Properties

Label 2592.2.i.q.865.1
Level $2592$
Weight $2$
Character 2592.865
Analytic conductor $20.697$
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] = [2592,2,Mod(865,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.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: \(20.6972242039\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2592.865
Dual form 2592.2.i.q.1729.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{7} +O(q^{10})\) \(q+(1.00000 + 1.73205i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(-2.00000 + 3.46410i) q^{11} +(1.00000 + 1.73205i) q^{13} +6.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{25} +(1.00000 - 1.73205i) q^{29} +(2.00000 + 3.46410i) q^{31} -8.00000 q^{35} -2.00000 q^{37} +(1.00000 + 1.73205i) q^{41} +(2.00000 - 3.46410i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(-4.50000 - 7.79423i) q^{49} -10.0000 q^{53} -8.00000 q^{55} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} -16.0000 q^{71} -6.00000 q^{73} +(-8.00000 - 13.8564i) q^{77} +(2.00000 - 3.46410i) q^{79} +(-6.00000 + 10.3923i) q^{83} +(6.00000 + 10.3923i) q^{85} -10.0000 q^{89} -8.00000 q^{91} +(4.00000 + 6.92820i) q^{95} +(7.00000 - 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 4 q^{7} - 4 q^{11} + 2 q^{13} + 12 q^{17} + 8 q^{19} + q^{25} + 2 q^{29} + 4 q^{31} - 16 q^{35} - 4 q^{37} + 2 q^{41} + 4 q^{43} - 8 q^{47} - 9 q^{49} - 20 q^{53} - 16 q^{55} + 4 q^{59} - 6 q^{61} - 4 q^{65} + 4 q^{67} - 32 q^{71} - 12 q^{73} - 16 q^{77} + 4 q^{79} - 12 q^{83} + 12 q^{85} - 20 q^{89} - 16 q^{91} + 8 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 1.73205i 0.185695 0.321634i −0.758115 0.652121i \(-0.773880\pi\)
0.943811 + 0.330487i \(0.107213\pi\)
\(30\) 0 0
\(31\) 2.00000 + 3.46410i 0.359211 + 0.622171i 0.987829 0.155543i \(-0.0497126\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.00000 + 1.73205i 0.156174 + 0.270501i 0.933486 0.358614i \(-0.116751\pi\)
−0.777312 + 0.629115i \(0.783417\pi\)
\(42\) 0 0
\(43\) 2.00000 3.46410i 0.304997 0.528271i −0.672264 0.740312i \(-0.734678\pi\)
0.977261 + 0.212041i \(0.0680112\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 + 6.92820i −0.583460 + 1.01058i 0.411606 + 0.911362i \(0.364968\pi\)
−0.995066 + 0.0992202i \(0.968365\pi\)
\(48\) 0 0
\(49\) −4.50000 7.79423i −0.642857 1.11346i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 + 3.46410i 0.260378 + 0.450988i 0.966342 0.257260i \(-0.0828195\pi\)
−0.705965 + 0.708247i \(0.749486\pi\)
\(60\) 0 0
\(61\) −3.00000 + 5.19615i −0.384111 + 0.665299i −0.991645 0.128994i \(-0.958825\pi\)
0.607535 + 0.794293i \(0.292159\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 + 3.46410i −0.248069 + 0.429669i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.00000 13.8564i −0.911685 1.57908i
\(78\) 0 0
\(79\) 2.00000 3.46410i 0.225018 0.389742i −0.731307 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.00000 + 10.3923i −0.658586 + 1.14070i 0.322396 + 0.946605i \(0.395512\pi\)
−0.980982 + 0.194099i \(0.937822\pi\)
\(84\) 0 0
\(85\) 6.00000 + 10.3923i 0.650791 + 1.12720i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 + 6.92820i 0.410391 + 0.710819i
\(96\) 0 0
\(97\) 7.00000 12.1244i 0.710742 1.23104i −0.253837 0.967247i \(-0.581693\pi\)
0.964579 0.263795i \(-0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 + 5.19615i −0.298511 + 0.517036i −0.975796 0.218685i \(-0.929823\pi\)
