# Properties

 Label 847.2.a.a.1.1 Level $847$ Weight $2$ Character 847.1 Self dual yes Analytic conductor $6.763$ 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] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$847 = 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 847.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: $$6.76332905120$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 847.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} -2.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} -4.00000 q^{15} -1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} +2.00000 q^{20} +2.00000 q^{21} -4.00000 q^{23} +6.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} +4.00000 q^{30} +10.0000 q^{31} -5.00000 q^{32} +4.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} -6.00000 q^{37} -8.00000 q^{39} -6.00000 q^{40} -4.00000 q^{41} -2.00000 q^{42} -12.0000 q^{43} -2.00000 q^{45} +4.00000 q^{46} -10.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -8.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +4.00000 q^{54} +3.00000 q^{56} -6.00000 q^{58} +2.00000 q^{59} +4.00000 q^{60} -10.0000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +8.00000 q^{65} +8.00000 q^{67} +4.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} -12.0000 q^{71} +3.00000 q^{72} +8.00000 q^{73} +6.00000 q^{74} -2.00000 q^{75} +8.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} -11.0000 q^{81} +4.00000 q^{82} -2.00000 q^{84} +8.00000 q^{85} +12.0000 q^{86} +12.0000 q^{87} -6.00000 q^{89} +2.00000 q^{90} -4.00000 q^{91} +4.00000 q^{92} +20.0000 q^{93} +10.0000 q^{94} -10.0000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107 −0.353553 0.935414i $$-0.615027\pi$$
−0.353553 + 0.935414i $$0.615027\pi$$
$$3$$ 2.00000 1.15470 0.577350 0.816497i $$-0.304087\pi$$
0.577350 + 0.816497i $$0.304087\pi$$
$$4$$ −1.00000 −0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ −2.00000 −0.816497
$$7$$ 1.00000 0.377964
$$8$$ 3.00000 1.06066
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 0 0
$$12$$ −2.00000 −0.577350
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ −4.00000 −1.03280
$$16$$ −1.00000 −0.250000
$$17$$ −4.00000 −0.970143 −0.485071 0.874475i $$-0.661206\pi$$
−0.485071 + 0.874475i $$0.661206\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 6.00000 1.22474
$$25$$ −1.00000 −0.200000
$$26$$ 4.00000 0.784465
$$27$$ −4.00000 −0.769800
$$28$$ −1.00000 −0.188982
$$29$$ 6.00000 1.11417 0.557086 0.830455i $$-0.311919\pi$$
0.557086 + 0.830455i $$0.311919\pi$$
$$30$$ 4.00000 0.730297
$$31$$ 10.0000 1.79605 0.898027 0.439941i $$-0.145001\pi$$
0.898027 + 0.439941i $$0.145001\pi$$
$$32$$ −5.00000 −0.883883
$$33$$ 0 0
$$34$$ 4.00000 0.685994
$$35$$ −2.00000 −0.338062
$$36$$ −1.00000 −0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ 0 0
$$39$$ −8.00000 −1.28103
$$40$$ −6.00000 −0.948683
$$41$$ −4.00000 −0.624695 −0.312348 0.949968i $$-0.601115\pi$$
−0.312348 + 0.949968i $$0.601115\pi$$
$$42$$ −2.00000 −0.308607
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ 4.00000 0.589768
$$47$$ −10.0000 −1.45865 −0.729325 0.684167i $$-0.760166\pi$$
−0.729325 + 0.684167i $$0.760166\pi$$
$$48$$ −2.00000 −0.288675
$$49$$ 1.00000 0.142857
$$50$$ 1.00000 0.141421
$$51$$ −8.00000 −1.12022
$$52$$ 4.00000 0.554700
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 4.00000 0.544331
$$55$$ 0 0
$$56$$ 3.00000 0.400892
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 2.00000 0.260378 0.130189 0.991489i $$-0.458442\pi$$
0.130189 + 0.991489i $$0.458442\pi$$
$$60$$ 4.00000 0.516398
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ −10.0000 −1.27000
$$63$$ 1.00000 0.125988
$$64$$ 7.00000 0.875000
$$65$$ 8.00000 0.992278
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 4.00000 0.485071
$$69$$ −8.00000 −0.963087
$$70$$ 2.00000 0.239046
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 3.00000 0.353553
$$73$$ 8.00000 0.936329 0.468165 0.883641i $$-0.344915\pi$$
0.468165 + 0.883641i $$0.344915\pi$$
$$74$$ 6.00000 0.697486
$$75$$ −2.00000 −0.230940
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 8.00000 0.905822
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 2.00000 0.223607
$$81$$ −11.0000 −1.22222
$$82$$ 4.00000 0.441726
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −2.00000 −0.218218
$$85$$ 8.00000 0.867722
$$86$$ 12.0000 1.29399
$$87$$ 12.0000 1.28654
