Properties

Label 2592.2.i.f.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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 864)
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.f.1729.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{5} +(0.500000 - 0.866025i) q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{5} +(0.500000 - 0.866025i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-0.500000 - 0.866025i) q^{13} +6.00000 q^{17} +5.00000 q^{19} +(-3.00000 - 5.19615i) q^{23} +(0.500000 - 0.866025i) q^{25} +(-4.00000 + 6.92820i) q^{29} +(4.00000 + 6.92820i) q^{31} -2.00000 q^{35} -5.00000 q^{37} +(-4.00000 - 6.92820i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(5.00000 - 8.66025i) q^{47} +(3.00000 + 5.19615i) q^{49} +4.00000 q^{53} -4.00000 q^{55} +(-7.00000 - 12.1244i) q^{59} +(-1.50000 + 2.59808i) q^{61} +(-1.00000 + 1.73205i) q^{65} +(-6.50000 - 11.2583i) q^{67} -4.00000 q^{71} +9.00000 q^{73} +(-1.00000 - 1.73205i) q^{77} +(5.50000 - 9.52628i) q^{79} +(6.00000 - 10.3923i) q^{83} +(-6.00000 - 10.3923i) q^{85} -2.00000 q^{89} -1.00000 q^{91} +(-5.00000 - 8.66025i) q^{95} +(-0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} + 2 q^{11} - q^{13} + 12 q^{17} + 10 q^{19} - 6 q^{23} + q^{25} - 8 q^{29} + 8 q^{31} - 4 q^{35} - 10 q^{37} - 8 q^{41} - 4 q^{43} + 10 q^{47} + 6 q^{49} + 8 q^{53} - 8 q^{55} - 14 q^{59} - 3 q^{61} - 2 q^{65} - 13 q^{67} - 8 q^{71} + 18 q^{73} - 2 q^{77} + 11 q^{79} + 12 q^{83} - 12 q^{85} - 4 q^{89} - 2 q^{91} - 10 q^{95} - q^{97}+O(q^{100}) \) 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 0.550990 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 0.500000 0.866025i 0.188982 0.327327i −0.755929 0.654654i \(-0.772814\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) −0.500000 0.866025i −0.138675 0.240192i 0.788320 0.615265i \(-0.210951\pi\)
−0.926995 + 0.375073i \(0.877618\pi\)
\(14\) 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
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 5.19615i −0.625543 1.08347i −0.988436 0.151642i \(-0.951544\pi\)
0.362892 0.931831i \(-0.381789\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.00000 + 6.92820i −0.742781 + 1.28654i 0.208443 + 0.978035i \(0.433160\pi\)
−0.951224 + 0.308500i \(0.900173\pi\)
\(30\) 0 0
\(31\) 4.00000 + 6.92820i 0.718421 + 1.24434i 0.961625 + 0.274367i \(0.0884683\pi\)
−0.243204 + 0.969975i \(0.578198\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 6.92820i −0.624695 1.08200i −0.988600 0.150567i \(-0.951890\pi\)
0.363905 0.931436i \(-0.381443\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.00000 8.66025i 0.729325 1.26323i −0.227844 0.973698i \(-0.573168\pi\)
0.957169 0.289530i \(-0.0934991\pi\)
\(48\) 0 0
\(49\) 3.00000 + 5.19615i 0.428571 + 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.00000 12.1244i −0.911322 1.57846i −0.812198 0.583382i \(-0.801729\pi\)
−0.0991242 0.995075i \(-0.531604\pi\)
\(60\) 0 0
\(61\) −1.50000 + 2.59808i −0.192055 + 0.332650i −0.945931 0.324367i \(-0.894849\pi\)
0.753876 + 0.657017i \(0.228182\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 + 1.73205i −0.124035 + 0.214834i
\(66\) 0 0
\(67\) −6.50000 11.2583i −0.794101 1.37542i −0.923408 0.383819i \(-0.874609\pi\)
0.129307 0.991605i \(-0.458725\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 9.00000 1.05337 0.526685 0.850060i \(-0.323435\pi\)
0.526685 + 0.850060i \(0.323435\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 1.73205i −0.113961 0.197386i
\(78\) 0 0
\(79\) 5.50000 9.52628i 0.618798 1.07179i −0.370907 0.928670i \(-0.620953\pi\)
0.989705 0.143120i \(-0.0457135\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 10.3923i 0.658586 1.14070i −0.322396 0.946605i \(-0.604488\pi\)
