# Properties

 Label 147.2.a.b.1.1 Level $147$ Weight $2$ Character 147.1 Self dual yes Analytic conductor $1.174$ 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] = [147,2,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$147 = 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 147.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: $$1.17380090971$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 147.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{9} +4.00000 q^{10} -2.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} -4.00000 q^{16} +2.00000 q^{18} -1.00000 q^{19} +4.00000 q^{20} -4.00000 q^{22} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} +4.00000 q^{29} -4.00000 q^{30} -9.00000 q^{31} -8.00000 q^{32} +2.00000 q^{33} +2.00000 q^{36} +3.00000 q^{37} -2.00000 q^{38} +1.00000 q^{39} +10.0000 q^{41} +5.00000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +6.00000 q^{47} +4.00000 q^{48} -2.00000 q^{50} -2.00000 q^{52} +12.0000 q^{53} -2.00000 q^{54} -4.00000 q^{55} +1.00000 q^{57} +8.00000 q^{58} +12.0000 q^{59} -4.00000 q^{60} -10.0000 q^{61} -18.0000 q^{62} -8.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} -5.00000 q^{67} -6.00000 q^{71} +3.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -2.00000 q^{76} +2.00000 q^{78} -1.00000 q^{79} -8.00000 q^{80} +1.00000 q^{81} +20.0000 q^{82} -6.00000 q^{83} +10.0000 q^{86} -4.00000 q^{87} -16.0000 q^{89} +4.00000 q^{90} +9.00000 q^{93} +12.0000 q^{94} -2.00000 q^{95} +8.00000 q^{96} +6.00000 q^{97} -2.00000 q^{99} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000 1.41421 0.707107 0.707107i $$-0.250000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$3$$ −1.00000 −0.577350
$$4$$ 2.00000 1.00000
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ −2.00000 −0.816497
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 4.00000 1.26491
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ −2.00000 −0.577350
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 0 0
$$15$$ −2.00000 −0.516398
$$16$$ −4.00000 −1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 2.00000 0.471405
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 4.00000 0.894427
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ −2.00000 −0.392232
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ 4.00000 0.742781 0.371391 0.928477i $$-0.378881\pi$$
0.371391 + 0.928477i $$0.378881\pi$$
$$30$$ −4.00000 −0.730297
$$31$$ −9.00000 −1.61645 −0.808224 0.588875i $$-0.799571\pi$$
−0.808224 + 0.588875i $$0.799571\pi$$
$$32$$ −8.00000 −1.41421
$$33$$ 2.00000 0.348155
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ 3.00000 0.493197 0.246598 0.969118i $$-0.420687\pi$$
0.246598 + 0.969118i $$0.420687\pi$$
$$38$$ −2.00000 −0.324443
$$39$$ 1.00000 0.160128
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 5.00000 0.762493 0.381246 0.924473i $$-0.375495\pi$$
0.381246 + 0.924473i $$0.375495\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 2.00000 0.298142
$$46$$ 0 0
$$47$$ 6.00000 0.875190 0.437595 0.899172i $$-0.355830\pi$$
0.437595 + 0.899172i $$0.355830\pi$$
$$48$$ 4.00000 0.577350
$$49$$ 0 0
$$50$$ −2.00000 −0.282843
$$51$$ 0 0
$$52$$ −2.00000 −0.277350
$$53$$ 12.0000 1.64833 0.824163 0.566352i $$-0.191646\pi$$
0.824163 + 0.566352i $$0.191646\pi$$
$$54$$ −2.00000 −0.272166
$$55$$ −4.00000 −0.539360
$$56$$ 0 0
$$57$$ 1.00000 0.132453
$$58$$ 8.00000 1.05045
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ −4.00000 −0.516398
$$61$$ −10.0000 −1.28037 −0.640184 0.768221i $$-0.721142\pi$$
−0.640184 + 0.768221i $$0.721142\pi$$
$$62$$ −18.0000 −2.28600
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ −2.00000 −0.248069
$$66$$ 4.00000 0.492366
$$67$$ −5.00000 −0.610847 −0.305424 0.952217i $$-0.598798\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ 3.00000 0.351123 0.175562 0.984468i $$-0.443826\pi$$
0.175562 + 0.984468i $$0.443826\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 1.00000 0.115470
$$76$$ −2.00000 −0.229416
$$77$$ 0 0
$$78$$ 2.00000 0.226455
$$79$$ −1.00000 −0.112509 −0.0562544 0.998416i $$-0.517916\pi$$
−0.0562544 + 0.998416i $$0.517916\pi$$
$$80$$ −8.00000 −0.894427
$$81$$ 1.00000 0.111111
$$82$$ 20.0000 2.20863
$$83$$ −6.00000 −0.658586 −0.329293 0.944228i $$-0.606810\pi$$
−0.329293 + 0.944228i $$0.606810\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 10.0000 1.07833
$$87$$ −4.00000 −0.428845
$$88$$ 0 0
$$89$$ −16.0000 −1.69600 −0.847998 0.529999i $$-0.822192\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 4.00000 0.421637
