# Properties

 Label 2704.2.a.o.1.2 Level $2704$ Weight $2$ Character 2704.1 Self dual yes Analytic conductor $21.592$ Analytic rank $1$ 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(1,2704)]

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, 0]))

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

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

chi := DirichletCharacter("2704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2704 = 2^{4} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2704.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: $$21.5915487066$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 2704.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## 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
$$5$$ 1.73205 0.774597 0.387298 0.921954i $$-0.373408\pi$$
0.387298 + 0.921954i $$0.373408\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 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
$$13$$ 0 0
$$14$$ 0 0
$$15$$ −3.46410 −0.894427
$$16$$ 0 0
$$17$$ 3.00000 0.727607 0.363803 0.931476i $$-0.381478\pi$$
0.363803 + 0.931476i $$0.381478\pi$$
$$18$$ 0 0
$$19$$ −3.46410 −0.794719 −0.397360 0.917663i $$-0.630073\pi$$
−0.397360 + 0.917663i $$0.630073\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −6.00000 −1.25109 −0.625543 0.780189i $$-0.715123\pi$$
−0.625543 + 0.780189i $$0.715123\pi$$
$$24$$ 0 0
$$25$$ −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
$$31$$ 3.46410 0.622171 0.311086 0.950382i $$-0.399307\pi$$
0.311086 + 0.950382i $$0.399307\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −8.66025 −1.42374 −0.711868 0.702313i $$-0.752151\pi$$
−0.711868 + 0.702313i $$0.752151\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −5.19615 −0.811503 −0.405751 0.913984i $$-0.632990\pi$$
−0.405751 + 0.913984i $$0.632990\pi$$
$$42$$ 0 0
$$43$$ 8.00000 1.21999 0.609994 0.792406i $$-0.291172\pi$$
0.609994 + 0.792406i $$0.291172\pi$$
$$44$$ 0 0
$$45$$ 1.73205 0.258199
$$46$$ 0 0
$$47$$ 3.46410 0.505291 0.252646 0.967559i $$-0.418699\pi$$
0.252646 + 0.967559i $$0.418699\pi$$
$$48$$ 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
$$57$$ 6.92820 0.917663
$$58$$ 0 0
$$59$$ −6.92820 −0.901975 −0.450988 0.892530i $$-0.648928\pi$$
−0.450988 + 0.892530i $$0.648928\pi$$
$$60$$ 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
$$67$$ −3.46410 −0.423207 −0.211604 0.977356i $$-0.567869\pi$$
−0.211604 + 0.977356i $$0.567869\pi$$
$$68$$ 0 0
$$69$$ 12.0000 1.44463
$$70$$ 0 0
$$71$$ 3.46410 0.411113 0.205557 0.978645i $$-0.434100\pi$$
0.205557 + 0.978645i $$0.434100\pi$$
$$72$$ 0 0
$$73$$ 1.73205 0.202721 0.101361 0.994850i $$-0.467680\pi$$
0.101361 + 0.994850i $$0.467680\pi$$
$$74$$ 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
$$83$$ 13.8564 1.52094 0.760469 0.649374i $$-0.224969\pi$$
0.760469 + 0.649374i $$0.224969\pi$$
$$84$$ 0 0
$$85$$ 5.19615 0.563602
$$86$$ 0 0
$$87$$ −6.00000 −0.643268
$$88$$ 0 0
$$89$$ 6.92820 0.734388 0.367194 0.930144i $$-0.380318\pi$$
0.367194 + 0.930144i $$0.380318\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −6.92820 −0.718421
$$94$$ 0 0
$$95$$ −6.00000 −0.615587
$$96$$ 0 0
$$97$$ −6.92820 −0.703452 −0.351726 0.936103i $$-0.614405\pi$$
−0.351726 + 0.936103i $$0.614405\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 3.00000 0.298511 0.149256 0.988799i $$-0.452312\pi$$
0.149256 + 0.988799i $$0.452312\pi$$
$$102$$ 0 0
