Properties

Label 3024.2.r.m.1009.2
Level $3024$
Weight $2$
Character 3024.1009
Analytic conductor $24.147$
Analytic rank $0$
Dimension $8$
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] = [3024,2,Mod(1009,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.r (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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2091141441.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{6} + 3x^{5} - 15x^{4} + 9x^{3} + 9x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1009.2
Root \(0.335492 - 1.69925i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1009
Dual form 3024.2.r.m.2017.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.164508 - 0.284936i) q^{5} +(-0.500000 - 0.866025i) q^{7} +O(q^{10})\) \(q+(0.164508 - 0.284936i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(0.664508 + 1.15096i) q^{11} +(-1.53937 + 2.66626i) q^{13} -7.35741 q^{17} +2.93671 q^{19} +(3.34321 - 5.79062i) q^{23} +(2.44587 + 4.23638i) q^{25} +(-3.88258 - 6.72483i) q^{29} +(-1.63555 + 2.83286i) q^{31} -0.329016 q^{35} +0.329016 q^{37} +(0.135552 - 0.234783i) q^{41} +(-5.48255 - 9.49606i) q^{43} +(-0.571014 - 0.989025i) q^{47} +(-0.500000 + 0.866025i) q^{49} -6.42828 q^{53} +0.437267 q^{55} +(-0.372170 + 0.644618i) q^{59} +(-4.42195 - 7.65904i) q^{61} +(0.506476 + 0.877243i) q^{65} +(4.28640 - 7.42426i) q^{67} +1.60769 q^{71} -13.4941 q^{73} +(0.664508 - 1.15096i) q^{77} +(-0.628926 - 1.08933i) q^{79} +(0.0316459 + 0.0548124i) q^{83} +(-1.21035 + 2.09639i) q^{85} -11.3071 q^{89} +3.07874 q^{91} +(0.483112 - 0.836774i) q^{95} +(5.51420 + 9.55087i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{5} - 4 q^{7} + 7 q^{11} + 3 q^{13} - 6 q^{17} + 8 q^{19} + 2 q^{23} - 5 q^{25} + 9 q^{29} - 3 q^{31} - 6 q^{35} + 6 q^{37} - 9 q^{41} - 8 q^{43} + 3 q^{47} - 4 q^{49} - 12 q^{53} + 56 q^{55} + 10 q^{59} + 20 q^{61} - q^{65} - 11 q^{67} - 6 q^{71} - 48 q^{73} + 7 q^{77} - 21 q^{79} + 8 q^{83} + 9 q^{85} - 12 q^{89} - 6 q^{91} + 36 q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(5\) 0.164508 0.284936i 0.0735702 0.127427i −0.826893 0.562359i \(-0.809894\pi\)
0.900464 + 0.434931i \(0.143227\pi\)
\(6\) 0 0
\(7\) −0.500000 0.866025i −0.188982 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.664508 + 1.15096i 0.200357 + 0.347028i 0.948643 0.316348i \(-0.102457\pi\)
−0.748287 + 0.663375i \(0.769123\pi\)
\(12\) 0 0
\(13\) −1.53937 + 2.66626i −0.426944 + 0.739489i −0.996600 0.0823948i \(-0.973743\pi\)
0.569656 + 0.821883i \(0.307076\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.35741 −1.78443 −0.892217 0.451606i \(-0.850851\pi\)
−0.892217 + 0.451606i \(0.850851\pi\)
\(18\) 0 0
\(19\) 2.93671 0.673727 0.336864 0.941553i \(-0.390634\pi\)
0.336864 + 0.941553i \(0.390634\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.34321 5.79062i 0.697108 1.20743i −0.272356 0.962196i \(-0.587803\pi\)
0.969465 0.245231i \(-0.0788637\pi\)
\(24\) 0 0
\(25\) 2.44587 + 4.23638i 0.489175 + 0.847276i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.88258 6.72483i −0.720977 1.24877i −0.960608 0.277906i \(-0.910360\pi\)
0.239631 0.970864i \(-0.422974\pi\)
\(30\) 0 0
\(31\) −1.63555 + 2.83286i −0.293754 + 0.508796i −0.974694 0.223542i \(-0.928238\pi\)
0.680940 + 0.732339i \(0.261571\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.329016 −0.0556138
\(36\) 0 0
\(37\) 0.329016 0.0540899 0.0270449 0.999634i \(-0.491390\pi\)
0.0270449 + 0.999634i \(0.491390\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.135552 0.234783i 0.0211696 0.0366669i −0.855247 0.518221i \(-0.826594\pi\)
0.876416 + 0.481555i \(0.159928\pi\)
\(42\) 0 0
\(43\) −5.48255 9.49606i −0.836081 1.44814i −0.893147 0.449765i \(-0.851508\pi\)
0.0570654 0.998370i \(-0.481826\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.571014 0.989025i −0.0832910 0.144264i 0.821371 0.570395i \(-0.193210\pi\)
−0.904662 + 0.426131i \(0.859876\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.42828 −0.882992 −0.441496 0.897263i \(-0.645552\pi\)
−0.441496 + 0.897263i \(0.645552\pi\)
\(54\) 0 0
\(55\) 0.437267 0.0589611
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.372170 + 0.644618i −0.0484525 + 0.0839221i −0.889234 0.457452i \(-0.848762\pi\)
0.840782 + 0.541374i \(0.182096\pi\)
\(60\) 0 0
\(61\) −4.42195 7.65904i −0.566173 0.980640i −0.996940 0.0781767i \(-0.975090\pi\)
0.430767 0.902463i \(-0.358243\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.506476 + 0.877243i 0.0628207 + 0.108809i
\(66\) 0 0
\(67\) 4.28640 7.42426i 0.523667 0.907018i −0.475954 0.879470i \(-0.657897\pi\)
0.999620 0.0275474i \(-0.00876971\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.60769 0.190798 0.0953990 0.995439i \(-0.469587\pi\)
