Properties

Label 240.2.f.a.49.2
Level $240$
Weight $2$
Character 240.49
Analytic conductor $1.916$
Analytic rank $0$
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] = [240,2,Mod(49,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("240.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.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: \(1.91640964851\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 240.49
Dual form 240.2.f.a.49.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −2.00000 + 1.00000i −0.894427 + 0.447214i
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 3.00000 4.00000i 0.600000 0.800000i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 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
\(35\) −2.00000 4.00000i −0.338062 0.676123i
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 2.00000 1.00000i 0.298142 0.149071i
\(46\) 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
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 4.00000 2.00000i 0.539360 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 2.00000i 0.251976i
\(64\) 0 0
\(65\) −6.00000 12.0000i −0.744208 1.48842i
\(66\) 0 0
\(67\) 8.00000i 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 4.00000i 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 4.00000 + 3.00000i 0.461880 + 0.346410i
\(76\) 0 0
\(77\) 4.00000i 0.455842i
\(78\) 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
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 0 0
\(85\) 2.00000 + 4.00000i 0.216930 + 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 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
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 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 2.00000i 0.390360 0.195180i
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 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
\(115\) −4.00000 8.00000i −0.373002 0.746004i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 2.00000i 0.180334i
\(124\) 0 0
\(125\) −2.00000 + 11.0000i −0.178885 + 0.983870i
\(126\) 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
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 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 \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000i 0.247436i
\(148\) 0 0
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) −16.0000 + 8.00000i −1.28515 + 0.642575i
\(156\) 0 0
\(157\) 22.0000i 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 16.0000i 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) 2.00000 + 4.00000i 0.155700 + 0.311400i
\(166\) 0 0
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.0000i 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 8.00000 + 6.00000i 0.604743 + 0.453557i
\(176\) 0 0
\(177\) 10.0000i 0.751646i
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 2.00000i 0.147844i
\(184\) 0 0
\(185\) 2.00000 + 4.00000i 0.147043 + 0.294086i
\(186\) 0 0
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 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
\(195\) 12.0000 6.00000i 0.859338 0.429669i
\(196\) 0 0
\(197\) 22.0000i 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 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
\(205\) −4.00000 + 2.00000i −0.279372 + 0.139686i
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 12.0000i 0.822226i
\(214\) 0 0
\(215\) −4.00000 8.00000i −0.272798 0.545595i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 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
\(227\) 28.0000i 1.85843i −0.369546 0.929213i \(-0.620487\pi\)
0.369546 0.929213i \(-0.379513\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 14.0000i 0.917170i −0.888650 0.458585i \(-0.848356\pi\)
0.888650 0.458585i \(-0.151644\pi\)
\(234\) 0 0
\(235\) 8.00000 + 16.0000i 0.521862 + 1.04372i
\(236\) 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
