# Properties

 Label 816.2.a.h.1.1 Level $816$ Weight $2$ Character 816.1 Self dual yes Analytic conductor $6.516$ 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] = [816,2,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("816.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$816 = 2^{4} \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 816.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: $$6.51579280494$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 102) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 816.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} -4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} -4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -6.00000 q^{13} -4.00000 q^{15} -1.00000 q^{17} -4.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} +11.0000 q^{25} +1.00000 q^{27} -4.00000 q^{29} +6.00000 q^{31} -8.00000 q^{35} -4.00000 q^{37} -6.00000 q^{39} -10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{45} -4.00000 q^{47} -3.00000 q^{49} -1.00000 q^{51} -2.00000 q^{53} -4.00000 q^{57} -12.0000 q^{59} -4.00000 q^{61} +2.00000 q^{63} +24.0000 q^{65} +12.0000 q^{67} -6.00000 q^{69} +6.00000 q^{71} +2.00000 q^{73} +11.0000 q^{75} -10.0000 q^{79} +1.00000 q^{81} +12.0000 q^{83} +4.00000 q^{85} -4.00000 q^{87} -2.00000 q^{89} -12.0000 q^{91} +6.00000 q^{93} +16.0000 q^{95} +6.00000 q^{97} +O(q^{100})$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 1.00000 0.577350
$$4$$ 0 0
$$5$$ −4.00000 −1.78885 −0.894427 0.447214i $$-0.852416\pi$$
−0.894427 + 0.447214i $$0.852416\pi$$
$$6$$ 0 0
$$7$$ 2.00000 0.755929 0.377964 0.925820i $$-0.376624\pi$$
0.377964 + 0.925820i $$0.376624\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −1.66410 −0.832050 0.554700i $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ 0 0
$$15$$ −4.00000 −1.03280
$$16$$ 0 0
$$17$$ −1.00000 −0.242536
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 2.00000 0.436436
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ 11.0000 2.20000
$$26$$ 0 0
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −4.00000 −0.742781 −0.371391 0.928477i $$-0.621119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 6.00000 1.07763 0.538816 0.842424i $$-0.318872\pi$$
0.538816 + 0.842424i $$0.318872\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −8.00000 −1.35225
$$36$$ 0 0
$$37$$ −4.00000 −0.657596 −0.328798 0.944400i $$-0.606644\pi$$
−0.328798 + 0.944400i $$0.606644\pi$$
$$38$$ 0 0
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −10.0000 −1.56174 −0.780869 0.624695i $$-0.785223\pi$$
−0.780869 + 0.624695i $$0.785223\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ −4.00000 −0.596285
$$46$$ 0 0
$$47$$ −4.00000 −0.583460 −0.291730 0.956501i $$-0.594231\pi$$
−0.291730 + 0.956501i $$0.594231\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 0 0
$$53$$ −2.00000 −0.274721 −0.137361 0.990521i $$-0.543862\pi$$
−0.137361 + 0.990521i $$0.543862\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ −4.00000 −0.512148 −0.256074 0.966657i $$-0.582429\pi$$
−0.256074 + 0.966657i $$0.582429\pi$$
$$62$$ 0 0
$$63$$ 2.00000 0.251976
$$64$$ 0 0
$$65$$ 24.0000 2.97683
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ 0 0
$$69$$ −6.00000 −0.722315
$$70$$ 0 0
$$71$$ 6.00000 0.712069 0.356034 0.934473i $$-0.384129\pi$$
0.356034 + 0.934473i $$0.384129\pi$$
$$72$$ 0 0
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ 0 0
$$75$$ 11.0000 1.27017
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 0 0
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 4.00000 0.433861
