# Properties

 Label 91.2.a.b.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)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} -2.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.00000 q^{12} +1.00000 q^{13} +6.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} +6.00000 q^{20} -2.00000 q^{21} +3.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} -2.00000 q^{28} -9.00000 q^{29} +5.00000 q^{31} -3.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -1.00000 q^{43} -3.00000 q^{45} +3.00000 q^{47} -8.00000 q^{48} +1.00000 q^{49} +12.0000 q^{51} -2.00000 q^{52} -9.00000 q^{53} +14.0000 q^{57} -12.0000 q^{60} -10.0000 q^{61} +1.00000 q^{63} -8.00000 q^{64} -3.00000 q^{65} +14.0000 q^{67} +12.0000 q^{68} -6.00000 q^{69} -6.00000 q^{71} +11.0000 q^{73} -8.00000 q^{75} +14.0000 q^{76} -1.00000 q^{79} -12.0000 q^{80} -11.0000 q^{81} +3.00000 q^{83} +4.00000 q^{84} +18.0000 q^{85} +18.0000 q^{87} +15.0000 q^{89} +1.00000 q^{91} -6.00000 q^{92} -10.0000 q^{93} +21.0000 q^{95} -1.00000 q^{97} +O(q^{100})$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ −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
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 4.00000 1.15470
$$13$$ 1.00000 0.277350
$$14$$ 0 0
$$15$$ 6.00000 1.54919
$$16$$ 4.00000 1.00000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ −7.00000 −1.60591 −0.802955 0.596040i $$-0.796740\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 6.00000 1.34164
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ 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
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ −2.00000 −0.377964
$$29$$ −9.00000 −1.67126 −0.835629 0.549294i $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 0 0
$$31$$ 5.00000 0.898027 0.449013 0.893525i $$-0.351776\pi$$
0.449013 + 0.893525i $$0.351776\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −3.00000 −0.507093
$$36$$ −2.00000 −0.333333
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ −2.00000 −0.320256
$$40$$ 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 $$0.524294\pi$$
$$44$$ 0 0
$$45$$ −3.00000 −0.447214
$$46$$ 0 0
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ −8.00000 −1.15470
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 12.0000 1.68034
$$52$$ −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
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 14.0000 1.85435
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −12.0000 −1.54919
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ −8.00000 −1.00000
$$65$$ −3.00000 −0.372104
$$66$$ 0 0
$$67$$ 14.0000 1.71037 0.855186 0.518321i $$-0.173443\pi$$
0.855186 + 0.518321i $$0.173443\pi$$
$$68$$ 12.0000 1.45521
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 0 0
$$75$$ −8.00000 −0.923760
$$76$$ 14.0000 1.60591
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ −12.0000 −1.34164
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 3.00000 0.329293 0.164646 0.986353i $$-0.447352\pi$$
0.164646 + 0.986353i $$0.447352\pi$$
$$84$$ 4.00000 0.436436
$$85$$ 18.0000 1.95237
$$86$$ 0 0
$$87$$ 18.0000 1.92980
$$88$$ 0 0
$$89$$ 15.0000 1.59000 0.794998 0.606612i $$-0.207472\pi$$
0.794998 + 0.606612i $$0.207472\pi$$
$$90$$ 0 0
$$91$$ 1.00000 0.104828
$$92$$ −6.00000 −0.625543
$$93$$ −10.0000 −1.03695
$$94$$ 0 0
$$95$$ 21.0000 2.15455
$$96$$ 0 0
$$97$$ −1.00000 −0.101535 −0.0507673 0.998711i $$-0.516167\pi$$
−0.0507673 + 0.998711i $$0.516167\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −8.00000 −0.800000
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ 6.00000 0.585540
