# Properties

 Label 3240.2.a.h.1.1 Level $3240$ Weight $2$ Character 3240.1 Self dual yes Analytic conductor $25.872$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3240 = 2^{3} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3240.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: $$25.8715302549$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 3240.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} -3.27492 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} -3.27492 q^{7} -6.27492 q^{11} -1.27492 q^{13} -2.00000 q^{17} +1.00000 q^{19} +7.27492 q^{23} +1.00000 q^{25} -6.27492 q^{29} +6.27492 q^{31} +3.27492 q^{35} -10.5498 q^{37} -7.54983 q^{41} +4.00000 q^{43} +1.27492 q^{47} +3.72508 q^{49} -0.725083 q^{53} +6.27492 q^{55} +13.0000 q^{59} +8.54983 q^{61} +1.27492 q^{65} +0.549834 q^{67} +8.27492 q^{71} +15.0997 q^{73} +20.5498 q^{77} +10.5498 q^{79} +2.54983 q^{83} +2.00000 q^{85} -12.8248 q^{89} +4.17525 q^{91} -1.00000 q^{95} +16.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + q^7 $$2 q - 2 q^{5} + q^{7} - 5 q^{11} + 5 q^{13} - 4 q^{17} + 2 q^{19} + 7 q^{23} + 2 q^{25} - 5 q^{29} + 5 q^{31} - q^{35} - 6 q^{37} + 8 q^{43} - 5 q^{47} + 15 q^{49} - 9 q^{53} + 5 q^{55} + 26 q^{59} + 2 q^{61} - 5 q^{65} - 14 q^{67} + 9 q^{71} + 26 q^{77} + 6 q^{79} - 10 q^{83} + 4 q^{85} - 3 q^{89} + 31 q^{91} - 2 q^{95} + 32 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 - 5 * q^11 + 5 * q^13 - 4 * q^17 + 2 * q^19 + 7 * q^23 + 2 * q^25 - 5 * q^29 + 5 * q^31 - q^35 - 6 * q^37 + 8 * q^43 - 5 * q^47 + 15 * q^49 - 9 * q^53 + 5 * q^55 + 26 * q^59 + 2 * q^61 - 5 * q^65 - 14 * q^67 + 9 * q^71 + 26 * q^77 + 6 * q^79 - 10 * q^83 + 4 * q^85 - 3 * q^89 + 31 * q^91 - 2 * q^95 + 32 * 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
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −3.27492 −1.23780 −0.618901 0.785469i $$-0.712422\pi$$
−0.618901 + 0.785469i $$0.712422\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −6.27492 −1.89196 −0.945979 0.324227i $$-0.894896\pi$$
−0.945979 + 0.324227i $$0.894896\pi$$
$$12$$ 0 0
$$13$$ −1.27492 −0.353598 −0.176799 0.984247i $$-0.556574\pi$$
−0.176799 + 0.984247i $$0.556574\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.27492 1.51693 0.758463 0.651717i $$-0.225951\pi$$
0.758463 + 0.651717i $$0.225951\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.27492 −1.16522 −0.582611 0.812751i $$-0.697969\pi$$
−0.582611 + 0.812751i $$0.697969\pi$$
$$30$$ 0 0
$$31$$ 6.27492 1.12701 0.563504 0.826113i $$-0.309453\pi$$
0.563504 + 0.826113i $$0.309453\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.27492 0.553562
$$36$$ 0 0
$$37$$ −10.5498 −1.73438 −0.867191 0.497976i $$-0.834077\pi$$
−0.867191 + 0.497976i $$0.834077\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.54983 −1.17909 −0.589543 0.807737i $$-0.700692\pi$$
−0.589543 + 0.807737i $$0.700692\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.27492 0.185966 0.0929829 0.995668i $$-0.470360\pi$$
0.0929829 + 0.995668i $$0.470360\pi$$
$$48$$ 0 0
$$49$$ 3.72508 0.532155
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −0.725083 −0.0995978 −0.0497989 0.998759i $$-0.515858\pi$$
−0.0497989 + 0.998759i $$0.515858\pi$$
$$54$$ 0 0
$$55$$ 6.27492 0.846110
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 13.0000 1.69246 0.846228 0.532821i $$-0.178868\pi$$
0.846228 + 0.532821i $$0.178868\pi$$
$$60$$ 0 0
$$61$$ 8.54983 1.09469 0.547347 0.836906i $$-0.315638\pi$$
0.547347 + 0.836906i $$0.315638\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.27492 0.158134
$$66$$ 0 0
$$67$$ 0.549834 0.0671730 0.0335865 0.999436i $$-0.489307\pi$$
0.0335865 + 0.999436i $$0.489307\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 8.27492 0.982052 0.491026 0.871145i $$-0.336622\pi$$
0.491026 + 0.871145i $$0.336622\pi$$
