# Properties

 Label 36.22.a.a.1.1 Level $36$ Weight $22$ Character 36.1 Self dual yes Analytic conductor $100.612$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,22,Mod(1,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 22, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("36.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 36.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: $$100.611843943$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 12) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 36.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.12681e7 q^{5} +2.81914e8 q^{7} +O(q^{10})$$ $$q+1.12681e7 q^{5} +2.81914e8 q^{7} +3.61721e10 q^{11} -4.49099e11 q^{13} -2.12186e12 q^{17} -4.60941e12 q^{19} -9.50953e13 q^{23} -3.49867e14 q^{25} +2.24574e15 q^{29} -3.15569e15 q^{31} +3.17663e15 q^{35} -1.81785e16 q^{37} +1.69650e17 q^{41} -1.58969e17 q^{43} +1.34697e17 q^{47} -4.79070e17 q^{49} +1.56374e16 q^{53} +4.07590e17 q^{55} -2.97724e18 q^{59} +3.60386e18 q^{61} -5.06048e18 q^{65} +2.10662e19 q^{67} -2.19801e19 q^{71} -1.70544e19 q^{73} +1.01974e19 q^{77} -1.15020e20 q^{79} +9.66285e19 q^{83} -2.39093e19 q^{85} -6.04276e19 q^{89} -1.26607e20 q^{91} -5.19392e19 q^{95} -4.07820e20 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 1.12681e7 0.516018 0.258009 0.966142i $$-0.416933\pi$$
0.258009 + 0.966142i $$0.416933\pi$$
$$6$$ 0 0
$$7$$ 2.81914e8 0.377214 0.188607 0.982053i $$-0.439603\pi$$
0.188607 + 0.982053i $$0.439603\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.61721e10 0.420485 0.210242 0.977649i $$-0.432575\pi$$
0.210242 + 0.977649i $$0.432575\pi$$
$$12$$ 0 0
$$13$$ −4.49099e11 −0.903517 −0.451759 0.892140i $$-0.649203\pi$$
−0.451759 + 0.892140i $$0.649203\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −2.12186e12 −0.255272 −0.127636 0.991821i $$-0.540739\pi$$
−0.127636 + 0.991821i $$0.540739\pi$$
$$18$$ 0 0
$$19$$ −4.60941e12 −0.172477 −0.0862387 0.996275i $$-0.527485\pi$$
−0.0862387 + 0.996275i $$0.527485\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −9.50953e13 −0.478648 −0.239324 0.970940i $$-0.576926\pi$$
−0.239324 + 0.970940i $$0.576926\pi$$
$$24$$ 0 0
$$25$$ −3.49867e14 −0.733725
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 2.24574e15 0.991245 0.495622 0.868538i $$-0.334940\pi$$
0.495622 + 0.868538i $$0.334940\pi$$
$$30$$ 0 0
$$31$$ −3.15569e15 −0.691508 −0.345754 0.938325i $$-0.612377\pi$$
−0.345754 + 0.938325i $$0.612377\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 3.17663e15 0.194649
$$36$$ 0 0
$$37$$ −1.81785e16 −0.621498 −0.310749 0.950492i $$-0.600580\pi$$
−0.310749 + 0.950492i $$0.600580\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 1.69650e17 1.97389 0.986944 0.161061i $$-0.0514916\pi$$
0.986944 + 0.161061i $$0.0514916\pi$$
$$42$$ 0 0
$$43$$ −1.58969e17 −1.12174 −0.560870 0.827904i $$-0.689533\pi$$
−0.560870 + 0.827904i $$0.689533\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.34697e17 0.373536 0.186768 0.982404i $$-0.440199\pi$$
0.186768 + 0.982404i $$0.440199\pi$$
$$48$$ 0 0
$$49$$ −4.79070e17 −0.857710
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 1.56374e16 0.0122819 0.00614097 0.999981i $$-0.498045\pi$$
0.00614097 + 0.999981i $$0.498045\pi$$
$$54$$ 0 0
$$55$$ 4.07590e17 0.216978
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −2.97724e18 −0.758347 −0.379173 0.925326i $$-0.623792\pi$$
−0.379173 + 0.925326i $$0.623792\pi$$
$$60$$ 0 0
$$61$$ 3.60386e18 0.646851 0.323425 0.946254i $$-0.395166\pi$$
0.323425 + 0.946254i $$0.395166\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −5.06048e18 −0.466232
$$66$$ 0 0
$$67$$ 2.10662e19 1.41189 0.705945 0.708267i $$-0.250523\pi$$
0.705945 + 0.708267i $$0.250523\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.19801e19 −0.801340 −0.400670 0.916222i $$-0.631223\pi$$
−0.400670 + 0.916222i $$0.631223\pi$$
$$72$$ 0 0
$$73$$ −1.70544e19 −0.464458 −0.232229 0.972661i $$-0.574602\pi$$
−0.232229 + 0.972661i $$0.574602\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.01974e19 0.158613
$$78$$ 0 0
