Properties

Label 504.2.r.e.337.1
Level $504$
Weight $2$
Character 504.337
Analytic conductor $4.024$
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] = [504,2,Mod(169,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.r (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: \(4.02446026187\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 337.1
Root \(0.335492 + 1.69925i\) of defining polynomial
Character \(\chi\) \(=\) 504.337
Dual form 504.2.r.e.169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.30385 - 1.14017i) q^{3} +(-0.164508 + 0.284936i) q^{5} +(0.500000 + 0.866025i) q^{7} +(0.400030 + 2.97321i) q^{9} +O(q^{10})\) \(q+(-1.30385 - 1.14017i) q^{3} +(-0.164508 + 0.284936i) q^{5} +(0.500000 + 0.866025i) q^{7} +(0.400030 + 2.97321i) q^{9} +(0.664508 + 1.15096i) q^{11} +(-1.53937 + 2.66626i) q^{13} +(0.539368 - 0.183946i) q^{15} +7.35741 q^{17} -2.93671 q^{19} +(0.335492 - 1.69925i) q^{21} +(3.34321 - 5.79062i) q^{23} +(2.44587 + 4.23638i) q^{25} +(2.86838 - 4.33271i) q^{27} +(3.88258 + 6.72483i) q^{29} +(1.63555 - 2.83286i) q^{31} +(0.445874 - 2.25833i) q^{33} -0.329016 q^{35} +0.329016 q^{37} +(5.04709 - 1.72126i) q^{39} +(-0.135552 + 0.234783i) q^{41} +(5.48255 + 9.49606i) q^{43} +(-0.912983 - 0.375134i) q^{45} +(-0.571014 - 0.989025i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-9.59293 - 8.38869i) q^{51} +6.42828 q^{53} -0.437267 q^{55} +(3.82902 + 3.34834i) q^{57} +(-0.372170 + 0.644618i) q^{59} +(-4.42195 - 7.65904i) q^{61} +(-2.37486 + 1.83304i) q^{63} +(-0.506476 - 0.877243i) q^{65} +(-4.28640 + 7.42426i) q^{67} +(-10.9613 + 3.73825i) q^{69} +1.60769 q^{71} -13.4941 q^{73} +(1.64114 - 8.31230i) q^{75} +(-0.664508 + 1.15096i) q^{77} +(0.628926 + 1.08933i) q^{79} +(-8.67995 + 2.37875i) q^{81} +(0.0316459 + 0.0548124i) q^{83} +(-1.21035 + 2.09639i) q^{85} +(2.60515 - 13.1949i) q^{87} +11.3071 q^{89} -3.07874 q^{91} +(-5.36245 + 1.82881i) q^{93} +(0.483112 - 0.836774i) q^{95} +(5.51420 + 9.55087i) q^{97} +(-3.15623 + 2.43614i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} - 3 q^{5} + 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} - 3 q^{5} + 4 q^{7} - q^{9} + 7 q^{11} + 3 q^{13} - 11 q^{15} + 6 q^{17} - 8 q^{19} + q^{21} + 2 q^{23} - 5 q^{25} + 11 q^{27} - 9 q^{29} + 3 q^{31} - 21 q^{33} - 6 q^{35} + 6 q^{37} + 2 q^{39} + 9 q^{41} + 8 q^{43} + 7 q^{45} + 3 q^{47} - 4 q^{49} - 18 q^{51} + 12 q^{53} - 56 q^{55} + 34 q^{57} + 10 q^{59} + 20 q^{61} - 2 q^{63} + q^{65} + 11 q^{67} - 17 q^{69} - 6 q^{71} - 48 q^{73} + 52 q^{75} - 7 q^{77} + 21 q^{79} - 25 q^{81} + 8 q^{83} + 9 q^{85} - 15 q^{87} + 12 q^{89} + 6 q^{91} + 29 q^{93} + 36 q^{95} + 16 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.30385 1.14017i −0.752776 0.658277i
\(4\) 0 0
\(5\) −0.164508 + 0.284936i −0.0735702 + 0.127427i −0.900464 0.434931i \(-0.856773\pi\)
0.826893 + 0.562359i \(0.190106\pi\)
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0.400030 + 2.97321i 0.133343 + 0.991070i
\(10\) 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
\(15\) 0.539368 0.183946i 0.139264 0.0474946i
\(16\) 0 0
\(17\) 7.35741 1.78443 0.892217 0.451606i \(-0.149149\pi\)
0.892217 + 0.451606i \(0.149149\pi\)
\(18\) 0 0
\(19\) −2.93671 −0.673727 −0.336864 0.941553i \(-0.609366\pi\)
−0.336864 + 0.941553i \(0.609366\pi\)
\(20\) 0 0
\(21\) 0.335492 1.69925i 0.0732104 0.370806i
\(22\) 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
\(27\) 2.86838 4.33271i 0.552021 0.833830i
\(28\) 0 0
\(29\) 3.88258 + 6.72483i 0.720977 + 1.24877i 0.960608 + 0.277906i \(0.0896402\pi\)
−0.239631 + 0.970864i \(0.577026\pi\)
\(30\) 0 0
\(31\) 1.63555 2.83286i 0.293754 0.508796i −0.680940 0.732339i \(-0.738429\pi\)
0.974694 + 0.223542i \(0.0717621\pi\)
\(32\) 0 0
\(33\) 0.445874 2.25833i 0.0776168 0.393124i
\(34\) 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
\(39\) 5.04709 1.72126i 0.808181 0.275622i
\(40\) 0 0
\(41\) −0.135552 + 0.234783i −0.0211696 + 0.0366669i −0.876416 0.481555i \(-0.840072\pi\)