0.980982 0.194099i \(-0.0621783\pi\)
\(84\) 0 0
\(85\) −6.00000 10.3923i −0.650791 1.12720i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.00000 8.66025i −0.512989 0.888523i
\(96\) 0 0
\(97\) −0.500000 + 0.866025i −0.0507673 + 0.0879316i −0.890292 0.455389i \(-0.849500\pi\)
0.839525 + 0.543321i \(0.182833\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) −1.50000 2.59808i −0.147799 0.255996i 0.782614 0.622507i \(-0.213886\pi\)
−0.930414 + 0.366511i \(0.880552\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.00000 + 8.66025i 0.470360 + 0.814688i 0.999425 0.0338931i \(-0.0107906\pi\)
−0.529065 + 0.848581i \(0.677457\pi\)
\(114\) 0 0
\(115\) −6.00000 + 10.3923i −0.559503 + 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.00000 5.19615i 0.275010 0.476331i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 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 \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 13.8564i −0.698963 1.21064i −0.968826 0.247741i \(-0.920312\pi\)
0.269863 0.962899i \(-0.413022\pi\)
\(132\) 0 0
\(133\) 2.50000 4.33013i 0.216777 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000 5.19615i 0.256307 0.443937i −0.708942 0.705266i \(-0.750827\pi\)
0.965250 + 0.261329i \(0.0841608\pi\)
\(138\) 0 0
\(139\) 2.50000 + 4.33013i 0.212047 + 0.367277i 0.952355 0.304991i \(-0.0986536\pi\)
−0.740308 + 0.672268i \(0.765320\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.00000 + 10.3923i 0.491539 + 0.851371i 0.999953 0.00974235i \(-0.00310113\pi\)
−0.508413 + 0.861113i \(0.669768\pi\)
\(150\) 0 0
\(151\) −4.50000 + 7.79423i −0.366205 + 0.634285i −0.988969 0.148124i \(-0.952676\pi\)
0.622764 + 0.782410i \(0.286010\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 13.8564i 0.642575 1.11297i
\(156\) 0 0
\(157\) −1.00000 1.73205i −0.0798087 0.138233i 0.823359 0.567521i \(-0.192098\pi\)
−0.903167 + 0.429289i \(0.858764\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.00000 12.1244i −0.541676 0.938211i −0.998808 0.0488118i \(-0.984457\pi\)
0.457132 0.889399i \(-0.348877\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.0000 20.7846i 0.912343 1.58022i 0.101598 0.994826i \(-0.467605\pi\)
0.810745 0.585399i \(-0.199062\pi\)
\(174\) 0 0
\(175\) −0.500000 0.866025i −0.0377964 0.0654654i
\(176\) 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
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.00000 + 8.66025i 0.367607 + 0.636715i
\(186\) 0 0
\(187\) 6.00000 10.3923i 0.438763 0.759961i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 5.19615i 0.217072 0.375980i −0.736839 0.676068i \(-0.763683\pi\)
0.953912 + 0.300088i \(0.0970159\pi\)
\(192\) 0 0
\(193\) 10.5000 + 18.1865i 0.755807 + 1.30910i 0.944972 + 0.327150i \(0.106088\pi\)
−0.189166 + 0.981945i \(0.560578\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 19.0000 1.34687 0.673437 0.739244i \(-0.264817\pi\)
0.673437 + 0.739244i \(0.264817\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.00000 + 6.92820i 0.280745 + 0.486265i
\(204\) 0 0
\(205\) −8.00000 + 13.8564i −0.558744 + 0.967773i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.00000 8.66025i 0.345857 0.599042i
\(210\) 0 0
\(211\) 0.500000 + 0.866025i 0.0344214 + 0.0596196i 0.882723 0.469894i \(-0.155708\pi\)
−0.848301 + 0.529514i \(0.822374\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.00000 5.19615i −0.201802 0.349531i
\(222\) 0 0
\(223\) 8.00000 13.8564i 0.535720 0.927894i −0.463409 0.886145i \(-0.653374\pi\)
0.999128 0.0417488i \(-0.0132929\pi\)
\(224\) 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
\(229\) −13.0000 22.5167i −0.859064 1.48794i −0.872823 0.488037i \(-0.837713\pi\)
