# Properties

 Label 162.2.a.a.1.1 Level $162$ Weight $2$ Character 162.1 Self dual yes Analytic conductor $1.294$ 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] = [162,2,Mod(1,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("162.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 162.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: $$1.29357651274$$ 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 $$\chi$$ $$=$$ 162.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} -4.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} -1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -4.00000 q^{19} -3.00000 q^{20} +4.00000 q^{25} +1.00000 q^{26} -4.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} +3.00000 q^{34} +12.0000 q^{35} -1.00000 q^{37} +4.00000 q^{38} +3.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} -12.0000 q^{47} +9.00000 q^{49} -4.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +4.00000 q^{56} -9.00000 q^{58} -1.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -12.0000 q^{70} -12.0000 q^{71} +11.0000 q^{73} +1.00000 q^{74} -4.00000 q^{76} -16.0000 q^{79} -3.00000 q^{80} -6.00000 q^{82} -12.0000 q^{83} +9.00000 q^{85} -8.00000 q^{86} -3.00000 q^{89} +4.00000 q^{91} +12.0000 q^{94} +12.0000 q^{95} +2.00000 q^{97} -9.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ −4.00000 −1.51186 −0.755929 0.654654i $$-0.772814\pi$$
−0.755929 + 0.654654i $$0.772814\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 0 0
$$10$$ 3.00000 0.948683
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 4.00000 1.06904
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −3.00000 −0.670820
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ −4.00000 −0.755929
$$29$$ 9.00000 1.67126 0.835629 0.549294i $$-0.185103\pi$$
0.835629 + 0.549294i $$0.185103\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 3.00000 0.514496
$$35$$ 12.0000 2.02837
$$36$$ 0 0
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ 3.00000 0.474342
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ 9.00000 1.28571
$$50$$ −4.00000 −0.565685
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 4.00000 0.534522
$$57$$ 0 0
$$58$$ −9.00000 −1.18176
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −1.00000 −0.128037 −0.0640184 0.997949i $$-0.520392\pi$$
−0.0640184 + 0.997949i $$0.520392\pi$$
$$62$$ 4.00000 0.508001
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 3.00000 0.372104
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −3.00000 −0.363803
$$69$$ 0 0
$$70$$ −12.0000 −1.43427
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 11.0000 1.28745 0.643726 0.765256i $$-0.277388\pi$$
0.643726 + 0.765256i $$0.277388\pi$$
$$74$$ 1.00000 0.116248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −16.0000 −1.80014 −0.900070 0.435745i $$-0.856485\pi$$
−0.900070 + 0.435745i $$0.856485\pi$$
$$80$$ −3.00000 −0.335410
$$81$$ 0 0
$$82$$ −6.00000 −0.662589
$$83$$ −12.0000 −1.31717 −0.658586 0.752506i $$-0.728845\pi$$
−0.658586 + 0.752506i $$0.728845\pi$$
$$84$$ 0 0
$$85$$ 9.00000 0.976187
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.00000 −0.317999 −0.159000 0.987279i $$-0.550827\pi$$
−0.159000 + 0.987279i $$0.550827\pi$$
$$90$$ 0 0
$$91$$ 4.00000 0.419314
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 12.0000 1.23771
$$95$$ 12.0000 1.23117
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ −9.00000 −0.909137
$$99$$ 0 0
$$100$$ 4.00000 0.400000
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 11.0000 1.05361 0.526804 0.849987i $$-0.323390\pi$$
0.526804 + 0.849987i $$0.323390\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −4.00000 −0.377964
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 9.00000 0.835629
