# Properties

 Label 2704.2.f.b.337.2 Level $2704$ Weight $2$ Character 2704.337 Analytic conductor $21.592$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(337,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("2704.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$21.5915487066$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 337.2 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2704.337 Dual form 2704.2.f.b.337.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} -3.46410i q^{15} -3.00000 q^{17} -3.46410i q^{19} +6.00000 q^{23} +2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} +3.46410i q^{31} +8.66025i q^{37} -5.19615i q^{41} -8.00000 q^{43} +1.73205i q^{45} -3.46410i q^{47} +7.00000 q^{49} +6.00000 q^{51} -3.00000 q^{53} +6.92820i q^{57} +6.92820i q^{59} +1.00000 q^{61} -3.46410i q^{67} -12.0000 q^{69} +3.46410i q^{71} -1.73205i q^{73} -4.00000 q^{75} -4.00000 q^{79} -11.0000 q^{81} +13.8564i q^{83} -5.19615i q^{85} -6.00000 q^{87} -6.92820i q^{89} -6.92820i q^{93} +6.00000 q^{95} -6.92820i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 + 2 * q^9 $$2 q - 4 q^{3} + 2 q^{9} - 6 q^{17} + 12 q^{23} + 4 q^{25} + 8 q^{27} + 6 q^{29} - 16 q^{43} + 14 q^{49} + 12 q^{51} - 6 q^{53} + 2 q^{61} - 24 q^{69} - 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} + 12 q^{95}+O(q^{100})$$ 2 * q - 4 * q^3 + 2 * q^9 - 6 * q^17 + 12 * q^23 + 4 * q^25 + 8 * q^27 + 6 * q^29 - 16 * q^43 + 14 * q^49 + 12 * q^51 - 6 * q^53 + 2 * q^61 - 24 * q^69 - 8 * q^75 - 8 * q^79 - 22 * q^81 - 12 * q^87 + 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$1185$$ $$2367$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ −2.00000 −1.15470 −0.577350 0.816497i $$-0.695913\pi$$
−0.577350 + 0.816497i $$0.695913\pi$$
$$4$$ 0 0
$$5$$ 1.73205i 0.774597i 0.921954 + 0.387298i $$0.126592\pi$$
−0.921954 + 0.387298i $$0.873408\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$12$$ 0 0
$$13$$ 0 0
$$14$$ 0 0
$$15$$ − 3.46410i − 0.894427i
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ − 3.46410i − 0.794719i −0.917663 0.397360i $$-0.869927\pi$$
0.917663 0.397360i $$-0.130073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 2.00000 0.400000
$$26$$ 0 0
$$27$$ 4.00000 0.769800
$$28$$ 0 0
$$29$$ 3.00000 0.557086 0.278543 0.960424i $$-0.410149\pi$$
0.278543 + 0.960424i $$0.410149\pi$$
$$30$$ 0 0
$$31$$ 3.46410i 0.622171i 0.950382 + 0.311086i $$0.100693\pi$$
−0.950382 + 0.311086i $$0.899307\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 8.66025i 1.42374i 0.702313 + 0.711868i $$0.252151\pi$$
−0.702313 + 0.711868i $$0.747849\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 5.19615i − 0.811503i −0.913984 0.405751i $$-0.867010\pi$$
0.913984 0.405751i $$-0.132990\pi$$
$$42$$ 0 0
$$43$$ −8.00000 −1.21999 −0.609994 0.792406i $$-0.708828\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 1.73205i 0.258199i
$$46$$ 0 0
$$47$$ − 3.46410i − 0.505291i −0.967559 0.252646i $$-0.918699\pi$$
0.967559 0.252646i $$-0.0813007\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 6.00000 0.840168
$$52$$ 0 0
$$53$$ −3.00000 −0.412082 −0.206041 0.978543i $$-0.566058\pi$$
−0.206041 + 0.978543i $$0.566058\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 6.92820i 0.917663i
$$58$$ 0 0
$$59$$ 6.92820i 0.901975i 0.892530 + 0.450988i $$0.148928\pi$$
−0.892530 + 0.450988i $$0.851072\pi$$
$$60$$ 0 0
$$61$$ 1.00000 0.128037 0.0640184 0.997949i $$-0.479608\pi$$
0.0640184 + 0.997949i $$0.479608\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 3.46410i − 0.423207i −0.977356 0.211604i $$-0.932131\pi$$
0.977356 0.211604i $$-0.0678686\pi$$
$$68$$ 0 0
$$69$$ −12.0000 −1.44463
$$70$$ 0 0
$$71$$ 3.46410i 0.411113i 0.978645 + 0.205557i $$0.0659005\pi$$
−0.978645 + 0.205557i $$0.934100\pi$$
$$72$$ 0 0
$$73$$ − 1.73205i − 0.202721i −0.994850 0.101361i $$-0.967680\pi$$
0.994850 0.101361i $$-0.0323196\pi$$
$$74$$ 0 0
$$75$$ −4.00000 −0.461880
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 −0.450035 −0.225018 0.974355i $$-0.572244\pi$$
