Properties

Label 684.2.k.e.577.1
Level $684$
Weight $2$
Character 684.577
Analytic conductor $5.462$
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] = [684,2,Mod(505,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, 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("684.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 684.k (of order \(3\), degree \(2\), 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: \(5.46176749826\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 684.577
Dual form 684.2.k.e.505.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.00000 + 3.46410i) q^{5} -3.00000 q^{7} +O(q^{10})\) \(q+(2.00000 + 3.46410i) q^{5} -3.00000 q^{7} -4.00000 q^{11} +(-2.50000 + 4.33013i) q^{13} +(-4.00000 - 1.73205i) q^{19} +(2.00000 - 3.46410i) q^{23} +(-5.50000 + 9.52628i) q^{25} +(4.00000 - 6.92820i) q^{29} +1.00000 q^{31} +(-6.00000 - 10.3923i) q^{35} -5.00000 q^{37} +(4.00000 + 6.92820i) q^{41} +(2.50000 + 4.33013i) q^{43} +(-4.00000 + 6.92820i) q^{47} +2.00000 q^{49} +(2.00000 - 3.46410i) q^{53} +(-8.00000 - 13.8564i) q^{55} +(6.00000 + 10.3923i) q^{59} +(0.500000 - 0.866025i) q^{61} -20.0000 q^{65} +(-1.50000 + 2.59808i) q^{67} +(8.00000 + 13.8564i) q^{71} +(7.50000 + 12.9904i) q^{73} +12.0000 q^{77} +(3.50000 + 6.06218i) q^{79} +(-6.00000 + 10.3923i) q^{89} +(7.50000 - 12.9904i) q^{91} +(-2.00000 - 17.3205i) q^{95} +(1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 6 q^{7} - 8 q^{11} - 5 q^{13} - 8 q^{19} + 4 q^{23} - 11 q^{25} + 8 q^{29} + 2 q^{31} - 12 q^{35} - 10 q^{37} + 8 q^{41} + 5 q^{43} - 8 q^{47} + 4 q^{49} + 4 q^{53} - 16 q^{55} + 12 q^{59} + q^{61} - 40 q^{65} - 3 q^{67} + 16 q^{71} + 15 q^{73} + 24 q^{77} + 7 q^{79} - 12 q^{89} + 15 q^{91} - 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 + 3.46410i 0.894427 + 1.54919i 0.834512 + 0.550990i \(0.185750\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −4.00000 1.73205i −0.917663 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 3.46410i 0.417029 0.722315i −0.578610 0.815604i \(-0.696405\pi\)
0.995639 + 0.0932891i \(0.0297381\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 6.92820i 0.742781 1.28654i −0.208443 0.978035i \(-0.566840\pi\)
0.951224 0.308500i \(-0.0998271\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 10.3923i −1.01419 1.75662i
\(36\) 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 \(0.0481100\pi\)
−0.363905 + 0.931436i \(0.618557\pi\)
\(42\) 0 0
\(43\) 2.50000 + 4.33013i 0.381246 + 0.660338i 0.991241 0.132068i \(-0.0421616\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 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
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.00000 3.46410i 0.274721 0.475831i −0.695344 0.718677i \(-0.744748\pi\)
0.970065 + 0.242846i \(0.0780811\pi\)
\(54\) 0 0
\(55\) −8.00000 13.8564i −1.07872 1.86840i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 + 10.3923i 0.781133 + 1.35296i 0.931282 + 0.364299i \(0.118692\pi\)
−0.150148 + 0.988663i \(0.547975\pi\)
\(60\) 0 0
\(61\) 0.500000 0.866025i 0.0640184 0.110883i −0.832240 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −20.0000 −2.48069
\(66\) 0 0
\(67\) −1.50000 + 2.59808i −0.183254 + 0.317406i −0.942987 0.332830i \(-0.891996\pi\)
0.759733 + 0.650236i \(0.225330\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 + 13.8564i 0.949425 + 1.64445i 0.746639 + 0.665230i \(0.231667\pi\)
0.202787 + 0.979223i \(0.435000\pi\)
\(72\) 0 0
\(73\) 7.50000 + 12.9904i 0.877809 + 1.52041i 0.853740 + 0.520699i \(0.174329\pi\)