0.855247 + 0.518221i \(0.173406\pi\)
\(42\) 0 0
\(43\) 5.48255 + 9.49606i 0.836081 + 1.44814i 0.893147 + 0.449765i \(0.148492\pi\)
−0.0570654 + 0.998370i \(0.518174\pi\)
\(44\) 0 0
\(45\) −0.912983 0.375134i −0.136099 0.0559216i
\(46\) 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
\(51\) −9.59293 8.38869i −1.34328 1.17465i
\(52\) 0 0
\(53\) 6.42828 0.882992 0.441496 0.897263i \(-0.354448\pi\)
0.441496 + 0.897263i \(0.354448\pi\)
\(54\) 0 0
\(55\) −0.437267 −0.0589611
\(56\) 0 0
\(57\) 3.82902 + 3.34834i 0.507166 + 0.443499i
\(58\) 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
\(63\) −2.37486 + 1.83304i −0.299204 + 0.230941i
\(64\) 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 \(0.342103\pi\)
−0.999620 + 0.0275474i \(0.991230\pi\)
\(68\) 0 0
\(69\) −10.9613 + 3.73825i −1.31959 + 0.450032i
\(70\) 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
\(75\) 1.64114 8.31230i 0.189503 0.959821i
\(76\) 0 0
\(77\) −0.664508 + 1.15096i −0.0757277 + 0.131164i
\(78\) 0 0
\(79\) 0.628926 + 1.08933i 0.0707597 + 0.122559i 0.899234 0.437467i \(-0.144124\pi\)
−0.828475 + 0.560026i \(0.810791\pi\)
\(80\) 0 0
\(81\) −8.67995 + 2.37875i −0.964439 + 0.264305i
\(82\) 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
\(87\) 2.60515 13.1949i 0.279302 1.41465i
\(88\) 0 0
\(89\) 11.3071 1.19855 0.599274 0.800544i \(-0.295456\pi\)
0.599274 + 0.800544i \(0.295456\pi\)
\(90\) 0 0
\(91\) −3.07874 −0.322739
\(92\) 0 0
\(93\) −5.36245 + 1.82881i −0.556060 + 0.189638i
\(94\) 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
\(99\) −3.15623 + 2.43614i −0.317213 + 0.244841i
\(100\) 0 0
\(101\) −7.12079 12.3336i −0.708546 1.22724i −0.965397 0.260786i \(-0.916018\pi\)
0.256851 0.966451i \(-0.417315\pi\)
\(102\) 0 0
\(103\) −5.29288 + 9.16753i −0.521522 + 0.903303i 0.478164 + 0.878270i \(0.341302\pi\)
−0.999687 + 0.0250330i \(0.992031\pi\)
\(104\) 0 0
\(105\) 0.428986 + 0.375134i 0.0418647 + 0.0366093i
\(106\) 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
\(111\) −0.428986 0.375134i −0.0407175 0.0356061i
\(112\) 0 0
\(113\) 1.94966 3.37691i 0.183409 0.317673i −0.759630 0.650355i \(-0.774620\pi\)
0.943039 + 0.332682i \(0.107954\pi\)
\(114\) 0 0
\(115\) 1.09997 + 1.90520i 0.102573 + 0.177661i
\(116\) 0 0
\(117\) −8.54315 3.51028i −0.789815 0.324525i
\(118\) 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
\(123\) 0.444430 0.151568i 0.0400729 0.0136665i
\(124\) 0 0
\(125\) −3.25454 −0.291095
\(126\) 0 0
\(127\) −7.00787 −0.621848 −0.310924 0.950435i \(-0.600638\pi\)
−0.310924 + 0.950435i \(0.600638\pi\)
\(128\) 0 0
\(129\) 3.67871 18.6324i 0.323892 1.64049i
\(130\) 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
\(135\) 0.762673 + 1.53007i 0.0656405 + 0.131688i
\(136\) 0 0
\(137\) 8.41926 + 14.5826i 0.719306 + 1.24587i 0.961275 + 0.275591i \(0.0888734\pi\)
−0.241969 + 0.970284i \(0.577793\pi\)
\(138\) 0 0
\(139\) 7.81032 13.5279i 0.662463 1.14742i −0.317504 0.948257i \(-0.602845\pi\)
0.979967 0.199162i \(-0.0638221\pi\)
\(140\) 0 0
\(141\) −0.383141 + 1.94059i −0.0322663 + 0.163427i
\(142\) 0 0
\(143\) −4.09169 −0.342164
\(144\) 0 0
\(145\) −2.55486 −0.212170
\(146\) 0 0
\(147\) 1.63934 0.559079i 0.135210 0.0461121i
\(148\) 0 0
\(149\) 7.53233 13.0464i 0.617073 1.06880i −0.372944 0.927854i \(-0.621652\pi\)
0.990017 0.140948i \(-0.0450149\pi\)
\(150\) 0 0
\(151\) 3.86714 + 6.69808i 0.314703 + 0.545082i 0.979374 0.202054i \(-0.0647617\pi\)
−0.664671 + 0.747136i \(0.731428\pi\)
\(152\) 0 0
\(153\) 2.94318 + 21.8751i 0.237942 + 1.76850i
\(154\) 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
\(159\) −8.38149 7.32932i −0.664695 0.581253i
\(160\) 0 0
\(161\) 6.68643 0.526964
\(162\) 0 0
\(163\) 13.4513 1.05359 0.526793 0.849993i \(-0.323394\pi\)
0.526793 + 0.849993i \(0.323394\pi\)
\(164\) 0 0
\(165\) 0.570129 + 0.498558i 0.0443845 + 0.0388127i
\(166\) 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
\(171\) −1.17477 8.73145i −0.0898370 0.667711i
