# Properties

 Label 3888.2.a.bk.1.3 Level $3888$ Weight $2$ Character 3888.1 Self dual yes Analytic conductor $31.046$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3888,2,Mod(1,3888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3888, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3888.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3888 = 2^{4} \cdot 3^{5}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3888.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: $$31.0458363059$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 243) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$1.87939$$ of defining polynomial Character $$\chi$$ $$=$$ 3888.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.87939 q^{5} +2.18479 q^{7} +O(q^{10})$$ $$q+3.87939 q^{5} +2.18479 q^{7} -0.162504 q^{11} +2.41147 q^{13} +3.00000 q^{17} -3.59627 q^{19} -2.83750 q^{23} +10.0496 q^{25} +6.71688 q^{29} +5.18479 q^{31} +8.47565 q^{35} -6.63816 q^{37} -5.80066 q^{41} +6.22668 q^{43} -7.39693 q^{47} -2.22668 q^{49} +1.40373 q^{53} -0.630415 q^{55} +5.12061 q^{59} -3.78106 q^{61} +9.35504 q^{65} +5.86484 q^{67} +15.3182 q^{71} +8.68004 q^{73} -0.355037 q^{77} +1.26857 q^{79} +8.47565 q^{83} +11.6382 q^{85} +7.72193 q^{89} +5.26857 q^{91} -13.9513 q^{95} -3.90673 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 6 q^{5} + 3 q^{7}+O(q^{10})$$ 3 * q + 6 * q^5 + 3 * q^7 $$3 q + 6 q^{5} + 3 q^{7} - 3 q^{11} - 3 q^{13} + 9 q^{17} + 3 q^{19} - 6 q^{23} + 3 q^{25} + 12 q^{29} + 12 q^{31} + 6 q^{35} - 3 q^{37} - 3 q^{41} + 12 q^{43} + 6 q^{47} + 18 q^{53} - 9 q^{55} + 21 q^{59} + 6 q^{61} + 3 q^{65} - 6 q^{67} + 9 q^{71} + 6 q^{73} + 24 q^{77} - 6 q^{79} + 6 q^{83} + 18 q^{85} + 6 q^{91} - 3 q^{95} + 15 q^{97}+O(q^{100})$$ 3 * q + 6 * q^5 + 3 * q^7 - 3 * q^11 - 3 * q^13 + 9 * q^17 + 3 * q^19 - 6 * q^23 + 3 * q^25 + 12 * q^29 + 12 * q^31 + 6 * q^35 - 3 * q^37 - 3 * q^41 + 12 * q^43 + 6 * q^47 + 18 * q^53 - 9 * q^55 + 21 * q^59 + 6 * q^61 + 3 * q^65 - 6 * q^67 + 9 * q^71 + 6 * q^73 + 24 * q^77 - 6 * q^79 + 6 * q^83 + 18 * q^85 + 6 * q^91 - 3 * q^95 + 15 * 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
$$5$$ 3.87939 1.73491 0.867457 0.497512i $$-0.165753\pi$$
0.867457 + 0.497512i $$0.165753\pi$$
$$6$$ 0 0
$$7$$ 2.18479 0.825774 0.412887 0.910782i $$-0.364520\pi$$
0.412887 + 0.910782i $$0.364520\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −0.162504 −0.0489967 −0.0244984 0.999700i $$-0.507799\pi$$
−0.0244984 + 0.999700i $$0.507799\pi$$
$$12$$ 0 0
$$13$$ 2.41147 0.668823 0.334411 0.942427i $$-0.391463\pi$$
0.334411 + 0.942427i $$0.391463\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −3.59627 −0.825040 −0.412520 0.910949i $$-0.635351\pi$$
−0.412520 + 0.910949i $$0.635351\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −2.83750 −0.591659 −0.295829 0.955241i $$-0.595596\pi$$
−0.295829 + 0.955241i $$0.595596\pi$$
$$24$$ 0 0
$$25$$ 10.0496 2.00993
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.71688 1.24729 0.623647 0.781706i $$-0.285650\pi$$
0.623647 + 0.781706i $$0.285650\pi$$
$$30$$ 0 0
$$31$$ 5.18479 0.931216 0.465608 0.884991i $$-0.345836\pi$$
0.465608 + 0.884991i $$0.345836\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 8.47565 1.43265
$$36$$ 0 0
$$37$$ −6.63816 −1.09131 −0.545653 0.838011i $$-0.683718\pi$$
−0.545653 + 0.838011i $$0.683718\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.80066 −0.905911 −0.452955 0.891533i $$-0.649630\pi$$
−0.452955 + 0.891533i $$0.649630\pi$$
$$42$$ 0 0
$$43$$ 6.22668 0.949560 0.474780 0.880105i $$-0.342528\pi$$
0.474780 + 0.880105i $$0.342528\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.39693 −1.07895 −0.539476 0.842001i $$-0.681378\pi$$
−0.539476 + 0.842001i $$0.681378\pi$$
$$48$$ 0 0
$$49$$ −2.22668 −0.318097
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.40373 0.192818 0.0964088 0.995342i $$-0.469264\pi$$
0.0964088 + 0.995342i $$0.469264\pi$$
$$54$$ 0 0
$$55$$ −0.630415 −0.0850051
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 5.12061 0.666647 0.333324 0.942812i $$-0.391830\pi$$
0.333324 + 0.942812i $$0.391830\pi$$
$$60$$ 0 0
$$61$$ −3.78106 −0.484115 −0.242058 0.970262i $$-0.577822\pi$$