0.0137585 0.999905i \(-0.495620\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) −20.0000 −1.30466
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 10.3923i −0.388108 0.672222i 0.604087 0.796918i \(-0.293538\pi\)
−0.992195 + 0.124696i \(0.960204\pi\)
\(240\) 0 0
\(241\) −6.50000 + 11.2583i −0.418702 + 0.725213i −0.995809 0.0914555i \(-0.970848\pi\)
0.577107 + 0.816668i \(0.304181\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.00000 10.3923i 0.383326 0.663940i
\(246\) 0 0
\(247\) −2.50000 4.33013i −0.159071 0.275519i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 + 3.46410i 0.124757 + 0.216085i 0.921638 0.388051i \(-0.126852\pi\)
−0.796881 + 0.604136i \(0.793518\pi\)
\(258\) 0 0
\(259\) −2.50000 + 4.33013i −0.155342 + 0.269061i
\(260\) 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
\(265\) −4.00000 6.92820i −0.245718 0.425596i
\(266\) 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
\(271\) 21.0000 1.27566 0.637830 0.770178i \(-0.279832\pi\)
0.637830 + 0.770178i \(0.279832\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 1.73205i −0.0603023 0.104447i
\(276\) 0 0
\(277\) 1.00000 1.73205i 0.0600842 0.104069i −0.834419 0.551131i \(-0.814196\pi\)
0.894503 + 0.447062i \(0.147530\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −12.0000 + 20.7846i −0.715860 + 1.23991i 0.246767 + 0.969075i \(0.420632\pi\)
−0.962627 + 0.270831i \(0.912702\pi\)
\(282\) 0 0
\(283\) 10.0000 + 17.3205i 0.594438 + 1.02960i 0.993626 + 0.112728i \(0.0359589\pi\)
−0.399188 + 0.916869i \(0.630708\pi\)
\(284\) 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
\(293\) 9.00000 + 15.5885i 0.525786 + 0.910687i 0.999549 + 0.0300351i \(0.00956192\pi\)
−0.473763 + 0.880652i \(0.657105\pi\)
\(294\) 0 0
\(295\) −14.0000 + 24.2487i −0.815112 + 1.41181i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 + 5.19615i −0.173494 + 0.300501i
\(300\) 0 0
\(301\) 2.00000 + 3.46410i 0.115278 + 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 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
\(311\) 15.0000 + 25.9808i 0.850572 + 1.47323i 0.880693 + 0.473688i \(0.157077\pi\)
−0.0301210 + 0.999546i \(0.509589\pi\)
\(312\) 0 0
\(313\) 8.50000 14.7224i 0.480448 0.832161i −0.519300 0.854592i \(-0.673807\pi\)
0.999748 + 0.0224310i \(0.00714060\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.00000 + 10.3923i −0.336994 + 0.583690i −0.983866 0.178908i \(-0.942743\pi\)
0.646872 + 0.762598i \(0.276077\pi\)
\(318\) 0 0
\(319\) 8.00000 + 13.8564i 0.447914 + 0.775810i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5.00000 8.66025i −0.275659 0.477455i
\(330\) 0 0
\(331\) 14.5000 25.1147i 0.796992 1.38043i −0.124574 0.992210i \(-0.539757\pi\)
0.921567 0.388221i \(-0.126910\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −13.0000 + 22.5167i −0.710266 + 1.23022i
\(336\) 0 0
\(337\) 15.5000 + 26.8468i 0.844339 + 1.46244i 0.886194 + 0.463314i \(0.153340\pi\)
−0.0418554 + 0.999124i \(0.513327\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.00000 + 3.46410i 0.107366 + 0.185963i 0.914702 0.404128i \(-0.132425\pi\)
−0.807337 + 0.590091i \(0.799092\pi\)
\(348\) 0 0
\(349\) 2.50000 4.33013i 0.133822 0.231786i −0.791325 0.611396i \(-0.790608\pi\)
0.925147 + 0.379610i \(0.123942\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.00000 + 3.46410i −0.106449 + 0.184376i −0.914329 0.404971i \(-0.867282\pi\)
0.807880 + 0.589347i \(0.200615\pi\)
\(354\) 0 0
\(355\) 4.00000 + 6.92820i 0.212298 + 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.00000 15.5885i −0.471082 0.815937i
\(366\) 0 0
\(367\) −7.50000 + 12.9904i −0.391497 + 0.678092i −0.992647 0.121044i \(-0.961376\pi\)
