# Properties

 Label 2200.2.a.a.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 440) 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)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} +O(q^{10})$$ $$q-3.00000 q^{3} -1.00000 q^{7} +6.00000 q^{9} -1.00000 q^{11} +6.00000 q^{13} -3.00000 q^{17} -5.00000 q^{19} +3.00000 q^{21} +2.00000 q^{23} -9.00000 q^{27} -5.00000 q^{29} +5.00000 q^{31} +3.00000 q^{33} +1.00000 q^{37} -18.0000 q^{39} -2.00000 q^{41} -12.0000 q^{43} +2.00000 q^{47} -6.00000 q^{49} +9.00000 q^{51} +13.0000 q^{53} +15.0000 q^{57} +2.00000 q^{59} +1.00000 q^{61} -6.00000 q^{63} -16.0000 q^{67} -6.00000 q^{69} +15.0000 q^{71} -10.0000 q^{73} +1.00000 q^{77} +2.00000 q^{79} +9.00000 q^{81} +14.0000 q^{83} +15.0000 q^{87} +9.00000 q^{89} -6.00000 q^{91} -15.0000 q^{93} +16.0000 q^{97} -6.00000 q^{99} +O(q^{100})$$

## 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 $$-0.833333\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.00000 −0.377964 −0.188982 0.981981i $$-0.560519\pi$$
−0.188982 + 0.981981i $$0.560519\pi$$
$$8$$ 0 0
$$9$$ 6.00000 2.00000
$$10$$ 0 0
$$11$$ −1.00000 −0.301511
$$12$$ 0 0
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −5.00000 −1.14708 −0.573539 0.819178i $$-0.694430\pi$$
−0.573539 + 0.819178i $$0.694430\pi$$
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ 0 0
$$23$$ 2.00000 0.417029 0.208514 0.978019i $$-0.433137\pi$$
0.208514 + 0.978019i $$0.433137\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −9.00000 −1.73205
$$28$$ 0 0
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 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
$$39$$ −18.0000 −2.88231
$$40$$ 0 0
$$41$$ −2.00000 −0.312348 −0.156174 0.987730i $$-0.549916\pi$$
−0.156174 + 0.987730i $$0.549916\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.00000 0.291730 0.145865 0.989305i $$-0.453403\pi$$
0.145865 + 0.989305i $$0.453403\pi$$
$$48$$ 0 0
$$49$$ −6.00000 −0.857143
$$50$$ 0 0
$$51$$ 9.00000 1.26025
$$52$$ 0 0
$$53$$ 13.0000 1.78569 0.892844 0.450367i $$-0.148707\pi$$
0.892844 + 0.450367i $$0.148707\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 15.0000 1.98680
$$58$$ 0 0
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ −6.00000 −0.755929
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −16.0000 −1.95471 −0.977356 0.211604i $$-0.932131\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 15.0000 1.78017 0.890086 0.455792i $$-0.150644\pi$$
0.890086 + 0.455792i $$0.150644\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.00000 0.113961
$$78$$ 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
$$83$$ 14.0000 1.53670 0.768350 0.640030i $$-0.221078\pi$$
0.768350 + 0.640030i $$0.221078\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 15.0000 1.60817
$$88$$ 0 0
$$89$$ 9.00000 0.953998 0.476999 0.878904i $$-0.341725\pi$$
0.476999 + 0.878904i $$0.341725\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ 0 0
$$93$$ −15.0000 −1.55543
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 16.0000 1.62455 0.812277 0.583272i $$-0.198228\pi$$
0.812277 + 0.583272i $$0.198228\pi$$
$$98$$ 0 0
$$99$$ −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
$$107$$ 4.00000 0.386695 0.193347 0.981130i $$-0.438066\pi$$
0.193347 + 0.981130i $$0.438066\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ 16.0000 1.50515 0.752577 0.658505i $$-0.228811\pi$$
