# Properties

 Label 2736.2.a.y.1.1 Level $2736$ Weight $2$ Character 2736.1 Self dual yes Analytic conductor $21.847$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(1,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.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: $$21.8470699930$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 228) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.37228 q^{5} +2.37228 q^{7} +O(q^{10})$$ $$q-4.37228 q^{5} +2.37228 q^{7} -4.37228 q^{11} +2.00000 q^{13} +4.37228 q^{17} -1.00000 q^{19} +2.74456 q^{23} +14.1168 q^{25} +8.74456 q^{29} -4.74456 q^{31} -10.3723 q^{35} -6.74456 q^{37} +2.37228 q^{43} -7.62772 q^{47} -1.37228 q^{49} -8.74456 q^{53} +19.1168 q^{55} -8.37228 q^{61} -8.74456 q^{65} -13.4891 q^{67} -12.0000 q^{71} +0.372281 q^{73} -10.3723 q^{77} -8.00000 q^{79} -2.74456 q^{83} -19.1168 q^{85} -3.25544 q^{89} +4.74456 q^{91} +4.37228 q^{95} +14.0000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} - q^{7}+O(q^{10})$$ 2 * q - 3 * q^5 - q^7 $$2 q - 3 q^{5} - q^{7} - 3 q^{11} + 4 q^{13} + 3 q^{17} - 2 q^{19} - 6 q^{23} + 11 q^{25} + 6 q^{29} + 2 q^{31} - 15 q^{35} - 2 q^{37} - q^{43} - 21 q^{47} + 3 q^{49} - 6 q^{53} + 21 q^{55} - 11 q^{61} - 6 q^{65} - 4 q^{67} - 24 q^{71} - 5 q^{73} - 15 q^{77} - 16 q^{79} + 6 q^{83} - 21 q^{85} - 18 q^{89} - 2 q^{91} + 3 q^{95} + 28 q^{97}+O(q^{100})$$ 2 * q - 3 * q^5 - q^7 - 3 * q^11 + 4 * q^13 + 3 * q^17 - 2 * q^19 - 6 * q^23 + 11 * q^25 + 6 * q^29 + 2 * q^31 - 15 * q^35 - 2 * q^37 - q^43 - 21 * q^47 + 3 * q^49 - 6 * q^53 + 21 * q^55 - 11 * q^61 - 6 * q^65 - 4 * q^67 - 24 * q^71 - 5 * q^73 - 15 * q^77 - 16 * q^79 + 6 * q^83 - 21 * q^85 - 18 * q^89 - 2 * q^91 + 3 * q^95 + 28 * q^97

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ −4.37228 −1.95534 −0.977672 0.210138i $$-0.932609\pi$$
−0.977672 + 0.210138i $$0.932609\pi$$
$$6$$ 0 0
$$7$$ 2.37228 0.896638 0.448319 0.893874i $$-0.352023\pi$$
0.448319 + 0.893874i $$0.352023\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −4.37228 −1.31829 −0.659146 0.752015i $$-0.729082\pi$$
−0.659146 + 0.752015i $$0.729082\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 4.37228 1.06043 0.530217 0.847862i $$-0.322110\pi$$
0.530217 + 0.847862i $$0.322110\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 2.74456 0.572281 0.286140 0.958188i $$-0.407628\pi$$
0.286140 + 0.958188i $$0.407628\pi$$
$$24$$ 0 0
$$25$$ 14.1168 2.82337
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 8.74456 1.62382 0.811912 0.583779i $$-0.198427\pi$$
0.811912 + 0.583779i $$0.198427\pi$$
$$30$$ 0 0
$$31$$ −4.74456 −0.852149 −0.426074 0.904688i $$-0.640104\pi$$
−0.426074 + 0.904688i $$0.640104\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −10.3723 −1.75324
$$36$$ 0 0
$$37$$ −6.74456 −1.10880 −0.554400 0.832251i $$-0.687052\pi$$
−0.554400 + 0.832251i $$0.687052\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 2.37228 0.361770 0.180885 0.983504i $$-0.442104\pi$$
0.180885 + 0.983504i $$0.442104\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.62772 −1.11262 −0.556309 0.830976i $$-0.687783\pi$$
−0.556309 + 0.830976i $$0.687783\pi$$
$$48$$ 0 0
$$49$$ −1.37228 −0.196040
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −8.74456 −1.20116 −0.600579 0.799565i $$-0.705063\pi$$
−0.600579 + 0.799565i $$0.705063\pi$$
$$54$$ 0 0
$$55$$ 19.1168 2.57771
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −8.37228 −1.07196 −0.535980 0.844230i $$-0.680058\pi$$
−0.535980 + 0.844230i $$0.680058\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −8.74456 −1.08463
$$66$$ 0 0
$$67$$ −13.4891 −1.64796 −0.823979 0.566620i $$-0.808251\pi$$
−0.823979 + 0.566620i $$0.808251\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −12.0000 −1.42414 −0.712069 0.702109i $$-0.752242\pi$$
−0.712069 + 0.702109i $$0.752242\pi$$
