Properties

Label 3840.2.d.x.2689.1
Level $3840$
Weight $2$
Character 3840.2689
Analytic conductor $30.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(2689,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.d (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: \(30.6625543762\)
Analytic rank: \(0\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3840.2689
Dual form 3840.2.d.x.2689.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +(-1.00000 - 2.00000i) q^{5} -2.00000i q^{7} +1.00000 q^{9} -2.00000i q^{11} +6.00000 q^{13} +(-1.00000 - 2.00000i) q^{15} -2.00000i q^{17} -2.00000i q^{21} -4.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +8.00000 q^{31} -2.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +2.00000 q^{37} +6.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +(-1.00000 - 2.00000i) q^{45} -8.00000i q^{47} +3.00000 q^{49} -2.00000i q^{51} -6.00000 q^{53} +(-4.00000 + 2.00000i) q^{55} +10.0000i q^{59} -2.00000i q^{61} -2.00000i q^{63} +(-6.00000 - 12.0000i) q^{65} -8.00000 q^{67} -4.00000i q^{69} +12.0000 q^{71} +4.00000i q^{73} +(-3.00000 + 4.00000i) q^{75} -4.00000 q^{77} +1.00000 q^{81} +4.00000 q^{83} +(-4.00000 + 2.00000i) q^{85} -10.0000 q^{89} -12.0000i q^{91} +8.00000 q^{93} +8.00000i q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 12 q^{13} - 2 q^{15} - 6 q^{25} + 2 q^{27} + 16 q^{31} - 8 q^{35} + 4 q^{37} + 12 q^{39} - 4 q^{41} - 8 q^{43} - 2 q^{45} + 6 q^{49} - 12 q^{53} - 8 q^{55} - 12 q^{65} - 16 q^{67} + 24 q^{71} - 6 q^{75} - 8 q^{77} + 2 q^{81} + 8 q^{83} - 8 q^{85} - 20 q^{89} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(1537\) \(2561\) \(2821\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 2.00000i −0.447214 0.894427i
\(6\) 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
\(11\) 2.00000i 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) −1.00000 2.00000i −0.258199 0.516398i
\(16\) 0 0
\(17\) 2.00000i 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) −3.00000 + 4.00000i −0.600000 + 0.800000i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 2.00000i 0.348155i
\(34\) 0 0
\(35\) −4.00000 + 2.00000i −0.676123 + 0.338062i
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 2.00000i −0.149071 0.298142i
\(46\) 0 0
\(47\) 8.00000i 1.16692i −0.812142 0.583460i \(-0.801699\pi\)
0.812142 0.583460i \(-0.198301\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 2.00000i 0.280056i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −4.00000 + 2.00000i −0.539360 + 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000i 1.30189i 0.759125 + 0.650945i \(0.225627\pi\)
−0.759125 + 0.650945i \(0.774373\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i −0.991769 0.128037i \(-0.959132\pi\)
0.991769 0.128037i \(-0.0408676\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −6.00000 12.0000i −0.744208 1.48842i
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 4.00000i 0.481543i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i 0.972217 + 0.234082i \(0.0752085\pi\)
−0.972217 + 0.234082i \(0.924791\pi\)
\(74\) 0 0
\(75\) −3.00000 + 4.00000i −0.346410 + 0.461880i
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −4.00000 + 2.00000i −0.433861 + 0.216930i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 12.0000i 1.25794i
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 8.00000i 0.796030i −0.917379 0.398015i \(-0.869699\pi\)
0.917379 0.398015i \(-0.130301\pi\)
\(102\) 0 0
\(103\) 14.0000i 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) −4.00000 + 2.00000i −0.390360 + 0.195180i
