# Properties

 Label 6552.2.a.bb.1.2 Level $6552$ Weight $2$ Character 6552.1 Self dual yes Analytic conductor $52.318$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6552,2,Mod(1,6552)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6552.a (trivial)

## 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: yes Analytic conductor: $$52.3179834043$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-2.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 6552.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.37228 q^{5} -1.00000 q^{7} +O(q^{10})$$ $$q+1.37228 q^{5} -1.00000 q^{7} -5.37228 q^{11} -1.00000 q^{13} +5.37228 q^{17} +7.37228 q^{19} -1.37228 q^{23} -3.11684 q^{25} -4.62772 q^{29} -10.7446 q^{31} -1.37228 q^{35} +5.37228 q^{37} +6.00000 q^{41} +7.37228 q^{43} -9.48913 q^{47} +1.00000 q^{49} +2.74456 q^{53} -7.37228 q^{55} +2.74456 q^{59} -12.1168 q^{61} -1.37228 q^{65} +12.0000 q^{67} -10.0000 q^{71} +9.37228 q^{73} +5.37228 q^{77} -14.7446 q^{79} +1.25544 q^{83} +7.37228 q^{85} -15.4891 q^{89} +1.00000 q^{91} +10.1168 q^{95} -15.4891 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - 2 * q^7 $$2 q - 3 q^{5} - 2 q^{7} - 5 q^{11} - 2 q^{13} + 5 q^{17} + 9 q^{19} + 3 q^{23} + 11 q^{25} - 15 q^{29} - 10 q^{31} + 3 q^{35} + 5 q^{37} + 12 q^{41} + 9 q^{43} + 4 q^{47} + 2 q^{49} - 6 q^{53} - 9 q^{55} - 6 q^{59} - 7 q^{61} + 3 q^{65} + 24 q^{67} - 20 q^{71} + 13 q^{73} + 5 q^{77} - 18 q^{79} + 14 q^{83} + 9 q^{85} - 8 q^{89} + 2 q^{91} + 3 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - 3 * q^5 - 2 * q^7 - 5 * q^11 - 2 * q^13 + 5 * q^17 + 9 * q^19 + 3 * q^23 + 11 * q^25 - 15 * q^29 - 10 * q^31 + 3 * q^35 + 5 * q^37 + 12 * q^41 + 9 * q^43 + 4 * q^47 + 2 * q^49 - 6 * q^53 - 9 * q^55 - 6 * q^59 - 7 * q^61 + 3 * q^65 + 24 * q^67 - 20 * q^71 + 13 * q^73 + 5 * q^77 - 18 * q^79 + 14 * q^83 + 9 * q^85 - 8 * q^89 + 2 * q^91 + 3 * q^95 - 8 * q^97

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.37228 0.613703 0.306851 0.951757i $$-0.400725\pi$$
0.306851 + 0.951757i $$0.400725\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −5.37228 −1.61980 −0.809902 0.586565i $$-0.800480\pi$$
−0.809902 + 0.586565i $$0.800480\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.37228 1.30297 0.651485 0.758662i $$-0.274146\pi$$
0.651485 + 0.758662i $$0.274146\pi$$
$$18$$ 0 0
$$19$$ 7.37228 1.69132 0.845659 0.533724i $$-0.179208\pi$$
0.845659 + 0.533724i $$0.179208\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.37228 −0.286140 −0.143070 0.989713i $$-0.545697\pi$$
−0.143070 + 0.989713i $$0.545697\pi$$
$$24$$ 0 0
$$25$$ −3.11684 −0.623369
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −4.62772 −0.859346 −0.429673 0.902985i $$-0.641371\pi$$
−0.429673 + 0.902985i $$0.641371\pi$$
$$30$$ 0 0
$$31$$ −10.7446 −1.92978 −0.964890 0.262654i $$-0.915402\pi$$
−0.964890 + 0.262654i $$0.915402\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.37228 −0.231958
$$36$$ 0 0
$$37$$ 5.37228 0.883198 0.441599 0.897213i $$-0.354411\pi$$
0.441599 + 0.897213i $$0.354411\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 7.37228 1.12426 0.562131 0.827048i $$-0.309982\pi$$
0.562131 + 0.827048i $$0.309982\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.48913 −1.38413 −0.692066 0.721835i $$-0.743299\pi$$
−0.692066 + 0.721835i $$0.743299\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 2.74456 0.376995 0.188497 0.982074i $$-0.439638\pi$$
0.188497 + 0.982074i $$0.439638\pi$$
$$54$$ 0 0
$$55$$ −7.37228 −0.994078
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 2.74456 0.357312 0.178656 0.983912i $$-0.442825\pi$$
0.178656 + 0.983912i $$0.442825\pi$$
$$60$$ 0 0
$$61$$ −12.1168 −1.55140 −0.775701 0.631100i $$-0.782604\pi$$
−0.775701 + 0.631100i $$0.782604\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1.37228 −0.170211
