Properties

Label 576.2.s.b.191.1
Level $576$
Weight $2$
Character 576.191
Analytic conductor $4.599$
Analytic rank $1$
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] = [576,2,Mod(191,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("576.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.s (of order \(6\), 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: \(4.59938315643\)
Analytic rank: \(1\)
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 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 191.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 576.191
Dual form 576.2.s.b.383.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +(-1.50000 + 0.866025i) q^{5} +(-1.50000 - 0.866025i) q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +(-1.50000 + 0.866025i) q^{5} +(-1.50000 - 0.866025i) q^{7} -3.00000 q^{9} +(-1.50000 + 2.59808i) q^{11} +(-2.50000 - 4.33013i) q^{13} +(-1.50000 - 2.59808i) q^{15} -6.92820i q^{17} -3.46410i q^{19} +(1.50000 - 2.59808i) q^{21} +(4.50000 + 7.79423i) q^{23} +(-1.00000 + 1.73205i) q^{25} -5.19615i q^{27} +(-1.50000 - 0.866025i) q^{29} +(-4.50000 + 2.59808i) q^{31} +(-4.50000 - 2.59808i) q^{33} +3.00000 q^{35} -2.00000 q^{37} +(7.50000 - 4.33013i) q^{39} +(-4.50000 + 2.59808i) q^{41} +(-4.50000 - 2.59808i) q^{43} +(4.50000 - 2.59808i) q^{45} +(1.50000 - 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} +12.0000 q^{51} -5.19615i q^{55} +6.00000 q^{57} +(1.50000 + 2.59808i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(4.50000 + 2.59808i) q^{63} +(7.50000 + 4.33013i) q^{65} +(-7.50000 + 4.33013i) q^{67} +(-13.5000 + 7.79423i) q^{69} -12.0000 q^{71} -2.00000 q^{73} +(-3.00000 - 1.73205i) q^{75} +(4.50000 - 2.59808i) q^{77} +(-7.50000 - 4.33013i) q^{79} +9.00000 q^{81} +(-7.50000 + 12.9904i) q^{83} +(6.00000 + 10.3923i) q^{85} +(1.50000 - 2.59808i) q^{87} +6.92820i q^{89} +8.66025i q^{91} +(-4.50000 - 7.79423i) q^{93} +(3.00000 + 5.19615i) q^{95} +(2.50000 - 4.33013i) q^{97} +(4.50000 - 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 3 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - 3 q^{7} - 6 q^{9} - 3 q^{11} - 5 q^{13} - 3 q^{15} + 3 q^{21} + 9 q^{23} - 2 q^{25} - 3 q^{29} - 9 q^{31} - 9 q^{33} + 6 q^{35} - 4 q^{37} + 15 q^{39} - 9 q^{41} - 9 q^{43} + 9 q^{45} + 3 q^{47} - 4 q^{49} + 24 q^{51} + 12 q^{57} + 3 q^{59} - q^{61} + 9 q^{63} + 15 q^{65} - 15 q^{67} - 27 q^{69} - 24 q^{71} - 4 q^{73} - 6 q^{75} + 9 q^{77} - 15 q^{79} + 18 q^{81} - 15 q^{83} + 12 q^{85} + 3 q^{87} - 9 q^{93} + 6 q^{95} + 5 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

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
\(3\) 1.73205i 1.00000i
\(4\) 0 0
\(5\) −1.50000 + 0.866025i −0.670820 + 0.387298i −0.796387 0.604787i \(-0.793258\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 0 0
\(7\) −1.50000 0.866025i −0.566947 0.327327i 0.188982 0.981981i \(-0.439481\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) −2.50000 4.33013i −0.693375 1.20096i −0.970725 0.240192i \(-0.922790\pi\)
0.277350 0.960769i \(-0.410544\pi\)
\(14\) 0 0
\(15\) −1.50000 2.59808i −0.387298 0.670820i
\(16\) 0 0
\(17\) 6.92820i 1.68034i −0.542326 0.840168i \(-0.682456\pi\)
0.542326 0.840168i \(-0.317544\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 1.50000 2.59808i 0.327327 0.566947i
\(22\) 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
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −1.50000 0.866025i −0.278543 0.160817i 0.354221 0.935162i \(-0.384746\pi\)
−0.632764 + 0.774345i \(0.718080\pi\)
\(30\) 0 0
\(31\) −4.50000 + 2.59808i −0.808224 + 0.466628i −0.846339 0.532645i \(-0.821198\pi\)
0.0381148 + 0.999273i \(0.487865\pi\)
\(32\) 0 0
\(33\) −4.50000 2.59808i −0.783349 0.452267i
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 7.50000 4.33013i 1.20096 0.693375i
\(40\) 0 0
\(41\) −4.50000 + 2.59808i −0.702782 + 0.405751i −0.808383 0.588657i \(-0.799657\pi\)
0.105601 + 0.994409i \(0.466323\pi\)
\(42\) 0 0
\(43\) −4.50000 2.59808i −0.686244 0.396203i 0.115960 0.993254i \(-0.463006\pi\)
−0.802203 + 0.597051i \(0.796339\pi\)
\(44\) 0 0
\(45\) 4.50000 2.59808i 0.670820 0.387298i
\(46\) 0 0
\(47\) 1.50000 2.59808i 0.218797 0.378968i −0.735643 0.677369i \(-0.763120\pi\)
0.954441 + 0.298401i \(0.0964533\pi\)
\(48\) 0 0
\(49\) −2.00000 3.46410i −0.285714 0.494872i
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 5.19615i 0.700649i
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) −0.500000 + 0.866025i −0.0640184 + 0.110883i −0.896258 0.443533i \(-0.853725\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 4.50000 + 2.59808i 0.566947 + 0.327327i
\(64\) 0 0
\(65\) 7.50000 + 4.33013i 0.930261 + 0.537086i
\(66\) 0 0
\(67\) −7.50000 + 4.33013i −0.916271 + 0.529009i −0.882443 0.470418i \(-0.844103\pi\)
−0.0338274 + 0.999428i \(0.510770\pi\)
\(68\) 0 0
