# Properties

 Label 2736.2.k.b.2431.2 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

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

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.00000 q^{5} +1.73205i q^{7} +O(q^{10})$$ $$q-3.00000 q^{5} +1.73205i q^{7} +5.19615i q^{11} +6.92820i q^{13} -3.00000 q^{17} +(4.00000 - 1.73205i) q^{19} -3.46410i q^{23} +4.00000 q^{25} -4.00000 q^{31} -5.19615i q^{35} -6.92820i q^{37} +6.92820i q^{41} +8.66025i q^{43} -8.66025i q^{47} +4.00000 q^{49} -6.92820i q^{53} -15.5885i q^{55} -12.0000 q^{59} +7.00000 q^{61} -20.7846i q^{65} +8.00000 q^{67} -12.0000 q^{71} -5.00000 q^{73} -9.00000 q^{77} -8.00000 q^{79} +3.46410i q^{83} +9.00000 q^{85} +6.92820i q^{89} -12.0000 q^{91} +(-12.0000 + 5.19615i) q^{95} -6.92820i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{5}+O(q^{10})$$ 2 * q - 6 * q^5 $$2 q - 6 q^{5} - 6 q^{17} + 8 q^{19} + 8 q^{25} - 8 q^{31} + 8 q^{49} - 24 q^{59} + 14 q^{61} + 16 q^{67} - 24 q^{71} - 10 q^{73} - 18 q^{77} - 16 q^{79} + 18 q^{85} - 24 q^{91} - 24 q^{95}+O(q^{100})$$ 2 * q - 6 * q^5 - 6 * q^17 + 8 * q^19 + 8 * q^25 - 8 * q^31 + 8 * q^49 - 24 * q^59 + 14 * q^61 + 16 * q^67 - 24 * q^71 - 10 * q^73 - 18 * q^77 - 16 * q^79 + 18 * q^85 - 24 * q^91 - 24 * q^95

## 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
$$5$$ −3.00000 −1.34164 −0.670820 0.741620i $$-0.734058\pi$$
−0.670820 + 0.741620i $$0.734058\pi$$
$$6$$ 0 0
$$7$$ 1.73205i 0.654654i 0.944911 + 0.327327i $$0.106148\pi$$
−0.944911 + 0.327327i $$0.893852\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.19615i 1.56670i 0.621582 + 0.783349i $$0.286490\pi$$
−0.621582 + 0.783349i $$0.713510\pi$$
$$12$$ 0 0
$$13$$ 6.92820i 1.92154i 0.277350 + 0.960769i $$0.410544\pi$$
−0.277350 + 0.960769i $$0.589456\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −3.00000 −0.727607 −0.363803 0.931476i $$-0.618522\pi$$
−0.363803 + 0.931476i $$0.618522\pi$$
$$18$$ 0 0
$$19$$ 4.00000 1.73205i 0.917663 0.397360i
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.46410i 0.722315i −0.932505 0.361158i $$-0.882382\pi$$
0.932505 0.361158i $$-0.117618\pi$$
$$24$$ 0 0
$$25$$ 4.00000 0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −4.00000 −0.718421 −0.359211 0.933257i $$-0.616954\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 5.19615i 0.878310i
$$36$$ 0 0
$$37$$ 6.92820i 1.13899i −0.821995 0.569495i $$-0.807139\pi$$
0.821995 0.569495i $$-0.192861\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820i 1.08200i 0.841021 + 0.541002i $$0.181955\pi$$
−0.841021 + 0.541002i $$0.818045\pi$$
$$42$$ 0 0
$$43$$ 8.66025i 1.32068i 0.750968 + 0.660338i $$0.229587\pi$$
−0.750968 + 0.660338i $$0.770413\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 8.66025i 1.26323i −0.775283 0.631614i $$-0.782393\pi$$
0.775283 0.631614i $$-0.217607\pi$$
$$48$$ 0 0
$$49$$ 4.00000 0.571429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.92820i 0.951662i −0.879537 0.475831i $$-0.842147\pi$$
0.879537 0.475831i $$-0.157853\pi$$
$$54$$ 0 0
$$55$$ 15.5885i 2.10195i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −12.0000 −1.56227 −0.781133 0.624364i $$-0.785358\pi$$
−0.781133 + 0.624364i $$0.785358\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 20.7846i 2.57801i
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ 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
$$73$$ −5.00000 −0.585206 −0.292603 0.956234i $$-0.594521\pi$$
−0.292603 + 0.956234i $$0.594521\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −9.00000 −1.02565
$$78$$ 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
$$83$$ 3.46410i 0.380235i 0.981761 + 0.190117i $$0.0608868\pi$$
−0.981761 + 0.190117i $$0.939113\pi$$
$$84$$ 0 0
$$85$$ 9.00000 0.976187
$$86$$ 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
