# Properties

 Label 4680.2.l.i.2809.7 Level $4680$ Weight $2$ Character 4680.2809 Analytic conductor $37.370$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4680.l (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: $$37.3699881460$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 4x^{9} + 2x^{8} + 16x^{7} - 15x^{6} - 40x^{5} - 75x^{4} + 400x^{3} + 250x^{2} - 2500x + 3125$$ x^10 - 4*x^9 + 2*x^8 + 16*x^7 - 15*x^6 - 40*x^5 - 75*x^4 + 400*x^3 + 250*x^2 - 2500*x + 3125 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 520) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 2809.7 Root $$1.64680 + 1.51263i$$ of defining polynomial Character $$\chi$$ $$=$$ 4680.2809 Dual form 4680.2.l.i.2809.8

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+(1.64680 - 1.51263i) q^{5} -3.54010i q^{7} +O(q^{10})$$ $$q+(1.64680 - 1.51263i) q^{5} -3.54010i q^{7} +3.86747 q^{11} +1.00000i q^{13} +1.29361i q^{17} -3.69006 q^{19} -8.38231i q^{23} +(0.423922 - 4.98200i) q^{25} -7.53566 q^{29} -8.36045 q^{31} +(-5.35484 - 5.82984i) q^{35} +1.04712i q^{37} +2.97032 q^{41} -4.96180i q^{43} +0.459905i q^{47} -5.53228 q^{49} -12.2279i q^{53} +(6.36896 - 5.85004i) q^{55} +3.28026 q^{59} +0.582376 q^{61} +(1.51263 + 1.64680i) q^{65} +8.15554i q^{67} +11.8315 q^{71} +11.7131i q^{73} -13.6912i q^{77} -3.87310 q^{79} +3.54453i q^{83} +(1.95674 + 2.13032i) q^{85} +1.46329 q^{89} +3.54010 q^{91} +(-6.07680 + 5.58168i) q^{95} -2.90278i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 4 q^{5}+O(q^{10})$$ 10 * q + 4 * q^5 $$10 q + 4 q^{5} - 16 q^{11} - 36 q^{19} + 12 q^{25} + 4 q^{29} - 8 q^{31} - 18 q^{35} + 24 q^{41} - 38 q^{49} + 32 q^{55} + 28 q^{59} + 20 q^{61} - 4 q^{65} - 36 q^{71} - 16 q^{79} + 8 q^{85} - 12 q^{89} - 52 q^{95}+O(q^{100})$$ 10 * q + 4 * q^5 - 16 * q^11 - 36 * q^19 + 12 * q^25 + 4 * q^29 - 8 * q^31 - 18 * q^35 + 24 * q^41 - 38 * q^49 + 32 * q^55 + 28 * q^59 + 20 * q^61 - 4 * q^65 - 36 * q^71 - 16 * q^79 + 8 * q^85 - 12 * q^89 - 52 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$$.

 $$n$$ $$937$$ $$1081$$ $$2081$$ $$2341$$ $$3511$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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
$$5$$ 1.64680 1.51263i 0.736473 0.676467i
$$6$$ 0 0
$$7$$ 3.54010i 1.33803i −0.743249 0.669015i $$-0.766716\pi$$
0.743249 0.669015i $$-0.233284\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 3.86747 1.16609 0.583043 0.812441i $$-0.301862\pi$$
0.583043 + 0.812441i $$0.301862\pi$$
$$12$$ 0 0
$$13$$ 1.00000i 0.277350i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.29361i 0.313746i 0.987619 + 0.156873i $$0.0501413\pi$$
−0.987619 + 0.156873i $$0.949859\pi$$
$$18$$ 0 0
$$19$$ −3.69006 −0.846558 −0.423279 0.905999i $$-0.639121\pi$$
−0.423279 + 0.905999i $$0.639121\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 8.38231i 1.74783i −0.486076 0.873916i $$-0.661572\pi$$
0.486076 0.873916i $$-0.338428\pi$$
$$24$$ 0 0
$$25$$ 0.423922 4.98200i 0.0847844 0.996399i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −7.53566 −1.39934 −0.699669 0.714467i $$-0.746669\pi$$
−0.699669 + 0.714467i $$0.746669\pi$$
$$30$$ 0 0
$$31$$ −8.36045 −1.50158 −0.750790 0.660541i $$-0.770327\pi$$
−0.750790 + 0.660541i $$0.770327\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −5.35484 5.82984i −0.905134 0.985423i
$$36$$ 0 0
$$37$$ 1.04712i 0.172145i 0.996289 + 0.0860725i $$0.0274317\pi$$