0.601150 + 0.799136i \(0.294709\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.00000 3.46410i 0.103835 0.179847i
\(372\) 0 0
\(373\) −5.50000 9.52628i −0.284779 0.493252i 0.687776 0.725923i \(-0.258587\pi\)
−0.972556 + 0.232671i \(0.925254\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.00000 + 13.8564i 0.408781 + 0.708029i 0.994753 0.102302i \(-0.0326207\pi\)
−0.585973 + 0.810331i \(0.699287\pi\)
\(384\) 0 0
\(385\) −2.00000 + 3.46410i −0.101929 + 0.176547i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.00000 + 15.5885i −0.456318 + 0.790366i −0.998763 0.0497253i \(-0.984165\pi\)
0.542445 + 0.840091i \(0.317499\pi\)
\(390\) 0 0
\(391\) −18.0000 31.1769i −0.910299 1.57668i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −22.0000 −1.10694
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 24.2487i −0.699127 1.21092i −0.968770 0.247962i \(-0.920239\pi\)
0.269643 0.962960i \(-0.413094\pi\)
\(402\) 0 0
\(403\) 4.00000 6.92820i 0.199254 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.00000 + 8.66025i −0.247841 + 0.429273i
\(408\) 0 0
\(409\) −6.50000 11.2583i −0.321404 0.556689i 0.659374 0.751815i \(-0.270822\pi\)
−0.980778 + 0.195127i \(0.937488\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −14.0000 −0.688895
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13.0000 + 22.5167i 0.635092 + 1.10001i 0.986496 + 0.163787i \(0.0523710\pi\)
−0.351404 + 0.936224i \(0.614296\pi\)
\(420\) 0 0
\(421\) 9.50000 16.4545i 0.463002 0.801942i −0.536107 0.844150i \(-0.680106\pi\)
0.999109 + 0.0422075i \(0.0134391\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) 1.50000 + 2.59808i 0.0725901 + 0.125730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −15.0000 25.9808i −0.717547 1.24283i
\(438\) 0 0
\(439\) −4.00000 + 6.92820i −0.190910 + 0.330665i −0.945552 0.325471i \(-0.894477\pi\)
0.754642 + 0.656136i \(0.227810\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.0000 + 31.1769i −0.855206 + 1.48126i 0.0212481 + 0.999774i \(0.493236\pi\)
−0.876454 + 0.481486i \(0.840097\pi\)
\(444\) 0 0
\(445\) 2.00000 + 3.46410i 0.0948091 + 0.164214i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.00000 + 1.73205i 0.0468807 + 0.0811998i
\(456\) 0 0
\(457\) −1.00000 + 1.73205i −0.0467780 + 0.0810219i −0.888466 0.458942i \(-0.848229\pi\)
0.841688 + 0.539964i \(0.181562\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −15.0000 + 25.9808i −0.698620 + 1.21004i 0.270326 + 0.962769i \(0.412869\pi\)
−0.968945 + 0.247276i \(0.920465\pi\)
\(462\) 0 0
\(463\) −9.50000 16.4545i −0.441502 0.764705i 0.556299 0.830982i \(-0.312221\pi\)
−0.997801 + 0.0662777i \(0.978888\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.00000 −0.0925490 −0.0462745 0.998929i \(-0.514735\pi\)
−0.0462745 + 0.998929i \(0.514735\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 + 6.92820i 0.183920 + 0.318559i
\(474\) 0 0
\(475\) 2.50000 4.33013i 0.114708 0.198680i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.00000 + 10.3923i −0.274147 + 0.474837i −0.969920 0.243426i \(-0.921729\pi\)
0.695773 + 0.718262i \(0.255062\pi\)
\(480\) 0 0
\(481\) 2.50000 + 4.33013i 0.113990 + 0.197437i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 11.0000 0.498458 0.249229 0.968445i \(-0.419823\pi\)
0.249229 + 0.968445i \(0.419823\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.00000 + 12.1244i 0.315906 + 0.547165i 0.979630 0.200813i \(-0.0643584\pi\)
−0.663724 + 0.747978i \(0.731025\pi\)
\(492\) 0 0
\(493\) −24.0000 + 41.5692i −1.08091 + 1.87218i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.00000 + 3.46410i −0.0897123 + 0.155386i
\(498\) 0 0
\(499\) −12.0000 20.7846i −0.537194 0.930447i −0.999054 0.0434940i \(-0.986151\pi\)