0.0953990 + 0.995439i \(0.469587\pi\)
\(72\) 0 0
\(73\) −13.4941 −1.57936 −0.789680 0.613519i \(-0.789754\pi\)
−0.789680 + 0.613519i \(0.789754\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.664508 1.15096i 0.0757277 0.131164i
\(78\) 0 0
\(79\) −0.628926 1.08933i −0.0707597 0.122559i 0.828475 0.560026i \(-0.189209\pi\)
−0.899234 + 0.437467i \(0.855876\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.0316459 + 0.0548124i 0.00347359 + 0.00601644i 0.867757 0.496989i \(-0.165561\pi\)
−0.864283 + 0.503005i \(0.832228\pi\)
\(84\) 0 0
\(85\) −1.21035 + 2.09639i −0.131281 + 0.227386i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.3071 −1.19855 −0.599274 0.800544i \(-0.704544\pi\)
−0.599274 + 0.800544i \(0.704544\pi\)
\(90\) 0 0
\(91\) 3.07874 0.322739
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.483112 0.836774i 0.0495662 0.0858512i
\(96\) 0 0
\(97\) 5.51420 + 9.55087i 0.559882 + 0.969744i 0.997506 + 0.0705859i \(0.0224869\pi\)
−0.437624 + 0.899158i \(0.644180\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.12079 + 12.3336i 0.708546 + 1.22724i 0.965397 + 0.260786i \(0.0839816\pi\)
−0.256851 + 0.966451i \(0.582685\pi\)
\(102\) 0 0
\(103\) 5.29288 9.16753i 0.521522 0.903303i −0.478164 0.878270i \(-0.658698\pi\)
0.999687 0.0250330i \(-0.00796907\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −15.5574 −1.50399 −0.751993 0.659171i \(-0.770907\pi\)
−0.751993 + 0.659171i \(0.770907\pi\)
\(108\) 0 0
\(109\) −14.3574 −1.37519 −0.687595 0.726094i \(-0.741334\pi\)
−0.687595 + 0.726094i \(0.741334\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.94966 + 3.37691i −0.183409 + 0.317673i −0.943039 0.332682i \(-0.892046\pi\)
0.759630 + 0.650355i \(0.225380\pi\)
\(114\) 0 0
\(115\) −1.09997 1.90520i −0.102573 0.177661i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.67871 + 6.37171i 0.337226 + 0.584093i
\(120\) 0 0
\(121\) 4.61686 7.99663i 0.419714 0.726967i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.25454 0.291095
\(126\) 0 0
\(127\) 7.00787 0.621848 0.310924 0.950435i \(-0.399362\pi\)
0.310924 + 0.950435i \(0.399362\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.33280 + 5.77258i −0.291188 + 0.504353i −0.974091 0.226156i \(-0.927384\pi\)
0.682903 + 0.730509i \(0.260717\pi\)
\(132\) 0 0
\(133\) −1.46835 2.54326i −0.127322 0.220529i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.41926 14.5826i −0.719306 1.24587i −0.961275 0.275591i \(-0.911127\pi\)
0.241969 0.970284i \(-0.422207\pi\)
\(138\) 0 0
\(139\) −7.81032 + 13.5279i −0.662463 + 1.14742i 0.317504 + 0.948257i \(0.397155\pi\)
−0.979967 + 0.199162i \(0.936178\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.09169 −0.342164
\(144\) 0 0
\(145\) −2.55486 −0.212170
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.53233 + 13.0464i −0.617073 + 1.06880i 0.372944 + 0.927854i \(0.378348\pi\)
−0.990017 + 0.140948i \(0.954985\pi\)
\(150\) 0 0
\(151\) −3.86714 6.69808i −0.314703 0.545082i 0.664671 0.747136i \(-0.268572\pi\)
−0.979374 + 0.202054i \(0.935238\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.538122 + 0.932055i 0.0432230 + 0.0748645i
\(156\) 0 0
\(157\) 4.35812 7.54849i 0.347816 0.602435i −0.638045 0.769999i \(-0.720257\pi\)
0.985861 + 0.167564i \(0.0535901\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.68643 −0.526964
\(162\) 0 0
\(163\) −13.4513 −1.05359 −0.526793 0.849993i \(-0.676606\pi\)
−0.526793 + 0.849993i \(0.676606\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.81804 10.0771i 0.450214 0.779793i −0.548185 0.836357i \(-0.684681\pi\)
0.998399 + 0.0565638i \(0.0180144\pi\)
\(168\) 0 0
\(169\) 1.76069 + 3.04961i 0.135438 + 0.234585i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.885831 + 1.53430i 0.0673485 + 0.116651i 0.897733 0.440539i \(-0.145213\pi\)
−0.830385 + 0.557190i \(0.811879\pi\)
\(174\) 0 0
\(175\) 2.44587 4.23638i 0.184891 0.320240i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.73139 −0.503128 −0.251564 0.967841i \(-0.580945\pi\)
−0.251564 + 0.967841i \(0.580945\pi\)
\(180\) 0 0
\(181\) −13.2711 −0.986433 −0.493217 0.869906i \(-0.664179\pi\)
−0.493217 + 0.869906i \(0.664179\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.0541257 0.0937484i 0.00397940 0.00689252i
\(186\) 0 0
\(187\) −4.88906 8.46810i −0.357523 0.619249i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.4083 18.0277i −0.753117 1.30444i −0.946305 0.323276i \(-0.895216\pi\)
0.193187 0.981162i \(-0.438117\pi\)
\(192\) 0 0
\(193\) 10.2585 17.7683i 0.738426 1.27899i −0.214778 0.976663i \(-0.568903\pi\)
0.953204 0.302328i \(-0.0977638\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.20781 −0.513535 −0.256768 0.966473i \(-0.582658\pi\)
