LMFDB - Embedded newform 2052.3.bl.a.145.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(145,2052)]

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, 2, 5]))

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

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

chi := DirichletCharacter("2052.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.bl (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			145.17
Character	$\chi$	$=$	2052.145
Dual form			2052.3.bl.a.1585.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84061 q^{5} +(3.31490 + 5.74157i) q^{7} +O(q^{10})$	$q-1.84061 q^{5} +(3.31490 + 5.74157i) q^{7} +(7.65648 + 13.2614i) q^{11} +(-2.19489 + 1.26722i) q^{13} +(12.9506 + 22.4311i) q^{17} +(7.42625 + 17.4886i) q^{19} +(2.54164 + 4.40226i) q^{23} -21.6122 q^{25} +14.8760i q^{29} +(-34.8501 - 20.1207i) q^{31} +(-6.10143 - 10.5680i) q^{35} -17.0230i q^{37} -53.8019i q^{41} +(25.2100 - 43.6650i) q^{43} -64.6082 q^{47} +(2.52290 - 4.36980i) q^{49} +(88.6168 + 51.1629i) q^{53} +(-14.0926 - 24.4091i) q^{55} +55.2023i q^{59} +83.1725 q^{61} +(4.03993 - 2.33245i) q^{65} +(-96.2160 + 55.5503i) q^{67} +(-83.8226 + 48.3950i) q^{71} +(-66.6462 - 115.435i) q^{73} +(-50.7609 + 87.9205i) q^{77} +(28.9694 + 16.7255i) q^{79} +(45.0486 + 78.0264i) q^{83} +(-23.8370 - 41.2869i) q^{85} +(26.4870 + 15.2923i) q^{89} +(-14.5517 - 8.40140i) q^{91} +(-13.6688 - 32.1896i) q^{95} +(-68.9293 - 39.7964i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - q^{7}+O(q^{10}) $ 80 * q - q^7	$ 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) $ 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	−1.84061			−0.368122			−0.184061	−	0.982915i	$-0.558924\pi$
$5$	−1.84061			−0.368122			−0.184061	+	0.982915i	$0.558924\pi$
$6$		0			0
$7$	3.31490	+	5.74157i	0.473557	+	0.820225i	0.999542	−	0.0302694i	$-0.00963651\pi$
$7$	3.31490	+	5.74157i	0.473557	+	0.820225i	−0.525985	+	0.850494i	$0.676303\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.65648	+	13.2614i	0.696044	+	1.20558i	0.969828	+	0.243792i	$0.0783913\pi$
$11$	7.65648	+	13.2614i	0.696044	+	1.20558i	−0.273784	+	0.961791i	$0.588275\pi$
$12$		0			0
$13$	−2.19489	+	1.26722i	−0.168838	+	0.0974784i	−0.582038	−	0.813162i	$-0.697744\pi$
$13$	−2.19489	+	1.26722i	−0.168838	+	0.0974784i	0.413200	+	0.910640i	$0.364411\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.9506	+	22.4311i	0.761800	+	1.31948i	0.941922	+	0.335833i	$0.109018\pi$
$17$	12.9506	+	22.4311i	0.761800	+	1.31948i	−0.180121	+	0.983644i	$0.557649\pi$
$18$		0			0
$19$	7.42625	+	17.4886i	0.390856	+	0.920452i
$19$	7.42625	+	17.4886i	0.390856	+	0.920452i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.54164	+	4.40226i	0.110506	+	0.191402i	0.915975	−	0.401236i	$-0.131419\pi$
$23$	2.54164	+	4.40226i	0.110506	+	0.191402i	−0.805468	+	0.592639i	$0.798086\pi$
$24$		0			0
$25$	−21.6122			−0.864487
$26$		0			0
$27$		0			0
$28$		0			0
$29$			14.8760i			0.512964i	0.966549	+	0.256482i	$0.0825635\pi$
$29$			14.8760i			0.512964i	−0.966549	+	0.256482i	$0.917437\pi$
$30$		0			0
$31$	−34.8501	−	20.1207i	−1.12420	−	0.649055i	−0.181728	−	0.983349i	$-0.558169\pi$
$31$	−34.8501	−	20.1207i	−1.12420	−	0.649055i	−0.942469	+	0.334294i	$0.891502\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.10143	−	10.5680i	−0.174326	−	0.301942i
$36$		0			0
$37$		−	17.0230i		−	0.460082i	−0.973181	−	0.230041i	$-0.926114\pi$
$37$		−	17.0230i		−	0.460082i	0.973181	−	0.230041i	$-0.0738861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	53.8019i		−	1.31224i	−0.754656	−	0.656121i	$-0.772196\pi$
$41$		−	53.8019i		−	1.31224i	0.754656	−	0.656121i	$-0.227804\pi$
$42$		0			0
$43$	25.2100	−	43.6650i	0.586279	−	1.01547i	−0.408435	−	0.912787i	$-0.633925\pi$
$43$	25.2100	−	43.6650i	0.586279	−	1.01547i	0.994715	−	0.102678i	$-0.0327412\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−64.6082			−1.37464			−0.687322	−	0.726353i	$-0.741214\pi$
$47$	−64.6082			−1.37464			−0.687322	+	0.726353i	$0.741214\pi$
$48$		0			0
$49$	2.52290	−	4.36980i	0.0514878	−	0.0891795i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	88.6168	+	51.1629i	1.67201	+	0.965338i	0.966509	+	0.256634i	$0.0826134\pi$
$53$	88.6168	+	51.1629i	1.67201	+	0.965338i	0.705506	+	0.708704i	$0.250720\pi$
$54$		0			0
$55$	−14.0926	−	24.4091i	−0.256229	−	0.443801i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			55.2023i			0.935632i	0.883826	+	0.467816i	$0.154959\pi$
$59$			55.2023i			0.935632i	−0.883826	+	0.467816i	$0.845041\pi$
$60$		0			0
$61$	83.1725			1.36348			0.681742	−	0.731593i	$-0.261223\pi$
$61$	83.1725			1.36348			0.681742	+	0.731593i	$0.261223\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.03993	−	2.33245i	0.0621527	−	0.0358839i
$66$		0			0
$67$	−96.2160	+	55.5503i	−1.43606	+	0.829110i	−0.997573	−	0.0696280i	$-0.977819\pi$
$67$	−96.2160	+	55.5503i	−1.43606	+	0.829110i	−0.438487	+	0.898738i	$0.644485\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−83.8226	+	48.3950i	−1.18060	+	0.681620i	−0.956153	−	0.292866i	$-0.905391\pi$
$71$	−83.8226	+	48.3950i	−1.18060	+	0.681620i	−0.224447	+	0.974486i	$0.572058\pi$
$72$		0			0
$73$	−66.6462	−	115.435i	−0.912961	−	1.58130i	−0.809859	−	0.586624i	$-0.800456\pi$
$73$	−66.6462	−	115.435i	−0.912961	−	1.58130i	−0.103102	−	0.994671i	$-0.532877\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−50.7609	+	87.9205i	−0.659232	+	1.14182i
$78$		0			0
$79$	28.9694	+	16.7255i	0.366701	+	0.211715i	0.672016	−	0.740537i	$-0.265429\pi$
$79$	28.9694	+	16.7255i	0.366701	+	0.211715i	−0.305315	+	0.952251i	$0.598762\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	45.0486	+	78.0264i	0.542754	+	0.940077i	0.998745	+	0.0500924i	$0.0159516\pi$
$83$	45.0486	+	78.0264i	0.542754	+	0.940077i	−0.455991	+	0.889984i	$0.650715\pi$
