Properties

 Label 1080.2.a.h.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)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} -2.00000 q^{7} +O(q^{10})$$ $$q+1.00000 q^{5} -2.00000 q^{7} -6.00000 q^{13} -7.00000 q^{17} +7.00000 q^{19} -7.00000 q^{23} +1.00000 q^{25} -6.00000 q^{29} +3.00000 q^{31} -2.00000 q^{35} -6.00000 q^{37} -4.00000 q^{41} +8.00000 q^{43} +4.00000 q^{47} -3.00000 q^{49} +5.00000 q^{53} -6.00000 q^{59} -3.00000 q^{61} -6.00000 q^{65} -10.0000 q^{67} -12.0000 q^{71} +16.0000 q^{73} +1.00000 q^{79} -9.00000 q^{83} -7.00000 q^{85} +4.00000 q^{89} +12.0000 q^{91} +7.00000 q^{95} -16.0000 q^{97} +O(q^{100})$$

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 $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −7.00000 −1.69775 −0.848875 0.528594i $$-0.822719\pi$$
−0.848875 + 0.528594i $$0.822719\pi$$
$$18$$ 0 0
$$19$$ 7.00000 1.60591 0.802955 0.596040i $$-0.203260\pi$$
0.802955 + 0.596040i $$0.203260\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −7.00000 −1.45960 −0.729800 0.683660i $$-0.760387\pi$$
−0.729800 + 0.683660i $$0.760387\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 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
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 4.00000 0.583460 0.291730 0.956501i $$-0.405769\pi$$
0.291730 + 0.956501i $$0.405769\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 5.00000 0.686803 0.343401 0.939189i $$-0.388421\pi$$
0.343401 + 0.939189i $$0.388421\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −6.00000 −0.781133 −0.390567 0.920575i $$-0.627721\pi$$
−0.390567 + 0.920575i $$0.627721\pi$$
$$60$$ 0 0
$$61$$ −3.00000 −0.384111 −0.192055 0.981384i $$-0.561515\pi$$
−0.192055 + 0.981384i $$0.561515\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −6.00000 −0.744208
$$66$$ 0 0
$$67$$ −10.0000 −1.22169 −0.610847 0.791748i $$-0.709171\pi$$
−0.610847 + 0.791748i $$0.709171\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 16.0000 1.87266 0.936329 0.351123i $$-0.114200\pi$$
0.936329 + 0.351123i $$0.114200\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 1.00000 0.112509 0.0562544 0.998416i $$-0.482084\pi$$
0.0562544 + 0.998416i $$0.482084\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −9.00000 −0.987878 −0.493939 0.869496i $$-0.664443\pi$$
−0.493939 + 0.869496i $$0.664443\pi$$
$$84$$ 0 0
$$85$$ −7.00000 −0.759257
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 4.00000 0.423999 0.212000 0.977270i $$-0.432002\pi$$
0.212000 + 0.977270i $$0.432002\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 7.00000 0.718185
$$96$$ 0 0
$$97$$ −16.0000 −1.62455 −0.812277 0.583272i $$-0.801772\pi$$
−0.812277 + 0.583272i $$0.801772\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 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
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ 5.00000 0.478913 0.239457 0.970907i $$-0.423031\pi$$
0.239457 + 0.970907i $$0.423031\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −7.00000 −0.652753
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 14.0000 1.28338
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ 14.0000 1.24230 0.621150 0.783692i $$-0.286666\pi$$
0.621150 + 0.783692i $$0.286666\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 0 0
$$133$$ −14.0000 −1.21395
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000 0.256307 0.128154 0.991754i $$-0.459095\pi$$
0.128154 + 0.991754i $$0.459095\pi$$
$$138$$ 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
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 3.00000 0.240966
$$156$$ 0 0
$$157$$ −20.0000 −1.59617 −0.798087 0.602542i $$-0.794154\pi$$
−0.798087 + 0.602542i $$0.794154\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 14.0000 1.10335
$$162$$ 0 0
$$163$$ 2.00000 0.156652 0.0783260 0.996928i $$-0.475042\pi$$
