# Properties

 Label 91.2.a.a.1.1 Level $91$ Weight $2$ Character 91.1 Self dual yes Analytic conductor $0.727$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [91,2,Mod(1,91)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(91, 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("91.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$91 = 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 91.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: $$0.726638658394$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 91.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -3.00000 q^{9} +6.00000 q^{10} -6.00000 q^{11} -1.00000 q^{13} +2.00000 q^{14} -4.00000 q^{16} +4.00000 q^{17} +6.00000 q^{18} +5.00000 q^{19} -6.00000 q^{20} +12.0000 q^{22} +3.00000 q^{23} +4.00000 q^{25} +2.00000 q^{26} -2.00000 q^{28} -5.00000 q^{29} -3.00000 q^{31} +8.00000 q^{32} -8.00000 q^{34} +3.00000 q^{35} -6.00000 q^{36} -4.00000 q^{37} -10.0000 q^{38} -6.00000 q^{41} -1.00000 q^{43} -12.0000 q^{44} +9.00000 q^{45} -6.00000 q^{46} +7.00000 q^{47} +1.00000 q^{49} -8.00000 q^{50} -2.00000 q^{52} -9.00000 q^{53} +18.0000 q^{55} +10.0000 q^{58} +8.00000 q^{59} -10.0000 q^{61} +6.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} +3.00000 q^{65} -6.00000 q^{67} +8.00000 q^{68} -6.00000 q^{70} -8.00000 q^{71} -13.0000 q^{73} +8.00000 q^{74} +10.0000 q^{76} +6.00000 q^{77} +3.00000 q^{79} +12.0000 q^{80} +9.00000 q^{81} +12.0000 q^{82} +15.0000 q^{83} -12.0000 q^{85} +2.00000 q^{86} +3.00000 q^{89} -18.0000 q^{90} +1.00000 q^{91} +6.00000 q^{92} -14.0000 q^{94} -15.0000 q^{95} +7.00000 q^{97} -2.00000 q^{98} +18.0000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −1.41421 −0.707107 0.707107i $$-0.750000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$4$$ 2.00000 1.00000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ −1.00000 −0.377964
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 6.00000 1.89737
$$11$$ −6.00000 −1.80907 −0.904534 0.426401i $$-0.859781\pi$$
−0.904534 + 0.426401i $$0.859781\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ 2.00000 0.534522
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ 4.00000 0.970143 0.485071 0.874475i $$-0.338794\pi$$
0.485071 + 0.874475i $$0.338794\pi$$
$$18$$ 6.00000 1.41421
$$19$$ 5.00000 1.14708 0.573539 0.819178i $$-0.305570\pi$$
0.573539 + 0.819178i $$0.305570\pi$$
$$20$$ −6.00000 −1.34164
$$21$$ 0 0
$$22$$ 12.0000 2.55841
$$23$$ 3.00000 0.625543 0.312772 0.949828i $$-0.398743\pi$$
0.312772 + 0.949828i $$0.398743\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 2.00000 0.392232
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ −5.00000 −0.928477 −0.464238 0.885710i $$-0.653672\pi$$
−0.464238 + 0.885710i $$0.653672\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 8.00000 1.41421
$$33$$ 0 0
$$34$$ −8.00000 −1.37199
$$35$$ 3.00000 0.507093
$$36$$ −6.00000 −1.00000
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ −10.0000 −1.62221
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ −1.00000 −0.152499 −0.0762493 0.997089i $$-0.524294\pi$$
−0.0762493 + 0.997089i $$0.524294\pi$$
$$44$$ −12.0000 −1.80907
$$45$$ 9.00000 1.34164
$$46$$ −6.00000 −0.884652
$$47$$ 7.00000 1.02105 0.510527 0.859861i $$-0.329450\pi$$
0.510527 + 0.859861i $$0.329450\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ −8.00000 −1.13137
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ 18.0000 2.42712
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 10.0000 1.31306
$$59$$ 8.00000 1.04151 0.520756 0.853706i $$-0.325650\pi$$
0.520756 + 0.853706i $$0.325650\pi$$
$$60$$ 0 0
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 6.00000 0.762001
$$63$$ 3.00000 0.377964
$$64$$ −8.00000 −1.00000
$$65$$ 3.00000 0.372104
$$66$$ 0 0
$$67$$ −6.00000 −0.733017 −0.366508 0.930415i $$-0.619447\pi$$
−0.366508 + 0.930415i $$0.619447\pi$$
$$68$$ 8.00000 0.970143
$$69$$ 0 0
$$70$$ −6.00000 −0.717137
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −13.0000 −1.52153 −0.760767 0.649025i $$-0.775177\pi$$
−0.760767 + 0.649025i $$0.775177\pi$$
$$74$$ 8.00000 0.929981
$$75$$ 0 0
$$76$$ 10.0000 1.14708
$$77$$ 6.00000 0.683763
$$78$$ 0 0
$$79$$ 3.00000 0.337526 0.168763 0.985657i $$-0.446023\pi$$
0.168763 + 0.985657i $$0.446023\pi$$