$84$		0			0
$85$	−23.8370	−	41.2869i	−0.280435	−	0.485728i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	26.4870	+	15.2923i	0.297606	+	0.171823i	0.641367	−	0.767234i	$-0.278368\pi$
$89$	26.4870	+	15.2923i	0.297606	+	0.171823i	−0.343761	+	0.939057i	$0.611701\pi$
$90$		0			0
$91$	−14.5517	−	8.40140i	−0.159908	−	0.0923231i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−13.6688	−	32.1896i	−0.143882	−	0.338838i
$96$		0			0
$97$	−68.9293	−	39.7964i	−0.710612	−	0.410272i	0.100676	−	0.994919i	$-0.467900\pi$
$97$	−68.9293	−	39.7964i	−0.710612	−	0.410272i	−0.811287	+	0.584647i	$0.801233\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−122.797			−1.21582			−0.607908	−	0.794007i	$-0.707991\pi$
$101$	−122.797			−1.21582			−0.607908	+	0.794007i	$0.707991\pi$
$102$		0			0
$103$	57.5120	+	33.2045i	0.558369	+	0.322374i	0.752490	−	0.658603i	$-0.228852\pi$
$103$	57.5120	+	33.2045i	0.558369	+	0.322374i	−0.194122	+	0.980977i	$0.562186\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.3363i		−	0.133984i	−0.997753	−	0.0669922i	$-0.978660\pi$
$107$		−	14.3363i		−	0.133984i	0.997753	−	0.0669922i	$-0.0213403\pi$
$108$		0			0
$109$	20.7750	−	11.9944i	0.190596	−	0.110041i	−0.401665	−	0.915786i	$-0.631569\pi$
$109$	20.7750	−	11.9944i	0.190596	−	0.110041i	0.592262	+	0.805746i	$0.298235\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−41.1529	−	23.7596i	−0.364185	−	0.210262i	0.306730	−	0.951797i	$-0.400765\pi$
$113$	−41.1529	−	23.7596i	−0.364185	−	0.210262i	−0.670915	+	0.741534i	$0.734098\pi$
$114$		0			0
$115$	−4.67817	−	8.10283i	−0.0406797	−	0.0704594i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−85.8599	+	148.714i	−0.721512	+	1.24969i
$120$		0			0
$121$	−56.7434	+	98.2824i	−0.468954	+	0.812251i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	85.7947			0.686358
$126$		0			0
$127$	145.228	+	83.8477i	1.14353	+	0.660218i	0.947302	−	0.320341i	$-0.103797\pi$
$127$	145.228	+	83.8477i	1.14353	+	0.660218i	0.196228	+	0.980558i	$0.437131\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−236.282			−1.80368			−0.901842	−	0.432067i	$-0.857784\pi$
$131$	−236.282			−1.80368			−0.901842	+	0.432067i	$0.857784\pi$
$132$		0			0
$133$	−75.7947	+	100.611i	−0.569885	+	0.756476i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	31.2336			0.227983			0.113991	−	0.993482i	$-0.463636\pi$
$137$	31.2336			0.227983			0.113991	+	0.993482i	$0.463636\pi$
$138$		0			0
$139$	−17.1870	−	29.7688i	−0.123648	−	0.214164i	0.797556	−	0.603245i	$-0.206126\pi$
$139$	−17.1870	−	29.7688i	−0.123648	−	0.214164i	−0.921203	+	0.389081i	$0.872793\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−33.6102	−	19.4049i	−0.235037	−	0.135698i
$144$		0			0
$145$		−	27.3808i		−	0.188833i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	224.036			1.50360			0.751799	−	0.659392i	$-0.229186\pi$
$149$	224.036			1.50360			0.751799	+	0.659392i	$0.229186\pi$
$150$		0			0
$151$	10.4885	−	6.05557i	0.0694606	−	0.0401031i	−0.464867	−	0.885380i	$-0.653898\pi$
$151$	10.4885	−	6.05557i	0.0694606	−	0.0401031i	0.534328	+	0.845277i	$0.320565\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	64.1454	+	37.0343i	0.413841	+	0.238931i
$156$		0			0
$157$	−195.910			−1.24783			−0.623916	−	0.781491i	$-0.714459\pi$
$157$	−195.910			−1.24783			−0.623916	+	0.781491i	$0.714459\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−16.8506	+	29.1861i	−0.104662	+	0.181280i
$162$		0			0
$163$	−314.591			−1.93001			−0.965003	−	0.262239i	$-0.915539\pi$
$163$	−314.591			−1.93001			−0.965003	+	0.262239i	$0.915539\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	89.2185	−	51.5103i	0.534242	−	0.308445i	−0.208500	−	0.978022i	$-0.566858\pi$
$167$	89.2185	−	51.5103i	0.534242	−	0.308445i	0.742742	+	0.669577i	$0.233525\pi$
$168$		0			0
$169$	−81.2883	+	140.795i	−0.480996	+	0.833109i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	35.1923	+	20.3183i	0.203423	+	0.117447i	0.598251	−	0.801309i	$-0.295862\pi$
$173$	35.1923	+	20.3183i	0.203423	+	0.117447i	−0.394828	+	0.918755i	$0.629196\pi$
$174$		0			0
$175$	−71.6421	−	124.088i	−0.409384	−	0.709073i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	74.3147i		−	0.415166i	−0.978217	−	0.207583i	$-0.933440\pi$
$179$		−	74.3147i		−	0.415166i	0.978217	−	0.207583i	$-0.0665597\pi$
$180$		0			0
$181$	−85.8009	−	49.5372i	−0.474038	−	0.273686i	0.243890	−	0.969803i	$-0.421576\pi$
$181$	−85.8009	−	49.5372i	−0.474038	−	0.273686i	−0.717929	+	0.696117i	$0.754910\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			31.3327i			0.169366i
$186$		0			0
$187$	−198.312	+	343.487i	−1.06049	+	1.83683i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	166.239	+	287.934i	0.870359	+	1.50751i	0.861626	+	0.507544i	$0.169447\pi$
$191$	166.239	+	287.934i	0.870359	+	1.50751i	0.00873349	+	0.999962i	$0.497220\pi$
$192$		0			0
$193$			264.461i			1.37027i	0.728418	+	0.685133i	$0.240256\pi$
$193$			264.461i			1.37027i	−0.728418	+	0.685133i	$0.759744\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−36.4851			−0.185203			−0.0926017	−	0.995703i	$-0.529518\pi$
$197$	−36.4851			−0.185203			−0.0926017	+	0.995703i	$0.529518\pi$
$198$		0			0
$199$	−151.188	+	261.866i	−0.759740	+	1.31591i	0.183243	+	0.983068i	$0.441341\pi$
$199$	−151.188	+	261.866i	−0.759740	+	1.31591i	−0.942983	+	0.332841i	$0.891993\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−85.4114	+	49.3123i	−0.420746	+	0.242918i
$204$		0			0
$205$			99.0282i			0.483064i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−175.064	+	232.384i	−0.837629	+	1.11188i
$210$		0			0
$211$		−	287.521i		−	1.36266i	−0.731976	−	0.681330i	$-0.761402\pi$
$211$		−	287.521i		−	1.36266i	0.731976	−	0.681330i	$-0.238598\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−46.4017	+	80.3702i	−0.215822	+	0.373815i