$$106$$ 0 0
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ −8.00000 −0.769800
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 4.00000 0.377964
$$113$$ 9.00000 0.846649 0.423324 0.905978i $$-0.360863\pi$$
0.423324 + 0.905978i $$0.360863\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ 18.0000 1.67126
$$117$$ 1.00000 0.0924500
$$118$$ 0 0
$$119$$ −6.00000 −0.550019
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 12.0000 1.08200
$$124$$ −10.0000 −0.898027
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 2.00000 0.176090
$$130$$ 0 0
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −7.00000 −0.606977
$$134$$ 0 0
$$135$$ −12.0000 −1.03280
$$136$$ 0 0
$$137$$ 6.00000 0.512615 0.256307 0.966595i $$-0.417494\pi$$
0.256307 + 0.966595i $$0.417494\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 6.00000 0.507093
$$141$$ −6.00000 −0.505291
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 4.00000 0.333333
$$145$$ 27.0000 2.24223
$$146$$ 0 0
$$147$$ −2.00000 −0.164957
$$148$$ −4.00000 −0.328798
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 0 0
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ −15.0000 −1.20483
$$156$$ 4.00000 0.320256
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 18.0000 1.42749
$$160$$ 0 0
$$161$$ 3.00000 0.236433
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 12.0000 0.937043
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −15.0000 −1.16073 −0.580367 0.814355i $$-0.697091\pi$$
−0.580367 + 0.814355i $$0.697091\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ −7.00000 −0.535303
$$172$$ 2.00000 0.152499
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 4.00000 0.302372
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 15.0000 1.12115 0.560576 0.828103i $$-0.310580\pi$$
0.560576 + 0.828103i $$0.310580\pi$$
$$180$$ 6.00000 0.447214
$$181$$ −16.0000 −1.18927 −0.594635 0.803996i $$-0.702704\pi$$
−0.594635 + 0.803996i $$0.702704\pi$$
$$182$$ 0 0
$$183$$ 20.0000 1.47844
$$184$$ 0 0
$$185$$ −6.00000 −0.441129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −6.00000 −0.437595
$$189$$ 4.00000 0.290957
$$190$$ 0 0
$$191$$ 24.0000 1.73658 0.868290 0.496058i $$-0.165220\pi$$
0.868290 + 0.496058i $$0.165220\pi$$
$$192$$ 16.0000 1.15470
$$193$$ −22.0000 −1.58359 −0.791797 0.610784i $$-0.790854\pi$$
−0.791797 + 0.610784i $$0.790854\pi$$
$$194$$ 0 0
$$195$$ 6.00000 0.429669
$$196$$ −2.00000 −0.142857
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ −28.0000 −1.97497
$$202$$ 0 0
$$203$$ −9.00000 −0.631676
$$204$$ −24.0000 −1.68034
$$205$$ 18.0000 1.25717
$$206$$ 0 0
$$207$$ 3.00000 0.208514
$$208$$ 4.00000 0.277350
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ 18.0000 1.23625
$$213$$ 12.0000 0.822226
$$214$$ 0 0
$$215$$ 3.00000 0.204598
$$216$$ 0 0
$$217$$ 5.00000 0.339422
$$218$$ 0 0
$$219$$ −22.0000 −1.48662
$$220$$ 0 0
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ −1.00000 −0.0669650 −0.0334825 0.999439i $$-0.510660\pi$$
−0.0334825 + 0.999439i $$0.510660\pi$$
$$224$$ 0 0
$$225$$ 4.00000 0.266667
$$226$$ 0 0
$$227$$ 24.0000 1.59294 0.796468 0.604681i $$-0.206699\pi$$
0.796468 + 0.604681i $$0.206699\pi$$
$$228$$ −28.0000 −1.85435
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.00000 −0.589610 −0.294805 0.955557i $$-0.595255\pi$$
−0.294805 + 0.955557i $$0.595255\pi$$
$$234$$ 0 0
$$235$$ −9.00000 −0.587095
$$236$$ 0 0
$$237$$ 2.00000 0.129914
$$238$$ 0 0
$$239$$ −12.0000 −0.776215 −0.388108 0.921614i $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 24.0000 1.54919
$$241$$ −1.00000 −0.0644157 −0.0322078 0.999481i $$-0.510254\pi$$
