# Properties

 Label 1444.2.a.b.1.1 Level $1444$ Weight $2$ Character 1444.1 Self dual yes Analytic conductor $11.530$ 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] = [1444,2,Mod(1,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1444.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1444.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: $$11.5303980519$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1444.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} -1.00000 q^{5} -2.00000 q^{9} -4.00000 q^{11} -1.00000 q^{13} +1.00000 q^{15} +3.00000 q^{17} +5.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +7.00000 q^{29} +4.00000 q^{31} +4.00000 q^{33} +10.0000 q^{37} +1.00000 q^{39} -5.00000 q^{41} -5.00000 q^{43} +2.00000 q^{45} -7.00000 q^{47} -7.00000 q^{49} -3.00000 q^{51} +11.0000 q^{53} +4.00000 q^{55} +3.00000 q^{59} +11.0000 q^{61} +1.00000 q^{65} -3.00000 q^{67} -5.00000 q^{69} +11.0000 q^{71} +15.0000 q^{73} +4.00000 q^{75} -13.0000 q^{79} +1.00000 q^{81} -3.00000 q^{85} -7.00000 q^{87} +3.00000 q^{89} -4.00000 q^{93} -5.00000 q^{97} +8.00000 q^{99} +O(q^{100})$$

## 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 $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214 −0.223607 0.974679i $$-0.571783\pi$$
−0.223607 + 0.974679i $$0.571783\pi$$
$$6$$ 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
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ 1.00000 0.258199
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ 0 0
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 5.00000 1.04257 0.521286 0.853382i $$-0.325452\pi$$
0.521286 + 0.853382i $$0.325452\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ 7.00000 1.29987 0.649934 0.759991i $$-0.274797\pi$$
0.649934 + 0.759991i $$0.274797\pi$$
$$30$$ 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
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 0 0
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ −5.00000 −0.780869 −0.390434 0.920631i $$-0.627675\pi$$
−0.390434 + 0.920631i $$0.627675\pi$$
$$42$$ 0 0
$$43$$ −5.00000 −0.762493 −0.381246 0.924473i $$-0.624505\pi$$
−0.381246 + 0.924473i $$0.624505\pi$$
$$44$$ 0 0
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ −7.00000 −1.02105 −0.510527 0.859861i $$-0.670550\pi$$
−0.510527 + 0.859861i $$0.670550\pi$$
$$48$$ 0 0
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ −3.00000 −0.420084
$$52$$ 0 0
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ 11.0000 1.40841 0.704203 0.709999i $$-0.251305\pi$$
0.704203 + 0.709999i $$0.251305\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ −3.00000 −0.366508 −0.183254 0.983066i $$-0.558663\pi$$
−0.183254 + 0.983066i $$0.558663\pi$$
$$68$$ 0 0
$$69$$ −5.00000 −0.601929
$$70$$ 0 0
$$71$$ 11.0000 1.30546 0.652730 0.757591i $$-0.273624\pi$$
0.652730 + 0.757591i $$0.273624\pi$$
$$72$$ 0 0
$$73$$ 15.0000 1.75562 0.877809 0.479012i $$-0.159005\pi$$
0.877809 + 0.479012i $$0.159005\pi$$
$$74$$ 0 0
$$75$$ 4.00000 0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −13.0000 −1.46261 −0.731307 0.682048i $$-0.761089\pi$$
−0.731307 + 0.682048i $$0.761089\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −3.00000 −0.325396
$$86$$ 0 0
$$87$$ −7.00000 −0.750479
$$88$$ 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
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ 0 0
$$99$$ 8.00000 0.804030
$$100$$ 0 0
$$101$$ −1.00000 −0.0995037 −0.0497519 0.998762i $$-0.515843\pi$$
−0.0497519 + 0.998762i $$0.515843\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 20.0000 1.93347 0.966736 0.255774i $$-0.0823304\pi$$