−0.996289 + 0.0860725i $$0.972568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 2.97032 0.463885 0.231943 0.972729i $$-0.425492\pi$$
0.231943 + 0.972729i $$0.425492\pi$$
$$42$$ 0 0
$$43$$ 4.96180i 0.756668i −0.925669 0.378334i $$-0.876497\pi$$
0.925669 0.378334i $$-0.123503\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0.459905i 0.0670840i 0.999437 + 0.0335420i $$0.0106788\pi$$
−0.999437 + 0.0335420i $$0.989321\pi$$
$$48$$ 0 0
$$49$$ −5.53228 −0.790325
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 12.2279i 1.67963i −0.542870 0.839817i $$-0.682662\pi$$
0.542870 0.839817i $$-0.317338\pi$$
$$54$$ 0 0
$$55$$ 6.36896 5.85004i 0.858790 0.788819i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.28026 0.427053 0.213526 0.976937i $$-0.431505\pi$$
0.213526 + 0.976937i $$0.431505\pi$$
$$60$$ 0 0
$$61$$ 0.582376 0.0745656 0.0372828 0.999305i $$-0.488130\pi$$
0.0372828 + 0.999305i $$0.488130\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 1.51263 + 1.64680i 0.187618 + 0.204261i
$$66$$ 0 0
$$67$$ 8.15554i 0.996358i 0.867074 + 0.498179i $$0.165998\pi$$
−0.867074 + 0.498179i $$0.834002\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 11.8315 1.40414 0.702068 0.712110i $$-0.252260\pi$$
0.702068 + 0.712110i $$0.252260\pi$$
$$72$$ 0 0
$$73$$ 11.7131i 1.37091i 0.728114 + 0.685456i $$0.240397\pi$$
−0.728114 + 0.685456i $$0.759603\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 13.6912i 1.56026i
$$78$$ 0 0
$$79$$ −3.87310 −0.435757 −0.217879 0.975976i $$-0.569914\pi$$
−0.217879 + 0.975976i $$0.569914\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.54453i 0.389062i 0.980896 + 0.194531i $$0.0623185\pi$$
−0.980896 + 0.194531i $$0.937681\pi$$
$$84$$ 0 0
$$85$$ 1.95674 + 2.13032i 0.212239 + 0.231065i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 1.46329 0.155109 0.0775544 0.996988i $$-0.475289\pi$$
0.0775544 + 0.996988i $$0.475289\pi$$
$$90$$ 0 0
$$91$$ 3.54010 0.371103
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −6.07680 + 5.58168i −0.623467 + 0.572669i
$$96$$ 0 0
$$97$$ 2.90278i 0.294733i −0.989082 0.147366i $$-0.952920\pi$$
0.989082 0.147366i $$-0.0470797\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −7.68443 −0.764629 −0.382315 0.924032i $$-0.624873\pi$$
−0.382315 + 0.924032i $$0.624873\pi$$
$$102$$ 0 0
$$103$$ 8.69788i 0.857028i 0.903535 + 0.428514i $$0.140963\pi$$
−0.903535 + 0.428514i $$0.859037\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 17.4031i 1.68242i −0.540705 0.841212i $$-0.681843\pi$$
0.540705 0.841212i $$-0.318157\pi$$
$$108$$ 0 0
$$109$$ −5.73608 −0.549417 −0.274708 0.961528i $$-0.588581\pi$$
−0.274708 + 0.961528i $$0.588581\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 16.6704i 1.56822i 0.620623 + 0.784109i $$0.286880\pi$$
−0.620623 + 0.784109i $$0.713120\pi$$
$$114$$ 0 0
$$115$$ −12.6793 13.8040i −1.18235 1.28723i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 4.57949 0.419801
$$120$$ 0 0
$$121$$ 3.95731 0.359756
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −6.83778 8.84560i −0.611590 0.791175i
$$126$$ 0 0
$$127$$ 15.4031i 1.36681i −0.730041 0.683403i $$-0.760499\pi$$
0.730041 0.683403i $$-0.239501\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 11.6956 1.02185 0.510926 0.859624i $$-0.329302\pi$$
0.510926 + 0.859624i $$0.329302\pi$$
$$132$$ 0 0
$$133$$ 13.0632i 1.13272i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 5.61690i 0.479884i −0.970787 0.239942i $$-0.922872\pi$$
0.970787 0.239942i $$-0.0771284\pi$$
$$138$$ 0 0
