# Properties

 Label 3120.2.a.u.1.1 Level $3120$ Weight $2$ Character 3120.1 Self dual yes Analytic conductor $24.913$ Analytic rank $0$ 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] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.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: $$24.9133254306$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3120.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} -1.00000 q^{17} +2.00000 q^{19} -3.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +6.00000 q^{31} +1.00000 q^{33} -3.00000 q^{35} +11.0000 q^{37} -1.00000 q^{39} -5.00000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +10.0000 q^{47} +2.00000 q^{49} -1.00000 q^{51} +11.0000 q^{53} +1.00000 q^{55} +2.00000 q^{57} -8.00000 q^{59} +13.0000 q^{61} -3.00000 q^{63} -1.00000 q^{65} -12.0000 q^{67} +3.00000 q^{69} +5.00000 q^{71} +10.0000 q^{73} +1.00000 q^{75} -3.00000 q^{77} +3.00000 q^{79} +1.00000 q^{81} +12.0000 q^{83} -1.00000 q^{85} -2.00000 q^{87} -15.0000 q^{89} +3.00000 q^{91} +6.00000 q^{93} +2.00000 q^{95} +17.0000 q^{97} +1.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −3.00000 −1.13389 −0.566947 0.823754i $$-0.691875\pi$$
−0.566947 + 0.823754i $$0.691875\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ 2.00000 0.458831 0.229416 0.973329i $$-0.426318\pi$$
0.229416 + 0.973329i $$0.426318\pi$$
$$20$$ 0 0
$$21$$ −3.00000 −0.654654
$$22$$ 0 0
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ 1.00000 0.174078
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ 0 0
$$37$$ 11.0000 1.80839 0.904194 0.427121i $$-0.140472\pi$$
0.904194 + 0.427121i $$0.140472\pi$$
$$38$$ 0 0
$$39$$ −1.00000 −0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ 0 0
$$45$$ 1.00000 0.149071
$$46$$ 0 0
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 0 0
$$49$$ 2.00000 0.285714
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 1.00000 0.134840
$$56$$ 0 0
$$57$$ 2.00000 0.264906
$$58$$ 0 0
$$59$$ −8.00000 −1.04151 −0.520756 0.853706i $$-0.674350\pi$$
−0.520756 + 0.853706i $$0.674350\pi$$
$$60$$ 0 0
$$61$$ 13.0000 1.66448 0.832240 0.554416i $$-0.187058\pi$$
0.832240 + 0.554416i $$0.187058\pi$$
$$62$$ 0 0
$$63$$ −3.00000 −0.377964
$$64$$ 0 0
$$65$$ −1.00000 −0.124035
$$66$$ 0 0
$$67$$ −12.0000 −1.46603 −0.733017 0.680211i $$-0.761888\pi$$
−0.733017 + 0.680211i $$0.761888\pi$$
$$68$$ 0 0
$$69$$ 3.00000 0.361158
$$70$$ 0 0
$$71$$ 5.00000 0.593391 0.296695 0.954972i $$-0.404115\pi$$
0.296695 + 0.954972i $$0.404115\pi$$
$$72$$ 0 0
$$73$$ 10.0000 1.17041 0.585206 0.810885i $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ 0 0
$$75$$ 1.00000 0.115470
$$76$$ 0 0
$$77$$ −3.00000 −0.341882
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ −1.00000 −0.108465
$$86$$ 0 0
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −15.0000 −1.59000 −0.794998 0.606612i $$-0.792528\pi$$
−0.794998 + 0.606612i $$0.792528\pi$$
$$90$$ 0 0
$$91$$ 3.00000 0.314485
$$92$$ 0 0
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 2.00000 0.205196
$$96$$ 0 0
$$97$$ 17.0000 1.72609 0.863044 0.505128i $$-0.168555\pi$$
0.863044 + 0.505128i $$0.168555\pi$$
$$98$$ 0 0
$$99$$ 1.00000 0.100504
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ −3.00000 −0.292770
$$106$$ 0 0
$$107$$ −9.00000 −0.870063 −0.435031 0.900415i $$-0.643263\pi$$
−0.435031 + 0.900415i $$0.643263\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 11.0000 1.04407
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ 3.00000 0.279751