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) −8.00000 + 4.00000i −0.746004 + 0.373002i
\(116\) 0 0
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 11.0000 + 2.00000i 0.983870 + 0.178885i
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 18.0000i 1.57267i −0.617802 0.786334i \(-0.711977\pi\)
0.617802 0.786334i \(-0.288023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −1.00000 2.00000i −0.0860663 0.172133i
\(136\) 0 0
\(137\) 18.0000i 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 20.0000i 1.69638i −0.529694 0.848189i \(-0.677693\pi\)
0.529694 0.848189i \(-0.322307\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 20.0000i 1.63846i 0.573462 + 0.819232i \(0.305600\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) −8.00000 16.0000i −0.642575 1.28515i
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) −4.00000 + 2.00000i −0.311400 + 0.155700i
\(166\) 0 0
\(167\) 12.0000i 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 8.00000 + 6.00000i 0.604743 + 0.453557i
\(176\) 0 0
\(177\) 10.0000i 0.751646i
\(178\) 0 0
\(179\) 10.0000i 0.747435i −0.927543 0.373718i \(-0.878083\pi\)
0.927543 0.373718i \(-0.121917\pi\)
\(180\) 0 0
\(181\) 2.00000i 0.148659i 0.997234 + 0.0743294i \(0.0236816\pi\)
−0.997234 + 0.0743294i \(0.976318\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) −2.00000 4.00000i −0.147043 0.294086i
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 2.00000i 0.145479i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i −0.989583 0.143963i \(-0.954015\pi\)
0.989583 0.143963i \(-0.0459847\pi\)
\(194\) 0 0
\(195\) −6.00000 12.0000i −0.429669 0.859338i
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.00000 + 4.00000i 0.139686 + 0.279372i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000i 0.826114i 0.910705 + 0.413057i \(0.135539\pi\)
−0.910705 + 0.413057i \(0.864461\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 4.00000 + 8.00000i 0.272798 + 0.545595i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 4.00000i 0.270295i
\(220\) 0 0
\(221\) 12.0000i 0.807207i
\(222\) 0 0
\(223\) 26.0000i 1.74109i −0.492090 0.870544i \(-0.663767\pi\)
0.492090 0.870544i \(-0.336233\pi\)
\(224\) 0 0
\(225\) −3.00000 + 4.00000i −0.200000 + 0.266667i
\(226\) 0 0
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 14.0000i 0.917170i 0.888650 + 0.458585i \(0.151644\pi\)
−0.888650 + 0.458585i \(0.848356\pi\)
\(234\) 0 0
\(235\) −16.0000 + 8.00000i −1.04372 + 0.521862i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.00000 6.00000i −0.191663 0.383326i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 18.0000i 1.13615i 0.822977 + 0.568075i \(0.192312\pi\)
−0.822977 + 0.568075i \(0.807688\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) −4.00000 + 2.00000i −0.250490 + 0.125245i
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.00000i 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 0 0
\(265\) 6.00000 + 12.0000i 0.368577 + 0.737154i
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 12.0000i 0.726273i
\(274\) 0 0
\(275\) 8.00000 + 6.00000i 0.482418 + 0.361814i
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000i 0.236113i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 8.00000i 0.468968i
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 20.0000 10.0000i 1.16445 0.582223i
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) 24.0000i 1.38796i
\(300\) 0 0
\(301\) 8.00000i 0.461112i
\(302\) 0 0
\(303\) 8.00000i 0.459588i
\(304\) 0 0