0.0240681 + 0.999710i \(0.492338\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 3.50000 + 6.06218i 0.393781 + 0.682048i 0.992945 0.118578i \(-0.0378336\pi\)
−0.599164 + 0.800626i \(0.704500\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 + 10.3923i −0.635999 + 1.10158i 0.350304 + 0.936636i \(0.386078\pi\)
−0.986303 + 0.164946i \(0.947255\pi\)
\(90\) 0 0
\(91\) 7.50000 12.9904i 0.786214 1.36176i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 17.3205i −0.205196 1.77705i
\(96\) 0 0
\(97\) 1.00000 + 1.73205i 0.101535 + 0.175863i 0.912317 0.409484i \(-0.134291\pi\)
−0.810782 + 0.585348i \(0.800958\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.00000 13.8564i 0.796030 1.37876i −0.126153 0.992011i \(-0.540263\pi\)
0.922183 0.386753i \(-0.126403\pi\)
\(102\) 0 0
\(103\) −15.0000 −1.47799 −0.738997 0.673709i \(-0.764700\pi\)
−0.738997 + 0.673709i \(0.764700\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −9.00000 15.5885i −0.862044 1.49310i −0.869953 0.493135i \(-0.835851\pi\)
0.00790932 0.999969i \(-0.497482\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.00000 −0.376288 −0.188144 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) 16.0000 1.49201
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) 6.00000 10.3923i 0.532414 0.922168i −0.466870 0.884326i \(-0.654618\pi\)
0.999284 0.0378419i \(-0.0120483\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 12.0000 + 5.19615i 1.04053 + 0.450564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000 6.92820i 0.341743 0.591916i −0.643013 0.765855i \(-0.722316\pi\)
0.984757 + 0.173939i \(0.0556494\pi\)
\(138\) 0 0
\(139\) 7.50000 12.9904i 0.636142 1.10183i −0.350130 0.936701i \(-0.613863\pi\)
0.986272 0.165129i \(-0.0528040\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.0000 17.3205i 0.836242 1.44841i
\(144\) 0 0
\(145\) 32.0000 2.65746
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 10.3923i −0.491539 0.851371i 0.508413 0.861113i \(-0.330232\pi\)
−0.999953 + 0.00974235i \(0.996899\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 + 3.46410i 0.160644 + 0.278243i
\(156\) 0 0
\(157\) 2.50000 + 4.33013i 0.199522 + 0.345582i 0.948373 0.317156i \(-0.102728\pi\)
−0.748852 + 0.662738i \(0.769394\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 + 10.3923i −0.472866 + 0.819028i
\(162\) 0 0
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 + 10.3923i −0.464294 + 0.804181i −0.999169 0.0407502i \(-0.987025\pi\)
0.534875 + 0.844931i \(0.320359\pi\)
\(168\) 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 0.810745 + 0.585399i \(0.199062\pi\)
0.101598 + 0.994826i \(0.467605\pi\)
\(174\) 0 0
\(175\) 16.5000 28.5788i 1.24728 2.16036i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −5.00000 + 8.66025i −0.371647 + 0.643712i −0.989819 0.142331i \(-0.954540\pi\)
0.618172 + 0.786043i \(0.287874\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.0000 17.3205i −0.735215 1.27343i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 0 0
\(193\) 4.50000 + 7.79423i 0.323917 + 0.561041i 0.981293 0.192522i \(-0.0616668\pi\)
−0.657376 + 0.753563i \(0.728333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.0000 1.42494 0.712470 0.701702i \(-0.247576\pi\)
0.712470 + 0.701702i \(0.247576\pi\)
\(198\) 0 0
\(199\) 3.50000 6.06218i 0.248108 0.429736i −0.714893 0.699234i \(-0.753524\pi\)
0.963001 + 0.269498i \(0.0868577\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.0000 + 20.7846i −0.842235 + 1.45879i
\(204\) 0 0
\(205\) −16.0000 + 27.7128i −1.11749 + 1.93555i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000 + 6.92820i 1.10674 + 0.479234i
