# Properties

 Label 234.2.a.d.1.1 Level $234$ Weight $2$ Character 234.1 Self dual yes Analytic conductor $1.868$ 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] = [234,2,Mod(1,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 234.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: $$1.86849940730$$ 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 $$\chi$$ $$=$$ 234.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{8} +2.00000 q^{10} +4.00000 q^{11} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.00000 q^{19} +2.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} -1.00000 q^{26} -2.00000 q^{28} +8.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{35} +6.00000 q^{37} -6.00000 q^{38} +2.00000 q^{40} -6.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -1.00000 q^{52} -12.0000 q^{53} +8.00000 q^{55} -2.00000 q^{56} +8.00000 q^{58} -4.00000 q^{59} +10.0000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -2.00000 q^{67} -4.00000 q^{70} +16.0000 q^{71} +14.0000 q^{73} +6.00000 q^{74} -6.00000 q^{76} -8.00000 q^{77} -4.00000 q^{79} +2.00000 q^{80} -6.00000 q^{82} +12.0000 q^{83} -8.00000 q^{86} +4.00000 q^{88} +6.00000 q^{89} +2.00000 q^{91} -4.00000 q^{92} -8.00000 q^{94} -12.0000 q^{95} -10.0000 q^{97} -3.00000 q^{98} +O(q^{100})$$

## 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
$$5$$ 2.00000 0.894427 0.447214 0.894427i $$-0.352416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ −2.00000 −0.755929 −0.377964 0.925820i $$-0.623376\pi$$
−0.377964 + 0.925820i $$0.623376\pi$$
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 2.00000 0.632456
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350
$$14$$ −2.00000 −0.534522
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 2.00000 0.447214
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ −4.00000 −0.834058 −0.417029 0.908893i $$-0.636929\pi$$
−0.417029 + 0.908893i $$0.636929\pi$$
$$24$$ 0 0
$$25$$ −1.00000 −0.200000
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ −2.00000 −0.377964
$$29$$ 8.00000 1.48556 0.742781 0.669534i $$-0.233506\pi$$
0.742781 + 0.669534i $$0.233506\pi$$
$$30$$ 0 0
$$31$$ −2.00000 −0.359211 −0.179605 0.983739i $$-0.557482\pi$$
−0.179605 + 0.983739i $$0.557482\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −4.00000 −0.676123
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −6.00000 −0.973329
$$39$$ 0 0
$$40$$ 2.00000 0.316228
$$41$$ −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 $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ −4.00000 −0.589768
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 0 0
$$49$$ −3.00000 −0.428571
$$50$$ −1.00000 −0.141421
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ −12.0000 −1.64833 −0.824163 0.566352i $$-0.808354\pi$$
−0.824163 + 0.566352i $$0.808354\pi$$
$$54$$ 0 0
$$55$$ 8.00000 1.07872
$$56$$ −2.00000 −0.267261
$$57$$ 0 0
$$58$$ 8.00000 1.05045
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −2.00000 −0.254000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ −2.00000 −0.248069
$$66$$ 0 0
$$67$$ −2.00000 −0.244339 −0.122169 0.992509i $$-0.538985\pi$$
−0.122169 + 0.992509i $$0.538985\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ −4.00000 −0.478091
$$71$$ 16.0000 1.89885 0.949425 0.313993i $$-0.101667\pi$$
0.949425 + 0.313993i $$0.101667\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ −6.00000 −0.688247
$$77$$ −8.00000 −0.911685
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 2.00000 0.223607
$$81$$ 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
$$85$$ 0 0
$$86$$ −8.00000 −0.862662
$$87$$ 0 0
$$88$$ 4.00000 0.426401
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ −4.00000 −0.417029
$$93$$ 0 0
$$94$$ −8.00000 −0.825137
$$95$$ −12.0000 −1.23117
$$96$$ 0 0
$$97$$ −10.0000 −1.01535 −0.507673 0.861550i $$-0.669494\pi$$
−0.507673 + 0.861550i $$0.669494\pi$$
$$98$$ −3.00000 −0.303046