0.677284 + 0.735721i \(0.263157\pi\)
\(102\) 0 0
\(103\) −6.00000 10.3923i −0.591198 1.02398i −0.994071 0.108729i \(-0.965322\pi\)
0.402874 0.915255i \(-0.368011\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.00000 + 1.73205i 0.0940721 + 0.162938i 0.909221 0.416314i \(-0.136678\pi\)
−0.815149 + 0.579252i \(0.803345\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 + 20.7846i −1.10004 + 1.90532i
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 3.46410i −0.174741 0.302660i 0.765331 0.643637i \(-0.222575\pi\)
−0.940072 + 0.340977i \(0.889242\pi\)
\(132\) 0 0
\(133\) −8.00000 + 13.8564i −0.693688 + 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 15.5885i 0.768922 1.33181i −0.169226 0.985577i \(-0.554127\pi\)
0.938148 0.346235i \(-0.112540\pi\)
\(138\) 0 0
\(139\) −10.0000 17.3205i −0.848189 1.46911i −0.882823 0.469706i \(-0.844360\pi\)
0.0346338 0.999400i \(-0.488974\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 + 15.5885i 0.737309 + 1.27706i 0.953703 + 0.300750i \(0.0972370\pi\)
−0.216394 + 0.976306i \(0.569430\pi\)
\(150\) 0 0
\(151\) 6.00000 10.3923i 0.488273 0.845714i −0.511636 0.859202i \(-0.670960\pi\)
0.999909 + 0.0134886i \(0.00429367\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 + 6.92820i −0.321288 + 0.556487i
\(156\) 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
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.00000 6.92820i −0.309529 0.536120i 0.668730 0.743505i \(-0.266838\pi\)
−0.978259 + 0.207385i \(0.933505\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.00000 + 5.19615i −0.228086 + 0.395056i −0.957241 0.289292i \(-0.906580\pi\)
0.729155 + 0.684349i \(0.239913\pi\)
\(174\) 0 0
\(175\) 2.00000 + 3.46410i 0.151186 + 0.261861i
\(176\) 0 0
\(177\) 0 0
\(178\) 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
\(183\) 0 0
\(184\) 0 0
\(185\) −2.00000 3.46410i −0.147043 0.254686i
\(186\) 0 0
\(187\) −12.0000 + 20.7846i −0.877527 + 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) −9.00000 15.5885i −0.647834 1.12208i −0.983639 0.180150i \(-0.942342\pi\)
0.335805 0.941932i \(-0.390992\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.00000 + 6.92820i 0.280745 + 0.486265i
\(204\) 0 0
\(205\) −2.00000 + 3.46410i −0.139686 + 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.00000 + 13.8564i −0.553372 + 0.958468i
\(210\) 0 0
\(211\) 10.0000 + 17.3205i 0.688428 + 1.19239i 0.972346 + 0.233544i \(0.0750324\pi\)
−0.283918 + 0.958849i \(0.591634\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.00000 + 10.3923i 0.403604 + 0.699062i
\(222\) 0 0
\(223\) −2.00000 + 3.46410i −0.133930 + 0.231973i −0.925188 0.379509i \(-0.876093\pi\)
0.791258 + 0.611482i \(0.209426\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.00000 + 10.3923i −0.398234 + 0.689761i −0.993508 0.113761i \(-0.963710\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(228\) 0 0
\(229\) 5.00000 + 8.66025i 0.330409 + 0.572286i 0.982592 0.185776i \(-0.0594799\pi\)
−0.652183 + 0.758062i \(0.726147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 7.00000 12.1244i 0.450910 0.780998i −0.547533 0.836784i \(-0.684433\pi\)
0.998443 + 0.0557856i \(0.0177663\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.00000 15.5885i 0.574989 0.995910i
\(246\) 0 0
\(247\) 4.00000 + 6.92820i 0.254514 + 0.440831i
\(248\) 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