−0.0322078 + 0.999481i $$0.510254\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 20.0000 1.28037
$$245$$ −3.00000 −0.191663
$$246$$ 0 0
$$247$$ −7.00000 −0.445399
$$248$$ 0 0
$$249$$ −6.00000 −0.380235
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ −2.00000 −0.125988
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −36.0000 −2.25441
$$256$$ 16.0000 1.00000
$$257$$ −24.0000 −1.49708 −0.748539 0.663090i $$-0.769245\pi$$
−0.748539 + 0.663090i $$0.769245\pi$$
$$258$$ 0 0
$$259$$ 2.00000 0.124274
$$260$$ 6.00000 0.372104
$$261$$ −9.00000 −0.557086
$$262$$ 0 0
$$263$$ 9.00000 0.554964 0.277482 0.960731i $$-0.410500\pi$$
0.277482 + 0.960731i $$0.410500\pi$$
$$264$$ 0 0
$$265$$ 27.0000 1.65860
$$266$$ 0 0
$$267$$ −30.0000 −1.83597
$$268$$ −28.0000 −1.71037
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −24.0000 −1.45521
$$273$$ −2.00000 −0.121046
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 12.0000 0.722315
$$277$$ −19.0000 −1.14160 −0.570800 0.821089i $$-0.693367\pi$$
−0.570800 + 0.821089i $$0.693367\pi$$
$$278$$ 0 0
$$279$$ 5.00000 0.299342
$$280$$ 0 0
$$281$$ 12.0000 0.715860 0.357930 0.933748i $$-0.383483\pi$$
0.357930 + 0.933748i $$0.383483\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 12.0000 0.712069
$$285$$ −42.0000 −2.48787
$$286$$ 0 0
$$287$$ −6.00000 −0.354169
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 2.00000 0.117242
$$292$$ −22.0000 −1.28745
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 3.00000 0.173494
$$300$$ 16.0000 0.923760
$$301$$ −1.00000 −0.0576390
$$302$$ 0 0
$$303$$ 0 0
$$304$$ −28.0000 −1.60591
$$305$$ 30.0000 1.71780
$$306$$ 0 0
$$307$$ 11.0000 0.627803 0.313902 0.949456i $$-0.398364\pi$$
0.313902 + 0.949456i $$0.398364\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ −12.0000 −0.680458 −0.340229 0.940343i $$-0.610505\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ 8.00000 0.452187 0.226093 0.974106i $$-0.427405\pi$$
0.226093 + 0.974106i $$0.427405\pi$$
$$314$$ 0 0
$$315$$ −3.00000 −0.169031
$$316$$ 2.00000 0.112509
$$317$$ 12.0000 0.673987 0.336994 0.941507i $$-0.390590\pi$$
0.336994 + 0.941507i $$0.390590\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 24.0000 1.34164
$$321$$ 24.0000 1.33955
$$322$$ 0 0
$$323$$ 42.0000 2.33694
$$324$$ 22.0000 1.22222
$$325$$ 4.00000 0.221880
$$326$$ 0 0
$$327$$ 32.0000 1.76960
$$328$$ 0 0
$$329$$ 3.00000 0.165395
$$330$$ 0 0
$$331$$ 26.0000 1.42909 0.714545 0.699590i $$-0.246634\pi$$
0.714545 + 0.699590i $$0.246634\pi$$
$$332$$ −6.00000 −0.329293
$$333$$ 2.00000 0.109599
$$334$$ 0 0
$$335$$ −42.0000 −2.29471
$$336$$ −8.00000 −0.436436
$$337$$ 5.00000 0.272367 0.136184 0.990684i $$-0.456516\pi$$
0.136184 + 0.990684i $$0.456516\pi$$
$$338$$ 0 0
$$339$$ −18.0000 −0.977626
$$340$$ −36.0000 −1.95237
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 18.0000 0.969087
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ −36.0000 −1.92980
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 0 0
$$351$$ 4.00000 0.213504
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 18.0000 0.955341
$$356$$ −30.0000 −1.59000
$$357$$ 12.0000 0.635107
$$358$$ 0 0
$$359$$ 18.0000 0.950004 0.475002 0.879985i $$-0.342447\pi$$
0.475002 + 0.879985i $$0.342447\pi$$
$$360$$ 0 0
$$361$$ 30.0000 1.57895
$$362$$ 0 0
$$363$$ 22.0000 1.15470
$$364$$ −2.00000 −0.104828
$$365$$ −33.0000 −1.72730
$$366$$ 0 0
$$367$$ −28.0000 −1.46159 −0.730794 0.682598i $$-0.760850\pi$$
−0.730794 + 0.682598i $$0.760850\pi$$
$$368$$ 12.0000 0.625543
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −9.00000 −0.467257
$$372$$ 20.0000 1.03695