$$86$$ 0 0
$$87$$ −4.00000 −0.428845
$$88$$ 0 0
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ −12.0000 −1.25794
$$92$$ 0 0
$$93$$ 6.00000 0.622171
$$94$$ 0 0
$$95$$ 16.0000 1.64157
$$96$$ 0 0
$$97$$ 6.00000 0.609208 0.304604 0.952479i $$-0.401476\pi$$
0.304604 + 0.952479i $$0.401476\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 14.0000 1.39305 0.696526 0.717532i $$-0.254728\pi$$
0.696526 + 0.717532i $$0.254728\pi$$
$$102$$ 0 0
$$103$$ −4.00000 −0.394132 −0.197066 0.980390i $$-0.563141\pi$$
−0.197066 + 0.980390i $$0.563141\pi$$
$$104$$ 0 0
$$105$$ −8.00000 −0.780720
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 16.0000 1.53252 0.766261 0.642529i $$-0.222115\pi$$
0.766261 + 0.642529i $$0.222115\pi$$
$$110$$ 0 0
$$111$$ −4.00000 −0.379663
$$112$$ 0 0
$$113$$ 2.00000 0.188144 0.0940721 0.995565i $$-0.470012\pi$$
0.0940721 + 0.995565i $$0.470012\pi$$
$$114$$ 0 0
$$115$$ 24.0000 2.23801
$$116$$ 0 0
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ −10.0000 −0.901670
$$124$$ 0 0
$$125$$ −24.0000 −2.14663
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 0 0
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 16.0000 1.39793 0.698963 0.715158i $$-0.253645\pi$$
0.698963 + 0.715158i $$0.253645\pi$$
$$132$$ 0 0
$$133$$ −8.00000 −0.693688
$$134$$ 0 0
$$135$$ −4.00000 −0.344265
$$136$$ 0 0
$$137$$ −6.00000 −0.512615 −0.256307 0.966595i $$-0.582506\pi$$
−0.256307 + 0.966595i $$0.582506\pi$$
$$138$$ 0 0
$$139$$ −8.00000 −0.678551 −0.339276 0.940687i $$-0.610182\pi$$
−0.339276 + 0.940687i $$0.610182\pi$$
$$140$$ 0 0
$$141$$ −4.00000 −0.336861
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 16.0000 1.32873
$$146$$ 0 0
$$147$$ −3.00000 −0.247436
$$148$$ 0 0
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ 0 0
$$151$$ 24.0000 1.95309 0.976546 0.215308i $$-0.0690756\pi$$
0.976546 + 0.215308i $$0.0690756\pi$$
$$152$$ 0 0
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ −24.0000 −1.92773
$$156$$ 0 0
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 0 0
$$159$$ −2.00000 −0.158610
$$160$$ 0 0
$$161$$ −12.0000 −0.945732
$$162$$ 0 0
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 2.00000 0.154765 0.0773823 0.997001i $$-0.475344\pi$$
0.0773823 + 0.997001i $$0.475344\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 0 0
$$173$$ −4.00000 −0.304114 −0.152057 0.988372i $$-0.548590\pi$$
−0.152057 + 0.988372i $$0.548590\pi$$
$$174$$ 0 0
$$175$$ 22.0000 1.66304
$$176$$ 0 0
$$177$$ −12.0000 −0.901975
$$178$$ 0 0
$$179$$ 12.0000 0.896922 0.448461 0.893802i $$-0.351972\pi$$
0.448461 + 0.893802i $$0.351972\pi$$
$$180$$ 0 0
$$181$$ −20.0000 −1.48659 −0.743294 0.668965i $$-0.766738\pi$$
−0.743294 + 0.668965i $$0.766738\pi$$
$$182$$ 0 0
$$183$$ −4.00000 −0.295689
$$184$$ 0 0
$$185$$ 16.0000 1.17634
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 2.00000 0.145479
$$190$$ 0 0
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ 6.00000 0.431889 0.215945 0.976406i $$-0.430717\pi$$
0.215945 + 0.976406i $$0.430717\pi$$
$$194$$ 0 0
$$195$$ 24.0000 1.71868
$$196$$ 0 0
$$197$$ 8.00000 0.569976 0.284988 0.958531i $$-0.408010\pi$$
0.284988 + 0.958531i $$0.408010\pi$$
$$198$$ 0 0
$$199$$ −14.0000 −0.992434 −0.496217 0.868199i $$-0.665278\pi$$
−0.496217 + 0.868199i $$0.665278\pi$$
$$200$$ 0 0
$$201$$ 12.0000 0.846415
$$202$$ 0 0
$$203$$ −8.00000 −0.561490
$$204$$ 0 0
$$205$$ 40.0000 2.79372
$$206$$ 0 0
$$207$$ −6.00000 −0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.00000 −0.550743 −0.275371 0.961338i $$-0.588801\pi$$
−0.275371 + 0.961338i $$0.588801\pi$$
$$212$$ 0 0
