LMFDB - Embedded newform 2052.3.m.a.881.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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, 2]))

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

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (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			881.16
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84834i q^{5} +(2.46982 + 4.27785i) q^{7} +O(q^{10})$	$q-1.84834i q^{5} +(2.46982 + 4.27785i) q^{7} +(10.4350 - 6.02464i) q^{11} +(-5.11176 - 8.85383i) q^{13} +(-16.2730 + 9.39522i) q^{17} +(8.26716 - 17.1071i) q^{19} +(-16.3016 + 9.41172i) q^{23} +21.5836 q^{25} -4.70293i q^{29} +(14.2830 - 24.7388i) q^{31} +(7.90692 - 4.56506i) q^{35} -52.8630 q^{37} +7.43066i q^{41} +(20.6996 - 35.8527i) q^{43} +4.31930i q^{47} +(12.3000 - 21.3043i) q^{49} +(-41.4703 - 23.9429i) q^{53} +(-11.1356 - 19.2874i) q^{55} +50.8643i q^{59} +23.0213 q^{61} +(-16.3649 + 9.44828i) q^{65} +(-19.3966 - 33.5959i) q^{67} +(-95.0990 + 54.9054i) q^{71} +(11.5505 + 20.0061i) q^{73} +(51.5450 + 29.7595i) q^{77} +(74.5143 - 129.062i) q^{79} +(64.2549 - 37.0976i) q^{83} +(17.3656 + 30.0781i) q^{85} +(9.22299 + 5.32490i) q^{89} +(25.2502 - 43.7346i) q^{91} +(-31.6198 - 15.2805i) q^{95} +(-50.3430 + 87.1966i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$		−	1.84834i		−	0.369668i	−0.982770	−	0.184834i	$-0.940825\pi$
$5$		−	1.84834i		−	0.369668i	0.982770	−	0.184834i	$-0.0591748\pi$
$6$		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
$13$	−5.11176	−	8.85383i	−0.393212	−	0.681064i	0.599659	−	0.800256i	$-0.295303\pi$
$13$	−5.11176	−	8.85383i	−0.393212	−	0.681064i	−0.992871	+	0.119192i	$0.961970\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−16.2730	+	9.39522i	−0.957235	+	0.552660i	−0.895321	−	0.445422i	$-0.853054\pi$
$17$	−16.2730	+	9.39522i	−0.957235	+	0.552660i	−0.0619139	+	0.998081i	$0.519720\pi$
$18$		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
$23$	−16.3016	+	9.41172i	−0.708764	+	0.409205i	−0.810603	−	0.585596i	$-0.800861\pi$
$23$	−16.3016	+	9.41172i	−0.708764	+	0.409205i	0.101839	+	0.994801i	$0.467527\pi$
$24$		0			0
$25$	21.5836			0.863345
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.70293i		−	0.162170i	−0.996707	−	0.0810850i	$-0.974161\pi$
$29$		−	4.70293i		−	0.162170i	0.996707	−	0.0810850i	$-0.0258385\pi$
$30$		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
$41$			7.43066i			0.181236i	0.995886	+	0.0906178i	$0.0288842\pi$
$41$			7.43066i			0.181236i	−0.995886	+	0.0906178i	$0.971116\pi$
$42$		0			0
$43$	20.6996	−	35.8527i	0.481385	−	0.833784i	−0.518386	−	0.855146i	$-0.673467\pi$
$43$	20.6996	−	35.8527i	0.481385	−	0.833784i	0.999772	+	0.0213626i	$0.00680046\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			4.31930i			0.0919000i	0.998944	+	0.0459500i	$0.0146315\pi$
$47$			4.31930i			0.0919000i	−0.998944	+	0.0459500i	$0.985369\pi$
$48$		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	$-0.482534\pi$
$53$	−41.4703	−	23.9429i	−0.782459	−	0.451753i	−0.837301	+	0.546742i	$0.815868\pi$
$54$		0			0
$55$	−11.1356	−	19.2874i	−0.202465	−	0.350680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			50.8643i			0.862107i	0.902326	+	0.431053i	$0.141858\pi$
$59$			50.8643i			0.862107i	−0.902326	+	0.431053i	$0.858142\pi$
$60$		0			0
$61$	23.0213			0.377399			0.188699	−	0.982035i	$-0.439573\pi$
$61$	23.0213			0.377399			0.188699	+	0.982035i	$0.439573\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−16.3649	+	9.44828i	−0.251768	+	0.145358i
$66$		0			0
$67$	−19.3966	−	33.5959i	−0.289502	−	0.501432i	0.684189	−	0.729305i	$-0.260156\pi$
$67$	−19.3966	−	33.5959i	−0.289502	−	0.501432i	−0.973691	+	0.227873i	$0.926823\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−95.0990	+	54.9054i	−1.33942	+	0.773316i	−0.986722	−	0.162420i	$-0.948070\pi$
$71$	−95.0990	+	54.9054i	−1.33942	+	0.773316i	−0.352701	+	0.935736i	$0.614737\pi$
