# Properties

 Label 3520.2.a.u.1.1 Level $3520$ Weight $2$ Character 3520.1 Self dual yes Analytic conductor $28.107$ 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] = [3520,2,Mod(1,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, 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("3520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3520 = 2^{6} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3520.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: $$28.1073415115$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3520.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{5} +2.00000 q^{7} -3.00000 q^{9} -1.00000 q^{11} -8.00000 q^{19} +8.00000 q^{23} +1.00000 q^{25} -10.0000 q^{29} -8.00000 q^{31} +2.00000 q^{35} +10.0000 q^{37} -2.00000 q^{41} -6.00000 q^{43} -3.00000 q^{45} +8.00000 q^{47} -3.00000 q^{49} -14.0000 q^{53} -1.00000 q^{55} -4.00000 q^{59} -10.0000 q^{61} -6.00000 q^{63} +4.00000 q^{67} -8.00000 q^{73} -2.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +10.0000 q^{83} +6.00000 q^{89} -8.00000 q^{95} -10.0000 q^{97} +3.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 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
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −8.00000 −1.83533 −0.917663 0.397360i $$-0.869927\pi$$
−0.917663 + 0.397360i $$0.869927\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −10.0000 −1.85695 −0.928477 0.371391i $$-0.878881\pi$$
−0.928477 + 0.371391i $$0.878881\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2.00000 0.338062
$$36$$ 0 0
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ 0 0
$$55$$ −1.00000 −0.134840
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ −6.00000 −0.755929
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −8.00000 −0.936329 −0.468165 0.883641i $$-0.655085\pi$$
−0.468165 + 0.883641i $$0.655085\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ 4.00000 0.450035 0.225018 0.974355i $$-0.427756\pi$$
0.225018 + 0.974355i $$0.427756\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 10.0000 1.09764 0.548821 0.835940i $$-0.315077\pi$$
0.548821 + 0.835940i $$0.315077\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ 0 0
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 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
$$107$$ 2.00000 0.193347 0.0966736 0.995316i $$-0.469180\pi$$
0.0966736 + 0.995316i $$0.469180\pi$$
$$108$$ 0 0
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −10.0000 −0.940721 −0.470360 0.882474i $$-0.655876\pi$$
−0.470360 + 0.882474i $$0.655876\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ −16.0000 −1.38738
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −18.0000 −1.53784 −0.768922 0.639343i $$-0.779207\pi$$
−0.768922 + 0.639343i $$0.779207\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −10.0000 −0.830455
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 12.0000 0.976546 0.488273 0.872691i $$-0.337627\pi$$
0.488273 + 0.872691i $$0.337627\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 16.0000 1.26098
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.0000 −0.773823 −0.386912 0.922117i $$-0.626458\pi$$
−0.386912 + 0.922117i $$0.626458\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 24.0000 1.83533
$$172$$ 0 0
$$173$$ 12.0000 0.912343 0.456172 0.889892i $$-0.349220\pi$$
0.456172 + 0.889892i $$0.349220\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −20.0000 −1.49487 −0.747435 0.664335i $$-0.768715\pi$$
−0.747435 + 0.664335i $$0.768715\pi$$
$$180$$ 0 0
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 10.0000 0.735215
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −8.00000 −0.575853 −0.287926 0.957653i $$-0.592966\pi$$
−0.287926 + 0.957653i $$0.592966\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 12.0000 0.854965 0.427482 0.904024i $$-0.359401\pi$$
0.427482 + 0.904024i $$0.359401\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −20.0000 −1.40372
$$204$$ 0 0
$$205$$ −2.00000 −0.139686
$$206$$ 0 0
$$207$$ −24.0000 −1.66812
$$208$$ 0 0
$$209$$ 8.00000 0.553372
$$210$$ 0 0
$$211$$ −16.0000 −1.10149 −0.550743 0.834675i $$-0.685655\pi$$
