# Properties

 Label 162.2.a.d.1.1 Level $162$ Weight $2$ Character 162.1 Self dual yes Analytic conductor $1.294$ 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] = [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: $$0$$ 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 $$-0.265942\pi$$
0.670820 + 0.741620i $$0.265942\pi$$
$$6$$ 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 $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 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 $$-0.814897\pi$$
−0.835629 + 0.549294i $$0.814897\pi$$
$$30$$ 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 $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 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 $$-0.160736\pi$$
0.875190 + 0.483779i $$0.160736\pi$$
$$48$$ 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 $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 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 $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 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 $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 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 $$-0.449173\pi$$
0.159000 + 0.987279i $$0.449173\pi$$
$$90$$ 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 $$-0.403510\pi$$
0.298511 + 0.954406i $$0.403510\pi$$
$$102$$ 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 $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 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 $$-0.250704\pi$$
0.705541 + 0.708669i $$0.250704\pi$$
$$114$$ 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 $$-0.324356\pi$$
0.524222 + 0.851581i $$0.324356\pi$$
$$132$$ 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 $$-0.625613\pi$$
−0.384461 + 0.923141i $$0.625613\pi$$
$$138$$ 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 $$-0.620181\pi$$
−0.368654 + 0.929567i $$0.620181\pi$$
$$150$$ 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 $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 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 $$-0.463620\pi$$
0.114043 + 0.993476i $$0.463620\pi$$
$$174$$ 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 $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 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 $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 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 $$-0.465917\pi$$
0.106871 + 0.994273i $$0.465917\pi$$
$$198$$ 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 $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 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 $$-0.741458\pi$$
−0.687878 + 0.725826i $$0.741458\pi$$
$$234$$ 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 $$-0.626871\pi$$
−0.388108 + 0.921614i $$0.626871\pi$$
$$240$$ 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 $$-0.226453\pi$$
0.757433 + 0.652913i $$0.226453\pi$$
$$252$$ 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 $$-0.345033\pi$$
0.467837 + 0.883815i $$0.345033\pi$$
$$258$$ 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 $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 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 $$-0.721147\pi$$
−0.640196 + 0.768211i $$0.721147\pi$$
$$270$$ 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 $$-0.201981\pi$$
0.805342 + 0.592810i $$0.201981\pi$$
$$282$$ 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 $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 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 $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 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 $$-0.700769\pi$$
−0.589739 + 0.807594i $$0.700769\pi$$
$$318$$ 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 $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ 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 $$-0.340992\pi$$
0.479022 + 0.877803i $$0.340992\pi$$
$$354$$ 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 $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 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 $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 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 $$-0.476134\pi$$
0.0749064 + 0.997191i $$0.476134\pi$$
$$402$$ 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 $$-0.300608\pi$$
0.586238 + 0.810139i $$0.300608\pi$$
$$420$$ 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 $$-0.406674\pi$$
0.289010 + 0.957326i $$0.406674\pi$$
$$432$$ 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 $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 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 $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 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 $$-0.637680\pi$$
−0.419172 + 0.907907i $$0.637680\pi$$
$$462$$ 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 $$-0.312606\pi$$
0.555294 + 0.831654i $$0.312606\pi$$