$216$		0			0
$217$		−	266.792i		−	1.22946i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−56.8503	−	32.8225i	−0.257241	−	0.148518i
$222$		0			0
$223$	358.751	+	207.125i	1.60875	+	0.928812i	0.989651	+	0.143498i	$0.0458352\pi$
$223$	358.751	+	207.125i	1.60875	+	0.928812i	0.619099	+	0.785313i	$0.287498\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	235.073	−	135.719i	1.03556	−	0.597882i	0.116989	−	0.993133i	$-0.462676\pi$
$227$	235.073	−	135.719i	1.03556	−	0.597882i	0.918573	+	0.395251i	$0.129343\pi$
$228$		0			0
$229$	−23.6064	+	40.8875i	−0.103085	+	0.178548i	−0.912954	−	0.408062i	$-0.866205\pi$
$229$	−23.6064	+	40.8875i	−0.103085	+	0.178548i	0.809869	+	0.586610i	$0.199538\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−140.000	−	242.488i	−0.600860	−	1.04072i	−0.992691	−	0.120682i	$-0.961492\pi$
$233$	−140.000	−	242.488i	−0.600860	−	1.04072i	0.391832	−	0.920037i	$-0.371842\pi$
$234$		0			0
$235$	118.918			0.506036
$236$		0			0
$237$		0			0
$238$		0			0
$239$	116.760	−	202.234i	0.488534	−	0.846166i	−0.511379	−	0.859356i	$-0.670865\pi$
$239$	116.760	−	202.234i	0.488534	−	0.846166i	0.999913	+	0.0131891i	$0.00419834\pi$
$240$		0			0
$241$			101.973i			0.423124i	0.977365	+	0.211562i	$0.0678551\pi$
$241$			101.973i			0.423124i	−0.977365	+	0.211562i	$0.932145\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.64368	+	8.04308i	−0.0189538	+	0.0328289i
$246$		0			0
$247$	−38.4617	−	28.9748i	−0.155715	−	0.117307i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	32.2790	−	55.9089i	0.128602	−	0.222745i	−0.794533	−	0.607220i	$-0.792284\pi$
$251$	32.2790	−	55.9089i	0.128602	−	0.222745i	0.923135	+	0.384476i	$0.125618\pi$
$252$		0			0
$253$	−38.9201	+	67.4116i	−0.153834	+	0.266449i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	117.520	−	67.8503i	0.457277	−	0.264009i	−0.253621	−	0.967304i	$-0.581622\pi$
$257$	117.520	−	67.8503i	0.457277	−	0.264009i	0.710899	+	0.703294i	$0.248288\pi$
$258$		0			0
$259$	97.7390	−	56.4296i	0.377371	−	0.217875i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−171.790	+	297.549i	−0.653195	+	1.13137i	0.329149	+	0.944278i	$0.393238\pi$
$263$	−171.790	+	297.549i	−0.653195	+	1.13137i	−0.982343	+	0.187088i	$0.940095\pi$
$264$		0			0
$265$	−163.109	−	94.1709i	−0.615505	−	0.355362i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−406.540	+	234.716i	−1.51130	+	0.872550i	−0.511388	+	0.859350i	$0.670869\pi$
$269$	−406.540	+	234.716i	−1.51130	+	0.872550i	−0.999913	+	0.0132004i	$0.995798\pi$
$270$		0			0
$271$	−5.21532	−	9.03320i	−0.0192447	−	0.0333328i	0.856243	−	0.516574i	$-0.172793\pi$
$271$	−5.21532	−	9.03320i	−0.0192447	−	0.0333328i	−0.875487	+	0.483241i	$0.839459\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−165.473	−	286.608i	−0.601720	−	1.04221i
$276$		0			0
$277$	−239.373	−	414.607i	−0.864163	−	1.49677i	−0.867875	−	0.496782i	$-0.834515\pi$
$277$	−239.373	−	414.607i	−0.864163	−	1.49677i	0.00371208	−	0.999993i	$-0.498818\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			260.184i			0.925923i	0.886378	+	0.462962i	$0.153213\pi$
$281$			260.184i			0.925923i	−0.886378	+	0.462962i	$0.846787\pi$
$282$		0			0
$283$	−270.948			−0.957414			−0.478707	−	0.877975i	$-0.658894\pi$
$283$	−270.948			−0.957414			−0.478707	+	0.877975i	$0.658894\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	308.907	−	178.348i	1.07633	−	0.621421i
$288$		0			0
$289$	−190.936	+	330.712i	−0.660680	+	1.14433i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	196.791	+	113.617i	0.671641	+	0.387772i	0.796698	−	0.604377i	$-0.206578\pi$
$293$	196.791	+	113.617i	0.671641	+	0.387772i	−0.125057	+	0.992150i	$0.539911\pi$
$294$		0			0
$295$		−	101.606i		−	0.344426i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.1572	−	6.44164i	−0.0373152	−	0.0215439i
$300$		0			0
$301$	334.274			1.11055
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−153.088			−0.501928
$306$		0			0
$307$	429.650	−	248.058i	1.39951	−	0.808008i	0.405170	−	0.914241i	$-0.367212\pi$
$307$	429.650	−	248.058i	1.39951	−	0.808008i	0.994341	+	0.106233i	$0.0338790\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	86.7171	−	150.198i	0.278833	−	0.482953i	−0.692262	−	0.721646i	$-0.743386\pi$
$311$	86.7171	−	150.198i	0.278833	−	0.482953i	0.971095	+	0.238693i	$0.0767191\pi$
$312$		0			0
$313$	−211.100			−0.674442			−0.337221	−	0.941426i	$-0.609487\pi$
$313$	−211.100			−0.674442			−0.337221	+	0.941426i	$0.609487\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			166.038i			0.523781i	0.965098	+	0.261890i	$0.0843459\pi$
$317$			166.038i			0.523781i	−0.965098	+	0.261890i	$0.915654\pi$
$318$		0			0
$319$	−197.276	+	113.897i	−0.618421	+	0.357045i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−296.114	+	393.067i	−0.916761	+	1.21693i
$324$		0			0
$325$	47.4363	−	27.3873i	0.145958	−	0.0842688i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−214.170	−	370.953i	−0.650972	−	1.12752i
$330$		0			0
$331$	182.184	−	105.184i	0.550404	−	0.317776i	−0.198881	−	0.980024i	$-0.563731\pi$
$331$	182.184	−	105.184i	0.550404	−	0.317776i	0.749285	+	0.662248i	$0.230397\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	177.096	−	102.246i	0.528645	−	0.305213i
$336$		0			0
$337$			206.471i			0.612673i	0.951923	+	0.306336i	$0.0991032\pi$
$337$			206.471i			0.612673i	−0.951923	+	0.306336i	$0.900897\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	616.215i		−	1.80708i
$342$		0			0
$343$	358.313			1.04464
$344$		0			0
$345$		0			0
$346$		0			0
$347$	152.509			0.439506			0.219753	−	0.975556i	$-0.429475\pi$
$347$	152.509			0.439506			0.219753	+	0.975556i	$0.429475\pi$
$348$		0			0
$349$	22.5332	+	39.0287i	0.0645651	+	0.111830i	0.896501	−	0.443042i	$-0.146101\pi$