−0.225018 + 0.974355i $$0.572244\pi$$
$$80$$ 0 0
$$81$$ −11.0000 −1.22222
$$82$$ 0 0
$$83$$ 13.8564i 1.52094i 0.649374 + 0.760469i $$0.275031\pi$$
−0.649374 + 0.760469i $$0.724969\pi$$
$$84$$ 0 0
$$85$$ − 5.19615i − 0.563602i
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ − 6.92820i − 0.734388i −0.930144 0.367194i $$-0.880318\pi$$
0.930144 0.367194i $$-0.119682\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ − 6.92820i − 0.718421i
$$94$$ 0 0
$$95$$ 6.00000 0.615587
$$96$$ 0 0
$$97$$ − 6.92820i − 0.703452i −0.936103 0.351726i $$-0.885595\pi$$
0.936103 0.351726i $$-0.114405\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −3.00000 −0.298511 −0.149256 0.988799i $$-0.547688\pi$$
−0.149256 + 0.988799i $$0.547688\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.985329 0.492665 0.870219i $$-0.336023\pi$$
0.492665 + 0.870219i $$0.336023\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −6.00000 −0.580042 −0.290021 0.957020i $$-0.593662\pi$$
−0.290021 + 0.957020i $$0.593662\pi$$
$$108$$ 0 0
$$109$$ 13.8564i 1.32720i 0.748086 + 0.663602i $$0.230973\pi$$
−0.748086 + 0.663602i $$0.769027\pi$$
$$110$$ 0 0
$$111$$ − 17.3205i − 1.64399i
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ 10.3923i 0.969087i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000 1.00000
$$122$$ 0 0
$$123$$ 10.3923i 0.937043i
$$124$$ 0 0
$$125$$ 12.1244i 1.08444i
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 16.0000 1.40872
$$130$$ 0 0
$$131$$ −18.0000 −1.57267 −0.786334 0.617802i $$-0.788023\pi$$
−0.786334 + 0.617802i $$0.788023\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 6.92820i 0.596285i
$$136$$ 0 0
$$137$$ 15.5885i 1.33181i 0.746036 + 0.665906i $$0.231955\pi$$
−0.746036 + 0.665906i $$0.768045\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 6.92820i 0.583460i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.19615i 0.431517i
$$146$$ 0 0
$$147$$ −14.0000 −1.15470
$$148$$ 0 0
$$149$$ 19.0526i 1.56085i 0.625252 + 0.780423i $$0.284996\pi$$
−0.625252 + 0.780423i $$0.715004\pi$$
$$150$$ 0 0
$$151$$ 17.3205i 1.40952i 0.709444 + 0.704761i $$0.248946\pi$$
−0.709444 + 0.704761i $$0.751054\pi$$
$$152$$ 0 0
$$153$$ −3.00000 −0.242536
$$154$$ 0 0
$$155$$ −6.00000 −0.481932
$$156$$ 0 0
$$157$$ −13.0000 −1.03751 −0.518756 0.854922i $$-0.673605\pi$$
−0.518756 + 0.854922i $$0.673605\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.7846i 1.62798i 0.580881 + 0.813988i $$0.302708\pi$$
−0.580881 + 0.813988i $$0.697292\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 13.8564i 1.07224i 0.844141 + 0.536120i $$0.180111\pi$$
−0.844141 + 0.536120i $$0.819889\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ − 3.46410i − 0.264906i
$$172$$ 0 0
$$173$$ 6.00000 0.456172 0.228086 0.973641i $$-0.426753\pi$$
0.228086 + 0.973641i $$0.426753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 13.8564i − 1.04151i
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 11.0000 0.817624 0.408812 0.912619i $$-0.365943\pi$$
0.408812 + 0.912619i $$0.365943\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ −15.0000 −1.10282
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −18.0000 −1.30243 −0.651217 0.758891i $$-0.725741\pi$$
−0.651217 + 0.758891i $$0.725741\pi$$
$$192$$ 0 0
$$193$$ − 5.19615i − 0.374027i −0.982357 0.187014i $$-0.940119\pi$$
0.982357 0.187014i $$-0.0598809\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 13.8564i 0.987228i 0.869681 + 0.493614i $$0.164324\pi$$
−0.869681 + 0.493614i $$0.835676\pi$$
$$198$$ 0 0
$$199$$ 2.00000 0.141776 0.0708881 0.997484i $$-0.477417\pi$$
0.0708881 + 0.997484i $$0.477417\pi$$
$$200$$ 0 0
$$201$$ 6.92820i 0.488678i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 9.00000 0.628587
$$206$$ 0 0
$$207$$ 6.00000 0.417029
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −10.0000 −0.688428 −0.344214 0.938891i $$-0.611855\pi$$
−0.344214 + 0.938891i $$0.611855\pi$$
$$212$$ 0 0
$$213$$ − 6.92820i − 0.474713i
$$214$$ 0 0
$$215$$ − 13.8564i − 0.944999i