\(210\) 0 0
\(211\) −10.5000 18.1865i −0.722850 1.25201i −0.959853 0.280504i \(-0.909498\pi\)
0.237003 0.971509i \(-0.423835\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −10.0000 + 17.3205i −0.681994 + 1.18125i
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 9.50000 + 16.4545i 0.636167 + 1.10187i 0.986267 + 0.165161i \(0.0528144\pi\)
−0.350100 + 0.936713i \(0.613852\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) −32.0000 −2.08745
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −0.500000 + 0.866025i −0.0322078 + 0.0557856i −0.881680 0.471848i \(-0.843587\pi\)
0.849472 + 0.527633i \(0.176921\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.00000 + 6.92820i 0.255551 + 0.442627i
\(246\) 0 0
\(247\) 17.5000 12.9904i 1.11350 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.00000 13.8564i 0.504956 0.874609i −0.495028 0.868877i \(-0.664842\pi\)
0.999984 0.00573163i \(-0.00182444\pi\)
\(252\) 0 0
\(253\) −8.00000 + 13.8564i −0.502956 + 0.871145i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.0000 + 17.3205i −0.623783 + 1.08042i 0.364992 + 0.931011i \(0.381072\pi\)
−0.988775 + 0.149413i \(0.952262\pi\)
\(258\) 0 0
\(259\) 15.0000 0.932055
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 10.3923i −0.365826 0.633630i 0.623082 0.782157i \(-0.285880\pi\)
−0.988908 + 0.148527i \(0.952547\pi\)
\(270\) 0 0
\(271\) −2.00000 3.46410i −0.121491 0.210429i 0.798865 0.601511i \(-0.205434\pi\)
−0.920356 + 0.391082i \(0.872101\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.0000 38.1051i 1.32665 2.29783i
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 17.3205i 0.596550 1.03325i −0.396776 0.917915i \(-0.629871\pi\)
0.993326 0.115339i \(-0.0367956\pi\)
\(282\) 0 0
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 20.7846i −0.708338 1.22688i
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −24.0000 + 41.5692i −1.39733 + 2.42025i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.0000 + 17.3205i 0.578315 + 1.00167i
\(300\) 0 0
\(301\) −7.50000 12.9904i −0.432293 0.748753i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) −13.0000 + 22.5167i −0.734803 + 1.27272i 0.220006 + 0.975499i \(0.429392\pi\)
−0.954810 + 0.297218i \(0.903941\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 10.3923i 0.336994 0.583690i −0.646872 0.762598i \(-0.723923\pi\)
0.983866 + 0.178908i \(0.0572566\pi\)
\(318\) 0 0
\(319\) −16.0000 + 27.7128i −0.895828 + 1.55162i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −27.5000 47.6314i −1.52543 2.64211i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 12.0000 20.7846i 0.661581 1.14589i
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −6.50000 11.2583i −0.354078 0.613280i 0.632882 0.774248i \(-0.281872\pi\)
−0.986960 + 0.160968i \(0.948538\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.0000 24.2487i −0.751559 1.30174i −0.947067 0.321037i \(-0.895969\pi\)
0.195507 0.980702i \(-0.437365\pi\)
\(348\) 0 0
\(349\) −5.00000 −0.267644 −0.133822 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.0000 1.91609 0.958043 0.286623i \(-0.0925328\pi\)
0.958043 + 0.286623i \(0.0925328\pi\)
\(354\) 0 0
\(355\) −32.0000 + 55.4256i −1.69838 + 2.94169i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −30.0000 + 51.9615i −1.57027 + 2.71979i
\(366\) 0 0
\(367\) 2.50000 4.33013i 0.130499 0.226031i −0.793370 0.608740i \(-0.791675\pi\)
0.923869 + 0.382709i \(0.125009\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 + 10.3923i −0.311504 + 0.539542i
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.0000 + 34.6410i 1.03005 + 1.78410i
