LMFDB - Embedded newform 2052.3.m.a.881.9

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			881.9
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.32

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.14879i q^{5} +(-6.67945 - 11.5692i) q^{7} +O(q^{10})$	$q-6.14879i q^{5} +(-6.67945 - 11.5692i) q^{7} +(-0.586623 + 0.338687i) q^{11} +(10.9877 + 19.0312i) q^{13} +(-10.9425 + 6.31763i) q^{17} +(7.16635 + 17.5967i) q^{19} +(-30.1014 + 17.3790i) q^{23} -12.8077 q^{25} +38.1321i q^{29} +(19.7392 - 34.1893i) q^{31} +(-71.1363 + 41.0706i) q^{35} -24.0074 q^{37} -1.00016i q^{41} +(5.90752 - 10.2321i) q^{43} +36.4056i q^{47} +(-64.7302 + 112.116i) q^{49} +(-29.2948 - 16.9134i) q^{53} +(2.08251 + 3.60702i) q^{55} -61.9016i q^{59} +99.9371 q^{61} +(117.019 - 67.5609i) q^{65} +(15.8247 + 27.4092i) q^{67} +(-104.178 + 60.1470i) q^{71} +(9.08945 + 15.7434i) q^{73} +(7.83664 + 4.52448i) q^{77} +(-4.10688 + 7.11332i) q^{79} +(139.075 - 80.2951i) q^{83} +(38.8458 + 67.2829i) q^{85} +(91.8708 + 53.0416i) q^{89} +(146.783 - 254.236i) q^{91} +(108.198 - 44.0644i) q^{95} +(-25.7098 + 44.5306i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	6.14879i		−	1.22976i	−0.788621	−	0.614879i	$-0.789205\pi$
$5$		−	6.14879i		−	1.22976i	0.788621	−	0.614879i	$-0.210795\pi$
$6$		0			0
$7$	−6.67945	−	11.5692i	−0.954208	−	1.65274i	−0.736171	−	0.676795i	$-0.763368\pi$
$7$	−6.67945	−	11.5692i	−0.954208	−	1.65274i	−0.218036	−	0.975941i	$-0.569965\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.586623	+	0.338687i	−0.0533293	+	0.0307897i	−0.526428	−	0.850220i	$-0.676469\pi$
$11$	−0.586623	+	0.338687i	−0.0533293	+	0.0307897i	0.473098	+	0.881010i	$0.343136\pi$
$12$		0			0
$13$	10.9877	+	19.0312i	0.845205	+	1.46394i	0.885443	+	0.464748i	$0.153855\pi$
$13$	10.9877	+	19.0312i	0.845205	+	1.46394i	−0.0402375	+	0.999190i	$0.512811\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−10.9425	+	6.31763i	−0.643674	+	0.371625i	−0.786028	−	0.618190i	$-0.787866\pi$
$17$	−10.9425	+	6.31763i	−0.643674	+	0.371625i	0.142354	+	0.989816i	$0.454533\pi$
$18$		0			0
$19$	7.16635	+	17.5967i	0.377176	+	0.926141i
$19$	7.16635	+	17.5967i	0.377176	+	0.926141i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−30.1014	+	17.3790i	−1.30876	+	0.755610i	−0.981888	−	0.189460i	$-0.939326\pi$
$23$	−30.1014	+	17.3790i	−1.30876	+	0.755610i	−0.326867	+	0.945070i	$0.605993\pi$
$24$		0			0
$25$	−12.8077			−0.512306
$26$		0			0
$27$		0			0
$28$		0			0
$29$			38.1321i			1.31490i	0.753498	+	0.657450i	$0.228365\pi$
$29$			38.1321i			1.31490i	−0.753498	+	0.657450i	$0.771635\pi$
$30$		0			0
$31$	19.7392	−	34.1893i	0.636749	−	1.10288i	−0.349393	−	0.936976i	$-0.613612\pi$
$31$	19.7392	−	34.1893i	0.636749	−	1.10288i	0.986142	−	0.165905i	$-0.0530545\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−71.1363	+	41.0706i	−2.03247	+	1.17345i
$36$		0			0
$37$	−24.0074			−0.648850			−0.324425	−	0.945911i	$-0.605171\pi$
$37$	−24.0074			−0.648850			−0.324425	+	0.945911i	$0.605171\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.00016i		−	0.0243941i	−0.999926	−	0.0121970i	$-0.996117\pi$