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 16.0000 1.59206 0.796030 0.605257i $$-0.206930\pi$$
0.796030 + 0.605257i $$0.206930\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ −12.0000 −1.16554
$$107$$ 20.0000 1.93347 0.966736 0.255774i $$-0.0823304\pi$$
0.966736 + 0.255774i $$0.0823304\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 0 0
$$112$$ −2.00000 −0.188982
$$113$$ −4.00000 −0.376288 −0.188144 0.982141i $$-0.560247\pi$$
−0.188144 + 0.982141i $$0.560247\pi$$
$$114$$ 0 0
$$115$$ −8.00000 −0.746004
$$116$$ 8.00000 0.742781
$$117$$ 0 0
$$118$$ −4.00000 −0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ 0 0
$$124$$ −2.00000 −0.179605
$$125$$ −12.0000 −1.07331
$$126$$ 0 0
$$127$$ −8.00000 −0.709885 −0.354943 0.934888i $$-0.615500\pi$$
−0.354943 + 0.934888i $$0.615500\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ −2.00000 −0.175412
$$131$$ −4.00000 −0.349482 −0.174741 0.984614i $$-0.555909\pi$$
−0.174741 + 0.984614i $$0.555909\pi$$
$$132$$ 0 0
$$133$$ 12.0000 1.04053
$$134$$ −2.00000 −0.172774
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ −20.0000 −1.69638 −0.848189 0.529694i $$-0.822307\pi$$
−0.848189 + 0.529694i $$0.822307\pi$$
$$140$$ −4.00000 −0.338062
$$141$$ 0 0
$$142$$ 16.0000 1.34269
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ 16.0000 1.32873
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 18.0000 1.46482 0.732410 0.680864i $$-0.238396\pi$$
0.732410 + 0.680864i $$0.238396\pi$$
$$152$$ −6.00000 −0.486664
$$153$$ 0 0
$$154$$ −8.00000 −0.644658
$$155$$ −4.00000 −0.321288
$$156$$ 0 0
$$157$$ 2.00000 0.159617 0.0798087 0.996810i $$-0.474569\pi$$
0.0798087 + 0.996810i $$0.474569\pi$$
$$158$$ −4.00000 −0.318223
$$159$$ 0 0
$$160$$ 2.00000 0.158114
$$161$$ 8.00000 0.630488
$$162$$ 0 0
$$163$$ −10.0000 −0.783260 −0.391630 0.920123i $$-0.628089\pi$$
−0.391630 + 0.920123i $$0.628089\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ 12.0000 0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −8.00000 −0.609994
$$173$$ −16.0000 −1.21646 −0.608229 0.793762i $$-0.708120\pi$$
−0.608229 + 0.793762i $$0.708120\pi$$
$$174$$ 0 0
$$175$$ 2.00000 0.151186
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 2.00000 0.148250
$$183$$ 0 0
$$184$$ −4.00000 −0.294884
$$185$$ 12.0000 0.882258
$$186$$ 0 0
$$187$$ 0 0
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ −12.0000 −0.870572
$$191$$ 4.00000 0.289430 0.144715 0.989473i $$-0.453773\pi$$
0.144715 + 0.989473i $$0.453773\pi$$
$$192$$ 0 0
$$193$$ 2.00000 0.143963 0.0719816 0.997406i $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −10.0000 −0.717958
$$195$$ 0 0
$$196$$ −3.00000 −0.214286
$$197$$ 18.0000 1.28245 0.641223 0.767354i $$-0.278427\pi$$
0.641223 + 0.767354i $$0.278427\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 0 0
$$202$$ 16.0000 1.12576
$$203$$ −16.0000 −1.12298
$$204$$ 0 0
$$205$$ −12.0000 −0.838116
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ −24.0000 −1.66011
$$210$$ 0 0
$$211$$ 8.00000 0.550743 0.275371 0.961338i $$-0.411199\pi$$
0.275371 + 0.961338i $$0.411199\pi$$
$$212$$ −12.0000 −0.824163
$$213$$ 0 0
$$214$$ 20.0000 1.36717
$$215$$ −16.0000 −1.09119
$$216$$ 0 0
$$217$$ 4.00000 0.271538
$$218$$ 10.0000 0.677285
$$219$$ 0 0
$$220$$ 8.00000 0.539360
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 6.00000 0.401790 0.200895 0.979613i $$-0.435615\pi$$
0.200895 + 0.979613i $$0.435615\pi$$
$$224$$ −2.00000 −0.133631
$$225$$ 0 0
$$226$$ −4.00000 −0.266076
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ −30.0000 −1.98246 −0.991228 0.132164i $$-0.957808\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ −8.00000 −0.527504
$$231$$ 0 0
$$232$$ 8.00000 0.525226
$$233$$ −20.0000 −1.31024 −0.655122 0.755523i $$-0.727383\pi$$
−0.655122 + 0.755523i $$0.727383\pi$$
$$234$$ 0 0
$$235$$ −16.0000 −1.04372
$$236$$ −4.00000 −0.260378
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 5.00000 0.321412
$$243$$ 0 0
$$244$$ 10.0000 0.640184