\(378\) 0 0
\(379\) 3.00000 0.154100 0.0770498 0.997027i \(-0.475450\pi\)
0.0770498 + 0.997027i \(0.475450\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.0000 + 17.3205i 0.510976 + 0.885037i 0.999919 + 0.0127209i \(0.00404928\pi\)
−0.488943 + 0.872316i \(0.662617\pi\)
\(384\) 0 0
\(385\) 24.0000 + 41.5692i 1.22315 + 2.11856i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.00000 + 10.3923i −0.304212 + 0.526911i −0.977086 0.212847i \(-0.931726\pi\)
0.672874 + 0.739758i \(0.265060\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −14.0000 + 24.2487i −0.704416 + 1.22009i
\(396\) 0 0
\(397\) −17.5000 30.3109i −0.878300 1.52126i −0.853206 0.521575i \(-0.825345\pi\)
−0.0250943 0.999685i \(-0.507989\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 3.46410i −0.0998752 0.172989i 0.811758 0.583994i \(-0.198511\pi\)
−0.911633 + 0.411005i \(0.865178\pi\)
\(402\) 0 0
\(403\) −2.50000 + 4.33013i −0.124534 + 0.215699i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −11.0000 + 19.0526i −0.543915 + 0.942088i 0.454759 + 0.890614i \(0.349725\pi\)
−0.998674 + 0.0514740i \(0.983608\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −18.0000 31.1769i −0.885722 1.53412i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −13.0000 22.5167i −0.633581 1.09739i −0.986814 0.161859i \(-0.948251\pi\)
0.353233 0.935536i \(-0.385082\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.50000 + 2.59808i −0.0725901 + 0.125730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 20.7846i 0.578020 1.00116i −0.417687 0.908591i \(-0.637159\pi\)
0.995706 0.0925683i \(-0.0295076\pi\)
\(432\) 0 0
\(433\) −5.50000 + 9.52628i −0.264313 + 0.457804i −0.967383 0.253317i \(-0.918479\pi\)
0.703070 + 0.711120i \(0.251812\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −14.0000 + 10.3923i −0.669711 + 0.497131i
\(438\) 0 0
\(439\) 18.5000 + 32.0429i 0.882957 + 1.52933i 0.848038 + 0.529936i \(0.177784\pi\)
0.0349192 + 0.999390i \(0.488883\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.0000 + 34.6410i −0.950229 + 1.64584i −0.205301 + 0.978699i \(0.565817\pi\)
−0.744927 + 0.667146i \(0.767516\pi\)
\(444\) 0 0
\(445\) −48.0000 −2.27542
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 0 0
\(451\) −16.0000 27.7128i −0.753411 1.30495i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 60.0000 2.81284
\(456\) 0 0
\(457\) −7.00000 −0.327446 −0.163723 0.986506i \(-0.552350\pi\)
−0.163723 + 0.986506i \(0.552350\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 + 3.46410i 0.0931493 + 0.161339i 0.908835 0.417156i \(-0.136973\pi\)
−0.815685 + 0.578496i \(0.803640\pi\)
\(462\) 0 0
\(463\) 31.0000 1.44069 0.720346 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 40.0000 1.85098 0.925490 0.378773i \(-0.123654\pi\)
0.925490 + 0.378773i \(0.123654\pi\)
\(468\) 0 0
\(469\) 4.50000 7.79423i 0.207791 0.359904i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.0000 17.3205i −0.459800 0.796398i
\(474\) 0 0
\(475\) 38.5000 28.5788i 1.76650 1.31129i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.0000 27.7128i 0.731059 1.26623i −0.225372 0.974273i \(-0.572360\pi\)
0.956431 0.291958i \(-0.0943068\pi\)
\(480\) 0 0
\(481\) 12.5000 21.6506i 0.569951 0.987184i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.00000 + 6.92820i −0.181631 + 0.314594i
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.00000 + 13.8564i 0.361035 + 0.625331i 0.988131 0.153611i \(-0.0490902\pi\)
−0.627096 + 0.778942i \(0.715757\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.0000 41.5692i −1.07655 1.86463i
\(498\) 0 0