$41$		−	1.00016i		−	0.0243941i	0.999926	−	0.0121970i	$-0.00388253\pi$
$42$		0			0
$43$	5.90752	−	10.2321i	0.137384	−	0.237956i	−0.789122	−	0.614237i	$-0.789464\pi$
$43$	5.90752	−	10.2321i	0.137384	−	0.237956i	0.926506	+	0.376281i	$0.122797\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			36.4056i			0.774588i	0.921956	+	0.387294i	$0.126590\pi$
$47$			36.4056i			0.774588i	−0.921956	+	0.387294i	$0.873410\pi$
$48$		0			0
$49$	−64.7302	+	112.116i	−1.32102	+	2.28808i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−29.2948	−	16.9134i	−0.552732	−	0.319120i	0.197491	−	0.980305i	$-0.436721\pi$
$53$	−29.2948	−	16.9134i	−0.552732	−	0.319120i	−0.750223	+	0.661185i	$0.770054\pi$
$54$		0			0
$55$	2.08251	+	3.60702i	0.0378639	+	0.0655822i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	61.9016i		−	1.04918i	−0.851355	−	0.524590i	$-0.824219\pi$
$59$		−	61.9016i		−	1.04918i	0.851355	−	0.524590i	$-0.175781\pi$
$60$		0			0
$61$	99.9371			1.63831			0.819156	−	0.573570i	$-0.194442\pi$
$61$	99.9371			1.63831			0.819156	+	0.573570i	$0.194442\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	117.019	−	67.5609i	1.80029	−	1.03940i
$66$		0			0
$67$	15.8247	+	27.4092i	0.236189	+	0.409092i	0.959618	−	0.281308i	$-0.0907681\pi$
$67$	15.8247	+	27.4092i	0.236189	+	0.409092i	−0.723428	+	0.690399i	$0.757435\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−104.178	+	60.1470i	−1.46729	+	0.847141i	−0.999330	−	0.0366080i	$-0.988345\pi$
$71$	−104.178	+	60.1470i	−1.46729	+	0.847141i	−0.467961	+	0.883749i	$0.655011\pi$
$72$		0			0
$73$	9.08945	+	15.7434i	0.124513	+	0.215663i	0.921542	−	0.388278i	$-0.126930\pi$
$73$	9.08945	+	15.7434i	0.124513	+	0.215663i	−0.797030	+	0.603940i	$0.793596\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.83664	+	4.52448i	0.101774	+	0.0587595i
$78$		0			0
$79$	−4.10688	+	7.11332i	−0.0519858	+	0.0900420i	−0.890847	−	0.454303i	$-0.849888\pi$
$79$	−4.10688	+	7.11332i	−0.0519858	+	0.0900420i	0.838861	+	0.544345i	$0.183222\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	139.075	−	80.2951i	1.67560	−	0.967410i	0.711195	−	0.702995i	$-0.248154\pi$
$83$	139.075	−	80.2951i	1.67560	−	0.967410i	0.964409	−	0.264416i	$-0.0851791\pi$
$84$		0			0
$85$	38.8458	+	67.2829i	0.457010	+	0.791564i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	91.8708	+	53.0416i	1.03226	+	0.595973i	0.917630	−	0.397436i	$-0.130100\pi$
$89$	91.8708	+	53.0416i	1.03226	+	0.595973i	0.114626	+	0.993409i	$0.463433\pi$
$90$		0			0
$91$	146.783	−	254.236i	1.61300	−	2.79380i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	108.198	−	44.0644i	1.13893	−	0.463836i
$96$		0			0
$97$	−25.7098	+	44.5306i	−0.265049	+	0.459078i	−0.967576	−	0.252578i	$-0.918721\pi$
$97$	−25.7098	+	44.5306i	−0.265049	+	0.459078i	0.702527	+	0.711657i	$0.252055\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	2.95928i		−	0.0292998i	−0.999893	−	0.0146499i	$-0.995337\pi$
$101$		−	2.95928i		−	0.0292998i	0.999893	−	0.0146499i	$-0.00466337\pi$
$102$		0			0
$103$	5.46277	−	9.46179i	0.0530366	−	0.0918620i	−0.838288	−	0.545227i	$-0.816443\pi$