−0.242058 + 0.970262i $$0.577822\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 9.35504 1.16035
$$66$$ 0 0
$$67$$ 5.86484 0.716504 0.358252 0.933625i $$-0.383373\pi$$
0.358252 + 0.933625i $$0.383373\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 15.3182 1.81794 0.908968 0.416866i $$-0.136872\pi$$
0.908968 + 0.416866i $$0.136872\pi$$
$$72$$ 0 0
$$73$$ 8.68004 1.01592 0.507961 0.861380i $$-0.330399\pi$$
0.507961 + 0.861380i $$0.330399\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −0.355037 −0.0404602
$$78$$ 0 0
$$79$$ 1.26857 0.142725 0.0713627 0.997450i $$-0.477265\pi$$
0.0713627 + 0.997450i $$0.477265\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 8.47565 0.930324 0.465162 0.885226i $$-0.345996\pi$$
0.465162 + 0.885226i $$0.345996\pi$$
$$84$$ 0 0
$$85$$ 11.6382 1.26234
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.72193 0.818523 0.409262 0.912417i $$-0.365786\pi$$
0.409262 + 0.912417i $$0.365786\pi$$
$$90$$ 0 0
$$91$$ 5.26857 0.552296
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −13.9513 −1.43137
$$96$$ 0 0
$$97$$ −3.90673 −0.396668 −0.198334 0.980134i $$-0.563553\pi$$
−0.198334 + 0.980134i $$0.563553\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −8.11381 −0.807354 −0.403677 0.914902i $$-0.632268\pi$$
−0.403677 + 0.914902i $$0.632268\pi$$
$$102$$ 0 0
$$103$$ −18.6459 −1.83723 −0.918617 0.395148i $$-0.870693\pi$$
−0.918617 + 0.395148i $$0.870693\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −7.59627 −0.734359 −0.367179 0.930150i $$-0.619676\pi$$
−0.367179 + 0.930150i $$0.619676\pi$$
$$108$$ 0 0
$$109$$ −15.6382 −1.49786 −0.748932 0.662647i $$-0.769433\pi$$
−0.748932 + 0.662647i $$0.769433\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.31315 −0.217603 −0.108801 0.994064i $$-0.534701\pi$$
−0.108801 + 0.994064i $$0.534701\pi$$
$$114$$ 0 0
$$115$$ −11.0077 −1.02648
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 6.55438 0.600839
$$120$$ 0 0
$$121$$ −10.9736 −0.997599
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 19.5895 1.75213
$$126$$ 0 0
$$127$$ −0.0418891 −0.00371705 −0.00185853 0.999998i $$-0.500592\pi$$
−0.00185853 + 0.999998i $$0.500592\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −18.3482 −1.60309 −0.801546 0.597933i $$-0.795989\pi$$
−0.801546 + 0.597933i $$0.795989\pi$$
$$132$$ 0 0
$$133$$ −7.85710 −0.681297
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 14.3131 1.22285 0.611427 0.791301i $$-0.290596\pi$$
0.611427 + 0.791301i $$0.290596\pi$$
$$138$$ 0 0
$$139$$ −10.4953 −0.890196 −0.445098 0.895482i $$-0.646831\pi$$
−0.445098 + 0.895482i $$0.646831\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −0.391874 −0.0327701
$$144$$ 0 0
$$145$$ 26.0574 2.16395
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.27126 −0.104146 −0.0520728 0.998643i $$-0.516583\pi$$
−0.0520728 + 0.998643i $$0.516583\pi$$
$$150$$ 0 0
$$151$$ −7.85710 −0.639401 −0.319701 0.947519i $$-0.603582\pi$$
−0.319701 + 0.947519i $$0.603582\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 20.1138 1.61558
$$156$$ 0 0
$$157$$ −12.3523 −0.985825 −0.492912 0.870079i $$-0.664068\pi$$
−0.492912 + 0.870079i $$0.664068\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −6.19934 −0.488576
$$162$$ 0 0
$$163$$ 13.7469 1.07674 0.538371 0.842708i $$-0.319040\pi$$
0.538371 + 0.842708i $$0.319040\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.71688 0.287621 0.143810 0.989605i $$-0.454064\pi$$
0.143810 + 0.989605i $$0.454064\pi$$
$$168$$ 0 0
$$169$$ −7.18479 −0.552676
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −1.55943 −0.118561 −0.0592806 0.998241i $$-0.518881\pi$$
−0.0592806 + 0.998241i $$0.518881\pi$$
$$174$$ 0 0
$$175$$ 21.9564 1.65974
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 12.1925 0.911313 0.455656 0.890156i $$-0.349405\pi$$
0.455656 + 0.890156i $$0.349405\pi$$
$$180$$ 0 0
$$181$$ −16.8726 −1.25413 −0.627064 0.778967i $$-0.715744\pi$$
−0.627064 + 0.778967i $$0.715744\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −25.7520 −1.89332
$$186$$ 0 0
$$187$$ −0.487511 −0.0356504
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 17.4757 1.26449 0.632247 0.774767i $$-0.282133\pi$$
0.632247 + 0.774767i $$0.282133\pi$$
$$192$$ 0 0