$$72$$ 0 0
$$73$$ 15.0997 1.76728 0.883641 0.468165i $$-0.155085\pi$$
0.883641 + 0.468165i $$0.155085\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 20.5498 2.34187
$$78$$ 0 0
$$79$$ 10.5498 1.18695 0.593475 0.804853i $$-0.297756\pi$$
0.593475 + 0.804853i $$0.297756\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 2.54983 0.279881 0.139940 0.990160i $$-0.455309\pi$$
0.139940 + 0.990160i $$0.455309\pi$$
$$84$$ 0 0
$$85$$ 2.00000 0.216930
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −12.8248 −1.35942 −0.679710 0.733481i $$-0.737895\pi$$
−0.679710 + 0.733481i $$0.737895\pi$$
$$90$$ 0 0
$$91$$ 4.17525 0.437685
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −9.72508 −0.967682 −0.483841 0.875156i $$-0.660759\pi$$
−0.483841 + 0.875156i $$0.660759\pi$$
$$102$$ 0 0
$$103$$ −5.82475 −0.573930 −0.286965 0.957941i $$-0.592646\pi$$
−0.286965 + 0.957941i $$0.592646\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −15.0997 −1.45974 −0.729870 0.683586i $$-0.760419\pi$$
−0.729870 + 0.683586i $$0.760419\pi$$
$$108$$ 0 0
$$109$$ −6.27492 −0.601028 −0.300514 0.953777i $$-0.597158\pi$$
−0.300514 + 0.953777i $$0.597158\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −7.27492 −0.678390
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.54983 0.600422
$$120$$ 0 0
$$121$$ 28.3746 2.57951
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 11.2749 1.00049 0.500244 0.865885i $$-0.333244\pi$$
0.500244 + 0.865885i $$0.333244\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1.54983 −0.135410 −0.0677048 0.997705i $$-0.521568\pi$$
−0.0677048 + 0.997705i $$0.521568\pi$$
$$132$$ 0 0
$$133$$ −3.27492 −0.283971
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −14.5498 −1.24308 −0.621538 0.783384i $$-0.713492\pi$$
−0.621538 + 0.783384i $$0.713492\pi$$
$$138$$ 0 0
$$139$$ −9.00000 −0.763370 −0.381685 0.924292i $$-0.624656\pi$$
−0.381685 + 0.924292i $$0.624656\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ 6.27492 0.521104
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −2.00000 −0.163846 −0.0819232 0.996639i $$-0.526106\pi$$
−0.0819232 + 0.996639i $$0.526106\pi$$
$$150$$ 0 0
$$151$$ −12.2749 −0.998919 −0.499459 0.866337i $$-0.666468\pi$$
−0.499459 + 0.866337i $$0.666468\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −6.27492 −0.504013
$$156$$ 0 0
$$157$$ 10.7251 0.855955 0.427977 0.903789i $$-0.359226\pi$$
0.427977 + 0.903789i $$0.359226\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −23.8248 −1.87765
$$162$$ 0 0
$$163$$ 20.5498 1.60959 0.804794 0.593555i $$-0.202276\pi$$
0.804794 + 0.593555i $$0.202276\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 7.45017 0.576511 0.288256 0.957554i $$-0.406925\pi$$
0.288256 + 0.957554i $$0.406925\pi$$
$$168$$ 0 0
$$169$$ −11.3746 −0.874968
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −8.72508 −0.663356 −0.331678 0.943393i $$-0.607615\pi$$
−0.331678 + 0.943393i $$0.607615\pi$$
$$174$$ 0 0
$$175$$ −3.27492 −0.247560
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 11.5498 0.863275 0.431638 0.902047i $$-0.357936\pi$$
0.431638 + 0.902047i $$0.357936\pi$$
$$180$$ 0 0
$$181$$ 16.2749 1.20971 0.604853 0.796337i $$-0.293232\pi$$
0.604853 + 0.796337i $$0.293232\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.5498 0.775639
$$186$$ 0 0
$$187$$ 12.5498 0.917735
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −19.3746 −1.40190 −0.700948 0.713212i $$-0.747239\pi$$
−0.700948 + 0.713212i $$0.747239\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 22.3746 1.59412 0.797062 0.603898i $$-0.206387\pi$$
0.797062 + 0.603898i $$0.206387\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 20.5498 1.44232
$$204$$ 0 0
$$205$$ 7.54983 0.527303
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.27492 −0.434045
$$210$$ 0 0