0.0783260 + 0.996928i $$0.475042\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19.0000 −1.47026 −0.735132 0.677924i $$-0.762880\pi$$
−0.735132 + 0.677924i $$0.762880\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −9.00000 −0.684257 −0.342129 0.939653i $$-0.611148\pi$$
−0.342129 + 0.939653i $$0.611148\pi$$
$$174$$ 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
$$181$$ 5.00000 0.371647 0.185824 0.982583i $$-0.440505\pi$$
0.185824 + 0.982583i $$0.440505\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2.00000 −0.144715 −0.0723575 0.997379i $$-0.523052\pi$$
−0.0723575 + 0.997379i $$0.523052\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 23.0000 1.63868 0.819341 0.573306i $$-0.194340\pi$$
0.819341 + 0.573306i $$0.194340\pi$$
$$198$$ 0 0
$$199$$ 24.0000 1.70131 0.850657 0.525720i $$-0.176204\pi$$
0.850657 + 0.525720i $$0.176204\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 12.0000 0.842235
$$204$$ 0 0
$$205$$ −4.00000 −0.279372
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −15.0000 −1.03264 −0.516321 0.856395i $$-0.672699\pi$$
−0.516321 + 0.856395i $$0.672699\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 8.00000 0.545595
$$216$$ 0 0
$$217$$ −6.00000 −0.407307
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 42.0000 2.82523
$$222$$ 0 0
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 27.0000 1.79205 0.896026 0.444001i $$-0.146441\pi$$
0.896026 + 0.444001i $$0.146441\pi$$
$$228$$ 0 0
$$229$$ 3.00000 0.198246 0.0991228 0.995075i $$-0.468396\pi$$
0.0991228 + 0.995075i $$0.468396\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 0 0
$$235$$ 4.00000 0.260931
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 18.0000 1.16432 0.582162 0.813073i $$-0.302207\pi$$
0.582162 + 0.813073i $$0.302207\pi$$
$$240$$ 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
$$247$$ −42.0000 −2.67240
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −16.0000 −1.00991 −0.504956 0.863145i $$-0.668491\pi$$
−0.504956 + 0.863145i $$0.668491\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −9.00000 −0.561405 −0.280702 0.959795i $$-0.590567\pi$$
−0.280702 + 0.959795i $$0.590567\pi$$
$$258$$ 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
$$265$$ 5.00000 0.307148
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −13.0000 −0.789694 −0.394847 0.918747i $$-0.629202\pi$$
−0.394847 + 0.918747i $$0.629202\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ 32.0000 1.88235
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 11.0000 0.642627 0.321313 0.946973i $$-0.395876\pi$$
0.321313 + 0.946973i $$0.395876\pi$$
$$294$$ 0 0
$$295$$ −6.00000 −0.349334
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 42.0000 2.42892
$$300$$ 0 0
$$301$$ −16.0000 −0.922225
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −26.0000 −1.48390 −0.741949 0.670456i $$-0.766098\pi$$
−0.741949 + 0.670456i $$0.766098\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −10.0000 −0.567048 −0.283524 0.958965i $$-0.591504\pi$$
−0.283524 + 0.958965i $$0.591504\pi$$
$$312$$ 0 0
$$313$$ 16.0000 0.904373 0.452187 0.891923i $$-0.350644\pi$$
0.452187 + 0.891923i $$0.350644\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.00000 0.0561656 0.0280828 0.999606i $$-0.491060\pi$$
0.0280828 + 0.999606i $$0.491060\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −49.0000 −2.72643
$$324$$ 0 0
$$325$$ −6.00000 −0.332820
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −10.0000 −0.546358
$$336$$ 0 0
$$337$$ 34.0000 1.85210 0.926049 0.377403i $$-0.123183\pi$$
0.926049 + 0.377403i $$0.123183\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 0 0
$$349$$ −35.0000 −1.87351 −0.936754 0.349990i $$-0.886185\pi$$
−0.936754 + 0.349990i $$0.886185\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ −12.0000 −0.636894
