# Properties

 Label 2888.2.a.e.1.1 Level $2888$ Weight $2$ Character 2888.1 Self dual yes Analytic conductor $23.061$ 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] = [2888,2,Mod(1,2888)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2888, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2888.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2888 = 2^{3} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2888.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: $$23.0607961037$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 152) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2888.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +3.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +3.00000 q^{5} -2.00000 q^{9} -4.00000 q^{11} -5.00000 q^{13} +3.00000 q^{15} -5.00000 q^{17} -1.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} +3.00000 q^{29} +4.00000 q^{31} -4.00000 q^{33} +2.00000 q^{37} -5.00000 q^{39} -5.00000 q^{41} -11.0000 q^{43} -6.00000 q^{45} -5.00000 q^{47} -7.00000 q^{49} -5.00000 q^{51} -9.00000 q^{53} -12.0000 q^{55} +13.0000 q^{59} -1.00000 q^{61} -15.0000 q^{65} -5.00000 q^{67} -1.00000 q^{69} +1.00000 q^{71} -9.00000 q^{73} +4.00000 q^{75} +17.0000 q^{79} +1.00000 q^{81} +16.0000 q^{83} -15.0000 q^{85} +3.00000 q^{87} +3.00000 q^{89} +4.00000 q^{93} -13.0000 q^{97} +8.00000 q^{99} +O(q^{100})$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 3.00000 1.34164 0.670820 0.741620i $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ 0 0
$$13$$ −5.00000 −1.38675 −0.693375 0.720577i $$-0.743877\pi$$
−0.693375 + 0.720577i $$0.743877\pi$$
$$14$$ 0 0
$$15$$ 3.00000 0.774597
$$16$$ 0 0
$$17$$ −5.00000 −1.21268 −0.606339 0.795206i $$-0.707363\pi$$
−0.606339 + 0.795206i $$0.707363\pi$$
$$18$$ 0 0
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1.00000 −0.208514 −0.104257 0.994550i $$-0.533247\pi$$
−0.104257 + 0.994550i $$0.533247\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 4.00000 0.718421 0.359211 0.933257i $$-0.383046\pi$$
0.359211 + 0.933257i $$0.383046\pi$$
$$32$$ 0 0
$$33$$ −4.00000 −0.696311
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −5.00000 −0.800641
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −11.0000 −1.67748 −0.838742 0.544529i $$-0.816708\pi$$
−0.838742 + 0.544529i $$0.816708\pi$$
$$44$$ 0 0
$$45$$ −6.00000 −0.894427
$$46$$ 0 0
$$47$$ −5.00000 −0.729325 −0.364662 0.931140i $$-0.618816\pi$$
−0.364662 + 0.931140i $$0.618816\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ −5.00000 −0.700140
$$52$$ 0 0
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ −12.0000 −1.61808
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 13.0000 1.69246 0.846228 0.532821i $$-0.178868\pi$$
0.846228 + 0.532821i $$0.178868\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −15.0000 −1.86052
$$66$$ 0 0
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 0 0
$$69$$ −1.00000 −0.120386
$$70$$ 0 0
$$71$$ 1.00000 0.118678 0.0593391 0.998238i $$-0.481101\pi$$
0.0593391 + 0.998238i $$0.481101\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 17.0000 1.91265 0.956325 0.292306i $$-0.0944227\pi$$
0.956325 + 0.292306i $$0.0944227\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ −15.0000 −1.62698
$$86$$ 0 0
$$87$$ 3.00000 0.321634
$$88$$ 0 0
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 4.00000 0.414781
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −13.0000 −1.31995 −0.659975 0.751288i $$-0.729433\pi$$