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −6.00000 + 3.00000i −0.383326 + 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 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
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.00000i 0.246651i 0.992366 + 0.123325i \(0.0393559\pi\)
−0.992366 + 0.123325i \(0.960644\pi\)
\(264\) 0 0
\(265\) −6.00000 12.0000i −0.368577 0.737154i
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 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
\(275\) −6.00000 + 8.00000i −0.361814 + 0.482418i
\(276\) 0 0
\(277\) 2.00000i 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 16.0000i 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000i 0.236113i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 0 0
\(295\) −20.0000 + 10.0000i −1.16445 + 0.582223i
\(296\) 0 0
\(297\) 2.00000i 0.116052i
\(298\) 0 0
\(299\) −24.0000 −1.38796
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 8.00000i 0.459588i
\(304\) 0 0
\(305\) −4.00000 + 2.00000i −0.229039 + 0.114520i
\(306\) 0 0
\(307\) 12.0000i 0.684876i 0.939540 + 0.342438i \(0.111253\pi\)
−0.939540 + 0.342438i \(0.888747\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 4.00000i 0.226093i −0.993590 0.113047i \(-0.963939\pi\)
0.993590 0.113047i \(-0.0360610\pi\)
\(314\) 0 0
\(315\) 2.00000 + 4.00000i 0.112687 + 0.225374i
\(316\) 0 0
\(317\) 2.00000i 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 24.0000 + 18.0000i 1.33128 + 0.998460i
\(326\) 0 0
\(327\) 10.0000i 0.553001i
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 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
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 8.00000 4.00000i 0.430706 0.215353i
\(346\) 0 0
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 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
\(355\) 24.0000 12.0000i 1.27379 0.636894i
\(356\) 0 0
\(357\) 4.00000i 0.211702i
\(358\) 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
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 4.00000 + 8.00000i 0.209370 + 0.418739i
\(366\) 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
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) 0 0
\(375\) −11.0000 2.00000i −0.568038 0.103280i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 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
\(387\) 4.00000i 0.203331i
\(388\) 0 0
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 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
\(403\) 48.0000i 2.39105i
\(404\) 0 0
\(405\) −2.00000 + 1.00000i −0.0993808 + 0.0496904i
\(406\) 0 0
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) 20.0000i 0.984136i
\(414\) 0 0
\(415\) −4.00000 8.00000i −0.196352 0.392705i
\(416\) 0 0
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) −8.00000 6.00000i −0.388057 0.291043i
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) 12.0000 0.579365
\(430\) 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
\(443\) 36.0000i 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) −20.0000 + 10.0000i −0.948091 + 0.474045i
\(446\) 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
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 24.0000 12.0000i 1.12514 0.562569i
\(456\) 0 0
\(457\) 32.0000i 1.49690i −0.663193 0.748448i \(-0.730799\pi\)
0.663193 0.748448i \(-0.269201\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 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
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 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
\(483\) 8.00000i 0.364013i
\(484\) 0 0
\(485\) −8.00000 16.0000i −0.363261 0.726523i
\(486\) 0 0
\(487\) 18.0000i 0.815658i −0.913058 0.407829i \(-0.866286\pi\)
0.913058 0.407829i \(-0.133714\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 16.0000 8.00000i 0.711991 0.355995i
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −14.0000 28.0000i −0.616914 1.23383i
\(516\) 0 0
\(517\) 16.0000i 0.703679i
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) −6.00000 + 8.00000i −0.261861 + 0.349149i