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 10.0000 0.527780 0.263890 0.964553i $$-0.414994\pi$$
0.263890 + 0.964553i $$0.414994\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 16.0000 0.837478
$$366$$ 0 0
$$367$$ −26.0000 −1.35719 −0.678594 0.734513i $$-0.737411\pi$$
−0.678594 + 0.734513i $$0.737411\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −10.0000 −0.519174
$$372$$ 0 0
$$373$$ 8.00000 0.414224 0.207112 0.978317i $$-0.433593\pi$$
0.207112 + 0.978317i $$0.433593\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ −35.0000 −1.79783 −0.898915 0.438124i $$-0.855643\pi$$
−0.898915 + 0.438124i $$0.855643\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −35.0000 −1.78842 −0.894208 0.447651i $$-0.852261\pi$$
−0.894208 + 0.447651i $$0.852261\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −24.0000 −1.21685 −0.608424 0.793612i $$-0.708198\pi$$
−0.608424 + 0.793612i $$0.708198\pi$$
$$390$$ 0 0
$$391$$ 49.0000 2.47804
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 1.00000 0.0503155
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 0 0
$$403$$ −18.0000 −0.896644
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 1.00000 0.0494468 0.0247234 0.999694i $$-0.492129\pi$$
0.0247234 + 0.999694i $$0.492129\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 12.0000 0.590481
$$414$$ 0 0
$$415$$ −9.00000 −0.441793
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 16.0000 0.781651 0.390826 0.920465i $$-0.372190\pi$$
0.390826 + 0.920465i $$0.372190\pi$$
$$420$$ 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
$$425$$ −7.00000 −0.339550
$$426$$ 0 0
$$427$$ 6.00000 0.290360
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 26.0000 1.25238 0.626188 0.779672i $$-0.284614\pi$$
0.626188 + 0.779672i $$0.284614\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −49.0000 −2.34399
$$438$$ 0 0
$$439$$ 19.0000 0.906821 0.453410 0.891302i $$-0.350207\pi$$
0.453410 + 0.891302i $$0.350207\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −9.00000 −0.427603 −0.213801 0.976877i $$-0.568585\pi$$
−0.213801 + 0.976877i $$0.568585\pi$$
$$444$$ 0 0
$$445$$ 4.00000 0.189618
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −14.0000 −0.660701 −0.330350 0.943858i $$-0.607167\pi$$
−0.330350 + 0.943858i $$0.607167\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 12.0000 0.562569
$$456$$ 0 0
$$457$$ 4.00000 0.187112 0.0935561 0.995614i $$-0.470177\pi$$
0.0935561 + 0.995614i $$0.470177\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −14.0000 −0.652045 −0.326023 0.945362i $$-0.605709\pi$$
−0.326023 + 0.945362i $$0.605709\pi$$
$$462$$ 0 0
$$463$$ −26.0000 −1.20832 −0.604161 0.796862i $$-0.706492\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −37.0000 −1.71216 −0.856078 0.516847i $$-0.827106\pi$$
−0.856078 + 0.516847i $$0.827106\pi$$
$$468$$ 0 0
$$469$$ 20.0000 0.923514
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 7.00000 0.321182
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −24.0000 −1.09659 −0.548294 0.836286i $$-0.684723\pi$$
−0.548294 + 0.836286i $$0.684723\pi$$
$$480$$ 0 0
$$481$$ 36.0000 1.64146
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −16.0000 −0.726523
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 34.0000 1.53440 0.767199 0.641409i $$-0.221650\pi$$
0.767199 + 0.641409i $$0.221650\pi$$
$$492$$ 0 0
$$493$$ 42.0000 1.89158
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 24.0000 1.07655
$$498$$ 0 0
$$499$$ 11.0000 0.492428 0.246214 0.969216i $$-0.420813\pi$$
0.246214 + 0.969216i $$0.420813\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 9.00000 0.401290 0.200645 0.979664i $$-0.435696\pi$$
0.200645 + 0.979664i $$0.435696\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −36.0000 −1.59567 −0.797836 0.602875i $$-0.794022\pi$$
−0.797836 + 0.602875i $$0.794022\pi$$
$$510$$ 0 0
