LMFDB - Embedded newform 2052.3.be.a.125.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(125,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2052.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.be (of order $6$, degree $2$, not 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:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.16
Character	$\chi$	$=$	2052.125
Dual form			2052.3.be.a.197.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.60071 + 0.924171i) q^{5} +(2.46982 + 4.27785i) q^{7} +O(q^{10})$	$q+(-1.60071 + 0.924171i) q^{5} +(2.46982 + 4.27785i) q^{7} +(10.4350 - 6.02464i) q^{11} +10.2235 q^{13} +(16.2730 + 9.39522i) q^{17} +(8.26716 + 17.1071i) q^{19} -18.8234i q^{23} +(-10.7918 + 18.6920i) q^{25} +(4.07286 + 2.35147i) q^{29} +(14.2830 - 24.7388i) q^{31} +(-7.90692 - 4.56506i) q^{35} -52.8630 q^{37} +(6.43514 - 3.71533i) q^{41} -41.3991 q^{43} +(-3.74062 - 2.15965i) q^{47} +(12.3000 - 21.3043i) q^{49} +(41.4703 - 23.9429i) q^{53} +(-11.1356 + 19.2874i) q^{55} +(44.0498 - 25.4322i) q^{59} +(-11.5107 + 19.9371i) q^{61} +(-16.3649 + 9.44828i) q^{65} +38.7932 q^{67} +(95.0990 + 54.9054i) q^{71} +(11.5505 - 20.0061i) q^{73} +(51.5450 + 29.7595i) q^{77} -149.029 q^{79} +(64.2549 - 37.0976i) q^{83} -34.7311 q^{85} +(-9.22299 + 5.32490i) q^{89} +(25.2502 + 43.7346i) q^{91} +(-29.0433 - 19.7433i) q^{95} +100.686 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.60071	+	0.924171i	−0.320142	+	0.184834i	−0.651456	−	0.758686i	$-0.725841\pi$
$5$	−1.60071	+	0.924171i	−0.320142	+	0.184834i	0.331314	+	0.943521i	$0.392508\pi$
$6$		0			0
$7$	2.46982	+	4.27785i	0.352831	+	0.611121i	0.986744	−	0.162283i	$-0.0518858\pi$
$7$	2.46982	+	4.27785i	0.352831	+	0.611121i	−0.633913	+	0.773404i	$0.718552\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.4350	−	6.02464i	0.948635	−	0.547694i	0.0559781	−	0.998432i	$-0.482172\pi$
$11$	10.4350	−	6.02464i	0.948635	−	0.547694i	0.892657	+	0.450738i	$0.148839\pi$
$12$		0			0
$13$	10.2235			0.786425			0.393212	−	0.919448i	$-0.371364\pi$
$13$	10.2235			0.786425			0.393212	+	0.919448i	$0.371364\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16.2730	+	9.39522i	0.957235	+	0.552660i	0.895321	−	0.445422i	$-0.146946\pi$
$17$	16.2730	+	9.39522i	0.957235	+	0.552660i	0.0619139	+	0.998081i	$0.480280\pi$
$18$		0			0
$19$	8.26716	+	17.1071i	0.435114	+	0.900375i
$19$	8.26716	+	17.1071i	0.435114	+	0.900375i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	18.8234i		−	0.818410i	−0.912442	−	0.409205i	$-0.865806\pi$
$23$		−	18.8234i		−	0.818410i	0.912442	−	0.409205i	$-0.134194\pi$
$24$		0			0
$25$	−10.7918	+	18.6920i	−0.431673	+	0.747679i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.07286	+	2.35147i	0.140443	+	0.0810850i	0.568575	−	0.822631i	$-0.307495\pi$
$29$	4.07286	+	2.35147i	0.140443	+	0.0810850i	−0.428132	+	0.903716i	$0.640828\pi$
$30$		0			0
$31$	14.2830	−	24.7388i	0.460740	−	0.798026i	−0.538258	−	0.842780i	$-0.680917\pi$
$31$	14.2830	−	24.7388i	0.460740	−	0.798026i	0.998998	+	0.0447545i	$0.0142506\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.90692	−	4.56506i	−0.225912	−	0.130430i
$36$		0			0
$37$	−52.8630			−1.42873			−0.714365	−	0.699773i	$-0.753285\pi$
$37$	−52.8630			−1.42873			−0.714365	+	0.699773i	$0.753285\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.43514	−	3.71533i	0.156955	−	0.0906178i	−0.419466	−	0.907771i	$-0.637783\pi$
$41$	6.43514	−	3.71533i	0.156955	−	0.0906178i	0.576420	+	0.817153i	$0.304449\pi$
$42$		0			0
$43$	−41.3991			−0.962771			−0.481385	−	0.876509i	$-0.659866\pi$
$43$	−41.3991			−0.962771			−0.481385	+	0.876509i	$0.659866\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.74062	−	2.15965i	−0.0795878	−	0.0459500i	0.459678	−	0.888086i	$-0.347965\pi$
$47$	−3.74062	−	2.15965i	−0.0795878	−	0.0459500i	−0.539266	+	0.842136i	$0.681298\pi$
$48$		0			0
$49$	12.3000	−	21.3043i	0.251021	−	0.434781i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	41.4703	−	23.9429i	0.782459	−	0.451753i	−0.0548421	−	0.998495i	$-0.517466\pi$
$53$	41.4703	−	23.9429i	0.782459	−	0.451753i	0.837301	+	0.546742i	$0.184132\pi$
$54$		0			0
$55$	−11.1356	+	19.2874i	−0.202465	+	0.350680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	44.0498	−	25.4322i	0.746606	−	0.431053i	−0.0778600	−	0.996964i	$-0.524809\pi$
$59$	44.0498	−	25.4322i	0.746606	−	0.431053i	0.824466	+	0.565911i	$0.191475\pi$
$60$		0			0
$61$	−11.5107	+	19.9371i	−0.188699	+	0.326837i	−0.944817	−	0.327599i	$-0.893761\pi$
$61$	−11.5107	+	19.9371i	−0.188699	+	0.326837i	0.756117	+	0.654436i	$0.227094\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−16.3649	+	9.44828i	−0.251768	+	0.145358i
$66$		0			0
$67$	38.7932			0.579004			0.289502	−	0.957177i	$-0.406510\pi$
$67$	38.7932			0.579004			0.289502	+	0.957177i	$0.406510\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	95.0990	+	54.9054i	1.33942	+	0.773316i	0.986722	−	0.162420i	$-0.0519301\pi$
$71$	95.0990	+	54.9054i	1.33942	+	0.773316i	0.352701	+	0.935736i	$0.385263\pi$
$72$		0			0
$73$	11.5505	−	20.0061i	0.158226	−	0.274056i	−0.776003	−	0.630730i	$-0.782756\pi$
$73$	11.5505	−	20.0061i	0.158226	−	0.274056i	0.934229	+	0.356673i	$0.116089\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	51.5450	+	29.7595i	0.669415	+	0.386487i
$78$		0			0
$79$	−149.029			−1.88644			−0.943218	−	0.332173i	$-0.892218\pi$
$79$	−149.029			−1.88644			−0.943218	+	0.332173i	$0.892218\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	64.2549	−	37.0976i	0.774156	−	0.446959i	−0.0601995	−	0.998186i	$-0.519174\pi$
$83$	64.2549	−	37.0976i	0.774156	−	0.446959i	0.834355	+	0.551227i	$0.185840\pi$
$84$		0			0
$85$	−34.7311			−0.408602