−0.550743 + 0.834675i $$0.685655\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −6.00000 −0.409197
$$216$$ 0 0
$$217$$ −16.0000 −1.08615
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ 0 0
$$225$$ −3.00000 −0.200000
$$226$$ 0 0
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ −6.00000 −0.396491 −0.198246 0.980152i $$-0.563524\pi$$
−0.198246 + 0.980152i $$0.563524\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4.00000 −0.262049 −0.131024 0.991379i $$-0.541827\pi$$
−0.131024 + 0.991379i $$0.541827\pi$$
$$234$$ 0 0
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −28.0000 −1.81117 −0.905585 0.424165i $$-0.860568\pi$$
−0.905585 + 0.424165i $$0.860568\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −4.00000 −0.252478 −0.126239 0.992000i $$-0.540291\pi$$
−0.126239 + 0.992000i $$0.540291\pi$$
$$252$$ 0 0
$$253$$ −8.00000 −0.502956
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 20.0000 1.24274
$$260$$ 0 0
$$261$$ 30.0000 1.85695
$$262$$ 0 0
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ −14.0000 −0.860013
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −14.0000 −0.853595 −0.426798 0.904347i $$-0.640358\pi$$
−0.426798 + 0.904347i $$0.640358\pi$$
$$270$$ 0 0
$$271$$ −12.0000 −0.728948 −0.364474 0.931214i $$-0.618751\pi$$
−0.364474 + 0.931214i $$0.618751\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.00000 −0.0603023
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 0 0
$$279$$ 24.0000 1.43684
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ 10.0000 0.594438 0.297219 0.954809i $$-0.403941\pi$$
0.297219 + 0.954809i $$0.403941\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −24.0000 −1.40209 −0.701047 0.713115i $$-0.747284\pi$$
−0.701047 + 0.713115i $$0.747284\pi$$
$$294$$ 0 0
$$295$$ −4.00000 −0.232889
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −10.0000 −0.572598
$$306$$ 0 0
$$307$$ 26.0000 1.48390 0.741949 0.670456i $$-0.233902\pi$$
0.741949 + 0.670456i $$0.233902\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −6.00000 −0.339140 −0.169570 0.985518i $$-0.554238\pi$$
−0.169570 + 0.985518i $$0.554238\pi$$
$$314$$ 0 0
$$315$$ −6.00000 −0.338062
$$316$$ 0 0
$$317$$ 6.00000 0.336994 0.168497 0.985702i $$-0.446109\pi$$
0.168497 + 0.985702i $$0.446109\pi$$
$$318$$ 0 0
$$319$$ 10.0000 0.559893
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ −30.0000 −1.64399
$$334$$ 0 0
$$335$$ 4.00000 0.218543
$$336$$ 0 0
$$337$$ 32.0000 1.74315 0.871576 0.490261i $$-0.163099\pi$$
0.871576 + 0.490261i $$0.163099\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 8.00000 0.433224
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.0000 0.536828 0.268414 0.963304i $$-0.413500\pi$$
0.268414 + 0.963304i $$0.413500\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 12.0000 0.633336 0.316668 0.948536i $$-0.397436\pi$$
0.316668 + 0.948536i $$0.397436\pi$$
$$360$$ 0 0
$$361$$ 45.0000 2.36842
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ 24.0000 1.25279 0.626395 0.779506i $$-0.284530\pi$$
0.626395 + 0.779506i $$0.284530\pi$$
$$368$$ 0 0
$$369$$ 6.00000 0.312348
$$370$$ 0 0
$$371$$ −28.0000 −1.45369
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −4.00000 −0.204390 −0.102195 0.994764i $$-0.532587\pi$$
−0.102195 + 0.994764i $$0.532587\pi$$
$$384$$ 0 0
$$385$$ −2.00000 −0.101929
$$386$$ 0 0
$$387$$ 18.0000 0.914991
$$388$$ 0 0
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 4.00000 0.201262
$$396$$ 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 $$-0.648376\pi$$
−0.449439 + 0.893311i $$0.648376\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 9.00000 0.447214
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ 0 0
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.00000 −0.393654
$$414$$ 0 0
$$415$$ 10.0000 0.490881
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 20.0000 0.977064 0.488532 0.872546i $$-0.337533\pi$$
0.488532 + 0.872546i $$0.337533\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ 0 0
$$423$$ −24.0000 −1.16692
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −20.0000 −0.967868
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 0 0