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 9.00000 0.933257
$$94$$ 12.0000 1.23771
$$95$$ −2.00000 −0.205196
$$96$$ 8.00000 0.816497
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ −2.00000 −0.201008
$$100$$ −2.00000 −0.200000
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 0 0
$$103$$ 7.00000 0.689730 0.344865 0.938652i $$-0.387925\pi$$
0.344865 + 0.938652i $$0.387925\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 24.0000 2.33109
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −2.00000 −0.192450
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ −8.00000 −0.762770
$$111$$ −3.00000 −0.284747
$$112$$ 0 0
$$113$$ 10.0000 0.940721 0.470360 0.882474i $$-0.344124\pi$$
0.470360 + 0.882474i $$0.344124\pi$$
$$114$$ 2.00000 0.187317
$$115$$ 0 0
$$116$$ 8.00000 0.742781
$$117$$ −1.00000 −0.0924500
$$118$$ 24.0000 2.20938
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ −20.0000 −1.81071
$$123$$ −10.0000 −0.901670
$$124$$ −18.0000 −1.61645
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −15.0000 −1.33103 −0.665517 0.746382i $$-0.731789\pi$$
−0.665517 + 0.746382i $$0.731789\pi$$
$$128$$ 0 0
$$129$$ −5.00000 −0.440225
$$130$$ −4.00000 −0.350823
$$131$$ 14.0000 1.22319 0.611593 0.791173i $$-0.290529\pi$$
0.611593 + 0.791173i $$0.290529\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −10.0000 −0.863868
$$135$$ −2.00000 −0.172133
$$136$$ 0 0
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 3.00000 0.254457 0.127228 0.991873i $$-0.459392\pi$$
0.127228 + 0.991873i $$0.459392\pi$$
$$140$$ 0 0
$$141$$ −6.00000 −0.505291
$$142$$ −12.0000 −1.00702
$$143$$ 2.00000 0.167248
$$144$$ −4.00000 −0.333333
$$145$$ 8.00000 0.664364
$$146$$ 6.00000 0.496564
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ −12.0000 −0.983078 −0.491539 0.870855i $$-0.663566\pi$$
−0.491539 + 0.870855i $$0.663566\pi$$
$$150$$ 2.00000 0.163299
$$151$$ −16.0000 −1.30206 −0.651031 0.759051i $$-0.725663\pi$$
−0.651031 + 0.759051i $$0.725663\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −18.0000 −1.44579
$$156$$ 2.00000 0.160128
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ −12.0000 −0.951662
$$160$$ −16.0000 −1.26491
$$161$$ 0 0
$$162$$ 2.00000 0.157135
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 20.0000 1.56174
$$165$$ 4.00000 0.311400
$$166$$ −12.0000 −0.931381
$$167$$ 14.0000 1.08335 0.541676 0.840587i $$-0.317790\pi$$
0.541676 + 0.840587i $$0.317790\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 10.0000 0.762493
$$173$$ −8.00000 −0.608229 −0.304114 0.952636i $$-0.598361\pi$$
−0.304114 + 0.952636i $$0.598361\pi$$
$$174$$ −8.00000 −0.606478
$$175$$ 0 0
$$176$$ 8.00000 0.603023
$$177$$ −12.0000 −0.901975
$$178$$ −32.0000 −2.39850
$$179$$ 2.00000 0.149487 0.0747435 0.997203i $$-0.476186\pi$$
0.0747435 + 0.997203i $$0.476186\pi$$
$$180$$ 4.00000 0.298142
$$181$$ −13.0000 −0.966282 −0.483141 0.875542i $$-0.660504\pi$$
−0.483141 + 0.875542i $$0.660504\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ 0 0
$$185$$ 6.00000 0.441129
$$186$$ 18.0000 1.31982
$$187$$ 0 0
$$188$$ 12.0000 0.875190
$$189$$ 0 0
$$190$$ −4.00000 −0.290191
$$191$$ 10.0000 0.723575 0.361787 0.932261i $$-0.382167\pi$$
0.361787 + 0.932261i $$0.382167\pi$$
$$192$$ 8.00000 0.577350
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 12.0000 0.861550
$$195$$ 2.00000 0.143223
$$196$$ 0 0
$$197$$ 16.0000 1.13995 0.569976 0.821661i $$-0.306952\pi$$
0.569976 + 0.821661i $$0.306952\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 5.00000 0.352673
$$202$$ −4.00000 −0.281439
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.0000 1.39686
$$206$$ 14.0000 0.975426
$$207$$ 0 0
$$208$$ 4.00000 0.277350
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 24.0000 1.64833
$$213$$ 6.00000 0.411113
$$214$$ −16.0000 −1.09374
$$215$$ 10.0000 0.681994
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 18.0000 1.21911
$$219$$ −3.00000 −0.202721
$$220$$ −8.00000 −0.539360
$$221$$ 0 0
$$222$$ −6.00000 −0.402694
$$223$$ −16.0000 −1.07144 −0.535720 0.844396i $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ 0 0
$$225$$ −1.00000 −0.0666667
$$226$$ 20.0000 1.33038
$$227$$ −18.0000 −1.19470 −0.597351 0.801980i $$-0.703780\pi$$
−0.597351 + 0.801980i $$0.703780\pi$$
$$228$$ 2.00000 0.132453
$$229$$ 19.0000 1.25556 0.627778 0.778393i $$-0.283965\pi$$
0.627778 + 0.778393i $$0.283965\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 12.0000 0.782794
