# Properties

 Label 2200.2.a.k.1.1 Level $2200$ Weight $2$ Character 2200.1 Self dual yes Analytic conductor $17.567$ 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] = [2200,2,Mod(1,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2200 = 2^{3} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2200.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: $$17.5670884447$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} +2.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} +2.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +6.00000 q^{17} +4.00000 q^{19} +6.00000 q^{21} -1.00000 q^{23} +9.00000 q^{27} -8.00000 q^{29} -7.00000 q^{31} -3.00000 q^{33} +1.00000 q^{37} +4.00000 q^{41} -6.00000 q^{43} +8.00000 q^{47} -3.00000 q^{49} +18.0000 q^{51} -2.00000 q^{53} +12.0000 q^{57} -1.00000 q^{59} +4.00000 q^{61} +12.0000 q^{63} +5.00000 q^{67} -3.00000 q^{69} +3.00000 q^{71} -16.0000 q^{73} -2.00000 q^{77} +2.00000 q^{79} +9.00000 q^{81} +2.00000 q^{83} -24.0000 q^{87} +15.0000 q^{89} -21.0000 q^{93} +7.00000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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
$$3$$ 3.00000 1.73205 0.866025 0.500000i $$-0.166667\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 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
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 6.00000 1.30931
$$22$$ 0 0
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 9.00000 1.73205
$$28$$ 0 0
$$29$$ −8.00000 −1.48556 −0.742781 0.669534i $$-0.766494\pi$$
−0.742781 + 0.669534i $$0.766494\pi$$
$$30$$ 0 0
$$31$$ −7.00000 −1.25724 −0.628619 0.777714i $$-0.716379\pi$$
−0.628619 + 0.777714i $$0.716379\pi$$
$$32$$ 0 0
$$33$$ −3.00000 −0.522233
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 1.00000 0.164399 0.0821995 0.996616i $$-0.473806\pi$$
0.0821995 + 0.996616i $$0.473806\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 4.00000 0.624695 0.312348 0.949968i $$-0.398885\pi$$
0.312348 + 0.949968i $$0.398885\pi$$
$$42$$ 0 0
$$43$$ −6.00000 −0.914991 −0.457496 0.889212i $$-0.651253\pi$$
−0.457496 + 0.889212i $$0.651253\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 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
$$51$$ 18.0000 2.52050
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 12.0000 1.58944
$$58$$ 0 0
$$59$$ −1.00000 −0.130189 −0.0650945 0.997879i $$-0.520735\pi$$
−0.0650945 + 0.997879i $$0.520735\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ 12.0000 1.51186
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 5.00000 0.610847 0.305424 0.952217i $$-0.401202\pi$$
0.305424 + 0.952217i $$0.401202\pi$$
$$68$$ 0 0
$$69$$ −3.00000 −0.361158
$$70$$ 0 0
$$71$$ 3.00000 0.356034 0.178017 0.984027i $$-0.443032\pi$$
0.178017 + 0.984027i $$0.443032\pi$$
$$72$$ 0 0
$$73$$ −16.0000 −1.87266 −0.936329 0.351123i $$-0.885800\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ 2.00000 0.225018 0.112509 0.993651i $$-0.464111\pi$$
0.112509 + 0.993651i $$0.464111\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 2.00000 0.219529 0.109764 0.993958i $$-0.464990\pi$$
0.109764 + 0.993958i $$0.464990\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −24.0000 −2.57307
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −21.0000 −2.17760