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.00000 + 1.73205i 0.0623783 + 0.108042i 0.895528 0.445005i \(-0.146798\pi\)
−0.833150 + 0.553047i \(0.813465\pi\)
\(258\) 0 0
\(259\) 4.00000 6.92820i 0.248548 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 20.7846i 0.739952 1.28163i −0.212565 0.977147i \(-0.568182\pi\)
0.952517 0.304487i \(-0.0984850\pi\)
\(264\) 0 0
\(265\) −10.0000 17.3205i −0.614295 1.06399i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 + 3.46410i 0.120605 + 0.208893i
\(276\) 0 0
\(277\) −11.0000 + 19.0526i −0.660926 + 1.14476i 0.319447 + 0.947604i \(0.396503\pi\)
−0.980373 + 0.197153i \(0.936830\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.00000 + 5.19615i −0.178965 + 0.309976i −0.941526 0.336939i \(-0.890608\pi\)
0.762561 + 0.646916i \(0.223942\pi\)
\(282\) 0 0
\(283\) 14.0000 + 24.2487i 0.832214 + 1.44144i 0.896279 + 0.443491i \(0.146260\pi\)
−0.0640654 + 0.997946i \(0.520407\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.00000 5.19615i −0.175262 0.303562i 0.764990 0.644042i \(-0.222744\pi\)
−0.940252 + 0.340480i \(0.889411\pi\)
\(294\) 0 0
\(295\) −4.00000 + 6.92820i −0.232889 + 0.403376i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 + 13.8564i 0.461112 + 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000 + 20.7846i 0.680458 + 1.17859i 0.974841 + 0.222900i \(0.0715523\pi\)
−0.294384 + 0.955687i \(0.595114\pi\)
\(312\) 0 0
\(313\) −5.00000 + 8.66025i −0.282617 + 0.489506i −0.972028 0.234863i \(-0.924536\pi\)
0.689412 + 0.724370i \(0.257869\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 15.5885i 0.505490 0.875535i −0.494489 0.869184i \(-0.664645\pi\)
0.999980 0.00635137i \(-0.00202172\pi\)
\(318\) 0 0
\(319\) 4.00000 + 6.92820i 0.223957 + 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −16.0000 27.7128i −0.882109 1.52786i
\(330\) 0 0
\(331\) 14.0000 24.2487i 0.769510 1.33283i −0.168320 0.985732i \(-0.553834\pi\)
0.937829 0.347097i \(-0.112833\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 + 6.92820i −0.218543 + 0.378528i
\(336\) 0 0
\(337\) −1.00000 1.73205i −0.0544735 0.0943508i 0.837503 0.546433i \(-0.184015\pi\)
−0.891976 + 0.452082i \(0.850681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.0000 17.3205i −0.536828 0.929814i −0.999072 0.0430610i \(-0.986289\pi\)
0.462244 0.886753i \(-0.347044\pi\)
\(348\) 0 0
\(349\) 13.0000 22.5167i 0.695874 1.20529i −0.274011 0.961727i \(-0.588351\pi\)
0.969885 0.243563i \(-0.0783162\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.00000 + 12.1244i −0.372572 + 0.645314i −0.989960 0.141344i \(-0.954858\pi\)
0.617388 + 0.786659i \(0.288191\pi\)
\(354\) 0 0
\(355\) −16.0000 27.7128i −0.849192 1.47084i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 10.3923i −0.314054 0.543958i
\(366\) 0 0
\(367\) −6.00000 + 10.3923i −0.313197 + 0.542474i −0.979053 0.203607i \(-0.934733\pi\)
0.665855 + 0.746081i \(0.268067\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.0000 34.6410i 1.03835 1.79847i
\(372\) 0 0
\(373\) 17.0000 + 29.4449i 0.880227 + 1.52460i 0.851089 + 0.525022i \(0.175943\pi\)
0.0291379 + 0.999575i \(0.490724\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.0000 27.7128i −0.817562 1.41606i −0.907474 0.420109i \(-0.861992\pi\)
0.0899119 0.995950i \(-0.471341\pi\)
\(384\) 0 0