$$373$$ 14.0000 0.724893 0.362446 0.932005i $$-0.381942\pi$$
0.362446 + 0.932005i $$0.381942\pi$$
$$374$$ 0 0
$$375$$ −6.00000 −0.309839
$$376$$ 0 0
$$377$$ −9.00000 −0.463524
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ −42.0000 −2.15455
$$381$$ 32.0000 1.63941
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −1.00000 −0.0508329
$$388$$ 2.00000 0.101535
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 24.0000 1.21064
$$394$$ 0 0
$$395$$ 3.00000 0.150946
$$396$$ 0 0
$$397$$ 11.0000 0.552074 0.276037 0.961147i $$-0.410979\pi$$
0.276037 + 0.961147i $$0.410979\pi$$
$$398$$ 0 0
$$399$$ 14.0000 0.700877
$$400$$ 16.0000 0.800000
$$401$$ 12.0000 0.599251 0.299626 0.954057i $$-0.403138\pi$$
0.299626 + 0.954057i $$0.403138\pi$$
$$402$$ 0 0
$$403$$ 5.00000 0.249068
$$404$$ 0 0
$$405$$ 33.0000 1.63978
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 23.0000 1.13728 0.568638 0.822588i $$-0.307470\pi$$
0.568638 + 0.822588i $$0.307470\pi$$
$$410$$ 0 0
$$411$$ −12.0000 −0.591916
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −9.00000 −0.441793
$$416$$ 0 0
$$417$$ 8.00000 0.391762
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ −12.0000 −0.585540
$$421$$ −10.0000 −0.487370 −0.243685 0.969854i $$-0.578356\pi$$
−0.243685 + 0.969854i $$0.578356\pi$$
$$422$$ 0 0
$$423$$ 3.00000 0.145865
$$424$$ 0 0
$$425$$ −24.0000 −1.16417
$$426$$ 0 0
$$427$$ −10.0000 −0.483934
$$428$$ 24.0000 1.16008
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −30.0000 −1.44505 −0.722525 0.691345i $$-0.757018\pi$$
−0.722525 + 0.691345i $$0.757018\pi$$
$$432$$ 16.0000 0.769800
$$433$$ −16.0000 −0.768911 −0.384455 0.923144i $$-0.625611\pi$$
−0.384455 + 0.923144i $$0.625611\pi$$
$$434$$ 0 0
$$435$$ −54.0000 −2.58910
$$436$$ 32.0000 1.53252
$$437$$ −21.0000 −1.00457
$$438$$ 0 0
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ 0 0
$$443$$ 39.0000 1.85295 0.926473 0.376361i $$-0.122825\pi$$
0.926473 + 0.376361i $$0.122825\pi$$
$$444$$ 8.00000 0.379663
$$445$$ −45.0000 −2.13320
$$446$$ 0 0
$$447$$ 0 0
$$448$$ −8.00000 −0.377964
$$449$$ −24.0000 −1.13263 −0.566315 0.824189i $$-0.691631\pi$$
−0.566315 + 0.824189i $$0.691631\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −18.0000 −0.846649
$$453$$ 20.0000 0.939682
$$454$$ 0 0
$$455$$ −3.00000 −0.140642
$$456$$ 0 0
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 0 0
$$459$$ −24.0000 −1.12022
$$460$$ 18.0000 0.839254
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −4.00000 −0.185896 −0.0929479 0.995671i $$-0.529629\pi$$
−0.0929479 + 0.995671i $$0.529629\pi$$
$$464$$ −36.0000 −1.67126
$$465$$ 30.0000 1.39122
$$466$$ 0 0
$$467$$ −36.0000 −1.66588 −0.832941 0.553362i $$-0.813345\pi$$
−0.832941 + 0.553362i $$0.813345\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 14.0000 0.646460
$$470$$ 0 0
$$471$$ −28.0000 −1.29017
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −28.0000 −1.28473
$$476$$ 12.0000 0.550019
$$477$$ −9.00000 −0.412082
$$478$$ 0 0
$$479$$ 9.00000 0.411220 0.205610 0.978634i $$-0.434082\pi$$
0.205610 + 0.978634i $$0.434082\pi$$
$$480$$ 0 0
$$481$$ 2.00000 0.0911922
$$482$$ 0 0
$$483$$ −6.00000 −0.273009
$$484$$ 22.0000 1.00000
$$485$$ 3.00000 0.136223
$$486$$ 0 0
$$487$$ 2.00000 0.0906287 0.0453143 0.998973i $$-0.485571\pi$$
0.0453143 + 0.998973i $$0.485571\pi$$
$$488$$ 0 0
$$489$$ 32.0000 1.44709
$$490$$ 0 0
$$491$$ −24.0000 −1.08310 −0.541552 0.840667i $$-0.682163\pi$$
−0.541552 + 0.840667i $$0.682163\pi$$
$$492$$ −24.0000 −1.08200
$$493$$ 54.0000 2.43204
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 20.0000 0.898027
$$497$$ −6.00000 −0.269137