$349$	22.5332	+	39.0287i	0.0645651	+	0.111830i	−0.831936	+	0.554872i	$0.812767\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	87.1767	+	150.995i	0.246960	+	0.427747i	0.962681	−	0.270640i	$-0.0872352\pi$
$353$	87.1767	+	150.995i	0.246960	+	0.427747i	−0.715721	+	0.698386i	$0.753902\pi$
$354$		0			0
$355$	154.285	−	89.0763i	0.434605	−	0.250919i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	85.1783	+	147.533i	0.237265	+	0.410956i	0.959929	−	0.280244i	$-0.0904155\pi$
$359$	85.1783	+	147.533i	0.237265	+	0.410956i	−0.722663	+	0.691200i	$0.757082\pi$
$360$		0			0
$361$	−250.701	+	259.749i	−0.694464	+	0.719528i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	122.669	+	212.470i	0.336081	+	0.582109i
$366$		0			0
$367$	−413.722			−1.12731			−0.563653	−	0.826011i	$-0.690605\pi$
$367$	−413.722			−1.12731			−0.563653	+	0.826011i	$0.690605\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			678.399i			1.82857i
$372$		0			0
$373$	37.5725	+	21.6925i	0.100731	+	0.0581568i	0.549519	−	0.835481i	$-0.314811\pi$
$373$	37.5725	+	21.6925i	0.100731	+	0.0581568i	−0.448788	+	0.893638i	$0.648144\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.8511	−	32.6511i	−0.0500029	−	0.0866076i
$378$		0			0
$379$		−	151.050i		−	0.398548i	−0.979944	−	0.199274i	$-0.936142\pi$
$379$		−	151.050i		−	0.398548i	0.979944	−	0.199274i	$-0.0638583\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			173.407i			0.452759i	0.974039	+	0.226379i	$0.0726889\pi$
$383$			173.407i			0.452759i	−0.974039	+	0.226379i	$0.927311\pi$
$384$		0			0
$385$	93.4309	−	161.827i	0.242678	−	0.420330i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−84.3857			−0.216930			−0.108465	−	0.994100i	$-0.534594\pi$
$389$	−84.3857			−0.216930			−0.108465	+	0.994100i	$0.534594\pi$
$390$		0			0
$391$	−65.8317	+	114.024i	−0.168367	+	0.291621i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−53.3212	−	30.7850i	−0.134990	−	0.0779368i
$396$		0			0
$397$	−277.537	−	480.708i	−0.699085	−	1.21085i	−0.968784	−	0.247906i	$-0.920258\pi$
$397$	−277.537	−	480.708i	−0.699085	−	1.21085i	0.269699	−	0.962945i	$-0.413076\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			313.889i			0.782765i	0.920228	+	0.391382i	$0.128003\pi$
$401$			313.889i			0.782765i	−0.920228	+	0.391382i	$0.871997\pi$
$402$		0			0
$403$	101.989			0.253075
$404$		0			0
$405$		0			0
$406$		0			0
$407$	225.750	−	130.337i	0.554667	−	0.320237i
$408$		0			0
$409$	−87.0297	+	50.2466i	−0.212787	+	0.122852i	−0.602606	−	0.798039i	$-0.705871\pi$
$409$	−87.0297	+	50.2466i	−0.212787	+	0.122852i	0.389819	+	0.920891i	$0.372538\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−316.948	+	182.990i	−0.767429	+	0.443075i
$414$		0			0
$415$	−82.9167	−	143.616i	−0.199799	−	0.346063i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	229.603	−	397.685i	0.547980	−	0.949129i	−0.450433	−	0.892810i	$-0.648730\pi$
$419$	229.603	−	397.685i	0.547980	−	0.949129i	0.998413	−	0.0563185i	$-0.0179362\pi$
$420$		0			0
$421$	−66.9362	−	38.6457i	−0.158993	−	0.0917949i	0.418392	−	0.908266i	$-0.362594\pi$
$421$	−66.9362	−	38.6457i	−0.158993	−	0.0917949i	−0.577386	+	0.816472i	$0.695927\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−279.891	−	484.785i	−0.658566	−	1.14067i
$426$		0			0
$427$	275.708	+	477.541i	0.645687	+	1.11836i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−353.010	−	203.810i	−0.819049	−	0.472878i	0.0310396	−	0.999518i	$-0.490118\pi$
$431$	−353.010	−	203.810i	−0.819049	−	0.472878i	−0.850088	+	0.526640i	$0.823452\pi$
$432$		0			0
$433$	539.985	+	311.760i	1.24708	+	0.720001i	0.970526	−	0.240998i	$-0.0774746\pi$
$433$	539.985	+	311.760i	1.24708	+	0.720001i	0.276553	+	0.960999i	$0.410808\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−58.1144	+	77.1421i	−0.132985	+	0.176526i
$438$		0			0
$439$	401.699	+	231.921i	0.915033	+	0.528294i	0.882047	−	0.471161i	$-0.156165\pi$
$439$	401.699	+	231.921i	0.915033	+	0.528294i	0.0329858	+	0.999456i	$0.489498\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	79.0533			0.178450			0.0892249	−	0.996012i	$-0.471561\pi$
$443$	79.0533			0.178450			0.0892249	+	0.996012i	$0.471561\pi$
$444$		0			0
$445$	−48.7521	−	28.1470i	−0.109555	−	0.0632518i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	368.400i		−	0.820491i	−0.911975	−	0.410245i	$-0.865443\pi$
$449$		−	368.400i		−	0.820491i	0.911975	−	0.410245i	$-0.134557\pi$
$450$		0			0
$451$	713.489	−	411.933i	1.58202	−	0.913377i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	26.7839	+	15.4637i	0.0588657	+	0.0339861i
$456$		0			0
$457$	−333.410	−	577.483i	−0.729562	−	1.26364i	−0.957068	−	0.289862i	$-0.906390\pi$
$457$	−333.410	−	577.483i	−0.729562	−	1.26364i	0.227506	−	0.973777i	$-0.426943\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−317.321	+	549.616i	−0.688332	+	1.19223i	0.284045	+	0.958811i	$0.408323\pi$
$461$	−317.321	+	549.616i	−0.688332	+	1.19223i	−0.972377	+	0.233415i	$0.925010\pi$
$462$		0			0
$463$	−187.291	+	324.397i	−0.404516	+	0.700642i	−0.994265	−	0.106944i	$-0.965893\pi$
$463$	−187.291	+	324.397i	−0.404516	+	0.700642i	0.589749	+	0.807587i	$0.299227\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−246.992			−0.528890			−0.264445	−	0.964401i	$-0.585189\pi$
$467$	−246.992			−0.528890			−0.264445	+	0.964401i	$0.585189\pi$
$468$		0			0
$469$	−637.893	−	368.287i	−1.36011	−	0.785261i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	772.080			1.63230
$474$		0			0
$475$	−160.497	−	377.966i	−0.337889	−	0.795718i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	508.875			1.06237			0.531185	−	0.847256i	$-0.321747\pi$
$479$	508.875			1.06237			0.531185	+	0.847256i	$0.321747\pi$
$480$		0			0
$481$	21.5719	+	37.3637i	0.0448481	+	0.0776791i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	126.872	+	73.2495i	0.261591	+	0.151030i