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −64.0000 −3.06154
$$438$$ 0 0
$$439$$ −16.0000 −0.763638 −0.381819 0.924237i $$-0.624702\pi$$
−0.381819 + 0.924237i $$0.624702\pi$$
$$440$$ 0 0
$$441$$ 9.00000 0.428571
$$442$$ 0 0
$$443$$ −24.0000 −1.14027 −0.570137 0.821549i $$-0.693110\pi$$
−0.570137 + 0.821549i $$0.693110\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.00000 0.283158 0.141579 0.989927i $$-0.454782\pi$$
0.141579 + 0.989927i $$0.454782\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 12.0000 0.561336 0.280668 0.959805i $$-0.409444\pi$$
0.280668 + 0.959805i $$0.409444\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.0000 0.465746 0.232873 0.972507i $$-0.425187\pi$$
0.232873 + 0.972507i $$0.425187\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ −8.00000 −0.367065
$$476$$ 0 0
$$477$$ 42.0000 1.92305
$$478$$ 0 0
$$479$$ −20.0000 −0.913823 −0.456912 0.889512i $$-0.651044\pi$$
−0.456912 + 0.889512i $$0.651044\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −10.0000 −0.454077
$$486$$ 0 0
$$487$$ 24.0000 1.08754 0.543772 0.839233i $$-0.316996\pi$$
0.543772 + 0.839233i $$0.316996\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4.00000 −0.180517 −0.0902587 0.995918i $$-0.528769\pi$$
−0.0902587 + 0.995918i $$0.528769\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 3.00000 0.134840
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 4.00000 0.179065 0.0895323 0.995984i $$-0.471463\pi$$
0.0895323 + 0.995984i $$0.471463\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 6.00000 0.267527 0.133763 0.991013i $$-0.457294\pi$$
0.133763 + 0.991013i $$0.457294\pi$$
$$504$$ 0 0
$$505$$ −14.0000 −0.622992
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −14.0000 −0.620539 −0.310270 0.950649i $$-0.600419\pi$$
−0.310270 + 0.950649i $$0.600419\pi$$
$$510$$ 0 0
$$511$$ −16.0000 −0.707798
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 22.0000 0.963837 0.481919 0.876216i $$-0.339940\pi$$
0.481919 + 0.876216i $$0.339940\pi$$
$$522$$ 0 0
$$523$$ 14.0000 0.612177 0.306089 0.952003i $$-0.400980\pi$$
0.306089 + 0.952003i $$0.400980\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 2.00000 0.0864675
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 42.0000 1.80572 0.902861 0.429934i $$-0.141463\pi$$
0.902861 + 0.429934i $$0.141463\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 2.00000 0.0856706
$$546$$ 0 0
$$547$$ 30.0000 1.28271 0.641354 0.767245i $$-0.278373\pi$$
0.641354 + 0.767245i $$0.278373\pi$$
$$548$$ 0 0
$$549$$ 30.0000 1.28037
$$550$$ 0 0
$$551$$ 80.0000 3.40811
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 12.0000 0.508456 0.254228 0.967144i $$-0.418179\pi$$
0.254228 + 0.967144i $$0.418179\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −42.0000 −1.77009 −0.885044 0.465506i $$-0.845872\pi$$
−0.885044 + 0.465506i $$0.845872\pi$$
$$564$$ 0 0
$$565$$ −10.0000 −0.420703
$$566$$ 0 0
$$567$$ 18.0000 0.755929
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −28.0000 −1.17176 −0.585882 0.810397i $$-0.699252\pi$$
−0.585882 + 0.810397i $$0.699252\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 8.00000 0.333623
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 20.0000 0.829740
$$582$$ 0 0
$$583$$ 14.0000 0.579821
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12.0000 0.495293 0.247647 0.968850i $$-0.420343\pi$$
0.247647 + 0.968850i $$0.420343\pi$$
$$588$$ 0 0
$$589$$ 64.0000 2.63707
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 8.00000 0.328521 0.164260 0.986417i $$-0.447476\pi$$
0.164260 + 0.986417i $$0.447476\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ −12.0000 −0.488678
$$604$$ 0 0
$$605$$ 1.00000 0.0406558
$$606$$ 0 0
$$607$$ −42.0000 −1.70473 −0.852364 0.522949i $$-0.824832\pi$$
−0.852364 + 0.522949i $$0.824832\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 4.00000 0.161558 0.0807792 0.996732i $$-0.474259\pi$$
0.0807792 + 0.996732i $$0.474259\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 30.0000 1.20775 0.603877 0.797077i $$-0.293622\pi$$
0.603877 + 0.797077i $$0.293622\pi$$
$$618$$ 0 0