$$193$$ −1.99226 −0.143406 −0.0717030 0.997426i $$-0.522843\pi$$
−0.0717030 + 0.997426i $$0.522843\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 21.1925 1.50991 0.754953 0.655779i $$-0.227660\pi$$
0.754953 + 0.655779i $$0.227660\pi$$
$$198$$ 0 0
$$199$$ 3.08378 0.218603 0.109302 0.994009i $$-0.465139\pi$$
0.109302 + 0.994009i $$0.465139\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 14.6750 1.02998
$$204$$ 0 0
$$205$$ −22.5030 −1.57168
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0.584407 0.0404243
$$210$$ 0 0
$$211$$ −1.00774 −0.0693757 −0.0346879 0.999398i $$-0.511044\pi$$
−0.0346879 + 0.999398i $$0.511044\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 24.1557 1.64740
$$216$$ 0 0
$$217$$ 11.3277 0.768974
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 7.23442 0.486640
$$222$$ 0 0
$$223$$ −18.2841 −1.22439 −0.612195 0.790707i $$-0.709713\pi$$
−0.612195 + 0.790707i $$0.709713\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2.64496 0.175552 0.0877762 0.996140i $$-0.472024\pi$$
0.0877762 + 0.996140i $$0.472024\pi$$
$$228$$ 0 0
$$229$$ −3.46286 −0.228832 −0.114416 0.993433i $$-0.536500\pi$$
−0.114416 + 0.993433i $$0.536500\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.12567 0.401306 0.200653 0.979662i $$-0.435694\pi$$
0.200653 + 0.979662i $$0.435694\pi$$
$$234$$ 0 0
$$235$$ −28.6955 −1.87189
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 28.9513 1.87270 0.936352 0.351062i $$-0.114179\pi$$
0.936352 + 0.351062i $$0.114179\pi$$
$$240$$ 0 0
$$241$$ 22.3259 1.43814 0.719070 0.694937i $$-0.244568\pi$$
0.719070 + 0.694937i $$0.244568\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −8.63816 −0.551872
$$246$$ 0 0
$$247$$ −8.67230 −0.551805
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 22.7219 1.43420 0.717098 0.696972i $$-0.245470\pi$$
0.717098 + 0.696972i $$0.245470\pi$$
$$252$$ 0 0
$$253$$ 0.461104 0.0289894
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −19.5895 −1.22196 −0.610978 0.791647i $$-0.709224\pi$$
−0.610978 + 0.791647i $$0.709224\pi$$
$$258$$ 0 0
$$259$$ −14.5030 −0.901172
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −17.8007 −1.09764 −0.548818 0.835942i $$-0.684922\pi$$
−0.548818 + 0.835942i $$0.684922\pi$$
$$264$$ 0 0
$$265$$ 5.44562 0.334522
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 22.7888 1.38946 0.694729 0.719272i $$-0.255524\pi$$
0.694729 + 0.719272i $$0.255524\pi$$
$$270$$ 0 0
$$271$$ 3.44562 0.209307 0.104653 0.994509i $$-0.466627\pi$$
0.104653 + 0.994509i $$0.466627\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.63310 −0.0984798
$$276$$ 0 0
$$277$$ −2.61350 −0.157030 −0.0785151 0.996913i $$-0.525018\pi$$
−0.0785151 + 0.996913i $$0.525018\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −13.6851 −0.816384 −0.408192 0.912896i $$-0.633841\pi$$
−0.408192 + 0.912896i $$0.633841\pi$$
$$282$$ 0 0
$$283$$ −22.8803 −1.36009 −0.680047 0.733169i $$-0.738041\pi$$
−0.680047 + 0.733169i $$0.738041\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.6732 −0.748078
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 24.2814 1.41853 0.709266 0.704941i $$-0.249026\pi$$
0.709266 + 0.704941i $$0.249026\pi$$
$$294$$ 0 0
$$295$$ 19.8648 1.15658
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −6.84255 −0.395715
$$300$$ 0 0
$$301$$ 13.6040 0.784122
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −14.6682 −0.839898
$$306$$ 0 0
$$307$$ 16.1489 0.921666 0.460833 0.887487i $$-0.347551\pi$$
0.460833 + 0.887487i $$0.347551\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −18.7101 −1.06095 −0.530475 0.847700i $$-0.677987\pi$$
−0.530475 + 0.847700i $$0.677987\pi$$
$$312$$ 0 0
$$313$$ 2.77332 0.156757 0.0783786 0.996924i $$-0.475026\pi$$
0.0783786 + 0.996924i $$0.475026\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 17.4825 0.981913 0.490956 0.871184i $$-0.336647\pi$$
0.490956 + 0.871184i $$0.336647\pi$$
$$318$$ 0 0
$$319$$ −1.09152 −0.0611133
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −10.7888 −0.600305
$$324$$ 0 0
$$325$$ 24.2344 1.34428
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −16.1607 −0.890971
$$330$$ 0 0
$$331$$ 32.4593 1.78413 0.892064 0.451910i $$-0.149257\pi$$