$486$		0			0
$487$			358.976i			0.737118i	0.929604	+	0.368559i	$0.120149\pi$
$487$			358.976i			0.737118i	−0.929604	+	0.368559i	$0.879851\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	497.970			1.01419			0.507097	−	0.861889i	$-0.330718\pi$
$491$	497.970			1.01419			0.507097	+	0.861889i	$0.330718\pi$
$492$		0			0
$493$	−333.684	+	192.653i	−0.676844	+	0.390776i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−555.727	−	320.849i	−1.11816	−	0.645572i
$498$		0			0
$499$	755.258			1.51354			0.756772	−	0.653679i	$-0.226776\pi$
$499$	755.258			1.51354			0.756772	+	0.653679i	$0.226776\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−34.5273	+	59.8030i	−0.0686427	+	0.118893i	−0.898304	−	0.439374i	$-0.855200\pi$
$503$	−34.5273	+	59.8030i	−0.0686427	+	0.118893i	0.829661	+	0.558267i	$0.188534\pi$
$504$		0			0
$505$	226.022			0.447568
$506$		0			0
$507$		0			0
$508$		0			0
$509$	358.646	−	207.064i	0.704608	−	0.406806i	−0.104453	−	0.994530i	$-0.533309\pi$
$509$	358.646	−	207.064i	0.704608	−	0.406806i	0.809061	+	0.587724i	$0.199976\pi$
$510$		0			0
$511$	441.850	−	765.308i	0.864678	−	1.49767i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−105.857	−	61.1165i	−0.205548	−	0.118673i
$516$		0			0
$517$	−494.672	−	856.796i	−0.956812	−	1.65725i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			395.573i			0.759258i	0.925139	+	0.379629i	$0.123948\pi$
$521$			395.573i			0.759258i	−0.925139	+	0.379629i	$0.876052\pi$
$522$		0			0
$523$	−430.932	−	248.799i	−0.823962	−	0.475715i	0.0278188	−	0.999613i	$-0.491144\pi$
$523$	−430.932	−	248.799i	−0.823962	−	0.475715i	−0.851781	+	0.523898i	$0.824477\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1042.30i		−	1.97780i
$528$		0			0
$529$	251.580	−	435.749i	0.475577	−	0.823723i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	68.1788	+	118.089i	0.127915	+	0.221556i
$534$		0			0
$535$			26.3876i			0.0493226i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	77.2662			0.143351
$540$		0			0
$541$	−116.908	+	202.490i	−0.216095	+	0.374288i	−0.953611	−	0.301042i	$-0.902666\pi$
$541$	−116.908	+	202.490i	−0.216095	+	0.374288i	0.737515	+	0.675330i	$0.235999\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−38.2386	+	22.0771i	−0.0701626	+	0.0405084i
$546$		0			0
$547$			382.913i			0.700023i	0.936745	+	0.350012i	$0.113822\pi$
$547$			382.913i			0.700023i	−0.936745	+	0.350012i	$0.886178\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−260.160	+	110.473i	−0.472159	+	0.200495i
$552$		0			0
$553$			221.773i			0.401036i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−440.741	+	763.386i	−0.791277	+	1.37053i	0.133900	+	0.990995i	$0.457250\pi$
$557$	−440.741	+	763.386i	−0.791277	+	1.37053i	−0.925177	+	0.379537i	$0.876083\pi$
$558$		0			0
$559$			127.786i			0.228598i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−51.5358	−	29.7542i	−0.0915379	−	0.0528494i	0.453532	−	0.891240i	$-0.350164\pi$
$563$	−51.5358	−	29.7542i	−0.0915379	−	0.0528494i	−0.545070	+	0.838390i	$0.683497\pi$
$564$		0			0
$565$	75.7464	+	43.7322i	0.134064	+	0.0774021i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	791.507	−	456.977i	1.39105	−	0.803123i	0.397618	−	0.917551i	$-0.369837\pi$
$569$	791.507	−	456.977i	1.39105	−	0.803123i	0.993432	+	0.114428i	$0.0365034\pi$
$570$		0			0
$571$	14.0632	−	24.3582i	0.0246291	−	0.0426588i	−0.853448	−	0.521178i	$-0.825493\pi$
$571$	14.0632	−	24.3582i	0.0246291	−	0.0426588i	0.878077	+	0.478519i	$0.158826\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−54.9304	−	95.1423i	−0.0955312	−	0.165465i
$576$		0			0
$577$	696.198			1.20658			0.603291	−	0.797521i	$-0.293856\pi$
$577$	696.198			1.20658			0.603291	+	0.797521i	$0.293856\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−298.663	+	517.299i	−0.514049	+	0.890360i
$582$		0			0
$583$			1566.91i			2.68767i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−124.726	+	216.032i	−0.212481	+	0.368027i	−0.952490	−	0.304569i	$-0.901488\pi$
$587$	−124.726	+	216.032i	−0.212481	+	0.368027i	0.740010	+	0.672596i	$0.234821\pi$
$588$		0			0
$589$	93.0772	−	758.901i	0.158026	−	1.28846i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−98.5298	+	170.659i	−0.166155	+	0.287789i	−0.937065	−	0.349156i	$-0.886468\pi$
$593$	−98.5298	+	170.659i	−0.166155	+	0.287789i	0.770910	+	0.636944i	$0.219802\pi$
$594$		0			0
$595$	158.034	−	273.724i	0.265604	−	0.460040i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	303.325	−	175.125i	0.506386	−	0.292362i	−0.224961	−	0.974368i	$-0.572225\pi$
$599$	303.325	−	175.125i	0.506386	−	0.292362i	0.731347	+	0.682006i	$0.238892\pi$
$600$		0			0
$601$	718.814	−	415.008i	1.19603	−	0.690528i	0.236362	−	0.971665i	$-0.424045\pi$
$601$	718.814	−	415.008i	1.19603	−	0.690528i	0.959668	+	0.281137i	$0.0907114\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	104.442	−	180.899i	0.172632	−	0.299007i
$606$		0			0
$607$	812.587	+	469.147i	1.33869	+	0.772895i	0.986614	−	0.163075i	$-0.0521412\pi$
$607$	812.587	+	469.147i	1.33869	+	0.772895i	0.352080	+	0.935970i	$0.385475\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	141.808	−	81.8728i	0.232091	−	0.133998i
$612$		0			0
$613$	89.8809	+	155.678i	0.146625	+	0.253961i	0.929978	−	0.367615i	$-0.119826\pi$
$613$	89.8809	+	155.678i	0.146625	+	0.253961i	−0.783353	+	0.621577i	$0.786492\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−60.7786	−	105.272i	−0.0985067	−	0.170619i	0.812560	−	0.582878i	$-0.198073\pi$
$617$	−60.7786	−	105.272i	−0.0985067	−	0.170619i	−0.911067	+	0.412259i	$0.864740\pi$
$618$		0			0
$619$	156.050	+	270.287i	0.252101	+	0.436651i	0.964104	−	0.265525i	$-0.0855452\pi$
