Properties

Label 1280.2.f.a.129.1
Level $1280$
Weight $2$
Character 1280.129
Analytic conductor $10.221$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(129,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1280.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(10.2208514587\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.129
Dual form 1280.2.f.a.129.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +(-2.00000 - 1.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +(-2.00000 - 1.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} -4.00000i q^{11} -4.00000 q^{13} +(4.00000 + 2.00000i) q^{15} -4.00000i q^{19} +4.00000i q^{21} -2.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +4.00000 q^{27} +2.00000i q^{29} +8.00000i q^{33} +(-2.00000 + 4.00000i) q^{35} -4.00000 q^{37} +8.00000 q^{39} -2.00000 q^{41} -6.00000 q^{43} +(-2.00000 - 1.00000i) q^{45} -6.00000i q^{47} +3.00000 q^{49} -4.00000 q^{53} +(-4.00000 + 8.00000i) q^{55} +8.00000i q^{57} +12.0000i q^{59} +10.0000i q^{61} -2.00000i q^{63} +(8.00000 + 4.00000i) q^{65} +14.0000 q^{67} +4.00000i q^{69} -8.00000 q^{71} +8.00000i q^{73} +(-6.00000 - 8.00000i) q^{75} -8.00000 q^{77} -16.0000 q^{79} -11.0000 q^{81} -2.00000 q^{83} -4.00000i q^{87} +6.00000 q^{89} +8.00000i q^{91} +(-4.00000 + 8.00000i) q^{95} +16.0000i q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 4 q^{5} + 2 q^{9} - 8 q^{13} + 8 q^{15} + 6 q^{25} + 8 q^{27} - 4 q^{35} - 8 q^{37} + 16 q^{39} - 4 q^{41} - 12 q^{43} - 4 q^{45} + 6 q^{49} - 8 q^{53} - 8 q^{55} + 16 q^{65} + 28 q^{67} - 16 q^{71} - 12 q^{75} - 16 q^{77} - 32 q^{79} - 22 q^{81} - 4 q^{83} + 12 q^{89} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) −2.00000 1.00000i −0.894427 0.447214i
\(6\) 0 0
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 4.00000 + 2.00000i 1.03280 + 0.516398i
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 4.00000i 0.872872i
\(22\) 0 0
\(23\) 2.00000i 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 8.00000i 1.39262i
\(34\) 0 0
\(35\) −2.00000 + 4.00000i −0.338062 + 0.676123i
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −2.00000 1.00000i −0.298142 0.149071i
\(46\) 0 0
\(47\) 6.00000i 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −4.00000 + 8.00000i −0.539360 + 1.07872i
\(56\) 0 0
\(57\) 8.00000i 1.05963i
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) 10.0000i 1.28037i 0.768221 + 0.640184i \(0.221142\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) 8.00000 + 4.00000i 0.992278 + 0.496139i
\(66\) 0 0
\(67\) 14.0000 1.71037 0.855186 0.518321i \(-0.173443\pi\)
0.855186 + 0.518321i \(0.173443\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) −6.00000 8.00000i −0.692820 0.923760i
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.00000i 0.428845i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 + 8.00000i −0.410391 + 0.820783i
\(96\) 0 0
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i 0.724066 + 0.689730i \(0.242271\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 0 0
\(105\) 4.00000 8.00000i 0.390360 0.780720i
\(106\) 0 0
\(107\) 10.0000 0.966736 0.483368 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 16.0000i 1.50515i −0.658505 0.752577i \(-0.728811\pi\)