\(69\) −13.5000 + 7.79423i −1.62521 + 0.938315i
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) −3.00000 1.73205i −0.346410 0.200000i
\(76\) 0 0
\(77\) 4.50000 2.59808i 0.512823 0.296078i
\(78\) 0 0
\(79\) −7.50000 4.33013i −0.843816 0.487177i 0.0147436 0.999891i \(-0.495307\pi\)
−0.858559 + 0.512714i \(0.828640\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −7.50000 + 12.9904i −0.823232 + 1.42588i 0.0800311 + 0.996792i \(0.474498\pi\)
−0.903263 + 0.429087i \(0.858835\pi\)
\(84\) 0 0
\(85\) 6.00000 + 10.3923i 0.650791 + 1.12720i
\(86\) 0 0
\(87\) 1.50000 2.59808i 0.160817 0.278543i
\(88\) 0 0
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) 8.66025i 0.907841i
\(92\) 0 0
\(93\) −4.50000 7.79423i −0.466628 0.808224i
\(94\) 0 0
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) 2.50000 4.33013i 0.253837 0.439658i −0.710742 0.703452i \(-0.751641\pi\)
0.964579 + 0.263795i \(0.0849741\pi\)
\(98\) 0 0
\(99\) 4.50000 7.79423i 0.452267 0.783349i
\(100\) 0 0
\(101\) 4.50000 + 2.59808i 0.447767 + 0.258518i 0.706887 0.707327i \(-0.250099\pi\)
−0.259120 + 0.965845i \(0.583432\pi\)
\(102\) 0 0
\(103\) 1.50000 0.866025i 0.147799 0.0853320i −0.424277 0.905533i \(-0.639472\pi\)
0.572076 + 0.820201i \(0.306138\pi\)
\(104\) 0 0
\(105\) 5.19615i 0.507093i
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 3.46410i 0.328798i
\(112\) 0 0
\(113\) −10.5000 + 6.06218i −0.987757 + 0.570282i −0.904603 0.426255i \(-0.859833\pi\)
−0.0831539 + 0.996537i \(0.526499\pi\)
\(114\) 0 0
\(115\) −13.5000 7.79423i −1.25888 0.726816i
\(116\) 0 0
\(117\) 7.50000 + 12.9904i 0.693375 + 1.20096i
\(118\) 0 0
\(119\) −6.00000 + 10.3923i −0.550019 + 0.952661i
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) −4.50000 7.79423i −0.405751 0.702782i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 10.3923i 0.922168i 0.887357 + 0.461084i \(0.152539\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 4.50000 7.79423i 0.396203 0.686244i
\(130\) 0 0
\(131\) 1.50000 + 2.59808i 0.131056 + 0.226995i 0.924084 0.382190i \(-0.124830\pi\)
−0.793028 + 0.609185i \(0.791497\pi\)
\(132\) 0 0
\(133\) −3.00000 + 5.19615i −0.260133 + 0.450564i
\(134\) 0 0
\(135\) 4.50000 + 7.79423i 0.387298 + 0.670820i
\(136\) 0 0
\(137\) 7.50000 + 4.33013i 0.640768 + 0.369948i 0.784910 0.619609i \(-0.212709\pi\)
−0.144142 + 0.989557i \(0.546042\pi\)
\(138\) 0 0
\(139\) −1.50000 + 0.866025i −0.127228 + 0.0734553i −0.562263 0.826958i \(-0.690069\pi\)
0.435035 + 0.900414i \(0.356736\pi\)
\(140\) 0 0
\(141\) 4.50000 + 2.59808i 0.378968 + 0.218797i
\(142\) 0 0
\(143\) 15.0000 1.25436
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 6.00000 3.46410i 0.494872 0.285714i
\(148\) 0 0
\(149\) −7.50000 + 4.33013i −0.614424 + 0.354738i −0.774695 0.632335i \(-0.782097\pi\)
0.160271 + 0.987073i \(0.448763\pi\)
\(150\) 0 0
\(151\) −7.50000 4.33013i −0.610341 0.352381i 0.162758 0.986666i \(-0.447961\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 20.7846i 1.68034i
\(154\) 0 0
\(155\) 4.50000 7.79423i 0.361449 0.626048i
\(156\) 0 0
\(157\) −0.500000 0.866025i −0.0399043 0.0691164i 0.845383 0.534160i \(-0.179372\pi\)
−0.885288 + 0.465044i \(0.846039\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.5885i 1.22854i
\(162\) 0 0
\(163\) 17.3205i 1.35665i −0.734763 0.678323i \(-0.762707\pi\)
0.734763 0.678323i \(-0.237293\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 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
\(169\) −6.00000 + 10.3923i −0.461538 + 0.799408i
\(170\) 0 0
\(171\) 10.3923i 0.794719i
\(172\) 0 0
\(173\) 16.5000 + 9.52628i 1.25447 + 0.724270i 0.971994 0.235004i \(-0.0755104\pi\)
0.282477 + 0.959274i \(0.408844\pi\)
\(174\) 0 0
\(175\) 3.00000 1.73205i 0.226779 0.130931i
\(176\) 0 0
\(177\) −4.50000 + 2.59808i −0.338241 + 0.195283i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −1.50000 0.866025i −0.110883 0.0640184i
\(184\) 0 0
\(185\) 3.00000 1.73205i 0.220564 0.127343i
\(186\) 0 0
\(187\) 18.0000 + 10.3923i 1.31629 + 0.759961i
\(188\) 0 0
\(189\) −4.50000 + 7.79423i −0.327327 + 0.566947i
\(190\) 0 0
\(191\) 1.50000 2.59808i 0.108536 0.187990i −0.806641 0.591041i \(-0.798717\pi\)
0.915177 + 0.403051i \(0.132050\pi\)
\(192\) 0 0
\(193\) 6.50000 + 11.2583i 0.467880 + 0.810392i 0.999326 0.0366998i \(-0.0116845\pi\)
−0.531446 + 0.847092i \(0.678351\pi\)
\(194\) 0 0
\(195\) −7.50000 + 12.9904i −0.537086 + 0.930261i
\(196\) 0 0
\(197\) 6.92820i 0.493614i 0.969065 + 0.246807i \(0.0793814\pi\)
−0.969065 + 0.246807i \(0.920619\pi\)
\(198\) 0 0
\(199\) 24.2487i 1.71895i −0.511182 0.859473i \(-0.670792\pi\)
0.511182 0.859473i \(-0.329208\pi\)
\(200\) 0 0
\(201\) −7.50000 12.9904i −0.529009 0.916271i
\(202\) 0 0
\(203\) 1.50000 + 2.59808i 0.105279 + 0.182349i
\(204\) 0 0
\(205\) 4.50000 7.79423i 0.314294 0.544373i
\(206\) 0 0
\(207\) −13.5000 23.3827i −0.938315 1.62521i
\(208\) 0 0