−0.659975 + 0.751288i $$0.729433\pi$$
$$98$$ 0 0
$$99$$ 8.00000 0.804030
$$100$$ 0 0
$$101$$ 19.0000 1.89057 0.945285 0.326245i $$-0.105783\pi$$
0.945285 + 0.326245i $$0.105783\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 0 0
$$109$$ 7.00000 0.670478 0.335239 0.942133i $$-0.391183\pi$$
0.335239 + 0.942133i $$0.391183\pi$$
$$110$$ 0 0
$$111$$ 2.00000 0.189832
$$112$$ 0 0
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ −3.00000 −0.279751
$$116$$ 0 0
$$117$$ 10.0000 0.924500
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ −5.00000 −0.450835
$$124$$ 0 0
$$125$$ −3.00000 −0.268328
$$126$$ 0 0
$$127$$ 15.0000 1.33103 0.665517 0.746382i $$-0.268211\pi$$
0.665517 + 0.746382i $$0.268211\pi$$
$$128$$ 0 0
$$129$$ −11.0000 −0.968496
$$130$$ 0 0
$$131$$ −15.0000 −1.31056 −0.655278 0.755388i $$-0.727449\pi$$
−0.655278 + 0.755388i $$0.727449\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −15.0000 −1.29099
$$136$$ 0 0
$$137$$ −5.00000 −0.427179 −0.213589 0.976924i $$-0.568515\pi$$
−0.213589 + 0.976924i $$0.568515\pi$$
$$138$$ 0 0
$$139$$ 15.0000 1.27228 0.636142 0.771572i $$-0.280529\pi$$
0.636142 + 0.771572i $$0.280529\pi$$
$$140$$ 0 0
$$141$$ −5.00000 −0.421076
$$142$$ 0 0
$$143$$ 20.0000 1.67248
$$144$$ 0 0
$$145$$ 9.00000 0.747409
$$146$$ 0 0
$$147$$ −7.00000 −0.577350
$$148$$ 0 0
$$149$$ −17.0000 −1.39269 −0.696347 0.717705i $$-0.745193\pi$$
−0.696347 + 0.717705i $$0.745193\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ 10.0000 0.808452
$$154$$ 0 0
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 0 0
$$159$$ −9.00000 −0.713746
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 0 0
$$165$$ −12.0000 −0.934199
$$166$$ 0 0
$$167$$ −5.00000 −0.386912 −0.193456 0.981109i $$-0.561970\pi$$
−0.193456 + 0.981109i $$0.561970\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −5.00000 −0.380143 −0.190071 0.981770i $$-0.560872\pi$$
−0.190071 + 0.981770i $$0.560872\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 13.0000 0.977140
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 7.00000 0.520306 0.260153 0.965567i $$-0.416227\pi$$
0.260153 + 0.965567i $$0.416227\pi$$
$$182$$ 0 0
$$183$$ −1.00000 −0.0739221
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 0 0
$$187$$ 20.0000 1.46254
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.0000 −1.15772 −0.578860 0.815427i $$-0.696502\pi$$
−0.578860 + 0.815427i $$0.696502\pi$$
$$192$$ 0 0
$$193$$ 15.0000 1.07972 0.539862 0.841754i $$-0.318476\pi$$
0.539862 + 0.841754i $$0.318476\pi$$
$$194$$ 0 0
$$195$$ −15.0000 −1.07417
$$196$$ 0 0
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 0 0
$$199$$ 3.00000 0.212664 0.106332 0.994331i $$-0.466089\pi$$
0.106332 + 0.994331i $$0.466089\pi$$
$$200$$ 0 0
$$201$$ −5.00000 −0.352673
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −15.0000 −1.04765
$$206$$ 0 0
$$207$$ 2.00000 0.139010
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 9.00000 0.619586 0.309793 0.950804i $$-0.399740\pi$$
0.309793 + 0.950804i $$0.399740\pi$$
$$212$$ 0 0
$$213$$ 1.00000 0.0685189
$$214$$ 0 0
$$215$$ −33.0000 −2.25058
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −9.00000 −0.608164
$$220$$ 0 0
$$221$$ 25.0000 1.68168
$$222$$ 0 0
$$223$$ −11.0000 −0.736614 −0.368307 0.929704i $$-0.620063\pi$$
−0.368307 + 0.929704i $$0.620063\pi$$
$$224$$ 0 0
$$225$$ −8.00000 −0.533333
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −22.0000 −1.45380 −0.726900 0.686743i $$-0.759040\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 11.0000 0.720634 0.360317 0.932830i $$-0.382669\pi$$