0.658505 0.752577i \(-0.271189\pi\)
\(114\) 0 0
\(115\) −2.00000 + 4.00000i −0.186501 + 0.373002i
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) −2.00000 11.0000i −0.178885 0.983870i
\(126\) 0 0
\(127\) 6.00000i 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) −8.00000 4.00000i −0.688530 0.344265i
\(136\) 0 0
\(137\) 8.00000i 0.683486i −0.939793 0.341743i \(-0.888983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) 12.0000i 1.01058i
\(142\) 0 0
\(143\) 16.0000i 1.33799i
\(144\) 0 0
\(145\) 2.00000 4.00000i 0.166091 0.332182i
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 18.0000i 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 8.00000 16.0000i 0.622799 1.24560i
\(166\) 0 0
\(167\) 18.0000i 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 0 0
\(175\) 8.00000 6.00000i 0.604743 0.453557i
\(176\) 0 0
\(177\) 24.0000i 1.80395i
\(178\) 0 0
\(179\) 4.00000i 0.298974i −0.988764 0.149487i \(-0.952238\pi\)
0.988764 0.149487i \(-0.0477622\pi\)
\(180\) 0 0
\(181\) 22.0000i 1.63525i 0.575753 + 0.817624i \(0.304709\pi\)
−0.575753 + 0.817624i \(0.695291\pi\)
\(182\) 0 0
\(183\) 20.0000i 1.47844i
\(184\) 0 0
\(185\) 8.00000 + 4.00000i 0.588172 + 0.294086i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.00000i 0.581914i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) 0 0
\(195\) −16.0000 8.00000i −1.14578 0.572892i
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −28.0000 −1.97497
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 4.00000 + 2.00000i 0.279372 + 0.139686i
\(206\) 0 0
\(207\) 2.00000i 0.139010i
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 4.00000i 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) 16.0000 1.09630
\(214\) 0 0
\(215\) 12.0000 + 6.00000i 0.818393 + 0.409197i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 16.0000i 1.08118i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 6.00000i 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 0 0
\(225\) 3.00000 + 4.00000i 0.200000 + 0.266667i
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) 6.00000i 0.396491i 0.980152 + 0.198246i \(0.0635244\pi\)
−0.980152 + 0.198246i \(0.936476\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) 24.0000i 1.57229i −0.618041 0.786146i \(-0.712073\pi\)
0.618041 0.786146i \(-0.287927\pi\)
\(234\) 0 0
\(235\) −6.00000 + 12.0000i −0.391397 + 0.782794i
\(236\) 0 0
\(237\) 32.0000 2.07862
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) −6.00000 3.00000i −0.383326 0.191663i
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 20.0000i 1.26239i −0.775625 0.631194i \(-0.782565\pi\)
0.775625 0.631194i \(-0.217435\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 8.00000i 0.497096i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) 30.0000i 1.84988i 0.380114 + 0.924940i \(0.375885\pi\)
−0.380114 + 0.924940i \(0.624115\pi\)
\(264\) 0 0
\(265\) 8.00000 + 4.00000i 0.491436 + 0.245718i
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 0 0
\(269\) 6.00000i 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 16.0000i 0.968364i
\(274\) 0 0
\(275\) 16.0000 12.0000i 0.964836 0.723627i
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 0 0