$$139$$ −7.32956 −0.621686 −0.310843 0.950461i $$-0.600611\pi$$
−0.310843 + 0.950461i $$0.600611\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 3.86747i 0.323414i
$$144$$ 0 0
$$145$$ −12.4098 + 11.3986i −1.03057 + 0.946606i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −21.8953 −1.79373 −0.896867 0.442300i $$-0.854163\pi$$
−0.896867 + 0.442300i $$0.854163\pi$$
$$150$$ 0 0
$$151$$ 23.5777 1.91872 0.959361 0.282181i $$-0.0910578\pi$$
0.959361 + 0.282181i $$0.0910578\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −13.7680 + 12.6462i −1.10587 + 1.01577i
$$156$$ 0 0
$$157$$ 14.6377i 1.16822i 0.811676 + 0.584109i $$0.198556\pi$$
−0.811676 + 0.584109i $$0.801444\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −29.6742 −2.33865
$$162$$ 0 0
$$163$$ 10.8697i 0.851378i −0.904870 0.425689i $$-0.860032\pi$$
0.904870 0.425689i $$-0.139968\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.34700i 0.413763i −0.978366 0.206882i $$-0.933669\pi$$
0.978366 0.206882i $$-0.0663315\pi$$
$$168$$ 0 0
$$169$$ −1.00000 −0.0769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0.126903i 0.00964825i 0.999988 + 0.00482412i $$0.00153557\pi$$
−0.999988 + 0.00482412i $$0.998464\pi$$
$$174$$ 0 0
$$175$$ −17.6367 1.50072i −1.33321 0.113444i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 5.23533 0.391307 0.195653 0.980673i $$-0.437317\pi$$
0.195653 + 0.980673i $$0.437317\pi$$
$$180$$ 0 0
$$181$$ −11.6455 −0.865606 −0.432803 0.901488i $$-0.642475\pi$$
−0.432803 + 0.901488i $$0.642475\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 1.58390 + 1.72440i 0.116450 + 0.126780i
$$186$$ 0 0
$$187$$ 5.00298i 0.365854i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −22.9420 −1.66003 −0.830014 0.557743i $$-0.811667\pi$$
−0.830014 + 0.557743i $$0.811667\pi$$
$$192$$ 0 0
$$193$$ 6.73792i 0.485006i −0.970151 0.242503i $$-0.922032\pi$$
0.970151 0.242503i $$-0.0779684\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.68886i 0.262821i −0.991328 0.131410i $$-0.958049\pi$$
0.991328 0.131410i $$-0.0419505\pi$$
$$198$$ 0 0
$$199$$ 6.95329 0.492906 0.246453 0.969155i $$-0.420735\pi$$
0.246453 + 0.969155i $$0.420735\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 26.6770i 1.87236i
$$204$$ 0 0
$$205$$ 4.89152 4.49298i 0.341639 0.313803i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −14.2712 −0.987159
$$210$$ 0 0
$$211$$ 11.2858 0.776949 0.388474 0.921459i $$-0.373002\pi$$
0.388474 + 0.921459i $$0.373002\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −7.50535 8.17111i −0.511861 0.557265i
$$216$$ 0 0
$$217$$ 29.5968i 2.00916i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −1.29361 −0.0870174
$$222$$ 0 0
$$223$$ 2.26547i 0.151707i −0.997119 0.0758535i $$-0.975832\pi$$
0.997119 0.0758535i $$-0.0241681\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 17.7131i 1.17566i −0.808985 0.587829i $$-0.799983\pi$$
0.808985 0.587829i $$-0.200017\pi$$
$$228$$ 0 0
$$229$$ −13.9313 −0.920608 −0.460304 0.887761i $$-0.652260\pi$$
−0.460304 + 0.887761i $$0.652260\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 10.0077i 0.655628i 0.944742 + 0.327814i $$0.106312\pi$$
−0.944742 + 0.327814i $$0.893688\pi$$
$$234$$ 0 0
$$235$$ 0.695664 + 0.757372i 0.0453801 + 0.0494055i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −8.82190 −0.570641 −0.285321 0.958432i $$-0.592100\pi$$
−0.285321 + 0.958432i $$0.592100\pi$$
$$240$$ 0 0
$$241$$ 8.52784 0.549327 0.274663 0.961540i $$-0.411434\pi$$