$$116$$ 0 0
$$117$$ −1.00000 −0.0924500
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ −5.00000 −0.450835
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −10.0000 −0.887357 −0.443678 0.896186i $$-0.646327\pi$$
−0.443678 + 0.896186i $$0.646327\pi$$
$$128$$ 0 0
$$129$$ −4.00000 −0.352180
$$130$$ 0 0
$$131$$ 6.00000 0.524222 0.262111 0.965038i $$-0.415581\pi$$
0.262111 + 0.965038i $$0.415581\pi$$
$$132$$ 0 0
$$133$$ −6.00000 −0.520266
$$134$$ 0 0
$$135$$ 1.00000 0.0860663
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 1.00000 0.0848189 0.0424094 0.999100i $$-0.486497\pi$$
0.0424094 + 0.999100i $$0.486497\pi$$
$$140$$ 0 0
$$141$$ 10.0000 0.842152
$$142$$ 0 0
$$143$$ −1.00000 −0.0836242
$$144$$ 0 0
$$145$$ −2.00000 −0.166091
$$146$$ 0 0
$$147$$ 2.00000 0.164957
$$148$$ 0 0
$$149$$ 13.0000 1.06500 0.532501 0.846430i $$-0.321252\pi$$
0.532501 + 0.846430i $$0.321252\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 6.00000 0.481932
$$156$$ 0 0
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 0 0
$$159$$ 11.0000 0.872357
$$160$$ 0 0
$$161$$ −9.00000 −0.709299
$$162$$ 0 0
$$163$$ 13.0000 1.01824 0.509119 0.860696i $$-0.329971\pi$$
0.509119 + 0.860696i $$0.329971\pi$$
$$164$$ 0 0
$$165$$ 1.00000 0.0778499
$$166$$ 0 0
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 2.00000 0.152944
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ −3.00000 −0.226779
$$176$$ 0 0
$$177$$ −8.00000 −0.601317
$$178$$ 0 0
$$179$$ −2.00000 −0.149487 −0.0747435 0.997203i $$-0.523814\pi$$
−0.0747435 + 0.997203i $$0.523814\pi$$
$$180$$ 0 0
$$181$$ −7.00000 −0.520306 −0.260153 0.965567i $$-0.583773\pi$$
−0.260153 + 0.965567i $$0.583773\pi$$
$$182$$ 0 0
$$183$$ 13.0000 0.960988
$$184$$ 0 0
$$185$$ 11.0000 0.808736
$$186$$ 0 0
$$187$$ −1.00000 −0.0731272
$$188$$ 0 0
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 0 0
$$193$$ −13.0000 −0.935760 −0.467880 0.883792i $$-0.654982\pi$$
−0.467880 + 0.883792i $$0.654982\pi$$
$$194$$ 0 0
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ 0 0
$$201$$ −12.0000 −0.846415
$$202$$ 0 0
$$203$$ 6.00000 0.421117
$$204$$ 0 0
$$205$$ −5.00000 −0.349215
$$206$$ 0 0
$$207$$ 3.00000 0.208514
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 5.00000 0.342594
$$214$$ 0 0
$$215$$ −4.00000 −0.272798
$$216$$ 0 0
$$217$$ −18.0000 −1.22192
$$218$$ 0 0
$$219$$ 10.0000 0.675737
$$220$$ 0 0
$$221$$ 1.00000 0.0672673
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 0 0
$$227$$ −22.0000 −1.46019 −0.730096 0.683345i $$-0.760525\pi$$
−0.730096 + 0.683345i $$0.760525\pi$$
$$228$$ 0 0
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ −27.0000 −1.76883 −0.884414 0.466702i $$-0.845442\pi$$
−0.884414 + 0.466702i $$0.845442\pi$$
$$234$$ 0 0
$$235$$ 10.0000 0.652328
$$236$$ 0 0
$$237$$ 3.00000 0.194871
$$238$$ 0 0
$$239$$ 13.0000 0.840900 0.420450 0.907316i $$-0.361872\pi$$
0.420450 + 0.907316i $$0.361872\pi$$
$$240$$ 0 0
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 2.00000 0.127775
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 3.00000 0.188608
$$254$$ 0 0
$$255$$ −1.00000 −0.0626224
$$256$$ 0 0
$$257$$ −18.0000 −1.12281 −0.561405 0.827541i $$-0.689739\pi$$
−0.561405 + 0.827541i $$0.689739\pi$$
$$258$$ 0 0
$$259$$ −33.0000 −2.05052
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 0 0
$$263$$ 8.00000 0.493301 0.246651 0.969104i $$-0.420670\pi$$
0.246651 + 0.969104i $$0.420670\pi$$
$$264$$ 0 0
$$265$$ 11.0000 0.675725
$$266$$ 0 0