$$213$$ 6.00000 0.411113
$$214$$ 0 0
$$215$$ −16.0000 −1.09119
$$216$$ 0 0
$$217$$ 12.0000 0.814613
$$218$$ 0 0
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 6.00000 0.403604
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 11.0000 0.733333
$$226$$ 0 0
$$227$$ −4.00000 −0.265489 −0.132745 0.991150i $$-0.542379\pi$$
−0.132745 + 0.991150i $$0.542379\pi$$
$$228$$ 0 0
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 16.0000 1.04372
$$236$$ 0 0
$$237$$ −10.0000 −0.649570
$$238$$ 0 0
$$239$$ 20.0000 1.29369 0.646846 0.762620i $$-0.276088\pi$$
0.646846 + 0.762620i $$0.276088\pi$$
$$240$$ 0 0
$$241$$ −18.0000 −1.15948 −0.579741 0.814801i $$-0.696846\pi$$
−0.579741 + 0.814801i $$0.696846\pi$$
$$242$$ 0 0
$$243$$ 1.00000 0.0641500
$$244$$ 0 0
$$245$$ 12.0000 0.766652
$$246$$ 0 0
$$247$$ 24.0000 1.52708
$$248$$ 0 0
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 4.00000 0.250490
$$256$$ 0 0
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ −8.00000 −0.497096
$$260$$ 0 0
$$261$$ −4.00000 −0.247594
$$262$$ 0 0
$$263$$ 12.0000 0.739952 0.369976 0.929041i $$-0.379366\pi$$
0.369976 + 0.929041i $$0.379366\pi$$
$$264$$ 0 0
$$265$$ 8.00000 0.491436
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ 0 0
$$269$$ −12.0000 −0.731653 −0.365826 0.930683i $$-0.619214\pi$$
−0.365826 + 0.930683i $$0.619214\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ 0 0
$$273$$ −12.0000 −0.726273
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −8.00000 −0.480673 −0.240337 0.970690i $$-0.577258\pi$$
−0.240337 + 0.970690i $$0.577258\pi$$
$$278$$ 0 0
$$279$$ 6.00000 0.359211
$$280$$ 0 0
$$281$$ −18.0000 −1.07379 −0.536895 0.843649i $$-0.680403\pi$$
−0.536895 + 0.843649i $$0.680403\pi$$
$$282$$ 0 0
$$283$$ −32.0000 −1.90220 −0.951101 0.308879i $$-0.900046\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ 0 0
$$285$$ 16.0000 0.947758
$$286$$ 0 0
$$287$$ −20.0000 −1.18056
$$288$$ 0 0
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 6.00000 0.351726
$$292$$ 0 0
$$293$$ 2.00000 0.116841 0.0584206 0.998292i $$-0.481394\pi$$
0.0584206 + 0.998292i $$0.481394\pi$$
$$294$$ 0 0
$$295$$ 48.0000 2.79467
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 36.0000 2.08193
$$300$$ 0 0
$$301$$ 8.00000 0.461112
$$302$$ 0 0
$$303$$ 14.0000 0.804279
$$304$$ 0 0
$$305$$ 16.0000 0.916157
$$306$$ 0 0
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ −4.00000 −0.227552
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ −8.00000 −0.450749
$$316$$ 0 0
$$317$$ −16.0000 −0.898650 −0.449325 0.893368i $$-0.648335\pi$$
−0.449325 + 0.893368i $$0.648335\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.00000 0.222566
$$324$$ 0 0
$$325$$ −66.0000 −3.66102
$$326$$ 0 0
$$327$$ 16.0000 0.884802
$$328$$ 0 0
$$329$$ −8.00000 −0.441054
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 0 0
$$333$$ −4.00000 −0.219199
$$334$$ 0 0
$$335$$ −48.0000 −2.62252
$$336$$ 0 0
$$337$$ −6.00000 −0.326841 −0.163420 0.986557i $$-0.552253\pi$$
−0.163420 + 0.986557i $$0.552253\pi$$
$$338$$ 0 0
$$339$$ 2.00000 0.108625
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ −20.0000 −1.07990
$$344$$ 0 0
$$345$$ 24.0000 1.29212
$$346$$ 0 0
$$347$$ 4.00000 0.214731 0.107366 0.994220i $$-0.465758\pi$$
0.107366 + 0.994220i $$0.465758\pi$$
$$348$$ 0 0
$$349$$ −30.0000 −1.60586 −0.802932 0.596071i $$-0.796728\pi$$
−0.802932 + 0.596071i $$0.796728\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 0 0
$$353$$ 14.0000 0.745145 0.372572 0.928003i $$-0.378476\pi$$
0.372572 + 0.928003i $$0.378476\pi$$
$$354$$ 0 0
$$355$$ −24.0000 −1.27379
$$356$$ 0 0