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.22299	+	5.32490i	−0.103629	+	0.0598303i	−0.550919	−	0.834559i	$-0.685723\pi$
$89$	−9.22299	+	5.32490i	−0.103629	+	0.0598303i	0.447290	+	0.894389i	$0.352389\pi$
$90$		0			0
$91$	25.2502	+	43.7346i	0.277475	+	0.480601i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−29.0433	−	19.7433i	−0.305718	−	0.207824i
$96$		0			0
$97$	100.686			1.03800			0.519000	−	0.854774i	$-0.326305\pi$
$97$	100.686			1.03800			0.519000	+	0.854774i	$0.326305\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	127.441	+	73.5780i	1.26179	+	0.728495i	0.973421	−	0.229025i	$-0.0735537\pi$
$101$	127.441	+	73.5780i	1.26179	+	0.728495i	0.288369	+	0.957519i	$0.406887\pi$
$102$		0			0
$103$	−28.4113	+	49.2098i	−0.275838	+	0.477765i	−0.970346	−	0.241720i	$-0.922289\pi$
$103$	−28.4113	+	49.2098i	−0.275838	+	0.477765i	0.694509	+	0.719484i	$0.255622\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			62.5758i			0.584821i	0.956293	+	0.292410i	$0.0944573\pi$
$107$			62.5758i			0.584821i	−0.956293	+	0.292410i	$0.905543\pi$
$108$		0			0
$109$	−32.2512	+	55.8606i	−0.295882	+	0.512483i	−0.975190	−	0.221371i	$-0.928947\pi$
$109$	−32.2512	+	55.8606i	−0.295882	+	0.512483i	0.679308	+	0.733854i	$0.262280\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	110.388	+	63.7326i	0.976885	+	0.564005i	0.901328	−	0.433137i	$-0.142593\pi$
$113$	110.388	+	63.7326i	0.976885	+	0.564005i	0.0755570	+	0.997141i	$0.475927\pi$
$114$		0			0
$115$	17.3961	+	30.1309i	0.151270	+	0.262008i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			92.8178i			0.779982i
$120$		0			0
$121$	12.0925	−	20.9449i	0.0999384	−	0.173098i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	86.1025i		−	0.688820i
$126$		0			0
$127$	118.378	+	205.036i	0.932109	+	1.61446i	0.779710	+	0.626141i	$0.215367\pi$
$127$	118.378	+	205.036i	0.932109	+	1.61446i	0.152399	+	0.988319i	$0.451300\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−217.986	+	125.854i	−1.66402	+	0.960720i	−0.693247	+	0.720700i	$0.743820\pi$
$131$	−217.986	+	125.854i	−1.66402	+	0.960720i	−0.970768	+	0.240019i	$0.922846\pi$
$132$		0			0
$133$	−52.7633	+	77.6171i	−0.396717	+	0.583587i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.56119	−	2.63340i	−0.0332934	−	0.0192219i	0.483261	−	0.875476i	$-0.339452\pi$
$137$	−4.56119	−	2.63340i	−0.0332934	−	0.0192219i	−0.516554	+	0.856254i	$0.672786\pi$
$138$		0			0
$139$	162.679			1.17035			0.585176	−	0.810906i	$-0.301025\pi$
$139$	162.679			1.17035			0.585176	+	0.810906i	$0.301025\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	106.682	−	61.5930i	0.746030	−	0.430720i
$144$		0			0
$145$	−8.69263			−0.0599492
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−61.8394	+	35.7030i	−0.415029	+	0.239617i	−0.692948	−	0.720987i	$-0.743689\pi$
$149$	−61.8394	+	35.7030i	−0.415029	+	0.239617i	0.277919	+	0.960605i	$0.410355\pi$
$150$		0			0
$151$	7.78566	+	13.4852i	0.0515607	+	0.0893057i	0.890654	−	0.454682i	$-0.150247\pi$
$151$	7.78566	+	13.4852i	0.0515607	+	0.0893057i	−0.839093	+	0.543988i	$0.816914\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			52.7996i			0.340642i
$156$		0			0
$157$	74.6100	+	129.228i	0.475223	+	0.823110i	0.999597	−	0.0283778i	$-0.00903416\pi$
$157$	74.6100	+	129.228i	0.475223	+	0.823110i	−0.524375	+	0.851488i	$0.675701\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	80.5238	−	46.4904i	0.500148	−	0.288760i
$162$		0			0
$163$	157.638			0.967101			0.483551	−	0.875316i	$-0.339347\pi$
$163$	157.638			0.967101			0.483551	+	0.875316i	$0.339347\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	246.867i		−	1.47825i	−0.673571	−	0.739123i	$-0.735240\pi$
$167$		−	246.867i		−	1.47825i	0.673571	−	0.739123i	$-0.264760\pi$
$168$		0			0
$169$	−64.4797			−0.381536
$170$		0			0
$171$		0			0
$172$		0			0
$173$			104.588i			0.604553i	0.953220	+	0.302276i	$0.0977465\pi$
$173$			104.588i			0.604553i	−0.953220	+	0.302276i	$0.902253\pi$
$174$		0			0
$175$	−106.615			−0.609230
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	238.780i		−	1.33397i	−0.745072	−	0.666985i	$-0.767585\pi$
$179$		−	238.780i		−	1.33397i	0.745072	−	0.666985i	$-0.232415\pi$
$180$		0			0
$181$	108.062	+	187.169i	0.597029	+	1.03408i	0.993257	+	0.115933i	$0.0369857\pi$
$181$	108.062	+	187.169i	0.597029	+	1.03408i	−0.396228	+	0.918152i	$0.629681\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	84.6185	−	48.8545i	0.457397	−	0.264078i
$186$		0			0
$187$	226.411			1.21075
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−150.835	+	87.0849i	−0.789714	+	0.455942i	−0.839862	−	0.542800i	$-0.817364\pi$
$191$	−150.835	+	87.0849i	−0.789714	+	0.455942i	0.0501477	+	0.998742i	$0.484031\pi$
$192$		0			0
$193$	21.5500	+	37.3257i	0.111658	+	0.193397i	0.916439	−	0.400175i	$-0.131051\pi$
$193$	21.5500	+	37.3257i	0.111658	+	0.193397i	−0.804781	+	0.593572i	$0.797717\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	229.707i		−	1.16603i	−0.812463	−	0.583013i	$-0.801874\pi$
$197$		−	229.707i		−	1.16603i	0.812463	−	0.583013i	$-0.198126\pi$
$198$		0			0
$199$	−32.8318	−	56.8663i	−0.164984	−	0.285760i	0.771666	−	0.636028i	$-0.219424\pi$
$199$	−32.8318	−	56.8663i	−0.164984	−	0.285760i	−0.936650	+	0.350268i	$0.886090\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			23.2308i			0.114437i
$204$		0			0
$205$	−6.86720	+	11.8943i	−0.0334985	+	0.0580211i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	189.332	+	128.706i	0.905895	+	0.615818i
$210$		0			0
$211$	77.2746	+	133.844i	0.366230	+	0.634329i	0.988973	−	0.148097i	$-0.0473149\pi$
$211$	77.2746	+	133.844i	0.366230	+	0.634329i	−0.622743	+	0.782427i	$0.713982\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	66.2681	−	38.2599i	0.308224	−	0.177953i