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −10.0000 −1.18678 −0.593391 0.804914i $$-0.702211\pi$$
−0.593391 + 0.804914i $$0.702211\pi$$
$$72$$ 0 0
$$73$$ 9.37228 1.09694 0.548471 0.836169i $$-0.315210\pi$$
0.548471 + 0.836169i $$0.315210\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 5.37228 0.612228
$$78$$ 0 0
$$79$$ −14.7446 −1.65889 −0.829446 0.558586i $$-0.811344\pi$$
−0.829446 + 0.558586i $$0.811344\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.25544 0.137802 0.0689011 0.997623i $$-0.478051\pi$$
0.0689011 + 0.997623i $$0.478051\pi$$
$$84$$ 0 0
$$85$$ 7.37228 0.799636
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −15.4891 −1.64184 −0.820922 0.571040i $$-0.806540\pi$$
−0.820922 + 0.571040i $$0.806540\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 10.1168 1.03797
$$96$$ 0 0
$$97$$ −15.4891 −1.57268 −0.786341 0.617792i $$-0.788027\pi$$
−0.786341 + 0.617792i $$0.788027\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 0 0
$$103$$ −6.11684 −0.602711 −0.301355 0.953512i $$-0.597439\pi$$
−0.301355 + 0.953512i $$0.597439\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −10.0000 −0.966736 −0.483368 0.875417i $$-0.660587\pi$$
−0.483368 + 0.875417i $$0.660587\pi$$
$$108$$ 0 0
$$109$$ 4.11684 0.394322 0.197161 0.980371i $$-0.436828\pi$$
0.197161 + 0.980371i $$0.436828\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 13.4891 1.26895 0.634475 0.772943i $$-0.281216\pi$$
0.634475 + 0.772943i $$0.281216\pi$$
$$114$$ 0 0
$$115$$ −1.88316 −0.175605
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −5.37228 −0.492476
$$120$$ 0 0
$$121$$ 17.8614 1.62376
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −11.1386 −0.996266
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −12.6277 −1.10329 −0.551644 0.834079i $$-0.685999\pi$$
−0.551644 + 0.834079i $$0.685999\pi$$
$$132$$ 0 0
$$133$$ −7.37228 −0.639258
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 12.8614 1.09882 0.549412 0.835552i $$-0.314852\pi$$
0.549412 + 0.835552i $$0.314852\pi$$
$$138$$ 0 0
$$139$$ −20.2337 −1.71620 −0.858100 0.513483i $$-0.828355\pi$$
−0.858100 + 0.513483i $$0.828355\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 5.37228 0.449253
$$144$$ 0 0
$$145$$ −6.35053 −0.527383
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.4891 −1.43276 −0.716382 0.697708i $$-0.754203\pi$$
−0.716382 + 0.697708i $$0.754203\pi$$
$$150$$ 0 0
$$151$$ −12.6277 −1.02763 −0.513815 0.857901i $$-0.671768\pi$$
−0.513815 + 0.857901i $$0.671768\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −14.7446 −1.18431
$$156$$ 0 0
$$157$$ 17.3723 1.38646 0.693229 0.720717i $$-0.256187\pi$$
0.693229 + 0.720717i $$0.256187\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1.37228 0.108151
$$162$$ 0 0
$$163$$ −2.74456 −0.214971 −0.107485 0.994207i $$-0.534280\pi$$
−0.107485 + 0.994207i $$0.534280\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1.88316 0.145723 0.0728615 0.997342i $$-0.476787\pi$$
0.0728615 + 0.997342i $$0.476787\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 0 0
$$175$$ 3.11684 0.235611
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −26.2337 −1.96080 −0.980399 0.197023i $$-0.936873\pi$$
−0.980399 + 0.197023i $$0.936873\pi$$
$$180$$ 0 0
$$181$$ −0.510875 −0.0379730 −0.0189865 0.999820i $$-0.506044\pi$$
−0.0189865 + 0.999820i $$0.506044\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 7.37228 0.542021
$$186$$ 0 0
$$187$$ −28.8614 −2.11056
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 21.6060 1.56335 0.781677 0.623684i $$-0.214365\pi$$
0.781677 + 0.623684i $$0.214365\pi$$
$$192$$ 0 0
$$193$$ 3.48913 0.251153 0.125576 0.992084i $$-0.459922\pi$$
0.125576 + 0.992084i $$0.459922\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −18.7446 −1.33549 −0.667747 0.744388i $$-0.732741\pi$$