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 3.46410i 0.234082i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 10.3923i 0.695920i 0.937509 + 0.347960i $$0.113126\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ 2.00000 0.133333
$$226$$ 0 0
$$227$$ 24.2487i 1.60944i 0.593652 + 0.804722i $$0.297686\pi$$
−0.593652 + 0.804722i $$0.702314\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 6.00000 0.391397
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ − 20.7846i − 1.34444i −0.740349 0.672222i $$-0.765340\pi$$
0.740349 0.672222i $$-0.234660\pi$$
$$240$$ 0 0
$$241$$ 1.73205i 0.111571i 0.998443 + 0.0557856i $$0.0177663\pi$$
−0.998443 + 0.0557856i $$0.982234\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ 12.1244i 0.774597i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ − 27.7128i − 1.75623i
$$250$$ 0 0
$$251$$ 18.0000 1.13615 0.568075 0.822977i $$-0.307688\pi$$
0.568075 + 0.822977i $$0.307688\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 10.3923i 0.650791i
$$256$$ 0 0
$$257$$ 3.00000 0.187135 0.0935674 0.995613i $$-0.470173\pi$$
0.0935674 + 0.995613i $$0.470173\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 3.00000 0.185695
$$262$$ 0 0
$$263$$ −12.0000 −0.739952 −0.369976 0.929041i $$-0.620634\pi$$
−0.369976 + 0.929041i $$0.620634\pi$$
$$264$$ 0 0
$$265$$ − 5.19615i − 0.319197i
$$266$$ 0 0
$$267$$ 13.8564i 0.847998i
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ − 20.7846i − 1.26258i −0.775549 0.631288i $$-0.782527\pi$$
0.775549 0.631288i $$-0.217473\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −7.00000 −0.420589 −0.210295 0.977638i $$-0.567442\pi$$
−0.210295 + 0.977638i $$0.567442\pi$$
$$278$$ 0 0
$$279$$ 3.46410i 0.207390i
$$280$$ 0 0
$$281$$ 22.5167i 1.34323i 0.740900 + 0.671616i $$0.234399\pi$$
−0.740900 + 0.671616i $$0.765601\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ −12.0000 −0.710819
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 13.8564i 0.812277i
$$292$$ 0 0
$$293$$ − 5.19615i − 0.303562i −0.988414 0.151781i $$-0.951499\pi$$
0.988414 0.151781i $$-0.0485009\pi$$
$$294$$ 0 0
$$295$$ −12.0000 −0.698667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 6.00000 0.344691
$$304$$ 0 0
$$305$$ 1.73205i 0.0991769i
$$306$$ 0 0
$$307$$ − 17.3205i − 0.988534i −0.869310 0.494267i $$-0.835437\pi$$
0.869310 0.494267i $$-0.164563\pi$$
$$308$$ 0 0
$$309$$ −20.0000 −1.13776
$$310$$ 0 0
$$311$$ 30.0000 1.70114 0.850572 0.525859i $$-0.176256\pi$$
0.850572 + 0.525859i $$0.176256\pi$$
$$312$$ 0 0
$$313$$ 10.0000 0.565233 0.282617 0.959233i $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.19615i 0.291845i 0.989296 + 0.145922i $$0.0466150\pi$$
−0.989296 + 0.145922i $$0.953385\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ 10.3923i 0.578243i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 27.7128i − 1.53252i
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ − 27.7128i − 1.52323i −0.648027 0.761617i $$-0.724406\pi$$
0.648027 0.761617i $$-0.275594\pi$$
$$332$$ 0 0
$$333$$ 8.66025i 0.474579i
$$334$$ 0 0
$$335$$ 6.00000 0.327815
$$336$$ 0 0
$$337$$ −23.0000 −1.25289 −0.626445 0.779466i $$-0.715491\pi$$
−0.626445 + 0.779466i $$0.715491\pi$$
$$338$$ 0 0
$$339$$ 30.0000 1.62938
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ − 20.7846i − 1.11901i
$$346$$ 0 0
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ 0 0
$$349$$ 13.8564i 0.741716i 0.928689 + 0.370858i $$0.120936\pi$$
−0.928689 + 0.370858i $$0.879064\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 32.9090i 1.75157i 0.482704 + 0.875784i $$0.339655\pi$$
−0.482704 + 0.875784i $$0.660345\pi$$
$$354$$ 0 0
$$355$$ −6.00000 −0.318447
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ − 6.92820i − 0.365657i −0.983145 0.182828i $$-0.941475\pi$$
0.983145 0.182828i $$-0.0585252\pi$$
$$360$$ 0 0
$$361$$ 7.00000 0.368421
$$362$$ 0 0
$$363$$ −22.0000 −1.15470
$$364$$ 0 0
$$365$$ 3.00000 0.157027
$$366$$ 0 0
$$367$$ 22.0000 1.14839 0.574195 0.818718i $$-0.305315\pi$$
0.574195 + 0.818718i $$0.305315\pi$$
$$368$$ 0 0
$$369$$ − 5.19615i − 0.270501i