$$267$$ −15.0000 −0.917985
$$268$$ 0 0
$$269$$ −4.00000 −0.243884 −0.121942 0.992537i $$-0.538912\pi$$
−0.121942 + 0.992537i $$0.538912\pi$$
$$270$$ 0 0
$$271$$ −22.0000 −1.33640 −0.668202 0.743980i $$-0.732936\pi$$
−0.668202 + 0.743980i $$0.732936\pi$$
$$272$$ 0 0
$$273$$ 3.00000 0.181568
$$274$$ 0 0
$$275$$ 1.00000 0.0603023
$$276$$ 0 0
$$277$$ −18.0000 −1.08152 −0.540758 0.841178i $$-0.681862\pi$$
−0.540758 + 0.841178i $$0.681862\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 0 0
$$283$$ 12.0000 0.713326 0.356663 0.934233i $$-0.383914\pi$$
0.356663 + 0.934233i $$0.383914\pi$$
$$284$$ 0 0
$$285$$ 2.00000 0.118470
$$286$$ 0 0
$$287$$ 15.0000 0.885422
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 17.0000 0.996558
$$292$$ 0 0
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 0 0
$$297$$ 1.00000 0.0580259
$$298$$ 0 0
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 13.0000 0.744378
$$306$$ 0 0
$$307$$ 5.00000 0.285365 0.142683 0.989769i $$-0.454427\pi$$
0.142683 + 0.989769i $$0.454427\pi$$
$$308$$ 0 0
$$309$$ 16.0000 0.910208
$$310$$ 0 0
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 0 0
$$317$$ 28.0000 1.57264 0.786318 0.617822i $$-0.211985\pi$$
0.786318 + 0.617822i $$0.211985\pi$$
$$318$$ 0 0
$$319$$ −2.00000 −0.111979
$$320$$ 0 0
$$321$$ −9.00000 −0.502331
$$322$$ 0 0
$$323$$ −2.00000 −0.111283
$$324$$ 0 0
$$325$$ −1.00000 −0.0554700
$$326$$ 0 0
$$327$$ −16.0000 −0.884802
$$328$$ 0 0
$$329$$ −30.0000 −1.65395
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 0 0
$$333$$ 11.0000 0.602796
$$334$$ 0 0
$$335$$ −12.0000 −0.655630
$$336$$ 0 0
$$337$$ 4.00000 0.217894 0.108947 0.994048i $$-0.465252\pi$$
0.108947 + 0.994048i $$0.465252\pi$$
$$338$$ 0 0
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 6.00000 0.324918
$$342$$ 0 0
$$343$$ 15.0000 0.809924
$$344$$ 0 0
$$345$$ 3.00000 0.161515
$$346$$ 0 0
$$347$$ 19.0000 1.01997 0.509987 0.860182i $$-0.329650\pi$$
0.509987 + 0.860182i $$0.329650\pi$$
$$348$$ 0 0
$$349$$ −8.00000 −0.428230 −0.214115 0.976808i $$-0.568687\pi$$
−0.214115 + 0.976808i $$0.568687\pi$$
$$350$$ 0 0
$$351$$ −1.00000 −0.0533761
$$352$$ 0 0
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ 5.00000 0.265372
$$356$$ 0 0
$$357$$ 3.00000 0.158777
$$358$$ 0 0
$$359$$ −24.0000 −1.26667 −0.633336 0.773877i $$-0.718315\pi$$
−0.633336 + 0.773877i $$0.718315\pi$$
$$360$$ 0 0
$$361$$ −15.0000 −0.789474
$$362$$ 0 0
$$363$$ −10.0000 −0.524864
$$364$$ 0 0
$$365$$ 10.0000 0.523424
$$366$$ 0 0
$$367$$ −36.0000 −1.87918 −0.939592 0.342296i $$-0.888796\pi$$
−0.939592 + 0.342296i $$0.888796\pi$$
$$368$$ 0 0
$$369$$ −5.00000 −0.260290
$$370$$ 0 0
$$371$$ −33.0000 −1.71327
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 1.00000 0.0516398
$$376$$ 0 0
$$377$$ 2.00000 0.103005
$$378$$ 0 0
$$379$$ 14.0000 0.719132 0.359566 0.933120i $$-0.382925\pi$$
0.359566 + 0.933120i $$0.382925\pi$$
$$380$$ 0 0
$$381$$ −10.0000 −0.512316
$$382$$ 0 0
$$383$$ 30.0000 1.53293 0.766464 0.642287i $$-0.222014\pi$$
0.766464 + 0.642287i $$0.222014\pi$$
$$384$$ 0 0
$$385$$ −3.00000 −0.152894
$$386$$ 0 0
$$387$$ −4.00000 −0.203331
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ −3.00000 −0.151717
$$392$$ 0 0
$$393$$ 6.00000 0.302660
$$394$$ 0 0
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ −29.0000 −1.45547 −0.727734 0.685859i $$-0.759427\pi$$
−0.727734 + 0.685859i $$0.759427\pi$$
$$398$$ 0 0
$$399$$ −6.00000 −0.300376
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −6.00000 −0.298881