0.461860 0.886953i \(-0.347182\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.00000 + 5.19615i 0.132973 + 0.230315i 0.924821 0.380402i \(-0.124214\pi\)
−0.791849 + 0.610718i \(0.790881\pi\)
\(510\) 0 0
\(511\) 4.50000 7.79423i 0.199068 0.344796i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.00000 + 5.19615i −0.132196 + 0.228970i
\(516\) 0 0
\(517\) −10.0000 17.3205i −0.439799 0.761755i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −15.0000 −0.655904 −0.327952 0.944694i \(-0.606358\pi\)
−0.327952 + 0.944694i \(0.606358\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 + 41.5692i 1.04546 + 1.81078i
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.00000 + 6.92820i −0.173259 + 0.300094i
\(534\) 0 0
\(535\) −2.00000 3.46410i −0.0864675 0.149766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 37.0000 1.59075 0.795377 0.606115i \(-0.207273\pi\)
0.795377 + 0.606115i \(0.207273\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 + 17.3205i 0.428353 + 0.741929i
\(546\) 0 0
\(547\) −7.50000 + 12.9904i −0.320677 + 0.555429i −0.980628 0.195880i \(-0.937244\pi\)
0.659951 + 0.751309i \(0.270577\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.0000 + 34.6410i −0.852029 + 1.47576i
\(552\) 0 0
\(553\) −5.50000 9.52628i −0.233884 0.405099i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000 + 20.7846i 0.505740 + 0.875967i 0.999978 + 0.00664037i \(0.00211371\pi\)
−0.494238 + 0.869326i \(0.664553\pi\)
\(564\) 0 0
\(565\) 10.0000 17.3205i 0.420703 0.728679i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.00000 8.66025i 0.209611 0.363057i −0.741981 0.670421i \(-0.766114\pi\)
0.951592 + 0.307364i \(0.0994469\pi\)
\(570\) 0 0
\(571\) 9.50000 + 16.4545i 0.397563 + 0.688599i 0.993425 0.114488i \(-0.0365228\pi\)
−0.595862 + 0.803087i \(0.703189\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 10.3923i −0.248922 0.431145i
\(582\) 0 0
\(583\) 4.00000 6.92820i 0.165663 0.286937i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000 5.19615i 0.123823 0.214468i −0.797449 0.603386i \(-0.793818\pi\)
0.921272 + 0.388918i \(0.127151\pi\)
\(588\) 0 0
\(589\) 20.0000 + 34.6410i 0.824086 + 1.42736i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.00000 6.92820i −0.163436 0.283079i 0.772663 0.634816i \(-0.218924\pi\)
−0.936099 + 0.351738i \(0.885591\pi\)
\(600\) 0 0
\(601\) 13.0000 22.5167i 0.530281 0.918474i −0.469095 0.883148i \(-0.655420\pi\)
0.999376 0.0353259i \(-0.0112469\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.00000 12.1244i 0.284590 0.492925i
\(606\) 0 0
\(607\) 20.5000 + 35.5070i 0.832069 + 1.44119i 0.896394 + 0.443257i \(0.146177\pi\)
−0.0643251 + 0.997929i \(0.520489\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −10.0000 −0.404557
\(612\) 0 0
\(613\) −9.00000 −0.363507 −0.181753 0.983344i \(-0.558177\pi\)
−0.181753 + 0.983344i \(0.558177\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21.0000 36.3731i −0.845428 1.46432i −0.885249 0.465118i \(-0.846012\pi\)
0.0398207 0.999207i \(-0.487321\pi\)
\(618\) 0 0
\(619\) −4.50000 + 7.79423i −0.180870 + 0.313276i −0.942177 0.335115i \(-0.891225\pi\)
0.761307 + 0.648392i \(0.224558\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.00000 + 1.73205i −0.0400642 + 0.0693932i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) 1.00000 0.0398094 0.0199047 0.999802i \(-0.493664\pi\)
0.0199047 + 0.999802i \(0.493664\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 20.0000 + 34.6410i 0.793676 + 1.37469i
\(636\) 0 0
\(637\) 3.00000 5.19615i 0.118864 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 + 3.46410i −0.0789953 + 0.136824i −0.902817 0.430026i \(-0.858505\pi\)