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ −6.00000 −0.603023
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 0 0
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −2.00000 −0.193347 −0.0966736 0.995316i $$-0.530820\pi$$
−0.0966736 + 0.995316i $$0.530820\pi$$
$$108$$ 0 0
$$109$$ −14.0000 −1.34096 −0.670478 0.741929i $$-0.733911\pi$$
−0.670478 + 0.741929i $$0.733911\pi$$
$$110$$ 0 0
$$111$$ 3.00000 0.284747
$$112$$ 0 0
$$113$$ 7.00000 0.658505 0.329252 0.944242i $$-0.393203\pi$$
0.329252 + 0.944242i $$0.393203\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ −18.0000 −1.58481
$$130$$ 0 0
$$131$$ −2.00000 −0.174741 −0.0873704 0.996176i $$-0.527846\pi$$
−0.0873704 + 0.996176i $$0.527846\pi$$
$$132$$ 0 0
$$133$$ 8.00000 0.693688
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 15.0000 1.28154 0.640768 0.767734i $$-0.278616\pi$$
0.640768 + 0.767734i $$0.278616\pi$$
$$138$$ 0 0
$$139$$ −22.0000 −1.86602 −0.933008 0.359856i $$-0.882826\pi$$
−0.933008 + 0.359856i $$0.882826\pi$$
$$140$$ 0 0
$$141$$ 24.0000 2.02116
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −9.00000 −0.742307
$$148$$ 0 0
$$149$$ 18.0000 1.47462 0.737309 0.675556i $$-0.236096\pi$$
0.737309 + 0.675556i $$0.236096\pi$$
$$150$$ 0 0
$$151$$ −18.0000 −1.46482 −0.732410 0.680864i $$-0.761604\pi$$
−0.732410 + 0.680864i $$0.761604\pi$$
$$152$$ 0 0
$$153$$ 36.0000 2.91043
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 24.0000 1.83533
$$172$$ 0 0
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −3.00000 −0.225494
$$178$$ 0 0
$$179$$ −5.00000 −0.373718 −0.186859 0.982387i $$-0.559831\pi$$
−0.186859 + 0.982387i $$0.559831\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 0 0
$$183$$ 12.0000 0.887066
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −6.00000 −0.438763
$$188$$ 0 0
$$189$$ 18.0000 1.30931
$$190$$ 0 0
$$191$$ −9.00000 −0.651217 −0.325609 0.945505i $$-0.605569\pi$$
−0.325609 + 0.945505i $$0.605569\pi$$
$$192$$ 0 0
$$193$$ −4.00000 −0.287926 −0.143963 0.989583i $$-0.545985\pi$$
−0.143963 + 0.989583i $$0.545985\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ 15.0000 1.05802
$$202$$ 0 0
$$203$$ −16.0000 −1.12298
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ −4.00000 −0.276686
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 9.00000 0.616670
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −14.0000 −0.950382
$$218$$ 0 0
$$219$$ −48.0000 −3.24354
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −29.0000 −1.94198 −0.970992 0.239113i $$-0.923143\pi$$
−0.970992 + 0.239113i $$0.923143\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 0 0
$$229$$ −21.0000 −1.38772 −0.693860 0.720110i $$-0.744091\pi$$
−0.693860 + 0.720110i $$0.744091\pi$$
$$230$$ 0 0
$$231$$ −6.00000 −0.394771
$$232$$ 0 0
$$233$$ 16.0000 1.04819 0.524097 0.851658i $$-0.324403\pi$$
0.524097 + 0.851658i $$0.324403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 6.00000 0.389742
$$238$$ 0 0
$$239$$ −2.00000 −0.129369 −0.0646846 0.997906i $$-0.520604\pi$$
−0.0646846 + 0.997906i $$0.520604\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ 0 0
$$251$$ −13.0000 −0.820553 −0.410276 0.911961i $$-0.634568\pi$$
−0.410276 + 0.911961i $$0.634568\pi$$
$$252$$ 0 0
$$253$$ 1.00000 0.0628695
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 10.0000 0.623783 0.311891 0.950118i $$-0.399037\pi$$
