# Properties

 Label 2736.2.k.f.2431.1 Level $2736$ Weight $2$ Character 2736.2431 Analytic conductor $21.847$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2431,2736)]

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([1, 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("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.k (of order $$2$$, degree $$1$$, 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: $$21.8470699930$$ 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: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2431.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2736.2431 Dual form 2736.2.k.f.2431.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.46410i q^{7} +O(q^{10})$$ $$q-3.46410i q^{7} +3.46410i q^{11} +3.46410i q^{13} +6.00000 q^{17} +(4.00000 + 1.73205i) q^{19} -5.00000 q^{25} +6.92820i q^{29} -10.0000 q^{31} +3.46410i q^{37} -6.92820i q^{41} -10.3923i q^{43} +6.92820i q^{47} -5.00000 q^{49} +13.8564i q^{53} +12.0000 q^{59} +10.0000 q^{61} -4.00000 q^{67} +12.0000 q^{71} -2.00000 q^{73} +12.0000 q^{77} +10.0000 q^{79} -3.46410i q^{83} +6.92820i q^{89} +12.0000 q^{91} +6.92820i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q + 12 q^{17} + 8 q^{19} - 10 q^{25} - 20 q^{31} - 10 q^{49} + 24 q^{59} + 20 q^{61} - 8 q^{67} + 24 q^{71} - 4 q^{73} + 24 q^{77} + 20 q^{79} + 24 q^{91}+O(q^{100})$$ 2 * q + 12 * q^17 + 8 * q^19 - 10 * q^25 - 20 * q^31 - 10 * q^49 + 24 * q^59 + 20 * q^61 - 8 * q^67 + 24 * q^71 - 4 * q^73 + 24 * q^77 + 20 * q^79 + 24 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$6$$ 0 0
$$7$$ 3.46410i 1.30931i −0.755929 0.654654i $$-0.772814\pi$$
0.755929 0.654654i $$-0.227186\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.46410i 1.04447i 0.852803 + 0.522233i $$0.174901\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ 3.46410i 0.960769i 0.877058 + 0.480384i $$0.159503\pi$$
−0.877058 + 0.480384i $$0.840497\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 0 0
$$19$$ 4.00000 + 1.73205i 0.917663 + 0.397360i
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ −5.00000 −1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.92820i 1.28654i 0.765641 + 0.643268i $$0.222422\pi$$
−0.765641 + 0.643268i $$0.777578\pi$$
$$30$$ 0 0
$$31$$ −10.0000 −1.79605 −0.898027 0.439941i $$-0.854999\pi$$
−0.898027 + 0.439941i $$0.854999\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.46410i 0.569495i 0.958603 + 0.284747i $$0.0919097\pi$$
−0.958603 + 0.284747i $$0.908090\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820i 1.08200i −0.841021 0.541002i $$-0.818045\pi$$
0.841021 0.541002i $$-0.181955\pi$$
$$42$$ 0 0
$$43$$ 10.3923i 1.58481i −0.609994 0.792406i $$-0.708828\pi$$
0.609994 0.792406i $$-0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.92820i 1.01058i 0.862949 + 0.505291i $$0.168615\pi$$
−0.862949 + 0.505291i $$0.831385\pi$$
$$48$$ 0 0
$$49$$ −5.00000 −0.714286
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 13.8564i 1.90332i 0.307148 + 0.951662i $$0.400625\pi$$
−0.307148 + 0.951662i $$0.599375\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 12.0000 1.56227 0.781133 0.624364i $$-0.214642\pi$$
0.781133 + 0.624364i $$0.214642\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 12.0000 1.36753
$$78$$ 0 0
$$79$$ 10.0000 1.12509 0.562544 0.826767i $$-0.309823\pi$$
0.562544 + 0.826767i $$0.309823\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.46410i 0.380235i −0.981761 0.190117i $$-0.939113\pi$$
0.981761 0.190117i $$-0.0608868\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.92820i 0.734388i 0.930144 + 0.367194i $$0.119682\pi$$
−0.930144 + 0.367194i $$0.880318\pi$$
$$90$$ 0 0
$$91$$ 12.0000 1.25794
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 6.92820i 0.703452i 0.936103 + 0.351726i $$0.114405\pi$$
−0.936103 + 0.351726i $$0.885595\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ −2.00000 −0.197066 −0.0985329 0.995134i $$-0.531415\pi$$
−0.0985329 + 0.995134i $$0.531415\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ 10.3923i 0.995402i 0.867349 + 0.497701i $$0.165822\pi$$