\(499\) 17.5000 + 30.3109i 0.783408 + 1.35690i 0.929946 + 0.367697i \(0.119854\pi\)
−0.146538 + 0.989205i \(0.546813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 + 20.7846i −0.535054 + 0.926740i 0.464107 + 0.885779i \(0.346375\pi\)
−0.999161 + 0.0409609i \(0.986958\pi\)
\(504\) 0 0
\(505\) 64.0000 2.84796
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) −22.5000 38.9711i −0.995341 1.72398i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −30.0000 51.9615i −1.32196 2.28970i
\(516\) 0 0
\(517\) 16.0000 27.7128i 0.703679 1.21881i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 0 0
\(523\) −14.5000 + 25.1147i −0.634041 + 1.09819i 0.352677 + 0.935745i \(0.385272\pi\)
−0.986718 + 0.162446i \(0.948062\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −40.0000 −1.73259
\(534\) 0 0
\(535\) 16.0000 + 27.7128i 0.691740 + 1.19813i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −8.00000 −0.344584
\(540\) 0 0
\(541\) 9.50000 16.4545i 0.408437 0.707433i −0.586278 0.810110i \(-0.699407\pi\)
0.994715 + 0.102677i \(0.0327407\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 36.0000 62.3538i 1.54207 2.67094i
\(546\) 0 0
\(547\) −3.50000 + 6.06218i −0.149649 + 0.259200i −0.931098 0.364770i \(-0.881148\pi\)
0.781449 + 0.623970i \(0.214481\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −28.0000 + 20.7846i −1.19284 + 0.885454i
\(552\) 0 0
\(553\) −10.5000 18.1865i −0.446505 0.773370i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.00000 6.92820i 0.169485 0.293557i −0.768754 0.639545i \(-0.779123\pi\)
0.938239 + 0.345988i \(0.112456\pi\)
\(558\) 0 0
\(559\) −25.0000 −1.05739
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −8.00000 13.8564i −0.336563 0.582943i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −8.00000 −0.335377 −0.167689 0.985840i \(-0.553630\pi\)
−0.167689 + 0.985840i \(0.553630\pi\)
\(570\) 0 0
\(571\) 41.0000 1.71580 0.857898 0.513820i \(-0.171770\pi\)
0.857898 + 0.513820i \(0.171770\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.0000 + 38.1051i 0.917463 + 1.58909i
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −8.00000 + 13.8564i −0.331326 + 0.573874i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 + 10.3923i 0.247647 + 0.428936i 0.962872 0.269957i \(-0.0870095\pi\)
−0.715226 + 0.698893i \(0.753676\pi\)
\(588\) 0 0
\(589\) −4.00000 1.73205i −0.164817 0.0713679i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.00000 + 3.46410i −0.0821302 + 0.142254i −0.904165 0.427184i \(-0.859506\pi\)
0.822035 + 0.569438i \(0.192839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18.0000 + 31.1769i −0.735460 + 1.27385i 0.219061 + 0.975711i \(0.429701\pi\)
−0.954521 + 0.298143i \(0.903633\pi\)
\(600\) 0 0
\(601\) 31.0000 1.26452 0.632258 0.774758i \(-0.282128\pi\)
0.632258 + 0.774758i \(0.282128\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.0000 + 17.3205i 0.406558 + 0.704179i
\(606\) 0 0
\(607\) 5.00000 0.202944 0.101472 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.0000 34.6410i −0.809113 1.40143i
\(612\) 0 0
\(613\) 13.0000 + 22.5167i 0.525065 + 0.909439i 0.999574 + 0.0291886i \(0.00929235\pi\)
−0.474509 + 0.880251i \(0.657374\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 10.3923i 0.241551 0.418378i −0.719605 0.694383i \(-0.755677\pi\)
0.961156 + 0.276005i \(0.0890106\pi\)
\(618\) 0 0
\(619\) 25.0000 1.00483 0.502417 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 18.0000 31.1769i 0.721155 1.24908i
\(624\) 0 0
\(625\) −20.5000 35.5070i −0.820000 1.42028i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 9.50000 16.4545i 0.378189 0.655043i −0.612610 0.790386i \(-0.709880\pi\)