−0.667747 + 0.744388i $$0.732741\pi$$
$$198$$ 0 0
$$199$$ −11.3723 −0.806160 −0.403080 0.915165i $$-0.632060\pi$$
−0.403080 + 0.915165i $$0.632060\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4.62772 0.324802
$$204$$ 0 0
$$205$$ 8.23369 0.575066
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −39.6060 −2.73960
$$210$$ 0 0
$$211$$ 6.11684 0.421101 0.210550 0.977583i $$-0.432474\pi$$
0.210550 + 0.977583i $$0.432474\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 10.1168 0.689963
$$216$$ 0 0
$$217$$ 10.7446 0.729388
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −5.37228 −0.361379
$$222$$ 0 0
$$223$$ 5.48913 0.367579 0.183790 0.982966i $$-0.441164\pi$$
0.183790 + 0.982966i $$0.441164\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −26.7446 −1.77510 −0.887549 0.460712i $$-0.847594\pi$$
−0.887549 + 0.460712i $$0.847594\pi$$
$$228$$ 0 0
$$229$$ −3.48913 −0.230568 −0.115284 0.993333i $$-0.536778\pi$$
−0.115284 + 0.993333i $$0.536778\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −12.0000 −0.786146 −0.393073 0.919507i $$-0.628588\pi$$
−0.393073 + 0.919507i $$0.628588\pi$$
$$234$$ 0 0
$$235$$ −13.0217 −0.849445
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 3.25544 0.210577 0.105288 0.994442i $$-0.466423\pi$$
0.105288 + 0.994442i $$0.466423\pi$$
$$240$$ 0 0
$$241$$ 19.4891 1.25540 0.627702 0.778453i $$-0.283995\pi$$
0.627702 + 0.778453i $$0.283995\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 1.37228 0.0876718
$$246$$ 0 0
$$247$$ −7.37228 −0.469087
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1.88316 0.118864 0.0594319 0.998232i $$-0.481071\pi$$
0.0594319 + 0.998232i $$0.481071\pi$$
$$252$$ 0 0
$$253$$ 7.37228 0.463491
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −23.4891 −1.46521 −0.732606 0.680653i $$-0.761696\pi$$
−0.732606 + 0.680653i $$0.761696\pi$$
$$258$$ 0 0
$$259$$ −5.37228 −0.333817
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −20.7446 −1.27916 −0.639582 0.768723i $$-0.720893\pi$$
−0.639582 + 0.768723i $$0.720893\pi$$
$$264$$ 0 0
$$265$$ 3.76631 0.231363
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −7.48913 −0.456620 −0.228310 0.973588i $$-0.573320\pi$$
−0.228310 + 0.973588i $$0.573320\pi$$
$$270$$ 0 0
$$271$$ 8.23369 0.500161 0.250080 0.968225i $$-0.419543\pi$$
0.250080 + 0.968225i $$0.419543\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 16.7446 1.00974
$$276$$ 0 0
$$277$$ 7.25544 0.435937 0.217968 0.975956i $$-0.430057\pi$$
0.217968 + 0.975956i $$0.430057\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −8.23369 −0.491181 −0.245590 0.969374i $$-0.578982\pi$$
−0.245590 + 0.969374i $$0.578982\pi$$
$$282$$ 0 0
$$283$$ 22.9783 1.36592 0.682958 0.730458i $$-0.260693\pi$$
0.682958 + 0.730458i $$0.260693\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ 11.8614 0.697730
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 15.4891 0.904884 0.452442 0.891794i $$-0.350553\pi$$
0.452442 + 0.891794i $$0.350553\pi$$
$$294$$ 0 0
$$295$$ 3.76631 0.219283
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.37228 0.0793611
$$300$$ 0 0
$$301$$ −7.37228 −0.424931
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −16.6277 −0.952100
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −4.00000 −0.226819 −0.113410 0.993548i $$-0.536177\pi$$
−0.113410 + 0.993548i $$0.536177\pi$$
$$312$$ 0 0
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 16.2337 0.911775 0.455887 0.890037i $$-0.349322\pi$$
0.455887 + 0.890037i $$0.349322\pi$$
$$318$$ 0 0
$$319$$ 24.8614 1.39197
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 39.6060 2.20374
$$324$$ 0 0
$$325$$ 3.11684 0.172891
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 9.48913 0.523152