$$511$$ −32.0000 −1.41560
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ 0 0
$$523$$ −18.0000 −0.787085 −0.393543 0.919306i $$-0.628751\pi$$
−0.393543 + 0.919306i $$0.628751\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −21.0000 −0.914774
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 24.0000 1.03956
$$534$$ 0 0
$$535$$ −4.00000 −0.172935
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 5.00000 0.214176
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −42.0000 −1.78926
$$552$$ 0 0
$$553$$ −2.00000 −0.0850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −34.0000 −1.44063 −0.720313 0.693649i $$-0.756002\pi$$
−0.720313 + 0.693649i $$0.756002\pi$$
$$558$$ 0 0
$$559$$ −48.0000 −2.03018
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ 6.00000 0.252422
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −22.0000 −0.922288 −0.461144 0.887325i $$-0.652561\pi$$
−0.461144 + 0.887325i $$0.652561\pi$$
$$570$$ 0 0
$$571$$ −17.0000 −0.711428 −0.355714 0.934595i $$-0.615762\pi$$
−0.355714 + 0.934595i $$0.615762\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −7.00000 −0.291920
$$576$$ 0 0
$$577$$ −20.0000 −0.832611 −0.416305 0.909225i $$-0.636675\pi$$
−0.416305 + 0.909225i $$0.636675\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 18.0000 0.746766
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9.00000 0.371470 0.185735 0.982600i $$-0.440533\pi$$
0.185735 + 0.982600i $$0.440533\pi$$
$$588$$ 0 0
$$589$$ 21.0000 0.865290
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −13.0000 −0.533846 −0.266923 0.963718i $$-0.586007\pi$$
−0.266923 + 0.963718i $$0.586007\pi$$
$$594$$ 0 0
$$595$$ 14.0000 0.573944
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −26.0000 −1.06233 −0.531166 0.847268i $$-0.678246\pi$$
−0.531166 + 0.847268i $$0.678246\pi$$
$$600$$ 0 0
$$601$$ 5.00000 0.203954 0.101977 0.994787i $$-0.467483\pi$$
0.101977 + 0.994787i $$0.467483\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −11.0000 −0.447214
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 29.0000 1.16750 0.583748 0.811935i $$-0.301586\pi$$
0.583748 + 0.811935i $$0.301586\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −8.00000 −0.320513
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 42.0000 1.67465
$$630$$ 0 0
$$631$$ −47.0000 −1.87104 −0.935520 0.353273i $$-0.885069\pi$$
−0.935520 + 0.353273i $$0.885069\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 14.0000 0.555573
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −48.0000 −1.89589 −0.947943 0.318440i $$-0.896841\pi$$
−0.947943 + 0.318440i $$0.896841\pi$$
$$642$$ 0 0
$$643$$ 42.0000 1.65632 0.828159 0.560493i $$-0.189388\pi$$
0.828159 + 0.560493i $$0.189388\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 9.00000 0.353827 0.176913 0.984226i $$-0.443389\pi$$
0.176913 + 0.984226i $$0.443389\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −3.00000 −0.117399 −0.0586995 0.998276i $$-0.518695\pi$$
−0.0586995 + 0.998276i $$0.518695\pi$$
$$654$$ 0 0
$$655$$ 14.0000 0.547025
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 46.0000 1.79191 0.895953 0.444149i $$-0.146494\pi$$
0.895953 + 0.444149i $$0.146494\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −14.0000 −0.542897
$$666$$ 0 0
$$667$$ 42.0000 1.62625
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 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 $$-0.555084\pi$$
−0.172188 + 0.985064i $$0.555084\pi$$
$$684$$ 0 0
$$685$$ 3.00000 0.114624
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −30.0000 −1.14291
$$690$$ 0 0
$$691$$ 1.00000 0.0380418 0.0190209 0.999819i $$-0.493945\pi$$
0.0190209 + 0.999819i $$0.493945\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 28.0000 1.06058
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 38.0000 1.43524 0.717620 0.696435i $$-0.245231\pi$$