\(209\) 9.00000 + 5.19615i 0.622543 + 0.359425i
\(210\) 0 0
\(211\) 16.5000 9.52628i 1.13591 0.655816i 0.190493 0.981689i \(-0.438991\pi\)
0.945414 + 0.325872i \(0.105658\pi\)
\(212\) 0 0
\(213\) 20.7846i 1.42414i
\(214\) 0 0
\(215\) 9.00000 0.613795
\(216\) 0 0
\(217\) 9.00000 0.610960
\(218\) 0 0
\(219\) 3.46410i 0.234082i
\(220\) 0 0
\(221\) −30.0000 + 17.3205i −2.01802 + 1.16510i
\(222\) 0 0
\(223\) 22.5000 + 12.9904i 1.50671 + 0.869900i 0.999970 + 0.00780243i \(0.00248362\pi\)
0.506742 + 0.862098i \(0.330850\pi\)
\(224\) 0 0
\(225\) 3.00000 5.19615i 0.200000 0.346410i
\(226\) 0 0
\(227\) −1.50000 + 2.59808i −0.0995585 + 0.172440i −0.911502 0.411296i \(-0.865076\pi\)
0.811943 + 0.583736i \(0.198410\pi\)
\(228\) 0 0
\(229\) −8.50000 14.7224i −0.561696 0.972886i −0.997349 0.0727709i \(-0.976816\pi\)
0.435653 0.900115i \(-0.356518\pi\)
\(230\) 0 0
\(231\) 4.50000 + 7.79423i 0.296078 + 0.512823i
\(232\) 0 0
\(233\) 13.8564i 0.907763i −0.891062 0.453882i \(-0.850039\pi\)
0.891062 0.453882i \(-0.149961\pi\)
\(234\) 0 0
\(235\) 5.19615i 0.338960i
\(236\) 0 0
\(237\) 7.50000 12.9904i 0.487177 0.843816i
\(238\) 0 0
\(239\) −7.50000 12.9904i −0.485135 0.840278i 0.514719 0.857359i \(-0.327896\pi\)
−0.999854 + 0.0170808i \(0.994563\pi\)
\(240\) 0 0
\(241\) 8.50000 14.7224i 0.547533 0.948355i −0.450910 0.892570i \(-0.648900\pi\)
0.998443 0.0557856i \(-0.0177663\pi\)
\(242\) 0 0
\(243\) 15.5885i 1.00000i
\(244\) 0 0
\(245\) 6.00000 + 3.46410i 0.383326 + 0.221313i
\(246\) 0 0
\(247\) −15.0000 + 8.66025i −0.954427 + 0.551039i
\(248\) 0 0
\(249\) −22.5000 12.9904i −1.42588 0.823232i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) 0 0
\(255\) −18.0000 + 10.3923i −1.12720 + 0.650791i
\(256\) 0 0
\(257\) 1.50000 0.866025i 0.0935674 0.0540212i −0.452486 0.891771i \(-0.649463\pi\)
0.546054 + 0.837750i \(0.316129\pi\)
\(258\) 0 0
\(259\) 3.00000 + 1.73205i 0.186411 + 0.107624i
\(260\) 0 0
\(261\) 4.50000 + 2.59808i 0.278543 + 0.160817i
\(262\) 0 0
\(263\) −4.50000 + 7.79423i −0.277482 + 0.480613i −0.970758 0.240059i \(-0.922833\pi\)
0.693276 + 0.720672i \(0.256167\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 0 0
\(269\) 27.7128i 1.68968i −0.535019 0.844840i \(-0.679696\pi\)
0.535019 0.844840i \(-0.320304\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i 0.676435 + 0.736502i \(0.263524\pi\)
−0.676435 + 0.736502i \(0.736476\pi\)
\(272\) 0 0
\(273\) −15.0000 −0.907841
\(274\) 0 0
\(275\) −3.00000 5.19615i −0.180907 0.313340i
\(276\) 0 0
\(277\) 9.50000 16.4545i 0.570800 0.988654i −0.425684 0.904872i \(-0.639967\pi\)
0.996484 0.0837823i \(-0.0267000\pi\)
\(278\) 0 0
\(279\) 13.5000 7.79423i 0.808224 0.466628i
\(280\) 0 0
\(281\) −16.5000 9.52628i −0.984307 0.568290i −0.0807396 0.996735i \(-0.525728\pi\)
−0.903568 + 0.428445i \(0.859062\pi\)
\(282\) 0 0
\(283\) −13.5000 + 7.79423i −0.802492 + 0.463319i −0.844342 0.535805i \(-0.820008\pi\)
0.0418500 + 0.999124i \(0.486675\pi\)
\(284\) 0 0
\(285\) −9.00000 + 5.19615i −0.533114 + 0.307794i
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −31.0000 −1.82353
\(290\) 0 0
\(291\) 7.50000 + 4.33013i 0.439658 + 0.253837i
\(292\) 0 0
\(293\) −1.50000 + 0.866025i −0.0876309 + 0.0505937i −0.543175 0.839619i \(-0.682778\pi\)
0.455544 + 0.890213i \(0.349445\pi\)
\(294\) 0 0
\(295\) −4.50000 2.59808i −0.262000 0.151266i
\(296\) 0 0
\(297\) 13.5000 + 7.79423i 0.783349 + 0.452267i
\(298\) 0 0
\(299\) 22.5000 38.9711i 1.30121 2.25376i
\(300\) 0 0
\(301\) 4.50000 + 7.79423i 0.259376 + 0.449252i
\(302\) 0 0
\(303\) −4.50000 + 7.79423i −0.258518 + 0.447767i
\(304\) 0 0
\(305\) 1.73205i 0.0991769i
\(306\) 0 0
\(307\) 10.3923i 0.593120i 0.955014 + 0.296560i \(0.0958395\pi\)
−0.955014 + 0.296560i \(0.904160\pi\)
\(308\) 0 0
\(309\) 1.50000 + 2.59808i 0.0853320 + 0.147799i
\(310\) 0 0
\(311\) −7.50000 12.9904i −0.425286 0.736617i 0.571161 0.820838i \(-0.306493\pi\)
−0.996447 + 0.0842210i \(0.973160\pi\)
\(312\) 0 0
\(313\) 0.500000 0.866025i 0.0282617 0.0489506i −0.851549 0.524276i \(-0.824336\pi\)
0.879810 + 0.475325i \(0.157669\pi\)
\(314\) 0 0
\(315\) −9.00000 −0.507093
\(316\) 0 0
\(317\) 10.5000 + 6.06218i 0.589739 + 0.340486i 0.764994 0.644037i \(-0.222742\pi\)
−0.175255 + 0.984523i \(0.556075\pi\)
\(318\) 0 0
\(319\) 4.50000 2.59808i 0.251952 0.145464i
\(320\) 0 0
\(321\) 20.7846i 1.16008i
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 10.0000 0.554700
\(326\) 0 0
\(327\) 24.2487i 1.34096i
\(328\) 0 0
\(329\) −4.50000 + 2.59808i −0.248093 + 0.143237i
\(330\) 0 0
\(331\) 1.50000 + 0.866025i 0.0824475 + 0.0476011i 0.540657 0.841243i \(-0.318176\pi\)
−0.458209 + 0.888844i \(0.651509\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 7.50000 12.9904i 0.409769 0.709740i
\(336\) 0 0
\(337\) −3.50000 6.06218i −0.190657 0.330228i 0.754811 0.655942i \(-0.227729\pi\)
−0.945468 + 0.325714i \(0.894395\pi\)
\(338\) 0 0