$72$		0			0
$73$	11.5505	+	20.0061i	0.158226	+	0.274056i	0.934229	−	0.356673i	$-0.116089\pi$
$73$	11.5505	+	20.0061i	0.158226	+	0.274056i	−0.776003	+	0.630730i	$0.782756\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	51.5450	+	29.7595i	0.669415	+	0.386487i
$78$		0			0
$79$	74.5143	−	129.062i	0.943218	−	1.63370i	0.183939	−	0.982938i	$-0.441115\pi$
$79$	74.5143	−	129.062i	0.943218	−	1.63370i	0.759280	−	0.650764i	$-0.225551\pi$
$80$		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
$85$	17.3656	+	30.0781i	0.204301	+	0.353859i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.22299	+	5.32490i	0.103629	+	0.0598303i	0.550919	−	0.834559i	$-0.314277\pi$
$89$	9.22299	+	5.32490i	0.103629	+	0.0598303i	−0.447290	+	0.894389i	$0.647611\pi$
$90$		0			0
$91$	25.2502	−	43.7346i	0.277475	−	0.480601i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−31.6198	−	15.2805i	−0.332840	−	0.160848i
$96$		0			0
$97$	−50.3430	+	87.1966i	−0.519000	+	0.898934i	0.480756	+	0.876854i	$0.340362\pi$
$97$	−50.3430	+	87.1966i	−0.519000	+	0.898934i	−0.999756	+	0.0220799i	$0.992971\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	147.156i		−	1.45699i	−0.685052	−	0.728495i	$-0.740220\pi$
$101$		−	147.156i		−	1.45699i	0.685052	−	0.728495i	$-0.259780\pi$
$102$		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	0.679308	−	0.733854i	$-0.262280\pi$
$109$	−32.2512	−	55.8606i	−0.295882	−	0.512483i	−0.975190	+	0.221371i	$0.928947\pi$
$110$		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
$119$	−80.3826	−	46.4089i	−0.675484	−	0.389991i
$120$		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	0.152399	−	0.988319i	$-0.451300\pi$
$127$	118.378	−	205.036i	0.932109	−	1.61446i	0.779710	−	0.626141i	$-0.215367\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	251.709i		−	1.92144i	−0.277521	−	0.960720i	$-0.589513\pi$
$131$		−	251.709i		−	1.92144i	0.277521	−	0.960720i	$-0.410487\pi$
$132$		0			0
$133$	93.6001	−	6.88581i	0.703760	−	0.0517730i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			5.26681i			0.0384439i	0.999815	+	0.0192219i	$0.00611891\pi$
$137$			5.26681i			0.0384439i	−0.999815	+	0.0192219i	$0.993881\pi$
$138$		0			0
$139$	−81.3395	−	140.884i	−0.585176	−	1.01355i	−0.994853	−	0.101324i	$-0.967692\pi$
$139$	−81.3395	−	140.884i	−0.585176	−	1.01355i	0.409677	−	0.912230i	$-0.365641\pi$
$140$		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
$149$		−	71.4060i		−	0.479235i	−0.970867	−	0.239617i	$-0.922978\pi$
$149$		−	71.4060i		−	0.479235i	0.970867	−	0.239617i	$-0.0770220\pi$
$150$		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
$155$	−45.7258	−	26.3998i	−0.295005	−	0.170321i
$156$		0			0
$157$	−149.220			−0.950445			−0.475223	−	0.879866i	$-0.657633\pi$
$157$	−149.220			−0.950445			−0.475223	+	0.879866i	$0.657633\pi$
$158$		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
$167$	−213.793	+	123.433i	−1.28020	+	0.739123i	−0.976884	−	0.213768i	$-0.931426\pi$
$167$	−213.793	+	123.433i	−1.28020	+	0.739123i	−0.303313	+	0.952891i	$0.598093\pi$
$168$		0			0
$169$	32.2398	−	55.8410i	0.190768	−	0.330420i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−90.5755	−	52.2938i	−0.523558	−	0.302276i	0.214831	−	0.976651i	$-0.431080\pi$
$173$	−90.5755	−	52.2938i	−0.523558	−	0.302276i	−0.738389	+	0.674375i	$0.764413\pi$
$174$		0			0
$175$	53.3076	+	92.3315i	0.304615	+	0.527608i
$176$		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	−0.396228	−	0.918152i	$-0.629681\pi$
$181$	108.062	−	187.169i	0.597029	−	1.03408i	0.993257	−	0.115933i	$-0.0369857\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			97.7090i			0.528157i
$186$		0			0
$187$	−113.206	+	196.078i	−0.605377	+	1.04854i
$188$		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
$193$	−43.1000			−0.223316			−0.111658	−	0.993747i	$-0.535616\pi$
$193$	−43.1000			−0.223316			−0.111658	+	0.993747i	$0.535616\pi$