$$468$$ 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 $$-0.411605\pi$$
0.274147 + 0.961688i $$0.411605\pi$$
$$480$$ 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 $$-0.268490\pi$$
0.664863 + 0.746965i $$0.268490\pi$$
$$510$$ 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 $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 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 $$-0.479756\pi$$
0.0635570 + 0.997978i $$0.479756\pi$$
$$558$$ 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 $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 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 $$-0.398191\pi$$
0.314416 + 0.949285i $$0.398191\pi$$
$$570$$ 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 $$-0.335061\pi$$
0.495293 + 0.868726i $$0.335061\pi$$
$$588$$ 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 $$-0.736969\pi$$
−0.677574 + 0.735455i $$0.736969\pi$$
$$594$$ 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 $$-0.236966\pi$$
0.735460 + 0.677568i $$0.236966\pi$$
$$600$$ 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 $$-0.480766\pi$$
0.0603877 + 0.998175i $$0.480766\pi$$
$$618$$ 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 $$-0.848388\pi$$
−0.888697 + 0.458496i $$0.848388\pi$$
$$642$$ 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 $$-0.750246\pi$$
−0.707653 + 0.706560i $$0.750246\pi$$
$$648$$ 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 $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 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 $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 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 $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ 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 $$-0.741841\pi$$
−0.688751 + 0.724998i $$0.741841\pi$$
$$684$$ 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 $$-0.0867150\pi$$
0.963122 + 0.269066i $$0.0867150\pi$$
$$702$$ 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 $$-0.570644\pi$$
−0.220119 + 0.975473i $$0.570644\pi$$
$$744$$ 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 $$-0.412356\pi$$
0.271875 + 0.962333i $$0.412356\pi$$
$$762$$ 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 $$-0.338615\pi$$
0.485561 + 0.874203i $$0.338615\pi$$
$$774$$ 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 $$-0.141170\pi$$
0.903256 + 0.429101i $$0.141170\pi$$
$$798$$ 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 $$-0.259547\pi$$
0.685583 + 0.727994i $$0.259547\pi$$
$$810$$ 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 $$-0.415700\pi$$
0.261752 + 0.965135i $$0.415700\pi$$
$$822$$ 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 $$-0.185721\pi$$
0.834562 + 0.550914i $$0.185721\pi$$
$$828$$ 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 $$-0.635969\pi$$
−0.414286 + 0.910147i $$0.635969\pi$$
$$840$$ 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 $$-0.267959\pi$$
0.666107 + 0.745856i $$0.267959\pi$$
$$858$$ 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 $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ 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 $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 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 $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 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 $$-0.630149\pi$$
−0.397578 + 0.917568i $$0.630149\pi$$
$$912$$ 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 $$-0.484329\pi$$
0.0492134 + 0.998788i $$0.484329\pi$$
$$930$$ 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 $$-0.354947\pi$$
0.440087 + 0.897955i $$0.354947\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 12.0000 0.389948 0.194974 0.980808i $$-0.437538\pi$$
0.194974 + 0.980808i $$0.437538\pi$$
$$948$$ 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 $$-0.546566\pi$$
−0.145769 + 0.989319i $$0.546566\pi$$
$$954$$ 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 $$-0.303971\pi$$
0.577647 + 0.816286i $$0.303971\pi$$
$$972$$ 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 $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 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 $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 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.d.1.1 yes 1
3.2 odd 2 162.2.a.a.1.1 1
4.3 odd 2 1296.2.a.l.1.1 1
5.2 odd 4 4050.2.c.n.649.2 2
5.3 odd 4 4050.2.c.n.649.1 2
5.4 even 2 4050.2.a.r.1.1 1
7.6 odd 2 7938.2.a.s.1.1 1
8.3 odd 2 5184.2.a.h.1.1 1
8.5 even 2 5184.2.a.c.1.1 1
9.2 odd 6 162.2.c.d.109.1 2
9.4 even 3 162.2.c.a.55.1 2
9.5 odd 6 162.2.c.d.55.1 2
9.7 even 3 162.2.c.a.109.1 2
12.11 even 2 1296.2.a.c.1.1 1
15.2 even 4 4050.2.c.g.649.1 2
15.8 even 4 4050.2.c.g.649.2 2
15.14 odd 2 4050.2.a.bh.1.1 1
21.20 even 2 7938.2.a.n.1.1 1
24.5 odd 2 5184.2.a.y.1.1 1
24.11 even 2 5184.2.a.bd.1.1 1
36.7 odd 6 1296.2.i.b.433.1 2
36.11 even 6 1296.2.i.n.433.1 2
36.23 even 6 1296.2.i.n.865.1 2
36.31 odd 6 1296.2.i.b.865.1 2

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