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 12.0000 1.10004
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 1.00000 0.0905357
$$123$$ 0 0
$$124$$ −4.00000 −0.359211
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 0 0
$$130$$ −3.00000 −0.263117
$$131$$ −12.0000 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ 16.0000 1.38738
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 3.00000 0.257248
$$137$$ 9.00000 0.768922 0.384461 0.923141i $$-0.374387\pi$$
0.384461 + 0.923141i $$0.374387\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 12.0000 1.01419
$$141$$ 0 0
$$142$$ 12.0000 1.00702
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −27.0000 −2.24223
$$146$$ −11.0000 −0.910366
$$147$$ 0 0
$$148$$ −1.00000 −0.0821995
$$149$$ 9.00000 0.737309 0.368654 0.929567i $$-0.379819\pi$$
0.368654 + 0.929567i $$0.379819\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 16.0000 1.27289
$$159$$ 0 0
$$160$$ 3.00000 0.237171
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 8.00000 0.626608 0.313304 0.949653i $$-0.398564\pi$$
0.313304 + 0.949653i $$0.398564\pi$$
$$164$$ 6.00000 0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 12.0000 0.928588 0.464294 0.885681i $$-0.346308\pi$$
0.464294 + 0.885681i $$0.346308\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ −9.00000 −0.690268
$$171$$ 0 0
$$172$$ 8.00000 0.609994
$$173$$ −3.00000 −0.228086 −0.114043 0.993476i $$-0.536380\pi$$
−0.114043 + 0.993476i $$0.536380\pi$$
$$174$$ 0 0
$$175$$ −16.0000 −1.20949
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 3.00000 0.224860
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −10.0000 −0.743294 −0.371647 0.928374i $$-0.621207\pi$$
−0.371647 + 0.928374i $$0.621207\pi$$
$$182$$ −4.00000 −0.296500
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 3.00000 0.220564
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ −12.0000 −0.870572
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −13.0000 −0.935760 −0.467880 0.883792i $$-0.654982\pi$$
−0.467880 + 0.883792i $$0.654982\pi$$
$$194$$ −2.00000 −0.143592
$$195$$ 0 0
$$196$$ 9.00000 0.642857
$$197$$ −3.00000 −0.213741 −0.106871 0.994273i $$-0.534083\pi$$
−0.106871 + 0.994273i $$0.534083\pi$$
$$198$$ 0 0
$$199$$ −4.00000 −0.283552 −0.141776 0.989899i $$-0.545281\pi$$
−0.141776 + 0.989899i $$0.545281\pi$$
$$200$$ −4.00000 −0.282843
$$201$$ 0 0
$$202$$ 6.00000 0.422159
$$203$$ −36.0000 −2.52670
$$204$$ 0 0
$$205$$ −18.0000 −1.25717
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 0 0
$$214$$ −12.0000 −0.820303
$$215$$ −24.0000 −1.63679
$$216$$ 0 0
$$217$$ 16.0000 1.08615
$$218$$ −11.0000 −0.745014
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.00000 0.201802
$$222$$ 0 0
$$223$$ 8.00000 0.535720 0.267860 0.963458i $$-0.413684\pi$$
0.267860 + 0.963458i $$0.413684\pi$$
$$224$$ 4.00000 0.267261
$$225$$ 0 0
$$226$$ 15.0000 0.997785
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 23.0000 1.51988 0.759941 0.649992i $$-0.225228\pi$$
0.759941 + 0.649992i $$0.225228\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −9.00000 −0.590879
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 36.0000 2.34838
$$236$$ 0 0
$$237$$ 0 0
$$238$$ −12.0000 −0.777844
$$239$$ 12.0000 0.776215 0.388108 0.921614i $$-0.373129\pi$$
0.388108 + 0.921614i $$0.373129\pi$$
$$240$$ 0 0
$$241$$ −13.0000 −0.837404 −0.418702 0.908124i $$-0.637515\pi$$
−0.418702 + 0.908124i $$0.637515\pi$$
$$242$$ 11.0000 0.707107
$$243$$ 0 0
$$244$$ −1.00000 −0.0640184
$$245$$ −27.0000 −1.72497
$$246$$ 0 0
$$247$$ 4.00000 0.254514
$$248$$ 4.00000 0.254000
$$249$$ 0 0
$$250$$ −3.00000 −0.189737
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 16.0000 1.00393