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 19.0000 0.983783 0.491891 0.870657i $$-0.336306\pi$$
0.491891 + 0.870657i $$0.336306\pi$$
$$374$$ 0 0
$$375$$ − 24.2487i − 1.25220i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 24.2487i 1.24557i 0.782392 + 0.622786i $$0.213999\pi$$
−0.782392 + 0.622786i $$0.786001\pi$$
$$380$$ 0 0
$$381$$ −4.00000 −0.204926
$$382$$ 0 0
$$383$$ − 20.7846i − 1.06204i −0.847358 0.531022i $$-0.821808\pi$$
0.847358 0.531022i $$-0.178192\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −8.00000 −0.406663
$$388$$ 0 0
$$389$$ −9.00000 −0.456318 −0.228159 0.973624i $$-0.573271\pi$$
−0.228159 + 0.973624i $$0.573271\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ 0 0
$$395$$ − 6.92820i − 0.348596i
$$396$$ 0 0
$$397$$ 13.8564i 0.695433i 0.937600 + 0.347717i $$0.113043\pi$$
−0.937600 + 0.347717i $$0.886957\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ − 1.73205i − 0.0864945i −0.999064 0.0432472i $$-0.986230\pi$$
0.999064 0.0432472i $$-0.0137703\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ − 19.0526i − 0.946729i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ − 15.5885i − 0.770800i −0.922750 0.385400i $$-0.874064\pi$$
0.922750 0.385400i $$-0.125936\pi$$
$$410$$ 0 0
$$411$$ − 31.1769i − 1.53784i
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −24.0000 −1.17811
$$416$$ 0 0
$$417$$ −8.00000 −0.391762
$$418$$ 0 0
$$419$$ −18.0000 −0.879358 −0.439679 0.898155i $$-0.644908\pi$$
−0.439679 + 0.898155i $$0.644908\pi$$
$$420$$ 0 0
$$421$$ − 15.5885i − 0.759735i −0.925041 0.379867i $$-0.875970\pi$$
0.925041 0.379867i $$-0.124030\pi$$
$$422$$ 0 0
$$423$$ − 3.46410i − 0.168430i
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.92820i 0.333720i 0.985981 + 0.166860i $$0.0533628\pi$$
−0.985981 + 0.166860i $$0.946637\pi$$
$$432$$ 0 0
$$433$$ −17.0000 −0.816968 −0.408484 0.912766i $$-0.633942\pi$$
−0.408484 + 0.912766i $$0.633942\pi$$
$$434$$ 0 0
$$435$$ − 10.3923i − 0.498273i
$$436$$ 0 0
$$437$$ − 20.7846i − 0.994263i
$$438$$ 0 0
$$439$$ 28.0000 1.33637 0.668184 0.743996i $$-0.267072\pi$$
0.668184 + 0.743996i $$0.267072\pi$$
$$440$$ 0 0
$$441$$ 7.00000 0.333333
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ 12.0000 0.568855
$$446$$ 0 0
$$447$$ − 38.1051i − 1.80231i
$$448$$ 0 0
$$449$$ 6.92820i 0.326962i 0.986546 + 0.163481i $$0.0522723\pi$$
−0.986546 + 0.163481i $$0.947728\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ − 34.6410i − 1.62758i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.73205i 0.0810219i 0.999179 + 0.0405110i $$0.0128986\pi$$
−0.999179 + 0.0405110i $$0.987101\pi$$
$$458$$ 0 0
$$459$$ −12.0000 −0.560112
$$460$$ 0 0
$$461$$ − 22.5167i − 1.04871i −0.851501 0.524353i $$-0.824307\pi$$
0.851501 0.524353i $$-0.175693\pi$$
$$462$$ 0 0
$$463$$ 13.8564i 0.643962i 0.946746 + 0.321981i $$0.104349\pi$$
−0.946746 + 0.321981i $$0.895651\pi$$
$$464$$ 0 0
$$465$$ 12.0000 0.556487
$$466$$ 0 0
$$467$$ −12.0000 −0.555294 −0.277647 0.960683i $$-0.589555\pi$$
−0.277647 + 0.960683i $$0.589555\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 26.0000 1.19802
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 6.92820i − 0.317888i
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ 0 0
$$479$$ − 24.2487i − 1.10795i −0.832533 0.553976i $$-0.813110\pi$$
0.832533 0.553976i $$-0.186890\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 12.0000 0.544892
$$486$$ 0 0
$$487$$ − 6.92820i − 0.313947i −0.987603 0.156973i $$-0.949826\pi$$
0.987603 0.156973i $$-0.0501737\pi$$
$$488$$ 0 0
$$489$$ − 41.5692i − 1.87983i
$$490$$ 0 0
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ 0 0
$$493$$ −9.00000 −0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 31.1769i 1.39567i 0.716258 + 0.697835i $$0.245853\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ − 27.7128i − 1.23812i
$$502$$ 0 0
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ − 5.19615i − 0.231226i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 19.0526i 0.844490i 0.906482 + 0.422245i $$0.138758\pi$$