\(305\) −4.00000 + 2.00000i −0.229039 + 0.114520i
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 14.0000i 0.796432i
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i 0.993590 + 0.113047i \(0.0360610\pi\)
−0.993590 + 0.113047i \(0.963939\pi\)
\(314\) 0 0
\(315\) −4.00000 + 2.00000i −0.225374 + 0.112687i
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −18.0000 + 24.0000i −0.998460 + 1.33128i
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 8.00000i 0.439720i 0.975531 + 0.219860i \(0.0705600\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 8.00000 + 16.0000i 0.437087 + 0.874173i
\(336\) 0 0
\(337\) 28.0000i 1.52526i 0.646837 + 0.762629i \(0.276092\pi\)
−0.646837 + 0.762629i \(0.723908\pi\)
\(338\) 0 0
\(339\) 6.00000i 0.325875i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) −8.00000 + 4.00000i −0.430706 + 0.215353i
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.535288i 0.963518 + 0.267644i \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 14.0000i 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 0 0
\(355\) −12.0000 24.0000i −0.636894 1.27379i
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 8.00000 4.00000i 0.418739 0.209370i
\(366\) 0 0
\(367\) 2.00000i 0.104399i 0.998637 + 0.0521996i \(0.0166232\pi\)
−0.998637 + 0.0521996i \(0.983377\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 12.0000i 0.623009i
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 11.0000 + 2.00000i 0.568038 + 0.103280i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 0 0
\(381\) 2.00000i 0.102463i
\(382\) 0 0
\(383\) 16.0000i 0.817562i −0.912633 0.408781i \(-0.865954\pi\)
0.912633 0.408781i \(-0.134046\pi\)
\(384\) 0 0
\(385\) 4.00000 + 8.00000i 0.203859 + 0.407718i
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 20.0000i 1.01404i −0.861934 0.507020i \(-0.830747\pi\)
0.861934 0.507020i \(-0.169253\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) 0 0
\(405\) −1.00000 2.00000i −0.0496904 0.0993808i
\(406\) 0 0
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 18.0000i 0.887875i
\(412\) 0 0
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) −4.00000 8.00000i −0.196352 0.392705i
\(416\) 0 0
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) 10.0000i 0.488532i 0.969708 + 0.244266i \(0.0785470\pi\)
−0.969708 + 0.244266i \(0.921453\pi\)
\(420\) 0 0
\(421\) 22.0000i 1.07221i 0.844150 + 0.536107i \(0.180106\pi\)
−0.844150 + 0.536107i \(0.819894\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) 8.00000 + 6.00000i 0.388057 + 0.291043i
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 12.0000i 0.579365i
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) 4.00000i 0.192228i −0.995370 0.0961139i \(-0.969359\pi\)
0.995370 0.0961139i \(-0.0306413\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 10.0000 + 20.0000i 0.474045 + 0.948091i
\(446\) 0 0
\(447\) 20.0000i 0.945968i
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 4.00000i 0.188353i
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) −24.0000 + 12.0000i −1.12514 + 0.562569i
\(456\) 0 0
\(457\) 32.0000i 1.49690i 0.663193 + 0.748448i \(0.269201\pi\)
−0.663193 + 0.748448i \(0.730799\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 12.0000i 0.558896i −0.960161 0.279448i \(-0.909849\pi\)
0.960161 0.279448i \(-0.0901514\pi\)
\(462\) 0 0
\(463\) 6.00000i 0.278844i −0.990233 0.139422i \(-0.955476\pi\)
0.990233 0.139422i \(-0.0445244\pi\)
\(464\) 0 0
\(465\) −8.00000 16.0000i −0.370991 0.741982i
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 16.0000i 0.738811i