$$404$$ 0 0
$$405$$ 1.00000 0.0496904
$$406$$ 0 0
$$407$$ 11.0000 0.545250
$$408$$ 0 0
$$409$$ 2.00000 0.0988936 0.0494468 0.998777i $$-0.484254\pi$$
0.0494468 + 0.998777i $$0.484254\pi$$
$$410$$ 0 0
$$411$$ 18.0000 0.887875
$$412$$ 0 0
$$413$$ 24.0000 1.18096
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 1.00000 0.0489702
$$418$$ 0 0
$$419$$ 26.0000 1.27018 0.635092 0.772437i $$-0.280962\pi$$
0.635092 + 0.772437i $$0.280962\pi$$
$$420$$ 0 0
$$421$$ 4.00000 0.194948 0.0974740 0.995238i $$-0.468924\pi$$
0.0974740 + 0.995238i $$0.468924\pi$$
$$422$$ 0 0
$$423$$ 10.0000 0.486217
$$424$$ 0 0
$$425$$ −1.00000 −0.0485071
$$426$$ 0 0
$$427$$ −39.0000 −1.88734
$$428$$ 0 0
$$429$$ −1.00000 −0.0482805
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 0 0
$$433$$ 4.00000 0.192228 0.0961139 0.995370i $$-0.469359\pi$$
0.0961139 + 0.995370i $$0.469359\pi$$
$$434$$ 0 0
$$435$$ −2.00000 −0.0958927
$$436$$ 0 0
$$437$$ 6.00000 0.287019
$$438$$ 0 0
$$439$$ 17.0000 0.811366 0.405683 0.914014i $$-0.367034\pi$$
0.405683 + 0.914014i $$0.367034\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ 0 0
$$443$$ −9.00000 −0.427603 −0.213801 0.976877i $$-0.568585\pi$$
−0.213801 + 0.976877i $$0.568585\pi$$
$$444$$ 0 0
$$445$$ −15.0000 −0.711068
$$446$$ 0 0
$$447$$ 13.0000 0.614879
$$448$$ 0 0
$$449$$ −13.0000 −0.613508 −0.306754 0.951789i $$-0.599243\pi$$
−0.306754 + 0.951789i $$0.599243\pi$$
$$450$$ 0 0
$$451$$ −5.00000 −0.235441
$$452$$ 0 0
$$453$$ −16.0000 −0.751746
$$454$$ 0 0
$$455$$ 3.00000 0.140642
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 15.0000 0.698620 0.349310 0.937007i $$-0.386416\pi$$
0.349310 + 0.937007i $$0.386416\pi$$
$$462$$ 0 0
$$463$$ −27.0000 −1.25480 −0.627398 0.778699i $$-0.715880\pi$$
−0.627398 + 0.778699i $$0.715880\pi$$
$$464$$ 0 0
$$465$$ 6.00000 0.278243
$$466$$ 0 0
$$467$$ 23.0000 1.06431 0.532157 0.846646i $$-0.321382\pi$$
0.532157 + 0.846646i $$0.321382\pi$$
$$468$$ 0 0
$$469$$ 36.0000 1.66233
$$470$$ 0 0
$$471$$ −10.0000 −0.460776
$$472$$ 0 0
$$473$$ −4.00000 −0.183920
$$474$$ 0 0
$$475$$ 2.00000 0.0917663
$$476$$ 0 0
$$477$$ 11.0000 0.503655
$$478$$ 0 0
$$479$$ −9.00000 −0.411220 −0.205610 0.978634i $$-0.565918\pi$$
−0.205610 + 0.978634i $$0.565918\pi$$
$$480$$ 0 0
$$481$$ −11.0000 −0.501557
$$482$$ 0 0
$$483$$ −9.00000 −0.409514
$$484$$ 0 0
$$485$$ 17.0000 0.771930
$$486$$ 0 0
$$487$$ 7.00000 0.317200 0.158600 0.987343i $$-0.449302\pi$$
0.158600 + 0.987343i $$0.449302\pi$$
$$488$$ 0 0
$$489$$ 13.0000 0.587880
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 0 0
$$493$$ 2.00000 0.0900755
$$494$$ 0 0
$$495$$ 1.00000 0.0449467
$$496$$ 0 0
$$497$$ −15.0000 −0.672842
$$498$$ 0 0
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ 0 0
$$501$$ 12.0000 0.536120
$$502$$ 0 0
$$503$$ −28.0000 −1.24846 −0.624229 0.781241i $$-0.714587\pi$$
−0.624229 + 0.781241i $$0.714587\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 1.00000 0.0444116
$$508$$ 0 0
$$509$$ −7.00000 −0.310270 −0.155135 0.987893i $$-0.549581\pi$$
−0.155135 + 0.987893i $$0.549581\pi$$
$$510$$ 0 0
$$511$$ −30.0000 −1.32712
$$512$$ 0 0
$$513$$ 2.00000 0.0883022
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 10.0000 0.439799
$$518$$ 0 0
$$519$$ −6.00000 −0.263371
$$520$$ 0 0
$$521$$ −14.0000 −0.613351 −0.306676 0.951814i $$-0.599217\pi$$
−0.306676 + 0.951814i $$0.599217\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 0 0
$$525$$ −3.00000 −0.130931
$$526$$ 0 0