$$330$$ 0 0
$$331$$ 24.2337 1.33200 0.666002 0.745950i $$-0.268004\pi$$
0.666002 + 0.745950i $$0.268004\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 16.4674 0.899709
$$336$$ 0 0
$$337$$ −6.62772 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 57.7228 3.12587
$$342$$ 0 0
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −28.7446 −1.54309 −0.771544 0.636175i $$-0.780515\pi$$
−0.771544 + 0.636175i $$0.780515\pi$$
$$348$$ 0 0
$$349$$ −6.23369 −0.333682 −0.166841 0.985984i $$-0.553357\pi$$
−0.166841 + 0.985984i $$0.553357\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −11.4891 −0.611504 −0.305752 0.952111i $$-0.598908\pi$$
−0.305752 + 0.952111i $$0.598908\pi$$
$$354$$ 0 0
$$355$$ −13.7228 −0.728331
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 27.4891 1.45082 0.725410 0.688317i $$-0.241650\pi$$
0.725410 + 0.688317i $$0.241650\pi$$
$$360$$ 0 0
$$361$$ 35.3505 1.86055
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.8614 0.673197
$$366$$ 0 0
$$367$$ 26.9783 1.40825 0.704127 0.710074i $$-0.251339\pi$$
0.704127 + 0.710074i $$0.251339\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.74456 −0.142491
$$372$$ 0 0
$$373$$ −36.9783 −1.91466 −0.957331 0.288995i $$-0.906679\pi$$
−0.957331 + 0.288995i $$0.906679\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 4.62772 0.238340
$$378$$ 0 0
$$379$$ 24.2337 1.24480 0.622400 0.782699i $$-0.286158\pi$$
0.622400 + 0.782699i $$0.286158\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 8.86141 0.452797 0.226398 0.974035i $$-0.427305\pi$$
0.226398 + 0.974035i $$0.427305\pi$$
$$384$$ 0 0
$$385$$ 7.37228 0.375726
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −14.7446 −0.747579 −0.373790 0.927514i $$-0.621942\pi$$
−0.373790 + 0.927514i $$0.621942\pi$$
$$390$$ 0 0
$$391$$ −7.37228 −0.372832
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −20.2337 −1.01807
$$396$$ 0 0
$$397$$ 26.2337 1.31663 0.658316 0.752742i $$-0.271269\pi$$
0.658316 + 0.752742i $$0.271269\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −24.2337 −1.21017 −0.605086 0.796160i $$-0.706861\pi$$
−0.605086 + 0.796160i $$0.706861\pi$$
$$402$$ 0 0
$$403$$ 10.7446 0.535225
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −28.8614 −1.43061
$$408$$ 0 0
$$409$$ 2.62772 0.129932 0.0649662 0.997887i $$-0.479306\pi$$
0.0649662 + 0.997887i $$0.479306\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −2.74456 −0.135051
$$414$$ 0 0
$$415$$ 1.72281 0.0845696
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −23.3723 −1.14181 −0.570905 0.821016i $$-0.693408\pi$$
−0.570905 + 0.821016i $$0.693408\pi$$
$$420$$ 0 0
$$421$$ −12.9783 −0.632521 −0.316261 0.948672i $$-0.602427\pi$$
−0.316261 + 0.948672i $$0.602427\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −16.7446 −0.812231
$$426$$ 0 0
$$427$$ 12.1168 0.586375
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −31.4891 −1.51678 −0.758389 0.651802i $$-0.774013\pi$$
−0.758389 + 0.651802i $$0.774013\pi$$
$$432$$ 0 0
$$433$$ 20.9783 1.00815 0.504075 0.863660i $$-0.331833\pi$$
0.504075 + 0.863660i $$0.331833\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −10.1168 −0.483954
$$438$$ 0 0
$$439$$ −20.6277 −0.984507 −0.492254 0.870452i $$-0.663827\pi$$
−0.492254 + 0.870452i $$0.663827\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −30.0000 −1.42534 −0.712672 0.701498i $$-0.752515\pi$$
−0.712672 + 0.701498i $$0.752515\pi$$
$$444$$ 0 0
$$445$$ −21.2554 −1.00760
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.11684 −0.0999000 −0.0499500 0.998752i $$-0.515906\pi$$
−0.0499500 + 0.998752i $$0.515906\pi$$
$$450$$ 0 0
$$451$$ −32.2337 −1.51783
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1.37228 0.0643335
$$456$$ 0 0
$$457$$ 12.7446 0.596165 0.298083 0.954540i $$-0.403653\pi$$