0.823821 + 0.566849i \(0.191838\pi\)
\(642\) 0 0
\(643\) −6.00000 10.3923i −0.236617 0.409832i 0.723124 0.690718i \(-0.242705\pi\)
−0.959741 + 0.280885i \(0.909372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.0000 + 38.1051i 0.860927 + 1.49117i 0.871036 + 0.491220i \(0.163449\pi\)
−0.0101092 + 0.999949i \(0.503218\pi\)
\(654\) 0 0
\(655\) −16.0000 + 27.7128i −0.625172 + 1.08283i
\(656\) 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
\(661\) −11.5000 19.9186i −0.447298 0.774743i 0.550911 0.834564i \(-0.314280\pi\)
−0.998209 + 0.0598209i \(0.980947\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.0000 −0.387783
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 + 5.19615i 0.115814 + 0.200595i
\(672\) 0 0
\(673\) −11.5000 + 19.9186i −0.443292 + 0.767805i −0.997932 0.0642860i \(-0.979523\pi\)
0.554639 + 0.832091i \(0.312856\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.00000 12.1244i 0.269032 0.465977i −0.699580 0.714554i \(-0.746630\pi\)
0.968612 + 0.248577i \(0.0799630\pi\)
\(678\) 0 0
\(679\) 0.500000 + 0.866025i 0.0191882 + 0.0332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.00000 3.46410i −0.0761939 0.131972i
\(690\) 0 0
\(691\) 2.00000 3.46410i 0.0760836 0.131781i −0.825473 0.564441i \(-0.809092\pi\)
0.901557 + 0.432660i \(0.142425\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.00000 8.66025i 0.189661 0.328502i
\(696\) 0 0
\(697\) −24.0000 41.5692i −0.909065 1.57455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −25.0000 −0.942893
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 11.5000 19.9186i 0.431892 0.748058i −0.565145 0.824992i \(-0.691180\pi\)
0.997036 + 0.0769337i \(0.0245130\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 41.5692i 0.898807 1.55678i
\(714\) 0 0
\(715\) 2.00000 + 3.46410i 0.0747958 + 0.129550i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −52.0000 −1.93927 −0.969636 0.244551i \(-0.921359\pi\)
−0.969636 + 0.244551i \(0.921359\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.00000 + 6.92820i 0.148556 + 0.257307i
\(726\) 0 0
\(727\) 24.0000 41.5692i 0.890111 1.54172i 0.0503692 0.998731i \(-0.483960\pi\)
0.839742 0.542986i \(-0.182706\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 + 20.7846i −0.443836 + 0.768747i
\(732\) 0 0
\(733\) 17.0000 + 29.4449i 0.627909 + 1.08757i 0.987971 + 0.154642i \(0.0494225\pi\)
−0.360061 + 0.932929i \(0.617244\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −26.0000 −0.957722
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.0000 + 36.3731i 0.770415 + 1.33440i 0.937336 + 0.348428i \(0.113284\pi\)
−0.166920 + 0.985970i \(0.553382\pi\)
\(744\) 0 0
\(745\) 12.0000 20.7846i 0.439646 0.761489i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.00000 1.73205i 0.0365392 0.0632878i
\(750\) 0 0
\(751\) −12.5000 21.6506i −0.456131 0.790043i 0.542621 0.839978i \(-0.317432\pi\)
−0.998752 + 0.0499348i \(0.984099\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.0000 0.655087
\(756\) 0 0
\(757\) 33.0000 1.19941 0.599703 0.800223i \(-0.295286\pi\)
0.599703 + 0.800223i \(0.295286\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 + 36.3731i 0.761249 + 1.31852i 0.942207 + 0.335032i \(0.108747\pi\)
−0.180957 + 0.983491i \(0.557920\pi\)
\(762\) 0 0
\(763\) −5.00000 + 8.66025i −0.181012 + 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.00000 + 12.1244i −0.252755 + 0.437785i
\(768\) 0 0
\(769\) 8.50000 + 14.7224i 0.306518 + 0.530904i 0.977598 0.210480i \(-0.0675028\pi\)
−0.671080 + 0.741385i \(0.734169\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0000 34.6410i −0.716574 1.24114i
\(780\) 0 0
\(781\) −4.00000 + 6.92820i −0.143131 + 0.247911i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 + 3.46410i −0.0713831 + 0.123639i