\(385\) 16.0000 27.7128i 0.815436 1.41238i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.0000 + 25.9808i −0.760530 + 1.31728i 0.182047 + 0.983290i \(0.441728\pi\)
−0.942578 + 0.333987i \(0.891606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.00000 + 8.66025i 0.249688 + 0.432472i 0.963439 0.267927i \(-0.0863386\pi\)
−0.713751 + 0.700399i \(0.753005\pi\)
\(402\) 0 0
\(403\) −4.00000 + 6.92820i −0.199254 + 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 6.92820i 0.198273 0.343418i
\(408\) 0 0
\(409\) −5.00000 8.66025i −0.247234 0.428222i 0.715523 0.698589i \(-0.246188\pi\)
−0.962757 + 0.270367i \(0.912855\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.0000 −0.787309
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.0000 + 17.3205i 0.488532 + 0.846162i 0.999913 0.0131919i \(-0.00419923\pi\)
−0.511381 + 0.859354i \(0.670866\pi\)
\(420\) 0 0
\(421\) −19.0000 + 32.9090i −0.926003 + 1.60388i −0.136064 + 0.990700i \(0.543445\pi\)
−0.789940 + 0.613185i \(0.789888\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) −12.0000 20.7846i −0.580721 1.00584i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −14.0000 + 24.2487i −0.668184 + 1.15733i 0.310228 + 0.950662i \(0.399595\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.00000 10.3923i 0.285069 0.493753i −0.687557 0.726130i \(-0.741317\pi\)
0.972626 + 0.232377i \(0.0746503\pi\)
\(444\) 0 0
\(445\) −10.0000 17.3205i −0.474045 0.821071i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 13.8564i −0.375046 0.649598i
\(456\) 0 0
\(457\) −13.0000 + 22.5167i −0.608114 + 1.05328i 0.383437 + 0.923567i \(0.374740\pi\)
−0.991551 + 0.129718i \(0.958593\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 + 5.19615i −0.139724 + 0.242009i −0.927392 0.374091i \(-0.877955\pi\)
0.787668 + 0.616100i \(0.211288\pi\)
\(462\) 0 0
\(463\) −10.0000 17.3205i −0.464739 0.804952i 0.534450 0.845200i \(-0.320519\pi\)
−0.999190 + 0.0402476i \(0.987185\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 + 13.8564i 0.367840 + 0.637118i
\(474\) 0 0
\(475\) 2.00000 3.46410i 0.0917663 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 20.7846i 0.548294 0.949673i −0.450098 0.892979i \(-0.648611\pi\)
0.998392 0.0566937i \(-0.0180558\pi\)
\(480\) 0 0
\(481\) −2.00000 3.46410i −0.0911922 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.0000 1.27141
\(486\) 0 0
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14.0000 24.2487i −0.631811 1.09433i −0.987181 0.159603i \(-0.948978\pi\)
0.355370 0.934726i \(-0.384355\pi\)
\(492\) 0 0
\(493\) 6.00000 10.3923i 0.270226 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 55.4256i 1.43540 2.48618i
\(498\) 0 0
\(499\) −6.00000 10.3923i −0.268597 0.465223i 0.699903 0.714238i \(-0.253227\pi\)
−0.968500 + 0.249015i \(0.919893\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.00000 + 15.5885i 0.398918 + 0.690946i 0.993593 0.113020i \(-0.0360525\pi\)
−0.594675 + 0.803966i \(0.702719\pi\)
\(510\) 0 0
\(511\) 12.0000 20.7846i 0.530849 0.919457i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.0000 20.7846i 0.528783 0.915879i
\(516\) 0 0
\(517\) −16.0000 27.7128i −0.703679 1.21881i
\(518\) 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
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 + 20.7846i 0.522728 + 0.905392i
\(528\) 0 0
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.00000 + 3.46410i −0.0866296 + 0.150047i
\(534\) 0 0