0.892064 + 0.451910i $$0.149257\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 22.7520 1.24307
$$336$$ 0 0
$$337$$ 8.28581 0.451357 0.225678 0.974202i $$-0.427540\pi$$
0.225678 + 0.974202i $$0.427540\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −0.842549 −0.0456266
$$342$$ 0 0
$$343$$ −20.1584 −1.08845
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −14.9632 −0.803265 −0.401632 0.915801i $$-0.631557\pi$$
−0.401632 + 0.915801i $$0.631557\pi$$
$$348$$ 0 0
$$349$$ 33.6459 1.80102 0.900512 0.434832i $$-0.143192\pi$$
0.900512 + 0.434832i $$0.143192\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −15.7537 −0.838486 −0.419243 0.907874i $$-0.637704\pi$$
−0.419243 + 0.907874i $$0.637704\pi$$
$$354$$ 0 0
$$355$$ 59.4252 3.15396
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −18.1257 −0.956636 −0.478318 0.878187i $$-0.658753\pi$$
−0.478318 + 0.878187i $$0.658753\pi$$
$$360$$ 0 0
$$361$$ −6.06687 −0.319309
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 33.6732 1.76254
$$366$$ 0 0
$$367$$ 19.1429 0.999251 0.499626 0.866241i $$-0.333471\pi$$
0.499626 + 0.866241i $$0.333471\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 3.06687 0.159224
$$372$$ 0 0
$$373$$ −15.2499 −0.789610 −0.394805 0.918765i $$-0.629188\pi$$
−0.394805 + 0.918765i $$0.629188\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 16.1976 0.834218
$$378$$ 0 0
$$379$$ −9.84760 −0.505837 −0.252919 0.967488i $$-0.581390\pi$$
−0.252919 + 0.967488i $$0.581390\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −28.3901 −1.45067 −0.725334 0.688397i $$-0.758315\pi$$
−0.725334 + 0.688397i $$0.758315\pi$$
$$384$$ 0 0
$$385$$ −1.37733 −0.0701950
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 10.8844 0.551863 0.275931 0.961177i $$-0.411014\pi$$
0.275931 + 0.961177i $$0.411014\pi$$
$$390$$ 0 0
$$391$$ −8.51249 −0.430495
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4.92127 0.247616
$$396$$ 0 0
$$397$$ −18.1070 −0.908764 −0.454382 0.890807i $$-0.650140\pi$$
−0.454382 + 0.890807i $$0.650140\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.43376 0.0715987 0.0357993 0.999359i $$-0.488602\pi$$
0.0357993 + 0.999359i $$0.488602\pi$$
$$402$$ 0 0
$$403$$ 12.5030 0.622818
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.07873 0.0534704
$$408$$ 0 0
$$409$$ 8.60401 0.425441 0.212720 0.977113i $$-0.431768\pi$$
0.212720 + 0.977113i $$0.431768\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 11.1875 0.550500
$$414$$ 0 0
$$415$$ 32.8803 1.61403
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 12.3114 0.601451 0.300725 0.953711i $$-0.402771\pi$$
0.300725 + 0.953711i $$0.402771\pi$$
$$420$$ 0 0
$$421$$ 11.1165 0.541785 0.270892 0.962610i $$-0.412681\pi$$
0.270892 + 0.962610i $$0.412681\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 30.1489 1.46244
$$426$$ 0 0
$$427$$ −8.26083 −0.399770
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 36.8958 1.77721 0.888604 0.458675i $$-0.151676\pi$$
0.888604 + 0.458675i $$0.151676\pi$$
$$432$$ 0 0
$$433$$ −37.9982 −1.82608 −0.913040 0.407871i $$-0.866271\pi$$
−0.913040 + 0.407871i $$0.866271\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 10.2044 0.488142
$$438$$ 0 0
$$439$$ −0.202029 −0.00964231 −0.00482115 0.999988i $$-0.501535\pi$$
−0.00482115 + 0.999988i $$0.501535\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −21.2294 −1.00864 −0.504319 0.863517i $$-0.668256\pi$$
−0.504319 + 0.863517i $$0.668256\pi$$
$$444$$ 0 0
$$445$$ 29.9564 1.42007
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −33.2594 −1.56961 −0.784804 0.619744i $$-0.787236\pi$$
−0.784804 + 0.619744i $$0.787236\pi$$
$$450$$ 0 0
$$451$$ 0.942629 0.0443867
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 20.4388 0.958186
$$456$$ 0 0
$$457$$ −0.0341483 −0.00159739 −0.000798695 1.00000i $$-0.500254\pi$$
−0.000798695 1.00000i $$0.500254\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.9864 −0.697986 −0.348993 0.937125i $$-0.613476\pi$$
−0.348993 + 0.937125i $$0.613476\pi$$
$$462$$ 0 0
$$463$$ 30.4424 1.41478 0.707390 0.706823i $$-0.249872\pi$$
0.707390 + 0.706823i $$0.249872\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0.510734 0.0236339 0.0118170 0.999930i $$-0.496238\pi$$
