# Properties

 Label 1080.2.a.e.1.1 Level $1080$ Weight $2$ Character 1080.1 Self dual yes Analytic conductor $8.624$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, 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("1080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.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: $$8.62384341830$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{5} +2.00000 q^{7} +O(q^{10})$$ $$q-1.00000 q^{5} +2.00000 q^{7} -4.00000 q^{11} -2.00000 q^{13} -5.00000 q^{17} -5.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +2.00000 q^{29} +7.00000 q^{31} -2.00000 q^{35} -6.00000 q^{37} +4.00000 q^{43} -4.00000 q^{47} -3.00000 q^{49} -9.00000 q^{53} +4.00000 q^{55} -14.0000 q^{59} -11.0000 q^{61} +2.00000 q^{65} +14.0000 q^{67} -12.0000 q^{73} -8.00000 q^{77} -3.00000 q^{79} +1.00000 q^{83} +5.00000 q^{85} -4.00000 q^{91} +5.00000 q^{95} +16.0000 q^{97} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −2.00000 −0.554700 −0.277350 0.960769i $$-0.589456\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ 0 0
$$31$$ 7.00000 1.25724 0.628619 0.777714i $$-0.283621\pi$$
0.628619 + 0.777714i $$0.283621\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −2.00000 −0.338062
$$36$$ 0 0
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −14.0000 −1.82264 −0.911322 0.411693i $$-0.864937\pi$$
−0.911322 + 0.411693i $$0.864937\pi$$
$$60$$ 0 0
$$61$$ −11.0000 −1.40841 −0.704203 0.709999i $$-0.748695\pi$$
−0.704203 + 0.709999i $$0.748695\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 2.00000 0.248069
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −12.0000 −1.40449 −0.702247 0.711934i $$-0.747820\pi$$
−0.702247 + 0.711934i $$0.747820\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ −3.00000 −0.337526 −0.168763 0.985657i $$-0.553977\pi$$
−0.168763 + 0.985657i $$0.553977\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 1.00000 0.109764 0.0548821 0.998493i $$-0.482522\pi$$
0.0548821 + 0.998493i $$0.482522\pi$$
$$84$$ 0 0
$$85$$ 5.00000 0.542326
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ −4.00000 −0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 5.00000 0.512989
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −12.0000 −1.19404 −0.597022 0.802225i $$-0.703650\pi$$
−0.597022 + 0.802225i $$0.703650\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −19.0000 −1.81987 −0.909935 0.414751i $$-0.863869\pi$$
−0.909935 + 0.414751i $$0.863869\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −6.00000 −0.564433 −0.282216 0.959351i $$-0.591070\pi$$
−0.282216 + 0.959351i $$0.591070\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −10.0000 −0.916698
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 6.00000 0.532414 0.266207 0.963916i $$-0.414230\pi$$
0.266207 + 0.963916i $$0.414230\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 18.0000 1.57267 0.786334 0.617802i $$-0.211977\pi$$
0.786334 + 0.617802i $$0.211977\pi$$
$$132$$ 0 0
$$133$$ −10.0000 −0.867110
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 17.0000 1.45241 0.726204 0.687479i $$-0.241283\pi$$
0.726204 + 0.687479i $$0.241283\pi$$
$$138$$ 0 0
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 16.0000 1.31077 0.655386 0.755295i $$-0.272506\pi$$
0.655386 + 0.755295i $$0.272506\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −7.00000 −0.562254
$$156$$ 0 0
$$157$$ 16.0000 1.27694 0.638470 0.769647i $$-0.279568\pi$$
0.638470 + 0.769647i $$0.279568\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ −14.0000 −1.09656 −0.548282 0.836293i $$-0.684718\pi$$
−0.548282 + 0.836293i $$0.684718\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 13.0000 0.988372 0.494186 0.869356i $$-0.335466\pi$$