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −4.00000 −0.419314
$$92$$ 4.00000 0.417029
$$93$$ 20.0000 2.07390
$$94$$ 10.0000 1.03142
$$95$$ 0 0
$$96$$ −10.0000 −1.02062
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 4.00000 0.398015 0.199007 0.979998i $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 8.00000 0.792118
$$103$$ 14.0000 1.37946 0.689730 0.724066i $$-0.257729\pi$$
0.689730 + 0.724066i $$0.257729\pi$$
$$104$$ −12.0000 −1.17670
$$105$$ −4.00000 −0.390360
$$106$$ 6.00000 0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 4.00000 0.384900
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −12.0000 −1.13899
$$112$$ −1.00000 −0.0944911
$$113$$ 18.0000 1.69330 0.846649 0.532152i $$-0.178617\pi$$
0.846649 + 0.532152i $$0.178617\pi$$
$$114$$ 0 0
$$115$$ 8.00000 0.746004
$$116$$ −6.00000 −0.557086
$$117$$ −4.00000 −0.369800
$$118$$ −2.00000 −0.184115
$$119$$ −4.00000 −0.366679
$$120$$ −12.0000 −1.09545
$$121$$ 0 0
$$122$$ 0 0
$$123$$ −8.00000 −0.721336
$$124$$ −10.0000 −0.898027
$$125$$ 12.0000 1.07331
$$126$$ −1.00000 −0.0890871
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 3.00000 0.265165
$$129$$ −24.0000 −2.11308
$$130$$ −8.00000 −0.701646
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −8.00000 −0.691095
$$135$$ 8.00000 0.688530
$$136$$ −12.0000 −1.02899
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 8.00000 0.681005
$$139$$ 8.00000 0.678551 0.339276 0.940687i $$-0.389818\pi$$
0.339276 + 0.940687i $$0.389818\pi$$
$$140$$ 2.00000 0.169031
$$141$$ −20.0000 −1.68430
$$142$$ 12.0000 1.00702
$$143$$ 0 0
$$144$$ −1.00000 −0.0833333
$$145$$ −12.0000 −0.996546
$$146$$ −8.00000 −0.662085
$$147$$ 2.00000 0.164957
$$148$$ 6.00000 0.493197
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 2.00000 0.163299
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 0 0
$$153$$ −4.00000 −0.323381
$$154$$ 0 0
$$155$$ −20.0000 −1.60644
$$156$$ 8.00000 0.640513
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 8.00000 0.636446
$$159$$ −12.0000 −0.951662
$$160$$ 10.0000 0.790569
$$161$$ −4.00000 −0.315244
$$162$$ 11.0000 0.864242
$$163$$ −8.00000 −0.626608 −0.313304 0.949653i $$-0.601436\pi$$
−0.313304 + 0.949653i $$0.601436\pi$$
$$164$$ 4.00000 0.312348
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 6.00000 0.462910
$$169$$ 3.00000 0.230769
$$170$$ −8.00000 −0.613572
$$171$$ 0 0
$$172$$ 12.0000 0.914991
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ −12.0000 −0.909718
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ 4.00000 0.300658
$$178$$ 6.00000 0.449719
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 2.00000 0.149071
$$181$$ 10.0000 0.743294 0.371647 0.928374i $$-0.378793\pi$$
0.371647 + 0.928374i $$0.378793\pi$$
$$182$$ 4.00000 0.296500
$$183$$ 0 0
$$184$$ −12.0000 −0.884652
$$185$$ 12.0000 0.882258
$$186$$ −20.0000 −1.46647
$$187$$ 0 0
$$188$$ 10.0000 0.729325
$$189$$ −4.00000 −0.290957
$$190$$ 0 0
$$191$$ 8.00000 0.578860 0.289430 0.957199i $$-0.406534\pi$$
0.289430 + 0.957199i $$0.406534\pi$$
$$192$$ 14.0000 1.01036
$$193$$ 14.0000 1.00774 0.503871 0.863779i $$-0.331909\pi$$
0.503871 + 0.863779i $$0.331909\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 16.0000 1.14578
$$196$$ −1.00000 −0.0714286
$$197$$ −22.0000 −1.56744 −0.783718 0.621117i $$-0.786679\pi$$
−0.783718 + 0.621117i $$0.786679\pi$$
$$198$$ 0 0
$$199$$ −18.0000 −1.27599 −0.637993 0.770042i $$-0.720235\pi$$
−0.637993 + 0.770042i $$0.720235\pi$$
$$200$$ −3.00000 −0.212132
$$201$$ 16.0000 1.12855
$$202$$ −4.00000 −0.281439
$$203$$ 6.00000 0.421117
$$204$$ 8.00000 0.560112
$$205$$ 8.00000 0.558744
$$206$$ −14.0000 −0.975426
$$207$$ −4.00000 −0.278019
$$208$$ 4.00000 0.277350
$$209$$ 0 0
$$210$$ 4.00000 0.276026
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −24.0000 −1.64445
$$214$$ 12.0000 0.820303
$$215$$ 24.0000 1.63679
$$216$$ −12.0000 −0.816497
$$217$$ 10.0000 0.678844
$$218$$ −14.0000 −0.948200
$$219$$ 16.0000 1.08118
$$220$$ 0 0
$$221$$ 16.0000 1.07628
$$222$$ 12.0000 0.805387
$$223$$ 22.0000 1.47323 0.736614 0.676313i $$-0.236423\pi$$
0.736614 + 0.676313i $$0.236423\pi$$
$$224$$ −5.00000 −0.334077
$$225$$ −1.00000 −0.0666667
$$226$$ −18.0000 −1.19734
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 18.0000 1.18947 0.594737 0.803921i $$-0.297256\pi$$
0.594737 + 0.803921i $$0.297256\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 18.0000 1.18176
