# Properties

 Label 5472.2.a.bs.1.1 Level $5472$ Weight $2$ Character 5472.1 Self dual yes Analytic conductor $43.694$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5472,2,Mod(1,5472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5472, 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("5472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5472 = 2^{5} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5472.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: $$43.6941399860$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.15317.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 5x + 2$$ x^4 - 2*x^3 - 4*x^2 + 5*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-0.329727$$ of defining polynomial Character $$\chi$$ $$=$$ 5472.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.40617 q^{5} +2.50407 q^{7} +O(q^{10})$$ $$q-3.40617 q^{5} +2.50407 q^{7} +1.40617 q^{11} +6.22101 q^{13} +3.16352 q^{17} +1.00000 q^{19} +7.91023 q^{23} +6.60197 q^{25} -6.22101 q^{29} +8.13124 q^{31} -8.52927 q^{35} -6.13124 q^{37} -1.68923 q^{41} +1.71694 q^{43} -9.84818 q^{47} -0.729644 q^{49} -2.78713 q^{53} -4.78964 q^{55} -9.25078 q^{59} +6.52927 q^{61} -21.1898 q^{65} +1.87233 q^{67} +13.0081 q^{71} -6.28663 q^{73} +3.52114 q^{77} -11.2543 q^{79} -1.80420 q^{83} -10.7755 q^{85} +7.57426 q^{89} +15.5778 q^{91} -3.40617 q^{95} +4.81233 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - q^{5} - q^{7}+O(q^{10})$$ 4 * q - q^5 - q^7 $$4 q - q^{5} - q^{7} - 7 q^{11} + 10 q^{13} - 5 q^{17} + 4 q^{19} + 8 q^{23} + 17 q^{25} - 10 q^{29} - 6 q^{31} - 5 q^{35} + 14 q^{37} + 2 q^{41} + 3 q^{43} + 3 q^{47} + 7 q^{49} - 4 q^{53} - 35 q^{55} - 20 q^{59} - 3 q^{61} + 12 q^{65} + 8 q^{67} + 30 q^{71} + 9 q^{73} + 7 q^{77} + 10 q^{79} - 4 q^{83} + 19 q^{85} + 16 q^{89} + 10 q^{91} - q^{95} - 6 q^{97}+O(q^{100})$$ 4 * q - q^5 - q^7 - 7 * q^11 + 10 * q^13 - 5 * q^17 + 4 * q^19 + 8 * q^23 + 17 * q^25 - 10 * q^29 - 6 * q^31 - 5 * q^35 + 14 * q^37 + 2 * q^41 + 3 * q^43 + 3 * q^47 + 7 * q^49 - 4 * q^53 - 35 * q^55 - 20 * q^59 - 3 * q^61 + 12 * q^65 + 8 * q^67 + 30 * q^71 + 9 * q^73 + 7 * q^77 + 10 * q^79 - 4 * q^83 + 19 * q^85 + 16 * q^89 + 10 * q^91 - q^95 - 6 * 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
$$5$$ −3.40617 −1.52328 −0.761642 0.647998i $$-0.775606\pi$$
−0.761642 + 0.647998i $$0.775606\pi$$
$$6$$ 0 0
$$7$$ 2.50407 0.946449 0.473224 0.880942i $$-0.343090\pi$$
0.473224 + 0.880942i $$0.343090\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.40617 0.423975 0.211988 0.977272i $$-0.432006\pi$$
0.211988 + 0.977272i $$0.432006\pi$$
$$12$$ 0 0
$$13$$ 6.22101 1.72540 0.862698 0.505719i $$-0.168773\pi$$
0.862698 + 0.505719i $$0.168773\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.16352 0.767267 0.383633 0.923485i $$-0.374673\pi$$
0.383633 + 0.923485i $$0.374673\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 7.91023 1.64940 0.824699 0.565572i $$-0.191345\pi$$
0.824699 + 0.565572i $$0.191345\pi$$
$$24$$ 0 0
$$25$$ 6.60197 1.32039
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.22101 −1.15521 −0.577606 0.816316i $$-0.696013\pi$$
−0.577606 + 0.816316i $$0.696013\pi$$
$$30$$ 0 0
$$31$$ 8.13124 1.46041 0.730207 0.683226i $$-0.239424\pi$$
0.730207 + 0.683226i $$0.239424\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −8.52927 −1.44171
$$36$$ 0 0
$$37$$ −6.13124 −1.00797 −0.503985 0.863712i $$-0.668133\pi$$
−0.503985 + 0.863712i $$0.668133\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.68923 −0.263813 −0.131906 0.991262i $$-0.542110\pi$$
−0.131906 + 0.991262i $$0.542110\pi$$
$$42$$ 0 0
$$43$$ 1.71694 0.261831 0.130915 0.991394i $$-0.458208\pi$$
0.130915 + 0.991394i $$0.458208\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −9.84818 −1.43650 −0.718252 0.695783i $$-0.755058\pi$$
−0.718252 + 0.695783i $$0.755058\pi$$
$$48$$ 0 0
$$49$$ −0.729644 −0.104235
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.78713 −0.382842 −0.191421 0.981508i $$-0.561310\pi$$
−0.191421 + 0.981508i $$0.561310\pi$$
$$54$$ 0 0
$$55$$ −4.78964 −0.645834