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ −8.00000 −0.347170
$$532$$ 0 0
$$533$$ 5.00000 0.216574
$$534$$ 0 0
$$535$$ −9.00000 −0.389104
$$536$$ 0 0
$$537$$ −2.00000 −0.0863064
$$538$$ 0 0
$$539$$ 2.00000 0.0861461
$$540$$ 0 0
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ −7.00000 −0.300399
$$544$$ 0 0
$$545$$ −16.0000 −0.685365
$$546$$ 0 0
$$547$$ 32.0000 1.36822 0.684111 0.729378i $$-0.260191\pi$$
0.684111 + 0.729378i $$0.260191\pi$$
$$548$$ 0 0
$$549$$ 13.0000 0.554826
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ −9.00000 −0.382719
$$554$$ 0 0
$$555$$ 11.0000 0.466924
$$556$$ 0 0
$$557$$ −30.0000 −1.27114 −0.635570 0.772043i $$-0.719235\pi$$
−0.635570 + 0.772043i $$0.719235\pi$$
$$558$$ 0 0
$$559$$ 4.00000 0.169182
$$560$$ 0 0
$$561$$ −1.00000 −0.0422200
$$562$$ 0 0
$$563$$ −21.0000 −0.885044 −0.442522 0.896758i $$-0.645916\pi$$
−0.442522 + 0.896758i $$0.645916\pi$$
$$564$$ 0 0
$$565$$ 14.0000 0.588984
$$566$$ 0 0
$$567$$ −3.00000 −0.125988
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 31.0000 1.29731 0.648655 0.761083i $$-0.275332\pi$$
0.648655 + 0.761083i $$0.275332\pi$$
$$572$$ 0 0
$$573$$ 8.00000 0.334205
$$574$$ 0 0
$$575$$ 3.00000 0.125109
$$576$$ 0 0
$$577$$ −19.0000 −0.790980 −0.395490 0.918470i $$-0.629425\pi$$
−0.395490 + 0.918470i $$0.629425\pi$$
$$578$$ 0 0
$$579$$ −13.0000 −0.540262
$$580$$ 0 0
$$581$$ −36.0000 −1.49353
$$582$$ 0 0
$$583$$ 11.0000 0.455573
$$584$$ 0 0
$$585$$ −1.00000 −0.0413449
$$586$$ 0 0
$$587$$ 18.0000 0.742940 0.371470 0.928445i $$-0.378854\pi$$
0.371470 + 0.928445i $$0.378854\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 4.00000 0.164260 0.0821302 0.996622i $$-0.473828\pi$$
0.0821302 + 0.996622i $$0.473828\pi$$
$$594$$ 0 0
$$595$$ 3.00000 0.122988
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −16.0000 −0.653742 −0.326871 0.945069i $$-0.605994\pi$$
−0.326871 + 0.945069i $$0.605994\pi$$
$$600$$ 0 0
$$601$$ −37.0000 −1.50926 −0.754631 0.656150i $$-0.772184\pi$$
−0.754631 + 0.656150i $$0.772184\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ 0 0
$$605$$ −10.0000 −0.406558
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 6.00000 0.243132
$$610$$ 0 0
$$611$$ −10.0000 −0.404557
$$612$$ 0 0
$$613$$ 13.0000 0.525065 0.262533 0.964923i $$-0.415442\pi$$
0.262533 + 0.964923i $$0.415442\pi$$
$$614$$ 0 0
$$615$$ −5.00000 −0.201619
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 34.0000 1.36658 0.683288 0.730149i $$-0.260549\pi$$
0.683288 + 0.730149i $$0.260549\pi$$
$$620$$ 0 0
$$621$$ 3.00000 0.120386
$$622$$ 0 0
$$623$$ 45.0000 1.80289
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 2.00000 0.0798723
$$628$$ 0 0
$$629$$ −11.0000 −0.438599
$$630$$ 0 0
$$631$$ −40.0000 −1.59237 −0.796187 0.605050i $$-0.793153\pi$$
−0.796187 + 0.605050i $$0.793153\pi$$
$$632$$ 0 0
$$633$$ 4.00000 0.158986
$$634$$ 0 0
$$635$$ −10.0000 −0.396838
$$636$$ 0 0
$$637$$ −2.00000 −0.0792429
$$638$$ 0 0
$$639$$ 5.00000 0.197797
$$640$$ 0 0
$$641$$ −12.0000 −0.473972 −0.236986 0.971513i $$-0.576159\pi$$
−0.236986 + 0.971513i $$0.576159\pi$$
$$642$$ 0 0
$$643$$ −15.0000 −0.591542 −0.295771 0.955259i $$-0.595577\pi$$
−0.295771 + 0.955259i $$0.595577\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 0 0
$$647$$ −47.0000 −1.84776 −0.923880 0.382682i $$-0.875001\pi$$
−0.923880 + 0.382682i $$0.875001\pi$$
$$648$$ 0 0
$$649$$ −8.00000 −0.314027
$$650$$ 0 0
$$651$$ −18.0000 −0.705476
$$652$$ 0 0
$$653$$ 22.0000 0.860927 0.430463 0.902608i $$-0.358350\pi$$