0.360317 + 0.932830i $$0.382669\pi$$
$$234$$ 0 0
$$235$$ −15.0000 −0.978492
$$236$$ 0 0
$$237$$ 17.0000 1.10427
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −21.0000 −1.35273 −0.676364 0.736567i $$-0.736446\pi$$
−0.676364 + 0.736567i $$0.736446\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ −21.0000 −1.34164
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 16.0000 1.01396
$$250$$ 0 0
$$251$$ −9.00000 −0.568075 −0.284037 0.958813i $$-0.591674\pi$$
−0.284037 + 0.958813i $$0.591674\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ 0 0
$$255$$ −15.0000 −0.939336
$$256$$ 0 0
$$257$$ −25.0000 −1.55946 −0.779729 0.626118i $$-0.784643\pi$$
−0.779729 + 0.626118i $$0.784643\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 0 0
$$263$$ −19.0000 −1.17159 −0.585795 0.810459i $$-0.699218\pi$$
−0.585795 + 0.810459i $$0.699218\pi$$
$$264$$ 0 0
$$265$$ −27.0000 −1.65860
$$266$$ 0 0
$$267$$ 3.00000 0.183597
$$268$$ 0 0
$$269$$ −17.0000 −1.03651 −0.518254 0.855227i $$-0.673418\pi$$
−0.518254 + 0.855227i $$0.673418\pi$$
$$270$$ 0 0
$$271$$ 29.0000 1.76162 0.880812 0.473466i $$-0.156997\pi$$
0.880812 + 0.473466i $$0.156997\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −16.0000 −0.964836
$$276$$ 0 0
$$277$$ −30.0000 −1.80253 −0.901263 0.433273i $$-0.857359\pi$$
−0.901263 + 0.433273i $$0.857359\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 15.0000 0.894825 0.447412 0.894328i $$-0.352346\pi$$
0.447412 + 0.894328i $$0.352346\pi$$
$$282$$ 0 0
$$283$$ −15.0000 −0.891657 −0.445829 0.895118i $$-0.647091\pi$$
−0.445829 + 0.895118i $$0.647091\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −13.0000 −0.762073
$$292$$ 0 0
$$293$$ 10.0000 0.584206 0.292103 0.956387i $$-0.405645\pi$$
0.292103 + 0.956387i $$0.405645\pi$$
$$294$$ 0 0
$$295$$ 39.0000 2.27067
$$296$$ 0 0
$$297$$ 20.0000 1.16052
$$298$$ 0 0
$$299$$ 5.00000 0.289157
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 19.0000 1.09152
$$304$$ 0 0
$$305$$ −3.00000 −0.171780
$$306$$ 0 0
$$307$$ −3.00000 −0.171219 −0.0856095 0.996329i $$-0.527284\pi$$
−0.0856095 + 0.996329i $$0.527284\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ −5.00000 −0.282617 −0.141308 0.989966i $$-0.545131\pi$$
−0.141308 + 0.989966i $$0.545131\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 27.0000 1.51647 0.758236 0.651981i $$-0.226062\pi$$
0.758236 + 0.651981i $$0.226062\pi$$
$$318$$ 0 0
$$319$$ −12.0000 −0.671871
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −20.0000 −1.10940
$$326$$ 0 0
$$327$$ 7.00000 0.387101
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ 0 0
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −15.0000 −0.819538
$$336$$ 0 0
$$337$$ 3.00000 0.163420 0.0817102 0.996656i $$-0.473962\pi$$
0.0817102 + 0.996656i $$0.473962\pi$$
$$338$$ 0 0
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ −3.00000 −0.161515
$$346$$ 0 0
$$347$$ 5.00000 0.268414 0.134207 0.990953i $$-0.457151\pi$$
0.134207 + 0.990953i $$0.457151\pi$$
$$348$$ 0 0
$$349$$ −14.0000 −0.749403 −0.374701 0.927146i $$-0.622255\pi$$
−0.374701 + 0.927146i $$0.622255\pi$$
$$350$$ 0 0
$$351$$ 25.0000 1.33440
$$352$$ 0 0
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 0 0
$$355$$ 3.00000 0.159223
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5.00000 0.263890 0.131945 0.991257i $$-0.457878\pi$$
0.131945 + 0.991257i $$0.457878\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 0 0