0.311891 + 0.950118i $$0.399037\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 0 0
$$261$$ −48.0000 −2.97113
$$262$$ 0 0
$$263$$ −14.0000 −0.863277 −0.431638 0.902047i $$-0.642064\pi$$
−0.431638 + 0.902047i $$0.642064\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 45.0000 2.75396
$$268$$ 0 0
$$269$$ 26.0000 1.58525 0.792624 0.609711i $$-0.208714\pi$$
0.792624 + 0.609711i $$0.208714\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ −42.0000 −2.51447
$$280$$ 0 0
$$281$$ 6.00000 0.357930 0.178965 0.983855i $$-0.442725\pi$$
0.178965 + 0.983855i $$0.442725\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 8.00000 0.472225
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 21.0000 1.23104
$$292$$ 0 0
$$293$$ −12.0000 −0.701047 −0.350524 0.936554i $$-0.613996\pi$$
−0.350524 + 0.936554i $$0.613996\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −9.00000 −0.522233
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −12.0000 −0.691669
$$302$$ 0 0
$$303$$ −30.0000 −1.72345
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ 48.0000 2.73062
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 9.00000 0.508710 0.254355 0.967111i $$-0.418137\pi$$
0.254355 + 0.967111i $$0.418137\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 15.0000 0.842484 0.421242 0.906948i $$-0.361594\pi$$
0.421242 + 0.906948i $$0.361594\pi$$
$$318$$ 0 0
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −6.00000 −0.334887
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −42.0000 −2.32261
$$328$$ 0 0
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −35.0000 −1.92377 −0.961887 0.273447i $$-0.911836\pi$$
−0.961887 + 0.273447i $$0.911836\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 10.0000 0.544735 0.272367 0.962193i $$-0.412193\pi$$
0.272367 + 0.962193i $$0.412193\pi$$
$$338$$ 0 0
$$339$$ 21.0000 1.14056
$$340$$ 0 0
$$341$$ 7.00000 0.379071
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −3.00000 −0.159674 −0.0798369 0.996808i $$-0.525440\pi$$
−0.0798369 + 0.996808i $$0.525440\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 36.0000 1.90532
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 3.00000 0.157459
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −33.0000 −1.72259 −0.861293 0.508109i $$-0.830345\pi$$
−0.861293 + 0.508109i $$0.830345\pi$$
$$368$$ 0 0
$$369$$ 24.0000 1.24939
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ 22.0000 1.13912 0.569558 0.821951i $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ −12.0000 −0.614779
$$382$$ 0 0
$$383$$ −1.00000 −0.0510976 −0.0255488 0.999674i $$-0.508133\pi$$
−0.0255488 + 0.999674i $$0.508133\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −36.0000 −1.82998
$$388$$ 0 0
$$389$$ 13.0000 0.659126 0.329563 0.944134i $$-0.393099\pi$$
0.329563 + 0.944134i $$0.393099\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 26.0000 1.30490 0.652451 0.757831i $$-0.273741\pi$$
0.652451 + 0.757831i $$0.273741\pi$$
$$398$$ 0 0
$$399$$ 24.0000 1.20150
$$400$$ 0 0
$$401$$ −6.00000 −0.299626 −0.149813 0.988714i $$-0.547867\pi$$
−0.149813 + 0.988714i $$0.547867\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.00000 −0.0495682
$$408$$ 0 0
$$409$$ 34.0000 1.68119 0.840596 0.541663i $$-0.182205\pi$$
0.840596 + 0.541663i $$0.182205\pi$$