0.430463 + 0.902608i $$0.358350\pi$$
$$654$$ 0 0
$$655$$ 6.00000 0.234439
$$656$$ 0 0
$$657$$ 10.0000 0.390137
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 4.00000 0.155582 0.0777910 0.996970i $$-0.475213\pi$$
0.0777910 + 0.996970i $$0.475213\pi$$
$$662$$ 0 0
$$663$$ 1.00000 0.0388368
$$664$$ 0 0
$$665$$ −6.00000 −0.232670
$$666$$ 0 0
$$667$$ −6.00000 −0.232321
$$668$$ 0 0
$$669$$ −8.00000 −0.309298
$$670$$ 0 0
$$671$$ 13.0000 0.501859
$$672$$ 0 0
$$673$$ −6.00000 −0.231283 −0.115642 0.993291i $$-0.536892\pi$$
−0.115642 + 0.993291i $$0.536892\pi$$
$$674$$ 0 0
$$675$$ 1.00000 0.0384900
$$676$$ 0 0
$$677$$ 3.00000 0.115299 0.0576497 0.998337i $$-0.481639\pi$$
0.0576497 + 0.998337i $$0.481639\pi$$
$$678$$ 0 0
$$679$$ −51.0000 −1.95720
$$680$$ 0 0
$$681$$ −22.0000 −0.843042
$$682$$ 0 0
$$683$$ 24.0000 0.918334 0.459167 0.888350i $$-0.348148\pi$$
0.459167 + 0.888350i $$0.348148\pi$$
$$684$$ 0 0
$$685$$ 18.0000 0.687745
$$686$$ 0 0
$$687$$ 18.0000 0.686743
$$688$$ 0 0
$$689$$ −11.0000 −0.419067
$$690$$ 0 0
$$691$$ −22.0000 −0.836919 −0.418460 0.908235i $$-0.637430\pi$$
−0.418460 + 0.908235i $$0.637430\pi$$
$$692$$ 0 0
$$693$$ −3.00000 −0.113961
$$694$$ 0 0
$$695$$ 1.00000 0.0379322
$$696$$ 0 0
$$697$$ 5.00000 0.189389
$$698$$ 0 0
$$699$$ −27.0000 −1.02123
$$700$$ 0 0
$$701$$ −20.0000 −0.755390 −0.377695 0.925930i $$-0.623283\pi$$
−0.377695 + 0.925930i $$0.623283\pi$$
$$702$$ 0 0
$$703$$ 22.0000 0.829746
$$704$$ 0 0
$$705$$ 10.0000 0.376622
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ 3.00000 0.112509
$$712$$ 0 0
$$713$$ 18.0000 0.674105
$$714$$ 0 0
$$715$$ −1.00000 −0.0373979
$$716$$ 0 0
$$717$$ 13.0000 0.485494
$$718$$ 0 0
$$719$$ 24.0000 0.895049 0.447524 0.894272i $$-0.352306\pi$$
0.447524 + 0.894272i $$0.352306\pi$$
$$720$$ 0 0
$$721$$ −48.0000 −1.78761
$$722$$ 0 0
$$723$$ −2.00000 −0.0743808
$$724$$ 0 0
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −38.0000 −1.40934 −0.704671 0.709534i $$-0.748905\pi$$
−0.704671 + 0.709534i $$0.748905\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ −49.0000 −1.80986 −0.904928 0.425564i $$-0.860076\pi$$
−0.904928 + 0.425564i $$0.860076\pi$$
$$734$$ 0 0
$$735$$ 2.00000 0.0737711
$$736$$ 0 0
$$737$$ −12.0000 −0.442026
$$738$$ 0 0
$$739$$ −10.0000 −0.367856 −0.183928 0.982940i $$-0.558881\pi$$
−0.183928 + 0.982940i $$0.558881\pi$$
$$740$$ 0 0
$$741$$ −2.00000 −0.0734718
$$742$$ 0 0
$$743$$ 34.0000 1.24734 0.623670 0.781688i $$-0.285641\pi$$
0.623670 + 0.781688i $$0.285641\pi$$
$$744$$ 0 0
$$745$$ 13.0000 0.476283
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 27.0000 0.986559
$$750$$ 0 0
$$751$$ 5.00000 0.182453 0.0912263 0.995830i $$-0.470921\pi$$
0.0912263 + 0.995830i $$0.470921\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ −8.00000 −0.290765 −0.145382 0.989376i $$-0.546441\pi$$
−0.145382 + 0.989376i $$0.546441\pi$$
$$758$$ 0 0
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 48.0000 1.73772
$$764$$ 0 0
$$765$$ −1.00000 −0.0361551
$$766$$ 0 0
$$767$$ 8.00000 0.288863
$$768$$ 0 0
$$769$$ −40.0000 −1.44244 −0.721218 0.692708i $$-0.756418\pi$$
−0.721218 + 0.692708i $$0.756418\pi$$
$$770$$ 0 0
$$771$$ −18.0000 −0.648254
$$772$$ 0 0
$$773$$ 36.0000 1.29483 0.647415 0.762138i $$-0.275850\pi$$
0.647415 + 0.762138i $$0.275850\pi$$
$$774$$ 0 0
$$775$$ 6.00000 0.215526
$$776$$ 0 0
$$777$$ −33.0000 −1.18387
$$778$$ 0 0
$$779$$ −10.0000 −0.358287
$$780$$ 0 0
$$781$$ 5.00000 0.178914