$$236$$ 24.0000 1.56227
$$237$$ 1.00000 0.0649570
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 8.00000 0.516398
$$241$$ −14.0000 −0.901819 −0.450910 0.892570i $$-0.648900\pi$$
−0.450910 + 0.892570i $$0.648900\pi$$
$$242$$ −14.0000 −0.899954
$$243$$ −1.00000 −0.0641500
$$244$$ −20.0000 −1.28037
$$245$$ 0 0
$$246$$ −20.0000 −1.27515
$$247$$ 1.00000 0.0636285
$$248$$ 0 0
$$249$$ 6.00000 0.380235
$$250$$ −24.0000 −1.51789
$$251$$ 8.00000 0.504956 0.252478 0.967603i $$-0.418755\pi$$
0.252478 + 0.967603i $$0.418755\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ −30.0000 −1.88237
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ −26.0000 −1.62184 −0.810918 0.585160i $$-0.801032\pi$$
−0.810918 + 0.585160i $$0.801032\pi$$
$$258$$ −10.0000 −0.622573
$$259$$ 0 0
$$260$$ −4.00000 −0.248069
$$261$$ 4.00000 0.247594
$$262$$ 28.0000 1.72985
$$263$$ 4.00000 0.246651 0.123325 0.992366i $$-0.460644\pi$$
0.123325 + 0.992366i $$0.460644\pi$$
$$264$$ 0 0
$$265$$ 24.0000 1.47431
$$266$$ 0 0
$$267$$ 16.0000 0.979184
$$268$$ −10.0000 −0.610847
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ −4.00000 −0.243432
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −24.0000 −1.44989
$$275$$ 2.00000 0.120605
$$276$$ 0 0
$$277$$ 13.0000 0.781094 0.390547 0.920583i $$-0.372286\pi$$
0.390547 + 0.920583i $$0.372286\pi$$
$$278$$ 6.00000 0.359856
$$279$$ −9.00000 −0.538816
$$280$$ 0 0
$$281$$ −4.00000 −0.238620 −0.119310 0.992857i $$-0.538068\pi$$
−0.119310 + 0.992857i $$0.538068\pi$$
$$282$$ −12.0000 −0.714590
$$283$$ 11.0000 0.653882 0.326941 0.945045i $$-0.393982\pi$$
0.326941 + 0.945045i $$0.393982\pi$$
$$284$$ −12.0000 −0.712069
$$285$$ 2.00000 0.118470
$$286$$ 4.00000 0.236525
$$287$$ 0 0
$$288$$ −8.00000 −0.471405
$$289$$ −17.0000 −1.00000
$$290$$ 16.0000 0.939552
$$291$$ −6.00000 −0.351726
$$292$$ 6.00000 0.351123
$$293$$ −8.00000 −0.467365 −0.233682 0.972313i $$-0.575078\pi$$
−0.233682 + 0.972313i $$0.575078\pi$$
$$294$$ 0 0
$$295$$ 24.0000 1.39733
$$296$$ 0 0
$$297$$ 2.00000 0.116052
$$298$$ −24.0000 −1.39028
$$299$$ 0 0
$$300$$ 2.00000 0.115470
$$301$$ 0 0
$$302$$ −32.0000 −1.84139
$$303$$ 2.00000 0.114897
$$304$$ 4.00000 0.229416
$$305$$ −20.0000 −1.14520
$$306$$ 0 0
$$307$$ 17.0000 0.970241 0.485121 0.874447i $$-0.338776\pi$$
0.485121 + 0.874447i $$0.338776\pi$$
$$308$$ 0 0
$$309$$ −7.00000 −0.398216
$$310$$ −36.0000 −2.04466
$$311$$ 6.00000 0.340229 0.170114 0.985424i $$-0.445586\pi$$
0.170114 + 0.985424i $$0.445586\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 28.0000 1.58013
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ 24.0000 1.34797 0.673987 0.738743i $$-0.264580\pi$$
0.673987 + 0.738743i $$0.264580\pi$$
$$318$$ −24.0000 −1.34585
$$319$$ −8.00000 −0.447914
$$320$$ −16.0000 −0.894427
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 2.00000 0.111111
$$325$$ 1.00000 0.0554700
$$326$$ 8.00000 0.443079
$$327$$ −9.00000 −0.497701
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 8.00000 0.440386
$$331$$ −25.0000 −1.37412 −0.687062 0.726599i $$-0.741100\pi$$
−0.687062 + 0.726599i $$0.741100\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 3.00000 0.164399
$$334$$ 28.0000 1.53209
$$335$$ −10.0000 −0.546358
$$336$$ 0 0
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ −24.0000 −1.30543
$$339$$ −10.0000 −0.543125
$$340$$ 0 0
$$341$$ 18.0000 0.974755
$$342$$ −2.00000 −0.108148
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −16.0000 −0.860165
$$347$$ 32.0000 1.71785 0.858925 0.512101i $$-0.171133\pi$$
0.858925 + 0.512101i $$0.171133\pi$$
$$348$$ −8.00000 −0.428845
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 0 0
$$351$$ 1.00000 0.0533761
$$352$$ 16.0000 0.852803
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ −24.0000 −1.27559
$$355$$ −12.0000 −0.636894
$$356$$ −32.0000 −1.69600
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 20.0000 1.05556 0.527780 0.849381i $$-0.323025\pi$$
0.527780 + 0.849381i $$0.323025\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −26.0000 −1.36653
$$363$$ 7.00000 0.367405
$$364$$ 0 0
$$365$$ 6.00000 0.314054
$$366$$ 20.0000 1.04542
$$367$$ 9.00000 0.469796 0.234898 0.972020i $$-0.424524\pi$$
0.234898 + 0.972020i $$0.424524\pi$$
$$368$$ 0 0
$$369$$ 10.0000 0.520579
$$370$$ 12.0000 0.623850
$$371$$ 0 0
$$372$$ 18.0000 0.933257
$$373$$ 23.0000 1.19089 0.595447 0.803394i $$-0.296975\pi$$
0.595447 + 0.803394i $$0.296975\pi$$
$$374$$ 0 0
$$375$$ 12.0000 0.619677