0.966736 + 0.255774i $$0.0823304\pi$$
$$108$$ 0 0
$$109$$ 3.00000 0.287348 0.143674 0.989625i $$-0.454108\pi$$
0.143674 + 0.989625i $$0.454108\pi$$
$$110$$ 0 0
$$111$$ −10.0000 −0.949158
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 0 0
$$115$$ −5.00000 −0.466252
$$116$$ 0 0
$$117$$ 2.00000 0.184900
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 0 0
$$123$$ 5.00000 0.450835
$$124$$ 0 0
$$125$$ 9.00000 0.804984
$$126$$ 0 0
$$127$$ −3.00000 −0.266207 −0.133103 0.991102i $$-0.542494\pi$$
−0.133103 + 0.991102i $$0.542494\pi$$
$$128$$ 0 0
$$129$$ 5.00000 0.440225
$$130$$ 0 0
$$131$$ 15.0000 1.31056 0.655278 0.755388i $$-0.272551\pi$$
0.655278 + 0.755388i $$0.272551\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −5.00000 −0.430331
$$136$$ 0 0
$$137$$ −5.00000 −0.427179 −0.213589 0.976924i $$-0.568515\pi$$
−0.213589 + 0.976924i $$0.568515\pi$$
$$138$$ 0 0
$$139$$ 9.00000 0.763370 0.381685 0.924292i $$-0.375344\pi$$
0.381685 + 0.924292i $$0.375344\pi$$
$$140$$ 0 0
$$141$$ 7.00000 0.589506
$$142$$ 0 0
$$143$$ 4.00000 0.334497
$$144$$ 0 0
$$145$$ −7.00000 −0.581318
$$146$$ 0 0
$$147$$ 7.00000 0.577350
$$148$$ 0 0
$$149$$ 3.00000 0.245770 0.122885 0.992421i $$-0.460785\pi$$
0.122885 + 0.992421i $$0.460785\pi$$
$$150$$ 0 0
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 7.00000 0.558661 0.279330 0.960195i $$-0.409888\pi$$
0.279330 + 0.960195i $$0.409888\pi$$
$$158$$ 0 0
$$159$$ −11.0000 −0.872357
$$160$$ 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
$$165$$ −4.00000 −0.311400
$$166$$ 0 0
$$167$$ −15.0000 −1.16073 −0.580367 0.814355i $$-0.697091\pi$$
−0.580367 + 0.814355i $$0.697091\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 15.0000 1.14043 0.570214 0.821496i $$-0.306860\pi$$
0.570214 + 0.821496i $$0.306860\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −3.00000 −0.225494
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −5.00000 −0.371647 −0.185824 0.982583i $$-0.559495\pi$$
−0.185824 + 0.982583i $$0.559495\pi$$
$$182$$ 0 0
$$183$$ −11.0000 −0.813143
$$184$$ 0 0
$$185$$ −10.0000 −0.735215
$$186$$ 0 0
$$187$$ −12.0000 −0.877527
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.0000 1.15772 0.578860 0.815427i $$-0.303498\pi$$
0.578860 + 0.815427i $$0.303498\pi$$
$$192$$ 0 0
$$193$$ 15.0000 1.07972 0.539862 0.841754i $$-0.318476\pi$$
0.539862 + 0.841754i $$0.318476\pi$$
$$194$$ 0 0
$$195$$ −1.00000 −0.0716115
$$196$$ 0 0
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ 0 0
$$199$$ −7.00000 −0.496217 −0.248108 0.968732i $$-0.579809\pi$$
−0.248108 + 0.968732i $$0.579809\pi$$
$$200$$ 0 0
$$201$$ 3.00000 0.211604
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 5.00000 0.349215
$$206$$ 0 0
$$207$$ −10.0000 −0.695048
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −9.00000 −0.619586 −0.309793 0.950804i $$-0.600260\pi$$
−0.309793 + 0.950804i $$0.600260\pi$$
$$212$$ 0 0
$$213$$ −11.0000 −0.753708
$$214$$ 0 0
$$215$$ 5.00000 0.340997
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −15.0000 −1.01361
$$220$$ 0 0
$$221$$ −3.00000 −0.201802
$$222$$ 0 0
$$223$$ −25.0000 −1.67412 −0.837062 0.547108i $$-0.815729\pi$$
−0.837062 + 0.547108i $$0.815729\pi$$
$$224$$ 0 0
$$225$$ 8.00000 0.533333
$$226$$ 0 0
$$227$$ −20.0000 −1.32745 −0.663723 0.747978i $$-0.731025\pi$$
−0.663723 + 0.747978i $$0.731025\pi$$
$$228$$ 0 0
$$229$$ 2.00000 0.132164 0.0660819 0.997814i $$-0.478950\pi$$
0.0660819 + 0.997814i $$0.478950\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −21.0000 −1.37576 −0.687878 0.725826i $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$234$$ 0 0