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 4.00000 0.261488
$$235$$ 20.0000 1.30466
$$236$$ −2.00000 −0.130189
$$237$$ −16.0000 −1.03931
$$238$$ 4.00000 0.259281
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 4.00000 0.258199
$$241$$ 20.0000 1.28831 0.644157 0.764894i $$-0.277208\pi$$
0.644157 + 0.764894i $$0.277208\pi$$
$$242$$ 0 0
$$243$$ −10.0000 −0.641500
$$244$$ 0 0
$$245$$ −2.00000 −0.127775
$$246$$ 8.00000 0.510061
$$247$$ 0 0
$$248$$ 30.0000 1.90500
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ −2.00000 −0.126239 −0.0631194 0.998006i $$-0.520105\pi$$
−0.0631194 + 0.998006i $$0.520105\pi$$
$$252$$ −1.00000 −0.0629941
$$253$$ 0 0
$$254$$ 8.00000 0.501965
$$255$$ 16.0000 1.00196
$$256$$ −17.0000 −1.06250
$$257$$ −14.0000 −0.873296 −0.436648 0.899632i $$-0.643834\pi$$
−0.436648 + 0.899632i $$0.643834\pi$$
$$258$$ 24.0000 1.49417
$$259$$ −6.00000 −0.372822
$$260$$ −8.00000 −0.496139
$$261$$ 6.00000 0.371391
$$262$$ 12.0000 0.741362
$$263$$ −8.00000 −0.493301 −0.246651 0.969104i $$-0.579330\pi$$
−0.246651 + 0.969104i $$0.579330\pi$$
$$264$$ 0 0
$$265$$ 12.0000 0.737154
$$266$$ 0 0
$$267$$ −12.0000 −0.734388
$$268$$ −8.00000 −0.488678
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ −8.00000 −0.486864
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 4.00000 0.242536
$$273$$ −8.00000 −0.484182
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ −22.0000 −1.32185 −0.660926 0.750451i $$-0.729836\pi$$
−0.660926 + 0.750451i $$0.729836\pi$$
$$278$$ −8.00000 −0.479808
$$279$$ 10.0000 0.598684
$$280$$ −6.00000 −0.358569
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 20.0000 1.19098
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 12.0000 0.712069
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −4.00000 −0.236113
$$288$$ −5.00000 −0.294628
$$289$$ −1.00000 −0.0588235
$$290$$ 12.0000 0.704664
$$291$$ −20.0000 −1.17242
$$292$$ −8.00000 −0.468165
$$293$$ 24.0000 1.40209 0.701047 0.713115i $$-0.252716\pi$$
0.701047 + 0.713115i $$0.252716\pi$$
$$294$$ −2.00000 −0.116642
$$295$$ −4.00000 −0.232889
$$296$$ −18.0000 −1.04623
$$297$$ 0 0
$$298$$ −10.0000 −0.579284
$$299$$ 16.0000 0.925304
$$300$$ 2.00000 0.115470
$$301$$ −12.0000 −0.691669
$$302$$ −16.0000 −0.920697
$$303$$ 8.00000 0.459588
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 4.00000 0.228665
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 28.0000 1.59286
$$310$$ 20.0000 1.13592
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ −24.0000 −1.35873
$$313$$ 2.00000 0.113047 0.0565233 0.998401i $$-0.481998\pi$$
0.0565233 + 0.998401i $$0.481998\pi$$
$$314$$ −14.0000 −0.790066
$$315$$ −2.00000 −0.112687
$$316$$ 8.00000 0.450035
$$317$$ −2.00000 −0.112331 −0.0561656 0.998421i $$-0.517887\pi$$
−0.0561656 + 0.998421i $$0.517887\pi$$
$$318$$ 12.0000 0.672927
$$319$$ 0 0
$$320$$ −14.0000 −0.782624
$$321$$ −24.0000 −1.33955
$$322$$ 4.00000 0.222911
$$323$$ 0 0
$$324$$ 11.0000 0.611111
$$325$$ 4.00000 0.221880
$$326$$ 8.00000 0.443079
$$327$$ 28.0000 1.54840
$$328$$ −12.0000 −0.662589
$$329$$ −10.0000 −0.551318
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ −16.0000 −0.874173
$$336$$ −2.00000 −0.109109
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −3.00000 −0.163178
$$339$$ 36.0000 1.95525
$$340$$ −8.00000 −0.433861
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ −36.0000 −1.94099
$$345$$ 16.0000 0.861411
$$346$$ 12.0000 0.645124
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ −12.0000 −0.643268
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 16.0000 0.854017
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 24.0000 1.27379
$$356$$ 6.00000 0.317999
$$357$$ −8.00000 −0.423405
$$358$$ −12.0000 −0.634220
$$359$$ −16.0000 −0.844448 −0.422224 0.906492i $$-0.638750\pi$$
−0.422224 + 0.906492i $$0.638750\pi$$
$$360$$ −6.00000 −0.316228
$$361$$ −19.0000 −1.00000
$$362$$ −10.0000 −0.525588
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ −16.0000 −0.837478
$$366$$ 0 0
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 4.00000 0.208514
$$369$$ −4.00000 −0.208232
$$370$$ −12.0000 −0.623850
$$371$$ −6.00000 −0.311504
$$372$$ −20.0000 −1.03695
$$373$$ 26.0000 1.34623 0.673114 0.739538i $$-0.264956\pi$$
0.673114 + 0.739538i $$0.264956\pi$$
$$374$$ 0 0
$$375$$ 24.0000 1.23935
$$376$$ −30.0000 −1.54713
$$377$$ −24.0000 −1.23606
$$378$$ 4.00000 0.205738