\(786\) 0 0
\(787\) 1.50000 + 2.59808i 0.0534692 + 0.0926114i 0.891521 0.452979i \(-0.149639\pi\)
−0.838052 + 0.545590i \(0.816305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) 3.00000 0.106533
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 + 20.7846i 0.425062 + 0.736229i 0.996426 0.0844678i \(-0.0269190\pi\)
−0.571364 + 0.820696i \(0.693586\pi\)
\(798\) 0 0
\(799\) 30.0000 51.9615i 1.06132 1.83827i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.00000 15.5885i 0.317603 0.550105i
\(804\) 0 0
\(805\) 6.00000 + 10.3923i 0.211472 + 0.366281i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.0000 19.0526i −0.385313 0.667382i
\(816\) 0 0
\(817\) −10.0000 + 17.3205i −0.349856 + 0.605968i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 + 1.73205i −0.0349002 + 0.0604490i −0.882948 0.469471i \(-0.844445\pi\)
0.848048 + 0.529920i \(0.177778\pi\)
\(822\) 0 0
\(823\) 21.5000 + 37.2391i 0.749443 + 1.29807i 0.948090 + 0.318002i \(0.103012\pi\)
−0.198647 + 0.980071i \(0.563655\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.00000 −0.0695468 −0.0347734 0.999395i \(-0.511071\pi\)
−0.0347734 + 0.999395i \(0.511071\pi\)
\(828\) 0 0
\(829\) 41.0000 1.42399 0.711994 0.702185i \(-0.247792\pi\)
0.711994 + 0.702185i \(0.247792\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.0000 + 31.1769i 0.623663 + 1.08022i
\(834\) 0 0
\(835\) −14.0000 + 24.2487i −0.484490 + 0.839161i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.00000 3.46410i 0.0690477 0.119594i −0.829435 0.558604i \(-0.811337\pi\)
0.898482 + 0.439010i \(0.144671\pi\)
\(840\) 0 0
\(841\) −17.5000 30.3109i −0.603448 1.04520i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −24.0000 −0.825625
\(846\) 0 0
\(847\) 7.00000 0.240523
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.0000 + 25.9808i 0.514193 + 0.890609i
\(852\) 0 0
\(853\) −7.50000 + 12.9904i −0.256795 + 0.444782i −0.965382 0.260842i \(-0.916000\pi\)
0.708586 + 0.705624i \(0.249333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.0000 + 24.2487i −0.478231 + 0.828320i −0.999689 0.0249570i \(-0.992055\pi\)
0.521458 + 0.853277i \(0.325388\pi\)
\(858\) 0 0
\(859\) −9.50000 16.4545i −0.324136 0.561420i 0.657201 0.753715i \(-0.271740\pi\)
−0.981337 + 0.192295i \(0.938407\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.0000 1.29354 0.646768 0.762687i \(-0.276120\pi\)
0.646768 + 0.762687i \(0.276120\pi\)
\(864\) 0 0
\(865\) −48.0000 −1.63205
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −11.0000 19.0526i −0.373149 0.646314i
\(870\) 0 0
\(871\) −6.50000 + 11.2583i −0.220244 + 0.381474i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6.00000 + 10.3923i −0.202837 + 0.351324i
\(876\) 0 0
\(877\) 8.50000 + 14.7224i 0.287025 + 0.497141i 0.973098 0.230391i \(-0.0740005\pi\)
−0.686074 + 0.727532i \(0.740667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 31.0000 1.04323 0.521617 0.853180i \(-0.325329\pi\)
0.521617 + 0.853180i \(0.325329\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −6.00000 10.3923i −0.201460 0.348939i 0.747539 0.664218i \(-0.231235\pi\)
−0.948999 + 0.315279i \(0.897902\pi\)
\(888\) 0 0
\(889\) −10.0000 + 17.3205i −0.335389 + 0.580911i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.0000 43.3013i 0.836593 1.44902i
\(894\) 0 0
\(895\) 12.0000 + 20.7846i 0.401116 + 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −64.0000 −2.13452
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.0000 + 32.9090i 0.631581 + 1.09393i
\(906\) 0 0
\(907\) −8.50000 + 14.7224i −0.282238 + 0.488850i −0.971936 0.235247i \(-0.924410\pi\)