\(526\) 0 0
\(527\) 16.0000i 0.696971i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) −12.0000 24.0000i −0.518805 1.03761i
\(536\) 0 0
\(537\) 10.0000i 0.431532i
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) 20.0000 10.0000i 0.856706 0.428353i
\(546\) 0 0
\(547\) 28.0000i 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −4.00000 + 2.00000i −0.169791 + 0.0848953i
\(556\) 0 0
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) 0 0
\(565\) −6.00000 12.0000i −0.252422 0.504844i
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 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
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) 6.00000 + 12.0000i 0.248069 + 0.496139i
\(586\) 0 0
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 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
\(595\) −8.00000 + 4.00000i −0.327968 + 0.163984i
\(596\) 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 \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 0 0
\(605\) 14.0000 7.00000i 0.569181 0.284590i
\(606\) 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
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) −2.00000 4.00000i −0.0806478 0.161296i
\(616\) 0 0
\(617\) 2.00000i 0.0805170i −0.999189 0.0402585i \(-0.987182\pi\)
0.999189 0.0402585i \(-0.0128181\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 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
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) −2.00000 4.00000i −0.0793676 0.158735i
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) 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
\(643\) 24.0000i 0.946468i 0.880937 + 0.473234i \(0.156913\pi\)
−0.880937 + 0.473234i \(0.843087\pi\)
\(644\) 0 0
\(645\) 8.00000 4.00000i 0.315000 0.157500i
\(646\) 0 0
\(647\) 48.0000i 1.88707i −0.331266 0.943537i \(-0.607476\pi\)
0.331266 0.943537i \(-0.392524\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) 0 0
\(653\) 26.0000i 1.01746i 0.860927 + 0.508729i \(0.169885\pi\)
−0.860927 + 0.508729i \(0.830115\pi\)
\(654\) 0 0
\(655\) −36.0000 + 18.0000i −1.40664 + 0.703318i
\(656\) 0 0
\(657\) 4.00000i 0.156055i
\(658\) 0 0
\(659\) −50.0000 −1.94772 −0.973862 0.227142i \(-0.927062\pi\)
−0.973862 + 0.227142i \(0.927062\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) 12.0000i 0.466041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 26.0000 1.00522
\(670\) 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
\(675\) −4.00000 3.00000i −0.153960 0.115470i
\(676\) 0 0
\(677\) 2.00000i 0.0768662i −0.999261 0.0384331i \(-0.987763\pi\)
0.999261 0.0384331i \(-0.0122367\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 28.0000 1.07296
\(682\) 0 0
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) 0 0
\(685\) −18.0000 36.0000i −0.687745 1.37549i
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 4.00000i 0.151947i
\(694\) 0 0
\(695\) 40.0000 20.0000i 1.51729 0.758643i
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 0 0
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) 32.0000 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −16.0000 + 8.00000i −0.602595 + 0.301297i
\(706\) 0 0
\(707\) 16.0000i 0.601742i
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 12.0000 + 24.0000i 0.448775 + 0.897549i
\(716\) 0 0
\(717\) 20.0000i 0.746914i
\(718\) 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
\(723\) 22.0000i 0.818189i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 18.0000i 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 14.0000i 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) −3.00000 6.00000i −0.110657 0.221313i
\(736\) 0 0
\(737\) 16.0000i 0.589368i
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 0 0
\(745\) −40.0000 + 20.0000i −1.46549 + 0.732743i
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 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
\(755\) −16.0000 + 8.00000i −0.582300 + 0.291150i
\(756\) 0 0