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ −16.0000 −0.819705
$$382$$ −8.00000 −0.409316
$$383$$ 2.00000 0.102195 0.0510976 0.998694i $$-0.483728\pi$$
0.0510976 + 0.998694i $$0.483728\pi$$
$$384$$ 6.00000 0.306186
$$385$$ 0 0
$$386$$ −14.0000 −0.712581
$$387$$ −12.0000 −0.609994
$$388$$ 10.0000 0.507673
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ −16.0000 −0.810191
$$391$$ 16.0000 0.809155
$$392$$ 3.00000 0.151523
$$393$$ −24.0000 −1.21064
$$394$$ 22.0000 1.10834
$$395$$ 16.0000 0.805047
$$396$$ 0 0
$$397$$ 22.0000 1.10415 0.552074 0.833795i $$-0.313837\pi$$
0.552074 + 0.833795i $$0.313837\pi$$
$$398$$ 18.0000 0.902258
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ −22.0000 −1.09863 −0.549314 0.835616i $$-0.685111\pi$$
−0.549314 + 0.835616i $$0.685111\pi$$
$$402$$ −16.0000 −0.798007
$$403$$ −40.0000 −1.99254
$$404$$ −4.00000 −0.199007
$$405$$ 22.0000 1.09319
$$406$$ −6.00000 −0.297775
$$407$$ 0 0
$$408$$ −24.0000 −1.18818
$$409$$ −24.0000 −1.18672 −0.593362 0.804936i $$-0.702200\pi$$
−0.593362 + 0.804936i $$0.702200\pi$$
$$410$$ −8.00000 −0.395092
$$411$$ −20.0000 −0.986527
$$412$$ −14.0000 −0.689730
$$413$$ 2.00000 0.0984136
$$414$$ 4.00000 0.196589
$$415$$ 0 0
$$416$$ 20.0000 0.980581
$$417$$ 16.0000 0.783523
$$418$$ 0 0
$$419$$ 2.00000 0.0977064 0.0488532 0.998806i $$-0.484443\pi$$
0.0488532 + 0.998806i $$0.484443\pi$$
$$420$$ 4.00000 0.195180
$$421$$ −14.0000 −0.682318 −0.341159 0.940006i $$-0.610819\pi$$
−0.341159 + 0.940006i $$0.610819\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ −10.0000 −0.486217
$$424$$ −18.0000 −0.874157
$$425$$ 4.00000 0.194029
$$426$$ 24.0000 1.16280
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ −24.0000 −1.15738
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 4.00000 0.192450
$$433$$ −26.0000 −1.24948 −0.624740 0.780833i $$-0.714795\pi$$
−0.624740 + 0.780833i $$0.714795\pi$$
$$434$$ −10.0000 −0.480015
$$435$$ −24.0000 −1.15071
$$436$$ −14.0000 −0.670478
$$437$$ 0 0
$$438$$ −16.0000 −0.764510
$$439$$ 20.0000 0.954548 0.477274 0.878755i $$-0.341625\pi$$
0.477274 + 0.878755i $$0.341625\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −16.0000 −0.761042
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 12.0000 0.569495
$$445$$ 12.0000 0.568855
$$446$$ −22.0000 −1.04173
$$447$$ 20.0000 0.945968
$$448$$ 7.00000 0.330719
$$449$$ −10.0000 −0.471929 −0.235965 0.971762i $$-0.575825\pi$$
−0.235965 + 0.971762i $$0.575825\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −18.0000 −0.846649
$$453$$ 32.0000 1.50349
$$454$$ 12.0000 0.563188
$$455$$ 8.00000 0.375046
$$456$$ 0 0
$$457$$ −18.0000 −0.842004 −0.421002 0.907060i $$-0.638322\pi$$
−0.421002 + 0.907060i $$0.638322\pi$$
$$458$$ −18.0000 −0.841085
$$459$$ 16.0000 0.746816
$$460$$ −8.00000 −0.373002
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ −40.0000 −1.85496
$$466$$ −18.0000 −0.833834
$$467$$ 30.0000 1.38823 0.694117 0.719862i $$-0.255795\pi$$
0.694117 + 0.719862i $$0.255795\pi$$
$$468$$ 4.00000 0.184900
$$469$$ 8.00000 0.369406
$$470$$ −20.0000 −0.922531
$$471$$ 28.0000 1.29017
$$472$$ 6.00000 0.276172
$$473$$ 0 0
$$474$$ 16.0000 0.734904
$$475$$ 0 0
$$476$$ 4.00000 0.183340
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ 4.00000 0.182765 0.0913823 0.995816i $$-0.470871\pi$$
0.0913823 + 0.995816i $$0.470871\pi$$
$$480$$ 20.0000 0.912871
$$481$$ 24.0000 1.09431
$$482$$ −20.0000 −0.910975
$$483$$ −8.00000 −0.364013
$$484$$ 0 0
$$485$$ 20.0000 0.908153
$$486$$ 10.0000 0.453609
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 0 0
$$489$$ −16.0000 −0.723545
$$490$$ 2.00000 0.0903508
$$491$$ −28.0000 −1.26362 −0.631811 0.775122i $$-0.717688\pi$$
−0.631811 + 0.775122i $$0.717688\pi$$
$$492$$ 8.00000 0.360668
$$493$$ −24.0000 −1.08091
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −10.0000 −0.449013
$$497$$ −12.0000 −0.538274
$$498$$ 0 0
$$499$$ −16.0000 −0.716258 −0.358129 0.933672i $$-0.616585\pi$$
−0.358129 + 0.933672i $$0.616585\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 2.00000 0.0892644
$$503$$ −4.00000 −0.178351 −0.0891756 0.996016i $$-0.528423\pi$$
−0.0891756 + 0.996016i $$0.528423\pi$$
$$504$$ 3.00000 0.133631
$$505$$ −8.00000 −0.355995
$$506$$ 0 0
$$507$$ 6.00000 0.266469
$$508$$ 8.00000 0.354943
$$509$$ 18.0000 0.797836 0.398918 0.916987i $$-0.369386\pi$$
0.398918 + 0.916987i $$0.369386\pi$$
$$510$$ −16.0000 −0.708492
$$511$$ 8.00000 0.353899