\(339\) −10.5000 18.1865i −0.570282 0.987757i
\(340\) 0 0
\(341\) 15.5885i 0.844162i
\(342\) 0 0
\(343\) 19.0526i 1.02874i
\(344\) 0 0
\(345\) 13.5000 23.3827i 0.726816 1.25888i
\(346\) 0 0
\(347\) −4.50000 7.79423i −0.241573 0.418416i 0.719590 0.694399i \(-0.244330\pi\)
−0.961162 + 0.275983i \(0.910997\pi\)
\(348\) 0 0
\(349\) −6.50000 + 11.2583i −0.347937 + 0.602645i −0.985883 0.167437i \(-0.946451\pi\)
0.637946 + 0.770081i \(0.279784\pi\)
\(350\) 0 0
\(351\) −22.5000 + 12.9904i −1.20096 + 0.693375i
\(352\) 0 0
\(353\) −4.50000 2.59808i −0.239511 0.138282i 0.375441 0.926846i \(-0.377491\pi\)
−0.614952 + 0.788565i \(0.710825\pi\)
\(354\) 0 0
\(355\) 18.0000 10.3923i 0.955341 0.551566i
\(356\) 0 0
\(357\) −18.0000 10.3923i −0.952661 0.550019i
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) −3.00000 + 1.73205i −0.157459 + 0.0909091i
\(364\) 0 0
\(365\) 3.00000 1.73205i 0.157027 0.0906597i
\(366\) 0 0
\(367\) 16.5000 + 9.52628i 0.861293 + 0.497268i 0.864445 0.502727i \(-0.167670\pi\)
−0.00315207 + 0.999995i \(0.501003\pi\)
\(368\) 0 0
\(369\) 13.5000 7.79423i 0.702782 0.405751i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.50000 + 9.52628i 0.284779 + 0.493252i 0.972556 0.232671i \(-0.0747464\pi\)
−0.687776 + 0.725923i \(0.741413\pi\)
\(374\) 0 0
\(375\) 21.0000 1.08444
\(376\) 0 0
\(377\) 8.66025i 0.446026i
\(378\) 0 0
\(379\) 17.3205i 0.889695i 0.895606 + 0.444847i \(0.146742\pi\)
−0.895606 + 0.444847i \(0.853258\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −13.5000 23.3827i −0.689818 1.19480i −0.971897 0.235408i \(-0.924357\pi\)
0.282079 0.959391i \(-0.408976\pi\)
\(384\) 0 0
\(385\) −4.50000 + 7.79423i −0.229341 + 0.397231i
\(386\) 0 0
\(387\) 13.5000 + 7.79423i 0.686244 + 0.396203i
\(388\) 0 0
\(389\) −31.5000 18.1865i −1.59711 0.922094i −0.992040 0.125924i \(-0.959810\pi\)
−0.605074 0.796170i \(-0.706856\pi\)
\(390\) 0 0
\(391\) 54.0000 31.1769i 2.73090 1.57668i
\(392\) 0 0
\(393\) −4.50000 + 2.59808i −0.226995 + 0.131056i
\(394\) 0 0
\(395\) 15.0000 0.754732
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) −9.00000 5.19615i −0.450564 0.260133i
\(400\) 0 0
\(401\) −16.5000 + 9.52628i −0.823971 + 0.475720i −0.851784 0.523893i \(-0.824479\pi\)
0.0278131 + 0.999613i \(0.491146\pi\)
\(402\) 0 0
\(403\) 22.5000 + 12.9904i 1.12080 + 0.647097i
\(404\) 0 0
\(405\) −13.5000 + 7.79423i −0.670820 + 0.387298i
\(406\) 0 0
\(407\) 3.00000 5.19615i 0.148704 0.257564i
\(408\) 0 0
\(409\) −15.5000 26.8468i −0.766426 1.32749i −0.939490 0.342578i \(-0.888700\pi\)
0.173064 0.984911i \(-0.444633\pi\)
\(410\) 0 0
\(411\) −7.50000 + 12.9904i −0.369948 + 0.640768i
\(412\) 0 0
\(413\) 5.19615i 0.255686i
\(414\) 0 0
\(415\) 25.9808i 1.27535i
\(416\) 0 0
\(417\) −1.50000 2.59808i −0.0734553 0.127228i
\(418\) 0 0
\(419\) 7.50000 + 12.9904i 0.366399 + 0.634622i 0.989000 0.147918i \(-0.0472572\pi\)
−0.622601 + 0.782540i \(0.713924\pi\)
\(420\) 0 0
\(421\) −8.50000 + 14.7224i −0.414265 + 0.717527i −0.995351 0.0963145i \(-0.969295\pi\)
0.581086 + 0.813842i \(0.302628\pi\)
\(422\) 0 0
\(423\) −4.50000 + 7.79423i −0.218797 + 0.378968i
\(424\) 0 0
\(425\) 12.0000 + 6.92820i 0.582086 + 0.336067i
\(426\) 0 0
\(427\) 1.50000 0.866025i 0.0725901 0.0419099i
\(428\) 0 0
\(429\) 25.9808i 1.25436i
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 5.19615i 0.249136i
\(436\) 0 0
\(437\) 27.0000 15.5885i 1.29159 0.745697i
\(438\) 0 0
\(439\) 4.50000 + 2.59808i 0.214773 + 0.123999i 0.603528 0.797342i \(-0.293761\pi\)
−0.388755 + 0.921341i \(0.627095\pi\)
\(440\) 0 0
\(441\) 6.00000 + 10.3923i 0.285714 + 0.494872i
\(442\) 0 0
\(443\) 4.50000 7.79423i 0.213801 0.370315i −0.739100 0.673596i \(-0.764749\pi\)
0.952901 + 0.303281i \(0.0980821\pi\)
\(444\) 0 0
\(445\) −6.00000 10.3923i −0.284427 0.492642i
\(446\) 0 0
\(447\) −7.50000 12.9904i −0.354738 0.614424i
\(448\) 0 0
\(449\) 20.7846i 0.980886i 0.871473 + 0.490443i \(0.163165\pi\)
−0.871473 + 0.490443i \(0.836835\pi\)
\(450\) 0 0
\(451\) 15.5885i 0.734032i
\(452\) 0 0
\(453\) 7.50000 12.9904i 0.352381 0.610341i
\(454\) 0 0
\(455\) −7.50000 12.9904i −0.351605 0.608998i
\(456\) 0 0
\(457\) 14.5000 25.1147i 0.678281 1.17482i −0.297217 0.954810i \(-0.596058\pi\)
0.975498 0.220008i \(-0.0706083\pi\)
\(458\) 0 0
\(459\) −36.0000 −1.68034
\(460\) 0 0
\(461\) 22.5000 + 12.9904i 1.04793 + 0.605022i 0.922069 0.387026i \(-0.126497\pi\)
0.125860 + 0.992048i \(0.459831\pi\)
\(462\) 0 0
\(463\) −16.5000 + 9.52628i −0.766820 + 0.442724i −0.831739 0.555167i \(-0.812654\pi\)
0.0649190 + 0.997891i \(0.479321\pi\)
\(464\) 0 0
\(465\) 13.5000 + 7.79423i 0.626048 + 0.361449i
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) 1.50000 0.866025i 0.0691164 0.0399043i
\(472\) 0 0
\(473\) 13.5000 7.79423i 0.620731 0.358379i
\(474\) 0 0
\(475\) 6.00000 + 3.46410i 0.275299 + 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.50000 12.9904i 0.342684 0.593546i −0.642246 0.766498i \(-0.721997\pi\)