$$211$$ −16.0997 −1.10835 −0.554173 0.832401i $$-0.686966\pi$$
−0.554173 + 0.832401i $$0.686966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ −20.5498 −1.39501
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.54983 0.171520
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 7.45017 0.494485 0.247242 0.968954i $$-0.420476\pi$$
0.247242 + 0.968954i $$0.420476\pi$$
$$228$$ 0 0
$$229$$ 8.54983 0.564989 0.282494 0.959269i $$-0.408838\pi$$
0.282494 + 0.959269i $$0.408838\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 9.09967 0.596139 0.298070 0.954544i $$-0.403657\pi$$
0.298070 + 0.954544i $$0.403657\pi$$
$$234$$ 0 0
$$235$$ −1.27492 −0.0831664
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 13.4502 0.870019 0.435009 0.900426i $$-0.356745\pi$$
0.435009 + 0.900426i $$0.356745\pi$$
$$240$$ 0 0
$$241$$ −24.2749 −1.56368 −0.781842 0.623476i $$-0.785720\pi$$
−0.781842 + 0.623476i $$0.785720\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.72508 −0.237987
$$246$$ 0 0
$$247$$ −1.27492 −0.0811210
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −18.7251 −1.18192 −0.590958 0.806702i $$-0.701250\pi$$
−0.590958 + 0.806702i $$0.701250\pi$$
$$252$$ 0 0
$$253$$ −45.6495 −2.86996
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −3.09967 −0.193352 −0.0966760 0.995316i $$-0.530821\pi$$
−0.0966760 + 0.995316i $$0.530821\pi$$
$$258$$ 0 0
$$259$$ 34.5498 2.14682
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0.725083 0.0447105 0.0223553 0.999750i $$-0.492884\pi$$
0.0223553 + 0.999750i $$0.492884\pi$$
$$264$$ 0 0
$$265$$ 0.725083 0.0445415
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 25.9244 1.58064 0.790320 0.612694i $$-0.209914\pi$$
0.790320 + 0.612694i $$0.209914\pi$$
$$270$$ 0 0
$$271$$ 13.4502 0.817039 0.408520 0.912750i $$-0.366045\pi$$
0.408520 + 0.912750i $$0.366045\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −6.27492 −0.378392
$$276$$ 0 0
$$277$$ 25.2749 1.51862 0.759311 0.650728i $$-0.225536\pi$$
0.759311 + 0.650728i $$0.225536\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6.17525 −0.368384 −0.184192 0.982890i $$-0.558967\pi$$
−0.184192 + 0.982890i $$0.558967\pi$$
$$282$$ 0 0
$$283$$ −11.4502 −0.680642 −0.340321 0.940309i $$-0.610536\pi$$
−0.340321 + 0.940309i $$0.610536\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 24.7251 1.45948
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −10.7251 −0.626566 −0.313283 0.949660i $$-0.601429\pi$$
−0.313283 + 0.949660i $$0.601429\pi$$
$$294$$ 0 0
$$295$$ −13.0000 −0.756889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −9.27492 −0.536382
$$300$$ 0 0
$$301$$ −13.0997 −0.755052
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −8.54983 −0.489562
$$306$$ 0 0
$$307$$ 1.09967 0.0627614 0.0313807 0.999508i $$-0.490010\pi$$
0.0313807 + 0.999508i $$0.490010\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 25.9244 1.47004 0.735020 0.678046i $$-0.237173\pi$$
0.735020 + 0.678046i $$0.237173\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.27492 −0.296269 −0.148134 0.988967i $$-0.547327\pi$$
−0.148134 + 0.988967i $$0.547327\pi$$
$$318$$ 0 0
$$319$$ 39.3746 2.20455
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −2.00000 −0.111283
$$324$$ 0 0
$$325$$ −1.27492 −0.0707197
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −4.17525 −0.230189
$$330$$ 0 0
$$331$$ −18.8248 −1.03470 −0.517351 0.855773i $$-0.673082\pi$$
−0.517351 + 0.855773i $$0.673082\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −0.549834 −0.0300407
$$336$$ 0 0
$$337$$ 5.45017 0.296889 0.148445 0.988921i $$-0.452573\pi$$
0.148445 + 0.988921i $$0.452573\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −39.3746 −2.13225
$$342$$ 0 0
$$343$$ 10.7251 0.579100
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −14.0000 −0.751559 −0.375780 0.926709i $$-0.622625\pi$$