$$79$$ −1.15020e20 −1.36675 −0.683375 0.730067i $$-0.739489\pi$$
−0.683375 + 0.730067i $$0.739489\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 9.66285e19 0.683574 0.341787 0.939777i $$-0.388968\pi$$
0.341787 + 0.939777i $$0.388968\pi$$
$$84$$ 0 0
$$85$$ −2.39093e19 −0.131725
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −6.04276e19 −0.205419 −0.102709 0.994711i $$-0.532751\pi$$
−0.102709 + 0.994711i $$0.532751\pi$$
$$90$$ 0 0
$$91$$ −1.26607e20 −0.340819
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −5.19392e19 −0.0890015
$$96$$ 0 0
$$97$$ −4.07820e20 −0.561520 −0.280760 0.959778i $$-0.590587\pi$$
−0.280760 + 0.959778i $$0.590587\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1.95076e21 −1.75724 −0.878618 0.477525i $$-0.841534\pi$$
−0.878618 + 0.477525i $$0.841534\pi$$
$$102$$ 0 0
$$103$$ 8.98058e20 0.658436 0.329218 0.944254i $$-0.393215\pi$$
0.329218 + 0.944254i $$0.393215\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −3.22013e21 −1.58250 −0.791250 0.611493i $$-0.790569\pi$$
−0.791250 + 0.611493i $$0.790569\pi$$
$$108$$ 0 0
$$109$$ −5.55319e20 −0.224680 −0.112340 0.993670i $$-0.535835\pi$$
−0.112340 + 0.993670i $$0.535835\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −4.00790e21 −1.11069 −0.555346 0.831620i $$-0.687414\pi$$
−0.555346 + 0.831620i $$0.687414\pi$$
$$114$$ 0 0
$$115$$ −1.07154e21 −0.246991
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −5.98182e20 −0.0962920
$$120$$ 0 0
$$121$$ −6.09183e21 −0.823193
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −9.31538e21 −0.894634
$$126$$ 0 0
$$127$$ −1.78249e22 −1.44906 −0.724532 0.689241i $$-0.757944\pi$$
−0.724532 + 0.689241i $$0.757944\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.43009e21 −0.318756 −0.159378 0.987218i $$-0.550949\pi$$
−0.159378 + 0.987218i $$0.550949\pi$$
$$132$$ 0 0
$$133$$ −1.29946e21 −0.0650608
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2.64038e22 −0.968502 −0.484251 0.874929i $$-0.660908\pi$$
−0.484251 + 0.874929i $$0.660908\pi$$
$$138$$ 0 0
$$139$$ −4.41795e22 −1.39176 −0.695882 0.718156i $$-0.744986\pi$$
−0.695882 + 0.718156i $$0.744986\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −1.62448e22 −0.379915
$$144$$ 0 0
$$145$$ 2.53052e22 0.511501
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −1.19466e23 −1.81464 −0.907319 0.420444i $$-0.861874\pi$$
−0.907319 + 0.420444i $$0.861874\pi$$
$$150$$ 0 0
$$151$$ 5.87257e21 0.0775479 0.0387739 0.999248i $$-0.487655\pi$$
0.0387739 + 0.999248i $$0.487655\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −3.55586e22 −0.356831
$$156$$ 0 0
$$157$$ 1.96848e23 1.72657 0.863286 0.504716i $$-0.168403\pi$$
0.863286 + 0.504716i $$0.168403\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −2.68087e22 −0.180553
$$162$$ 0 0
$$163$$ −1.93738e23 −1.14616 −0.573079 0.819500i $$-0.694251\pi$$
−0.573079 + 0.819500i $$0.694251\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2.08785e23 0.957585 0.478792 0.877928i $$-0.341075\pi$$
0.478792 + 0.877928i $$0.341075\pi$$
$$168$$ 0 0
$$169$$ −4.53750e22 −0.183656
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 2.48568e23 0.786975 0.393488 0.919330i $$-0.371268\pi$$
0.393488 + 0.919330i $$0.371268\pi$$
$$174$$ 0 0
$$175$$ −9.86325e22 −0.276771
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 7.15674e23 1.58401 0.792006 0.610513i $$-0.209037\pi$$
0.792006 + 0.610513i $$0.209037\pi$$
$$180$$ 0 0
$$181$$ 2.74196e22 0.0540052 0.0270026 0.999635i $$-0.491404\pi$$
0.0270026 + 0.999635i $$0.491404\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −2.04837e23 −0.320705
$$186$$ 0 0
$$187$$ −7.67521e22 −0.107338
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 1.35303e24 1.51516 0.757580 0.652742i $$-0.226382\pi$$
0.757580 + 0.652742i $$0.226382\pi$$
$$192$$ 0 0
$$193$$ 1.27470e24 1.27955 0.639774 0.768563i $$-0.279028\pi$$
0.639774 + 0.768563i $$0.279028\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 1.36579e23 0.110532 0.0552659 0.998472i $$-0.482399\pi$$
0.0552659 + 0.998472i $$0.482399\pi$$
$$198$$ 0 0
$$199$$ 1.10960e24 0.807625 0.403813 0.914842i $$-0.367685\pi$$