$103$	5.46277	−	9.46179i	0.0530366	−	0.0918620i	0.891325	+	0.453365i	$0.149777\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.7588i			0.137933i	0.997619	+	0.0689664i	$0.0219701\pi$
$107$			14.7588i			0.137933i	−0.997619	+	0.0689664i	$0.978030\pi$
$108$		0			0
$109$	16.1494	+	27.9716i	0.148160	+	0.256620i	0.930547	−	0.366172i	$-0.119332\pi$
$109$	16.1494	+	27.9716i	0.148160	+	0.256620i	−0.782388	+	0.622792i	$0.785998\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	134.566	+	77.6917i	1.19085	+	0.687537i	0.958499	−	0.285095i	$-0.0920253\pi$
$113$	134.566	+	77.6917i	1.19085	+	0.687537i	0.232350	+	0.972632i	$0.425359\pi$
$114$		0			0
$115$	106.860	+	185.087i	0.929218	+	1.60945i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	146.179	+	84.3967i	1.22840	+	0.709216i
$120$		0			0
$121$	−60.2706	+	104.392i	−0.498104	+	0.862741i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	74.9682i		−	0.599745i
$126$		0			0
$127$	6.01913	−	10.4254i	0.0473947	−	0.0820900i	−0.841355	−	0.540483i	$-0.818241\pi$
$127$	6.01913	−	10.4254i	0.0473947	−	0.0820900i	0.888750	+	0.458393i	$0.151575\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			10.5931i			0.0808632i	0.999182	+	0.0404316i	$0.0128733\pi$
$131$			10.5931i			0.0808632i	−0.999182	+	0.0404316i	$0.987127\pi$
$132$		0			0
$133$	155.711	−	200.445i	1.17076	−	1.50710i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	18.4865i		−	0.134938i	−0.997721	−	0.0674688i	$-0.978508\pi$
$137$		−	18.4865i		−	0.134938i	0.997721	−	0.0674688i	$-0.0214923\pi$
$138$		0			0
$139$	63.6164	+	110.187i	0.457672	+	0.792711i	0.998837	−	0.0482048i	$-0.0153500\pi$
$139$	63.6164	+	110.187i	0.457672	+	0.792711i	−0.541165	+	0.840916i	$0.682017\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.8912	−	7.44275i	−0.0901484	−	0.0520472i
$144$		0			0
$145$	234.467			1.61701
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	199.546i		−	1.33923i	−0.742707	−	0.669616i	$-0.766459\pi$
$149$		−	199.546i		−	1.33923i	0.742707	−	0.669616i	$-0.233541\pi$
$150$		0			0
$151$	−86.0705	−	149.078i	−0.570003	−	0.987274i	−0.996565	−	0.0828158i	$-0.973609\pi$
$151$	−86.0705	−	149.078i	−0.570003	−	0.987274i	0.426562	−	0.904458i	$-0.359725\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−210.223	−	121.372i	−1.35628	−	0.783047i
$156$		0			0
$157$	−126.114			−0.803271			−0.401636	−	0.915800i	$-0.631558\pi$
$157$	−126.114			−0.803271			−0.401636	+	0.915800i	$0.631558\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	402.121	+	232.165i	2.49765	+	1.44202i
$162$		0			0
$163$	−130.019			−0.797661			−0.398831	−	0.917025i	$-0.630584\pi$
$163$	−130.019			−0.797661			−0.398831	+	0.917025i	$0.630584\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.1623	−	8.75398i	0.0907924	−	0.0524190i	−0.453916	−	0.891044i	$-0.649974\pi$
$167$	15.1623	−	8.75398i	0.0907924	−	0.0524190i	0.544709	+	0.838625i	$0.316640\pi$
$168$		0			0
$169$	−156.958	+	271.859i	−0.928744	+	1.60863i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−225.015	−	129.913i	−1.30067	−	0.750940i	−0.320148	−	0.947367i	$-0.603733\pi$