$$235$$ 7.00000 0.456630
$$236$$ 0 0
$$237$$ 13.0000 0.844441
$$238$$ 0 0
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ 19.0000 1.22390 0.611949 0.790897i $$-0.290386\pi$$
0.611949 + 0.790897i $$0.290386\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 7.00000 0.447214
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −31.0000 −1.95670 −0.978351 0.206951i $$-0.933646\pi$$
−0.978351 + 0.206951i $$0.933646\pi$$
$$252$$ 0 0
$$253$$ −20.0000 −1.25739
$$254$$ 0 0
$$255$$ 3.00000 0.187867
$$256$$ 0 0
$$257$$ 23.0000 1.43470 0.717350 0.696713i $$-0.245355\pi$$
0.717350 + 0.696713i $$0.245355\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −14.0000 −0.866578
$$262$$ 0 0
$$263$$ −9.00000 −0.554964 −0.277482 0.960731i $$-0.589500\pi$$
−0.277482 + 0.960731i $$0.589500\pi$$
$$264$$ 0 0
$$265$$ −11.0000 −0.675725
$$266$$ 0 0
$$267$$ −3.00000 −0.183597
$$268$$ 0 0
$$269$$ 27.0000 1.64622 0.823110 0.567883i $$-0.192237\pi$$
0.823110 + 0.567883i $$0.192237\pi$$
$$270$$ 0 0
$$271$$ 31.0000 1.88312 0.941558 0.336851i $$-0.109362\pi$$
0.941558 + 0.336851i $$0.109362\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 16.0000 0.964836
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ 0 0
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 7.00000 0.417585 0.208792 0.977960i $$-0.433047\pi$$
0.208792 + 0.977960i $$0.433047\pi$$
$$282$$ 0 0
$$283$$ −9.00000 −0.534994 −0.267497 0.963559i $$-0.586197\pi$$
−0.267497 + 0.963559i $$0.586197\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 5.00000 0.293105
$$292$$ 0 0
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ 0 0
$$295$$ −3.00000 −0.174667
$$296$$ 0 0
$$297$$ −20.0000 −1.16052
$$298$$ 0 0
$$299$$ −5.00000 −0.289157
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 1.00000 0.0574485
$$304$$ 0 0
$$305$$ −11.0000 −0.629858
$$306$$ 0 0
$$307$$ 27.0000 1.54097 0.770486 0.637457i $$-0.220014\pi$$
0.770486 + 0.637457i $$0.220014\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.0000 −1.13410 −0.567048 0.823685i $$-0.691915\pi$$
−0.567048 + 0.823685i $$0.691915\pi$$
$$312$$ 0 0
$$313$$ 11.0000 0.621757 0.310878 0.950450i $$-0.399377\pi$$
0.310878 + 0.950450i $$0.399377\pi$$
$$314$$ 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
$$319$$ −28.0000 −1.56770
$$320$$ 0 0
$$321$$ −20.0000 −1.11629
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 4.00000 0.221880
$$326$$ 0 0
$$327$$ −3.00000 −0.165900
$$328$$ 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
$$333$$ −20.0000 −1.09599
$$334$$ 0 0
$$335$$ 3.00000 0.163908
$$336$$ 0 0
$$337$$ −5.00000 −0.272367 −0.136184 0.990684i $$-0.543484\pi$$
−0.136184 + 0.990684i $$0.543484\pi$$
$$338$$ 0 0
$$339$$ −14.0000 −0.760376
$$340$$ 0 0
$$341$$ −16.0000 −0.866449
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 5.00000 0.269191
$$346$$ 0 0
$$347$$ −5.00000 −0.268414 −0.134207 0.990953i $$-0.542849\pi$$
−0.134207 + 0.990953i $$0.542849\pi$$
$$348$$ 0 0
$$349$$ 10.0000 0.535288 0.267644 0.963518i $$-0.413755\pi$$
0.267644 + 0.963518i $$0.413755\pi$$
$$350$$ 0 0
$$351$$ −5.00000 −0.266880
$$352$$ 0 0
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ 0 0
$$355$$ −11.0000 −0.583819
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.0000 0.791670 0.395835 0.918322i $$-0.370455\pi$$
0.395835 + 0.918322i $$0.370455\pi$$
$$360$$ 0 0
$$361$$ 0 0
$$362$$ 0 0
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ −15.0000 −0.785136
$$366$$ 0 0
$$367$$ 25.0000 1.30499 0.652495 0.757793i $$-0.273722\pi$$
0.652495 + 0.757793i $$0.273722\pi$$
$$368$$ 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