0.494186 + 0.869356i $$0.335466\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ −19.0000 −1.41226 −0.706129 0.708083i $$-0.749560\pi$$
−0.706129 + 0.708083i $$0.749560\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 20.0000 1.46254
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −14.0000 −1.01300 −0.506502 0.862239i $$-0.669062\pi$$
−0.506502 + 0.862239i $$0.669062\pi$$
$$192$$ 0 0
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5.00000 0.356235 0.178118 0.984009i $$-0.442999\pi$$
0.178118 + 0.984009i $$0.442999\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 4.00000 0.280745
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 20.0000 1.38343
$$210$$ 0 0
$$211$$ −19.0000 −1.30801 −0.654007 0.756489i $$-0.726913\pi$$
−0.654007 + 0.756489i $$0.726913\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ 14.0000 0.950382
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 10.0000 0.672673
$$222$$ 0 0
$$223$$ −10.0000 −0.669650 −0.334825 0.942280i $$-0.608677\pi$$
−0.334825 + 0.942280i $$0.608677\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −3.00000 −0.199117 −0.0995585 0.995032i $$-0.531743\pi$$
−0.0995585 + 0.995032i $$0.531743\pi$$
$$228$$ 0 0
$$229$$ −29.0000 −1.91637 −0.958187 0.286143i $$-0.907627\pi$$
−0.958187 + 0.286143i $$0.907627\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 11.0000 0.708572 0.354286 0.935137i $$-0.384724\pi$$
0.354286 + 0.935137i $$0.384724\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3.00000 0.191663
$$246$$ 0 0
$$247$$ 10.0000 0.636285
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 24.0000 1.51487 0.757433 0.652913i $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −27.0000 −1.68421 −0.842107 0.539311i $$-0.818685\pi$$
−0.842107 + 0.539311i $$0.818685\pi$$
$$258$$ 0 0
$$259$$ −12.0000 −0.745644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 0 0
$$265$$ 9.00000 0.552866
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −9.00000 −0.546711 −0.273356 0.961913i $$-0.588134\pi$$
−0.273356 + 0.961913i $$0.588134\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −4.00000 −0.240337 −0.120168 0.992754i $$-0.538343\pi$$
−0.120168 + 0.992754i $$0.538343\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 14.0000 0.835170 0.417585 0.908638i $$-0.362877\pi$$
0.417585 + 0.908638i $$0.362877\pi$$
$$282$$ 0 0
$$283$$ 8.00000 0.475551 0.237775 0.971320i $$-0.423582\pi$$
0.237775 + 0.971320i $$0.423582\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −7.00000 −0.408944 −0.204472 0.978872i $$-0.565548\pi$$
−0.204472 + 0.978872i $$0.565548\pi$$
$$294$$ 0 0
$$295$$ 14.0000 0.815112
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 2.00000 0.115663
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 11.0000 0.629858
$$306$$ 0 0
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 22.0000 1.24751 0.623753 0.781622i $$-0.285607\pi$$
0.623753 + 0.781622i $$0.285607\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 27.0000 1.51647 0.758236 0.651981i $$-0.226062\pi$$
0.758236 + 0.651981i $$0.226062\pi$$
$$318$$ 0 0
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 25.0000 1.39104
$$324$$ 0 0
$$325$$ −2.00000 −0.110940
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −14.0000 −0.764902
$$336$$ 0 0
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −28.0000 −1.51629
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −20.0000 −1.07366 −0.536828 0.843692i $$-0.680378\pi$$
−0.536828 + 0.843692i $$0.680378\pi$$
$$348$$ 0 0
$$349$$ −3.00000 −0.160586 −0.0802932 0.996771i $$-0.525586\pi$$
−0.0802932 + 0.996771i $$0.525586\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −26.0000 −1.38384 −0.691920 0.721974i $$-0.743235\pi$$
−0.691920 + 0.721974i $$0.743235\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −18.0000 −0.950004 −0.475002 0.879985i $$-0.657553\pi$$