0.274663 + 0.961540i $$0.411434\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −9.11057 + 8.36827i −0.582053 + 0.534629i
$$246$$ 0 0
$$247$$ 3.69006i 0.234793i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 29.1307 1.83871 0.919357 0.393425i $$-0.128710\pi$$
0.919357 + 0.393425i $$0.128710\pi$$
$$252$$ 0 0
$$253$$ 32.4183i 2.03812i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 4.57949i 0.285661i −0.989747 0.142830i $$-0.954380\pi$$
0.989747 0.142830i $$-0.0456203\pi$$
$$258$$ 0 0
$$259$$ 3.70690 0.230335
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 14.7654i 0.910474i −0.890370 0.455237i $$-0.849554\pi$$
0.890370 0.455237i $$-0.150446\pi$$
$$264$$ 0 0
$$265$$ −18.4963 20.1370i −1.13622 1.23700i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 16.0392 0.977930 0.488965 0.872303i $$-0.337375\pi$$
0.488965 + 0.872303i $$0.337375\pi$$
$$270$$ 0 0
$$271$$ 0.405316 0.0246212 0.0123106 0.999924i $$-0.496081\pi$$
0.0123106 + 0.999924i $$0.496081\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.63951 19.2677i 0.0988659 1.16189i
$$276$$ 0 0
$$277$$ 8.36607i 0.502669i −0.967900 0.251334i $$-0.919131\pi$$
0.967900 0.251334i $$-0.0808694\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 17.7676 1.05993 0.529963 0.848021i $$-0.322206\pi$$
0.529963 + 0.848021i $$0.322206\pi$$
$$282$$ 0 0
$$283$$ 10.6979i 0.635923i 0.948104 + 0.317961i $$0.102998\pi$$
−0.948104 + 0.317961i $$0.897002\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 10.5152i 0.620693i
$$288$$ 0 0
$$289$$ 15.3266 0.901564
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 13.0383i 0.761703i −0.924636 0.380851i $$-0.875631\pi$$
0.924636 0.380851i $$-0.124369\pi$$
$$294$$ 0 0
$$295$$ 5.40194 4.96180i 0.314513 0.288887i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 8.38231 0.484762
$$300$$ 0 0
$$301$$ −17.5653 −1.01244
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0.959058 0.880917i 0.0549155 0.0504412i
$$306$$ 0 0
$$307$$ 16.0724i 0.917299i −0.888617 0.458649i $$-0.848333\pi$$
0.888617 0.458649i $$-0.151667\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.40532 −0.363213 −0.181606 0.983371i $$-0.558130\pi$$
−0.181606 + 0.983371i $$0.558130\pi$$
$$312$$ 0 0
$$313$$ 9.69215i 0.547833i 0.961753 + 0.273916i $$0.0883192\pi$$
−0.961753 + 0.273916i $$0.911681\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 12.6417i 0.710031i −0.934860 0.355016i $$-0.884476\pi$$
0.934860 0.355016i $$-0.115524\pi$$
$$318$$ 0 0
$$319$$ −29.1439 −1.63175
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 4.77349i 0.265604i
$$324$$ 0 0
$$325$$ 4.98200 + 0.423922i 0.276351 + 0.0235150i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1.62811 0.0897604
$$330$$ 0 0
$$331$$ 21.3813 1.17522 0.587610 0.809144i $$-0.300069\pi$$
0.587610 + 0.809144i $$0.300069\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 12.3363 + 13.4306i 0.674004 + 0.733791i
$$336$$ 0 0
$$337$$ 14.2046i 0.773771i −0.922128 0.386886i $$-0.873551\pi$$
0.922128 0.386886i $$-0.126449\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −32.3338 −1.75097
$$342$$ 0 0
$$343$$ 5.19589i 0.280551i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 8.40945i 0.451443i 0.974192 + 0.225722i $$0.0724740\pi$$
−0.974192 + 0.225722i $$0.927526\pi$$
$$348$$ 0 0
$$349$$ −12.8579 −0.688268 −0.344134 0.938921i $$-0.611827\pi$$
−0.344134 + 0.938921i $$0.611827\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 20.4947i 1.09082i 0.838168 + 0.545412i $$0.183627\pi$$