\(172\) 0 0
\(173\) −0.885831 1.53430i −0.0673485 0.116651i 0.830385 0.557190i \(-0.188121\pi\)
−0.897733 + 0.440539i \(0.854787\pi\)
\(174\) 0 0
\(175\) −2.44587 + 4.23638i −0.184891 + 0.320240i
\(176\) 0 0
\(177\) 1.22023 0.416146i 0.0917178 0.0312794i
\(178\) 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
\(183\) −2.96706 + 15.0280i −0.219331 + 1.11090i
\(184\) 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
\(189\) 5.18643 + 0.317738i 0.377257 + 0.0231121i
\(190\) 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
\(195\) −0.339838 + 1.72126i −0.0243363 + 0.123262i
\(196\) 0 0
\(197\) 7.20781 0.513535 0.256768 0.966473i \(-0.417342\pi\)
0.256768 + 0.966473i \(0.417342\pi\)
\(198\) 0 0
\(199\) −14.3276 −1.01566 −0.507828 0.861458i \(-0.669552\pi\)
−0.507828 + 0.861458i \(0.669552\pi\)
\(200\) 0 0
\(201\) 14.0537 4.79288i 0.991273 0.338063i
\(202\) 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
\(207\) 18.5541 + 7.62366i 1.28960 + 0.529881i
\(208\) 0 0
\(209\) −1.95147 3.38004i −0.134986 0.233802i
\(210\) 0 0
\(211\) 3.68897 6.38948i 0.253959 0.439870i −0.710653 0.703543i \(-0.751600\pi\)
0.964612 + 0.263672i \(0.0849338\pi\)
\(212\) 0 0
\(213\) −2.09618 1.83304i −0.143628 0.125598i
\(214\) 0 0
\(215\) −3.60769 −0.246043
\(216\) 0 0
\(217\) 3.27110 0.222057
\(218\) 0 0
\(219\) 17.5942 + 15.3855i 1.18890 + 1.03966i
\(220\) 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 \(0.185665\pi\)
0.0596502 + 0.998219i \(0.481001\pi\)
\(224\) 0 0
\(225\) −11.6172 + 8.96678i −0.774481 + 0.597785i
\(226\) 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
\(231\) 2.17871 0.743025i 0.143348 0.0488875i
\(232\) 0 0
\(233\) 4.10855 0.269160 0.134580 0.990903i \(-0.457031\pi\)
0.134580 + 0.990903i \(0.457031\pi\)
\(234\) 0 0
\(235\) 0.375745 0.0245109
\(236\) 0 0
\(237\) 0.422000 2.13740i 0.0274118 0.138839i
\(238\) 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
\(243\) 14.0295 + 6.79509i 0.899992 + 0.435905i
\(244\) 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
\(249\) 0.0212339 0.107549i 0.00134564 0.00681561i
\(250\) 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
\(255\) 3.96835 1.35337i 0.248508 0.0847511i
\(256\) 0 0
\(257\) −0.254753 + 0.441245i −0.0158911 + 0.0275241i −0.873862 0.486175i \(-0.838392\pi\)
0.857971 + 0.513699i \(0.171725\pi\)
\(258\) 0 0
\(259\) 0.164508 + 0.284936i 0.0102220 + 0.0177051i
\(260\) 0 0
\(261\) −18.4412 + 14.2339i −1.14148 + 0.881054i
\(262\) 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
\(267\) −14.7427 12.8920i −0.902238 0.788976i
\(268\) 0 0
\(269\) −4.16505 −0.253947 −0.126974 0.991906i \(-0.540526\pi\)
−0.126974 + 0.991906i \(0.540526\pi\)
\(270\) 0 0
\(271\) −13.0230 −0.791092 −0.395546 0.918446i \(-0.629445\pi\)
−0.395546 + 0.918446i \(0.629445\pi\)
\(272\) 0 0
\(273\) 4.01420 + 3.51028i 0.242950 + 0.212452i
\(274\) 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
\(279\) 9.07695 + 3.72961i 0.543423 + 0.223286i
\(280\) 0 0
\(281\) −16.1488 27.9706i −0.963359 1.66859i −0.713960 0.700186i \(-0.753100\pi\)
−0.249399 0.968401i \(-0.580233\pi\)
\(282\) 0 0
\(283\) 7.06383 12.2349i 0.419901 0.727290i −0.576028 0.817430i \(-0.695398\pi\)
0.995929 + 0.0901399i \(0.0287314\pi\)
\(284\) 0 0
\(285\) −1.58397 + 0.540196i −0.0938261 + 0.0319984i
\(286\) 0 0
\(287\) −0.271104 −0.0160027
\(288\) 0 0
\(289\) 37.1315 2.18421
\(290\) 0 0
\(291\) 3.69994 18.7400i 0.216894 1.09856i
\(292\) 0 0
\(293\) −8.33818 + 14.4422i −0.487122 + 0.843720i −0.999890 0.0148072i \(-0.995287\pi\)
0.512769 + 0.858527i \(0.328620\pi\)
\(294\) 0 0
\(295\) −0.122450 0.212090i −0.00712931 0.0123483i
\(296\) 0 0
\(297\) 6.89285 + 0.422279i 0.399963 + 0.0245031i
\(298\) 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
\(303\) −4.77794 + 24.2000i −0.274486 + 1.39025i
\(304\) 0 0
\(305\) 2.90978 0.166614
\(306\) 0 0
\(307\) 27.0345 1.54294 0.771469 0.636267i \(-0.219522\pi\)
0.771469 + 0.636267i \(0.219522\pi\)
\(308\) 0 0