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ 3.00000 0.154100 0.0770498 0.997027i $$-0.475450\pi$$
0.0770498 + 0.997027i $$0.475450\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 15.0000 0.768473
$$382$$ 20.0000 1.02329
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 22.0000 1.11977
$$387$$ 5.00000 0.254164
$$388$$ 12.0000 0.609208
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 4.00000 0.202548
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −14.0000 −0.706207
$$394$$ 32.0000 1.61214
$$395$$ −2.00000 −0.100631
$$396$$ −4.00000 −0.201008
$$397$$ 9.00000 0.451697 0.225849 0.974162i $$-0.427485\pi$$
0.225849 + 0.974162i $$0.427485\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −36.0000 −1.79775 −0.898877 0.438201i $$-0.855616\pi$$
−0.898877 + 0.438201i $$0.855616\pi$$
$$402$$ 10.0000 0.498755
$$403$$ 9.00000 0.448322
$$404$$ −4.00000 −0.199007
$$405$$ 2.00000 0.0993808
$$406$$ 0 0
$$407$$ −6.00000 −0.297409
$$408$$ 0 0
$$409$$ −5.00000 −0.247234 −0.123617 0.992330i $$-0.539449\pi$$
−0.123617 + 0.992330i $$0.539449\pi$$
$$410$$ 40.0000 1.97546
$$411$$ 12.0000 0.591916
$$412$$ 14.0000 0.689730
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −12.0000 −0.589057
$$416$$ 8.00000 0.392232
$$417$$ −3.00000 −0.146911
$$418$$ 4.00000 0.195646
$$419$$ −30.0000 −1.46560 −0.732798 0.680446i $$-0.761786\pi$$
−0.732798 + 0.680446i $$0.761786\pi$$
$$420$$ 0 0
$$421$$ −7.00000 −0.341159 −0.170580 0.985344i $$-0.554564\pi$$
−0.170580 + 0.985344i $$0.554564\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 6.00000 0.291730
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 12.0000 0.581402
$$427$$ 0 0
$$428$$ −16.0000 −0.773389
$$429$$ −2.00000 −0.0965609
$$430$$ 20.0000 0.964486
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −31.0000 −1.48976 −0.744882 0.667196i $$-0.767494\pi$$
−0.744882 + 0.667196i $$0.767494\pi$$
$$434$$ 0 0
$$435$$ −8.00000 −0.383571
$$436$$ 18.0000 0.862044
$$437$$ 0 0
$$438$$ −6.00000 −0.286691
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ −32.0000 −1.51695
$$446$$ −32.0000 −1.51524
$$447$$ 12.0000 0.567581
$$448$$ 0 0
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ −2.00000 −0.0942809
$$451$$ −20.0000 −0.941763
$$452$$ 20.0000 0.940721
$$453$$ 16.0000 0.751746
$$454$$ −36.0000 −1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −11.0000 −0.514558 −0.257279 0.966337i $$-0.582826\pi$$
−0.257279 + 0.966337i $$0.582826\pi$$
$$458$$ 38.0000 1.77562
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −20.0000 −0.931493 −0.465746 0.884918i $$-0.654214\pi$$
−0.465746 + 0.884918i $$0.654214\pi$$
$$462$$ 0 0
$$463$$ −17.0000 −0.790057 −0.395029 0.918669i $$-0.629265\pi$$
−0.395029 + 0.918669i $$0.629265\pi$$
$$464$$ −16.0000 −0.742781
$$465$$ 18.0000 0.834730
$$466$$ 12.0000 0.555889
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ −2.00000 −0.0924500
$$469$$ 0 0
$$470$$ 24.0000 1.10704
$$471$$ −14.0000 −0.645086
$$472$$ 0 0
$$473$$ −10.0000 −0.459800
$$474$$ 2.00000 0.0918630
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 12.0000 0.549442
$$478$$ 12.0000 0.548867
$$479$$ 28.0000 1.27935 0.639676 0.768644i $$-0.279068\pi$$
0.639676 + 0.768644i $$0.279068\pi$$
$$480$$ 16.0000 0.730297
$$481$$ −3.00000 −0.136788
$$482$$ −28.0000 −1.27537
$$483$$ 0 0
$$484$$ −14.0000 −0.636364
$$485$$ 12.0000 0.544892
$$486$$ −2.00000 −0.0907218
$$487$$ 31.0000 1.40474 0.702372 0.711810i $$-0.252124\pi$$
0.702372 + 0.711810i $$0.252124\pi$$
$$488$$ 0 0
$$489$$ −4.00000 −0.180886
$$490$$ 0 0
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ −20.0000 −0.901670
$$493$$ 0 0
$$494$$ 2.00000 0.0899843
$$495$$ −4.00000 −0.179787
$$496$$ 36.0000 1.61645
$$497$$ 0 0
$$498$$ 12.0000 0.537733
$$499$$ 37.0000 1.65635 0.828174 0.560471i $$-0.189380\pi$$
0.828174 + 0.560471i $$0.189380\pi$$
$$500$$ −24.0000 −1.07331
$$501$$ −14.0000 −0.625474
$$502$$ 16.0000 0.714115
$$503$$ 42.0000 1.87269 0.936344 0.351085i $$-0.114187\pi$$
0.936344 + 0.351085i $$0.114187\pi$$
$$504$$ 0 0
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ 12.0000 0.532939
$$508$$ −30.0000 −1.33103
$$509$$ −2.00000 −0.0886484 −0.0443242 0.999017i $$-0.514113\pi$$
−0.0443242 + 0.999017i $$0.514113\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 32.0000 1.41421
$$513$$ 1.00000 0.0441511
$$514$$ −52.0000 −2.29362
$$515$$ 14.0000 0.616914
$$516$$ −10.0000 −0.440225
$$517$$ −12.0000 −0.527759
$$518$$ 0 0
$$519$$ 8.00000 0.351161
$$520$$ 0 0
$$521$$ −12.0000 −0.525730 −0.262865 0.964833i $$-0.584667\pi$$