$173$	−225.015	−	129.913i	−1.30067	−	0.750940i	−0.980518	+	0.196427i	$0.937066\pi$
$174$		0			0
$175$	85.5482	+	148.174i	0.488847	+	0.846707i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			120.898i			0.675407i	0.941252	+	0.337704i	$0.109650\pi$
$179$			120.898i			0.675407i	−0.941252	+	0.337704i	$0.890350\pi$
$180$		0			0
$181$	−110.148	+	190.783i	−0.608555	+	1.05405i	0.382924	+	0.923780i	$0.374917\pi$
$181$	−110.148	+	190.783i	−0.608555	+	1.05405i	−0.991479	+	0.130268i	$0.958416\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			147.617i			0.797928i
$186$		0			0
$187$	4.27940	−	7.41213i	0.0228845	−	0.0396371i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	87.7696	−	50.6738i	0.459527	−	0.265308i	−0.252319	−	0.967644i	$-0.581193\pi$
$191$	87.7696	−	50.6738i	0.459527	−	0.265308i	0.711845	+	0.702336i	$0.247860\pi$
$192$		0			0
$193$	−319.750			−1.65673			−0.828367	−	0.560185i	$-0.810730\pi$
$193$	−319.750			−1.65673			−0.828367	+	0.560185i	$0.810730\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			362.670i			1.84096i	0.390786	+	0.920482i	$0.372203\pi$
$197$			362.670i			1.84096i	−0.390786	+	0.920482i	$0.627797\pi$
$198$		0			0
$199$	−63.0213	+	109.156i	−0.316690	+	0.548523i	−0.979795	−	0.200003i	$-0.935905\pi$
$199$	−63.0213	+	109.156i	−0.316690	+	0.548523i	0.663105	+	0.748526i	$0.269238\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	441.156	−	254.702i	2.17318	−	1.25469i
$204$		0			0
$205$	−6.14976			−0.0299988
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.1637	−	7.89546i	−0.0486302	−	0.0377773i
$210$		0			0
$211$	52.7903			0.250191			0.125095	−	0.992145i	$-0.460076\pi$
$211$	52.7903			0.250191			0.125095	+	0.992145i	$0.460076\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−62.9152	−	36.3241i	−0.292629	−	0.168949i
$216$		0			0
$217$	−527.389			−2.43036
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−240.464	−	138.832i	−1.08807	−	0.628200i
$222$		0			0
$223$	−0.0946169	+	0.163881i	−0.000424291	+	0.000734893i	−0.866237	−	0.499633i	$-0.833468\pi$
$223$	−0.0946169	+	0.163881i	−0.000424291	+	0.000734893i	0.865813	+	0.500367i	$0.166802\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	17.0193	−	9.82612i	0.0749751	−	0.0432869i	−0.462044	−	0.886857i	$-0.652884\pi$
$227$	17.0193	−	9.82612i	0.0749751	−	0.0432869i	0.537019	+	0.843570i	$0.319550\pi$
$228$		0			0
$229$	175.467	−	303.918i	0.766233	−	1.32715i	−0.173359	−	0.984859i	$-0.555462\pi$
$229$	175.467	−	303.918i	0.766233	−	1.32715i	0.939592	−	0.342296i	$-0.111204\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−212.611	+	122.751i	−0.912494	+	0.526829i	−0.881233	−	0.472683i	$-0.843286\pi$
$233$	−212.611	+	122.751i	−0.912494	+	0.526829i	−0.0312613	+	0.999511i	$0.509952\pi$
$234$		0			0
$235$	223.851			0.952556
$236$		0			0
$237$		0			0
$238$		0			0
$239$	215.150	+	124.217i	0.900210	+	0.519736i	0.877268	−	0.480000i	$-0.159363\pi$
$239$	215.150	+	124.217i	0.900210	+	0.519736i	0.0229416	+	0.999737i	$0.492697\pi$
$240$		0			0
$241$	−14.4503			−0.0599599			−0.0299800	−	0.999550i	$-0.509544\pi$