−0.838168 + 0.545412i $$0.816373\pi$$
$$354$$ 0 0
$$355$$ 19.4841 17.8966i 1.03411 0.949852i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −5.48437 −0.289454 −0.144727 0.989472i $$-0.546230\pi$$
−0.144727 + 0.989472i $$0.546230\pi$$
$$360$$ 0 0
$$361$$ −5.38346 −0.283340
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 17.7175 + 19.2891i 0.927377 + 1.00964i
$$366$$ 0 0
$$367$$ 21.3267i 1.11325i −0.830765 0.556623i $$-0.812097\pi$$
0.830765 0.556623i $$-0.187903\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −43.2880 −2.24740
$$372$$ 0 0
$$373$$ 7.79670i 0.403698i −0.979417 0.201849i $$-0.935305\pi$$
0.979417 0.201849i $$-0.0646950\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 7.53566i 0.388106i
$$378$$ 0 0
$$379$$ 30.9542 1.59001 0.795006 0.606601i $$-0.207468\pi$$
0.795006 + 0.606601i $$0.207468\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 16.3396i 0.834914i −0.908697 0.417457i $$-0.862921\pi$$
0.908697 0.417457i $$-0.137079\pi$$
$$384$$ 0 0
$$385$$ −20.7097 22.5467i −1.05546 1.14909i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 6.00000 0.304212 0.152106 0.988364i $$-0.451394\pi$$
0.152106 + 0.988364i $$0.451394\pi$$
$$390$$ 0 0
$$391$$ 10.8434 0.548375
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −6.37823 + 5.85855i −0.320924 + 0.294776i
$$396$$ 0 0
$$397$$ 32.8000i 1.64619i 0.567906 + 0.823093i $$0.307754\pi$$
−0.567906 + 0.823093i $$0.692246\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.7379 0.935727 0.467864 0.883801i $$-0.345024\pi$$
0.467864 + 0.883801i $$0.345024\pi$$
$$402$$ 0 0
$$403$$ 8.36045i 0.416463i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 4.04969i 0.200736i
$$408$$ 0 0
$$409$$ 37.5574 1.85710 0.928548 0.371213i $$-0.121058\pi$$
0.928548 + 0.371213i $$0.121058\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 11.6124i 0.571410i
$$414$$ 0 0
$$415$$ 5.36155 + 5.83714i 0.263188 + 0.286534i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −15.0870 −0.737049 −0.368524 0.929618i $$-0.620137\pi$$
−0.368524 + 0.929618i $$0.620137\pi$$
$$420$$ 0 0
$$421$$ 12.3233 0.600600 0.300300 0.953845i $$-0.402913\pi$$
0.300300 + 0.953845i $$0.402913\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6.44474 + 0.548388i 0.312616 + 0.0266007i
$$426$$ 0 0
$$427$$ 2.06167i 0.0997710i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 2.14265 0.103208 0.0516038 0.998668i $$-0.483567\pi$$
0.0516038 + 0.998668i $$0.483567\pi$$
$$432$$ 0 0
$$433$$ 14.4918i 0.696433i 0.937414 + 0.348217i $$0.113213\pi$$
−0.937414 + 0.348217i $$0.886787\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 30.9312i 1.47964i
$$438$$ 0 0
$$439$$ −5.82917 −0.278211 −0.139106 0.990278i $$-0.544423\pi$$
−0.139106 + 0.990278i $$0.544423\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 4.06340i 0.193058i 0.995330 + 0.0965291i $$0.0307741\pi$$
−0.995330 + 0.0965291i $$0.969226\pi$$
$$444$$ 0 0
$$445$$ 2.40976 2.21342i 0.114233 0.104926i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −29.1218 −1.37434 −0.687172 0.726495i $$-0.741148\pi$$
−0.687172 + 0.726495i $$0.741148\pi$$
$$450$$ 0 0
$$451$$ 11.4876 0.540930
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 5.82984 5.35484i 0.273307 0.251039i
$$456$$ 0 0
$$457$$ 5.73494i 0.268269i −0.990963 0.134135i $$-0.957175\pi$$
0.990963 0.134135i $$-0.0428254\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 20.9824 0.977249 0.488624 0.872494i $$-0.337499\pi$$