−0.256768 + 0.966473i \(0.582658\pi\)
\(198\) 0 0
\(199\) 14.3276 1.01566 0.507828 0.861458i \(-0.330448\pi\)
0.507828 + 0.861458i \(0.330448\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.88258 + 6.72483i −0.272504 + 0.471991i
\(204\) 0 0
\(205\) −0.0445987 0.0772471i −0.00311491 0.00539517i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.95147 + 3.38004i 0.134986 + 0.233802i
\(210\) 0 0
\(211\) −3.68897 + 6.38948i −0.253959 + 0.439870i −0.964612 0.263672i \(-0.915066\pi\)
0.710653 + 0.703543i \(0.248400\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.60769 −0.246043
\(216\) 0 0
\(217\) 3.27110 0.222057
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.3258 19.6168i 0.761854 1.31957i
\(222\) 0 0
\(223\) −13.3549 23.1313i −0.894308 1.54899i −0.834658 0.550768i \(-0.814335\pi\)
−0.0596502 0.998219i \(-0.518999\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.46386 + 14.6598i 0.561766 + 0.973007i 0.997342 + 0.0728556i \(0.0232112\pi\)
−0.435576 + 0.900152i \(0.643455\pi\)
\(228\) 0 0
\(229\) 8.22580 14.2475i 0.543576 0.941501i −0.455119 0.890431i \(-0.650403\pi\)
0.998695 0.0510706i \(-0.0162633\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.10855 −0.269160 −0.134580 0.990903i \(-0.542969\pi\)
−0.134580 + 0.990903i \(0.542969\pi\)
\(234\) 0 0
\(235\) −0.375745 −0.0245109
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.4182 + 18.0448i −0.673895 + 1.16722i 0.302896 + 0.953024i \(0.402047\pi\)
−0.976791 + 0.214197i \(0.931287\pi\)
\(240\) 0 0
\(241\) −11.3477 19.6548i −0.730969 1.26608i −0.956470 0.291832i \(-0.905735\pi\)
0.225501 0.974243i \(-0.427598\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.164508 + 0.284936i 0.0105100 + 0.0182039i
\(246\) 0 0
\(247\) −4.52067 + 7.83004i −0.287644 + 0.498213i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00030 0.252497 0.126248 0.991999i \(-0.459706\pi\)
0.126248 + 0.991999i \(0.459706\pi\)
\(252\) 0 0
\(253\) 8.88637 0.558681
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.254753 0.441245i 0.0158911 0.0275241i −0.857971 0.513699i \(-0.828275\pi\)
0.873862 + 0.486175i \(0.161608\pi\)
\(258\) 0 0
\(259\) −0.164508 0.284936i −0.0102220 0.0177051i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.56344 + 2.70796i 0.0964059 + 0.166980i 0.910194 0.414181i \(-0.135932\pi\)
−0.813789 + 0.581161i \(0.802599\pi\)
\(264\) 0 0
\(265\) −1.05750 + 1.83165i −0.0649619 + 0.112517i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.16505 0.253947 0.126974 0.991906i \(-0.459474\pi\)
0.126974 + 0.991906i \(0.459474\pi\)
\(270\) 0 0
\(271\) 13.0230 0.791092 0.395546 0.918446i \(-0.370555\pi\)
0.395546 + 0.918446i \(0.370555\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.25061 + 5.63021i −0.196019 + 0.339515i
\(276\) 0 0
\(277\) 0.347709 + 0.602250i 0.0208918 + 0.0361857i 0.876282 0.481798i \(-0.160016\pi\)
−0.855390 + 0.517984i \(0.826683\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.1488 + 27.9706i 0.963359 + 1.66859i 0.713960 + 0.700186i \(0.246900\pi\)
0.249399 + 0.968401i \(0.419767\pi\)
\(282\) 0 0
\(283\) −7.06383 + 12.2349i −0.419901 + 0.727290i −0.995929 0.0901399i \(-0.971269\pi\)
0.576028 + 0.817430i \(0.304602\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.271104 −0.0160027
\(288\) 0 0
\(289\) 37.1315 2.18421
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.33818 14.4422i 0.487122 0.843720i −0.512769 0.858527i \(-0.671380\pi\)
0.999890 + 0.0148072i \(0.00471344\pi\)
\(294\) 0 0
\(295\) 0.122450 + 0.212090i 0.00712931 + 0.0123483i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.2929 + 17.8278i 0.595252 + 1.03101i
\(300\) 0 0
\(301\) −5.48255 + 9.49606i −0.316009 + 0.547344i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.90978 −0.166614
\(306\) 0 0
\(307\) −27.0345 −1.54294 −0.771469 0.636267i \(-0.780478\pi\)
−0.771469 + 0.636267i \(0.780478\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.7377 20.3302i 0.665581 1.15282i −0.313546 0.949573i \(-0.601517\pi\)
0.979127 0.203248i \(-0.0651496\pi\)
\(312\) 0 0
\(313\) 0.364597 + 0.631501i 0.0206083 + 0.0356946i 0.876146 0.482047i \(-0.160106\pi\)
−0.855537 + 0.517741i \(0.826773\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.7398 + 28.9943i 0.940203 + 1.62848i 0.765082 + 0.643933i \(0.222699\pi\)
0.175122 + 0.984547i \(0.443968\pi\)
\(318\) 0 0
\(319\) 5.16001 8.93741i 0.288905 0.500399i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −21.6066 −1.20222
\(324\) 0 0
\(325\) −15.0604 −0.835401
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.571014 + 0.989025i −0.0314810 + 0.0545267i
\(330\) 0 0