0.984930 + 0.172953i \(0.0553307\pi\)
\(480\) 0 0
\(481\) 5.00000 + 8.66025i 0.227980 + 0.394874i
\(482\) 0 0
\(483\) 27.0000 1.22854
\(484\) 0 0
\(485\) 8.66025i 0.393242i
\(486\) 0 0
\(487\) 3.46410i 0.156973i −0.996915 0.0784867i \(-0.974991\pi\)
0.996915 0.0784867i \(-0.0250088\pi\)
\(488\) 0 0
\(489\) 30.0000 1.35665
\(490\) 0 0
\(491\) −10.5000 18.1865i −0.473858 0.820747i 0.525694 0.850674i \(-0.323806\pi\)
−0.999552 + 0.0299272i \(0.990472\pi\)
\(492\) 0 0
\(493\) −6.00000 + 10.3923i −0.270226 + 0.468046i
\(494\) 0 0
\(495\) 15.5885i 0.700649i
\(496\) 0 0
\(497\) 18.0000 + 10.3923i 0.807410 + 0.466159i
\(498\) 0 0
\(499\) −13.5000 + 7.79423i −0.604343 + 0.348918i −0.770748 0.637140i \(-0.780117\pi\)
0.166405 + 0.986057i \(0.446784\pi\)
\(500\) 0 0
\(501\) 4.50000 2.59808i 0.201045 0.116073i
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) −18.0000 10.3923i −0.799408 0.461538i
\(508\) 0 0
\(509\) −31.5000 + 18.1865i −1.39621 + 0.806104i −0.993993 0.109439i \(-0.965094\pi\)
−0.402219 + 0.915543i \(0.631761\pi\)
\(510\) 0 0
\(511\) 3.00000 + 1.73205i 0.132712 + 0.0766214i
\(512\) 0 0
\(513\) −18.0000 −0.794719
\(514\) 0 0
\(515\) −1.50000 + 2.59808i −0.0660979 + 0.114485i
\(516\) 0 0
\(517\) 4.50000 + 7.79423i 0.197910 + 0.342790i
\(518\) 0 0
\(519\) −16.5000 + 28.5788i −0.724270 + 1.25447i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 17.3205i 0.757373i 0.925525 + 0.378686i \(0.123624\pi\)
−0.925525 + 0.378686i \(0.876376\pi\)
\(524\) 0 0
\(525\) 3.00000 + 5.19615i 0.130931 + 0.226779i
\(526\) 0 0
\(527\) 18.0000 + 31.1769i 0.784092 + 1.35809i
\(528\) 0 0
\(529\) −29.0000 + 50.2295i −1.26087 + 2.18389i
\(530\) 0 0
\(531\) −4.50000 7.79423i −0.195283 0.338241i
\(532\) 0 0
\(533\) 22.5000 + 12.9904i 0.974583 + 0.562676i
\(534\) 0 0
\(535\) −18.0000 + 10.3923i −0.778208 + 0.449299i
\(536\) 0 0
\(537\) 20.7846i 0.896922i
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) 17.3205i 0.743294i
\(544\) 0 0
\(545\) −21.0000 + 12.1244i −0.899541 + 0.519350i
\(546\) 0 0
\(547\) −34.5000 19.9186i −1.47511 0.851657i −0.475507 0.879712i \(-0.657735\pi\)
−0.999606 + 0.0280547i \(0.991069\pi\)
\(548\) 0 0
\(549\) 1.50000 2.59808i 0.0640184 0.110883i
\(550\) 0 0
\(551\) −3.00000 + 5.19615i −0.127804 + 0.221364i
\(552\) 0 0
\(553\) 7.50000 + 12.9904i 0.318932 + 0.552407i
\(554\) 0 0
\(555\) 3.00000 + 5.19615i 0.127343 + 0.220564i
\(556\) 0 0
\(557\) 27.7128i 1.17423i 0.809504 + 0.587115i \(0.199736\pi\)
−0.809504 + 0.587115i \(0.800264\pi\)
\(558\) 0 0
\(559\) 25.9808i 1.09887i
\(560\) 0 0
\(561\) −18.0000 + 31.1769i −0.759961 + 1.31629i
\(562\) 0 0
\(563\) −4.50000 7.79423i −0.189652 0.328488i 0.755482 0.655169i \(-0.227403\pi\)
−0.945134 + 0.326682i \(0.894069\pi\)
\(564\) 0 0
\(565\) 10.5000 18.1865i 0.441738 0.765113i
\(566\) 0 0
\(567\) −13.5000 7.79423i −0.566947 0.327327i
\(568\) 0 0
\(569\) 1.50000 + 0.866025i 0.0628833 + 0.0363057i 0.531112 0.847302i \(-0.321774\pi\)
−0.468229 + 0.883607i \(0.655108\pi\)
\(570\) 0 0
\(571\) 28.5000 16.4545i 1.19269 0.688599i 0.233773 0.972291i \(-0.424893\pi\)
0.958915 + 0.283693i \(0.0915595\pi\)
\(572\) 0 0
\(573\) 4.50000 + 2.59808i 0.187990 + 0.108536i
\(574\) 0 0
\(575\) −18.0000 −0.750652
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) −19.5000 + 11.2583i −0.810392 + 0.467880i
\(580\) 0 0
\(581\) 22.5000 12.9904i 0.933457 0.538932i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −22.5000 12.9904i −0.930261 0.537086i
\(586\) 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
\(589\) 9.00000 + 15.5885i 0.370839 + 0.642311i
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) 34.6410i 1.42254i −0.702921 0.711268i \(-0.748121\pi\)
0.702921 0.711268i \(-0.251879\pi\)
\(594\) 0 0
\(595\) 20.7846i 0.852086i
\(596\) 0 0
\(597\) 42.0000 1.71895
\(598\) 0 0
\(599\) 22.5000 + 38.9711i 0.919325 + 1.59232i 0.800443 + 0.599409i \(0.204598\pi\)
0.118882 + 0.992908i \(0.462069\pi\)
\(600\) 0 0
\(601\) −5.50000 + 9.52628i −0.224350 + 0.388585i −0.956124 0.292962i \(-0.905359\pi\)
0.731774 + 0.681547i \(0.238692\pi\)
\(602\) 0 0
\(603\) 22.5000 12.9904i 0.916271 0.529009i
\(604\) 0 0
\(605\) −3.00000 1.73205i −0.121967 0.0704179i
\(606\) 0 0
\(607\) 1.50000 0.866025i 0.0608831 0.0351509i −0.469249 0.883066i \(-0.655475\pi\)
0.530133 + 0.847915i \(0.322142\pi\)
\(608\) 0 0
\(609\) −4.50000 + 2.59808i −0.182349 + 0.105279i
\(610\) 0 0
\(611\) −15.0000 −0.606835
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) 13.5000 + 7.79423i 0.544373 + 0.314294i
\(616\) 0 0
\(617\) 19.5000 11.2583i 0.785040 0.453243i −0.0531732 0.998585i \(-0.516934\pi\)
0.838214 + 0.545342i \(0.183600\pi\)
\(618\) 0 0
\(619\) 25.5000 + 14.7224i 1.02493 + 0.591744i 0.915529 0.402253i \(-0.131773\pi\)
0.109403 + 0.993997i \(0.465106\pi\)
\(620\) 0 0
