# Properties

 Label 9072.2.a.bp.1.2 Level $9072$ Weight $2$ Character 9072.1 Self dual yes Analytic conductor $72.440$ 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] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9072 = 2^{4} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9072.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: $$72.4402847137$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1134) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 9072.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.73205 q^{5} +1.00000 q^{7} +O(q^{10})$$ $$q+3.73205 q^{5} +1.00000 q^{7} +4.19615 q^{11} +0.464102 q^{13} +7.00000 q^{17} +2.73205 q^{19} -6.19615 q^{23} +8.92820 q^{25} +8.46410 q^{29} +2.19615 q^{31} +3.73205 q^{35} -6.66025 q^{37} +9.46410 q^{41} -5.46410 q^{43} -1.26795 q^{47} +1.00000 q^{49} -2.53590 q^{53} +15.6603 q^{55} +6.19615 q^{59} -9.92820 q^{61} +1.73205 q^{65} +3.26795 q^{67} -13.4641 q^{71} +11.7321 q^{73} +4.19615 q^{77} -15.1244 q^{79} +14.5885 q^{83} +26.1244 q^{85} -3.92820 q^{89} +0.464102 q^{91} +10.1962 q^{95} -2.92820 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q + 4 * q^5 + 2 * q^7 $$2 q + 4 q^{5} + 2 q^{7} - 2 q^{11} - 6 q^{13} + 14 q^{17} + 2 q^{19} - 2 q^{23} + 4 q^{25} + 10 q^{29} - 6 q^{31} + 4 q^{35} + 4 q^{37} + 12 q^{41} - 4 q^{43} - 6 q^{47} + 2 q^{49} - 12 q^{53} + 14 q^{55} + 2 q^{59} - 6 q^{61} + 10 q^{67} - 20 q^{71} + 20 q^{73} - 2 q^{77} - 6 q^{79} - 2 q^{83} + 28 q^{85} + 6 q^{89} - 6 q^{91} + 10 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q + 4 * q^5 + 2 * q^7 - 2 * q^11 - 6 * q^13 + 14 * q^17 + 2 * q^19 - 2 * q^23 + 4 * q^25 + 10 * q^29 - 6 * q^31 + 4 * q^35 + 4 * q^37 + 12 * q^41 - 4 * q^43 - 6 * q^47 + 2 * q^49 - 12 * q^53 + 14 * q^55 + 2 * q^59 - 6 * q^61 + 10 * q^67 - 20 * q^71 + 20 * q^73 - 2 * q^77 - 6 * q^79 - 2 * q^83 + 28 * q^85 + 6 * q^89 - 6 * q^91 + 10 * 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
$$5$$ 3.73205 1.66902 0.834512 0.550990i $$-0.185750\pi$$
0.834512 + 0.550990i $$0.185750\pi$$
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 4.19615 1.26519 0.632594 0.774484i $$-0.281990\pi$$
0.632594 + 0.774484i $$0.281990\pi$$
$$12$$ 0 0
$$13$$ 0.464102 0.128719 0.0643593 0.997927i $$-0.479500\pi$$
0.0643593 + 0.997927i $$0.479500\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.00000 1.69775 0.848875 0.528594i $$-0.177281\pi$$
0.848875 + 0.528594i $$0.177281\pi$$
$$18$$ 0 0
$$19$$ 2.73205 0.626775 0.313388 0.949625i $$-0.398536\pi$$
0.313388 + 0.949625i $$0.398536\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.19615 −1.29199 −0.645994 0.763343i $$-0.723557\pi$$
−0.645994 + 0.763343i $$0.723557\pi$$
$$24$$ 0 0
$$25$$ 8.92820 1.78564
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 8.46410 1.57174 0.785872 0.618389i $$-0.212214\pi$$
0.785872 + 0.618389i $$0.212214\pi$$
$$30$$ 0 0
$$31$$ 2.19615 0.394441 0.197220 0.980359i $$-0.436809\pi$$
0.197220 + 0.980359i $$0.436809\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.73205 0.630832
$$36$$ 0 0
$$37$$ −6.66025 −1.09494 −0.547470 0.836826i $$-0.684409\pi$$
−0.547470 + 0.836826i $$0.684409\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.46410 1.47804 0.739022 0.673681i $$-0.235288\pi$$
0.739022 + 0.673681i $$0.235288\pi$$
$$42$$ 0 0
$$43$$ −5.46410 −0.833268 −0.416634 0.909074i $$-0.636790\pi$$
−0.416634 + 0.909074i $$0.636790\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −1.26795 −0.184949 −0.0924747 0.995715i $$-0.529478\pi$$
−0.0924747 + 0.995715i $$0.529478\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −2.53590 −0.348332 −0.174166 0.984716i $$-0.555723\pi$$
−0.174166 + 0.984716i $$0.555723\pi$$
$$54$$ 0 0
$$55$$ 15.6603 2.11163
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 6.19615 0.806670 0.403335 0.915052i $$-0.367851\pi$$
0.403335 + 0.915052i $$0.367851\pi$$
$$60$$ 0 0
$$61$$ −9.92820 −1.27118 −0.635588 0.772028i $$-0.719242\pi$$
