Properties

Label 1620.2.i.k.541.1
Level $1620$
Weight $2$
Character 1620.541
Analytic conductor $12.936$
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] = [1620,2,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), 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: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.2.i.k.1081.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{5} +(2.00000 - 3.46410i) q^{7} +(-3.00000 + 5.19615i) q^{11} +(2.00000 + 3.46410i) q^{13} -3.00000 q^{17} -7.00000 q^{19} +(4.50000 + 7.79423i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(3.50000 + 6.06218i) q^{31} +4.00000 q^{35} +2.00000 q^{37} +(-3.00000 - 5.19615i) q^{41} +(-1.00000 + 1.73205i) q^{43} +(-4.50000 - 7.79423i) q^{49} +9.00000 q^{53} -6.00000 q^{55} +(6.00000 + 10.3923i) q^{59} +(3.50000 - 6.06218i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(-1.00000 - 1.73205i) q^{67} +6.00000 q^{71} +2.00000 q^{73} +(12.0000 + 20.7846i) q^{77} +(0.500000 - 0.866025i) q^{79} +(-4.50000 + 7.79423i) q^{83} +(-1.50000 - 2.59808i) q^{85} -6.00000 q^{89} +16.0000 q^{91} +(-3.50000 - 6.06218i) q^{95} +(-4.00000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + 4 q^{7} - 6 q^{11} + 4 q^{13} - 6 q^{17} - 14 q^{19} + 9 q^{23} - q^{25} + 7 q^{31} + 8 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{43} - 9 q^{49} + 18 q^{53} - 12 q^{55} + 12 q^{59} + 7 q^{61} - 4 q^{65} - 2 q^{67} + 12 q^{71} + 4 q^{73} + 24 q^{77} + q^{79} - 9 q^{83} - 3 q^{85} - 12 q^{89} + 32 q^{91} - 7 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) 2.00000 3.46410i 0.755929 1.30931i −0.188982 0.981981i \(-0.560519\pi\)
0.944911 0.327327i \(-0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 + 5.19615i −0.904534 + 1.56670i −0.0829925 + 0.996550i \(0.526448\pi\)
−0.821541 + 0.570149i \(0.806886\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.50000 + 7.79423i 0.938315 + 1.62521i 0.768613 + 0.639713i \(0.220947\pi\)
0.169701 + 0.985496i \(0.445720\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 3.50000 + 6.06218i 0.628619 + 1.08880i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 5.19615i −0.468521 0.811503i 0.530831 0.847477i \(-0.321880\pi\)
−0.999353 + 0.0359748i \(0.988546\pi\)
\(42\) 0 0
\(43\) −1.00000 + 1.73205i −0.152499 + 0.264135i −0.932145 0.362084i \(-0.882065\pi\)
0.779647 + 0.626219i \(0.215399\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −4.50000 7.79423i −0.642857 1.11346i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) 3.50000 6.06218i 0.448129 0.776182i −0.550135 0.835076i \(-0.685424\pi\)
0.998264 + 0.0588933i \(0.0187572\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 + 3.46410i −0.248069 + 0.429669i
\(66\) 0 0
\(67\) −1.00000 1.73205i −0.122169 0.211604i 0.798454 0.602056i \(-0.205652\pi\)
−0.920623 + 0.390453i \(0.872318\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 + 20.7846i 1.36753 + 2.36863i
\(78\) 0 0
\(79\) 0.500000 0.866025i 0.0562544 0.0974355i −0.836527 0.547926i \(-0.815418\pi\)
0.892781 + 0.450490i \(0.148751\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.50000 + 7.79423i −0.493939 + 0.855528i −0.999976 0.00698436i \(-0.997777\pi\)
0.506036 + 0.862512i \(0.331110\pi\)
\(84\) 0 0
\(85\) −1.50000 2.59808i −0.162698 0.281801i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.50000 6.06218i −0.359092 0.621966i
\(96\) 0 0