$216$		0			0
$217$	141.105			0.650254
$218$		0			0
$219$		0			0
$220$		0			0
$221$	166.367	+	96.0522i	0.752793	+	0.434625i
$222$		0			0
$223$	130.874			0.586878			0.293439	−	0.955978i	$-0.405200\pi$
$223$	130.874			0.586878			0.293439	+	0.955978i	$0.405200\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−140.131	+	80.9047i	−0.617318	+	0.356409i	−0.775824	−	0.630949i	$-0.782666\pi$
$227$	−140.131	+	80.9047i	−0.617318	+	0.356409i	0.158506	+	0.987358i	$0.449332\pi$
$228$		0			0
$229$	168.547	−	291.933i	0.736015	−	1.27482i	−0.218262	−	0.975890i	$-0.570039\pi$
$229$	168.547	−	291.933i	0.736015	−	1.27482i	0.954277	−	0.298925i	$-0.0966280\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−7.44103	−	4.29608i	−0.0319357	−	0.0184381i	0.483947	−	0.875097i	$-0.339203\pi$
$233$	−7.44103	−	4.29608i	−0.0319357	−	0.0184381i	−0.515883	+	0.856659i	$0.672536\pi$
$234$		0			0
$235$	7.98355			0.0339725
$236$		0			0
$237$		0			0
$238$		0			0
$239$	216.237	+	124.844i	0.904756	+	0.522361i	0.878740	−	0.477300i	$-0.158385\pi$
$239$	216.237	+	124.844i	0.904756	+	0.522361i	0.0260161	+	0.999662i	$0.491718\pi$
$240$		0			0
$241$	−138.916	+	240.609i	−0.576414	+	0.998378i	0.419473	+	0.907768i	$0.362215\pi$
$241$	−138.916	+	240.609i	−0.576414	+	0.998378i	−0.995886	+	0.0906101i	$0.971118\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			45.4693i			0.185589i
$246$		0			0
$247$	84.5195	+	174.895i	0.342184	+	0.708077i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	413.219	−	238.572i	1.64629	−	0.950486i	0.667761	−	0.744376i	$-0.267253\pi$
$251$	413.219	−	238.572i	1.64629	−	0.950486i	0.978529	−	0.206110i	$-0.0660805\pi$
$252$		0			0
$253$	−113.404	−	196.422i	−0.448239	−	0.776372i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			176.278i			0.685908i	0.939352	+	0.342954i	$0.111428\pi$
$257$			176.278i			0.685908i	−0.939352	+	0.342954i	$0.888572\pi$
$258$		0			0
$259$	−130.562	−	226.140i	−0.504100	−	0.873127i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			188.617i			0.717177i	0.933496	+	0.358588i	$0.116742\pi$
$263$			188.617i			0.717177i	−0.933496	+	0.358588i	$0.883258\pi$
$264$		0			0
$265$	−44.2547	+	76.6513i	−0.166999	+	0.289250i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−356.589	−	205.877i	−1.32561	−	0.765340i	−0.340991	−	0.940067i	$-0.610763\pi$
$269$	−356.589	−	205.877i	−1.32561	−	0.765340i	−0.984617	+	0.174726i	$0.944096\pi$
$270$		0			0
$271$	24.3004	−	42.0896i	0.0896695	−	0.155312i	−0.817702	−	0.575642i	$-0.804752\pi$
$271$	24.3004	−	42.0896i	0.0896695	−	0.155312i	0.907371	+	0.420330i	$0.138086\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			260.067i			0.945699i
$276$		0			0
$277$	−76.9499	−	133.281i	−0.277798	−	0.481159i	0.693040	−	0.720900i	$-0.256271\pi$
$277$	−76.9499	−	133.281i	−0.277798	−	0.481159i	−0.970837	+	0.239740i	$0.922938\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−116.407	−	67.2076i	−0.414260	−	0.239173i	0.278359	−	0.960477i	$-0.410210\pi$
$281$	−116.407	−	67.2076i	−0.414260	−	0.239173i	−0.692618	+	0.721304i	$0.743543\pi$
$282$		0			0
$283$	31.8317	+	55.1341i	0.112480	+	0.194820i	0.916769	−	0.399417i	$-0.130787\pi$
$283$	31.8317	+	55.1341i	0.112480	+	0.194820i	−0.804290	+	0.594237i	$0.797454\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	31.7872	+	18.3524i	0.110757	+	0.0639455i
$288$		0			0
$289$	32.0401	+	55.4951i	0.110866	+	0.192025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−186.208	−	107.507i	−0.635523	−	0.366919i	0.147365	−	0.989082i	$-0.452921\pi$
$293$	−186.208	−	107.507i	−0.635523	−	0.366919i	−0.782888	+	0.622163i	$0.786254\pi$
$294$		0			0
$295$	−47.0073	+	81.4191i	−0.159347	+	0.275997i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	192.442i		−	0.643618i
$300$		0			0
$301$	−102.248	−	177.099i	−0.339695	−	0.588369i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	42.5513i		−	0.139512i
$306$		0			0
$307$	−137.106	+	237.474i	−0.446599	+	0.773532i	−0.998162	−	0.0606011i	$-0.980698\pi$
$307$	−137.106	+	237.474i	−0.446599	+	0.773532i	0.551563	+	0.834133i	$0.314032\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	395.495	+	228.339i	1.27169	+	0.734209i	0.975305	−	0.220861i	$-0.0708866\pi$
$311$	395.495	+	228.339i	1.27169	+	0.734209i	0.296382	+	0.955070i	$0.404220\pi$
$312$		0			0
$313$	−49.7646	+	86.1947i	−0.158992	+	0.275383i	−0.934506	−	0.355949i	$-0.884158\pi$
$313$	−49.7646	+	86.1947i	−0.158992	+	0.275383i	0.775513	+	0.631331i	$0.217491\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−107.225	−	61.9065i	−0.338250	−	0.195289i	0.321248	−	0.946995i	$-0.395898\pi$
$317$	−107.225	−	61.9065i	−0.338250	−	0.195289i	−0.659498	+	0.751706i	$0.729231\pi$
$318$		0			0
$319$	56.6669			0.177639
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−26.1937	+	356.056i	−0.0810951	+	1.10234i
$324$		0			0
$325$	−110.330	+	191.098i	−0.339478	+	0.587993i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	21.3358i		−	0.0648503i
$330$		0			0
$331$	284.794	+	493.278i	0.860406	+	1.49027i	0.871538	+	0.490328i	$0.163123\pi$
$331$	284.794	+	493.278i	0.860406	+	1.49027i	−0.0111320	+	0.999938i	$0.503544\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−62.0968	+	35.8516i	−0.185364	+	0.107020i
$336$		0			0
$337$	88.6716	+	153.584i	0.263120	+	0.455738i	0.967069	−	0.254513i	$-0.0819150\pi$
$337$	88.6716	+	153.584i	0.263120	+	0.455738i	−0.703949	+	0.710250i	$0.748582\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	344.199i		−	1.00938i
$342$		0			0
$343$	363.557			1.05993
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−336.292	+	194.158i	−0.969140	+	0.559533i	−0.898974	−	0.438002i	$-0.855686\pi$