\(331\) −4.92051 8.52257i −0.270456 0.468443i 0.698523 0.715588i \(-0.253841\pi\)
−0.968979 + 0.247145i \(0.920508\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.41029 2.44270i −0.0770525 0.133459i
\(336\) 0 0
\(337\) 3.93490 6.81545i 0.214348 0.371261i −0.738723 0.674009i \(-0.764571\pi\)
0.953071 + 0.302748i \(0.0979041\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.34735 −0.235422
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.14721 14.1114i 0.437365 0.757539i −0.560120 0.828411i \(-0.689245\pi\)
0.997485 + 0.0708727i \(0.0225784\pi\)
\(348\) 0 0
\(349\) 1.79219 + 3.10416i 0.0959336 + 0.166162i 0.909998 0.414613i \(-0.136083\pi\)
−0.814064 + 0.580775i \(0.802750\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.80329 16.9798i −0.521776 0.903743i −0.999679 0.0253304i \(-0.991936\pi\)
0.477903 0.878413i \(-0.341397\pi\)
\(354\) 0 0
\(355\) 0.264478 0.458089i 0.0140370 0.0243129i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.8040 −0.886882 −0.443441 0.896303i \(-0.646243\pi\)
−0.443441 + 0.896303i \(0.646243\pi\)
\(360\) 0 0
\(361\) −10.3757 −0.546092
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.21988 + 3.84494i −0.116194 + 0.201254i
\(366\) 0 0
\(367\) 16.8160 + 29.1262i 0.877790 + 1.52038i 0.853760 + 0.520666i \(0.174316\pi\)
0.0240298 + 0.999711i \(0.492350\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.21414 + 5.56705i 0.166870 + 0.289027i
\(372\) 0 0
\(373\) 17.2159 29.8189i 0.891407 1.54396i 0.0532169 0.998583i \(-0.483053\pi\)
0.838190 0.545379i \(-0.183614\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 23.9069 1.23127
\(378\) 0 0
\(379\) −18.0913 −0.929287 −0.464644 0.885498i \(-0.653818\pi\)
−0.464644 + 0.885498i \(0.653818\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.03792 8.72594i 0.257426 0.445875i −0.708126 0.706086i \(-0.750459\pi\)
0.965552 + 0.260212i \(0.0837923\pi\)
\(384\) 0 0
\(385\) −0.218634 0.378684i −0.0111426 0.0192995i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.715236 1.23882i −0.0362639 0.0628109i 0.847324 0.531077i \(-0.178212\pi\)
−0.883588 + 0.468266i \(0.844879\pi\)
\(390\) 0 0
\(391\) −24.5974 + 42.6040i −1.24394 + 2.15457i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.413853 −0.0208232
\(396\) 0 0
\(397\) −10.5433 −0.529152 −0.264576 0.964365i \(-0.585232\pi\)
−0.264576 + 0.964365i \(0.585232\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.804044 1.39265i 0.0401521 0.0695454i −0.845251 0.534369i \(-0.820549\pi\)
0.885403 + 0.464824i \(0.153882\pi\)
\(402\) 0 0
\(403\) −5.03543 8.72162i −0.250833 0.434455i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.218634 + 0.378684i 0.0108373 + 0.0187707i
\(408\) 0 0
\(409\) −13.0174 + 22.5468i −0.643670 + 1.11487i 0.340937 + 0.940086i \(0.389256\pi\)
−0.984607 + 0.174783i \(0.944077\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.744341 0.0366266
\(414\) 0 0
\(415\) 0.0208240 0.00102221
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.00882 + 3.47938i −0.0981372 + 0.169979i −0.910914 0.412597i \(-0.864622\pi\)
0.812776 + 0.582576i \(0.197955\pi\)
\(420\) 0 0
\(421\) −2.63431 4.56275i −0.128388 0.222375i 0.794664 0.607049i \(-0.207647\pi\)
−0.923052 + 0.384675i \(0.874314\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −17.9953 31.1688i −0.872901 1.51191i
\(426\) 0 0
\(427\) −4.42195 + 7.65904i −0.213993 + 0.370647i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.7343 −0.854230 −0.427115 0.904197i \(-0.640470\pi\)
−0.427115 + 0.904197i \(0.640470\pi\)
\(432\) 0 0
\(433\) −4.76835 −0.229152 −0.114576 0.993414i \(-0.536551\pi\)
−0.114576 + 0.993414i \(0.536551\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.81804 17.0054i 0.469661 0.813476i
\(438\) 0 0
\(439\) 18.1134 + 31.3733i 0.864506 + 1.49737i 0.867537 + 0.497373i \(0.165702\pi\)
−0.00303091 + 0.999995i \(0.500965\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.81211 13.5310i −0.371164 0.642876i 0.618581 0.785721i \(-0.287708\pi\)
−0.989745 + 0.142846i \(0.954375\pi\)
\(444\) 0 0
\(445\) −1.86010 + 3.22179i −0.0881773 + 0.152728i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.4142 0.727441 0.363721 0.931508i \(-0.381506\pi\)
0.363721 + 0.931508i \(0.381506\pi\)
\(450\) 0 0
\(451\) 0.360301 0.0169659
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.506476 0.877243i 0.0237440 0.0411258i
\(456\) 0 0
\(457\) 4.71214 + 8.16166i 0.220424 + 0.381786i 0.954937 0.296809i \(-0.0959224\pi\)
−0.734512 + 0.678595i \(0.762589\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.0406 + 34.7113i 0.933383 + 1.61667i 0.777491 + 0.628893i \(0.216492\pi\)
0.155892 + 0.987774i \(0.450175\pi\)
\(462\) 0 0
\(463\) 10.9717 19.0036i 0.509900 0.883172i −0.490035 0.871703i \(-0.663016\pi\)