−0.635588 + 0.772028i $$0.719242\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.73205 0.214834
$$66$$ 0 0
$$67$$ 3.26795 0.399244 0.199622 0.979873i $$-0.436029\pi$$
0.199622 + 0.979873i $$0.436029\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −13.4641 −1.59789 −0.798947 0.601401i $$-0.794609\pi$$
−0.798947 + 0.601401i $$0.794609\pi$$
$$72$$ 0 0
$$73$$ 11.7321 1.37313 0.686566 0.727067i $$-0.259117\pi$$
0.686566 + 0.727067i $$0.259117\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 4.19615 0.478196
$$78$$ 0 0
$$79$$ −15.1244 −1.70162 −0.850811 0.525471i $$-0.823889\pi$$
−0.850811 + 0.525471i $$0.823889\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 14.5885 1.60129 0.800646 0.599138i $$-0.204490\pi$$
0.800646 + 0.599138i $$0.204490\pi$$
$$84$$ 0 0
$$85$$ 26.1244 2.83358
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.92820 −0.416389 −0.208194 0.978087i $$-0.566759\pi$$
−0.208194 + 0.978087i $$0.566759\pi$$
$$90$$ 0 0
$$91$$ 0.464102 0.0486511
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 10.1962 1.04610
$$96$$ 0 0
$$97$$ −2.92820 −0.297314 −0.148657 0.988889i $$-0.547495\pi$$
−0.148657 + 0.988889i $$0.547495\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.92820 −0.490375 −0.245187 0.969476i $$-0.578849\pi$$
−0.245187 + 0.969476i $$0.578849\pi$$
$$102$$ 0 0
$$103$$ −12.3923 −1.22105 −0.610525 0.791997i $$-0.709042\pi$$
−0.610525 + 0.791997i $$0.709042\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ −7.19615 −0.689266 −0.344633 0.938737i $$-0.611997\pi$$
−0.344633 + 0.938737i $$0.611997\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.26795 −0.213351 −0.106675 0.994294i $$-0.534021\pi$$
−0.106675 + 0.994294i $$0.534021\pi$$
$$114$$ 0 0
$$115$$ −23.1244 −2.15636
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 7.00000 0.641689
$$120$$ 0 0
$$121$$ 6.60770 0.600700
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 14.6603 1.31125
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −17.4641 −1.52585 −0.762923 0.646490i $$-0.776236\pi$$
−0.762923 + 0.646490i $$0.776236\pi$$
$$132$$ 0 0
$$133$$ 2.73205 0.236899
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −11.7321 −1.00234 −0.501168 0.865350i $$-0.667096\pi$$
−0.501168 + 0.865350i $$0.667096\pi$$
$$138$$ 0 0
$$139$$ 6.73205 0.571005 0.285503 0.958378i $$-0.407839\pi$$
0.285503 + 0.958378i $$0.407839\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 1.94744 0.162853
$$144$$ 0 0
$$145$$ 31.5885 2.62328
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 9.00000 0.737309 0.368654 0.929567i $$-0.379819\pi$$
0.368654 + 0.929567i $$0.379819\pi$$
$$150$$ 0 0
$$151$$ 16.1962 1.31802 0.659012 0.752132i $$-0.270975\pi$$
0.659012 + 0.752132i $$0.270975\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 8.19615 0.658331
$$156$$ 0 0
$$157$$ −1.00000 −0.0798087 −0.0399043 0.999204i $$-0.512705\pi$$
−0.0399043 + 0.999204i $$0.512705\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.19615 −0.488325
$$162$$ 0 0
$$163$$ 6.53590 0.511931 0.255966 0.966686i $$-0.417607\pi$$
0.255966 + 0.966686i $$0.417607\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 12.1962 0.943767 0.471883 0.881661i $$-0.343574\pi$$
0.471883 + 0.881661i $$0.343574\pi$$
$$168$$ 0 0
$$169$$ −12.7846 −0.983432
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 9.73205 0.739914 0.369957 0.929049i $$-0.379372\pi$$
0.369957 + 0.929049i $$0.379372\pi$$
$$174$$ 0 0
$$175$$ 8.92820 0.674909
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −8.19615 −0.612609 −0.306305 0.951934i $$-0.599093\pi$$
−0.306305 + 0.951934i $$0.599093\pi$$
$$180$$ 0 0
$$181$$ 4.39230 0.326477 0.163239 0.986587i $$-0.447806\pi$$
0.163239 + 0.986587i $$0.447806\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −24.8564 −1.82748
$$186$$ 0 0
$$187$$ 29.3731 2.14797
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −11.6603 −0.843706 −0.421853 0.906664i $$-0.638620\pi$$
−0.421853 + 0.906664i $$0.638620\pi$$
$$192$$ 0 0
$$193$$ −8.85641 −0.637498 −0.318749 0.947839i $$-0.603263\pi$$