$$357$$ −2.00000 −0.105851
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ −11.0000 −0.577350
$$364$$ 0 0
$$365$$ −8.00000 −0.418739
$$366$$ 0 0
$$367$$ −10.0000 −0.521996 −0.260998 0.965339i $$-0.584052\pi$$
−0.260998 + 0.965339i $$0.584052\pi$$
$$368$$ 0 0
$$369$$ −10.0000 −0.520579
$$370$$ 0 0
$$371$$ −4.00000 −0.207670
$$372$$ 0 0
$$373$$ −14.0000 −0.724893 −0.362446 0.932005i $$-0.618058\pi$$
−0.362446 + 0.932005i $$0.618058\pi$$
$$374$$ 0 0
$$375$$ −24.0000 −1.23935
$$376$$ 0 0
$$377$$ 24.0000 1.23606
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ −8.00000 −0.409852
$$382$$ 0 0
$$383$$ −28.0000 −1.43073 −0.715367 0.698749i $$-0.753740\pi$$
−0.715367 + 0.698749i $$0.753740\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 4.00000 0.203331
$$388$$ 0 0
$$389$$ −14.0000 −0.709828 −0.354914 0.934899i $$-0.615490\pi$$
−0.354914 + 0.934899i $$0.615490\pi$$
$$390$$ 0 0
$$391$$ 6.00000 0.303433
$$392$$ 0 0
$$393$$ 16.0000 0.807093
$$394$$ 0 0
$$395$$ 40.0000 2.01262
$$396$$ 0 0
$$397$$ 20.0000 1.00377 0.501886 0.864934i $$-0.332640\pi$$
0.501886 + 0.864934i $$0.332640\pi$$
$$398$$ 0 0
$$399$$ −8.00000 −0.400501
$$400$$ 0 0
$$401$$ 30.0000 1.49813 0.749064 0.662497i $$-0.230503\pi$$
0.749064 + 0.662497i $$0.230503\pi$$
$$402$$ 0 0
$$403$$ −36.0000 −1.79329
$$404$$ 0 0
$$405$$ −4.00000 −0.198762
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 0 0
$$411$$ −6.00000 −0.295958
$$412$$ 0 0
$$413$$ −24.0000 −1.18096
$$414$$ 0 0
$$415$$ −48.0000 −2.35623
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ 12.0000 0.586238 0.293119 0.956076i $$-0.405307\pi$$
0.293119 + 0.956076i $$0.405307\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 0 0
$$423$$ −4.00000 −0.194487
$$424$$ 0 0
$$425$$ −11.0000 −0.533578
$$426$$ 0 0
$$427$$ −8.00000 −0.387147
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −14.0000 −0.674356 −0.337178 0.941441i $$-0.609472\pi$$
−0.337178 + 0.941441i $$0.609472\pi$$
$$432$$ 0 0
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ 0 0
$$435$$ 16.0000 0.767141
$$436$$ 0 0
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ 10.0000 0.477274 0.238637 0.971109i $$-0.423299\pi$$
0.238637 + 0.971109i $$0.423299\pi$$
$$440$$ 0 0
$$441$$ −3.00000 −0.142857
$$442$$ 0 0
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 8.00000 0.379236
$$446$$ 0 0
$$447$$ −6.00000 −0.283790
$$448$$ 0 0
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 24.0000 1.12762
$$454$$ 0 0
$$455$$ 48.0000 2.25027
$$456$$ 0 0
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ 0 0
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ −10.0000 −0.465746 −0.232873 0.972507i $$-0.574813\pi$$
−0.232873 + 0.972507i $$0.574813\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ 36.0000 1.66588 0.832941 0.553362i $$-0.186655\pi$$
0.832941 + 0.553362i $$0.186655\pi$$
$$468$$ 0 0
$$469$$ 24.0000 1.10822
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −44.0000 −2.01886
$$476$$ 0 0
$$477$$ −2.00000 −0.0915737
$$478$$ 0 0
$$479$$ −10.0000 −0.456912 −0.228456 0.973554i $$-0.573368\pi$$
−0.228456 + 0.973554i $$0.573368\pi$$
$$480$$ 0 0
$$481$$ 24.0000 1.09431
$$482$$ 0 0
$$483$$ −12.0000 −0.546019
$$484$$ 0 0
$$485$$ −24.0000 −1.08978
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 0 0
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ 20.0000 0.902587 0.451294 0.892375i $$-0.350963\pi$$
0.451294 + 0.892375i $$0.350963\pi$$
$$492$$ 0 0
$$493$$ 4.00000 0.180151
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 12.0000 0.538274
$$498$$ 0 0