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ −27.0000 −1.41324
$$366$$ 0 0
$$367$$ −5.00000 −0.260998 −0.130499 0.991448i $$-0.541658\pi$$
−0.130499 + 0.991448i $$0.541658\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ −3.00000 −0.154919
$$376$$ 0 0
$$377$$ −15.0000 −0.772539
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 15.0000 0.768473
$$382$$ 0 0
$$383$$ −15.0000 −0.766464 −0.383232 0.923652i $$-0.625189\pi$$
−0.383232 + 0.923652i $$0.625189\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 22.0000 1.11832
$$388$$ 0 0
$$389$$ −17.0000 −0.861934 −0.430967 0.902368i $$-0.641828\pi$$
−0.430967 + 0.902368i $$0.641828\pi$$
$$390$$ 0 0
$$391$$ 5.00000 0.252861
$$392$$ 0 0
$$393$$ −15.0000 −0.756650
$$394$$ 0 0
$$395$$ 51.0000 2.56609
$$396$$ 0 0
$$397$$ 35.0000 1.75660 0.878300 0.478110i $$-0.158678\pi$$
0.878300 + 0.478110i $$0.158678\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 11.0000 0.549314 0.274657 0.961542i $$-0.411436\pi$$
0.274657 + 0.961542i $$0.411436\pi$$
$$402$$ 0 0
$$403$$ −20.0000 −0.996271
$$404$$ 0 0
$$405$$ 3.00000 0.149071
$$406$$ 0 0
$$407$$ −8.00000 −0.396545
$$408$$ 0 0
$$409$$ 7.00000 0.346128 0.173064 0.984911i $$-0.444633\pi$$
0.173064 + 0.984911i $$0.444633\pi$$
$$410$$ 0 0
$$411$$ −5.00000 −0.246632
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 48.0000 2.35623
$$416$$ 0 0
$$417$$ 15.0000 0.734553
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −29.0000 −1.41337 −0.706687 0.707527i $$-0.749811\pi$$
−0.706687 + 0.707527i $$0.749811\pi$$
$$422$$ 0 0
$$423$$ 10.0000 0.486217
$$424$$ 0 0
$$425$$ −20.0000 −0.970143
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 20.0000 0.965609
$$430$$ 0 0
$$431$$ −9.00000 −0.433515 −0.216757 0.976226i $$-0.569548\pi$$
−0.216757 + 0.976226i $$0.569548\pi$$
$$432$$ 0 0
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 0 0
$$435$$ 9.00000 0.431517
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −7.00000 −0.334092 −0.167046 0.985949i $$-0.553423\pi$$
−0.167046 + 0.985949i $$0.553423\pi$$
$$440$$ 0 0
$$441$$ 14.0000 0.666667
$$442$$ 0 0
$$443$$ 15.0000 0.712672 0.356336 0.934358i $$-0.384026\pi$$
0.356336 + 0.934358i $$0.384026\pi$$
$$444$$ 0 0
$$445$$ 9.00000 0.426641
$$446$$ 0 0
$$447$$ −17.0000 −0.804072
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 20.0000 0.941763
$$452$$ 0 0
$$453$$ 16.0000 0.751746
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ 25.0000 1.16690
$$460$$ 0 0
$$461$$ −1.00000 −0.0465746 −0.0232873 0.999729i $$-0.507413\pi$$
−0.0232873 + 0.999729i $$0.507413\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 12.0000 0.556487
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −13.0000 −0.599008
$$472$$ 0 0
$$473$$ 44.0000 2.02312
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 18.0000 0.824163
$$478$$ 0 0
$$479$$ 27.0000 1.23366 0.616831 0.787096i $$-0.288416\pi$$
0.616831 + 0.787096i $$0.288416\pi$$
$$480$$ 0 0
$$481$$ −10.0000 −0.455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −39.0000 −1.77090
$$486$$ 0 0
$$487$$ −40.0000 −1.81257 −0.906287 0.422664i $$-0.861095\pi$$
−0.906287 + 0.422664i $$0.861095\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ 9.00000 0.406164 0.203082 0.979162i $$-0.434904\pi$$
0.203082 + 0.979162i $$0.434904\pi$$
$$492$$ 0 0
$$493$$ −15.0000 −0.675566
$$494$$ 0 0
$$495$$ 24.0000 1.07872
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −35.0000 −1.56682 −0.783408 0.621508i $$-0.786520\pi$$
−0.783408 + 0.621508i $$0.786520\pi$$