$$95$$ −12.0000 + 5.19615i −1.23117 + 0.533114i
$$96$$ 0 0
$$97$$ 6.92820i 0.703452i −0.936103 0.351726i $$-0.885595\pi$$
0.936103 0.351726i $$-0.114405\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 −0.597022 −0.298511 0.954406i $$-0.596490\pi$$
−0.298511 + 0.954406i $$0.596490\pi$$
$$102$$ 0 0
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 13.8564i 1.32720i 0.748086 + 0.663602i $$0.230973\pi$$
−0.748086 + 0.663602i $$0.769027\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 13.8564i 1.30350i 0.758433 + 0.651751i $$0.225965\pi$$
−0.758433 + 0.651751i $$0.774035\pi$$
$$114$$ 0 0
$$115$$ 10.3923i 0.969087i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 5.19615i 0.476331i
$$120$$ 0 0
$$121$$ −16.0000 −1.45455
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3.00000 0.268328
$$126$$ 0 0
$$127$$ −16.0000 −1.41977 −0.709885 0.704317i $$-0.751253\pi$$
−0.709885 + 0.704317i $$0.751253\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 15.5885i 1.36197i −0.732297 0.680985i $$-0.761552\pi$$
0.732297 0.680985i $$-0.238448\pi$$
$$132$$ 0 0
$$133$$ 3.00000 + 6.92820i 0.260133 + 0.600751i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 21.0000 1.79415 0.897076 0.441877i $$-0.145687\pi$$
0.897076 + 0.441877i $$0.145687\pi$$
$$138$$ 0 0
$$139$$ 19.0526i 1.61602i −0.589171 0.808008i $$-0.700546\pi$$
0.589171 0.808008i $$-0.299454\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −36.0000 −3.01047
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 9.00000 0.737309 0.368654 0.929567i $$-0.379819\pi$$
0.368654 + 0.929567i $$0.379819\pi$$
$$150$$ 0 0
$$151$$ −20.0000 −1.62758 −0.813788 0.581161i $$-0.802599\pi$$
−0.813788 + 0.581161i $$0.802599\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 12.0000 0.963863
$$156$$ 0 0
$$157$$ −2.00000 −0.159617 −0.0798087 0.996810i $$-0.525431\pi$$
−0.0798087 + 0.996810i $$0.525431\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 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
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −35.0000 −2.69231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ 6.92820i 0.523723i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 13.8564i 1.02994i −0.857209 0.514969i $$-0.827803\pi$$
0.857209 0.514969i $$-0.172197\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 20.7846i 1.52811i
$$186$$ 0 0
$$187$$ 15.5885i 1.13994i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.1244i 0.877288i 0.898661 + 0.438644i $$0.144541\pi$$
−0.898661 + 0.438644i $$0.855459\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ 0 0
$$199$$ 19.0526i 1.35060i −0.737543 0.675300i $$-0.764014\pi$$
0.737543 0.675300i $$-0.235986\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 20.7846i 1.45166i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 9.00000 + 20.7846i 0.622543 + 1.43770i
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 25.9808i 1.77187i
$$216$$ 0 0
$$217$$ 6.92820i 0.470317i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 20.7846i 1.39812i
$$222$$ 0 0
$$223$$ 4.00000 0.267860 0.133930 0.990991i $$-0.457240\pi$$
0.133930 + 0.990991i $$0.457240\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ −13.0000 −0.859064 −0.429532 0.903052i $$-0.641321\pi$$
−0.429532 + 0.903052i $$0.641321\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.0000 1.37576 0.687878 0.725826i $$-0.258542\pi$$
0.687878 + 0.725826i $$0.258542\pi$$
$$234$$ 0 0
$$235$$ 25.9808i 1.69480i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 22.5167i 1.45648i −0.685321 0.728241i $$-0.740338\pi$$
0.685321 0.728241i $$-0.259662\pi$$
$$240$$ 0 0
$$241$$ 6.92820i 0.446285i −0.974786 0.223142i $$-0.928369\pi$$
0.974786 0.223142i $$-0.0716315\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −12.0000 −0.766652