−0.262865 + 0.964833i $$0.584667\pi$$
$$522$$ 8.00000 0.350150
$$523$$ −31.0000 −1.35554 −0.677768 0.735276i $$-0.737052\pi$$
−0.677768 + 0.735276i $$0.737052\pi$$
$$524$$ 28.0000 1.22319
$$525$$ 0 0
$$526$$ 8.00000 0.348817
$$527$$ 0 0
$$528$$ −8.00000 −0.348155
$$529$$ −23.0000 −1.00000
$$530$$ 48.0000 2.08499
$$531$$ 12.0000 0.520756
$$532$$ 0 0
$$533$$ −10.0000 −0.433148
$$534$$ 32.0000 1.38478
$$535$$ −16.0000 −0.691740
$$536$$ 0 0
$$537$$ −2.00000 −0.0863064
$$538$$ −12.0000 −0.517357
$$539$$ 0 0
$$540$$ −4.00000 −0.172133
$$541$$ −19.0000 −0.816874 −0.408437 0.912787i $$-0.633926\pi$$
−0.408437 + 0.912787i $$0.633926\pi$$
$$542$$ −32.0000 −1.37452
$$543$$ 13.0000 0.557883
$$544$$ 0 0
$$545$$ 18.0000 0.771035
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ −24.0000 −1.02523
$$549$$ −10.0000 −0.426790
$$550$$ 4.00000 0.170561
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ −6.00000 −0.254686
$$556$$ 6.00000 0.254457
$$557$$ −2.00000 −0.0847427 −0.0423714 0.999102i $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ −18.0000 −0.762001
$$559$$ −5.00000 −0.211477
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −8.00000 −0.337460
$$563$$ 26.0000 1.09577 0.547885 0.836554i $$-0.315433\pi$$
0.547885 + 0.836554i $$0.315433\pi$$
$$564$$ −12.0000 −0.505291
$$565$$ 20.0000 0.841406
$$566$$ 22.0000 0.924729
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −26.0000 −1.08998 −0.544988 0.838444i $$-0.683466\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 4.00000 0.167542
$$571$$ −19.0000 −0.795125 −0.397563 0.917575i $$-0.630144\pi$$
−0.397563 + 0.917575i $$0.630144\pi$$
$$572$$ 4.00000 0.167248
$$573$$ −10.0000 −0.417756
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ 17.0000 0.707719 0.353860 0.935299i $$-0.384869\pi$$
0.353860 + 0.935299i $$0.384869\pi$$
$$578$$ −34.0000 −1.41421
$$579$$ −11.0000 −0.457144
$$580$$ 16.0000 0.664364
$$581$$ 0 0
$$582$$ −12.0000 −0.497416
$$583$$ −24.0000 −0.993978
$$584$$ 0 0
$$585$$ −2.00000 −0.0826898
$$586$$ −16.0000 −0.660954
$$587$$ −16.0000 −0.660391 −0.330195 0.943913i $$-0.607115\pi$$
−0.330195 + 0.943913i $$0.607115\pi$$
$$588$$ 0 0
$$589$$ 9.00000 0.370839
$$590$$ 48.0000 1.97613
$$591$$ −16.0000 −0.658152
$$592$$ −12.0000 −0.493197
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ −24.0000 −0.983078
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 12.0000 0.490307 0.245153 0.969484i $$-0.421162\pi$$
0.245153 + 0.969484i $$0.421162\pi$$
$$600$$ 0 0
$$601$$ 9.00000 0.367118 0.183559 0.983009i $$-0.441238\pi$$
0.183559 + 0.983009i $$0.441238\pi$$
$$602$$ 0 0
$$603$$ −5.00000 −0.203616
$$604$$ −32.0000 −1.30206
$$605$$ −14.0000 −0.569181
$$606$$ 4.00000 0.162489
$$607$$ −23.0000 −0.933541 −0.466771 0.884378i $$-0.654583\pi$$
−0.466771 + 0.884378i $$0.654583\pi$$
$$608$$ 8.00000 0.324443
$$609$$ 0 0
$$610$$ −40.0000 −1.61955
$$611$$ −6.00000 −0.242734
$$612$$ 0 0
$$613$$ 34.0000 1.37325 0.686624 0.727013i $$-0.259092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 34.0000 1.37213
$$615$$ −20.0000 −0.806478
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ −14.0000 −0.563163
$$619$$ 29.0000 1.16561 0.582804 0.812613i $$-0.301955\pi$$
0.582804 + 0.812613i $$0.301955\pi$$
$$620$$ −36.0000 −1.44579
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ 0 0
$$624$$ −4.00000 −0.160128
$$625$$ −19.0000 −0.760000
$$626$$ 2.00000 0.0799361
$$627$$ −2.00000 −0.0798723
$$628$$ 28.0000 1.11732
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ −4.00000 −0.158986
$$634$$ 48.0000 1.90632
$$635$$ −30.0000 −1.19051
$$636$$ −24.0000 −0.951662
$$637$$ 0 0
$$638$$ −16.0000 −0.633446
$$639$$ −6.00000 −0.237356
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 16.0000 0.631470
$$643$$ 19.0000 0.749287 0.374643 0.927169i $$-0.377765\pi$$
0.374643 + 0.927169i $$0.377765\pi$$
$$644$$ 0 0
$$645$$ −10.0000 −0.393750
$$646$$ 0 0
$$647$$ −2.00000 −0.0786281 −0.0393141 0.999227i $$-0.512517\pi$$
−0.0393141 + 0.999227i $$0.512517\pi$$
$$648$$ 0 0
$$649$$ −24.0000 −0.942082
$$650$$ 2.00000 0.0784465
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ −18.0000 −0.703856
$$655$$ 28.0000 1.09405
$$656$$ −40.0000 −1.56174
$$657$$ 3.00000 0.117041
$$658$$ 0 0
$$659$$ 36.0000 1.40236 0.701180 0.712984i $$-0.252657\pi$$
0.701180 + 0.712984i $$0.252657\pi$$
$$660$$ 8.00000 0.311400
$$661$$ 41.0000 1.59472 0.797358 0.603507i $$-0.206231\pi$$
0.797358 + 0.603507i $$0.206231\pi$$
$$662$$ −50.0000 −1.94331