\(535\) −4.00000 6.92820i −0.172935 0.299532i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 14.0000 + 24.2487i 0.599694 + 1.03870i
\(546\) 0 0
\(547\) 6.00000 10.3923i 0.256541 0.444343i −0.708772 0.705438i \(-0.750750\pi\)
0.965313 + 0.261095i \(0.0840836\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.00000 6.92820i 0.170406 0.295151i
\(552\) 0 0
\(553\) 8.00000 + 13.8564i 0.340195 + 0.589234i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 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 0.711445 0.702742i \(-0.248041\pi\)
−0.964315 + 0.264758i \(0.914708\pi\)
\(564\) 0 0
\(565\) −2.00000 + 3.46410i −0.0841406 + 0.145736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −23.0000 + 39.8372i −0.964210 + 1.67006i −0.252488 + 0.967600i \(0.581249\pi\)
−0.711722 + 0.702461i \(0.752085\pi\)
\(570\) 0 0
\(571\) 22.0000 + 38.1051i 0.920671 + 1.59465i 0.798379 + 0.602155i \(0.205691\pi\)
0.122292 + 0.992494i \(0.460975\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −24.0000 41.5692i −0.995688 1.72458i
\(582\) 0 0
\(583\) 20.0000 34.6410i 0.828315 1.43468i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000 31.1769i 0.742940 1.28681i −0.208212 0.978084i \(-0.566764\pi\)
0.951151 0.308725i \(-0.0999023\pi\)
\(588\) 0 0
\(589\) 8.00000 + 13.8564i 0.329634 + 0.570943i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) −48.0000 −1.96781
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.00000 + 13.8564i 0.326871 + 0.566157i 0.981889 0.189456i \(-0.0606724\pi\)
−0.655018 + 0.755613i \(0.727339\pi\)
\(600\) 0 0
\(601\) −5.00000 + 8.66025i −0.203954 + 0.353259i −0.949799 0.312861i \(-0.898713\pi\)
0.745845 + 0.666120i \(0.232046\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 8.66025i 0.203279 0.352089i
\(606\) 0 0
\(607\) −10.0000 17.3205i −0.405887 0.703018i 0.588537 0.808470i \(-0.299704\pi\)
−0.994424 + 0.105453i \(0.966371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.0000 + 36.3731i 0.845428 + 1.46432i 0.885249 + 0.465118i \(0.153988\pi\)
−0.0398207 + 0.999207i \(0.512679\pi\)
\(618\) 0 0
\(619\) 6.00000 10.3923i 0.241160 0.417702i −0.719885 0.694094i \(-0.755805\pi\)
0.961045 + 0.276392i \(0.0891387\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 20.0000 34.6410i 0.801283 1.38786i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.0000 + 34.6410i 0.793676 + 1.37469i
\(636\) 0 0
\(637\) 9.00000 15.5885i 0.356593 0.617637i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.00000 8.66025i 0.197488 0.342059i −0.750225 0.661182i \(-0.770055\pi\)
0.947713 + 0.319123i \(0.103388\pi\)
\(642\) 0 0
\(643\) 6.00000 + 10.3923i 0.236617 + 0.409832i 0.959741 0.280885i \(-0.0906280\pi\)
−0.723124 + 0.690718i \(0.757295\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 48.0000 1.88707 0.943537 0.331266i \(-0.107476\pi\)
0.943537 + 0.331266i \(0.107476\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.00000 + 8.66025i 0.195665 + 0.338902i 0.947118 0.320884i \(-0.103980\pi\)
−0.751453 + 0.659786i \(0.770647\pi\)
\(654\) 0 0
\(655\) 4.00000 6.92820i 0.156293 0.270707i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18.0000 + 31.1769i −0.701180 + 1.21448i 0.266872 + 0.963732i \(0.414010\pi\)
−0.968052 + 0.250748i \(0.919323\pi\)
\(660\) 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
\(665\) −32.0000 −1.24091
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 20.7846i −0.463255 0.802381i