$194$		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	−0.936650	−	0.350268i	$-0.886090\pi$
$199$	−32.8318	+	56.8663i	−0.164984	+	0.285760i	0.771666	+	0.636028i	$0.219424\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	20.1184	−	11.6154i	0.0991055	−	0.0572186i
$204$		0			0
$205$	13.7344			0.0669970
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.7966	−	228.319i	−0.0803666	−	1.09244i
$210$		0			0
$211$	−154.549			−0.732460			−0.366230	−	0.930524i	$-0.619352\pi$
$211$	−154.549			−0.732460			−0.366230	+	0.930524i	$0.619352\pi$
$212$		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
$223$	−65.4369	+	113.340i	−0.293439	+	0.508251i	−0.974621	−	0.223863i	$-0.928133\pi$
$223$	−65.4369	+	113.340i	−0.293439	+	0.508251i	0.681182	+	0.732115i	$0.261466\pi$
$224$		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	$-0.660797\pi$
$233$	7.44103	−	4.29608i	0.0319357	−	0.0184381i	0.515883	+	0.856659i	$0.327464\pi$
$234$		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
$241$	277.831			1.15283			0.576414	−	0.817158i	$-0.304452\pi$
$241$	277.831			1.15283			0.576414	+	0.817158i	$0.304452\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−39.3776	−	22.7346i	−0.160725	−	0.0927945i
$246$		0			0
$247$	−193.723	+	14.2515i	−0.784305	+	0.0576984i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−413.219	−	238.572i	−1.64629	−	0.950486i	−0.978529	−	0.206110i	$-0.933919\pi$
$251$	−413.219	−	238.572i	−1.64629	−	0.950486i	−0.667761	−	0.744376i	$-0.732747\pi$
$252$		0			0
$253$	−113.404	+	196.422i	−0.448239	+	0.776372i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	152.662	−	88.1392i	0.594014	−	0.342954i	−0.172669	−	0.984980i	$-0.555239\pi$
$257$	152.662	−	88.1392i	0.594014	−	0.342954i	0.766683	+	0.642026i	$0.221906\pi$
$258$		0			0
$259$	−130.562	−	226.140i	−0.504100	−	0.873127i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−163.347	−	94.3087i	−0.621093	−	0.358588i	0.156201	−	0.987725i	$-0.450075\pi$
$263$	−163.347	−	94.3087i	−0.621093	−	0.358588i	−0.777294	+	0.629137i	$0.783408\pi$
$264$		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	$-0.389237\pi$
$269$	356.589	−	205.877i	1.32561	−	0.765340i	0.984617	+	0.174726i	$0.0559041\pi$
$270$		0			0
$271$	24.3004	+	42.0896i	0.0896695	+	0.155312i	0.907371	−	0.420330i	$-0.138086\pi$
$271$	24.3004	+	42.0896i	0.0896695	+	0.155312i	−0.817702	+	0.575642i	$0.804752\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	225.225	−	130.034i	0.818999	−	0.472849i
$276$		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
$281$			134.415i			0.478346i	0.970977	+	0.239173i	$0.0768763\pi$
$281$			134.415i			0.478346i	−0.970977	+	0.239173i	$0.923124\pi$
$282$		0			0
$283$	−63.6634			−0.224959			−0.112480	−	0.993654i	$-0.535879\pi$
$283$	−63.6634			−0.224959			−0.112480	+	0.993654i	$0.535879\pi$
$284$		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
$295$	94.0146			0.318694
$296$		0			0
$297$		0			0
$298$		0			0
$299$	166.659	+	96.2209i	0.557390	+	0.321809i
$300$		0			0
$301$	204.496			0.679390
$302$		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	0.551563	−	0.834133i	$-0.314032\pi$
$307$	−137.106	−	237.474i	−0.446599	−	0.773532i	−0.998162	+	0.0606011i	$0.980698\pi$
$308$		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
$313$	99.5291			0.317984			0.158992	−	0.987280i	$-0.449176\pi$
$313$	99.5291			0.317984			0.158992	+	0.987280i	$0.449176\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			123.813i			0.390577i	0.980746	+	0.195289i	$0.0625644\pi$
$317$			123.813i			0.390577i	−0.980746	+	0.195289i	$0.937436\pi$
$318$		0			0
$319$	−28.3335	−	49.0750i	−0.0888197	−	0.153840i
$320$		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
$329$	−18.4773	+	10.6679i	−0.0561620	+	0.0324252i
$330$		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
$337$	−177.343			−0.526241			−0.263120	−	0.964763i	$-0.584752\pi$