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −9.25078 −1.20435 −0.602174 0.798365i $$-0.705699\pi$$
−0.602174 + 0.798365i $$0.705699\pi$$
$$60$$ 0 0
$$61$$ 6.52927 0.835988 0.417994 0.908450i $$-0.362733\pi$$
0.417994 + 0.908450i $$0.362733\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −21.1898 −2.62827
$$66$$ 0 0
$$67$$ 1.87233 0.228741 0.114370 0.993438i $$-0.463515\pi$$
0.114370 + 0.993438i $$0.463515\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 13.0081 1.54378 0.771891 0.635755i $$-0.219311\pi$$
0.771891 + 0.635755i $$0.219311\pi$$
$$72$$ 0 0
$$73$$ −6.28663 −0.735794 −0.367897 0.929867i $$-0.619922\pi$$
−0.367897 + 0.929867i $$0.619922\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.52114 0.401271
$$78$$ 0 0
$$79$$ −11.2543 −1.26621 −0.633106 0.774065i $$-0.718220\pi$$
−0.633106 + 0.774065i $$0.718220\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −1.80420 −0.198036 −0.0990182 0.995086i $$-0.531570\pi$$
−0.0990182 + 0.995086i $$0.531570\pi$$
$$84$$ 0 0
$$85$$ −10.7755 −1.16877
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.57426 0.802870 0.401435 0.915888i $$-0.368512\pi$$
0.401435 + 0.915888i $$0.368512\pi$$
$$90$$ 0 0
$$91$$ 15.5778 1.63300
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −3.40617 −0.349465
$$96$$ 0 0
$$97$$ 4.81233 0.488618 0.244309 0.969697i $$-0.421439\pi$$
0.244309 + 0.969697i $$0.421439\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 16.2462 1.61656 0.808279 0.588799i $$-0.200399\pi$$
0.808279 + 0.588799i $$0.200399\pi$$
$$102$$ 0 0
$$103$$ 7.68923 0.757642 0.378821 0.925470i $$-0.376330\pi$$
0.378821 + 0.925470i $$0.376330\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 11.8886 1.14931 0.574657 0.818394i $$-0.305135\pi$$
0.574657 + 0.818394i $$0.305135\pi$$
$$108$$ 0 0
$$109$$ −5.65489 −0.541640 −0.270820 0.962630i $$-0.587295\pi$$
−0.270820 + 0.962630i $$0.587295\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 8.81233 0.828995 0.414497 0.910051i $$-0.363957\pi$$
0.414497 + 0.910051i $$0.363957\pi$$
$$114$$ 0 0
$$115$$ −26.9436 −2.51250
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 7.92167 0.726179
$$120$$ 0 0
$$121$$ −9.02270 −0.820245
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −5.45657 −0.488051
$$126$$ 0 0
$$127$$ 12.0504 1.06930 0.534650 0.845073i $$-0.320443\pi$$
0.534650 + 0.845073i $$0.320443\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.920878 −0.0804575 −0.0402287 0.999190i $$-0.512809\pi$$
−0.0402287 + 0.999190i $$0.512809\pi$$
$$132$$ 0 0
$$133$$ 2.50407 0.217130
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0.525705 0.0449140 0.0224570 0.999748i $$-0.492851\pi$$
0.0224570 + 0.999748i $$0.492851\pi$$
$$138$$ 0 0
$$139$$ −4.52927 −0.384168 −0.192084 0.981379i $$-0.561525\pi$$
−0.192084 + 0.981379i $$0.561525\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.74777 0.731525
$$144$$ 0 0
$$145$$ 21.1898 1.75972
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 18.9713 1.55419 0.777094 0.629384i $$-0.216693\pi$$
0.777094 + 0.629384i $$0.216693\pi$$
$$150$$ 0 0
$$151$$ 6.55698 0.533600 0.266800 0.963752i $$-0.414034\pi$$
0.266800 + 0.963752i $$0.414034\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −27.6964 −2.22463
$$156$$ 0 0
$$157$$ 17.5651 1.40185 0.700925 0.713235i $$-0.252771\pi$$
0.700925 + 0.713235i $$0.252771\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 19.8078 1.56107
$$162$$ 0 0
$$163$$ −22.9436 −1.79708 −0.898540 0.438892i $$-0.855371\pi$$
−0.898540 + 0.438892i $$0.855371\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.24621 −0.483346 −0.241673 0.970358i $$-0.577696\pi$$
−0.241673 + 0.970358i $$0.577696\pi$$
$$168$$ 0 0
$$169$$ 25.7009 1.97699
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 16.9436 1.28820 0.644098 0.764943i $$-0.277233\pi$$
0.644098 + 0.764943i $$0.277233\pi$$
$$174$$ 0 0
$$175$$ 16.5318 1.24969
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −18.2462 −1.36379 −0.681893 0.731452i $$-0.738843\pi$$
−0.681893 + 0.731452i $$0.738843\pi$$
$$180$$ 0 0
$$181$$ 10.9273 0.812220 0.406110 0.913824i $$-0.366885\pi$$