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 14.0000 0.617514
$$515$$ −28.0000 −1.23383
$$516$$ 24.0000 1.05654
$$517$$ 0 0
$$518$$ 6.00000 0.263625
$$519$$ −24.0000 −1.05348
$$520$$ 24.0000 1.05247
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 12.0000 0.524222
$$525$$ −2.00000 −0.0872872
$$526$$ 8.00000 0.348817
$$527$$ −40.0000 −1.74243
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −12.0000 −0.521247
$$531$$ 2.00000 0.0867926
$$532$$ 0 0
$$533$$ 16.0000 0.693037
$$534$$ 12.0000 0.519291
$$535$$ 24.0000 1.03761
$$536$$ 24.0000 1.03664
$$537$$ 24.0000 1.03568
$$538$$ −10.0000 −0.431131
$$539$$ 0 0
$$540$$ −8.00000 −0.344265
$$541$$ 26.0000 1.11783 0.558914 0.829226i $$-0.311218\pi$$
0.558914 + 0.829226i $$0.311218\pi$$
$$542$$ −4.00000 −0.171815
$$543$$ 20.0000 0.858282
$$544$$ 20.0000 0.857493
$$545$$ −28.0000 −1.19939
$$546$$ 8.00000 0.342368
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ −24.0000 −1.02151
$$553$$ −8.00000 −0.340195
$$554$$ 22.0000 0.934690
$$555$$ 24.0000 1.01874
$$556$$ −8.00000 −0.339276
$$557$$ −22.0000 −0.932170 −0.466085 0.884740i $$-0.654336\pi$$
−0.466085 + 0.884740i $$0.654336\pi$$
$$558$$ −10.0000 −0.423334
$$559$$ 48.0000 2.03018
$$560$$ 2.00000 0.0845154
$$561$$ 0 0
$$562$$ 6.00000 0.253095
$$563$$ 32.0000 1.34864 0.674320 0.738440i $$-0.264437\pi$$
0.674320 + 0.738440i $$0.264437\pi$$
$$564$$ 20.0000 0.842152
$$565$$ −36.0000 −1.51453
$$566$$ 4.00000 0.168133
$$567$$ −11.0000 −0.461957
$$568$$ −36.0000 −1.51053
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ −20.0000 −0.836974 −0.418487 0.908223i $$-0.637439\pi$$
−0.418487 + 0.908223i $$0.637439\pi$$
$$572$$ 0 0
$$573$$ 16.0000 0.668410
$$574$$ 4.00000 0.166957
$$575$$ 4.00000 0.166812
$$576$$ 7.00000 0.291667
$$577$$ −18.0000 −0.749350 −0.374675 0.927156i $$-0.622246\pi$$
−0.374675 + 0.927156i $$0.622246\pi$$
$$578$$ 1.00000 0.0415945
$$579$$ 28.0000 1.16364
$$580$$ 12.0000 0.498273
$$581$$ 0 0
$$582$$ 20.0000 0.829027
$$583$$ 0 0
$$584$$ 24.0000 0.993127
$$585$$ 8.00000 0.330759
$$586$$ −24.0000 −0.991431
$$587$$ −2.00000 −0.0825488 −0.0412744 0.999148i $$-0.513142\pi$$
−0.0412744 + 0.999148i $$0.513142\pi$$
$$588$$ −2.00000 −0.0824786
$$589$$ 0 0
$$590$$ 4.00000 0.164677
$$591$$ −44.0000 −1.80992
$$592$$ 6.00000 0.246598
$$593$$ −32.0000 −1.31408 −0.657041 0.753855i $$-0.728192\pi$$
−0.657041 + 0.753855i $$0.728192\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ −10.0000 −0.409616
$$597$$ −36.0000 −1.47338
$$598$$ −16.0000 −0.654289
$$599$$ −20.0000 −0.817178 −0.408589 0.912719i $$-0.633979\pi$$
−0.408589 + 0.912719i $$0.633979\pi$$
$$600$$ −6.00000 −0.244949
$$601$$ 28.0000 1.14214 0.571072 0.820900i $$-0.306528\pi$$
0.571072 + 0.820900i $$0.306528\pi$$
$$602$$ 12.0000 0.489083
$$603$$ 8.00000 0.325785
$$604$$ −16.0000 −0.651031
$$605$$ 0 0
$$606$$ −8.00000 −0.324978
$$607$$ 40.0000 1.62355 0.811775 0.583970i $$-0.198502\pi$$
0.811775 + 0.583970i $$0.198502\pi$$
$$608$$ 0 0
$$609$$ 12.0000 0.486265
$$610$$ 0 0
$$611$$ 40.0000 1.61823
$$612$$ 4.00000 0.161690
$$613$$ −26.0000 −1.05013 −0.525065 0.851062i $$-0.675959\pi$$
−0.525065 + 0.851062i $$0.675959\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 16.0000 0.645182
$$616$$ 0 0
$$617$$ 6.00000 0.241551 0.120775 0.992680i $$-0.461462\pi$$
0.120775 + 0.992680i $$0.461462\pi$$
$$618$$ −28.0000 −1.12633
$$619$$ 14.0000 0.562708 0.281354 0.959604i $$-0.409217\pi$$
0.281354 + 0.959604i $$0.409217\pi$$
$$620$$ 20.0000 0.803219
$$621$$ 16.0000 0.642058
$$622$$ 18.0000 0.721734
$$623$$ −6.00000 −0.240385
$$624$$ 8.00000 0.320256
$$625$$ −19.0000 −0.760000
$$626$$ −2.00000 −0.0799361
$$627$$ 0 0
$$628$$ −14.0000 −0.558661
$$629$$ 24.0000 0.956943
$$630$$ 2.00000 0.0796819
$$631$$ −8.00000 −0.318475 −0.159237 0.987240i $$-0.550904\pi$$
−0.159237 + 0.987240i $$0.550904\pi$$
$$632$$ −24.0000 −0.954669
$$633$$ 24.0000 0.953914
$$634$$ 2.00000 0.0794301
$$635$$ 16.0000 0.634941
$$636$$ 12.0000 0.475831
$$637$$ −4.00000 −0.158486
$$638$$ 0 0
$$639$$ −12.0000 −0.474713
$$640$$ −6.00000 −0.237171
$$641$$ −18.0000 −0.710957 −0.355479 0.934684i $$-0.615682\pi$$
−0.355479 + 0.934684i $$0.615682\pi$$
$$642$$ 24.0000 0.947204
$$643$$ 14.0000 0.552106 0.276053 0.961142i $$-0.410973\pi$$
0.276053 + 0.961142i $$0.410973\pi$$
$$644$$ 4.00000 0.157622
$$645$$ 48.0000 1.89000
$$646$$ 0 0
$$647$$ −22.0000 −0.864909 −0.432455 0.901656i $$-0.642352\pi$$
