# Properties

 Label 48.22.a.b.1.1 Level $48$ Weight $22$ Character 48.1 Self dual yes Analytic conductor $134.149$ 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] = [48,22,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$134.149125258$$ Analytic rank: $$0$$ 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 $$\chi$$ $$=$$ 48.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-59049.0 q^{3} -1.12681e7 q^{5} -2.81914e8 q^{7} +3.48678e9 q^{9} +O(q^{10})$$ $$q-59049.0 q^{3} -1.12681e7 q^{5} -2.81914e8 q^{7} +3.48678e9 q^{9} +3.61721e10 q^{11} -4.49099e11 q^{13} +6.65369e11 q^{15} +2.12186e12 q^{17} +4.60941e12 q^{19} +1.66467e13 q^{21} -9.50953e13 q^{23} -3.49867e14 q^{25} -2.05891e14 q^{27} -2.24574e15 q^{29} +3.15569e15 q^{31} -2.13593e15 q^{33} +3.17663e15 q^{35} -1.81785e16 q^{37} +2.65188e16 q^{39} -1.69650e17 q^{41} +1.58969e17 q^{43} -3.92894e16 q^{45} +1.34697e17 q^{47} -4.79070e17 q^{49} -1.25294e17 q^{51} -1.56374e16 q^{53} -4.07590e17 q^{55} -2.72181e17 q^{57} -2.97724e18 q^{59} +3.60386e18 q^{61} -9.82974e17 q^{63} +5.06048e18 q^{65} -2.10662e19 q^{67} +5.61528e18 q^{69} -2.19801e19 q^{71} -1.70544e19 q^{73} +2.06593e19 q^{75} -1.01974e19 q^{77} +1.15020e20 q^{79} +1.21577e19 q^{81} +9.66285e19 q^{83} -2.39093e19 q^{85} +1.32609e20 q^{87} +6.04276e19 q^{89} +1.26607e20 q^{91} -1.86341e20 q^{93} -5.19392e19 q^{95} -4.07820e20 q^{97} +1.26124e20 q^{99} +O(q^{100})$$

## 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
$$3$$ −59049.0 −0.577350
$$4$$ 0 0
$$5$$ −1.12681e7 −0.516018 −0.258009 0.966142i $$-0.583067\pi$$
−0.258009 + 0.966142i $$0.583067\pi$$
$$6$$ 0 0
$$7$$ −2.81914e8 −0.377214 −0.188607 0.982053i $$-0.560397\pi$$
−0.188607 + 0.982053i $$0.560397\pi$$
$$8$$ 0 0
$$9$$ 3.48678e9 0.333333
$$10$$ 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
$$15$$ 6.65369e11 0.297923
$$16$$ 0 0
$$17$$ 2.12186e12 0.255272 0.127636 0.991821i $$-0.459261\pi$$
0.127636 + 0.991821i $$0.459261\pi$$
$$18$$ 0 0
$$19$$ 4.60941e12 0.172477 0.0862387 0.996275i $$-0.472515\pi$$
0.0862387 + 0.996275i $$0.472515\pi$$
$$20$$ 0 0
$$21$$ 1.66467e13 0.217784
$$22$$ 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
$$27$$ −2.05891e14 −0.192450
$$28$$ 0 0
$$29$$ −2.24574e15 −0.991245 −0.495622 0.868538i $$-0.665060\pi$$
−0.495622 + 0.868538i $$0.665060\pi$$
$$30$$ 0 0
$$31$$ 3.15569e15 0.691508 0.345754 0.938325i $$-0.387623\pi$$
0.345754 + 0.938325i $$0.387623\pi$$
$$32$$ 0 0
$$33$$ −2.13593e15 −0.242767
$$34$$ 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
$$39$$ 2.65188e16 0.521646
$$40$$ 0 0
$$41$$ −1.69650e17 −1.97389 −0.986944 0.161061i $$-0.948508\pi$$
−0.986944 + 0.161061i $$0.948508\pi$$
$$42$$ 0 0
$$43$$ 1.58969e17 1.12174 0.560870 0.827904i $$-0.310467\pi$$
0.560870 + 0.827904i $$0.310467\pi$$
$$44$$ 0 0
$$45$$ −3.92894e16 −0.172006
$$46$$ 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
$$51$$ −1.25294e17 −0.147381
$$52$$ 0 0
$$53$$ −1.56374e16 −0.0122819 −0.00614097 0.999981i $$-0.501955\pi$$
−0.00614097 + 0.999981i $$0.501955\pi$$
$$54$$ 0 0
$$55$$ −4.07590e17 −0.216978
$$56$$ 0 0
$$57$$ −2.72181e17 −0.0995799
$$58$$ 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
$$63$$ −9.82974e17 −0.125738
$$64$$ 0 0
$$65$$ 5.06048e18 0.466232
$$66$$ 0 0
$$67$$ −2.10662e19 −1.41189 −0.705945 0.708267i $$-0.749477\pi$$
−0.705945 + 0.708267i $$0.749477\pi$$