0.488624 + 0.872494i $$0.337499\pi$$
$$462$$ 0 0
$$463$$ 0.847438i 0.0393838i 0.999806 + 0.0196919i $$0.00626853\pi$$
−0.999806 + 0.0196919i $$0.993731\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.2776i 1.30853i 0.756264 + 0.654266i $$0.227022\pi$$
−0.756264 + 0.654266i $$0.772978\pi$$
$$468$$ 0 0
$$469$$ 28.8714 1.33316
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 19.1896i 0.882339i
$$474$$ 0 0
$$475$$ −1.56430 + 18.3839i −0.0717749 + 0.843510i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −13.9191 −0.635981 −0.317990 0.948094i $$-0.603008\pi$$
−0.317990 + 0.948094i $$0.603008\pi$$
$$480$$ 0 0
$$481$$ −1.04712 −0.0477445
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −4.39082 4.78031i −0.199377 0.217063i
$$486$$ 0 0
$$487$$ 23.3189i 1.05668i 0.849033 + 0.528340i $$0.177185\pi$$
−0.849033 + 0.528340i $$0.822815\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.99557 0.360835 0.180417 0.983590i $$-0.442255\pi$$
0.180417 + 0.983590i $$0.442255\pi$$
$$492$$ 0 0
$$493$$ 9.74818i 0.439036i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 41.8845i 1.87878i
$$498$$ 0 0
$$499$$ −39.7191 −1.77807 −0.889035 0.457840i $$-0.848623\pi$$
−0.889035 + 0.457840i $$0.848623\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 2.69788i 0.120293i −0.998190 0.0601463i $$-0.980843\pi$$
0.998190 0.0601463i $$-0.0191567\pi$$
$$504$$ 0 0
$$505$$ −12.6547 + 11.6237i −0.563129 + 0.517247i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −26.6547 −1.18145 −0.590725 0.806873i $$-0.701158\pi$$
−0.590725 + 0.806873i $$0.701158\pi$$
$$510$$ 0 0
$$511$$ 41.4654 1.83432
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 13.1566 + 14.3237i 0.579751 + 0.631177i
$$516$$ 0 0
$$517$$ 1.77867i 0.0782257i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −19.3099 −0.845982 −0.422991 0.906134i $$-0.639020\pi$$
−0.422991 + 0.906134i $$0.639020\pi$$
$$522$$ 0 0
$$523$$ 2.37105i 0.103679i 0.998655 + 0.0518395i $$0.0165084\pi$$
−0.998655 + 0.0518395i $$0.983492\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 10.8151i 0.471114i
$$528$$ 0 0
$$529$$ −47.2631 −2.05492
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 2.97032i 0.128659i
$$534$$ 0 0
$$535$$ −26.3244 28.6595i −1.13810 1.23906i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −21.3959 −0.921587
$$540$$ 0 0
$$541$$ −4.50886 −0.193851 −0.0969256 0.995292i $$-0.530901\pi$$
−0.0969256 + 0.995292i $$0.530901\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −9.44619 + 8.67654i −0.404630 + 0.371662i
$$546$$ 0 0
$$547$$ 18.9179i 0.808870i 0.914567 + 0.404435i $$0.132532\pi$$
−0.914567 + 0.404435i $$0.867468\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 27.8070 1.18462
$$552$$ 0 0
$$553$$ 13.7111i 0.583057i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 20.3285i 0.861347i −0.902508 0.430674i $$-0.858276\pi$$
0.902508 0.430674i $$-0.141724\pi$$
$$558$$ 0 0
$$559$$ 4.96180 0.209862
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 45.6544i 1.92410i −0.272866 0.962052i $$-0.587972\pi$$
0.272866 0.962052i $$-0.412028\pi$$
$$564$$ 0 0
$$565$$ 25.2161 + 27.4528i 1.06085 + 1.15495i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 38.8564 1.62894 0.814472 0.580203i $$-0.197027\pi$$
0.814472 + 0.580203i $$0.197027\pi$$
$$570$$ 0 0
$$571$$ −43.0103 −1.79993 −0.899963 0.435966i $$-0.856407\pi$$
−0.899963 + 0.435966i $$0.856407\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −41.7606 3.55345i −1.74154 0.148189i