0.406110 + 0.913824i $$0.366885\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 20.8840 1.53542
$$186$$ 0 0
$$187$$ 4.44844 0.325302
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 15.5122 1.12242 0.561212 0.827672i $$-0.310335\pi$$
0.561212 + 0.827672i $$0.310335\pi$$
$$192$$ 0 0
$$193$$ −9.56512 −0.688512 −0.344256 0.938876i $$-0.611869\pi$$
−0.344256 + 0.938876i $$0.611869\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −19.1394 −1.36362 −0.681812 0.731527i $$-0.738808\pi$$
−0.681812 + 0.731527i $$0.738808\pi$$
$$198$$ 0 0
$$199$$ −0.133749 −0.00948125 −0.00474062 0.999989i $$-0.501509\pi$$
−0.00474062 + 0.999989i $$0.501509\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −15.5778 −1.09335
$$204$$ 0 0
$$205$$ 5.75379 0.401862
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 1.40617 0.0972666
$$210$$ 0 0
$$211$$ −23.0117 −1.58419 −0.792095 0.610397i $$-0.791010\pi$$
−0.792095 + 0.610397i $$0.791010\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −5.84818 −0.398843
$$216$$ 0 0
$$217$$ 20.3612 1.38221
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 19.6803 1.32384
$$222$$ 0 0
$$223$$ 7.51471 0.503222 0.251611 0.967828i $$-0.419040\pi$$
0.251611 + 0.967828i $$0.419040\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −5.81690 −0.386081 −0.193041 0.981191i $$-0.561835\pi$$
−0.193041 + 0.981191i $$0.561835\pi$$
$$228$$ 0 0
$$229$$ −26.1702 −1.72938 −0.864688 0.502309i $$-0.832484\pi$$
−0.864688 + 0.502309i $$0.832484\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2.22352 −0.145667 −0.0728337 0.997344i $$-0.523204\pi$$
−0.0728337 + 0.997344i $$0.523204\pi$$
$$234$$ 0 0
$$235$$ 33.5445 2.18820
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −8.12873 −0.525804 −0.262902 0.964823i $$-0.584680\pi$$
−0.262902 + 0.964823i $$0.584680\pi$$
$$240$$ 0 0
$$241$$ 25.9517 1.67170 0.835848 0.548960i $$-0.184976\pi$$
0.835848 + 0.548960i $$0.184976\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 2.48529 0.158779
$$246$$ 0 0
$$247$$ 6.22101 0.395833
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1.53741 0.0970403 0.0485202 0.998822i $$-0.484549\pi$$
0.0485202 + 0.998822i $$0.484549\pi$$
$$252$$ 0 0
$$253$$ 11.1231 0.699304
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0.370318 0.0230998 0.0115499 0.999933i $$-0.496323\pi$$
0.0115499 + 0.999933i $$0.496323\pi$$
$$258$$ 0 0
$$259$$ −15.3530 −0.953992
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 18.9067 1.16584 0.582919 0.812530i $$-0.301910\pi$$
0.582919 + 0.812530i $$0.301910\pi$$
$$264$$ 0 0
$$265$$ 9.49342 0.583176
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 15.1898 0.926138 0.463069 0.886322i $$-0.346748\pi$$
0.463069 + 0.886322i $$0.346748\pi$$
$$270$$ 0 0
$$271$$ −28.4027 −1.72534 −0.862669 0.505768i $$-0.831209\pi$$
−0.862669 + 0.505768i $$0.831209\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 9.28347 0.559814
$$276$$ 0 0
$$277$$ 28.4647 1.71028 0.855139 0.518398i $$-0.173471\pi$$
0.855139 + 0.518398i $$0.173471\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 27.1394 1.61900 0.809500 0.587120i $$-0.199738\pi$$
0.809500 + 0.587120i $$0.199738\pi$$
$$282$$ 0 0
$$283$$ −26.9117 −1.59974 −0.799868 0.600175i $$-0.795097\pi$$
−0.799868 + 0.600175i $$0.795097\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.22994 −0.249685
$$288$$ 0 0
$$289$$ −6.99213 −0.411302
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −10.2714 −0.600062 −0.300031 0.953929i $$-0.596997\pi$$
−0.300031 + 0.953929i $$0.596997\pi$$
$$294$$ 0 0
$$295$$ 31.5097 1.83457
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 49.2096 2.84587
$$300$$ 0 0
$$301$$ 4.29933 0.247809
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −22.2398 −1.27345
$$306$$ 0 0
$$307$$ 5.05854 0.288706 0.144353 0.989526i $$-0.453890\pi$$
0.144353 + 0.989526i $$0.453890\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 13.4222 0.761105 0.380553 0.924759i $$-0.375734\pi$$
0.380553 + 0.924759i $$0.375734\pi$$
$$312$$ 0 0
$$313$$ 6.94001 0.392272 0.196136 0.980577i $$-0.437161\pi$$