$$68$$ 0 0
$$69$$ 5.61528e18 0.276348
$$70$$ 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
$$75$$ 2.06593e19 0.423616
$$76$$ 0 0
$$77$$ −1.01974e19 −0.158613
$$78$$ 0 0
$$79$$ 1.15020e20 1.36675 0.683375 0.730067i $$-0.260511\pi$$
0.683375 + 0.730067i $$0.260511\pi$$
$$80$$ 0 0
$$81$$ 1.21577e19 0.111111
$$82$$ 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
$$87$$ 1.32609e20 0.572295
$$88$$ 0 0
$$89$$ 6.04276e19 0.205419 0.102709 0.994711i $$-0.467249\pi$$
0.102709 + 0.994711i $$0.467249\pi$$
$$90$$ 0 0
$$91$$ 1.26607e20 0.340819
$$92$$ 0 0
$$93$$ −1.86341e20 −0.399242
$$94$$ 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
$$99$$ 1.26124e20 0.140162
$$100$$ 0 0
$$101$$ 1.95076e21 1.75724 0.878618 0.477525i $$-0.158466\pi$$
0.878618 + 0.477525i $$0.158466\pi$$
$$102$$ 0 0
$$103$$ −8.98058e20 −0.658436 −0.329218 0.944254i $$-0.606785\pi$$
−0.329218 + 0.944254i $$0.606785\pi$$
$$104$$ 0 0
$$105$$ −1.87577e20 −0.112381
$$106$$ 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
$$111$$ 1.07342e21 0.358822
$$112$$ 0 0
$$113$$ 4.00790e21 1.11069 0.555346 0.831620i $$-0.312586\pi$$
0.555346 + 0.831620i $$0.312586\pi$$
$$114$$ 0 0
$$115$$ 1.07154e21 0.246991
$$116$$ 0 0
$$117$$ −1.56591e21 −0.301172
$$118$$ 0 0
$$119$$ −5.98182e20 −0.0962920
$$120$$ 0 0
$$121$$ −6.09183e21 −0.823193
$$122$$ 0 0
$$123$$ 1.00176e22 1.13963
$$124$$ 0 0
$$125$$ 9.31538e21 0.894634
$$126$$ 0 0
$$127$$ 1.78249e22 1.44906 0.724532 0.689241i $$-0.242056\pi$$
0.724532 + 0.689241i $$0.242056\pi$$
$$128$$ 0 0
$$129$$ −9.38693e21 −0.647637
$$130$$ 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
$$135$$ 2.32000e21 0.0993078
$$136$$ 0 0
$$137$$ 2.64038e22 0.968502 0.484251 0.874929i $$-0.339092\pi$$
0.484251 + 0.874929i $$0.339092\pi$$
$$138$$ 0 0
$$139$$ 4.41795e22 1.39176 0.695882 0.718156i $$-0.255014\pi$$
0.695882 + 0.718156i $$0.255014\pi$$
$$140$$ 0 0
$$141$$ −7.95375e21 −0.215661
$$142$$ 0 0
$$143$$ −1.62448e22 −0.379915
$$144$$ 0 0
$$145$$ 2.53052e22 0.511501
$$146$$ 0 0
$$147$$ 2.82886e22 0.495199
$$148$$ 0 0
$$149$$ 1.19466e23 1.81464 0.907319 0.420444i $$-0.138126\pi$$
0.907319 + 0.420444i $$0.138126\pi$$
$$150$$ 0 0
$$151$$ −5.87257e21 −0.0775479 −0.0387739 0.999248i $$-0.512345\pi$$
−0.0387739 + 0.999248i $$0.512345\pi$$
$$152$$ 0 0
$$153$$ 7.39846e21 0.0850906
$$154$$ 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
$$159$$ 9.23371e20 0.00709099
$$160$$ 0 0
$$161$$ 2.68087e22 0.180553
$$162$$ 0 0
$$163$$ 1.93738e23 1.14616 0.573079 0.819500i $$-0.305749\pi$$
0.573079 + 0.819500i $$0.305749\pi$$
$$164$$ 0 0
$$165$$ 2.40678e22 0.125272
$$166$$ 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
$$171$$ 1.60720e22 0.0574925
$$172$$ 0 0
$$173$$ −2.48568e23 −0.786975 −0.393488 0.919330i $$-0.628732\pi$$
−0.393488 + 0.919330i $$0.628732\pi$$
$$174$$ 0 0
$$175$$ 9.86325e22 0.276771
$$176$$ 0 0
$$177$$ 1.75803e23 0.437832
$$178$$ 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
$$183$$ −2.12804e23 −0.373459
$$184$$ 0 0
$$185$$ 2.04837e23 0.320705
$$186$$ 0 0
$$187$$ 7.67521e22 0.107338
$$188$$ 0 0
$$189$$ 5.80436e22 0.0725948
$$190$$ 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
$$195$$ −2.98816e23 −0.269179
$$196$$ 0 0
$$197$$ −1.36579e23 −0.110532 −0.0552659 0.998472i $$-0.517601\pi$$
−0.0552659 + 0.998472i $$0.517601\pi$$
$$198$$ 0 0