\(97\) −4.00000 + 6.92820i −0.406138 + 0.703452i −0.994453 0.105180i \(-0.966458\pi\)
0.588315 + 0.808632i \(0.299792\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 + 10.3923i −0.597022 + 1.03407i 0.396236 + 0.918149i \(0.370316\pi\)
−0.993258 + 0.115924i \(0.963017\pi\)
\(102\) 0 0
\(103\) 2.00000 + 3.46410i 0.197066 + 0.341328i 0.947576 0.319531i \(-0.103525\pi\)
−0.750510 + 0.660859i \(0.770192\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.00000 5.19615i −0.282216 0.488813i 0.689714 0.724082i \(-0.257736\pi\)
−0.971930 + 0.235269i \(0.924403\pi\)
\(114\) 0 0
\(115\) −4.50000 + 7.79423i −0.419627 + 0.726816i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 + 10.3923i −0.550019 + 0.952661i
\(120\) 0 0
\(121\) −12.5000 21.6506i −1.13636 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 + 5.19615i 0.262111 + 0.453990i 0.966803 0.255524i \(-0.0822479\pi\)
−0.704692 + 0.709514i \(0.748915\pi\)
\(132\) 0 0
\(133\) −14.0000 + 24.2487i −1.21395 + 2.10263i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.50000 7.79423i 0.384461 0.665906i −0.607233 0.794524i \(-0.707721\pi\)
0.991694 + 0.128618i \(0.0410540\pi\)
\(138\) 0 0
\(139\) 2.00000 + 3.46410i 0.169638 + 0.293821i 0.938293 0.345843i \(-0.112407\pi\)
−0.768655 + 0.639664i \(0.779074\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −24.0000 −2.00698
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.00000 + 5.19615i 0.245770 + 0.425685i 0.962348 0.271821i \(-0.0876260\pi\)
−0.716578 + 0.697507i \(0.754293\pi\)
\(150\) 0 0
\(151\) 8.00000 13.8564i 0.651031 1.12762i −0.331842 0.943335i \(-0.607670\pi\)
0.982873 0.184284i \(-0.0589965\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.50000 + 6.06218i −0.281127 + 0.486926i
\(156\) 0 0
\(157\) 5.00000 + 8.66025i 0.399043 + 0.691164i 0.993608 0.112884i \(-0.0360089\pi\)
−0.594565 + 0.804048i \(0.702676\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 36.0000 2.83720
\(162\) 0 0
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.50000 2.59808i −0.116073 0.201045i 0.802135 0.597143i \(-0.203697\pi\)
−0.918208 + 0.396098i \(0.870364\pi\)
\(168\) 0 0
\(169\) −1.50000 + 2.59808i −0.115385 + 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.50000 7.79423i 0.342129 0.592584i −0.642699 0.766119i \(-0.722185\pi\)
0.984828 + 0.173534i \(0.0555188\pi\)
\(174\) 0 0
\(175\) 2.00000 + 3.46410i 0.151186 + 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 + 1.73205i 0.0735215 + 0.127343i
\(186\) 0 0
\(187\) 9.00000 15.5885i 0.658145 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 + 15.5885i −0.651217 + 1.12794i 0.331611 + 0.943416i \(0.392408\pi\)
−0.982828 + 0.184525i \(0.940925\pi\)
\(192\) 0 0
\(193\) −4.00000 6.92820i −0.287926 0.498703i 0.685388 0.728178i \(-0.259632\pi\)
−0.973315 + 0.229475i \(0.926299\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 5.19615i 0.209529 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 21.0000 36.3731i 1.45260 2.51598i
\(210\) 0 0
\(211\) 6.50000 + 11.2583i 0.447478 + 0.775055i 0.998221 0.0596196i \(-0.0189888\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 28.0000 1.90076
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 10.3923i −0.403604 0.699062i
\(222\) 0 0
\(223\) −7.00000 + 12.1244i −0.468755 + 0.811907i −0.999362 0.0357107i \(-0.988630\pi\)