$$80$$ 12.0000 1.34164
$$81$$ 9.00000 1.00000
$$82$$ 12.0000 1.32518
$$83$$ 15.0000 1.64646 0.823232 0.567705i $$-0.192169\pi$$
0.823232 + 0.567705i $$0.192169\pi$$
$$84$$ 0 0
$$85$$ −12.0000 −1.30158
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 3.00000 0.317999 0.159000 0.987279i $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ −18.0000 −1.89737
$$91$$ 1.00000 0.104828
$$92$$ 6.00000 0.625543
$$93$$ 0 0
$$94$$ −14.0000 −1.44399
$$95$$ −15.0000 −1.53897
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ −2.00000 −0.202031
$$99$$ 18.0000 1.80907
$$100$$ 8.00000 0.800000
$$101$$ −14.0000 −1.39305 −0.696526 0.717532i $$-0.745272\pi$$
−0.696526 + 0.717532i $$0.745272\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 18.0000 1.74831
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ −36.0000 −3.43247
$$111$$ 0 0
$$112$$ 4.00000 0.377964
$$113$$ −3.00000 −0.282216 −0.141108 0.989994i $$-0.545067\pi$$
−0.141108 + 0.989994i $$0.545067\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ −10.0000 −0.928477
$$117$$ 3.00000 0.277350
$$118$$ −16.0000 −1.47292
$$119$$ −4.00000 −0.366679
$$120$$ 0 0
$$121$$ 25.0000 2.27273
$$122$$ 20.0000 1.81071
$$123$$ 0 0
$$124$$ −6.00000 −0.538816
$$125$$ 3.00000 0.268328
$$126$$ −6.00000 −0.534522
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ −6.00000 −0.526235
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 0 0
$$133$$ −5.00000 −0.433555
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ −18.0000 −1.52674 −0.763370 0.645961i $$-0.776457\pi$$
−0.763370 + 0.645961i $$0.776457\pi$$
$$140$$ 6.00000 0.507093
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ 6.00000 0.501745
$$144$$ 12.0000 1.00000
$$145$$ 15.0000 1.24568
$$146$$ 26.0000 2.15178
$$147$$ 0 0
$$148$$ −8.00000 −0.657596
$$149$$ −18.0000 −1.47462 −0.737309 0.675556i $$-0.763904\pi$$
−0.737309 + 0.675556i $$0.763904\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ −12.0000 −0.970143
$$154$$ −12.0000 −0.966988
$$155$$ 9.00000 0.722897
$$156$$ 0 0
$$157$$ −8.00000 −0.638470 −0.319235 0.947676i $$-0.603426\pi$$
−0.319235 + 0.947676i $$0.603426\pi$$
$$158$$ −6.00000 −0.477334
$$159$$ 0 0
$$160$$ −24.0000 −1.89737
$$161$$ −3.00000 −0.236433
$$162$$ −18.0000 −1.41421
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ −12.0000 −0.937043
$$165$$ 0 0
$$166$$ −30.0000 −2.32845
$$167$$ 5.00000 0.386912 0.193456 0.981109i $$-0.438030\pi$$
0.193456 + 0.981109i $$0.438030\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 24.0000 1.84072
$$171$$ −15.0000 −1.14708
$$172$$ −2.00000 −0.152499
$$173$$ −8.00000 −0.608229 −0.304114 0.952636i $$-0.598361\pi$$
−0.304114 + 0.952636i $$0.598361\pi$$
$$174$$ 0 0
$$175$$ −4.00000 −0.302372
$$176$$ 24.0000 1.80907
$$177$$ 0 0
$$178$$ −6.00000 −0.449719
$$179$$ 23.0000 1.71910 0.859550 0.511051i $$-0.170744\pi$$
0.859550 + 0.511051i $$0.170744\pi$$
$$180$$ 18.0000 1.34164
$$181$$ 14.0000 1.04061 0.520306 0.853980i $$-0.325818\pi$$
0.520306 + 0.853980i $$0.325818\pi$$
$$182$$ −2.00000 −0.148250
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ −24.0000 −1.75505
$$188$$ 14.0000 1.02105
$$189$$ 0 0
$$190$$ 30.0000 2.17643
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 0 0
$$193$$ 22.0000 1.58359 0.791797 0.610784i $$-0.209146\pi$$
0.791797 + 0.610784i $$0.209146\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 2.00000 0.142857
$$197$$ 2.00000 0.142494 0.0712470 0.997459i $$-0.477302\pi$$
0.0712470 + 0.997459i $$0.477302\pi$$
$$198$$ −36.0000 −2.55841
$$199$$ 4.00000 0.283552 0.141776 0.989899i $$-0.454719\pi$$
0.141776 + 0.989899i $$0.454719\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 28.0000 1.97007
$$203$$ 5.00000 0.350931
$$204$$ 0 0
$$205$$ 18.0000 1.25717
$$206$$ 8.00000 0.557386
$$207$$ −9.00000 −0.625543
$$208$$ 4.00000 0.277350
$$209$$ −30.0000 −2.07514
$$210$$ 0 0
$$211$$ −5.00000 −0.344214 −0.172107 0.985078i $$-0.555058\pi$$
−0.172107 + 0.985078i $$0.555058\pi$$
$$212$$ −18.0000 −1.23625
$$213$$ 0 0
$$214$$ 8.00000 0.546869
$$215$$ 3.00000 0.204598
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ 4.00000 0.270914
$$219$$ 0 0
$$220$$ 36.0000 2.42712
$$221$$ −4.00000 −0.269069
$$222$$ 0 0
$$223$$ 15.0000 1.00447 0.502237 0.864730i $$-0.332510\pi$$
0.502237 + 0.864730i $$0.332510\pi$$