$$199$$ −1.10960e24 −0.807625 −0.403813 0.914842i $$-0.632315\pi$$
−0.403813 + 0.914842i $$0.632315\pi$$
$$200$$ 0 0
$$201$$ 1.24394e24 0.815155
$$202$$ 0 0
$$203$$ 6.33107e23 0.373911
$$204$$ 0 0
$$205$$ 1.91163e24 1.01856
$$206$$ 0 0
$$207$$ −3.31577e23 −0.159549
$$208$$ 0 0
$$209$$ 1.66732e23 0.0725241
$$210$$ 0 0
$$211$$ 4.25779e23 0.167579 0.0837893 0.996483i $$-0.473298\pi$$
0.0837893 + 0.996483i $$0.473298\pi$$
$$212$$ 0 0
$$213$$ 1.29790e24 0.462654
$$214$$ 0 0
$$215$$ −1.79127e24 −0.578839
$$216$$ 0 0
$$217$$ −8.89635e23 −0.260846
$$218$$ 0 0
$$219$$ 1.00705e24 0.268155
$$220$$ 0 0
$$221$$ −9.52924e23 −0.230642
$$222$$ 0 0
$$223$$ 1.06482e23 0.0234463 0.0117231 0.999931i $$-0.496268\pi$$
0.0117231 + 0.999931i $$0.496268\pi$$
$$224$$ 0 0
$$225$$ −1.21991e24 −0.244575
$$226$$ 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
$$231$$ 6.02148e23 0.0915750
$$232$$ 0 0
$$233$$ −6.84067e24 −0.950302 −0.475151 0.879904i $$-0.657607\pi$$
−0.475151 + 0.879904i $$0.657607\pi$$
$$234$$ 0 0
$$235$$ −1.51778e24 −0.192751
$$236$$ 0 0
$$237$$ −6.79182e24 −0.789094
$$238$$ 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
$$243$$ −7.17898e23 −0.0641500
$$244$$ 0 0
$$245$$ 5.39821e24 0.442594
$$246$$ 0 0
$$247$$ −2.07008e24 −0.155836
$$248$$ 0 0
$$249$$ −5.70582e24 −0.394661
$$250$$ 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
$$255$$ 1.41182e24 0.0760514
$$256$$ 0 0
$$257$$ 1.76815e25 0.877445 0.438723 0.898622i $$-0.355431\pi$$
0.438723 + 0.898622i $$0.355431\pi$$
$$258$$ 0 0
$$259$$ 5.12478e24 0.234438
$$260$$ 0 0
$$261$$ −7.83042e24 −0.330415
$$262$$ 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
$$267$$ −3.56819e24 −0.118599
$$268$$ 0 0
$$269$$ 2.09117e25 0.642674 0.321337 0.946965i $$-0.395868\pi$$
0.321337 + 0.946965i $$0.395868\pi$$
$$270$$ 0 0
$$271$$ −1.57378e25 −0.447474 −0.223737 0.974650i $$-0.571826\pi$$
−0.223737 + 0.974650i $$0.571826\pi$$
$$272$$ 0 0
$$273$$ −7.47603e24 −0.196772
$$274$$ 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
$$279$$ 1.10032e25 0.230503
$$280$$ 0 0
$$281$$ −5.51786e25 −1.07239 −0.536197 0.844093i $$-0.680140\pi$$
−0.536197 + 0.844093i $$0.680140\pi$$
$$282$$ 0 0
$$283$$ −1.45585e25 −0.262639 −0.131320 0.991340i $$-0.541921\pi$$
−0.131320 + 0.991340i $$0.541921\pi$$
$$284$$ 0 0
$$285$$ 3.06696e24 0.0513851
$$286$$ 0 0
$$287$$ 4.78267e25 0.744578
$$288$$ 0 0
$$289$$ −6.45896e25 −0.934836
$$290$$ 0 0
$$291$$ 2.40814e25 0.324194
$$292$$ 0 0
$$293$$ −2.86675e25 −0.359153 −0.179576 0.983744i $$-0.557473\pi$$
−0.179576 + 0.983744i $$0.557473\pi$$
$$294$$ 0 0
$$295$$ 3.35478e25 0.391321
$$296$$ 0 0
$$297$$ −7.44751e24 −0.0809223
$$298$$ 0 0
$$299$$ 4.27072e25 0.432467
$$300$$ 0 0
$$301$$ −4.48155e25 −0.423136
$$302$$ 0 0
$$303$$ −1.15191e26 −1.01454
$$304$$ 0 0
$$305$$ −4.06086e25 −0.333787
$$306$$ 0 0
$$307$$ −8.31398e25 −0.638052 −0.319026 0.947746i $$-0.603356\pi$$
−0.319026 + 0.947746i $$0.603356\pi$$
$$308$$ 0 0
$$309$$ 5.30294e25 0.380148
$$310$$ 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
$$315$$ 1.10762e25 0.0648831
$$316$$ 0 0
$$317$$ 8.51585e25 0.466773 0.233387 0.972384i $$-0.425019\pi$$
0.233387 + 0.972384i $$0.425019\pi$$
$$318$$ 0 0
$$319$$ −8.12332e25 −0.416803
$$320$$ 0 0
$$321$$ 1.90145e26 0.913657
$$322$$ 0 0
$$323$$ 9.78051e24 0.0440286
$$324$$ 0 0
$$325$$ 1.57125e26 0.662933
$$326$$ 0 0