$$245$$ −6.00000 −0.383326
$$246$$ 0 0
$$247$$ 6.00000 0.381771
$$248$$ −2.00000 −0.127000
$$249$$ 0 0
$$250$$ −12.0000 −0.758947
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ −16.0000 −1.00591
$$254$$ −8.00000 −0.501965
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −12.0000 −0.748539 −0.374270 0.927320i $$-0.622107\pi$$
−0.374270 + 0.927320i $$0.622107\pi$$
$$258$$ 0 0
$$259$$ −12.0000 −0.745644
$$260$$ −2.00000 −0.124035
$$261$$ 0 0
$$262$$ −4.00000 −0.247121
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −24.0000 −1.47431
$$266$$ 12.0000 0.735767
$$267$$ 0 0
$$268$$ −2.00000 −0.122169
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 22.0000 1.33640 0.668202 0.743980i $$-0.267064\pi$$
0.668202 + 0.743980i $$0.267064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 18.0000 1.08742
$$275$$ −4.00000 −0.241209
$$276$$ 0 0
$$277$$ −6.00000 −0.360505 −0.180253 0.983620i $$-0.557691\pi$$
−0.180253 + 0.983620i $$0.557691\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 0 0
$$280$$ −4.00000 −0.239046
$$281$$ 10.0000 0.596550 0.298275 0.954480i $$-0.403589\pi$$
0.298275 + 0.954480i $$0.403589\pi$$
$$282$$ 0 0
$$283$$ −20.0000 −1.18888 −0.594438 0.804141i $$-0.702626\pi$$
−0.594438 + 0.804141i $$0.702626\pi$$
$$284$$ 16.0000 0.949425
$$285$$ 0 0
$$286$$ −4.00000 −0.236525
$$287$$ 12.0000 0.708338
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 16.0000 0.939552
$$291$$ 0 0
$$292$$ 14.0000 0.819288
$$293$$ −2.00000 −0.116841 −0.0584206 0.998292i $$-0.518606\pi$$
−0.0584206 + 0.998292i $$0.518606\pi$$
$$294$$ 0 0
$$295$$ −8.00000 −0.465778
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ 4.00000 0.231326
$$300$$ 0 0
$$301$$ 16.0000 0.922225
$$302$$ 18.0000 1.03578
$$303$$ 0 0
$$304$$ −6.00000 −0.344124
$$305$$ 20.0000 1.14520
$$306$$ 0 0
$$307$$ 2.00000 0.114146 0.0570730 0.998370i $$-0.481823\pi$$
0.0570730 + 0.998370i $$0.481823\pi$$
$$308$$ −8.00000 −0.455842
$$309$$ 0 0
$$310$$ −4.00000 −0.227185
$$311$$ 12.0000 0.680458 0.340229 0.940343i $$-0.389495\pi$$
0.340229 + 0.940343i $$0.389495\pi$$
$$312$$ 0 0
$$313$$ 26.0000 1.46961 0.734803 0.678280i $$-0.237274\pi$$
0.734803 + 0.678280i $$0.237274\pi$$
$$314$$ 2.00000 0.112867
$$315$$ 0 0
$$316$$ −4.00000 −0.225018
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 32.0000 1.79166
$$320$$ 2.00000 0.111803
$$321$$ 0 0
$$322$$ 8.00000 0.445823
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ −10.0000 −0.553849
$$327$$ 0 0
$$328$$ −6.00000 −0.331295
$$329$$ 16.0000 0.882109
$$330$$ 0 0
$$331$$ −10.0000 −0.549650 −0.274825 0.961494i $$-0.588620\pi$$
−0.274825 + 0.961494i $$0.588620\pi$$
$$332$$ 12.0000 0.658586
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −34.0000 −1.85210 −0.926049 0.377403i $$-0.876817\pi$$
−0.926049 + 0.377403i $$0.876817\pi$$
$$338$$ 1.00000 0.0543928
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −8.00000 −0.433224
$$342$$ 0 0
$$343$$ 20.0000 1.07990
$$344$$ −8.00000 −0.431331
$$345$$ 0 0
$$346$$ −16.0000 −0.860165
$$347$$ 24.0000 1.28839 0.644194 0.764862i $$-0.277193\pi$$
0.644194 + 0.764862i $$0.277193\pi$$
$$348$$ 0 0
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 2.00000 0.106904
$$351$$ 0 0
$$352$$ 4.00000 0.213201
$$353$$ −14.0000 −0.745145 −0.372572 0.928003i $$-0.621524\pi$$
−0.372572 + 0.928003i $$0.621524\pi$$
$$354$$ 0 0
$$355$$ 32.0000 1.69838
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ −12.0000 −0.634220
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ −18.0000 −0.946059
$$363$$ 0 0
$$364$$ 2.00000 0.104828
$$365$$ 28.0000 1.46559
$$366$$ 0 0
$$367$$ 8.00000 0.417597 0.208798 0.977959i $$-0.433045\pi$$
0.208798 + 0.977959i $$0.433045\pi$$
$$368$$ −4.00000 −0.208514
$$369$$ 0 0
$$370$$ 12.0000 0.623850
$$371$$ 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
$$376$$ −8.00000 −0.412568
$$377$$ −8.00000 −0.412021
$$378$$ 0 0
$$379$$ 10.0000 0.513665 0.256833 0.966456i $$-0.417321\pi$$
0.256833 + 0.966456i $$0.417321\pi$$