−0.318749 + 0.947839i $$0.603263\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −25.7846 −1.83708 −0.918539 0.395331i $$-0.870630\pi$$
−0.918539 + 0.395331i $$0.870630\pi$$
$$198$$ 0 0
$$199$$ −5.12436 −0.363256 −0.181628 0.983367i $$-0.558137\pi$$
−0.181628 + 0.983367i $$0.558137\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 8.46410 0.594063
$$204$$ 0 0
$$205$$ 35.3205 2.46689
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 11.4641 0.792988
$$210$$ 0 0
$$211$$ 20.7321 1.42725 0.713627 0.700526i $$-0.247051\pi$$
0.713627 + 0.700526i $$0.247051\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −20.3923 −1.39074
$$216$$ 0 0
$$217$$ 2.19615 0.149085
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.24871 0.218532
$$222$$ 0 0
$$223$$ 18.5359 1.24126 0.620628 0.784105i $$-0.286878\pi$$
0.620628 + 0.784105i $$0.286878\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5.07180 0.336627 0.168313 0.985734i $$-0.446168\pi$$
0.168313 + 0.985734i $$0.446168\pi$$
$$228$$ 0 0
$$229$$ 4.46410 0.294996 0.147498 0.989062i $$-0.452878\pi$$
0.147498 + 0.989062i $$0.452878\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −13.1962 −0.864509 −0.432254 0.901752i $$-0.642282\pi$$
−0.432254 + 0.901752i $$0.642282\pi$$
$$234$$ 0 0
$$235$$ −4.73205 −0.308685
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −28.0526 −1.81457 −0.907285 0.420517i $$-0.861849\pi$$
−0.907285 + 0.420517i $$0.861849\pi$$
$$240$$ 0 0
$$241$$ −17.7321 −1.14222 −0.571111 0.820873i $$-0.693487\pi$$
−0.571111 + 0.820873i $$0.693487\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3.73205 0.238432
$$246$$ 0 0
$$247$$ 1.26795 0.0806777
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −16.0526 −1.01323 −0.506614 0.862173i $$-0.669103\pi$$
−0.506614 + 0.862173i $$0.669103\pi$$
$$252$$ 0 0
$$253$$ −26.0000 −1.63461
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −0.464102 −0.0289499 −0.0144749 0.999895i $$-0.504608\pi$$
−0.0144749 + 0.999895i $$0.504608\pi$$
$$258$$ 0 0
$$259$$ −6.66025 −0.413848
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 23.6603 1.45895 0.729477 0.684005i $$-0.239764\pi$$
0.729477 + 0.684005i $$0.239764\pi$$
$$264$$ 0 0
$$265$$ −9.46410 −0.581375
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −25.5885 −1.56016 −0.780078 0.625682i $$-0.784821\pi$$
−0.780078 + 0.625682i $$0.784821\pi$$
$$270$$ 0 0
$$271$$ −25.5167 −1.55003 −0.775013 0.631945i $$-0.782257\pi$$
−0.775013 + 0.631945i $$0.782257\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 37.4641 2.25917
$$276$$ 0 0
$$277$$ 22.7846 1.36899 0.684497 0.729015i $$-0.260022\pi$$
0.684497 + 0.729015i $$0.260022\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −2.80385 −0.167264 −0.0836318 0.996497i $$-0.526652\pi$$
−0.0836318 + 0.996497i $$0.526652\pi$$
$$282$$ 0 0
$$283$$ 19.3205 1.14848 0.574242 0.818685i $$-0.305297\pi$$
0.574242 + 0.818685i $$0.305297\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 9.46410 0.558648
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 20.6603 1.20698 0.603492 0.797369i $$-0.293775\pi$$
0.603492 + 0.797369i $$0.293775\pi$$
$$294$$ 0 0
$$295$$ 23.1244 1.34635
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2.87564 −0.166303
$$300$$ 0 0
$$301$$ −5.46410 −0.314946
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −37.0526 −2.12162
$$306$$ 0 0
$$307$$ −5.85641 −0.334243 −0.167121 0.985936i $$-0.553447\pi$$
−0.167121 + 0.985936i $$0.553447\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0.196152 0.0111228 0.00556139 0.999985i $$-0.498230\pi$$
0.00556139 + 0.999985i $$0.498230\pi$$
$$312$$ 0 0
$$313$$ −5.58846 −0.315878 −0.157939 0.987449i $$-0.550485\pi$$
−0.157939 + 0.987449i $$0.550485\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10.6077 −0.595788 −0.297894 0.954599i $$-0.596284\pi$$
−0.297894 + 0.954599i $$0.596284\pi$$
$$318$$ 0 0
$$319$$ 35.5167 1.98855
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 19.1244 1.06411
$$324$$ 0 0
$$325$$ 4.14359 0.229845