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) 16.0000 8.00000i 0.726523 0.363261i
\(486\) 0 0
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 18.0000i 0.812329i 0.913800 + 0.406164i \(0.133134\pi\)
−0.913800 + 0.406164i \(0.866866\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.00000 + 2.00000i −0.179787 + 0.0898933i
\(496\) 0 0
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 24.0000i 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) 0 0
\(505\) −16.0000 + 8.00000i −0.711991 + 0.355995i
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −28.0000 + 14.0000i −1.23383 + 0.616914i
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 8.00000 + 6.00000i 0.349149 + 0.261861i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 10.0000i 0.433963i
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 12.0000 + 24.0000i 0.518805 + 1.03761i
\(536\) 0 0
\(537\) 10.0000i 0.431532i
\(538\) 0 0
\(539\) 6.00000i 0.258438i
\(540\) 0 0
\(541\) 38.0000i 1.63375i 0.576816 + 0.816874i \(0.304295\pi\)
−0.576816 + 0.816874i \(0.695705\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) 20.0000 10.0000i 0.856706 0.428353i
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 2.00000i 0.0853579i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.00000 4.00000i −0.0848953 0.169791i
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 44.0000 1.85438 0.927189 0.374593i \(-0.122217\pi\)
0.927189 + 0.374593i \(0.122217\pi\)
\(564\) 0 0
\(565\) 12.0000 6.00000i 0.504844 0.252422i
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) 8.00000i 0.334790i 0.985890 + 0.167395i \(0.0535355\pi\)
−0.985890 + 0.167395i \(0.946465\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) 16.0000 + 12.0000i 0.667246 + 0.500435i
\(576\) 0 0
\(577\) 32.0000i 1.33218i −0.745873 0.666089i \(-0.767967\pi\)
0.745873 0.666089i \(-0.232033\pi\)
\(578\) 0 0
\(579\) 4.00000i 0.166234i
\(580\) 0 0
\(581\) 8.00000i 0.331896i
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) −6.00000 12.0000i −0.248069 0.496139i
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 4.00000 + 8.00000i 0.163984 + 0.327968i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −7.00000 14.0000i −0.284590 0.569181i
\(606\) 0 0
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 48.0000i 1.94187i
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 2.00000 + 4.00000i 0.0806478 + 0.161296i
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) 4.00000i 0.160514i
\(622\) 0 0
\(623\) 20.0000i 0.801283i
\(624\) 0 0
\(625\) −7.00000 24.0000i −0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.00000i 0.159490i
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) 4.00000 2.00000i 0.158735 0.0793676i
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 4.00000 + 8.00000i 0.157500 + 0.315000i
\(646\) 0 0
\(647\) 48.0000i 1.88707i 0.331266 + 0.943537i \(0.392524\pi\)
−0.331266 + 0.943537i \(0.607476\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) −36.0000 + 18.0000i −1.40664 + 0.703318i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) 50.0000i 1.94772i 0.227142 + 0.973862i \(0.427062\pi\)
−0.227142 + 0.973862i \(0.572938\pi\)
\(660\) 0 0
\(661\) 2.00000i 0.0777910i 0.999243 + 0.0388955i \(0.0123839\pi\)
−0.999243 + 0.0388955i \(0.987616\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 26.0000i 1.00522i
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 36.0000i 1.38770i 0.720121 + 0.693849i \(0.244086\pi\)
−0.720121 + 0.693849i \(0.755914\pi\)