$$500$$ 0 0
$$501$$ −5.00000 −0.223384
$$502$$ 0 0
$$503$$ 15.0000 0.668817 0.334408 0.942428i $$-0.391463\pi$$
0.334408 + 0.942428i $$0.391463\pi$$
$$504$$ 0 0
$$505$$ 57.0000 2.53647
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ 0 0
$$509$$ −5.00000 −0.221621 −0.110811 0.993842i $$-0.535345\pi$$
−0.110811 + 0.993842i $$0.535345\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 20.0000 0.879599
$$518$$ 0 0
$$519$$ −5.00000 −0.219476
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 0 0
$$523$$ −29.0000 −1.26808 −0.634041 0.773300i $$-0.718605\pi$$
−0.634041 + 0.773300i $$0.718605\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −20.0000 −0.871214
$$528$$ 0 0
$$529$$ −22.0000 −0.956522
$$530$$ 0 0
$$531$$ −26.0000 −1.12830
$$532$$ 0 0
$$533$$ 25.0000 1.08287
$$534$$ 0 0
$$535$$ −36.0000 −1.55642
$$536$$ 0 0
$$537$$ −12.0000 −0.517838
$$538$$ 0 0
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ 11.0000 0.472927 0.236463 0.971640i $$-0.424012\pi$$
0.236463 + 0.971640i $$0.424012\pi$$
$$542$$ 0 0
$$543$$ 7.00000 0.300399
$$544$$ 0 0
$$545$$ 21.0000 0.899541
$$546$$ 0 0
$$547$$ 7.00000 0.299298 0.149649 0.988739i $$-0.452186\pi$$
0.149649 + 0.988739i $$0.452186\pi$$
$$548$$ 0 0
$$549$$ 2.00000 0.0853579
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ −33.0000 −1.39825 −0.699127 0.714997i $$-0.746428\pi$$
−0.699127 + 0.714997i $$0.746428\pi$$
$$558$$ 0 0
$$559$$ 55.0000 2.32625
$$560$$ 0 0
$$561$$ 20.0000 0.844401
$$562$$ 0 0
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 18.0000 0.757266
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ −4.00000 −0.167395 −0.0836974 0.996491i $$-0.526673\pi$$
−0.0836974 + 0.996491i $$0.526673\pi$$
$$572$$ 0 0
$$573$$ −16.0000 −0.668410
$$574$$ 0 0
$$575$$ −4.00000 −0.166812
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 15.0000 0.623379
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ 30.0000 1.24035
$$586$$ 0 0
$$587$$ 25.0000 1.03186 0.515930 0.856631i $$-0.327446\pi$$
0.515930 + 0.856631i $$0.327446\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 0 0
$$593$$ 35.0000 1.43728 0.718639 0.695383i $$-0.244765\pi$$
0.718639 + 0.695383i $$0.244765\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 3.00000 0.122782
$$598$$ 0 0
$$599$$ −25.0000 −1.02147 −0.510736 0.859738i $$-0.670627\pi$$
−0.510736 + 0.859738i $$0.670627\pi$$
$$600$$ 0 0
$$601$$ −34.0000 −1.38689 −0.693444 0.720510i $$-0.743908\pi$$
−0.693444 + 0.720510i $$0.743908\pi$$
$$602$$ 0 0
$$603$$ 10.0000 0.407231
$$604$$ 0 0
$$605$$ 15.0000 0.609837
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 25.0000 1.01139
$$612$$ 0 0
$$613$$ −1.00000 −0.0403896 −0.0201948 0.999796i $$-0.506429\pi$$
−0.0201948 + 0.999796i $$0.506429\pi$$
$$614$$ 0 0
$$615$$ −15.0000 −0.604858
$$616$$ 0 0
$$617$$ 3.00000 0.120775 0.0603877 0.998175i $$-0.480766\pi$$
0.0603877 + 0.998175i $$0.480766\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 5.00000 0.200643
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −10.0000 −0.398726
$$630$$ 0 0
$$631$$ −29.0000 −1.15447 −0.577236 0.816577i $$-0.695869\pi$$
−0.577236 + 0.816577i $$0.695869\pi$$
$$632$$ 0 0
$$633$$ 9.00000 0.357718
$$634$$ 0 0
$$635$$ 45.0000 1.78577
$$636$$ 0 0
$$637$$ 35.0000 1.38675
$$638$$ 0 0
$$639$$ −2.00000 −0.0791188
$$640$$ 0 0
$$641$$ −9.00000 −0.355479 −0.177739 0.984078i $$-0.556878\pi$$
−0.177739 + 0.984078i $$0.556878\pi$$
$$642$$ 0 0
$$643$$ 5.00000 0.197181 0.0985904 0.995128i $$-0.468567\pi$$