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −1.26795 −0.0699043
$$330$$ 0 0
$$331$$ 8.39230 0.461283 0.230641 0.973039i $$-0.425918\pi$$
0.230641 + 0.973039i $$0.425918\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 12.1962 0.666347
$$336$$ 0 0
$$337$$ 4.39230 0.239264 0.119632 0.992818i $$-0.461829\pi$$
0.119632 + 0.992818i $$0.461829\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 9.21539 0.499041
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −14.5359 −0.780328 −0.390164 0.920745i $$-0.627582\pi$$
−0.390164 + 0.920745i $$0.627582\pi$$
$$348$$ 0 0
$$349$$ −5.46410 −0.292487 −0.146243 0.989249i $$-0.546718\pi$$
−0.146243 + 0.989249i $$0.546718\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ −50.2487 −2.66692
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2.92820 −0.154545 −0.0772723 0.997010i $$-0.524621\pi$$
−0.0772723 + 0.997010i $$0.524621\pi$$
$$360$$ 0 0
$$361$$ −11.5359 −0.607153
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 43.7846 2.29179
$$366$$ 0 0
$$367$$ 13.1244 0.685086 0.342543 0.939502i $$-0.388712\pi$$
0.342543 + 0.939502i $$0.388712\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −2.53590 −0.131657
$$372$$ 0 0
$$373$$ −33.8564 −1.75302 −0.876509 0.481385i $$-0.840134\pi$$
−0.876509 + 0.481385i $$0.840134\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 3.92820 0.202313
$$378$$ 0 0
$$379$$ −17.5167 −0.899770 −0.449885 0.893086i $$-0.648535\pi$$
−0.449885 + 0.893086i $$0.648535\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 35.7128 1.82484 0.912420 0.409256i $$-0.134212\pi$$
0.912420 + 0.409256i $$0.134212\pi$$
$$384$$ 0 0
$$385$$ 15.6603 0.798120
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −6.53590 −0.331383 −0.165692 0.986178i $$-0.552986\pi$$
−0.165692 + 0.986178i $$0.552986\pi$$
$$390$$ 0 0
$$391$$ −43.3731 −2.19347
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −56.4449 −2.84005
$$396$$ 0 0
$$397$$ −21.0000 −1.05396 −0.526980 0.849878i $$-0.676676\pi$$
−0.526980 + 0.849878i $$0.676676\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −34.5167 −1.72368 −0.861840 0.507180i $$-0.830688\pi$$
−0.861840 + 0.507180i $$0.830688\pi$$
$$402$$ 0 0
$$403$$ 1.01924 0.0507719
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −27.9474 −1.38530
$$408$$ 0 0
$$409$$ 34.6603 1.71384 0.856920 0.515450i $$-0.172375\pi$$
0.856920 + 0.515450i $$0.172375\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 6.19615 0.304893
$$414$$ 0 0
$$415$$ 54.4449 2.67259
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 2.53590 0.123887 0.0619434 0.998080i $$-0.480270\pi$$
0.0619434 + 0.998080i $$0.480270\pi$$
$$420$$ 0 0
$$421$$ −24.1244 −1.17575 −0.587875 0.808952i $$-0.700035\pi$$
−0.587875 + 0.808952i $$0.700035\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 62.4974 3.03157
$$426$$ 0 0
$$427$$ −9.92820 −0.480459
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 21.4641 1.03389 0.516945 0.856019i $$-0.327069\pi$$
0.516945 + 0.856019i $$0.327069\pi$$
$$432$$ 0 0
$$433$$ 12.2679 0.589560 0.294780 0.955565i $$-0.404754\pi$$
0.294780 + 0.955565i $$0.404754\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −16.9282 −0.809786
$$438$$ 0 0
$$439$$ 11.3205 0.540298 0.270149 0.962818i $$-0.412927\pi$$
0.270149 + 0.962818i $$0.412927\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.7321 0.889987 0.444993 0.895534i $$-0.353206\pi$$
0.444993 + 0.895534i $$0.353206\pi$$
$$444$$ 0 0
$$445$$ −14.6603 −0.694963
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −11.8564 −0.559538 −0.279769 0.960067i $$-0.590258\pi$$
−0.279769 + 0.960067i $$0.590258\pi$$
$$450$$ 0 0
$$451$$ 39.7128 1.87000
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 1.73205 0.0811998
$$456$$ 0 0
$$457$$ 20.8564 0.975622 0.487811 0.872949i $$-0.337796\pi$$
0.487811 + 0.872949i $$0.337796\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 34.7846 1.62008 0.810040 0.586374i $$-0.199445\pi$$
0.810040 + 0.586374i $$0.199445\pi$$
$$462$$ 0 0