0.530607 + 0.847618i \(0.321964\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.50000 12.9904i 0.497792 0.862202i −0.502204 0.864749i \(-0.667477\pi\)
0.999997 + 0.00254715i \(0.000810783\pi\)
\(228\) 0 0
\(229\) 12.5000 + 21.6506i 0.826023 + 1.43071i 0.901135 + 0.433539i \(0.142735\pi\)
−0.0751115 + 0.997175i \(0.523931\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.00000 15.5885i −0.582162 1.00833i −0.995223 0.0976302i \(-0.968874\pi\)
0.413061 0.910703i \(-0.364460\pi\)
\(240\) 0 0
\(241\) 12.5000 21.6506i 0.805196 1.39464i −0.110963 0.993825i \(-0.535394\pi\)
0.916159 0.400815i \(-0.131273\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.50000 7.79423i 0.287494 0.497955i
\(246\) 0 0
\(247\) −14.0000 24.2487i −0.890799 1.54291i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −54.0000 −3.39495
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.50000 12.9904i −0.467837 0.810318i 0.531487 0.847066i \(-0.321633\pi\)
−0.999325 + 0.0367485i \(0.988300\pi\)
\(258\) 0 0
\(259\) 4.00000 6.92820i 0.248548 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 10.3923i 0.369976 0.640817i −0.619586 0.784929i \(-0.712699\pi\)
0.989561 + 0.144112i \(0.0460326\pi\)
\(264\) 0 0
\(265\) 4.50000 + 7.79423i 0.276433 + 0.478796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.00000 5.19615i −0.180907 0.313340i
\(276\) 0 0
\(277\) 11.0000 19.0526i 0.660926 1.14476i −0.319447 0.947604i \(-0.603497\pi\)
0.980373 0.197153i \(-0.0631696\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 20.7846i 0.715860 1.23991i −0.246767 0.969075i \(-0.579368\pi\)
0.962627 0.270831i \(-0.0872985\pi\)
\(282\) 0 0
\(283\) −1.00000 1.73205i −0.0594438 0.102960i 0.834772 0.550596i \(-0.185599\pi\)
−0.894216 + 0.447636i \(0.852266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.50000 12.9904i −0.438155 0.758906i 0.559393 0.828903i \(-0.311034\pi\)
−0.997547 + 0.0699967i \(0.977701\pi\)
\(294\) 0 0
\(295\) −6.00000 + 10.3923i −0.349334 + 0.605063i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −18.0000 + 31.1769i −1.04097 + 1.80301i
\(300\) 0 0
\(301\) 4.00000 + 6.92820i 0.230556 + 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) −10.0000 + 17.3205i −0.565233 + 0.979013i 0.431795 + 0.901972i \(0.357881\pi\)
−0.997028 + 0.0770410i \(0.975453\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.50000 + 7.79423i −0.252745 + 0.437767i −0.964281 0.264883i \(-0.914667\pi\)
0.711535 + 0.702650i \(0.248000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 21.0000 1.16847
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 + 6.92820i −0.219860 + 0.380808i −0.954765 0.297361i \(-0.903893\pi\)
0.734905 + 0.678170i \(0.237227\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 1.73205i 0.0546358 0.0946320i
\(336\) 0 0
\(337\) 14.0000 + 24.2487i 0.762629 + 1.32091i 0.941491 + 0.337037i \(0.109425\pi\)
−0.178863 + 0.983874i \(0.557242\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −42.0000 −2.27443
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 + 10.3923i 0.322097 + 0.557888i 0.980921 0.194409i \(-0.0622790\pi\)
−0.658824 + 0.752297i \(0.728946\pi\)
\(348\) 0 0
\(349\) 9.50000 16.4545i 0.508523 0.880788i −0.491428 0.870918i \(-0.663525\pi\)
0.999951 0.00987003i \(-0.00314178\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.00000 + 15.5885i −0.479022 + 0.829690i −0.999711 0.0240566i \(-0.992342\pi\)
0.520689 + 0.853746i \(0.325675\pi\)