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ 28.0000 1.08335
$$669$$ 16.0000 0.618596
$$670$$ −20.0000 −0.772667
$$671$$ 20.0000 0.772091
$$672$$ 0 0
$$673$$ −41.0000 −1.58043 −0.790217 0.612827i $$-0.790032\pi$$
−0.790217 + 0.612827i $$0.790032\pi$$
$$674$$ 26.0000 1.00148
$$675$$ 1.00000 0.0384900
$$676$$ −24.0000 −0.923077
$$677$$ −12.0000 −0.461197 −0.230599 0.973049i $$-0.574068\pi$$
−0.230599 + 0.973049i $$0.574068\pi$$
$$678$$ −20.0000 −0.768095
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ 36.0000 1.37851
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −2.00000 −0.0764719
$$685$$ −24.0000 −0.916993
$$686$$ 0 0
$$687$$ −19.0000 −0.724895
$$688$$ −20.0000 −0.762493
$$689$$ −12.0000 −0.457164
$$690$$ 0 0
$$691$$ 37.0000 1.40755 0.703773 0.710425i $$-0.251497\pi$$
0.703773 + 0.710425i $$0.251497\pi$$
$$692$$ −16.0000 −0.608229
$$693$$ 0 0
$$694$$ 64.0000 2.42941
$$695$$ 6.00000 0.227593
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 28.0000 1.05982
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 2.00000 0.0754851
$$703$$ −3.00000 −0.113147
$$704$$ 16.0000 0.603023
$$705$$ −12.0000 −0.451946
$$706$$ −68.0000 −2.55921
$$707$$ 0 0
$$708$$ −24.0000 −0.901975
$$709$$ 30.0000 1.12667 0.563337 0.826227i $$-0.309517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ −24.0000 −0.900704
$$711$$ −1.00000 −0.0375029
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 4.00000 0.149592
$$716$$ 4.00000 0.149487
$$717$$ −6.00000 −0.224074
$$718$$ 40.0000 1.49279
$$719$$ 18.0000 0.671287 0.335643 0.941989i $$-0.391046\pi$$
0.335643 + 0.941989i $$0.391046\pi$$
$$720$$ −8.00000 −0.298142
$$721$$ 0 0
$$722$$ −36.0000 −1.33978
$$723$$ 14.0000 0.520666
$$724$$ −26.0000 −0.966282
$$725$$ −4.00000 −0.148556
$$726$$ 14.0000 0.519589
$$727$$ 13.0000 0.482143 0.241072 0.970507i $$-0.422501\pi$$
0.241072 + 0.970507i $$0.422501\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 12.0000 0.444140
$$731$$ 0 0
$$732$$ 20.0000 0.739221
$$733$$ 15.0000 0.554038 0.277019 0.960864i $$-0.410654\pi$$
0.277019 + 0.960864i $$0.410654\pi$$
$$734$$ 18.0000 0.664392
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 10.0000 0.368355
$$738$$ 20.0000 0.736210
$$739$$ −15.0000 −0.551784 −0.275892 0.961189i $$-0.588973\pi$$
−0.275892 + 0.961189i $$0.588973\pi$$
$$740$$ 12.0000 0.441129
$$741$$ −1.00000 −0.0367359
$$742$$ 0 0
$$743$$ 42.0000 1.54083 0.770415 0.637542i $$-0.220049\pi$$
0.770415 + 0.637542i $$0.220049\pi$$
$$744$$ 0 0
$$745$$ −24.0000 −0.879292
$$746$$ 46.0000 1.68418
$$747$$ −6.00000 −0.219529
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 24.0000 0.876356
$$751$$ 13.0000 0.474377 0.237188 0.971464i $$-0.423774\pi$$
0.237188 + 0.971464i $$0.423774\pi$$
$$752$$ −24.0000 −0.875190
$$753$$ −8.00000 −0.291536
$$754$$ −8.00000 −0.291343
$$755$$ −32.0000 −1.16460
$$756$$ 0 0
$$757$$ −22.0000 −0.799604 −0.399802 0.916602i $$-0.630921\pi$$
−0.399802 + 0.916602i $$0.630921\pi$$
$$758$$ 6.00000 0.217930
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 48.0000 1.74000 0.869999 0.493053i $$-0.164119\pi$$
0.869999 + 0.493053i $$0.164119\pi$$
$$762$$ 30.0000 1.08679
$$763$$ 0 0
$$764$$ 20.0000 0.723575
$$765$$ 0 0
$$766$$ 24.0000 0.867155
$$767$$ −12.0000 −0.433295
$$768$$ −16.0000 −0.577350
$$769$$ 49.0000 1.76699 0.883493 0.468445i $$-0.155186\pi$$
0.883493 + 0.468445i $$0.155186\pi$$
$$770$$ 0 0
$$771$$ 26.0000 0.936367
$$772$$ 22.0000 0.791797
$$773$$ 34.0000 1.22290 0.611448 0.791285i $$-0.290588\pi$$
0.611448 + 0.791285i $$0.290588\pi$$
$$774$$ 10.0000 0.359443
$$775$$ 9.00000 0.323290
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −12.0000 −0.430221
$$779$$ −10.0000 −0.358287
$$780$$ 4.00000 0.143223
$$781$$ 12.0000 0.429394
$$782$$ 0 0
$$783$$ −4.00000 −0.142948
$$784$$ 0 0
$$785$$ 28.0000 0.999363
$$786$$ −28.0000 −0.998727
$$787$$ −40.0000 −1.42585 −0.712923 0.701242i $$-0.752629\pi$$
−0.712923 + 0.701242i $$0.752629\pi$$
$$788$$ 32.0000 1.13995
$$789$$ −4.00000 −0.142404
$$790$$ −4.00000 −0.142314
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 10.0000 0.355110
$$794$$ 18.0000 0.638796
$$795$$ −24.0000 −0.851192
$$796$$ 0 0
$$797$$ 8.00000 0.283375 0.141687 0.989911i $$-0.454747\pi$$
0.141687 + 0.989911i $$0.454747\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 8.00000 0.282843
$$801$$ −16.0000 −0.565332
$$802$$ −72.0000 −2.54241
$$803$$ −6.00000 −0.211735
$$804$$ 10.0000 0.352673
$$805$$ 0 0
$$806$$ 18.0000 0.634023
$$807$$ 6.00000 0.211210
$$808$$ 0 0