0.298083 + 0.954540i $$0.403653\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12.1168 0.564338 0.282169 0.959365i $$-0.408946\pi$$
0.282169 + 0.959365i $$0.408946\pi$$
$$462$$ 0 0
$$463$$ −2.11684 −0.0983781 −0.0491890 0.998789i $$-0.515664\pi$$
−0.0491890 + 0.998789i $$0.515664\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18.3505 0.849161 0.424581 0.905390i $$-0.360422\pi$$
0.424581 + 0.905390i $$0.360422\pi$$
$$468$$ 0 0
$$469$$ −12.0000 −0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −39.6060 −1.82108
$$474$$ 0 0
$$475$$ −22.9783 −1.05431
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 11.6060 0.530290 0.265145 0.964209i $$-0.414580\pi$$
0.265145 + 0.964209i $$0.414580\pi$$
$$480$$ 0 0
$$481$$ −5.37228 −0.244955
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −21.2554 −0.965160
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 19.2554 0.868986 0.434493 0.900675i $$-0.356928\pi$$
0.434493 + 0.900675i $$0.356928\pi$$
$$492$$ 0 0
$$493$$ −24.8614 −1.11970
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 10.0000 0.448561
$$498$$ 0 0
$$499$$ −30.9783 −1.38678 −0.693388 0.720564i $$-0.743883\pi$$
−0.693388 + 0.720564i $$0.743883\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −26.9783 −1.20290 −0.601450 0.798910i $$-0.705410\pi$$
−0.601450 + 0.798910i $$0.705410\pi$$
$$504$$ 0 0
$$505$$ 2.74456 0.122131
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −5.37228 −0.238122 −0.119061 0.992887i $$-0.537988\pi$$
−0.119061 + 0.992887i $$0.537988\pi$$
$$510$$ 0 0
$$511$$ −9.37228 −0.414605
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −8.39403 −0.369885
$$516$$ 0 0
$$517$$ 50.9783 2.24202
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 44.1168 1.93279 0.966397 0.257054i $$-0.0827519\pi$$
0.966397 + 0.257054i $$0.0827519\pi$$
$$522$$ 0 0
$$523$$ −33.7228 −1.47460 −0.737298 0.675568i $$-0.763899\pi$$
−0.737298 + 0.675568i $$0.763899\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −57.7228 −2.51445
$$528$$ 0 0
$$529$$ −21.1168 −0.918124
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −6.00000 −0.259889
$$534$$ 0 0
$$535$$ −13.7228 −0.593289
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −5.37228 −0.231401
$$540$$ 0 0
$$541$$ −5.37228 −0.230972 −0.115486 0.993309i $$-0.536843\pi$$
−0.115486 + 0.993309i $$0.536843\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 5.64947 0.241997
$$546$$ 0 0
$$547$$ −30.9783 −1.32453 −0.662267 0.749268i $$-0.730406\pi$$
−0.662267 + 0.749268i $$0.730406\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −34.1168 −1.45343
$$552$$ 0 0
$$553$$ 14.7446 0.627003
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −2.74456 −0.116291 −0.0581454 0.998308i $$-0.518519\pi$$
−0.0581454 + 0.998308i $$0.518519\pi$$
$$558$$ 0 0
$$559$$ −7.37228 −0.311814
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −6.11684 −0.257794 −0.128897 0.991658i $$-0.541144\pi$$
−0.128897 + 0.991658i $$0.541144\pi$$
$$564$$ 0 0
$$565$$ 18.5109 0.778758
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −2.74456 −0.115058 −0.0575290 0.998344i $$-0.518322\pi$$
−0.0575290 + 0.998344i $$0.518322\pi$$
$$570$$ 0 0
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 4.27719 0.178371
$$576$$ 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
$$581$$ −1.25544 −0.0520843
$$582$$ 0 0
$$583$$ −14.7446 −0.610657
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 37.7228 1.55699 0.778494 0.627653i $$-0.215984\pi$$
0.778494 + 0.627653i $$0.215984\pi$$
$$588$$ 0 0
$$589$$ −79.2119 −3.26387
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 8.74456 0.359096 0.179548 0.983749i $$-0.442536\pi$$
0.179548 + 0.983749i $$0.442536\pi$$
$$594$$ 0 0
$$595$$ −7.37228 −0.302234
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 27.8832 1.13927 0.569637 0.821896i $$-0.307084\pi$$