−0.906482 + 0.422245i $$0.861242\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ − 13.8564i − 0.611775i
$$514$$ 0 0
$$515$$ 17.3205i 0.763233i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −12.0000 −0.526742
$$520$$ 0 0
$$521$$ 9.00000 0.394297 0.197149 0.980374i $$-0.436832\pi$$
0.197149 + 0.980374i $$0.436832\pi$$
$$522$$ 0 0
$$523$$ 16.0000 0.699631 0.349816 0.936819i $$-0.386244\pi$$
0.349816 + 0.936819i $$0.386244\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 10.3923i − 0.452696i
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 6.92820i 0.300658i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 10.3923i − 0.449299i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 29.4449i − 1.26593i −0.774179 0.632967i $$-0.781837\pi$$
0.774179 0.632967i $$-0.218163\pi$$
$$542$$ 0 0
$$543$$ −22.0000 −0.944110
$$544$$ 0 0
$$545$$ −24.0000 −1.02805
$$546$$ 0 0
$$547$$ 22.0000 0.940652 0.470326 0.882493i $$-0.344136\pi$$
0.470326 + 0.882493i $$0.344136\pi$$
$$548$$ 0 0
$$549$$ 1.00000 0.0426790
$$550$$ 0 0
$$551$$ − 10.3923i − 0.442727i
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 30.0000 1.27343
$$556$$ 0 0
$$557$$ − 15.5885i − 0.660504i −0.943893 0.330252i $$-0.892866\pi$$
0.943893 0.330252i $$-0.107134\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ − 25.9808i − 1.09302i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −42.0000 −1.76073 −0.880366 0.474295i $$-0.842703\pi$$
−0.880366 + 0.474295i $$0.842703\pi$$
$$570$$ 0 0
$$571$$ −40.0000 −1.67395 −0.836974 0.547243i $$-0.815677\pi$$
−0.836974 + 0.547243i $$0.815677\pi$$
$$572$$ 0 0
$$573$$ 36.0000 1.50392
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ 19.0526i 0.793168i 0.917998 + 0.396584i $$0.129805\pi$$
−0.917998 + 0.396584i $$0.870195\pi$$
$$578$$ 0 0
$$579$$ 10.3923i 0.431889i
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 20.7846i 0.857873i 0.903335 + 0.428936i $$0.141112\pi$$
−0.903335 + 0.428936i $$0.858888\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ − 27.7128i − 1.13995i
$$592$$ 0 0
$$593$$ 25.9808i 1.06690i 0.845831 + 0.533451i $$0.179105\pi$$
−0.845831 + 0.533451i $$0.820895\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −4.00000 −0.163709
$$598$$ 0 0
$$599$$ −30.0000 −1.22577 −0.612883 0.790173i $$-0.709990\pi$$
−0.612883 + 0.790173i $$0.709990\pi$$
$$600$$ 0 0
$$601$$ 25.0000 1.01977 0.509886 0.860242i $$-0.329688\pi$$
0.509886 + 0.860242i $$0.329688\pi$$
$$602$$ 0 0
$$603$$ − 3.46410i − 0.141069i
$$604$$ 0 0
$$605$$ 19.0526i 0.774597i
$$606$$ 0 0
$$607$$ 34.0000 1.38002 0.690009 0.723801i $$-0.257607\pi$$
0.690009 + 0.723801i $$0.257607\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 12.1244i − 0.489698i −0.969561 0.244849i $$-0.921262\pi$$
0.969561 0.244849i $$-0.0787384\pi$$
$$614$$ 0 0
$$615$$ −18.0000 −0.725830
$$616$$ 0 0
$$617$$ 22.5167i 0.906487i 0.891387 + 0.453243i $$0.149733\pi$$
−0.891387 + 0.453243i $$0.850267\pi$$
$$618$$ 0 0
$$619$$ − 20.7846i − 0.835404i −0.908584 0.417702i $$-0.862836\pi$$
0.908584 0.417702i $$-0.137164\pi$$
$$620$$ 0 0
$$621$$ 24.0000 0.963087
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ − 25.9808i − 1.03592i
$$630$$ 0 0
$$631$$ − 48.4974i − 1.93065i −0.261048 0.965326i $$-0.584068\pi$$
0.261048 0.965326i $$-0.415932\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ 3.46410i 0.137469i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 3.46410i 0.137038i
$$640$$ 0 0
$$641$$ 33.0000 1.30342 0.651711 0.758468i $$-0.274052\pi$$
0.651711 + 0.758468i $$0.274052\pi$$
$$642$$ 0 0
$$643$$ 13.8564i 0.546443i 0.961951 + 0.273222i $$0.0880892\pi$$
−0.961951 + 0.273222i $$0.911911\pi$$
$$644$$ 0 0
$$645$$ 27.7128i 1.09119i
$$646$$ 0 0
$$647$$ −18.0000 −0.707653 −0.353827 0.935311i $$-0.615120\pi$$
−0.353827 + 0.935311i $$0.615120\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.0000 −1.17399 −0.586995 0.809590i $$-0.699689\pi$$
−0.586995 + 0.809590i $$0.699689\pi$$
$$654$$ 0 0
$$655$$ − 31.1769i − 1.21818i
$$656$$ 0 0
$$657$$ − 1.73205i − 0.0675737i