−0.432455 + 0.901656i $$0.642352\pi$$
$$648$$ −33.0000 −1.29636
$$649$$ 0 0
$$650$$ −4.00000 −0.156893
$$651$$ 20.0000 0.783862
$$652$$ 8.00000 0.313304
$$653$$ −26.0000 −1.01746 −0.508729 0.860927i $$-0.669885\pi$$
−0.508729 + 0.860927i $$0.669885\pi$$
$$654$$ −28.0000 −1.09489
$$655$$ 24.0000 0.937758
$$656$$ 4.00000 0.156174
$$657$$ 8.00000 0.312110
$$658$$ 10.0000 0.389841
$$659$$ −4.00000 −0.155818 −0.0779089 0.996960i $$-0.524824\pi$$
−0.0779089 + 0.996960i $$0.524824\pi$$
$$660$$ 0 0
$$661$$ 22.0000 0.855701 0.427850 0.903850i $$-0.359271\pi$$
0.427850 + 0.903850i $$0.359271\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 32.0000 1.24278
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ −24.0000 −0.929284
$$668$$ 0 0
$$669$$ 44.0000 1.70114
$$670$$ 16.0000 0.618134
$$671$$ 0 0
$$672$$ −10.0000 −0.385758
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 4.00000 0.153960
$$676$$ −3.00000 −0.115385
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ −36.0000 −1.38257
$$679$$ −10.0000 −0.383765
$$680$$ 24.0000 0.920358
$$681$$ −24.0000 −0.919682
$$682$$ 0 0
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 0 0
$$685$$ 20.0000 0.764161
$$686$$ −1.00000 −0.0381802
$$687$$ 36.0000 1.37349
$$688$$ 12.0000 0.457496
$$689$$ 24.0000 0.914327
$$690$$ −16.0000 −0.609110
$$691$$ −46.0000 −1.74992 −0.874961 0.484193i $$-0.839113\pi$$
−0.874961 + 0.484193i $$0.839113\pi$$
$$692$$ 12.0000 0.456172
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ −16.0000 −0.606915
$$696$$ 36.0000 1.36458
$$697$$ 16.0000 0.606043
$$698$$ 0 0
$$699$$ 36.0000 1.36165
$$700$$ 1.00000 0.0377964
$$701$$ 22.0000 0.830929 0.415464 0.909610i $$-0.363619\pi$$
0.415464 + 0.909610i $$0.363619\pi$$
$$702$$ −16.0000 −0.603881
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 40.0000 1.50649
$$706$$ 30.0000 1.12906
$$707$$ 4.00000 0.150435
$$708$$ −4.00000 −0.150329
$$709$$ −34.0000 −1.27690 −0.638448 0.769665i $$-0.720423\pi$$
−0.638448 + 0.769665i $$0.720423\pi$$
$$710$$ −24.0000 −0.900704
$$711$$ −8.00000 −0.300023
$$712$$ −18.0000 −0.674579
$$713$$ −40.0000 −1.49801
$$714$$ 8.00000 0.299392
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 16.0000 0.597115
$$719$$ −6.00000 −0.223762 −0.111881 0.993722i $$-0.535688\pi$$
−0.111881 + 0.993722i $$0.535688\pi$$
$$720$$ 2.00000 0.0745356
$$721$$ 14.0000 0.521387
$$722$$ 19.0000 0.707107
$$723$$ 40.0000 1.48762
$$724$$ −10.0000 −0.371647
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 18.0000 0.667583 0.333792 0.942647i $$-0.391672\pi$$
0.333792 + 0.942647i $$0.391672\pi$$
$$728$$ −12.0000 −0.444750
$$729$$ 13.0000 0.481481
$$730$$ 16.0000 0.592187
$$731$$ 48.0000 1.77534
$$732$$ 0 0
$$733$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$734$$ −22.0000 −0.812035
$$735$$ −4.00000 −0.147542
$$736$$ 20.0000 0.737210
$$737$$ 0 0
$$738$$ 4.00000 0.147242
$$739$$ 4.00000 0.147142 0.0735712 0.997290i $$-0.476560\pi$$
0.0735712 + 0.997290i $$0.476560\pi$$
$$740$$ −12.0000 −0.441129
$$741$$ 0 0
$$742$$ 6.00000 0.220267
$$743$$ 8.00000 0.293492 0.146746 0.989174i $$-0.453120\pi$$
0.146746 + 0.989174i $$0.453120\pi$$
$$744$$ 60.0000 2.19971
$$745$$ −20.0000 −0.732743
$$746$$ −26.0000 −0.951928
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ −24.0000 −0.876356
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ 10.0000 0.364662
$$753$$ −4.00000 −0.145768
$$754$$ 24.0000 0.874028
$$755$$ −32.0000 −1.16460
$$756$$ 4.00000 0.145479
$$757$$ −10.0000 −0.363456 −0.181728 0.983349i $$-0.558169\pi$$
−0.181728 + 0.983349i $$0.558169\pi$$
$$758$$ 8.00000 0.290573
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −48.0000 −1.74000 −0.869999 0.493053i $$-0.835881\pi$$
−0.869999 + 0.493053i $$0.835881\pi$$
$$762$$ 16.0000 0.579619
$$763$$ 14.0000 0.506834
$$764$$ −8.00000 −0.289430
$$765$$ 8.00000 0.289241
$$766$$ −2.00000 −0.0722629
$$767$$ −8.00000 −0.288863
$$768$$ −34.0000 −1.22687
$$769$$ −32.0000 −1.15395 −0.576975 0.816762i $$-0.695767\pi$$
−0.576975 + 0.816762i $$0.695767\pi$$
$$770$$ 0 0
$$771$$ −28.0000 −1.00840
$$772$$ −14.0000 −0.503871
$$773$$ −30.0000 −1.07903 −0.539513 0.841978i $$-0.681391\pi$$
−0.539513 + 0.841978i $$0.681391\pi$$
$$774$$ 12.0000 0.431331
$$775$$ −10.0000 −0.359211
$$776$$ −30.0000 −1.07694
$$777$$ −12.0000 −0.430498
$$778$$ −6.00000 −0.215110
$$779$$ 0 0
$$780$$ −16.0000 −0.572892
$$781$$ 0 0
$$782$$ −16.0000 −0.572159
$$783$$ −24.0000 −0.857690