\(309\) 17.3536 5.91828i 0.987213 0.336679i
\(310\) 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
\(315\) −0.131616 0.978233i −0.00741573 0.0551172i
\(316\) 0 0
\(317\) −16.7398 28.9943i −0.940203 1.62848i −0.765082 0.643933i \(-0.777301\pi\)
−0.175122 0.984547i \(-0.556032\pi\)
\(318\) 0 0
\(319\) −5.16001 + 8.93741i −0.288905 + 0.500399i
\(320\) 0 0
\(321\) 20.2844 + 17.7380i 1.13216 + 0.990039i
\(322\) 0 0
\(323\) −21.6066 −1.20222
\(324\) 0 0
\(325\) −15.0604 −0.835401
\(326\) 0 0
\(327\) 18.7199 + 16.3699i 1.03521 + 0.905256i
\(328\) 0 0
\(329\) 0.571014 0.989025i 0.0314810 0.0545267i
\(330\) 0 0
\(331\) 4.92051 + 8.52257i 0.270456 + 0.468443i 0.968979 0.247145i \(-0.0794924\pi\)
−0.698523 + 0.715588i \(0.746159\pi\)
\(332\) 0 0
\(333\) 0.131616 + 0.978233i 0.00721252 + 0.0536068i
\(334\) 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
\(339\) −6.39231 + 2.18003i −0.347182 + 0.118403i
\(340\) 0 0
\(341\) 4.34735 0.235422
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.738063 3.73825i 0.0397360 0.201260i
\(346\) 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
\(351\) 7.13665 + 14.3175i 0.380926 + 0.764212i
\(352\) 0 0
\(353\) 9.80329 + 16.9798i 0.521776 + 0.903743i 0.999679 + 0.0253304i \(0.00806378\pi\)
−0.477903 + 0.878413i \(0.658603\pi\)
\(354\) 0 0
\(355\) −0.264478 + 0.458089i −0.0140370 + 0.0243129i
\(356\) 0 0
\(357\) 2.46835 12.5021i 0.130639 0.661680i
\(358\) 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
\(363\) −15.1372 + 5.16238i −0.794496 + 0.270955i
\(364\) 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 \(-0.825684\pi\)
−0.0240298 0.999711i \(-0.507650\pi\)
\(368\) 0 0
\(369\) −0.752282 0.309104i −0.0391623 0.0160913i
\(370\) 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
\(375\) 4.24342 + 3.71073i 0.219129 + 0.191621i
\(376\) 0 0
\(377\) −23.9069 −1.23127
\(378\) 0 0
\(379\) 18.0913 0.929287 0.464644 0.885498i \(-0.346182\pi\)
0.464644 + 0.885498i \(0.346182\pi\)
\(380\) 0 0
\(381\) 9.13719 + 7.99016i 0.468112 + 0.409348i
\(382\) 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
\(387\) −26.0406 + 20.0995i −1.32372 + 1.02171i
\(388\) 0 0
\(389\) 0.715236 + 1.23882i 0.0362639 + 0.0628109i 0.883588 0.468266i \(-0.155121\pi\)
−0.847324 + 0.531077i \(0.821788\pi\)
\(390\) 0 0
\(391\) 24.5974 42.6040i 1.24394 2.15457i
\(392\) 0 0
\(393\) 10.9272 3.72660i 0.551203 0.187982i
\(394\) 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
\(399\) −0.985242 + 4.99020i −0.0493238 + 0.249822i
\(400\) 0 0
\(401\) −0.804044 + 1.39265i −0.0401521 + 0.0695454i −0.885403 0.464824i \(-0.846118\pi\)
0.845251 + 0.534369i \(0.179451\pi\)
\(402\) 0 0
\(403\) 5.03543 + 8.72162i 0.250833 + 0.434455i
\(404\) 0 0
\(405\) 0.750130 2.86455i 0.0372743 0.142341i
\(406\) 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
\(411\) 5.64919 28.6128i 0.278654 1.41137i
\(412\) 0 0
\(413\) −0.744341 −0.0366266
\(414\) 0 0
\(415\) −0.0208240 −0.00102221
\(416\) 0 0
\(417\) −25.6075 + 8.73318i −1.25401 + 0.427666i
\(418\) 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
\(423\) 2.71216 2.09338i 0.131870 0.101784i
\(424\) 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
\(429\) 5.33493 + 4.66522i 0.257573 + 0.225239i
\(430\) 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
\(435\) 3.33115 + 2.91297i 0.159716 + 0.139666i
\(436\) 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 \(-0.834298\pi\)
0.00303091 0.999995i \(-0.499035\pi\)
\(440\) 0 0
\(441\) −2.77489 1.14017i −0.132138 0.0542938i
\(442\) 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
\(447\) −24.6961 + 8.42235i −1.16808 + 0.398363i
\(448\) 0 0
\(449\) −15.4142 −0.727441 −0.363721 0.931508i \(-0.618494\pi\)
−0.363721 + 0.931508i \(0.618494\pi\)
\(450\) 0 0
\(451\) −0.360301 −0.0169659
\(452\) 0 0
\(453\) 2.59479 13.1425i 0.121914 0.617486i
\(454\) 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
\(459\) 21.1039 31.8775i 0.985045 1.48792i
\(460\) 0 0