−0.475002 + 0.879985i $$0.657553\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 12.0000 0.628109
$$366$$ 0 0
$$367$$ −22.0000 −1.14839 −0.574195 0.818718i $$-0.694685\pi$$
−0.574195 + 0.818718i $$0.694685\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −18.0000 −0.934513
$$372$$ 0 0
$$373$$ −32.0000 −1.65690 −0.828449 0.560065i $$-0.810776\pi$$
−0.828449 + 0.560065i $$0.810776\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ −23.0000 −1.18143 −0.590715 0.806880i $$-0.701154\pi$$
−0.590715 + 0.806880i $$0.701154\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 27.0000 1.37964 0.689818 0.723983i $$-0.257691\pi$$
0.689818 + 0.723983i $$0.257691\pi$$
$$384$$ 0 0
$$385$$ 8.00000 0.407718
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 36.0000 1.82527 0.912636 0.408773i $$-0.134043\pi$$
0.912636 + 0.408773i $$0.134043\pi$$
$$390$$ 0 0
$$391$$ 5.00000 0.252861
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −14.0000 −0.697390
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −28.0000 −1.37779
$$414$$ 0 0
$$415$$ −1.00000 −0.0490881
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ 3.00000 0.146211 0.0731055 0.997324i $$-0.476709\pi$$
0.0731055 + 0.997324i $$0.476709\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −5.00000 −0.242536
$$426$$ 0 0
$$427$$ −22.0000 −1.06465
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −14.0000 −0.674356 −0.337178 0.941441i $$-0.609472\pi$$
−0.337178 + 0.941441i $$0.609472\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 5.00000 0.239182
$$438$$ 0 0
$$439$$ −17.0000 −0.811366 −0.405683 0.914014i $$-0.632966\pi$$
−0.405683 + 0.914014i $$0.632966\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −39.0000 −1.85295 −0.926473 0.376361i $$-0.877175\pi$$
−0.926473 + 0.376361i $$0.877175\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ 28.0000 1.30978 0.654892 0.755722i $$-0.272714\pi$$
0.654892 + 0.755722i $$0.272714\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −42.0000 −1.95614 −0.978068 0.208288i $$-0.933211\pi$$
−0.978068 + 0.208288i $$0.933211\pi$$
$$462$$ 0 0
$$463$$ −18.0000 −0.836531 −0.418265 0.908325i $$-0.637362\pi$$
−0.418265 + 0.908325i $$0.637362\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 13.0000 0.601568 0.300784 0.953692i $$-0.402752\pi$$
0.300784 + 0.953692i $$0.402752\pi$$
$$468$$ 0 0
$$469$$ 28.0000 1.29292
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ −5.00000 −0.229416
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −28.0000 −1.27935 −0.639676 0.768644i $$-0.720932\pi$$
−0.639676 + 0.768644i $$0.720932\pi$$
$$480$$ 0 0
$$481$$ 12.0000 0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −16.0000 −0.726523
$$486$$ 0 0
$$487$$ 40.0000 1.81257 0.906287 0.422664i $$-0.138905\pi$$
0.906287 + 0.422664i $$0.138905\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 6.00000 0.270776 0.135388 0.990793i $$-0.456772\pi$$
0.135388 + 0.990793i $$0.456772\pi$$
$$492$$ 0 0
$$493$$ −10.0000 −0.450377
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −25.0000 −1.11915 −0.559577 0.828778i $$-0.689036\pi$$
−0.559577 + 0.828778i $$0.689036\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 15.0000 0.668817 0.334408 0.942428i $$-0.391463\pi$$
0.334408 + 0.942428i $$0.391463\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −28.0000 −1.24108 −0.620539 0.784176i $$-0.713086\pi$$
−0.620539 + 0.784176i $$0.713086\pi$$
$$510$$ 0 0
$$511$$ −24.0000 −1.06170
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −4.00000 −0.176261
$$516$$ 0 0
$$517$$ 16.0000 0.703679
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −40.0000 −1.75243 −0.876216 0.481919i $$-0.839940\pi$$
−0.876216 + 0.481919i $$0.839940\pi$$
$$522$$ 0 0