0.403813 + 0.914842i $$0.367685\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 6.33107e23 0.373911
$$204$$ 0 0
$$205$$ 1.91163e24 1.01856
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1.66732e23 −0.0725241
$$210$$ 0 0
$$211$$ −4.25779e23 −0.167579 −0.0837893 0.996483i $$-0.526702\pi$$
−0.0837893 + 0.996483i $$0.526702\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −1.79127e24 −0.578839
$$216$$ 0 0
$$217$$ −8.89635e23 −0.260846
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 9.52924e23 0.230642
$$222$$ 0 0
$$223$$ −1.06482e23 −0.0234463 −0.0117231 0.999931i $$-0.503732\pi$$
−0.0117231 + 0.999931i $$0.503732\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 5.78572e24 1.05703 0.528513 0.848925i $$-0.322750\pi$$
0.528513 + 0.848925i $$0.322750\pi$$
$$228$$ 0 0
$$229$$ −7.95470e24 −1.32541 −0.662707 0.748879i $$-0.730592\pi$$
−0.662707 + 0.748879i $$0.730592\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.84067e24 0.950302 0.475151 0.879904i $$-0.342393\pi$$
0.475151 + 0.879904i $$0.342393\pi$$
$$234$$ 0 0
$$235$$ 1.51778e24 0.192751
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1.71539e25 −1.82467 −0.912335 0.409445i $$-0.865722\pi$$
−0.912335 + 0.409445i $$0.865722\pi$$
$$240$$ 0 0
$$241$$ −6.00591e24 −0.585329 −0.292664 0.956215i $$-0.594542\pi$$
−0.292664 + 0.956215i $$0.594542\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −5.39821e24 −0.442594
$$246$$ 0 0
$$247$$ 2.07008e24 0.155836
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −1.48883e25 −0.946829 −0.473415 0.880840i $$-0.656979\pi$$
−0.473415 + 0.880840i $$0.656979\pi$$
$$252$$ 0 0
$$253$$ −3.43979e24 −0.201264
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −1.76815e25 −0.877445 −0.438723 0.898622i $$-0.644569\pi$$
−0.438723 + 0.898622i $$0.644569\pi$$
$$258$$ 0 0
$$259$$ −5.12478e24 −0.234438
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3.25801e25 −1.26887 −0.634434 0.772977i $$-0.718767\pi$$
−0.634434 + 0.772977i $$0.718767\pi$$
$$264$$ 0 0
$$265$$ 1.76203e23 0.00633771
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −2.09117e25 −0.642674 −0.321337 0.946965i $$-0.604132\pi$$
−0.321337 + 0.946965i $$0.604132\pi$$
$$270$$ 0 0
$$271$$ 1.57378e25 0.447474 0.223737 0.974650i $$-0.428174\pi$$
0.223737 + 0.974650i $$0.428174\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1.26554e25 −0.308520
$$276$$ 0 0
$$277$$ −1.23463e25 −0.278934 −0.139467 0.990227i $$-0.544539\pi$$
−0.139467 + 0.990227i $$0.544539\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5.51786e25 1.07239 0.536197 0.844093i $$-0.319860\pi$$
0.536197 + 0.844093i $$0.319860\pi$$
$$282$$ 0 0
$$283$$ 1.45585e25 0.262639 0.131320 0.991340i $$-0.458079\pi$$
0.131320 + 0.991340i $$0.458079\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 4.78267e25 0.744578
$$288$$ 0 0
$$289$$ −6.45896e25 −0.934836
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.86675e25 0.359153 0.179576 0.983744i $$-0.442527\pi$$
0.179576 + 0.983744i $$0.442527\pi$$
$$294$$ 0 0
$$295$$ −3.35478e25 −0.391321
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 4.27072e25 0.432467
$$300$$ 0 0
$$301$$ −4.48155e25 −0.423136
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 4.06086e25 0.333787
$$306$$ 0 0
$$307$$ 8.31398e25 0.638052 0.319026 0.947746i $$-0.396644\pi$$
0.319026 + 0.947746i $$0.396644\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 1.80388e26 1.20843 0.604217 0.796820i $$-0.293486\pi$$
0.604217 + 0.796820i $$0.293486\pi$$
$$312$$ 0 0
$$313$$ 2.33157e26 1.46027 0.730133 0.683305i $$-0.239458\pi$$
0.730133 + 0.683305i $$0.239458\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −8.51585e25 −0.466773 −0.233387 0.972384i $$-0.574981\pi$$
−0.233387 + 0.972384i $$0.574981\pi$$
$$318$$ 0 0
$$319$$ 8.12332e25 0.416803
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 9.78051e24 0.0440286
$$324$$ 0 0
$$325$$ 1.57125e26 0.662933
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 3.79731e25 0.140903
$$330$$ 0 0
$$331$$ 5.68513e25 0.197946 0.0989730 0.995090i $$-0.468444\pi$$