\(674\) 0 0
\(675\) −3.00000 + 4.00000i −0.115470 + 0.153960i
\(676\) 0 0
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 0 0
\(685\) −36.0000 + 18.0000i −1.37549 + 0.687745i
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 8.00000i 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −40.0000 + 20.0000i −1.51729 + 0.758643i
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 0 0
\(699\) 14.0000i 0.529529i
\(700\) 0 0
\(701\) 32.0000i 1.20862i −0.796748 0.604312i \(-0.793448\pi\)
0.796748 0.604312i \(-0.206552\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −16.0000 + 8.00000i −0.602595 + 0.301297i
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 30.0000i 1.12667i −0.826227 0.563337i \(-0.809517\pi\)
0.826227 0.563337i \(-0.190483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) −24.0000 + 12.0000i −0.897549 + 0.448775i
\(716\) 0 0
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 0 0
\(723\) 22.0000 0.818189
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.0000i 0.667583i 0.942647 + 0.333792i \(0.108328\pi\)
−0.942647 + 0.333792i \(0.891672\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000i 0.295891i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) −3.00000 6.00000i −0.110657 0.221313i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) 40.0000i 1.47142i −0.677295 0.735712i \(-0.736848\pi\)
0.677295 0.735712i \(-0.263152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000i 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 40.0000 20.0000i 1.46549 0.732743i
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 24.0000i 0.876941i
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 18.0000i 0.655956i
\(754\) 0 0
\(755\) 8.00000 + 16.0000i 0.291150 + 0.582300i
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 20.0000 0.724049
\(764\) 0 0
\(765\) −4.00000 + 2.00000i −0.144620 + 0.0723102i
\(766\) 0 0
\(767\) 60.0000i 2.16647i
\(768\) 0 0
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 18.0000i 0.648254i
\(772\) 0 0
\(773\) 54.0000 1.94225 0.971123 0.238581i \(-0.0766824\pi\)
0.971123 + 0.238581i \(0.0766824\pi\)
\(774\) 0 0
\(775\) −24.0000 + 32.0000i −0.862105 + 1.14947i
\(776\) 0 0
\(777\) 4.00000i 0.143499i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 24.0000i 0.858788i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.0000 + 44.0000i 0.785214 + 1.57043i
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) 4.00000i 0.142404i
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 6.00000 + 12.0000i 0.212798 + 0.425596i
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 8.00000 + 16.0000i 0.281963 + 0.563926i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 52.0000i 1.82597i −0.407997 0.912983i \(-0.633772\pi\)
0.407997 0.912983i \(-0.366228\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 16.0000 + 32.0000i 0.560456 + 1.12091i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 12.0000i 0.419314i
\(820\) 0 0
\(821\) 8.00000i 0.279202i −0.990208 0.139601i \(-0.955418\pi\)
0.990208 0.139601i \(-0.0445820\pi\)
\(822\) 0 0
\(823\) 6.00000i 0.209147i 0.994517 + 0.104573i \(0.0333477\pi\)
−0.994517 + 0.104573i \(0.966652\pi\)
\(824\) 0 0
\(825\) 8.00000 + 6.00000i 0.278524 + 0.208893i
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 30.0000i 1.04194i −0.853574 0.520972i \(-0.825570\pi\)
0.853574 0.520972i \(-0.174430\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) −24.0000 + 12.0000i −0.830554 + 0.415277i
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) 0 0