−0.375780 + 0.926709i $$0.622625\pi$$
$$348$$ 0 0
$$349$$ 12.8248 0.686493 0.343247 0.939245i $$-0.388473\pi$$
0.343247 + 0.939245i $$0.388473\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.54983 −0.348613 −0.174306 0.984691i $$-0.555768\pi$$
−0.174306 + 0.984691i $$0.555768\pi$$
$$354$$ 0 0
$$355$$ −8.27492 −0.439187
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −0.824752 −0.0435287 −0.0217644 0.999763i $$-0.506928\pi$$
−0.0217644 + 0.999763i $$0.506928\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −15.0997 −0.790353
$$366$$ 0 0
$$367$$ 12.5498 0.655096 0.327548 0.944835i $$-0.393778\pi$$
0.327548 + 0.944835i $$0.393778\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 2.37459 0.123282
$$372$$ 0 0
$$373$$ 31.0997 1.61028 0.805140 0.593085i $$-0.202090\pi$$
0.805140 + 0.593085i $$0.202090\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 8.00000 0.412021
$$378$$ 0 0
$$379$$ 9.27492 0.476420 0.238210 0.971214i $$-0.423439\pi$$
0.238210 + 0.971214i $$0.423439\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.37459 −0.121336 −0.0606678 0.998158i $$-0.519323\pi$$
−0.0606678 + 0.998158i $$0.519323\pi$$
$$384$$ 0 0
$$385$$ −20.5498 −1.04732
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 16.5498 0.839110 0.419555 0.907730i $$-0.362186\pi$$
0.419555 + 0.907730i $$0.362186\pi$$
$$390$$ 0 0
$$391$$ −14.5498 −0.735817
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −10.5498 −0.530820
$$396$$ 0 0
$$397$$ 7.09967 0.356322 0.178161 0.984001i $$-0.442985\pi$$
0.178161 + 0.984001i $$0.442985\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −23.2749 −1.16229 −0.581147 0.813799i $$-0.697396\pi$$
−0.581147 + 0.813799i $$0.697396\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 66.1993 3.28138
$$408$$ 0 0
$$409$$ −29.8248 −1.47474 −0.737370 0.675490i $$-0.763932\pi$$
−0.737370 + 0.675490i $$0.763932\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −42.5739 −2.09493
$$414$$ 0 0
$$415$$ −2.54983 −0.125166
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 9.09967 0.444548 0.222274 0.974984i $$-0.428652\pi$$
0.222274 + 0.974984i $$0.428652\pi$$
$$420$$ 0 0
$$421$$ −32.2749 −1.57298 −0.786492 0.617601i $$-0.788105\pi$$
−0.786492 + 0.617601i $$0.788105\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −2.00000 −0.0970143
$$426$$ 0 0
$$427$$ −28.0000 −1.35501
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.27492 0.302252 0.151126 0.988514i $$-0.451710\pi$$
0.151126 + 0.988514i $$0.451710\pi$$
$$432$$ 0 0
$$433$$ 12.5498 0.603107 0.301553 0.953449i $$-0.402495\pi$$
0.301553 + 0.953449i $$0.402495\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.27492 0.348006
$$438$$ 0 0
$$439$$ −25.3746 −1.21106 −0.605531 0.795821i $$-0.707039\pi$$
−0.605531 + 0.795821i $$0.707039\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −10.5498 −0.501238 −0.250619 0.968086i $$-0.580634\pi$$
−0.250619 + 0.968086i $$0.580634\pi$$
$$444$$ 0 0
$$445$$ 12.8248 0.607952
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −1.54983 −0.0731412 −0.0365706 0.999331i $$-0.511643\pi$$
−0.0365706 + 0.999331i $$0.511643\pi$$
$$450$$ 0 0
$$451$$ 47.3746 2.23078
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −4.17525 −0.195739
$$456$$ 0 0
$$457$$ 16.0000 0.748448 0.374224 0.927338i $$-0.377909\pi$$
0.374224 + 0.927338i $$0.377909\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 23.3746 1.08866 0.544332 0.838870i $$-0.316783\pi$$
0.544332 + 0.838870i $$0.316783\pi$$
$$462$$ 0 0
$$463$$ 41.4743 1.92747 0.963736 0.266857i $$-0.0859853\pi$$
0.963736 + 0.266857i $$0.0859853\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 40.1993 1.86020 0.930102 0.367302i $$-0.119718\pi$$
0.930102 + 0.367302i $$0.119718\pi$$
$$468$$ 0 0
$$469$$ −1.80066 −0.0831469