\(285\) 8.00000 16.0000i 0.473879 0.947758i
\(286\) 0 0
\(287\) 4.00000i 0.236113i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 32.0000i 1.87587i
\(292\) 0 0
\(293\) 28.0000 1.63578 0.817889 0.575376i \(-0.195144\pi\)
0.817889 + 0.575376i \(0.195144\pi\)
\(294\) 0 0
\(295\) 12.0000 24.0000i 0.698667 1.39733i
\(296\) 0 0
\(297\) 16.0000i 0.928414i
\(298\) 0 0
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 12.0000i 0.691669i
\(302\) 0 0
\(303\) 12.0000i 0.689382i
\(304\) 0 0
\(305\) 10.0000 20.0000i 0.572598 1.14520i
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 28.0000i 1.59286i
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i 0.974106 + 0.226093i \(0.0725954\pi\)
−0.974106 + 0.226093i \(0.927405\pi\)
\(314\) 0 0
\(315\) −2.00000 + 4.00000i −0.112687 + 0.225374i
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −12.0000 16.0000i −0.665640 0.887520i
\(326\) 0 0
\(327\) 12.0000i 0.663602i
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 28.0000i 1.53902i 0.638635 + 0.769510i \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −28.0000 14.0000i −1.52980 0.764902i
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 0 0
\(339\) 32.0000i 1.73800i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 4.00000 8.00000i 0.215353 0.430706i
\(346\) 0 0
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) 0 0
\(349\) 2.00000i 0.107058i 0.998566 + 0.0535288i \(0.0170469\pi\)
−0.998566 + 0.0535288i \(0.982953\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 16.0000 + 8.00000i 0.849192 + 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) 8.00000 16.0000i 0.418739 0.837478i
\(366\) 0 0
\(367\) 10.0000i 0.521996i 0.965339 + 0.260998i \(0.0840516\pi\)
−0.965339 + 0.260998i \(0.915948\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 8.00000i 0.415339i
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) 4.00000 + 22.0000i 0.206559 + 1.13608i
\(376\) 0 0
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) 20.0000i 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 12.0000i 0.614779i
\(382\) 0 0
\(383\) 6.00000i 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 0 0
\(385\) 16.0000 + 8.00000i 0.815436 + 0.407718i
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) 30.0000i 1.52106i 0.649303 + 0.760530i \(0.275061\pi\)
−0.649303 + 0.760530i \(0.724939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 24.0000i 1.21064i
\(394\) 0 0
\(395\) 32.0000 + 16.0000i 1.61009 + 0.805047i
\(396\) 0 0
\(397\) −36.0000 −1.80679 −0.903394 0.428811i \(-0.858933\pi\)
−0.903394 + 0.428811i \(0.858933\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 22.0000 + 11.0000i 1.09319 + 0.546594i
\(406\) 0 0
\(407\) 16.0000i 0.793091i
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 16.0000i 0.789222i
\(412\) 0 0
\(413\) 24.0000 1.18096
\(414\) 0 0
\(415\) 4.00000 + 2.00000i 0.196352 + 0.0981761i
\(416\) 0 0
\(417\) 8.00000i 0.391762i
\(418\) 0 0
\(419\) 12.0000i 0.586238i 0.956076 + 0.293119i \(0.0946933\pi\)
−0.956076 + 0.293119i \(0.905307\pi\)
\(420\) 0 0
\(421\) 2.00000i 0.0974740i −0.998812 0.0487370i \(-0.984480\pi\)
0.998812 0.0487370i \(-0.0155196\pi\)
\(422\) 0 0