0.569637 + 0.821896i $$0.307084\pi$$
$$600$$ 0 0
$$601$$ 39.4891 1.61080 0.805398 0.592735i $$-0.201952\pi$$
0.805398 + 0.592735i $$0.201952\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 24.5109 0.996509
$$606$$ 0 0
$$607$$ −10.1168 −0.410630 −0.205315 0.978696i $$-0.565822\pi$$
−0.205315 + 0.978696i $$0.565822\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 9.48913 0.383889
$$612$$ 0 0
$$613$$ −5.37228 −0.216984 −0.108492 0.994097i $$-0.534602\pi$$
−0.108492 + 0.994097i $$0.534602\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.8614 −0.517781 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$618$$ 0 0
$$619$$ 3.37228 0.135543 0.0677717 0.997701i $$-0.478411\pi$$
0.0677717 + 0.997701i $$0.478411\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 15.4891 0.620559
$$624$$ 0 0
$$625$$ 0.298936 0.0119574
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 28.8614 1.15078
$$630$$ 0 0
$$631$$ −23.6060 −0.939739 −0.469869 0.882736i $$-0.655699\pi$$
−0.469869 + 0.882736i $$0.655699\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 10.9783 0.435659
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −2.74456 −0.108404 −0.0542019 0.998530i $$-0.517261\pi$$
−0.0542019 + 0.998530i $$0.517261\pi$$
$$642$$ 0 0
$$643$$ −9.88316 −0.389754 −0.194877 0.980828i $$-0.562431\pi$$
−0.194877 + 0.980828i $$0.562431\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.4674 1.11917 0.559584 0.828774i $$-0.310961\pi$$
0.559584 + 0.828774i $$0.310961\pi$$
$$648$$ 0 0
$$649$$ −14.7446 −0.578775
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −35.8397 −1.40251 −0.701257 0.712908i $$-0.747377\pi$$
−0.701257 + 0.712908i $$0.747377\pi$$
$$654$$ 0 0
$$655$$ −17.3288 −0.677092
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 39.7228 1.54738 0.773691 0.633564i $$-0.218409\pi$$
0.773691 + 0.633564i $$0.218409\pi$$
$$660$$ 0 0
$$661$$ 10.2337 0.398044 0.199022 0.979995i $$-0.436223\pi$$
0.199022 + 0.979995i $$0.436223\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −10.1168 −0.392314
$$666$$ 0 0
$$667$$ 6.35053 0.245894
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 65.0951 2.51297
$$672$$ 0 0
$$673$$ 27.8832 1.07482 0.537408 0.843322i $$-0.319403\pi$$
0.537408 + 0.843322i $$0.319403\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ 15.4891 0.594418
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −37.6060 −1.43895 −0.719476 0.694517i $$-0.755618\pi$$
−0.719476 + 0.694517i $$0.755618\pi$$
$$684$$ 0 0
$$685$$ 17.6495 0.674352
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −2.74456 −0.104560
$$690$$ 0 0
$$691$$ −34.9783 −1.33064 −0.665318 0.746560i $$-0.731704\pi$$
−0.665318 + 0.746560i $$0.731704\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −27.7663 −1.05324
$$696$$ 0 0
$$697$$ 32.2337 1.22094
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −25.2554 −0.953885 −0.476942 0.878935i $$-0.658255\pi$$
−0.476942 + 0.878935i $$0.658255\pi$$
$$702$$ 0 0
$$703$$ 39.6060 1.49377
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −2.00000 −0.0752177
$$708$$ 0 0
$$709$$ 12.5109 0.469856 0.234928 0.972013i $$-0.424515\pi$$
0.234928 + 0.972013i $$0.424515\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 14.7446 0.552188
$$714$$ 0 0
$$715$$ 7.37228 0.275708
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −10.9783 −0.409420 −0.204710 0.978823i $$-0.565625\pi$$
−0.204710 + 0.978823i $$0.565625\pi$$
$$720$$ 0 0
$$721$$ 6.11684 0.227803
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 14.4239 0.535689
$$726$$ 0 0
$$727$$ 2.35053 0.0871764 0.0435882 0.999050i $$-0.486121\pi$$
0.0435882 + 0.999050i $$0.486121\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 39.6060 1.46488
$$732$$ 0 0
$$733$$ 35.2554 1.30219 0.651095 0.758997i $$-0.274310\pi$$