0.717620 + 0.696435i $$0.245231\pi$$
$$702$$ 0 0
$$703$$ −42.0000 −1.58406
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.00000 0.300871
$$708$$ 0 0
$$709$$ −14.0000 −0.525781 −0.262891 0.964826i $$-0.584676\pi$$
−0.262891 + 0.964826i $$0.584676\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −21.0000 −0.786456
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 30.0000 1.11881 0.559406 0.828894i $$-0.311029\pi$$
0.559406 + 0.828894i $$0.311029\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −56.0000 −2.07123
$$732$$ 0 0
$$733$$ −16.0000 −0.590973 −0.295487 0.955347i $$-0.595482\pi$$
−0.295487 + 0.955347i $$0.595482\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −35.0000 −1.28750 −0.643748 0.765238i $$-0.722621\pi$$
−0.643748 + 0.765238i $$0.722621\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 8.00000 0.292314
$$750$$ 0 0
$$751$$ −15.0000 −0.547358 −0.273679 0.961821i $$-0.588241\pi$$
−0.273679 + 0.961821i $$0.588241\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 0 0
$$763$$ −10.0000 −0.362024
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 36.0000 1.29988
$$768$$ 0 0
$$769$$ −27.0000 −0.973645 −0.486822 0.873501i $$-0.661844\pi$$
−0.486822 + 0.873501i $$0.661844\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 5.00000 0.179838 0.0899188 0.995949i $$-0.471339\pi$$
0.0899188 + 0.995949i $$0.471339\pi$$
$$774$$ 0 0
$$775$$ 3.00000 0.107763
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −28.0000 −1.00320
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −20.0000 −0.713831
$$786$$ 0 0
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −12.0000 −0.426671
$$792$$ 0 0
$$793$$ 18.0000 0.639199
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −25.0000 −0.885545 −0.442773 0.896634i $$-0.646005\pi$$
−0.442773 + 0.896634i $$0.646005\pi$$
$$798$$ 0 0
$$799$$ −28.0000 −0.990569
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 14.0000 0.493435
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −12.0000 −0.421898 −0.210949 0.977497i $$-0.567655\pi$$
−0.210949 + 0.977497i $$0.567655\pi$$
$$810$$ 0 0
$$811$$ 36.0000 1.26413 0.632065 0.774915i $$-0.282207\pi$$
0.632065 + 0.774915i $$0.282207\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 2.00000 0.0700569
$$816$$ 0 0
$$817$$ 56.0000 1.95919
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 46.0000 1.60541 0.802706 0.596376i $$-0.203393\pi$$
0.802706 + 0.596376i $$0.203393\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 17.0000 0.591148 0.295574 0.955320i $$-0.404489\pi$$
0.295574 + 0.955320i $$0.404489\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 21.0000 0.727607
$$834$$ 0 0
$$835$$ −19.0000 −0.657522
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 23.0000 0.791224
$$846$$ 0 0
$$847$$ 22.0000 0.755929
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 42.0000 1.43974
$$852$$ 0 0
$$853$$ −20.0000 −0.684787 −0.342393 0.939557i $$-0.611238\pi$$
−0.342393 + 0.939557i $$0.611238\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 49.0000 1.67381 0.836904 0.547350i $$-0.184363\pi$$
0.836904 + 0.547350i $$0.184363\pi$$
$$858$$ 0 0
$$859$$ 21.0000 0.716511 0.358255 0.933624i $$-0.383372\pi$$
0.358255 + 0.933624i $$0.383372\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 17.0000 0.578687 0.289343 0.957225i $$-0.406563\pi$$
0.289343 + 0.957225i $$0.406563\pi$$
$$864$$ 0 0
$$865$$ −9.00000 −0.306009
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 60.0000 2.03302
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.00000 −0.0676123
$$876$$ 0 0
$$877$$ 52.0000 1.75592 0.877958 0.478738i $$-0.158906\pi$$
0.877958 + 0.478738i $$0.158906\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.00000 −0.0673817 −0.0336909 0.999432i $$-0.510726\pi$$
−0.0336909 + 0.999432i $$0.510726\pi$$
$$882$$ 0 0