$$72$$ 0 0
$$73$$ 0.372281 0.0435722 0.0217861 0.999763i $$-0.493065\pi$$
0.0217861 + 0.999763i $$0.493065\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −10.3723 −1.18203
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −2.74456 −0.301255 −0.150627 0.988591i $$-0.548129\pi$$
−0.150627 + 0.988591i $$0.548129\pi$$
$$84$$ 0 0
$$85$$ −19.1168 −2.07351
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −3.25544 −0.345076 −0.172538 0.985003i $$-0.555197\pi$$
−0.172538 + 0.985003i $$0.555197\pi$$
$$90$$ 0 0
$$91$$ 4.74456 0.497365
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 4.37228 0.448587
$$96$$ 0 0
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 11.4891 1.14321 0.571605 0.820529i $$-0.306321\pi$$
0.571605 + 0.820529i $$0.306321\pi$$
$$102$$ 0 0
$$103$$ 12.7446 1.25576 0.627880 0.778311i $$-0.283923\pi$$
0.627880 + 0.778311i $$0.283923\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 3.25544 0.314715 0.157358 0.987542i $$-0.449703\pi$$
0.157358 + 0.987542i $$0.449703\pi$$
$$108$$ 0 0
$$109$$ 16.2337 1.55491 0.777453 0.628941i $$-0.216511\pi$$
0.777453 + 0.628941i $$0.216511\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −3.25544 −0.306246 −0.153123 0.988207i $$-0.548933\pi$$
−0.153123 + 0.988207i $$0.548933\pi$$
$$114$$ 0 0
$$115$$ −12.0000 −1.11901
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 10.3723 0.950825
$$120$$ 0 0
$$121$$ 8.11684 0.737895
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −39.8614 −3.56531
$$126$$ 0 0
$$127$$ −22.2337 −1.97292 −0.986460 0.164000i $$-0.947560\pi$$
−0.986460 + 0.164000i $$0.947560\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 9.86141 0.861595 0.430798 0.902449i $$-0.358232\pi$$
0.430798 + 0.902449i $$0.358232\pi$$
$$132$$ 0 0
$$133$$ −2.37228 −0.205703
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 4.37228 0.373549 0.186775 0.982403i $$-0.440197\pi$$
0.186775 + 0.982403i $$0.440197\pi$$
$$138$$ 0 0
$$139$$ −9.62772 −0.816612 −0.408306 0.912845i $$-0.633880\pi$$
−0.408306 + 0.912845i $$0.633880\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −8.74456 −0.731257
$$144$$ 0 0
$$145$$ −38.2337 −3.17513
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 4.37228 0.358191 0.179096 0.983832i $$-0.442683\pi$$
0.179096 + 0.983832i $$0.442683\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 20.7446 1.66624
$$156$$ 0 0
$$157$$ −15.4891 −1.23617 −0.618083 0.786113i $$-0.712091\pi$$
−0.618083 + 0.786113i $$0.712091\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.51087 0.513129
$$162$$ 0 0
$$163$$ −13.4891 −1.05655 −0.528275 0.849073i $$-0.677161\pi$$
−0.528275 + 0.849073i $$0.677161\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −14.2337 −1.10144 −0.550718 0.834691i $$-0.685646\pi$$
−0.550718 + 0.834691i $$0.685646\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −20.7446 −1.57718 −0.788590 0.614919i $$-0.789189\pi$$
−0.788590 + 0.614919i $$0.789189\pi$$
$$174$$ 0 0
$$175$$ 33.4891 2.53154
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 3.25544 0.243323 0.121661 0.992572i $$-0.461178\pi$$
0.121661 + 0.992572i $$0.461178\pi$$
$$180$$ 0 0
$$181$$ −22.0000 −1.63525 −0.817624 0.575753i $$-0.804709\pi$$
−0.817624 + 0.575753i $$0.804709\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 29.4891 2.16808
$$186$$ 0 0
$$187$$ −19.1168 −1.39796
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −16.3723 −1.18466 −0.592328 0.805697i $$-0.701791\pi$$
−0.592328 + 0.805697i $$0.701791\pi$$
$$192$$ 0 0
$$193$$ 5.25544 0.378295 0.189147 0.981949i $$-0.439428\pi$$
0.189147 + 0.981949i $$0.439428\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −11.4891 −0.818566 −0.409283 0.912407i $$-0.634221\pi$$
−0.409283 + 0.912407i $$0.634221\pi$$
$$198$$ 0 0
$$199$$ −3.11684 −0.220947 −0.110474 0.993879i $$-0.535237\pi$$