0.0989730 + 0.995090i $$0.468444\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 2.37376e26 0.728561
$$336$$ 0 0
$$337$$ 2.08109e26 0.600034 0.300017 0.953934i $$-0.403008\pi$$
0.300017 + 0.953934i $$0.403008\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −1.14148e26 −0.290768
$$342$$ 0 0
$$343$$ −2.92519e26 −0.700754
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 5.50754e26 1.16815 0.584075 0.811700i $$-0.301457\pi$$
0.584075 + 0.811700i $$0.301457\pi$$
$$348$$ 0 0
$$349$$ −2.04674e25 −0.0408692 −0.0204346 0.999791i $$-0.506505\pi$$
−0.0204346 + 0.999791i $$0.506505\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −5.92034e26 −1.04885 −0.524424 0.851458i $$-0.675719\pi$$
−0.524424 + 0.851458i $$0.675719\pi$$
$$354$$ 0 0
$$355$$ −2.47674e26 −0.413506
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 5.22516e26 0.775547 0.387773 0.921755i $$-0.373244\pi$$
0.387773 + 0.921755i $$0.373244\pi$$
$$360$$ 0 0
$$361$$ −6.92963e26 −0.970252
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.92171e26 −0.239669
$$366$$ 0 0
$$367$$ −1.15652e27 −1.36195 −0.680974 0.732308i $$-0.738443\pi$$
−0.680974 + 0.732308i $$0.738443\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 4.40840e24 0.00463292
$$372$$ 0 0
$$373$$ 2.94328e26 0.292340 0.146170 0.989259i $$-0.453305\pi$$
0.146170 + 0.989259i $$0.453305\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −1.00856e27 −0.895607
$$378$$ 0 0
$$379$$ 1.24279e27 1.04397 0.521984 0.852955i $$-0.325192\pi$$
0.521984 + 0.852955i $$0.325192\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −2.18301e27 −1.64236 −0.821181 0.570668i $$-0.806684\pi$$
−0.821181 + 0.570668i $$0.806684\pi$$
$$384$$ 0 0
$$385$$ 1.14905e26 0.0818470
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −1.52211e27 −0.972690 −0.486345 0.873767i $$-0.661670\pi$$
−0.486345 + 0.873767i $$0.661670\pi$$
$$390$$ 0 0
$$391$$ 2.01779e26 0.122185
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −1.29606e27 −0.705269
$$396$$ 0 0
$$397$$ −6.32566e26 −0.326441 −0.163221 0.986590i $$-0.552188\pi$$
−0.163221 + 0.986590i $$0.552188\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.76087e26 0.174692 0.0873459 0.996178i $$-0.472161\pi$$
0.0873459 + 0.996178i $$0.472161\pi$$
$$402$$ 0 0
$$403$$ 1.41722e27 0.624789
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6.57554e26 −0.261331
$$408$$ 0 0
$$409$$ −3.84109e27 −1.44997 −0.724986 0.688764i $$-0.758154\pi$$
−0.724986 + 0.688764i $$0.758154\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −8.39326e26 −0.286059
$$414$$ 0 0
$$415$$ 1.08882e27 0.352737
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 2.15090e27 0.630048 0.315024 0.949084i $$-0.397987\pi$$
0.315024 + 0.949084i $$0.397987\pi$$
$$420$$ 0 0
$$421$$ −1.87739e27 −0.523109 −0.261554 0.965189i $$-0.584235\pi$$
−0.261554 + 0.965189i $$0.584235\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 7.42369e26 0.187299
$$426$$ 0 0
$$427$$ 1.01598e27 0.244001
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5.68016e27 1.23694 0.618471 0.785807i $$-0.287752\pi$$
0.618471 + 0.785807i $$0.287752\pi$$
$$432$$ 0 0
$$433$$ −2.55781e27 −0.530574 −0.265287 0.964170i $$-0.585467\pi$$
−0.265287 + 0.964170i $$0.585467\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 4.38333e26 0.0825560
$$438$$ 0 0
$$439$$ 4.04971e27 0.727020 0.363510 0.931590i $$-0.381578\pi$$
0.363510 + 0.931590i $$0.381578\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 1.15784e28 1.88978 0.944890 0.327388i $$-0.106169\pi$$
0.944890 + 0.327388i $$0.106169\pi$$
$$444$$ 0 0
$$445$$ −6.80903e26 −0.106000
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 1.09657e27 0.155399 0.0776997 0.996977i $$-0.475242\pi$$
0.0776997 + 0.996977i $$0.475242\pi$$
$$450$$ 0 0
$$451$$ 6.13658e27 0.829990
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1.42662e27 −0.175869
$$456$$ 0 0
$$457$$ −1.08169e27 −0.127345 −0.0636725 0.997971i $$-0.520281\pi$$
−0.0636725 + 0.997971i $$0.520281\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1.20305e27 −0.129248 −0.0646238 0.997910i $$-0.520585\pi$$