$$883$$ −26.0000 −0.874970 −0.437485 0.899226i $$-0.644131\pi$$
−0.437485 + 0.899226i $$0.644131\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −39.0000 −1.30949 −0.654746 0.755849i $$-0.727224\pi$$
−0.654746 + 0.755849i $$0.727224\pi$$
$$888$$ 0 0
$$889$$ −28.0000 −0.939090
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 28.0000 0.936984
$$894$$ 0 0
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −18.0000 −0.600334
$$900$$ 0 0
$$901$$ −35.0000 −1.16602
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 5.00000 0.166206
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 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
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −28.0000 −0.924641
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 72.0000 2.36991
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 36.0000 1.18112 0.590561 0.806993i $$-0.298907\pi$$
0.590561 + 0.806993i $$0.298907\pi$$
$$930$$ 0 0
$$931$$ −21.0000 −0.688247
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −16.0000 −0.522697 −0.261349 0.965244i $$-0.584167\pi$$
−0.261349 + 0.965244i $$0.584167\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 50.0000 1.62995 0.814977 0.579494i $$-0.196750\pi$$
0.814977 + 0.579494i $$0.196750\pi$$
$$942$$ 0 0
$$943$$ 28.0000 0.911805
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −13.0000 −0.422443 −0.211222 0.977438i $$-0.567744\pi$$
−0.211222 + 0.977438i $$0.567744\pi$$
$$948$$ 0 0
$$949$$ −96.0000 −3.11629
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ −2.00000 −0.0647185
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −6.00000 −0.193750
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 6.00000 0.193147
$$966$$ 0 0
$$967$$ 50.0000 1.60789 0.803946 0.594703i $$-0.202730\pi$$
0.803946 + 0.594703i $$0.202730\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −34.0000 −1.09111 −0.545556 0.838074i $$-0.683681\pi$$
−0.545556 + 0.838074i $$0.683681\pi$$
$$972$$ 0 0
$$973$$ −24.0000 −0.769405
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −22.0000 −0.703842 −0.351921 0.936030i $$-0.614471\pi$$
−0.351921 + 0.936030i $$0.614471\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −51.0000 −1.62665 −0.813324 0.581811i $$-0.802344\pi$$
−0.813324 + 0.581811i $$0.802344\pi$$
$$984$$ 0 0
$$985$$ 23.0000 0.732841
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −56.0000 −1.78070
$$990$$ 0 0
$$991$$ 29.0000 0.921215 0.460608 0.887604i $$-0.347632\pi$$
0.460608 + 0.887604i $$0.347632\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 24.0000 0.760851
$$996$$ 0 0
$$997$$ −12.0000 −0.380044 −0.190022 0.981780i $$-0.560856\pi$$
−0.190022 + 0.981780i $$0.560856\pi$$
$$998$$ 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.h.1.1 yes 1
3.2 odd 2 1080.2.a.b.1.1 1
4.3 odd 2 2160.2.a.u.1.1 1
5.2 odd 4 5400.2.f.o.649.1 2
5.3 odd 4 5400.2.f.o.649.2 2
5.4 even 2 5400.2.a.bi.1.1 1
8.3 odd 2 8640.2.a.x.1.1 1
8.5 even 2 8640.2.a.h.1.1 1
9.2 odd 6 3240.2.q.v.1081.1 2
9.4 even 3 3240.2.q.i.2161.1 2
9.5 odd 6 3240.2.q.v.2161.1 2
9.7 even 3 3240.2.q.i.1081.1 2
12.11 even 2 2160.2.a.i.1.1 1
15.2 even 4 5400.2.f.n.649.1 2
15.8 even 4 5400.2.f.n.649.2 2
15.14 odd 2 5400.2.a.bh.1.1 1
24.5 odd 2 8640.2.a.bm.1.1 1
24.11 even 2 8640.2.a.ca.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.a.b.1.1 1 3.2 odd 2
1080.2.a.h.1.1 yes 1 1.1 even 1 trivial
2160.2.a.i.1.1 1 12.11 even 2
2160.2.a.u.1.1 1 4.3 odd 2
3240.2.q.i.1081.1 2 9.7 even 3
3240.2.q.i.2161.1 2 9.4 even 3
3240.2.q.v.1081.1 2 9.2 odd 6
3240.2.q.v.2161.1 2 9.5 odd 6
5400.2.a.bh.1.1 1 15.14 odd 2
5400.2.a.bi.1.1 1 5.4 even 2
5400.2.f.n.649.1 2 15.2 even 4
5400.2.f.n.649.2 2 15.8 even 4
5400.2.f.o.649.1 2 5.2 odd 4
5400.2.f.o.649.2 2 5.3 odd 4
8640.2.a.h.1.1 1 8.5 even 2
8640.2.a.x.1.1 1 8.3 odd 2
8640.2.a.bm.1.1 1 24.5 odd 2
8640.2.a.ca.1.1 1 24.11 even 2