−0.110474 + 0.993879i $$0.535237\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 20.7446 1.45598
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 4.37228 0.302437
$$210$$ 0 0
$$211$$ 4.00000 0.275371 0.137686 0.990476i $$-0.456034\pi$$
0.137686 + 0.990476i $$0.456034\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −10.3723 −0.707384
$$216$$ 0 0
$$217$$ −11.2554 −0.764069
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 8.74456 0.588223
$$222$$ 0 0
$$223$$ 9.48913 0.635439 0.317719 0.948185i $$-0.397083\pi$$
0.317719 + 0.948185i $$0.397083\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 26.2337 1.74119 0.870596 0.491999i $$-0.163734\pi$$
0.870596 + 0.491999i $$0.163734\pi$$
$$228$$ 0 0
$$229$$ 0.372281 0.0246010 0.0123005 0.999924i $$-0.496085\pi$$
0.0123005 + 0.999924i $$0.496085\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −4.37228 −0.286438 −0.143219 0.989691i $$-0.545745\pi$$
−0.143219 + 0.989691i $$0.545745\pi$$
$$234$$ 0 0
$$235$$ 33.3505 2.17555
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −9.86141 −0.637881 −0.318941 0.947775i $$-0.603327\pi$$
−0.318941 + 0.947775i $$0.603327\pi$$
$$240$$ 0 0
$$241$$ 10.7446 0.692118 0.346059 0.938213i $$-0.387520\pi$$
0.346059 + 0.938213i $$0.387520\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 6.00000 0.383326
$$246$$ 0 0
$$247$$ −2.00000 −0.127257
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 4.37228 0.275976 0.137988 0.990434i $$-0.455936\pi$$
0.137988 + 0.990434i $$0.455936\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −5.48913 −0.342402 −0.171201 0.985236i $$-0.554765\pi$$
−0.171201 + 0.985236i $$0.554765\pi$$
$$258$$ 0 0
$$259$$ −16.0000 −0.994192
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −2.13859 −0.131871 −0.0659357 0.997824i $$-0.521003\pi$$
−0.0659357 + 0.997824i $$0.521003\pi$$
$$264$$ 0 0
$$265$$ 38.2337 2.34868
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.25544 −0.198488 −0.0992438 0.995063i $$-0.531642\pi$$
−0.0992438 + 0.995063i $$0.531642\pi$$
$$270$$ 0 0
$$271$$ 4.00000 0.242983 0.121491 0.992592i $$-0.461232\pi$$
0.121491 + 0.992592i $$0.461232\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −61.7228 −3.72203
$$276$$ 0 0
$$277$$ 6.88316 0.413569 0.206784 0.978387i $$-0.433700\pi$$
0.206784 + 0.978387i $$0.433700\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 15.2554 0.910063 0.455032 0.890475i $$-0.349628\pi$$
0.455032 + 0.890475i $$0.349628\pi$$
$$282$$ 0 0
$$283$$ 8.88316 0.528049 0.264024 0.964516i $$-0.414950\pi$$
0.264024 + 0.964516i $$0.414950\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 2.11684 0.124520
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6.51087 −0.380369 −0.190185 0.981748i $$-0.560909\pi$$
−0.190185 + 0.981748i $$0.560909\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.48913 0.317444
$$300$$ 0 0
$$301$$ 5.62772 0.324376
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 36.6060 2.09605
$$306$$ 0 0
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 9.86141 0.559189 0.279595 0.960118i $$-0.409800\pi$$
0.279595 + 0.960118i $$0.409800\pi$$
$$312$$ 0 0
$$313$$ 14.0000 0.791327 0.395663 0.918396i $$-0.370515\pi$$
0.395663 + 0.918396i $$0.370515\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.51087 −0.365687 −0.182844 0.983142i $$-0.558530\pi$$
−0.182844 + 0.983142i $$0.558530\pi$$
$$318$$ 0 0
$$319$$ −38.2337 −2.14068
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −4.37228 −0.243280
$$324$$ 0 0
$$325$$ 28.2337 1.56612
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −18.0951 −0.997615
$$330$$ 0 0
$$331$$ 4.00000 0.219860 0.109930 0.993939i $$-0.464937\pi$$
0.109930 + 0.993939i $$0.464937\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 58.9783 3.22233
$$336$$ 0 0
$$337$$ −3.48913 −0.190065 −0.0950324 0.995474i $$-0.530295\pi$$