$$499$$ −32.0000 −1.43252 −0.716258 0.697835i $$-0.754147\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 2.00000 0.0893534
$$502$$ 0 0
$$503$$ 26.0000 1.15928 0.579641 0.814872i $$-0.303193\pi$$
0.579641 + 0.814872i $$0.303193\pi$$
$$504$$ 0 0
$$505$$ −56.0000 −2.49197
$$506$$ 0 0
$$507$$ 23.0000 1.02147
$$508$$ 0 0
$$509$$ 26.0000 1.15243 0.576215 0.817298i $$-0.304529\pi$$
0.576215 + 0.817298i $$0.304529\pi$$
$$510$$ 0 0
$$511$$ 4.00000 0.176950
$$512$$ 0 0
$$513$$ −4.00000 −0.176604
$$514$$ 0 0
$$515$$ 16.0000 0.705044
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −4.00000 −0.175581
$$520$$ 0 0
$$521$$ −10.0000 −0.438108 −0.219054 0.975713i $$-0.570297\pi$$
−0.219054 + 0.975713i $$0.570297\pi$$
$$522$$ 0 0
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 0 0
$$525$$ 22.0000 0.960159
$$526$$ 0 0
$$527$$ −6.00000 −0.261364
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −12.0000 −0.520756
$$532$$ 0 0
$$533$$ 60.0000 2.59889
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 12.0000 0.517838
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −12.0000 −0.515920 −0.257960 0.966156i $$-0.583050\pi$$
−0.257960 + 0.966156i $$0.583050\pi$$
$$542$$ 0 0
$$543$$ −20.0000 −0.858282
$$544$$ 0 0
$$545$$ −64.0000 −2.74146
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ 0 0
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ 16.0000 0.681623
$$552$$ 0 0
$$553$$ −20.0000 −0.850487
$$554$$ 0 0
$$555$$ 16.0000 0.679162
$$556$$ 0 0
$$557$$ 14.0000 0.593199 0.296600 0.955002i $$-0.404147\pi$$
0.296600 + 0.955002i $$0.404147\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 4.00000 0.168580 0.0842900 0.996441i $$-0.473138\pi$$
0.0842900 + 0.996441i $$0.473138\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ 0 0
$$567$$ 2.00000 0.0839921
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 0 0
$$573$$ 4.00000 0.167102
$$574$$ 0 0
$$575$$ −66.0000 −2.75239
$$576$$ 0 0
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ 0 0
$$579$$ 6.00000 0.249351
$$580$$ 0 0
$$581$$ 24.0000 0.995688
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 24.0000 0.992278
$$586$$ 0 0
$$587$$ −36.0000 −1.48588 −0.742940 0.669359i $$-0.766569\pi$$
−0.742940 + 0.669359i $$0.766569\pi$$
$$588$$ 0 0
$$589$$ −24.0000 −0.988903
$$590$$ 0 0
$$591$$ 8.00000 0.329076
$$592$$ 0 0
$$593$$ −30.0000 −1.23195 −0.615976 0.787765i $$-0.711238\pi$$
−0.615976 + 0.787765i $$0.711238\pi$$
$$594$$ 0 0
$$595$$ 8.00000 0.327968
$$596$$ 0 0
$$597$$ −14.0000 −0.572982
$$598$$ 0 0
$$599$$ 16.0000 0.653742 0.326871 0.945069i $$-0.394006\pi$$
0.326871 + 0.945069i $$0.394006\pi$$
$$600$$ 0 0
$$601$$ −38.0000 −1.55005 −0.775026 0.631929i $$-0.782263\pi$$
−0.775026 + 0.631929i $$0.782263\pi$$
$$602$$ 0 0
$$603$$ 12.0000 0.488678
$$604$$ 0 0
$$605$$ 44.0000 1.78885
$$606$$ 0 0
$$607$$ 38.0000 1.54237 0.771186 0.636610i $$-0.219664\pi$$
0.771186 + 0.636610i $$0.219664\pi$$
$$608$$ 0 0
$$609$$ −8.00000 −0.324176
$$610$$ 0 0
$$611$$ 24.0000 0.970936
$$612$$ 0 0
$$613$$ 30.0000 1.21169 0.605844 0.795583i $$-0.292835\pi$$
0.605844 + 0.795583i $$0.292835\pi$$
$$614$$ 0 0
$$615$$ 40.0000 1.61296
$$616$$ 0 0
$$617$$ −6.00000 −0.241551 −0.120775 0.992680i $$-0.538538\pi$$
−0.120775 + 0.992680i $$0.538538\pi$$
$$618$$ 0 0
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ −6.00000 −0.240772
$$622$$ 0 0
$$623$$ −4.00000 −0.160257
$$624$$ 0 0
$$625$$ 41.0000 1.64000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ −20.0000 −0.796187 −0.398094 0.917345i $$-0.630328\pi$$
−0.398094 + 0.917345i $$0.630328\pi$$
$$632$$ 0 0
$$633$$ −8.00000 −0.317971
$$634$$ 0 0