$$327$$ 3.27910e25 0.129719
$$328$$ 0 0
$$329$$ −3.79731e25 −0.140903
$$330$$ 0 0
$$331$$ −5.68513e25 −0.197946 −0.0989730 0.995090i $$-0.531556\pi$$
−0.0989730 + 0.995090i $$0.531556\pi$$
$$332$$ 0 0
$$333$$ −6.33845e25 −0.207166
$$334$$ 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
$$339$$ −2.36663e26 −0.641258
$$340$$ 0 0
$$341$$ 1.14148e26 0.290768
$$342$$ 0 0
$$343$$ 2.92519e26 0.700754
$$344$$ 0 0
$$345$$ −6.32735e25 −0.142601
$$346$$ 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
$$351$$ 9.24654e25 0.173882
$$352$$ 0 0
$$353$$ 5.92034e26 1.04885 0.524424 0.851458i $$-0.324281\pi$$
0.524424 + 0.851458i $$0.324281\pi$$
$$354$$ 0 0
$$355$$ 2.47674e26 0.413506
$$356$$ 0 0
$$357$$ 3.53220e25 0.0555942
$$358$$ 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
$$363$$ 3.59716e26 0.475270
$$364$$ 0 0
$$365$$ 1.92171e26 0.239669
$$366$$ 0 0
$$367$$ 1.15652e27 1.36195 0.680974 0.732308i $$-0.261557\pi$$
0.680974 + 0.732308i $$0.261557\pi$$
$$368$$ 0 0
$$369$$ −5.91532e26 −0.657963
$$370$$ 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
$$375$$ −5.50064e26 −0.516517
$$376$$ 0 0
$$377$$ 1.00856e27 0.895607
$$378$$ 0 0
$$379$$ −1.24279e27 −1.04397 −0.521984 0.852955i $$-0.674808\pi$$
−0.521984 + 0.852955i $$0.674808\pi$$
$$380$$ 0 0
$$381$$ −1.05254e27 −0.836618
$$382$$ 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
$$387$$ 5.54289e26 0.373914
$$388$$ 0 0
$$389$$ 1.52211e27 0.972690 0.486345 0.873767i $$-0.338330\pi$$
0.486345 + 0.873767i $$0.338330\pi$$
$$390$$ 0 0
$$391$$ −2.01779e26 −0.122185
$$392$$ 0 0
$$393$$ 3.20642e26 0.184034
$$394$$ 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
$$399$$ 7.67316e25 0.0375629
$$400$$ 0 0
$$401$$ −3.76087e26 −0.174692 −0.0873459 0.996178i $$-0.527839\pi$$
−0.0873459 + 0.996178i $$0.527839\pi$$
$$402$$ 0 0
$$403$$ −1.41722e27 −0.624789
$$404$$ 0 0
$$405$$ −1.36994e26 −0.0573354
$$406$$ 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
$$411$$ −1.55912e27 −0.559165
$$412$$ 0 0
$$413$$ 8.39326e26 0.286059
$$414$$ 0 0
$$415$$ −1.08882e27 −0.352737
$$416$$ 0 0
$$417$$ −2.60876e27 −0.803535
$$418$$ 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
$$423$$ 4.69661e26 0.124512
$$424$$ 0 0
$$425$$ −7.42369e26 −0.187299
$$426$$ 0 0
$$427$$ −1.01598e27 −0.244001
$$428$$ 0 0
$$429$$ 9.59241e26 0.219344
$$430$$ 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
$$435$$ −1.49425e27 −0.295315
$$436$$ 0 0
$$437$$ −4.38333e26 −0.0825560
$$438$$ 0 0
$$439$$ −4.04971e27 −0.727020 −0.363510 0.931590i $$-0.618422\pi$$
−0.363510 + 0.931590i $$0.618422\pi$$
$$440$$ 0 0
$$441$$ −1.67041e27 −0.285903
$$442$$ 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
$$447$$ −7.05437e27 −1.04768
$$448$$ 0 0
$$449$$ −1.09657e27 −0.155399 −0.0776997 0.996977i $$-0.524758\pi$$
−0.0776997 + 0.996977i $$0.524758\pi$$
$$450$$ 0 0
$$451$$ −6.13658e27 −0.829990
$$452$$ 0 0
$$453$$ 3.46769e26 0.0447723
$$454$$ 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
$$459$$ −4.36872e26 −0.0491271
$$460$$ 0 0
$$461$$ 1.20305e27 0.129248 0.0646238 0.997910i $$-0.479415\pi$$
0.0646238 + 0.997910i $$0.479415\pi$$
$$462$$ 0 0
$$463$$ −9.71985e27 −0.997836 −0.498918 0.866649i $$-0.666269\pi$$
−0.498918 + 0.866649i $$0.666269\pi$$
$$464$$ 0 0
$$465$$ 2.09970e27 0.206016
$$466$$ 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