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −15.0000 −0.935674 −0.467837 0.883815i $$-0.654967\pi$$
−0.467837 + 0.883815i $$0.654967\pi$$
$$258$$ 0 0
$$259$$ 4.00000 0.248548
$$260$$ 3.00000 0.186052
$$261$$ 0 0
$$262$$ 12.0000 0.741362
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 18.0000 1.10573
$$266$$ −16.0000 −0.981023
$$267$$ 0 0
$$268$$ −4.00000 −0.244339
$$269$$ 21.0000 1.28039 0.640196 0.768211i $$-0.278853\pi$$
0.640196 + 0.768211i $$0.278853\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −3.00000 −0.181902
$$273$$ 0 0
$$274$$ −9.00000 −0.543710
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ −12.0000 −0.717137
$$281$$ −27.0000 −1.61068 −0.805342 0.592810i $$-0.798019\pi$$
−0.805342 + 0.592810i $$0.798019\pi$$
$$282$$ 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
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 27.0000 1.58549
$$291$$ 0 0
$$292$$ 11.0000 0.643726
$$293$$ 9.00000 0.525786 0.262893 0.964825i $$-0.415323\pi$$
0.262893 + 0.964825i $$0.415323\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1.00000 0.0581238
$$297$$ 0 0
$$298$$ −9.00000 −0.521356
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −32.0000 −1.84445
$$302$$ −8.00000 −0.460348
$$303$$ 0 0
$$304$$ −4.00000 −0.229416
$$305$$ 3.00000 0.171780
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −12.0000 −0.681554
$$311$$ 24.0000 1.36092 0.680458 0.732787i $$-0.261781\pi$$
0.680458 + 0.732787i $$0.261781\pi$$
$$312$$ 0 0
$$313$$ 23.0000 1.30004 0.650018 0.759918i $$-0.274761\pi$$
0.650018 + 0.759918i $$0.274761\pi$$
$$314$$ 13.0000 0.733632
$$315$$ 0 0
$$316$$ −16.0000 −0.900070
$$317$$ 21.0000 1.17948 0.589739 0.807594i $$-0.299231\pi$$
0.589739 + 0.807594i $$0.299231\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ −3.00000 −0.167705
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 12.0000 0.667698
$$324$$ 0 0
$$325$$ −4.00000 −0.221880
$$326$$ −8.00000 −0.443079
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 48.0000 2.64633
$$330$$ 0 0
$$331$$ 20.0000 1.09930 0.549650 0.835395i $$-0.314761\pi$$
0.549650 + 0.835395i $$0.314761\pi$$
$$332$$ −12.0000 −0.658586
$$333$$ 0 0
$$334$$ −12.0000 −0.656611
$$335$$ 12.0000 0.655630
$$336$$ 0 0
$$337$$ 2.00000 0.108947 0.0544735 0.998515i $$-0.482652\pi$$
0.0544735 + 0.998515i $$0.482652\pi$$
$$338$$ 12.0000 0.652714
$$339$$ 0 0
$$340$$ 9.00000 0.488094
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −8.00000 −0.431959
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ 3.00000 0.161281
$$347$$ 12.0000 0.644194 0.322097 0.946707i $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 0 0
$$349$$ 14.0000 0.749403 0.374701 0.927146i $$-0.377745\pi$$
0.374701 + 0.927146i $$0.377745\pi$$
$$350$$ 16.0000 0.855236
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 36.0000 1.91068
$$356$$ −3.00000 −0.159000
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 10.0000 0.525588
$$363$$ 0 0
$$364$$ 4.00000 0.209657
$$365$$ −33.0000 −1.72730
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ −3.00000 −0.155963
$$371$$ 24.0000 1.24602
$$372$$ 0 0
$$373$$ −10.0000 −0.517780 −0.258890 0.965907i $$-0.583357\pi$$
−0.258890 + 0.965907i $$0.583357\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 12.0000 0.618853
$$377$$ −9.00000 −0.463524
$$378$$ 0 0
$$379$$ −28.0000 −1.43826 −0.719132 0.694874i $$-0.755460\pi$$
−0.719132 + 0.694874i $$0.755460\pi$$
$$380$$ 12.0000 0.615587
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 13.0000 0.661683
$$387$$ 0 0
$$388$$ 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
$$391$$ 0 0
$$392$$ −9.00000 −0.454569
$$393$$ 0 0
$$394$$ 3.00000 0.151138
$$395$$ 48.0000 2.41514
$$396$$ 0 0
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ 4.00000 0.200502