$$224$$ −8.00000 −0.534522
$$225$$ −12.0000 −0.800000
$$226$$ 6.00000 0.399114
$$227$$ 20.0000 1.32745 0.663723 0.747978i $$-0.268975\pi$$
0.663723 + 0.747978i $$0.268975\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 18.0000 1.18688
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 15.0000 0.982683 0.491341 0.870967i $$-0.336507\pi$$
0.491341 + 0.870967i $$0.336507\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ −21.0000 −1.36989
$$236$$ 16.0000 1.04151
$$237$$ 0 0
$$238$$ 8.00000 0.518563
$$239$$ −4.00000 −0.258738 −0.129369 0.991596i $$-0.541295\pi$$
−0.129369 + 0.991596i $$0.541295\pi$$
$$240$$ 0 0
$$241$$ −17.0000 −1.09507 −0.547533 0.836784i $$-0.684433\pi$$
−0.547533 + 0.836784i $$0.684433\pi$$
$$242$$ −50.0000 −3.21412
$$243$$ 0 0
$$244$$ −20.0000 −1.28037
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 0 0
$$249$$ 0 0
$$250$$ −6.00000 −0.379473
$$251$$ −26.0000 −1.64111 −0.820553 0.571571i $$-0.806334\pi$$
−0.820553 + 0.571571i $$0.806334\pi$$
$$252$$ 6.00000 0.377964
$$253$$ −18.0000 −1.13165
$$254$$ 8.00000 0.501965
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 6.00000 0.372104
$$261$$ 15.0000 0.928477
$$262$$ −16.0000 −0.988483
$$263$$ −15.0000 −0.924940 −0.462470 0.886635i $$-0.653037\pi$$
−0.462470 + 0.886635i $$0.653037\pi$$
$$264$$ 0 0
$$265$$ 27.0000 1.65860
$$266$$ 10.0000 0.613139
$$267$$ 0 0
$$268$$ −12.0000 −0.733017
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ −16.0000 −0.970143
$$273$$ 0 0
$$274$$ −8.00000 −0.483298
$$275$$ −24.0000 −1.44725
$$276$$ 0 0
$$277$$ 1.00000 0.0600842 0.0300421 0.999549i $$-0.490436\pi$$
0.0300421 + 0.999549i $$0.490436\pi$$
$$278$$ 36.0000 2.15914
$$279$$ 9.00000 0.538816
$$280$$ 0 0
$$281$$ −30.0000 −1.78965 −0.894825 0.446417i $$-0.852700\pi$$
−0.894825 + 0.446417i $$0.852700\pi$$
$$282$$ 0 0
$$283$$ 16.0000 0.951101 0.475551 0.879688i $$-0.342249\pi$$
0.475551 + 0.879688i $$0.342249\pi$$
$$284$$ −16.0000 −0.949425
$$285$$ 0 0
$$286$$ −12.0000 −0.709575
$$287$$ 6.00000 0.354169
$$288$$ −24.0000 −1.41421
$$289$$ −1.00000 −0.0588235
$$290$$ −30.0000 −1.76166
$$291$$ 0 0
$$292$$ −26.0000 −1.52153
$$293$$ −19.0000 −1.10999 −0.554996 0.831853i $$-0.687280\pi$$
−0.554996 + 0.831853i $$0.687280\pi$$
$$294$$ 0 0
$$295$$ −24.0000 −1.39733
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 36.0000 2.08542
$$299$$ −3.00000 −0.173494
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −20.0000 −1.14708
$$305$$ 30.0000 1.71780
$$306$$ 24.0000 1.37199
$$307$$ −33.0000 −1.88341 −0.941705 0.336440i $$-0.890777\pi$$
−0.941705 + 0.336440i $$0.890777\pi$$
$$308$$ 12.0000 0.683763
$$309$$ 0 0
$$310$$ −18.0000 −1.02233
$$311$$ −6.00000 −0.340229 −0.170114 0.985424i $$-0.554414\pi$$
−0.170114 + 0.985424i $$0.554414\pi$$
$$312$$ 0 0
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ 16.0000 0.902932
$$315$$ −9.00000 −0.507093
$$316$$ 6.00000 0.337526
$$317$$ −24.0000 −1.34797 −0.673987 0.738743i $$-0.735420\pi$$
−0.673987 + 0.738743i $$0.735420\pi$$
$$318$$ 0 0
$$319$$ 30.0000 1.67968
$$320$$ 24.0000 1.34164
$$321$$ 0 0
$$322$$ 6.00000 0.334367
$$323$$ 20.0000 1.11283
$$324$$ 18.0000 1.00000
$$325$$ −4.00000 −0.221880
$$326$$ 8.00000 0.443079
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −7.00000 −0.385922
$$330$$ 0 0
$$331$$ 22.0000 1.20923 0.604615 0.796518i $$-0.293327\pi$$
0.604615 + 0.796518i $$0.293327\pi$$
$$332$$ 30.0000 1.64646
$$333$$ 12.0000 0.657596
$$334$$ −10.0000 −0.547176
$$335$$ 18.0000 0.983445
$$336$$ 0 0
$$337$$ 17.0000 0.926049 0.463025 0.886345i $$-0.346764\pi$$
0.463025 + 0.886345i $$0.346764\pi$$
$$338$$ −2.00000 −0.108786
$$339$$ 0 0
$$340$$ −24.0000 −1.30158
$$341$$ 18.0000 0.974755
$$342$$ 30.0000 1.62221
$$343$$ −1.00000 −0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 16.0000 0.860165
$$347$$ −32.0000 −1.71785 −0.858925 0.512101i $$-0.828867\pi$$
−0.858925 + 0.512101i $$0.828867\pi$$
$$348$$ 0 0
$$349$$ 11.0000 0.588817 0.294408 0.955680i $$-0.404877\pi$$
0.294408 + 0.955680i $$0.404877\pi$$
$$350$$ 8.00000 0.427618
$$351$$ 0 0
$$352$$ −48.0000 −2.55841
$$353$$ −10.0000 −0.532246 −0.266123 0.963939i $$-0.585743\pi$$
−0.266123 + 0.963939i $$0.585743\pi$$
$$354$$ 0 0
$$355$$ 24.0000 1.27379
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −46.0000 −2.43118