\(354\) 0 0
\(355\) 3.00000 + 5.19615i 0.159223 + 0.275783i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.00000 + 1.73205i 0.0523424 + 0.0906597i
\(366\) 0 0
\(367\) 2.00000 3.46410i 0.104399 0.180825i −0.809093 0.587680i \(-0.800041\pi\)
0.913493 + 0.406855i \(0.133375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 18.0000 31.1769i 0.934513 1.61862i
\(372\) 0 0
\(373\) 2.00000 + 3.46410i 0.103556 + 0.179364i 0.913147 0.407630i \(-0.133645\pi\)
−0.809591 + 0.586994i \(0.800311\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.50000 + 7.79423i 0.229939 + 0.398266i 0.957790 0.287469i \(-0.0928139\pi\)
−0.727851 + 0.685736i \(0.759481\pi\)
\(384\) 0 0
\(385\) −12.0000 + 20.7846i −0.611577 + 1.05928i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.0000 25.9808i 0.760530 1.31728i −0.182047 0.983290i \(-0.558272\pi\)
0.942578 0.333987i \(-0.108394\pi\)
\(390\) 0 0
\(391\) −13.5000 23.3827i −0.682724 1.18251i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 5.19615i −0.149813 0.259483i 0.781345 0.624099i \(-0.214534\pi\)
−0.931158 + 0.364615i \(0.881200\pi\)
\(402\) 0 0
\(403\) −14.0000 + 24.2487i −0.697390 + 1.20791i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 + 10.3923i −0.297409 + 0.515127i
\(408\) 0 0
\(409\) −8.50000 14.7224i −0.420298 0.727977i 0.575670 0.817682i \(-0.304741\pi\)
−0.995968 + 0.0897044i \(0.971408\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 48.0000 2.36193
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.0000 25.9808i −0.732798 1.26924i −0.955683 0.294398i \(-0.904881\pi\)
0.222885 0.974845i \(-0.428453\pi\)
\(420\) 0 0
\(421\) 18.5000 32.0429i 0.901635 1.56168i 0.0762630 0.997088i \(-0.475701\pi\)
0.825372 0.564590i \(-0.190966\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.50000 2.59808i 0.0727607 0.126025i
\(426\) 0 0
\(427\) −14.0000 24.2487i −0.677507 1.17348i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −31.5000 54.5596i −1.50685 2.60994i
\(438\) 0 0
\(439\) 9.50000 16.4545i 0.453410 0.785330i −0.545185 0.838316i \(-0.683541\pi\)
0.998595 + 0.0529862i \(0.0168739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.5000 23.3827i 0.641404 1.11094i −0.343715 0.939074i \(-0.611685\pi\)
0.985119 0.171871i \(-0.0549812\pi\)
\(444\) 0 0
\(445\) −3.00000 5.19615i −0.142214 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 + 13.8564i 0.375046 + 0.649598i
\(456\) 0 0
\(457\) −4.00000 + 6.92820i −0.187112 + 0.324088i −0.944286 0.329125i \(-0.893246\pi\)
0.757174 + 0.653213i \(0.226579\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 5.19615i 0.139724 0.242009i −0.787668 0.616100i \(-0.788712\pi\)
0.927392 + 0.374091i \(0.122045\pi\)
\(462\) 0 0
\(463\) 11.0000 + 19.0526i 0.511213 + 0.885448i 0.999916 + 0.0129968i \(0.00413714\pi\)
−0.488702 + 0.872451i \(0.662530\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 10.3923i −0.275880 0.477839i
\(474\) 0 0
\(475\) 3.50000 6.06218i 0.160591 0.278152i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.00000 + 5.19615i −0.137073 + 0.237418i −0.926388 0.376571i \(-0.877103\pi\)
0.789314 + 0.613990i \(0.210436\pi\)
\(480\) 0 0
\(481\) 4.00000 + 6.92820i 0.182384 + 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.0000 + 25.9808i 0.676941 + 1.17250i 0.975898 + 0.218229i \(0.0700279\pi\)
−0.298957 + 0.954267i \(0.596639\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 20.7846i 0.538274 0.932317i