0.0985904 + 0.995128i $$0.468567\pi$$
$$644$$ 0 0
$$645$$ −33.0000 −1.29937
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −52.0000 −2.04118
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ −45.0000 −1.75830
$$656$$ 0 0
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ 19.0000 0.740135 0.370067 0.929005i $$-0.379335\pi$$
0.370067 + 0.929005i $$0.379335\pi$$
$$660$$ 0 0
$$661$$ −29.0000 −1.12797 −0.563985 0.825785i $$-0.690732\pi$$
−0.563985 + 0.825785i $$0.690732\pi$$
$$662$$ 0 0
$$663$$ 25.0000 0.970920
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3.00000 −0.116160
$$668$$ 0 0
$$669$$ −11.0000 −0.425285
$$670$$ 0 0
$$671$$ 4.00000 0.154418
$$672$$ 0 0
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ 0 0
$$677$$ −30.0000 −1.15299 −0.576497 0.817099i $$-0.695581\pi$$
−0.576497 + 0.817099i $$0.695581\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ −15.0000 −0.573121
$$686$$ 0 0
$$687$$ −22.0000 −0.839352
$$688$$ 0 0
$$689$$ 45.0000 1.71436
$$690$$ 0 0
$$691$$ 36.0000 1.36950 0.684752 0.728776i $$-0.259910\pi$$
0.684752 + 0.728776i $$0.259910\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 45.0000 1.70695
$$696$$ 0 0
$$697$$ 25.0000 0.946943
$$698$$ 0 0
$$699$$ 11.0000 0.416058
$$700$$ 0 0
$$701$$ 23.0000 0.868698 0.434349 0.900745i $$-0.356978\pi$$
0.434349 + 0.900745i $$0.356978\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −15.0000 −0.564933
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −17.0000 −0.638448 −0.319224 0.947679i $$-0.603422\pi$$
−0.319224 + 0.947679i $$0.603422\pi$$
$$710$$ 0 0
$$711$$ −34.0000 −1.27510
$$712$$ 0 0
$$713$$ −4.00000 −0.149801
$$714$$ 0 0
$$715$$ 60.0000 2.24387
$$716$$ 0 0
$$717$$ 12.0000 0.448148
$$718$$ 0 0
$$719$$ −27.0000 −1.00693 −0.503465 0.864016i $$-0.667942\pi$$
−0.503465 + 0.864016i $$0.667942\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −21.0000 −0.780998
$$724$$ 0 0
$$725$$ 12.0000 0.445669
$$726$$ 0 0
$$727$$ 13.0000 0.482143 0.241072 0.970507i $$-0.422501\pi$$
0.241072 + 0.970507i $$0.422501\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 55.0000 2.03425
$$732$$ 0 0
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 0 0
$$735$$ −21.0000 −0.774597
$$736$$ 0 0
$$737$$ 20.0000 0.736709
$$738$$ 0 0
$$739$$ 37.0000 1.36107 0.680534 0.732717i $$-0.261748\pi$$
0.680534 + 0.732717i $$0.261748\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 29.0000 1.06391 0.531953 0.846774i $$-0.321458\pi$$
0.531953 + 0.846774i $$0.321458\pi$$
$$744$$ 0 0
$$745$$ −51.0000 −1.86850
$$746$$ 0 0
$$747$$ −32.0000 −1.17082
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 31.0000 1.13121 0.565603 0.824678i $$-0.308643\pi$$
0.565603 + 0.824678i $$0.308643\pi$$
$$752$$ 0 0
$$753$$ −9.00000 −0.327978
$$754$$ 0 0
$$755$$ 48.0000 1.74690
$$756$$ 0 0
$$757$$ 35.0000 1.27210 0.636048 0.771649i $$-0.280568\pi$$
0.636048 + 0.771649i $$0.280568\pi$$
$$758$$ 0 0
$$759$$ 4.00000 0.145191
$$760$$ 0 0
$$761$$ 38.0000 1.37750 0.688749 0.724999i $$-0.258160\pi$$
0.688749 + 0.724999i $$0.258160\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 30.0000 1.08465
$$766$$ 0 0
$$767$$ −65.0000 −2.34701
$$768$$ 0 0
$$769$$ 23.0000 0.829401 0.414701 0.909958i $$-0.363886\pi$$
0.414701 + 0.909958i $$0.363886\pi$$
$$770$$ 0 0
$$771$$ −25.0000 −0.900353
$$772$$ 0 0
$$773$$ −25.0000 −0.899188 −0.449594 0.893233i $$-0.648431\pi$$
−0.449594 + 0.893233i $$0.648431\pi$$
$$774$$ 0 0
$$775$$ 16.0000 0.574737
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −4.00000 −0.143131