\(672\) 0 0
\(673\) −1.00000 + 1.73205i −0.0385472 + 0.0667657i −0.884655 0.466246i \(-0.845606\pi\)
0.846108 + 0.533011i \(0.178940\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.00000 + 12.1244i −0.269032 + 0.465977i −0.968612 0.248577i \(-0.920037\pi\)
0.699580 + 0.714554i \(0.253370\pi\)
\(678\) 0 0
\(679\) 28.0000 + 48.4974i 1.07454 + 1.86116i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10.0000 17.3205i −0.380970 0.659859i
\(690\) 0 0
\(691\) −26.0000 + 45.0333i −0.989087 + 1.71315i −0.366947 + 0.930242i \(0.619597\pi\)
−0.622139 + 0.782907i \(0.713736\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 20.0000 34.6410i 0.758643 1.31401i
\(696\) 0 0
\(697\) 6.00000 + 10.3923i 0.227266 + 0.393637i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 20.7846i −0.451306 0.781686i
\(708\) 0 0
\(709\) −11.0000 + 19.0526i −0.413114 + 0.715534i −0.995228 0.0975728i \(-0.968892\pi\)
0.582115 + 0.813107i \(0.302225\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −8.00000 13.8564i −0.299183 0.518200i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.00000 1.73205i −0.0371391 0.0643268i
\(726\) 0 0
\(727\) 6.00000 10.3923i 0.222528 0.385429i −0.733047 0.680178i \(-0.761903\pi\)
0.955575 + 0.294749i \(0.0952359\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000 20.7846i 0.443836 0.768747i
\(732\) 0 0
\(733\) −7.00000 12.1244i −0.258551 0.447823i 0.707303 0.706910i \(-0.249912\pi\)
−0.965854 + 0.259087i \(0.916578\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 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
\(745\) −18.0000 + 31.1769i −0.659469 + 1.14223i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000 13.8564i 0.292314 0.506302i
\(750\) 0 0
\(751\) 2.00000 + 3.46410i 0.0729810 + 0.126407i 0.900207 0.435463i \(-0.143415\pi\)
−0.827225 + 0.561870i \(0.810082\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.0000 25.9808i −0.543750 0.941802i −0.998684 0.0512772i \(-0.983671\pi\)
0.454935 0.890525i \(-0.349663\pi\)
\(762\) 0 0
\(763\) −28.0000 + 48.4974i −1.01367 + 1.75572i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.00000 + 6.92820i −0.144432 + 0.250163i
\(768\) 0 0
\(769\) 7.00000 + 12.1244i 0.252426 + 0.437215i 0.964193 0.265200i \(-0.0854381\pi\)
−0.711767 + 0.702416i \(0.752105\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.00000 + 6.92820i 0.143315 + 0.248229i
\(780\) 0 0
\(781\) 32.0000 55.4256i 1.14505 1.98328i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10.0000 + 17.3205i −0.356915 + 0.618195i
\(786\) 0 0
\(787\) 6.00000 + 10.3923i 0.213877 + 0.370446i 0.952925 0.303207i \(-0.0980575\pi\)
−0.739048 + 0.673653i \(0.764724\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.00000 5.19615i −0.106265 0.184057i 0.807989 0.589197i \(-0.200556\pi\)
−0.914255 + 0.405140i \(0.867223\pi\)
\(798\) 0 0
\(799\) −24.0000 + 41.5692i −0.849059 + 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.0000 20.7846i 0.423471 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.00000 + 6.92820i 0.140114 + 0.242684i
\(816\) 0 0
\(817\) 8.00000 13.8564i 0.279885 0.484774i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.0000 22.5167i 0.453703 0.785837i −0.544909 0.838495i \(-0.683436\pi\)
0.998613 + 0.0526580i \(0.0167693\pi\)
\(822\) 0 0
\(823\) −2.00000 3.46410i −0.0697156 0.120751i 0.829060 0.559159i \(-0.188876\pi\)