$$103$$ −10.0000 −0.985329 −0.492665 0.870219i $$-0.663977\pi$$
−0.492665 + 0.870219i $$0.663977\pi$$
$$104$$ 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
$$109$$ 13.8564 1.32720 0.663602 0.748086i $$-0.269027\pi$$
0.663602 + 0.748086i $$0.269027\pi$$
$$110$$ 0 0
$$111$$ 17.3205 1.64399
$$112$$ 0 0
$$113$$ −15.0000 −1.41108 −0.705541 0.708669i $$-0.749296\pi$$
−0.705541 + 0.708669i $$0.749296\pi$$
$$114$$ 0 0
$$115$$ −10.3923 −0.969087
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 10.3923 0.937043
$$124$$ 0 0
$$125$$ −12.1244 −1.08444
$$126$$ 0 0
$$127$$ −2.00000 −0.177471 −0.0887357 0.996055i $$-0.528283\pi$$
−0.0887357 + 0.996055i $$0.528283\pi$$
$$128$$ 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
$$135$$ 6.92820 0.596285
$$136$$ 0 0
$$137$$ −15.5885 −1.33181 −0.665906 0.746036i $$-0.731955\pi$$
−0.665906 + 0.746036i $$0.731955\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ −6.92820 −0.583460
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 5.19615 0.431517
$$146$$ 0 0
$$147$$ 14.0000 1.15470
$$148$$ 0 0
$$149$$ 19.0526 1.56085 0.780423 0.625252i $$-0.215004\pi$$
0.780423 + 0.625252i $$0.215004\pi$$
$$150$$ 0 0
$$151$$ −17.3205 −1.40952 −0.704761 0.709444i $$-0.748946\pi$$
−0.704761 + 0.709444i $$0.748946\pi$$
$$152$$ 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
$$163$$ −20.7846 −1.62798 −0.813988 0.580881i $$-0.802708\pi$$
−0.813988 + 0.580881i $$0.802708\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −13.8564 −1.07224 −0.536120 0.844141i $$-0.680111\pi$$
−0.536120 + 0.844141i $$0.680111\pi$$
$$168$$ 0 0
$$169$$ 0 0
$$170$$ 0 0
$$171$$ −3.46410 −0.264906
$$172$$ 0 0
$$173$$ −6.00000 −0.456172 −0.228086 0.973641i $$-0.573247\pi$$
−0.228086 + 0.973641i $$0.573247\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 13.8564 1.04151
$$178$$ 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 $$-0.634057\pi$$
−0.408812 + 0.912619i $$0.634057\pi$$
$$182$$ 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
$$193$$ 5.19615 0.374027 0.187014 0.982357i $$-0.440119\pi$$
0.187014 + 0.982357i $$0.440119\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 13.8564 0.987228 0.493614 0.869681i $$-0.335676\pi$$
0.493614 + 0.869681i $$0.335676\pi$$
$$198$$ 0 0
$$199$$ −2.00000 −0.141776 −0.0708881 0.997484i $$-0.522583\pi$$
−0.0708881 + 0.997484i $$0.522583\pi$$
$$200$$ 0 0
$$201$$ 6.92820 0.488678
$$202$$ 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
$$213$$ −6.92820 −0.474713
$$214$$ 0 0
$$215$$ 13.8564 0.944999
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −3.46410 −0.234082
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 10.3923 0.695920 0.347960 0.937509i $$-0.386874\pi$$
0.347960 + 0.937509i $$0.386874\pi$$
$$224$$ 0 0
$$225$$ −2.00000 −0.133333
$$226$$ 0 0
$$227$$ 24.2487 1.60944 0.804722 0.593652i $$-0.202314\pi$$
0.804722 + 0.593652i $$0.202314\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$230$$ 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
$$235$$ 6.00000 0.391397
$$236$$ 0 0
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 0 0
$$241$$ −1.73205 −0.111571 −0.0557856 0.998443i $$-0.517766\pi$$
−0.0557856 + 0.998443i $$0.517766\pi$$
$$242$$ 0 0
$$243$$ 10.0000 0.641500
$$244$$ 0 0
$$245$$ −12.1244 −0.774597
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ −27.7128 −1.75623
$$250$$ 0 0
$$251$$ −18.0000 −1.13615 −0.568075 0.822977i $$-0.692312\pi$$