\(423\) 6.00000i 0.291730i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 0 0
\(429\) 32.0000i 1.54497i
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) −4.00000 + 8.00000i −0.191785 + 0.383571i
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) −12.0000 6.00000i −0.568855 0.284427i
\(446\) 0 0
\(447\) 36.0000i 1.70274i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 8.00000i 0.376705i
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) 8.00000 16.0000i 0.375046 0.750092i
\(456\) 0 0
\(457\) 24.0000i 1.12267i 0.827588 + 0.561336i \(0.189713\pi\)
−0.827588 + 0.561336i \(0.810287\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000i 0.838344i 0.907907 + 0.419172i \(0.137680\pi\)
−0.907907 + 0.419172i \(0.862320\pi\)
\(462\) 0 0
\(463\) 26.0000i 1.20832i 0.796862 + 0.604161i \(0.206492\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) 28.0000i 1.29292i
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) 24.0000i 1.10352i
\(474\) 0 0
\(475\) 16.0000 12.0000i 0.734130 0.550598i
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) 16.0000 32.0000i 0.726523 1.45305i
\(486\) 0 0
\(487\) 2.00000i 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 4.00000i 0.180517i −0.995918 0.0902587i \(-0.971231\pi\)
0.995918 0.0902587i \(-0.0287694\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 + 8.00000i −0.179787 + 0.359573i
\(496\) 0 0
\(497\) 16.0000i 0.717698i
\(498\) 0 0
\(499\) 4.00000i 0.179065i −0.995984 0.0895323i \(-0.971463\pi\)
0.995984 0.0895323i \(-0.0285372\pi\)
\(500\) 0 0
\(501\) 36.0000i 1.60836i
\(502\) 0 0
\(503\) 34.0000i 1.51599i −0.652263 0.757993i \(-0.726180\pi\)
0.652263 0.757993i \(-0.273820\pi\)
\(504\) 0 0
\(505\) 6.00000 12.0000i 0.266996 0.533993i
\(506\) 0 0
\(507\) −6.00000 −0.266469
\(508\) 0 0
\(509\) 34.0000i 1.50702i 0.657434 + 0.753512i \(0.271642\pi\)
−0.657434 + 0.753512i \(0.728358\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 16.0000i 0.706417i
\(514\) 0 0
\(515\) 14.0000 28.0000i 0.616914 1.23383i
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) 0 0
\(525\) −16.0000 + 12.0000i −0.698297 + 0.523723i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −20.0000 10.0000i −0.864675 0.432338i
\(536\) 0 0
\(537\) 8.00000i 0.345225i
\(538\) 0 0
\(539\) 12.0000i 0.516877i
\(540\) 0 0
\(541\) 2.00000i 0.0859867i 0.999075 + 0.0429934i \(0.0136894\pi\)
−0.999075 + 0.0429934i \(0.986311\pi\)
\(542\) 0 0
\(543\) 44.0000i 1.88822i
\(544\) 0 0
\(545\) −6.00000 + 12.0000i −0.257012 + 0.514024i
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 0 0
\(549\) 10.0000i 0.426790i
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) −16.0000 8.00000i −0.679162 0.339581i
\(556\) 0 0
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) 0 0
\(565\) −16.0000 + 32.0000i −0.673125 + 1.34625i
\(566\) 0 0
\(567\) 22.0000i 0.923913i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 8.00000 6.00000i 0.333623 0.250217i
\(576\) 0 0
\(577\) 32.0000i 1.33218i 0.745873 + 0.666089i \(0.232033\pi\)
−0.745873 + 0.666089i \(0.767967\pi\)
\(578\) 0 0
\(579\) 32.0000i 1.32987i
\(580\) 0 0
\(581\) 4.00000i 0.165948i
\(582\) 0 0
\(583\) 16.0000i 0.662652i
\(584\) 0 0
\(585\) 8.00000 + 4.00000i 0.330759 + 0.165380i