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −25.0997 −1.15408
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 4.27492 0.195326 0.0976630 0.995220i $$-0.468863\pi$$
0.0976630 + 0.995220i $$0.468863\pi$$
$$480$$ 0 0
$$481$$ 13.4502 0.613275
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −16.0000 −0.726523
$$486$$ 0 0
$$487$$ −37.2749 −1.68909 −0.844544 0.535486i $$-0.820128\pi$$
−0.844544 + 0.535486i $$0.820128\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −15.0000 −0.676941 −0.338470 0.940977i $$-0.609909\pi$$
−0.338470 + 0.940977i $$0.609909\pi$$
$$492$$ 0 0
$$493$$ 12.5498 0.565216
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −27.0997 −1.21559
$$498$$ 0 0
$$499$$ −29.5498 −1.32283 −0.661416 0.750019i $$-0.730044\pi$$
−0.661416 + 0.750019i $$0.730044\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 9.72508 0.432761
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −28.1993 −1.24991 −0.624957 0.780659i $$-0.714883\pi$$
−0.624957 + 0.780659i $$0.714883\pi$$
$$510$$ 0 0
$$511$$ −49.4502 −2.18755
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 5.82475 0.256669
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 26.9244 1.17958 0.589790 0.807557i $$-0.299210\pi$$
0.589790 + 0.807557i $$0.299210\pi$$
$$522$$ 0 0
$$523$$ −39.6495 −1.73375 −0.866876 0.498524i $$-0.833876\pi$$
−0.866876 + 0.498524i $$0.833876\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −12.5498 −0.546679
$$528$$ 0 0
$$529$$ 29.9244 1.30106
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 9.62541 0.416923
$$534$$ 0 0
$$535$$ 15.0997 0.652816
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −23.3746 −1.00681
$$540$$ 0 0
$$541$$ 43.9244 1.88846 0.944229 0.329289i $$-0.106809\pi$$
0.944229 + 0.329289i $$0.106809\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 6.27492 0.268788
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.27492 −0.267320
$$552$$ 0 0
$$553$$ −34.5498 −1.46921
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −32.9244 −1.39505 −0.697526 0.716559i $$-0.745716\pi$$
−0.697526 + 0.716559i $$0.745716\pi$$
$$558$$ 0 0
$$559$$ −5.09967 −0.215693
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −37.0997 −1.56356 −0.781782 0.623551i $$-0.785689\pi$$
−0.781782 + 0.623551i $$0.785689\pi$$
$$564$$ 0 0
$$565$$ −6.00000 −0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 26.4502 1.10885 0.554424 0.832234i $$-0.312938\pi$$
0.554424 + 0.832234i $$0.312938\pi$$
$$570$$ 0 0
$$571$$ −1.17525 −0.0491826 −0.0245913 0.999698i $$-0.507828\pi$$
−0.0245913 + 0.999698i $$0.507828\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 7.27492 0.303385
$$576$$ 0 0
$$577$$ −11.4502 −0.476677 −0.238338 0.971182i $$-0.576603\pi$$
−0.238338 + 0.971182i $$0.576603\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −8.35050 −0.346437
$$582$$ 0 0
$$583$$ 4.54983 0.188435
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 34.5498 1.42602 0.713012 0.701152i $$-0.247330\pi$$
0.713012 + 0.701152i $$0.247330\pi$$
$$588$$ 0 0
$$589$$ 6.27492 0.258553
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 48.1993 1.97931 0.989655 0.143469i $$-0.0458258\pi$$
0.989655 + 0.143469i $$0.0458258\pi$$
$$594$$ 0 0
$$595$$ −6.54983 −0.268517
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 15.7251 0.642509 0.321255 0.946993i $$-0.395895\pi$$
0.321255 + 0.946993i $$0.395895\pi$$
$$600$$ 0 0
$$601$$ −1.90033 −0.0775161 −0.0387581 0.999249i $$-0.512340\pi$$
−0.0387581 + 0.999249i $$0.512340\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −28.3746 −1.15359
$$606$$ 0 0
$$607$$ −24.0000 −0.974130 −0.487065 0.873366i $$-0.661933\pi$$
−0.487065 + 0.873366i $$0.661933\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1.62541 −0.0657572
$$612$$ 0 0
$$613$$ −46.0241 −1.85890 −0.929448 0.368954i $$-0.879716\pi$$