−0.0950324 + 0.995474i $$0.530295\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 20.7446 1.12338
$$342$$ 0 0
$$343$$ −19.8614 −1.07242
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.11684 −0.0599553 −0.0299777 0.999551i $$-0.509544\pi$$
−0.0299777 + 0.999551i $$0.509544\pi$$
$$348$$ 0 0
$$349$$ 24.3723 1.30462 0.652309 0.757953i $$-0.273800\pi$$
0.652309 + 0.757953i $$0.273800\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 0 0
$$355$$ 52.4674 2.78468
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −33.8614 −1.78714 −0.893568 0.448927i $$-0.851806\pi$$
−0.893568 + 0.448927i $$0.851806\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −1.62772 −0.0851987
$$366$$ 0 0
$$367$$ −2.51087 −0.131067 −0.0655333 0.997850i $$-0.520875\pi$$
−0.0655333 + 0.997850i $$0.520875\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −20.7446 −1.07700
$$372$$ 0 0
$$373$$ 4.23369 0.219212 0.109606 0.993975i $$-0.465041\pi$$
0.109606 + 0.993975i $$0.465041\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 17.4891 0.900736
$$378$$ 0 0
$$379$$ 35.7228 1.83496 0.917479 0.397785i $$-0.130221\pi$$
0.917479 + 0.397785i $$0.130221\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −26.2337 −1.34048 −0.670239 0.742145i $$-0.733809\pi$$
−0.670239 + 0.742145i $$0.733809\pi$$
$$384$$ 0 0
$$385$$ 45.3505 2.31128
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 19.6277 0.995165 0.497582 0.867417i $$-0.334221\pi$$
0.497582 + 0.867417i $$0.334221\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 34.9783 1.75995
$$396$$ 0 0
$$397$$ −10.6060 −0.532298 −0.266149 0.963932i $$-0.585751\pi$$
−0.266149 + 0.963932i $$0.585751\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 17.4891 0.873365 0.436683 0.899616i $$-0.356153\pi$$
0.436683 + 0.899616i $$0.356153\pi$$
$$402$$ 0 0
$$403$$ −9.48913 −0.472687
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 29.4891 1.46172
$$408$$ 0 0
$$409$$ −38.4674 −1.90209 −0.951045 0.309053i $$-0.899988\pi$$
−0.951045 + 0.309053i $$0.899988\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 37.7228 1.84288 0.921440 0.388521i $$-0.127014\pi$$
0.921440 + 0.388521i $$0.127014\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 61.7228 2.99400
$$426$$ 0 0
$$427$$ −19.8614 −0.961161
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 8.74456 0.421211 0.210605 0.977571i $$-0.432456\pi$$
0.210605 + 0.977571i $$0.432456\pi$$
$$432$$ 0 0
$$433$$ −15.4891 −0.744360 −0.372180 0.928161i $$-0.621390\pi$$
−0.372180 + 0.928161i $$0.621390\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −2.74456 −0.131290
$$438$$ 0 0
$$439$$ 30.2337 1.44298 0.721488 0.692427i $$-0.243459\pi$$
0.721488 + 0.692427i $$0.243459\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7.62772 −0.362404 −0.181202 0.983446i $$-0.557999\pi$$
−0.181202 + 0.983446i $$0.557999\pi$$
$$444$$ 0 0
$$445$$ 14.2337 0.674742
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −22.9783 −1.08441 −0.542205 0.840246i $$-0.682411\pi$$
−0.542205 + 0.840246i $$0.682411\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −20.7446 −0.972520
$$456$$ 0 0
$$457$$ −13.8614 −0.648409 −0.324205 0.945987i $$-0.605097\pi$$
−0.324205 + 0.945987i $$0.605097\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 39.3505 1.83274 0.916368 0.400336i $$-0.131107\pi$$
0.916368 + 0.400336i $$0.131107\pi$$
$$462$$ 0 0
$$463$$ −17.3505 −0.806348 −0.403174 0.915123i $$-0.632093\pi$$
−0.403174 + 0.915123i $$0.632093\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −13.1168 −0.606975 −0.303488 0.952835i $$-0.598151\pi$$
−0.303488 + 0.952835i $$0.598151\pi$$
$$468$$ 0 0
$$469$$ −32.0000 −1.47762
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −10.3723 −0.476918
$$474$$ 0 0
$$475$$ −14.1168 −0.647725