$$471$$ −1.16237e28 −0.996836
$$472$$ 0 0
$$473$$ 5.75022e27 0.471675
$$474$$ 0 0
$$475$$ −1.61268e27 −0.126551
$$476$$ 0 0
$$477$$ −5.45242e25 −0.00409398
$$478$$ 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
$$483$$ −1.58303e27 −0.104242
$$484$$ 0 0
$$485$$ 4.59535e27 0.289755
$$486$$ 0 0
$$487$$ 7.92907e27 0.478815 0.239408 0.970919i $$-0.423047\pi$$
0.239408 + 0.970919i $$0.423047\pi$$
$$488$$ 0 0
$$489$$ −1.14400e28 −0.661735
$$490$$ 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
$$495$$ −1.42118e27 −0.0723260
$$496$$ 0 0
$$497$$ 6.19650e27 0.302276
$$498$$ 0 0
$$499$$ 7.12364e27 0.333155 0.166578 0.986028i $$-0.446728\pi$$
0.166578 + 0.986028i $$0.446728\pi$$
$$500$$ 0 0
$$501$$ −1.23286e28 −0.552862
$$502$$ 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
$$507$$ 2.67935e27 0.106034
$$508$$ 0 0
$$509$$ 4.53859e28 1.72339 0.861696 0.507425i $$-0.169402\pi$$
0.861696 + 0.507425i $$0.169402\pi$$
$$510$$ 0 0
$$511$$ 4.80788e27 0.175200
$$512$$ 0 0
$$513$$ −9.49036e26 −0.0331933
$$514$$ 0 0
$$515$$ 1.01194e28 0.339765
$$516$$ 0 0
$$517$$ 4.87229e27 0.157066
$$518$$ 0 0
$$519$$ 1.46777e28 0.454360
$$520$$ 0 0
$$521$$ 4.13279e28 1.22870 0.614352 0.789032i $$-0.289417\pi$$
0.614352 + 0.789032i $$0.289417\pi$$
$$522$$ 0 0
$$523$$ 2.57605e28 0.735675 0.367837 0.929890i $$-0.380098\pi$$
0.367837 + 0.929890i $$0.380098\pi$$
$$524$$ 0 0
$$525$$ −5.82415e27 −0.159794
$$526$$ 0 0
$$527$$ 6.69594e27 0.176522
$$528$$ 0 0
$$529$$ −3.04285e28 −0.770896
$$530$$ 0 0
$$531$$ −1.03810e28 −0.252782
$$532$$ 0 0
$$533$$ 7.61895e28 1.78344
$$534$$ 0 0
$$535$$ 3.62847e28 0.816599
$$536$$ 0 0
$$537$$ −4.22598e28 −0.914530
$$538$$ 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
$$543$$ −1.61910e27 −0.0311799
$$544$$ 0 0
$$545$$ 6.25739e27 0.115939
$$546$$ 0 0
$$547$$ −7.69499e28 −1.37196 −0.685979 0.727621i $$-0.740626\pi$$
−0.685979 + 0.727621i $$0.740626\pi$$
$$548$$ 0 0
$$549$$ 1.25659e28 0.215617
$$550$$ 0 0
$$551$$ −1.03515e28 −0.170967
$$552$$ 0 0
$$553$$ −3.24258e28 −0.515557
$$554$$ 0 0
$$555$$ −1.20954e28 −0.185159
$$556$$ 0 0
$$557$$ 9.27070e28 1.36657 0.683287 0.730150i $$-0.260550\pi$$
0.683287 + 0.730150i $$0.260550\pi$$
$$558$$ 0 0
$$559$$ −7.13926e28 −1.01351
$$560$$ 0 0
$$561$$ −4.53213e27 −0.0619716
$$562$$ 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
$$567$$ −3.42742e27 −0.0419126
$$568$$ 0 0
$$569$$ −1.10807e28 −0.130583 −0.0652917 0.997866i $$-0.520798\pi$$
−0.0652917 + 0.997866i $$0.520798\pi$$
$$570$$ 0 0
$$571$$ −1.89713e28 −0.215485 −0.107743 0.994179i $$-0.534362\pi$$
−0.107743 + 0.994179i $$0.534362\pi$$
$$572$$ 0 0
$$573$$ −7.98953e28 −0.874778
$$574$$ 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
$$579$$ −7.52698e28 −0.738748
$$580$$ 0 0
$$581$$ −2.72409e28 −0.257853
$$582$$ 0 0
$$583$$ −5.65636e26 −0.00516437
$$584$$ 0 0
$$585$$ 1.76448e28 0.155411
$$586$$ 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
$$591$$ 8.06483e27 0.0638155
$$592$$ 0 0
$$593$$ −2.26566e29 −1.73029 −0.865147 0.501519i $$-0.832775\pi$$
−0.865147 + 0.501519i $$0.832775\pi$$
$$594$$ 0 0
$$595$$ 6.74037e27 0.0496885
$$596$$ 0 0
$$597$$ 6.55209e28 0.466283
$$598$$ 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
$$603$$ −7.34533e28 −0.470630
$$604$$ 0 0
$$605$$ 6.86433e28 0.424783
$$606$$ 0 0
$$607$$ −1.39496e29 −0.833834 −0.416917 0.908945i $$-0.636889\pi$$