−0.568075 + 0.822977i $$0.692312\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −10.3923 −0.650791
$$256$$ 0 0
$$257$$ −3.00000 −0.187135 −0.0935674 0.995613i $$-0.529827\pi$$
−0.0935674 + 0.995613i $$0.529827\pi$$
$$258$$ 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
$$265$$ −5.19615 −0.319197
$$266$$ 0 0
$$267$$ −13.8564 −0.847998
$$268$$ 0 0
$$269$$ −6.00000 −0.365826 −0.182913 0.983129i $$-0.558553\pi$$
−0.182913 + 0.983129i $$0.558553\pi$$
$$270$$ 0 0
$$271$$ 20.7846 1.26258 0.631288 0.775549i $$-0.282527\pi$$
0.631288 + 0.775549i $$0.282527\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7.00000 0.420589 0.210295 0.977638i $$-0.432558\pi$$
0.210295 + 0.977638i $$0.432558\pi$$
$$278$$ 0 0
$$279$$ 3.46410 0.207390
$$280$$ 0 0
$$281$$ −22.5167 −1.34323 −0.671616 0.740900i $$-0.734399\pi$$
−0.671616 + 0.740900i $$0.734399\pi$$
$$282$$ 0 0
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 0 0
$$285$$ 12.0000 0.710819
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 13.8564 0.812277
$$292$$ 0 0
$$293$$ 5.19615 0.303562 0.151781 0.988414i $$-0.451499\pi$$
0.151781 + 0.988414i $$0.451499\pi$$
$$294$$ 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
$$305$$ 1.73205 0.0991769
$$306$$ 0 0
$$307$$ 17.3205 0.988534 0.494267 0.869310i $$-0.335437\pi$$
0.494267 + 0.869310i $$0.335437\pi$$
$$308$$ 0 0
$$309$$ 20.0000 1.13776
$$310$$ 0 0
$$311$$ −30.0000 −1.70114 −0.850572 0.525859i $$-0.823744\pi$$
−0.850572 + 0.525859i $$0.823744\pi$$
$$312$$ 0 0
$$313$$ 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
$$317$$ 5.19615 0.291845 0.145922 0.989296i $$-0.453385\pi$$
0.145922 + 0.989296i $$0.453385\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 12.0000 0.669775
$$322$$ 0 0
$$323$$ −10.3923 −0.578243
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −27.7128 −1.53252
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −27.7128 −1.52323 −0.761617 0.648027i $$-0.775594\pi$$
−0.761617 + 0.648027i $$0.775594\pi$$
$$332$$ 0 0
$$333$$ −8.66025 −0.474579
$$334$$ 0 0
$$335$$ −6.00000 −0.327815
$$336$$ 0 0
$$337$$ 23.0000 1.25289 0.626445 0.779466i $$-0.284509\pi$$
0.626445 + 0.779466i $$0.284509\pi$$
$$338$$ 0 0
$$339$$ 30.0000 1.62938
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 20.7846 1.11901
$$346$$ 0 0
$$347$$ 30.0000 1.61048 0.805242 0.592946i $$-0.202035\pi$$
0.805242 + 0.592946i $$0.202035\pi$$
$$348$$ 0 0
$$349$$ −13.8564 −0.741716 −0.370858 0.928689i $$-0.620936\pi$$
−0.370858 + 0.928689i $$0.620936\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 32.9090 1.75157 0.875784 0.482704i $$-0.160345\pi$$
0.875784 + 0.482704i $$0.160345\pi$$
$$354$$ 0 0
$$355$$ 6.00000 0.318447
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 6.92820 0.365657 0.182828 0.983145i $$-0.441475\pi$$
0.182828 + 0.983145i $$0.441475\pi$$
$$360$$ 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
$$369$$ −5.19615 −0.270501
$$370$$ 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
$$375$$ 24.2487 1.25220
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 24.2487 1.24557 0.622786 0.782392i $$-0.286001\pi$$
0.622786 + 0.782392i $$0.286001\pi$$
$$380$$ 0 0
$$381$$ 4.00000 0.204926
$$382$$ 0 0
$$383$$ −20.7846 −1.06204 −0.531022 0.847358i $$-0.678192\pi$$
−0.531022 + 0.847358i $$0.678192\pi$$
$$384$$ 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 $$-0.426729\pi$$
0.228159 + 0.973624i $$0.426729\pi$$