$347$	−336.292	+	194.158i	−0.969140	+	0.559533i	−0.0701662	+	0.997535i	$0.522353\pi$
$348$		0			0
$349$	−165.385	−	286.456i	−0.473884	−	0.820791i	0.525669	−	0.850689i	$-0.323815\pi$
$349$	−165.385	−	286.456i	−0.473884	−	0.820791i	−0.999553	+	0.0298982i	$0.990482\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	152.631	−	88.1216i	0.432383	−	0.249636i	−0.267979	−	0.963425i	$-0.586356\pi$
$353$	152.631	−	88.1216i	0.432383	−	0.249636i	0.700361	+	0.713789i	$0.253022\pi$
$354$		0			0
$355$	−202.968			−0.571741
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−208.764	−	120.530i	−0.581515	−	0.335738i	0.180220	−	0.983626i	$-0.442319\pi$
$359$	−208.764	−	120.530i	−0.581515	−	0.335738i	−0.761735	+	0.647888i	$0.775652\pi$
$360$		0			0
$361$	−224.308	+	282.855i	−0.621352	+	0.783532i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			42.6987i			0.116983i
$366$		0			0
$367$	−95.1953	+	164.883i	−0.259388	+	0.449273i	−0.966078	−	0.258250i	$-0.916854\pi$
$367$	−95.1953	+	164.883i	−0.259388	+	0.449273i	0.706690	+	0.707523i	$0.250187\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	204.848	+	118.269i	0.552151	+	0.318785i
$372$		0			0
$373$	143.800	−	249.069i	0.385523	−	0.667745i	−0.606319	−	0.795222i	$-0.707354\pi$
$373$	143.800	−	249.069i	0.385523	−	0.667745i	0.991842	+	0.127477i	$0.0406878\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	41.6390	+	24.0403i	0.110448	+	0.0637673i
$378$		0			0
$379$	127.708			0.336961			0.168481	−	0.985705i	$-0.446114\pi$
$379$	127.708			0.336961			0.168481	+	0.985705i	$0.446114\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	335.294	−	193.582i	0.875441	−	0.505436i	0.00628842	−	0.999980i	$-0.497998\pi$
$383$	335.294	−	193.582i	0.875441	−	0.505436i	0.869152	+	0.494544i	$0.164665\pi$
$384$		0			0
$385$	−110.011			−0.285744
$386$		0			0
$387$		0			0
$388$		0			0
$389$	16.4521	+	9.49861i	0.0422933	+	0.0244180i	0.520998	−	0.853558i	$-0.325560\pi$
$389$	16.4521	+	9.49861i	0.0422933	+	0.0244180i	−0.478704	+	0.877976i	$0.658893\pi$
$390$		0			0
$391$	176.850	−	306.314i	0.452302	−	0.783411i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	238.552	−	137.728i	0.603928	−	0.348678i
$396$		0			0
$397$	112.212	−	194.357i	0.282650	−	0.489564i	−0.689387	−	0.724394i	$-0.742120\pi$
$397$	112.212	−	194.357i	0.282650	−	0.489564i	0.972037	+	0.234829i	$0.0754531\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−152.211	+	87.8792i	−0.379579	+	0.219150i	−0.677635	−	0.735398i	$-0.736995\pi$
$401$	−152.211	+	87.8792i	−0.379579	+	0.219150i	0.298056	+	0.954548i	$0.403662\pi$
$402$		0			0
$403$	146.022	−	252.918i	0.362338	−	0.627587i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−551.625	+	318.481i	−1.35534	+	0.782508i
$408$		0			0
$409$	−634.354			−1.55099			−0.775494	−	0.631355i	$-0.782499\pi$
$409$	−634.354			−1.55099			−0.775494	+	0.631355i	$0.782499\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	217.590	+	125.625i	0.526852	+	0.304178i
$414$		0			0
$415$	−68.5690	+	118.765i	−0.165227	+	0.286181i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	329.899	+	190.467i	0.787348	+	0.454575i	0.839028	−	0.544088i	$-0.183124\pi$
$419$	329.899	+	190.467i	0.787348	+	0.454575i	−0.0516803	+	0.998664i	$0.516458\pi$
$420$		0			0
$421$	269.508			0.640161			0.320080	−	0.947390i	$-0.396290\pi$
$421$	269.508			0.640161			0.320080	+	0.947390i	$0.396290\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−351.230	+	202.783i	−0.826424	+	0.477136i
$426$		0			0
$427$	−113.717			−0.266316
$428$		0			0
$429$		0			0
$430$		0			0
$431$	601.019	−	346.999i	1.39448	−	0.805101i	0.400669	−	0.916223i	$-0.368778\pi$
$431$	601.019	−	346.999i	1.39448	−	0.805101i	0.993807	+	0.111122i	$0.0354443\pi$
$432$		0			0
$433$	−324.494	−	562.041i	−0.749409	−	1.29802i	−0.948106	−	0.317954i	$-0.897004\pi$
$433$	−324.494	−	562.041i	−0.749409	−	1.29802i	0.198697	−	0.980061i	$-0.436329\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	322.015	−	155.616i	0.736877	−	0.356102i
$438$		0			0
$439$	326.884			0.744611			0.372306	−	0.928110i	$-0.378567\pi$
$439$	326.884			0.744611			0.372306	+	0.928110i	$0.378567\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−697.089	−	402.464i	−1.57356	−	0.908497i	−0.995727	−	0.0923447i	$-0.970564\pi$
$443$	−697.089	−	402.464i	−1.57356	−	0.908497i	−0.577836	−	0.816153i	$-0.696103\pi$
$444$		0			0
$445$	9.84223	−	17.0472i	0.0221174	−	0.0383084i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			201.130i			0.447951i	0.974595	+	0.223975i	$0.0719035\pi$
$449$			201.130i			0.447951i	−0.974595	+	0.223975i	$0.928096\pi$
$450$		0			0
$451$	44.7670	−	77.5388i	0.0992617	−	0.171926i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−80.8366	−	46.6710i	−0.177663	−	0.102574i
$456$		0			0
$457$	−211.138	−	365.702i	−0.462010	−	0.800224i	0.537051	−	0.843549i	$-0.319538\pi$
$457$	−211.138	−	365.702i	−0.462010	−	0.800224i	−0.999061	+	0.0433255i	$0.986205\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	364.726i		−	0.791163i	−0.918431	−	0.395582i	$-0.870543\pi$
$461$		−	364.726i		−	0.791163i	0.918431	−	0.395582i	$-0.129457\pi$
$462$		0			0
$463$	168.041	−	291.055i	0.362939	−	0.628629i	−0.625504	−	0.780221i	$-0.715107\pi$
$463$	168.041	−	291.055i	0.362939	−	0.628629i	0.988443	+	0.151592i	$0.0484400\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			33.3762i			0.0714693i	0.999361	+	0.0357347i	$0.0113771\pi$
$467$			33.3762i			0.0714693i	−0.999361	+	0.0357347i	$0.988623\pi$
$468$		0			0
$469$	95.8122	+	165.952i	0.204290	+	0.353841i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−431.999	+	249.415i	−0.913317	+	0.527304i
$474$		0			0