0.752577 + 0.658505i $$0.228811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 36.0000 3.32820
$$118$$ 0 0
$$119$$ 3.00000 0.275010
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ 0 0
$$123$$ 6.00000 0.541002
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 8.00000 0.709885 0.354943 0.934888i $$-0.384500\pi$$
0.354943 + 0.934888i $$0.384500\pi$$
$$128$$ 0 0
$$129$$ 36.0000 3.16962
$$130$$ 0 0
$$131$$ 7.00000 0.611593 0.305796 0.952097i $$-0.401077\pi$$
0.305796 + 0.952097i $$0.401077\pi$$
$$132$$ 0 0
$$133$$ 5.00000 0.433555
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ −6.00000 −0.501745
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 18.0000 1.48461
$$148$$ 0 0
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ 0 0
$$153$$ −18.0000 −1.45521
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −7.00000 −0.558661 −0.279330 0.960195i $$-0.590112\pi$$
−0.279330 + 0.960195i $$0.590112\pi$$
$$158$$ 0 0
$$159$$ −39.0000 −3.09290
$$160$$ 0 0
$$161$$ −2.00000 −0.157622
$$162$$ 0 0
$$163$$ 1.00000 0.0783260 0.0391630 0.999233i $$-0.487531\pi$$
0.0391630 + 0.999233i $$0.487531\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −1.00000 −0.0773823 −0.0386912 0.999251i $$-0.512319\pi$$
−0.0386912 + 0.999251i $$0.512319\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −30.0000 −2.29416
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −6.00000 −0.450988
$$178$$ 0 0
$$179$$ 4.00000 0.298974 0.149487 0.988764i $$-0.452238\pi$$
0.149487 + 0.988764i $$0.452238\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ −3.00000 −0.221766
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.00000 0.219382
$$188$$ 0 0
$$189$$ 9.00000 0.654654
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −7.00000 −0.503871 −0.251936 0.967744i $$-0.581067\pi$$
−0.251936 + 0.967744i $$0.581067\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 24.0000 1.70993 0.854965 0.518686i $$-0.173579\pi$$
0.854965 + 0.518686i $$0.173579\pi$$
$$198$$ 0 0
$$199$$ −1.00000 −0.0708881 −0.0354441 0.999372i $$-0.511285\pi$$
−0.0354441 + 0.999372i $$0.511285\pi$$
$$200$$ 0 0
$$201$$ 48.0000 3.38566
$$202$$ 0 0
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 12.0000 0.834058
$$208$$ 0 0
$$209$$ 5.00000 0.345857
$$210$$ 0 0
$$211$$ −1.00000 −0.0688428 −0.0344214 0.999407i $$-0.510959\pi$$
−0.0344214 + 0.999407i $$0.510959\pi$$
$$212$$ 0 0
$$213$$ −45.0000 −3.08335
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −5.00000 −0.339422
$$218$$ 0 0
$$219$$ 30.0000 2.02721
$$220$$ 0 0
$$221$$ −18.0000 −1.21081
$$222$$ 0 0
$$223$$ −26.0000 −1.74109 −0.870544 0.492090i $$-0.836233\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 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
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ −3.00000 −0.197386
$$232$$ 0 0
$$233$$ −11.0000 −0.720634 −0.360317 0.932830i $$-0.617331\pi$$
−0.360317 + 0.932830i $$0.617331\pi$$
$$234$$ 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
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −30.0000 −1.90885
$$248$$ 0 0
$$249$$ −42.0000 −2.66164
$$250$$ 0 0
$$251$$ −10.0000 −0.631194 −0.315597 0.948893i $$-0.602205\pi$$
−0.315597 + 0.948893i $$0.602205\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 22.0000 1.37232 0.686161 0.727450i $$-0.259294\pi$$
0.686161 + 0.727450i $$0.259294\pi$$
$$258$$ 0 0