$$782$$ 0 0
$$783$$ −15.0000 −0.536056
$$784$$ 0 0
$$785$$ −39.0000 −1.39197
$$786$$ 0 0
$$787$$ 20.0000 0.712923 0.356462 0.934310i $$-0.383983\pi$$
0.356462 + 0.934310i $$0.383983\pi$$
$$788$$ 0 0
$$789$$ −19.0000 −0.676418
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 5.00000 0.177555
$$794$$ 0 0
$$795$$ −27.0000 −0.957591
$$796$$ 0 0
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ 25.0000 0.884436
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ 36.0000 1.27041
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −17.0000 −0.598428
$$808$$ 0 0
$$809$$ −42.0000 −1.47664 −0.738321 0.674450i $$-0.764381\pi$$
−0.738321 + 0.674450i $$0.764381\pi$$
$$810$$ 0 0
$$811$$ −21.0000 −0.737410 −0.368705 0.929547i $$-0.620199\pi$$
−0.368705 + 0.929547i $$0.620199\pi$$
$$812$$ 0 0
$$813$$ 29.0000 1.01707
$$814$$ 0 0
$$815$$ −12.0000 −0.420342
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 11.0000 0.383903 0.191951 0.981404i $$-0.438518\pi$$
0.191951 + 0.981404i $$0.438518\pi$$
$$822$$ 0 0
$$823$$ −5.00000 −0.174289 −0.0871445 0.996196i $$-0.527774\pi$$
−0.0871445 + 0.996196i $$0.527774\pi$$
$$824$$ 0 0
$$825$$ −16.0000 −0.557048
$$826$$ 0 0
$$827$$ 15.0000 0.521601 0.260801 0.965393i $$-0.416014\pi$$
0.260801 + 0.965393i $$0.416014\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ −30.0000 −1.04069
$$832$$ 0 0
$$833$$ 35.0000 1.21268
$$834$$ 0 0
$$835$$ −15.0000 −0.519096
$$836$$ 0 0
$$837$$ −20.0000 −0.691301
$$838$$ 0 0
$$839$$ −35.0000 −1.20833 −0.604167 0.796858i $$-0.706494\pi$$
−0.604167 + 0.796858i $$0.706494\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 15.0000 0.516627
$$844$$ 0 0
$$845$$ 36.0000 1.23844
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −15.0000 −0.514799
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ −21.0000 −0.719026 −0.359513 0.933140i $$-0.617057\pi$$
−0.359513 + 0.933140i $$0.617057\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 7.00000 0.239115 0.119558 0.992827i $$-0.461852\pi$$
0.119558 + 0.992827i $$0.461852\pi$$
$$858$$ 0 0
$$859$$ −13.0000 −0.443554 −0.221777 0.975097i $$-0.571186\pi$$
−0.221777 + 0.975097i $$0.571186\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −36.0000 −1.22545 −0.612727 0.790295i $$-0.709928\pi$$
−0.612727 + 0.790295i $$0.709928\pi$$
$$864$$ 0 0
$$865$$ −15.0000 −0.510015
$$866$$ 0 0
$$867$$ 8.00000 0.271694
$$868$$ 0 0
$$869$$ −68.0000 −2.30674
$$870$$ 0 0
$$871$$ 25.0000 0.847093
$$872$$ 0 0
$$873$$ 26.0000 0.879967
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 47.0000 1.58708 0.793539 0.608520i $$-0.208236\pi$$
0.793539 + 0.608520i $$0.208236\pi$$
$$878$$ 0 0
$$879$$ 10.0000 0.337292
$$880$$ 0 0
$$881$$ 14.0000 0.471672 0.235836 0.971793i $$-0.424217\pi$$
0.235836 + 0.971793i $$0.424217\pi$$
$$882$$ 0 0
$$883$$ 55.0000 1.85090 0.925449 0.378873i $$-0.123688\pi$$
0.925449 + 0.378873i $$0.123688\pi$$
$$884$$ 0 0
$$885$$ 39.0000 1.31097
$$886$$ 0 0
$$887$$ −45.0000 −1.51095 −0.755476 0.655176i $$-0.772594\pi$$
−0.755476 + 0.655176i $$0.772594\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ 0 0
$$897$$ 5.00000 0.166945
$$898$$ 0 0
$$899$$ 12.0000 0.400222
$$900$$ 0 0
$$901$$ 45.0000 1.49917
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 21.0000 0.698064
$$906$$ 0 0
$$907$$ −35.0000 −1.16216 −0.581078 0.813848i $$-0.697369\pi$$
−0.581078 + 0.813848i $$0.697369\pi$$
$$908$$ 0 0
$$909$$ −38.0000 −1.26038
$$910$$ 0 0
$$911$$ −56.0000 −1.85536 −0.927681 0.373373i $$-0.878201\pi$$