\(757\) 2.00000i 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) 0 0
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 20.0000i 0.724049i
\(764\) 0 0
\(765\) −2.00000 4.00000i −0.0723102 0.144620i
\(766\) 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
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 54.0000i 1.94225i −0.238581 0.971123i \(-0.576682\pi\)
0.238581 0.971123i \(-0.423318\pi\)
\(774\) 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
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 22.0000 + 44.0000i 0.785214 + 1.57043i
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 0 0
\(789\) −4.00000 −0.142404
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 12.0000 6.00000i 0.425596 0.212798i
\(796\) 0 0
\(797\) 2.00000i 0.0708436i −0.999372 0.0354218i \(-0.988723\pi\)
0.999372 0.0354218i \(-0.0112775\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 0 0
\(803\) 8.00000i 0.282314i
\(804\) 0 0
\(805\) 16.0000 8.00000i 0.563926 0.281963i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 0 0
\(815\) 16.0000 + 32.0000i 0.560456 + 1.12091i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 0 0
\(823\) 6.00000i 0.209147i −0.994517 0.104573i \(-0.966652\pi\)
0.994517 0.104573i \(-0.0333477\pi\)
\(824\) 0 0
\(825\) −8.00000 6.00000i −0.278524 0.208893i
\(826\) 0 0
\(827\) 28.0000i 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) −12.0000 24.0000i −0.415277 0.830554i
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) 46.0000 23.0000i 1.58245 0.791224i
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 0 0
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 14.0000i 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.0000i 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 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
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 8.00000i 0.270759i
\(874\) 0 0
\(875\) −22.0000 4.00000i −0.743736 0.135225i
\(876\) 0 0
\(877\) 22.0000i 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 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
\(883\) 24.0000i 0.807664i 0.914833 + 0.403832i \(0.132322\pi\)
−0.914833 + 0.403832i \(0.867678\pi\)
\(884\) 0 0
\(885\) −10.0000 20.0000i −0.336146 0.672293i
\(886\) 0 0
\(887\) 12.0000i 0.402921i 0.979497 + 0.201460i \(0.0645687\pi\)
−0.979497 + 0.201460i \(0.935431\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 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
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) −4.00000 + 2.00000i −0.132964 + 0.0664822i
\(906\) 0 0
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 0 0
\(909\) 8.00000 0.265343
\(910\) 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
\(915\) −2.00000 4.00000i −0.0661180 0.132236i
\(916\) 0 0
\(917\) 36.0000i 1.18882i
\(918\) 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
\(923\) 72.0000i 2.36991i
\(924\) 0 0
\(925\) −8.00000 6.00000i −0.263038 0.197279i
\(926\) 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
\(933\) 12.0000i 0.392862i
\(934\) 0 0
\(935\) −4.00000 8.00000i −0.130814 0.261628i
\(936\) 0 0
\(937\) 8.00000i 0.261349i 0.991425 + 0.130674i \(0.0417142\pi\)
−0.991425 + 0.130674i \(0.958286\pi\)
\(938\) 0 0
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) −28.0000 −0.912774 −0.456387 0.889781i \(-0.650857\pi\)
−0.456387 + 0.889781i \(0.650857\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) 0 0
\(945\) −4.00000 + 2.00000i −0.130120 + 0.0650600i
\(946\) 0 0
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 46.0000i 1.49009i 0.667016 + 0.745043i \(0.267571\pi\)
−0.667016 + 0.745043i \(0.732429\pi\)
\(954\) 0 0
\(955\) 24.0000 12.0000i 0.776622 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 4.00000 + 8.00000i 0.128765 + 0.257529i
\(966\) 0 0
\(967\) 38.0000i 1.22200i −0.791632 0.610999i \(-0.790768\pi\)
0.791632 0.610999i \(-0.209232\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 0 0
\(973\) 40.0000i 1.28234i