$$259$$ −1.00000 −0.0621370
$$260$$ 0 0
$$261$$ −30.0000 −1.85695
$$262$$ 0 0
$$263$$ 13.0000 0.801614 0.400807 0.916162i $$-0.368730\pi$$
0.400807 + 0.916162i $$0.368730\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −27.0000 −1.65237
$$268$$ 0 0
$$269$$ 20.0000 1.21942 0.609711 0.792624i $$-0.291286\pi$$
0.609711 + 0.792624i $$0.291286\pi$$
$$270$$ 0 0
$$271$$ 24.0000 1.45790 0.728948 0.684569i $$-0.240010\pi$$
0.728948 + 0.684569i $$0.240010\pi$$
$$272$$ 0 0
$$273$$ 18.0000 1.08941
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −16.0000 −0.961347 −0.480673 0.876900i $$-0.659608\pi$$
−0.480673 + 0.876900i $$0.659608\pi$$
$$278$$ 0 0
$$279$$ 30.0000 1.79605
$$280$$ 0 0
$$281$$ 18.0000 1.07379 0.536895 0.843649i $$-0.319597\pi$$
0.536895 + 0.843649i $$0.319597\pi$$
$$282$$ 0 0
$$283$$ −10.0000 −0.594438 −0.297219 0.954809i $$-0.596059\pi$$
−0.297219 + 0.954809i $$0.596059\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 2.00000 0.118056
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ −48.0000 −2.81381
$$292$$ 0 0
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 9.00000 0.522233
$$298$$ 0 0
$$299$$ 12.0000 0.693978
$$300$$ 0 0
$$301$$ 12.0000 0.691669
$$302$$ 0 0
$$303$$ 30.0000 1.72345
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ 0 0
$$309$$ −48.0000 −2.73062
$$310$$ 0 0
$$311$$ 21.0000 1.19080 0.595400 0.803429i $$-0.296993\pi$$
0.595400 + 0.803429i $$0.296993\pi$$
$$312$$ 0 0
$$313$$ 30.0000 1.69570 0.847850 0.530236i $$-0.177897\pi$$
0.847850 + 0.530236i $$0.177897\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.00000 0.168497 0.0842484 0.996445i $$-0.473151\pi$$
0.0842484 + 0.996445i $$0.473151\pi$$
$$318$$ 0 0
$$319$$ 5.00000 0.279946
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 15.0000 0.834622
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −30.0000 −1.65900
$$328$$ 0 0
$$329$$ −2.00000 −0.110264
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ 6.00000 0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −29.0000 −1.57973 −0.789865 0.613280i $$-0.789850\pi$$
−0.789865 + 0.613280i $$0.789850\pi$$
$$338$$ 0 0
$$339$$ −48.0000 −2.60700
$$340$$ 0 0
$$341$$ −5.00000 −0.270765
$$342$$ 0 0
$$343$$ 13.0000 0.701934
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000 0.107366 0.0536828 0.998558i $$-0.482904\pi$$
0.0536828 + 0.998558i $$0.482904\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −54.0000 −2.88231
$$352$$ 0 0
$$353$$ 18.0000 0.958043 0.479022 0.877803i $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −9.00000 −0.476331
$$358$$ 0 0
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ 0 0
$$363$$ −3.00000 −0.157459
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 12.0000 0.626395 0.313197 0.949688i $$-0.398600\pi$$
0.313197 + 0.949688i $$0.398600\pi$$
$$368$$ 0 0
$$369$$ −12.0000 −0.624695
$$370$$ 0 0
$$371$$ −13.0000 −0.674926
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −30.0000 −1.54508
$$378$$ 0 0
$$379$$ −2.00000 −0.102733 −0.0513665 0.998680i $$-0.516358\pi$$
−0.0513665 + 0.998680i $$0.516358\pi$$
$$380$$ 0 0
$$381$$ −24.0000 −1.22956
$$382$$ 0 0
$$383$$ 14.0000 0.715367 0.357683 0.933843i $$-0.383567\pi$$
0.357683 + 0.933843i $$0.383567\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −72.0000 −3.65997
$$388$$ 0 0
$$389$$ −2.00000 −0.101404 −0.0507020 0.998714i $$-0.516146\pi$$