$475$	−408.984	−	30.0874i	−0.861019	−	0.0633420i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−417.401	−	240.987i	−0.871401	−	0.503104i	−0.00358759	−	0.999994i	$-0.501142\pi$
$479$	−417.401	−	240.987i	−0.871401	−	0.503104i	−0.867814	+	0.496890i	$0.834475\pi$
$480$		0			0
$481$	−540.446			−1.12359
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−161.169	+	93.0510i	−0.332308	+	0.191858i
$486$		0			0
$487$	−887.516			−1.82241			−0.911207	−	0.411948i	$-0.864849\pi$
$487$	−887.516			−1.82241			−0.911207	+	0.411948i	$0.864849\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	806.106	−	465.406i	1.64176	−	0.947873i	0.661557	−	0.749895i	$-0.269896\pi$
$491$	806.106	−	465.406i	1.64176	−	0.947873i	0.980206	−	0.197978i	$-0.0634375\pi$
$492$		0			0
$493$	44.1851	+	76.5308i	0.0896249	+	0.155235i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			542.425i			1.09140i
$498$		0			0
$499$	326.171	+	564.945i	0.653649	+	1.13215i	0.982231	+	0.187678i	$0.0600962\pi$
$499$	326.171	+	564.945i	0.653649	+	1.13215i	−0.328581	+	0.944476i	$0.606570\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−264.025	+	152.435i	−0.524901	+	0.303052i	−0.738937	−	0.673774i	$-0.764672\pi$
$503$	−264.025	+	152.435i	−0.524901	+	0.303052i	0.214037	+	0.976826i	$0.431339\pi$
$504$		0			0
$505$	−271.994			−0.538603
$506$		0			0
$507$		0			0
$508$		0			0
$509$			51.6476i			0.101469i	0.998712	+	0.0507344i	$0.0161562\pi$
$509$			51.6476i			0.101469i	−0.998712	+	0.0507344i	$0.983844\pi$
$510$		0			0
$511$	114.111			0.223309
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	105.027i		−	0.203937i
$516$		0			0
$517$	−52.0445			−0.100666
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	733.507i		−	1.40788i	−0.710258	−	0.703941i	$-0.751422\pi$
$521$		−	733.507i		−	1.40788i	0.710258	−	0.703941i	$-0.248578\pi$
$522$		0			0
$523$	376.499	+	652.115i	0.719883	+	1.24687i	0.961046	+	0.276389i	$0.0891378\pi$
$523$	376.499	+	652.115i	0.719883	+	1.24687i	−0.241163	+	0.970485i	$0.577529\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	464.853	−	268.383i	0.882074	−	0.509265i
$528$		0			0
$529$	174.678			0.330204
$530$		0			0
$531$		0			0
$532$		0			0
$533$	65.7897	−	37.9837i	0.123433	−	0.0712640i
$534$		0			0
$535$	−57.8307	−	100.166i	−0.108095	−	0.187226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	296.413i		−	0.549931i
$540$		0			0
$541$	−161.332	−	279.436i	−0.298211	−	0.516517i	0.677516	−	0.735508i	$-0.263057\pi$
$541$	−161.332	−	279.436i	−0.298211	−	0.516517i	−0.975727	+	0.218992i	$0.929723\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	119.222i		−	0.218757i
$546$		0			0
$547$	−54.0558	+	93.6274i	−0.0988223	+	0.171165i	−0.911197	−	0.411970i	$-0.864841\pi$
$547$	−54.0558	+	93.6274i	−0.0988223	+	0.171165i	0.812375	+	0.583135i	$0.198174\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.55586	+	89.1149i	−0.0118981	+	0.161733i
$552$		0			0
$553$	−368.073	−	637.521i	−0.665593	−	1.15284i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−605.693	+	349.697i	−1.08742	+	0.627822i	−0.932888	−	0.360165i	$-0.882720\pi$
$557$	−605.693	+	349.697i	−1.08742	+	0.627822i	−0.154532	+	0.987988i	$0.549387\pi$
$558$		0			0
$559$	−423.245			−0.757146
$560$		0			0
$561$		0			0
$562$		0			0
$563$	973.582	+	562.098i	1.72928	+	0.998398i	0.892928	+	0.450199i	$0.148647\pi$
$563$	973.582	+	562.098i	1.72928	+	0.998398i	0.836348	+	0.548199i	$0.184686\pi$
$564$		0			0
$565$	−235.599			−0.416990
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−76.7723	+	44.3245i	−0.134925	+	0.0778989i	−0.565943	−	0.824444i	$-0.691488\pi$
$569$	−76.7723	+	44.3245i	−0.134925	+	0.0778989i	0.431018	+	0.902343i	$0.358154\pi$
$570$		0			0
$571$	−387.994	+	672.025i	−0.679499	+	1.17693i	0.295633	+	0.955302i	$0.404469\pi$
$571$	−387.994	+	672.025i	−0.679499	+	1.17693i	−0.975132	+	0.221625i	$0.928864\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	351.847	+	203.139i	0.611908	+	0.353285i
$576$		0			0
$577$	749.821			1.29952			0.649758	−	0.760141i	$-0.274870\pi$
$577$	749.821			1.29952			0.649758	+	0.760141i	$0.274870\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	317.396	+	183.248i	0.546292	+	0.315402i
$582$		0			0
$583$	288.495	−	499.687i	0.494845	−	0.857097i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	104.373i		−	0.177807i	−0.996040	−	0.0889036i	$-0.971664\pi$
$587$		−	104.373i		−	0.177807i	0.996040	−	0.0889036i	$-0.0283363\pi$
$588$		0			0
$589$	541.290	+	39.8207i	0.918998	+	0.0676073i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−785.885	+	453.731i	−1.32527	+	0.765145i	−0.984564	−	0.175025i	$-0.944000\pi$
$593$	−785.885	+	453.731i	−1.32527	+	0.765145i	−0.340706	+	0.940170i	$0.610666\pi$
$594$		0			0
$595$	−85.7795	−	148.574i	−0.144167	−	0.249705i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	97.6804i		−	0.163072i	−0.996670	−	0.0815362i	$-0.974017\pi$
$599$		−	97.6804i		−	0.163072i	0.996670	−	0.0815362i	$-0.0259826\pi$
$600$		0			0
$601$	−456.977	−	791.508i	−0.760361	−	1.31698i	−0.942664	−	0.333742i	$-0.891689\pi$
$601$	−456.977	−	791.508i	−0.760361	−	1.31698i	0.182303	−	0.983242i	$-0.441645\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			44.7023i			0.0738881i
$606$		0			0
$607$	107.749	−	186.627i	0.177511	−	0.307457i	−0.763517	−	0.645788i	$-0.776529\pi$
$607$	107.749	−	186.627i	0.177511	−	0.307457i	0.941027	+	0.338331i	$0.109862\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−38.2424	−	22.0792i	−0.0625898	−	0.0361362i
$612$		0			0
$613$	401.592	−	695.578i	0.655126	−	1.13471i	−0.326736	−	0.945116i	$-0.605949\pi$
$613$	401.592	−	695.578i	0.655126	−	1.13471i	0.981862	−	0.189596i	$-0.0607178\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			384.264i			0.622794i	0.950280	+	0.311397i	$0.100797\pi$