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 3.76631 0.172087 0.0860436 0.996291i $$-0.472578\pi$$
0.0860436 + 0.996291i $$0.472578\pi$$
$$480$$ 0 0
$$481$$ −13.4891 −0.615051
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −61.2119 −2.77949
$$486$$ 0 0
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −38.7446 −1.74852 −0.874259 0.485460i $$-0.838652\pi$$
−0.874259 + 0.485460i $$0.838652\pi$$
$$492$$ 0 0
$$493$$ 38.2337 1.72196
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −28.4674 −1.27694
$$498$$ 0 0
$$499$$ −17.3505 −0.776716 −0.388358 0.921508i $$-0.626958\pi$$
−0.388358 + 0.921508i $$0.626958\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 14.7446 0.657428 0.328714 0.944430i $$-0.393385\pi$$
0.328714 + 0.944430i $$0.393385\pi$$
$$504$$ 0 0
$$505$$ −50.2337 −2.23537
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 3.25544 0.144295 0.0721474 0.997394i $$-0.477015\pi$$
0.0721474 + 0.997394i $$0.477015\pi$$
$$510$$ 0 0
$$511$$ 0.883156 0.0390685
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −55.7228 −2.45544
$$516$$ 0 0
$$517$$ 33.3505 1.46675
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 17.4891 0.766212 0.383106 0.923704i $$-0.374854\pi$$
0.383106 + 0.923704i $$0.374854\pi$$
$$522$$ 0 0
$$523$$ 7.25544 0.317258 0.158629 0.987338i $$-0.449293\pi$$
0.158629 + 0.987338i $$0.449293\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −20.7446 −0.903647
$$528$$ 0 0
$$529$$ −15.4674 −0.672495
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −14.2337 −0.615376
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −1.86141 −0.0800281 −0.0400141 0.999199i $$-0.512740\pi$$
−0.0400141 + 0.999199i $$0.512740\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −70.9783 −3.04037
$$546$$ 0 0
$$547$$ 7.25544 0.310220 0.155110 0.987897i $$-0.450427\pi$$
0.155110 + 0.987897i $$0.450427\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −8.74456 −0.372531
$$552$$ 0 0
$$553$$ −18.9783 −0.807037
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 21.8614 0.926298 0.463149 0.886281i $$-0.346720\pi$$
0.463149 + 0.886281i $$0.346720\pi$$
$$558$$ 0 0
$$559$$ 4.74456 0.200674
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 1.02175 0.0430616 0.0215308 0.999768i $$-0.493146\pi$$
0.0215308 + 0.999768i $$0.493146\pi$$
$$564$$ 0 0
$$565$$ 14.2337 0.598816
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 32.7446 1.37272 0.686362 0.727260i $$-0.259207\pi$$
0.686362 + 0.727260i $$0.259207\pi$$
$$570$$ 0 0
$$571$$ 26.9783 1.12900 0.564502 0.825431i $$-0.309068\pi$$
0.564502 + 0.825431i $$0.309068\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 38.7446 1.61576
$$576$$ 0 0
$$577$$ 21.1168 0.879106 0.439553 0.898217i $$-0.355137\pi$$
0.439553 + 0.898217i $$0.355137\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.51087 −0.270117
$$582$$ 0 0
$$583$$ 38.2337 1.58348
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −6.60597 −0.272658 −0.136329 0.990664i $$-0.543530\pi$$
−0.136329 + 0.990664i $$0.543530\pi$$
$$588$$ 0 0
$$589$$ 4.74456 0.195496
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 16.9783 0.697213 0.348607 0.937269i $$-0.386655\pi$$
0.348607 + 0.937269i $$0.386655\pi$$
$$594$$ 0 0
$$595$$ −45.3505 −1.85919
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ −1.25544 −0.0512104 −0.0256052 0.999672i $$-0.508151\pi$$
−0.0256052 + 0.999672i $$0.508151\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −35.4891 −1.44284
$$606$$ 0 0
$$607$$ 19.2554 0.781554 0.390777 0.920485i $$-0.372206\pi$$
0.390777 + 0.920485i $$0.372206\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −15.2554 −0.617169
$$612$$ 0 0
$$613$$ −29.1168 −1.17602 −0.588009 0.808854i $$-0.700088\pi$$
−0.588009 + 0.808854i $$0.700088\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 10.8832 0.438139 0.219070 0.975709i $$-0.429698\pi$$