0.0118170 + 0.999930i $$0.496238\pi$$
$$468$$ 0 0
$$469$$ 12.8135 0.591670
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −1.01186 −0.0465254
$$474$$ 0 0
$$475$$ −36.1411 −1.65827
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 15.4507 0.705959 0.352980 0.935631i $$-0.385168\pi$$
0.352980 + 0.935631i $$0.385168\pi$$
$$480$$ 0 0
$$481$$ −16.0077 −0.729890
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −15.1557 −0.688185
$$486$$ 0 0
$$487$$ −29.5107 −1.33726 −0.668629 0.743596i $$-0.733119\pi$$
−0.668629 + 0.743596i $$0.733119\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −2.15745 −0.0973644 −0.0486822 0.998814i $$-0.515502\pi$$
−0.0486822 + 0.998814i $$0.515502\pi$$
$$492$$ 0 0
$$493$$ 20.1506 0.907539
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 33.4671 1.50120
$$498$$ 0 0
$$499$$ −7.49525 −0.335534 −0.167767 0.985827i $$-0.553656\pi$$
−0.167767 + 0.985827i $$0.553656\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −28.5963 −1.27504 −0.637522 0.770432i $$-0.720041\pi$$
−0.637522 + 0.770432i $$0.720041\pi$$
$$504$$ 0 0
$$505$$ −31.4766 −1.40069
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −1.69190 −0.0749923 −0.0374962 0.999297i $$-0.511938\pi$$
−0.0374962 + 0.999297i $$0.511938\pi$$
$$510$$ 0 0
$$511$$ 18.9641 0.838922
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −72.3346 −3.18744
$$516$$ 0 0
$$517$$ 1.20203 0.0528652
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −22.4037 −0.981525 −0.490763 0.871293i $$-0.663282\pi$$
−0.490763 + 0.871293i $$0.663282\pi$$
$$522$$ 0 0
$$523$$ −2.42871 −0.106200 −0.0531000 0.998589i $$-0.516910\pi$$
−0.0531000 + 0.998589i $$0.516910\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.5544 0.677559
$$528$$ 0 0
$$529$$ −14.9486 −0.649940
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −13.9881 −0.605894
$$534$$ 0 0
$$535$$ −29.4688 −1.27405
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0.361844 0.0155857
$$540$$ 0 0
$$541$$ 38.9394 1.67414 0.837069 0.547098i $$-0.184267\pi$$
0.837069 + 0.547098i $$0.184267\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −60.6664 −2.59866
$$546$$ 0 0
$$547$$ 14.6723 0.627342 0.313671 0.949532i $$-0.398441\pi$$
0.313671 + 0.949532i $$0.398441\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −24.1557 −1.02907
$$552$$ 0 0
$$553$$ 2.77156 0.117859
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −11.1070 −0.470619 −0.235309 0.971921i $$-0.575610\pi$$
−0.235309 + 0.971921i $$0.575610\pi$$
$$558$$ 0 0
$$559$$ 15.0155 0.635087
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −16.3081 −0.687304 −0.343652 0.939097i $$-0.611664\pi$$
−0.343652 + 0.939097i $$0.611664\pi$$
$$564$$ 0 0
$$565$$ −8.97359 −0.377522
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 36.0164 1.50989 0.754943 0.655790i $$-0.227664\pi$$
0.754943 + 0.655790i $$0.227664\pi$$
$$570$$ 0 0
$$571$$ −39.1584 −1.63873 −0.819364 0.573274i $$-0.805673\pi$$
−0.819364 + 0.573274i $$0.805673\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −28.5158 −1.18919
$$576$$ 0 0
$$577$$ 11.8057 0.491478 0.245739 0.969336i $$-0.420969\pi$$
0.245739 + 0.969336i $$0.420969\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.5175 0.768237
$$582$$ 0 0
$$583$$ −0.228112 −0.00944744
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 39.9614 1.64938 0.824692 0.565582i $$-0.191349\pi$$
0.824692 + 0.565582i $$0.191349\pi$$
$$588$$ 0 0
$$589$$ −18.6459 −0.768291
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −29.2995 −1.20319 −0.601594 0.798802i $$-0.705467\pi$$
−0.601594 + 0.798802i $$0.705467\pi$$
$$594$$ 0 0
$$595$$ 25.4270 1.04240
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −10.0719 −0.411527 −0.205764 0.978602i $$-0.565968\pi$$
−0.205764 + 0.978602i $$0.565968\pi$$
$$600$$ 0 0
$$601$$ −30.4192 −1.24083 −0.620413 0.784275i $$-0.713035\pi$$
−0.620413 + 0.784275i $$0.713035\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −42.5708 −1.73075
$$606$$ 0 0
$$607$$ 23.0743 0.936556 0.468278 0.883581i $$-0.344875\pi$$
0.468278 + 0.883581i $$0.344875\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −17.8375 −0.721628
$$612$$ 0 0
$$613$$ 0.765578 0.0309214 0.0154607 0.999880i $$-0.495079\pi$$