$337$	−177.343			−0.526241			−0.263120	+	0.964763i	$0.584752\pi$
$338$		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
$347$		−	388.316i		−	1.11907i	−0.828808	−	0.559533i	$-0.810980\pi$
$347$		−	388.316i		−	1.11907i	0.828808	−	0.559533i	$-0.189020\pi$
$348$		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
$355$	101.484	+	175.775i	0.285870	+	0.495142i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	208.764	−	120.530i	0.581515	−	0.335738i	−0.180220	−	0.983626i	$-0.557681\pi$
$359$	208.764	−	120.530i	0.581515	−	0.335738i	0.761735	+	0.647888i	$0.224348\pi$
$360$		0			0
$361$	−224.308	−	282.855i	−0.621352	−	0.783532i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	36.9781	−	21.3493i	0.101310	−	0.0584913i
$366$		0			0
$367$	190.391			0.518775			0.259388	−	0.965773i	$-0.416479\pi$
$367$	190.391			0.518775			0.259388	+	0.965773i	$0.416479\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	236.538i		−	0.637569i
$372$		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
$383$			387.164i			1.01087i	0.862864	+	0.505436i	$0.168668\pi$
$383$			387.164i			1.01087i	−0.862864	+	0.505436i	$0.831332\pi$
$384$		0			0
$385$	55.0057	−	95.2727i	0.142872	−	0.247462i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	18.9972i		−	0.0488361i	−0.999702	−	0.0244180i	$-0.992227\pi$
$389$		−	18.9972i		−	0.0488361i	0.999702	−	0.0244180i	$-0.00777327\pi$
$390$		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	0.972037	−	0.234829i	$-0.0754531\pi$
$397$	112.212	+	194.357i	0.282650	+	0.489564i	−0.689387	+	0.724394i	$0.742120\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	175.758i		−	0.438300i	−0.975691	−	0.219150i	$-0.929672\pi$
$401$		−	175.758i		−	0.438300i	0.975691	−	0.219150i	$-0.0703284\pi$
$402$		0			0
$403$	−292.044			−0.724675
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−551.625	+	318.481i	−1.35534	+	0.782508i
$408$		0			0
$409$	317.177	+	549.367i	0.775494	+	1.34320i	0.934516	+	0.355921i	$0.115833\pi$
$409$	317.177	+	549.367i	0.775494	+	1.34320i	−0.159022	+	0.987275i	$0.550834\pi$
$410$		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
$421$	−134.754	+	233.400i	−0.320080	+	0.554395i	−0.980504	−	0.196498i	$-0.937043\pi$
$421$	−134.754	+	233.400i	−0.320080	+	0.554395i	0.660424	+	0.750893i	$0.270377\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−351.230	+	202.783i	−0.826424	+	0.477136i
$426$		0			0
$427$	56.8585	+	98.4817i	0.133158	+	0.230636i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−601.019	−	346.999i	−1.39448	−	0.805101i	−0.400669	−	0.916223i	$-0.631222\pi$
$431$	−601.019	−	346.999i	−1.39448	−	0.805101i	−0.993807	+	0.111122i	$0.964556\pi$
$432$		0			0
$433$	−324.494	+	562.041i	−0.749409	+	1.29802i	0.198697	+	0.980061i	$0.436329\pi$
$433$	−324.494	+	562.041i	−0.749409	+	1.29802i	−0.948106	+	0.317954i	$0.897004\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.2397	+	356.681i	0.0600452	+	0.816205i
$438$		0			0
$439$	−163.442	+	283.090i	−0.372306	+	0.644852i	−0.989920	−	0.141629i	$-0.954766\pi$
$439$	−163.442	+	283.090i	−0.372306	+	0.644852i	0.617614	+	0.786481i	$0.288099\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			804.929i			1.81699i	0.417891	+	0.908497i	$0.362770\pi$
$443$			804.929i			1.81699i	−0.417891	+	0.908497i	$0.637230\pi$
$444$		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
$461$	315.862	+	182.363i	0.685167	+	0.395582i	0.801799	−	0.597594i	$-0.203876\pi$
$461$	315.862	+	182.363i	0.685167	+	0.395582i	−0.116632	+	0.993175i	$0.537210\pi$
$462$		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
$473$		−	498.830i		−	1.05461i
$474$		0			0
$475$	178.435	−	369.234i	0.375654	−	0.777335i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			481.973i			1.00621i	0.864226	+	0.503104i	$0.167809\pi$
$479$			481.973i			1.00621i	−0.864226	+	0.503104i	$0.832191\pi$
$480$		0			0
$481$	270.223	+	468.040i	0.561795	+	0.973057i
$482$		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