$$619$$ 36.0000 1.44696 0.723481 0.690344i $$-0.242541\pi$$
0.723481 + 0.690344i $$0.242541\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −10.0000 −0.396838
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 30.0000 1.18493 0.592464 0.805597i $$-0.298155\pi$$
0.592464 + 0.805597i $$0.298155\pi$$
$$642$$ 0 0
$$643$$ 40.0000 1.57745 0.788723 0.614749i $$-0.210743\pi$$
0.788723 + 0.614749i $$0.210743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 12.0000 0.471769 0.235884 0.971781i $$-0.424201\pi$$
0.235884 + 0.971781i $$0.424201\pi$$
$$648$$ 0 0
$$649$$ 4.00000 0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ 0 0
$$655$$ 16.0000 0.625172
$$656$$ 0 0
$$657$$ 24.0000 0.936329
$$658$$ 0 0
$$659$$ −20.0000 −0.779089 −0.389545 0.921008i $$-0.627368\pi$$
−0.389545 + 0.921008i $$0.627368\pi$$
$$660$$ 0 0
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −16.0000 −0.620453
$$666$$ 0 0
$$667$$ −80.0000 −3.09761
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 10.0000 0.386046
$$672$$ 0 0
$$673$$ 8.00000 0.308377 0.154189 0.988041i $$-0.450724\pi$$
0.154189 + 0.988041i $$0.450724\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 24.0000 0.922395 0.461197 0.887298i $$-0.347420\pi$$
0.461197 + 0.887298i $$0.347420\pi$$
$$678$$ 0 0
$$679$$ −20.0000 −0.767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 4.00000 0.152167 0.0760836 0.997101i $$-0.475758\pi$$
0.0760836 + 0.997101i $$0.475758\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ −4.00000 −0.151729
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 46.0000 1.73740 0.868698 0.495342i $$-0.164957\pi$$
0.868698 + 0.495342i $$0.164957\pi$$
$$702$$ 0 0
$$703$$ −80.0000 −3.01726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −28.0000 −1.05305
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ 0 0
$$713$$ −64.0000 −2.39682
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −16.0000 −0.596699 −0.298350 0.954457i $$-0.596436\pi$$
−0.298350 + 0.954457i $$0.596436\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −10.0000 −0.371391
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −6.00000 −0.220119 −0.110059 0.993925i $$-0.535104\pi$$
−0.110059 + 0.993925i $$0.535104\pi$$
$$744$$ 0 0
$$745$$ −6.00000 −0.219823
$$746$$ 0 0
$$747$$ −30.0000 −1.09764
$$748$$ 0 0
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −32.0000 −1.16770 −0.583848 0.811863i $$-0.698454\pi$$
−0.583848 + 0.811863i $$0.698454\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 12.0000 0.436725
$$756$$ 0 0
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 4.00000 0.144810
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 16.0000 0.573259
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 14.0000 0.499681
$$786$$ 0 0
$$787$$ 18.0000 0.641631 0.320815 0.947142i $$-0.396043\pi$$
0.320815 + 0.947142i $$0.396043\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −20.0000 −0.711118
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38.0000 1.34603 0.673015 0.739629i $$-0.264999\pi$$
0.673015 + 0.739629i $$0.264999\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −18.0000 −0.635999
$$802$$ 0 0
$$803$$ 8.00000 0.282314
$$804$$ 0 0
$$805$$ 16.0000 0.563926
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ −20.0000 −0.702295 −0.351147 0.936320i $$-0.614208\pi$$
−0.351147 + 0.936320i $$0.614208\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −26.0000 −0.907406 −0.453703 0.891153i $$-0.649897\pi$$
−0.453703 + 0.891153i $$0.649897\pi$$
$$822$$ 0 0
$$823$$ 28.0000 0.976019 0.488009 0.872838i $$-0.337723\pi$$
0.488009 + 0.872838i $$0.337723\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −22.0000 −0.765015 −0.382507 0.923952i $$-0.624939\pi$$
−0.382507 + 0.923952i $$0.624939\pi$$
$$828$$ 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
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −10.0000 −0.346064
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −13.0000 −0.447214
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 80.0000 2.74236
$$852$$ 0 0
$$853$$ −4.00000 −0.136957 −0.0684787 0.997653i $$-0.521815\pi$$
−0.0684787 + 0.997653i $$0.521815\pi$$
$$854$$ 0 0