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ 6.00000 0.315789
$$362$$ −28.0000 −1.47165
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 39.0000 2.04135
$$366$$ 0 0
$$367$$ 14.0000 0.730794 0.365397 0.930852i $$-0.380933\pi$$
0.365397 + 0.930852i $$0.380933\pi$$
$$368$$ −12.0000 −0.625543
$$369$$ 18.0000 0.937043
$$370$$ −24.0000 −1.24770
$$371$$ 9.00000 0.467257
$$372$$ 0 0
$$373$$ 30.0000 1.55334 0.776671 0.629907i $$-0.216907\pi$$
0.776671 + 0.629907i $$0.216907\pi$$
$$374$$ 48.0000 2.48202
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 5.00000 0.257513
$$378$$ 0 0
$$379$$ −6.00000 −0.308199 −0.154100 0.988055i $$-0.549248\pi$$
−0.154100 + 0.988055i $$0.549248\pi$$
$$380$$ −30.0000 −1.53897
$$381$$ 0 0
$$382$$ 16.0000 0.818631
$$383$$ −36.0000 −1.83951 −0.919757 0.392488i $$-0.871614\pi$$
−0.919757 + 0.392488i $$0.871614\pi$$
$$384$$ 0 0
$$385$$ −18.0000 −0.917365
$$386$$ −44.0000 −2.23954
$$387$$ 3.00000 0.152499
$$388$$ 14.0000 0.710742
$$389$$ 30.0000 1.52106 0.760530 0.649303i $$-0.224939\pi$$
0.760530 + 0.649303i $$0.224939\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4.00000 −0.201517
$$395$$ −9.00000 −0.452839
$$396$$ 36.0000 1.80907
$$397$$ −13.0000 −0.652451 −0.326226 0.945292i $$-0.605777\pi$$
−0.326226 + 0.945292i $$0.605777\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ 0 0
$$400$$ −16.0000 −0.800000
$$401$$ −32.0000 −1.59800 −0.799002 0.601329i $$-0.794638\pi$$
−0.799002 + 0.601329i $$0.794638\pi$$
$$402$$ 0 0
$$403$$ 3.00000 0.149441
$$404$$ −28.0000 −1.39305
$$405$$ −27.0000 −1.34164
$$406$$ −10.0000 −0.496292
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ −13.0000 −0.642809 −0.321404 0.946942i $$-0.604155\pi$$
−0.321404 + 0.946942i $$0.604155\pi$$
$$410$$ −36.0000 −1.77791
$$411$$ 0 0
$$412$$ −8.00000 −0.394132
$$413$$ −8.00000 −0.393654
$$414$$ 18.0000 0.884652
$$415$$ −45.0000 −2.20896
$$416$$ −8.00000 −0.392232
$$417$$ 0 0
$$418$$ 60.0000 2.93470
$$419$$ −10.0000 −0.488532 −0.244266 0.969708i $$-0.578547\pi$$
−0.244266 + 0.969708i $$0.578547\pi$$
$$420$$ 0 0
$$421$$ −12.0000 −0.584844 −0.292422 0.956289i $$-0.594461\pi$$
−0.292422 + 0.956289i $$0.594461\pi$$
$$422$$ 10.0000 0.486792
$$423$$ −21.0000 −1.02105
$$424$$ 0 0
$$425$$ 16.0000 0.776114
$$426$$ 0 0
$$427$$ 10.0000 0.483934
$$428$$ −8.00000 −0.386695
$$429$$ 0 0
$$430$$ −6.00000 −0.289346
$$431$$ 6.00000 0.289010 0.144505 0.989504i $$-0.453841\pi$$
0.144505 + 0.989504i $$0.453841\pi$$
$$432$$ 0 0
$$433$$ 12.0000 0.576683 0.288342 0.957528i $$-0.406896\pi$$
0.288342 + 0.957528i $$0.406896\pi$$
$$434$$ −6.00000 −0.288009
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 15.0000 0.717547
$$438$$ 0 0
$$439$$ −22.0000 −1.05000 −0.525001 0.851101i $$-0.675935\pi$$
−0.525001 + 0.851101i $$0.675935\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 8.00000 0.380521
$$443$$ 19.0000 0.902717 0.451359 0.892343i $$-0.350940\pi$$
0.451359 + 0.892343i $$0.350940\pi$$
$$444$$ 0 0
$$445$$ −9.00000 −0.426641
$$446$$ −30.0000 −1.42054
$$447$$ 0 0
$$448$$ 8.00000 0.377964
$$449$$ 36.0000 1.69895 0.849473 0.527633i $$-0.176920\pi$$
0.849473 + 0.527633i $$0.176920\pi$$
$$450$$ 24.0000 1.13137
$$451$$ 36.0000 1.69517
$$452$$ −6.00000 −0.282216
$$453$$ 0 0
$$454$$ −40.0000 −1.87729
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$458$$ −28.0000 −1.30835
$$459$$ 0 0
$$460$$ −18.0000 −0.839254
$$461$$ −22.0000 −1.02464 −0.512321 0.858794i $$-0.671214\pi$$
−0.512321 + 0.858794i $$0.671214\pi$$
$$462$$ 0 0
$$463$$ −14.0000 −0.650635 −0.325318 0.945605i $$-0.605471\pi$$
−0.325318 + 0.945605i $$0.605471\pi$$
$$464$$ 20.0000 0.928477
$$465$$ 0 0
$$466$$ −30.0000 −1.38972
$$467$$ −22.0000 −1.01804 −0.509019 0.860755i $$-0.669992\pi$$
−0.509019 + 0.860755i $$0.669992\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 6.00000 0.277054
$$470$$ 42.0000 1.93732
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 6.00000 0.275880
$$474$$ 0 0
$$475$$ 20.0000 0.917663
$$476$$ −8.00000 −0.366679
$$477$$ 27.0000 1.23625
$$478$$ 8.00000 0.365911
$$479$$ −11.0000 −0.502603 −0.251301 0.967909i $$-0.580859\pi$$
−0.251301 + 0.967909i $$0.580859\pi$$
$$480$$ 0 0
$$481$$ 4.00000 0.182384
$$482$$ 34.0000 1.54866
$$483$$ 0 0
$$484$$ 50.0000 2.27273