$$523$$ 10.0000 0.437269 0.218635 0.975807i $$-0.429840\pi$$
0.218635 + 0.975807i $$0.429840\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −35.0000 −1.52462
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −12.0000 −0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 12.0000 0.516877
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 19.0000 0.813871
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −10.0000 −0.426014
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 10.0000 0.423714 0.211857 0.977301i $$-0.432049\pi$$
0.211857 + 0.977301i $$0.432049\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −4.00000 −0.168580 −0.0842900 0.996441i $$-0.526862\pi$$
−0.0842900 + 0.996441i $$0.526862\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 10.0000 0.419222 0.209611 0.977785i $$-0.432780\pi$$
0.209611 + 0.977785i $$0.432780\pi$$
$$570$$ 0 0
$$571$$ −5.00000 −0.209243 −0.104622 0.994512i $$-0.533363\pi$$
−0.104622 + 0.994512i $$0.533363\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −1.00000 −0.0417029
$$576$$ 0 0
$$577$$ −16.0000 −0.666089 −0.333044 0.942911i $$-0.608076\pi$$
−0.333044 + 0.942911i $$0.608076\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.00000 0.0829740
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 15.0000 0.619116 0.309558 0.950881i $$-0.399819\pi$$
0.309558 + 0.950881i $$0.399819\pi$$
$$588$$ 0 0
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 9.00000 0.369586 0.184793 0.982777i $$-0.440839\pi$$
0.184793 + 0.982777i $$0.440839\pi$$
$$594$$ 0 0
$$595$$ 10.0000 0.409960
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −18.0000 −0.735460 −0.367730 0.929933i $$-0.619865\pi$$
−0.367730 + 0.929933i $$0.619865\pi$$
$$600$$ 0 0
$$601$$ −11.0000 −0.448699 −0.224350 0.974509i $$-0.572026\pi$$
−0.224350 + 0.974509i $$0.572026\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ −36.0000 −1.46119 −0.730597 0.682808i $$-0.760758\pi$$
−0.730597 + 0.682808i $$0.760758\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −33.0000 −1.32853 −0.664265 0.747497i $$-0.731255\pi$$
−0.664265 + 0.747497i $$0.731255\pi$$
$$618$$ 0 0
$$619$$ 32.0000 1.28619 0.643094 0.765787i $$-0.277650\pi$$
0.643094 + 0.765787i $$0.277650\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 30.0000 1.19618
$$630$$ 0 0
$$631$$ 13.0000 0.517522 0.258761 0.965941i $$-0.416686\pi$$
0.258761 + 0.965941i $$0.416686\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −6.00000 −0.238103
$$636$$ 0 0
$$637$$ 6.00000 0.237729
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 6.00000 0.236617 0.118308 0.992977i $$-0.462253\pi$$
0.118308 + 0.992977i $$0.462253\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −33.0000 −1.29736 −0.648682 0.761060i $$-0.724679\pi$$
−0.648682 + 0.761060i $$0.724679\pi$$
$$648$$ 0 0
$$649$$ 56.0000 2.19819
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −41.0000 −1.60445 −0.802227 0.597019i $$-0.796352\pi$$
−0.802227 + 0.597019i $$0.796352\pi$$
$$654$$ 0 0
$$655$$ −18.0000 −0.703318
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −18.0000 −0.701180 −0.350590 0.936529i $$-0.614019\pi$$
−0.350590 + 0.936529i $$0.614019\pi$$
$$660$$ 0 0
$$661$$ −42.0000 −1.63361 −0.816805 0.576913i $$-0.804257\pi$$
−0.816805 + 0.576913i $$0.804257\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10.0000 0.387783
$$666$$ 0 0
$$667$$ −2.00000 −0.0774403
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 44.0000 1.69860
$$672$$ 0 0
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.00000 0.0768662 0.0384331 0.999261i $$-0.487763\pi$$
0.0384331 + 0.999261i $$0.487763\pi$$
$$678$$ 0 0
$$679$$ 32.0000 1.22805
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 9.00000 0.344375 0.172188 0.985064i $$-0.444916\pi$$
0.172188 + 0.985064i $$0.444916\pi$$