$$390$$ 0 0
$$391$$ −18.0000 −0.910299
$$392$$ 0 0
$$393$$ 36.0000 1.81596
$$394$$ 0 0
$$395$$ −6.92820 −0.348596
$$396$$ 0 0
$$397$$ −13.8564 −0.695433 −0.347717 0.937600i $$-0.613043\pi$$
−0.347717 + 0.937600i $$0.613043\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 1.73205 0.0864945 0.0432472 0.999064i $$-0.486230\pi$$
0.0432472 + 0.999064i $$0.486230\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ −19.0526 −0.946729
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −15.5885 −0.770800 −0.385400 0.922750i $$-0.625936\pi$$
−0.385400 + 0.922750i $$0.625936\pi$$
$$410$$ 0 0
$$411$$ 31.1769 1.53784
$$412$$ 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
$$421$$ −15.5885 −0.759735 −0.379867 0.925041i $$-0.624030\pi$$
−0.379867 + 0.925041i $$0.624030\pi$$
$$422$$ 0 0
$$423$$ 3.46410 0.168430
$$424$$ 0 0
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 6.92820 0.333720 0.166860 0.985981i $$-0.446637\pi$$
0.166860 + 0.985981i $$0.446637\pi$$
$$432$$ 0 0
$$433$$ 17.0000 0.816968 0.408484 0.912766i $$-0.366058\pi$$
0.408484 + 0.912766i $$0.366058\pi$$
$$434$$ 0 0
$$435$$ −10.3923 −0.498273
$$436$$ 0 0
$$437$$ 20.7846 0.994263
$$438$$ 0 0
$$439$$ −28.0000 −1.33637 −0.668184 0.743996i $$-0.732928\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ −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
$$447$$ −38.1051 −1.80231
$$448$$ 0 0
$$449$$ −6.92820 −0.326962 −0.163481 0.986546i $$-0.552272\pi$$
−0.163481 + 0.986546i $$0.552272\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 34.6410 1.62758
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1.73205 0.0810219 0.0405110 0.999179i $$-0.487101\pi$$
0.0405110 + 0.999179i $$0.487101\pi$$
$$458$$ 0 0
$$459$$ 12.0000 0.560112
$$460$$ 0 0
$$461$$ −22.5167 −1.04871 −0.524353 0.851501i $$-0.675693\pi$$
−0.524353 + 0.851501i $$0.675693\pi$$
$$462$$ 0 0
$$463$$ −13.8564 −0.643962 −0.321981 0.946746i $$-0.604349\pi$$
−0.321981 + 0.946746i $$0.604349\pi$$
$$464$$ 0 0
$$465$$ −12.0000 −0.556487
$$466$$ 0 0
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 26.0000 1.19802
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 6.92820 0.317888
$$476$$ 0 0
$$477$$ −3.00000 −0.137361
$$478$$ 0 0
$$479$$ 24.2487 1.10795 0.553976 0.832533i $$-0.313110\pi$$
0.553976 + 0.832533i $$0.313110\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −12.0000 −0.544892
$$486$$ 0 0
$$487$$ −6.92820 −0.313947 −0.156973 0.987603i $$-0.550174\pi$$
−0.156973 + 0.987603i $$0.550174\pi$$
$$488$$ 0 0
$$489$$ 41.5692 1.87983
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 0 0
$$493$$ 9.00000 0.405340
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 31.1769 1.39567 0.697835 0.716258i $$-0.254147\pi$$
0.697835 + 0.716258i $$0.254147\pi$$
$$500$$ 0 0
$$501$$ 27.7128 1.23812
$$502$$ 0 0
$$503$$ −36.0000 −1.60516 −0.802580 0.596544i $$-0.796540\pi$$
−0.802580 + 0.596544i $$0.796540\pi$$
$$504$$ 0 0
$$505$$ 5.19615 0.231226
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 19.0526 0.844490 0.422245 0.906482i $$-0.361242\pi$$
0.422245 + 0.906482i $$0.361242\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −13.8564 −0.611775
$$514$$ 0 0
$$515$$ −17.3205 −0.763233
$$516$$ 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
$$527$$ 10.3923 0.452696
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ −6.92820 −0.300658
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −10.3923 −0.449299