$491$			930.811i			1.89575i	0.318649	+	0.947873i	$0.396771\pi$
$491$			930.811i			1.89575i	−0.318649	+	0.947873i	$0.603229\pi$
$492$		0			0
$493$	44.1851	+	76.5308i	0.0896249	+	0.155235i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−469.754	−	271.213i	−0.945179	−	0.545699i
$498$		0			0
$499$	−652.342			−1.30730			−0.653649	−	0.756798i	$-0.726763\pi$
$499$	−652.342			−1.30730			−0.653649	+	0.756798i	$0.726763\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	264.025	+	152.435i	0.524901	+	0.303052i	0.738937	−	0.673774i	$-0.235328\pi$
$503$	264.025	+	152.435i	0.524901	+	0.303052i	−0.214037	+	0.976826i	$0.568661\pi$
$504$		0			0
$505$	−271.994			−0.538603
$506$		0			0
$507$		0			0
$508$		0			0
$509$	44.7282	−	25.8238i	0.0878746	−	0.0507344i	−0.455419	−	0.890277i	$-0.650510\pi$
$509$	44.7282	−	25.8238i	0.0878746	−	0.0507344i	0.543293	+	0.839543i	$0.317177\pi$
$510$		0			0
$511$	−57.0554	+	98.8228i	−0.111654	+	0.193391i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	90.9565	+	52.5137i	0.176614	+	0.101968i
$516$		0			0
$517$	26.0222	+	45.0718i	0.0503331	+	0.0871795i
$518$		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	−0.241163	−	0.970485i	$-0.577529\pi$
$523$	376.499	−	652.115i	0.719883	−	1.24687i	0.961046	−	0.276389i	$-0.0891378\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			536.766i			1.01853i
$528$		0			0
$529$	−87.3391	+	151.276i	−0.165102	+	0.285965i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	65.7897	−	37.9837i	0.123433	−	0.0712640i
$534$		0			0
$535$	115.661			0.216190
$536$		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	−0.975727	−	0.218992i	$-0.929723\pi$
$541$	−161.332	+	279.436i	−0.298211	+	0.516517i	0.677516	+	0.735508i	$0.263057\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−103.250	+	59.6112i	−0.189449	+	0.109378i
$546$		0			0
$547$	108.112			0.197645			0.0988223	−	0.995105i	$-0.468492\pi$
$547$	108.112			0.197645			0.0988223	+	0.995105i	$0.468492\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−80.4537	−	38.8799i	−0.146014	−	0.0705625i
$552$		0			0
$553$	736.146			1.33119
$554$		0			0
$555$		0			0
$556$		0			0
$557$	605.693	+	349.697i	1.08742	+	0.627822i	0.932888	−	0.360165i	$-0.117280\pi$
$557$	605.693	+	349.697i	1.08742	+	0.627822i	0.154532	+	0.987988i	$0.450613\pi$
$558$		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
$565$	117.800	−	204.035i	0.208495	−	0.361124i
$566$		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
$583$	−576.989			−0.989690
$584$		0			0
$585$		0			0
$586$		0			0
$587$	90.3895	+	52.1864i	0.153986	+	0.0889036i	0.575013	−	0.818144i	$-0.304997\pi$
$587$	90.3895	+	52.1864i	0.153986	+	0.0889036i	−0.421027	+	0.907048i	$0.638330\pi$
$588$		0			0
$589$	−305.130	−	448.860i	−0.518048	−	0.762072i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	785.885	+	453.731i	1.32527	+	0.765145i	0.984564	−	0.175025i	$-0.0560005\pi$
$593$	785.885	+	453.731i	1.32527	+	0.765145i	0.340706	+	0.940170i	$0.389334\pi$
$594$		0			0
$595$	−85.7795	+	148.574i	−0.144167	+	0.249705i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−84.5937	+	48.8402i	−0.141225	+	0.0815362i	−0.568948	−	0.822374i	$-0.692649\pi$
$599$	−84.5937	+	48.8402i	−0.141225	+	0.0815362i	0.427723	+	0.903910i	$0.359316\pi$
$600$		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
$605$	−38.7133	−	22.3512i	−0.0639890	−	0.0369441i
$606$		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	0.981862	+	0.189596i	$0.0607178\pi$
$613$	401.592	+	695.578i	0.655126	+	1.13471i	−0.326736	+	0.945116i	$0.605949\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	332.782	−	192.132i	0.539355	−	0.311397i	−0.205462	−	0.978665i	$-0.565870\pi$
$617$	332.782	−	192.132i	0.539355	−	0.311397i	0.744818	+	0.667268i	$0.232536\pi$
$618$		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
$623$			52.6061i			0.0844399i
$624$		0			0