$$410$$ 0 0
$$411$$ 45.0000 2.21969
$$412$$ 0 0
$$413$$ −2.00000 −0.0984136
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −66.0000 −3.23203
$$418$$ 0 0
$$419$$ −28.0000 −1.36789 −0.683945 0.729534i $$-0.739737\pi$$
−0.683945 + 0.729534i $$0.739737\pi$$
$$420$$ 0 0
$$421$$ 30.0000 1.46211 0.731055 0.682318i $$-0.239028\pi$$
0.731055 + 0.682318i $$0.239028\pi$$
$$422$$ 0 0
$$423$$ 48.0000 2.33384
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 8.00000 0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −26.0000 −1.25238 −0.626188 0.779672i $$-0.715386\pi$$
−0.626188 + 0.779672i $$0.715386\pi$$
$$432$$ 0 0
$$433$$ −13.0000 −0.624740 −0.312370 0.949960i $$-0.601123\pi$$
−0.312370 + 0.949960i $$0.601123\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −4.00000 −0.191346
$$438$$ 0 0
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ −18.0000 −0.857143
$$442$$ 0 0
$$443$$ 9.00000 0.427603 0.213801 0.976877i $$-0.431415\pi$$
0.213801 + 0.976877i $$0.431415\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 54.0000 2.55411
$$448$$ 0 0
$$449$$ −21.0000 −0.991051 −0.495526 0.868593i $$-0.665025\pi$$
−0.495526 + 0.868593i $$0.665025\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ 0 0
$$453$$ −54.0000 −2.53714
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −12.0000 −0.561336 −0.280668 0.959805i $$-0.590556\pi$$
−0.280668 + 0.959805i $$0.590556\pi$$
$$458$$ 0 0
$$459$$ 54.0000 2.52050
$$460$$ 0 0
$$461$$ −28.0000 −1.30409 −0.652045 0.758180i $$-0.726089\pi$$
−0.652045 + 0.758180i $$0.726089\pi$$
$$462$$ 0 0
$$463$$ −27.0000 −1.25480 −0.627398 0.778699i $$-0.715880\pi$$
−0.627398 + 0.778699i $$0.715880\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 33.0000 1.52706 0.763529 0.645774i $$-0.223465\pi$$
0.763529 + 0.645774i $$0.223465\pi$$
$$468$$ 0 0
$$469$$ 10.0000 0.461757
$$470$$ 0 0
$$471$$ 33.0000 1.52056
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −12.0000 −0.549442
$$478$$ 0 0
$$479$$ 8.00000 0.365529 0.182765 0.983157i $$-0.441495\pi$$
0.182765 + 0.983157i $$0.441495\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −9.00000 −0.407829 −0.203914 0.978989i $$-0.565366\pi$$
−0.203914 + 0.978989i $$0.565366\pi$$
$$488$$ 0 0
$$489$$ 12.0000 0.542659
$$490$$ 0 0
$$491$$ 8.00000 0.361035 0.180517 0.983572i $$-0.442223\pi$$
0.180517 + 0.983572i $$0.442223\pi$$
$$492$$ 0 0
$$493$$ −48.0000 −2.16181
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 6.00000 0.269137
$$498$$ 0 0
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ −48.0000 −2.14448
$$502$$ 0 0
$$503$$ 30.0000 1.33763 0.668817 0.743427i $$-0.266801\pi$$
0.668817 + 0.743427i $$0.266801\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −39.0000 −1.73205
$$508$$ 0 0
$$509$$ −13.0000 −0.576215 −0.288107 0.957598i $$-0.593026\pi$$
−0.288107 + 0.957598i $$0.593026\pi$$
$$510$$ 0 0
$$511$$ −32.0000 −1.41560
$$512$$ 0 0
$$513$$ 36.0000 1.58944
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −8.00000 −0.351840
$$518$$ 0 0
$$519$$ −54.0000 −2.37034
$$520$$ 0 0
$$521$$ 37.0000 1.62100 0.810500 0.585739i $$-0.199196\pi$$
0.810500 + 0.585739i $$0.199196\pi$$
$$522$$ 0 0
$$523$$ −44.0000 −1.92399 −0.961993 0.273075i $$-0.911959\pi$$
−0.961993 + 0.273075i $$0.911959\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −42.0000 −1.82955