\(498\) 0 0
\(499\) −2.50000 4.33013i −0.111915 0.193843i 0.804627 0.593780i \(-0.202365\pi\)
−0.916542 + 0.399937i \(0.869032\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −33.0000 −1.47140 −0.735699 0.677309i \(-0.763146\pi\)
−0.735699 + 0.677309i \(0.763146\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 + 10.3923i 0.265945 + 0.460631i 0.967811 0.251679i \(-0.0809826\pi\)
−0.701866 + 0.712309i \(0.747649\pi\)
\(510\) 0 0
\(511\) 4.00000 6.92820i 0.176950 0.306486i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.00000 + 3.46410i −0.0881305 + 0.152647i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.5000 18.1865i −0.457387 0.792218i
\(528\) 0 0
\(529\) −29.0000 + 50.2295i −1.26087 + 2.18389i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 20.7846i 0.519778 0.900281i
\(534\) 0 0
\(535\) −6.00000 10.3923i −0.259403 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 54.0000 2.32594
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.50000 + 4.33013i 0.107088 + 0.185482i
\(546\) 0 0
\(547\) −13.0000 + 22.5167i −0.555840 + 0.962743i 0.441998 + 0.897016i \(0.354270\pi\)
−0.997838 + 0.0657267i \(0.979063\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.00000 3.46410i −0.0850487 0.147309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 6.00000 + 10.3923i 0.252870 + 0.437983i 0.964315 0.264758i \(-0.0852922\pi\)
−0.711445 + 0.702742i \(0.751959\pi\)
\(564\) 0 0
\(565\) 3.00000 5.19615i 0.126211 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 + 31.1769i −0.754599 + 1.30700i 0.190974 + 0.981595i \(0.438835\pi\)
−0.945573 + 0.325409i \(0.894498\pi\)
\(570\) 0 0
\(571\) 3.50000 + 6.06218i 0.146470 + 0.253694i 0.929921 0.367760i \(-0.119875\pi\)
−0.783450 + 0.621455i \(0.786542\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.00000 −0.375326
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 18.0000 + 31.1769i 0.746766 + 1.29344i
\(582\) 0 0
\(583\) −27.0000 + 46.7654i −1.11823 + 1.93682i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.5000 + 33.7750i −0.804851 + 1.39404i 0.111540 + 0.993760i \(0.464422\pi\)
−0.916392 + 0.400283i \(0.868912\pi\)
\(588\) 0 0
\(589\) −24.5000 42.4352i −1.00950 1.74851i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −21.0000 −0.862367 −0.431183 0.902264i \(-0.641904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18.0000 31.1769i −0.735460 1.27385i −0.954521 0.298143i \(-0.903633\pi\)
0.219061 0.975711i \(-0.429701\pi\)
\(600\) 0 0
\(601\) 3.50000 6.06218i 0.142768 0.247281i −0.785770 0.618519i \(-0.787733\pi\)
0.928538 + 0.371237i \(0.121066\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.5000 21.6506i 0.508197 0.880223i
\(606\) 0 0
\(607\) 11.0000 + 19.0526i 0.446476 + 0.773320i 0.998154 0.0607380i \(-0.0193454\pi\)
−0.551678 + 0.834058i \(0.686012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.50000 + 12.9904i 0.301939 + 0.522973i 0.976575 0.215177i \(-0.0690329\pi\)
−0.674636 + 0.738150i \(0.735700\pi\)
\(618\) 0 0
\(619\) −10.0000 + 17.3205i −0.401934 + 0.696170i −0.993959 0.109749i \(-0.964995\pi\)
0.592025 + 0.805919i \(0.298329\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 + 20.7846i −0.480770 + 0.832718i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00000 + 1.73205i 0.0396838 + 0.0687343i
\(636\) 0 0