$$399$$ 0 0
$$400$$ 4.00000 0.200000
$$401$$ −3.00000 −0.149813 −0.0749064 0.997191i $$-0.523866\pi$$
−0.0749064 + 0.997191i $$0.523866\pi$$
$$402$$ 0 0
$$403$$ 4.00000 0.199254
$$404$$ −6.00000 −0.298511
$$405$$ 0 0
$$406$$ 36.0000 1.78665
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −25.0000 −1.23617 −0.618085 0.786111i $$-0.712091\pi$$
−0.618085 + 0.786111i $$0.712091\pi$$
$$410$$ 18.0000 0.888957
$$411$$ 0 0
$$412$$ −4.00000 −0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 36.0000 1.76717
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −24.0000 −1.17248 −0.586238 0.810139i $$-0.699392\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ −13.0000 −0.633581 −0.316791 0.948495i $$-0.602605\pi$$
−0.316791 + 0.948495i $$0.602605\pi$$
$$422$$ −8.00000 −0.389434
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ 4.00000 0.193574
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 24.0000 1.15738
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ −16.0000 −0.768025
$$435$$ 0 0
$$436$$ 11.0000 0.526804
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −3.00000 −0.142695
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 9.00000 0.426641
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ −4.00000 −0.188982
$$449$$ −18.0000 −0.849473 −0.424736 0.905317i $$-0.639633\pi$$
−0.424736 + 0.905317i $$0.639633\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −15.0000 −0.705541
$$453$$ 0 0
$$454$$ 12.0000 0.563188
$$455$$ −12.0000 −0.562569
$$456$$ 0 0
$$457$$ −1.00000 −0.0467780 −0.0233890 0.999726i $$-0.507446\pi$$
−0.0233890 + 0.999726i $$0.507446\pi$$
$$458$$ −23.0000 −1.07472
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 18.0000 0.838344 0.419172 0.907907i $$-0.362320\pi$$
0.419172 + 0.907907i $$0.362320\pi$$
$$462$$ 0 0
$$463$$ 8.00000 0.371792 0.185896 0.982569i $$-0.440481\pi$$
0.185896 + 0.982569i $$0.440481\pi$$
$$464$$ 9.00000 0.417815
$$465$$ 0 0
$$466$$ −21.0000 −0.972806
$$467$$ −24.0000 −1.11059 −0.555294 0.831654i $$-0.687394\pi$$
−0.555294 + 0.831654i $$0.687394\pi$$
$$468$$ 0 0
$$469$$ 16.0000 0.738811
$$470$$ −36.0000 −1.66056
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −16.0000 −0.734130
$$476$$ 12.0000 0.550019
$$477$$ 0 0
$$478$$ −12.0000 −0.548867
$$479$$ −12.0000 −0.548294 −0.274147 0.961688i $$-0.588395\pi$$
−0.274147 + 0.961688i $$0.588395\pi$$
$$480$$ 0 0
$$481$$ 1.00000 0.0455961
$$482$$ 13.0000 0.592134
$$483$$ 0 0
$$484$$ −11.0000 −0.500000
$$485$$ −6.00000 −0.272446
$$486$$ 0 0
$$487$$ −4.00000 −0.181257 −0.0906287 0.995885i $$-0.528888\pi$$
−0.0906287 + 0.995885i $$0.528888\pi$$
$$488$$ 1.00000 0.0452679
$$489$$ 0 0
$$490$$ 27.0000 1.21974
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ −27.0000 −1.21602
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ −4.00000 −0.179605
$$497$$ 48.0000 2.15309
$$498$$ 0 0
$$499$$ −40.0000 −1.79065 −0.895323 0.445418i $$-0.853055\pi$$
−0.895323 + 0.445418i $$0.853055\pi$$
$$500$$ 3.00000 0.134164
$$501$$ 0 0
$$502$$ 24.0000 1.07117
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ 0 0
$$508$$ −16.0000 −0.709885
$$509$$ −30.0000 −1.32973 −0.664863 0.746965i $$-0.731510\pi$$
−0.664863 + 0.746965i $$0.731510\pi$$
$$510$$ 0 0
$$511$$ −44.0000 −1.94645
$$512$$ −1.00000 −0.0441942
$$513$$ 0 0
$$514$$ 15.0000 0.661622
$$515$$ 12.0000 0.528783
$$516$$ 0 0
$$517$$ 0 0
$$518$$ −4.00000 −0.175750
$$519$$ 0 0
$$520$$ −3.00000 −0.131559
$$521$$ 6.00000 0.262865 0.131432 0.991325i $$-0.458042\pi$$
0.131432 + 0.991325i $$0.458042\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ −12.0000 −0.524222
$$525$$ 0 0
$$526$$ −12.0000 −0.523225
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ −18.0000 −0.781870
$$531$$ 0 0