$$635$$ 32.0000 1.26988
$$636$$ 0 0
$$637$$ 18.0000 0.713186
$$638$$ 0 0
$$639$$ 6.00000 0.237356
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 0 0
$$643$$ −4.00000 −0.157745 −0.0788723 0.996885i $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 0 0
$$645$$ −16.0000 −0.629999
$$646$$ 0 0
$$647$$ −28.0000 −1.10079 −0.550397 0.834903i $$-0.685524\pi$$
−0.550397 + 0.834903i $$0.685524\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 12.0000 0.470317
$$652$$ 0 0
$$653$$ 20.0000 0.782660 0.391330 0.920250i $$-0.372015\pi$$
0.391330 + 0.920250i $$0.372015\pi$$
$$654$$ 0 0
$$655$$ −64.0000 −2.50069
$$656$$ 0 0
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ −12.0000 −0.467454 −0.233727 0.972302i $$-0.575092\pi$$
−0.233727 + 0.972302i $$0.575092\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 0 0
$$663$$ 6.00000 0.233021
$$664$$ 0 0
$$665$$ 32.0000 1.24091
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 4.00000 0.154649
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −26.0000 −1.00223 −0.501113 0.865382i $$-0.667076\pi$$
−0.501113 + 0.865382i $$0.667076\pi$$
$$674$$ 0 0
$$675$$ 11.0000 0.423390
$$676$$ 0 0
$$677$$ 8.00000 0.307465 0.153732 0.988113i $$-0.450871\pi$$
0.153732 + 0.988113i $$0.450871\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 0 0
$$681$$ −4.00000 −0.153280
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 24.0000 0.916993
$$686$$ 0 0
$$687$$ −2.00000 −0.0763048
$$688$$ 0 0
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −16.0000 −0.608669 −0.304334 0.952565i $$-0.598434\pi$$
−0.304334 + 0.952565i $$0.598434\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 32.0000 1.21383
$$696$$ 0 0
$$697$$ 10.0000 0.378777
$$698$$ 0 0
$$699$$ −6.00000 −0.226941
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 16.0000 0.603451
$$704$$ 0 0
$$705$$ 16.0000 0.602595
$$706$$ 0 0
$$707$$ 28.0000 1.05305
$$708$$ 0 0
$$709$$ −16.0000 −0.600893 −0.300446 0.953799i $$-0.597136\pi$$
−0.300446 + 0.953799i $$0.597136\pi$$
$$710$$ 0 0
$$711$$ −10.0000 −0.375029
$$712$$ 0 0
$$713$$ −36.0000 −1.34821
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.0000 0.746914
$$718$$ 0 0
$$719$$ −42.0000 −1.56634 −0.783168 0.621810i $$-0.786397\pi$$
−0.783168 + 0.621810i $$0.786397\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 0 0
$$723$$ −18.0000 −0.669427
$$724$$ 0 0
$$725$$ −44.0000 −1.63412
$$726$$ 0 0
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ 0 0
$$733$$ 18.0000 0.664845 0.332423 0.943131i $$-0.392134\pi$$
0.332423 + 0.943131i $$0.392134\pi$$
$$734$$ 0 0
$$735$$ 12.0000 0.442627
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 0 0
$$741$$ 24.0000 0.881662
$$742$$ 0 0
$$743$$ −38.0000 −1.39408 −0.697042 0.717030i $$-0.745501\pi$$
−0.697042 + 0.717030i $$0.745501\pi$$
$$744$$ 0 0
$$745$$ 24.0000 0.879292
$$746$$ 0 0
$$747$$ 12.0000 0.439057
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 34.0000 1.24068 0.620339 0.784334i $$-0.286995\pi$$
0.620339 + 0.784334i $$0.286995\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ 0 0
$$755$$ −96.0000 −3.49380
$$756$$ 0 0
$$757$$ 14.0000 0.508839 0.254419 0.967094i $$-0.418116\pi$$
0.254419 + 0.967094i $$0.418116\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 0 0
$$763$$ 32.0000 1.15848
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ 0 0
$$767$$ 72.0000 2.59977
$$768$$ 0 0
$$769$$ 26.0000 0.937584 0.468792 0.883309i $$-0.344689\pi$$
0.468792 + 0.883309i $$0.344689\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.648254
$$772$$ 0 0
$$773$$ 42.0000 1.51064 0.755318 0.655359i $$-0.227483\pi$$