−0.867349 + 0.497701i $$0.834178\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.92820i 0.651751i 0.945413 + 0.325875i $$0.105659\pi$$
−0.945413 + 0.325875i $$0.894341\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 20.7846i 1.90532i
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2.00000 0.177471 0.0887357 0.996055i $$-0.471717\pi$$
0.0887357 + 0.996055i $$0.471717\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 10.3923i 0.907980i −0.891007 0.453990i $$-0.850000\pi$$
0.891007 0.453990i $$-0.150000\pi$$
$$132$$ 0 0
$$133$$ 6.00000 13.8564i 0.520266 1.20150i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 18.0000 1.53784 0.768922 0.639343i $$-0.220793\pi$$
0.768922 + 0.639343i $$0.220793\pi$$
$$138$$ 0 0
$$139$$ 3.46410i 0.293821i −0.989150 0.146911i $$-0.953067\pi$$
0.989150 0.146911i $$-0.0469330\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −12.0000 −1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −24.0000 −1.96616 −0.983078 0.183186i $$-0.941359\pi$$
−0.983078 + 0.183186i $$0.941359\pi$$
$$150$$ 0 0
$$151$$ 10.0000 0.813788 0.406894 0.913475i $$-0.366612\pi$$
0.406894 + 0.913475i $$0.366612\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 3.46410i 0.271329i −0.990755 0.135665i $$-0.956683\pi$$
0.990755 0.135665i $$-0.0433170\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −12.0000 −0.928588 −0.464294 0.885681i $$-0.653692\pi$$
−0.464294 + 0.885681i $$0.653692\pi$$
$$168$$ 0 0
$$169$$ 1.00000 0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 13.8564i 1.05348i 0.850026 + 0.526742i $$0.176586\pi$$
−0.850026 + 0.526742i $$0.823414\pi$$
$$174$$ 0 0
$$175$$ 17.3205i 1.30931i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 3.46410i 0.257485i −0.991678 0.128742i $$-0.958906\pi$$
0.991678 0.128742i $$-0.0410940\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 20.7846i 1.51992i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.8564i 1.00261i 0.865269 + 0.501307i $$0.167147\pi$$
−0.865269 + 0.501307i $$0.832853\pi$$
$$192$$ 0 0
$$193$$ 13.8564i 0.997406i 0.866773 + 0.498703i $$0.166190\pi$$
−0.866773 + 0.498703i $$0.833810\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −12.0000 −0.854965 −0.427482 0.904024i $$-0.640599\pi$$
−0.427482 + 0.904024i $$0.640599\pi$$
$$198$$ 0 0
$$199$$ 3.46410i 0.245564i 0.992434 + 0.122782i $$0.0391815\pi$$
−0.992434 + 0.122782i $$0.960818\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 24.0000 1.68447
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −6.00000 + 13.8564i −0.415029 + 0.958468i
$$210$$ 0 0
$$211$$ −4.00000 −0.275371 −0.137686 0.990476i $$-0.543966\pi$$
−0.137686 + 0.990476i $$0.543966\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 34.6410i 2.35159i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.7846i 1.39812i
$$222$$ 0 0
$$223$$ −2.00000 −0.133930 −0.0669650 0.997755i $$-0.521332\pi$$
−0.0669650 + 0.997755i $$0.521332\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 26.0000 1.71813 0.859064 0.511868i $$-0.171046\pi$$
0.859064 + 0.511868i $$0.171046\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 18.0000 1.17922 0.589610 0.807688i $$-0.299282\pi$$
0.589610 + 0.807688i $$0.299282\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.92820i 0.448148i 0.974572 + 0.224074i $$0.0719358\pi$$
−0.974572 + 0.224074i $$0.928064\pi$$
$$240$$ 0 0
$$241$$ 27.7128i 1.78514i −0.450910 0.892570i $$-0.648900\pi$$
0.450910 0.892570i $$-0.351100\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −6.00000 + 13.8564i −0.381771 + 0.881662i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 31.1769i 1.96787i −0.178529 0.983935i $$-0.557134\pi$$
0.178529 0.983935i $$-0.442866\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 20.7846i 1.29651i 0.761424 + 0.648254i $$0.224501\pi$$
−0.761424 + 0.648254i $$0.775499\pi$$
$$258$$ 0 0
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 6.92820i 0.427211i −0.976920 0.213606i $$-0.931479\pi$$
0.976920 0.213606i $$-0.0685208\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 24.2487i 1.47300i −0.676435 0.736502i $$-0.736476\pi$$