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 29.4449 1.26593 0.632967 0.774179i $$-0.281837\pi$$
0.632967 + 0.774179i $$0.281837\pi$$
$$542$$ 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
$$551$$ −10.3923 −0.442727
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 30.0000 1.27343
$$556$$ 0 0
$$557$$ 15.5885 0.660504 0.330252 0.943893i $$-0.392866\pi$$
0.330252 + 0.943893i $$0.392866\pi$$
$$558$$ 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
$$565$$ −25.9808 −1.09302
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 42.0000 1.76073 0.880366 0.474295i $$-0.157297\pi$$
0.880366 + 0.474295i $$0.157297\pi$$
$$570$$ 0 0
$$571$$ 40.0000 1.67395 0.836974 0.547243i $$-0.184323\pi$$
0.836974 + 0.547243i $$0.184323\pi$$
$$572$$ 0 0
$$573$$ 36.0000 1.50392
$$574$$ 0 0
$$575$$ 12.0000 0.500435
$$576$$ 0 0
$$577$$ 19.0526 0.793168 0.396584 0.917998i $$-0.370195\pi$$
0.396584 + 0.917998i $$0.370195\pi$$
$$578$$ 0 0
$$579$$ −10.3923 −0.431889
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 20.7846 0.857873 0.428936 0.903335i $$-0.358888\pi$$
0.428936 + 0.903335i $$0.358888\pi$$
$$588$$ 0 0
$$589$$ −12.0000 −0.494451
$$590$$ 0 0
$$591$$ −27.7128 −1.13995
$$592$$ 0 0
$$593$$ −25.9808 −1.06690 −0.533451 0.845831i $$-0.679105\pi$$
−0.533451 + 0.845831i $$0.679105\pi$$
$$594$$ 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
$$603$$ −3.46410 −0.141069
$$604$$ 0 0
$$605$$ −19.0526 −0.774597
$$606$$ 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
$$613$$ −12.1244 −0.489698 −0.244849 0.969561i $$-0.578738\pi$$
−0.244849 + 0.969561i $$0.578738\pi$$
$$614$$ 0 0
$$615$$ 18.0000 0.725830
$$616$$ 0 0
$$617$$ 22.5167 0.906487 0.453243 0.891387i $$-0.350267\pi$$
0.453243 + 0.891387i $$0.350267\pi$$
$$618$$ 0 0
$$619$$ 20.7846 0.835404 0.417702 0.908584i $$-0.362836\pi$$
0.417702 + 0.908584i $$0.362836\pi$$
$$620$$ 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
$$629$$ −25.9808 −1.03592
$$630$$ 0 0
$$631$$ 48.4974 1.93065 0.965326 0.261048i $$-0.0840679\pi$$
0.965326 + 0.261048i $$0.0840679\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ 0 0
$$635$$ −3.46410 −0.137469
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 3.46410 0.137038
$$640$$ 0 0
$$641$$ −33.0000 −1.30342 −0.651711 0.758468i $$-0.725948\pi$$
−0.651711 + 0.758468i $$0.725948\pi$$
$$642$$ 0 0
$$643$$ 13.8564 0.546443 0.273222 0.961951i $$-0.411911\pi$$
0.273222 + 0.961951i $$0.411911\pi$$
$$644$$ 0 0
$$645$$ −27.7128 −1.09119
$$646$$ 0 0
$$647$$ 18.0000 0.707653 0.353827 0.935311i $$-0.384880\pi$$
0.353827 + 0.935311i $$0.384880\pi$$
$$648$$ 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
$$655$$ −31.1769 −1.21818
$$656$$ 0 0
$$657$$ 1.73205 0.0675737
$$658$$ 0 0
$$659$$ 12.0000 0.467454 0.233727 0.972302i $$-0.424908\pi$$
0.233727 + 0.972302i $$0.424908\pi$$
$$660$$ 0 0
$$661$$ 46.7654 1.81896 0.909481 0.415745i $$-0.136479\pi$$
0.909481 + 0.415745i $$0.136479\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −18.0000 −0.696963
$$668$$ 0 0
$$669$$ −20.7846 −0.803579
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 19.0000 0.732396 0.366198 0.930537i $$-0.380659\pi$$
0.366198 + 0.930537i $$0.380659\pi$$
$$674$$ 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
$$681$$ −48.4974 −1.85843
$$682$$ 0 0
$$683$$ −24.2487 −0.927851 −0.463926 0.885874i $$-0.653559\pi$$
−0.463926 + 0.885874i $$0.653559\pi$$