0.196136 + 0.980577i $$0.437161\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −5.60448 −0.314779 −0.157389 0.987537i $$-0.550308\pi$$
−0.157389 + 0.987537i $$0.550308\pi$$
$$318$$ 0 0
$$319$$ −8.74777 −0.489781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 3.16352 0.176023
$$324$$ 0 0
$$325$$ 41.0709 2.27820
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −24.6605 −1.35958
$$330$$ 0 0
$$331$$ 4.11140 0.225983 0.112992 0.993596i $$-0.463957\pi$$
0.112992 + 0.993596i $$0.463957\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −6.37745 −0.348437
$$336$$ 0 0
$$337$$ −0.632801 −0.0344709 −0.0172354 0.999851i $$-0.505486\pi$$
−0.0172354 + 0.999851i $$0.505486\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 11.4339 0.619179
$$342$$ 0 0
$$343$$ −19.3556 −1.04510
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 3.02871 0.162590 0.0812949 0.996690i $$-0.474094\pi$$
0.0812949 + 0.996690i $$0.474094\pi$$
$$348$$ 0 0
$$349$$ 19.0954 1.02215 0.511076 0.859535i $$-0.329247\pi$$
0.511076 + 0.859535i $$0.329247\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −15.2025 −0.809147 −0.404573 0.914506i $$-0.632580\pi$$
−0.404573 + 0.914506i $$0.632580\pi$$
$$354$$ 0 0
$$355$$ −44.3079 −2.35162
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 34.9411 1.84412 0.922059 0.387048i $$-0.126505\pi$$
0.922059 + 0.387048i $$0.126505\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 21.4133 1.12082
$$366$$ 0 0
$$367$$ 7.60839 0.397155 0.198577 0.980085i $$-0.436368\pi$$
0.198577 + 0.980085i $$0.436368\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −6.97916 −0.362340
$$372$$ 0 0
$$373$$ 9.38739 0.486060 0.243030 0.970019i $$-0.421859\pi$$
0.243030 + 0.970019i $$0.421859\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −38.7009 −1.99320
$$378$$ 0 0
$$379$$ 11.0046 0.565267 0.282633 0.959228i $$-0.408792\pi$$
0.282633 + 0.959228i $$0.408792\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −20.6923 −1.05733 −0.528665 0.848831i $$-0.677307\pi$$
−0.528665 + 0.848831i $$0.677307\pi$$
$$384$$ 0 0
$$385$$ −11.9936 −0.611249
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 14.4306 0.731659 0.365830 0.930682i $$-0.380785\pi$$
0.365830 + 0.930682i $$0.380785\pi$$
$$390$$ 0 0
$$391$$ 25.0242 1.26553
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 38.3342 1.92880
$$396$$ 0 0
$$397$$ −14.2739 −0.716388 −0.358194 0.933647i $$-0.616607\pi$$
−0.358194 + 0.933647i $$0.616607\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −5.38559 −0.268943 −0.134472 0.990917i $$-0.542934\pi$$
−0.134472 + 0.990917i $$0.542934\pi$$
$$402$$ 0 0
$$403$$ 50.5845 2.51979
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −8.62155 −0.427354
$$408$$ 0 0
$$409$$ 5.07983 0.251182 0.125591 0.992082i $$-0.459917\pi$$
0.125591 + 0.992082i $$0.459917\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −23.1646 −1.13985
$$414$$ 0 0
$$415$$ 6.14539 0.301666
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −4.30576 −0.210350 −0.105175 0.994454i $$-0.533540\pi$$
−0.105175 + 0.994454i $$0.533540\pi$$
$$420$$ 0 0
$$421$$ −17.8670 −0.870782 −0.435391 0.900241i $$-0.643390\pi$$
−0.435391 + 0.900241i $$0.643390\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 20.8855 1.01309
$$426$$ 0 0
$$427$$ 16.3497 0.791219
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 18.1958 0.876461 0.438230 0.898863i $$-0.355605\pi$$
0.438230 + 0.898863i $$0.355605\pi$$
$$432$$ 0 0
$$433$$ −16.6074 −0.798100 −0.399050 0.916929i $$-0.630660\pi$$
−0.399050 + 0.916929i $$0.630660\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 7.91023 0.378398
$$438$$ 0 0
$$439$$ −11.6084 −0.554038 −0.277019 0.960864i $$-0.589347\pi$$
−0.277019 + 0.960864i $$0.589347\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −23.8140 −1.13144 −0.565720 0.824598i $$-0.691402\pi$$
−0.565720 + 0.824598i $$0.691402\pi$$
$$444$$ 0 0
$$445$$ −25.7992 −1.22300
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −3.72336 −0.175716 −0.0878582 0.996133i $$-0.528002\pi$$
−0.0878582 + 0.996133i $$0.528002\pi$$
$$450$$ 0 0