$$246$$ 0 0
$$247$$ 12.0000 + 27.7128i 0.763542 + 1.76332i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 1.73205i 0.109326i −0.998505 0.0546630i $$-0.982592\pi$$
0.998505 0.0546630i $$-0.0174085\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 27.7128i 1.72868i −0.502910 0.864339i $$-0.667737\pi$$
0.502910 0.864339i $$-0.332263\pi$$
$$258$$ 0 0
$$259$$ 12.0000 0.745644
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 19.0526i 1.17483i 0.809285 + 0.587416i $$0.199855\pi$$
−0.809285 + 0.587416i $$0.800145\pi$$
$$264$$ 0 0
$$265$$ 20.7846i 1.27679i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 6.92820i 0.422420i 0.977441 + 0.211210i $$0.0677404\pi$$
−0.977441 + 0.211210i $$0.932260\pi$$
$$270$$ 0 0
$$271$$ 10.3923i 0.631288i −0.948878 0.315644i $$-0.897780\pi$$
0.948878 0.315644i $$-0.102220\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 20.7846i 1.25336i
$$276$$ 0 0
$$277$$ 23.0000 1.38194 0.690968 0.722885i $$-0.257185\pi$$
0.690968 + 0.722885i $$0.257185\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 13.8564i 0.826604i −0.910594 0.413302i $$-0.864375\pi$$
0.910594 0.413302i $$-0.135625\pi$$
$$282$$ 0 0
$$283$$ 8.66025i 0.514799i 0.966305 + 0.257399i $$0.0828656\pi$$
−0.966305 + 0.257399i $$0.917134\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −8.00000 −0.470588
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 13.8564i 0.809500i 0.914427 + 0.404750i $$0.132641\pi$$
−0.914427 + 0.404750i $$0.867359\pi$$
$$294$$ 0 0
$$295$$ 36.0000 2.09600
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −15.0000 −0.864586
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −21.0000 −1.20246
$$306$$ 0 0
$$307$$ −32.0000 −1.82634 −0.913168 0.407583i $$-0.866372\pi$$
−0.913168 + 0.407583i $$0.866372\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 19.0526i 1.08037i 0.841546 + 0.540186i $$0.181646\pi$$
−0.841546 + 0.540186i $$0.818354\pi$$
$$312$$ 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
$$317$$ 34.6410i 1.94563i 0.231577 + 0.972817i $$0.425612\pi$$
−0.231577 + 0.972817i $$0.574388\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −12.0000 + 5.19615i −0.667698 + 0.289122i
$$324$$ 0 0
$$325$$ 27.7128i 1.53723i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 15.0000 0.826977
$$330$$ 0 0
$$331$$ −8.00000 −0.439720 −0.219860 0.975531i $$-0.570560\pi$$
−0.219860 + 0.975531i $$0.570560\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −24.0000 −1.31126
$$336$$ 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
$$341$$ 20.7846i 1.12555i
$$342$$ 0 0
$$343$$ 19.0526i 1.02874i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 19.0526i 1.02279i 0.859344 + 0.511397i $$0.170872\pi$$
−0.859344 + 0.511397i $$0.829128\pi$$
$$348$$ 0 0
$$349$$ −13.0000 −0.695874 −0.347937 0.937518i $$-0.613118\pi$$
−0.347937 + 0.937518i $$0.613118\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −18.0000 −0.958043 −0.479022 0.877803i $$-0.659008\pi$$
−0.479022 + 0.877803i $$0.659008\pi$$
$$354$$ 0 0
$$355$$ 36.0000 1.91068
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 15.5885i 0.822727i −0.911471 0.411364i $$-0.865053\pi$$
0.911471 0.411364i $$-0.134947\pi$$
$$360$$ 0 0
$$361$$ 13.0000 13.8564i 0.684211 0.729285i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 15.0000 0.785136
$$366$$ 0 0
$$367$$ 10.3923i 0.542474i −0.962513 0.271237i $$-0.912567\pi$$
0.962513 0.271237i $$-0.0874327\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 12.0000 0.623009
$$372$$ 0 0
$$373$$ 13.8564i 0.717458i −0.933442 0.358729i $$-0.883210\pi$$
0.933442 0.358729i $$-0.116790\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −8.00000 −0.410932 −0.205466 0.978664i $$-0.565871\pi$$