\(637\) 18.0000 31.1769i 0.713186 1.23527i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 2.00000 + 3.46410i 0.0788723 + 0.136611i 0.902764 0.430137i \(-0.141535\pi\)
−0.823891 + 0.566748i \(0.808201\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 0 0
\(649\) −72.0000 −2.82625
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.50000 7.79423i −0.176099 0.305012i 0.764442 0.644692i \(-0.223014\pi\)
−0.940541 + 0.339680i \(0.889681\pi\)
\(654\) 0 0
\(655\) −3.00000 + 5.19615i −0.117220 + 0.203030i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 20.7846i 0.467454 0.809653i −0.531855 0.846836i \(-0.678505\pi\)
0.999309 + 0.0371821i \(0.0118382\pi\)
\(660\) 0 0
\(661\) −7.00000 12.1244i −0.272268 0.471583i 0.697174 0.716902i \(-0.254441\pi\)
−0.969442 + 0.245319i \(0.921107\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −28.0000 −1.08579
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 21.0000 + 36.3731i 0.810696 + 1.40417i
\(672\) 0 0
\(673\) 14.0000 24.2487i 0.539660 0.934719i −0.459262 0.888301i \(-0.651886\pi\)
0.998922 0.0464181i \(-0.0147807\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −15.0000 + 25.9808i −0.576497 + 0.998522i 0.419380 + 0.907811i \(0.362247\pi\)
−0.995877 + 0.0907112i \(0.971086\pi\)
\(678\) 0 0
\(679\) 16.0000 + 27.7128i 0.614024 + 1.06352i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −39.0000 −1.49229 −0.746147 0.665782i \(-0.768098\pi\)
−0.746147 + 0.665782i \(0.768098\pi\)
\(684\) 0 0
\(685\) 9.00000 0.343872
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.0000 + 31.1769i 0.685745 + 1.18775i
\(690\) 0 0
\(691\) 18.5000 32.0429i 0.703773 1.21897i −0.263359 0.964698i \(-0.584830\pi\)
0.967132 0.254273i \(-0.0818362\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.00000 + 3.46410i −0.0758643 + 0.131401i
\(696\) 0 0
\(697\) 9.00000 + 15.5885i 0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 + 41.5692i 0.902613 + 1.56337i
\(708\) 0 0
\(709\) 23.0000 39.8372i 0.863783 1.49612i −0.00446726 0.999990i \(-0.501422\pi\)
0.868250 0.496126i \(-0.165245\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −31.5000 + 54.5596i −1.17968 + 2.04327i
\(714\) 0 0
\(715\) −12.0000 20.7846i −0.448775 0.777300i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 14.0000 24.2487i 0.519231 0.899335i −0.480519 0.876984i \(-0.659552\pi\)
0.999750 0.0223506i \(-0.00711500\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 5.19615i 0.110959 0.192187i
\(732\) 0 0
\(733\) 20.0000 + 34.6410i 0.738717 + 1.27950i 0.953073 + 0.302740i \(0.0979013\pi\)
−0.214356 + 0.976756i \(0.568765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.0000 20.7846i −0.440237 0.762513i 0.557470 0.830197i \(-0.311772\pi\)
−0.997707 + 0.0676840i \(0.978439\pi\)
\(744\) 0 0
\(745\) −3.00000 + 5.19615i −0.109911 + 0.190372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 + 41.5692i −0.876941 + 1.51891i
\(750\) 0 0
\(751\) −11.5000 19.9186i −0.419641 0.726839i 0.576262 0.817265i \(-0.304511\pi\)
−0.995903 + 0.0904254i \(0.971177\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 14.0000 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 10.0000 17.3205i 0.362024 0.627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 + 41.5692i −0.866590 + 1.50098i
\(768\) 0 0
\(769\) −2.50000 4.33013i −0.0901523 0.156148i 0.817423 0.576038i \(-0.195402\pi\)