$$451$$ −2.37533 −0.111850
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −53.0607 −2.48752
$$456$$ 0 0
$$457$$ 26.7719 1.25234 0.626169 0.779688i $$-0.284622\pi$$
0.626169 + 0.779688i $$0.284622\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −7.76735 −0.361761 −0.180881 0.983505i $$-0.557895\pi$$
−0.180881 + 0.983505i $$0.557895\pi$$
$$462$$ 0 0
$$463$$ −13.7928 −0.641004 −0.320502 0.947248i $$-0.603852\pi$$
−0.320502 + 0.947248i $$0.603852\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −27.7282 −1.28311 −0.641554 0.767078i $$-0.721710\pi$$
−0.641554 + 0.767078i $$0.721710\pi$$
$$468$$ 0 0
$$469$$ 4.68843 0.216492
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 2.41430 0.111010
$$474$$ 0 0
$$475$$ 6.60197 0.302919
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −14.0163 −0.640420 −0.320210 0.947347i $$-0.603753\pi$$
−0.320210 + 0.947347i $$0.603753\pi$$
$$480$$ 0 0
$$481$$ −38.1425 −1.73915
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −16.3916 −0.744304
$$486$$ 0 0
$$487$$ −37.5764 −1.70275 −0.851374 0.524559i $$-0.824230\pi$$
−0.851374 + 0.524559i $$0.824230\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −5.24309 −0.236617 −0.118309 0.992977i $$-0.537747\pi$$
−0.118309 + 0.992977i $$0.537747\pi$$
$$492$$ 0 0
$$493$$ −19.6803 −0.886356
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 32.5733 1.46111
$$498$$ 0 0
$$499$$ 29.0542 1.30065 0.650323 0.759658i $$-0.274633\pi$$
0.650323 + 0.759658i $$0.274633\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 31.5512 1.40680 0.703399 0.710796i $$-0.251665\pi$$
0.703399 + 0.710796i $$0.251665\pi$$
$$504$$ 0 0
$$505$$ −55.3373 −2.46248
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −17.5097 −0.776104 −0.388052 0.921638i $$-0.626852\pi$$
−0.388052 + 0.921638i $$0.626852\pi$$
$$510$$ 0 0
$$511$$ −15.7421 −0.696391
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −26.1908 −1.15410
$$516$$ 0 0
$$517$$ −13.8482 −0.609042
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 37.8780 1.65947 0.829733 0.558161i $$-0.188493\pi$$
0.829733 + 0.558161i $$0.188493\pi$$
$$522$$ 0 0
$$523$$ 28.3739 1.24070 0.620352 0.784324i $$-0.286990\pi$$
0.620352 + 0.784324i $$0.286990\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 25.7234 1.12053
$$528$$ 0 0
$$529$$ 39.5718 1.72051
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −10.5087 −0.455182
$$534$$ 0 0
$$535$$ −40.4945 −1.75073
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.02600 −0.0441930
$$540$$ 0 0
$$541$$ 38.5868 1.65898 0.829488 0.558524i $$-0.188632\pi$$
0.829488 + 0.558524i $$0.188632\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 19.2615 0.825071
$$546$$ 0 0
$$547$$ −13.3947 −0.572717 −0.286359 0.958123i $$-0.592445\pi$$
−0.286359 + 0.958123i $$0.592445\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −6.22101 −0.265024
$$552$$ 0 0
$$553$$ −28.1816 −1.19841
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −4.46471 −0.189176 −0.0945879 0.995517i $$-0.530153\pi$$
−0.0945879 + 0.995517i $$0.530153\pi$$
$$558$$ 0 0
$$559$$ 10.6811 0.451762
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 22.1171 0.932124 0.466062 0.884752i $$-0.345672\pi$$
0.466062 + 0.884752i $$0.345672\pi$$
$$564$$ 0 0
$$565$$ −30.0163 −1.26279
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.70238 −0.280978 −0.140489 0.990082i $$-0.544868\pi$$
−0.140489 + 0.990082i $$0.544868\pi$$
$$570$$ 0 0
$$571$$ −6.37533 −0.266799 −0.133400 0.991062i $$-0.542589\pi$$
−0.133400 + 0.991062i $$0.542589\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 52.2231 2.17785
$$576$$ 0 0
$$577$$ 44.5654 1.85528 0.927641 0.373474i $$-0.121834\pi$$
0.927641 + 0.373474i $$0.121834\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4.51783 −0.187431
$$582$$ 0 0
$$583$$ −3.91917 −0.162315
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 29.6311 1.22301 0.611503 0.791242i $$-0.290565\pi$$
0.611503 + 0.791242i $$0.290565\pi$$
$$588$$ 0 0
$$589$$ 8.13124 0.335042
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −43.6572 −1.79279 −0.896393 0.443259i $$-0.853822\pi$$