$617$			384.264i			0.622794i	−0.950280	+	0.311397i	$0.899203\pi$
$618$		0			0
$619$	−55.6379	−	96.3677i	−0.0898835	−	0.155683i	0.817578	−	0.575818i	$-0.195316\pi$
$619$	−55.6379	−	96.3677i	−0.0898835	−	0.155683i	−0.907462	+	0.420135i	$0.861983\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−45.5582	−	26.3030i	−0.0731271	−	0.0422200i
$624$		0			0
$625$	−190.222	−	329.474i	−0.304355	−	0.527159i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−860.240	−	496.660i	−1.36763	−	0.789602i
$630$		0			0
$631$	−396.481	−	686.725i	−0.628337	−	1.08831i	−0.987885	−	0.155186i	$-0.950402\pi$
$631$	−396.481	−	686.725i	−0.628337	−	1.08831i	0.359548	−	0.933127i	$-0.382931\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−378.977	−	218.803i	−0.596815	−	0.344571i
$636$		0			0
$637$	125.749	−	217.804i	0.197409	−	0.341922i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			536.996i			0.837747i	0.908045	+	0.418873i	$0.137575\pi$
$641$			536.996i			0.837747i	−0.908045	+	0.418873i	$0.862425\pi$
$642$		0			0
$643$	251.413	+	435.461i	0.391001	+	0.677233i	0.992582	−	0.121579i	$-0.0387957\pi$
$643$	251.413	+	435.461i	0.391001	+	0.677233i	−0.601581	+	0.798812i	$0.705462\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	42.7301i		−	0.0660435i	−0.999455	−	0.0330217i	$-0.989487\pi$
$647$		−	42.7301i		−	0.0660435i	0.999455	−	0.0330217i	$-0.0105131\pi$
$648$		0			0
$649$	306.439	−	530.768i	0.472171	−	0.817824i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−539.034	−	311.211i	−0.825473	−	0.476587i	0.0268270	−	0.999640i	$-0.491460\pi$
$653$	−539.034	−	311.211i	−0.825473	−	0.476587i	−0.852300	+	0.523053i	$0.824793\pi$
$654$		0			0
$655$	232.622	−	402.913i	0.355148	−	0.615134i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−797.682	−	460.542i	−1.21044	−	0.698850i	−0.247587	−	0.968866i	$-0.579638\pi$
$659$	−797.682	−	460.542i	−1.21044	−	0.698850i	−0.962856	+	0.270016i	$0.912971\pi$
$660$		0			0
$661$	−111.337			−0.168437			−0.0842183	−	0.996447i	$-0.526839\pi$
$661$	−111.337			−0.168437			−0.0842183	+	0.996447i	$0.526839\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.7273	−	173.005i	0.0191389	−	0.260158i
$666$		0			0
$667$	44.2627	−	76.6652i	0.0663608	−	0.114940i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			277.390i			0.413399i
$672$		0			0
$673$	223.538	+	387.179i	0.332152	+	0.575304i	0.982934	−	0.183961i	$-0.0588921\pi$
$673$	223.538	+	387.179i	0.332152	+	0.575304i	−0.650782	+	0.759265i	$0.725559\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−202.443	+	116.881i	−0.299030	+	0.172645i	−0.642007	−	0.766699i	$-0.721898\pi$
$677$	−202.443	+	116.881i	−0.299030	+	0.172645i	0.342977	+	0.939344i	$0.388565\pi$
$678$		0			0
$679$	248.676	+	430.719i	0.366238	+	0.634343i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			496.474i			0.726902i	0.931613	+	0.363451i	$0.118402\pi$
$683$			496.474i			0.726902i	−0.931613	+	0.363451i	$0.881598\pi$
$684$		0			0
$685$	9.73487			0.0142115
$686$		0			0
$687$		0			0
$688$		0			0
$689$	423.973	−	244.781i	0.615345	−	0.355270i
$690$		0			0
$691$	237.938	+	412.120i	0.344338	+	0.596411i	0.985233	−	0.171217i	$-0.0547701\pi$
$691$	237.938	+	412.120i	0.344338	+	0.596411i	−0.640895	+	0.767628i	$0.721437\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−260.402	+	150.343i	−0.374679	+	0.216321i
$696$		0			0
$697$	139.625			0.200323
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−484.715	−	279.850i	−0.691462	−	0.399216i	0.112697	−	0.993629i	$-0.464051\pi$
$701$	−484.715	−	279.850i	−0.691462	−	0.399216i	−0.804160	+	0.594413i	$0.797384\pi$
$702$		0			0
$703$	−437.027	−	904.335i	−0.621661	−	1.28639i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			726.896i			1.02814i
$708$		0			0
$709$	−345.343	+	598.151i	−0.487084	+	0.843654i	−0.999890	−	0.0148505i	$-0.995273\pi$
$709$	−345.343	+	598.151i	−0.487084	+	0.843654i	0.512806	+	0.858505i	$0.328606\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−465.669	−	268.854i	−0.653113	−	0.377075i
$714$		0			0
$715$	−113.845	+	197.185i	−0.159224	+	0.275784i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−673.685	−	388.952i	−0.936975	−	0.540963i	−0.0479641	−	0.998849i	$-0.515273\pi$
$719$	−673.685	−	388.952i	−0.936975	−	0.540963i	−0.889011	+	0.457886i	$0.848607\pi$
$720$		0			0
$721$	−280.682			−0.389296
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−87.9071	+	50.7532i	−0.121251	+	0.0700044i
$726$		0			0
$727$	−421.433			−0.579688			−0.289844	−	0.957074i	$-0.593603\pi$
$727$	−421.433			−0.579688			−0.289844	+	0.957074i	$0.593603\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−673.688	−	388.954i	−0.921597	−	0.532085i
$732$		0			0
$733$	216.673	−	375.289i	0.295598	−	0.511990i	−0.679526	−	0.733651i	$-0.737815\pi$
$733$	216.673	−	375.289i	0.295598	−	0.511990i	0.975124	+	0.221661i	$0.0711479\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	404.807	−	233.715i	0.549263	−	0.317117i
$738$		0			0
$739$	354.109	−	613.335i	0.479174	−	0.829953i	−0.520541	−	0.853837i	$-0.674270\pi$
$739$	354.109	−	613.335i	0.479174	−	0.829953i	0.999715	+	0.0238836i	$0.00760311\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−706.541	+	407.921i	−0.950930	+	0.549019i	−0.893370	−	0.449323i	$-0.851665\pi$
$743$	−706.541	+	407.921i	−0.950930	+	0.549019i	−0.0575600	+	0.998342i	$0.518332\pi$
$744$		0			0
$745$	65.9913	−	114.300i	0.0885790	−	0.153423i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−267.690	+	154.551i	−0.357396	+	0.206343i
$750$		0			0
$751$	−1082.49			−1.44140			−0.720700	−	0.693247i	$-0.756180\pi$
$751$	−1082.49			−1.44140			−0.720700	+	0.693247i	$0.756180\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.9252	−	14.3906i	−0.0330135	−	0.0190604i