−0.416917 + 0.908945i $$0.636889\pi$$
$$608$$ 0 0
$$609$$ −3.73843e28 −0.215878
$$610$$ 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
$$615$$ −1.12880e29 −0.588068
$$616$$ 0 0
$$617$$ −1.93960e29 −0.976604 −0.488302 0.872675i $$-0.662384\pi$$
−0.488302 + 0.872675i $$0.662384\pi$$
$$618$$ 0 0
$$619$$ −1.29272e29 −0.629150 −0.314575 0.949233i $$-0.601862\pi$$
−0.314575 + 0.949233i $$0.601862\pi$$
$$620$$ 0 0
$$621$$ 1.95793e28 0.0921159
$$622$$ 0 0
$$623$$ −1.70354e28 −0.0774868
$$624$$ 0 0
$$625$$ 6.18632e28 0.272077
$$626$$ 0 0
$$627$$ −9.84535e27 −0.0418718
$$628$$ 0 0
$$629$$ −3.85722e28 −0.158651
$$630$$ 0 0
$$631$$ −1.99938e29 −0.795401 −0.397701 0.917515i $$-0.630192\pi$$
−0.397701 + 0.917515i $$0.630192\pi$$
$$632$$ 0 0
$$633$$ −2.51418e28 −0.0967516
$$634$$ 0 0
$$635$$ −2.00852e29 −0.747744
$$636$$ 0 0
$$637$$ 2.15150e29 0.774956
$$638$$ 0 0
$$639$$ −7.66398e28 −0.267113
$$640$$ 0 0
$$641$$ −1.98808e29 −0.670539 −0.335269 0.942122i $$-0.608827\pi$$
−0.335269 + 0.942122i $$0.608827\pi$$
$$642$$ 0 0
$$643$$ 1.68097e29 0.548712 0.274356 0.961628i $$-0.411535\pi$$
0.274356 + 0.961628i $$0.411535\pi$$
$$644$$ 0 0
$$645$$ 1.05773e29 0.334193
$$646$$ 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
$$651$$ 5.25320e28 0.150600
$$652$$ 0 0
$$653$$ 5.07947e29 1.41003 0.705017 0.709190i $$-0.250939\pi$$
0.705017 + 0.709190i $$0.250939\pi$$
$$654$$ 0 0
$$655$$ 6.11868e28 0.164484
$$656$$ 0 0
$$657$$ −5.94651e28 −0.154819
$$658$$ 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
$$663$$ 5.62692e28 0.133161
$$664$$ 0 0
$$665$$ 1.46424e28 0.0335726
$$666$$ 0 0
$$667$$ 2.13559e29 0.474458
$$668$$ 0 0
$$669$$ −6.28764e27 −0.0135367
$$670$$ 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
$$675$$ 7.20346e28 0.141205
$$676$$ 0 0
$$677$$ −2.94132e27 −0.00558936 −0.00279468 0.999996i $$-0.500890\pi$$
−0.00279468 + 0.999996i $$0.500890\pi$$
$$678$$ 0 0
$$679$$ 1.14970e29 0.211813
$$680$$ 0 0
$$681$$ −3.41641e29 −0.610274
$$682$$ 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
$$687$$ 4.69717e29 0.765228
$$688$$ 0 0
$$689$$ 7.02272e27 0.0110970
$$690$$ 0 0
$$691$$ 7.10396e29 1.08888 0.544442 0.838799i $$-0.316742\pi$$
0.544442 + 0.838799i $$0.316742\pi$$
$$692$$ 0 0
$$693$$ −3.55562e28 −0.0528709
$$694$$ 0 0
$$695$$ −4.97819e29 −0.718176
$$696$$ 0 0
$$697$$ −3.59973e29 −0.503878
$$698$$ 0 0
$$699$$ 4.03935e29 0.548657
$$700$$ 0 0
$$701$$ 8.10616e29 1.06850 0.534252 0.845325i $$-0.320593\pi$$
0.534252 + 0.845325i $$0.320593\pi$$
$$702$$ 0 0
$$703$$ −8.37921e28 −0.107194
$$704$$ 0 0
$$705$$ 8.96236e28 0.111285
$$706$$ 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
$$711$$ 4.01050e29 0.455584
$$712$$ 0 0
$$713$$ −3.00092e29 −0.330989
$$714$$ 0 0
$$715$$ 1.83048e29 0.196043
$$716$$ 0 0
$$717$$ 1.01292e30 1.05347
$$718$$ 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
$$723$$ 3.54643e29 0.337940
$$724$$ 0 0
$$725$$ 7.85712e29 0.727301
$$726$$ 0 0
$$727$$ 1.06635e30 0.958933 0.479467 0.877560i $$-0.340830\pi$$
0.479467 + 0.877560i $$0.340830\pi$$
$$728$$ 0 0
$$729$$ 4.23912e28 0.0370370
$$730$$ 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
$$735$$ −3.18759e29 −0.255532
$$736$$ 0 0
$$737$$ −7.62008e29 −0.593678
$$738$$ 0 0
$$739$$ −1.80814e29 −0.136920 −0.0684598 0.997654i $$-0.521808\pi$$
−0.0684598 + 0.997654i $$0.521808\pi$$
$$740$$ 0 0