$$576$$ 0 0
$$577$$ 13.7724i 0.573354i −0.958027 0.286677i $$-0.907449\pi$$
0.958027 0.286677i $$-0.0925507\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 12.5480 0.520577
$$582$$ 0 0
$$583$$ 47.2911i 1.95860i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 9.48516i 0.391494i 0.980654 + 0.195747i $$0.0627132\pi$$
−0.980654 + 0.195747i $$0.937287\pi$$
$$588$$ 0 0
$$589$$ 30.8505 1.27117
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 23.1923i 0.952392i 0.879339 + 0.476196i $$0.157985\pi$$
−0.879339 + 0.476196i $$0.842015\pi$$
$$594$$ 0 0
$$595$$ 7.54152 6.92706i 0.309172 0.283982i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 3.85607 0.157555 0.0787774 0.996892i $$-0.474898\pi$$
0.0787774 + 0.996892i $$0.474898\pi$$
$$600$$ 0 0
$$601$$ −37.4047 −1.52577 −0.762884 0.646535i $$-0.776218\pi$$
−0.762884 + 0.646535i $$0.776218\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 6.51692 5.98594i 0.264950 0.243363i
$$606$$ 0 0
$$607$$ 0.384705i 0.0156147i 0.999970 + 0.00780734i $$0.00248518\pi$$
−0.999970 + 0.00780734i $$0.997515\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −0.459905 −0.0186057
$$612$$ 0 0
$$613$$ 32.4602i 1.31106i −0.755171 0.655528i $$-0.772446\pi$$
0.755171 0.655528i $$-0.227554\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 14.5650i 0.586364i −0.956057 0.293182i $$-0.905286\pi$$
0.956057 0.293182i $$-0.0947143\pi$$
$$618$$ 0 0
$$619$$ −12.7941 −0.514236 −0.257118 0.966380i $$-0.582773\pi$$
−0.257118 + 0.966380i $$0.582773\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 5.18020i 0.207540i
$$624$$ 0 0
$$625$$ −24.6406 4.22396i −0.985623 0.168958i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −1.35456 −0.0540098
$$630$$ 0 0
$$631$$ 43.6282 1.73681 0.868405 0.495856i $$-0.165146\pi$$
0.868405 + 0.495856i $$0.165146\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −23.2992 25.3659i −0.924600 1.00662i
$$636$$ 0 0
$$637$$ 5.53228i 0.219197i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 22.9206 0.905310 0.452655 0.891686i $$-0.350477\pi$$
0.452655 + 0.891686i $$0.350477\pi$$
$$642$$ 0 0
$$643$$ 37.6848i 1.48614i −0.669212 0.743071i $$-0.733368\pi$$
0.669212 0.743071i $$-0.266632\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 22.9044i 0.900464i 0.892912 + 0.450232i $$0.148659\pi$$
−0.892912 + 0.450232i $$0.851341\pi$$
$$648$$ 0 0
$$649$$ 12.6863 0.497980
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 39.7742i 1.55648i 0.627964 + 0.778242i $$0.283888\pi$$
−0.627964 + 0.778242i $$0.716112\pi$$
$$654$$ 0 0
$$655$$ 19.2604 17.6911i 0.752567 0.691250i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 17.7393 0.691026 0.345513 0.938414i $$-0.387705\pi$$
0.345513 + 0.938414i $$0.387705\pi$$
$$660$$ 0 0
$$661$$ −22.0617 −0.858099 −0.429050 0.903281i $$-0.641151\pi$$
−0.429050 + 0.903281i $$0.641151\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 19.7597 + 21.5125i 0.766248 + 0.834217i
$$666$$ 0 0
$$667$$ 63.1663i 2.44581i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2.25232 0.0869499
$$672$$ 0 0
$$673$$ 15.2439i 0.587610i −0.955865 0.293805i $$-0.905078\pi$$
0.955865 0.293805i $$-0.0949216\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 42.7334i 1.64238i 0.570655 + 0.821190i $$0.306689\pi$$
−0.570655 + 0.821190i $$0.693311\pi$$
$$678$$ 0 0
$$679$$ −10.2761 −0.394361
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 29.3308i 1.12231i −0.827710 0.561156i $$-0.810357\pi$$