0.676435 0.736502i $$-0.263524\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 17.3205i 1.04447i
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 20.7846i 1.23991i −0.784639 0.619953i $$-0.787152\pi$$
0.784639 0.619953i $$-0.212848\pi$$
$$282$$ 0 0
$$283$$ 10.3923i 0.617758i 0.951101 + 0.308879i $$0.0999539\pi$$
−0.951101 + 0.308879i $$0.900046\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −24.0000 −1.41668
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 13.8564i 0.809500i −0.914427 0.404750i $$-0.867359\pi$$
0.914427 0.404750i $$-0.132641\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ −36.0000 −2.07501
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 4.00000 0.228292 0.114146 0.993464i $$-0.463587\pi$$
0.114146 + 0.993464i $$0.463587\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 13.8564i 0.785725i 0.919597 + 0.392862i $$0.128515\pi$$
−0.919597 + 0.392862i $$0.871485\pi$$
$$312$$ 0 0
$$313$$ −26.0000 −1.46961 −0.734803 0.678280i $$-0.762726\pi$$
−0.734803 + 0.678280i $$0.762726\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ −24.0000 −1.34374
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 24.0000 + 10.3923i 1.33540 + 0.578243i
$$324$$ 0 0
$$325$$ 17.3205i 0.960769i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 24.0000 1.32316
$$330$$ 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
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 6.92820i 0.377403i −0.982034 0.188702i $$-0.939572\pi$$
0.982034 0.188702i $$-0.0604279\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 34.6410i 1.87592i
$$342$$ 0 0
$$343$$ 6.92820i 0.374088i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 10.3923i 0.557888i −0.960307 0.278944i $$-0.910016\pi$$
0.960307 0.278944i $$-0.0899844\pi$$
$$348$$ 0 0
$$349$$ −10.0000 −0.535288 −0.267644 0.963518i $$-0.586245\pi$$
−0.267644 + 0.963518i $$0.586245\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 20.7846i 1.09697i 0.836160 + 0.548485i $$0.184795\pi$$
−0.836160 + 0.548485i $$0.815205\pi$$
$$360$$ 0 0
$$361$$ 13.0000 + 13.8564i 0.684211 + 0.729285i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 3.46410i 0.180825i 0.995904 + 0.0904123i $$0.0288185\pi$$
−0.995904 + 0.0904123i $$0.971182\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 48.0000 2.49204
$$372$$ 0 0
$$373$$ 38.1051i 1.97301i −0.163737 0.986504i $$-0.552355\pi$$
0.163737 0.986504i $$-0.447645\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −24.0000 −1.23606
$$378$$ 0 0
$$379$$ 16.0000 0.821865 0.410932 0.911666i $$-0.365203\pi$$
0.410932 + 0.911666i $$0.365203\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12.0000 −0.613171 −0.306586 0.951843i $$-0.599187\pi$$
−0.306586 + 0.951843i $$0.599187\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 20.7846i 1.03793i −0.854794 0.518967i $$-0.826317\pi$$
0.854794 0.518967i $$-0.173683\pi$$
$$402$$ 0 0
$$403$$ 34.6410i 1.72559i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −12.0000 −0.594818
$$408$$ 0 0
$$409$$ 34.6410i 1.71289i −0.516240 0.856444i $$-0.672669\pi$$
0.516240 0.856444i $$-0.327331\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 41.5692i 2.04549i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 17.3205i 0.846162i 0.906092 + 0.423081i $$0.139051\pi$$
−0.906092 + 0.423081i $$0.860949\pi$$
$$420$$ 0 0
$$421$$ 24.2487i 1.18181i 0.806741 + 0.590905i $$0.201229\pi$$
−0.806741 + 0.590905i $$0.798771\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −30.0000 −1.45521
$$426$$ 0 0
$$427$$ 34.6410i 1.67640i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 20.7846i 0.998845i 0.866359 + 0.499422i $$0.166454\pi$$
−0.866359 + 0.499422i $$0.833546\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −2.00000 −0.0954548 −0.0477274 0.998860i $$-0.515198\pi$$
−0.0477274 + 0.998860i $$0.515198\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 17.3205i 0.822922i 0.911427 + 0.411461i $$0.134981\pi$$
−0.911427 + 0.411461i $$0.865019\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 34.6410i 1.63481i 0.576063 + 0.817405i $$0.304588\pi$$