$625$	380.444			0.608710
$626$		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	0.359548	+	0.933127i	$0.382931\pi$
$631$	−396.481	+	686.725i	−0.628337	+	1.08831i	−0.987885	+	0.155186i	$0.950402\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−378.977	−	218.803i	−0.596815	−	0.344571i
$636$		0			0
$637$	−251.499			−0.394818
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−465.052	−	268.498i	−0.725510	−	0.418873i	0.0912673	−	0.995826i	$-0.470908\pi$
$641$	−465.052	−	268.498i	−0.725510	−	0.418873i	−0.816777	+	0.576953i	$0.804242\pi$
$642$		0			0
$643$	−502.827			−0.782001			−0.391001	−	0.920390i	$-0.627871\pi$
$643$	−502.827			−0.782001			−0.391001	+	0.920390i	$0.627871\pi$
$644$		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
$655$	−465.243			−0.710295
$656$		0			0
$657$		0			0
$658$		0			0
$659$			921.084i			1.39770i	0.715269	+	0.698850i	$0.246304\pi$
$659$			921.084i			1.39770i	−0.715269	+	0.698850i	$0.753696\pi$
$660$		0			0
$661$	55.6683	+	96.4203i	0.0842183	+	0.145870i	0.905058	−	0.425288i	$-0.139827\pi$
$661$	55.6683	+	96.4203i	0.0842183	+	0.145870i	−0.820840	+	0.571159i	$0.806494\pi$
$662$		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
$671$	240.227	−	138.695i	0.358014	−	0.206699i
$672$		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
$679$	−497.352			−0.732477
$680$		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
$689$			489.561i			0.710539i
$690$		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
$697$	−69.8126	−	120.919i	−0.100162	−	0.173485i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	484.715	−	279.850i	0.691462	−	0.399216i	−0.112697	−	0.993629i	$-0.535949\pi$
$701$	484.715	−	279.850i	0.691462	−	0.399216i	0.804160	+	0.594413i	$0.202616\pi$
$702$		0			0
$703$	−437.027	+	904.335i	−0.621661	+	1.28639i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	629.510	−	363.448i	0.890397	−	0.514071i
$708$		0			0
$709$	690.685			0.974168			0.487084	−	0.873355i	$-0.338061\pi$
$709$	690.685			0.974168			0.487084	+	0.873355i	$0.338061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			537.709i			0.754150i
$714$		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	$-0.484727\pi$
$719$	673.685	−	388.952i	0.936975	−	0.540963i	0.889011	+	0.457886i	$0.151393\pi$
$720$		0			0
$721$	−280.682			−0.389296
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	101.506i		−	0.140009i
$726$		0			0
$727$	210.717	−	364.972i	0.289844	−	0.502025i	−0.683928	−	0.729549i	$-0.739730\pi$
$727$	210.717	−	364.972i	0.289844	−	0.502025i	0.973772	+	0.227525i	$0.0730632\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			777.908i			1.06417i
$732$		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	0.999715	−	0.0238836i	$-0.00760311\pi$
$739$	354.109	+	613.335i	0.479174	+	0.829953i	−0.520541	+	0.853837i	$0.674270\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	815.843i		−	1.09804i	−0.835810	−	0.549019i	$-0.815001\pi$
$743$		−	815.843i		−	1.09804i	0.835810	−	0.549019i	$-0.184999\pi$
$744$		0			0
$745$	−131.983			−0.177158
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−267.690	+	154.551i	−0.357396	+	0.206343i
$750$		0			0
$751$	541.246	+	937.465i	0.720700	+	1.24829i	0.960720	+	0.277521i	$0.0895129\pi$
$751$	541.246	+	937.465i	0.720700	+	1.24829i	−0.240019	+	0.970768i	$0.577154\pi$
$752$		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	0.952498	−	0.304546i	$-0.0985046\pi$
$757$	160.866	+	278.628i	0.212505	+	0.368069i	−0.739993	+	0.672614i	$0.765171\pi$
$758$		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
$763$	159.309	−	275.931i	0.208793	−	0.361640i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	450.344	−	260.006i	0.587150	−	0.338991i
$768$		0			0
$769$	420.271	+	727.930i	0.546516	+	0.946593i	0.998510	+	0.0545721i	$0.0173795\pi$