$$375$$ −9.00000 −0.464758
$$376$$ 0 0
$$377$$ −7.00000 −0.360518
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 3.00000 0.153695
$$382$$ 0 0
$$383$$ −29.0000 −1.48183 −0.740915 0.671598i $$-0.765608\pi$$
−0.740915 + 0.671598i $$0.765608\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 10.0000 0.508329
$$388$$ 0 0
$$389$$ 3.00000 0.152106 0.0760530 0.997104i $$-0.475768\pi$$
0.0760530 + 0.997104i $$0.475768\pi$$
$$390$$ 0 0
$$391$$ 15.0000 0.758583
$$392$$ 0 0
$$393$$ −15.0000 −0.756650
$$394$$ 0 0
$$395$$ 13.0000 0.654101
$$396$$ 0 0
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 19.0000 0.948815 0.474407 0.880305i $$-0.342662\pi$$
0.474407 + 0.880305i $$0.342662\pi$$
$$402$$ 0 0
$$403$$ −4.00000 −0.199254
$$404$$ 0 0
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −40.0000 −1.98273
$$408$$ 0 0
$$409$$ −17.0000 −0.840596 −0.420298 0.907386i $$-0.638074\pi$$
−0.420298 + 0.907386i $$0.638074\pi$$
$$410$$ 0 0
$$411$$ 5.00000 0.246632
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −9.00000 −0.440732
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ −1.00000 −0.0487370 −0.0243685 0.999703i $$-0.507758\pi$$
−0.0243685 + 0.999703i $$0.507758\pi$$
$$422$$ 0 0
$$423$$ 14.0000 0.680703
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ −4.00000 −0.193122
$$430$$ 0 0
$$431$$ 21.0000 1.01153 0.505767 0.862670i $$-0.331209\pi$$
0.505767 + 0.862670i $$0.331209\pi$$
$$432$$ 0 0
$$433$$ −25.0000 −1.20142 −0.600712 0.799466i $$-0.705116\pi$$
−0.600712 + 0.799466i $$0.705116\pi$$
$$434$$ 0 0
$$435$$ 7.00000 0.335624
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −13.0000 −0.620456 −0.310228 0.950662i $$-0.600405\pi$$
−0.310228 + 0.950662i $$0.600405\pi$$
$$440$$ 0 0
$$441$$ 14.0000 0.666667
$$442$$ 0 0
$$443$$ 25.0000 1.18779 0.593893 0.804544i $$-0.297590\pi$$
0.593893 + 0.804544i $$0.297590\pi$$
$$444$$ 0 0
$$445$$ −3.00000 −0.142214
$$446$$ 0 0
$$447$$ −3.00000 −0.141895
$$448$$ 0 0
$$449$$ 22.0000 1.03824 0.519122 0.854700i $$-0.326259\pi$$
0.519122 + 0.854700i $$0.326259\pi$$
$$450$$ 0 0
$$451$$ 20.0000 0.941763
$$452$$ 0 0
$$453$$ 16.0000 0.751746
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −22.0000 −1.02912 −0.514558 0.857455i $$-0.672044\pi$$
−0.514558 + 0.857455i $$0.672044\pi$$
$$458$$ 0 0
$$459$$ 15.0000 0.700140
$$460$$ 0 0
$$461$$ 11.0000 0.512321 0.256161 0.966634i $$-0.417542\pi$$
0.256161 + 0.966634i $$0.417542\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ 0 0
$$465$$ 4.00000 0.185496
$$466$$ 0 0
$$467$$ −20.0000 −0.925490 −0.462745 0.886492i $$-0.653135\pi$$
−0.462745 + 0.886492i $$0.653135\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −7.00000 −0.322543
$$472$$ 0 0
$$473$$ 20.0000 0.919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −22.0000 −1.00731
$$478$$ 0 0
$$479$$ −23.0000 −1.05090 −0.525448 0.850825i $$-0.676102\pi$$
−0.525448 + 0.850825i $$0.676102\pi$$
$$480$$ 0 0
$$481$$ −10.0000 −0.455961
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 5.00000 0.227038
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ −1.00000 −0.0451294 −0.0225647 0.999745i $$-0.507183\pi$$
−0.0225647 + 0.999745i $$0.507183\pi$$
$$492$$ 0 0
$$493$$ 21.0000 0.945792
$$494$$ 0 0
$$495$$ −8.00000 −0.359573
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −5.00000 −0.223831 −0.111915 0.993718i $$-0.535699\pi$$
−0.111915 + 0.993718i $$0.535699\pi$$
$$500$$ 0 0
$$501$$ 15.0000 0.670151
$$502$$ 0 0
$$503$$ 21.0000 0.936344 0.468172 0.883637i $$-0.344913\pi$$
0.468172 + 0.883637i $$0.344913\pi$$
$$504$$ 0 0