−0.0646238 + 0.997910i $$0.520585\pi$$
$$462$$ 0 0
$$463$$ 9.71985e27 0.997836 0.498918 0.866649i $$-0.333731\pi$$
0.498918 + 0.866649i $$0.333731\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −1.01684e27 −0.0953729 −0.0476865 0.998862i $$-0.515185\pi$$
−0.0476865 + 0.998862i $$0.515185\pi$$
$$468$$ 0 0
$$469$$ 5.93886e27 0.532584
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5.75022e27 −0.471675
$$474$$ 0 0
$$475$$ 1.61268e27 0.126551
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 5.80013e27 0.416788 0.208394 0.978045i $$-0.433176\pi$$
0.208394 + 0.978045i $$0.433176\pi$$
$$480$$ 0 0
$$481$$ 8.16394e27 0.561535
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.59535e27 −0.289755
$$486$$ 0 0
$$487$$ −7.92907e27 −0.478815 −0.239408 0.970919i $$-0.576953\pi$$
−0.239408 + 0.970919i $$0.576953\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 1.94270e28 1.07659 0.538294 0.842757i $$-0.319069\pi$$
0.538294 + 0.842757i $$0.319069\pi$$
$$492$$ 0 0
$$493$$ −4.76515e27 −0.253037
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −6.19650e27 −0.302276
$$498$$ 0 0
$$499$$ −7.12364e27 −0.333155 −0.166578 0.986028i $$-0.553272\pi$$
−0.166578 + 0.986028i $$0.553272\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.58499e28 1.54179 0.770893 0.636964i $$-0.219810\pi$$
0.770893 + 0.636964i $$0.219810\pi$$
$$504$$ 0 0
$$505$$ −2.19814e28 −0.906766
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −4.53859e28 −1.72339 −0.861696 0.507425i $$-0.830598\pi$$
−0.861696 + 0.507425i $$0.830598\pi$$
$$510$$ 0 0
$$511$$ −4.80788e27 −0.175200
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 1.01194e28 0.339765
$$516$$ 0 0
$$517$$ 4.87229e27 0.157066
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −4.13279e28 −1.22870 −0.614352 0.789032i $$-0.710583\pi$$
−0.614352 + 0.789032i $$0.710583\pi$$
$$522$$ 0 0
$$523$$ −2.57605e28 −0.735675 −0.367837 0.929890i $$-0.619902\pi$$
−0.367837 + 0.929890i $$0.619902\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 6.69594e27 0.176522
$$528$$ 0 0
$$529$$ −3.04285e28 −0.770896
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −7.61895e28 −1.78344
$$534$$ 0 0
$$535$$ −3.62847e28 −0.816599
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.73290e28 −0.360654
$$540$$ 0 0
$$541$$ 5.00618e28 1.00216 0.501078 0.865402i $$-0.332937\pi$$
0.501078 + 0.865402i $$0.332937\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6.25739e27 −0.115939
$$546$$ 0 0
$$547$$ 7.69499e28 1.37196 0.685979 0.727621i $$-0.259374\pi$$
0.685979 + 0.727621i $$0.259374\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −1.03515e28 −0.170967
$$552$$ 0 0
$$553$$ −3.24258e28 −0.515557
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −9.27070e28 −1.36657 −0.683287 0.730150i $$-0.739450\pi$$
−0.683287 + 0.730150i $$0.739450\pi$$
$$558$$ 0 0
$$559$$ 7.13926e28 1.01351
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 3.62203e28 0.477104 0.238552 0.971130i $$-0.423327\pi$$
0.238552 + 0.971130i $$0.423327\pi$$
$$564$$ 0 0
$$565$$ −4.51614e28 −0.573137
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 1.10807e28 0.130583 0.0652917 0.997866i $$-0.479202\pi$$
0.0652917 + 0.997866i $$0.479202\pi$$
$$570$$ 0 0
$$571$$ 1.89713e28 0.215485 0.107743 0.994179i $$-0.465638\pi$$
0.107743 + 0.994179i $$0.465638\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 3.32707e28 0.351196
$$576$$ 0 0
$$577$$ −1.05096e29 −1.06965 −0.534825 0.844963i $$-0.679623\pi$$
−0.534825 + 0.844963i $$0.679623\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 2.72409e28 0.257853
$$582$$ 0 0
$$583$$ 5.65636e26 0.00516437
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 1.07068e29 0.909825 0.454912 0.890536i $$-0.349671\pi$$
0.454912 + 0.890536i $$0.349671\pi$$
$$588$$ 0 0
$$589$$ 1.45459e28 0.119269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 2.26566e29 1.73029 0.865147 0.501519i $$-0.167225\pi$$
0.865147 + 0.501519i $$0.167225\pi$$
$$594$$ 0 0
$$595$$ −6.74037e27 −0.0496885
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 1.58466e29 1.08882 0.544408 0.838821i $$-0.316755\pi$$