\(586\) 0 0
\(587\) −38.0000 −1.56843 −0.784214 0.620491i \(-0.786934\pi\)
−0.784214 + 0.620491i \(0.786934\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 0 0
\(593\) 16.0000i 0.657041i 0.944497 + 0.328521i \(0.106550\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 0 0
\(603\) 14.0000 0.570124
\(604\) 0 0
\(605\) 10.0000 + 5.00000i 0.406558 + 0.203279i
\(606\) 0 0
\(607\) 10.0000i 0.405887i 0.979190 + 0.202944i \(0.0650509\pi\)
−0.979190 + 0.202944i \(0.934949\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 24.0000i 0.970936i
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) −8.00000 4.00000i −0.322591 0.161296i
\(616\) 0 0
\(617\) 8.00000i 0.322068i 0.986949 + 0.161034i \(0.0514829\pi\)
−0.986949 + 0.161034i \(0.948517\pi\)
\(618\) 0 0
\(619\) 4.00000i 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 8.00000i 0.321029i
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 32.0000 1.27796
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 8.00000i 0.317971i
\(634\) 0 0
\(635\) −6.00000 + 12.0000i −0.238103 + 0.476205i
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) −18.0000 −0.709851 −0.354925 0.934895i \(-0.615494\pi\)
−0.354925 + 0.934895i \(0.615494\pi\)
\(644\) 0 0
\(645\) −24.0000 12.0000i −0.944999 0.472500i
\(646\) 0 0
\(647\) 2.00000i 0.0786281i −0.999227 0.0393141i \(-0.987483\pi\)
0.999227 0.0393141i \(-0.0125173\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) 12.0000 24.0000i 0.468879 0.937758i
\(656\) 0 0
\(657\) 8.00000i 0.312110i
\(658\) 0 0
\(659\) 36.0000i 1.40236i −0.712984 0.701180i \(-0.752657\pi\)
0.712984 0.701180i \(-0.247343\pi\)
\(660\) 0 0
\(661\) 14.0000i 0.544537i 0.962221 + 0.272268i \(0.0877739\pi\)
−0.962221 + 0.272268i \(0.912226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 + 8.00000i 0.620453 + 0.310227i
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 12.0000i 0.463947i
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 16.0000i 0.616755i 0.951264 + 0.308377i \(0.0997859\pi\)
−0.951264 + 0.308377i \(0.900214\pi\)
\(674\) 0 0
\(675\) 12.0000 + 16.0000i 0.461880 + 0.615840i
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) −38.0000 −1.45403 −0.727015 0.686622i \(-0.759093\pi\)
−0.727015 + 0.686622i \(0.759093\pi\)
\(684\) 0 0
\(685\) −8.00000 + 16.0000i −0.305664 + 0.611329i
\(686\) 0 0
\(687\) 12.0000i 0.457829i
\(688\) 0 0
\(689\) 16.0000 0.609551
\(690\) 0 0
\(691\) 36.0000i 1.36950i −0.728776 0.684752i \(-0.759910\pi\)
0.728776 0.684752i \(-0.240090\pi\)
\(692\) 0 0
\(693\) −8.00000 −0.303895
\(694\) 0 0
\(695\) −4.00000 + 8.00000i −0.151729 + 0.303457i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 48.0000i 1.81553i
\(700\) 0 0
\(701\) 22.0000i 0.830929i −0.909610 0.415464i \(-0.863619\pi\)
0.909610 0.415464i \(-0.136381\pi\)
\(702\) 0 0
\(703\) 16.0000i 0.603451i
\(704\) 0 0
\(705\) 12.0000 24.0000i 0.451946 0.903892i
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 38.0000i 1.42712i 0.700594 + 0.713560i \(0.252918\pi\)
−0.700594 + 0.713560i \(0.747082\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 16.0000 32.0000i 0.598366 1.19673i
\(716\) 0 0
\(717\) −32.0000 −1.19506
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 44.0000 1.63638