$$463$$ 32.5885 1.51451 0.757257 0.653117i $$-0.226539\pi$$
0.757257 + 0.653117i $$0.226539\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −14.5885 −0.675073 −0.337537 0.941312i $$-0.609594\pi$$
−0.337537 + 0.941312i $$0.609594\pi$$
$$468$$ 0 0
$$469$$ 3.26795 0.150900
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −22.9282 −1.05424
$$474$$ 0 0
$$475$$ 24.3923 1.11920
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −23.5167 −1.07450 −0.537252 0.843422i $$-0.680538\pi$$
−0.537252 + 0.843422i $$0.680538\pi$$
$$480$$ 0 0
$$481$$ −3.09103 −0.140939
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.9282 −0.496224
$$486$$ 0 0
$$487$$ 28.5885 1.29547 0.647733 0.761867i $$-0.275717\pi$$
0.647733 + 0.761867i $$0.275717\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −33.4641 −1.51021 −0.755107 0.655602i $$-0.772415\pi$$
−0.755107 + 0.655602i $$0.772415\pi$$
$$492$$ 0 0
$$493$$ 59.2487 2.66843
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −13.4641 −0.603947
$$498$$ 0 0
$$499$$ −30.1962 −1.35177 −0.675883 0.737009i $$-0.736237\pi$$
−0.675883 + 0.737009i $$0.736237\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −1.94744 −0.0868321 −0.0434161 0.999057i $$-0.513824\pi$$
−0.0434161 + 0.999057i $$0.513824\pi$$
$$504$$ 0 0
$$505$$ −18.3923 −0.818447
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 4.14359 0.183662 0.0918308 0.995775i $$-0.470728\pi$$
0.0918308 + 0.995775i $$0.470728\pi$$
$$510$$ 0 0
$$511$$ 11.7321 0.518995
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −46.2487 −2.03796
$$516$$ 0 0
$$517$$ −5.32051 −0.233996
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −30.0000 −1.31432 −0.657162 0.753749i $$-0.728243\pi$$
−0.657162 + 0.753749i $$0.728243\pi$$
$$522$$ 0 0
$$523$$ 29.1769 1.27582 0.637909 0.770112i $$-0.279800\pi$$
0.637909 + 0.770112i $$0.279800\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.3731 0.669661
$$528$$ 0 0
$$529$$ 15.3923 0.669231
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 4.39230 0.190252
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 4.19615 0.180741
$$540$$ 0 0
$$541$$ 20.6603 0.888254 0.444127 0.895964i $$-0.353514\pi$$
0.444127 + 0.895964i $$0.353514\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −26.8564 −1.15040
$$546$$ 0 0
$$547$$ 19.2679 0.823838 0.411919 0.911220i $$-0.364859\pi$$
0.411919 + 0.911220i $$0.364859\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 23.1244 0.985131
$$552$$ 0 0
$$553$$ −15.1244 −0.643153
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10.0718 0.426756 0.213378 0.976970i $$-0.431553\pi$$
0.213378 + 0.976970i $$0.431553\pi$$
$$558$$ 0 0
$$559$$ −2.53590 −0.107257
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 35.7128 1.50512 0.752558 0.658526i $$-0.228820\pi$$
0.752558 + 0.658526i $$0.228820\pi$$
$$564$$ 0 0
$$565$$ −8.46410 −0.356087
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 12.8038 0.536765 0.268383 0.963312i $$-0.413511\pi$$
0.268383 + 0.963312i $$0.413511\pi$$
$$570$$ 0 0
$$571$$ 19.2679 0.806339 0.403169 0.915125i $$-0.367909\pi$$
0.403169 + 0.915125i $$0.367909\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −55.3205 −2.30702
$$576$$ 0 0
$$577$$ 7.33975 0.305558 0.152779 0.988260i $$-0.451178\pi$$
0.152779 + 0.988260i $$0.451178\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 14.5885 0.605231
$$582$$ 0 0
$$583$$ −10.6410 −0.440706
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −11.2679 −0.465078 −0.232539 0.972587i $$-0.574703\pi$$
−0.232539 + 0.972587i $$0.574703\pi$$
$$588$$ 0 0
$$589$$ 6.00000 0.247226
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 40.1769 1.64987 0.824934 0.565229i $$-0.191212\pi$$
0.824934 + 0.565229i $$0.191212\pi$$
$$594$$ 0 0
$$595$$ 26.1244 1.07099
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −9.12436 −0.372811 −0.186406 0.982473i $$-0.559684\pi$$
−0.186406 + 0.982473i $$0.559684\pi$$
$$600$$ 0 0
$$601$$ 8.80385 0.359116 0.179558 0.983747i $$-0.442533\pi$$