0.689698 + 0.724097i \(0.257743\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 41.5692i 0.795155 1.37725i −0.127585 0.991828i \(-0.540723\pi\)
0.922740 0.385422i \(-0.125944\pi\)
\(912\) 0 0
\(913\) −12.0000 20.7846i −0.397142 0.687870i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −16.0000 −0.528367
\(918\) 0 0
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.00000 + 3.46410i 0.0658308 + 0.114022i
\(924\) 0 0
\(925\) −2.50000 + 4.33013i −0.0821995 + 0.142374i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.00000 + 10.3923i −0.196854 + 0.340960i −0.947507 0.319736i \(-0.896406\pi\)
0.750653 + 0.660697i \(0.229739\pi\)
\(930\) 0 0
\(931\) 15.0000 + 25.9808i 0.491605 + 0.851485i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −25.0000 −0.816714 −0.408357 0.912822i \(-0.633898\pi\)
−0.408357 + 0.912822i \(0.633898\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.0000 19.0526i −0.358590 0.621096i 0.629136 0.777295i \(-0.283409\pi\)
−0.987725 + 0.156200i \(0.950076\pi\)
\(942\) 0 0
\(943\) −24.0000 + 41.5692i −0.781548 + 1.35368i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.00000 5.19615i 0.0974869 0.168852i −0.813157 0.582045i \(-0.802253\pi\)
0.910644 + 0.413192i \(0.135586\pi\)
\(948\) 0 0
\(949\) −4.50000 7.79423i −0.146076 0.253011i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.00000 5.19615i −0.0968751 0.167793i
\(960\) 0 0
\(961\) −16.5000 + 28.5788i −0.532258 + 0.921898i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 21.0000 36.3731i 0.676014 1.17089i
\(966\) 0 0
\(967\) 13.5000 + 23.3827i 0.434131 + 0.751936i 0.997224 0.0744567i \(-0.0237223\pi\)
−0.563094 + 0.826393i \(0.690389\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 34.0000 1.09111 0.545556 0.838074i \(-0.316319\pi\)
0.545556 + 0.838074i \(0.316319\pi\)
\(972\) 0 0
\(973\) 5.00000 0.160293
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.0000 38.1051i −0.703842 1.21909i −0.967108 0.254367i \(-0.918133\pi\)
0.263265 0.964723i \(-0.415201\pi\)
\(978\) 0 0
\(979\) −2.00000 + 3.46410i −0.0639203 + 0.110713i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21.0000 + 36.3731i −0.669796 + 1.16012i 0.308165 + 0.951333i \(0.400285\pi\)
−0.977961 + 0.208788i \(0.933048\pi\)
\(984\) 0 0
\(985\) −2.00000 3.46410i −0.0637253 0.110375i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 7.00000 0.222362 0.111181 0.993800i \(-0.464537\pi\)
0.111181 + 0.993800i \(0.464537\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −19.0000 32.9090i −0.602340 1.04328i
\(996\) 0 0
\(997\) 27.0000 46.7654i 0.855099 1.48107i −0.0214550 0.999770i \(-0.506830\pi\)
0.876554 0.481304i \(-0.159837\pi\)
\(998\) 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.f.865.1 2
3.2 odd 2 2592.2.i.u.865.1 2
4.3 odd 2 2592.2.i.d.865.1 2
9.2 odd 6 864.2.a.b.1.1 1
9.4 even 3 inner 2592.2.i.f.1729.1 2
9.5 odd 6 2592.2.i.u.1729.1 2
9.7 even 3 864.2.a.j.1.1 yes 1
12.11 even 2 2592.2.i.s.865.1 2
36.7 odd 6 864.2.a.k.1.1 yes 1
36.11 even 6 864.2.a.c.1.1 yes 1
36.23 even 6 2592.2.i.s.1729.1 2
36.31 odd 6 2592.2.i.d.1729.1 2
72.11 even 6 1728.2.a.w.1.1 1
72.29 odd 6 1728.2.a.v.1.1 1
72.43 odd 6 1728.2.a.g.1.1 1
72.61 even 6 1728.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.b.1.1 1 9.2 odd 6
864.2.a.c.1.1 yes 1 36.11 even 6
864.2.a.j.1.1 yes 1 9.7 even 3
864.2.a.k.1.1 yes 1 36.7 odd 6
1728.2.a.f.1.1 1 72.61 even 6
1728.2.a.g.1.1 1 72.43 odd 6
1728.2.a.v.1.1 1 72.29 odd 6
1728.2.a.w.1.1 1 72.11 even 6
2592.2.i.d.865.1 2 4.3 odd 2
2592.2.i.d.1729.1 2 36.31 odd 6
2592.2.i.f.865.1 2 1.1 even 1 trivial
2592.2.i.f.1729.1 2 9.4 even 3 inner
2592.2.i.s.865.1 2 12.11 even 2
2592.2.i.s.1729.1 2 36.23 even 6
2592.2.i.u.865.1 2 3.2 odd 2
2592.2.i.u.1729.1 2 9.5 odd 6