$$684$$ 0 0
$$685$$ −27.0000 −1.03162
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −13.8564 −0.527123 −0.263561 0.964643i $$-0.584897\pi$$
−0.263561 + 0.964643i $$0.584897\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 6.92820 0.262802
$$696$$ 0 0
$$697$$ −15.5885 −0.590455
$$698$$ 0 0
$$699$$ 12.0000 0.453882
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ 0 0
$$703$$ 30.0000 1.13147
$$704$$ 0 0
$$705$$ −12.0000 −0.451946
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −5.19615 −0.195146 −0.0975728 0.995228i $$-0.531108\pi$$
−0.0975728 + 0.995228i $$0.531108\pi$$
$$710$$ 0 0
$$711$$ −4.00000 −0.150012
$$712$$ 0 0
$$713$$ −20.7846 −0.778390
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 41.5692 1.55243
$$718$$ 0 0
$$719$$ −48.0000 −1.79010 −0.895049 0.445968i $$-0.852860\pi$$
−0.895049 + 0.445968i $$0.852860\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 3.46410 0.128831
$$724$$ 0 0
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ −32.0000 −1.18681 −0.593407 0.804902i $$-0.702218\pi$$
−0.593407 + 0.804902i $$0.702218\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 24.0000 0.887672
$$732$$ 0 0
$$733$$ −12.1244 −0.447823 −0.223912 0.974609i $$-0.571883\pi$$
−0.223912 + 0.974609i $$0.571883\pi$$
$$734$$ 0 0
$$735$$ 24.2487 0.894427
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −20.7846 −0.764574 −0.382287 0.924044i $$-0.624863\pi$$
−0.382287 + 0.924044i $$0.624863\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 34.6410 1.27086 0.635428 0.772160i $$-0.280824\pi$$
0.635428 + 0.772160i $$0.280824\pi$$
$$744$$ 0 0
$$745$$ 33.0000 1.20903
$$746$$ 0 0
$$747$$ 13.8564 0.506979
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −16.0000 −0.583848 −0.291924 0.956441i $$-0.594295\pi$$
−0.291924 + 0.956441i $$0.594295\pi$$
$$752$$ 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
$$761$$ 34.6410 1.25574 0.627868 0.778320i $$-0.283928\pi$$
0.627868 + 0.778320i $$0.283928\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 5.19615 0.187867
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 6.92820 0.249837 0.124919 0.992167i $$-0.460133\pi$$
0.124919 + 0.992167i $$0.460133\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 0 0
$$773$$ 34.6410 1.24595 0.622975 0.782241i $$-0.285924\pi$$
0.622975 + 0.782241i $$0.285924\pi$$
$$774$$ 0 0
$$775$$ −6.92820 −0.248868
$$776$$ 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
$$785$$ −22.5167 −0.803654
$$786$$ 0 0
$$787$$ −38.1051 −1.35830 −0.679150 0.733999i $$-0.737652\pi$$
−0.679150 + 0.733999i $$0.737652\pi$$
$$788$$ 0 0
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 10.3923 0.368577
$$796$$ 0 0
$$797$$ −42.0000 −1.48772 −0.743858 0.668338i $$-0.767006\pi$$
−0.743858 + 0.668338i $$0.767006\pi$$
$$798$$ 0 0
$$799$$ 10.3923 0.367653
$$800$$ 0 0
$$801$$ 6.92820 0.244796
$$802$$ 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
$$811$$ −38.1051 −1.33805 −0.669026 0.743239i $$-0.733288\pi$$
−0.669026 + 0.743239i $$0.733288\pi$$
$$812$$ 0 0
$$813$$ −41.5692 −1.45790
$$814$$ 0 0
$$815$$ −36.0000 −1.26102
$$816$$ 0 0
$$817$$ −27.7128 −0.969549
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −41.5692 −1.45078 −0.725388 0.688340i $$-0.758340\pi$$
−0.725388 + 0.688340i $$0.758340\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −20.7846 −0.722752 −0.361376 0.932420i $$-0.617693\pi$$
−0.361376 + 0.932420i $$0.617693\pi$$