−0.898776 + 0.438408i \(0.855543\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −27.0000 46.7654i −0.935495 1.62032i
\(834\) 0 0
\(835\) 8.00000 13.8564i 0.276851 0.479521i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.0000 + 27.7128i −0.552381 + 0.956753i 0.445721 + 0.895172i \(0.352947\pi\)
−0.998102 + 0.0615805i \(0.980386\pi\)
\(840\) 0 0
\(841\) 12.5000 + 21.6506i 0.431034 + 0.746574i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 9.00000 15.5885i 0.308154 0.533739i −0.669804 0.742538i \(-0.733622\pi\)
0.977959 + 0.208799i \(0.0669554\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.00000 + 12.1244i −0.239115 + 0.414160i −0.960461 0.278416i \(-0.910191\pi\)
0.721345 + 0.692576i \(0.243524\pi\)
\(858\) 0 0
\(859\) 2.00000 + 3.46410i 0.0682391 + 0.118194i 0.898126 0.439738i \(-0.144929\pi\)
−0.829887 + 0.557931i \(0.811595\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.00000 + 13.8564i 0.271381 + 0.470046i
\(870\) 0 0
\(871\) −4.00000 + 6.92820i −0.135535 + 0.234753i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −24.0000 + 41.5692i −0.811348 + 1.40530i
\(876\) 0 0
\(877\) −11.0000 19.0526i −0.371444 0.643359i 0.618344 0.785907i \(-0.287804\pi\)
−0.989788 + 0.142548i \(0.954470\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12.0000 20.7846i −0.402921 0.697879i 0.591156 0.806557i \(-0.298672\pi\)
−0.994077 + 0.108678i \(0.965338\pi\)
\(888\) 0 0
\(889\) −40.0000 + 69.2820i −1.34156 + 2.32364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −16.0000 + 27.7128i −0.535420 + 0.927374i
\(894\) 0 0
\(895\) −12.0000 20.7846i −0.401116 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.0000 17.3205i −0.332411 0.575753i
\(906\) 0 0
\(907\) 10.0000 17.3205i 0.332045 0.575118i −0.650868 0.759191i \(-0.725595\pi\)
0.982913 + 0.184073i \(0.0589282\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(912\) 0 0
\(913\) −24.0000 41.5692i −0.794284 1.37574i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.0000 27.7128i −0.526646 0.912178i
\(924\) 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
\(931\) −18.0000 31.1769i −0.589926 1.02178i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −48.0000 −1.56977
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.0000 + 50.2295i 0.945373 + 1.63743i 0.755003 + 0.655722i \(0.227636\pi\)
0.190370 + 0.981712i \(0.439031\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.00000 10.3923i 0.194974 0.337705i −0.751918 0.659256i \(-0.770871\pi\)
0.946892 + 0.321552i \(0.104204\pi\)
\(948\) 0 0
\(949\) −6.00000 10.3923i −0.194768 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 36.0000 + 62.3538i 1.16250 + 2.01351i
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 18.0000 31.1769i 0.579441 1.00362i
\(966\) 0 0
\(967\) −6.00000 10.3923i −0.192947 0.334194i 0.753279 0.657702i \(-0.228471\pi\)
−0.946226 + 0.323508i \(0.895138\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 52.0000 1.66876 0.834380 0.551190i \(-0.185826\pi\)
0.834380 + 0.551190i \(0.185826\pi\)
\(972\) 0 0
\(973\) 80.0000 2.56468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11.0000 19.0526i −0.351921 0.609545i 0.634665 0.772787i \(-0.281138\pi\)
−0.986586 + 0.163242i \(0.947805\pi\)
\(978\) 0 0
\(979\) 20.0000 34.6410i 0.639203 1.10713i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12.0000 + 20.7846i −0.382741 + 0.662926i −0.991453 0.130465i \(-0.958353\pi\)