\(845\) −23.0000 46.0000i −0.791224 1.58245i
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 0 0
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 8.00000i 0.274236i
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000i 1.43469i 0.696717 + 0.717346i \(0.254643\pi\)
−0.696717 + 0.717346i \(0.745357\pi\)
\(858\) 0 0
\(859\) 20.0000i 0.682391i 0.939992 + 0.341196i \(0.110832\pi\)
−0.939992 + 0.341196i \(0.889168\pi\)
\(860\) 0 0
\(861\) 4.00000i 0.136320i
\(862\) 0 0
\(863\) 36.0000i 1.22545i −0.790295 0.612727i \(-0.790072\pi\)
0.790295 0.612727i \(-0.209928\pi\)
\(864\) 0 0
\(865\) 14.0000 + 28.0000i 0.476014 + 0.952029i
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 8.00000i 0.270759i
\(874\) 0 0
\(875\) 4.00000 22.0000i 0.135225 0.743736i
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 0 0
\(885\) 20.0000 10.0000i 0.672293 0.336146i
\(886\) 0 0
\(887\) 12.0000i 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 2.00000i 0.0670025i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −20.0000 + 10.0000i −0.668526 + 0.334263i
\(896\) 0 0
\(897\) 24.0000i 0.801337i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000i 0.399778i
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) 4.00000 2.00000i 0.132964 0.0664822i
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) 8.00000i 0.265343i
\(910\) 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
\(915\) −4.00000 + 2.00000i −0.132236 + 0.0661180i
\(916\) 0 0
\(917\) −36.0000 −1.18882
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −6.00000 + 8.00000i −0.197279 + 0.263038i
\(926\) 0 0
\(927\) 14.0000i 0.459820i
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 12.0000 0.392862
\(934\) 0 0
\(935\) 4.00000 + 8.00000i 0.130814 + 0.261628i
\(936\) 0 0
\(937\) 8.00000i 0.261349i −0.991425 0.130674i \(-0.958286\pi\)
0.991425 0.130674i \(-0.0417142\pi\)
\(938\) 0 0
\(939\) 4.00000i 0.130535i
\(940\) 0 0
\(941\) 28.0000i 0.912774i 0.889781 + 0.456387i \(0.150857\pi\)
−0.889781 + 0.456387i \(0.849143\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) 0 0
\(945\) −4.00000 + 2.00000i −0.130120 + 0.0650600i
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 24.0000i 0.779073i
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) 46.0000i 1.49009i −0.667016 0.745043i \(-0.732429\pi\)
0.667016 0.745043i \(-0.267571\pi\)
\(954\) 0 0
\(955\) 12.0000 + 24.0000i 0.388311 + 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −8.00000 + 4.00000i −0.257529 + 0.128765i
\(966\) 0 0
\(967\) 38.0000i 1.22200i 0.791632 + 0.610999i \(0.209232\pi\)
−0.791632 + 0.610999i \(0.790768\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000i 0.577647i 0.957382 + 0.288824i \(0.0932642\pi\)
−0.957382 + 0.288824i \(0.906736\pi\)
\(972\) 0 0
\(973\) −40.0000 −1.28234
\(974\) 0 0
\(975\) −18.0000 + 24.0000i −0.576461 + 0.768615i
\(976\) 0 0
\(977\) 42.0000i 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 0 0
\(979\) 20.0000i 0.639203i
\(980\) 0 0
\(981\) 10.0000i 0.319275i
\(982\) 0 0
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) −22.0000 44.0000i −0.700978 1.40196i
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) 16.0000i 0.508770i
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 8.00000i 0.253872i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −18.0000 −0.570066 −0.285033 0.958518i \(-0.592005\pi\)
−0.285033 + 0.958518i \(0.592005\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 3840.2.d.x.2689.1 2
4.3 odd 2 3840.2.d.g.2689.1 2