0.990799 + 0.135343i \(0.0432136\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 48.0000 1.90482
\(636\) 0 0
\(637\) −5.00000 + 8.66025i −0.198107 + 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 + 20.7846i 0.473972 + 0.820943i 0.999556 0.0297987i \(-0.00948663\pi\)
−0.525584 + 0.850741i \(0.676153\pi\)
\(642\) 0 0
\(643\) −0.500000 0.866025i −0.0197181 0.0341527i 0.855998 0.516979i \(-0.172944\pi\)
−0.875716 + 0.482826i \(0.839610\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) 0 0
\(649\) −24.0000 41.5692i −0.942082 1.63173i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.0000 + 17.3205i −0.389545 + 0.674711i −0.992388 0.123148i \(-0.960701\pi\)
0.602844 + 0.797859i \(0.294034\pi\)
\(660\) 0 0
\(661\) 13.0000 22.5167i 0.505641 0.875797i −0.494337 0.869270i \(-0.664589\pi\)
0.999979 0.00652642i \(-0.00207744\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 + 51.9615i 0.232670 + 2.01498i
\(666\) 0 0
\(667\) −16.0000 27.7128i −0.619522 1.07304i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 + 3.46410i −0.0772091 + 0.133730i
\(672\) 0 0
\(673\) 35.0000 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −32.0000 −1.22986 −0.614930 0.788582i \(-0.710816\pi\)
−0.614930 + 0.788582i \(0.710816\pi\)
\(678\) 0 0
\(679\) −3.00000 5.19615i −0.115129 0.199410i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 32.0000 1.22266
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0000 + 17.3205i 0.380970 + 0.659859i
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 60.0000 2.27593
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 17.3205i −0.377695 0.654187i 0.613032 0.790058i \(-0.289950\pi\)
−0.990726 + 0.135872i \(0.956616\pi\)
\(702\) 0 0
\(703\) 20.0000 + 8.66025i 0.754314 + 0.326628i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −24.0000 + 41.5692i −0.902613 + 1.56337i
\(708\) 0 0
\(709\) −16.5000 + 28.5788i −0.619671 + 1.07330i 0.369875 + 0.929081i \(0.379400\pi\)
−0.989546 + 0.144219i \(0.953933\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.00000 3.46410i 0.0749006 0.129732i
\(714\) 0 0
\(715\) 80.0000 2.99183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −14.0000 24.2487i −0.522112 0.904324i −0.999669 0.0257237i \(-0.991811\pi\)
0.477557 0.878601i \(-0.341522\pi\)
\(720\) 0 0
\(721\) 45.0000 1.67589
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 44.0000 + 76.2102i 1.63412 + 2.83038i
\(726\) 0 0
\(727\) −6.50000 11.2583i −0.241072 0.417548i 0.719948 0.694028i \(-0.244166\pi\)
−0.961020 + 0.276479i \(0.910832\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −30.0000 −1.10808 −0.554038 0.832492i \(-0.686914\pi\)
−0.554038 + 0.832492i \(0.686914\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000 10.3923i 0.221013 0.382805i
\(738\) 0 0
\(739\) 6.50000 + 11.2583i 0.239106 + 0.414144i 0.960458 0.278425i \(-0.0898122\pi\)
−0.721352 + 0.692569i \(0.756479\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10.0000 17.3205i −0.366864 0.635428i 0.622209 0.782851i \(-0.286235\pi\)
−0.989073 + 0.147423i \(0.952902\pi\)
\(744\) 0 0
\(745\) 24.0000 41.5692i 0.879292 1.52298i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −24.0000 −0.876941
\(750\) 0 0
\(751\) 15.5000 26.8468i 0.565603 0.979653i −0.431390 0.902165i \(-0.641977\pi\)
0.996993 0.0774878i \(-0.0246899\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.00000 13.8564i −0.291150 0.504286i
\(756\) 0 0
\(757\) 17.5000 + 30.3109i 0.636048 + 1.10167i 0.986292 + 0.165009i \(0.0527654\pi\)