−0.896393 + 0.443259i $$0.853822\pi$$
$$594$$ 0 0
$$595$$ −26.9825 −1.10618
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 12.5661 0.513438 0.256719 0.966486i $$-0.417359\pi$$
0.256719 + 0.966486i $$0.417359\pi$$
$$600$$ 0 0
$$601$$ −37.9213 −1.54684 −0.773421 0.633893i $$-0.781456\pi$$
−0.773421 + 0.633893i $$0.781456\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 30.7328 1.24947
$$606$$ 0 0
$$607$$ 1.93332 0.0784710 0.0392355 0.999230i $$-0.487508\pi$$
0.0392355 + 0.999230i $$0.487508\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −61.2656 −2.47854
$$612$$ 0 0
$$613$$ 18.4306 0.744404 0.372202 0.928152i $$-0.378603\pi$$
0.372202 + 0.928152i $$0.378603\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.5314 −1.22915 −0.614574 0.788859i $$-0.710672\pi$$
−0.614574 + 0.788859i $$0.710672\pi$$
$$618$$ 0 0
$$619$$ 28.9094 1.16197 0.580984 0.813915i $$-0.302668\pi$$
0.580984 + 0.813915i $$0.302668\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 18.9665 0.759875
$$624$$ 0 0
$$625$$ −14.4238 −0.576954
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −19.3963 −0.773382
$$630$$ 0 0
$$631$$ 32.5314 1.29505 0.647527 0.762042i $$-0.275803\pi$$
0.647527 + 0.762042i $$0.275803\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −41.0457 −1.62885
$$636$$ 0 0
$$637$$ −4.53912 −0.179846
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.1312 1.19011 0.595056 0.803684i $$-0.297130\pi$$
0.595056 + 0.803684i $$0.297130\pi$$
$$642$$ 0 0
$$643$$ 20.1752 0.795633 0.397817 0.917465i $$-0.369768\pi$$
0.397817 + 0.917465i $$0.369768\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −14.2741 −0.561174 −0.280587 0.959829i $$-0.590529\pi$$
−0.280587 + 0.959829i $$0.590529\pi$$
$$648$$ 0 0
$$649$$ −13.0081 −0.510614
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −9.21538 −0.360626 −0.180313 0.983609i $$-0.557711\pi$$
−0.180313 + 0.983609i $$0.557711\pi$$
$$654$$ 0 0
$$655$$ 3.13666 0.122560
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 12.1511 0.473339 0.236669 0.971590i $$-0.423944\pi$$
0.236669 + 0.971590i $$0.423944\pi$$
$$660$$ 0 0
$$661$$ 13.6924 0.532574 0.266287 0.963894i $$-0.414203\pi$$
0.266287 + 0.963894i $$0.414203\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −8.52927 −0.330751
$$666$$ 0 0
$$667$$ −49.2096 −1.90540
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 9.18124 0.354438
$$672$$ 0 0
$$673$$ 40.5341 1.56247 0.781237 0.624234i $$-0.214589\pi$$
0.781237 + 0.624234i $$0.214589\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −11.5653 −0.444492 −0.222246 0.974991i $$-0.571339\pi$$
−0.222246 + 0.974991i $$0.571339\pi$$
$$678$$ 0 0
$$679$$ 12.0504 0.462452
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −44.2625 −1.69366 −0.846828 0.531866i $$-0.821491\pi$$
−0.846828 + 0.531866i $$0.821491\pi$$
$$684$$ 0 0
$$685$$ −1.79064 −0.0684168
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −17.3387 −0.660554
$$690$$ 0 0
$$691$$ −32.6676 −1.24274 −0.621368 0.783519i $$-0.713423\pi$$
−0.621368 + 0.783519i $$0.713423\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 15.4275 0.585197
$$696$$ 0 0
$$697$$ −5.34391 −0.202415
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8.73150 0.329784 0.164892 0.986312i $$-0.447272\pi$$
0.164892 + 0.986312i $$0.447272\pi$$
$$702$$ 0 0
$$703$$ −6.13124 −0.231244
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 40.6816 1.52999
$$708$$ 0 0
$$709$$ 14.3058 0.537264 0.268632 0.963243i $$-0.413428\pi$$
0.268632 + 0.963243i $$0.413428\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 64.3200 2.40880
$$714$$ 0 0
$$715$$ −29.7964 −1.11432
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 21.4959 0.801663 0.400831 0.916152i $$-0.368721\pi$$
0.400831 + 0.916152i $$0.368721\pi$$
$$720$$ 0 0
$$721$$ 19.2543 0.717069
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −41.0709 −1.52533
$$726$$ 0 0
$$727$$ −47.6002 −1.76539 −0.882696 0.469944i $$-0.844274\pi$$
−0.882696 + 0.469944i $$0.844274\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 5.43158 0.200894
$$732$$ 0 0