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ − 46.7654i − 1.81896i −0.415745 0.909481i $$-0.636479\pi$$
0.415745 0.909481i $$-0.363521\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 18.0000 0.696963
$$668$$ 0 0
$$669$$ − 20.7846i − 0.803579i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 0 0
$$675$$ 8.00000 0.307920
$$676$$ 0 0
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ − 48.4974i − 1.85843i
$$682$$ 0 0
$$683$$ 24.2487i 0.927851i 0.885874 + 0.463926i $$0.153559\pi$$
−0.885874 + 0.463926i $$0.846441\pi$$
$$684$$ 0 0
$$685$$ −27.0000 −1.03162
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 13.8564i − 0.527123i −0.964643 0.263561i $$-0.915103\pi$$
0.964643 0.263561i $$-0.0848971\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 6.92820i 0.262802i
$$696$$ 0 0
$$697$$ 15.5885i 0.590455i
$$698$$ 0 0
$$699$$ −12.0000 −0.453882
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 0 0
$$703$$ 30.0000 1.13147
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 5.19615i 0.195146i 0.995228 + 0.0975728i $$0.0311079\pi$$
−0.995228 + 0.0975728i $$0.968892\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ 20.7846i 0.778390i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 41.5692i 1.55243i
$$718$$ 0 0
$$719$$ 48.0000 1.79010 0.895049 0.445968i $$-0.147140\pi$$
0.895049 + 0.445968i $$0.147140\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ − 3.46410i − 0.128831i
$$724$$ 0 0
$$725$$ 6.00000 0.222834
$$726$$ 0 0
$$727$$ 32.0000 1.18681 0.593407 0.804902i $$-0.297782\pi$$
0.593407 + 0.804902i $$0.297782\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ − 12.1244i − 0.447823i −0.974609 0.223912i $$-0.928117\pi$$
0.974609 0.223912i $$-0.0718827\pi$$
$$734$$ 0 0
$$735$$ − 24.2487i − 0.894427i
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 20.7846i 0.764574i 0.924044 + 0.382287i $$0.124863\pi$$
−0.924044 + 0.382287i $$0.875137\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 34.6410i 1.27086i 0.772160 + 0.635428i $$0.219176\pi$$
−0.772160 + 0.635428i $$0.780824\pi$$
$$744$$ 0 0
$$745$$ −33.0000 −1.20903
$$746$$ 0 0
$$747$$ 13.8564i 0.506979i
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 16.0000 0.583848 0.291924 0.956441i $$-0.405705\pi$$
0.291924 + 0.956441i $$0.405705\pi$$
$$752$$ 0 0
$$753$$ −36.0000 −1.31191
$$754$$ 0 0
$$755$$ −30.0000 −1.09181
$$756$$ 0 0
$$757$$ −26.0000 −0.944986 −0.472493 0.881334i $$-0.656646\pi$$
−0.472493 + 0.881334i $$0.656646\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 34.6410i − 1.25574i −0.778320 0.627868i $$-0.783928\pi$$
0.778320 0.627868i $$-0.216072\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ − 5.19615i − 0.187867i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 6.92820i 0.249837i 0.992167 + 0.124919i $$0.0398670\pi$$
−0.992167 + 0.124919i $$0.960133\pi$$
$$770$$ 0 0
$$771$$ −6.00000 −0.216085
$$772$$ 0 0
$$773$$ 34.6410i 1.24595i 0.782241 + 0.622975i $$0.214076\pi$$
−0.782241 + 0.622975i $$0.785924\pi$$
$$774$$ 0 0
$$775$$ 6.92820i 0.248868i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −18.0000 −0.644917
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 12.0000 0.428845
$$784$$ 0 0
$$785$$ − 22.5167i − 0.803654i
$$786$$ 0 0
$$787$$ 38.1051i 1.35830i 0.733999 + 0.679150i $$0.237652\pi$$
−0.733999 + 0.679150i $$0.762348\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 10.3923i 0.368577i
$$796$$ 0 0
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ 10.3923i 0.367653i
$$800$$ 0 0
$$801$$ − 6.92820i − 0.244796i
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 12.0000 0.422420
$$808$$ 0 0
$$809$$ 33.0000 1.16022 0.580109 0.814539i $$-0.303010\pi$$
0.580109 + 0.814539i $$0.303010\pi$$
$$810$$ 0 0
$$811$$ − 38.1051i − 1.33805i −0.743239 0.669026i $$-0.766712\pi$$
0.743239 0.669026i $$-0.233288\pi$$
$$812$$ 0 0
$$813$$ 41.5692i 1.45790i
$$814$$ 0 0
$$815$$ −36.0000 −1.26102
$$816$$ 0 0
$$817$$ 27.7128i 0.969549i
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 41.5692i − 1.45078i −0.688340 0.725388i $$-0.741660\pi$$
0.688340 0.725388i $$-0.258340\pi$$