0.651095 + 0.758997i $$0.274310\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −64.4674 −2.37469
$$738$$ 0 0
$$739$$ −5.25544 −0.193324 −0.0966622 0.995317i $$-0.530817\pi$$
−0.0966622 + 0.995317i $$0.530817\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 15.2554 0.559668 0.279834 0.960048i $$-0.409721\pi$$
0.279834 + 0.960048i $$0.409721\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 10.0000 0.365392
$$750$$ 0 0
$$751$$ −34.9783 −1.27637 −0.638187 0.769881i $$-0.720315\pi$$
−0.638187 + 0.769881i $$0.720315\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −17.3288 −0.630659
$$756$$ 0 0
$$757$$ 11.7228 0.426073 0.213036 0.977044i $$-0.431665\pi$$
0.213036 + 0.977044i $$0.431665\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −11.7228 −0.424952 −0.212476 0.977166i $$-0.568153\pi$$
−0.212476 + 0.977166i $$0.568153\pi$$
$$762$$ 0 0
$$763$$ −4.11684 −0.149040
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −2.74456 −0.0991004
$$768$$ 0 0
$$769$$ −41.8397 −1.50878 −0.754388 0.656428i $$-0.772066\pi$$
−0.754388 + 0.656428i $$0.772066\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −38.8614 −1.39775 −0.698874 0.715245i $$-0.746315\pi$$
−0.698874 + 0.715245i $$0.746315\pi$$
$$774$$ 0 0
$$775$$ 33.4891 1.20296
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 44.2337 1.58484
$$780$$ 0 0
$$781$$ 53.7228 1.92235
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 23.8397 0.850874
$$786$$ 0 0
$$787$$ −36.6277 −1.30564 −0.652819 0.757514i $$-0.726414\pi$$
−0.652819 + 0.757514i $$0.726414\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −13.4891 −0.479618
$$792$$ 0 0
$$793$$ 12.1168 0.430282
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38.2337 1.35431 0.677153 0.735842i $$-0.263213\pi$$
0.677153 + 0.735842i $$0.263213\pi$$
$$798$$ 0 0
$$799$$ −50.9783 −1.80348
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −50.3505 −1.77683
$$804$$ 0 0
$$805$$ 1.88316 0.0663725
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 47.2119 1.65988 0.829942 0.557850i $$-0.188374\pi$$
0.829942 + 0.557850i $$0.188374\pi$$
$$810$$ 0 0
$$811$$ 38.3505 1.34667 0.673335 0.739338i $$-0.264861\pi$$
0.673335 + 0.739338i $$0.264861\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −3.76631 −0.131928
$$816$$ 0 0
$$817$$ 54.3505 1.90148
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.9783 1.91875 0.959377 0.282127i $$-0.0910399\pi$$
0.959377 + 0.282127i $$0.0910399\pi$$
$$822$$ 0 0
$$823$$ 33.7228 1.17550 0.587752 0.809041i $$-0.300013\pi$$
0.587752 + 0.809041i $$0.300013\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −24.3505 −0.846751 −0.423375 0.905954i $$-0.639155\pi$$
−0.423375 + 0.905954i $$0.639155\pi$$
$$828$$ 0 0
$$829$$ −50.6277 −1.75837 −0.879187 0.476477i $$-0.841913\pi$$
−0.879187 + 0.476477i $$0.841913\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 5.37228 0.186139
$$834$$ 0 0
$$835$$ 2.58422 0.0894306
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 9.48913 0.327601 0.163800 0.986493i $$-0.447625\pi$$
0.163800 + 0.986493i $$0.447625\pi$$
$$840$$ 0 0
$$841$$ −7.58422 −0.261525
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1.37228 0.0472079
$$846$$ 0 0
$$847$$ −17.8614 −0.613725
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −7.37228 −0.252719
$$852$$ 0 0
$$853$$ 39.7228 1.36008 0.680042 0.733174i $$-0.261962\pi$$
0.680042 + 0.733174i $$0.261962\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −12.9783 −0.443329 −0.221664 0.975123i $$-0.571149\pi$$
−0.221664 + 0.975123i $$0.571149\pi$$
$$858$$ 0 0
$$859$$ −8.23369 −0.280930 −0.140465 0.990086i $$-0.544860\pi$$
−0.140465 + 0.990086i $$0.544860\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 41.2119 1.40287 0.701435 0.712733i $$-0.252543\pi$$