\(621\) 40.5000 23.3827i 1.62521 0.938315i
\(622\) 0 0
\(623\) 6.00000 10.3923i 0.240385 0.416359i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) −9.00000 + 15.5885i −0.359425 + 0.622543i
\(628\) 0 0
\(629\) 13.8564i 0.552491i
\(630\) 0 0
\(631\) 17.3205i 0.689519i −0.938691 0.344759i \(-0.887961\pi\)
0.938691 0.344759i \(-0.112039\pi\)
\(632\) 0 0
\(633\) 16.5000 + 28.5788i 0.655816 + 1.13591i
\(634\) 0 0
\(635\) −9.00000 15.5885i −0.357154 0.618609i
\(636\) 0 0
\(637\) −10.0000 + 17.3205i −0.396214 + 0.686264i
\(638\) 0 0
\(639\) 36.0000 1.42414
\(640\) 0 0
\(641\) 25.5000 + 14.7224i 1.00719 + 0.581501i 0.910368 0.413801i \(-0.135799\pi\)
0.0968219 + 0.995302i \(0.469132\pi\)
\(642\) 0 0
\(643\) 4.50000 2.59808i 0.177463 0.102458i −0.408637 0.912697i \(-0.633996\pi\)
0.586100 + 0.810239i \(0.300663\pi\)
\(644\) 0 0
\(645\) 15.5885i 0.613795i
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 15.5885i 0.610960i
\(652\) 0 0
\(653\) −13.5000 + 7.79423i −0.528296 + 0.305012i −0.740322 0.672252i \(-0.765327\pi\)
0.212026 + 0.977264i \(0.431994\pi\)
\(654\) 0 0
\(655\) −4.50000 2.59808i −0.175830 0.101515i
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) −1.50000 + 2.59808i −0.0584317 + 0.101207i −0.893762 0.448542i \(-0.851943\pi\)
0.835330 + 0.549749i \(0.185277\pi\)
\(660\) 0 0
\(661\) −24.5000 42.4352i −0.952940 1.65054i −0.739014 0.673690i \(-0.764708\pi\)
−0.213925 0.976850i \(-0.568625\pi\)
\(662\) 0 0
\(663\) −30.0000 51.9615i −1.16510 2.01802i
\(664\) 0 0
\(665\) 10.3923i 0.402996i
\(666\) 0 0
\(667\) 15.5885i 0.603587i
\(668\) 0 0
\(669\) −22.5000 + 38.9711i −0.869900 + 1.50671i
\(670\) 0 0
\(671\) −1.50000 2.59808i −0.0579069 0.100298i
\(672\) 0 0
\(673\) 18.5000 32.0429i 0.713123 1.23516i −0.250557 0.968102i \(-0.580614\pi\)
0.963679 0.267063i \(-0.0860531\pi\)
\(674\) 0 0
\(675\) 9.00000 + 5.19615i 0.346410 + 0.200000i
\(676\) 0 0
\(677\) −31.5000 18.1865i −1.21064 0.698965i −0.247744 0.968826i \(-0.579689\pi\)
−0.962899 + 0.269860i \(0.913022\pi\)
\(678\) 0 0
\(679\) −7.50000 + 4.33013i −0.287824 + 0.166175i
\(680\) 0 0
\(681\) −4.50000 2.59808i −0.172440 0.0995585i
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −15.0000 −0.573121
\(686\) 0 0
\(687\) 25.5000 14.7224i 0.972886 0.561696i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −4.50000 2.59808i −0.171188 0.0988355i 0.411958 0.911203i \(-0.364845\pi\)
−0.583146 + 0.812367i \(0.698178\pi\)
\(692\) 0 0
\(693\) −13.5000 + 7.79423i −0.512823 + 0.296078i
\(694\) 0 0
\(695\) 1.50000 2.59808i 0.0568982 0.0985506i
\(696\) 0 0
\(697\) 18.0000 + 31.1769i 0.681799 + 1.18091i
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 13.8564i 0.523349i 0.965156 + 0.261675i \(0.0842747\pi\)
−0.965156 + 0.261675i \(0.915725\pi\)
\(702\) 0 0
\(703\) 6.92820i 0.261302i
\(704\) 0 0
\(705\) −9.00000 −0.338960
\(706\) 0 0
\(707\) −4.50000 7.79423i −0.169240 0.293132i
\(708\) 0 0
\(709\) 9.50000 16.4545i 0.356780 0.617961i −0.630641 0.776075i \(-0.717208\pi\)
0.987421 + 0.158114i \(0.0505412\pi\)
\(710\) 0 0
\(711\) 22.5000 + 12.9904i 0.843816 + 0.487177i
\(712\) 0 0
\(713\) −40.5000 23.3827i −1.51674 0.875688i
\(714\) 0 0
\(715\) −22.5000 + 12.9904i −0.841452 + 0.485813i
\(716\) 0 0
\(717\) 22.5000 12.9904i 0.840278 0.485135i
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 25.5000 + 14.7224i 0.948355 + 0.547533i
\(724\) 0 0
\(725\) 3.00000 1.73205i 0.111417 0.0643268i
\(726\) 0 0
\(727\) −31.5000 18.1865i −1.16827 0.674501i −0.214998 0.976614i \(-0.568975\pi\)
−0.953272 + 0.302113i \(0.902308\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −18.0000 + 31.1769i −0.665754 + 1.15312i
\(732\) 0 0
\(733\) −2.50000 4.33013i −0.0923396 0.159937i 0.816156 0.577832i \(-0.196101\pi\)
−0.908495 + 0.417895i \(0.862768\pi\)
\(734\) 0 0
\(735\) −6.00000 + 10.3923i −0.221313 + 0.383326i
\(736\) 0 0
\(737\) 25.9808i 0.957014i
\(738\) 0 0
\(739\) 31.1769i 1.14686i 0.819254 + 0.573431i \(0.194388\pi\)
−0.819254 + 0.573431i \(0.805612\pi\)
\(740\) 0 0
\(741\) −15.0000 25.9808i −0.551039 0.954427i
\(742\) 0 0
\(743\) −13.5000 23.3827i −0.495267 0.857828i 0.504718 0.863284i \(-0.331596\pi\)
−0.999985 + 0.00545664i \(0.998263\pi\)
\(744\) 0 0
\(745\) 7.50000 12.9904i 0.274779 0.475931i
\(746\) 0 0
\(747\) 22.5000 38.9711i 0.823232 1.42588i
\(748\) 0 0
\(749\) −18.0000 10.3923i −0.657706 0.379727i
\(750\) 0 0
\(751\) −10.5000 + 6.06218i −0.383150 + 0.221212i −0.679188 0.733964i \(-0.737668\pi\)
0.296038 + 0.955176i \(0.404335\pi\)
\(752\) 0 0
\(753\) 20.7846i 0.757433i
\(754\) 0 0
\(755\) 15.0000 0.545906
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) 46.7654i 1.69748i
\(760\) 0 0
\(761\) 19.5000 11.2583i 0.706874 0.408114i −0.103028 0.994678i \(-0.532853\pi\)
0.809903 + 0.586564i \(0.199520\pi\)
\(762\) 0 0
\(763\) −21.0000 12.1244i −0.760251 0.438931i
\(764\) 0 0