0.999934 0.0114690i \(-0.00365078\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.9808 −0.970878 −0.485439 0.874271i \(-0.661340\pi\)
−0.485439 + 0.874271i \(0.661340\pi\)
\(468\) 0 0
\(469\) −8.57280 −0.395855
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.28640 12.6204i 0.335029 0.580287i
\(474\) 0 0
\(475\) 7.18282 + 12.4410i 0.329570 + 0.570833i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.65194 + 11.5215i 0.303935 + 0.526431i 0.977024 0.213131i \(-0.0683661\pi\)
−0.673089 + 0.739562i \(0.735033\pi\)
\(480\) 0 0
\(481\) −0.506476 + 0.877243i −0.0230933 + 0.0399988i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.62852 0.164762
\(486\) 0 0
\(487\) 38.5519 1.74695 0.873477 0.486865i \(-0.161860\pi\)
0.873477 + 0.486865i \(0.161860\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.25959 5.64578i 0.147103 0.254791i −0.783052 0.621956i \(-0.786338\pi\)
0.930156 + 0.367165i \(0.119672\pi\)
\(492\) 0 0
\(493\) 28.5658 + 49.4774i 1.28654 + 2.22835i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.803846 1.39230i −0.0360574 0.0624533i
\(498\) 0 0
\(499\) 14.3259 24.8132i 0.641316 1.11079i −0.343823 0.939034i \(-0.611722\pi\)
0.985139 0.171758i \(-0.0549446\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.2801 0.725892 0.362946 0.931810i \(-0.381771\pi\)
0.362946 + 0.931810i \(0.381771\pi\)
\(504\) 0 0
\(505\) 4.68571 0.208511
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.2296 22.9143i 0.586391 1.01566i −0.408309 0.912844i \(-0.633882\pi\)
0.994700 0.102815i \(-0.0327851\pi\)
\(510\) 0 0
\(511\) 6.74703 + 11.6862i 0.298471 + 0.516967i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.74144 3.01626i −0.0767370 0.132912i
\(516\) 0 0
\(517\) 0.758887 1.31443i 0.0333758 0.0578086i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.72559 0.338464 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(522\) 0 0
\(523\) 16.9546 0.741375 0.370687 0.928758i \(-0.379122\pi\)
0.370687 + 0.928758i \(0.379122\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0334 20.8425i 0.524184 0.907914i
\(528\) 0 0
\(529\) −10.8542 18.8000i −0.471920 0.817390i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.417328 + 0.722833i 0.0180765 + 0.0313094i
\(534\) 0 0
\(535\) −2.55931 + 4.43285i −0.110648 + 0.191649i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.32902 −0.0572448
\(540\) 0 0
\(541\) 21.9353 0.943072 0.471536 0.881847i \(-0.343700\pi\)
0.471536 + 0.881847i \(0.343700\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.36191 + 4.09094i −0.101173 + 0.175237i
\(546\) 0 0
\(547\) 21.4034 + 37.0718i 0.915144 + 1.58508i 0.806691 + 0.590974i \(0.201256\pi\)
0.108453 + 0.994102i \(0.465410\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −11.4020 19.7489i −0.485742 0.841330i
\(552\) 0 0
\(553\) −0.628926 + 1.08933i −0.0267447 + 0.0463231i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.5498 −1.54866 −0.774332 0.632780i \(-0.781914\pi\)
−0.774332 + 0.632780i \(0.781914\pi\)
\(558\) 0 0
\(559\) 33.7587 1.42784
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.8582 + 34.3955i −0.836925 + 1.44960i 0.0555277 + 0.998457i \(0.482316\pi\)
−0.892453 + 0.451140i \(0.851017\pi\)
\(564\) 0 0
\(565\) 0.641469 + 1.11106i 0.0269868 + 0.0467425i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.4173 + 26.7035i 0.646325 + 1.11947i 0.983994 + 0.178203i \(0.0570283\pi\)
−0.337669 + 0.941265i \(0.609638\pi\)
\(570\) 0 0
\(571\) −19.4009 + 33.6033i −0.811901 + 1.40625i 0.0996310 + 0.995024i \(0.468234\pi\)
−0.911532 + 0.411229i \(0.865100\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 32.7083 1.36403
\(576\) 0 0
\(577\) −2.10713 −0.0877211 −0.0438606 0.999038i \(-0.513966\pi\)
−0.0438606 + 0.999038i \(0.513966\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.0316459 0.0548124i 0.00131289 0.00227400i
\(582\) 0 0
\(583\) −4.27164 7.39870i −0.176913 0.306423i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.46457 6.00081i −0.142998 0.247680i 0.785626 0.618701i \(-0.212341\pi\)
−0.928624 + 0.371022i \(0.879008\pi\)
\(588\) 0 0
\(589\) −4.80314 + 8.31928i −0.197910 + 0.342790i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.0323 0.411977 0.205989 0.978554i \(-0.433959\pi\)
0.205989 + 0.978554i \(0.433959\pi\)
\(594\) 0 0
\(595\) 2.42070 0.0992392
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21.6645 37.5240i 0.885187 1.53319i 0.0396877 0.999212i \(-0.487364\pi\)
0.845499 0.533977i \(-0.179303\pi\)
\(600\) 0 0
\(601\) 23.2578 + 40.2837i 0.948707 + 1.64321i 0.748154 + 0.663525i \(0.230941\pi\)
0.200553 + 0.979683i \(0.435726\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.51902 2.63102i −0.0617569 0.106966i