$$532$$ 16.0000 0.693688
$$533$$ −6.00000 −0.259889
$$534$$ 0 0
$$535$$ −36.0000 −1.55642
$$536$$ 4.00000 0.172774
$$537$$ 0 0
$$538$$ −21.0000 −0.905374
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −1.00000 −0.0429934 −0.0214967 0.999769i $$-0.506843\pi$$
−0.0214967 + 0.999769i $$0.506843\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ 3.00000 0.128624
$$545$$ −33.0000 −1.41356
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 9.00000 0.384461
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −36.0000 −1.53365
$$552$$ 0 0
$$553$$ 64.0000 2.72156
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 12.0000 0.507093
$$561$$ 0 0
$$562$$ 27.0000 1.13893
$$563$$ 12.0000 0.505740 0.252870 0.967500i $$-0.418626\pi$$
0.252870 + 0.967500i $$0.418626\pi$$
$$564$$ 0 0
$$565$$ 45.0000 1.89316
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 12.0000 0.503509
$$569$$ −15.0000 −0.628833 −0.314416 0.949285i $$-0.601809\pi$$
−0.314416 + 0.949285i $$0.601809\pi$$
$$570$$ 0 0
$$571$$ −16.0000 −0.669579 −0.334790 0.942293i $$-0.608665\pi$$
−0.334790 + 0.942293i $$0.608665\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 24.0000 1.00174
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −25.0000 −1.04076 −0.520382 0.853934i $$-0.674210\pi$$
−0.520382 + 0.853934i $$0.674210\pi$$
$$578$$ 8.00000 0.332756
$$579$$ 0 0
$$580$$ −27.0000 −1.12111
$$581$$ 48.0000 1.99138
$$582$$ 0 0
$$583$$ 0 0
$$584$$ −11.0000 −0.455183
$$585$$ 0 0
$$586$$ −9.00000 −0.371787
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ 16.0000 0.659269
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −1.00000 −0.0410997
$$593$$ 33.0000 1.35515 0.677574 0.735455i $$-0.263031\pi$$
0.677574 + 0.735455i $$0.263031\pi$$
$$594$$ 0 0
$$595$$ −36.0000 −1.47586
$$596$$ 9.00000 0.368654
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −36.0000 −1.47092 −0.735460 0.677568i $$-0.763034\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 35.0000 1.42768 0.713840 0.700309i $$-0.246954\pi$$
0.713840 + 0.700309i $$0.246954\pi$$
$$602$$ 32.0000 1.30422
$$603$$ 0 0
$$604$$ 8.00000 0.325515
$$605$$ 33.0000 1.34164
$$606$$ 0 0
$$607$$ 20.0000 0.811775 0.405887 0.913923i $$-0.366962\pi$$
0.405887 + 0.913923i $$0.366962\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ −3.00000 −0.121466
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ −20.0000 −0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −3.00000 −0.120775 −0.0603877 0.998175i $$-0.519234\pi$$
−0.0603877 + 0.998175i $$0.519234\pi$$
$$618$$ 0 0
$$619$$ 8.00000 0.321547 0.160774 0.986991i $$-0.448601\pi$$
0.160774 + 0.986991i $$0.448601\pi$$
$$620$$ 12.0000 0.481932
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 12.0000 0.480770
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ −23.0000 −0.919265
$$627$$ 0 0
$$628$$ −13.0000 −0.518756
$$629$$ 3.00000 0.119618
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 16.0000 0.636446
$$633$$ 0 0
$$634$$ −21.0000 −0.834017
$$635$$ 48.0000 1.90482
$$636$$ 0 0
$$637$$ −9.00000 −0.356593
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 3.00000 0.118585
$$641$$ 45.0000 1.77739 0.888697 0.458496i $$-0.151612\pi$$
0.888697 + 0.458496i $$0.151612\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −12.0000 −0.472134
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 4.00000 0.156893
$$651$$ 0 0
$$652$$ 8.00000 0.313304
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 0 0
$$655$$ 36.0000 1.40664
$$656$$ 6.00000 0.234261
$$657$$ 0 0
$$658$$ −48.0000 −1.87123
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 23.0000 0.894596 0.447298 0.894385i $$-0.352386\pi$$
0.447298 + 0.894385i $$0.352386\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ −48.0000 −1.86136