$$485$$ −21.0000 −0.953561
$$486$$ 0 0
$$487$$ −26.0000 −1.17817 −0.589086 0.808070i $$-0.700512\pi$$
−0.589086 + 0.808070i $$0.700512\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 6.00000 0.271052
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ −20.0000 −0.900755
$$494$$ 10.0000 0.449921
$$495$$ −54.0000 −2.42712
$$496$$ 12.0000 0.538816
$$497$$ 8.00000 0.358849
$$498$$ 0 0
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ 6.00000 0.268328
$$501$$ 0 0
$$502$$ 52.0000 2.32087
$$503$$ 2.00000 0.0891756 0.0445878 0.999005i $$-0.485803\pi$$
0.0445878 + 0.999005i $$0.485803\pi$$
$$504$$ 0 0
$$505$$ 42.0000 1.86898
$$506$$ 36.0000 1.60040
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ −19.0000 −0.842160 −0.421080 0.907023i $$-0.638349\pi$$
−0.421080 + 0.907023i $$0.638349\pi$$
$$510$$ 0 0
$$511$$ 13.0000 0.575086
$$512$$ −32.0000 −1.41421
$$513$$ 0 0
$$514$$ 4.00000 0.176432
$$515$$ 12.0000 0.528783
$$516$$ 0 0
$$517$$ −42.0000 −1.84716
$$518$$ −8.00000 −0.351500
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 40.0000 1.75243 0.876216 0.481919i $$-0.160060\pi$$
0.876216 + 0.481919i $$0.160060\pi$$
$$522$$ −30.0000 −1.31306
$$523$$ 10.0000 0.437269 0.218635 0.975807i $$-0.429840\pi$$
0.218635 + 0.975807i $$0.429840\pi$$
$$524$$ 16.0000 0.698963
$$525$$ 0 0
$$526$$ 30.0000 1.30806
$$527$$ −12.0000 −0.522728
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ −54.0000 −2.34561
$$531$$ −24.0000 −1.04151
$$532$$ −10.0000 −0.433555
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −6.00000 −0.258438
$$540$$ 0 0
$$541$$ −40.0000 −1.71973 −0.859867 0.510518i $$-0.829454\pi$$
−0.859867 + 0.510518i $$0.829454\pi$$
$$542$$ −16.0000 −0.687259
$$543$$ 0 0
$$544$$ 32.0000 1.37199
$$545$$ 6.00000 0.257012
$$546$$ 0 0
$$547$$ −7.00000 −0.299298 −0.149649 0.988739i $$-0.547814\pi$$
−0.149649 + 0.988739i $$0.547814\pi$$
$$548$$ 8.00000 0.341743
$$549$$ 30.0000 1.28037
$$550$$ 48.0000 2.04673
$$551$$ −25.0000 −1.06504
$$552$$ 0 0
$$553$$ −3.00000 −0.127573
$$554$$ −2.00000 −0.0849719
$$555$$ 0 0
$$556$$ −36.0000 −1.52674
$$557$$ −12.0000 −0.508456 −0.254228 0.967144i $$-0.581821\pi$$
−0.254228 + 0.967144i $$0.581821\pi$$
$$558$$ −18.0000 −0.762001
$$559$$ 1.00000 0.0422955
$$560$$ −12.0000 −0.507093
$$561$$ 0 0
$$562$$ 60.0000 2.53095
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ 9.00000 0.378633
$$566$$ −32.0000 −1.34506
$$567$$ −9.00000 −0.377964
$$568$$ 0 0
$$569$$ 7.00000 0.293455 0.146728 0.989177i $$-0.453126\pi$$
0.146728 + 0.989177i $$0.453126\pi$$
$$570$$ 0 0
$$571$$ −17.0000 −0.711428 −0.355714 0.934595i $$-0.615762\pi$$
−0.355714 + 0.934595i $$0.615762\pi$$
$$572$$ 12.0000 0.501745
$$573$$ 0 0
$$574$$ −12.0000 −0.500870
$$575$$ 12.0000 0.500435
$$576$$ 24.0000 1.00000
$$577$$ 2.00000 0.0832611 0.0416305 0.999133i $$-0.486745\pi$$
0.0416305 + 0.999133i $$0.486745\pi$$
$$578$$ 2.00000 0.0831890
$$579$$ 0 0
$$580$$ 30.0000 1.24568
$$581$$ −15.0000 −0.622305
$$582$$ 0 0
$$583$$ 54.0000 2.23645
$$584$$ 0 0
$$585$$ −9.00000 −0.372104
$$586$$ 38.0000 1.56977
$$587$$ 39.0000 1.60970 0.804851 0.593477i $$-0.202245\pi$$
0.804851 + 0.593477i $$0.202245\pi$$
$$588$$ 0 0
$$589$$ −15.0000 −0.618064
$$590$$ 48.0000 1.97613
$$591$$ 0 0
$$592$$ 16.0000 0.657596
$$593$$ −27.0000 −1.10876 −0.554379 0.832265i $$-0.687044\pi$$
−0.554379 + 0.832265i $$0.687044\pi$$
$$594$$ 0 0
$$595$$ 12.0000 0.491952
$$596$$ −36.0000 −1.47462
$$597$$ 0 0
$$598$$ 6.00000 0.245358
$$599$$ 11.0000 0.449448 0.224724 0.974422i $$-0.427852\pi$$
0.224724 + 0.974422i $$0.427852\pi$$
$$600$$ 0 0
$$601$$ −10.0000 −0.407909 −0.203954 0.978980i $$-0.565379\pi$$
−0.203954 + 0.978980i $$0.565379\pi$$
$$602$$ −2.00000 −0.0815139
$$603$$ 18.0000 0.733017
$$604$$ 0 0
$$605$$ −75.0000 −3.04918
$$606$$ 0 0
$$607$$ 2.00000 0.0811775 0.0405887 0.999176i $$-0.487077\pi$$
0.0405887 + 0.999176i $$0.487077\pi$$
$$608$$ 40.0000 1.62221
$$609$$ 0 0
$$610$$ −60.0000 −2.42933
$$611$$ −7.00000 −0.283190
$$612$$ −24.0000 −0.970143
$$613$$ 8.00000 0.323117 0.161558 0.986863i $$-0.448348\pi$$
0.161558 + 0.986863i $$0.448348\pi$$
$$614$$ 66.0000 2.66354
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −30.0000 −1.20775 −0.603877 0.797077i $$-0.706378\pi$$
−0.603877 + 0.797077i $$0.706378\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 18.0000 0.722897