$$505$$ 1.00000 0.0444994
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ 0 0
$$509$$ 15.0000 0.664863 0.332432 0.943127i $$-0.392131\pi$$
0.332432 + 0.943127i $$0.392131\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 28.0000 1.23144
$$518$$ 0 0
$$519$$ −15.0000 −0.658427
$$520$$ 0 0
$$521$$ 26.0000 1.13908 0.569540 0.821963i $$-0.307121\pi$$
0.569540 + 0.821963i $$0.307121\pi$$
$$522$$ 0 0
$$523$$ −19.0000 −0.830812 −0.415406 0.909636i $$-0.636360\pi$$
−0.415406 + 0.909636i $$0.636360\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 2.00000 0.0869565
$$530$$ 0 0
$$531$$ −6.00000 −0.260378
$$532$$ 0 0
$$533$$ 5.00000 0.216574
$$534$$ 0 0
$$535$$ −20.0000 −0.864675
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −1.00000 −0.0429934 −0.0214967 0.999769i $$-0.506843\pi$$
−0.0214967 + 0.999769i $$0.506843\pi$$
$$542$$ 0 0
$$543$$ 5.00000 0.214571
$$544$$ 0 0
$$545$$ −3.00000 −0.128506
$$546$$ 0 0
$$547$$ 25.0000 1.06892 0.534461 0.845193i $$-0.320514\pi$$
0.534461 + 0.845193i $$0.320514\pi$$
$$548$$ 0 0
$$549$$ −22.0000 −0.938937
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 10.0000 0.424476
$$556$$ 0 0
$$557$$ −5.00000 −0.211857 −0.105928 0.994374i $$-0.533781\pi$$
−0.105928 + 0.994374i $$0.533781\pi$$
$$558$$ 0 0
$$559$$ 5.00000 0.211477
$$560$$ 0 0
$$561$$ 12.0000 0.506640
$$562$$ 0 0
$$563$$ 36.0000 1.51722 0.758610 0.651546i $$-0.225879\pi$$
0.758610 + 0.651546i $$0.225879\pi$$
$$564$$ 0 0
$$565$$ −14.0000 −0.588984
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 22.0000 0.922288 0.461144 0.887325i $$-0.347439\pi$$
0.461144 + 0.887325i $$0.347439\pi$$
$$570$$ 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
$$575$$ −20.0000 −0.834058
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ −15.0000 −0.623379
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −44.0000 −1.82229
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ 0 0
$$587$$ 15.0000 0.619116 0.309558 0.950881i $$-0.399819\pi$$
0.309558 + 0.950881i $$0.399819\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 0 0
$$593$$ −5.00000 −0.205325 −0.102663 0.994716i $$-0.532736\pi$$
−0.102663 + 0.994716i $$0.532736\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 7.00000 0.286491
$$598$$ 0 0
$$599$$ 45.0000 1.83865 0.919325 0.393499i $$-0.128735\pi$$
0.919325 + 0.393499i $$0.128735\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 6.00000 0.244339
$$604$$ 0 0
$$605$$ −5.00000 −0.203279
$$606$$ 0 0
$$607$$ 32.0000 1.29884 0.649420 0.760430i $$-0.275012\pi$$
0.649420 + 0.760430i $$0.275012\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 7.00000 0.283190
$$612$$ 0 0
$$613$$ −29.0000 −1.17130 −0.585649 0.810564i $$-0.699160\pi$$
−0.585649 + 0.810564i $$0.699160\pi$$
$$614$$ 0 0
$$615$$ −5.00000 −0.201619
$$616$$ 0 0
$$617$$ −45.0000 −1.81163 −0.905816 0.423672i $$-0.860741\pi$$
−0.905816 + 0.423672i $$0.860741\pi$$
$$618$$ 0 0
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 25.0000 1.00322
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 11.0000 0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 30.0000 1.19618
$$630$$ 0 0
$$631$$ 41.0000 1.63218 0.816092 0.577922i $$-0.196136\pi$$
0.816092 + 0.577922i $$0.196136\pi$$
$$632$$ 0 0
$$633$$ 9.00000 0.357718
$$634$$ 0 0
$$635$$ 3.00000 0.119051
$$636$$ 0 0
$$637$$ 7.00000 0.277350
$$638$$ 0 0
$$639$$ −22.0000 −0.870307
$$640$$ 0 0
$$641$$ 39.0000 1.54041 0.770204 0.637798i $$-0.220155\pi$$
0.770204 + 0.637798i $$0.220155\pi$$
$$642$$ 0 0
$$643$$ 19.0000 0.749287 0.374643 0.927169i $$-0.377765\pi$$
0.374643 + 0.927169i $$0.377765\pi$$
$$644$$ 0 0