−0.350244 + 0.936659i \(0.613901\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 27.0000 + 46.7654i 0.977466 + 1.69302i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −60.0000 −2.16647
\(768\) 0 0
\(769\) 8.50000 14.7224i 0.306518 0.530904i −0.671080 0.741385i \(-0.734169\pi\)
0.977598 + 0.210480i \(0.0675028\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.0000 20.7846i 0.431610 0.747570i −0.565402 0.824815i \(-0.691279\pi\)
0.997012 + 0.0772449i \(0.0246123\pi\)
\(774\) 0 0
\(775\) −5.50000 + 9.52628i −0.197566 + 0.342194i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.00000 34.6410i −0.143315 1.24114i
\(780\) 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
\(787\) −23.0000 −0.819861 −0.409931 0.912117i \(-0.634447\pi\)
−0.409931 + 0.912117i \(0.634447\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 2.50000 + 4.33013i 0.0887776 + 0.153767i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 48.0000 1.70025 0.850124 0.526583i \(-0.176527\pi\)
0.850124 + 0.526583i \(0.176527\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30.0000 51.9615i −1.05868 1.83368i
\(804\) 0 0
\(805\) −48.0000 −1.69178
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −12.0000 + 20.7846i −0.421377 + 0.729846i −0.996074 0.0885196i \(-0.971786\pi\)
0.574697 + 0.818366i \(0.305120\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 3.46410i −0.0700569 0.121342i
\(816\) 0 0
\(817\) −2.50000 21.6506i −0.0874639 0.757460i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.0000 20.7846i 0.418803 0.725388i −0.577016 0.816733i \(-0.695783\pi\)
0.995819 + 0.0913446i \(0.0291165\pi\)
\(822\) 0 0
\(823\) −22.0000 + 38.1051i −0.766872 + 1.32826i 0.172379 + 0.985031i \(0.444854\pi\)
−0.939251 + 0.343230i \(0.888479\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.00000 + 6.92820i −0.139094 + 0.240917i −0.927154 0.374681i \(-0.877752\pi\)
0.788060 + 0.615598i \(0.211086\pi\)
\(828\) 0 0
\(829\) 21.0000 0.729360 0.364680 0.931133i \(-0.381178\pi\)
0.364680 + 0.931133i \(0.381178\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −10.0000 17.3205i −0.345238 0.597970i 0.640159 0.768243i \(-0.278869\pi\)
−0.985397 + 0.170272i \(0.945535\pi\)
\(840\) 0 0
\(841\) −17.5000 30.3109i −0.603448 1.04520i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 24.0000 41.5692i 0.825625 1.43002i
\(846\) 0 0
\(847\) −15.0000 −0.515406
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.0000 + 17.3205i −0.342796 + 0.593739i
\(852\) 0 0
\(853\) −19.5000 33.7750i −0.667667 1.15643i −0.978555 0.205987i \(-0.933960\pi\)
0.310887 0.950447i \(-0.399374\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) 26.5000 45.8993i 0.904168 1.56607i 0.0821386 0.996621i \(-0.473825\pi\)
0.822030 0.569445i \(-0.192842\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −48.0000 + 83.1384i −1.63205 + 2.82679i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −14.0000 24.2487i −0.474917 0.822581i
\(870\) 0 0
\(871\) −7.50000 12.9904i −0.254128 0.440162i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 72.0000 2.43404
\(876\) 0 0
\(877\) 26.5000 + 45.8993i 0.894841 + 1.54991i 0.834001 + 0.551763i \(0.186045\pi\)
0.0608407 + 0.998147i \(0.480622\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) −27.5000 + 47.6314i −0.925449 + 1.60292i −0.134611 + 0.990899i \(0.542978\pi\)
−0.790838 + 0.612026i \(0.790355\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.0000 17.3205i 0.335767 0.581566i −0.647865 0.761755i \(-0.724338\pi\)
0.983632 + 0.180190i \(0.0576711\pi\)
\(888\) 0 0
