\) 0 0
\(969\) −4.19615 −0.134800
\(970\) 0 0
\(971\) −15.7128 −0.504248 −0.252124 0.967695i \(-0.581129\pi\)
−0.252124 + 0.967695i \(0.581129\pi\)
\(972\) 0 0
\(973\) 22.9282 0.735044
\(974\) 0 0
\(975\) 9.85641 0.315658
\(976\) 0 0
\(977\) 52.7321 1.68705 0.843524 0.537092i \(-0.180477\pi\)
0.843524 + 0.537092i \(0.180477\pi\)
\(978\) 0 0
\(979\) −32.7846 −1.04780
\(980\) 0 0
\(981\) 3.46410 0.110600
\(982\) 0 0
\(983\) −7.71281 −0.246001 −0.123000 0.992407i \(-0.539252\pi\)
−0.123000 + 0.992407i \(0.539252\pi\)
\(984\) 0 0
\(985\) −27.3205 −0.870504
\(986\) 0 0
\(987\) 2.73205 0.0869621
\(988\) 0 0
\(989\) 25.8564 0.822186
\(990\) 0 0
\(991\) −22.6410 −0.719216 −0.359608 0.933104i \(-0.617090\pi\)
−0.359608 + 0.933104i \(0.617090\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 52.7846 1.67338
\(996\) 0 0
\(997\) 9.32051 0.295183 0.147592 0.989048i \(-0.452848\pi\)
0.147592 + 0.989048i \(0.452848\pi\)
\(998\) 0 0
\(999\) 7.46410 0.236154
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 6384.2.a.bj.1.1 2
4.3 odd 2 1596.2.a.i.1.1 2
12.11 even 2 4788.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1596.2.a.i.1.1 2 4.3 odd 2
4788.2.a.j.1.2 2 12.11 even 2
6384.2.a.bj.1.1 2 1.1 even 1 trivial