0.544408 + 0.838821i $$0.316755\pi$$
$$600$$ 0 0
$$601$$ −1.75299e29 −1.16304 −0.581522 0.813530i $$-0.697543\pi$$
−0.581522 + 0.813530i $$0.697543\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −6.86433e28 −0.424783
$$606$$ 0 0
$$607$$ 1.39496e29 0.833834 0.416917 0.908945i $$-0.363111\pi$$
0.416917 + 0.908945i $$0.363111\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6.04924e28 −0.337496
$$612$$ 0 0
$$613$$ 1.79163e29 0.965858 0.482929 0.875660i $$-0.339573\pi$$
0.482929 + 0.875660i $$0.339573\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 1.93960e29 0.976604 0.488302 0.872675i $$-0.337616\pi$$
0.488302 + 0.872675i $$0.337616\pi$$
$$618$$ 0 0
$$619$$ 1.29272e29 0.629150 0.314575 0.949233i $$-0.398138\pi$$
0.314575 + 0.949233i $$0.398138\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −1.70354e28 −0.0774868
$$624$$ 0 0
$$625$$ 6.18632e28 0.272077
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3.85722e28 0.158651
$$630$$ 0 0
$$631$$ 1.99938e29 0.795401 0.397701 0.917515i $$-0.369808\pi$$
0.397701 + 0.917515i $$0.369808\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −2.00852e29 −0.747744
$$636$$ 0 0
$$637$$ 2.15150e29 0.774956
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.98808e29 0.670539 0.335269 0.942122i $$-0.391173\pi$$
0.335269 + 0.942122i $$0.391173\pi$$
$$642$$ 0 0
$$643$$ −1.68097e29 −0.548712 −0.274356 0.961628i $$-0.588465\pi$$
−0.274356 + 0.961628i $$0.588465\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −8.67978e27 −0.0265469 −0.0132735 0.999912i $$-0.504225\pi$$
−0.0132735 + 0.999912i $$0.504225\pi$$
$$648$$ 0 0
$$649$$ −1.07693e29 −0.318873
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −5.07947e29 −1.41003 −0.705017 0.709190i $$-0.749061\pi$$
−0.705017 + 0.709190i $$0.749061\pi$$
$$654$$ 0 0
$$655$$ −6.11868e28 −0.164484
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3.42808e28 −0.0864477 −0.0432239 0.999065i $$-0.513763\pi$$
−0.0432239 + 0.999065i $$0.513763\pi$$
$$660$$ 0 0
$$661$$ 5.03536e28 0.123003 0.0615015 0.998107i $$-0.480411\pi$$
0.0615015 + 0.998107i $$0.480411\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −1.46424e28 −0.0335726
$$666$$ 0 0
$$667$$ −2.13559e29 −0.474458
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1.30359e29 0.271991
$$672$$ 0 0
$$673$$ −4.22880e29 −0.855182 −0.427591 0.903972i $$-0.640638\pi$$
−0.427591 + 0.903972i $$0.640638\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 2.94132e27 0.00558936 0.00279468 0.999996i $$-0.499110\pi$$
0.00279468 + 0.999996i $$0.499110\pi$$
$$678$$ 0 0
$$679$$ −1.14970e29 −0.211813
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 2.09159e29 0.362293 0.181147 0.983456i $$-0.442019\pi$$
0.181147 + 0.983456i $$0.442019\pi$$
$$684$$ 0 0
$$685$$ −2.97520e29 −0.499765
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −7.02272e27 −0.0110970
$$690$$ 0 0
$$691$$ −7.10396e29 −1.08888 −0.544442 0.838799i $$-0.683258\pi$$
−0.544442 + 0.838799i $$0.683258\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −4.97819e29 −0.718176
$$696$$ 0 0
$$697$$ −3.59973e29 −0.503878
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8.10616e29 −1.06850 −0.534252 0.845325i $$-0.679407\pi$$
−0.534252 + 0.845325i $$0.679407\pi$$
$$702$$ 0 0
$$703$$ 8.37921e28 0.107194
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −5.49948e29 −0.662854
$$708$$ 0 0
$$709$$ 1.31827e30 1.54248 0.771241 0.636544i $$-0.219637\pi$$
0.771241 + 0.636544i $$0.219637\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 3.00092e29 0.330989
$$714$$ 0 0
$$715$$ −1.83048e29 −0.196043
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −1.04204e30 −1.05252 −0.526262 0.850323i $$-0.676407\pi$$
−0.526262 + 0.850323i $$0.676407\pi$$
$$720$$ 0 0
$$721$$ 2.53175e29 0.248371
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −7.85712e29 −0.727301
$$726$$ 0 0
$$727$$ −1.06635e30 −0.958933 −0.479467 0.877560i $$-0.659170\pi$$
−0.479467 + 0.877560i $$0.659170\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 3.37309e29 0.286349
$$732$$ 0 0
$$733$$ 2.08122e30 1.71682 0.858412 0.512961i $$-0.171451\pi$$