−0.929448 + 0.368954i $$0.879716\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 43.6495 1.75726 0.878631 0.477501i $$-0.158457\pi$$
0.878631 + 0.477501i $$0.158457\pi$$
$$618$$ 0 0
$$619$$ −18.3746 −0.738537 −0.369268 0.929323i $$-0.620392\pi$$
−0.369268 + 0.929323i $$0.620392\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 42.0000 1.68269
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 21.0997 0.841299
$$630$$ 0 0
$$631$$ 24.4743 0.974305 0.487152 0.873317i $$-0.338036\pi$$
0.487152 + 0.873317i $$0.338036\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −11.2749 −0.447431
$$636$$ 0 0
$$637$$ −4.74917 −0.188169
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −33.9244 −1.33993 −0.669967 0.742391i $$-0.733692\pi$$
−0.669967 + 0.742391i $$0.733692\pi$$
$$642$$ 0 0
$$643$$ −33.0997 −1.30532 −0.652662 0.757649i $$-0.726348\pi$$
−0.652662 + 0.757649i $$0.726348\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 28.5498 1.12241 0.561205 0.827677i $$-0.310338\pi$$
0.561205 + 0.827677i $$0.310338\pi$$
$$648$$ 0 0
$$649$$ −81.5739 −3.20206
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −9.45017 −0.369814 −0.184907 0.982756i $$-0.559198\pi$$
−0.184907 + 0.982756i $$0.559198\pi$$
$$654$$ 0 0
$$655$$ 1.54983 0.0605570
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 17.2749 0.672935 0.336468 0.941695i $$-0.390768\pi$$
0.336468 + 0.941695i $$0.390768\pi$$
$$660$$ 0 0
$$661$$ −39.9244 −1.55288 −0.776440 0.630191i $$-0.782977\pi$$
−0.776440 + 0.630191i $$0.782977\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 3.27492 0.126996
$$666$$ 0 0
$$667$$ −45.6495 −1.76756
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −53.6495 −2.07112
$$672$$ 0 0
$$673$$ −8.90033 −0.343083 −0.171541 0.985177i $$-0.554875\pi$$
−0.171541 + 0.985177i $$0.554875\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 26.9244 1.03479 0.517395 0.855747i $$-0.326902\pi$$
0.517395 + 0.855747i $$0.326902\pi$$
$$678$$ 0 0
$$679$$ −52.3987 −2.01088
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 1.45017 0.0554890 0.0277445 0.999615i $$-0.491168\pi$$
0.0277445 + 0.999615i $$0.491168\pi$$
$$684$$ 0 0
$$685$$ 14.5498 0.555921
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0.924421 0.0352176
$$690$$ 0 0
$$691$$ −7.82475 −0.297668 −0.148834 0.988862i $$-0.547552\pi$$
−0.148834 + 0.988862i $$0.547552\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 9.00000 0.341389
$$696$$ 0 0
$$697$$ 15.0997 0.571941
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 33.3746 1.26054 0.630270 0.776376i $$-0.282944\pi$$
0.630270 + 0.776376i $$0.282944\pi$$
$$702$$ 0 0
$$703$$ −10.5498 −0.397895
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 31.8488 1.19780
$$708$$ 0 0
$$709$$ −0.900331 −0.0338126 −0.0169063 0.999857i $$-0.505382\pi$$
−0.0169063 + 0.999857i $$0.505382\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 45.6495 1.70959
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 44.8248 1.67168 0.835841 0.548972i $$-0.184981\pi$$
0.835841 + 0.548972i $$0.184981\pi$$
$$720$$ 0 0
$$721$$ 19.0756 0.710412
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −6.27492 −0.233045
$$726$$ 0 0
$$727$$ −30.3746 −1.12653 −0.563266 0.826276i $$-0.690455\pi$$
−0.563266 + 0.826276i $$0.690455\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −3.45017 −0.127088
$$738$$ 0 0
$$739$$ 15.9244 0.585789 0.292895 0.956145i $$-0.405381\pi$$
0.292895 + 0.956145i $$0.405381\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 28.5498 1.04739 0.523696 0.851905i $$-0.324553\pi$$
0.523696 + 0.851905i $$0.324553\pi$$
$$744$$ 0 0
$$745$$ 2.00000 0.0732743
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 49.4502 1.80687
$$750$$ 0 0
$$751$$ 34.5498 1.26074 0.630371 0.776294i $$-0.282903\pi$$