0.608712 + 0.793391i \(0.291686\pi\)
\(984\) 0 0
\(985\) 22.0000 + 38.1051i 0.700978 + 1.21413i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 6.92820i −0.126809 0.219639i
\(996\) 0 0
\(997\) 9.00000 15.5885i 0.285033 0.493691i −0.687584 0.726105i \(-0.741329\pi\)
0.972617 + 0.232413i \(0.0746622\pi\)
\(998\) 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 2592.2.i.q.865.1 2
3.2 odd 2 2592.2.i.b.865.1 2
4.3 odd 2 2592.2.i.w.865.1 2
9.2 odd 6 96.2.a.a.1.1 1
9.4 even 3 inner 2592.2.i.q.1729.1 2
9.5 odd 6 2592.2.i.b.1729.1 2
9.7 even 3 288.2.a.c.1.1 1
12.11 even 2 2592.2.i.h.865.1 2
36.7 odd 6 288.2.a.b.1.1 1
36.11 even 6 96.2.a.b.1.1 yes 1
36.23 even 6 2592.2.i.h.1729.1 2
36.31 odd 6 2592.2.i.w.1729.1 2
45.2 even 12 2400.2.f.a.1249.2 2
45.7 odd 12 7200.2.f.x.6049.2 2
45.29 odd 6 2400.2.a.r.1.1 1
45.34 even 6 7200.2.a.e.1.1 1
45.38 even 12 2400.2.f.a.1249.1 2
45.43 odd 12 7200.2.f.x.6049.1 2
63.20 even 6 4704.2.a.t.1.1 1
72.11 even 6 192.2.a.a.1.1 1
72.29 odd 6 192.2.a.c.1.1 1
72.43 odd 6 576.2.a.g.1.1 1
72.61 even 6 576.2.a.h.1.1 1
144.11 even 12 768.2.d.h.385.1 2
144.29 odd 12 768.2.d.a.385.1 2
144.43 odd 12 2304.2.d.s.1153.1 2
144.61 even 12 2304.2.d.c.1153.2 2
144.83 even 12 768.2.d.h.385.2 2
144.101 odd 12 768.2.d.a.385.2 2
144.115 odd 12 2304.2.d.s.1153.2 2
144.133 even 12 2304.2.d.c.1153.1 2
180.7 even 12 7200.2.f.f.6049.1 2
180.43 even 12 7200.2.f.f.6049.2 2
180.47 odd 12 2400.2.f.r.1249.1 2
180.79 odd 6 7200.2.a.bx.1.1 1
180.83 odd 12 2400.2.f.r.1249.2 2
180.119 even 6 2400.2.a.q.1.1 1
252.83 odd 6 4704.2.a.e.1.1 1
360.29 odd 6 4800.2.a.f.1.1 1
360.83 odd 12 4800.2.f.e.3649.1 2
360.173 even 12 4800.2.f.bh.3649.2 2
360.227 odd 12 4800.2.f.e.3649.2 2
360.299 even 6 4800.2.a.co.1.1 1
360.317 even 12 4800.2.f.bh.3649.1 2
504.83 odd 6 9408.2.a.ct.1.1 1
504.461 even 6 9408.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.a.a.1.1 1 9.2 odd 6
96.2.a.b.1.1 yes 1 36.11 even 6
192.2.a.a.1.1 1 72.11 even 6
192.2.a.c.1.1 1 72.29 odd 6
288.2.a.b.1.1 1 36.7 odd 6
288.2.a.c.1.1 1 9.7 even 3
576.2.a.g.1.1 1 72.43 odd 6
576.2.a.h.1.1 1 72.61 even 6
768.2.d.a.385.1 2 144.29 odd 12
768.2.d.a.385.2 2 144.101 odd 12
768.2.d.h.385.1 2 144.11 even 12
768.2.d.h.385.2 2 144.83 even 12
2304.2.d.c.1153.1 2 144.133 even 12
2304.2.d.c.1153.2 2 144.61 even 12
2304.2.d.s.1153.1 2 144.43 odd 12
2304.2.d.s.1153.2 2 144.115 odd 12
2400.2.a.q.1.1 1 180.119 even 6
2400.2.a.r.1.1 1 45.29 odd 6
2400.2.f.a.1249.1 2 45.38 even 12
2400.2.f.a.1249.2 2 45.2 even 12
2400.2.f.r.1249.1 2 180.47 odd 12
2400.2.f.r.1249.2 2 180.83 odd 12
2592.2.i.b.865.1 2 3.2 odd 2
2592.2.i.b.1729.1 2 9.5 odd 6
2592.2.i.h.865.1 2 12.11 even 2
2592.2.i.h.1729.1 2 36.23 even 6
2592.2.i.q.865.1 2 1.1 even 1 trivial
2592.2.i.q.1729.1 2 9.4 even 3 inner
2592.2.i.w.865.1 2 4.3 odd 2
2592.2.i.w.1729.1 2 36.31 odd 6
4704.2.a.e.1.1 1 252.83 odd 6
4704.2.a.t.1.1 1 63.20 even 6
4800.2.a.f.1.1 1 360.29 odd 6
4800.2.a.co.1.1 1 360.299 even 6
4800.2.f.e.3649.1 2 360.83 odd 12
4800.2.f.e.3649.2 2 360.227 odd 12
4800.2.f.bh.3649.1 2 360.317 even 12
4800.2.f.bh.3649.2 2 360.173 even 12
7200.2.a.e.1.1 1 45.34 even 6
7200.2.a.bx.1.1 1 180.79 odd 6
7200.2.f.f.6049.1 2 180.7 even 12
7200.2.f.f.6049.2 2 180.43 even 12
7200.2.f.x.6049.1 2 45.43 odd 12
7200.2.f.x.6049.2 2 45.7 odd 12
9408.2.a.bj.1.1 1 504.461 even 6
9408.2.a.ct.1.1 1 504.83 odd 6