\(889\) −18.0000 + 31.1769i −0.603701 + 1.04564i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 28.0000 20.7846i 0.936984 0.695530i
\(894\) 0 0
\(895\) 24.0000 + 41.5692i 0.802232 + 1.38951i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.00000 6.92820i 0.133407 0.231069i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −40.0000 −1.32964
\(906\) 0 0
\(907\) 20.0000 + 34.6410i 0.664089 + 1.15024i 0.979531 + 0.201291i \(0.0645138\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −25.0000 −0.824674 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −80.0000 −2.63323
\(924\) 0 0
\(925\) 27.5000 47.6314i 0.904194 1.56611i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 14.0000 + 24.2487i 0.459325 + 0.795574i 0.998925 0.0463469i \(-0.0147580\pi\)
−0.539600 + 0.841921i \(0.681425\pi\)
\(930\) 0 0
\(931\) −8.00000 3.46410i −0.262189 0.113531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.50000 11.2583i 0.212346 0.367794i −0.740102 0.672494i \(-0.765223\pi\)
0.952448 + 0.304700i \(0.0985564\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8.00000 + 13.8564i −0.260793 + 0.451706i −0.966453 0.256844i \(-0.917317\pi\)
0.705660 + 0.708550i \(0.250651\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 + 41.5692i 0.779895 + 1.35082i 0.932002 + 0.362454i \(0.118061\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(948\) 0 0
\(949\) −75.0000 −2.43460
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −10.0000 17.3205i −0.323932 0.561066i 0.657364 0.753573i \(-0.271671\pi\)
−0.981296 + 0.192507i \(0.938338\pi\)
\(954\) 0 0
\(955\) 8.00000 + 13.8564i 0.258874 + 0.448383i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 + 20.7846i −0.387500 + 0.671170i
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −18.0000 + 31.1769i −0.579441 + 1.00362i
\(966\) 0 0
\(967\) −13.5000 23.3827i −0.434131 0.751936i 0.563094 0.826393i \(-0.309611\pi\)
−0.997224 + 0.0744567i \(0.976278\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0000 + 34.6410i 0.641831 + 1.11168i 0.985024 + 0.172418i \(0.0551581\pi\)
−0.343193 + 0.939265i \(0.611509\pi\)
\(972\) 0 0
\(973\) −22.5000 + 38.9711i −0.721317 + 1.24936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.00000 0.255943 0.127971 0.991778i \(-0.459153\pi\)
0.127971 + 0.991778i \(0.459153\pi\)
\(978\) 0 0
\(979\) 24.0000 41.5692i 0.767043 1.32856i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 18.0000 + 31.1769i 0.574111 + 0.994389i 0.996138 + 0.0878058i \(0.0279855\pi\)
−0.422027 + 0.906583i \(0.638681\pi\)
\(984\) 0 0
\(985\) 40.0000 + 69.2820i 1.27451 + 2.20751i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.0000 0.635963
\(990\) 0 0
\(991\) 2.50000 + 4.33013i 0.0794151 + 0.137551i 0.902998 0.429645i \(-0.141361\pi\)
−0.823583 + 0.567196i \(0.808028\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 28.0000 0.887660
\(996\) 0 0
\(997\) 26.5000 45.8993i 0.839263 1.45365i −0.0512480 0.998686i \(-0.516320\pi\)
0.890511 0.454961i \(-0.150347\pi\)
\(998\) 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 684.2.k.e.577.1 yes 2
3.2 odd 2 684.2.k.a.577.1 yes 2
4.3 odd 2 2736.2.s.r.577.1 2
12.11 even 2 2736.2.s.b.577.1 2
19.11 even 3 inner 684.2.k.e.505.1 yes 2
57.11 odd 6 684.2.k.a.505.1 2
76.11 odd 6 2736.2.s.r.1873.1 2
228.11 even 6 2736.2.s.b.1873.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.2.k.a.505.1 2 57.11 odd 6
684.2.k.a.577.1 yes 2 3.2 odd 2
684.2.k.e.505.1 yes 2 19.11 even 3 inner
684.2.k.e.577.1 yes 2 1.1 even 1 trivial
2736.2.s.b.577.1 2 12.11 even 2
2736.2.s.b.1873.1 2 228.11 even 6
2736.2.s.r.577.1 2 4.3 odd 2
2736.2.s.r.1873.1 2 76.11 odd 6