$241$	−14.4503			−0.0599599			−0.0299800	+	0.999550i	$0.509544\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	689.378	+	398.013i	2.81379	+	1.62454i
$246$		0			0
$247$	−256.145	+	329.731i	−1.03702	+	1.33494i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	39.7299	+	22.9381i	0.158286	+	0.0913867i	0.577051	−	0.816708i	$-0.304203\pi$
$251$	39.7299	+	22.9381i	0.158286	+	0.0913867i	−0.418765	+	0.908095i	$0.637537\pi$
$252$		0			0
$253$	11.7721	−	20.3899i	0.0465300	−	0.0805923i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−377.517	+	217.960i	−1.46894	+	0.848093i	−0.999394	−	0.0348156i	$-0.988916\pi$
$257$	−377.517	+	217.960i	−1.46894	+	0.848093i	−0.469546	+	0.882908i	$0.655582\pi$
$258$		0			0
$259$	160.357	+	277.746i	0.619137	+	1.07238i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	178.922	+	103.300i	0.680311	+	0.392778i	0.799972	−	0.600037i	$-0.204848\pi$
$263$	178.922	+	103.300i	0.680311	+	0.392778i	−0.119661	+	0.992815i	$0.538181\pi$
$264$		0			0
$265$	−103.997	+	180.128i	−0.392441	+	0.679727i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	250.230	−	144.470i	0.930224	−	0.537065i	0.0433415	−	0.999060i	$-0.486200\pi$
$269$	250.230	−	144.470i	0.930224	−	0.537065i	0.886882	+	0.461995i	$0.152866\pi$
$270$		0			0
$271$	147.862	+	256.105i	0.545618	+	0.945037i	0.998568	+	0.0535016i	$0.0170382\pi$
$271$	147.862	+	256.105i	0.545618	+	0.945037i	−0.452950	+	0.891536i	$0.649628\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.51326	−	4.33778i	0.0273210	−	0.0157738i
$276$		0			0
$277$	−46.2692	−	80.1406i	−0.167037	−	0.289316i	0.770340	−	0.637633i	$-0.220087\pi$
$277$	−46.2692	−	80.1406i	−0.167037	−	0.289316i	−0.937377	+	0.348317i	$0.886753\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			52.7052i			0.187563i	0.995593	+	0.0937815i	$0.0298955\pi$
$281$			52.7052i			0.187563i	−0.995593	+	0.0937815i	$0.970104\pi$
$282$		0			0
$283$	378.564			1.33768			0.668842	−	0.743405i	$-0.266790\pi$
$283$	378.564			1.33768			0.668842	+	0.743405i	$0.266790\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−11.5710	+	6.68050i	−0.0403170	+	0.0232770i
$288$		0			0
$289$	−64.6750	+	112.020i	−0.223789	+	0.387614i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	413.513	+	238.742i	1.41131	+	0.814818i	0.995512	−	0.0946404i	$-0.0301701\pi$
$293$	413.513	+	238.742i	1.41131	+	0.814818i	0.415795	+	0.909458i	$0.363503\pi$
$294$		0			0
$295$	−380.620			−1.29024
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−661.488	−	381.910i	−2.21233	−	1.27729i
$300$		0			0
$301$	−157.836			−0.524372
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	614.492i		−	2.01473i
$306$		0			0
$307$	24.3451	+	42.1669i	0.0792999	+	0.137351i	0.902948	−	0.429750i	$-0.141398\pi$
$307$	24.3451	+	42.1669i	0.0792999	+	0.137351i	−0.823648	+	0.567101i	$0.808065\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−58.5917	−	33.8279i	−0.188398	−	0.108771i	0.402835	−	0.915273i	$-0.368025\pi$
$311$	−58.5917	−	33.8279i	−0.188398	−	0.108771i	−0.591232	+	0.806501i	$0.701358\pi$
$312$		0			0
$313$	−2.73622			−0.00874190			−0.00437095	−	0.999990i	$-0.501391\pi$