$$741$$ 1.22236e29 0.0899721
$$742$$ 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
$$747$$ 3.36923e29 0.227858
$$748$$ 0 0
$$749$$ 9.07800e29 0.596941
$$750$$ 0 0
$$751$$ 9.22610e29 0.589928 0.294964 0.955508i $$-0.404692\pi$$
0.294964 + 0.955508i $$0.404692\pi$$
$$752$$ 0 0
$$753$$ 8.79141e29 0.546652
$$754$$ 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
$$759$$ 2.03116e29 0.116200
$$760$$ 0 0
$$761$$ −3.51722e30 −1.95732 −0.978658 0.205496i $$-0.934119\pi$$
−0.978658 + 0.205496i $$0.934119\pi$$
$$762$$ 0 0
$$763$$ 1.56552e29 0.0847524
$$764$$ 0 0
$$765$$ −8.33666e28 −0.0439083
$$766$$ 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
$$771$$ −1.04407e30 −0.506593
$$772$$ 0 0
$$773$$ −4.21079e30 −1.98829 −0.994143 0.108071i $$-0.965533\pi$$
−0.994143 + 0.108071i $$0.965533\pi$$
$$774$$ 0 0
$$775$$ −1.10407e30 −0.507376
$$776$$ 0 0
$$777$$ −3.02613e29 −0.135353
$$778$$ 0 0
$$779$$ −7.81985e29 −0.340451
$$780$$ 0 0
$$781$$ −7.95066e29 −0.336951
$$782$$ 0 0
$$783$$ 4.62378e29 0.190765
$$784$$ 0 0
$$785$$ −2.21810e30 −0.890943
$$786$$ 0 0
$$787$$ −3.65263e29 −0.142847 −0.0714235 0.997446i $$-0.522754\pi$$
−0.0714235 + 0.997446i $$0.522754\pi$$
$$788$$ 0 0
$$789$$ 1.92382e30 0.732581
$$790$$ 0 0
$$791$$ −1.12988e30 −0.418968
$$792$$ 0 0
$$793$$ −1.61849e30 −0.584441
$$794$$ 0 0
$$795$$ −1.04046e28 −0.00365908
$$796$$ 0 0
$$797$$ −3.15540e30 −1.08079 −0.540396 0.841411i $$-0.681726\pi$$
−0.540396 + 0.841411i $$0.681726\pi$$
$$798$$ 0 0
$$799$$ 2.85809e29 0.0953531
$$800$$ 0 0
$$801$$ 2.10698e29 0.0684729
$$802$$ 0 0
$$803$$ −6.16894e29 −0.195298
$$804$$ 0 0
$$805$$ −3.02083e29 −0.0931686
$$806$$ 0 0
$$807$$ −1.23482e30 −0.371048
$$808$$ 0 0
$$809$$ −1.77681e30 −0.520214 −0.260107 0.965580i $$-0.583758\pi$$
−0.260107 + 0.965580i $$0.583758\pi$$
$$810$$ 0 0
$$811$$ −3.55440e30 −1.01402 −0.507010 0.861940i $$-0.669249\pi$$
−0.507010 + 0.861940i $$0.669249\pi$$
$$812$$ 0 0
$$813$$ 9.29304e29 0.258349
$$814$$ 0 0
$$815$$ −2.18305e30 −0.591439
$$816$$ 0 0
$$817$$ 7.32751e29 0.193475
$$818$$ 0 0
$$819$$ 4.41452e29 0.113606
$$820$$ 0 0
$$821$$ 2.53502e30 0.635884 0.317942 0.948110i $$-0.397008\pi$$
0.317942 + 0.948110i $$0.397008\pi$$
$$822$$ 0 0
$$823$$ 1.71174e30 0.418543 0.209272 0.977858i $$-0.432891\pi$$
0.209272 + 0.977858i $$0.432891\pi$$
$$824$$ 0 0
$$825$$ 7.47290e29 0.178124
$$826$$ 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
$$831$$ 7.29039e29 0.161042
$$832$$ 0 0
$$833$$ −1.01652e30 −0.218949
$$834$$ 0 0
$$835$$ −2.35261e30 −0.494131
$$836$$ 0 0
$$837$$ −6.49729e29 −0.133081
$$838$$ 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
$$843$$ 3.25824e30 0.619147
$$844$$ 0 0
$$845$$ 5.11290e29 0.0947701
$$846$$ 0 0
$$847$$ 1.71737e30 0.310519
$$848$$ 0 0
$$849$$ 8.59666e29 0.151635
$$850$$ 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
$$855$$ −1.81101e29 −0.0296672
$$856$$ 0 0
$$857$$ 2.63232e29 0.0420766 0.0210383 0.999779i $$-0.493303\pi$$
0.0210383 + 0.999779i $$0.493303\pi$$
$$858$$ 0 0
$$859$$ −2.57894e30 −0.402266 −0.201133 0.979564i $$-0.564462\pi$$
−0.201133 + 0.979564i $$0.564462\pi$$
$$860$$ 0 0
$$861$$ −2.82412e30 −0.429882
$$862$$ 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
$$867$$ 3.81395e30 0.539728
$$868$$ 0 0
$$869$$ 4.16052e30 0.574698
$$870$$ 0 0
$$871$$ 9.46080e30 1.27567
$$872$$ 0 0
$$873$$ −1.42198e30 −0.187173