0.179558 + 0.983747i $$0.442533\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 24.6603 1.00258
$$606$$ 0 0
$$607$$ −6.58846 −0.267417 −0.133709 0.991021i $$-0.542689\pi$$
−0.133709 + 0.991021i $$0.542689\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −0.588457 −0.0238064
$$612$$ 0 0
$$613$$ −14.7846 −0.597145 −0.298572 0.954387i $$-0.596510\pi$$
−0.298572 + 0.954387i $$0.596510\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −39.9808 −1.60956 −0.804782 0.593570i $$-0.797718\pi$$
−0.804782 + 0.593570i $$0.797718\pi$$
$$618$$ 0 0
$$619$$ −31.7128 −1.27465 −0.637323 0.770597i $$-0.719958\pi$$
−0.637323 + 0.770597i $$0.719958\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −3.92820 −0.157380
$$624$$ 0 0
$$625$$ 10.0718 0.402872
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −46.6218 −1.85893
$$630$$ 0 0
$$631$$ −13.6603 −0.543806 −0.271903 0.962325i $$-0.587653\pi$$
−0.271903 + 0.962325i $$0.587653\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −44.7846 −1.77722
$$636$$ 0 0
$$637$$ 0.464102 0.0183884
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −19.4449 −0.768026 −0.384013 0.923328i $$-0.625458\pi$$
−0.384013 + 0.923328i $$0.625458\pi$$
$$642$$ 0 0
$$643$$ 40.5885 1.60065 0.800326 0.599565i $$-0.204660\pi$$
0.800326 + 0.599565i $$0.204660\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −16.3923 −0.644448 −0.322224 0.946663i $$-0.604430\pi$$
−0.322224 + 0.946663i $$0.604430\pi$$
$$648$$ 0 0
$$649$$ 26.0000 1.02059
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 18.2487 0.714127 0.357064 0.934080i $$-0.383778\pi$$
0.357064 + 0.934080i $$0.383778\pi$$
$$654$$ 0 0
$$655$$ −65.1769 −2.54667
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 15.6077 0.607989 0.303995 0.952674i $$-0.401679\pi$$
0.303995 + 0.952674i $$0.401679\pi$$
$$660$$ 0 0
$$661$$ 14.8564 0.577847 0.288924 0.957352i $$-0.406703\pi$$
0.288924 + 0.957352i $$0.406703\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10.1962 0.395390
$$666$$ 0 0
$$667$$ −52.4449 −2.03067
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −41.6603 −1.60828
$$672$$ 0 0
$$673$$ 16.3205 0.629109 0.314555 0.949239i $$-0.398145\pi$$
0.314555 + 0.949239i $$0.398145\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 36.0000 1.38359 0.691796 0.722093i $$-0.256820\pi$$
0.691796 + 0.722093i $$0.256820\pi$$
$$678$$ 0 0
$$679$$ −2.92820 −0.112374
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −25.8564 −0.989368 −0.494684 0.869073i $$-0.664716\pi$$
−0.494684 + 0.869073i $$0.664716\pi$$
$$684$$ 0 0
$$685$$ −43.7846 −1.67292
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1.17691 −0.0448369
$$690$$ 0 0
$$691$$ 28.0000 1.06517 0.532585 0.846376i $$-0.321221\pi$$
0.532585 + 0.846376i $$0.321221\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 25.1244 0.953021
$$696$$ 0 0
$$697$$ 66.2487 2.50935
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 6.60770 0.249569 0.124785 0.992184i $$-0.460176\pi$$
0.124785 + 0.992184i $$0.460176\pi$$
$$702$$ 0 0
$$703$$ −18.1962 −0.686281
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −4.92820 −0.185344
$$708$$ 0 0
$$709$$ −28.1244 −1.05623 −0.528116 0.849172i $$-0.677101\pi$$
−0.528116 + 0.849172i $$0.677101\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −13.6077 −0.509612
$$714$$ 0 0
$$715$$ 7.26795 0.271806
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.53590 0.0945731 0.0472865 0.998881i $$-0.484943\pi$$
0.0472865 + 0.998881i $$0.484943\pi$$
$$720$$ 0 0
$$721$$ −12.3923 −0.461514
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 75.5692 2.80657
$$726$$ 0 0
$$727$$ −16.6795 −0.618608 −0.309304 0.950963i $$-0.600096\pi$$
−0.309304 + 0.950963i $$0.600096\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −38.2487 −1.41468
$$732$$ 0 0
$$733$$ 12.6795 0.468328 0.234164 0.972197i $$-0.424765\pi$$
0.234164 + 0.972197i $$0.424765\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 13.7128 0.505118
$$738$$ 0 0
$$739$$ 16.7321 0.615498 0.307749 0.951468i $$-0.400424\pi$$