0.755318 + 0.655359i $$0.227483\pi$$
$$774$$ 0 0
$$775$$ 66.0000 2.37079
$$776$$ 0 0
$$777$$ −8.00000 −0.286998
$$778$$ 0 0
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −4.00000 −0.142948
$$784$$ 0 0
$$785$$ −24.0000 −0.856597
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 12.0000 0.427211
$$790$$ 0 0
$$791$$ 4.00000 0.142224
$$792$$ 0 0
$$793$$ 24.0000 0.852265
$$794$$ 0 0
$$795$$ 8.00000 0.283731
$$796$$ 0 0
$$797$$ 46.0000 1.62940 0.814702 0.579880i $$-0.196901\pi$$
0.814702 + 0.579880i $$0.196901\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 48.0000 1.69178
$$806$$ 0 0
$$807$$ −12.0000 −0.422420
$$808$$ 0 0
$$809$$ −26.0000 −0.914111 −0.457056 0.889438i $$-0.651096\pi$$
−0.457056 + 0.889438i $$0.651096\pi$$
$$810$$ 0 0
$$811$$ 12.0000 0.421377 0.210688 0.977553i $$-0.432429\pi$$
0.210688 + 0.977553i $$0.432429\pi$$
$$812$$ 0 0
$$813$$ 16.0000 0.561144
$$814$$ 0 0
$$815$$ 48.0000 1.68137
$$816$$ 0 0
$$817$$ −16.0000 −0.559769
$$818$$ 0 0
$$819$$ −12.0000 −0.419314
$$820$$ 0 0
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ −18.0000 −0.627441 −0.313720 0.949515i $$-0.601575\pi$$
−0.313720 + 0.949515i $$0.601575\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −16.0000 −0.556375 −0.278187 0.960527i $$-0.589734\pi$$
−0.278187 + 0.960527i $$0.589734\pi$$
$$828$$ 0 0
$$829$$ 6.00000 0.208389 0.104194 0.994557i $$-0.466774\pi$$
0.104194 + 0.994557i $$0.466774\pi$$
$$830$$ 0 0
$$831$$ −8.00000 −0.277517
$$832$$ 0 0
$$833$$ 3.00000 0.103944
$$834$$ 0 0
$$835$$ −8.00000 −0.276851
$$836$$ 0 0
$$837$$ 6.00000 0.207390
$$838$$ 0 0
$$839$$ 30.0000 1.03572 0.517858 0.855467i $$-0.326730\pi$$
0.517858 + 0.855467i $$0.326730\pi$$
$$840$$ 0 0
$$841$$ −13.0000 −0.448276
$$842$$ 0 0
$$843$$ −18.0000 −0.619953
$$844$$ 0 0
$$845$$ −92.0000 −3.16490
$$846$$ 0 0
$$847$$ −22.0000 −0.755929
$$848$$ 0 0
$$849$$ −32.0000 −1.09824
$$850$$ 0 0
$$851$$ 24.0000 0.822709
$$852$$ 0 0
$$853$$ −16.0000 −0.547830 −0.273915 0.961754i $$-0.588319\pi$$
−0.273915 + 0.961754i $$0.588319\pi$$
$$854$$ 0 0
$$855$$ 16.0000 0.547188
$$856$$ 0 0
$$857$$ −10.0000 −0.341593 −0.170797 0.985306i $$-0.554634\pi$$
−0.170797 + 0.985306i $$0.554634\pi$$
$$858$$ 0 0
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ −20.0000 −0.681598
$$862$$ 0 0
$$863$$ 48.0000 1.63394 0.816970 0.576681i $$-0.195652\pi$$
0.816970 + 0.576681i $$0.195652\pi$$
$$864$$ 0 0
$$865$$ 16.0000 0.544016
$$866$$ 0 0
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −72.0000 −2.43963
$$872$$ 0 0
$$873$$ 6.00000 0.203069
$$874$$ 0 0
$$875$$ −48.0000 −1.62270
$$876$$ 0 0
$$877$$ −24.0000 −0.810422 −0.405211 0.914223i $$-0.632802\pi$$
−0.405211 + 0.914223i $$0.632802\pi$$
$$878$$ 0 0
$$879$$ 2.00000 0.0674583
$$880$$ 0 0
$$881$$ 50.0000 1.68454 0.842271 0.539054i $$-0.181218\pi$$
0.842271 + 0.539054i $$0.181218\pi$$
$$882$$ 0 0
$$883$$ 52.0000 1.74994 0.874970 0.484178i $$-0.160881\pi$$
0.874970 + 0.484178i $$0.160881\pi$$
$$884$$ 0 0
$$885$$ 48.0000 1.61350
$$886$$ 0 0
$$887$$ −18.0000 −0.604381 −0.302190 0.953248i $$-0.597718\pi$$
−0.302190 + 0.953248i $$0.597718\pi$$
$$888$$ 0 0
$$889$$ −16.0000 −0.536623
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 16.0000 0.535420
$$894$$ 0 0
$$895$$ −48.0000 −1.60446
$$896$$ 0 0
$$897$$ 36.0000 1.20201
$$898$$ 0 0
$$899$$ −24.0000 −0.800445
$$900$$ 0 0
$$901$$ 2.00000 0.0666297
$$902$$ 0 0
$$903$$ 8.00000 0.266223
$$904$$ 0 0
$$905$$ 80.0000 2.65929
$$906$$ 0 0
$$907$$ −24.0000 −0.796907 −0.398453 0.917189i $$-0.630453\pi$$
−0.398453 + 0.917189i $$0.630453\pi$$
$$908$$ 0 0