−0.0507020 + 0.998714i $$0.516146\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ 0 0
$$393$$ −21.0000 −1.05931
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −22.0000 −1.10415 −0.552074 0.833795i $$-0.686163\pi$$
−0.552074 + 0.833795i $$0.686163\pi$$
$$398$$ 0 0
$$399$$ −15.0000 −0.750939
$$400$$ 0 0
$$401$$ −21.0000 −1.04869 −0.524345 0.851506i $$-0.675690\pi$$
−0.524345 + 0.851506i $$0.675690\pi$$
$$402$$ 0 0
$$403$$ 30.0000 1.49441
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −1.00000 −0.0495682
$$408$$ 0 0
$$409$$ −32.0000 −1.58230 −0.791149 0.611623i $$-0.790517\pi$$
−0.791149 + 0.611623i $$0.790517\pi$$
$$410$$ 0 0
$$411$$ 36.0000 1.77575
$$412$$ 0 0
$$413$$ −2.00000 −0.0984136
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 12.0000 0.587643
$$418$$ 0 0
$$419$$ −4.00000 −0.195413 −0.0977064 0.995215i $$-0.531151\pi$$
−0.0977064 + 0.995215i $$0.531151\pi$$
$$420$$ 0 0
$$421$$ −36.0000 −1.75453 −0.877266 0.480004i $$-0.840635\pi$$
−0.877266 + 0.480004i $$0.840635\pi$$
$$422$$ 0 0
$$423$$ 12.0000 0.583460
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.00000 −0.0483934
$$428$$ 0 0
$$429$$ 18.0000 0.869048
$$430$$ 0 0
$$431$$ −8.00000 −0.385346 −0.192673 0.981263i $$-0.561716\pi$$
−0.192673 + 0.981263i $$0.561716\pi$$
$$432$$ 0 0
$$433$$ −4.00000 −0.192228 −0.0961139 0.995370i $$-0.530641\pi$$
−0.0961139 + 0.995370i $$0.530641\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −10.0000 −0.478365
$$438$$ 0 0
$$439$$ 16.0000 0.763638 0.381819 0.924237i $$-0.375298\pi$$
0.381819 + 0.924237i $$0.375298\pi$$
$$440$$ 0 0
$$441$$ −36.0000 −1.71429
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 45.0000 2.12843
$$448$$ 0 0
$$449$$ −6.00000 −0.283158 −0.141579 0.989927i $$-0.545218\pi$$
−0.141579 + 0.989927i $$0.545218\pi$$
$$450$$ 0 0
$$451$$ 2.00000 0.0941763
$$452$$ 0 0
$$453$$ −54.0000 −2.53714
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −33.0000 −1.54367 −0.771837 0.635820i $$-0.780662\pi$$
−0.771837 + 0.635820i $$0.780662\pi$$
$$458$$ 0 0
$$459$$ 27.0000 1.26025
$$460$$ 0 0
$$461$$ 5.00000 0.232873 0.116437 0.993198i $$-0.462853\pi$$
0.116437 + 0.993198i $$0.462853\pi$$
$$462$$ 0 0
$$463$$ −6.00000 −0.278844 −0.139422 0.990233i $$-0.544524\pi$$
−0.139422 + 0.990233i $$0.544524\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −27.0000 −1.24941 −0.624705 0.780860i $$-0.714781\pi$$
−0.624705 + 0.780860i $$0.714781\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ 0 0
$$471$$ 21.0000 0.967629
$$472$$ 0 0
$$473$$ 12.0000 0.551761
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 78.0000 3.57137
$$478$$ 0 0
$$479$$ 20.0000 0.913823 0.456912 0.889512i $$-0.348956\pi$$
0.456912 + 0.889512i $$0.348956\pi$$
$$480$$ 0 0
$$481$$ 6.00000 0.273576
$$482$$ 0 0
$$483$$ 6.00000 0.273009
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000 0.543772 0.271886 0.962329i $$-0.412353\pi$$
0.271886 + 0.962329i $$0.412353\pi$$
$$488$$ 0 0
$$489$$ −3.00000 −0.135665
$$490$$ 0 0
$$491$$ −13.0000 −0.586682 −0.293341 0.956008i $$-0.594767\pi$$
−0.293341 + 0.956008i $$0.594767\pi$$
$$492$$ 0 0
$$493$$ 15.0000 0.675566
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −15.0000 −0.672842
$$498$$ 0 0
$$499$$ −8.00000 −0.358129 −0.179065 0.983837i $$-0.557307\pi$$
−0.179065 + 0.983837i $$0.557307\pi$$
$$500$$ 0 0
$$501$$ 3.00000 0.134030
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −69.0000 −3.06440