$756$		0			0
$757$	160.866	−	278.628i	0.212505	−	0.368069i	−0.739993	−	0.672614i	$-0.765171\pi$
$757$	160.866	−	278.628i	0.212505	−	0.368069i	0.952498	+	0.304546i	$0.0985046\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−958.373	−	553.317i	−1.25936	−	0.727092i	−0.286410	−	0.958107i	$-0.592462\pi$
$761$	−958.373	−	553.317i	−1.25936	−	0.727092i	−0.972950	+	0.231015i	$0.925795\pi$
$762$		0			0
$763$	−318.618			−0.417585
$764$		0			0
$765$		0			0
$766$		0			0
$767$	450.344	−	260.006i	0.587150	−	0.338991i
$768$		0			0
$769$	−840.541			−1.09303			−0.546516	−	0.837449i	$-0.684046\pi$
$769$	−840.541			−1.09303			−0.546516	+	0.837449i	$0.684046\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−676.445	+	390.545i	−0.875090	+	0.505233i	−0.869036	−	0.494748i	$-0.835260\pi$
$773$	−676.445	+	390.545i	−0.875090	+	0.505233i	−0.00605359	+	0.999982i	$0.501927\pi$
$774$		0			0
$775$	308.278	+	533.953i	0.397778	+	0.688972i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	116.759	+	79.3715i	0.149883	+	0.101889i
$780$		0			0
$781$	1323.14			1.69416
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−238.858	−	137.905i	−0.304278	−	0.175675i
$786$		0			0
$787$	−383.786	+	664.737i	−0.487657	+	0.844647i	−0.999899	−	0.0141942i	$-0.995482\pi$
$787$	−383.786	+	664.737i	−0.487657	+	0.844647i	0.512242	+	0.858841i	$0.328815\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			629.631i			0.795993i
$792$		0			0
$793$	−117.680	+	203.827i	−0.148398	+	0.257033i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	796.725	+	459.990i	0.999655	+	0.577151i	0.908146	−	0.418653i	$-0.137498\pi$
$797$	796.725	+	459.990i	0.999655	+	0.577151i	0.0915091	+	0.995804i	$0.470831\pi$
$798$		0			0
$799$	−40.5808	−	70.2880i	−0.0507894	−	0.0879699i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	278.351i		−	0.346639i
$804$		0			0
$805$	−85.9302	+	148.835i	−0.106746	+	0.184889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1141.94i		−	1.41155i	−0.708438	−	0.705773i	$-0.750600\pi$
$809$		−	1141.94i		−	1.41155i	0.708438	−	0.705773i	$-0.249400\pi$
$810$		0			0
$811$	−618.024	−	1070.45i	−0.762052	−	1.31991i	−0.941791	−	0.336198i	$-0.890859\pi$
$811$	−618.024	−	1070.45i	−0.762052	−	1.31991i	0.179739	−	0.983714i	$-0.442475\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−252.332	+	145.684i	−0.309610	+	0.178753i
$816$		0			0
$817$	−342.253	−	708.221i	−0.418915	−	0.866855i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1061.61	−	612.923i	−1.29308	−	0.746557i	−0.313877	−	0.949464i	$-0.601628\pi$
$821$	−1061.61	−	612.923i	−1.29308	−	0.746557i	−0.979198	+	0.202906i	$0.934961\pi$
$822$		0			0
$823$	266.361			0.323646			0.161823	−	0.986820i	$-0.448263\pi$
$823$	266.361			0.323646			0.161823	+	0.986820i	$0.448263\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	725.041	−	418.603i	0.876712	−	0.506170i	0.00713916	−	0.999975i	$-0.497728\pi$
$827$	725.041	−	418.603i	0.876712	−	0.506170i	0.869573	+	0.493805i	$0.164394\pi$
$828$		0			0
$829$	−425.019			−0.512689			−0.256344	−	0.966586i	$-0.582518\pi$
$829$	−425.019			−0.512689			−0.256344	+	0.966586i	$0.582518\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	400.316	−	231.123i	0.480572	−	0.277458i
$834$		0			0
$835$	228.147	+	395.163i	0.273230	+	0.473249i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	242.391i		−	0.288904i	−0.989512	−	0.144452i	$-0.953858\pi$
$839$		−	242.391i		−	0.288904i	0.989512	−	0.144452i	$-0.0461420\pi$
$840$		0			0
$841$	−409.441	−	709.173i	−0.486850	−	0.843250i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	103.213	−	59.5902i	0.122146	−	0.0705210i
$846$		0			0
$847$	119.465			0.141045
$848$		0			0
$849$		0			0
$850$		0			0
$851$			995.064i			1.16929i
$852$		0			0
$853$	1433.19			1.68017			0.840087	−	0.542452i	$-0.182504\pi$
$853$	1433.19			1.68017			0.840087	+	0.542452i	$0.182504\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1339.15i		−	1.56261i	−0.624152	−	0.781303i	$-0.714555\pi$
$857$		−	1339.15i		−	1.56261i	0.624152	−	0.781303i	$-0.285445\pi$
$858$		0			0
$859$	355.159			0.413457			0.206728	−	0.978398i	$-0.433718\pi$
$859$	355.159			0.413457			0.206728	+	0.978398i	$0.433718\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	71.3787i		−	0.0827100i	−0.999145	−	0.0413550i	$-0.986833\pi$
$863$		−	71.3787i		−	0.0827100i	0.999145	−	0.0413550i	$-0.0131675\pi$
$864$		0			0
$865$	−96.6568	−	167.415i	−0.111742	−	0.193543i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1555.11	+	897.843i	−1.78954	+	1.03319i
$870$		0			0
$871$	396.604			0.455343
$872$		0			0
$873$		0			0
$874$		0			0
$875$	368.333	−	212.657i	0.420952	−	0.243037i
$876$		0			0
$877$	−363.008	−	628.749i	−0.413920	−	0.716931i	0.581394	−	0.813622i	$-0.302508\pi$
$877$	−363.008	−	628.749i	−0.413920	−	0.716931i	−0.995314	+	0.0966909i	$0.969174\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	738.131i		−	0.837833i	−0.908025	−	0.418916i	$-0.862410\pi$
$881$		−	738.131i		−	0.837833i	0.908025	−	0.418916i	$-0.137590\pi$
$882$		0			0
$883$	172.011	+	297.931i	0.194803	+	0.337408i	0.946836	−	0.321717i	$-0.104260\pi$
$883$	172.011	+	297.931i	0.194803	+	0.337408i	−0.752033	+	0.659125i	$0.770927\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	688.643i		−	0.776373i	−0.921581	−	0.388186i	$-0.873102\pi$
$887$		−	688.643i		−	0.776373i	0.921581	−	0.388186i	$-0.126898\pi$
$888$		0			0
$889$	−584.743	+	1012.80i	−0.657754	+	1.13926i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.02108	−	81.8456i	0.00674253	−	0.0916524i
$894$		0			0
$895$	220.674	+	382.219i	0.246563	+	0.427060i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	116.345	−	67.1718i	0.129416	−	0.0747183i
$900$		0			0