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 12.0000 0.464294
$$669$$ 0 0
$$670$$ −12.0000 −0.463600
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 11.0000 0.424019 0.212009 0.977268i $$-0.431999\pi$$
0.212009 + 0.977268i $$0.431999\pi$$
$$674$$ −2.00000 −0.0770371
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ −8.00000 −0.307012
$$680$$ −9.00000 −0.345134
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 36.0000 1.37750 0.688751 0.724998i $$-0.258159\pi$$
0.688751 + 0.724998i $$0.258159\pi$$
$$684$$ 0 0
$$685$$ −27.0000 −1.03162
$$686$$ 8.00000 0.305441
$$687$$ 0 0
$$688$$ 8.00000 0.304997
$$689$$ 6.00000 0.228582
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −3.00000 −0.114043
$$693$$ 0 0
$$694$$ −12.0000 −0.455514
$$695$$ −60.0000 −2.27593
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ −14.0000 −0.529908
$$699$$ 0 0
$$700$$ −16.0000 −0.604743
$$701$$ −51.0000 −1.92624 −0.963122 0.269066i $$-0.913285\pi$$
−0.963122 + 0.269066i $$0.913285\pi$$
$$702$$ 0 0
$$703$$ 4.00000 0.150863
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 18.0000 0.677439
$$707$$ 24.0000 0.902613
$$708$$ 0 0
$$709$$ 47.0000 1.76512 0.882561 0.470198i $$-0.155817\pi$$
0.882561 + 0.470198i $$0.155817\pi$$
$$710$$ −36.0000 −1.35106
$$711$$ 0 0
$$712$$ 3.00000 0.112430
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 12.0000 0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 16.0000 0.595871
$$722$$ 3.00000 0.111648
$$723$$ 0 0
$$724$$ −10.0000 −0.371647
$$725$$ 36.0000 1.33701
$$726$$ 0 0
$$727$$ −28.0000 −1.03846 −0.519231 0.854634i $$-0.673782\pi$$
−0.519231 + 0.854634i $$0.673782\pi$$
$$728$$ −4.00000 −0.148250
$$729$$ 0 0
$$730$$ 33.0000 1.22138
$$731$$ −24.0000 −0.887672
$$732$$ 0 0
$$733$$ 14.0000 0.517102 0.258551 0.965998i $$-0.416755\pi$$
0.258551 + 0.965998i $$0.416755\pi$$
$$734$$ −8.00000 −0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 3.00000 0.110282
$$741$$ 0 0
$$742$$ −24.0000 −0.881068
$$743$$ 12.0000 0.440237 0.220119 0.975473i $$-0.429356\pi$$
0.220119 + 0.975473i $$0.429356\pi$$
$$744$$ 0 0
$$745$$ −27.0000 −0.989203
$$746$$ 10.0000 0.366126
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −48.0000 −1.75388
$$750$$ 0 0
$$751$$ 20.0000 0.729810 0.364905 0.931045i $$-0.381101\pi$$
0.364905 + 0.931045i $$0.381101\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ 9.00000 0.327761
$$755$$ −24.0000 −0.873449
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 28.0000 1.01701
$$759$$ 0 0
$$760$$ −12.0000 −0.435286
$$761$$ −15.0000 −0.543750 −0.271875 0.962333i $$-0.587644\pi$$
−0.271875 + 0.962333i $$0.587644\pi$$
$$762$$ 0 0
$$763$$ −44.0000 −1.59291
$$764$$ 12.0000 0.434145
$$765$$ 0 0
$$766$$ 12.0000 0.433578
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −37.0000 −1.33425 −0.667127 0.744944i $$-0.732476\pi$$
−0.667127 + 0.744944i $$0.732476\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −13.0000 −0.467880
$$773$$ −27.0000 −0.971123 −0.485561 0.874203i $$-0.661385\pi$$
−0.485561 + 0.874203i $$0.661385\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ −2.00000 −0.0717958
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 9.00000 0.321429
$$785$$ 39.0000 1.39197
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ −3.00000 −0.106871
$$789$$ 0 0
$$790$$ −48.0000 −1.70776
$$791$$ 60.0000 2.13335
$$792$$ 0 0
$$793$$ 1.00000 0.0355110
$$794$$ 25.0000 0.887217
$$795$$ 0 0
$$796$$ −4.00000 −0.141776
$$797$$ −51.0000 −1.80651 −0.903256 0.429101i $$-0.858830\pi$$
−0.903256 + 0.429101i $$0.858830\pi$$
$$798$$ 0 0
$$799$$ 36.0000 1.27359
$$800$$ −4.00000 −0.141421
$$801$$ 0 0
$$802$$ 3.00000 0.105934
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −4.00000 −0.140894