$313$	−2.73622			−0.00874190			−0.00437095	+	0.999990i	$0.501391\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	413.748i		−	1.30520i	−0.757703	−	0.652600i	$-0.773678\pi$
$317$		−	413.748i		−	1.30520i	0.757703	−	0.652600i	$-0.226322\pi$
$318$		0			0
$319$	−12.9148	−	22.3692i	−0.0404854	−	0.0701228i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−189.587	−	147.277i	−0.586956	−	0.455965i
$324$		0			0
$325$	−140.726	−	243.745i	−0.433004	−	0.749985i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	421.182	−	243.170i	1.28019	−	0.739117i
$330$		0			0
$331$	198.804	+	344.339i	0.600617	+	1.04030i	0.992728	+	0.120382i	$0.0384118\pi$
$331$	198.804	+	344.339i	0.600617	+	1.04030i	−0.392110	+	0.919918i	$0.628255\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	168.533	−	97.3027i	0.503084	−	0.290456i
$336$		0			0
$337$	−34.1929			−0.101463			−0.0507313	−	0.998712i	$-0.516155\pi$
$337$	−34.1929			−0.101463			−0.0507313	+	0.998712i	$0.516155\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			26.7416i			0.0784212i
$342$		0			0
$343$	1074.86			3.13371
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	119.082i		−	0.343177i	−0.985169	−	0.171588i	$-0.945110\pi$
$347$		−	119.082i		−	0.343177i	0.985169	−	0.171588i	$-0.0548899\pi$
$348$		0			0
$349$	196.268	+	339.946i	0.562373	+	0.974059i	0.997289	+	0.0735875i	$0.0234448\pi$
$349$	196.268	+	339.946i	0.562373	+	0.974059i	−0.434916	+	0.900471i	$0.643222\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	285.889	−	165.058i	0.809884	−	0.467587i	−0.0370318	−	0.999314i	$-0.511790\pi$
$353$	285.889	−	165.058i	0.809884	−	0.467587i	0.846915	+	0.531728i	$0.178457\pi$
$354$		0			0
$355$	369.832	+	640.567i	1.04178	+	1.80441i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	85.9500	−	49.6232i	0.239415	−	0.138226i	−0.375493	−	0.926825i	$-0.622527\pi$
$359$	85.9500	−	49.6232i	0.239415	−	0.138226i	0.614908	+	0.788599i	$0.289193\pi$
$360$		0			0
$361$	−258.287	+	252.208i	−0.715476	+	0.698638i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	96.8028	−	55.8891i	0.265213	−	0.153121i
$366$		0			0
$367$	66.6391			0.181578			0.0907889	−	0.995870i	$-0.471061\pi$
$367$	66.6391			0.181578			0.0907889	+	0.995870i	$0.471061\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			451.888i			1.21803i
$372$		0			0
$373$	163.975	−	284.013i	0.439611	−	0.761429i	−0.558048	−	0.829809i	$-0.688449\pi$
$373$	163.975	−	284.013i	0.439611	−	0.761429i	0.997659	+	0.0683793i	$0.0217828\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−725.700	+	418.983i	−1.92493	+	1.11136i
$378$		0			0
$379$	−588.049			−1.55158			−0.775790	−	0.630991i	$-0.782649\pi$
$379$	−588.049			−1.55158			−0.775790	+	0.630991i	$0.782649\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	363.143i		−	0.948153i	−0.880484	−	0.474076i	$-0.842782\pi$
$383$		−	363.143i		−	0.948153i	0.880484	−	0.474076i	$-0.157218\pi$
$384$		0			0
$385$	27.8201	−	48.1859i	0.0722600	−	0.125158i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			419.346i			1.07801i	0.842302	+	0.539006i	$0.181200\pi$
$389$			419.346i			1.07801i	−0.842302	+	0.539006i	$0.818800\pi$