$619$	156.050	+	270.287i	0.252101	+	0.436651i	−0.712003	+	0.702176i	$0.752212\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			202.769i			0.325472i
$624$		0			0
$625$	382.390			0.611823
$626$		0			0
$627$		0			0
$628$		0			0
$629$	381.846	−	220.459i	0.607068	−	0.350491i
$630$		0			0
$631$	−445.955	+	772.417i	−0.706744	+	1.22412i	0.259315	+	0.965793i	$0.416503\pi$
$631$	−445.955	+	772.417i	−0.706744	+	1.22412i	−0.966059	+	0.258323i	$0.916830\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−267.309	−	154.331i	−0.420958	−	0.243040i
$636$		0			0
$637$			12.7883i			0.0200758i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−418.445	−	241.589i	−0.652800	−	0.376894i	0.136728	−	0.990609i	$-0.456341\pi$
$641$	−418.445	−	241.589i	−0.652800	−	0.376894i	−0.789528	+	0.613714i	$0.789675\pi$
$642$		0			0
$643$	337.986			0.525640			0.262820	−	0.964845i	$-0.415347\pi$
$643$	337.986			0.525640			0.262820	+	0.964845i	$0.415347\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−556.371			−0.859924			−0.429962	−	0.902847i	$-0.641473\pi$
$647$	−556.371			−0.859924			−0.429962	+	0.902847i	$0.641473\pi$
$648$		0			0
$649$	−732.061	+	422.655i	−1.12798	+	0.651241i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.4095	+	19.7619i	−0.0174725	+	0.0302632i	−0.874629	−	0.484792i	$-0.838895\pi$
$653$	−11.4095	+	19.7619i	−0.0174725	+	0.0302632i	0.857157	+	0.515055i	$0.172229\pi$
$654$		0			0
$655$	434.903			0.663975
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	249.124i		−	0.378034i	−0.981974	−	0.189017i	$-0.939470\pi$
$659$		−	249.124i		−	0.378034i	0.981974	−	0.189017i	$-0.0605301\pi$
$660$		0			0
$661$	821.543	−	474.318i	1.24288	−	0.717577i	0.273200	−	0.961957i	$-0.411918\pi$
$661$	821.543	−	474.318i	1.24288	−	0.717577i	0.969679	+	0.244381i	$0.0785847\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	139.508	−	185.186i	0.209787	−	0.278475i
$666$		0			0
$667$	−65.4878	+	37.8094i	−0.0981826	+	0.0566858i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	636.809	+	1102.99i	0.949044	+	1.64379i
$672$		0			0
$673$	498.486	−	287.801i	0.740692	−	0.427639i	−0.0816286	−	0.996663i	$-0.526012\pi$
$673$	498.486	−	287.801i	0.740692	−	0.427639i	0.822321	+	0.569024i	$0.192679\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	773.523	−	446.594i	1.14257	−	0.659666i	0.195508	−	0.980702i	$-0.437364\pi$
$677$	773.523	−	446.594i	1.14257	−	0.659666i	0.947067	+	0.321036i	$0.104031\pi$
$678$		0			0
$679$		−	527.684i		−	0.777148i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	711.769i		−	1.04212i	−0.853519	−	0.521061i	$-0.825536\pi$
$683$		−	711.769i		−	1.04212i	0.853519	−	0.521061i	$-0.174464\pi$
$684$		0			0
$685$	−57.4889			−0.0839253
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−259.338			−0.376398
$690$		0			0
$691$	−25.8916	−	44.8455i	−0.0374697	−	0.0648995i	0.846682	−	0.532099i	$-0.178596\pi$
$691$	−25.8916	−	44.8455i	−0.0374697	−	0.0648995i	−0.884152	+	0.467199i	$0.845263\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	31.6345	+	54.7926i	0.0455173	+	0.0788383i
$696$		0			0
$697$	1206.84	−	696.767i	1.73147	−	0.999666i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−448.023	−	775.998i	−0.639120	−	1.10699i	−0.985626	−	0.168940i	$-0.945966\pi$
$701$	−448.023	−	775.998i	−0.639120	−	1.10699i	0.346507	−	0.938048i	$-0.387368\pi$
$702$		0			0
$703$	297.709	−	126.417i	0.423484	−	0.179826i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−407.061	−	705.050i	−0.575758	−	0.997242i
$708$		0			0
$709$	−610.213			−0.860667			−0.430334	−	0.902670i	$-0.641604\pi$
$709$	−610.213			−0.860667			−0.430334	+	0.902670i	$0.641604\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	204.559i		−	0.286899i
$714$		0			0
$715$	61.8633	+	35.7168i	0.0865220	+	0.0499535i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.9204	−	32.7710i	−0.0263148	−	0.0455786i	0.852568	−	0.522616i	$-0.175044\pi$
$719$	−18.9204	−	32.7710i	−0.0263148	−	0.0455786i	−0.878883	+	0.477038i	$0.841711\pi$
$720$		0			0
$721$			440.279i			0.610650i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	321.502i		−	0.443451i
$726$		0			0
$727$	−89.4565	+	154.943i	−0.123049	+	0.213127i	−0.920969	−	0.389637i	$-0.872601\pi$
$727$	−89.4565	+	154.943i	−0.123049	+	0.213127i	0.797920	+	0.602764i	$0.205934\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1305.94			1.78651
$732$		0			0
$733$	220.426	−	381.790i	0.300718	−	0.520859i	−0.675581	−	0.737286i	$-0.736107\pi$
$733$	220.426	−	381.790i	0.300718	−	0.520859i	0.976299	+	0.216427i	$0.0694403\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1473.35	−	850.640i	−1.99912	−	1.15419i
$738$		0			0
$739$	129.576	+	224.433i	0.175340	+	0.303698i	0.940279	−	0.340405i	$-0.110564\pi$
$739$	129.576	+	224.433i	0.175340	+	0.303698i	−0.764939	+	0.644103i	$0.777231\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	673.856i		−	0.906939i	−0.891272	−	0.453470i	$-0.850186\pi$
$743$		−	673.856i		−	0.906939i	0.891272	−	0.453470i	$-0.149814\pi$
$744$		0			0
$745$	−412.363			−0.553507
$746$		0			0
$747$		0			0
$748$		0			0
$749$	82.3131	−	47.5235i	0.109897	−	0.0634493i
$750$		0			0
$751$	−879.453	+	507.753i	−1.17104	+	0.676102i	−0.953926	−	0.300043i	$-0.902999\pi$
$751$	−879.453	+	507.753i	−1.17104	+	0.676102i	−0.217118	+	0.976145i	$0.569665\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19.3053	+	11.1459i	−0.0255699	+	0.0147628i
$756$		0			0
$757$	190.298	+	329.606i	0.251385	+	0.435411i	0.963907	−	0.266238i	$-0.0857807\pi$
$757$	190.298	+	329.606i	0.251385	+	0.435411i	−0.712523	+	0.701649i	$0.752447\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	246.211	−	426.449i	0.323536	−	0.560380i	−0.657679	−	0.753298i	$-0.728462\pi$
$761$	246.211	−	426.449i	0.323536	−	0.560380i	0.981215	+	0.192918i	$0.0617951\pi$
$762$		0			0