\(974\) 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
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 16.0000i 0.510321i −0.966899 0.255160i \(-0.917872\pi\)
0.966899 0.255160i \(-0.0821283\pi\)
\(984\) 0 0
\(985\) 22.0000 + 44.0000i 0.700978 + 1.40196i
\(986\) 0 0
\(987\) 16.0000i 0.509286i
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 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
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 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 240.2.f.a.49.2 2
3.2 odd 2 720.2.f.f.289.1 2
4.3 odd 2 30.2.c.a.19.2 yes 2
5.2 odd 4 1200.2.a.m.1.1 1
5.3 odd 4 1200.2.a.g.1.1 1
5.4 even 2 inner 240.2.f.a.49.1 2
8.3 odd 2 960.2.f.h.769.2 2
8.5 even 2 960.2.f.i.769.1 2
12.11 even 2 90.2.c.a.19.1 2
15.2 even 4 3600.2.a.o.1.1 1
15.8 even 4 3600.2.a.bg.1.1 1
15.14 odd 2 720.2.f.f.289.2 2
16.3 odd 4 3840.2.d.y.2689.2 2
16.5 even 4 3840.2.d.x.2689.1 2
16.11 odd 4 3840.2.d.g.2689.1 2
16.13 even 4 3840.2.d.j.2689.2 2
20.3 even 4 150.2.a.c.1.1 1
20.7 even 4 150.2.a.a.1.1 1
20.19 odd 2 30.2.c.a.19.1 2
24.5 odd 2 2880.2.f.c.1729.2 2
24.11 even 2 2880.2.f.e.1729.2 2
28.3 even 6 1470.2.n.a.79.1 4
28.11 odd 6 1470.2.n.h.79.1 4
28.19 even 6 1470.2.n.a.949.2 4
28.23 odd 6 1470.2.n.h.949.2 4
28.27 even 2 1470.2.g.g.589.2 2
36.7 odd 6 810.2.i.e.109.2 4
36.11 even 6 810.2.i.b.109.1 4
36.23 even 6 810.2.i.b.379.2 4
36.31 odd 6 810.2.i.e.379.1 4
40.3 even 4 4800.2.a.l.1.1 1
40.13 odd 4 4800.2.a.cj.1.1 1
40.19 odd 2 960.2.f.h.769.1 2
40.27 even 4 4800.2.a.cg.1.1 1
40.29 even 2 960.2.f.i.769.2 2
40.37 odd 4 4800.2.a.m.1.1 1
60.23 odd 4 450.2.a.b.1.1 1
60.47 odd 4 450.2.a.f.1.1 1
60.59 even 2 90.2.c.a.19.2 2
80.19 odd 4 3840.2.d.g.2689.2 2
80.29 even 4 3840.2.d.x.2689.2 2
80.59 odd 4 3840.2.d.y.2689.1 2
80.69 even 4 3840.2.d.j.2689.1 2
120.29 odd 2 2880.2.f.c.1729.1 2
120.59 even 2 2880.2.f.e.1729.1 2
140.19 even 6 1470.2.n.a.949.1 4
140.27 odd 4 7350.2.a.bg.1.1 1
140.39 odd 6 1470.2.n.h.79.2 4
140.59 even 6 1470.2.n.a.79.2 4
140.79 odd 6 1470.2.n.h.949.1 4
140.83 odd 4 7350.2.a.cc.1.1 1
140.139 even 2 1470.2.g.g.589.1 2
180.59 even 6 810.2.i.b.379.1 4
180.79 odd 6 810.2.i.e.109.1 4
180.119 even 6 810.2.i.b.109.2 4
180.139 odd 6 810.2.i.e.379.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.c.a.19.1 2 20.19 odd 2
30.2.c.a.19.2 yes 2 4.3 odd 2
90.2.c.a.19.1 2 12.11 even 2
90.2.c.a.19.2 2 60.59 even 2
150.2.a.a.1.1 1 20.7 even 4
150.2.a.c.1.1 1 20.3 even 4
240.2.f.a.49.1 2 5.4 even 2 inner
240.2.f.a.49.2 2 1.1 even 1 trivial
450.2.a.b.1.1 1 60.23 odd 4
450.2.a.f.1.1 1 60.47 odd 4
720.2.f.f.289.1 2 3.2 odd 2
720.2.f.f.289.2 2 15.14 odd 2
810.2.i.b.109.1 4 36.11 even 6
810.2.i.b.109.2 4 180.119 even 6
810.2.i.b.379.1 4 180.59 even 6
810.2.i.b.379.2 4 36.23 even 6
810.2.i.e.109.1 4 180.79 odd 6
810.2.i.e.109.2 4 36.7 odd 6
810.2.i.e.379.1 4 36.31 odd 6
810.2.i.e.379.2 4 180.139 odd 6
960.2.f.h.769.1 2 40.19 odd 2
960.2.f.h.769.2 2 8.3 odd 2
960.2.f.i.769.1 2 8.5 even 2
960.2.f.i.769.2 2 40.29 even 2
1200.2.a.g.1.1 1 5.3 odd 4
1200.2.a.m.1.1 1 5.2 odd 4
1470.2.g.g.589.1 2 140.139 even 2
1470.2.g.g.589.2 2 28.27 even 2
1470.2.n.a.79.1 4 28.3 even 6
1470.2.n.a.79.2 4 140.59 even 6
1470.2.n.a.949.1 4 140.19 even 6
1470.2.n.a.949.2 4 28.19 even 6
1470.2.n.h.79.1 4 28.11 odd 6
1470.2.n.h.79.2 4 140.39 odd 6
1470.2.n.h.949.1 4 140.79 odd 6
1470.2.n.h.949.2 4 28.23 odd 6
2880.2.f.c.1729.1 2 120.29 odd 2
2880.2.f.c.1729.2 2 24.5 odd 2
2880.2.f.e.1729.1 2 120.59 even 2
2880.2.f.e.1729.2 2 24.11 even 2
3600.2.a.o.1.1 1 15.2 even 4
3600.2.a.bg.1.1 1 15.8 even 4
3840.2.d.g.2689.1 2 16.11 odd 4
3840.2.d.g.2689.2 2 80.19 odd 4
3840.2.d.j.2689.1 2 80.69 even 4
3840.2.d.j.2689.2 2 16.13 even 4
3840.2.d.x.2689.1 2 16.5 even 4
3840.2.d.x.2689.2 2 80.29 even 4
3840.2.d.y.2689.1 2 80.59 odd 4
3840.2.d.y.2689.2 2 16.3 odd 4
4800.2.a.l.1.1 1 40.3 even 4
4800.2.a.m.1.1 1 40.37 odd 4
4800.2.a.cg.1.1 1 40.27 even 4
4800.2.a.cj.1.1 1 40.13 odd 4
7350.2.a.bg.1.1 1 140.27 odd 4
7350.2.a.cc.1.1 1 140.83 odd 4