$$498$$ 0 0
$$499$$ 14.0000 0.626726 0.313363 0.949633i $$-0.398544\pi$$
0.313363 + 0.949633i $$0.398544\pi$$
$$500$$ −6.00000 −0.268328
$$501$$ 30.0000 1.34030
$$502$$ 0 0
$$503$$ −6.00000 −0.267527 −0.133763 0.991013i $$-0.542706\pi$$
−0.133763 + 0.991013i $$0.542706\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −2.00000 −0.0888231
$$508$$ 32.0000 1.41977
$$509$$ 21.0000 0.930809 0.465404 0.885098i $$-0.345909\pi$$
0.465404 + 0.885098i $$0.345909\pi$$
$$510$$ 0 0
$$511$$ 11.0000 0.486611
$$512$$ 0 0
$$513$$ −28.0000 −1.23623
$$514$$ 0 0
$$515$$ 12.0000 0.528783
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ 2.00000 0.0874539 0.0437269 0.999044i $$-0.486077\pi$$
0.0437269 + 0.999044i $$0.486077\pi$$
$$524$$ 24.0000 1.04844
$$525$$ −8.00000 −0.349149
$$526$$ 0 0
$$527$$ −30.0000 −1.30682
$$528$$ 0 0
$$529$$ −14.0000 −0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 14.0000 0.606977
$$533$$ −6.00000 −0.259889
$$534$$ 0 0
$$535$$ 36.0000 1.55642
$$536$$ 0 0
$$537$$ −30.0000 −1.29460
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 24.0000 1.03280
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 0 0
$$543$$ 32.0000 1.37325
$$544$$ 0 0
$$545$$ 48.0000 2.05609
$$546$$ 0 0
$$547$$ 17.0000 0.726868 0.363434 0.931620i $$-0.381604\pi$$
0.363434 + 0.931620i $$0.381604\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ −10.0000 −0.426790
$$550$$ 0 0
$$551$$ 63.0000 2.68389
$$552$$ 0 0
$$553$$ −1.00000 −0.0425243
$$554$$ 0 0
$$555$$ 12.0000 0.509372
$$556$$ 8.00000 0.339276
$$557$$ −12.0000 −0.508456 −0.254228 0.967144i $$-0.581821\pi$$
−0.254228 + 0.967144i $$0.581821\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ −12.0000 −0.507093
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 12.0000 0.505291
$$565$$ −27.0000 −1.13590
$$566$$ 0 0
$$567$$ −11.0000 −0.461957
$$568$$ 0 0
$$569$$ 27.0000 1.13190 0.565949 0.824440i $$-0.308510\pi$$
0.565949 + 0.824440i $$0.308510\pi$$
$$570$$ 0 0
$$571$$ 23.0000 0.962520 0.481260 0.876578i $$-0.340179\pi$$
0.481260 + 0.876578i $$0.340179\pi$$
$$572$$ 0 0
$$573$$ −48.0000 −2.00523
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ −8.00000 −0.333333
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 0 0
$$579$$ 44.0000 1.82858
$$580$$ −54.0000 −2.24223
$$581$$ 3.00000 0.124461
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −3.00000 −0.124035
$$586$$ 0 0
$$587$$ −9.00000 −0.371470 −0.185735 0.982600i $$-0.559467\pi$$
−0.185735 + 0.982600i $$0.559467\pi$$
$$588$$ 4.00000 0.164957
$$589$$ −35.0000 −1.44215
$$590$$ 0 0
$$591$$ −12.0000 −0.493614
$$592$$ 8.00000 0.328798
$$593$$ 9.00000 0.369586 0.184793 0.982777i $$-0.440839\pi$$
0.184793 + 0.982777i $$0.440839\pi$$
$$594$$ 0 0
$$595$$ 18.0000 0.737928
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ 15.0000 0.612883 0.306442 0.951889i $$-0.400862\pi$$
0.306442 + 0.951889i $$0.400862\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 14.0000 0.570124
$$604$$ 20.0000 0.813788
$$605$$ 33.0000 1.34164
$$606$$ 0 0
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 0 0
$$609$$ 18.0000 0.729397
$$610$$ 0 0
$$611$$ 3.00000 0.121367
$$612$$ 12.0000 0.485071
$$613$$ −16.0000 −0.646234 −0.323117 0.946359i $$-0.604731\pi$$
−0.323117 + 0.946359i $$0.604731\pi$$
$$614$$ 0 0
$$615$$ −36.0000 −1.45166
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ −28.0000 −1.12542 −0.562708 0.826656i $$-0.690240\pi$$
−0.562708 + 0.826656i $$0.690240\pi$$
$$620$$ 30.0000 1.20483
$$621$$ 12.0000 0.481543
$$622$$ 0 0
$$623$$ 15.0000 0.600962
$$624$$ −8.00000 −0.320256
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −28.0000 −1.11732