$$380$$ −12.0000 −0.615587
$$381$$ 0 0
$$382$$ 4.00000 0.204658
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −16.0000 −0.815436
$$386$$ 2.00000 0.101797
$$387$$ 0 0
$$388$$ −10.0000 −0.507673
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ −3.00000 −0.151523
$$393$$ 0 0
$$394$$ 18.0000 0.906827
$$395$$ −8.00000 −0.402524
$$396$$ 0 0
$$397$$ −10.0000 −0.501886 −0.250943 0.968002i $$-0.580741\pi$$
−0.250943 + 0.968002i $$0.580741\pi$$
$$398$$ 8.00000 0.401004
$$399$$ 0 0
$$400$$ −1.00000 −0.0500000
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 0 0
$$403$$ 2.00000 0.0996271
$$404$$ 16.0000 0.796030
$$405$$ 0 0
$$406$$ −16.0000 −0.794067
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ −12.0000 −0.592638
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 8.00000 0.393654
$$414$$ 0 0
$$415$$ 24.0000 1.17811
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ −24.0000 −1.17388
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 8.00000 0.389434
$$423$$ 0 0
$$424$$ −12.0000 −0.582772
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −20.0000 −0.967868
$$428$$ 20.0000 0.966736
$$429$$ 0 0
$$430$$ −16.0000 −0.771589
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 4.00000 0.192006
$$435$$ 0 0
$$436$$ 10.0000 0.478913
$$437$$ 24.0000 1.14808
$$438$$ 0 0
$$439$$ −4.00000 −0.190910 −0.0954548 0.995434i $$-0.530431\pi$$
−0.0954548 + 0.995434i $$0.530431\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −36.0000 −1.71041 −0.855206 0.518289i $$-0.826569\pi$$
−0.855206 + 0.518289i $$0.826569\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ 6.00000 0.284108
$$447$$ 0 0
$$448$$ −2.00000 −0.0944911
$$449$$ −26.0000 −1.22702 −0.613508 0.789689i $$-0.710242\pi$$
−0.613508 + 0.789689i $$0.710242\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ −4.00000 −0.188144
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 4.00000 0.187523
$$456$$ 0 0
$$457$$ −2.00000 −0.0935561 −0.0467780 0.998905i $$-0.514895\pi$$
−0.0467780 + 0.998905i $$0.514895\pi$$
$$458$$ −30.0000 −1.40181
$$459$$ 0 0
$$460$$ −8.00000 −0.373002
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ 22.0000 1.02243 0.511213 0.859454i $$-0.329196\pi$$
0.511213 + 0.859454i $$0.329196\pi$$
$$464$$ 8.00000 0.371391
$$465$$ 0 0
$$466$$ −20.0000 −0.926482
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ 0 0
$$469$$ 4.00000 0.184703
$$470$$ −16.0000 −0.738025
$$471$$ 0 0
$$472$$ −4.00000 −0.184115
$$473$$ −32.0000 −1.47136
$$474$$ 0 0
$$475$$ 6.00000 0.275299
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 24.0000 1.09773
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ 6.00000 0.273293
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ −20.0000 −0.908153
$$486$$ 0 0
$$487$$ 38.0000 1.72194 0.860972 0.508652i $$-0.169856\pi$$
0.860972 + 0.508652i $$0.169856\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 0 0
$$490$$ −6.00000 −0.271052
$$491$$ 16.0000 0.722070 0.361035 0.932552i $$-0.382424\pi$$
0.361035 + 0.932552i $$0.382424\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 6.00000 0.269953
$$495$$ 0 0
$$496$$ −2.00000 −0.0898027
$$497$$ −32.0000 −1.43540
$$498$$ 0 0
$$499$$ 22.0000 0.984855 0.492428 0.870353i $$-0.336110\pi$$
0.492428 + 0.870353i $$0.336110\pi$$
$$500$$ −12.0000 −0.536656
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −32.0000 −1.42681 −0.713405 0.700752i $$-0.752848\pi$$
−0.713405 + 0.700752i $$0.752848\pi$$
$$504$$ 0 0
$$505$$ 32.0000 1.42398
$$506$$ −16.0000 −0.711287
$$507$$ 0 0
$$508$$ −8.00000 −0.354943
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −28.0000 −1.23865
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −12.0000 −0.529297
$$515$$ 8.00000 0.352522
$$516$$ 0 0
$$517$$ −32.0000 −1.40736
$$518$$ −12.0000 −0.527250
$$519$$ 0 0
$$520$$ −2.00000 −0.0877058
$$521$$ 36.0000 1.57719 0.788594 0.614914i $$-0.210809\pi$$
0.788594 + 0.614914i $$0.210809\pi$$
$$522$$ 0 0