−0.205466 + 0.978664i $$0.565871\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 12.0000 0.613171 0.306586 0.951843i $$-0.400813\pi$$
0.306586 + 0.951843i $$0.400813\pi$$
$$384$$ 0 0
$$385$$ 27.0000 1.37605
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −3.00000 −0.152106 −0.0760530 0.997104i $$-0.524232\pi$$
−0.0760530 + 0.997104i $$0.524232\pi$$
$$390$$ 0 0
$$391$$ 10.3923i 0.525561i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 24.0000 1.20757
$$396$$ 0 0
$$397$$ −25.0000 −1.25471 −0.627357 0.778732i $$-0.715863\pi$$
−0.627357 + 0.778732i $$0.715863\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 13.8564i 0.691956i −0.938243 0.345978i $$-0.887547\pi$$
0.938243 0.345978i $$-0.112453\pi$$
$$402$$ 0 0
$$403$$ 27.7128i 1.38047i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 36.0000 1.78445
$$408$$ 0 0
$$409$$ 6.92820i 0.342578i 0.985221 + 0.171289i $$0.0547931\pi$$
−0.985221 + 0.171289i $$0.945207\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 20.7846i 1.02274i
$$414$$ 0 0
$$415$$ 10.3923i 0.510138i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 24.2487i 1.18463i −0.805708 0.592314i $$-0.798215\pi$$
0.805708 0.592314i $$-0.201785\pi$$
$$420$$ 0 0
$$421$$ 34.6410i 1.68830i 0.536107 + 0.844150i $$0.319894\pi$$
−0.536107 + 0.844150i $$0.680106\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −12.0000 −0.582086
$$426$$ 0 0
$$427$$ 12.1244i 0.586739i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 13.8564i 0.665896i 0.942945 + 0.332948i $$0.108043\pi$$
−0.942945 + 0.332948i $$0.891957\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.00000 13.8564i −0.287019 0.662842i
$$438$$ 0 0
$$439$$ −8.00000 −0.381819 −0.190910 0.981608i $$-0.561144\pi$$
−0.190910 + 0.981608i $$0.561144\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 5.19615i 0.246877i 0.992352 + 0.123438i $$0.0393921\pi$$
−0.992352 + 0.123438i $$0.960608\pi$$
$$444$$ 0 0
$$445$$ 20.7846i 0.985285i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 13.8564i 0.653924i 0.945037 + 0.326962i $$0.106025\pi$$
−0.945037 + 0.326962i $$0.893975\pi$$
$$450$$ 0 0
$$451$$ −36.0000 −1.69517
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 36.0000 1.68771
$$456$$ 0 0
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.00000 0.419172 0.209586 0.977790i $$-0.432788\pi$$
0.209586 + 0.977790i $$0.432788\pi$$
$$462$$ 0 0
$$463$$ 19.0526i 0.885448i −0.896658 0.442724i $$-0.854012\pi$$
0.896658 0.442724i $$-0.145988\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 32.9090i 1.52285i 0.648256 + 0.761423i $$0.275499\pi$$
−0.648256 + 0.761423i $$0.724501\pi$$
$$468$$ 0 0
$$469$$ 13.8564i 0.639829i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −45.0000 −2.06910
$$474$$ 0 0
$$475$$ 16.0000 6.92820i 0.734130 0.317888i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 17.3205i 0.791394i −0.918381 0.395697i $$-0.870503\pi$$
0.918381 0.395697i $$-0.129497\pi$$
$$480$$ 0 0
$$481$$ 48.0000 2.18861
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 20.7846i 0.943781i
$$486$$ 0 0
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 10.3923i 0.468998i −0.972116 0.234499i $$-0.924655\pi$$
0.972116 0.234499i $$-0.0753450\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 20.7846i 0.932317i
$$498$$ 0 0
$$499$$ 36.3731i 1.62828i 0.580667 + 0.814141i $$0.302792\pi$$
−0.580667 + 0.814141i $$0.697208\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.46410i 0.154457i −0.997013 0.0772283i $$-0.975393\pi$$
0.997013 0.0772283i $$-0.0246070\pi$$
$$504$$ 0 0
$$505$$ 18.0000 0.800989
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 6.92820i 0.307087i 0.988142 + 0.153544i $$0.0490686\pi$$
−0.988142 + 0.153544i $$0.950931\pi$$
$$510$$ 0 0
$$511$$ 8.66025i 0.383107i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 24.0000 1.05757