$763$	137.734	+	79.5207i	0.180516	+	0.104221i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−69.9534	−	121.163i	−0.0912039	−	0.157970i
$768$		0			0
$769$	−140.073	−	242.613i	−0.182149	−	0.315491i	0.760463	−	0.649381i	$-0.224972\pi$
$769$	−140.073	−	242.613i	−0.182149	−	0.315491i	−0.942612	+	0.333890i	$0.891639\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−765.269	−	441.828i	−0.989999	−	0.571576i	−0.0847249	−	0.996404i	$-0.527001\pi$
$773$	−765.269	−	441.828i	−0.989999	−	0.571576i	−0.905274	+	0.424828i	$0.860334\pi$
$774$		0			0
$775$	753.186	+	434.852i	0.971853	+	0.561100i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	940.919	−	399.547i	1.20786	−	0.512897i
$780$		0			0
$781$	−1283.57	−	741.071i	−1.64350	−	0.948875i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	360.593			0.459354
$786$		0			0
$787$	−119.325	−	68.8925i	−0.151620	−	0.0875381i	0.422270	−	0.906470i	$-0.361233\pi$
$787$	−119.325	−	68.8925i	−0.151620	−	0.0875381i	−0.573891	+	0.818932i	$0.694567\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	315.043i		−	0.398285i
$792$		0			0
$793$	−182.554	+	105.398i	−0.230207	+	0.132910i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	658.196	+	380.010i	0.825842	+	0.476800i	0.852427	−	0.522846i	$-0.175130\pi$
$797$	658.196	+	380.010i	0.825842	+	0.476800i	−0.0265846	+	0.999647i	$0.508463\pi$
$798$		0			0
$799$	−836.716	−	1449.23i	−1.04720	−	1.81381i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1020.55	−	1767.64i	1.27092	−	2.20130i
$804$		0			0
$805$	31.0153	−	53.7201i	0.0385283	−	0.0667330i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	195.804			0.242032			0.121016	−	0.992651i	$-0.461385\pi$
$809$	195.804			0.242032			0.121016	+	0.992651i	$0.461385\pi$
$810$		0			0
$811$	−177.853	−	102.683i	−0.219301	−	0.126613i	0.386326	−	0.922362i	$-0.373744\pi$
$811$	−177.853	−	102.683i	−0.219301	−	0.126613i	−0.605626	+	0.795749i	$0.707077\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	579.039			0.710477
$816$		0			0
$817$	950.855	+	116.620i	1.16384	+	0.142742i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1122.25			−1.36693			−0.683465	−	0.729983i	$-0.739528\pi$
$821$	−1122.25			−1.36693			−0.683465	+	0.729983i	$0.739528\pi$
$822$		0			0
$823$	83.4857	+	144.601i	0.101441	+	0.175700i	0.912278	−	0.409571i	$-0.134321\pi$
$823$	83.4857	+	144.601i	0.101441	+	0.175700i	−0.810838	+	0.585271i	$0.800988\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1227.81	+	708.877i	1.48466	+	0.857167i	0.999848	−	0.0174521i	$-0.00555547\pi$
$827$	1227.81	+	708.877i	1.48466	+	0.857167i	0.484810	+	0.874620i	$0.338889\pi$
$828$		0			0
$829$		−	1069.64i		−	1.29027i	−0.764068	−	0.645136i	$-0.776801\pi$
$829$		−	1069.64i		−	1.29027i	0.764068	−	0.645136i	$-0.223199\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	130.693			0.156894
$834$		0			0
$835$	−164.216	+	94.8103i	−0.196666	+	0.113545i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	901.098	+	520.249i	1.07401	+	0.620082i	0.929275	−	0.369387i	$-0.120432\pi$
$839$	901.098	+	520.249i	1.07401	+	0.620082i	0.144739	+	0.989470i	$0.453766\pi$
$840$		0			0
$841$	619.706			0.736868
$842$		0			0
$843$		0			0
$844$		0			0
$845$	149.620	−	259.149i	0.177065	−	0.306686i
$846$		0			0
$847$	−752.394			−0.888305
$848$		0			0
$849$		0			0
$850$		0			0
$851$	74.9398	−	43.2665i	0.0880609	−	0.0508420i
$852$		0			0
$853$	29.6199	−	51.3032i	0.0347244	−	0.0601445i	−0.848141	−	0.529771i	$-0.822278\pi$
$853$	29.6199	−	51.3032i	0.0347244	−	0.0601445i	0.882865	+	0.469626i	$0.155611\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−431.664	−	249.221i	−0.503692	−	0.290807i	0.226545	−	0.974001i	$-0.427257\pi$
$857$	−431.664	−	249.221i	−0.503692	−	0.290807i	−0.730237	+	0.683194i	$0.760590\pi$
$858$		0			0
$859$	3.18867	+	5.52293i	0.00371207	+	0.00642949i	0.867875	−	0.496782i	$-0.165485\pi$
$859$	3.18867	+	5.52293i	0.00371207	+	0.00642949i	−0.864163	+	0.503211i	$0.832152\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1182.43i		−	1.37014i	−0.728475	−	0.685072i	$-0.759771\pi$
$863$		−	1182.43i		−	1.37014i	0.728475	−	0.685072i	$-0.240229\pi$
$864$		0			0
$865$	−64.7752	−	37.3980i	−0.0748846	−	0.0432346i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			512.233i			0.589451i
$870$		0			0
$871$	140.789	−	243.854i	0.161641	−	0.279970i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	284.401	+	492.596i	0.325029	+	0.562967i
$876$		0			0
$877$			70.7531i			0.0806762i	0.999186	+	0.0403381i	$0.0128435\pi$
$877$			70.7531i			0.0806762i	−0.999186	+	0.0403381i	$0.987156\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1012.09			−1.14880			−0.574399	−	0.818576i	$-0.694764\pi$
$881$	−1012.09			−1.14880			−0.574399	+	0.818576i	$0.694764\pi$
$882$		0			0
$883$	−222.537	+	385.445i	−0.252024	+	0.436518i	−0.964083	−	0.265602i	$-0.914429\pi$
$883$	−222.537	+	385.445i	−0.252024	+	0.436518i	0.712059	+	0.702119i	$0.247763\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	352.559	−	203.550i	0.397474	−	0.229481i	−0.287920	−	0.957655i	$-0.592964\pi$
$887$	352.559	−	203.550i	0.397474	−	0.229481i	0.685393	+	0.728173i	$0.259630\pi$
$888$		0			0
$889$			1111.79i			1.25060i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−479.797	−	1129.91i	−0.537287	−	1.26529i
$894$		0			0
$895$			136.784i			0.152832i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	299.315	−	518.429i	0.332942	−	0.576673i
$900$		0			0
$901$			2650.36i			2.94158i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	157.926	+	91.1786i	0.174504	+	0.100750i
$906$		0			0
$907$	301.076	+	173.826i	0.331947	+	0.191650i	0.656705	−	0.754147i	$-0.271950\pi$
$907$	301.076	+	173.826i	0.331947	+	0.191650i	−0.324758	+	0.945797i	$0.605283\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−261.179	+	150.792i	−0.286695	+	0.165523i	−0.636450	−	0.771318i	$-0.719598\pi$