\(606\) 0 0
\(607\) 18.9227 32.7751i 0.768048 1.33030i −0.170572 0.985345i \(-0.554561\pi\)
0.938620 0.344953i \(-0.112105\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.51600 0.142242
\(612\) 0 0
\(613\) 30.2811 1.22304 0.611522 0.791228i \(-0.290558\pi\)
0.611522 + 0.791228i \(0.290558\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.08770 + 14.0083i −0.325599 + 0.563954i −0.981633 0.190777i \(-0.938899\pi\)
0.656035 + 0.754731i \(0.272233\pi\)
\(618\) 0 0
\(619\) −12.9377 22.4088i −0.520012 0.900687i −0.999729 0.0232638i \(-0.992594\pi\)
0.479718 0.877423i \(-0.340739\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.65354 + 9.79221i 0.226504 + 0.392317i
\(624\) 0 0
\(625\) −11.6940 + 20.2546i −0.467759 + 0.810182i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.42070 −0.0965198
\(630\) 0 0
\(631\) −29.2969 −1.16629 −0.583146 0.812368i \(-0.698178\pi\)
−0.583146 + 0.812368i \(0.698178\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.15285 1.99679i 0.0457495 0.0792404i
\(636\) 0 0
\(637\) −1.53937 2.66626i −0.0609920 0.105641i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.82630 15.2876i −0.348618 0.603824i 0.637386 0.770545i \(-0.280016\pi\)
−0.986004 + 0.166720i \(0.946682\pi\)
\(642\) 0 0
\(643\) −14.5426 + 25.1885i −0.573504 + 0.993338i 0.422698 + 0.906270i \(0.361083\pi\)
−0.996202 + 0.0870676i \(0.972250\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.8747 1.56764 0.783819 0.620989i \(-0.213269\pi\)
0.783819 + 0.620989i \(0.213269\pi\)
\(648\) 0 0
\(649\) −0.989241 −0.0388311
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.08986 12.2800i 0.277448 0.480553i −0.693302 0.720647i \(-0.743845\pi\)
0.970750 + 0.240094i \(0.0771782\pi\)
\(654\) 0 0
\(655\) 1.09654 + 1.89927i 0.0428455 + 0.0742106i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11.3592 + 19.6747i 0.442491 + 0.766418i 0.997874 0.0651775i \(-0.0207614\pi\)
−0.555382 + 0.831595i \(0.687428\pi\)
\(660\) 0 0
\(661\) −20.0052 + 34.6500i −0.778111 + 1.34773i 0.154919 + 0.987927i \(0.450488\pi\)
−0.933030 + 0.359800i \(0.882845\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.966223 −0.0374685
\(666\) 0 0
\(667\) −51.9212 −2.01040
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.87684 10.1790i 0.226873 0.392956i
\(672\) 0 0
\(673\) 0.913881 + 1.58289i 0.0352275 + 0.0610159i 0.883102 0.469182i \(-0.155451\pi\)
−0.847874 + 0.530198i \(0.822118\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −14.2246 24.6376i −0.546694 0.946902i −0.998498 0.0547845i \(-0.982553\pi\)
0.451804 0.892117i \(-0.350781\pi\)
\(678\) 0 0
\(679\) 5.51420 9.55087i 0.211616 0.366529i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.868335 0.0332259 0.0166130 0.999862i \(-0.494712\pi\)
0.0166130 + 0.999862i \(0.494712\pi\)
\(684\) 0 0
\(685\) −5.54014 −0.211678
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.89549 17.1395i 0.376988 0.652962i
\(690\) 0 0
\(691\) −8.71932 15.1023i −0.331699 0.574519i 0.651146 0.758952i \(-0.274288\pi\)
−0.982845 + 0.184433i \(0.940955\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.56972 + 4.45088i 0.0974750 + 0.168832i
\(696\) 0 0
\(697\) −0.997310 + 1.72739i −0.0377758 + 0.0654296i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.23113 −0.348655 −0.174327 0.984688i \(-0.555775\pi\)
−0.174327 + 0.984688i \(0.555775\pi\)
\(702\) 0 0
\(703\) 0.966223 0.0364418
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.12079 12.3336i 0.267805 0.463852i
\(708\) 0 0
\(709\) −10.8231 18.7461i −0.406469 0.704025i 0.588022 0.808845i \(-0.299907\pi\)
−0.994491 + 0.104820i \(0.966573\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.9360 + 18.9417i 0.409556 + 0.709373i
\(714\) 0 0
\(715\) −0.673115 + 1.16587i −0.0251731 + 0.0436010i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 48.6526 1.81444 0.907218 0.420661i \(-0.138202\pi\)
0.907218 + 0.420661i \(0.138202\pi\)
\(720\) 0 0
\(721\) −10.5858 −0.394234
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 18.9926 32.8962i 0.705368 1.22173i
\(726\) 0 0
\(727\) 1.11417 + 1.92980i 0.0413222 + 0.0715722i 0.885947 0.463787i \(-0.153510\pi\)
−0.844625 + 0.535359i \(0.820176\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 40.3374 + 69.8664i 1.49193 + 2.58410i
\(732\) 0 0
\(733\) −19.0129 + 32.9313i −0.702258 + 1.21635i 0.265415 + 0.964134i \(0.414491\pi\)
−0.967672 + 0.252211i \(0.918842\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.3934 0.419681
\(738\) 0 0
\(739\) 48.6048 1.78796 0.893978 0.448112i \(-0.147903\pi\)
0.893978 + 0.448112i \(0.147903\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.2700 36.8408i 0.780322 1.35156i −0.151432 0.988468i \(-0.548388\pi\)