$$828$$ 0 0
$$829$$ −25.0000 −0.868286 −0.434143 0.900844i $$-0.642949\pi$$
−0.434143 + 0.900844i $$0.642949\pi$$
$$830$$ 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
$$837$$ 13.8564 0.478947
$$838$$ 0 0
$$839$$ −45.0333 −1.55472 −0.777361 0.629054i $$-0.783442\pi$$
−0.777361 + 0.629054i $$0.783442\pi$$
$$840$$ 0 0
$$841$$ −20.0000 −0.689655
$$842$$ 0 0
$$843$$ 45.0333 1.55103
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ −8.00000 −0.274559
$$850$$ 0 0
$$851$$ 51.9615 1.78122
$$852$$ 0 0
$$853$$ 25.9808 0.889564 0.444782 0.895639i $$-0.353281\pi$$
0.444782 + 0.895639i $$0.353281\pi$$
$$854$$ 0 0
$$855$$ −6.00000 −0.205196
$$856$$ 0 0
$$857$$ −3.00000 −0.102478 −0.0512390 0.998686i $$-0.516317\pi$$
−0.0512390 + 0.998686i $$0.516317\pi$$
$$858$$ 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
$$863$$ −27.7128 −0.943355 −0.471678 0.881771i $$-0.656351\pi$$
−0.471678 + 0.881771i $$0.656351\pi$$
$$864$$ 0 0
$$865$$ −10.3923 −0.353349
$$866$$ 0 0
$$867$$ 16.0000 0.543388
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −6.92820 −0.234484
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12.1244 −0.409410 −0.204705 0.978824i $$-0.565624\pi$$
−0.204705 + 0.978824i $$0.565624\pi$$
$$878$$ 0 0
$$879$$ −10.3923 −0.350524
$$880$$ 0 0
$$881$$ −27.0000 −0.909653 −0.454827 0.890580i $$-0.650299\pi$$
−0.454827 + 0.890580i $$0.650299\pi$$
$$882$$ 0 0
$$883$$ 10.0000 0.336527 0.168263 0.985742i $$-0.446184\pi$$
0.168263 + 0.985742i $$0.446184\pi$$
$$884$$ 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
$$899$$ 10.3923 0.346603
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −19.0526 −0.633328
$$906$$ 0 0
$$907$$ 28.0000 0.929725 0.464862 0.885383i $$-0.346104\pi$$
0.464862 + 0.885383i $$0.346104\pi$$
$$908$$ 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
$$915$$ −3.46410 −0.114520
$$916$$ 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
$$921$$ −34.6410 −1.14146
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 17.3205 0.569495
$$926$$ 0 0
$$927$$ −10.0000 −0.328443
$$928$$ 0 0
$$929$$ 46.7654 1.53432 0.767161 0.641455i $$-0.221669\pi$$
0.767161 + 0.641455i $$0.221669\pi$$
$$930$$ 0 0
$$931$$ 24.2487 0.794719
$$932$$ 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
$$941$$ −20.7846 −0.677559 −0.338779 0.940866i $$-0.610014\pi$$
−0.338779 + 0.940866i $$0.610014\pi$$
$$942$$ 0 0
$$943$$ 31.1769 1.01526
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −17.3205 −0.562841 −0.281420 0.959585i $$-0.590806\pi$$
−0.281420 + 0.959585i $$0.590806\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ −10.3923 −0.336994
$$952$$ 0 0
$$953$$ 6.00000 0.194359 0.0971795 0.995267i $$-0.469018\pi$$
0.0971795 + 0.995267i $$0.469018\pi$$
$$954$$ 0 0
$$955$$ −31.1769 −1.00886
$$956$$ 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
$$967$$ 58.8897 1.89377 0.946883 0.321578i $$-0.104213\pi$$
0.946883 + 0.321578i $$0.104213\pi$$
$$968$$ 0 0
$$969$$ 20.7846 0.667698
$$970$$ 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
$$977$$ 43.3013 1.38533 0.692665 0.721259i $$-0.256436\pi$$
0.692665 + 0.721259i $$0.256436\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 13.8564 0.442401
$$982$$ 0 0
$$983$$ −51.9615 −1.65732 −0.828658 0.559756i $$-0.810895\pi$$
−0.828658 + 0.559756i $$0.810895\pi$$
$$984$$ 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
$$993$$ 55.4256 1.75888
$$994$$ 0 0