$$909$$ 14.0000 0.464351
$$910$$ 0 0
$$911$$ −26.0000 −0.861418 −0.430709 0.902491i $$-0.641737\pi$$
−0.430709 + 0.902491i $$0.641737\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 16.0000 0.528944
$$916$$ 0 0
$$917$$ 32.0000 1.05673
$$918$$ 0 0
$$919$$ −16.0000 −0.527791 −0.263896 0.964551i $$-0.585007\pi$$
−0.263896 + 0.964551i $$0.585007\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 0 0
$$923$$ −36.0000 −1.18495
$$924$$ 0 0
$$925$$ −44.0000 −1.44671
$$926$$ 0 0
$$927$$ −4.00000 −0.131377
$$928$$ 0 0
$$929$$ −34.0000 −1.11550 −0.557752 0.830008i $$-0.688336\pi$$
−0.557752 + 0.830008i $$0.688336\pi$$
$$930$$ 0 0
$$931$$ 12.0000 0.393284
$$932$$ 0 0
$$933$$ −30.0000 −0.982156
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 0 0
$$939$$ −26.0000 −0.848478
$$940$$ 0 0
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ 60.0000 1.95387
$$944$$ 0 0
$$945$$ −8.00000 −0.260240
$$946$$ 0 0
$$947$$ 24.0000 0.779895 0.389948 0.920837i $$-0.372493\pi$$
0.389948 + 0.920837i $$0.372493\pi$$
$$948$$ 0 0
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ −16.0000 −0.518836
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 0 0
$$955$$ −16.0000 −0.517748
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −12.0000 −0.387500
$$960$$ 0 0
$$961$$ 5.00000 0.161290
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −24.0000 −0.772587
$$966$$ 0 0
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ 0 0
$$969$$ 4.00000 0.128499
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 0 0
$$973$$ −16.0000 −0.512936
$$974$$ 0 0
$$975$$ −66.0000 −2.11369
$$976$$ 0 0
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 16.0000 0.510841
$$982$$ 0 0
$$983$$ 38.0000 1.21201 0.606006 0.795460i $$-0.292771\pi$$
0.606006 + 0.795460i $$0.292771\pi$$
$$984$$ 0 0
$$985$$ −32.0000 −1.01960
$$986$$ 0 0
$$987$$ −8.00000 −0.254643
$$988$$ 0 0
$$989$$ −24.0000 −0.763156
$$990$$ 0 0
$$991$$ 34.0000 1.08005 0.540023 0.841650i $$-0.318416\pi$$
0.540023 + 0.841650i $$0.318416\pi$$
$$992$$ 0 0
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 56.0000 1.77532
$$996$$ 0 0
$$997$$ −20.0000 −0.633406 −0.316703 0.948525i $$-0.602576\pi$$
−0.316703 + 0.948525i $$0.602576\pi$$
$$998$$ 0 0
$$999$$ −4.00000 −0.126554
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 816.2.a.h.1.1 1
3.2 odd 2 2448.2.a.t.1.1 1
4.3 odd 2 102.2.a.a.1.1 1
8.3 odd 2 3264.2.a.bf.1.1 1
8.5 even 2 3264.2.a.p.1.1 1
12.11 even 2 306.2.a.d.1.1 1
20.3 even 4 2550.2.d.q.2449.2 2
20.7 even 4 2550.2.d.q.2449.1 2
20.19 odd 2 2550.2.a.be.1.1 1
24.5 odd 2 9792.2.a.b.1.1 1
24.11 even 2 9792.2.a.a.1.1 1
28.27 even 2 4998.2.a.x.1.1 1
60.59 even 2 7650.2.a.z.1.1 1
68.15 odd 8 1734.2.f.g.1483.2 4
68.19 odd 8 1734.2.f.g.1483.1 4
68.43 odd 8 1734.2.f.g.829.1 4
68.47 odd 4 1734.2.b.d.577.1 2
68.55 odd 4 1734.2.b.d.577.2 2
68.59 odd 8 1734.2.f.g.829.2 4
68.67 odd 2 1734.2.a.h.1.1 1
204.203 even 2 5202.2.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.a.1.1 1 4.3 odd 2
306.2.a.d.1.1 1 12.11 even 2
816.2.a.h.1.1 1 1.1 even 1 trivial
1734.2.a.h.1.1 1 68.67 odd 2
1734.2.b.d.577.1 2 68.47 odd 4
1734.2.b.d.577.2 2 68.55 odd 4
1734.2.f.g.829.1 4 68.43 odd 8
1734.2.f.g.829.2 4 68.59 odd 8
1734.2.f.g.1483.1 4 68.19 odd 8
1734.2.f.g.1483.2 4 68.15 odd 8
2448.2.a.t.1.1 1 3.2 odd 2
2550.2.a.be.1.1 1 20.19 odd 2
2550.2.d.q.2449.1 2 20.7 even 4
2550.2.d.q.2449.2 2 20.3 even 4
3264.2.a.p.1.1 1 8.5 even 2
3264.2.a.bf.1.1 1 8.3 odd 2
4998.2.a.x.1.1 1 28.27 even 2
5202.2.a.g.1.1 1 204.203 even 2
7650.2.a.z.1.1 1 60.59 even 2
9792.2.a.a.1.1 1 24.11 even 2
9792.2.a.b.1.1 1 24.5 odd 2