0.0154607 + 0.999880i $$0.495079\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −9.28817 −0.373928 −0.186964 0.982367i $$-0.559865\pi$$
−0.186964 + 0.982367i $$0.559865\pi$$
$$618$$ 0 0
$$619$$ 35.0823 1.41008 0.705039 0.709168i $$-0.250929\pi$$
0.705039 + 0.709168i $$0.250929\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 16.8708 0.675915
$$624$$ 0 0
$$625$$ 25.7469 1.02988
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −19.9145 −0.794042
$$630$$ 0 0
$$631$$ −35.7621 −1.42367 −0.711833 0.702349i $$-0.752135\pi$$
−0.711833 + 0.702349i $$0.752135\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −0.162504 −0.00644877
$$636$$ 0 0
$$637$$ −5.36959 −0.212751
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 2.92633 0.115583 0.0577915 0.998329i $$-0.481594\pi$$
0.0577915 + 0.998329i $$0.481594\pi$$
$$642$$ 0 0
$$643$$ 20.2517 0.798647 0.399324 0.916810i $$-0.369245\pi$$
0.399324 + 0.916810i $$0.369245\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −10.7219 −0.421523 −0.210761 0.977538i $$-0.567594\pi$$
−0.210761 + 0.977538i $$0.567594\pi$$
$$648$$ 0 0
$$649$$ −0.832119 −0.0326635
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 35.7270 1.39811 0.699053 0.715070i $$-0.253605\pi$$
0.699053 + 0.715070i $$0.253605\pi$$
$$654$$ 0 0
$$655$$ −71.1799 −2.78123
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −30.8658 −1.20236 −0.601180 0.799114i $$-0.705302\pi$$
−0.601180 + 0.799114i $$0.705302\pi$$
$$660$$ 0 0
$$661$$ −9.84793 −0.383040 −0.191520 0.981489i $$-0.561342\pi$$
−0.191520 + 0.981489i $$0.561342\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −30.4807 −1.18199
$$666$$ 0 0
$$667$$ −19.0591 −0.737972
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0.614437 0.0237201
$$672$$ 0 0
$$673$$ −19.6973 −0.759274 −0.379637 0.925135i $$-0.623951\pi$$
−0.379637 + 0.925135i $$0.623951\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −28.4570 −1.09369 −0.546845 0.837234i $$-0.684171\pi$$
−0.546845 + 0.837234i $$0.684171\pi$$
$$678$$ 0 0
$$679$$ −8.53539 −0.327558
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.5107 −0.478710 −0.239355 0.970932i $$-0.576936\pi$$
−0.239355 + 0.970932i $$0.576936\pi$$
$$684$$ 0 0
$$685$$ 55.5262 2.12155
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 3.38507 0.128961
$$690$$ 0 0
$$691$$ −42.6255 −1.62155 −0.810775 0.585358i $$-0.800954\pi$$
−0.810775 + 0.585358i $$0.800954\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −40.7151 −1.54441
$$696$$ 0 0
$$697$$ −17.4020 −0.659147
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 51.7701 1.95533 0.977665 0.210167i $$-0.0674008\pi$$
0.977665 + 0.210167i $$0.0674008\pi$$
$$702$$ 0 0
$$703$$ 23.8726 0.900371
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −17.7270 −0.666692
$$708$$ 0 0
$$709$$ 15.1584 0.569285 0.284643 0.958634i $$-0.408125\pi$$
0.284643 + 0.958634i $$0.408125\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −14.7118 −0.550962
$$714$$ 0 0
$$715$$ −1.52023 −0.0568534
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 2.61493 0.0975206 0.0487603 0.998811i $$-0.484473\pi$$
0.0487603 + 0.998811i $$0.484473\pi$$
$$720$$ 0 0
$$721$$ −40.7374 −1.51714
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 67.5022 2.50697
$$726$$ 0 0
$$727$$ 4.09926 0.152033 0.0760166 0.997107i $$-0.475780\pi$$
0.0760166 + 0.997107i $$0.475780\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 18.6800 0.690906
$$732$$ 0 0
$$733$$ −38.2080 −1.41125 −0.705623 0.708588i $$-0.749333\pi$$
−0.705623 + 0.708588i $$0.749333\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −0.953058 −0.0351064
$$738$$ 0 0
$$739$$ 24.2094 0.890559 0.445279 0.895392i $$-0.353104\pi$$
0.445279 + 0.895392i $$0.353104\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 3.31139 0.121483 0.0607416 0.998154i $$-0.480653\pi$$
0.0607416 + 0.998154i $$0.480653\pi$$
$$744$$ 0 0
$$745$$ −4.93170 −0.180684
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −16.5963 −0.606414
$$750$$ 0 0
$$751$$ −13.7110 −0.500322 −0.250161 0.968204i $$-0.580484\pi$$
−0.250161 + 0.968204i $$0.580484\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −30.4807 −1.10931
$$756$$ 0 0
$$757$$ 12.3833 0.450079 0.225040 0.974350i $$-0.427749\pi$$