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ 4.00000 0.140546
$$811$$ −32.0000 −1.12367 −0.561836 0.827249i $$-0.689905\pi$$
−0.561836 + 0.827249i $$0.689905\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ −12.0000 −0.420600
$$815$$ 8.00000 0.280228
$$816$$ 0 0
$$817$$ −5.00000 −0.174928
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ 40.0000 1.39686
$$821$$ 2.00000 0.0698005 0.0349002 0.999391i $$-0.488889\pi$$
0.0349002 + 0.999391i $$0.488889\pi$$
$$822$$ 24.0000 0.837096
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ −2.00000 −0.0696311
$$826$$ 0 0
$$827$$ −30.0000 −1.04320 −0.521601 0.853189i $$-0.674665\pi$$
−0.521601 + 0.853189i $$0.674665\pi$$
$$828$$ 0 0
$$829$$ −41.0000 −1.42399 −0.711994 0.702185i $$-0.752208\pi$$
−0.711994 + 0.702185i $$0.752208\pi$$
$$830$$ −24.0000 −0.833052
$$831$$ −13.0000 −0.450965
$$832$$ 8.00000 0.277350
$$833$$ 0 0
$$834$$ −6.00000 −0.207763
$$835$$ 28.0000 0.968980
$$836$$ 4.00000 0.138343
$$837$$ 9.00000 0.311086
$$838$$ −60.0000 −2.07267
$$839$$ 44.0000 1.51905 0.759524 0.650479i $$-0.225432\pi$$
0.759524 + 0.650479i $$0.225432\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ −14.0000 −0.482472
$$843$$ 4.00000 0.137767
$$844$$ 8.00000 0.275371
$$845$$ −24.0000 −0.825625
$$846$$ 12.0000 0.412568
$$847$$ 0 0
$$848$$ −48.0000 −1.64833
$$849$$ −11.0000 −0.377519
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 12.0000 0.411113
$$853$$ −35.0000 −1.19838 −0.599189 0.800608i $$-0.704510\pi$$
−0.599189 + 0.800608i $$0.704510\pi$$
$$854$$ 0 0
$$855$$ −2.00000 −0.0683986
$$856$$ 0 0
$$857$$ 32.0000 1.09310 0.546550 0.837427i $$-0.315941\pi$$
0.546550 + 0.837427i $$0.315941\pi$$
$$858$$ −4.00000 −0.136558
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 20.0000 0.681994
$$861$$ 0 0
$$862$$ −36.0000 −1.22616
$$863$$ −54.0000 −1.83818 −0.919091 0.394046i $$-0.871075\pi$$
−0.919091 + 0.394046i $$0.871075\pi$$
$$864$$ 8.00000 0.272166
$$865$$ −16.0000 −0.544016
$$866$$ −62.0000 −2.10685
$$867$$ 17.0000 0.577350
$$868$$ 0 0
$$869$$ 2.00000 0.0678454
$$870$$ −16.0000 −0.542451
$$871$$ 5.00000 0.169419
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ 0 0
$$876$$ −6.00000 −0.202721
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ 0 0
$$879$$ 8.00000 0.269833
$$880$$ 16.0000 0.539360
$$881$$ −24.0000 −0.808581 −0.404290 0.914631i $$-0.632481\pi$$
−0.404290 + 0.914631i $$0.632481\pi$$
$$882$$ 0 0
$$883$$ −13.0000 −0.437485 −0.218742 0.975783i $$-0.570195\pi$$
−0.218742 + 0.975783i $$0.570195\pi$$
$$884$$ 0 0
$$885$$ −24.0000 −0.806751
$$886$$ 24.0000 0.806296
$$887$$ 34.0000 1.14161 0.570804 0.821086i $$-0.306632\pi$$
0.570804 + 0.821086i $$0.306632\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −64.0000 −2.14528
$$891$$ −2.00000 −0.0670025
$$892$$ −32.0000 −1.07144
$$893$$ −6.00000 −0.200782
$$894$$ 24.0000 0.802680
$$895$$ 4.00000 0.133705
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −36.0000 −1.20134
$$899$$ −36.0000 −1.20067
$$900$$ −2.00000 −0.0666667
$$901$$ 0 0
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −26.0000 −0.864269
$$906$$ 32.0000 1.06313
$$907$$ −37.0000 −1.22856 −0.614282 0.789086i $$-0.710554\pi$$
−0.614282 + 0.789086i $$0.710554\pi$$
$$908$$ −36.0000 −1.19470
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ −24.0000 −0.795155 −0.397578 0.917568i $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 12.0000 0.397142
$$914$$ −22.0000 −0.727695
$$915$$ 20.0000 0.661180
$$916$$ 38.0000 1.25556
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 23.0000 0.758700 0.379350 0.925253i $$-0.376148\pi$$
0.379350 + 0.925253i $$0.376148\pi$$
$$920$$ 0 0
$$921$$ −17.0000 −0.560169
$$922$$ −40.0000 −1.31733
$$923$$ 6.00000 0.197492
$$924$$ 0 0
$$925$$ −3.00000 −0.0986394
$$926$$ −34.0000 −1.11731
$$927$$ 7.00000 0.229910
$$928$$ −32.0000 −1.05045
$$929$$ −14.0000 −0.459325 −0.229663 0.973270i $$-0.573762\pi$$
−0.229663 + 0.973270i $$0.573762\pi$$
$$930$$ 36.0000 1.18049
$$931$$ 0 0
$$932$$ 12.0000 0.393073
$$933$$ −6.00000 −0.196431
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −15.0000 −0.490029 −0.245014 0.969519i $$-0.578793\pi$$
−0.245014 + 0.969519i $$0.578793\pi$$
$$938$$ 0 0
$$939$$ −1.00000 −0.0326338
$$940$$ 24.0000 0.782794
$$941$$ 4.00000 0.130396 0.0651981 0.997872i $$-0.479232\pi$$
0.0651981 + 0.997872i $$0.479232\pi$$
$$942$$ −28.0000 −0.912289
$$943$$ 0 0
$$944$$ −48.0000 −1.56227
$$945$$ 0 0
$$946$$ −20.0000 −0.650256
$$947$$ −10.0000 −0.324956 −0.162478 0.986712i $$-0.551949\pi$$