$$508$$ 0 0
$$509$$ 8.00000 0.354594 0.177297 0.984157i $$-0.443265\pi$$
0.177297 + 0.984157i $$0.443265\pi$$
$$510$$ 0 0
$$511$$ 10.0000 0.442374
$$512$$ 0 0
$$513$$ 45.0000 1.98680
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −2.00000 −0.0879599
$$518$$ 0 0
$$519$$ −18.0000 −0.790112
$$520$$ 0 0
$$521$$ −26.0000 −1.13908 −0.569540 0.821963i $$-0.692879\pi$$
−0.569540 + 0.821963i $$0.692879\pi$$
$$522$$ 0 0
$$523$$ 4.00000 0.174908 0.0874539 0.996169i $$-0.472127\pi$$
0.0874539 + 0.996169i $$0.472127\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −15.0000 −0.653410
$$528$$ 0 0
$$529$$ −19.0000 −0.826087
$$530$$ 0 0
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 15.0000 0.644900 0.322450 0.946586i $$-0.395494\pi$$
0.322450 + 0.946586i $$0.395494\pi$$
$$542$$ 0 0
$$543$$ −66.0000 −2.83233
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 36.0000 1.53925 0.769624 0.638497i $$-0.220443\pi$$
0.769624 + 0.638497i $$0.220443\pi$$
$$548$$ 0 0
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 25.0000 1.06504
$$552$$ 0 0
$$553$$ −2.00000 −0.0850487
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 30.0000 1.27114 0.635570 0.772043i $$-0.280765\pi$$
0.635570 + 0.772043i $$0.280765\pi$$
$$558$$ 0 0
$$559$$ −72.0000 −3.04528
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 0 0
$$563$$ 18.0000 0.758610 0.379305 0.925272i $$-0.376163\pi$$
0.379305 + 0.925272i $$0.376163\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −9.00000 −0.377964
$$568$$ 0 0
$$569$$ 36.0000 1.50920 0.754599 0.656186i $$-0.227831\pi$$
0.754599 + 0.656186i $$0.227831\pi$$
$$570$$ 0 0
$$571$$ −19.0000 −0.795125 −0.397563 0.917575i $$-0.630144\pi$$
−0.397563 + 0.917575i $$0.630144\pi$$
$$572$$ 0 0
$$573$$ −36.0000 −1.50392
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 14.0000 0.582828 0.291414 0.956597i $$-0.405874\pi$$
0.291414 + 0.956597i $$0.405874\pi$$
$$578$$ 0 0
$$579$$ 21.0000 0.872730
$$580$$ 0 0
$$581$$ −14.0000 −0.580818
$$582$$ 0 0
$$583$$ −13.0000 −0.538405
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 39.0000 1.60970 0.804851 0.593477i $$-0.202245\pi$$
0.804851 + 0.593477i $$0.202245\pi$$
$$588$$ 0 0
$$589$$ −25.0000 −1.03011
$$590$$ 0 0
$$591$$ −72.0000 −2.96168
$$592$$ 0 0
$$593$$ 22.0000 0.903432 0.451716 0.892162i $$-0.350812\pi$$
0.451716 + 0.892162i $$0.350812\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.00000 0.122782
$$598$$ 0 0
$$599$$ −27.0000 −1.10319 −0.551595 0.834112i $$-0.685981\pi$$
−0.551595 + 0.834112i $$0.685981\pi$$
$$600$$ 0 0
$$601$$ 14.0000 0.571072 0.285536 0.958368i $$-0.407828\pi$$
0.285536 + 0.958368i $$0.407828\pi$$
$$602$$ 0 0
$$603$$ −96.0000 −3.90942
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 21.0000 0.852364 0.426182 0.904638i $$-0.359858\pi$$
0.426182 + 0.904638i $$0.359858\pi$$
$$608$$ 0 0
$$609$$ −15.0000 −0.607831
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 10.0000 0.403896 0.201948 0.979396i $$-0.435273\pi$$
0.201948 + 0.979396i $$0.435273\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −12.0000 −0.483102 −0.241551 0.970388i $$-0.577656\pi$$
−0.241551 + 0.970388i $$0.577656\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ −18.0000 −0.722315
$$622$$ 0 0
$$623$$ −9.00000 −0.360577
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −15.0000 −0.599042
$$628$$ 0 0
$$629$$ −3.00000 −0.119618
$$630$$ 0 0