0.701435 + 0.712733i $$0.252543\pi$$
$$864$$ 0 0
$$865$$ −30.1902 −1.02650
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 79.2119 2.68708
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 11.1386 0.376553
$$876$$ 0 0
$$877$$ 8.97825 0.303174 0.151587 0.988444i $$-0.451562\pi$$
0.151587 + 0.988444i $$0.451562\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −26.6277 −0.897111 −0.448555 0.893755i $$-0.648061\pi$$
−0.448555 + 0.893755i $$0.648061\pi$$
$$882$$ 0 0
$$883$$ 20.8614 0.702042 0.351021 0.936368i $$-0.385835\pi$$
0.351021 + 0.936368i $$0.385835\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −37.7228 −1.26661 −0.633304 0.773903i $$-0.718302\pi$$
−0.633304 + 0.773903i $$0.718302\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −69.9565 −2.34101
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 49.7228 1.65835
$$900$$ 0 0
$$901$$ 14.7446 0.491213
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −0.701064 −0.0233041
$$906$$ 0 0
$$907$$ 20.0000 0.664089 0.332045 0.943264i $$-0.392262\pi$$
0.332045 + 0.943264i $$0.392262\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −14.6277 −0.484638 −0.242319 0.970197i $$-0.577908\pi$$
−0.242319 + 0.970197i $$0.577908\pi$$
$$912$$ 0 0
$$913$$ −6.74456 −0.223212
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 12.6277 0.417004
$$918$$ 0 0
$$919$$ 52.2337 1.72303 0.861515 0.507732i $$-0.169516\pi$$
0.861515 + 0.507732i $$0.169516\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 10.0000 0.329154
$$924$$ 0 0
$$925$$ −16.7446 −0.550558
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −7.48913 −0.245710 −0.122855 0.992425i $$-0.539205\pi$$
−0.122855 + 0.992425i $$0.539205\pi$$
$$930$$ 0 0
$$931$$ 7.37228 0.241617
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −39.6060 −1.29525
$$936$$ 0 0
$$937$$ −27.4891 −0.898031 −0.449015 0.893524i $$-0.648225\pi$$
−0.449015 + 0.893524i $$0.648225\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −18.0000 −0.586783 −0.293392 0.955992i $$-0.594784\pi$$
−0.293392 + 0.955992i $$0.594784\pi$$
$$942$$ 0 0
$$943$$ −8.23369 −0.268126
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 21.6060 0.702100 0.351050 0.936357i $$-0.385825\pi$$
0.351050 + 0.936357i $$0.385825\pi$$
$$948$$ 0 0
$$949$$ −9.37228 −0.304237
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 45.9565 1.48868 0.744339 0.667802i $$-0.232765\pi$$
0.744339 + 0.667802i $$0.232765\pi$$
$$954$$ 0 0
$$955$$ 29.6495 0.959434
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.8614 −0.415316
$$960$$ 0 0
$$961$$ 84.4456 2.72405
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 4.78806 0.154133
$$966$$ 0 0
$$967$$ −29.8832 −0.960978 −0.480489 0.877001i $$-0.659541\pi$$
−0.480489 + 0.877001i $$0.659541\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 0 0
$$973$$ 20.2337 0.648662
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 14.1168 0.451638 0.225819 0.974169i $$-0.427494\pi$$
0.225819 + 0.974169i $$0.427494\pi$$
$$978$$ 0 0
$$979$$ 83.2119 2.65947
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −27.3723 −0.873040 −0.436520 0.899695i $$-0.643789\pi$$
−0.436520 + 0.899695i $$0.643789\pi$$
$$984$$ 0 0
$$985$$ −25.7228 −0.819597
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −10.1168 −0.321697
$$990$$ 0 0
$$991$$ 52.7011 1.67410 0.837052 0.547123i $$-0.184277\pi$$
0.837052 + 0.547123i $$0.184277\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −15.6060 −0.494742
$$996$$ 0 0
$$997$$ 30.4674 0.964911 0.482456 0.875920i $$-0.339745\pi$$
0.482456 + 0.875920i $$0.339745\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6552.2.a.bb.1.2 2
3.2 odd 2 6552.2.a.bm.1.1 yes 2

By twisted newform
Twist Min Dim Char Parity Ord Type
6552.2.a.bb.1.2 2 1.1 even 1 trivial
6552.2.a.bm.1.1 yes 2 3.2 odd 2