\(724\) 0 0
\(725\) −8.00000 + 6.00000i −0.297113 + 0.222834i
\(726\) 0 0
\(727\) 18.0000i 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −36.0000 −1.32969 −0.664845 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) 0 0
\(735\) 12.0000 + 6.00000i 0.442627 + 0.221313i
\(736\) 0 0
\(737\) 56.0000i 2.06279i
\(738\) 0 0
\(739\) 12.0000i 0.441427i 0.975339 + 0.220714i \(0.0708386\pi\)
−0.975339 + 0.220714i \(0.929161\pi\)
\(740\) 0 0
\(741\) 32.0000i 1.17555i
\(742\) 0 0
\(743\) 30.0000i 1.10059i 0.834969 + 0.550297i \(0.185485\pi\)
−0.834969 + 0.550297i \(0.814515\pi\)
\(744\) 0 0
\(745\) −18.0000 + 36.0000i −0.659469 + 1.31894i
\(746\) 0 0
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) 20.0000i 0.730784i
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 40.0000i 1.45768i
\(754\) 0 0
\(755\) −16.0000 8.00000i −0.582300 0.291150i
\(756\) 0 0
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 0 0
\(763\) −12.0000 −0.434429
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 48.0000i 1.73318i
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.0000 −0.719350 −0.359675 0.933078i \(-0.617112\pi\)
−0.359675 + 0.933078i \(0.617112\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 16.0000i 0.573997i
\(778\) 0 0
\(779\) 8.00000i 0.286630i
\(780\) 0 0
\(781\) 32.0000i 1.14505i
\(782\) 0 0
\(783\) 8.00000i 0.285897i
\(784\) 0 0
\(785\) 8.00000 + 4.00000i 0.285532 + 0.142766i
\(786\) 0 0
\(787\) −2.00000 −0.0712923 −0.0356462 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) 0 0
\(789\) 60.0000i 2.13606i
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) 0 0
\(793\) 40.0000i 1.42044i
\(794\) 0 0
\(795\) −16.0000 8.00000i −0.567462 0.283731i
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 32.0000 1.12926
\(804\) 0 0
\(805\) 8.00000 + 4.00000i 0.281963 + 0.140981i
\(806\) 0 0
\(807\) 12.0000i 0.422420i
\(808\) 0 0
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 36.0000i 1.26413i −0.774915 0.632065i \(-0.782207\pi\)
0.774915 0.632065i \(-0.217793\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 4.00000 + 2.00000i 0.140114 + 0.0700569i
\(816\) 0 0
\(817\) 24.0000i 0.839654i
\(818\) 0 0
\(819\) 8.00000i 0.279543i
\(820\) 0 0
\(821\) 18.0000i 0.628204i −0.949389 0.314102i \(-0.898297\pi\)
0.949389 0.314102i \(-0.101703\pi\)
\(822\) 0 0
\(823\) 2.00000i 0.0697156i −0.999392 0.0348578i \(-0.988902\pi\)
0.999392 0.0348578i \(-0.0110978\pi\)
\(824\) 0 0
\(825\) −32.0000 + 24.0000i −1.11410 + 0.835573i
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 0 0
\(829\) 10.0000i 0.347314i 0.984806 + 0.173657i \(0.0555585\pi\)
−0.984806 + 0.173657i \(0.944442\pi\)
\(830\) 0 0
\(831\) −24.0000 −0.832551
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −18.0000 + 36.0000i −0.622916 + 1.24583i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 36.0000 1.23991
\(844\) 0 0
\(845\) −6.00000 3.00000i −0.206406 0.103203i
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 8.00000i 0.274236i
\(852\) 0 0
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 0 0
\(855\) −4.00000 + 8.00000i −0.136797 + 0.273594i
\(856\) 0 0
\(857\) 8.00000i 0.273275i 0.990621 + 0.136637i \(0.0436295\pi\)