$$629$$ −12.0000 −0.478471
$$630$$ 0 0
$$631$$ 2.00000 0.0796187 0.0398094 0.999207i $$-0.487325\pi$$
0.0398094 + 0.999207i $$0.487325\pi$$
$$632$$ 0 0
$$633$$ −46.0000 −1.82834
$$634$$ 0 0
$$635$$ 48.0000 1.90482
$$636$$ −36.0000 −1.42749
$$637$$ 1.00000 0.0396214
$$638$$ 0 0
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ −39.0000 −1.54041 −0.770204 0.637798i $$-0.779845\pi$$
−0.770204 + 0.637798i $$0.779845\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ −6.00000 −0.236433
$$645$$ −6.00000 −0.236250
$$646$$ 0 0
$$647$$ −48.0000 −1.88707 −0.943537 0.331266i $$-0.892524\pi$$
−0.943537 + 0.331266i $$0.892524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −10.0000 −0.391931
$$652$$ 32.0000 1.25322
$$653$$ −6.00000 −0.234798 −0.117399 0.993085i $$-0.537456\pi$$
−0.117399 + 0.993085i $$0.537456\pi$$
$$654$$ 0 0
$$655$$ 36.0000 1.40664
$$656$$ −24.0000 −0.937043
$$657$$ 11.0000 0.429151
$$658$$ 0 0
$$659$$ −3.00000 −0.116863 −0.0584317 0.998291i $$-0.518610\pi$$
−0.0584317 + 0.998291i $$0.518610\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 12.0000 0.466041
$$664$$ 0 0
$$665$$ 21.0000 0.814345
$$666$$ 0 0
$$667$$ −27.0000 −1.04544
$$668$$ 30.0000 1.16073
$$669$$ 2.00000 0.0773245
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 0 0
$$675$$ 16.0000 0.615840
$$676$$ −2.00000 −0.0769231
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ −1.00000 −0.0383765
$$680$$ 0 0
$$681$$ −48.0000 −1.83936
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 14.0000 0.535303
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ −28.0000 −1.06827
$$688$$ −4.00000 −0.152499
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ −37.0000 −1.40755 −0.703773 0.710425i $$-0.748503\pi$$
−0.703773 + 0.710425i $$0.748503\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000 0.455186
$$696$$ 0 0
$$697$$ 36.0000 1.36360
$$698$$ 0 0
$$699$$ 18.0000 0.680823
$$700$$ −8.00000 −0.302372
$$701$$ 9.00000 0.339925 0.169963 0.985451i $$-0.445635\pi$$
0.169963 + 0.985451i $$0.445635\pi$$
$$702$$ 0 0
$$703$$ −14.0000 −0.528020
$$704$$ 0 0
$$705$$ 18.0000 0.677919
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 8.00000 0.300446 0.150223 0.988652i $$-0.452001\pi$$
0.150223 + 0.988652i $$0.452001\pi$$
$$710$$ 0 0
$$711$$ −1.00000 −0.0375029
$$712$$ 0 0
$$713$$ 15.0000 0.561754
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −30.0000 −1.12115
$$717$$ 24.0000 0.896296
$$718$$ 0 0
$$719$$ 12.0000 0.447524 0.223762 0.974644i $$-0.428166\pi$$
0.223762 + 0.974644i $$0.428166\pi$$
$$720$$ −12.0000 −0.447214
$$721$$ −4.00000 −0.148968
$$722$$ 0 0
$$723$$ 2.00000 0.0743808
$$724$$ 32.0000 1.18927
$$725$$ −36.0000 −1.33701
$$726$$ 0 0
$$727$$ 26.0000 0.964287 0.482143 0.876092i $$-0.339858\pi$$
0.482143 + 0.876092i $$0.339858\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 6.00000 0.221918
$$732$$ −40.0000 −1.47844
$$733$$ −49.0000 −1.80986 −0.904928 0.425564i $$-0.860076\pi$$
−0.904928 + 0.425564i $$0.860076\pi$$
$$734$$ 0 0
$$735$$ 6.00000 0.221313
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 12.0000 0.441129
$$741$$ 14.0000 0.514303
$$742$$ 0 0
$$743$$ 48.0000 1.76095 0.880475 0.474093i $$-0.157224\pi$$
0.880475 + 0.474093i $$0.157224\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 3.00000 0.109764
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ −13.0000 −0.474377 −0.237188 0.971464i $$-0.576226\pi$$
−0.237188 + 0.971464i $$0.576226\pi$$
$$752$$ 12.0000 0.437595
$$753$$ −24.0000 −0.874609
$$754$$ 0 0
$$755$$ 30.0000 1.09181
$$756$$ −8.00000 −0.290957
$$757$$ −43.0000 −1.56286 −0.781431 0.623992i $$-0.785510\pi$$