0.931754 0.363090i \(-0.118278\pi\)
\(744\) 0 0
\(745\) 2.47826 + 4.29247i 0.0907963 + 0.157264i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.77868 + 13.4731i 0.284227 + 0.492295i
\(750\) 0 0
\(751\) 4.62893 8.01754i 0.168912 0.292564i −0.769126 0.639098i \(-0.779308\pi\)
0.938038 + 0.346534i \(0.112641\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.54470 −0.0926111
\(756\) 0 0
\(757\) 41.0363 1.49149 0.745744 0.666232i \(-0.232094\pi\)
0.745744 + 0.666232i \(0.232094\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −19.1859 + 33.2309i −0.695487 + 1.20462i 0.274529 + 0.961579i \(0.411478\pi\)
−0.970016 + 0.243040i \(0.921855\pi\)
\(762\) 0 0
\(763\) 7.17871 + 12.4339i 0.259887 + 0.450137i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.14581 1.98461i −0.0413730 0.0716601i
\(768\) 0 0
\(769\) 0.149772 0.259412i 0.00540090 0.00935463i −0.863312 0.504670i \(-0.831614\pi\)
0.868713 + 0.495315i \(0.164947\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −40.2941 −1.44928 −0.724639 0.689128i \(-0.757994\pi\)
−0.724639 + 0.689128i \(0.757994\pi\)
\(774\) 0 0
\(775\) −16.0014 −0.574788
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.398076 0.689488i 0.0142626 0.0247035i
\(780\) 0 0
\(781\) 1.06832 + 1.85039i 0.0382276 + 0.0662122i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.43389 2.48357i −0.0511777 0.0886425i
\(786\) 0 0
\(787\) 12.2565 21.2289i 0.436897 0.756728i −0.560551 0.828120i \(-0.689411\pi\)
0.997448 + 0.0713920i \(0.0227441\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.89932 0.138644
\(792\) 0 0
\(793\) 27.2280 0.966896
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26.9140 + 46.6164i −0.953343 + 1.65124i −0.215228 + 0.976564i \(0.569049\pi\)
−0.738115 + 0.674675i \(0.764284\pi\)
\(798\) 0 0
\(799\) 4.20119 + 7.27667i 0.148627 + 0.257430i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −8.96691 15.5311i −0.316435 0.548082i
\(804\) 0 0
\(805\) −1.09997 + 1.90520i −0.0387689 + 0.0671496i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −7.44011 −0.261580 −0.130790 0.991410i \(-0.541751\pi\)
−0.130790 + 0.991410i \(0.541751\pi\)
\(810\) 0 0
\(811\) −2.94854 −0.103537 −0.0517687 0.998659i \(-0.516486\pi\)
−0.0517687 + 0.998659i \(0.516486\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.21284 + 3.83276i −0.0775125 + 0.134256i
\(816\) 0 0
\(817\) −16.1007 27.8872i −0.563291 0.975648i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.44331 14.6242i −0.294674 0.510390i 0.680235 0.732994i \(-0.261878\pi\)
−0.974909 + 0.222604i \(0.928544\pi\)
\(822\) 0 0
\(823\) 24.3829 42.2325i 0.849935 1.47213i −0.0313301 0.999509i \(-0.509974\pi\)
0.881265 0.472622i \(-0.156692\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.4541 −0.502618 −0.251309 0.967907i \(-0.580861\pi\)
−0.251309 + 0.967907i \(0.580861\pi\)
\(828\) 0 0
\(829\) −32.5659 −1.13106 −0.565530 0.824728i \(-0.691328\pi\)
−0.565530 + 0.824728i \(0.691328\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.67871 6.37171i 0.127460 0.220767i
\(834\) 0 0
\(835\) −1.91423 3.31554i −0.0662446 0.114739i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 24.4412 + 42.3334i 0.843803 + 1.46151i 0.886656 + 0.462429i \(0.153022\pi\)
−0.0428531 + 0.999081i \(0.513645\pi\)
\(840\) 0 0
\(841\) −15.6489 + 27.1047i −0.539617 + 0.934644i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.15859 0.0398567
\(846\) 0 0
\(847\) −9.23372 −0.317274
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.09997 1.90520i 0.0377065 0.0653096i
\(852\) 0 0
\(853\) 10.5285 + 18.2360i 0.360491 + 0.624388i 0.988042 0.154187i \(-0.0492760\pi\)
−0.627551 + 0.778575i \(0.715943\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.7290 + 32.4396i 0.639771 + 1.10812i 0.985483 + 0.169776i \(0.0543043\pi\)
−0.345711 + 0.938341i \(0.612362\pi\)
\(858\) 0 0
\(859\) 13.2123 22.8844i 0.450799 0.780807i −0.547637 0.836716i \(-0.684472\pi\)
0.998436 + 0.0559093i \(0.0178058\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29.1888 −0.993597 −0.496798 0.867866i \(-0.665491\pi\)
−0.496798 + 0.867866i \(0.665491\pi\)
\(864\) 0 0
\(865\) 0.582905 0.0198194
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.835853 1.44774i 0.0283544 0.0491112i
\(870\) 0 0
\(871\) 13.1967 + 22.8573i 0.447153 + 0.774491i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.62727 2.81852i −0.0550118 0.0952832i
\(876\) 0 0
\(877\) −5.70263 + 9.87725i −0.192564 + 0.333531i −0.946099 0.323877i \(-0.895014\pi\)
0.753535 + 0.657408i \(0.228347\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.7109 0.563005 0.281503 0.959560i \(-0.409167\pi\)
0.281503 + 0.959560i \(0.409167\pi\)