5.4 even 2 3840.2.d.j.2689.1 2
8.3 odd 2 3840.2.d.y.2689.2 2
8.5 even 2 3840.2.d.j.2689.2 2
16.3 odd 4 30.2.c.a.19.2 yes 2
16.5 even 4 960.2.f.i.769.1 2
16.11 odd 4 960.2.f.h.769.2 2
16.13 even 4 240.2.f.a.49.2 2
20.19 odd 2 3840.2.d.y.2689.1 2
40.19 odd 2 3840.2.d.g.2689.2 2
40.29 even 2 inner 3840.2.d.x.2689.2 2
48.5 odd 4 2880.2.f.c.1729.2 2
48.11 even 4 2880.2.f.e.1729.2 2
48.29 odd 4 720.2.f.f.289.1 2
48.35 even 4 90.2.c.a.19.1 2
80.3 even 4 150.2.a.c.1.1 1
80.13 odd 4 1200.2.a.g.1.1 1
80.19 odd 4 30.2.c.a.19.1 2
80.27 even 4 4800.2.a.cg.1.1 1
80.29 even 4 240.2.f.a.49.1 2
80.37 odd 4 4800.2.a.m.1.1 1
80.43 even 4 4800.2.a.l.1.1 1
80.53 odd 4 4800.2.a.cj.1.1 1
80.59 odd 4 960.2.f.h.769.1 2
80.67 even 4 150.2.a.a.1.1 1
80.69 even 4 960.2.f.i.769.2 2
80.77 odd 4 1200.2.a.m.1.1 1
112.3 even 12 1470.2.n.a.79.1 4
112.19 even 12 1470.2.n.a.949.2 4
112.51 odd 12 1470.2.n.h.949.2 4
112.67 odd 12 1470.2.n.h.79.1 4
112.83 even 4 1470.2.g.g.589.2 2
144.67 odd 12 810.2.i.e.379.1 4
144.83 even 12 810.2.i.b.109.1 4
144.115 odd 12 810.2.i.e.109.2 4
144.131 even 12 810.2.i.b.379.2 4
240.29 odd 4 720.2.f.f.289.2 2
240.59 even 4 2880.2.f.e.1729.1 2
240.77 even 4 3600.2.a.o.1.1 1
240.83 odd 4 450.2.a.b.1.1 1
240.149 odd 4 2880.2.f.c.1729.1 2
240.173 even 4 3600.2.a.bg.1.1 1
240.179 even 4 90.2.c.a.19.2 2
240.227 odd 4 450.2.a.f.1.1 1
560.19 even 12 1470.2.n.a.949.1 4
560.83 odd 4 7350.2.a.cc.1.1 1
560.179 odd 12 1470.2.n.h.79.2 4
560.307 odd 4 7350.2.a.bg.1.1 1
560.339 even 12 1470.2.n.a.79.2 4
560.419 even 4 1470.2.g.g.589.1 2
560.499 odd 12 1470.2.n.h.949.1 4
720.259 odd 12 810.2.i.e.109.1 4
720.419 even 12 810.2.i.b.379.1 4
720.499 odd 12 810.2.i.e.379.2 4
720.659 even 12 810.2.i.b.109.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.c.a.19.1 2 80.19 odd 4
30.2.c.a.19.2 yes 2 16.3 odd 4
90.2.c.a.19.1 2 48.35 even 4
90.2.c.a.19.2 2 240.179 even 4
150.2.a.a.1.1 1 80.67 even 4
150.2.a.c.1.1 1 80.3 even 4
240.2.f.a.49.1 2 80.29 even 4
240.2.f.a.49.2 2 16.13 even 4
450.2.a.b.1.1 1 240.83 odd 4
450.2.a.f.1.1 1 240.227 odd 4
720.2.f.f.289.1 2 48.29 odd 4
720.2.f.f.289.2 2 240.29 odd 4
810.2.i.b.109.1 4 144.83 even 12
810.2.i.b.109.2 4 720.659 even 12
810.2.i.b.379.1 4 720.419 even 12
810.2.i.b.379.2 4 144.131 even 12
810.2.i.e.109.1 4 720.259 odd 12
810.2.i.e.109.2 4 144.115 odd 12
810.2.i.e.379.1 4 144.67 odd 12
810.2.i.e.379.2 4 720.499 odd 12
960.2.f.h.769.1 2 80.59 odd 4
960.2.f.h.769.2 2 16.11 odd 4
960.2.f.i.769.1 2 16.5 even 4
960.2.f.i.769.2 2 80.69 even 4
1200.2.a.g.1.1 1 80.13 odd 4
1200.2.a.m.1.1 1 80.77 odd 4
1470.2.g.g.589.1 2 560.419 even 4
1470.2.g.g.589.2 2 112.83 even 4
1470.2.n.a.79.1 4 112.3 even 12
1470.2.n.a.79.2 4 560.339 even 12
1470.2.n.a.949.1 4 560.19 even 12
1470.2.n.a.949.2 4 112.19 even 12
1470.2.n.h.79.1 4 112.67 odd 12
1470.2.n.h.79.2 4 560.179 odd 12
1470.2.n.h.949.1 4 560.499 odd 12
1470.2.n.h.949.2 4 112.51 odd 12
2880.2.f.c.1729.1 2 240.149 odd 4
2880.2.f.c.1729.2 2 48.5 odd 4
2880.2.f.e.1729.1 2 240.59 even 4
2880.2.f.e.1729.2 2 48.11 even 4
3600.2.a.o.1.1 1 240.77 even 4
3600.2.a.bg.1.1 1 240.173 even 4
3840.2.d.g.2689.1 2 4.3 odd 2
3840.2.d.g.2689.2 2 40.19 odd 2
3840.2.d.j.2689.1 2 5.4 even 2
3840.2.d.j.2689.2 2 8.5 even 2
3840.2.d.x.2689.1 2 1.1 even 1 trivial
3840.2.d.x.2689.2 2 40.29 even 2 inner
3840.2.d.y.2689.1 2 20.19 odd 2
3840.2.d.y.2689.2 2 8.3 odd 2
4800.2.a.l.1.1 1 80.43 even 4
4800.2.a.m.1.1 1 80.37 odd 4
4800.2.a.cg.1.1 1 80.27 even 4
4800.2.a.cj.1.1 1 80.53 odd 4
7350.2.a.bg.1.1 1 560.307 odd 4
7350.2.a.cc.1.1 1 560.83 odd 4