−0.907575 + 0.419890i \(0.862069\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.00000 0.323708 0.161854 0.986815i \(-0.448253\pi\)
0.161854 + 0.986815i \(0.448253\pi\)
\(774\) 0 0
\(775\) −7.00000 −0.251447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 21.0000 + 36.3731i 0.752403 + 1.30320i
\(780\) 0 0
\(781\) −18.0000 + 31.1769i −0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.00000 + 8.66025i −0.178458 + 0.309098i
\(786\) 0 0
\(787\) −13.0000 22.5167i −0.463400 0.802632i 0.535728 0.844391i \(-0.320037\pi\)
−0.999128 + 0.0417585i \(0.986704\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 28.0000 0.994309
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.50000 2.59808i −0.0531327 0.0920286i 0.838236 0.545308i \(-0.183587\pi\)
−0.891368 + 0.453279i \(0.850254\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.00000 + 10.3923i −0.211735 + 0.366736i
\(804\) 0 0
\(805\) 18.0000 + 31.1769i 0.634417 + 1.09884i
\(806\) 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
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.0000 19.0526i −0.385313 0.667382i
\(816\) 0 0
\(817\) 7.00000 12.1244i 0.244899 0.424178i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 25.9808i 0.523504 0.906735i −0.476122 0.879379i \(-0.657958\pi\)
0.999626 0.0273557i \(-0.00870868\pi\)
\(822\) 0 0
\(823\) 11.0000 + 19.0526i 0.383436 + 0.664130i 0.991551 0.129719i \(-0.0414074\pi\)
−0.608115 + 0.793849i \(0.708074\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00000 0.104320 0.0521601 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 13.5000 + 23.3827i 0.467747 + 0.810162i
\(834\) 0 0
\(835\) 1.50000 2.59808i 0.0519096 0.0899101i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.00000 10.3923i 0.207143 0.358782i −0.743670 0.668546i \(-0.766917\pi\)
0.950813 + 0.309764i \(0.100250\pi\)
\(840\) 0 0
\(841\) 14.5000 + 25.1147i 0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −100.000 −3.43604
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.00000 + 15.5885i 0.308516 + 0.534365i
\(852\) 0 0
\(853\) −16.0000 + 27.7128i −0.547830 + 0.948869i 0.450593 + 0.892729i \(0.351212\pi\)
−0.998423 + 0.0561393i \(0.982121\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.5000 33.7750i 0.666107 1.15373i −0.312877 0.949794i \(-0.601293\pi\)
0.978984 0.203938i \(-0.0653741\pi\)
\(858\) 0 0
\(859\) 6.50000 + 11.2583i 0.221777 + 0.384129i 0.955348 0.295484i \(-0.0954809\pi\)
−0.733571 + 0.679613i \(0.762148\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.0000 0.919091 0.459545 0.888154i \(-0.348012\pi\)
0.459545 + 0.888154i \(0.348012\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.00000 + 5.19615i 0.101768 + 0.176267i
\(870\) 0 0
\(871\) 4.00000 6.92820i 0.135535 0.234753i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.00000 + 3.46410i −0.0676123 + 0.117108i
\(876\) 0 0
\(877\) −1.00000 1.73205i −0.0337676 0.0584872i 0.848648 0.528958i \(-0.177417\pi\)
−0.882415 + 0.470471i \(0.844084\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.50000 2.59808i −0.0503651 0.0872349i 0.839744 0.542983i \(-0.182705\pi\)
−0.890109 + 0.455748i \(0.849372\pi\)
\(888\) 0 0
\(889\) 4.00000 6.92820i 0.134156 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.50000 + 4.33013i 0.0831028 + 0.143938i
\(906\) 0 0
\(907\) −22.0000 + 38.1051i −0.730498 + 1.26526i 0.226173 + 0.974087i \(0.427379\pi\)