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ −3.00000 −0.120192
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ −44.0000 −1.75859
$$627$$ 0 0
$$628$$ −16.0000 −0.638470
$$629$$ −16.0000 −0.637962
$$630$$ 18.0000 0.717137
$$631$$ 22.0000 0.875806 0.437903 0.899022i $$-0.355721\pi$$
0.437903 + 0.899022i $$0.355721\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 48.0000 1.90632
$$635$$ 12.0000 0.476205
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ −60.0000 −2.37542
$$639$$ 24.0000 0.949425
$$640$$ 0 0
$$641$$ 9.00000 0.355479 0.177739 0.984078i $$-0.443122\pi$$
0.177739 + 0.984078i $$0.443122\pi$$
$$642$$ 0 0
$$643$$ 8.00000 0.315489 0.157745 0.987480i $$-0.449578\pi$$
0.157745 + 0.987480i $$0.449578\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ 0 0
$$646$$ −40.0000 −1.57378
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 0 0
$$649$$ −48.0000 −1.88416
$$650$$ 8.00000 0.313786
$$651$$ 0 0
$$652$$ −8.00000 −0.313304
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ −24.0000 −0.937758
$$656$$ 24.0000 0.937043
$$657$$ 39.0000 1.52153
$$658$$ 14.0000 0.545777
$$659$$ 17.0000 0.662226 0.331113 0.943591i $$-0.392576\pi$$
0.331113 + 0.943591i $$0.392576\pi$$
$$660$$ 0 0
$$661$$ −33.0000 −1.28355 −0.641776 0.766892i $$-0.721802\pi$$
−0.641776 + 0.766892i $$0.721802\pi$$
$$662$$ −44.0000 −1.71011
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 15.0000 0.581675
$$666$$ −24.0000 −0.929981
$$667$$ −15.0000 −0.580802
$$668$$ 10.0000 0.386912
$$669$$ 0 0
$$670$$ −36.0000 −1.39080
$$671$$ 60.0000 2.31627
$$672$$ 0 0
$$673$$ 1.00000 0.0385472 0.0192736 0.999814i $$-0.493865\pi$$
0.0192736 + 0.999814i $$0.493865\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 2.00000 0.0769231
$$677$$ 22.0000 0.845529 0.422764 0.906240i $$-0.361060\pi$$
0.422764 + 0.906240i $$0.361060\pi$$
$$678$$ 0 0
$$679$$ −7.00000 −0.268635
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −36.0000 −1.37851
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −30.0000 −1.14708
$$685$$ −12.0000 −0.458496
$$686$$ 2.00000 0.0763604
$$687$$ 0 0
$$688$$ 4.00000 0.152499
$$689$$ 9.00000 0.342873
$$690$$ 0 0
$$691$$ 11.0000 0.418460 0.209230 0.977866i $$-0.432904\pi$$
0.209230 + 0.977866i $$0.432904\pi$$
$$692$$ −16.0000 −0.608229
$$693$$ −18.0000 −0.683763
$$694$$ 64.0000 2.42941
$$695$$ 54.0000 2.04834
$$696$$ 0 0
$$697$$ −24.0000 −0.909065
$$698$$ −22.0000 −0.832712
$$699$$ 0 0
$$700$$ −8.00000 −0.302372
$$701$$ −27.0000 −1.01978 −0.509888 0.860241i $$-0.670313\pi$$
−0.509888 + 0.860241i $$0.670313\pi$$
$$702$$ 0 0
$$703$$ −20.0000 −0.754314
$$704$$ 48.0000 1.80907
$$705$$ 0 0
$$706$$ 20.0000 0.752710
$$707$$ 14.0000 0.526524
$$708$$ 0 0
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ −48.0000 −1.80141
$$711$$ −9.00000 −0.337526
$$712$$ 0 0
$$713$$ −9.00000 −0.337053
$$714$$ 0 0
$$715$$ −18.0000 −0.673162
$$716$$ 46.0000 1.71910
$$717$$ 0 0
$$718$$ −40.0000 −1.49279
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ −36.0000 −1.34164
$$721$$ 4.00000 0.148968
$$722$$ −12.0000 −0.446594
$$723$$ 0 0
$$724$$ 28.0000 1.04061
$$725$$ −20.0000 −0.742781
$$726$$ 0 0
$$727$$ 46.0000 1.70605 0.853023 0.521874i $$-0.174767\pi$$
0.853023 + 0.521874i $$0.174767\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ −78.0000 −2.88691
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ 51.0000 1.88373 0.941864 0.335994i $$-0.109072\pi$$
0.941864 + 0.335994i $$0.109072\pi$$
$$734$$ −28.0000 −1.03350
$$735$$ 0 0
$$736$$ 24.0000 0.884652
$$737$$ 36.0000 1.32608
$$738$$ −36.0000 −1.32518
$$739$$ −26.0000 −0.956425 −0.478213 0.878244i $$-0.658715\pi$$
−0.478213 + 0.878244i $$0.658715\pi$$
$$740$$ 24.0000 0.882258
$$741$$ 0 0
$$742$$ −18.0000 −0.660801
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ 54.0000 1.97841
$$746$$ −60.0000 −2.19676
$$747$$ −45.0000 −1.64646
$$748$$ −48.0000 −1.75505
$$749$$ 4.00000 0.146157
$$750$$ 0 0
$$751$$ −17.0000 −0.620339 −0.310169 0.950681i $$-0.600386\pi$$
−0.310169 + 0.950681i $$0.600386\pi$$
$$752$$ −28.0000 −1.02105
$$753$$ 0 0
$$754$$ −10.0000 −0.364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −15.0000 −0.545184 −0.272592 0.962130i $$-0.587881\pi$$
−0.272592 + 0.962130i $$0.587881\pi$$