0.219070 + 0.975709i $$0.429698\pi$$
$$618$$ 0 0
$$619$$ −6.97825 −0.280480 −0.140240 0.990118i $$-0.544787\pi$$
−0.140240 + 0.990118i $$0.544787\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −7.72281 −0.309408
$$624$$ 0 0
$$625$$ 103.701 4.14804
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −29.4891 −1.17581
$$630$$ 0 0
$$631$$ −4.13859 −0.164755 −0.0823774 0.996601i $$-0.526251\pi$$
−0.0823774 + 0.996601i $$0.526251\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 97.2119 3.85774
$$636$$ 0 0
$$637$$ −2.74456 −0.108744
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −14.2337 −0.562197 −0.281098 0.959679i $$-0.590699\pi$$
−0.281098 + 0.959679i $$0.590699\pi$$
$$642$$ 0 0
$$643$$ 37.3505 1.47296 0.736481 0.676459i $$-0.236486\pi$$
0.736481 + 0.676459i $$0.236486\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −21.8614 −0.859461 −0.429730 0.902957i $$-0.641391\pi$$
−0.429730 + 0.902957i $$0.641391\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −30.6060 −1.19770 −0.598852 0.800860i $$-0.704376\pi$$
−0.598852 + 0.800860i $$0.704376\pi$$
$$654$$ 0 0
$$655$$ −43.1168 −1.68471
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −8.74456 −0.340640 −0.170320 0.985389i $$-0.554480\pi$$
−0.170320 + 0.985389i $$0.554480\pi$$
$$660$$ 0 0
$$661$$ −24.2337 −0.942581 −0.471291 0.881978i $$-0.656212\pi$$
−0.471291 + 0.881978i $$0.656212\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 10.3723 0.402220
$$666$$ 0 0
$$667$$ 24.0000 0.929284
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36.6060 1.41316
$$672$$ 0 0
$$673$$ −36.2337 −1.39671 −0.698353 0.715753i $$-0.746083\pi$$
−0.698353 + 0.715753i $$0.746083\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −8.74456 −0.336081 −0.168040 0.985780i $$-0.553744\pi$$
−0.168040 + 0.985780i $$0.553744\pi$$
$$678$$ 0 0
$$679$$ 33.2119 1.27456
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 20.7446 0.793769 0.396884 0.917869i $$-0.370091\pi$$
0.396884 + 0.917869i $$0.370091\pi$$
$$684$$ 0 0
$$685$$ −19.1168 −0.730417
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −17.4891 −0.666283
$$690$$ 0 0
$$691$$ −29.3505 −1.11655 −0.558273 0.829657i $$-0.688536\pi$$
−0.558273 + 0.829657i $$0.688536\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 42.0951 1.59676
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −12.5109 −0.472529 −0.236265 0.971689i $$-0.575923\pi$$
−0.236265 + 0.971689i $$0.575923\pi$$
$$702$$ 0 0
$$703$$ 6.74456 0.254376
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 27.2554 1.02505
$$708$$ 0 0
$$709$$ −50.4674 −1.89534 −0.947671 0.319248i $$-0.896570\pi$$
−0.947671 + 0.319248i $$0.896570\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −13.0217 −0.487668
$$714$$ 0 0
$$715$$ 38.2337 1.42986
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −10.8832 −0.405873 −0.202937 0.979192i $$-0.565049\pi$$
−0.202937 + 0.979192i $$0.565049\pi$$
$$720$$ 0 0
$$721$$ 30.2337 1.12596
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 123.446 4.58466
$$726$$ 0 0
$$727$$ 39.5842 1.46810 0.734049 0.679097i $$-0.237628\pi$$
0.734049 + 0.679097i $$0.237628\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 10.3723 0.383633
$$732$$ 0 0
$$733$$ 2.00000 0.0738717 0.0369358 0.999318i $$-0.488240\pi$$
0.0369358 + 0.999318i $$0.488240\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 58.9783 2.17249
$$738$$ 0 0
$$739$$ −44.6060 −1.64086 −0.820429 0.571749i $$-0.806265\pi$$
−0.820429 + 0.571749i $$0.806265\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 40.4674 1.48460 0.742302 0.670065i $$-0.233734\pi$$
0.742302 + 0.670065i $$0.233734\pi$$
$$744$$ 0 0
$$745$$ −19.1168 −0.700387
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 7.72281 0.282185
$$750$$ 0 0
$$751$$ 2.97825 0.108678 0.0543390 0.998523i $$-0.482695\pi$$