0.858412 + 0.512961i $$0.171451\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 7.62008e29 0.593678
$$738$$ 0 0
$$739$$ 1.80814e29 0.136920 0.0684598 0.997654i $$-0.478192\pi$$
0.0684598 + 0.997654i $$0.478192\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −4.87745e29 −0.348988 −0.174494 0.984658i $$-0.555829\pi$$
−0.174494 + 0.984658i $$0.555829\pi$$
$$744$$ 0 0
$$745$$ −1.34616e30 −0.936386
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −9.07800e29 −0.596941
$$750$$ 0 0
$$751$$ −9.22610e29 −0.589928 −0.294964 0.955508i $$-0.595308\pi$$
−0.294964 + 0.955508i $$0.595308\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 6.61726e28 0.0400161
$$756$$ 0 0
$$757$$ 1.92184e30 1.13035 0.565173 0.824972i $$-0.308809\pi$$
0.565173 + 0.824972i $$0.308809\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 3.51722e30 1.95732 0.978658 0.205496i $$-0.0658806\pi$$
0.978658 + 0.205496i $$0.0658806\pi$$
$$762$$ 0 0
$$763$$ −1.56552e29 −0.0847524
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1.33707e30 0.685180
$$768$$ 0 0
$$769$$ 1.78079e29 0.0887946 0.0443973 0.999014i $$-0.485863\pi$$
0.0443973 + 0.999014i $$0.485863\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 4.21079e30 1.98829 0.994143 0.108071i $$-0.0344673\pi$$
0.994143 + 0.108071i $$0.0344673\pi$$
$$774$$ 0 0
$$775$$ 1.10407e30 0.507376
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −7.81985e29 −0.340451
$$780$$ 0 0
$$781$$ −7.95066e29 −0.336951
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 2.21810e30 0.890943
$$786$$ 0 0
$$787$$ 3.65263e29 0.142847 0.0714235 0.997446i $$-0.477246\pi$$
0.0714235 + 0.997446i $$0.477246\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −1.12988e30 −0.418968
$$792$$ 0 0
$$793$$ −1.61849e30 −0.584441
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 3.15540e30 1.08079 0.540396 0.841411i $$-0.318274\pi$$
0.540396 + 0.841411i $$0.318274\pi$$
$$798$$ 0 0
$$799$$ −2.85809e29 −0.0953531
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6.16894e29 −0.195298
$$804$$ 0 0
$$805$$ −3.02083e29 −0.0931686
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 1.77681e30 0.520214 0.260107 0.965580i $$-0.416242\pi$$
0.260107 + 0.965580i $$0.416242\pi$$
$$810$$ 0 0
$$811$$ 3.55440e30 1.01402 0.507010 0.861940i $$-0.330751\pi$$
0.507010 + 0.861940i $$0.330751\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −2.18305e30 −0.591439
$$816$$ 0 0
$$817$$ 7.32751e29 0.193475
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −2.53502e30 −0.635884 −0.317942 0.948110i $$-0.602992\pi$$
−0.317942 + 0.948110i $$0.602992\pi$$
$$822$$ 0 0
$$823$$ −1.71174e30 −0.418543 −0.209272 0.977858i $$-0.567109\pi$$
−0.209272 + 0.977858i $$0.567109\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.61940e30 −1.53819 −0.769097 0.639132i $$-0.779294\pi$$
−0.769097 + 0.639132i $$0.779294\pi$$
$$828$$ 0 0
$$829$$ 6.52629e30 1.47858 0.739288 0.673389i $$-0.235162\pi$$
0.739288 + 0.673389i $$0.235162\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 1.01652e30 0.218949
$$834$$ 0 0
$$835$$ 2.35261e30 0.494131
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −3.94428e30 −0.787894 −0.393947 0.919133i $$-0.628891\pi$$
−0.393947 + 0.919133i $$0.628891\pi$$
$$840$$ 0 0
$$841$$ −8.94839e28 −0.0174336
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −5.11290e29 −0.0947701
$$846$$ 0 0
$$847$$ −1.71737e30 −0.310519
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 1.72869e30 0.297479
$$852$$ 0 0
$$853$$ −1.15149e31 −1.93329 −0.966644 0.256124i $$-0.917554\pi$$
−0.966644 + 0.256124i $$0.917554\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −2.63232e29 −0.0420766 −0.0210383 0.999779i $$-0.506697\pi$$
−0.0210383 + 0.999779i $$0.506697\pi$$
$$858$$ 0 0
$$859$$ 2.57894e30 0.402266 0.201133 0.979564i $$-0.435538\pi$$
0.201133 + 0.979564i $$0.435538\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 3.33890e30 0.496009 0.248005 0.968759i $$-0.420225\pi$$
0.248005 + 0.968759i $$0.420225\pi$$
$$864$$ 0 0
$$865$$ 2.80089e30 0.406094