\(882\) 0 0
\(883\) −11.8347 −0.398268 −0.199134 0.979972i \(-0.563813\pi\)
−0.199134 + 0.979972i \(0.563813\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17.8800 + 30.9691i −0.600353 + 1.03984i 0.392414 + 0.919789i \(0.371640\pi\)
−0.992767 + 0.120053i \(0.961693\pi\)
\(888\) 0 0
\(889\) −3.50394 6.06899i −0.117518 0.203548i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.67690 2.90448i −0.0561154 0.0971947i
\(894\) 0 0
\(895\) −1.10737 + 1.91801i −0.0370152 + 0.0641122i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 25.4007 0.847159
\(900\) 0 0
\(901\) 47.2955 1.57564
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.18320 + 3.78142i −0.0725721 + 0.125698i
\(906\) 0 0
\(907\) −8.32308 14.4160i −0.276363 0.478675i 0.694115 0.719864i \(-0.255796\pi\)
−0.970478 + 0.241189i \(0.922463\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9.88258 17.1171i −0.327425 0.567116i 0.654575 0.755997i \(-0.272847\pi\)
−0.982000 + 0.188881i \(0.939514\pi\)
\(912\) 0 0
\(913\) −0.0420579 + 0.0728465i −0.00139191 + 0.00241087i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.66560 0.220118
\(918\) 0 0
\(919\) 10.2384 0.337734 0.168867 0.985639i \(-0.445989\pi\)
0.168867 + 0.985639i \(0.445989\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.47483 + 4.28653i −0.0814600 + 0.141093i
\(924\) 0 0
\(925\) 0.804731 + 1.39384i 0.0264594 + 0.0458290i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17.5842 30.4567i −0.576917 0.999250i −0.995830 0.0912240i \(-0.970922\pi\)
0.418913 0.908026i \(-0.362411\pi\)
\(930\) 0 0
\(931\) −1.46835 + 2.54326i −0.0481234 + 0.0833521i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.21715 −0.105212
\(936\) 0 0
\(937\) −31.5829 −1.03177 −0.515884 0.856659i \(-0.672536\pi\)
−0.515884 + 0.856659i \(0.672536\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.49641 11.2521i 0.211777 0.366808i −0.740494 0.672063i \(-0.765408\pi\)
0.952271 + 0.305255i \(0.0987416\pi\)
\(942\) 0 0
\(943\) −0.906357 1.56986i −0.0295150 0.0511216i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.7246 + 30.6998i 0.575971 + 0.997610i 0.995935 + 0.0900699i \(0.0287090\pi\)
−0.419965 + 0.907540i \(0.637958\pi\)
\(948\) 0 0
\(949\) 20.7723 35.9787i 0.674298 1.16792i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 40.0040 1.29586 0.647928 0.761702i \(-0.275636\pi\)
0.647928 + 0.761702i \(0.275636\pi\)
\(954\) 0 0
\(955\) −6.84898 −0.221628
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.41926 + 14.5826i −0.271872 + 0.470896i
\(960\) 0 0
\(961\) 10.1499 + 17.5802i 0.327417 + 0.567104i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.37522 5.84605i −0.108652 0.188191i
\(966\) 0 0
\(967\) 3.03814 5.26222i 0.0977001 0.169222i −0.813032 0.582219i \(-0.802185\pi\)
0.910732 + 0.412997i \(0.135518\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 57.1397 1.83370 0.916850 0.399232i \(-0.130723\pi\)
0.916850 + 0.399232i \(0.130723\pi\)
\(972\) 0 0
\(973\) 15.6206 0.500775
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.91064 + 10.2375i −0.189098 + 0.327528i −0.944950 0.327215i \(-0.893890\pi\)
0.755852 + 0.654743i \(0.227223\pi\)
\(978\) 0 0
\(979\) −7.51364 13.0140i −0.240137 0.415929i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9.26336 16.0446i −0.295455 0.511744i 0.679635 0.733550i \(-0.262138\pi\)
−0.975091 + 0.221806i \(0.928805\pi\)
\(984\) 0 0
\(985\) −1.18574 + 2.05377i −0.0377809 + 0.0654384i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −73.3174 −2.33136
\(990\) 0 0
\(991\) −12.5408 −0.398371 −0.199186 0.979962i \(-0.563830\pi\)
−0.199186 + 0.979962i \(0.563830\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.35700 4.08245i 0.0747220 0.129422i
\(996\) 0 0
\(997\) −7.94209 13.7561i −0.251528 0.435660i 0.712418 0.701755i \(-0.247600\pi\)
−0.963947 + 0.266095i \(0.914267\pi\)
\(998\) 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 3024.2.r.m.1009.2 8
3.2 odd 2 1008.2.r.l.337.4 8
4.3 odd 2 1512.2.r.e.1009.2 8
9.2 odd 6 1008.2.r.l.673.4 8
9.4 even 3 9072.2.a.cg.1.3 4
9.5 odd 6 9072.2.a.cj.1.2 4
9.7 even 3 inner 3024.2.r.m.2017.2 8
12.11 even 2 504.2.r.e.337.1 yes 8
36.7 odd 6 1512.2.r.e.505.2 8
36.11 even 6 504.2.r.e.169.1 8
36.23 even 6 4536.2.a.z.1.2 4
36.31 odd 6 4536.2.a.y.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.e.169.1 8 36.11 even 6
504.2.r.e.337.1 yes 8 12.11 even 2
1008.2.r.l.337.4 8 3.2 odd 2
1008.2.r.l.673.4 8 9.2 odd 6
1512.2.r.e.505.2 8 36.7 odd 6
1512.2.r.e.1009.2 8 4.3 odd 2
3024.2.r.m.1009.2 8 1.1 even 1 trivial
3024.2.r.m.2017.2 8 9.7 even 3 inner
4536.2.a.y.1.3 4 36.31 odd 6
4536.2.a.z.1.2 4 36.23 even 6
9072.2.a.cg.1.3 4 9.4 even 3
9072.2.a.cj.1.2 4 9.5 odd 6