\(461\) −20.0406 34.7113i −0.933383 1.61667i −0.777491 0.628893i \(-0.783508\pi\)
−0.155892 0.987774i \(-0.549825\pi\)
\(462\) 0 0
\(463\) −10.9717 + 19.0036i −0.509900 + 0.883172i 0.490035 + 0.871703i \(0.336984\pi\)
−0.999934 + 0.0114690i \(0.996349\pi\)
\(464\) 0 0
\(465\) 0.361072 1.82881i 0.0167443 0.0848089i
\(466\) 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
\(471\) −14.2889 + 4.87307i −0.658396 + 0.224539i
\(472\) 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
\(477\) 2.57150 + 19.1126i 0.117741 + 0.875107i
\(478\) 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
\(483\) −8.71807 7.62366i −0.396686 0.346888i
\(484\) 0 0
\(485\) −3.62852 −0.164762
\(486\) 0 0
\(487\) −38.5519 −1.74695 −0.873477 0.486865i \(-0.838140\pi\)
−0.873477 + 0.486865i \(0.838140\pi\)
\(488\) 0 0
\(489\) −17.5384 15.3367i −0.793115 0.693552i
\(490\) 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
\(495\) −0.174920 1.30009i −0.00786206 0.0584346i
\(496\) 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 \(0.388278\pi\)
−0.985139 + 0.171758i \(0.945055\pi\)
\(500\) 0 0
\(501\) −19.0755 + 6.50550i −0.852230 + 0.290644i
\(502\) 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
\(507\) 1.18140 5.98370i 0.0524676 0.265746i
\(508\) 0 0
\(509\) −13.2296 + 22.9143i −0.586391 + 1.01566i 0.408309 + 0.912844i \(0.366118\pi\)
−0.994700 + 0.102815i \(0.967215\pi\)
\(510\) 0 0
\(511\) −6.74703 11.6862i −0.298471 0.516967i
\(512\) 0 0
\(513\) −8.42361 + 12.7239i −0.371911 + 0.561774i
\(514\) 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
\(519\) −0.594379 + 3.01049i −0.0260903 + 0.132146i
\(520\) 0 0
\(521\) −7.72559 −0.338464 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(522\) 0 0
\(523\) −16.9546 −0.741375 −0.370687 0.928758i \(-0.620878\pi\)
−0.370687 + 0.928758i \(0.620878\pi\)
\(524\) 0 0
\(525\) 8.01923 2.73488i 0.349988 0.119360i
\(526\) 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
\(531\) −2.06546 0.848674i −0.0896335 0.0368293i
\(532\) 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
\(537\) 8.77669 + 7.67492i 0.378742 + 0.331197i
\(538\) 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
\(543\) 17.3035 + 15.1313i 0.742563 + 0.649346i
\(544\) 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 \(-0.798744\pi\)
−0.108453 0.994102i \(-0.534590\pi\)
\(548\) 0 0
\(549\) 21.0030 16.2112i 0.896387 0.691879i
\(550\) 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
\(555\) 0.177461 0.0605211i 0.00753278 0.00256898i
\(556\) 0 0
\(557\) 36.5498 1.54866 0.774332 0.632780i \(-0.218086\pi\)
0.774332 + 0.632780i \(0.218086\pi\)
\(558\) 0 0
\(559\) −33.7587 −1.42784
\(560\) 0 0
\(561\) 3.28048 16.6155i 0.138502 0.701505i
\(562\) 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
\(567\) −6.40003 6.32769i −0.268776 0.265738i
\(568\) 0 0
\(569\) −15.4173 26.7035i −0.646325 1.11947i −0.983994 0.178203i \(-0.942972\pi\)
0.337669 0.941265i \(-0.390362\pi\)
\(570\) 0 0
\(571\) 19.4009 33.6033i 0.811901 1.40625i −0.0996310 0.995024i \(-0.531766\pi\)
0.911532 0.411229i \(-0.134900\pi\)
\(572\) 0 0
\(573\) −6.98380 + 35.3725i −0.291752 + 1.47771i
\(574\) 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
\(579\) −33.6344 + 11.4707i −1.39780 + 0.476705i
\(580\) 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
\(585\) 2.40562 1.85678i 0.0994602 0.0767686i
\(586\) 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
\(591\) −9.39788 8.21812i −0.386577 0.338048i
\(592\) 0 0
\(593\) −10.0323 −0.411977 −0.205989 0.978554i \(-0.566041\pi\)
−0.205989 + 0.978554i \(0.566041\pi\)
\(594\) 0 0
\(595\) −2.42070 −0.0992392
\(596\) 0 0
\(597\) 18.6810 + 16.3359i 0.764562 + 0.668583i
\(598\) 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
\(603\) −23.7886 9.77444i −0.968746 0.398046i
\(604\) 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 \(0.445439\pi\)
−0.938620 + 0.344953i \(0.887895\pi\)
\(608\) 0 0
\(609\) 12.7297 4.34134i 0.515835 0.175920i