−0.956671 + 0.291172i \(0.905955\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 20.7846i 0.397578 0.688625i −0.595849 0.803097i \(-0.703184\pi\)
0.993426 + 0.114472i \(0.0365176\pi\)
\(912\) 0 0
\(913\) −27.0000 46.7654i −0.893570 1.54771i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.0000 + 20.7846i 0.394985 + 0.684134i
\(924\) 0 0
\(925\) −1.00000 + 1.73205i −0.0328798 + 0.0569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.00000 + 5.19615i −0.0984268 + 0.170480i −0.911034 0.412332i \(-0.864714\pi\)
0.812607 + 0.582812i \(0.198048\pi\)
\(930\) 0 0
\(931\) 31.5000 + 54.5596i 1.03237 + 1.78812i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 44.0000 1.43742 0.718709 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.0000 + 36.3731i 0.684580 + 1.18573i 0.973568 + 0.228395i \(0.0733479\pi\)
−0.288988 + 0.957333i \(0.593319\pi\)
\(942\) 0 0
\(943\) 27.0000 46.7654i 0.879241 1.52289i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.50000 2.59808i 0.0487435 0.0844261i −0.840624 0.541619i \(-0.817812\pi\)
0.889368 + 0.457193i \(0.151145\pi\)
\(948\) 0 0
\(949\) 4.00000 + 6.92820i 0.129845 + 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 31.1769i −0.581250 1.00676i
\(960\) 0 0
\(961\) −9.00000 + 15.5885i −0.290323 + 0.502853i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.00000 6.92820i 0.128765 0.223027i
\(966\) 0 0
\(967\) 11.0000 + 19.0526i 0.353736 + 0.612689i 0.986901 0.161328i \(-0.0515777\pi\)
−0.633165 + 0.774017i \(0.718244\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0000 + 46.7654i 0.863807 + 1.49616i 0.868227 + 0.496167i \(0.165259\pi\)
−0.00442082 + 0.999990i \(0.501407\pi\)
\(978\) 0 0
\(979\) 18.0000 31.1769i 0.575282 0.996419i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16.5000 28.5788i 0.526268 0.911523i −0.473263 0.880921i \(-0.656924\pi\)
0.999532 0.0306024i \(-0.00974257\pi\)
\(984\) 0 0
\(985\) 7.50000 + 12.9904i 0.238970 + 0.413908i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.0000 + 17.3205i 0.317021 + 0.549097i
\(996\) 0 0
\(997\) −13.0000 + 22.5167i −0.411714 + 0.713110i −0.995077 0.0991016i \(-0.968403\pi\)
0.583363 + 0.812211i \(0.301736\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1620.2.i.k.541.1 2
3.2 odd 2 1620.2.i.f.541.1 2
9.2 odd 6 540.2.a.d.1.1 yes 1
9.4 even 3 inner 1620.2.i.k.1081.1 2
9.5 odd 6 1620.2.i.f.1081.1 2
9.7 even 3 540.2.a.a.1.1 1
36.7 odd 6 2160.2.a.k.1.1 1
36.11 even 6 2160.2.a.x.1.1 1
45.2 even 12 2700.2.d.b.649.1 2
45.7 odd 12 2700.2.d.l.649.1 2
45.29 odd 6 2700.2.a.r.1.1 1
45.34 even 6 2700.2.a.t.1.1 1
45.38 even 12 2700.2.d.b.649.2 2
45.43 odd 12 2700.2.d.l.649.2 2
72.11 even 6 8640.2.a.bc.1.1 1
72.29 odd 6 8640.2.a.b.1.1 1
72.43 odd 6 8640.2.a.ch.1.1 1
72.61 even 6 8640.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
540.2.a.a.1.1 1 9.7 even 3
540.2.a.d.1.1 yes 1 9.2 odd 6
1620.2.i.f.541.1 2 3.2 odd 2
1620.2.i.f.1081.1 2 9.5 odd 6
1620.2.i.k.541.1 2 1.1 even 1 trivial
1620.2.i.k.1081.1 2 9.4 even 3 inner
2160.2.a.k.1.1 1 36.7 odd 6
2160.2.a.x.1.1 1 36.11 even 6
2700.2.a.r.1.1 1 45.29 odd 6
2700.2.a.t.1.1 1 45.34 even 6
2700.2.d.b.649.1 2 45.2 even 12
2700.2.d.b.649.2 2 45.38 even 12
2700.2.d.l.649.1 2 45.7 odd 12
2700.2.d.l.649.2 2 45.43 odd 12
8640.2.a.b.1.1 1 72.29 odd 6
8640.2.a.bc.1.1 1 72.11 even 6
8640.2.a.be.1.1 1 72.61 even 6
8640.2.a.ch.1.1 1 72.43 odd 6