−0.990621 + 0.136637i \(0.956370\pi\)
\(858\) 0 0
\(859\) 44.0000i 1.50126i 0.660722 + 0.750630i \(0.270250\pi\)
−0.660722 + 0.750630i \(0.729750\pi\)
\(860\) 0 0
\(861\) 8.00000i 0.272639i
\(862\) 0 0
\(863\) 42.0000i 1.42970i 0.699280 + 0.714848i \(0.253504\pi\)
−0.699280 + 0.714848i \(0.746496\pi\)
\(864\) 0 0
\(865\) −24.0000 12.0000i −0.816024 0.408012i
\(866\) 0 0
\(867\) −34.0000 −1.15470
\(868\) 0 0
\(869\) 64.0000i 2.17105i
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) 0 0
\(873\) 16.0000i 0.541518i
\(874\) 0 0
\(875\) −22.0000 + 4.00000i −0.743736 + 0.135225i
\(876\) 0 0
\(877\) 44.0000 1.48577 0.742887 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(878\) 0 0
\(879\) −56.0000 −1.88883
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) −50.0000 −1.68263 −0.841317 0.540542i \(-0.818219\pi\)
−0.841317 + 0.540542i \(0.818219\pi\)
\(884\) 0 0
\(885\) −24.0000 + 48.0000i −0.806751 + 1.61350i
\(886\) 0 0
\(887\) 18.0000i 0.604381i −0.953248 0.302190i \(-0.902282\pi\)
0.953248 0.302190i \(-0.0977178\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 44.0000i 1.47406i
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −4.00000 + 8.00000i −0.133705 + 0.267411i
\(896\) 0 0
\(897\) 16.0000i 0.534224i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 24.0000i 0.798670i
\(904\) 0 0
\(905\) 22.0000 44.0000i 0.731305 1.46261i
\(906\) 0 0
\(907\) −22.0000 −0.730498 −0.365249 0.930910i \(-0.619016\pi\)
−0.365249 + 0.930910i \(0.619016\pi\)
\(908\) 0 0
\(909\) 6.00000i 0.199007i
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 8.00000i 0.264761i
\(914\) 0 0
\(915\) −20.0000 + 40.0000i −0.661180 + 1.32236i
\(916\) 0 0
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −12.0000 16.0000i −0.394558 0.526077i
\(926\) 0 0
\(927\) 14.0000i 0.459820i
\(928\) 0 0
\(929\) −38.0000 −1.24674 −0.623370 0.781927i \(-0.714237\pi\)
−0.623370 + 0.781927i \(0.714237\pi\)
\(930\) 0 0
\(931\) 12.0000i 0.393284i
\(932\) 0 0
\(933\) 48.0000 1.57145
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 40.0000i 1.30674i 0.757037 + 0.653372i \(0.226646\pi\)
−0.757037 + 0.653372i \(0.773354\pi\)
\(938\) 0 0
\(939\) 16.0000i 0.522140i
\(940\) 0 0
\(941\) 14.0000i 0.456387i −0.973616 0.228193i \(-0.926718\pi\)
0.973616 0.228193i \(-0.0732819\pi\)
\(942\) 0 0
\(943\) 4.00000i 0.130258i
\(944\) 0 0
\(945\) −8.00000 + 16.0000i −0.260240 + 0.520480i
\(946\) 0 0
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 0 0
\(949\) 32.0000i 1.03876i
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 24.0000i 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −16.0000 −0.517207
\(958\) 0 0
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 10.0000 0.322245
\(964\) 0 0
\(965\) 16.0000 32.0000i 0.515058 1.03012i
\(966\) 0 0
\(967\) 50.0000i 1.60789i −0.594703 0.803946i \(-0.702730\pi\)
0.594703 0.803946i \(-0.297270\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000i 1.15529i −0.816286 0.577647i \(-0.803971\pi\)
0.816286 0.577647i \(-0.196029\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) 24.0000 + 32.0000i 0.768615 + 1.02482i
\(976\) 0 0
\(977\) 32.0000i 1.02377i 0.859054 + 0.511885i \(0.171053\pi\)