−0.927681 + 0.373373i $$0.878201\pi$$
$$912$$ 0 0
$$913$$ −64.0000 −2.11809
$$914$$ 0 0
$$915$$ −3.00000 −0.0991769
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −3.00000 −0.0988534
$$922$$ 0 0
$$923$$ −5.00000 −0.164577
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 7.00000 0.229663 0.114831 0.993385i $$-0.463367\pi$$
0.114831 + 0.993385i $$0.463367\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 12.0000 0.392862
$$934$$ 0 0
$$935$$ 60.0000 1.96221
$$936$$ 0 0
$$937$$ −17.0000 −0.555366 −0.277683 0.960673i $$-0.589566\pi$$
−0.277683 + 0.960673i $$0.589566\pi$$
$$938$$ 0 0
$$939$$ −5.00000 −0.163169
$$940$$ 0 0
$$941$$ 39.0000 1.27136 0.635682 0.771951i $$-0.280719\pi$$
0.635682 + 0.771951i $$0.280719\pi$$
$$942$$ 0 0
$$943$$ 5.00000 0.162822
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −35.0000 −1.13735 −0.568674 0.822563i $$-0.692543\pi$$
−0.568674 + 0.822563i $$0.692543\pi$$
$$948$$ 0 0
$$949$$ 45.0000 1.46076
$$950$$ 0 0
$$951$$ 27.0000 0.875535
$$952$$ 0 0
$$953$$ 15.0000 0.485898 0.242949 0.970039i $$-0.421885\pi$$
0.242949 + 0.970039i $$0.421885\pi$$
$$954$$ 0 0
$$955$$ −48.0000 −1.55324
$$956$$ 0 0
$$957$$ −12.0000 −0.387905
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ 24.0000 0.773389
$$964$$ 0 0
$$965$$ 45.0000 1.44860
$$966$$ 0 0
$$967$$ −23.0000 −0.739630 −0.369815 0.929105i $$-0.620579\pi$$
−0.369815 + 0.929105i $$0.620579\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −3.00000 −0.0962746 −0.0481373 0.998841i $$-0.515328\pi$$
−0.0481373 + 0.998841i $$0.515328\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −20.0000 −0.640513
$$976$$ 0 0
$$977$$ 30.0000 0.959785 0.479893 0.877327i $$-0.340676\pi$$
0.479893 + 0.877327i $$0.340676\pi$$
$$978$$ 0 0
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −14.0000 −0.446986
$$982$$ 0 0
$$983$$ 25.0000 0.797376 0.398688 0.917087i $$-0.369466\pi$$
0.398688 + 0.917087i $$0.369466\pi$$
$$984$$ 0 0
$$985$$ 30.0000 0.955879
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 11.0000 0.349780
$$990$$ 0 0
$$991$$ −55.0000 −1.74713 −0.873566 0.486705i $$-0.838199\pi$$
−0.873566 + 0.486705i $$0.838199\pi$$
$$992$$ 0 0
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 9.00000 0.285319
$$996$$ 0 0
$$997$$ −25.0000 −0.791758 −0.395879 0.918303i $$-0.629560\pi$$
−0.395879 + 0.918303i $$0.629560\pi$$
$$998$$ 0 0
$$999$$ −10.0000 −0.316386
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 2888.2.a.e.1.1 1
4.3 odd 2 5776.2.a.h.1.1 1
19.7 even 3 152.2.i.a.49.1 2
19.11 even 3 152.2.i.a.121.1 yes 2
19.18 odd 2 2888.2.a.c.1.1 1
57.11 odd 6 1368.2.s.g.577.1 2
57.26 odd 6 1368.2.s.g.505.1 2
76.7 odd 6 304.2.i.b.49.1 2
76.11 odd 6 304.2.i.b.273.1 2
76.75 even 2 5776.2.a.o.1.1 1
152.11 odd 6 1216.2.i.e.577.1 2
152.45 even 6 1216.2.i.i.961.1 2
152.83 odd 6 1216.2.i.e.961.1 2
152.125 even 6 1216.2.i.i.577.1 2
228.11 even 6 2736.2.s.q.577.1 2
228.83 even 6 2736.2.s.q.1873.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.i.a.49.1 2 19.7 even 3
152.2.i.a.121.1 yes 2 19.11 even 3
304.2.i.b.49.1 2 76.7 odd 6
304.2.i.b.273.1 2 76.11 odd 6
1216.2.i.e.577.1 2 152.11 odd 6
1216.2.i.e.961.1 2 152.83 odd 6
1216.2.i.i.577.1 2 152.125 even 6
1216.2.i.i.961.1 2 152.45 even 6
1368.2.s.g.505.1 2 57.26 odd 6
1368.2.s.g.577.1 2 57.11 odd 6
2736.2.s.q.577.1 2 228.11 even 6
2736.2.s.q.1873.1 2 228.83 even 6
2888.2.a.c.1.1 1 19.18 odd 2
2888.2.a.e.1.1 1 1.1 even 1 trivial
5776.2.a.h.1.1 1 4.3 odd 2
5776.2.a.o.1.1 1 76.75 even 2