\(610\) 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
\(615\) −0.0299250 + 0.151568i −0.00120669 + 0.00611183i
\(616\) 0 0
\(617\) 8.08770 14.0083i 0.325599 0.563954i −0.656035 0.754731i \(-0.727767\pi\)
0.981633 + 0.190777i \(0.0611008\pi\)
\(618\) 0 0
\(619\) 12.9377 + 22.4088i 0.520012 + 0.900687i 0.999729 + 0.0232638i \(0.00740578\pi\)
−0.479718 + 0.877423i \(0.659261\pi\)
\(620\) 0 0
\(621\) −15.4994 31.0949i −0.621971 1.24779i
\(622\) 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
\(627\) −1.30940 + 6.63205i −0.0522925 + 0.264859i
\(628\) 0 0
\(629\) 2.42070 0.0965198
\(630\) 0 0
\(631\) 29.2969 1.16629 0.583146 0.812368i \(-0.301822\pi\)
0.583146 + 0.812368i \(0.301822\pi\)
\(632\) 0 0
\(633\) −12.0949 + 4.12485i −0.480731 + 0.163948i
\(634\) 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
\(639\) 0.643125 + 4.78001i 0.0254416 + 0.189094i
\(640\) 0 0
\(641\) 8.82630 + 15.2876i 0.348618 + 0.603824i 0.986004 0.166720i \(-0.0533177\pi\)
−0.637386 + 0.770545i \(0.719984\pi\)
\(642\) 0 0
\(643\) 14.5426 25.1885i 0.573504 0.993338i −0.422698 0.906270i \(-0.638917\pi\)
0.996202 0.0870676i \(-0.0277496\pi\)
\(644\) 0 0
\(645\) 4.70388 + 4.11338i 0.185215 + 0.161964i
\(646\) 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
\(651\) −4.26502 3.72961i −0.167159 0.146175i
\(652\) 0 0
\(653\) −7.08986 + 12.2800i −0.277448 + 0.480553i −0.970750 0.240094i \(-0.922822\pi\)
0.693302 + 0.720647i \(0.256155\pi\)
\(654\) 0 0
\(655\) −1.09654 1.89927i −0.0428455 0.0742106i
\(656\) 0 0
\(657\) −5.39803 40.1207i −0.210597 1.56526i
\(658\) 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
\(663\) 37.1335 12.6640i 1.44215 0.491829i
\(664\) 0 0
\(665\) 0.966223 0.0374685
\(666\) 0 0
\(667\) 51.9212 2.01040
\(668\) 0 0
\(669\) 8.96091 45.3865i 0.346449 1.75474i
\(670\) 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
\(675\) 25.3707 + 1.55430i 0.976519 + 0.0598249i
\(676\) 0 0
\(677\) 14.2246 + 24.6376i 0.546694 + 0.946902i 0.998498 + 0.0547845i \(0.0174472\pi\)
−0.451804 + 0.892117i \(0.649219\pi\)
\(678\) 0 0
\(679\) −5.51420 + 9.55087i −0.211616 + 0.366529i
\(680\) 0 0
\(681\) 5.67912 28.7644i 0.217624 1.10225i
\(682\) 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
\(687\) −26.9697 + 9.19775i −1.02896 + 0.350916i
\(688\) 0 0
\(689\) −9.89549 + 17.1395i −0.376988 + 0.652962i
\(690\) 0 0
\(691\) 8.71932 + 15.1023i 0.331699 + 0.574519i 0.982845 0.184433i \(-0.0590450\pi\)
−0.651146 + 0.758952i \(0.725712\pi\)
\(692\) 0 0
\(693\) −3.68787 1.51530i −0.140091 0.0575616i
\(694\) 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
\(699\) −5.35692 4.68444i −0.202617 0.177182i
\(700\) 0 0
\(701\) 9.23113 0.348655 0.174327 0.984688i \(-0.444225\pi\)
0.174327 + 0.984688i \(0.444225\pi\)
\(702\) 0 0
\(703\) −0.966223 −0.0364418
\(704\) 0 0
\(705\) −0.489914 0.428413i −0.0184512 0.0161350i
\(706\) 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
\(711\) −2.98722 + 2.30570i −0.112030 + 0.0864703i
\(712\) 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
\(717\) 34.1578 11.6492i 1.27565 0.435046i
\(718\) 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
\(723\) −7.61412 + 38.5651i −0.283172 + 1.43425i
\(724\) 0 0
\(725\) −18.9926 + 32.8962i −0.705368 + 1.22173i
\(726\) 0 0
\(727\) −1.11417 1.92980i −0.0413222 0.0715722i 0.844625 0.535359i \(-0.179824\pi\)
−0.885947 + 0.463787i \(0.846490\pi\)
\(728\) 0 0
\(729\) −10.5447 24.8557i −0.390546 0.920583i
\(730\) 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
\(735\) −0.110382 + 0.559079i −0.00407151 + 0.0206220i
\(736\) 0 0
\(737\) −11.3934 −0.419681
\(738\) 0 0
\(739\) −48.6048 −1.78796 −0.893978 0.448112i \(-0.852097\pi\)
−0.893978 + 0.448112i \(0.852097\pi\)
\(740\) 0 0
\(741\) −14.8218 + 5.05483i −0.544494 + 0.185694i
\(742\) 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
\(747\) −0.150309 + 0.116017i −0.00549953 + 0.00424482i
\(748\) 0 0
\(749\) −7.77868 13.4731i −0.284227 0.492295i