$$784$$ −1.00000 −0.0357143
$$785$$ −28.0000 −0.999363
$$786$$ 24.0000 0.856052
$$787$$ −16.0000 −0.570338 −0.285169 0.958477i $$-0.592050\pi$$
−0.285169 + 0.958477i $$0.592050\pi$$
$$788$$ 22.0000 0.783718
$$789$$ −16.0000 −0.569615
$$790$$ −16.0000 −0.569254
$$791$$ 18.0000 0.640006
$$792$$ 0 0
$$793$$ 0 0
$$794$$ −22.0000 −0.780751
$$795$$ 24.0000 0.851192
$$796$$ 18.0000 0.637993
$$797$$ 14.0000 0.495905 0.247953 0.968772i $$-0.420242\pi$$
0.247953 + 0.968772i $$0.420242\pi$$
$$798$$ 0 0
$$799$$ 40.0000 1.41510
$$800$$ 5.00000 0.176777
$$801$$ −6.00000 −0.212000
$$802$$ 22.0000 0.776847
$$803$$ 0 0
$$804$$ −16.0000 −0.564276
$$805$$ 8.00000 0.281963
$$806$$ 40.0000 1.40894
$$807$$ 20.0000 0.704033
$$808$$ 12.0000 0.422159
$$809$$ 30.0000 1.05474 0.527372 0.849635i $$-0.323177\pi$$
0.527372 + 0.849635i $$0.323177\pi$$
$$810$$ −22.0000 −0.773001
$$811$$ −28.0000 −0.983213 −0.491606 0.870817i $$-0.663590\pi$$
−0.491606 + 0.870817i $$0.663590\pi$$
$$812$$ −6.00000 −0.210559
$$813$$ 8.00000 0.280572
$$814$$ 0 0
$$815$$ 16.0000 0.560456
$$816$$ 8.00000 0.280056
$$817$$ 0 0
$$818$$ 24.0000 0.839140
$$819$$ −4.00000 −0.139771
$$820$$ −8.00000 −0.279372
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ 20.0000 0.697580
$$823$$ 24.0000 0.836587 0.418294 0.908312i $$-0.362628\pi$$
0.418294 + 0.908312i $$0.362628\pi$$
$$824$$ 42.0000 1.46314
$$825$$ 0 0
$$826$$ −2.00000 −0.0695889
$$827$$ 28.0000 0.973655 0.486828 0.873498i $$-0.338154\pi$$
0.486828 + 0.873498i $$0.338154\pi$$
$$828$$ 4.00000 0.139010
$$829$$ −2.00000 −0.0694629 −0.0347314 0.999397i $$-0.511058\pi$$
−0.0347314 + 0.999397i $$0.511058\pi$$
$$830$$ 0 0
$$831$$ −44.0000 −1.52634
$$832$$ −28.0000 −0.970725
$$833$$ −4.00000 −0.138592
$$834$$ −16.0000 −0.554035
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −40.0000 −1.38260
$$838$$ −2.00000 −0.0690889
$$839$$ 34.0000 1.17381 0.586905 0.809656i $$-0.300346\pi$$
0.586905 + 0.809656i $$0.300346\pi$$
$$840$$ −12.0000 −0.414039
$$841$$ 7.00000 0.241379
$$842$$ 14.0000 0.482472
$$843$$ −12.0000 −0.413302
$$844$$ −12.0000 −0.413057
$$845$$ −6.00000 −0.206406
$$846$$ 10.0000 0.343807
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ −8.00000 −0.274559
$$850$$ −4.00000 −0.137199
$$851$$ 24.0000 0.822709
$$852$$ 24.0000 0.822226
$$853$$ −44.0000 −1.50653 −0.753266 0.657716i $$-0.771523\pi$$
−0.753266 + 0.657716i $$0.771523\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −36.0000 −1.23045
$$857$$ −56.0000 −1.91292 −0.956462 0.291858i $$-0.905727\pi$$
−0.956462 + 0.291858i $$0.905727\pi$$
$$858$$ 0 0
$$859$$ 6.00000 0.204717 0.102359 0.994748i $$-0.467361\pi$$
0.102359 + 0.994748i $$0.467361\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ −8.00000 −0.272639
$$862$$ 0 0
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 20.0000 0.680414
$$865$$ 24.0000 0.816024
$$866$$ 26.0000 0.883516
$$867$$ −2.00000 −0.0679236
$$868$$ −10.0000 −0.339422
$$869$$ 0 0
$$870$$ 24.0000 0.813676
$$871$$ −32.0000 −1.08428
$$872$$ 42.0000 1.42230
$$873$$ −10.0000 −0.338449
$$874$$ 0 0
$$875$$ 12.0000 0.405674
$$876$$ −16.0000 −0.540590
$$877$$ −42.0000 −1.41824 −0.709120 0.705088i $$-0.750907\pi$$
−0.709120 + 0.705088i $$0.750907\pi$$
$$878$$ −20.0000 −0.674967
$$879$$ 48.0000 1.61900
$$880$$ 0 0
$$881$$ −34.0000 −1.14549 −0.572745 0.819734i $$-0.694121\pi$$
−0.572745 + 0.819734i $$0.694121\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ 28.0000 0.942275 0.471138 0.882060i $$-0.343844\pi$$
0.471138 + 0.882060i $$0.343844\pi$$
$$884$$ −16.0000 −0.538138
$$885$$ −8.00000 −0.268917
$$886$$ −4.00000 −0.134383
$$887$$ 28.0000 0.940148 0.470074 0.882627i $$-0.344227\pi$$
0.470074 + 0.882627i $$0.344227\pi$$
$$888$$ −36.0000 −1.20808
$$889$$ −8.00000 −0.268311
$$890$$ −12.0000 −0.402241
$$891$$ 0 0
$$892$$ −22.0000 −0.736614
$$893$$ 0 0
$$894$$ −20.0000 −0.668900
$$895$$ −24.0000 −0.802232
$$896$$ 3.00000 0.100223
$$897$$ 32.0000 1.06845
$$898$$ 10.0000 0.333704
$$899$$ 60.0000 2.00111
$$900$$ 1.00000 0.0333333
$$901$$ 24.0000 0.799556
$$902$$ 0 0
$$903$$ −24.0000 −0.798670
$$904$$ 54.0000 1.79601
$$905$$ −20.0000 −0.664822
$$906$$ −32.0000 −1.06313
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 12.0000 0.398234
$$909$$ 4.00000 0.132672
$$910$$ −8.00000 −0.265197
$$911$$ 36.0000 1.19273 0.596367 0.802712i $$-0.296610\pi$$
0.596367 + 0.802712i $$0.296610\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 18.0000 0.595387