$$822$$ 0 0
$$823$$ 4.00000 0.139431 0.0697156 0.997567i $$-0.477791\pi$$
0.0697156 + 0.997567i $$0.477791\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 20.7846i 0.722752i 0.932420 + 0.361376i $$0.117693\pi$$
−0.932420 + 0.361376i $$0.882307\pi$$
$$828$$ 0 0
$$829$$ 25.0000 0.868286 0.434143 0.900844i $$-0.357051\pi$$
0.434143 + 0.900844i $$0.357051\pi$$
$$830$$ 0 0
$$831$$ 14.0000 0.485655
$$832$$ 0 0
$$833$$ −21.0000 −0.727607
$$834$$ 0 0
$$835$$ −24.0000 −0.830554
$$836$$ 0 0
$$837$$ 13.8564i 0.478947i
$$838$$ 0 0
$$839$$ 45.0333i 1.55472i 0.629054 + 0.777361i $$0.283442\pi$$
−0.629054 + 0.777361i $$0.716558\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ − 45.0333i − 1.55103i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 8.00000 0.274559
$$850$$ 0 0
$$851$$ 51.9615i 1.78122i
$$852$$ 0 0
$$853$$ − 25.9808i − 0.889564i −0.895639 0.444782i $$-0.853281\pi$$
0.895639 0.444782i $$-0.146719\pi$$
$$854$$ 0 0
$$855$$ 6.00000 0.205196
$$856$$ 0 0
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ 14.0000 0.477674 0.238837 0.971060i $$-0.423234\pi$$
0.238837 + 0.971060i $$0.423234\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ − 27.7128i − 0.943355i −0.881771 0.471678i $$-0.843649\pi$$
0.881771 0.471678i $$-0.156351\pi$$
$$864$$ 0 0
$$865$$ 10.3923i 0.353349i
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ − 6.92820i − 0.234484i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 12.1244i − 0.409410i −0.978824 0.204705i $$-0.934376\pi$$
0.978824 0.204705i $$-0.0656236\pi$$
$$878$$ 0 0
$$879$$ 10.3923i 0.350524i
$$880$$ 0 0
$$881$$ 27.0000 0.909653 0.454827 0.890580i $$-0.349701\pi$$
0.454827 + 0.890580i $$0.349701\pi$$
$$882$$ 0 0
$$883$$ −10.0000 −0.336527 −0.168263 0.985742i $$-0.553816\pi$$
−0.168263 + 0.985742i $$0.553816\pi$$
$$884$$ 0 0
$$885$$ 24.0000 0.806751
$$886$$ 0 0
$$887$$ −36.0000 −1.20876 −0.604381 0.796696i $$-0.706579\pi$$
−0.604381 + 0.796696i $$0.706579\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −12.0000 −0.401565
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 10.3923i 0.346603i
$$900$$ 0 0
$$901$$ 9.00000 0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 19.0526i 0.633328i
$$906$$ 0 0
$$907$$ −28.0000 −0.929725 −0.464862 0.885383i $$-0.653896\pi$$
−0.464862 + 0.885383i $$0.653896\pi$$
$$908$$ 0 0
$$909$$ −3.00000 −0.0995037
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ − 3.46410i − 0.114520i
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −22.0000 −0.725713 −0.362857 0.931845i $$-0.618198\pi$$
−0.362857 + 0.931845i $$0.618198\pi$$
$$920$$ 0 0
$$921$$ 34.6410i 1.14146i
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 17.3205i 0.569495i
$$926$$ 0 0
$$927$$ 10.0000 0.328443
$$928$$ 0 0
$$929$$ 46.7654i 1.53432i 0.641455 + 0.767161i $$0.278331\pi$$
−0.641455 + 0.767161i $$0.721669\pi$$
$$930$$ 0 0
$$931$$ − 24.2487i − 0.794719i
$$932$$ 0 0
$$933$$ −60.0000 −1.96431
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 7.00000 0.228680 0.114340 0.993442i $$-0.463525\pi$$
0.114340 + 0.993442i $$0.463525\pi$$
$$938$$ 0 0
$$939$$ −20.0000 −0.652675
$$940$$ 0 0
$$941$$ − 20.7846i − 0.677559i −0.940866 0.338779i $$-0.889986\pi$$
0.940866 0.338779i $$-0.110014\pi$$
$$942$$ 0 0
$$943$$ − 31.1769i − 1.01526i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 17.3205i 0.562841i 0.959585 + 0.281420i $$0.0908056\pi$$
−0.959585 + 0.281420i $$0.909194\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ − 10.3923i − 0.336994i
$$952$$ 0 0
$$953$$ −6.00000 −0.194359 −0.0971795 0.995267i $$-0.530982\pi$$
−0.0971795 + 0.995267i $$0.530982\pi$$
$$954$$ 0 0
$$955$$ − 31.1769i − 1.00886i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 19.0000 0.612903
$$962$$ 0 0
$$963$$ −6.00000 −0.193347
$$964$$ 0 0
$$965$$ 9.00000 0.289720
$$966$$ 0 0
$$967$$ 58.8897i 1.89377i 0.321578 + 0.946883i $$0.395787\pi$$
−0.321578 + 0.946883i $$0.604213\pi$$
$$968$$ 0 0
$$969$$ − 20.7846i − 0.667698i
$$970$$ 0 0
$$971$$ −6.00000 −0.192549 −0.0962746 0.995355i $$-0.530693\pi$$