\(750\) 0 0
\(751\) −4.62893 + 8.01754i −0.168912 + 0.292564i −0.938038 0.346534i \(-0.887359\pi\)
0.769126 + 0.639098i \(0.220692\pi\)
\(752\) 0 0
\(753\) −5.21577 4.56102i −0.190073 0.166213i
\(754\) 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
\(759\) −11.5865 10.1320i −0.420562 0.367767i
\(760\) 0 0
\(761\) 19.1859 33.2309i 0.695487 1.20462i −0.274529 0.961579i \(-0.588522\pi\)
0.970016 0.243040i \(-0.0781448\pi\)
\(762\) 0 0
\(763\) −7.17871 12.4339i −0.259887 0.450137i
\(764\) 0 0
\(765\) −6.71719 2.76001i −0.242861 0.0997885i
\(766\) 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
\(771\) 0.835252 0.284854i 0.0300809 0.0102588i
\(772\) 0 0
\(773\) 40.2941 1.44928 0.724639 0.689128i \(-0.242006\pi\)
0.724639 + 0.689128i \(0.242006\pi\)
\(774\) 0 0
\(775\) 16.0014 0.574788
\(776\) 0 0
\(777\) 0.110382 0.559079i 0.00395994 0.0200569i
\(778\) 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
\(783\) 40.2735 + 2.46729i 1.43926 + 0.0881738i
\(784\) 0 0
\(785\) 1.43389 + 2.48357i 0.0511777 + 0.0886425i
\(786\) 0 0
\(787\) −12.2565 + 21.2289i −0.436897 + 0.756728i −0.997448 0.0713920i \(-0.977256\pi\)
0.560551 + 0.828120i \(0.310589\pi\)
\(788\) 0 0
\(789\) 1.04904 5.31335i 0.0373470 0.189160i
\(790\) 0 0
\(791\) 3.89932 0.138644
\(792\) 0 0
\(793\) 27.2280 0.966896
\(794\) 0 0
\(795\) 3.46721 1.18246i 0.122969 0.0419374i
\(796\) 0 0
\(797\) 26.9140 46.6164i 0.953343 1.65124i 0.215228 0.976564i \(-0.430951\pi\)
0.738115 0.674675i \(-0.235716\pi\)
\(798\) 0 0
\(799\) −4.20119 7.27667i −0.148627 0.257430i
\(800\) 0 0
\(801\) 4.52317 + 33.6183i 0.159818 + 1.18784i
\(802\) 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
\(807\) 5.43058 + 4.74886i 0.191165 + 0.167168i
\(808\) 0 0
\(809\) 7.44011 0.261580 0.130790 0.991410i \(-0.458249\pi\)
0.130790 + 0.991410i \(0.458249\pi\)
\(810\) 0 0
\(811\) 2.94854 0.103537 0.0517687 0.998659i \(-0.483514\pi\)
0.0517687 + 0.998659i \(0.483514\pi\)
\(812\) 0 0
\(813\) 16.9800 + 14.8484i 0.595515 + 0.520758i
\(814\) 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
\(819\) −1.23159 9.15373i −0.0430351 0.319857i
\(820\) 0 0
\(821\) 8.44331 + 14.6242i 0.294674 + 0.510390i 0.974909 0.222604i \(-0.0714557\pi\)
−0.680235 + 0.732994i \(0.738122\pi\)
\(822\) 0 0
\(823\) −24.3829 + 42.2325i −0.849935 + 1.47213i 0.0313301 + 0.999509i \(0.490026\pi\)
−0.881265 + 0.472622i \(0.843308\pi\)
\(824\) 0 0
\(825\) 10.6577 3.63469i 0.371053 0.126544i
\(826\) 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
\(831\) 0.233307 1.18169i 0.00809334 0.0409923i
\(832\) 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
\(837\) −7.58256 15.2121i −0.262092 0.525807i
\(838\) 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
\(843\) −10.8356 + 54.8818i −0.373199 + 1.89023i
\(844\) 0 0
\(845\) −1.15859 −0.0398567
\(846\) 0 0
\(847\) 9.23372 0.317274
\(848\) 0 0
\(849\) −23.1600 + 7.89848i −0.794850 + 0.271075i
\(850\) 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
\(855\) 2.68116 + 1.10166i 0.0916939 + 0.0376759i
\(856\) 0 0
\(857\) −18.7290 32.4396i −0.639771 1.10812i −0.985483 0.169776i \(-0.945696\pi\)
0.345711 0.938341i \(-0.387638\pi\)
\(858\) 0 0
\(859\) −13.2123 + 22.8844i −0.450799 + 0.780807i −0.998436 0.0559093i \(-0.982194\pi\)
0.547637 + 0.836716i \(0.315528\pi\)
\(860\) 0 0
\(861\) 0.353477 + 0.309104i 0.0120465 + 0.0105342i
\(862\) 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
\(867\) −48.4138 42.3362i −1.64422 1.43781i
\(868\) 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
\(873\) −26.1909 + 20.2155i −0.886428 + 0.684191i
\(874\) 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
\(879\) 27.3382 9.32341i 0.922095 0.314471i
\(880\) 0 0
\(881\) −16.7109 −0.563005 −0.281503 0.959560i \(-0.590833\pi\)
−0.281503 + 0.959560i \(0.590833\pi\)
\(882\) 0 0
\(883\) 11.8347 0.398268 0.199134 0.979972i \(-0.436187\pi\)
0.199134 + 0.979972i \(0.436187\pi\)
\(884\) 0 0
\(885\) −0.0821620 + 0.416146i −0.00276184 + 0.0139886i