$$758$$ 12.0000 0.435860
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 9.00000 0.326250 0.163125 0.986605i $$-0.447843\pi$$
0.163125 + 0.986605i $$0.447843\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ −16.0000 −0.578860
$$765$$ 36.0000 1.30158
$$766$$ 72.0000 2.60147
$$767$$ −8.00000 −0.288863
$$768$$ 0 0
$$769$$ −35.0000 −1.26213 −0.631066 0.775729i $$-0.717382\pi$$
−0.631066 + 0.775729i $$0.717382\pi$$
$$770$$ 36.0000 1.29735
$$771$$ 0 0
$$772$$ 44.0000 1.58359
$$773$$ 54.0000 1.94225 0.971123 0.238581i $$-0.0766824\pi$$
0.971123 + 0.238581i $$0.0766824\pi$$
$$774$$ −6.00000 −0.215666
$$775$$ −12.0000 −0.431053
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −60.0000 −2.15110
$$779$$ −30.0000 −1.07486
$$780$$ 0 0
$$781$$ 48.0000 1.71758
$$782$$ −24.0000 −0.858238
$$783$$ 0 0
$$784$$ −4.00000 −0.142857
$$785$$ 24.0000 0.856597
$$786$$ 0 0
$$787$$ 37.0000 1.31891 0.659454 0.751745i $$-0.270788\pi$$
0.659454 + 0.751745i $$0.270788\pi$$
$$788$$ 4.00000 0.142494
$$789$$ 0 0
$$790$$ 18.0000 0.640411
$$791$$ 3.00000 0.106668
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 26.0000 0.922705
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 28.0000 0.990569
$$800$$ 32.0000 1.13137
$$801$$ −9.00000 −0.317999
$$802$$ 64.0000 2.25992
$$803$$ 78.0000 2.75256
$$804$$ 0 0
$$805$$ 9.00000 0.317208
$$806$$ −6.00000 −0.211341
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −31.0000 −1.08990 −0.544951 0.838468i $$-0.683452\pi$$
−0.544951 + 0.838468i $$0.683452\pi$$
$$810$$ 54.0000 1.89737
$$811$$ −52.0000 −1.82597 −0.912983 0.407997i $$-0.866228\pi$$
−0.912983 + 0.407997i $$0.866228\pi$$
$$812$$ 10.0000 0.350931
$$813$$ 0 0
$$814$$ −48.0000 −1.68240
$$815$$ 12.0000 0.420342
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ 26.0000 0.909069
$$819$$ −3.00000 −0.104828
$$820$$ 36.0000 1.25717
$$821$$ −6.00000 −0.209401 −0.104701 0.994504i $$-0.533388\pi$$
−0.104701 + 0.994504i $$0.533388\pi$$
$$822$$ 0 0
$$823$$ −32.0000 −1.11545 −0.557725 0.830026i $$-0.688326\pi$$
−0.557725 + 0.830026i $$0.688326\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 16.0000 0.556711
$$827$$ 4.00000 0.139094 0.0695468 0.997579i $$-0.477845\pi$$
0.0695468 + 0.997579i $$0.477845\pi$$
$$828$$ −18.0000 −0.625543
$$829$$ −10.0000 −0.347314 −0.173657 0.984806i $$-0.555558\pi$$
−0.173657 + 0.984806i $$0.555558\pi$$
$$830$$ 90.0000 3.12395
$$831$$ 0 0
$$832$$ 8.00000 0.277350
$$833$$ 4.00000 0.138592
$$834$$ 0 0
$$835$$ −15.0000 −0.519096
$$836$$ −60.0000 −2.07514
$$837$$ 0 0
$$838$$ 20.0000 0.690889
$$839$$ −8.00000 −0.276191 −0.138095 0.990419i $$-0.544098\pi$$
−0.138095 + 0.990419i $$0.544098\pi$$
$$840$$ 0 0
$$841$$ −4.00000 −0.137931
$$842$$ 24.0000 0.827095
$$843$$ 0 0
$$844$$ −10.0000 −0.344214
$$845$$ −3.00000 −0.103203
$$846$$ 42.0000 1.44399
$$847$$ −25.0000 −0.859010
$$848$$ 36.0000 1.23625
$$849$$ 0 0
$$850$$ −32.0000 −1.09759
$$851$$ −12.0000 −0.411355
$$852$$ 0 0
$$853$$ 45.0000 1.54077 0.770385 0.637579i $$-0.220064\pi$$
0.770385 + 0.637579i $$0.220064\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 45.0000 1.53897
$$856$$ 0 0
$$857$$ −6.00000 −0.204956 −0.102478 0.994735i $$-0.532677\pi$$
−0.102478 + 0.994735i $$0.532677\pi$$
$$858$$ 0 0
$$859$$ −2.00000 −0.0682391 −0.0341196 0.999418i $$-0.510863\pi$$
−0.0341196 + 0.999418i $$0.510863\pi$$
$$860$$ 6.00000 0.204598
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 0 0
$$865$$ 24.0000 0.816024
$$866$$ −24.0000 −0.815553
$$867$$ 0 0
$$868$$ 6.00000 0.203653
$$869$$ −18.0000 −0.610608
$$870$$ 0 0
$$871$$ 6.00000 0.203302
$$872$$ 0 0
$$873$$ −21.0000 −0.710742
$$874$$ −30.0000 −1.01477
$$875$$ −3.00000 −0.101419
$$876$$ 0 0
$$877$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$878$$ 44.0000 1.48493
$$879$$ 0 0
$$880$$ −72.0000 −2.42712
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 6.00000 0.202031
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ −8.00000 −0.269069
$$885$$ 0 0
$$886$$ −38.0000 −1.27663
$$887$$ −12.0000 −0.402921 −0.201460 0.979497i $$-0.564569\pi$$
−0.201460 + 0.979497i $$0.564569\pi$$
$$888$$ 0 0
$$889$$ 4.00000 0.134156
$$890$$ 18.0000 0.603361
$$891$$ −54.0000 −1.80907
$$892$$ 30.0000 1.00447
$$893$$ 35.0000 1.17123
$$894$$ 0 0
$$895$$ −69.0000 −2.30642