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −15.0000 −0.647298
$$538$$ 0 0
$$539$$ 3.00000 0.129219
$$540$$ 0 0
$$541$$ 12.0000 0.515920 0.257960 0.966156i $$-0.416950\pi$$
0.257960 + 0.966156i $$0.416950\pi$$
$$542$$ 0 0
$$543$$ −15.0000 −0.643712
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 24.0000 1.02430
$$550$$ 0 0
$$551$$ −32.0000 −1.36325
$$552$$ 0 0
$$553$$ 4.00000 0.170097
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −18.0000 −0.759961
$$562$$ 0 0
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 18.0000 0.755929
$$568$$ 0 0
$$569$$ −24.0000 −1.00613 −0.503066 0.864248i $$-0.667795\pi$$
−0.503066 + 0.864248i $$0.667795\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ −27.0000 −1.12794
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 23.0000 0.957503 0.478751 0.877951i $$-0.341090\pi$$
0.478751 + 0.877951i $$0.341090\pi$$
$$578$$ 0 0
$$579$$ −12.0000 −0.498703
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ 0 0
$$583$$ 2.00000 0.0828315
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ 0 0
$$589$$ −28.0000 −1.15372
$$590$$ 0 0
$$591$$ −18.0000 −0.740421
$$592$$ 0 0
$$593$$ 4.00000 0.164260 0.0821302 0.996622i $$-0.473828\pi$$
0.0821302 + 0.996622i $$0.473828\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 24.0000 0.982255
$$598$$ 0 0
$$599$$ 48.0000 1.96123 0.980613 0.195952i $$-0.0627798\pi$$
0.980613 + 0.195952i $$0.0627798\pi$$
$$600$$ 0 0
$$601$$ 38.0000 1.55005 0.775026 0.631929i $$-0.217737\pi$$
0.775026 + 0.631929i $$0.217737\pi$$
$$602$$ 0 0
$$603$$ 30.0000 1.22169
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 18.0000 0.730597 0.365299 0.930890i $$-0.380967\pi$$
0.365299 + 0.930890i $$0.380967\pi$$
$$608$$ 0 0
$$609$$ −48.0000 −1.94506
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 16.0000 0.646234 0.323117 0.946359i $$-0.395269\pi$$
0.323117 + 0.946359i $$0.395269\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ 0 0
$$619$$ −3.00000 −0.120580 −0.0602901 0.998181i $$-0.519203\pi$$
−0.0602901 + 0.998181i $$0.519203\pi$$
$$620$$ 0 0
$$621$$ −9.00000 −0.361158
$$622$$ 0 0
$$623$$ 30.0000 1.20192
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −12.0000 −0.479234
$$628$$ 0 0
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 9.00000 0.358284 0.179142 0.983823i $$-0.442668\pi$$
0.179142 + 0.983823i $$0.442668\pi$$
$$632$$ 0 0
$$633$$ 60.0000 2.38479
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 18.0000 0.712069
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ −7.00000 −0.276053 −0.138027 0.990429i $$-0.544076\pi$$
−0.138027 + 0.990429i $$0.544076\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −15.0000 −0.589711 −0.294855 0.955542i $$-0.595271\pi$$
−0.294855 + 0.955542i $$0.595271\pi$$
$$648$$ 0 0
$$649$$ 1.00000 0.0392534
$$650$$ 0 0
$$651$$ −42.0000 −1.64611
$$652$$ 0 0
$$653$$ −11.0000 −0.430463 −0.215232 0.976563i $$-0.569051\pi$$
−0.215232 + 0.976563i $$0.569051\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −96.0000 −3.74532
$$658$$ 0 0
$$659$$ 22.0000 0.856998 0.428499 0.903542i $$-0.359042\pi$$
0.428499 + 0.903542i $$0.359042\pi$$
$$660$$ 0 0
$$661$$ −7.00000 −0.272268 −0.136134 0.990690i $$-0.543468\pi$$
−0.136134 + 0.990690i $$0.543468\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 8.00000 0.309761
$$668$$ 0 0