−0.576063 + 0.817405i $$0.695412\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 12.0000 0.558896 0.279448 0.960161i $$-0.409849\pi$$
0.279448 + 0.960161i $$0.409849\pi$$
$$462$$ 0 0
$$463$$ 3.46410i 0.160990i 0.996755 + 0.0804952i $$0.0256502\pi$$
−0.996755 + 0.0804952i $$0.974350\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 38.1051i 1.76329i 0.471909 + 0.881647i $$0.343565\pi$$
−0.471909 + 0.881647i $$0.656435\pi$$
$$468$$ 0 0
$$469$$ 13.8564i 0.639829i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 36.0000 1.65528
$$474$$ 0 0
$$475$$ −20.0000 8.66025i −0.917663 0.397360i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 20.7846i 0.949673i −0.880074 0.474837i $$-0.842507\pi$$
0.880074 0.474837i $$-0.157493\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −34.0000 −1.54069 −0.770344 0.637629i $$-0.779915\pi$$
−0.770344 + 0.637629i $$0.779915\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 17.3205i 0.781664i 0.920462 + 0.390832i $$0.127813\pi$$
−0.920462 + 0.390832i $$0.872187\pi$$
$$492$$ 0 0
$$493$$ 41.5692i 1.87218i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 41.5692i 1.86463i
$$498$$ 0 0
$$499$$ 31.1769i 1.39567i 0.716258 + 0.697835i $$0.245853\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 13.8564i 0.617827i 0.951090 + 0.308913i $$0.0999653\pi$$
−0.951090 + 0.308913i $$0.900035\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6.92820i 0.307087i −0.988142 0.153544i $$-0.950931\pi$$
0.988142 0.153544i $$-0.0490686\pi$$
$$510$$ 0 0
$$511$$ 6.92820i 0.306486i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −24.0000 −1.05552
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 20.7846i 0.910590i −0.890341 0.455295i $$-0.849534\pi$$
0.890341 0.455295i $$-0.150466\pi$$
$$522$$ 0 0
$$523$$ −16.0000 −0.699631 −0.349816 0.936819i $$-0.613756\pi$$
−0.349816 + 0.936819i $$0.613756\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −60.0000 −2.61364
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 24.0000 1.03956
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 17.3205i 0.746047i
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −32.0000 −1.36822 −0.684111 0.729378i $$-0.739809\pi$$
−0.684111 + 0.729378i $$0.739809\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 + 27.7128i −0.511217 + 1.18061i
$$552$$ 0 0
$$553$$ 34.6410i 1.47309i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −12.0000 −0.508456 −0.254228 0.967144i $$-0.581821\pi$$
−0.254228 + 0.967144i $$0.581821\pi$$
$$558$$ 0 0
$$559$$ 36.0000 1.52264
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −12.0000 −0.505740 −0.252870 0.967500i $$-0.581374\pi$$
−0.252870 + 0.967500i $$0.581374\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 20.7846i 0.871336i −0.900107 0.435668i $$-0.856512\pi$$
0.900107 0.435668i $$-0.143488\pi$$
$$570$$ 0 0
$$571$$ 24.2487i 1.01478i −0.861717 0.507388i $$-0.830611\pi$$
0.861717 0.507388i $$-0.169389\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 10.0000 0.416305 0.208153 0.978096i $$-0.433255\pi$$
0.208153 + 0.978096i $$0.433255\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −12.0000 −0.497844
$$582$$ 0 0
$$583$$ −48.0000 −1.98796
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 10.3923i 0.428936i 0.976731 + 0.214468i $$0.0688018\pi$$
−0.976731 + 0.214468i $$0.931198\pi$$
$$588$$ 0 0
$$589$$ −40.0000 17.3205i −1.64817 0.713679i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −12.0000 −0.490307 −0.245153 0.969484i $$-0.578838\pi$$
−0.245153 + 0.969484i $$0.578838\pi$$
$$600$$ 0 0
$$601$$ 20.7846i 0.847822i −0.905704 0.423911i $$-0.860657\pi$$
0.905704 0.423911i $$-0.139343\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −22.0000 −0.892952 −0.446476 0.894795i $$-0.647321\pi$$
−0.446476 + 0.894795i $$0.647321\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 10.0000 0.403896 0.201948 0.979396i $$-0.435273\pi$$
0.201948 + 0.979396i $$0.435273\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −42.0000 −1.69086 −0.845428 0.534089i $$-0.820655\pi$$
−0.845428 + 0.534089i $$0.820655\pi$$
$$618$$ 0 0