$769$	420.271	+	727.930i	0.546516	+	0.946593i	−0.451994	+	0.892021i	$0.649287\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	676.445	+	390.545i	0.875090	+	0.505233i	0.869036	−	0.494748i	$-0.164740\pi$
$773$	676.445	+	390.545i	0.875090	+	0.505233i	0.00605359	+	0.999982i	$0.498073\pi$
$774$		0			0
$775$	308.278	−	533.953i	0.397778	−	0.688972i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	127.117	+	61.4304i	0.163180	+	0.0788581i
$780$		0			0
$781$	−661.571	+	1145.87i	−0.847081	+	1.46719i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			275.809i			0.351350i
$786$		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
$803$	241.059	+	139.176i	0.300198	+	0.173319i
$804$		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	0.179739	+	0.983714i	$0.442475\pi$
$811$	−618.024	+	1070.45i	−0.762052	+	1.31991i	−0.941791	+	0.336198i	$0.890859\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	291.368i		−	0.357507i
$816$		0			0
$817$	−442.210	−	650.510i	−0.541261	−	0.796218i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			1225.85i			1.49311i	0.665321	+	0.746557i	$0.268295\pi$
$821$			1225.85i			1.49311i	−0.665321	+	0.746557i	$0.731705\pi$
$822$		0			0
$823$	−133.180	−	230.675i	−0.161823	−	0.280286i	0.773700	−	0.633553i	$-0.218404\pi$
$823$	−133.180	−	230.675i	−0.161823	−	0.280286i	−0.935522	+	0.353267i	$0.885071\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−725.041	−	418.603i	−0.876712	−	0.506170i	−0.00713916	−	0.999975i	$-0.502272\pi$
$827$	−725.041	−	418.603i	−0.876712	−	0.506170i	−0.869573	+	0.493805i	$0.835606\pi$
$828$		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
$833$			462.245i			0.554916i
$834$		0			0
$835$	228.147	+	395.163i	0.273230	+	0.473249i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	209.917	+	121.195i	0.250199	+	0.144452i	0.619855	−	0.784716i	$-0.287191\pi$
$839$	209.917	+	121.195i	0.250199	+	0.144452i	−0.369657	+	0.929168i	$0.620525\pi$
$840$		0			0
$841$	818.882			0.973701
$842$		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
$851$	861.751	−	497.532i	1.01263	−	0.584644i
$852$		0			0
$853$	−716.594	+	1241.18i	−0.840087	+	1.45507i	0.0497333	+	0.998763i	$0.484163\pi$
$853$	−716.594	+	1241.18i	−0.840087	+	1.45507i	−0.889820	+	0.456311i	$0.849170\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1159.74	+	669.577i	1.35326	+	0.781303i	0.988704	−	0.149880i	$-0.0478887\pi$
$857$	1159.74	+	669.577i	1.35326	+	0.781303i	0.364552	+	0.931183i	$0.381222\pi$
$858$		0			0
$859$	−177.580	−	307.577i	−0.206728	−	0.358064i	0.743954	−	0.668231i	$-0.232948\pi$
$859$	−177.580	−	307.577i	−0.206728	−	0.358064i	−0.950682	+	0.310167i	$0.899615\pi$
$860$		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
$869$		−	1795.69i		−	2.06638i
$870$		0			0
$871$	−198.302	+	343.469i	−0.227671	+	0.394338i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	368.333	−	212.657i	0.420952	−	0.243037i
$876$		0			0
$877$	726.017			0.827841			0.413920	−	0.910313i	$-0.364159\pi$
$877$	726.017			0.827841			0.413920	+	0.910313i	$0.364159\pi$
$878$		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	−0.752033	−	0.659125i	$-0.770927\pi$
$883$	172.011	−	297.931i	0.194803	−	0.337408i	0.946836	+	0.321717i	$0.104260\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−596.382	+	344.321i	−0.672359	+	0.388186i	−0.796970	−	0.604019i	$-0.793565\pi$
$887$	−596.382	+	344.321i	−0.672359	+	0.388186i	0.124611	+	0.992206i	$0.460232\pi$
$888$		0			0
$889$	1169.49			1.31551
$890$		0			0
$891$		0			0
$892$		0			0
$893$	73.8909	+	35.7084i	0.0827445	+	0.0399870i
$894$		0			0
$895$	−441.348			−0.493126
$896$		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
$907$	−279.863	+	484.737i	−0.308559	+	0.534440i	−0.978047	−	0.208382i	$-0.933180\pi$
$907$	−279.863	+	484.737i	−0.308559	+	0.534440i	0.669488	+	0.742823i	$0.266513\pi$
$908$		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
$925$	−1140.98			−1.23349
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1138.90	−	657.544i	−1.22594	−	0.707797i	−0.259762	−	0.965673i	$-0.583644\pi$