$$516$$ 0 0
$$517$$ 45.0000 1.97910
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 13.8564i 0.607060i 0.952822 + 0.303530i $$0.0981653\pi$$
−0.952822 + 0.303530i $$0.901835\pi$$
$$522$$ 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
$$527$$ 12.0000 0.522728
$$528$$ 0 0
$$529$$ 11.0000 0.478261
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −48.0000 −2.07911
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 20.7846i 0.895257i
$$540$$ 0 0
$$541$$ 7.00000 0.300954 0.150477 0.988614i $$-0.451919\pi$$
0.150477 + 0.988614i $$0.451919\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 41.5692i 1.78063i
$$546$$ 0 0
$$547$$ 28.0000 1.19719 0.598597 0.801050i $$-0.295725\pi$$
0.598597 + 0.801050i $$0.295725\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 13.8564i 0.589234i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −3.00000 −0.127114 −0.0635570 0.997978i $$-0.520244\pi$$
−0.0635570 + 0.997978i $$0.520244\pi$$
$$558$$ 0 0
$$559$$ −60.0000 −2.53773
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −36.0000 −1.51722 −0.758610 0.651546i $$-0.774121\pi$$
−0.758610 + 0.651546i $$0.774121\pi$$
$$564$$ 0 0
$$565$$ 41.5692i 1.74883i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.92820i 0.290445i −0.989399 0.145223i $$-0.953610\pi$$
0.989399 0.145223i $$-0.0463899\pi$$
$$570$$ 0 0
$$571$$ 45.0333i 1.88459i −0.334790 0.942293i $$-0.608665\pi$$
0.334790 0.942293i $$-0.391335\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 13.8564i 0.577852i
$$576$$ 0 0
$$577$$ −5.00000 −0.208153 −0.104076 0.994569i $$-0.533189\pi$$
−0.104076 + 0.994569i $$0.533189\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.00000 −0.248922
$$582$$ 0 0
$$583$$ 36.0000 1.49097
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 25.9808i 1.07234i 0.844110 + 0.536170i $$0.180130\pi$$
−0.844110 + 0.536170i $$0.819870\pi$$
$$588$$ 0 0
$$589$$ −16.0000 + 6.92820i −0.659269 + 0.285472i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 15.5885i 0.639064i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −24.0000 −0.980613 −0.490307 0.871550i $$-0.663115\pi$$
−0.490307 + 0.871550i $$0.663115\pi$$
$$600$$ 0 0
$$601$$ 34.6410i 1.41304i 0.707695 + 0.706518i $$0.249735\pi$$
−0.707695 + 0.706518i $$0.750265\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 48.0000 1.95148
$$606$$ 0 0
$$607$$ −4.00000 −0.162355 −0.0811775 0.996700i $$-0.525868\pi$$
−0.0811775 + 0.996700i $$0.525868\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 60.0000 2.42734
$$612$$ 0 0
$$613$$ 31.0000 1.25208 0.626039 0.779792i $$-0.284675\pi$$
0.626039 + 0.779792i $$0.284675\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.00000 0.362326 0.181163 0.983453i $$-0.442014\pi$$
0.181163 + 0.983453i $$0.442014\pi$$
$$618$$ 0 0
$$619$$ 10.3923i 0.417702i 0.977947 + 0.208851i $$0.0669724\pi$$
−0.977947 + 0.208851i $$0.933028\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −12.0000 −0.480770
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 20.7846i 0.828737i
$$630$$ 0 0
$$631$$ 8.66025i 0.344759i 0.985031 + 0.172380i $$0.0551456\pi$$
−0.985031 + 0.172380i $$0.944854\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 48.0000 1.90482
$$636$$ 0 0
$$637$$ 27.7128i 1.09802i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 20.7846i 0.820943i 0.911873 + 0.410471i $$0.134636\pi$$
−0.911873 + 0.410471i $$0.865364\pi$$
$$642$$ 0 0
$$643$$ 36.3731i 1.43441i 0.696860 + 0.717207i $$0.254580\pi$$
−0.696860 + 0.717207i $$0.745420\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 25.9808i 1.02141i 0.859756 + 0.510705i $$0.170615\pi$$
−0.859756 + 0.510705i $$0.829385\pi$$
$$648$$ 0 0
$$649$$ 62.3538i 2.44760i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 21.0000 0.821794 0.410897 0.911682i $$-0.365216\pi$$