$$669$$ −87.0000 −3.36361
$$670$$ 0 0
$$671$$ −4.00000 −0.154418
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 30.0000 1.15299 0.576497 0.817099i $$-0.304419\pi$$
0.576497 + 0.817099i $$0.304419\pi$$
$$678$$ 0 0
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ −54.0000 −2.06928
$$682$$ 0 0
$$683$$ −8.00000 −0.306111 −0.153056 0.988218i $$-0.548911\pi$$
−0.153056 + 0.988218i $$0.548911\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −63.0000 −2.40360
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −29.0000 −1.10321 −0.551606 0.834105i $$-0.685985\pi$$
−0.551606 + 0.834105i $$0.685985\pi$$
$$692$$ 0 0
$$693$$ −12.0000 −0.455842
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 24.0000 0.909065
$$698$$ 0 0
$$699$$ 48.0000 1.81553
$$700$$ 0 0
$$701$$ −10.0000 −0.377695 −0.188847 0.982006i $$-0.560475\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −20.0000 −0.752177
$$708$$ 0 0
$$709$$ 19.0000 0.713560 0.356780 0.934188i $$-0.383875\pi$$
0.356780 + 0.934188i $$0.383875\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 7.00000 0.262152
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −6.00000 −0.224074
$$718$$ 0 0
$$719$$ −23.0000 −0.857755 −0.428878 0.903363i $$-0.641091\pi$$
−0.428878 + 0.903363i $$0.641091\pi$$
$$720$$ 0 0
$$721$$ 32.0000 1.19174
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 35.0000 1.29808 0.649039 0.760755i $$-0.275171\pi$$
0.649039 + 0.760755i $$0.275171\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ −36.0000 −1.33151
$$732$$ 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
$$737$$ −5.00000 −0.184177
$$738$$ 0 0
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ 0 0
$$753$$ −39.0000 −1.42124
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 0 0
$$759$$ 3.00000 0.108893
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ −28.0000 −1.01367
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 16.0000 0.576975 0.288487 0.957484i $$-0.406848\pi$$
0.288487 + 0.957484i $$0.406848\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ 0 0
$$773$$ 6.00000 0.215805 0.107903 0.994161i $$-0.465587\pi$$
0.107903 + 0.994161i $$0.465587\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 6.00000 0.215249
$$778$$ 0 0
$$779$$ 16.0000 0.573259
$$780$$ 0 0
$$781$$ −3.00000 −0.107348
$$782$$ 0 0
$$783$$ −72.0000 −2.57307
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −24.0000 −0.855508 −0.427754 0.903895i $$-0.640695\pi$$
−0.427754 + 0.903895i $$0.640695\pi$$
$$788$$ 0 0
$$789$$ −42.0000 −1.49524
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −1.00000 −0.0354218 −0.0177109 0.999843i $$-0.505638\pi$$
−0.0177109 + 0.999843i $$0.505638\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ 90.0000 3.17999
$$802$$ 0 0
$$803$$ 16.0000 0.564628
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 78.0000 2.74573
$$808$$ 0 0
$$809$$ 12.0000 0.421898 0.210949 0.977497i $$-0.432345\pi$$
0.210949 + 0.977497i $$0.432345\pi$$
$$810$$ 0 0
$$811$$ −26.0000 −0.912983 −0.456492 0.889728i $$-0.650894\pi$$
−0.456492 + 0.889728i $$0.650894\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −24.0000 −0.839654
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 0 0
$$823$$ 31.0000 1.08059 0.540296 0.841475i $$-0.318312\pi$$