0.0543390 + 0.998523i $$0.482695\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 34.9783 1.27299
$$756$$ 0 0
$$757$$ 32.0951 1.16652 0.583258 0.812287i $$-0.301778\pi$$
0.583258 + 0.812287i $$0.301778\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −41.5842 −1.50743 −0.753713 0.657203i $$-0.771739\pi$$
−0.753713 + 0.657203i $$0.771739\pi$$
$$762$$ 0 0
$$763$$ 38.5109 1.39419
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 21.1168 0.761493 0.380746 0.924679i $$-0.375667\pi$$
0.380746 + 0.924679i $$0.375667\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −41.4891 −1.49226 −0.746130 0.665800i $$-0.768090\pi$$
−0.746130 + 0.665800i $$0.768090\pi$$
$$774$$ 0 0
$$775$$ −66.9783 −2.40593
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 52.4674 1.87743
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 67.7228 2.41713
$$786$$ 0 0
$$787$$ 21.4891 0.766005 0.383002 0.923747i $$-0.374890\pi$$
0.383002 + 0.923747i $$0.374890\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7.72281 −0.274592
$$792$$ 0 0
$$793$$ −16.7446 −0.594617
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 6.51087 0.230627 0.115314 0.993329i $$-0.463213\pi$$
0.115314 + 0.993329i $$0.463213\pi$$
$$798$$ 0 0
$$799$$ −33.3505 −1.17986
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −1.62772 −0.0574409
$$804$$ 0 0
$$805$$ −28.4674 −1.00334
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 19.6277 0.690074 0.345037 0.938589i $$-0.387866\pi$$
0.345037 + 0.938589i $$0.387866\pi$$
$$810$$ 0 0
$$811$$ −34.2337 −1.20211 −0.601054 0.799209i $$-0.705252\pi$$
−0.601054 + 0.799209i $$0.705252\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 58.9783 2.06592
$$816$$ 0 0
$$817$$ −2.37228 −0.0829956
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −41.5842 −1.45130 −0.725650 0.688064i $$-0.758461\pi$$
−0.725650 + 0.688064i $$0.758461\pi$$
$$822$$ 0 0
$$823$$ 37.3505 1.30196 0.650979 0.759096i $$-0.274359\pi$$
0.650979 + 0.759096i $$0.274359\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −17.4891 −0.608156 −0.304078 0.952647i $$-0.598348\pi$$
−0.304078 + 0.952647i $$0.598348\pi$$
$$828$$ 0 0
$$829$$ −7.76631 −0.269735 −0.134868 0.990864i $$-0.543061\pi$$
−0.134868 + 0.990864i $$0.543061\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −6.00000 −0.207888
$$834$$ 0 0
$$835$$ 62.2337 2.15369
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 8.74456 0.301896 0.150948 0.988542i $$-0.451767\pi$$
0.150948 + 0.988542i $$0.451767\pi$$
$$840$$ 0 0
$$841$$ 47.4674 1.63681
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 39.3505 1.35370
$$846$$ 0 0
$$847$$ 19.2554 0.661625
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −18.5109 −0.634545
$$852$$ 0 0
$$853$$ 31.4891 1.07817 0.539084 0.842252i $$-0.318771\pi$$
0.539084 + 0.842252i $$0.318771\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −10.9783 −0.375010 −0.187505 0.982264i $$-0.560040\pi$$
−0.187505 + 0.982264i $$0.560040\pi$$
$$858$$ 0 0
$$859$$ −44.6060 −1.52194 −0.760968 0.648789i $$-0.775276\pi$$
−0.760968 + 0.648789i $$0.775276\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 1.02175 0.0347808 0.0173904 0.999849i $$-0.494464\pi$$
0.0173904 + 0.999849i $$0.494464\pi$$
$$864$$ 0 0
$$865$$ 90.7011 3.08393
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 34.9783 1.18656
$$870$$ 0 0
$$871$$ −26.9783 −0.914123
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −94.5625 −3.19679
$$876$$ 0 0
$$877$$ −16.5109 −0.557533 −0.278766 0.960359i $$-0.589925\pi$$
−0.278766 + 0.960359i $$0.589925\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 56.8397 1.91498 0.957488 0.288472i $$-0.0931472\pi$$
0.957488 + 0.288472i $$0.0931472\pi$$
$$882$$ 0 0
$$883$$ 1.35053 0.0454490 0.0227245 0.999742i $$-0.492766\pi$$