$$995$$ −3.46410 −0.109819
$$996$$ 0 0
$$997$$ 17.0000 0.538395 0.269198 0.963085i $$-0.413241\pi$$
0.269198 + 0.963085i $$0.413241\pi$$
$$998$$ 0 0
$$999$$ −34.6410 −1.09599
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.a.o.1.2 2
4.3 odd 2 169.2.a.a.1.1 2
12.11 even 2 1521.2.a.k.1.2 2
13.2 odd 12 208.2.w.b.17.1 2
13.5 odd 4 2704.2.f.b.337.1 2
13.7 odd 12 208.2.w.b.49.1 2
13.8 odd 4 2704.2.f.b.337.2 2
13.12 even 2 inner 2704.2.a.o.1.1 2
20.19 odd 2 4225.2.a.v.1.2 2
28.27 even 2 8281.2.a.q.1.1 2
39.2 even 12 1872.2.by.d.433.1 2
39.20 even 12 1872.2.by.d.1297.1 2
52.3 odd 6 169.2.c.a.22.2 4
52.7 even 12 13.2.e.a.10.1 yes 2
52.11 even 12 169.2.e.a.147.1 2
52.15 even 12 13.2.e.a.4.1 2
52.19 even 12 169.2.e.a.23.1 2
52.23 odd 6 169.2.c.a.22.1 4
52.31 even 4 169.2.b.a.168.2 2
52.35 odd 6 169.2.c.a.146.2 4
52.43 odd 6 169.2.c.a.146.1 4
52.47 even 4 169.2.b.a.168.1 2
52.51 odd 2 169.2.a.a.1.2 2
104.59 even 12 832.2.w.d.257.1 2
104.67 even 12 832.2.w.d.641.1 2
104.85 odd 12 832.2.w.a.257.1 2
104.93 odd 12 832.2.w.a.641.1 2
156.47 odd 4 1521.2.b.a.1351.2 2
156.59 odd 12 117.2.q.c.10.1 2
156.83 odd 4 1521.2.b.a.1351.1 2
156.119 odd 12 117.2.q.c.82.1 2
156.155 even 2 1521.2.a.k.1.1 2
260.7 odd 12 325.2.m.a.49.1 4
260.59 even 12 325.2.n.a.101.1 2
260.67 odd 12 325.2.m.a.199.2 4
260.119 even 12 325.2.n.a.251.1 2
260.163 odd 12 325.2.m.a.49.2 4
260.223 odd 12 325.2.m.a.199.1 4
260.259 odd 2 4225.2.a.v.1.1 2
364.59 odd 12 637.2.k.c.569.1 2
364.67 even 12 637.2.u.c.30.1 2
364.111 odd 12 637.2.q.a.491.1 2
364.163 even 12 637.2.u.c.361.1 2
364.171 odd 12 637.2.u.b.30.1 2
364.215 odd 12 637.2.u.b.361.1 2
364.223 odd 12 637.2.q.a.589.1 2
364.275 even 12 637.2.k.a.459.1 2
364.319 even 12 637.2.k.a.569.1 2
364.327 odd 12 637.2.k.c.459.1 2
364.363 even 2 8281.2.a.q.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 52.15 even 12
13.2.e.a.10.1 yes 2 52.7 even 12
117.2.q.c.10.1 2 156.59 odd 12
117.2.q.c.82.1 2 156.119 odd 12
169.2.a.a.1.1 2 4.3 odd 2
169.2.a.a.1.2 2 52.51 odd 2
169.2.b.a.168.1 2 52.47 even 4
169.2.b.a.168.2 2 52.31 even 4
169.2.c.a.22.1 4 52.23 odd 6
169.2.c.a.22.2 4 52.3 odd 6
169.2.c.a.146.1 4 52.43 odd 6
169.2.c.a.146.2 4 52.35 odd 6
169.2.e.a.23.1 2 52.19 even 12
169.2.e.a.147.1 2 52.11 even 12
208.2.w.b.17.1 2 13.2 odd 12
208.2.w.b.49.1 2 13.7 odd 12
325.2.m.a.49.1 4 260.7 odd 12
325.2.m.a.49.2 4 260.163 odd 12
325.2.m.a.199.1 4 260.223 odd 12
325.2.m.a.199.2 4 260.67 odd 12
325.2.n.a.101.1 2 260.59 even 12
325.2.n.a.251.1 2 260.119 even 12
637.2.k.a.459.1 2 364.275 even 12
637.2.k.a.569.1 2 364.319 even 12
637.2.k.c.459.1 2 364.327 odd 12
637.2.k.c.569.1 2 364.59 odd 12
637.2.q.a.491.1 2 364.111 odd 12
637.2.q.a.589.1 2 364.223 odd 12
637.2.u.b.30.1 2 364.171 odd 12
637.2.u.b.361.1 2 364.215 odd 12
637.2.u.c.30.1 2 364.67 even 12
637.2.u.c.361.1 2 364.163 even 12
832.2.w.a.257.1 2 104.85 odd 12
832.2.w.a.641.1 2 104.93 odd 12
832.2.w.d.257.1 2 104.59 even 12
832.2.w.d.641.1 2 104.67 even 12
1521.2.a.k.1.1 2 156.155 even 2
1521.2.a.k.1.2 2 12.11 even 2
1521.2.b.a.1351.1 2 156.83 odd 4
1521.2.b.a.1351.2 2 156.47 odd 4
1872.2.by.d.433.1 2 39.2 even 12
1872.2.by.d.1297.1 2 39.20 even 12
2704.2.a.o.1.1 2 13.12 even 2 inner
2704.2.a.o.1.2 2 1.1 even 1 trivial
2704.2.f.b.337.1 2 13.5 odd 4
2704.2.f.b.337.2 2 13.8 odd 4
4225.2.a.v.1.1 2 260.259 odd 2
4225.2.a.v.1.2 2 20.19 odd 2
8281.2.a.q.1.1 2 28.27 even 2
8281.2.a.q.1.2 2 364.363 even 2