$$631$$ 15.0000 0.597141 0.298570 0.954388i $$-0.403490\pi$$
0.298570 + 0.954388i $$0.403490\pi$$
$$632$$ 0 0
$$633$$ 3.00000 0.119239
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −36.0000 −1.42637
$$638$$ 0 0
$$639$$ 90.0000 3.56034
$$640$$ 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
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ −2.00000 −0.0785069
$$650$$ 0 0
$$651$$ 15.0000 0.587896
$$652$$ 0 0
$$653$$ 19.0000 0.743527 0.371764 0.928327i $$-0.378753\pi$$
0.371764 + 0.928327i $$0.378753\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −60.0000 −2.34082
$$658$$ 0 0
$$659$$ 49.0000 1.90877 0.954384 0.298580i $$-0.0965131\pi$$
0.954384 + 0.298580i $$0.0965131\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ 0 0
$$663$$ 54.0000 2.09719
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −10.0000 −0.387202
$$668$$ 0 0
$$669$$ 78.0000 3.01565
$$670$$ 0 0
$$671$$ −1.00000 −0.0386046
$$672$$ 0 0
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ −16.0000 −0.614024
$$680$$ 0 0
$$681$$ 54.0000 2.06928
$$682$$ 0 0
$$683$$ 19.0000 0.727015 0.363507 0.931591i $$-0.381579\pi$$
0.363507 + 0.931591i $$0.381579\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −18.0000 −0.686743
$$688$$ 0 0
$$689$$ 78.0000 2.97156
$$690$$ 0 0
$$691$$ 34.0000 1.29342 0.646710 0.762736i $$-0.276144\pi$$
0.646710 + 0.762736i $$0.276144\pi$$
$$692$$ 0 0
$$693$$ 6.00000 0.227921
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 6.00000 0.227266
$$698$$ 0 0
$$699$$ 33.0000 1.24817
$$700$$ 0 0
$$701$$ −31.0000 −1.17085 −0.585427 0.810725i $$-0.699073\pi$$
−0.585427 + 0.810725i $$0.699073\pi$$
$$702$$ 0 0
$$703$$ −5.00000 −0.188579
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 10.0000 0.376089
$$708$$ 0 0
$$709$$ 46.0000 1.72757 0.863783 0.503864i $$-0.168089\pi$$
0.863783 + 0.503864i $$0.168089\pi$$
$$710$$ 0 0
$$711$$ 12.0000 0.450035
$$712$$ 0 0
$$713$$ 10.0000 0.374503
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 6.00000 0.224074
$$718$$ 0 0
$$719$$ 1.00000 0.0372937 0.0186469 0.999826i $$-0.494064\pi$$
0.0186469 + 0.999826i $$0.494064\pi$$
$$720$$ 0 0
$$721$$ −16.0000 −0.595871
$$722$$ 0 0
$$723$$ −54.0000 −2.00828
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 36.0000 1.33151
$$732$$ 0 0
$$733$$ 32.0000 1.18195 0.590973 0.806691i $$-0.298744\pi$$
0.590973 + 0.806691i $$0.298744\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 16.0000 0.589368
$$738$$ 0 0
$$739$$ −8.00000 −0.294285 −0.147142 0.989115i $$-0.547008\pi$$
−0.147142 + 0.989115i $$0.547008\pi$$
$$740$$ 0 0
$$741$$ 90.0000 3.30623
$$742$$ 0 0
$$743$$ −33.0000 −1.21065 −0.605326 0.795977i $$-0.706957\pi$$
−0.605326 + 0.795977i $$0.706957\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 84.0000 3.07340
$$748$$ 0 0
$$749$$ −4.00000 −0.146157
$$750$$ 0 0
$$751$$ −7.00000 −0.255434 −0.127717 0.991811i $$-0.540765\pi$$
−0.127717 + 0.991811i $$0.540765\pi$$
$$752$$ 0 0
$$753$$ 30.0000 1.09326
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 6.00000 0.217786
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ −10.0000 −0.362024
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 12.0000 0.433295
$$768$$ 0 0
$$769$$ 10.0000 0.360609 0.180305 0.983611i $$-0.442292\pi$$
0.180305 + 0.983611i $$0.442292\pi$$
$$770$$ 0 0
$$771$$ −66.0000 −2.37693
$$772$$ 0 0
$$773$$ 9.00000 0.323708 0.161854 0.986815i $$-0.448253\pi$$