$$523$$ −12.0000 −0.524723 −0.262362 0.964970i $$-0.584501\pi$$
−0.262362 + 0.964970i $$0.584501\pi$$
$$524$$ −4.00000 −0.174741
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −7.00000 −0.304348
$$530$$ −24.0000 −1.04249
$$531$$ 0 0
$$532$$ 12.0000 0.520266
$$533$$ 6.00000 0.259889
$$534$$ 0 0
$$535$$ 40.0000 1.72935
$$536$$ −2.00000 −0.0863868
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −12.0000 −0.516877
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 22.0000 0.944981
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 20.0000 0.856706
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ 18.0000 0.768922
$$549$$ 0 0
$$550$$ −4.00000 −0.170561
$$551$$ −48.0000 −2.04487
$$552$$ 0 0
$$553$$ 8.00000 0.340195
$$554$$ −6.00000 −0.254916
$$555$$ 0 0
$$556$$ −20.0000 −0.848189
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ 0 0
$$559$$ 8.00000 0.338364
$$560$$ −4.00000 −0.169031
$$561$$ 0 0
$$562$$ 10.0000 0.421825
$$563$$ 20.0000 0.842900 0.421450 0.906852i $$-0.361521\pi$$
0.421450 + 0.906852i $$0.361521\pi$$
$$564$$ 0 0
$$565$$ −8.00000 −0.336563
$$566$$ −20.0000 −0.840663
$$567$$ 0 0
$$568$$ 16.0000 0.671345
$$569$$ 4.00000 0.167689 0.0838444 0.996479i $$-0.473280\pi$$
0.0838444 + 0.996479i $$0.473280\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ −4.00000 −0.167248
$$573$$ 0 0
$$574$$ 12.0000 0.500870
$$575$$ 4.00000 0.166812
$$576$$ 0 0
$$577$$ −30.0000 −1.24892 −0.624458 0.781058i $$-0.714680\pi$$
−0.624458 + 0.781058i $$0.714680\pi$$
$$578$$ −17.0000 −0.707107
$$579$$ 0 0
$$580$$ 16.0000 0.664364
$$581$$ −24.0000 −0.995688
$$582$$ 0 0
$$583$$ −48.0000 −1.98796
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ −2.00000 −0.0826192
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ −8.00000 −0.329355
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ 38.0000 1.56047 0.780236 0.625485i $$-0.215099\pi$$
0.780236 + 0.625485i $$0.215099\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ 4.00000 0.163572
$$599$$ 40.0000 1.63436 0.817178 0.576386i $$-0.195537\pi$$
0.817178 + 0.576386i $$0.195537\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 16.0000 0.652111
$$603$$ 0 0
$$604$$ 18.0000 0.732410
$$605$$ 10.0000 0.406558
$$606$$ 0 0
$$607$$ 24.0000 0.974130 0.487065 0.873366i $$-0.338067\pi$$
0.487065 + 0.873366i $$0.338067\pi$$
$$608$$ −6.00000 −0.243332
$$609$$ 0 0
$$610$$ 20.0000 0.809776
$$611$$ 8.00000 0.323645
$$612$$ 0 0
$$613$$ −34.0000 −1.37325 −0.686624 0.727013i $$-0.740908\pi$$
−0.686624 + 0.727013i $$0.740908\pi$$
$$614$$ 2.00000 0.0807134
$$615$$ 0 0
$$616$$ −8.00000 −0.322329
$$617$$ −14.0000 −0.563619 −0.281809 0.959470i $$-0.590935\pi$$
−0.281809 + 0.959470i $$0.590935\pi$$
$$618$$ 0 0
$$619$$ 34.0000 1.36658 0.683288 0.730149i $$-0.260549\pi$$
0.683288 + 0.730149i $$0.260549\pi$$
$$620$$ −4.00000 −0.160644
$$621$$ 0 0
$$622$$ 12.0000 0.481156
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ −19.0000 −0.760000
$$626$$ 26.0000 1.03917
$$627$$ 0 0
$$628$$ 2.00000 0.0798087
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −14.0000 −0.557331 −0.278666 0.960388i $$-0.589892\pi$$
−0.278666 + 0.960388i $$0.589892\pi$$
$$632$$ −4.00000 −0.159111
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ −16.0000 −0.634941
$$636$$ 0 0
$$637$$ 3.00000 0.118864
$$638$$ 32.0000 1.26689
$$639$$ 0 0
$$640$$ 2.00000 0.0790569
$$641$$ 40.0000 1.57991 0.789953 0.613168i $$-0.210105\pi$$
0.789953 + 0.613168i $$0.210105\pi$$
$$642$$ 0 0
$$643$$ −10.0000 −0.394362 −0.197181 0.980367i $$-0.563179\pi$$
−0.197181 + 0.980367i $$0.563179\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −8.00000 −0.314512 −0.157256 0.987558i $$-0.550265\pi$$
−0.157256 + 0.987558i $$0.550265\pi$$
$$648$$ 0 0
$$649$$ −16.0000 −0.628055
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ −10.0000 −0.391630
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ −8.00000 −0.312586
$$656$$ −6.00000 −0.234261
$$657$$ 0 0
$$658$$ 16.0000 0.623745