$$645$$ −5.00000 −0.196875
$$646$$ 0 0
$$647$$ 32.0000 1.25805 0.629025 0.777385i $$-0.283454\pi$$
0.629025 + 0.777385i $$0.283454\pi$$
$$648$$ 0 0
$$649$$ −12.0000 −0.471041
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ −15.0000 −0.586098
$$656$$ 0 0
$$657$$ −30.0000 −1.17041
$$658$$ 0 0
$$659$$ 5.00000 0.194772 0.0973862 0.995247i $$-0.468952\pi$$
0.0973862 + 0.995247i $$0.468952\pi$$
$$660$$ 0 0
$$661$$ 31.0000 1.20576 0.602880 0.797832i $$-0.294020\pi$$
0.602880 + 0.797832i $$0.294020\pi$$
$$662$$ 0 0
$$663$$ 3.00000 0.116510
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 35.0000 1.35521
$$668$$ 0 0
$$669$$ 25.0000 0.966556
$$670$$ 0 0
$$671$$ −44.0000 −1.69860
$$672$$ 0 0
$$673$$ −10.0000 −0.385472 −0.192736 0.981251i $$-0.561736\pi$$
−0.192736 + 0.981251i $$0.561736\pi$$
$$674$$ 0 0
$$675$$ −20.0000 −0.769800
$$676$$ 0 0
$$677$$ 10.0000 0.384331 0.192166 0.981363i $$-0.438449\pi$$
0.192166 + 0.981363i $$0.438449\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 20.0000 0.766402
$$682$$ 0 0
$$683$$ 16.0000 0.612223 0.306111 0.951996i $$-0.400972\pi$$
0.306111 + 0.951996i $$0.400972\pi$$
$$684$$ 0 0
$$685$$ 5.00000 0.191040
$$686$$ 0 0
$$687$$ −2.00000 −0.0763048
$$688$$ 0 0
$$689$$ −11.0000 −0.419067
$$690$$ 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
$$695$$ −9.00000 −0.341389
$$696$$ 0 0
$$697$$ −15.0000 −0.568166
$$698$$ 0 0
$$699$$ 21.0000 0.794293
$$700$$ 0 0
$$701$$ 35.0000 1.32193 0.660966 0.750416i $$-0.270147\pi$$
0.660966 + 0.750416i $$0.270147\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ −7.00000 −0.263635
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 3.00000 0.112667 0.0563337 0.998412i $$-0.482059\pi$$
0.0563337 + 0.998412i $$0.482059\pi$$
$$710$$ 0 0
$$711$$ 26.0000 0.975076
$$712$$ 0 0
$$713$$ 20.0000 0.749006
$$714$$ 0 0
$$715$$ −4.00000 −0.149592
$$716$$ 0 0
$$717$$ −12.0000 −0.448148
$$718$$ 0 0
$$719$$ 7.00000 0.261056 0.130528 0.991445i $$-0.458333\pi$$
0.130528 + 0.991445i $$0.458333\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −19.0000 −0.706618
$$724$$ 0 0
$$725$$ −28.0000 −1.03989
$$726$$ 0 0
$$727$$ 7.00000 0.259616 0.129808 0.991539i $$-0.458564\pi$$
0.129808 + 0.991539i $$0.458564\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −15.0000 −0.554795
$$732$$ 0 0
$$733$$ −30.0000 −1.10808 −0.554038 0.832492i $$-0.686914\pi$$
−0.554038 + 0.832492i $$0.686914\pi$$
$$734$$ 0 0
$$735$$ −7.00000 −0.258199
$$736$$ 0 0
$$737$$ 12.0000 0.442026
$$738$$ 0 0
$$739$$ −13.0000 −0.478213 −0.239106 0.970993i $$-0.576854\pi$$
−0.239106 + 0.970993i $$0.576854\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −25.0000 −0.917161 −0.458581 0.888653i $$-0.651642\pi$$
−0.458581 + 0.888653i $$0.651642\pi$$
$$744$$ 0 0
$$745$$ −3.00000 −0.109911
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 29.0000 1.05823 0.529113 0.848552i $$-0.322525\pi$$
0.529113 + 0.848552i $$0.322525\pi$$
$$752$$ 0 0
$$753$$ 31.0000 1.12970
$$754$$ 0 0
$$755$$ 16.0000 0.582300
$$756$$ 0 0
$$757$$ −17.0000 −0.617876 −0.308938 0.951082i $$-0.599973\pi$$
−0.308938 + 0.951082i $$0.599973\pi$$
$$758$$ 0 0
$$759$$ 20.0000 0.725954
$$760$$ 0 0
$$761$$ 30.0000 1.08750 0.543750 0.839248i $$-0.317004\pi$$
0.543750 + 0.839248i $$0.317004\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 6.00000 0.216930
$$766$$ 0 0
$$767$$ −3.00000 −0.108324
$$768$$ 0 0
$$769$$ 7.00000 0.252426 0.126213 0.992003i $$-0.459718\pi$$
0.126213 + 0.992003i $$0.459718\pi$$
$$770$$ 0 0
$$771$$ −23.0000 −0.828325
$$772$$ 0 0
$$773$$ −21.0000 −0.755318 −0.377659 0.925945i $$-0.623271\pi$$