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −4.16052e30 −0.574698
$$870$$ 0 0
$$871$$ −9.46080e30 −1.27567
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −2.62614e30 −0.337468
$$876$$ 0 0
$$877$$ −1.94355e30 −0.243838 −0.121919 0.992540i $$-0.538905\pi$$
−0.121919 + 0.992540i $$0.538905\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −3.05871e30 −0.365840 −0.182920 0.983128i $$-0.558555\pi$$
−0.182920 + 0.983128i $$0.558555\pi$$
$$882$$ 0 0
$$883$$ −1.12080e31 −1.30901 −0.654503 0.756060i $$-0.727122\pi$$
−0.654503 + 0.756060i $$0.727122\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −5.65586e30 −0.629942 −0.314971 0.949101i $$-0.601995\pi$$
−0.314971 + 0.949101i $$0.601995\pi$$
$$888$$ 0 0
$$889$$ −5.02509e30 −0.546607
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −6.20875e29 −0.0644264
$$894$$ 0 0
$$895$$ 8.06428e30 0.817380
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −7.08687e30 −0.685453
$$900$$ 0 0
$$901$$ −3.31803e28 −0.00313523
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3.08966e29 0.0278677
$$906$$ 0 0
$$907$$ −1.28972e31 −1.13663 −0.568314 0.822812i $$-0.692404\pi$$
−0.568314 + 0.822812i $$0.692404\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 1.11830e31 0.941058 0.470529 0.882384i $$-0.344063\pi$$
0.470529 + 0.882384i $$0.344063\pi$$
$$912$$ 0 0
$$913$$ 3.49525e30 0.287432
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1.53082e30 −0.120239
$$918$$ 0 0
$$919$$ 1.99044e31 1.52804 0.764022 0.645190i $$-0.223222\pi$$
0.764022 + 0.645190i $$0.223222\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 9.87123e30 0.724025
$$924$$ 0 0
$$925$$ 6.36006e30 0.456009
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −2.46924e31 −1.69199 −0.845995 0.533190i $$-0.820993\pi$$
−0.845995 + 0.533190i $$0.820993\pi$$
$$930$$ 0 0
$$931$$ 2.20823e30 0.147936
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ −8.64849e29 −0.0553883
$$936$$ 0 0
$$937$$ 2.80473e31 1.75641 0.878204 0.478286i $$-0.158742\pi$$
0.878204 + 0.478286i $$0.158742\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −8.99005e30 −0.538358 −0.269179 0.963090i $$-0.586752\pi$$
−0.269179 + 0.963090i $$0.586752\pi$$
$$942$$ 0 0
$$943$$ −1.61329e31 −0.944799
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.54170e31 1.42380 0.711901 0.702280i $$-0.247834\pi$$
0.711901 + 0.702280i $$0.247834\pi$$
$$948$$ 0 0
$$949$$ 7.65911e30 0.419646
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 9.08338e30 0.476181 0.238090 0.971243i $$-0.423479\pi$$
0.238090 + 0.971243i $$0.423479\pi$$
$$954$$ 0 0
$$955$$ 1.52461e31 0.781851
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −7.44360e30 −0.365332
$$960$$ 0 0
$$961$$ −1.08671e31 −0.521817
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 1.43635e31 0.660271
$$966$$ 0 0
$$967$$ −1.32705e31 −0.596909 −0.298455 0.954424i $$-0.596471\pi$$
−0.298455 + 0.954424i $$0.596471\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 3.95025e31 1.70146 0.850732 0.525600i $$-0.176159\pi$$
0.850732 + 0.525600i $$0.176159\pi$$
$$972$$ 0 0
$$973$$ −1.24548e31 −0.524992
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −1.04812e31 −0.423172 −0.211586 0.977359i $$-0.567863\pi$$
−0.211586 + 0.977359i $$0.567863\pi$$
$$978$$ 0 0
$$979$$ −2.18579e30 −0.0863755
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 4.33201e31 1.64013 0.820063 0.572273i $$-0.193938\pi$$
0.820063 + 0.572273i $$0.193938\pi$$
$$984$$ 0 0
$$985$$ 1.53898e30 0.0570364
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.51172e31 0.536919
$$990$$ 0 0
$$991$$ 2.71476e31 0.943968 0.471984 0.881607i $$-0.343538\pi$$
0.471984 + 0.881607i $$0.343538\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1.25031e31 0.416750
$$996$$ 0 0
$$997$$ −5.18699e31 −1.69284 −0.846421 0.532515i $$-0.821247\pi$$
−0.846421 + 0.532515i $$0.821247\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 36.22.a.a.1.1 1
3.2 odd 2 12.22.a.a.1.1 1
12.11 even 2 48.22.a.b.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
12.22.a.a.1.1 1 3.2 odd 2
36.22.a.a.1.1 1 1.1 even 1 trivial
48.22.a.b.1.1 1 12.11 even 2