$$659$$ 48.0000 1.86981 0.934907 0.354892i $$-0.115482\pi$$
0.934907 + 0.354892i $$0.115482\pi$$
$$660$$ 0 0
$$661$$ 14.0000 0.544537 0.272268 0.962221i $$-0.412226\pi$$
0.272268 + 0.962221i $$0.412226\pi$$
$$662$$ −10.0000 −0.388661
$$663$$ 0 0
$$664$$ 12.0000 0.465690
$$665$$ 24.0000 0.930680
$$666$$ 0 0
$$667$$ −32.0000 −1.23904
$$668$$ 0 0
$$669$$ 0 0
$$670$$ −4.00000 −0.154533
$$671$$ 40.0000 1.54418
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ −34.0000 −1.30963
$$675$$ 0 0
$$676$$ 1.00000 0.0384615
$$677$$ 12.0000 0.461197 0.230599 0.973049i $$-0.425932\pi$$
0.230599 + 0.973049i $$0.425932\pi$$
$$678$$ 0 0
$$679$$ 20.0000 0.767530
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −8.00000 −0.306336
$$683$$ 4.00000 0.153056 0.0765279 0.997067i $$-0.475617\pi$$
0.0765279 + 0.997067i $$0.475617\pi$$
$$684$$ 0 0
$$685$$ 36.0000 1.37549
$$686$$ 20.0000 0.763604
$$687$$ 0 0
$$688$$ −8.00000 −0.304997
$$689$$ 12.0000 0.457164
$$690$$ 0 0
$$691$$ −26.0000 −0.989087 −0.494543 0.869153i $$-0.664665\pi$$
−0.494543 + 0.869153i $$0.664665\pi$$
$$692$$ −16.0000 −0.608229
$$693$$ 0 0
$$694$$ 24.0000 0.911028
$$695$$ −40.0000 −1.51729
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 22.0000 0.832712
$$699$$ 0 0
$$700$$ 2.00000 0.0755929
$$701$$ 12.0000 0.453234 0.226617 0.973984i $$-0.427233\pi$$
0.226617 + 0.973984i $$0.427233\pi$$
$$702$$ 0 0
$$703$$ −36.0000 −1.35777
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ −14.0000 −0.526897
$$707$$ −32.0000 −1.20348
$$708$$ 0 0
$$709$$ −22.0000 −0.826227 −0.413114 0.910679i $$-0.635559\pi$$
−0.413114 + 0.910679i $$0.635559\pi$$
$$710$$ 32.0000 1.20094
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ −8.00000 −0.299183
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −28.0000 −1.04422 −0.522112 0.852877i $$-0.674856\pi$$
−0.522112 + 0.852877i $$0.674856\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 17.0000 0.632674
$$723$$ 0 0
$$724$$ −18.0000 −0.668965
$$725$$ −8.00000 −0.297113
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 2.00000 0.0741249
$$729$$ 0 0
$$730$$ 28.0000 1.03633
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −22.0000 −0.812589 −0.406294 0.913742i $$-0.633179\pi$$
−0.406294 + 0.913742i $$0.633179\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ −4.00000 −0.147442
$$737$$ −8.00000 −0.294684
$$738$$ 0 0
$$739$$ −54.0000 −1.98642 −0.993211 0.116326i $$-0.962888\pi$$
−0.993211 + 0.116326i $$0.962888\pi$$
$$740$$ 12.0000 0.441129
$$741$$ 0 0
$$742$$ 24.0000 0.881068
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ 12.0000 0.439646
$$746$$ −10.0000 −0.366126
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −40.0000 −1.46157
$$750$$ 0 0
$$751$$ −12.0000 −0.437886 −0.218943 0.975738i $$-0.570261\pi$$
−0.218943 + 0.975738i $$0.570261\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ 0 0
$$754$$ −8.00000 −0.291343
$$755$$ 36.0000 1.31017
$$756$$ 0 0
$$757$$ −18.0000 −0.654221 −0.327111 0.944986i $$-0.606075\pi$$
−0.327111 + 0.944986i $$0.606075\pi$$
$$758$$ 10.0000 0.363216
$$759$$ 0 0
$$760$$ −12.0000 −0.435286
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ −20.0000 −0.724049
$$764$$ 4.00000 0.144715
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.00000 0.144432
$$768$$ 0 0
$$769$$ −30.0000 −1.08183 −0.540914 0.841078i $$-0.681921\pi$$
−0.540914 + 0.841078i $$0.681921\pi$$
$$770$$ −16.0000 −0.576600
$$771$$ 0 0
$$772$$ 2.00000 0.0719816
$$773$$ 14.0000 0.503545 0.251773 0.967786i $$-0.418987\pi$$
0.251773 + 0.967786i $$0.418987\pi$$
$$774$$ 0 0
$$775$$ 2.00000 0.0718421
$$776$$ −10.0000 −0.358979
$$777$$ 0 0
$$778$$ 12.0000 0.430221
$$779$$ 36.0000 1.28983
$$780$$ 0 0
$$781$$ 64.0000 2.29010
$$782$$ 0 0
$$783$$ 0 0
$$784$$ −3.00000 −0.107143
$$785$$ 4.00000 0.142766
$$786$$ 0 0
$$787$$ −22.0000 −0.784215 −0.392108 0.919919i $$-0.628254\pi$$
−0.392108 + 0.919919i $$0.628254\pi$$
$$788$$ 18.0000 0.641223
$$789$$ 0 0