0.410897 + 0.911682i $$0.365216\pi$$
$$654$$ 0 0
$$655$$ 46.7654i 1.82727i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −48.0000 −1.86981 −0.934907 0.354892i $$-0.884518\pi$$
−0.934907 + 0.354892i $$0.884518\pi$$
$$660$$ 0 0
$$661$$ 41.5692i 1.61686i 0.588596 + 0.808428i $$0.299681\pi$$
−0.588596 + 0.808428i $$0.700319\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −9.00000 20.7846i −0.349005 0.805993i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 36.3731i 1.40417i
$$672$$ 0 0
$$673$$ 34.6410i 1.33531i 0.744469 + 0.667657i $$0.232703\pi$$
−0.744469 + 0.667657i $$0.767297\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 27.7128i 1.06509i 0.846402 + 0.532545i $$0.178764\pi$$
−0.846402 + 0.532545i $$0.821236\pi$$
$$678$$ 0 0
$$679$$ 12.0000 0.460518
$$680$$ 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
$$685$$ −63.0000 −2.40711
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 48.0000 1.82865
$$690$$ 0 0
$$691$$ 1.73205i 0.0658903i 0.999457 + 0.0329452i $$0.0104887\pi$$
−0.999457 + 0.0329452i $$0.989511\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 57.1577i 2.16811i
$$696$$ 0 0
$$697$$ 20.7846i 0.787273i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −30.0000 −1.13308 −0.566542 0.824033i $$-0.691719\pi$$
−0.566542 + 0.824033i $$0.691719\pi$$
$$702$$ 0 0
$$703$$ −12.0000 27.7128i −0.452589 1.04521i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 10.3923i 0.390843i
$$708$$ 0 0
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 13.8564i 0.518927i
$$714$$ 0 0
$$715$$ 108.000 4.03897
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 1.73205i 0.0645946i −0.999478 0.0322973i $$-0.989718\pi$$
0.999478 0.0322973i $$-0.0102823\pi$$
$$720$$ 0 0
$$721$$ 13.8564i 0.516040i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 25.9808i 0.963573i −0.876289 0.481787i $$-0.839988\pi$$
0.876289 0.481787i $$-0.160012\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 25.9808i 0.960933i
$$732$$ 0 0
$$733$$ −2.00000 −0.0738717 −0.0369358 0.999318i $$-0.511760\pi$$
−0.0369358 + 0.999318i $$0.511760\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 41.5692i 1.53122i
$$738$$ 0 0
$$739$$ 12.1244i 0.446002i −0.974818 0.223001i $$-0.928415\pi$$
0.974818 0.223001i $$-0.0715853\pi$$
$$740$$ 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
$$745$$ −27.0000 −0.989203
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 40.0000 1.45962 0.729810 0.683650i $$-0.239608\pi$$
0.729810 + 0.683650i $$0.239608\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 60.0000 2.18362
$$756$$ 0 0
$$757$$ −13.0000 −0.472493 −0.236247 0.971693i $$-0.575917\pi$$
−0.236247 + 0.971693i $$0.575917\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −3.00000 −0.108750 −0.0543750 0.998521i $$-0.517317\pi$$
−0.0543750 + 0.998521i $$0.517317\pi$$
$$762$$ 0 0
$$763$$ −24.0000 −0.868858
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 83.1384i 3.00196i
$$768$$ 0 0
$$769$$ −5.00000 −0.180305 −0.0901523 0.995928i $$-0.528735\pi$$
−0.0901523 + 0.995928i $$0.528735\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 34.6410i 1.24595i −0.782241 0.622975i $$-0.785924\pi$$
0.782241 0.622975i $$-0.214076\pi$$
$$774$$ 0 0
$$775$$ −16.0000 −0.574737
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 12.0000 + 27.7128i 0.429945 + 0.992915i
$$780$$ 0 0
$$781$$ 62.3538i 2.23120i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 6.00000 0.214149
$$786$$ 0 0
$$787$$ 8.00000 0.285169 0.142585 0.989783i $$-0.454459\pi$$
0.142585 + 0.989783i $$0.454459\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24.0000 −0.853342
$$792$$ 0 0
$$793$$ 48.4974i 1.72219i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 25.9808i 0.919133i
$$800$$ 0 0
$$801$$