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −72.0000 −2.40267
$$899$$ 15.0000 0.500278
$$900$$ −24.0000 −0.800000
$$901$$ −36.0000 −1.19933
$$902$$ −72.0000 −2.39734
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −42.0000 −1.39613
$$906$$ 0 0
$$907$$ 7.00000 0.232431 0.116216 0.993224i $$-0.462924\pi$$
0.116216 + 0.993224i $$0.462924\pi$$
$$908$$ 40.0000 1.32745
$$909$$ 42.0000 1.39305
$$910$$ 6.00000 0.198898
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ 0 0
$$913$$ −90.0000 −2.97857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 28.0000 0.925146
$$917$$ −8.00000 −0.264183
$$918$$ 0 0
$$919$$ 8.00000 0.263896 0.131948 0.991257i $$-0.457877\pi$$
0.131948 + 0.991257i $$0.457877\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 44.0000 1.44906
$$923$$ 8.00000 0.263323
$$924$$ 0 0
$$925$$ −16.0000 −0.526077
$$926$$ 28.0000 0.920137
$$927$$ 12.0000 0.394132
$$928$$ −40.0000 −1.31306
$$929$$ 5.00000 0.164045 0.0820223 0.996630i $$-0.473862\pi$$
0.0820223 + 0.996630i $$0.473862\pi$$
$$930$$ 0 0
$$931$$ 5.00000 0.163868
$$932$$ 30.0000 0.982683
$$933$$ 0 0
$$934$$ 44.0000 1.43972
$$935$$ 72.0000 2.35465
$$936$$ 0 0
$$937$$ −8.00000 −0.261349 −0.130674 0.991425i $$-0.541714\pi$$
−0.130674 + 0.991425i $$0.541714\pi$$
$$938$$ −12.0000 −0.391814
$$939$$ 0 0
$$940$$ −42.0000 −1.36989
$$941$$ 55.0000 1.79295 0.896474 0.443096i $$-0.146120\pi$$
0.896474 + 0.443096i $$0.146120\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ −32.0000 −1.04151
$$945$$ 0 0
$$946$$ −12.0000 −0.390154
$$947$$ −18.0000 −0.584921 −0.292461 0.956278i $$-0.594474\pi$$
−0.292461 + 0.956278i $$0.594474\pi$$
$$948$$ 0 0
$$949$$ 13.0000 0.421998
$$950$$ −40.0000 −1.29777
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −39.0000 −1.26333 −0.631667 0.775240i $$-0.717629\pi$$
−0.631667 + 0.775240i $$0.717629\pi$$
$$954$$ −54.0000 −1.74831
$$955$$ 24.0000 0.776622
$$956$$ −8.00000 −0.258738
$$957$$ 0 0
$$958$$ 22.0000 0.710788
$$959$$ −4.00000 −0.129167
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −8.00000 −0.257930
$$963$$ 12.0000 0.386695
$$964$$ −34.0000 −1.09507
$$965$$ −66.0000 −2.12462
$$966$$ 0 0
$$967$$ 22.0000 0.707472 0.353736 0.935345i $$-0.384911\pi$$
0.353736 + 0.935345i $$0.384911\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 42.0000 1.34854
$$971$$ 38.0000 1.21948 0.609739 0.792602i $$-0.291274\pi$$
0.609739 + 0.792602i $$0.291274\pi$$
$$972$$ 0 0
$$973$$ 18.0000 0.577054
$$974$$ 52.0000 1.66619
$$975$$ 0 0
$$976$$ 40.0000 1.28037
$$977$$ 10.0000 0.319928 0.159964 0.987123i $$-0.448862\pi$$
0.159964 + 0.987123i $$0.448862\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ −6.00000 −0.191663
$$981$$ 6.00000 0.191565
$$982$$ 24.0000 0.765871
$$983$$ 17.0000 0.542216 0.271108 0.962549i $$-0.412610\pi$$
0.271108 + 0.962549i $$0.412610\pi$$
$$984$$ 0 0
$$985$$ −6.00000 −0.191176
$$986$$ 40.0000 1.27386
$$987$$ 0 0
$$988$$ −10.0000 −0.318142
$$989$$ −3.00000 −0.0953945
$$990$$ 108.000 3.43247
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ −24.0000 −0.762001
$$993$$ 0 0
$$994$$ −16.0000 −0.507489
$$995$$ −12.0000 −0.380426
$$996$$ 0 0
$$997$$ −28.0000 −0.886769 −0.443384 0.896332i $$-0.646222\pi$$
−0.443384 + 0.896332i $$0.646222\pi$$
$$998$$ 32.0000 1.01294
$$999$$ 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 91.2.a.a.1.1 1
3.2 odd 2 819.2.a.f.1.1 1
4.3 odd 2 1456.2.a.g.1.1 1
5.4 even 2 2275.2.a.h.1.1 1
7.2 even 3 637.2.e.e.508.1 2
7.3 odd 6 637.2.e.d.79.1 2
7.4 even 3 637.2.e.e.79.1 2
7.5 odd 6 637.2.e.d.508.1 2
7.6 odd 2 637.2.a.a.1.1 1
8.3 odd 2 5824.2.a.t.1.1 1
8.5 even 2 5824.2.a.s.1.1 1
13.5 odd 4 1183.2.c.b.337.2 2
13.8 odd 4 1183.2.c.b.337.1 2
13.12 even 2 1183.2.a.b.1.1 1
21.20 even 2 5733.2.a.l.1.1 1
91.90 odd 2 8281.2.a.l.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.a.1.1 1 1.1 even 1 trivial
637.2.a.a.1.1 1 7.6 odd 2
637.2.e.d.79.1 2 7.3 odd 6
637.2.e.d.508.1 2 7.5 odd 6
637.2.e.e.79.1 2 7.4 even 3
637.2.e.e.508.1 2 7.2 even 3
819.2.a.f.1.1 1 3.2 odd 2
1183.2.a.b.1.1 1 13.12 even 2
1183.2.c.b.337.1 2 13.8 odd 4
1183.2.c.b.337.2 2 13.5 odd 4
1456.2.a.g.1.1 1 4.3 odd 2
2275.2.a.h.1.1 1 5.4 even 2
5733.2.a.l.1.1 1 21.20 even 2
5824.2.a.s.1.1 1 8.5 even 2
5824.2.a.t.1.1 1 8.3 odd 2
8281.2.a.l.1.1 1 91.90 odd 2