$$790$$ −8.00000 −0.284627
$$791$$ 8.00000 0.284447
$$792$$ 0 0
$$793$$ −10.0000 −0.355110
$$794$$ −10.0000 −0.354887
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ 12.0000 0.425062 0.212531 0.977154i $$-0.431829\pi$$
0.212531 + 0.977154i $$0.431829\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −1.00000 −0.0353553
$$801$$ 0 0
$$802$$ 6.00000 0.211867
$$803$$ 56.0000 1.97620
$$804$$ 0 0
$$805$$ 16.0000 0.563926
$$806$$ 2.00000 0.0704470
$$807$$ 0 0
$$808$$ 16.0000 0.562878
$$809$$ −36.0000 −1.26569 −0.632846 0.774277i $$-0.718114\pi$$
−0.632846 + 0.774277i $$0.718114\pi$$
$$810$$ 0 0
$$811$$ 50.0000 1.75574 0.877869 0.478901i $$-0.158965\pi$$
0.877869 + 0.478901i $$0.158965\pi$$
$$812$$ −16.0000 −0.561490
$$813$$ 0 0
$$814$$ 24.0000 0.841200
$$815$$ −20.0000 −0.700569
$$816$$ 0 0
$$817$$ 48.0000 1.67931
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ −12.0000 −0.419058
$$821$$ −34.0000 −1.18661 −0.593304 0.804978i $$-0.702177\pi$$
−0.593304 + 0.804978i $$0.702177\pi$$
$$822$$ 0 0
$$823$$ −12.0000 −0.418294 −0.209147 0.977884i $$-0.567069\pi$$
−0.209147 + 0.977884i $$0.567069\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 8.00000 0.278356
$$827$$ 12.0000 0.417281 0.208640 0.977992i $$-0.433096\pi$$
0.208640 + 0.977992i $$0.433096\pi$$
$$828$$ 0 0
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 24.0000 0.833052
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −24.0000 −0.830057
$$837$$ 0 0
$$838$$ −36.0000 −1.24360
$$839$$ −16.0000 −0.552381 −0.276191 0.961103i $$-0.589072\pi$$
−0.276191 + 0.961103i $$0.589072\pi$$
$$840$$ 0 0
$$841$$ 35.0000 1.20690
$$842$$ 10.0000 0.344623
$$843$$ 0 0
$$844$$ 8.00000 0.275371
$$845$$ 2.00000 0.0688021
$$846$$ 0 0
$$847$$ −10.0000 −0.343604
$$848$$ −12.0000 −0.412082
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −24.0000 −0.822709
$$852$$ 0 0
$$853$$ −26.0000 −0.890223 −0.445112 0.895475i $$-0.646836\pi$$
−0.445112 + 0.895475i $$0.646836\pi$$
$$854$$ −20.0000 −0.684386
$$855$$ 0 0
$$856$$ 20.0000 0.683586
$$857$$ 28.0000 0.956462 0.478231 0.878234i $$-0.341278\pi$$
0.478231 + 0.878234i $$0.341278\pi$$
$$858$$ 0 0
$$859$$ 8.00000 0.272956 0.136478 0.990643i $$-0.456422\pi$$
0.136478 + 0.990643i $$0.456422\pi$$
$$860$$ −16.0000 −0.545595
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −8.00000 −0.272323 −0.136162 0.990687i $$-0.543477\pi$$
−0.136162 + 0.990687i $$0.543477\pi$$
$$864$$ 0 0
$$865$$ −32.0000 −1.08803
$$866$$ −2.00000 −0.0679628
$$867$$ 0 0
$$868$$ 4.00000 0.135769
$$869$$ −16.0000 −0.542763
$$870$$ 0 0
$$871$$ 2.00000 0.0677674
$$872$$ 10.0000 0.338643
$$873$$ 0 0
$$874$$ 24.0000 0.811812
$$875$$ 24.0000 0.811348
$$876$$ 0 0
$$877$$ 6.00000 0.202606 0.101303 0.994856i $$-0.467699\pi$$
0.101303 + 0.994856i $$0.467699\pi$$
$$878$$ −4.00000 −0.134993
$$879$$ 0 0
$$880$$ 8.00000 0.269680
$$881$$ −36.0000 −1.21287 −0.606435 0.795133i $$-0.707401\pi$$
−0.606435 + 0.795133i $$0.707401\pi$$
$$882$$ 0 0
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ −36.0000 −1.20944
$$887$$ −44.0000 −1.47738 −0.738688 0.674048i $$-0.764554\pi$$
−0.738688 + 0.674048i $$0.764554\pi$$
$$888$$ 0 0
$$889$$ 16.0000 0.536623
$$890$$ 12.0000 0.402241
$$891$$ 0 0
$$892$$ 6.00000 0.200895
$$893$$ 48.0000 1.60626
$$894$$ 0 0
$$895$$ −24.0000 −0.802232
$$896$$ −2.00000 −0.0668153
$$897$$ 0 0
$$898$$ −26.0000 −0.867631
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ 0 0
$$902$$ −24.0000 −0.799113
$$903$$ 0 0
$$904$$ −4.00000 −0.133038
$$905$$ −36.0000 −1.19668
$$906$$ 0 0
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 4.00000 0.132599
$$911$$ −12.0000 −0.397578 −0.198789 0.980042i $$-0.563701\pi$$
−0.198789 + 0.980042i $$0.563701\pi$$
$$912$$ 0 0
$$913$$ 48.0000 1.58857
$$914$$ −2.00000 −0.0661541
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ 8.00000 0.264183
$$918$$ 0 0
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ −8.00000 −0.263752
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ −16.0000 −0.526646