0.630371 + 0.776294i $$0.282903\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 12.2749 0.446730
$$756$$ 0 0
$$757$$ 34.3746 1.24937 0.624683 0.780879i $$-0.285228\pi$$
0.624683 + 0.780879i $$0.285228\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −49.5498 −1.79618 −0.898090 0.439812i $$-0.855045\pi$$
−0.898090 + 0.439812i $$0.855045\pi$$
$$762$$ 0 0
$$763$$ 20.5498 0.743954
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −16.5739 −0.598450
$$768$$ 0 0
$$769$$ −16.2749 −0.586889 −0.293444 0.955976i $$-0.594802\pi$$
−0.293444 + 0.955976i $$0.594802\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −31.6495 −1.13835 −0.569177 0.822215i $$-0.692738\pi$$
−0.569177 + 0.822215i $$0.692738\pi$$
$$774$$ 0 0
$$775$$ 6.27492 0.225402
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −7.54983 −0.270501
$$780$$ 0 0
$$781$$ −51.9244 −1.85800
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −10.7251 −0.382795
$$786$$ 0 0
$$787$$ −34.0000 −1.21197 −0.605985 0.795476i $$-0.707221\pi$$
−0.605985 + 0.795476i $$0.707221\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −19.6495 −0.698656
$$792$$ 0 0
$$793$$ −10.9003 −0.387082
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 25.4502 0.901491 0.450746 0.892652i $$-0.351158\pi$$
0.450746 + 0.892652i $$0.351158\pi$$
$$798$$ 0 0
$$799$$ −2.54983 −0.0902067
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −94.7492 −3.34363
$$804$$ 0 0
$$805$$ 23.8248 0.839712
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −16.6495 −0.585365 −0.292683 0.956210i $$-0.594548\pi$$
−0.292683 + 0.956210i $$0.594548\pi$$
$$810$$ 0 0
$$811$$ 13.1752 0.462646 0.231323 0.972877i $$-0.425695\pi$$
0.231323 + 0.972877i $$0.425695\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −20.5498 −0.719829
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −4.82475 −0.168385 −0.0841925 0.996450i $$-0.526831\pi$$
−0.0841925 + 0.996450i $$0.526831\pi$$
$$822$$ 0 0
$$823$$ 23.4502 0.817421 0.408711 0.912664i $$-0.365979\pi$$
0.408711 + 0.912664i $$0.365979\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −21.0997 −0.733707 −0.366854 0.930279i $$-0.619565\pi$$
−0.366854 + 0.930279i $$0.619565\pi$$
$$828$$ 0 0
$$829$$ −13.1752 −0.457595 −0.228798 0.973474i $$-0.573479\pi$$
−0.228798 + 0.973474i $$0.573479\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −7.45017 −0.258133
$$834$$ 0 0
$$835$$ −7.45017 −0.257824
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −52.4743 −1.81161 −0.905806 0.423692i $$-0.860734\pi$$
−0.905806 + 0.423692i $$0.860734\pi$$
$$840$$ 0 0
$$841$$ 10.3746 0.357744
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 11.3746 0.391298
$$846$$ 0 0
$$847$$ −92.9244 −3.19292
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −76.7492 −2.63093
$$852$$ 0 0
$$853$$ −49.2990 −1.68797 −0.843983 0.536370i $$-0.819795\pi$$
−0.843983 + 0.536370i $$0.819795\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 52.7492 1.80188 0.900939 0.433946i $$-0.142879\pi$$
0.900939 + 0.433946i $$0.142879\pi$$
$$858$$ 0 0
$$859$$ 52.4743 1.79040 0.895199 0.445666i $$-0.147033\pi$$
0.895199 + 0.445666i $$0.147033\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −23.2749 −0.792287 −0.396144 0.918189i $$-0.629652\pi$$
−0.396144 + 0.918189i $$0.629652\pi$$
$$864$$ 0 0
$$865$$ 8.72508 0.296662
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −66.1993 −2.24566
$$870$$ 0 0
$$871$$ −0.700993 −0.0237523
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 3.27492 0.110712
$$876$$ 0 0
$$877$$ −13.2749 −0.448262 −0.224131 0.974559i $$-0.571954\pi$$
−0.224131 + 0.974559i $$0.571954\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −18.6254 −0.627506 −0.313753 0.949505i $$-0.601586\pi$$
−0.313753 + 0.949505i $$0.601586\pi$$
$$882$$ 0 0
$$883$$ 32.1993 1.08359 0.541797 0.840509i $$-0.317744\pi$$