0.161854 + 0.986815i $$0.448253\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 3.00000 0.107624
$$778$$ 0 0
$$779$$ 10.0000 0.358287
$$780$$ 0 0
$$781$$ −15.0000 −0.536742
$$782$$ 0 0
$$783$$ 45.0000 1.60817
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 48.0000 1.71102 0.855508 0.517790i $$-0.173245\pi$$
0.855508 + 0.517790i $$0.173245\pi$$
$$788$$ 0 0
$$789$$ −39.0000 −1.38844
$$790$$ 0 0
$$791$$ −16.0000 −0.568895
$$792$$ 0 0
$$793$$ 6.00000 0.213066
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −34.0000 −1.20434 −0.602171 0.798367i $$-0.705697\pi$$
−0.602171 + 0.798367i $$0.705697\pi$$
$$798$$ 0 0
$$799$$ −6.00000 −0.212265
$$800$$ 0 0
$$801$$ 54.0000 1.90800
$$802$$ 0 0
$$803$$ 10.0000 0.352892
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −60.0000 −2.11210
$$808$$ 0 0
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 1.00000 0.0351147 0.0175574 0.999846i $$-0.494411\pi$$
0.0175574 + 0.999846i $$0.494411\pi$$
$$812$$ 0 0
$$813$$ −72.0000 −2.52515
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 60.0000 2.09913
$$818$$ 0 0
$$819$$ −36.0000 −1.25794
$$820$$ 0 0
$$821$$ 50.0000 1.74501 0.872506 0.488603i $$-0.162493\pi$$
0.872506 + 0.488603i $$0.162493\pi$$
$$822$$ 0 0
$$823$$ −44.0000 −1.53374 −0.766872 0.641800i $$-0.778188\pi$$
−0.766872 + 0.641800i $$0.778188\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 52.0000 1.80822 0.904109 0.427303i $$-0.140536\pi$$
0.904109 + 0.427303i $$0.140536\pi$$
$$828$$ 0 0
$$829$$ 4.00000 0.138926 0.0694629 0.997585i $$-0.477871\pi$$
0.0694629 + 0.997585i $$0.477871\pi$$
$$830$$ 0 0
$$831$$ 48.0000 1.66510
$$832$$ 0 0
$$833$$ 18.0000 0.623663
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −45.0000 −1.55543
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 0 0
$$843$$ −54.0000 −1.85986
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −1.00000 −0.0343604
$$848$$ 0 0
$$849$$ 30.0000 1.02960
$$850$$ 0 0
$$851$$ 2.00000 0.0685591
$$852$$ 0 0
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 21.0000 0.717346 0.358673 0.933463i $$-0.383229\pi$$
0.358673 + 0.933463i $$0.383229\pi$$
$$858$$ 0 0
$$859$$ −22.0000 −0.750630 −0.375315 0.926897i $$-0.622466\pi$$
−0.375315 + 0.926897i $$0.622466\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 46.0000 1.56586 0.782929 0.622111i $$-0.213725\pi$$
0.782929 + 0.622111i $$0.213725\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 24.0000 0.815083
$$868$$ 0 0
$$869$$ −2.00000 −0.0678454
$$870$$ 0 0
$$871$$ −96.0000 −3.25284
$$872$$ 0 0
$$873$$ 96.0000 3.24911
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ 0 0
$$879$$ 18.0000 0.607125
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 0 0
$$883$$ −5.00000 −0.168263 −0.0841317 0.996455i $$-0.526812\pi$$
−0.0841317 + 0.996455i $$0.526812\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 40.0000 1.34307 0.671534 0.740973i $$-0.265636\pi$$
0.671534 + 0.740973i $$0.265636\pi$$
$$888$$ 0 0
$$889$$ −8.00000 −0.268311
$$890$$ 0 0
$$891$$ −9.00000 −0.301511
$$892$$ 0 0
$$893$$ −10.0000 −0.334637
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −36.0000 −1.20201
$$898$$ 0 0
$$899$$ −25.0000 −0.833797
$$900$$ 0 0
$$901$$ −39.0000 −1.29928
$$902$$ 0 0
$$903$$ −36.0000 −1.19800
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 29.0000 0.962929 0.481465 0.876466i $$-0.340105\pi$$