0.827710 0.561156i $$-0.189643\pi$$
$$684$$ 0 0
$$685$$ −8.49627 9.24993i −0.324626 0.353421i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 12.2279 0.465846
$$690$$ 0 0
$$691$$ −7.31373 −0.278228 −0.139114 0.990276i $$-0.544425\pi$$
−0.139114 + 0.990276i $$0.544425\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −12.0704 + 11.0869i −0.457855 + 0.420550i
$$696$$ 0 0
$$697$$ 3.84242i 0.145542i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 11.1477 0.421044 0.210522 0.977589i $$-0.432484\pi$$
0.210522 + 0.977589i $$0.432484\pi$$
$$702$$ 0 0
$$703$$ 3.86393i 0.145731i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 27.2036i 1.02310i
$$708$$ 0 0
$$709$$ −36.1499 −1.35764 −0.678820 0.734305i $$-0.737508\pi$$
−0.678820 + 0.734305i $$0.737508\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 70.0799i 2.62451i
$$714$$ 0 0
$$715$$ 5.85004 + 6.36896i 0.218779 + 0.238186i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 49.4135 1.84281 0.921406 0.388601i $$-0.127041\pi$$
0.921406 + 0.388601i $$0.127041\pi$$
$$720$$ 0 0
$$721$$ 30.7913 1.14673
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −3.19453 + 37.5427i −0.118642 + 1.39430i
$$726$$ 0 0
$$727$$ 8.82270i 0.327216i 0.986525 + 0.163608i $$0.0523132\pi$$
−0.986525 + 0.163608i $$0.947687\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 6.41862 0.237401
$$732$$ 0 0
$$733$$ 36.9109i 1.36334i 0.731662 + 0.681668i $$0.238745\pi$$
−0.731662 + 0.681668i $$0.761255\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 31.5413i 1.16184i
$$738$$ 0 0
$$739$$ −39.9336 −1.46898 −0.734491 0.678619i $$-0.762579\pi$$
−0.734491 + 0.678619i $$0.762579\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 32.1600i 1.17984i −0.807463 0.589918i $$-0.799160\pi$$
0.807463 0.589918i $$-0.200840\pi$$
$$744$$ 0 0
$$745$$ −36.0573 + 33.1194i −1.32104 + 1.21340i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −61.6087 −2.25113
$$750$$ 0 0
$$751$$ 10.1167 0.369162 0.184581 0.982817i $$-0.440907\pi$$
0.184581 + 0.982817i $$0.440907\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 38.8278 35.6642i 1.41309 1.29795i
$$756$$ 0 0
$$757$$ 51.0892i 1.85687i 0.371498 + 0.928434i $$0.378844\pi$$
−0.371498 + 0.928434i $$0.621156\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 20.3683 0.738349 0.369175 0.929360i $$-0.379640\pi$$
0.369175 + 0.929360i $$0.379640\pi$$
$$762$$ 0 0
$$763$$ 20.3063i 0.735136i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3.28026i 0.118443i
$$768$$ 0 0
$$769$$ 10.1849 0.367276 0.183638 0.982994i $$-0.441213\pi$$
0.183638 + 0.982994i $$0.441213\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 14.3663i 0.516719i −0.966049 0.258360i $$-0.916818\pi$$
0.966049 0.258360i $$-0.0831820\pi$$
$$774$$ 0 0
$$775$$ −3.54418 + 41.6517i −0.127311 + 1.49617i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −10.9606 −0.392706
$$780$$ 0 0
$$781$$ 45.7578 1.63734
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 22.1414 + 24.1054i 0.790261 + 0.860360i
$$786$$ 0 0
$$787$$ 23.1526i 0.825300i 0.910890 + 0.412650i $$0.135397\pi$$
−0.910890 + 0.412650i $$0.864603\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 59.0148 2.09832
$$792$$ 0 0
$$793$$ 0.582376i 0.0206808i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 38.4364i 1.36149i −0.732522 0.680744i $$-0.761657\pi$$
0.732522 0.680744i $$-0.238343\pi$$
$$798$$ 0 0
$$799$$ −0.594936 −0.0210473
$$800$$ 0