0.225040 + 0.974350i $$0.427749\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 7.58265 0.274871 0.137435 0.990511i $$-0.456114\pi$$
0.137435 + 0.990511i $$0.456114\pi$$
$$762$$ 0 0
$$763$$ −34.1661 −1.23690
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.3482 0.445869
$$768$$ 0 0
$$769$$ 3.21719 0.116015 0.0580073 0.998316i $$-0.481525\pi$$
0.0580073 + 0.998316i $$0.481525\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −0.184468 −0.00663486 −0.00331743 0.999994i $$-0.501056\pi$$
−0.00331743 + 0.999994i $$0.501056\pi$$
$$774$$ 0 0
$$775$$ 52.1052 1.87168
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 20.8607 0.747413
$$780$$ 0 0
$$781$$ −2.48927 −0.0890729
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −47.9195 −1.71032
$$786$$ 0 0
$$787$$ 0.478016 0.0170394 0.00851971 0.999964i $$-0.497288\pi$$
0.00851971 + 0.999964i $$0.497288\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −5.05375 −0.179691
$$792$$ 0 0
$$793$$ −9.11793 −0.323787
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 14.4989 0.513576 0.256788 0.966468i $$-0.417336\pi$$
0.256788 + 0.966468i $$0.417336\pi$$
$$798$$ 0 0
$$799$$ −22.1908 −0.785053
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1.41054 −0.0497769
$$804$$ 0 0
$$805$$ −24.0496 −0.847638
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −14.8743 −0.522954 −0.261477 0.965210i $$-0.584210\pi$$
−0.261477 + 0.965210i $$0.584210\pi$$
$$810$$ 0 0
$$811$$ −21.5963 −0.758347 −0.379174 0.925325i $$-0.623792\pi$$
−0.379174 + 0.925325i $$0.623792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 53.3296 1.86805
$$816$$ 0 0
$$817$$ −22.3928 −0.783425
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.28136 −0.114520 −0.0572602 0.998359i $$-0.518236\pi$$
−0.0572602 + 0.998359i $$0.518236\pi$$
$$822$$ 0 0
$$823$$ −13.7314 −0.478648 −0.239324 0.970940i $$-0.576926\pi$$
−0.239324 + 0.970940i $$0.576926\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20.2327 −0.703559 −0.351779 0.936083i $$-0.614423\pi$$
−0.351779 + 0.936083i $$0.614423\pi$$
$$828$$ 0 0
$$829$$ 25.5276 0.886612 0.443306 0.896370i $$-0.353806\pi$$
0.443306 + 0.896370i $$0.353806\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6.68004 −0.231450
$$834$$ 0 0
$$835$$ 14.4192 0.498998
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −17.3800 −0.600025 −0.300012 0.953935i $$-0.596991\pi$$
−0.300012 + 0.953935i $$0.596991\pi$$
$$840$$ 0 0
$$841$$ 16.1165 0.555741
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −27.8726 −0.958846
$$846$$ 0 0
$$847$$ −23.9750 −0.823792
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 18.8357 0.645681
$$852$$ 0 0
$$853$$ 13.3027 0.455476 0.227738 0.973722i $$-0.426867\pi$$
0.227738 + 0.973722i $$0.426867\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −19.9213 −0.680498 −0.340249 0.940335i $$-0.610511\pi$$
−0.340249 + 0.940335i $$0.610511\pi$$
$$858$$ 0 0
$$859$$ −26.3446 −0.898866 −0.449433 0.893314i $$-0.648374\pi$$
−0.449433 + 0.893314i $$0.648374\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −38.2995 −1.30373 −0.651866 0.758334i $$-0.726013\pi$$
−0.651866 + 0.758334i $$0.726013\pi$$
$$864$$ 0 0
$$865$$ −6.04963 −0.205694
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −0.206148 −0.00699308
$$870$$ 0 0
$$871$$ 14.1429 0.479214
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 42.7989 1.44687
$$876$$ 0 0
$$877$$ −27.0574 −0.913662 −0.456831 0.889553i $$-0.651016\pi$$
−0.456831 + 0.889553i $$0.651016\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.5776 1.03019 0.515093 0.857134i $$-0.327757\pi$$
0.515093 + 0.857134i $$0.327757\pi$$
$$882$$ 0 0
$$883$$ −44.1052 −1.48426 −0.742130 0.670256i $$-0.766184\pi$$
−0.742130 + 0.670256i $$0.766184\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 7.88444 0.264734 0.132367 0.991201i $$-0.457742\pi$$
0.132367 + 0.991201i $$0.457742\pi$$
$$888$$ 0 0
$$889$$ −0.0915189 −0.00306945
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 26.6013 0.890179
$$894$$ 0 0
$$895$$ 47.2995 1.58105
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 34.8256 1.16150
$$900$$ 0 0
$$901$$ 4.21120 0.140295
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −65.4552 −2.17581