−0.859054 + 0.511885i \(0.828947\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767043i
\(980\) 0 0
\(981\) 6.00000i 0.191565i
\(982\) 0 0
\(983\) 46.0000i 1.46717i 0.679597 + 0.733586i \(0.262155\pi\)
−0.679597 + 0.733586i \(0.737845\pi\)
\(984\) 0 0
\(985\) −24.0000 12.0000i −0.764704 0.382352i
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) 12.0000i 0.381578i
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 56.0000i 1.77711i
\(994\) 0 0
\(995\) 16.0000 + 8.00000i 0.507234 + 0.253617i
\(996\) 0 0
\(997\) 12.0000 0.380044 0.190022 0.981780i \(-0.439144\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(998\) 0 0
\(999\) −16.0000 −0.506218
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 1280.2.f.a.129.1 2
4.3 odd 2 1280.2.f.e.129.1 2
5.4 even 2 1280.2.f.f.129.1 2
8.3 odd 2 1280.2.f.b.129.2 2
8.5 even 2 1280.2.f.f.129.2 2
16.3 odd 4 80.2.c.a.49.2 2
16.5 even 4 320.2.c.c.129.2 2
16.11 odd 4 320.2.c.b.129.1 2
16.13 even 4 40.2.c.a.9.1 2
20.19 odd 2 1280.2.f.b.129.1 2
40.19 odd 2 1280.2.f.e.129.2 2
40.29 even 2 inner 1280.2.f.a.129.2 2
48.5 odd 4 2880.2.f.h.1729.2 2
48.11 even 4 2880.2.f.i.1729.2 2
48.29 odd 4 360.2.f.c.289.1 2
48.35 even 4 720.2.f.e.289.1 2
80.3 even 4 400.2.a.b.1.1 1
80.13 odd 4 200.2.a.d.1.1 1
80.19 odd 4 80.2.c.a.49.1 2
80.27 even 4 1600.2.a.d.1.1 1
80.29 even 4 40.2.c.a.9.2 yes 2
80.37 odd 4 1600.2.a.v.1.1 1
80.43 even 4 1600.2.a.u.1.1 1
80.53 odd 4 1600.2.a.f.1.1 1
80.59 odd 4 320.2.c.b.129.2 2
80.67 even 4 400.2.a.g.1.1 1
80.69 even 4 320.2.c.c.129.1 2
80.77 odd 4 200.2.a.b.1.1 1
112.13 odd 4 1960.2.g.b.1569.2 2
240.29 odd 4 360.2.f.c.289.2 2
240.59 even 4 2880.2.f.i.1729.1 2
240.77 even 4 1800.2.a.j.1.1 1
240.83 odd 4 3600.2.a.k.1.1 1
240.149 odd 4 2880.2.f.h.1729.1 2
240.173 even 4 1800.2.a.s.1.1 1
240.179 even 4 720.2.f.e.289.2 2
240.227 odd 4 3600.2.a.bb.1.1 1
560.13 even 4 9800.2.a.d.1.1 1
560.237 even 4 9800.2.a.bf.1.1 1
560.349 odd 4 1960.2.g.b.1569.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.c.a.9.1 2 16.13 even 4
40.2.c.a.9.2 yes 2 80.29 even 4
80.2.c.a.49.1 2 80.19 odd 4
80.2.c.a.49.2 2 16.3 odd 4
200.2.a.b.1.1 1 80.77 odd 4
200.2.a.d.1.1 1 80.13 odd 4
320.2.c.b.129.1 2 16.11 odd 4
320.2.c.b.129.2 2 80.59 odd 4
320.2.c.c.129.1 2 80.69 even 4
320.2.c.c.129.2 2 16.5 even 4
360.2.f.c.289.1 2 48.29 odd 4
360.2.f.c.289.2 2 240.29 odd 4
400.2.a.b.1.1 1 80.3 even 4
400.2.a.g.1.1 1 80.67 even 4
720.2.f.e.289.1 2 48.35 even 4
720.2.f.e.289.2 2 240.179 even 4
1280.2.f.a.129.1 2 1.1 even 1 trivial
1280.2.f.a.129.2 2 40.29 even 2 inner
1280.2.f.b.129.1 2 20.19 odd 2
1280.2.f.b.129.2 2 8.3 odd 2
1280.2.f.e.129.1 2 4.3 odd 2
1280.2.f.e.129.2 2 40.19 odd 2
1280.2.f.f.129.1 2 5.4 even 2
1280.2.f.f.129.2 2 8.5 even 2
1600.2.a.d.1.1 1 80.27 even 4
1600.2.a.f.1.1 1 80.53 odd 4
1600.2.a.u.1.1 1 80.43 even 4
1600.2.a.v.1.1 1 80.37 odd 4
1800.2.a.j.1.1 1 240.77 even 4
1800.2.a.s.1.1 1 240.173 even 4
1960.2.g.b.1569.1 2 560.349 odd 4
1960.2.g.b.1569.2 2 112.13 odd 4
2880.2.f.h.1729.1 2 240.149 odd 4
2880.2.f.h.1729.2 2 48.5 odd 4
2880.2.f.i.1729.1 2 240.59 even 4
2880.2.f.i.1729.2 2 48.11 even 4
3600.2.a.k.1.1 1 240.83 odd 4
3600.2.a.bb.1.1 1 240.227 odd 4
9800.2.a.d.1.1 1 560.13 even 4
9800.2.a.bf.1.1 1 560.237 even 4