$$619$$ 45.0333i 1.81004i 0.425367 + 0.905021i $$0.360145\pi$$
−0.425367 + 0.905021i $$0.639855\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 24.0000 0.961540
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 20.7846i 0.828737i
$$630$$ 0 0
$$631$$ 17.3205i 0.689519i −0.938691 0.344759i $$-0.887961\pi$$
0.938691 0.344759i $$-0.112039\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 17.3205i 0.686264i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 6.92820i 0.273648i 0.990595 + 0.136824i $$0.0436894\pi$$
−0.990595 + 0.136824i $$0.956311\pi$$
$$642$$ 0 0
$$643$$ 31.1769i 1.22950i −0.788723 0.614749i $$-0.789257\pi$$
0.788723 0.614749i $$-0.210743\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 13.8564i 0.544752i −0.962191 0.272376i $$-0.912191\pi$$
0.962191 0.272376i $$-0.0878094\pi$$
$$648$$ 0 0
$$649$$ 41.5692i 1.63173i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 36.0000 1.40879 0.704394 0.709809i $$-0.251219\pi$$
0.704394 + 0.709809i $$0.251219\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 24.2487i 0.943166i −0.881822 0.471583i $$-0.843683\pi$$
0.881822 0.471583i $$-0.156317\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 34.6410i 1.33730i
$$672$$ 0 0
$$673$$ 34.6410i 1.33531i −0.744469 0.667657i $$-0.767297\pi$$
0.744469 0.667657i $$-0.232703\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 24.0000 0.921035
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −48.0000 −1.82865
$$690$$ 0 0
$$691$$ 31.1769i 1.18603i 0.805193 + 0.593013i $$0.202062\pi$$
−0.805193 + 0.593013i $$0.797938\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 41.5692i 1.57455i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 36.0000 1.35970 0.679851 0.733351i $$-0.262045\pi$$
0.679851 + 0.733351i $$0.262045\pi$$
$$702$$ 0 0
$$703$$ −6.00000 + 13.8564i −0.226294 + 0.522604i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 22.0000 0.826227 0.413114 0.910679i $$-0.364441\pi$$
0.413114 + 0.910679i $$0.364441\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 34.6410i 1.29189i −0.763383 0.645946i $$-0.776463\pi$$
0.763383 0.645946i $$-0.223537\pi$$
$$720$$ 0 0
$$721$$ 6.92820i 0.258020i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 34.6410i 1.28654i
$$726$$ 0 0
$$727$$ 10.3923i 0.385429i −0.981255 0.192715i $$-0.938271\pi$$
0.981255 0.192715i $$-0.0617292\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 62.3538i 2.30624i
$$732$$ 0 0
$$733$$ −26.0000 −0.960332 −0.480166 0.877178i $$-0.659424\pi$$
−0.480166 + 0.877178i $$0.659424\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 13.8564i 0.510407i
$$738$$ 0 0
$$739$$ 10.3923i 0.382287i 0.981562 + 0.191144i $$0.0612196\pi$$
−0.981562 + 0.191144i $$0.938780\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 41.5692i 1.51891i
$$750$$ 0 0
$$751$$ −38.0000 −1.38664 −0.693320 0.720630i $$-0.743853\pi$$
−0.693320 + 0.720630i $$0.743853\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −30.0000 −1.08750 −0.543750 0.839248i $$-0.682996\pi$$
−0.543750 + 0.839248i $$0.682996\pi$$
$$762$$ 0 0
$$763$$ 36.0000 1.30329
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 41.5692i 1.50098i
$$768$$ 0 0
$$769$$ −2.00000 −0.0721218 −0.0360609 0.999350i $$-0.511481\pi$$
−0.0360609 + 0.999350i $$0.511481\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 34.6410i 1.24595i 0.782241 + 0.622975i $$0.214076\pi$$
−0.782241 + 0.622975i $$0.785924\pi$$
$$774$$ 0 0
$$775$$ 50.0000 1.79605
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.0000 27.7128i 0.429945 0.992915i
$$780$$ 0 0
$$781$$ 41.5692i 1.48746i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 32.0000 1.14068 0.570338 0.821410i $$-0.306812\pi$$
0.570338 + 0.821410i $$0.306812\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 24.0000 0.853342
$$792$$ 0 0
$$793$$ 34.6410i 1.23014i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 48.4974i 1.71787i −0.512087 0.858933i $$-0.671128\pi$$
0.512087 0.858933i $$-0.328872\pi$$
$$798$$ 0 0
$$799$$ 41.5692i 1.47061i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 6.92820i 0.244491i
$$804$$ 0 0
$$805$$ 0 0