\(765\) −18.0000 31.1769i −0.650791 1.12720i
\(766\) 0 0
\(767\) 7.50000 12.9904i 0.270809 0.469055i
\(768\) 0 0
\(769\) −11.5000 19.9186i −0.414701 0.718283i 0.580696 0.814120i \(-0.302780\pi\)
−0.995397 + 0.0958377i \(0.969447\pi\)
\(770\) 0 0
\(771\) 1.50000 + 2.59808i 0.0540212 + 0.0935674i
\(772\) 0 0
\(773\) 27.7128i 0.996761i 0.866959 + 0.498380i \(0.166072\pi\)
−0.866959 + 0.498380i \(0.833928\pi\)
\(774\) 0 0
\(775\) 10.3923i 0.373303i
\(776\) 0 0
\(777\) −3.00000 + 5.19615i −0.107624 + 0.186411i
\(778\) 0 0
\(779\) 9.00000 + 15.5885i 0.322458 + 0.558514i
\(780\) 0 0
\(781\) 18.0000 31.1769i 0.644091 1.11560i
\(782\) 0 0
\(783\) −4.50000 + 7.79423i −0.160817 + 0.278543i
\(784\) 0 0
\(785\) 1.50000 + 0.866025i 0.0535373 + 0.0309098i
\(786\) 0 0
\(787\) −25.5000 + 14.7224i −0.908977 + 0.524798i −0.880102 0.474785i \(-0.842526\pi\)
−0.0288750 + 0.999583i \(0.509192\pi\)
\(788\) 0 0
\(789\) −13.5000 7.79423i −0.480613 0.277482i
\(790\) 0 0
\(791\) 21.0000 0.746674
\(792\) 0 0
\(793\) 5.00000 0.177555
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.5000 16.4545i 1.00952 0.582848i 0.0984702 0.995140i \(-0.468605\pi\)
0.911052 + 0.412292i \(0.135272\pi\)
\(798\) 0 0
\(799\) −18.0000 10.3923i −0.636794 0.367653i
\(800\) 0 0
\(801\) 20.7846i 0.734388i
\(802\) 0 0
\(803\) 3.00000 5.19615i 0.105868 0.183368i
\(804\) 0 0
\(805\) 13.5000 + 23.3827i 0.475812 + 0.824131i
\(806\) 0 0
\(807\) 48.0000 1.68968
\(808\) 0 0
\(809\) 48.4974i 1.70508i 0.522663 + 0.852539i \(0.324939\pi\)
−0.522663 + 0.852539i \(0.675061\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i −0.836881 0.547385i \(-0.815623\pi\)
0.836881 0.547385i \(-0.184377\pi\)
\(812\) 0 0
\(813\) −42.0000 −1.47300
\(814\) 0 0
\(815\) 15.0000 + 25.9808i 0.525427 + 0.910066i
\(816\) 0 0
\(817\) −9.00000 + 15.5885i −0.314870 + 0.545371i
\(818\) 0 0
\(819\) 25.9808i 0.907841i
\(820\) 0 0
\(821\) −25.5000 14.7224i −0.889956 0.513816i −0.0160280 0.999872i \(-0.505102\pi\)
−0.873928 + 0.486055i \(0.838435\pi\)
\(822\) 0 0
\(823\) −28.5000 + 16.4545i −0.993448 + 0.573567i −0.906303 0.422628i \(-0.861108\pi\)
−0.0871445 + 0.996196i \(0.527774\pi\)
\(824\) 0 0
\(825\) 9.00000 5.19615i 0.313340 0.180907i
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 28.5000 + 16.4545i 0.988654 + 0.570800i
\(832\) 0 0
\(833\) −24.0000 + 13.8564i −0.831551 + 0.480096i
\(834\) 0 0
\(835\) 4.50000 + 2.59808i 0.155729 + 0.0899101i
\(836\) 0 0
\(837\) 13.5000 + 23.3827i 0.466628 + 0.808224i
\(838\) 0 0
\(839\) −22.5000 + 38.9711i −0.776786 + 1.34543i 0.156999 + 0.987599i \(0.449818\pi\)
−0.933785 + 0.357834i \(0.883515\pi\)
\(840\) 0 0
\(841\) −13.0000 22.5167i −0.448276 0.776437i
\(842\) 0 0
\(843\) 16.5000 28.5788i 0.568290 0.984307i
\(844\) 0 0
\(845\) 20.7846i 0.715012i
\(846\) 0 0
\(847\) 3.46410i 0.119028i
\(848\) 0 0
\(849\) −13.5000 23.3827i −0.463319 0.802492i
\(850\) 0 0
\(851\) −9.00000 15.5885i −0.308516 0.534365i
\(852\) 0 0
\(853\) −24.5000 + 42.4352i −0.838864 + 1.45296i 0.0519811 + 0.998648i \(0.483446\pi\)
−0.890845 + 0.454307i \(0.849887\pi\)
\(854\) 0 0
\(855\) −9.00000 15.5885i −0.307794 0.533114i
\(856\) 0 0
\(857\) −28.5000 16.4545i −0.973541 0.562074i −0.0732274 0.997315i \(-0.523330\pi\)
−0.900314 + 0.435241i \(0.856663\pi\)
\(858\) 0 0
\(859\) −19.5000 + 11.2583i −0.665331 + 0.384129i −0.794305 0.607519i \(-0.792165\pi\)
0.128974 + 0.991648i \(0.458832\pi\)
\(860\) 0 0
\(861\) 15.5885i 0.531253i
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −33.0000 −1.12203
\(866\) 0 0
\(867\) 53.6936i 1.82353i
\(868\) 0 0
\(869\) 22.5000 12.9904i 0.763260 0.440668i
\(870\) 0 0
\(871\) 37.5000 + 21.6506i 1.27064 + 0.733604i
\(872\) 0 0
\(873\) −7.50000 + 12.9904i −0.253837 + 0.439658i
\(874\) 0 0
\(875\) −10.5000 + 18.1865i −0.354965 + 0.614817i
\(876\) 0 0
\(877\) −6.50000 11.2583i −0.219489 0.380167i 0.735163 0.677891i \(-0.237106\pi\)
−0.954652 + 0.297724i \(0.903772\pi\)
\(878\) 0 0
\(879\) −1.50000 2.59808i −0.0505937 0.0876309i
\(880\) 0 0
\(881\) 20.7846i 0.700251i 0.936703 + 0.350126i \(0.113861\pi\)
−0.936703 + 0.350126i \(0.886139\pi\)
\(882\) 0 0
\(883\) 17.3205i 0.582882i −0.956589 0.291441i \(-0.905865\pi\)
0.956589 0.291441i \(-0.0941346\pi\)
\(884\) 0 0
\(885\) 4.50000 7.79423i 0.151266 0.262000i
\(886\) 0 0
\(887\) 28.5000 + 49.3634i 0.956936 + 1.65746i 0.729873 + 0.683582i \(0.239579\pi\)
0.227063 + 0.973880i \(0.427088\pi\)
\(888\) 0 0
\(889\) 9.00000 15.5885i 0.301850 0.522820i
\(890\) 0 0
\(891\) −13.5000 + 23.3827i −0.452267 + 0.783349i
\(892\) 0 0
\(893\) −9.00000 5.19615i −0.301174 0.173883i
\(894\) 0 0
\(895\) 18.0000 10.3923i 0.601674 0.347376i
\(896\) 0 0
\(897\) 67.5000 + 38.9711i 2.25376 + 1.30121i
\(898\) 0 0
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −13.5000 + 7.79423i −0.449252 + 0.259376i
\(904\) 0 0
\(905\) 15.0000 8.66025i 0.498617 0.287877i
\(906\) 0 0