−0.162478 + 0.986712i $$0.551949\pi$$
$$948$$ 2.00000 0.0649570
$$949$$ −3.00000 −0.0973841
$$950$$ 2.00000 0.0648886
$$951$$ −24.0000 −0.778253
$$952$$ 0 0
$$953$$ 44.0000 1.42530 0.712650 0.701520i $$-0.247495\pi$$
0.712650 + 0.701520i $$0.247495\pi$$
$$954$$ 24.0000 0.777029
$$955$$ 20.0000 0.647185
$$956$$ 12.0000 0.388108
$$957$$ 8.00000 0.258603
$$958$$ 56.0000 1.80928
$$959$$ 0 0
$$960$$ 16.0000 0.516398
$$961$$ 50.0000 1.61290
$$962$$ −6.00000 −0.193448
$$963$$ −8.00000 −0.257796
$$964$$ −28.0000 −0.901819
$$965$$ 22.0000 0.708205
$$966$$ 0 0
$$967$$ 19.0000 0.610999 0.305499 0.952192i $$-0.401177\pi$$
0.305499 + 0.952192i $$0.401177\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 24.0000 0.770594
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ −2.00000 −0.0641500
$$973$$ 0 0
$$974$$ 62.0000 1.98661
$$975$$ −1.00000 −0.0320256
$$976$$ 40.0000 1.28037
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ −8.00000 −0.255812
$$979$$ 32.0000 1.02272
$$980$$ 0 0
$$981$$ 9.00000 0.287348
$$982$$ −56.0000 −1.78703
$$983$$ −36.0000 −1.14822 −0.574111 0.818778i $$-0.694652\pi$$
−0.574111 + 0.818778i $$0.694652\pi$$
$$984$$ 0 0
$$985$$ 32.0000 1.01960
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 2.00000 0.0636285
$$989$$ 0 0
$$990$$ −8.00000 −0.254257
$$991$$ 17.0000 0.540023 0.270011 0.962857i $$-0.412973\pi$$
0.270011 + 0.962857i $$0.412973\pi$$
$$992$$ 72.0000 2.28600
$$993$$ 25.0000 0.793351
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ −19.0000 −0.601736 −0.300868 0.953666i $$-0.597276\pi$$
−0.300868 + 0.953666i $$0.597276\pi$$
$$998$$ 74.0000 2.34243
$$999$$ −3.00000 −0.0949158
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 147.2.a.b.1.1 1
3.2 odd 2 441.2.a.a.1.1 1
4.3 odd 2 2352.2.a.w.1.1 1
5.4 even 2 3675.2.a.c.1.1 1
7.2 even 3 147.2.e.a.67.1 2
7.3 odd 6 21.2.e.a.16.1 yes 2
7.4 even 3 147.2.e.a.79.1 2
7.5 odd 6 21.2.e.a.4.1 2
7.6 odd 2 147.2.a.c.1.1 1
8.3 odd 2 9408.2.a.k.1.1 1
8.5 even 2 9408.2.a.bz.1.1 1
12.11 even 2 7056.2.a.m.1.1 1
21.2 odd 6 441.2.e.e.361.1 2
21.5 even 6 63.2.e.b.46.1 2
21.11 odd 6 441.2.e.e.226.1 2
21.17 even 6 63.2.e.b.37.1 2
21.20 even 2 441.2.a.b.1.1 1
28.3 even 6 336.2.q.f.289.1 2
28.11 odd 6 2352.2.q.c.961.1 2
28.19 even 6 336.2.q.f.193.1 2
28.23 odd 6 2352.2.q.c.1537.1 2
28.27 even 2 2352.2.a.d.1.1 1
35.3 even 12 525.2.r.e.499.2 4
35.12 even 12 525.2.r.e.424.2 4
35.17 even 12 525.2.r.e.499.1 4
35.19 odd 6 525.2.i.e.151.1 2
35.24 odd 6 525.2.i.e.226.1 2
35.33 even 12 525.2.r.e.424.1 4
35.34 odd 2 3675.2.a.a.1.1 1
56.3 even 6 1344.2.q.c.961.1 2
56.5 odd 6 1344.2.q.m.193.1 2
56.13 odd 2 9408.2.a.bg.1.1 1
56.19 even 6 1344.2.q.c.193.1 2
56.27 even 2 9408.2.a.cv.1.1 1
56.45 odd 6 1344.2.q.m.961.1 2
63.5 even 6 567.2.h.a.298.1 2
63.31 odd 6 567.2.g.a.541.1 2
63.38 even 6 567.2.h.a.352.1 2
63.40 odd 6 567.2.h.f.298.1 2
63.47 even 6 567.2.g.f.109.1 2
63.52 odd 6 567.2.h.f.352.1 2
63.59 even 6 567.2.g.f.541.1 2
63.61 odd 6 567.2.g.a.109.1 2
84.47 odd 6 1008.2.s.d.865.1 2
84.59 odd 6 1008.2.s.d.289.1 2
84.83 odd 2 7056.2.a.bp.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.2.e.a.4.1 2 7.5 odd 6
21.2.e.a.16.1 yes 2 7.3 odd 6
63.2.e.b.37.1 2 21.17 even 6
63.2.e.b.46.1 2 21.5 even 6
147.2.a.b.1.1 1 1.1 even 1 trivial
147.2.a.c.1.1 1 7.6 odd 2
147.2.e.a.67.1 2 7.2 even 3
147.2.e.a.79.1 2 7.4 even 3
336.2.q.f.193.1 2 28.19 even 6
336.2.q.f.289.1 2 28.3 even 6
441.2.a.a.1.1 1 3.2 odd 2
441.2.a.b.1.1 1 21.20 even 2
441.2.e.e.226.1 2 21.11 odd 6
441.2.e.e.361.1 2 21.2 odd 6
525.2.i.e.151.1 2 35.19 odd 6
525.2.i.e.226.1 2 35.24 odd 6
525.2.r.e.424.1 4 35.33 even 12
525.2.r.e.424.2 4 35.12 even 12
525.2.r.e.499.1 4 35.17 even 12
525.2.r.e.499.2 4 35.3 even 12
567.2.g.a.109.1 2 63.61 odd 6
567.2.g.a.541.1 2 63.31 odd 6
567.2.g.f.109.1 2 63.47 even 6
567.2.g.f.541.1 2 63.59 even 6
567.2.h.a.298.1 2 63.5 even 6
567.2.h.a.352.1 2 63.38 even 6
567.2.h.f.298.1 2 63.40 odd 6
567.2.h.f.352.1 2 63.52 odd 6
1008.2.s.d.289.1 2 84.59 odd 6
1008.2.s.d.865.1 2 84.47 odd 6
1344.2.q.c.193.1 2 56.19 even 6
1344.2.q.c.961.1 2 56.3 even 6
1344.2.q.m.193.1 2 56.5 odd 6
1344.2.q.m.961.1 2 56.45 odd 6
2352.2.a.d.1.1 1 28.27 even 2
2352.2.a.w.1.1 1 4.3 odd 2
2352.2.q.c.961.1 2 28.11 odd 6
2352.2.q.c.1537.1 2 28.23 odd 6
3675.2.a.a.1.1 1 35.34 odd 2
3675.2.a.c.1.1 1 5.4 even 2
7056.2.a.m.1.1 1 12.11 even 2
7056.2.a.bp.1.1 1 84.83 odd 2
9408.2.a.k.1.1 1 8.3 odd 2
9408.2.a.bg.1.1 1 56.13 odd 2
9408.2.a.bz.1.1 1 8.5 even 2
9408.2.a.cv.1.1 1 56.27 even 2