$$684$$ 0 0
$$685$$ −17.0000 −0.649537
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 18.0000 0.685745
$$690$$ 0 0
$$691$$ −19.0000 −0.722794 −0.361397 0.932412i $$-0.617700\pi$$
−0.361397 + 0.932412i $$0.617700\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −12.0000 −0.455186
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 10.0000 0.377695 0.188847 0.982006i $$-0.439525\pi$$
0.188847 + 0.982006i $$0.439525\pi$$
$$702$$ 0 0
$$703$$ 30.0000 1.13147
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −24.0000 −0.902613
$$708$$ 0 0
$$709$$ 26.0000 0.976450 0.488225 0.872718i $$-0.337644\pi$$
0.488225 + 0.872718i $$0.337644\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −7.00000 −0.262152
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −46.0000 −1.71551 −0.857755 0.514058i $$-0.828142\pi$$
−0.857755 + 0.514058i $$0.828142\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 2.00000 0.0742781
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −20.0000 −0.739727
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −56.0000 −2.06279
$$738$$ 0 0
$$739$$ −15.0000 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −16.0000 −0.586195
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 24.0000 0.876941
$$750$$ 0 0
$$751$$ −3.00000 −0.109472 −0.0547358 0.998501i $$-0.517432\pi$$
−0.0547358 + 0.998501i $$0.517432\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −8.00000 −0.291150
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ −38.0000 −1.37569
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 28.0000 1.01102
$$768$$ 0 0
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 15.0000 0.539513 0.269756 0.962929i $$-0.413057\pi$$
0.269756 + 0.962929i $$0.413057\pi$$
$$774$$ 0 0
$$775$$ 7.00000 0.251447
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −16.0000 −0.571064
$$786$$ 0 0
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 22.0000 0.781243
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −3.00000 −0.106265 −0.0531327 0.998587i $$-0.516921\pi$$
−0.0531327 + 0.998587i $$0.516921\pi$$
$$798$$ 0 0
$$799$$ 20.0000 0.707549
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 48.0000 1.69388
$$804$$ 0 0
$$805$$ 2.00000 0.0704907
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 24.0000 0.843795 0.421898 0.906644i $$-0.361364\pi$$
0.421898 + 0.906644i $$0.361364\pi$$
$$810$$ 0 0
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 14.0000 0.490399
$$816$$ 0 0
$$817$$ −20.0000 −0.699711
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ 0 0
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 39.0000 1.35616 0.678081 0.734987i $$-0.262812\pi$$
0.678081 + 0.734987i $$0.262812\pi$$
$$828$$ 0 0
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 15.0000 0.519719
$$834$$ 0 0
$$835$$ −3.00000 −0.103819
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 9.00000 0.309609
$$846$$ 0 0
$$847$$ 10.0000 0.343604
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ 0 0
$$853$$ −28.0000 −0.958702 −0.479351 0.877623i $$-0.659128\pi$$
−0.479351 + 0.877623i $$0.659128\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −21.0000 −0.717346 −0.358673 0.933463i $$-0.616771\pi$$
−0.358673 + 0.933463i $$0.616771\pi$$
$$858$$ 0 0
$$859$$ 41.0000 1.39890 0.699451 0.714681i $$-0.253428\pi$$
0.699451 + 0.714681i $$0.253428\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −33.0000 −1.12333 −0.561667 0.827364i $$-0.689840\pi$$
−0.561667 + 0.827364i $$0.689840\pi$$
$$864$$ 0 0
$$865$$ −13.0000 −0.442013
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 12.0000 0.407072
$$870$$ 0 0
$$871$$ −28.0000 −0.948744
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 54.0000 1.81931 0.909653 0.415369i $$-0.136347\pi$$
0.909653 + 0.415369i $$0.136347\pi$$