−0.781431 + 0.623992i $$0.785510\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ −16.0000 −0.579239
$$764$$ −48.0000 −1.73658
$$765$$ 18.0000 0.650791
$$766$$ 0 0
$$767$$ 0 0
$$768$$ −32.0000 −1.15470
$$769$$ 5.00000 0.180305 0.0901523 0.995928i $$-0.471265\pi$$
0.0901523 + 0.995928i $$0.471265\pi$$
$$770$$ 0 0
$$771$$ 48.0000 1.72868
$$772$$ 44.0000 1.58359
$$773$$ 30.0000 1.07903 0.539513 0.841978i $$-0.318609\pi$$
0.539513 + 0.841978i $$0.318609\pi$$
$$774$$ 0 0
$$775$$ 20.0000 0.718421
$$776$$ 0 0
$$777$$ −4.00000 −0.143499
$$778$$ 0 0
$$779$$ 42.0000 1.50481
$$780$$ −12.0000 −0.429669
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −36.0000 −1.28654
$$784$$ 4.00000 0.142857
$$785$$ −42.0000 −1.49904
$$786$$ 0 0
$$787$$ 5.00000 0.178231 0.0891154 0.996021i $$-0.471596\pi$$
0.0891154 + 0.996021i $$0.471596\pi$$
$$788$$ −12.0000 −0.427482
$$789$$ −18.0000 −0.640817
$$790$$ 0 0
$$791$$ 9.00000 0.320003
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ 0 0
$$795$$ −54.0000 −1.91518
$$796$$ −4.00000 −0.141776
$$797$$ 54.0000 1.91278 0.956389 0.292096i $$-0.0943526\pi$$
0.956389 + 0.292096i $$0.0943526\pi$$
$$798$$ 0 0
$$799$$ −18.0000 −0.636794
$$800$$ 0 0
$$801$$ 15.0000 0.529999
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 56.0000 1.97497
$$805$$ −9.00000 −0.317208
$$806$$ 0 0
$$807$$ 48.0000 1.68968
$$808$$ 0 0
$$809$$ −3.00000 −0.105474 −0.0527372 0.998608i $$-0.516795\pi$$
−0.0527372 + 0.998608i $$0.516795\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 18.0000 0.631676
$$813$$ 32.0000 1.12229
$$814$$ 0 0
$$815$$ 48.0000 1.68137
$$816$$ 48.0000 1.68034
$$817$$ 7.00000 0.244899
$$818$$ 0 0
$$819$$ 1.00000 0.0349428
$$820$$ −36.0000 −1.25717
$$821$$ −54.0000 −1.88461 −0.942306 0.334751i $$-0.891348\pi$$
−0.942306 + 0.334751i $$0.891348\pi$$
$$822$$ 0 0
$$823$$ 32.0000 1.11545 0.557725 0.830026i $$-0.311674\pi$$
0.557725 + 0.830026i $$0.311674\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ −6.00000 −0.208514
$$829$$ 20.0000 0.694629 0.347314 0.937749i $$-0.387094\pi$$
0.347314 + 0.937749i $$0.387094\pi$$
$$830$$ 0 0
$$831$$ 38.0000 1.31821
$$832$$ −8.00000 −0.277350
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 45.0000 1.55729
$$836$$ 0 0
$$837$$ 20.0000 0.691301
$$838$$ 0 0
$$839$$ −12.0000 −0.414286 −0.207143 0.978311i $$-0.566417\pi$$
−0.207143 + 0.978311i $$0.566417\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 0 0
$$843$$ −24.0000 −0.826604
$$844$$ −46.0000 −1.58339
$$845$$ −3.00000 −0.103203
$$846$$ 0 0
$$847$$ −11.0000 −0.377964
$$848$$ −36.0000 −1.23625
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ −24.0000 −0.822226
$$853$$ 17.0000 0.582069 0.291034 0.956713i $$-0.406001\pi$$
0.291034 + 0.956713i $$0.406001\pi$$
$$854$$ 0 0
$$855$$ 21.0000 0.718185
$$856$$ 0 0
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 50.0000 1.70598 0.852989 0.521929i $$-0.174787\pi$$
0.852989 + 0.521929i $$0.174787\pi$$
$$860$$ −6.00000 −0.204598
$$861$$ 12.0000 0.408959
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ −18.0000 −0.612018
$$866$$ 0 0
$$867$$ −38.0000 −1.29055
$$868$$ −10.0000 −0.339422
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 14.0000 0.474372
$$872$$ 0 0
$$873$$ −1.00000 −0.0338449
$$874$$ 0 0
$$875$$ 3.00000 0.101419
$$876$$ 44.0000 1.48662
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ −18.0000 −0.607125
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 0 0
$$883$$ −16.0000 −0.538443 −0.269221 0.963078i $$-0.586766\pi$$