$901$	899.795			0.998663
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−345.953	−	199.736i	−0.382268	−	0.220703i
$906$		0			0
$907$	559.726			0.617118			0.308559	−	0.951205i	$-0.400153\pi$
$907$	559.726			0.617118			0.308559	+	0.951205i	$0.400153\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−24.3919	+	14.0827i	−0.0267749	+	0.0154585i	−0.513328	−	0.858193i	$-0.671587\pi$
$911$	−24.3919	+	14.0827i	−0.0267749	+	0.0154585i	0.486553	+	0.873651i	$0.338254\pi$
$912$		0			0
$913$	446.999	−	774.225i	0.489594	−	0.848001i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1076.77	−	621.674i	−1.17423	−	0.677943i
$918$		0			0
$919$	212.578			0.231314			0.115657	−	0.993289i	$-0.463103\pi$
$919$	212.578			0.231314			0.115657	+	0.993289i	$0.463103\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	972.246	+	561.327i	1.05335	+	0.608155i
$924$		0			0
$925$	570.488	−	988.115i	0.616744	−	1.06823i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			1315.09i			1.41559i	0.706416	+	0.707797i	$0.250311\pi$
$929$			1315.09i			1.41559i	−0.706416	+	0.707797i	$0.749689\pi$
$930$		0			0
$931$	466.141	+	34.2923i	0.500689	+	0.0368338i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−362.419	+	209.243i	−0.387614	+	0.223789i
$936$		0			0
$937$	−201.216	−	348.516i	−0.214745	−	0.371948i	0.738449	−	0.674309i	$-0.235559\pi$
$937$	−201.216	−	348.516i	−0.214745	−	0.371948i	−0.953194	+	0.302361i	$0.902225\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	905.907i		−	0.962707i	−0.876527	−	0.481353i	$-0.840145\pi$
$941$		−	905.907i		−	0.962707i	0.876527	−	0.481353i	$-0.159855\pi$
$942$		0			0
$943$	−69.9353	−	121.131i	−0.0741625	−	0.128453i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1486.69i			1.56989i	0.619563	+	0.784947i	$0.287310\pi$
$947$			1486.69i			1.56989i	−0.619563	+	0.784947i	$0.712690\pi$
$948$		0			0
$949$	118.087	−	204.533i	0.124433	−	0.215525i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	910.211	+	525.511i	0.955101	+	0.551428i	0.894662	−	0.446744i	$-0.147417\pi$
$953$	910.211	+	525.511i	0.955101	+	0.551428i	0.0604392	+	0.998172i	$0.480750\pi$
$954$		0			0
$955$	160.963	−	278.795i	0.168547	−	0.291932i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	26.0161i		−	0.0271284i
$960$		0			0
$961$	72.4944	+	125.564i	0.0754364	+	0.130660i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−68.9906	−	39.8317i	−0.0714928	−	0.0412764i
$966$		0			0
$967$	578.495	+	1001.98i	0.598237	+	1.03618i	0.993081	+	0.117429i	$0.0374651\pi$
$967$	578.495	+	1001.98i	0.598237	+	1.03618i	−0.394845	+	0.918748i	$0.629202\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	453.947	+	262.086i	0.467504	+	0.269914i	0.715194	−	0.698926i	$-0.246338\pi$
$971$	453.947	+	262.086i	0.467504	+	0.269914i	−0.247690	+	0.968839i	$0.579671\pi$
$972$		0			0
$973$	401.787	+	695.916i	0.412936	+	0.715227i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	379.565	+	219.142i	0.388501	+	0.224301i	0.681510	−	0.731809i	$-0.261323\pi$
$977$	379.565	+	219.142i	0.388501	+	0.224301i	−0.293010	+	0.956109i	$0.594657\pi$
$978$		0			0
$979$	−64.1612	+	111.130i	−0.0655375	+	0.113514i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			802.644i			0.816525i	0.912865	+	0.408262i	$0.133865\pi$
$983$			802.644i			0.816525i	−0.912865	+	0.408262i	$0.866135\pi$
$984$		0			0
$985$	212.289	+	367.695i	0.215522	+	0.373294i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			779.274i			0.787942i
$990$		0			0
$991$	−767.110	+	1328.67i	−0.774077	+	1.34074i	0.161236	+	0.986916i	$0.448452\pi$
$991$	−767.110	+	1328.67i	−0.774077	+	1.34074i	−0.935312	+	0.353824i	$0.884881\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	105.108	+	60.6843i	0.105637	+	0.0609893i
$996$		0			0
$997$	−464.248	+	804.100i	−0.465644	+	0.806520i	−0.999230	−	0.0392260i	$-0.987511\pi$
$997$	−464.248	+	804.100i	−0.465644	+	0.806520i	0.533586	+	0.845746i	$0.320844\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.be.a.125.16		80
3.2	odd	2		684.3.be.a.581.11	yes	80
9.2	odd	6		2052.3.m.a.1493.25		80
9.7	even	3		684.3.m.a.353.17	✓	80
19.7	even	3		2052.3.m.a.881.16		80
57.26	odd	6		684.3.m.a.653.17	yes	80
171.7	even	3		684.3.be.a.425.11	yes	80
171.83	odd	6	inner	2052.3.be.a.197.16		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.17	✓	80	9.7	even	3
684.3.m.a.653.17	yes	80	57.26	odd	6
684.3.be.a.425.11	yes	80	171.7	even	3
684.3.be.a.581.11	yes	80	3.2	odd	2
2052.3.m.a.881.16		80	19.7	even	3
2052.3.m.a.1493.25		80	9.2	odd	6
2052.3.be.a.125.16		80	1.1	even	1	trivial
2052.3.be.a.197.16		80	171.83	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.be (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.60071 + 0.924171i) q^{5} +(2.46982 + 4.27785i) q^{7} +O(q^{10})\)	\(q+(-1.60071 + 0.924171i) q^{5} +(2.46982 + 4.27785i) q^{7} +(10.4350 - 6.02464i) q^{11} +10.2235 q^{13} +(16.2730 + 9.39522i) q^{17} +(8.26716 + 17.1071i) q^{19} -18.8234i q^{23} +(-10.7918 + 18.6920i) q^{25} +(4.07286 + 2.35147i) q^{29} +(14.2830 - 24.7388i) q^{31} +(-7.90692 - 4.56506i) q^{35} -52.8630 q^{37} +(6.43514 - 3.71533i) q^{41} -41.3991 q^{43} +(-3.74062 - 2.15965i) q^{47} +(12.3000 - 21.3043i) q^{49} +(41.4703 - 23.9429i) q^{53} +(-11.1356 + 19.2874i) q^{55} +(44.0498 - 25.4322i) q^{59} +(-11.5107 + 19.9371i) q^{61} +(-16.3649 + 9.44828i) q^{65} +38.7932 q^{67} +(95.0990 + 54.9054i) q^{71} +(11.5505 - 20.0061i) q^{73} +(51.5450 + 29.7595i) q^{77} -149.029 q^{79} +(64.2549 - 37.0976i) q^{83} -34.7311 q^{85} +(-9.22299 + 5.32490i) q^{89} +(25.2502 + 43.7346i) q^{91} +(-29.0433 - 19.7433i) q^{95} +100.686 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

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