0.0227245 + 0.999742i $$0.492766\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −41.4891 −1.39307 −0.696534 0.717524i $$-0.745276\pi$$
−0.696534 + 0.717524i $$0.745276\pi$$
$$888$$ 0 0
$$889$$ −52.7446 −1.76900
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 7.62772 0.255252
$$894$$ 0 0
$$895$$ −14.2337 −0.475780
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −41.4891 −1.38374
$$900$$ 0 0
$$901$$ −38.2337 −1.27375
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 96.1902 3.19747
$$906$$ 0 0
$$907$$ −5.76631 −0.191467 −0.0957336 0.995407i $$-0.530520\pi$$
−0.0957336 + 0.995407i $$0.530520\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −38.2337 −1.26674 −0.633369 0.773850i $$-0.718329\pi$$
−0.633369 + 0.773850i $$0.718329\pi$$
$$912$$ 0 0
$$913$$ 12.0000 0.397142
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 23.3940 0.772539
$$918$$ 0 0
$$919$$ −32.0000 −1.05558 −0.527791 0.849374i $$-0.676980\pi$$
−0.527791 + 0.849374i $$0.676980\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −24.0000 −0.789970
$$924$$ 0 0
$$925$$ −95.2119 −3.13055
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −11.4891 −0.376946 −0.188473 0.982078i $$-0.560354\pi$$
−0.188473 + 0.982078i $$0.560354\pi$$
$$930$$ 0 0
$$931$$ 1.37228 0.0449747
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 83.5842 2.73350
$$936$$ 0 0
$$937$$ 16.8397 0.550128 0.275064 0.961426i $$-0.411301\pi$$
0.275064 + 0.961426i $$0.411301\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −27.2554 −0.888502 −0.444251 0.895902i $$-0.646530\pi$$
−0.444251 + 0.895902i $$0.646530\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 13.7228 0.445932 0.222966 0.974826i $$-0.428426\pi$$
0.222966 + 0.974826i $$0.428426\pi$$
$$948$$ 0 0
$$949$$ 0.744563 0.0241695
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −16.4674 −0.533431 −0.266715 0.963775i $$-0.585938\pi$$
−0.266715 + 0.963775i $$0.585938\pi$$
$$954$$ 0 0
$$955$$ 71.5842 2.31641
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.3723 0.334938
$$960$$ 0 0
$$961$$ −8.48913 −0.273843
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −22.9783 −0.739696
$$966$$ 0 0
$$967$$ 50.9783 1.63935 0.819675 0.572829i $$-0.194154\pi$$
0.819675 + 0.572829i $$0.194154\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −1.02175 −0.0327895 −0.0163947 0.999866i $$-0.505219\pi$$
−0.0163947 + 0.999866i $$0.505219\pi$$
$$972$$ 0 0
$$973$$ −22.8397 −0.732206
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −7.72281 −0.247075 −0.123537 0.992340i $$-0.539424\pi$$
−0.123537 + 0.992340i $$0.539424\pi$$
$$978$$ 0 0
$$979$$ 14.2337 0.454911
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −6.51087 −0.207665 −0.103832 0.994595i $$-0.533111\pi$$
−0.103832 + 0.994595i $$0.533111\pi$$
$$984$$ 0 0
$$985$$ 50.2337 1.60058
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 6.51087 0.207034
$$990$$ 0 0
$$991$$ 37.9565 1.20573 0.602864 0.797844i $$-0.294026\pi$$
0.602864 + 0.797844i $$0.294026\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 13.6277 0.432028
$$996$$ 0 0
$$997$$ 18.8832 0.598036 0.299018 0.954248i $$-0.403341\pi$$
0.299018 + 0.954248i $$0.403341\pi$$
$$998$$ 0 0
$$999$$ 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 2736.2.a.y.1.1 2
3.2 odd 2 912.2.a.n.1.2 2
4.3 odd 2 684.2.a.d.1.1 2
12.11 even 2 228.2.a.c.1.2 2
24.5 odd 2 3648.2.a.bq.1.1 2
24.11 even 2 3648.2.a.bk.1.1 2
60.23 odd 4 5700.2.f.m.3649.4 4
60.47 odd 4 5700.2.f.m.3649.1 4
60.59 even 2 5700.2.a.t.1.2 2
228.227 odd 2 4332.2.a.i.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
228.2.a.c.1.2 2 12.11 even 2
684.2.a.d.1.1 2 4.3 odd 2
912.2.a.n.1.2 2 3.2 odd 2
2736.2.a.y.1.1 2 1.1 even 1 trivial
3648.2.a.bk.1.1 2 24.11 even 2
3648.2.a.bq.1.1 2 24.5 odd 2
4332.2.a.i.1.2 2 228.227 odd 2
5700.2.a.t.1.2 2 60.59 even 2
5700.2.f.m.3649.1 4 60.47 odd 4
5700.2.f.m.3649.4 4 60.23 odd 4