0.307749 + 0.951468i $$0.400424\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 19.6077 0.719337 0.359668 0.933080i $$-0.382890\pi$$
0.359668 + 0.933080i $$0.382890\pi$$
$$744$$ 0 0
$$745$$ 33.5885 1.23059
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 49.8564 1.81929 0.909643 0.415391i $$-0.136355\pi$$
0.909643 + 0.415391i $$0.136355\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 60.4449 2.19981
$$756$$ 0 0
$$757$$ −20.7846 −0.755429 −0.377715 0.925922i $$-0.623290\pi$$
−0.377715 + 0.925922i $$0.623290\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 37.0000 1.34125 0.670624 0.741797i $$-0.266026\pi$$
0.670624 + 0.741797i $$0.266026\pi$$
$$762$$ 0 0
$$763$$ −7.19615 −0.260518
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 2.87564 0.103833
$$768$$ 0 0
$$769$$ −35.5885 −1.28335 −0.641676 0.766976i $$-0.721761\pi$$
−0.641676 + 0.766976i $$0.721761\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 20.1244 0.723823 0.361911 0.932213i $$-0.382124\pi$$
0.361911 + 0.932213i $$0.382124\pi$$
$$774$$ 0 0
$$775$$ 19.6077 0.704329
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 25.8564 0.926402
$$780$$ 0 0
$$781$$ −56.4974 −2.02164
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −3.73205 −0.133203
$$786$$ 0 0
$$787$$ 7.60770 0.271185 0.135593 0.990765i $$-0.456706\pi$$
0.135593 + 0.990765i $$0.456706\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −2.26795 −0.0806390
$$792$$ 0 0
$$793$$ −4.60770 −0.163624
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 29.4449 1.04299 0.521495 0.853254i $$-0.325374\pi$$
0.521495 + 0.853254i $$0.325374\pi$$
$$798$$ 0 0
$$799$$ −8.87564 −0.313998
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 49.2295 1.73727
$$804$$ 0 0
$$805$$ −23.1244 −0.815026
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 7.87564 0.276893 0.138446 0.990370i $$-0.455789\pi$$
0.138446 + 0.990370i $$0.455789\pi$$
$$810$$ 0 0
$$811$$ −7.80385 −0.274030 −0.137015 0.990569i $$-0.543751\pi$$
−0.137015 + 0.990569i $$0.543751\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 24.3923 0.854425
$$816$$ 0 0
$$817$$ −14.9282 −0.522272
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 12.0718 0.421309 0.210654 0.977561i $$-0.432441\pi$$
0.210654 + 0.977561i $$0.432441\pi$$
$$822$$ 0 0
$$823$$ 40.7846 1.42166 0.710831 0.703363i $$-0.248319\pi$$
0.710831 + 0.703363i $$0.248319\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −11.3205 −0.393653 −0.196826 0.980438i $$-0.563064\pi$$
−0.196826 + 0.980438i $$0.563064\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 7.00000 0.242536
$$834$$ 0 0
$$835$$ 45.5167 1.57517
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 5.46410 0.188642 0.0943209 0.995542i $$-0.469932\pi$$
0.0943209 + 0.995542i $$0.469932\pi$$
$$840$$ 0 0
$$841$$ 42.6410 1.47038
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −47.7128 −1.64137
$$846$$ 0 0
$$847$$ 6.60770 0.227043
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 41.2679 1.41465
$$852$$ 0 0
$$853$$ 5.71281 0.195603 0.0978015 0.995206i $$-0.468819\pi$$
0.0978015 + 0.995206i $$0.468819\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 10.8564 0.370848 0.185424 0.982659i $$-0.440634\pi$$
0.185424 + 0.982659i $$0.440634\pi$$
$$858$$ 0 0
$$859$$ −3.60770 −0.123093 −0.0615465 0.998104i $$-0.519603\pi$$
−0.0615465 + 0.998104i $$0.519603\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −17.1244 −0.582920 −0.291460 0.956583i $$-0.594141\pi$$
−0.291460 + 0.956583i $$0.594141\pi$$
$$864$$ 0 0
$$865$$ 36.3205 1.23493
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −63.4641 −2.15287
$$870$$ 0 0
$$871$$ 1.51666 0.0513901
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 14.6603 0.495607
$$876$$ 0 0
$$877$$ 8.51666 0.287587 0.143794 0.989608i $$-0.454070\pi$$
0.143794 + 0.989608i $$0.454070\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −18.2487 −0.614815 −0.307407 0.951578i $$-0.599461\pi$$
−0.307407 + 0.951578i $$0.599461\pi$$