$$733$$ −49.7239 −1.83659 −0.918297 0.395892i $$-0.870435\pi$$
−0.918297 + 0.395892i $$0.870435\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 2.63280 0.0969805
$$738$$ 0 0
$$739$$ 35.5283 1.30693 0.653464 0.756957i $$-0.273315\pi$$
0.653464 + 0.756957i $$0.273315\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −20.3612 −0.746979 −0.373490 0.927634i $$-0.621839\pi$$
−0.373490 + 0.927634i $$0.621839\pi$$
$$744$$ 0 0
$$745$$ −64.6194 −2.36747
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 29.7699 1.08777
$$750$$ 0 0
$$751$$ 18.9868 0.692840 0.346420 0.938080i $$-0.387397\pi$$
0.346420 + 0.938080i $$0.387397\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −22.3342 −0.812824
$$756$$ 0 0
$$757$$ −22.4738 −0.816826 −0.408413 0.912797i $$-0.633918\pi$$
−0.408413 + 0.912797i $$0.633918\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 1.40973 0.0511028 0.0255514 0.999674i $$-0.491866\pi$$
0.0255514 + 0.999674i $$0.491866\pi$$
$$762$$ 0 0
$$763$$ −14.1602 −0.512634
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −57.5492 −2.07798
$$768$$ 0 0
$$769$$ −0.114116 −0.00411512 −0.00205756 0.999998i $$-0.500655\pi$$
−0.00205756 + 0.999998i $$0.500655\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 46.1799 1.66097 0.830487 0.557038i $$-0.188062\pi$$
0.830487 + 0.557038i $$0.188062\pi$$
$$774$$ 0 0
$$775$$ 53.6822 1.92832
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −1.68923 −0.0605228
$$780$$ 0 0
$$781$$ 18.2916 0.654525
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −59.8297 −2.13541
$$786$$ 0 0
$$787$$ 12.6847 0.452159 0.226080 0.974109i $$-0.427409\pi$$
0.226080 + 0.974109i $$0.427409\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 22.0667 0.784601
$$792$$ 0 0
$$793$$ 40.6186 1.44241
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −39.2796 −1.39135 −0.695677 0.718355i $$-0.744895\pi$$
−0.695677 + 0.718355i $$0.744895\pi$$
$$798$$ 0 0
$$799$$ −31.1549 −1.10218
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −8.84004 −0.311958
$$804$$ 0 0
$$805$$ −67.4685 −2.37795
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1.09896 0.0386374 0.0193187 0.999813i $$-0.493850\pi$$
0.0193187 + 0.999813i $$0.493850\pi$$
$$810$$ 0 0
$$811$$ 36.5627 1.28389 0.641944 0.766751i $$-0.278128\pi$$
0.641944 + 0.766751i $$0.278128\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 78.1496 2.73746
$$816$$ 0 0
$$817$$ 1.71694 0.0600681
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 17.8823 0.624097 0.312049 0.950066i $$-0.398985\pi$$
0.312049 + 0.950066i $$0.398985\pi$$
$$822$$ 0 0
$$823$$ −8.31328 −0.289783 −0.144891 0.989448i $$-0.546283\pi$$
−0.144891 + 0.989448i $$0.546283\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −0.280204 −0.00974363 −0.00487182 0.999988i $$-0.501551\pi$$
−0.00487182 + 0.999988i $$0.501551\pi$$
$$828$$ 0 0
$$829$$ 6.04148 0.209829 0.104915 0.994481i $$-0.466543\pi$$
0.104915 + 0.994481i $$0.466543\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −2.30824 −0.0799759
$$834$$ 0 0
$$835$$ 21.2756 0.736274
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −29.5942 −1.02171 −0.510853 0.859668i $$-0.670670\pi$$
−0.510853 + 0.859668i $$0.670670\pi$$
$$840$$ 0 0
$$841$$ 9.70093 0.334515
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −87.5416 −3.01152
$$846$$ 0 0
$$847$$ −22.5934 −0.776320
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −48.4996 −1.66254
$$852$$ 0 0
$$853$$ 15.2331 0.521570 0.260785 0.965397i $$-0.416019\pi$$
0.260785 + 0.965397i $$0.416019\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 2.25535 0.0770412 0.0385206 0.999258i $$-0.487735\pi$$
0.0385206 + 0.999258i $$0.487735\pi$$
$$858$$ 0 0
$$859$$ −1.91686 −0.0654025 −0.0327013 0.999465i $$-0.510411\pi$$
−0.0327013 + 0.999465i $$0.510411\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 35.8530 1.22045 0.610225 0.792228i $$-0.291079\pi$$
0.610225 + 0.792228i $$0.291079\pi$$
$$864$$ 0 0
$$865$$ −57.7126 −1.96229
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −15.8255 −0.536843
$$870$$ 0 0
$$871$$ 11.6478 0.394669