$$855$$ 24.0000 0.820783
$$856$$ 0 0
$$857$$ 48.0000 1.63965 0.819824 0.572615i $$-0.194071\pi$$
0.819824 + 0.572615i $$0.194071\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 52.0000 1.77010 0.885050 0.465495i $$-0.154124\pi$$
0.885050 + 0.465495i $$0.154124\pi$$
$$864$$ 0 0
$$865$$ 12.0000 0.408012
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.00000 −0.135691
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 30.0000 1.01535
$$874$$ 0 0
$$875$$ 2.00000 0.0676123
$$876$$ 0 0
$$877$$ −20.0000 −0.675352 −0.337676 0.941262i $$-0.609641\pi$$
−0.337676 + 0.941262i $$0.609641\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −22.0000 −0.741199 −0.370599 0.928793i $$-0.620848\pi$$
−0.370599 + 0.928793i $$0.620848\pi$$
$$882$$ 0 0
$$883$$ 56.0000 1.88455 0.942275 0.334840i $$-0.108682\pi$$
0.942275 + 0.334840i $$0.108682\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −26.0000 −0.872995 −0.436497 0.899706i $$-0.643781\pi$$
−0.436497 + 0.899706i $$0.643781\pi$$
$$888$$ 0 0
$$889$$ −20.0000 −0.670778
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ 0 0
$$893$$ −64.0000 −2.14168
$$894$$ 0 0
$$895$$ −20.0000 −0.668526
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 80.0000 2.66815
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 14.0000 0.465376
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 0 0
$$909$$ 42.0000 1.39305
$$910$$ 0 0
$$911$$ −48.0000 −1.59031 −0.795155 0.606406i $$-0.792611\pi$$
−0.795155 + 0.606406i $$0.792611\pi$$
$$912$$ 0 0
$$913$$ −10.0000 −0.330952
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 32.0000 1.05673
$$918$$ 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
$$925$$ 10.0000 0.328798
$$926$$ 0 0
$$927$$ −12.0000 −0.394132
$$928$$ 0 0
$$929$$ 2.00000 0.0656179 0.0328089 0.999462i $$-0.489555\pi$$
0.0328089 + 0.999462i $$0.489555\pi$$
$$930$$ 0 0
$$931$$ 24.0000 0.786568
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −20.0000 −0.653372 −0.326686 0.945133i $$-0.605932\pi$$
−0.326686 + 0.945133i $$0.605932\pi$$
$$938$$ 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
$$943$$ −16.0000 −0.521032
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −52.0000 −1.68977 −0.844886 0.534946i $$-0.820332\pi$$
−0.844886 + 0.534946i $$0.820332\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −52.0000 −1.68445 −0.842223 0.539130i $$-0.818753\pi$$
−0.842223 + 0.539130i $$0.818753\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 0 0
$$965$$ −8.00000 −0.257529
$$966$$ 0 0
$$967$$ 14.0000 0.450210 0.225105 0.974335i $$-0.427728\pi$$
0.225105 + 0.974335i $$0.427728\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 20.0000 0.641831 0.320915 0.947108i $$-0.396010\pi$$
0.320915 + 0.947108i $$0.396010\pi$$
$$972$$ 0 0
$$973$$ −8.00000 −0.256468
$$974$$ 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
$$979$$ −6.00000 −0.191761
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 0 0
$$983$$ 44.0000 1.40338 0.701691 0.712481i $$-0.252429\pi$$
0.701691 + 0.712481i $$0.252429\pi$$
$$984$$ 0 0
$$985$$ 12.0000 0.382352
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ 4.00000 0.126681 0.0633406 0.997992i $$-0.479825\pi$$
0.0633406 + 0.997992i $$0.479825\pi$$
$$998$$ 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 3520.2.a.u.1.1 1
4.3 odd 2 3520.2.a.t.1.1 1
8.3 odd 2 440.2.a.a.1.1 1
8.5 even 2 880.2.a.f.1.1 1
24.5 odd 2 7920.2.a.bg.1.1 1
24.11 even 2 3960.2.a.p.1.1 1
40.3 even 4 2200.2.b.d.1849.2 2
40.13 odd 4 4400.2.b.m.4049.1 2
40.19 odd 2 2200.2.a.f.1.1 1
40.27 even 4 2200.2.b.d.1849.1 2
40.29 even 2 4400.2.a.o.1.1 1
40.37 odd 4 4400.2.b.m.4049.2 2
88.21 odd 2 9680.2.a.n.1.1 1
88.43 even 2 4840.2.a.c.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.a.1.1 1 8.3 odd 2
880.2.a.f.1.1 1 8.5 even 2
2200.2.a.f.1.1 1 40.19 odd 2
2200.2.b.d.1849.1 2 40.27 even 4
2200.2.b.d.1849.2 2 40.3 even 4
3520.2.a.t.1.1 1 4.3 odd 2
3520.2.a.u.1.1 1 1.1 even 1 trivial
3960.2.a.p.1.1 1 24.11 even 2
4400.2.a.o.1.1 1 40.29 even 2
4400.2.b.m.4049.1 2 40.13 odd 4
4400.2.b.m.4049.2 2 40.37 odd 4
4840.2.a.c.1.1 1 88.43 even 2
7920.2.a.bg.1.1 1 24.5 odd 2
9680.2.a.n.1.1 1 88.21 odd 2