\(907\) −28.5000 16.4545i −0.946327 0.546362i −0.0543890 0.998520i \(-0.517321\pi\)
−0.891938 + 0.452158i \(0.850654\pi\)
\(908\) 0 0
\(909\) −13.5000 7.79423i −0.447767 0.258518i
\(910\) 0 0
\(911\) −16.5000 + 28.5788i −0.546669 + 0.946859i 0.451830 + 0.892104i \(0.350771\pi\)
−0.998500 + 0.0547553i \(0.982562\pi\)
\(912\) 0 0
\(913\) −22.5000 38.9711i −0.744641 1.28976i
\(914\) 0 0
\(915\) 3.00000 0.0991769
\(916\) 0 0
\(917\) 5.19615i 0.171592i
\(918\) 0 0
\(919\) 10.3923i 0.342811i 0.985201 + 0.171405i \(0.0548307\pi\)
−0.985201 + 0.171405i \(0.945169\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) 30.0000 + 51.9615i 0.987462 + 1.71033i
\(924\) 0 0
\(925\) 2.00000 3.46410i 0.0657596 0.113899i
\(926\) 0 0
\(927\) −4.50000 + 2.59808i −0.147799 + 0.0853320i
\(928\) 0 0
\(929\) −16.5000 9.52628i −0.541347 0.312547i 0.204277 0.978913i \(-0.434516\pi\)
−0.745625 + 0.666366i \(0.767849\pi\)
\(930\) 0 0
\(931\) −12.0000 + 6.92820i −0.393284 + 0.227063i
\(932\) 0 0
\(933\) 22.5000 12.9904i 0.736617 0.425286i
\(934\) 0 0
\(935\) −36.0000 −1.17733
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 1.50000 + 0.866025i 0.0489506 + 0.0282617i
\(940\) 0 0
\(941\) 46.5000 26.8468i 1.51586 0.875180i 0.516030 0.856571i \(-0.327409\pi\)
0.999827 0.0186097i \(-0.00592398\pi\)
\(942\) 0 0
\(943\) −40.5000 23.3827i −1.31886 0.761445i
\(944\) 0 0
\(945\) 15.5885i 0.507093i
\(946\) 0 0
\(947\) −13.5000 + 23.3827i −0.438691 + 0.759835i −0.997589 0.0694014i \(-0.977891\pi\)
0.558898 + 0.829237i \(0.311224\pi\)
\(948\) 0 0
\(949\) 5.00000 + 8.66025i 0.162307 + 0.281124i
\(950\) 0 0
\(951\) −10.5000 + 18.1865i −0.340486 + 0.589739i
\(952\) 0 0
\(953\) 48.4974i 1.57099i 0.618871 + 0.785493i \(0.287590\pi\)
−0.618871 + 0.785493i \(0.712410\pi\)
\(954\) 0 0
\(955\) 5.19615i 0.168144i
\(956\) 0 0
\(957\) 4.50000 + 7.79423i 0.145464 + 0.251952i
\(958\) 0 0
\(959\) −7.50000 12.9904i −0.242188 0.419481i
\(960\) 0 0
\(961\) −2.00000 + 3.46410i −0.0645161 + 0.111745i
\(962\) 0 0
\(963\) −36.0000 −1.16008
\(964\) 0 0
\(965\) −19.5000 11.2583i −0.627727 0.362418i
\(966\) 0 0
\(967\) 7.50000 4.33013i 0.241184 0.139247i −0.374537 0.927212i \(-0.622198\pi\)
0.615721 + 0.787964i \(0.288865\pi\)
\(968\) 0 0
\(969\) 41.5692i 1.33540i
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 0 0
\(975\) 17.3205i 0.554700i
\(976\) 0 0
\(977\) 13.5000 7.79423i 0.431903 0.249359i −0.268254 0.963348i \(-0.586447\pi\)
0.700157 + 0.713989i \(0.253113\pi\)
\(978\) 0 0
\(979\) −18.0000 10.3923i −0.575282 0.332140i
\(980\) 0 0
\(981\) −42.0000 −1.34096
\(982\) 0 0
\(983\) 7.50000 12.9904i 0.239213 0.414329i −0.721276 0.692648i \(-0.756444\pi\)
0.960489 + 0.278319i \(0.0897773\pi\)
\(984\) 0 0
\(985\) −6.00000 10.3923i −0.191176 0.331126i
\(986\) 0 0
\(987\) −4.50000 7.79423i −0.143237 0.248093i
\(988\) 0 0
\(989\) 46.7654i 1.48705i
\(990\) 0 0
\(991\) 45.0333i 1.43053i −0.698853 0.715265i \(-0.746306\pi\)
0.698853 0.715265i \(-0.253694\pi\)
\(992\) 0 0
\(993\) −1.50000 + 2.59808i −0.0476011 + 0.0824475i
\(994\) 0 0
\(995\) 21.0000 + 36.3731i 0.665745 + 1.15310i
\(996\) 0 0
\(997\) 5.50000 9.52628i 0.174187 0.301700i −0.765693 0.643206i \(-0.777604\pi\)
0.939880 + 0.341506i \(0.110937\pi\)
\(998\) 0 0
\(999\) 10.3923i 0.328798i
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 576.2.s.b.191.1 2
3.2 odd 2 1728.2.s.c.575.1 2
4.3 odd 2 576.2.s.c.191.1 2
8.3 odd 2 144.2.s.c.47.1 yes 2
8.5 even 2 144.2.s.b.47.1 2
9.2 odd 6 5184.2.c.b.5183.2 2
9.4 even 3 1728.2.s.d.1151.1 2
9.5 odd 6 576.2.s.c.383.1 2
9.7 even 3 5184.2.c.d.5183.1 2
12.11 even 2 1728.2.s.d.575.1 2
24.5 odd 2 432.2.s.a.143.1 2
24.11 even 2 432.2.s.b.143.1 2
36.7 odd 6 5184.2.c.b.5183.1 2
36.11 even 6 5184.2.c.d.5183.2 2
36.23 even 6 inner 576.2.s.b.383.1 2
36.31 odd 6 1728.2.s.c.1151.1 2
72.5 odd 6 144.2.s.c.95.1 yes 2
72.11 even 6 1296.2.c.a.1295.1 2
72.13 even 6 432.2.s.b.287.1 2
72.29 odd 6 1296.2.c.c.1295.1 2
72.43 odd 6 1296.2.c.c.1295.2 2
72.59 even 6 144.2.s.b.95.1 yes 2
72.61 even 6 1296.2.c.a.1295.2 2
72.67 odd 6 432.2.s.a.287.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.2.s.b.47.1 2 8.5 even 2
144.2.s.b.95.1 yes 2 72.59 even 6
144.2.s.c.47.1 yes 2 8.3 odd 2
144.2.s.c.95.1 yes 2 72.5 odd 6
432.2.s.a.143.1 2 24.5 odd 2
432.2.s.a.287.1 2 72.67 odd 6
432.2.s.b.143.1 2 24.11 even 2
432.2.s.b.287.1 2 72.13 even 6
576.2.s.b.191.1 2 1.1 even 1 trivial
576.2.s.b.383.1 2 36.23 even 6 inner
576.2.s.c.191.1 2 4.3 odd 2
576.2.s.c.383.1 2 9.5 odd 6
1296.2.c.a.1295.1 2 72.11 even 6
1296.2.c.a.1295.2 2 72.61 even 6
1296.2.c.c.1295.1 2 72.29 odd 6
1296.2.c.c.1295.2 2 72.43 odd 6
1728.2.s.c.575.1 2 3.2 odd 2
1728.2.s.c.1151.1 2 36.31 odd 6
1728.2.s.d.575.1 2 12.11 even 2
1728.2.s.d.1151.1 2 9.4 even 3
5184.2.c.b.5183.1 2 36.7 odd 6
5184.2.c.b.5183.2 2 9.2 odd 6
5184.2.c.d.5183.1 2 9.7 even 3
5184.2.c.d.5183.2 2 36.11 even 6