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −13.6636 −0.461915
$$876$$ 0 0
$$877$$ 48.2323 1.62869 0.814344 0.580383i $$-0.197097\pi$$
0.814344 + 0.580383i $$0.197097\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −4.77336 −0.160819 −0.0804094 0.996762i $$-0.525623\pi$$
−0.0804094 + 0.996762i $$0.525623\pi$$
$$882$$ 0 0
$$883$$ −22.2144 −0.747573 −0.373787 0.927515i $$-0.621941\pi$$
−0.373787 + 0.927515i $$0.621941\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −42.1729 −1.41603 −0.708014 0.706198i $$-0.750409\pi$$
−0.708014 + 0.706198i $$0.750409\pi$$
$$888$$ 0 0
$$889$$ 30.1750 1.01204
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −9.84818 −0.329557
$$894$$ 0 0
$$895$$ 62.1496 2.07743
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −50.5845 −1.68709
$$900$$ 0 0
$$901$$ −8.81714 −0.293742
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −37.2202 −1.23724
$$906$$ 0 0
$$907$$ −26.7564 −0.888430 −0.444215 0.895920i $$-0.646517\pi$$
−0.444215 + 0.895920i $$0.646517\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 11.1281 0.368691 0.184346 0.982861i $$-0.440983\pi$$
0.184346 + 0.982861i $$0.440983\pi$$
$$912$$ 0 0
$$913$$ −2.53700 −0.0839625
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −2.30594 −0.0761489
$$918$$ 0 0
$$919$$ −5.45195 −0.179843 −0.0899216 0.995949i $$-0.528662\pi$$
−0.0899216 + 0.995949i $$0.528662\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 80.9237 2.66364
$$924$$ 0 0
$$925$$ −40.4783 −1.33092
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1.94402 0.0637813 0.0318906 0.999491i $$-0.489847\pi$$
0.0318906 + 0.999491i $$0.489847\pi$$
$$930$$ 0 0
$$931$$ −0.729644 −0.0239131
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −15.1521 −0.495527
$$936$$ 0 0
$$937$$ −11.2897 −0.368820 −0.184410 0.982849i $$-0.559037\pi$$
−0.184410 + 0.982849i $$0.559037\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −51.4229 −1.67634 −0.838170 0.545409i $$-0.816374\pi$$
−0.838170 + 0.545409i $$0.816374\pi$$
$$942$$ 0 0
$$943$$ −13.3622 −0.435133
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1.94045 0.0630563 0.0315281 0.999503i $$-0.489963\pi$$
0.0315281 + 0.999503i $$0.489963\pi$$
$$948$$ 0 0
$$949$$ −39.1092 −1.26954
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 23.4523 0.759693 0.379847 0.925049i $$-0.375977\pi$$
0.379847 + 0.925049i $$0.375977\pi$$
$$954$$ 0 0
$$955$$ −52.8371 −1.70977
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 1.31640 0.0425088
$$960$$ 0 0
$$961$$ 35.1171 1.13281
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 32.5804 1.04880
$$966$$ 0 0
$$967$$ 59.0194 1.89794 0.948968 0.315373i $$-0.102130\pi$$
0.948968 + 0.315373i $$0.102130\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −39.7743 −1.27642 −0.638209 0.769863i $$-0.720324\pi$$
−0.638209 + 0.769863i $$0.720324\pi$$
$$972$$ 0 0
$$973$$ −11.3416 −0.363595
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 41.9538 1.34222 0.671111 0.741357i $$-0.265817\pi$$
0.671111 + 0.741357i $$0.265817\pi$$
$$978$$ 0 0
$$979$$ 10.6507 0.340397
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −7.93132 −0.252970 −0.126485 0.991969i $$-0.540370\pi$$
−0.126485 + 0.991969i $$0.540370\pi$$
$$984$$ 0 0
$$985$$ 65.1919 2.07719
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 13.5814 0.431863
$$990$$ 0 0
$$991$$ 24.8408 0.789093 0.394546 0.918876i $$-0.370902\pi$$
0.394546 + 0.918876i $$0.370902\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0.455573 0.0144426
$$996$$ 0 0
$$997$$ 28.1157 0.890433 0.445216 0.895423i $$-0.353127\pi$$
0.445216 + 0.895423i $$0.353127\pi$$
$$998$$ 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 5472.2.a.bs.1.1 4
3.2 odd 2 608.2.a.j.1.2 yes 4
4.3 odd 2 5472.2.a.bt.1.1 4
12.11 even 2 608.2.a.i.1.4 4
24.5 odd 2 1216.2.a.w.1.3 4
24.11 even 2 1216.2.a.x.1.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
608.2.a.i.1.4 4 12.11 even 2
608.2.a.j.1.2 yes 4 3.2 odd 2
1216.2.a.w.1.3 4 24.5 odd 2
1216.2.a.x.1.1 4 24.11 even 2
5472.2.a.bs.1.1 4 1.1 even 1 trivial
5472.2.a.bt.1.1 4 4.3 odd 2