0.540296 + 0.841475i $$0.318312\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 16.0000 0.556375 0.278187 0.960527i $$-0.410266\pi$$
0.278187 + 0.960527i $$0.410266\pi$$
$$828$$ 0 0
$$829$$ −35.0000 −1.21560 −0.607800 0.794090i $$-0.707948\pi$$
−0.607800 + 0.794090i $$0.707948\pi$$
$$830$$ 0 0
$$831$$ 6.00000 0.208138
$$832$$ 0 0
$$833$$ −18.0000 −0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −63.0000 −2.17760
$$838$$ 0 0
$$839$$ 21.0000 0.725001 0.362500 0.931984i $$-0.381923\pi$$
0.362500 + 0.931984i $$0.381923\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 0 0
$$843$$ 18.0000 0.619953
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 2.00000 0.0687208
$$848$$ 0 0
$$849$$ −12.0000 −0.411839
$$850$$ 0 0
$$851$$ −1.00000 −0.0342796
$$852$$ 0 0
$$853$$ 26.0000 0.890223 0.445112 0.895475i $$-0.353164\pi$$
0.445112 + 0.895475i $$0.353164\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 24.0000 0.819824 0.409912 0.912125i $$-0.365559\pi$$
0.409912 + 0.912125i $$0.365559\pi$$
$$858$$ 0 0
$$859$$ 11.0000 0.375315 0.187658 0.982235i $$-0.439910\pi$$
0.187658 + 0.982235i $$0.439910\pi$$
$$860$$ 0 0
$$861$$ 24.0000 0.817918
$$862$$ 0 0
$$863$$ 16.0000 0.544646 0.272323 0.962206i $$-0.412208\pi$$
0.272323 + 0.962206i $$0.412208\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 57.0000 1.93582
$$868$$ 0 0
$$869$$ −2.00000 −0.0678454
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 42.0000 1.42148
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −56.0000 −1.89099 −0.945493 0.325643i $$-0.894419\pi$$
−0.945493 + 0.325643i $$0.894419\pi$$
$$878$$ 0 0
$$879$$ −36.0000 −1.21425
$$880$$ 0 0
$$881$$ −19.0000 −0.640126 −0.320063 0.947396i $$-0.603704\pi$$
−0.320063 + 0.947396i $$0.603704\pi$$
$$882$$ 0 0
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 34.0000 1.14161 0.570804 0.821086i $$-0.306632\pi$$
0.570804 + 0.821086i $$0.306632\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ 0 0
$$893$$ 32.0000 1.07084
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 56.0000 1.86770
$$900$$ 0 0
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ −36.0000 −1.19800
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 0 0
$$909$$ −60.0000 −1.99007
$$910$$ 0 0
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ −2.00000 −0.0661903
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −4.00000 −0.132092
$$918$$ 0 0
$$919$$ −50.0000 −1.64935 −0.824674 0.565608i $$-0.808641\pi$$
−0.824674 + 0.565608i $$0.808641\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 96.0000 3.15305
$$928$$ 0 0
$$929$$ −22.0000 −0.721797 −0.360898 0.932605i $$-0.617530\pi$$
−0.360898 + 0.932605i $$0.617530\pi$$
$$930$$ 0 0
$$931$$ −12.0000 −0.393284
$$932$$ 0 0
$$933$$ −36.0000 −1.17859
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −4.00000 −0.130674 −0.0653372 0.997863i $$-0.520812\pi$$
−0.0653372 + 0.997863i $$0.520812\pi$$
$$938$$ 0 0
$$939$$ 27.0000 0.881112
$$940$$ 0 0
$$941$$ 10.0000 0.325991 0.162995 0.986627i $$-0.447884\pi$$
0.162995 + 0.986627i $$0.447884\pi$$
$$942$$ 0 0
$$943$$ −4.00000 −0.130258
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −47.0000 −1.52729 −0.763647 0.645634i $$-0.776593\pi$$
−0.763647 + 0.645634i $$0.776593\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 45.0000 1.45922
$$952$$ 0 0
$$953$$ 34.0000 1.10137 0.550684 0.834714i $$-0.314367\pi$$