$$924$$ 0 0
$$925$$ −6.00000 −0.197279
$$926$$ 22.0000 0.722965
$$927$$ 0 0
$$928$$ 8.00000 0.262613
$$929$$ −30.0000 −0.984268 −0.492134 0.870519i $$-0.663783\pi$$
−0.492134 + 0.870519i $$0.663783\pi$$
$$930$$ 0 0
$$931$$ 18.0000 0.589926
$$932$$ −20.0000 −0.655122
$$933$$ 0 0
$$934$$ 8.00000 0.261768
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 30.0000 0.980057 0.490029 0.871706i $$-0.336986\pi$$
0.490029 + 0.871706i $$0.336986\pi$$
$$938$$ 4.00000 0.130605
$$939$$ 0 0
$$940$$ −16.0000 −0.521862
$$941$$ −26.0000 −0.847576 −0.423788 0.905761i $$-0.639300\pi$$
−0.423788 + 0.905761i $$0.639300\pi$$
$$942$$ 0 0
$$943$$ 24.0000 0.781548
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −32.0000 −1.04041
$$947$$ −36.0000 −1.16984 −0.584921 0.811090i $$-0.698875\pi$$
−0.584921 + 0.811090i $$0.698875\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 6.00000 0.194666
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −28.0000 −0.907009 −0.453504 0.891254i $$-0.649826\pi$$
−0.453504 + 0.891254i $$0.649826\pi$$
$$954$$ 0 0
$$955$$ 8.00000 0.258874
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −36.0000 −1.16250
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ −6.00000 −0.193448
$$963$$ 0 0
$$964$$ 6.00000 0.193247
$$965$$ 4.00000 0.128765
$$966$$ 0 0
$$967$$ −58.0000 −1.86515 −0.932577 0.360971i $$-0.882445\pi$$
−0.932577 + 0.360971i $$0.882445\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 0 0
$$970$$ −20.0000 −0.642161
$$971$$ −48.0000 −1.54039 −0.770197 0.637806i $$-0.779842\pi$$
−0.770197 + 0.637806i $$0.779842\pi$$
$$972$$ 0 0
$$973$$ 40.0000 1.28234
$$974$$ 38.0000 1.21760
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 0 0
$$979$$ 24.0000 0.767043
$$980$$ −6.00000 −0.191663
$$981$$ 0 0
$$982$$ 16.0000 0.510581
$$983$$ 32.0000 1.02064 0.510321 0.859984i $$-0.329527\pi$$
0.510321 + 0.859984i $$0.329527\pi$$
$$984$$ 0 0
$$985$$ 36.0000 1.14706
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 6.00000 0.190885
$$989$$ 32.0000 1.01754
$$990$$ 0 0
$$991$$ −48.0000 −1.52477 −0.762385 0.647124i $$-0.775972\pi$$
−0.762385 + 0.647124i $$0.775972\pi$$
$$992$$ −2.00000 −0.0635001
$$993$$ 0 0
$$994$$ −32.0000 −1.01498
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ 38.0000 1.20347 0.601736 0.798695i $$-0.294476\pi$$
0.601736 + 0.798695i $$0.294476\pi$$
$$998$$ 22.0000 0.696398
$$999$$ 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 234.2.a.d.1.1 yes 1
3.2 odd 2 234.2.a.a.1.1 1
4.3 odd 2 1872.2.a.p.1.1 1
5.2 odd 4 5850.2.e.bd.5149.2 2
5.3 odd 4 5850.2.e.bd.5149.1 2
5.4 even 2 5850.2.a.v.1.1 1
8.3 odd 2 7488.2.a.s.1.1 1
8.5 even 2 7488.2.a.j.1.1 1
9.2 odd 6 2106.2.e.z.1405.1 2
9.4 even 3 2106.2.e.e.703.1 2
9.5 odd 6 2106.2.e.z.703.1 2
9.7 even 3 2106.2.e.e.1405.1 2
12.11 even 2 1872.2.a.g.1.1 1
13.5 odd 4 3042.2.b.b.1351.1 2
13.8 odd 4 3042.2.b.b.1351.2 2
13.12 even 2 3042.2.a.b.1.1 1
15.2 even 4 5850.2.e.d.5149.1 2
15.8 even 4 5850.2.e.d.5149.2 2
15.14 odd 2 5850.2.a.bv.1.1 1
24.5 odd 2 7488.2.a.bp.1.1 1
24.11 even 2 7488.2.a.bu.1.1 1
39.5 even 4 3042.2.b.c.1351.2 2
39.8 even 4 3042.2.b.c.1351.1 2
39.38 odd 2 3042.2.a.o.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
234.2.a.a.1.1 1 3.2 odd 2
234.2.a.d.1.1 yes 1 1.1 even 1 trivial
1872.2.a.g.1.1 1 12.11 even 2
1872.2.a.p.1.1 1 4.3 odd 2
2106.2.e.e.703.1 2 9.4 even 3
2106.2.e.e.1405.1 2 9.7 even 3
2106.2.e.z.703.1 2 9.5 odd 6
2106.2.e.z.1405.1 2 9.2 odd 6
3042.2.a.b.1.1 1 13.12 even 2
3042.2.a.o.1.1 1 39.38 odd 2
3042.2.b.b.1351.1 2 13.5 odd 4
3042.2.b.b.1351.2 2 13.8 odd 4
3042.2.b.c.1351.1 2 39.8 even 4
3042.2.b.c.1351.2 2 39.5 even 4
5850.2.a.v.1.1 1 5.4 even 2
5850.2.a.bv.1.1 1 15.14 odd 2
5850.2.e.d.5149.1 2 15.2 even 4
5850.2.e.d.5149.2 2 15.8 even 4
5850.2.e.bd.5149.1 2 5.3 odd 4
5850.2.e.bd.5149.2 2 5.2 odd 4
7488.2.a.j.1.1 1 8.5 even 2
7488.2.a.s.1.1 1 8.3 odd 2
7488.2.a.bp.1.1 1 24.5 odd 2
7488.2.a.bu.1.1 1 24.11 even 2