$$882$$ 0 0
$$883$$ −22.0000 −0.740359 −0.370179 0.928960i $$-0.620704\pi$$
−0.370179 + 0.928960i $$0.620704\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 23.0000 0.772264 0.386132 0.922443i $$-0.373811\pi$$
0.386132 + 0.922443i $$0.373811\pi$$
$$888$$ 0 0
$$889$$ 12.0000 0.402467
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ 0 0
$$895$$ −4.00000 −0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 14.0000 0.466926
$$900$$ 0 0
$$901$$ 45.0000 1.49917
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.0000 0.631581
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 2.00000 0.0662630 0.0331315 0.999451i $$-0.489452\pi$$
0.0331315 + 0.999451i $$0.489452\pi$$
$$912$$ 0 0
$$913$$ −4.00000 −0.132381
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 36.0000 1.18882
$$918$$ 0 0
$$919$$ 32.0000 1.05558 0.527791 0.849374i $$-0.323020\pi$$
0.527791 + 0.849374i $$0.323020\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 8.00000 0.262471 0.131236 0.991351i $$-0.458106\pi$$
0.131236 + 0.991351i $$0.458106\pi$$
$$930$$ 0 0
$$931$$ 15.0000 0.491605
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −20.0000 −0.654070
$$936$$ 0 0
$$937$$ −56.0000 −1.82944 −0.914720 0.404088i $$-0.867589\pi$$
−0.914720 + 0.404088i $$0.867589\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −34.0000 −1.10837 −0.554184 0.832394i $$-0.686970\pi$$
−0.554184 + 0.832394i $$0.686970\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 53.0000 1.72227 0.861134 0.508378i $$-0.169755\pi$$
0.861134 + 0.508378i $$0.169755\pi$$
$$948$$ 0 0
$$949$$ 24.0000 0.779073
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ 14.0000 0.453029
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 34.0000 1.09792
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −10.0000 −0.321911
$$966$$ 0 0
$$967$$ −30.0000 −0.964735 −0.482367 0.875969i $$-0.660223\pi$$
−0.482367 + 0.875969i $$0.660223\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −14.0000 −0.449281 −0.224641 0.974442i $$-0.572121\pi$$
−0.224641 + 0.974442i $$0.572121\pi$$
$$972$$ 0 0
$$973$$ 24.0000 0.769405
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 3.00000 0.0956851 0.0478426 0.998855i $$-0.484765\pi$$
0.0478426 + 0.998855i $$0.484765\pi$$
$$984$$ 0 0
$$985$$ −5.00000 −0.159313
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4.00000 −0.127193
$$990$$ 0 0
$$991$$ −55.0000 −1.74713 −0.873566 0.486705i $$-0.838199\pi$$
−0.873566 + 0.486705i $$0.838199\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ −48.0000 −1.52018 −0.760088 0.649821i $$-0.774844\pi$$
−0.760088 + 0.649821i $$0.774844\pi$$
$$998$$ 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 1080.2.a.e.1.1 1
3.2 odd 2 1080.2.a.l.1.1 yes 1
4.3 odd 2 2160.2.a.e.1.1 1
5.2 odd 4 5400.2.f.f.649.2 2
5.3 odd 4 5400.2.f.f.649.1 2
5.4 even 2 5400.2.a.j.1.1 1
8.3 odd 2 8640.2.a.bi.1.1 1
8.5 even 2 8640.2.a.cd.1.1 1
9.2 odd 6 3240.2.q.b.1081.1 2
9.4 even 3 3240.2.q.p.2161.1 2
9.5 odd 6 3240.2.q.b.2161.1 2
9.7 even 3 3240.2.q.p.1081.1 2
12.11 even 2 2160.2.a.m.1.1 1
15.2 even 4 5400.2.f.x.649.2 2
15.8 even 4 5400.2.f.x.649.1 2
15.14 odd 2 5400.2.a.q.1.1 1
24.5 odd 2 8640.2.a.t.1.1 1
24.11 even 2 8640.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.a.e.1.1 1 1.1 even 1 trivial
1080.2.a.l.1.1 yes 1 3.2 odd 2
2160.2.a.e.1.1 1 4.3 odd 2
2160.2.a.m.1.1 1 12.11 even 2
3240.2.q.b.1081.1 2 9.2 odd 6
3240.2.q.b.2161.1 2 9.5 odd 6
3240.2.q.p.1081.1 2 9.7 even 3
3240.2.q.p.2161.1 2 9.4 even 3
5400.2.a.j.1.1 1 5.4 even 2
5400.2.a.q.1.1 1 15.14 odd 2
5400.2.f.f.649.1 2 5.3 odd 4
5400.2.f.f.649.2 2 5.2 odd 4
5400.2.f.x.649.1 2 15.8 even 4
5400.2.f.x.649.2 2 15.2 even 4
8640.2.a.k.1.1 1 24.11 even 2
8640.2.a.t.1.1 1 24.5 odd 2
8640.2.a.bi.1.1 1 8.3 odd 2
8640.2.a.cd.1.1 1 8.5 even 2