−0.269221 + 0.963078i $$0.586766\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −24.0000 −0.805841 −0.402921 0.915235i $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 2.00000 0.0669650
$$893$$ −21.0000 −0.702738
$$894$$ 0 0
$$895$$ −45.0000 −1.50418
$$896$$ 0 0
$$897$$ −6.00000 −0.200334
$$898$$ 0 0
$$899$$ −45.0000 −1.50083
$$900$$ −8.00000 −0.266667
$$901$$ 54.0000 1.79900
$$902$$ 0 0
$$903$$ 2.00000 0.0665558
$$904$$ 0 0
$$905$$ 48.0000 1.59557
$$906$$ 0 0
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ −48.0000 −1.59294
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −15.0000 −0.496972 −0.248486 0.968635i $$-0.579933\pi$$
−0.248486 + 0.968635i $$0.579933\pi$$
$$912$$ 56.0000 1.85435
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −60.0000 −1.98354
$$916$$ −28.0000 −0.925146
$$917$$ −12.0000 −0.396275
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ −22.0000 −0.724925
$$922$$ 0 0
$$923$$ −6.00000 −0.197492
$$924$$ 0 0
$$925$$ 8.00000 0.263038
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ −3.00000 −0.0984268 −0.0492134 0.998788i $$-0.515671\pi$$
−0.0492134 + 0.998788i $$0.515671\pi$$
$$930$$ 0 0
$$931$$ −7.00000 −0.229416
$$932$$ 18.0000 0.589610
$$933$$ 24.0000 0.785725
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 2.00000 0.0653372 0.0326686 0.999466i $$-0.489599\pi$$
0.0326686 + 0.999466i $$0.489599\pi$$
$$938$$ 0 0
$$939$$ −16.0000 −0.522140
$$940$$ 18.0000 0.587095
$$941$$ 15.0000 0.488986 0.244493 0.969651i $$-0.421378\pi$$
0.244493 + 0.969651i $$0.421378\pi$$
$$942$$ 0 0
$$943$$ −18.0000 −0.586161
$$944$$ 0 0
$$945$$ −12.0000 −0.390360
$$946$$ 0 0
$$947$$ 54.0000 1.75476 0.877382 0.479792i $$-0.159288\pi$$
0.877382 + 0.479792i $$0.159288\pi$$
$$948$$ −4.00000 −0.129914
$$949$$ 11.0000 0.357075
$$950$$ 0 0
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ 21.0000 0.680257 0.340128 0.940379i $$-0.389529\pi$$
0.340128 + 0.940379i $$0.389529\pi$$
$$954$$ 0 0
$$955$$ −72.0000 −2.32987
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 6.00000 0.193750
$$960$$ −48.0000 −1.54919
$$961$$ −6.00000 −0.193548
$$962$$ 0 0
$$963$$ −12.0000 −0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 66.0000 2.12462
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ 0 0
$$969$$ −84.0000 −2.69847
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ −20.0000 −0.641500
$$973$$ −4.00000 −0.128234
$$974$$ 0 0
$$975$$ −8.00000 −0.256205
$$976$$ −40.0000 −1.28037
$$977$$ 48.0000 1.53566 0.767828 0.640656i $$-0.221338\pi$$
0.767828 + 0.640656i $$0.221338\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 6.00000 0.191663
$$981$$ −16.0000 −0.510841
$$982$$ 0 0
$$983$$ 57.0000 1.81802 0.909009 0.416777i $$-0.136840\pi$$
0.909009 + 0.416777i $$0.136840\pi$$
$$984$$ 0 0
$$985$$ −18.0000 −0.573528
$$986$$ 0 0
$$987$$ −6.00000 −0.190982
$$988$$ 14.0000 0.445399
$$989$$ −3.00000 −0.0953945
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ −52.0000 −1.65017
$$994$$ 0 0
$$995$$ −6.00000 −0.190213
$$996$$ 12.0000 0.380235
$$997$$ 8.00000 0.253363 0.126681 0.991943i $$-0.459567\pi$$
0.126681 + 0.991943i $$0.459567\pi$$
$$998$$ 0 0
$$999$$ 8.00000 0.253109
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.b.1.1 1
3.2 odd 2 819.2.a.c.1.1 1
4.3 odd 2 1456.2.a.k.1.1 1
5.4 even 2 2275.2.a.d.1.1 1
7.2 even 3 637.2.e.c.508.1 2
7.3 odd 6 637.2.e.b.79.1 2
7.4 even 3 637.2.e.c.79.1 2
7.5 odd 6 637.2.e.b.508.1 2
7.6 odd 2 637.2.a.b.1.1 1
8.3 odd 2 5824.2.a.f.1.1 1
8.5 even 2 5824.2.a.bd.1.1 1
13.5 odd 4 1183.2.c.a.337.2 2
13.8 odd 4 1183.2.c.a.337.1 2
13.12 even 2 1183.2.a.a.1.1 1
21.20 even 2 5733.2.a.f.1.1 1
91.90 odd 2 8281.2.a.h.1.1 1

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