0.550684 + 0.834714i $$0.314367\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 24.0000 0.775810
$$958$$ 0 0
$$959$$ 30.0000 0.968751
$$960$$ 0 0
$$961$$ 18.0000 0.580645
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8.00000 0.257263 0.128631 0.991692i $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 0 0
$$969$$ 72.0000 2.31297
$$970$$ 0 0
$$971$$ 53.0000 1.70085 0.850425 0.526096i $$-0.176345\pi$$
0.850425 + 0.526096i $$0.176345\pi$$
$$972$$ 0 0
$$973$$ −44.0000 −1.41058
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 11.0000 0.351921 0.175961 0.984397i $$-0.443697\pi$$
0.175961 + 0.984397i $$0.443697\pi$$
$$978$$ 0 0
$$979$$ −15.0000 −0.479402
$$980$$ 0 0
$$981$$ −84.0000 −2.68191
$$982$$ 0 0
$$983$$ −25.0000 −0.797376 −0.398688 0.917087i $$-0.630534\pi$$
−0.398688 + 0.917087i $$0.630534\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 48.0000 1.52786
$$988$$ 0 0
$$989$$ 6.00000 0.190789
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ −105.000 −3.33207
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −2.00000 −0.0633406 −0.0316703 0.999498i $$-0.510083\pi$$
−0.0316703 + 0.999498i $$0.510083\pi$$
$$998$$ 0 0
$$999$$ 9.00000 0.284747
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 2200.2.a.k.1.1 1
4.3 odd 2 4400.2.a.a.1.1 1
5.2 odd 4 2200.2.b.a.1849.1 2
5.3 odd 4 2200.2.b.a.1849.2 2
5.4 even 2 88.2.a.a.1.1 1
15.14 odd 2 792.2.a.g.1.1 1
20.3 even 4 4400.2.b.b.4049.1 2
20.7 even 4 4400.2.b.b.4049.2 2
20.19 odd 2 176.2.a.c.1.1 1
35.34 odd 2 4312.2.a.l.1.1 1
40.19 odd 2 704.2.a.b.1.1 1
40.29 even 2 704.2.a.l.1.1 1
55.4 even 10 968.2.i.j.753.1 4
55.9 even 10 968.2.i.j.81.1 4
55.14 even 10 968.2.i.j.9.1 4
55.19 odd 10 968.2.i.i.9.1 4
55.24 odd 10 968.2.i.i.81.1 4
55.29 odd 10 968.2.i.i.753.1 4
55.39 odd 10 968.2.i.i.729.1 4
55.49 even 10 968.2.i.j.729.1 4
55.54 odd 2 968.2.a.a.1.1 1
60.59 even 2 1584.2.a.q.1.1 1
80.19 odd 4 2816.2.c.d.1409.2 2
80.29 even 4 2816.2.c.i.1409.1 2
80.59 odd 4 2816.2.c.d.1409.1 2
80.69 even 4 2816.2.c.i.1409.2 2
120.29 odd 2 6336.2.a.h.1.1 1
120.59 even 2 6336.2.a.k.1.1 1
140.139 even 2 8624.2.a.c.1.1 1
165.164 even 2 8712.2.a.x.1.1 1
220.219 even 2 1936.2.a.l.1.1 1
440.109 odd 2 7744.2.a.bk.1.1 1
440.219 even 2 7744.2.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.a.1.1 1 5.4 even 2
176.2.a.c.1.1 1 20.19 odd 2
704.2.a.b.1.1 1 40.19 odd 2
704.2.a.l.1.1 1 40.29 even 2
792.2.a.g.1.1 1 15.14 odd 2
968.2.a.a.1.1 1 55.54 odd 2
968.2.i.i.9.1 4 55.19 odd 10
968.2.i.i.81.1 4 55.24 odd 10
968.2.i.i.729.1 4 55.39 odd 10
968.2.i.i.753.1 4 55.29 odd 10
968.2.i.j.9.1 4 55.14 even 10
968.2.i.j.81.1 4 55.9 even 10
968.2.i.j.729.1 4 55.49 even 10
968.2.i.j.753.1 4 55.4 even 10
1584.2.a.q.1.1 1 60.59 even 2
1936.2.a.l.1.1 1 220.219 even 2
2200.2.a.k.1.1 1 1.1 even 1 trivial
2200.2.b.a.1849.1 2 5.2 odd 4
2200.2.b.a.1849.2 2 5.3 odd 4
2816.2.c.d.1409.1 2 80.59 odd 4
2816.2.c.d.1409.2 2 80.19 odd 4
2816.2.c.i.1409.1 2 80.29 even 4
2816.2.c.i.1409.2 2 80.69 even 4
4312.2.a.l.1.1 1 35.34 odd 2
4400.2.a.a.1.1 1 4.3 odd 2
4400.2.b.b.4049.1 2 20.3 even 4
4400.2.b.b.4049.2 2 20.7 even 4
6336.2.a.h.1.1 1 120.29 odd 2
6336.2.a.k.1.1 1 120.59 even 2
7744.2.a.b.1.1 1 440.219 even 2
7744.2.a.bk.1.1 1 440.109 odd 2
8624.2.a.c.1.1 1 140.139 even 2
8712.2.a.x.1.1 1 165.164 even 2