$$915$$ 0 0
$$916$$ −18.0000 −0.594737
$$917$$ −12.0000 −0.396275
$$918$$ −16.0000 −0.528079
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 24.0000 0.791257
$$921$$ −40.0000 −1.31804
$$922$$ 0 0
$$923$$ 48.0000 1.57994
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ −4.00000 −0.131448
$$927$$ 14.0000 0.459820
$$928$$ −30.0000 −0.984798
$$929$$ 6.00000 0.196854 0.0984268 0.995144i $$-0.468619\pi$$
0.0984268 + 0.995144i $$0.468619\pi$$
$$930$$ 40.0000 1.31165
$$931$$ 0 0
$$932$$ −18.0000 −0.589610
$$933$$ −36.0000 −1.17859
$$934$$ −30.0000 −0.981630
$$935$$ 0 0
$$936$$ −12.0000 −0.392232
$$937$$ 16.0000 0.522697 0.261349 0.965244i $$-0.415833\pi$$
0.261349 + 0.965244i $$0.415833\pi$$
$$938$$ −8.00000 −0.261209
$$939$$ 4.00000 0.130535
$$940$$ −20.0000 −0.652328
$$941$$ 24.0000 0.782378 0.391189 0.920310i $$-0.372064\pi$$
0.391189 + 0.920310i $$0.372064\pi$$
$$942$$ −28.0000 −0.912289
$$943$$ 16.0000 0.521032
$$944$$ −2.00000 −0.0650945
$$945$$ 8.00000 0.260240
$$946$$ 0 0
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 16.0000 0.519656
$$949$$ −32.0000 −1.03876
$$950$$ 0 0
$$951$$ −4.00000 −0.129709
$$952$$ −12.0000 −0.388922
$$953$$ −34.0000 −1.10137 −0.550684 0.834714i $$-0.685633\pi$$
−0.550684 + 0.834714i $$0.685633\pi$$
$$954$$ 6.00000 0.194257
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ 0 0
$$958$$ −4.00000 −0.129234
$$959$$ −10.0000 −0.322917
$$960$$ −28.0000 −0.903696
$$961$$ 69.0000 2.22581
$$962$$ −24.0000 −0.773791
$$963$$ −12.0000 −0.386695
$$964$$ −20.0000 −0.644157
$$965$$ −28.0000 −0.901352
$$966$$ 8.00000 0.257396
$$967$$ −40.0000 −1.28631 −0.643157 0.765735i $$-0.722376\pi$$
−0.643157 + 0.765735i $$0.722376\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ −20.0000 −0.642161
$$971$$ 14.0000 0.449281 0.224641 0.974442i $$-0.427879\pi$$
0.224641 + 0.974442i $$0.427879\pi$$
$$972$$ 10.0000 0.320750
$$973$$ 8.00000 0.256468
$$974$$ 28.0000 0.897178
$$975$$ 8.00000 0.256205
$$976$$ 0 0
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 16.0000 0.511624
$$979$$ 0 0
$$980$$ 2.00000 0.0638877
$$981$$ 14.0000 0.446986
$$982$$ 28.0000 0.893516
$$983$$ 54.0000 1.72233 0.861166 0.508323i $$-0.169735\pi$$
0.861166 + 0.508323i $$0.169735\pi$$
$$984$$ −24.0000 −0.765092
$$985$$ 44.0000 1.40196
$$986$$ 24.0000 0.764316
$$987$$ −20.0000 −0.636607
$$988$$ 0 0
$$989$$ 48.0000 1.52631
$$990$$ 0 0
$$991$$ 52.0000 1.65183 0.825917 0.563791i $$-0.190658\pi$$
0.825917 + 0.563791i $$0.190658\pi$$
$$992$$ −50.0000 −1.58750
$$993$$ −40.0000 −1.26936
$$994$$ 12.0000 0.380617
$$995$$ 36.0000 1.14128
$$996$$ 0 0
$$997$$ −20.0000 −0.633406 −0.316703 0.948525i $$-0.602576\pi$$
−0.316703 + 0.948525i $$0.602576\pi$$
$$998$$ 16.0000 0.506471
$$999$$ 24.0000 0.759326
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 847.2.a.a.1.1 1
3.2 odd 2 7623.2.a.n.1.1 1
7.6 odd 2 5929.2.a.b.1.1 1
11.2 odd 10 847.2.f.e.323.1 4
11.3 even 5 847.2.f.k.372.1 4
11.4 even 5 847.2.f.k.148.1 4
11.5 even 5 847.2.f.k.729.1 4
11.6 odd 10 847.2.f.e.729.1 4
11.7 odd 10 847.2.f.e.148.1 4
11.8 odd 10 847.2.f.e.372.1 4
11.9 even 5 847.2.f.k.323.1 4
11.10 odd 2 77.2.a.c.1.1 1
33.32 even 2 693.2.a.a.1.1 1
44.43 even 2 1232.2.a.a.1.1 1
55.32 even 4 1925.2.b.d.1849.2 2
55.43 even 4 1925.2.b.d.1849.1 2
55.54 odd 2 1925.2.a.c.1.1 1
77.10 even 6 539.2.e.b.177.1 2
77.32 odd 6 539.2.e.a.177.1 2
77.54 even 6 539.2.e.b.67.1 2
77.65 odd 6 539.2.e.a.67.1 2
77.76 even 2 539.2.a.d.1.1 1
88.21 odd 2 4928.2.a.g.1.1 1
88.43 even 2 4928.2.a.bi.1.1 1
231.230 odd 2 4851.2.a.a.1.1 1
308.307 odd 2 8624.2.a.bc.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.c.1.1 1 11.10 odd 2
539.2.a.d.1.1 1 77.76 even 2
539.2.e.a.67.1 2 77.65 odd 6
539.2.e.a.177.1 2 77.32 odd 6
539.2.e.b.67.1 2 77.54 even 6
539.2.e.b.177.1 2 77.10 even 6
693.2.a.a.1.1 1 33.32 even 2
847.2.a.a.1.1 1 1.1 even 1 trivial
847.2.f.e.148.1 4 11.7 odd 10
847.2.f.e.323.1 4 11.2 odd 10
847.2.f.e.372.1 4 11.8 odd 10
847.2.f.e.729.1 4 11.6 odd 10
847.2.f.k.148.1 4 11.4 even 5
847.2.f.k.323.1 4 11.9 even 5
847.2.f.k.372.1 4 11.3 even 5
847.2.f.k.729.1 4 11.5 even 5
1232.2.a.a.1.1 1 44.43 even 2
1925.2.a.c.1.1 1 55.54 odd 2
1925.2.b.d.1849.1 2 55.43 even 4
1925.2.b.d.1849.2 2 55.32 even 4
4851.2.a.a.1.1 1 231.230 odd 2
4928.2.a.g.1.1 1 88.21 odd 2
4928.2.a.bi.1.1 1 88.43 even 2
5929.2.a.b.1.1 1 7.6 odd 2
7623.2.a.n.1.1 1 3.2 odd 2
8624.2.a.bc.1.1 1 308.307 odd 2