$$807$$ 0 0
$$808$$ 6.00000 0.211079
$$809$$ −39.0000 −1.37117 −0.685583 0.727994i $$-0.740453\pi$$
−0.685583 + 0.727994i $$0.740453\pi$$
$$810$$ 0 0
$$811$$ −16.0000 −0.561836 −0.280918 0.959732i $$-0.590639\pi$$
−0.280918 + 0.959732i $$0.590639\pi$$
$$812$$ −36.0000 −1.26335
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −24.0000 −0.840683
$$816$$ 0 0
$$817$$ −32.0000 −1.11954
$$818$$ 25.0000 0.874105
$$819$$ 0 0
$$820$$ −18.0000 −0.628587
$$821$$ −15.0000 −0.523504 −0.261752 0.965135i $$-0.584300\pi$$
−0.261752 + 0.965135i $$0.584300\pi$$
$$822$$ 0 0
$$823$$ 44.0000 1.53374 0.766872 0.641800i $$-0.221812\pi$$
0.766872 + 0.641800i $$0.221812\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −48.0000 −1.66912 −0.834562 0.550914i $$-0.814279\pi$$
−0.834562 + 0.550914i $$0.814279\pi$$
$$828$$ 0 0
$$829$$ 14.0000 0.486240 0.243120 0.969996i $$-0.421829\pi$$
0.243120 + 0.969996i $$0.421829\pi$$
$$830$$ −36.0000 −1.24958
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ −27.0000 −0.935495
$$834$$ 0 0
$$835$$ −36.0000 −1.24583
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 24.0000 0.829066
$$839$$ 24.0000 0.828572 0.414286 0.910147i $$-0.364031\pi$$
0.414286 + 0.910147i $$0.364031\pi$$
$$840$$ 0 0
$$841$$ 52.0000 1.79310
$$842$$ 13.0000 0.448010
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 36.0000 1.23844
$$846$$ 0 0
$$847$$ 44.0000 1.51186
$$848$$ −6.00000 −0.206041
$$849$$ 0 0
$$850$$ 12.0000 0.411597
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −10.0000 −0.342393 −0.171197 0.985237i $$-0.554763\pi$$
−0.171197 + 0.985237i $$0.554763\pi$$
$$854$$ −4.00000 −0.136877
$$855$$ 0 0
$$856$$ −12.0000 −0.410152
$$857$$ −39.0000 −1.33221 −0.666107 0.745856i $$-0.732041\pi$$
−0.666107 + 0.745856i $$0.732041\pi$$
$$858$$ 0 0
$$859$$ −52.0000 −1.77422 −0.887109 0.461561i $$-0.847290\pi$$
−0.887109 + 0.461561i $$0.847290\pi$$
$$860$$ −24.0000 −0.818393
$$861$$ 0 0
$$862$$ 12.0000 0.408722
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ 0 0
$$865$$ 9.00000 0.306009
$$866$$ −11.0000 −0.373795
$$867$$ 0 0
$$868$$ 16.0000 0.543075
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 4.00000 0.135535
$$872$$ −11.0000 −0.372507
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −12.0000 −0.405674
$$876$$ 0 0
$$877$$ −25.0000 −0.844190 −0.422095 0.906552i $$-0.638705\pi$$
−0.422095 + 0.906552i $$0.638705\pi$$
$$878$$ 28.0000 0.944954
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 30.0000 1.01073 0.505363 0.862907i $$-0.331359\pi$$
0.505363 + 0.862907i $$0.331359\pi$$
$$882$$ 0 0
$$883$$ −4.00000 −0.134611 −0.0673054 0.997732i $$-0.521440\pi$$
−0.0673054 + 0.997732i $$0.521440\pi$$
$$884$$ 3.00000 0.100901
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ 0 0
$$889$$ 64.0000 2.14649
$$890$$ −9.00000 −0.301681
$$891$$ 0 0
$$892$$ 8.00000 0.267860
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ −36.0000 −1.20335
$$896$$ 4.00000 0.133631
$$897$$ 0 0
$$898$$ 18.0000 0.600668
$$899$$ −36.0000 −1.20067
$$900$$ 0 0
$$901$$ 18.0000 0.599667
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 15.0000 0.498893
$$905$$ 30.0000 0.997234
$$906$$ 0 0
$$907$$ −16.0000 −0.531271 −0.265636 0.964073i $$-0.585582\pi$$
−0.265636 + 0.964073i $$0.585582\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 12.0000 0.397796
$$911$$ 24.0000 0.795155 0.397578 0.917568i $$-0.369851\pi$$
0.397578 + 0.917568i $$0.369851\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 1.00000 0.0330771
$$915$$ 0 0
$$916$$ 23.0000 0.759941
$$917$$ 48.0000 1.58510
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −18.0000 −0.592798
$$923$$ 12.0000 0.394985
$$924$$ 0 0
$$925$$ −4.00000 −0.131519
$$926$$ −8.00000 −0.262896
$$927$$ 0 0