$929$	−1138.90	−	657.544i	−1.22594	−	0.707797i	−0.966178	+	0.257875i	$0.916978\pi$
$930$		0			0
$931$	−262.769	−	386.544i	−0.282243	−	0.415192i
$932$		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	−0.953194	−	0.302361i	$-0.902225\pi$
$937$	−201.216	+	348.516i	−0.214745	+	0.371948i	0.738449	+	0.674309i	$0.235559\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−784.539	+	452.954i	−0.833729	+	0.481353i	−0.855128	−	0.518418i	$-0.826521\pi$
$941$	−784.539	+	452.954i	−0.833729	+	0.481353i	0.0213991	+	0.999771i	$0.493188\pi$
$942$		0			0
$943$	−69.9353	−	121.131i	−0.0741625	−	0.128453i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1287.51	−	743.345i	−1.35957	−	0.784947i	−0.370003	−	0.929031i	$-0.620643\pi$
$947$	−1287.51	−	743.345i	−1.35957	−	0.784947i	−0.989566	+	0.144083i	$0.953977\pi$
$948$		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	$-0.852583\pi$
$953$	−910.211	+	525.511i	−0.955101	+	0.551428i	−0.0604392	+	0.998172i	$0.519250\pi$
$954$		0			0
$955$	160.963	+	278.795i	0.168547	+	0.291932i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−22.5306	+	13.0081i	−0.0234939	+	0.0135642i
$960$		0			0
$961$	72.4944	+	125.564i	0.0754364	+	0.130660i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			79.6635i			0.0825528i
$966$		0			0
$967$	−1156.99			−1.19647			−0.598237	−	0.801319i	$-0.704132\pi$
$967$	−1156.99			−1.19647			−0.598237	+	0.801319i	$0.704132\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−453.947	+	262.086i	−0.467504	+	0.269914i	−0.715194	−	0.698926i	$-0.753662\pi$
$971$	−453.947	+	262.086i	−0.467504	+	0.269914i	0.247690	+	0.968839i	$0.420329\pi$
$972$		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
$979$	128.322			0.131075
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−695.110	−	401.322i	−0.707131	−	0.408262i	0.102867	−	0.994695i	$-0.467198\pi$
$983$	−695.110	−	401.322i	−0.707131	−	0.408262i	−0.809998	+	0.586433i	$0.800532\pi$
$984$		0			0
$985$	−424.577			−0.431043
$986$		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	−0.935312	−	0.353824i	$-0.884881\pi$
$991$	−767.110	−	1328.67i	−0.774077	−	1.34074i	0.161236	−	0.986916i	$-0.448452\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	105.108	+	60.6843i	0.105637	+	0.0609893i
$996$		0			0
$997$	928.495			0.931289			0.465644	−	0.884972i	$-0.345823\pi$
$997$	928.495			0.931289			0.465644	+	0.884972i	$0.345823\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.17	✓	80	171.106	even	3
684.3.m.a.653.17	yes	80	3.2	odd	2
684.3.be.a.425.11	yes	80	9.7	even	3
684.3.be.a.581.11	yes	80	57.11	odd	6
2052.3.m.a.881.16		80	1.1	even	1	trivial
2052.3.m.a.1493.25		80	171.11	odd	6	inner
2052.3.be.a.125.16		80	19.11	even	3
2052.3.be.a.197.16		80	9.2	odd	6

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.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.84834i q^{5} +(2.46982 + 4.27785i) q^{7} +O(q^{10})\)	\(q-1.84834i q^{5} +(2.46982 + 4.27785i) q^{7} +(10.4350 - 6.02464i) q^{11} +(-5.11176 - 8.85383i) q^{13} +(-16.2730 + 9.39522i) q^{17} +(8.26716 - 17.1071i) q^{19} +(-16.3016 + 9.41172i) q^{23} +21.5836 q^{25} -4.70293i q^{29} +(14.2830 - 24.7388i) q^{31} +(7.90692 - 4.56506i) q^{35} -52.8630 q^{37} +7.43066i q^{41} +(20.6996 - 35.8527i) q^{43} +4.31930i q^{47} +(12.3000 - 21.3043i) q^{49} +(-41.4703 - 23.9429i) q^{53} +(-11.1356 - 19.2874i) q^{55} +50.8643i q^{59} +23.0213 q^{61} +(-16.3649 + 9.44828i) q^{65} +(-19.3966 - 33.5959i) q^{67} +(-95.0990 + 54.9054i) q^{71} +(11.5505 + 20.0061i) q^{73} +(51.5450 + 29.7595i) q^{77} +(74.5143 - 129.062i) q^{79} +(64.2549 - 37.0976i) q^{83} +(17.3656 + 30.0781i) q^{85} +(9.22299 + 5.32490i) q^{89} +(25.2502 - 43.7346i) q^{91} +(-31.6198 - 15.2805i) q^{95} +(-50.3430 + 87.1966i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

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