−0.377659 + 0.925945i $$0.623271\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −44.0000 −1.57444
$$782$$ 0 0
$$783$$ 35.0000 1.25080
$$784$$ 0 0
$$785$$ −7.00000 −0.249841
$$786$$ 0 0
$$787$$ 52.0000 1.85360 0.926800 0.375555i $$-0.122548\pi$$
0.926800 + 0.375555i $$0.122548\pi$$
$$788$$ 0 0
$$789$$ 9.00000 0.320408
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −11.0000 −0.390621
$$794$$ 0 0
$$795$$ 11.0000 0.390130
$$796$$ 0 0
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ −21.0000 −0.742927
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 0 0
$$803$$ −60.0000 −2.11735
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −27.0000 −0.950445
$$808$$ 0 0
$$809$$ −18.0000 −0.632846 −0.316423 0.948618i $$-0.602482\pi$$
−0.316423 + 0.948618i $$0.602482\pi$$
$$810$$ 0 0
$$811$$ 21.0000 0.737410 0.368705 0.929547i $$-0.379801\pi$$
0.368705 + 0.929547i $$0.379801\pi$$
$$812$$ 0 0
$$813$$ −31.0000 −1.08722
$$814$$ 0 0
$$815$$ 4.00000 0.140114
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −9.00000 −0.314102 −0.157051 0.987590i $$-0.550199\pi$$
−0.157051 + 0.987590i $$0.550199\pi$$
$$822$$ 0 0
$$823$$ −31.0000 −1.08059 −0.540296 0.841475i $$-0.681688\pi$$
−0.540296 + 0.841475i $$0.681688\pi$$
$$824$$ 0 0
$$825$$ −16.0000 −0.557048
$$826$$ 0 0
$$827$$ −7.00000 −0.243414 −0.121707 0.992566i $$-0.538837\pi$$
−0.121707 + 0.992566i $$0.538837\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ −10.0000 −0.346896
$$832$$ 0 0
$$833$$ −21.0000 −0.727607
$$834$$ 0 0
$$835$$ 15.0000 0.519096
$$836$$ 0 0
$$837$$ 20.0000 0.691301
$$838$$ 0 0
$$839$$ −25.0000 −0.863096 −0.431548 0.902090i $$-0.642032\pi$$
−0.431548 + 0.902090i $$0.642032\pi$$
$$840$$ 0 0
$$841$$ 20.0000 0.689655
$$842$$ 0 0
$$843$$ −7.00000 −0.241093
$$844$$ 0 0
$$845$$ 12.0000 0.412813
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 9.00000 0.308879
$$850$$ 0 0
$$851$$ 50.0000 1.71398
$$852$$ 0 0
$$853$$ 39.0000 1.33533 0.667667 0.744460i $$-0.267293\pi$$
0.667667 + 0.744460i $$0.267293\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 15.0000 0.512390 0.256195 0.966625i $$-0.417531\pi$$
0.256195 + 0.966625i $$0.417531\pi$$
$$858$$ 0 0
$$859$$ 13.0000 0.443554 0.221777 0.975097i $$-0.428814\pi$$
0.221777 + 0.975097i $$0.428814\pi$$
$$860$$ 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
$$869$$ 52.0000 1.76398
$$870$$ 0 0
$$871$$ 3.00000 0.101651
$$872$$ 0 0
$$873$$ 10.0000 0.338449
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −5.00000 −0.168838 −0.0844190 0.996430i $$-0.526903\pi$$
−0.0844190 + 0.996430i $$0.526903\pi$$
$$878$$ 0 0
$$879$$ 30.0000 1.01187
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ 25.0000 0.841317 0.420658 0.907219i $$-0.361799\pi$$
0.420658 + 0.907219i $$0.361799\pi$$
$$884$$ 0 0
$$885$$ 3.00000 0.100844
$$886$$ 0 0
$$887$$ 25.0000 0.839418 0.419709 0.907659i $$-0.362132\pi$$
0.419709 + 0.907659i $$0.362132\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 12.0000 0.401116
$$896$$ 0 0
$$897$$ 5.00000 0.166945
$$898$$ 0 0
$$899$$ 28.0000 0.933852
$$900$$ 0 0
$$901$$ 33.0000 1.09939
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 5.00000 0.166206
$$906$$ 0 0
$$907$$ 35.0000 1.16216 0.581078 0.813848i $$-0.302631\pi$$
0.581078 + 0.813848i $$0.302631\pi$$
$$908$$ 0 0
$$909$$ 2.00000 0.0663358
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 11.0000 0.363649
$$916$$ 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
$$921$$ −27.0000 −0.889680
$$922$$ 0 0
$$923$$ −11.0000 −0.362069
$$924$$ 0 0
$$925$$ −40.0000 −1.31519