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ 0 0
$$785$$ −10.0000 −0.356915
$$786$$ 0 0
$$787$$ −52.0000 −1.85360 −0.926800 0.375555i $$-0.877452\pi$$
−0.926800 + 0.375555i $$0.877452\pi$$
$$788$$ 0 0
$$789$$ 8.00000 0.284808
$$790$$ 0 0
$$791$$ −42.0000 −1.49335
$$792$$ 0 0
$$793$$ −13.0000 −0.461644
$$794$$ 0 0
$$795$$ 11.0000 0.390130
$$796$$ 0 0
$$797$$ −47.0000 −1.66483 −0.832413 0.554156i $$-0.813041\pi$$
−0.832413 + 0.554156i $$0.813041\pi$$
$$798$$ 0 0
$$799$$ −10.0000 −0.353775
$$800$$ 0 0
$$801$$ −15.0000 −0.529999
$$802$$ 0 0
$$803$$ 10.0000 0.352892
$$804$$ 0 0
$$805$$ −9.00000 −0.317208
$$806$$ 0 0
$$807$$ −4.00000 −0.140807
$$808$$ 0 0
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ −36.0000 −1.26413 −0.632065 0.774915i $$-0.717793\pi$$
−0.632065 + 0.774915i $$0.717793\pi$$
$$812$$ 0 0
$$813$$ −22.0000 −0.771574
$$814$$ 0 0
$$815$$ 13.0000 0.455370
$$816$$ 0 0
$$817$$ −8.00000 −0.279885
$$818$$ 0 0
$$819$$ 3.00000 0.104828
$$820$$ 0 0
$$821$$ 27.0000 0.942306 0.471153 0.882051i $$-0.343838\pi$$
0.471153 + 0.882051i $$0.343838\pi$$
$$822$$ 0 0
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ 0 0
$$825$$ 1.00000 0.0348155
$$826$$ 0 0
$$827$$ −26.0000 −0.904109 −0.452054 0.891990i $$-0.649309\pi$$
−0.452054 + 0.891990i $$0.649309\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ −18.0000 −0.624413
$$832$$ 0 0
$$833$$ −2.00000 −0.0692959
$$834$$ 0 0
$$835$$ 12.0000 0.415277
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 0 0
$$839$$ 5.00000 0.172619 0.0863096 0.996268i $$-0.472493\pi$$
0.0863096 + 0.996268i $$0.472493\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ 0 0
$$843$$ 30.0000 1.03325
$$844$$ 0 0
$$845$$ 1.00000 0.0344010
$$846$$ 0 0
$$847$$ 30.0000 1.03081
$$848$$ 0 0
$$849$$ 12.0000 0.411839
$$850$$ 0 0
$$851$$ 33.0000 1.13123
$$852$$ 0 0
$$853$$ 45.0000 1.54077 0.770385 0.637579i $$-0.220064\pi$$
0.770385 + 0.637579i $$0.220064\pi$$
$$854$$ 0 0
$$855$$ 2.00000 0.0683986
$$856$$ 0 0
$$857$$ 29.0000 0.990621 0.495311 0.868716i $$-0.335054\pi$$
0.495311 + 0.868716i $$0.335054\pi$$
$$858$$ 0 0
$$859$$ 29.0000 0.989467 0.494734 0.869045i $$-0.335266\pi$$
0.494734 + 0.869045i $$0.335266\pi$$
$$860$$ 0 0
$$861$$ 15.0000 0.511199
$$862$$ 0 0
$$863$$ −34.0000 −1.15737 −0.578687 0.815550i $$-0.696435\pi$$
−0.578687 + 0.815550i $$0.696435\pi$$
$$864$$ 0 0
$$865$$ −6.00000 −0.204006
$$866$$ 0 0
$$867$$ −16.0000 −0.543388
$$868$$ 0 0
$$869$$ 3.00000 0.101768
$$870$$ 0 0
$$871$$ 12.0000 0.406604
$$872$$ 0 0
$$873$$ 17.0000 0.575363
$$874$$ 0 0
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ 18.0000 0.607817 0.303908 0.952701i $$-0.401708\pi$$
0.303908 + 0.952701i $$0.401708\pi$$
$$878$$ 0 0
$$879$$ 24.0000 0.809500
$$880$$ 0 0
$$881$$ 2.00000 0.0673817 0.0336909 0.999432i $$-0.489274\pi$$
0.0336909 + 0.999432i $$0.489274\pi$$
$$882$$ 0 0
$$883$$ 16.0000 0.538443 0.269221 0.963078i $$-0.413234\pi$$
0.269221 + 0.963078i $$0.413234\pi$$
$$884$$ 0 0
$$885$$ −8.00000 −0.268917
$$886$$ 0 0
$$887$$ 21.0000 0.705111 0.352555 0.935791i $$-0.385313\pi$$
0.352555 + 0.935791i $$0.385313\pi$$
$$888$$ 0 0
$$889$$ 30.0000 1.00617
$$890$$ 0 0
$$891$$ 1.00000 0.0335013
$$892$$ 0 0
$$893$$ 20.0000 0.669274
$$894$$ 0 0
$$895$$ −2.00000 −0.0668526
$$896$$ 0 0
$$897$$ −3.00000 −0.100167
$$898$$ 0 0
$$899$$ −12.0000 −0.400222
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ 0 0
$$903$$ 12.0000 0.399335
$$904$$ 0 0
$$905$$ −7.00000 −0.232688
$$906$$ 0 0
$$907$$ −6.00000 −0.199227 −0.0996134 0.995026i $$-0.531761\pi$$