$911$	−261.179	+	150.792i	−0.286695	+	0.165523i	0.349755	+	0.936841i	$0.386265\pi$
$912$		0			0
$913$	−689.827	+	1194.81i	−0.755560	+	1.30867i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−783.252	−	1356.63i	−0.854146	−	1.47943i
$918$		0			0
$919$	1364.04			1.48427			0.742135	−	0.670250i	$-0.233813\pi$
$919$	1364.04			1.48427			0.742135	+	0.670250i	$0.233813\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	122.654	−	212.443i	0.132886	−	0.230166i
$924$		0			0
$925$			367.905i			0.397735i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−817.772	+	1416.42i	−0.880271	+	1.52467i	−0.0292311	+	0.999573i	$0.509306\pi$
$929$	−817.772	+	1416.42i	−0.880271	+	1.52467i	−0.851040	+	0.525101i	$0.824027\pi$
$930$		0			0
$931$	95.1573	+	11.6708i	0.102210	+	0.0125358i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	365.015	−	632.224i	0.390390	−	0.676176i
$936$		0			0
$937$	339.778	−	588.513i	0.362624	−	0.628083i	−0.625768	−	0.780009i	$-0.715214\pi$
$937$	339.778	−	588.513i	0.362624	−	0.628083i	0.988392	+	0.151927i	$0.0485477\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−964.094	+	556.620i	−1.02454	+	0.591520i	−0.915416	−	0.402508i	$-0.868138\pi$
$941$	−964.094	+	556.620i	−1.02454	+	0.591520i	−0.109126	+	0.994028i	$0.534805\pi$
$942$		0			0
$943$	236.850	−	136.745i	0.251166	−	0.145011i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−738.701	+	1279.47i	−0.780043	+	1.35107i	0.151872	+	0.988400i	$0.451470\pi$
$947$	−738.701	+	1279.47i	−0.780043	+	1.35107i	−0.931916	+	0.362675i	$0.881864\pi$
$948$		0			0
$949$	292.562	+	168.911i	0.308284	+	0.177988i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−856.935	+	494.751i	−0.899197	+	0.519152i	−0.876940	−	0.480601i	$-0.840419\pi$
$953$	−856.935	+	494.751i	−0.899197	+	0.519152i	−0.0222574	+	0.999752i	$0.507085\pi$
$954$		0			0
$955$	−305.980	−	529.973i	−0.320398	−	0.554946i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	103.536	+	179.330i	0.107963	+	0.186997i
$960$		0			0
$961$	329.186	+	570.167i	0.342546	+	0.593306i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	486.770i		−	0.504425i
$966$		0			0
$967$	837.470			0.866050			0.433025	−	0.901382i	$-0.357446\pi$
$967$	837.470			0.866050			0.433025	+	0.901382i	$0.357446\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−107.490	+	62.0594i	−0.110700	+	0.0639129i	−0.554328	−	0.832298i	$-0.687025\pi$
$971$	−107.490	+	62.0594i	−0.110700	+	0.0639129i	0.443628	+	0.896211i	$0.353691\pi$
$972$		0			0
$973$	113.946	−	197.361i	0.117108	−	0.202837i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	444.691	+	256.743i	0.455160	+	0.262787i	0.710007	−	0.704195i	$-0.248692\pi$
$977$	444.691	+	256.743i	0.455160	+	0.262787i	−0.254847	+	0.966981i	$0.582025\pi$
$978$		0			0
$979$			468.339i			0.478386i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	694.081	+	400.728i	0.706085	+	0.407658i	0.809610	−	0.586969i	$-0.199679\pi$
$983$	694.081	+	400.728i	0.706085	+	0.407658i	−0.103525	+	0.994627i	$0.533012\pi$
$984$		0			0
$985$	67.1547			0.0681773
$986$		0			0
$987$		0			0
$988$		0			0
$989$	256.299			0.259150
$990$		0			0
$991$	1285.82	−	742.366i	1.29749	−	0.749108i	0.317522	−	0.948251i	$-0.397149\pi$
$991$	1285.82	−	742.366i	1.29749	−	0.749108i	0.979970	+	0.199143i	$0.0638159\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	278.278	−	481.992i	0.279677	−	0.484414i
$996$		0			0
$997$	−772.803			−0.775128			−0.387564	−	0.921843i	$-0.626683\pi$
$997$	−772.803			−0.775128			−0.387564	+	0.921843i	$0.626683\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.bl.a.145.17		80
3.2	odd	2		684.3.bl.a.373.12	yes	80
9.2	odd	6		684.3.s.a.601.3	yes	80
9.7	even	3		2052.3.s.a.829.24		80
19.8	odd	6		2052.3.s.a.901.24		80
57.8	even	6		684.3.s.a.445.3	✓	80
171.65	even	6		684.3.bl.a.673.12	yes	80
171.160	odd	6	inner	2052.3.bl.a.1585.17		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.s.a.445.3	✓	80	57.8	even	6
684.3.s.a.601.3	yes	80	9.2	odd	6
684.3.bl.a.373.12	yes	80	3.2	odd	2
684.3.bl.a.673.12	yes	80	171.65	even	6
2052.3.s.a.829.24		80	9.7	even	3
2052.3.s.a.901.24		80	19.8	odd	6
2052.3.bl.a.145.17		80	1.1	even	1	trivial
2052.3.bl.a.1585.17		80	171.160	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.bl (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.84061 q^{5} +(3.31490 + 5.74157i) q^{7} +O(q^{10})\)	\(q-1.84061 q^{5} +(3.31490 + 5.74157i) q^{7} +(7.65648 + 13.2614i) q^{11} +(-2.19489 + 1.26722i) q^{13} +(12.9506 + 22.4311i) q^{17} +(7.42625 + 17.4886i) q^{19} +(2.54164 + 4.40226i) q^{23} -21.6122 q^{25} +14.8760i q^{29} +(-34.8501 - 20.1207i) q^{31} +(-6.10143 - 10.5680i) q^{35} -17.0230i q^{37} -53.8019i q^{41} +(25.2100 - 43.6650i) q^{43} -64.6082 q^{47} +(2.52290 - 4.36980i) q^{49} +(88.6168 + 51.1629i) q^{53} +(-14.0926 - 24.4091i) q^{55} +55.2023i q^{59} +83.1725 q^{61} +(4.03993 - 2.33245i) q^{65} +(-96.2160 + 55.5503i) q^{67} +(-83.8226 + 48.3950i) q^{71} +(-66.6462 - 115.435i) q^{73} +(-50.7609 + 87.9205i) q^{77} +(28.9694 + 16.7255i) q^{79} +(45.0486 + 78.0264i) q^{83} +(-23.8370 - 41.2869i) q^{85} +(26.4870 + 15.2923i) q^{89} +(-14.5517 - 8.40140i) q^{91} +(-13.6688 - 32.1896i) q^{95} +(-68.9293 - 39.7964i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - q^{7}+O(q^{10}) \) 80 * q - q^7	\( 80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97}+O(q^{100}) \) 80 * q - q^7 + 6 * q^11 - 15 * q^13 + 21 * q^17 - 20 * q^19 - 24 * q^23 + 400 * q^25 + 24 * q^31 + 54 * q^35 + 76 * q^43 - 24 * q^47 - 267 * q^49 + 36 * q^53 + 14 * q^61 - 288 * q^65 - 21 * q^67 + 81 * q^71 + 55 * q^73 - 30 * q^77 - 51 * q^79 + 93 * q^83 - 216 * q^89 + 96 * q^91 + 432 * q^95 + 90 * q^97

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