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 23.0000 0.754606 0.377303 0.926090i $$-0.376852\pi$$
0.377303 + 0.926090i $$0.376852\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 20.0000 0.654771
$$934$$ 0 0
$$935$$ 12.0000 0.392442
$$936$$ 0 0
$$937$$ 23.0000 0.751377 0.375689 0.926746i $$-0.377406\pi$$
0.375689 + 0.926746i $$0.377406\pi$$
$$938$$ 0 0
$$939$$ −11.0000 −0.358971
$$940$$ 0 0
$$941$$ 19.0000 0.619382 0.309691 0.950837i $$-0.399774\pi$$
0.309691 + 0.950837i $$0.399774\pi$$
$$942$$ 0 0
$$943$$ −25.0000 −0.814112
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 0 0
$$949$$ −15.0000 −0.486921
$$950$$ 0 0
$$951$$ −15.0000 −0.486408
$$952$$ 0 0
$$953$$ −25.0000 −0.809829 −0.404915 0.914354i $$-0.632699\pi$$
−0.404915 + 0.914354i $$0.632699\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ 28.0000 0.905111
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ 0 0
$$963$$ −40.0000 −1.28898
$$964$$ 0 0
$$965$$ −15.0000 −0.482867
$$966$$ 0 0
$$967$$ −45.0000 −1.44710 −0.723551 0.690271i $$-0.757491\pi$$
−0.723551 + 0.690271i $$0.757491\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 35.0000 1.12320 0.561602 0.827408i $$-0.310185\pi$$
0.561602 + 0.827408i $$0.310185\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −4.00000 −0.128103
$$976$$ 0 0
$$977$$ −10.0000 −0.319928 −0.159964 0.987123i $$-0.551138\pi$$
−0.159964 + 0.987123i $$0.551138\pi$$
$$978$$ 0 0
$$979$$ −12.0000 −0.383522
$$980$$ 0 0
$$981$$ −6.00000 −0.191565
$$982$$ 0 0
$$983$$ −45.0000 −1.43528 −0.717639 0.696416i $$-0.754777\pi$$
−0.717639 + 0.696416i $$0.754777\pi$$
$$984$$ 0 0
$$985$$ −2.00000 −0.0637253
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −25.0000 −0.794954
$$990$$ 0 0
$$991$$ −5.00000 −0.158830 −0.0794151 0.996842i $$-0.525305\pi$$
−0.0794151 + 0.996842i $$0.525305\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 7.00000 0.221915
$$996$$ 0 0
$$997$$ −53.0000 −1.67853 −0.839263 0.543725i $$-0.817013\pi$$
−0.839263 + 0.543725i $$0.817013\pi$$
$$998$$ 0 0
$$999$$ 50.0000 1.58193
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 1444.2.a.b.1.1 1
4.3 odd 2 5776.2.a.k.1.1 1
19.7 even 3 76.2.e.a.49.1 yes 2
19.8 odd 6 1444.2.e.b.653.1 2
19.11 even 3 76.2.e.a.45.1 2
19.12 odd 6 1444.2.e.b.429.1 2
19.18 odd 2 1444.2.a.c.1.1 1
57.11 odd 6 684.2.k.b.577.1 2
57.26 odd 6 684.2.k.b.505.1 2
76.7 odd 6 304.2.i.a.49.1 2
76.11 odd 6 304.2.i.a.273.1 2
76.75 even 2 5776.2.a.f.1.1 1
95.7 odd 12 1900.2.s.a.49.1 4
95.49 even 6 1900.2.i.a.501.1 2
95.64 even 6 1900.2.i.a.201.1 2
95.68 odd 12 1900.2.s.a.349.1 4
95.83 odd 12 1900.2.s.a.49.2 4
95.87 odd 12 1900.2.s.a.349.2 4
152.11 odd 6 1216.2.i.g.577.1 2
152.45 even 6 1216.2.i.c.961.1 2
152.83 odd 6 1216.2.i.g.961.1 2
152.125 even 6 1216.2.i.c.577.1 2
228.11 even 6 2736.2.s.g.577.1 2
228.83 even 6 2736.2.s.g.1873.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
76.2.e.a.45.1 2 19.11 even 3
76.2.e.a.49.1 yes 2 19.7 even 3
304.2.i.a.49.1 2 76.7 odd 6
304.2.i.a.273.1 2 76.11 odd 6
684.2.k.b.505.1 2 57.26 odd 6
684.2.k.b.577.1 2 57.11 odd 6
1216.2.i.c.577.1 2 152.125 even 6
1216.2.i.c.961.1 2 152.45 even 6
1216.2.i.g.577.1 2 152.11 odd 6
1216.2.i.g.961.1 2 152.83 odd 6
1444.2.a.b.1.1 1 1.1 even 1 trivial
1444.2.a.c.1.1 1 19.18 odd 2
1444.2.e.b.429.1 2 19.12 odd 6
1444.2.e.b.653.1 2 19.8 odd 6
1900.2.i.a.201.1 2 95.64 even 6
1900.2.i.a.501.1 2 95.49 even 6
1900.2.s.a.49.1 4 95.7 odd 12
1900.2.s.a.49.2 4 95.83 odd 12
1900.2.s.a.349.1 4 95.68 odd 12
1900.2.s.a.349.2 4 95.87 odd 12
2736.2.s.g.577.1 2 228.11 even 6
2736.2.s.g.1873.1 2 228.83 even 6
5776.2.a.f.1.1 1 76.75 even 2
5776.2.a.k.1.1 1 4.3 odd 2