$$906$$ 0 0
$$907$$ 12.9067 0.428561 0.214280 0.976772i $$-0.431259\pi$$
0.214280 + 0.976772i $$0.431259\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −21.1857 −0.701914 −0.350957 0.936391i $$-0.614144\pi$$
−0.350957 + 0.936391i $$0.614144\pi$$
$$912$$ 0 0
$$913$$ −1.37733 −0.0455828
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −40.0871 −1.32379
$$918$$ 0 0
$$919$$ −31.4688 −1.03806 −0.519031 0.854756i $$-0.673707\pi$$
−0.519031 + 0.854756i $$0.673707\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 36.9394 1.21588
$$924$$ 0 0
$$925$$ −66.7110 −2.19344
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −2.32676 −0.0763386 −0.0381693 0.999271i $$-0.512153\pi$$
−0.0381693 + 0.999271i $$0.512153\pi$$
$$930$$ 0 0
$$931$$ 8.00774 0.262443
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −1.89124 −0.0618503
$$936$$ 0 0
$$937$$ −10.9982 −0.359297 −0.179649 0.983731i $$-0.557496\pi$$
−0.179649 + 0.983731i $$0.557496\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −24.1037 −0.785758 −0.392879 0.919590i $$-0.628521\pi$$
−0.392879 + 0.919590i $$0.628521\pi$$
$$942$$ 0 0
$$943$$ 16.4593 0.535990
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 11.9195 0.387332 0.193666 0.981067i $$-0.437962\pi$$
0.193666 + 0.981067i $$0.437962\pi$$
$$948$$ 0 0
$$949$$ 20.9317 0.679472
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −36.8289 −1.19301 −0.596503 0.802611i $$-0.703444\pi$$
−0.596503 + 0.802611i $$0.703444\pi$$
$$954$$ 0 0
$$955$$ 67.7948 2.19379
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 31.2713 1.00980
$$960$$ 0 0
$$961$$ −4.11793 −0.132836
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −7.72874 −0.248797
$$966$$ 0 0
$$967$$ −53.5604 −1.72239 −0.861193 0.508279i $$-0.830282\pi$$
−0.861193 + 0.508279i $$0.830282\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −53.2327 −1.70832 −0.854159 0.520012i $$-0.825927\pi$$
−0.854159 + 0.520012i $$0.825927\pi$$
$$972$$ 0 0
$$973$$ −22.9299 −0.735100
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 13.4570 0.430527 0.215264 0.976556i $$-0.430939\pi$$
0.215264 + 0.976556i $$0.430939\pi$$
$$978$$ 0 0
$$979$$ −1.25484 −0.0401050
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 10.2412 0.326644 0.163322 0.986573i $$-0.447779\pi$$
0.163322 + 0.986573i $$0.447779\pi$$
$$984$$ 0 0
$$985$$ 82.2140 2.61956
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −17.6682 −0.561816
$$990$$ 0 0
$$991$$ −2.00000 −0.0635321 −0.0317660 0.999495i $$-0.510113\pi$$
−0.0317660 + 0.999495i $$0.510113\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 11.9632 0.379258
$$996$$ 0 0
$$997$$ 38.5016 1.21936 0.609678 0.792649i $$-0.291299\pi$$
0.609678 + 0.792649i $$0.291299\pi$$
$$998$$ 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 3888.2.a.bk.1.3 3
3.2 odd 2 3888.2.a.bd.1.1 3
4.3 odd 2 243.2.a.f.1.1 yes 3
12.11 even 2 243.2.a.e.1.3 3
20.19 odd 2 6075.2.a.bq.1.3 3
36.7 odd 6 243.2.c.e.82.3 6
36.11 even 6 243.2.c.f.82.1 6
36.23 even 6 243.2.c.f.163.1 6
36.31 odd 6 243.2.c.e.163.3 6
60.59 even 2 6075.2.a.bv.1.1 3
108.7 odd 18 729.2.e.b.325.1 6
108.11 even 18 729.2.e.h.82.1 6
108.23 even 18 729.2.e.g.406.1 6
108.31 odd 18 729.2.e.b.406.1 6
108.43 odd 18 729.2.e.c.82.1 6
108.47 even 18 729.2.e.g.325.1 6
108.59 even 18 729.2.e.h.649.1 6
108.67 odd 18 729.2.e.i.163.1 6
108.79 odd 18 729.2.e.i.568.1 6
108.83 even 18 729.2.e.a.568.1 6
108.95 even 18 729.2.e.a.163.1 6
108.103 odd 18 729.2.e.c.649.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.e.1.3 3 12.11 even 2
243.2.a.f.1.1 yes 3 4.3 odd 2
243.2.c.e.82.3 6 36.7 odd 6
243.2.c.e.163.3 6 36.31 odd 6
243.2.c.f.82.1 6 36.11 even 6
243.2.c.f.163.1 6 36.23 even 6
729.2.e.a.163.1 6 108.95 even 18
729.2.e.a.568.1 6 108.83 even 18
729.2.e.b.325.1 6 108.7 odd 18
729.2.e.b.406.1 6 108.31 odd 18
729.2.e.c.82.1 6 108.43 odd 18
729.2.e.c.649.1 6 108.103 odd 18
729.2.e.g.325.1 6 108.47 even 18
729.2.e.g.406.1 6 108.23 even 18
729.2.e.h.82.1 6 108.11 even 18
729.2.e.h.649.1 6 108.59 even 18
729.2.e.i.163.1 6 108.67 odd 18
729.2.e.i.568.1 6 108.79 odd 18
3888.2.a.bd.1.1 3 3.2 odd 2
3888.2.a.bk.1.3 3 1.1 even 1 trivial
6075.2.a.bq.1.3 3 20.19 odd 2
6075.2.a.bv.1.1 3 60.59 even 2