−0.0996134 + 0.995026i $$0.531761\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −44.0000 −1.45779 −0.728893 0.684628i $$-0.759965\pi$$
−0.728893 + 0.684628i $$0.759965\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 13.0000 0.429767
$$916$$ 0 0
$$917$$ −18.0000 −0.594412
$$918$$ 0 0
$$919$$ −37.0000 −1.22052 −0.610259 0.792202i $$-0.708935\pi$$
−0.610259 + 0.792202i $$0.708935\pi$$
$$920$$ 0 0
$$921$$ 5.00000 0.164756
$$922$$ 0 0
$$923$$ −5.00000 −0.164577
$$924$$ 0 0
$$925$$ 11.0000 0.361678
$$926$$ 0 0
$$927$$ 16.0000 0.525509
$$928$$ 0 0
$$929$$ 1.00000 0.0328089 0.0164045 0.999865i $$-0.494778\pi$$
0.0164045 + 0.999865i $$0.494778\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 0 0
$$933$$ −24.0000 −0.785725
$$934$$ 0 0
$$935$$ −1.00000 −0.0327035
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 0 0
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ 37.0000 1.20617 0.603083 0.797679i $$-0.293939\pi$$
0.603083 + 0.797679i $$0.293939\pi$$
$$942$$ 0 0
$$943$$ −15.0000 −0.488467
$$944$$ 0 0
$$945$$ −3.00000 −0.0975900
$$946$$ 0 0
$$947$$ 24.0000 0.779895 0.389948 0.920837i $$-0.372493\pi$$
0.389948 + 0.920837i $$0.372493\pi$$
$$948$$ 0 0
$$949$$ −10.0000 −0.324614
$$950$$ 0 0
$$951$$ 28.0000 0.907962
$$952$$ 0 0
$$953$$ −1.00000 −0.0323932 −0.0161966 0.999869i $$-0.505156\pi$$
−0.0161966 + 0.999869i $$0.505156\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ 0 0
$$957$$ −2.00000 −0.0646508
$$958$$ 0 0
$$959$$ −54.0000 −1.74375
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ −9.00000 −0.290021
$$964$$ 0 0
$$965$$ −13.0000 −0.418485
$$966$$ 0 0
$$967$$ 16.0000 0.514525 0.257263 0.966342i $$-0.417179\pi$$
0.257263 + 0.966342i $$0.417179\pi$$
$$968$$ 0 0
$$969$$ −2.00000 −0.0642493
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 0 0
$$973$$ −3.00000 −0.0961756
$$974$$ 0 0
$$975$$ −1.00000 −0.0320256
$$976$$ 0 0
$$977$$ 32.0000 1.02377 0.511885 0.859054i $$-0.328947\pi$$
0.511885 + 0.859054i $$0.328947\pi$$
$$978$$ 0 0
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ 12.0000 0.382741 0.191370 0.981518i $$-0.438707\pi$$
0.191370 + 0.981518i $$0.438707\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −30.0000 −0.954911
$$988$$ 0 0
$$989$$ −12.0000 −0.381578
$$990$$ 0 0
$$991$$ 25.0000 0.794151 0.397076 0.917786i $$-0.370025\pi$$
0.397076 + 0.917786i $$0.370025\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −4.00000 −0.126809
$$996$$ 0 0
$$997$$ −36.0000 −1.14013 −0.570066 0.821599i $$-0.693082\pi$$
−0.570066 + 0.821599i $$0.693082\pi$$
$$998$$ 0 0
$$999$$ 11.0000 0.348025
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 3120.2.a.u.1.1 1
3.2 odd 2 9360.2.a.d.1.1 1
4.3 odd 2 195.2.a.b.1.1 1
12.11 even 2 585.2.a.b.1.1 1
20.3 even 4 975.2.c.a.274.1 2
20.7 even 4 975.2.c.a.274.2 2
20.19 odd 2 975.2.a.c.1.1 1
28.27 even 2 9555.2.a.v.1.1 1
52.51 odd 2 2535.2.a.a.1.1 1
60.23 odd 4 2925.2.c.c.2224.2 2
60.47 odd 4 2925.2.c.c.2224.1 2
60.59 even 2 2925.2.a.q.1.1 1
156.155 even 2 7605.2.a.u.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.b.1.1 1 4.3 odd 2
585.2.a.b.1.1 1 12.11 even 2
975.2.a.c.1.1 1 20.19 odd 2
975.2.c.a.274.1 2 20.3 even 4
975.2.c.a.274.2 2 20.7 even 4
2535.2.a.a.1.1 1 52.51 odd 2
2925.2.a.q.1.1 1 60.59 even 2
2925.2.c.c.2224.1 2 60.47 odd 4
2925.2.c.c.2224.2 2 60.23 odd 4
3120.2.a.u.1.1 1 1.1 even 1 trivial
7605.2.a.u.1.1 1 156.155 even 2
9360.2.a.d.1.1 1 3.2 odd 2
9555.2.a.v.1.1 1 28.27 even 2