−0.0962746 + 0.995355i $$0.530693\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 43.3013i 1.38533i 0.721259 + 0.692665i $$0.243564\pi$$
−0.721259 + 0.692665i $$0.756436\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 13.8564i 0.442401i
$$982$$ 0 0
$$983$$ 51.9615i 1.65732i 0.559756 + 0.828658i $$0.310895\pi$$
−0.559756 + 0.828658i $$0.689105\pi$$
$$984$$ 0 0
$$985$$ −24.0000 −0.764704
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −48.0000 −1.52631
$$990$$ 0 0
$$991$$ −2.00000 −0.0635321 −0.0317660 0.999495i $$-0.510113\pi$$
−0.0317660 + 0.999495i $$0.510113\pi$$
$$992$$ 0 0
$$993$$ 55.4256i 1.75888i
$$994$$ 0 0
$$995$$ 3.46410i 0.109819i
$$996$$ 0 0
$$997$$ 17.0000 0.538395 0.269198 0.963085i $$-0.413241\pi$$
0.269198 + 0.963085i $$0.413241\pi$$
$$998$$ 0 0
$$999$$ 34.6410i 1.09599i
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 2704.2.f.b.337.2 2
4.3 odd 2 169.2.b.a.168.1 2
12.11 even 2 1521.2.b.a.1351.2 2
13.5 odd 4 2704.2.a.o.1.2 2
13.8 odd 4 2704.2.a.o.1.1 2
13.9 even 3 208.2.w.b.49.1 2
13.10 even 6 208.2.w.b.17.1 2
13.12 even 2 inner 2704.2.f.b.337.1 2
39.23 odd 6 1872.2.by.d.433.1 2
39.35 odd 6 1872.2.by.d.1297.1 2
52.3 odd 6 169.2.e.a.147.1 2
52.7 even 12 169.2.c.a.146.1 4
52.11 even 12 169.2.c.a.22.1 4
52.15 even 12 169.2.c.a.22.2 4
52.19 even 12 169.2.c.a.146.2 4
52.23 odd 6 13.2.e.a.4.1 2
52.31 even 4 169.2.a.a.1.1 2
52.35 odd 6 13.2.e.a.10.1 yes 2
52.43 odd 6 169.2.e.a.23.1 2
52.47 even 4 169.2.a.a.1.2 2
52.51 odd 2 169.2.b.a.168.2 2
104.35 odd 6 832.2.w.d.257.1 2
104.61 even 6 832.2.w.a.257.1 2
104.75 odd 6 832.2.w.d.641.1 2
104.101 even 6 832.2.w.a.641.1 2
156.23 even 6 117.2.q.c.82.1 2
156.35 even 6 117.2.q.c.10.1 2
156.47 odd 4 1521.2.a.k.1.1 2
156.83 odd 4 1521.2.a.k.1.2 2
156.155 even 2 1521.2.b.a.1351.1 2
260.23 even 12 325.2.m.a.199.1 4
260.87 even 12 325.2.m.a.49.1 4
260.99 even 4 4225.2.a.v.1.1 2
260.127 even 12 325.2.m.a.199.2 4
260.139 odd 6 325.2.n.a.101.1 2
260.179 odd 6 325.2.n.a.251.1 2
260.239 even 4 4225.2.a.v.1.2 2
260.243 even 12 325.2.m.a.49.2 4
364.23 odd 6 637.2.k.a.459.1 2
364.75 even 6 637.2.k.c.459.1 2
364.83 odd 4 8281.2.a.q.1.1 2
364.87 even 6 637.2.k.c.569.1 2
364.139 even 6 637.2.q.a.491.1 2
364.179 odd 6 637.2.u.c.30.1 2
364.191 odd 6 637.2.u.c.361.1 2
364.243 even 6 637.2.u.b.361.1 2
364.283 even 6 637.2.u.b.30.1 2
364.307 odd 4 8281.2.a.q.1.2 2
364.335 even 6 637.2.q.a.589.1 2
364.347 odd 6 637.2.k.a.569.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 52.23 odd 6
13.2.e.a.10.1 yes 2 52.35 odd 6
117.2.q.c.10.1 2 156.35 even 6
117.2.q.c.82.1 2 156.23 even 6
169.2.a.a.1.1 2 52.31 even 4
169.2.a.a.1.2 2 52.47 even 4
169.2.b.a.168.1 2 4.3 odd 2
169.2.b.a.168.2 2 52.51 odd 2
169.2.c.a.22.1 4 52.11 even 12
169.2.c.a.22.2 4 52.15 even 12
169.2.c.a.146.1 4 52.7 even 12
169.2.c.a.146.2 4 52.19 even 12
169.2.e.a.23.1 2 52.43 odd 6
169.2.e.a.147.1 2 52.3 odd 6
208.2.w.b.17.1 2 13.10 even 6
208.2.w.b.49.1 2 13.9 even 3
325.2.m.a.49.1 4 260.87 even 12
325.2.m.a.49.2 4 260.243 even 12
325.2.m.a.199.1 4 260.23 even 12
325.2.m.a.199.2 4 260.127 even 12
325.2.n.a.101.1 2 260.139 odd 6
325.2.n.a.251.1 2 260.179 odd 6
637.2.k.a.459.1 2 364.23 odd 6
637.2.k.a.569.1 2 364.347 odd 6
637.2.k.c.459.1 2 364.75 even 6
637.2.k.c.569.1 2 364.87 even 6
637.2.q.a.491.1 2 364.139 even 6
637.2.q.a.589.1 2 364.335 even 6
637.2.u.b.30.1 2 364.283 even 6
637.2.u.b.361.1 2 364.243 even 6
637.2.u.c.30.1 2 364.179 odd 6
637.2.u.c.361.1 2 364.191 odd 6
832.2.w.a.257.1 2 104.61 even 6
832.2.w.a.641.1 2 104.101 even 6
832.2.w.d.257.1 2 104.35 odd 6
832.2.w.d.641.1 2 104.75 odd 6
1521.2.a.k.1.1 2 156.47 odd 4
1521.2.a.k.1.2 2 156.83 odd 4
1521.2.b.a.1351.1 2 156.155 even 2
1521.2.b.a.1351.2 2 12.11 even 2
1872.2.by.d.433.1 2 39.23 odd 6
1872.2.by.d.1297.1 2 39.35 odd 6
2704.2.a.o.1.1 2 13.8 odd 4
2704.2.a.o.1.2 2 13.5 odd 4
2704.2.f.b.337.1 2 13.12 even 2 inner
2704.2.f.b.337.2 2 1.1 even 1 trivial
4225.2.a.v.1.1 2 260.99 even 4
4225.2.a.v.1.2 2 260.239 even 4
8281.2.a.q.1.1 2 364.83 odd 4
8281.2.a.q.1.2 2 364.307 odd 4