$$882$$ 0 0
$$883$$ −7.66025 −0.257788 −0.128894 0.991658i $$-0.541143\pi$$
−0.128894 + 0.991658i $$0.541143\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −2.44486 −0.0820905 −0.0410452 0.999157i $$-0.513069\pi$$
−0.0410452 + 0.999157i $$0.513069\pi$$
$$888$$ 0 0
$$889$$ −12.0000 −0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −3.46410 −0.115922
$$894$$ 0 0
$$895$$ −30.5885 −1.02246
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 18.5885 0.619960
$$900$$ 0 0
$$901$$ −17.7513 −0.591381
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 16.3923 0.544899
$$906$$ 0 0
$$907$$ 36.0000 1.19536 0.597680 0.801735i $$-0.296089\pi$$
0.597680 + 0.801735i $$0.296089\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 6.24871 0.207029 0.103515 0.994628i $$-0.466991\pi$$
0.103515 + 0.994628i $$0.466991\pi$$
$$912$$ 0 0
$$913$$ 61.2154 2.02593
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −17.4641 −0.576715
$$918$$ 0 0
$$919$$ 24.9808 0.824039 0.412020 0.911175i $$-0.364824\pi$$
0.412020 + 0.911175i $$0.364824\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −6.24871 −0.205679
$$924$$ 0 0
$$925$$ −59.4641 −1.95517
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 45.4974 1.49272 0.746361 0.665541i $$-0.231799\pi$$
0.746361 + 0.665541i $$0.231799\pi$$
$$930$$ 0 0
$$931$$ 2.73205 0.0895393
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 109.622 3.58502
$$936$$ 0 0
$$937$$ 53.8372 1.75878 0.879392 0.476099i $$-0.157950\pi$$
0.879392 + 0.476099i $$0.157950\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −9.87564 −0.321937 −0.160968 0.986960i $$-0.551462\pi$$
−0.160968 + 0.986960i $$0.551462\pi$$
$$942$$ 0 0
$$943$$ −58.6410 −1.90961
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −2.24871 −0.0730733 −0.0365366 0.999332i $$-0.511633\pi$$
−0.0365366 + 0.999332i $$0.511633\pi$$
$$948$$ 0 0
$$949$$ 5.44486 0.176748
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −10.4115 −0.337263 −0.168631 0.985679i $$-0.553935\pi$$
−0.168631 + 0.985679i $$0.553935\pi$$
$$954$$ 0 0
$$955$$ −43.5167 −1.40817
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −11.7321 −0.378848
$$960$$ 0 0
$$961$$ −26.1769 −0.844417
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −33.0526 −1.06400
$$966$$ 0 0
$$967$$ −13.6603 −0.439284 −0.219642 0.975581i $$-0.570489\pi$$
−0.219642 + 0.975581i $$0.570489\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −33.1244 −1.06301 −0.531506 0.847055i $$-0.678374\pi$$
−0.531506 + 0.847055i $$0.678374\pi$$
$$972$$ 0 0
$$973$$ 6.73205 0.215820
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.28719 0.0731736 0.0365868 0.999330i $$-0.488351\pi$$
0.0365868 + 0.999330i $$0.488351\pi$$
$$978$$ 0 0
$$979$$ −16.4833 −0.526810
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 10.6410 0.339396 0.169698 0.985496i $$-0.445721\pi$$
0.169698 + 0.985496i $$0.445721\pi$$
$$984$$ 0 0
$$985$$ −96.2295 −3.06613
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 33.8564 1.07657
$$990$$ 0 0
$$991$$ 10.3397 0.328453 0.164226 0.986423i $$-0.447487\pi$$
0.164226 + 0.986423i $$0.447487\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −19.1244 −0.606283
$$996$$ 0 0
$$997$$ −7.24871 −0.229569 −0.114784 0.993390i $$-0.536618\pi$$
−0.114784 + 0.993390i $$0.536618\pi$$
$$998$$ 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 9072.2.a.bp.1.2 2
3.2 odd 2 9072.2.a.y.1.1 2
4.3 odd 2 1134.2.a.l.1.2 2
12.11 even 2 1134.2.a.m.1.1 yes 2
28.27 even 2 7938.2.a.bg.1.1 2
36.7 odd 6 1134.2.f.s.757.1 4
36.11 even 6 1134.2.f.r.757.2 4
36.23 even 6 1134.2.f.r.379.2 4
36.31 odd 6 1134.2.f.s.379.1 4
84.83 odd 2 7938.2.a.bt.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.2 2 4.3 odd 2
1134.2.a.m.1.1 yes 2 12.11 even 2
1134.2.f.r.379.2 4 36.23 even 6
1134.2.f.r.757.2 4 36.11 even 6
1134.2.f.s.379.1 4 36.31 odd 6
1134.2.f.s.757.1 4 36.7 odd 6
7938.2.a.bg.1.1 2 28.27 even 2
7938.2.a.bt.1.2 2 84.83 odd 2
9072.2.a.y.1.1 2 3.2 odd 2
9072.2.a.bp.1.2 2 1.1 even 1 trivial