$$874$$ 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
$$879$$ 1.69278e30 0.207357
$$880$$ 0 0
$$881$$ 3.05871e30 0.365840 0.182920 0.983128i $$-0.441445\pi$$
0.182920 + 0.983128i $$0.441445\pi$$
$$882$$ 0 0
$$883$$ 1.12080e31 1.30901 0.654503 0.756060i $$-0.272878\pi$$
0.654503 + 0.756060i $$0.272878\pi$$
$$884$$ 0 0
$$885$$ −1.98097e30 −0.225929
$$886$$ 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
$$891$$ 4.39768e29 0.0467205
$$892$$ 0 0
$$893$$ 6.20875e29 0.0644264
$$894$$ 0 0
$$895$$ −8.06428e30 −0.817380
$$896$$ 0 0
$$897$$ −2.52181e30 −0.249685
$$898$$ 0 0
$$899$$ −7.08687e30 −0.685453
$$900$$ 0 0
$$901$$ −3.31803e28 −0.00313523
$$902$$ 0 0
$$903$$ 2.64631e30 0.244298
$$904$$ 0 0
$$905$$ −3.08966e29 −0.0278677
$$906$$ 0 0
$$907$$ 1.28972e31 1.13663 0.568314 0.822812i $$-0.307596\pi$$
0.568314 + 0.822812i $$0.307596\pi$$
$$908$$ 0 0
$$909$$ 6.80189e30 0.585745
$$910$$ 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
$$915$$ 2.39790e30 0.192712
$$916$$ 0 0
$$917$$ 1.53082e30 0.120239
$$918$$ 0 0
$$919$$ −1.99044e31 −1.52804 −0.764022 0.645190i $$-0.776778\pi$$
−0.764022 + 0.645190i $$0.776778\pi$$
$$920$$ 0 0
$$921$$ 4.90932e30 0.368379
$$922$$ 0 0
$$923$$ 9.87123e30 0.724025
$$924$$ 0 0
$$925$$ 6.36006e30 0.456009
$$926$$ 0 0
$$927$$ −3.13134e30 −0.219479
$$928$$ 0 0
$$929$$ 2.46924e31 1.69199 0.845995 0.533190i $$-0.179007\pi$$
0.845995 + 0.533190i $$0.179007\pi$$
$$930$$ 0 0
$$931$$ −2.20823e30 −0.147936
$$932$$ 0 0
$$933$$ −1.06517e31 −0.697690
$$934$$ 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
$$939$$ −1.37677e31 −0.843086
$$940$$ 0 0
$$941$$ 8.99005e30 0.538358 0.269179 0.963090i $$-0.413248\pi$$
0.269179 + 0.963090i $$0.413248\pi$$
$$942$$ 0 0
$$943$$ 1.61329e31 0.944799
$$944$$ 0 0
$$945$$ −6.54041e29 −0.0374603
$$946$$ 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
$$951$$ −5.02852e30 −0.269492
$$952$$ 0 0
$$953$$ −9.08338e30 −0.476181 −0.238090 0.971243i $$-0.576521\pi$$
−0.238090 + 0.971243i $$0.576521\pi$$
$$954$$ 0 0
$$955$$ −1.52461e31 −0.781851
$$956$$ 0 0
$$957$$ 4.79674e30 0.240642
$$958$$ 0 0
$$959$$ −7.44360e30 −0.365332
$$960$$ 0 0
$$961$$ −1.08671e31 −0.521817
$$962$$ 0 0
$$963$$ −1.12279e31 −0.527500
$$964$$ 0 0
$$965$$ −1.43635e31 −0.660271
$$966$$ 0 0
$$967$$ 1.32705e31 0.596909 0.298455 0.954424i $$-0.403529\pi$$
0.298455 + 0.954424i $$0.403529\pi$$
$$968$$ 0 0
$$969$$ −5.77529e29 −0.0254199
$$970$$ 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
$$975$$ −9.27807e30 −0.382745
$$976$$ 0 0
$$977$$ 1.04812e31 0.423172 0.211586 0.977359i $$-0.432137\pi$$
0.211586 + 0.977359i $$0.432137\pi$$
$$978$$ 0 0
$$979$$ 2.18579e30 0.0863755
$$980$$ 0 0
$$981$$ −1.93628e30 −0.0748933
$$982$$ 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
$$987$$ 2.24227e30 0.0813502
$$988$$ 0 0
$$989$$ −1.51172e31 −0.536919
$$990$$ 0 0
$$991$$ −2.71476e31 −0.943968 −0.471984 0.881607i $$-0.656462\pi$$
−0.471984 + 0.881607i $$0.656462\pi$$
$$992$$ 0 0
$$993$$ 3.35701e30 0.114284
$$994$$ 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
$$999$$ 3.74279e30 0.119607
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 48.22.a.b.1.1 1
4.3 odd 2 12.22.a.a.1.1 1
12.11 even 2 36.22.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
12.22.a.a.1.1 1 4.3 odd 2
36.22.a.a.1.1 1 12.11 even 2
48.22.a.b.1.1 1 1.1 even 1 trivial