0.481465 + 0.876466i $$0.340105\pi$$
$$908$$ 0 0
$$909$$ −60.0000 −1.99007
$$910$$ 0 0
$$911$$ 3.00000 0.0993944 0.0496972 0.998764i $$-0.484174\pi$$
0.0496972 + 0.998764i $$0.484174\pi$$
$$912$$ 0 0
$$913$$ −14.0000 −0.463332
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −7.00000 −0.231160
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ −66.0000 −2.17477
$$922$$ 0 0
$$923$$ 90.0000 2.96239
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 96.0000 3.15305
$$928$$ 0 0
$$929$$ 59.0000 1.93573 0.967864 0.251476i $$-0.0809159\pi$$
0.967864 + 0.251476i $$0.0809159\pi$$
$$930$$ 0 0
$$931$$ 30.0000 0.983210
$$932$$ 0 0
$$933$$ −63.0000 −2.06253
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ −90.0000 −2.93704
$$940$$ 0 0
$$941$$ −41.0000 −1.33656 −0.668281 0.743909i $$-0.732970\pi$$
−0.668281 + 0.743909i $$0.732970\pi$$
$$942$$ 0 0
$$943$$ −4.00000 −0.130258
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −41.0000 −1.33232 −0.666160 0.745808i $$-0.732063\pi$$
−0.666160 + 0.745808i $$0.732063\pi$$
$$948$$ 0 0
$$949$$ −60.0000 −1.94768
$$950$$ 0 0
$$951$$ −9.00000 −0.291845
$$952$$ 0 0
$$953$$ −29.0000 −0.939402 −0.469701 0.882826i $$-0.655638\pi$$
−0.469701 + 0.882826i $$0.655638\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −15.0000 −0.484881
$$958$$ 0 0
$$959$$ 12.0000 0.387500
$$960$$ 0 0
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ 24.0000 0.773389
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −55.0000 −1.76868 −0.884340 0.466843i $$-0.845391\pi$$
−0.884340 + 0.466843i $$0.845391\pi$$
$$968$$ 0 0
$$969$$ −45.0000 −1.44561
$$970$$ 0 0
$$971$$ 44.0000 1.41203 0.706014 0.708198i $$-0.250492\pi$$
0.706014 + 0.708198i $$0.250492\pi$$
$$972$$ 0 0
$$973$$ 4.00000 0.128234
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 8.00000 0.255943 0.127971 0.991778i $$-0.459153\pi$$
0.127971 + 0.991778i $$0.459153\pi$$
$$978$$ 0 0
$$979$$ −9.00000 −0.287641
$$980$$ 0 0
$$981$$ 60.0000 1.91565
$$982$$ 0 0
$$983$$ 14.0000 0.446531 0.223265 0.974758i $$-0.428328\pi$$
0.223265 + 0.974758i $$0.428328\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 6.00000 0.190982
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 16.0000 0.508257 0.254128 0.967170i $$-0.418211\pi$$
0.254128 + 0.967170i $$0.418211\pi$$
$$992$$ 0 0
$$993$$ 60.0000 1.90404
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −8.00000 −0.253363 −0.126681 0.991943i $$-0.540433\pi$$
−0.126681 + 0.991943i $$0.540433\pi$$
$$998$$ 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.a.1.1 1
4.3 odd 2 4400.2.a.be.1.1 1
5.2 odd 4 2200.2.b.b.1849.2 2
5.3 odd 4 2200.2.b.b.1849.1 2
5.4 even 2 440.2.a.d.1.1 1
15.14 odd 2 3960.2.a.f.1.1 1
20.3 even 4 4400.2.b.a.4049.2 2
20.7 even 4 4400.2.b.a.4049.1 2
20.19 odd 2 880.2.a.a.1.1 1
40.19 odd 2 3520.2.a.bh.1.1 1
40.29 even 2 3520.2.a.a.1.1 1
55.54 odd 2 4840.2.a.i.1.1 1
60.59 even 2 7920.2.a.e.1.1 1
220.219 even 2 9680.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.a.d.1.1 1 5.4 even 2
880.2.a.a.1.1 1 20.19 odd 2
2200.2.a.a.1.1 1 1.1 even 1 trivial
2200.2.b.b.1849.1 2 5.3 odd 4
2200.2.b.b.1849.2 2 5.2 odd 4
3520.2.a.a.1.1 1 40.29 even 2
3520.2.a.bh.1.1 1 40.19 odd 2
3960.2.a.f.1.1 1 15.14 odd 2
4400.2.a.be.1.1 1 4.3 odd 2
4400.2.b.a.4049.1 2 20.7 even 4
4400.2.b.a.4049.2 2 20.3 even 4
4840.2.a.i.1.1 1 55.54 odd 2
7920.2.a.e.1.1 1 60.59 even 2
9680.2.a.a.1.1 1 220.219 even 2