0.541797 + 0.840509i $$0.317744\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −0.175248 −0.00588426 −0.00294213 0.999996i $$-0.500937\pi$$
−0.00294213 + 0.999996i $$0.500937\pi$$
$$888$$ 0 0
$$889$$ −36.9244 −1.23841
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 1.27492 0.0426635
$$894$$ 0 0
$$895$$ −11.5498 −0.386068
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −39.3746 −1.31322
$$900$$ 0 0
$$901$$ 1.45017 0.0483120
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −16.2749 −0.540997
$$906$$ 0 0
$$907$$ 51.0997 1.69674 0.848368 0.529406i $$-0.177585\pi$$
0.848368 + 0.529406i $$0.177585\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 27.0241 0.895348 0.447674 0.894197i $$-0.352253\pi$$
0.447674 + 0.894197i $$0.352253\pi$$
$$912$$ 0 0
$$913$$ −16.0000 −0.529523
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 5.07558 0.167610
$$918$$ 0 0
$$919$$ 11.9244 0.393350 0.196675 0.980469i $$-0.436986\pi$$
0.196675 + 0.980469i $$0.436986\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −10.5498 −0.347252
$$924$$ 0 0
$$925$$ −10.5498 −0.346876
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 60.1238 1.97260 0.986298 0.164972i $$-0.0527534\pi$$
0.986298 + 0.164972i $$0.0527534\pi$$
$$930$$ 0 0
$$931$$ 3.72508 0.122085
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −12.5498 −0.410423
$$936$$ 0 0
$$937$$ −0.549834 −0.0179623 −0.00898115 0.999960i $$-0.502859\pi$$
−0.00898115 + 0.999960i $$0.502859\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −7.09967 −0.231443 −0.115721 0.993282i $$-0.536918\pi$$
−0.115721 + 0.993282i $$0.536918\pi$$
$$942$$ 0 0
$$943$$ −54.9244 −1.78859
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.4502 0.892011 0.446005 0.895030i $$-0.352846\pi$$
0.446005 + 0.895030i $$0.352846\pi$$
$$948$$ 0 0
$$949$$ −19.2508 −0.624908
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 33.0997 1.07220 0.536102 0.844153i $$-0.319896\pi$$
0.536102 + 0.844153i $$0.319896\pi$$
$$954$$ 0 0
$$955$$ 19.3746 0.626947
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 47.6495 1.53868
$$960$$ 0 0
$$961$$ 8.37459 0.270148
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −4.00000 −0.128765
$$966$$ 0 0
$$967$$ 10.3505 0.332850 0.166425 0.986054i $$-0.446778\pi$$
0.166425 + 0.986054i $$0.446778\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −36.6495 −1.17614 −0.588069 0.808811i $$-0.700112\pi$$
−0.588069 + 0.808811i $$0.700112\pi$$
$$972$$ 0 0
$$973$$ 29.4743 0.944901
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −3.64950 −0.116758 −0.0583790 0.998294i $$-0.518593\pi$$
−0.0583790 + 0.998294i $$0.518593\pi$$
$$978$$ 0 0
$$979$$ 80.4743 2.57197
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 46.7492 1.49107 0.745534 0.666468i $$-0.232195\pi$$
0.745534 + 0.666468i $$0.232195\pi$$
$$984$$ 0 0
$$985$$ −22.3746 −0.712914
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 29.0997 0.925316
$$990$$ 0 0
$$991$$ −1.72508 −0.0547991 −0.0273995 0.999625i $$-0.508723\pi$$
−0.0273995 + 0.999625i $$0.508723\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −8.00000 −0.253617
$$996$$ 0 0
$$997$$ 8.37459 0.265226 0.132613 0.991168i $$-0.457663\pi$$
0.132613 + 0.991168i $$0.457663\pi$$
$$998$$ 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 3240.2.a.h.1.1 2
3.2 odd 2 3240.2.a.n.1.1 yes 2
4.3 odd 2 6480.2.a.bf.1.2 2
9.2 odd 6 3240.2.q.y.1081.2 4
9.4 even 3 3240.2.q.be.2161.2 4
9.5 odd 6 3240.2.q.y.2161.2 4
9.7 even 3 3240.2.q.be.1081.2 4
12.11 even 2 6480.2.a.bm.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
3240.2.a.h.1.1 2 1.1 even 1 trivial
3240.2.a.n.1.1 yes 2 3.2 odd 2
3240.2.q.y.1081.2 4 9.2 odd 6
3240.2.q.y.2161.2 4 9.5 odd 6
3240.2.q.be.1081.2 4 9.7 even 3
3240.2.q.be.2161.2 4 9.4 even 3
6480.2.a.bf.1.2 2 4.3 odd 2
6480.2.a.bm.1.2 2 12.11 even 2