$$928$$ −9.00000 −0.295439
$$929$$ −3.00000 −0.0984268 −0.0492134 0.998788i $$-0.515671\pi$$
−0.0492134 + 0.998788i $$0.515671\pi$$
$$930$$ 0 0
$$931$$ −36.0000 −1.17985
$$932$$ 21.0000 0.687878
$$933$$ 0 0
$$934$$ 24.0000 0.785304
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −37.0000 −1.20874 −0.604369 0.796705i $$-0.706575\pi$$
−0.604369 + 0.796705i $$0.706575\pi$$
$$938$$ −16.0000 −0.522419
$$939$$ 0 0
$$940$$ 36.0000 1.17419
$$941$$ −27.0000 −0.880175 −0.440087 0.897955i $$-0.645053\pi$$
−0.440087 + 0.897955i $$0.645053\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −12.0000 −0.389948 −0.194974 0.980808i $$-0.562462\pi$$
−0.194974 + 0.980808i $$0.562462\pi$$
$$948$$ 0 0
$$949$$ −11.0000 −0.357075
$$950$$ 16.0000 0.519109
$$951$$ 0 0
$$952$$ −12.0000 −0.388922
$$953$$ 9.00000 0.291539 0.145769 0.989319i $$-0.453434\pi$$
0.145769 + 0.989319i $$0.453434\pi$$
$$954$$ 0 0
$$955$$ −36.0000 −1.16493
$$956$$ 12.0000 0.388108
$$957$$ 0 0
$$958$$ 12.0000 0.387702
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −15.0000 −0.483871
$$962$$ −1.00000 −0.0322413
$$963$$ 0 0
$$964$$ −13.0000 −0.418702
$$965$$ 39.0000 1.25545
$$966$$ 0 0
$$967$$ −4.00000 −0.128631 −0.0643157 0.997930i $$-0.520486\pi$$
−0.0643157 + 0.997930i $$0.520486\pi$$
$$968$$ 11.0000 0.353553
$$969$$ 0 0
$$970$$ 6.00000 0.192648
$$971$$ −36.0000 −1.15529 −0.577647 0.816286i $$-0.696029\pi$$
−0.577647 + 0.816286i $$0.696029\pi$$
$$972$$ 0 0
$$973$$ −80.0000 −2.56468
$$974$$ 4.00000 0.128168
$$975$$ 0 0
$$976$$ −1.00000 −0.0320092
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ −27.0000 −0.862483
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 48.0000 1.53096 0.765481 0.643458i $$-0.222501\pi$$
0.765481 + 0.643458i $$0.222501\pi$$
$$984$$ 0 0
$$985$$ 9.00000 0.286764
$$986$$ 27.0000 0.859855
$$987$$ 0 0
$$988$$ 4.00000 0.127257
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −52.0000 −1.65183 −0.825917 0.563791i $$-0.809342\pi$$
−0.825917 + 0.563791i $$0.809342\pi$$
$$992$$ 4.00000 0.127000
$$993$$ 0 0
$$994$$ −48.0000 −1.52247
$$995$$ 12.0000 0.380426
$$996$$ 0 0
$$997$$ −37.0000 −1.17180 −0.585901 0.810383i $$-0.699259\pi$$
−0.585901 + 0.810383i $$0.699259\pi$$
$$998$$ 40.0000 1.26618
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 162.2.a.a.1.1 1
3.2 odd 2 162.2.a.d.1.1 yes 1
4.3 odd 2 1296.2.a.c.1.1 1
5.2 odd 4 4050.2.c.g.649.1 2
5.3 odd 4 4050.2.c.g.649.2 2
5.4 even 2 4050.2.a.bh.1.1 1
7.6 odd 2 7938.2.a.n.1.1 1
8.3 odd 2 5184.2.a.bd.1.1 1
8.5 even 2 5184.2.a.y.1.1 1
9.2 odd 6 162.2.c.a.109.1 2
9.4 even 3 162.2.c.d.55.1 2
9.5 odd 6 162.2.c.a.55.1 2
9.7 even 3 162.2.c.d.109.1 2
12.11 even 2 1296.2.a.l.1.1 1
15.2 even 4 4050.2.c.n.649.2 2
15.8 even 4 4050.2.c.n.649.1 2
15.14 odd 2 4050.2.a.r.1.1 1
21.20 even 2 7938.2.a.s.1.1 1
24.5 odd 2 5184.2.a.c.1.1 1
24.11 even 2 5184.2.a.h.1.1 1
36.7 odd 6 1296.2.i.n.433.1 2
36.11 even 6 1296.2.i.b.433.1 2
36.23 even 6 1296.2.i.b.865.1 2
36.31 odd 6 1296.2.i.n.865.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.a.a.1.1 1 1.1 even 1 trivial
162.2.a.d.1.1 yes 1 3.2 odd 2
162.2.c.a.55.1 2 9.5 odd 6
162.2.c.a.109.1 2 9.2 odd 6
162.2.c.d.55.1 2 9.4 even 3
162.2.c.d.109.1 2 9.7 even 3
1296.2.a.c.1.1 1 4.3 odd 2
1296.2.a.l.1.1 1 12.11 even 2
1296.2.i.b.433.1 2 36.11 even 6
1296.2.i.b.865.1 2 36.23 even 6
1296.2.i.n.433.1 2 36.7 odd 6
1296.2.i.n.865.1 2 36.31 odd 6
4050.2.a.r.1.1 1 15.14 odd 2
4050.2.a.bh.1.1 1 5.4 even 2
4050.2.c.g.649.1 2 5.2 odd 4
4050.2.c.g.649.2 2 5.3 odd 4
4050.2.c.n.649.1 2 15.8 even 4
4050.2.c.n.649.2 2 15.2 even 4
5184.2.a.c.1.1 1 24.5 odd 2
5184.2.a.h.1.1 1 24.11 even 2
5184.2.a.y.1.1 1 8.5 even 2
5184.2.a.bd.1.1 1 8.3 odd 2
7938.2.a.n.1.1 1 7.6 odd 2
7938.2.a.s.1.1 1 21.20 even 2