\(886\) 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
\(891\) −8.50574 8.40960i −0.284953 0.281732i
\(892\) 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
\(897\) 6.90636 34.9803i 0.230597 1.16796i
\(898\) 0 0
\(899\) 25.4007 0.847159
\(900\) 0 0
\(901\) 47.2955 1.57564
\(902\) 0 0
\(903\) 17.9755 6.13037i 0.598188 0.204006i
\(904\) 0 0
\(905\) 2.18320 3.78142i 0.0725721 0.125698i
\(906\) 0 0
\(907\) 8.32308 + 14.4160i 0.276363 + 0.478675i 0.970478 0.241189i \(-0.0775374\pi\)
−0.694115 + 0.719864i \(0.744204\pi\)
\(908\) 0 0
\(909\) 33.8218 26.1054i 1.12180 0.865862i
\(910\) 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
\(915\) −3.79391 3.31764i −0.125423 0.109678i
\(916\) 0 0
\(917\) −6.66560 −0.220118
\(918\) 0 0
\(919\) −10.2384 −0.337734 −0.168867 0.985639i \(-0.554011\pi\)
−0.168867 + 0.985639i \(0.554011\pi\)
\(920\) 0 0
\(921\) −35.2488 30.8238i −1.16149 1.01568i
\(922\) 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
\(927\) −29.3743 12.0695i −0.964778 0.396416i
\(928\) 0 0
\(929\) 17.5842 + 30.4567i 0.576917 + 0.999250i 0.995830 + 0.0912240i \(0.0290779\pi\)
−0.418913 + 0.908026i \(0.637589\pi\)
\(930\) 0 0
\(931\) 1.46835 2.54326i 0.0481234 0.0833521i
\(932\) 0 0
\(933\) −38.4840 + 13.1246i −1.25991 + 0.429679i
\(934\) 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
\(939\) 0.244639 1.23908i 0.00798349 0.0404359i
\(940\) 0 0
\(941\) −6.49641 + 11.2521i −0.211777 + 0.366808i −0.952271 0.305255i \(-0.901258\pi\)
0.740494 + 0.672063i \(0.234592\pi\)
\(942\) 0 0
\(943\) 0.906357 + 1.56986i 0.0295150 + 0.0511216i
\(944\) 0 0
\(945\) −0.943743 + 1.42553i −0.0307000 + 0.0463725i
\(946\) 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
\(951\) −11.2322 + 56.8903i −0.364228 + 1.84479i
\(952\) 0 0
\(953\) −40.0040 −1.29586 −0.647928 0.761702i \(-0.724364\pi\)
−0.647928 + 0.761702i \(0.724364\pi\)
\(954\) 0 0
\(955\) 6.84898 0.221628
\(956\) 0 0
\(957\) 16.9180 5.76972i 0.546882 0.186508i
\(958\) 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
\(963\) −6.22341 46.2553i −0.200546 1.49056i
\(964\) 0 0
\(965\) 3.37522 + 5.84605i 0.108652 + 0.188191i
\(966\) 0 0
\(967\) −3.03814 + 5.26222i −0.0977001 + 0.169222i −0.910732 0.412997i \(-0.864482\pi\)
0.813032 + 0.582219i \(0.197815\pi\)
\(968\) 0 0
\(969\) 28.1716 + 24.6351i 0.905004 + 0.791395i
\(970\) 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
\(975\) 19.6365 + 17.1714i 0.628870 + 0.549925i
\(976\) 0 0
\(977\) 5.91064 10.2375i 0.189098 0.327528i −0.755852 0.654743i \(-0.772777\pi\)
0.944950 + 0.327215i \(0.106110\pi\)
\(978\) 0 0
\(979\) 7.51364 + 13.0140i 0.240137 + 0.415929i
\(980\) 0 0
\(981\) −5.74339 42.6876i −0.183372 1.36291i
\(982\) 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
\(987\) −1.87217 + 0.638484i −0.0595918 + 0.0203232i
\(988\) 0 0
\(989\) 73.3174 2.33136
\(990\) 0 0
\(991\) 12.5408 0.398371 0.199186 0.979962i \(-0.436170\pi\)
0.199186 + 0.979962i \(0.436170\pi\)
\(992\) 0 0
\(993\) 3.30158 16.7223i 0.104773 0.530667i
\(994\) 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
\(999\) 0.943743 1.42553i 0.0298587 0.0451018i
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 504.2.r.e.337.1 yes 8
3.2 odd 2 1512.2.r.e.1009.2 8
4.3 odd 2 1008.2.r.l.337.4 8
9.2 odd 6 1512.2.r.e.505.2 8
9.4 even 3 4536.2.a.z.1.2 4
9.5 odd 6 4536.2.a.y.1.3 4
9.7 even 3 inner 504.2.r.e.169.1 8
12.11 even 2 3024.2.r.m.1009.2 8
36.7 odd 6 1008.2.r.l.673.4 8
36.11 even 6 3024.2.r.m.2017.2 8
36.23 even 6 9072.2.a.cg.1.3 4
36.31 odd 6 9072.2.a.cj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.e.169.1 8 9.7 even 3 inner
504.2.r.e.337.1 yes 8 1.1 even 1 trivial
1008.2.r.l.337.4 8 4.3 odd 2
1008.2.r.l.673.4 8 36.7 odd 6
1512.2.r.e.505.2 8 9.2 odd 6
1512.2.r.e.1009.2 8 3.2 odd 2
3024.2.r.m.1009.2 8 12.11 even 2
3024.2.r.m.2017.2 8 36.11 even 6
4536.2.a.y.1.3 4 9.5 odd 6
4536.2.a.z.1.2 4 9.4 even 3
9072.2.a.cg.1.3 4 36.23 even 6
9072.2.a.cj.1.2 4 36.31 odd 6