$390$		0			0
$391$	219.589	−	380.339i	0.561608	−	0.972733i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	43.7383	+	25.2523i	0.110730	+	0.0639300i
$396$		0			0
$397$	18.4059	+	31.8800i	0.0463625	+	0.0803022i	0.888275	−	0.459311i	$-0.151904\pi$
$397$	18.4059	+	31.8800i	0.0463625	+	0.0803022i	−0.841913	+	0.539613i	$0.818570\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			553.230i			1.37963i	0.723988	+	0.689813i	$0.242307\pi$
$401$			553.230i			1.37963i	−0.723988	+	0.689813i	$0.757693\pi$
$402$		0			0
$403$	867.552			2.15273
$404$		0			0
$405$		0			0
$406$		0			0
$407$	14.0833	−	8.13100i	0.0346027	−	0.0199779i
$408$		0			0
$409$	114.062	+	197.561i	0.278880	+	0.483035i	0.971107	−	0.238646i	$-0.0767034\pi$
$409$	114.062	+	197.561i	0.278880	+	0.483035i	−0.692227	+	0.721680i	$0.743370\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−716.149	+	413.469i	−1.73402	+	1.00114i
$414$		0			0
$415$	−493.718	−	855.144i	−1.18968	−	2.06059i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	427.551	+	246.847i	1.02041	+	0.589133i	0.914222	−	0.405213i	$-0.132803\pi$
$419$	427.551	+	246.847i	1.02041	+	0.589133i	0.106187	+	0.994346i	$0.466136\pi$
$420$		0			0
$421$	−87.0805	+	150.828i	−0.206842	+	0.358261i	−0.950718	−	0.310057i	$-0.899652\pi$
$421$	−87.0805	+	150.828i	−0.206842	+	0.358261i	0.743876	+	0.668318i	$0.232985\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	140.147	−	80.9141i	0.329758	−	0.190386i
$426$		0			0
$427$	−667.525	−	1156.19i	−1.56329	−	2.70770i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−183.318	−	105.838i	−0.425331	−	0.245565i	0.272025	−	0.962290i	$-0.412307\pi$
$431$	−183.318	−	105.838i	−0.425331	−	0.245565i	−0.697356	+	0.716725i	$0.745640\pi$
$432$		0			0
$433$	98.5927	−	170.768i	0.227697	−	0.394382i	−0.729428	−	0.684057i	$-0.760214\pi$
$433$	98.5927	−	170.768i	0.227697	−	0.394382i	0.957125	+	0.289675i	$0.0935472\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−521.530	−	405.140i	−1.19343	−	0.927094i
$438$		0			0
$439$	307.405	−	532.441i	0.700239	−	1.21285i	−0.268143	−	0.963379i	$-0.586410\pi$
$439$	307.405	−	532.441i	0.700239	−	1.21285i	0.968382	−	0.249471i	$-0.0802566\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			556.406i			1.25599i	0.778216	+	0.627997i	$0.216125\pi$
$443$			556.406i			1.25599i	−0.778216	+	0.627997i	$0.783875\pi$
$444$		0			0
$445$	326.142	−	564.894i	0.732903	−	1.26943i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			257.610i			0.573742i	0.957969	+	0.286871i	$0.0926150\pi$
$449$			257.610i			0.573742i	−0.957969	+	0.286871i	$0.907385\pi$
$450$		0			0
$451$	0.338740	+	0.586715i	0.000751086	+	0.00130092i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1563.24	−	902.540i	−3.43570	−	1.98360i
$456$		0			0
$457$	215.637	+	373.494i	0.471853	+	0.817273i	0.999481	−	0.0322025i	$-0.0102521\pi$
$457$	215.637	+	373.494i	0.471853	+	0.817273i	−0.527629	+	0.849475i	$0.676919\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−131.196	−	75.7463i	−0.284591	−	0.164309i	0.350909	−	0.936410i	$-0.385873\pi$
$461$	−131.196	−	75.7463i	−0.284591	−	0.164309i	−0.635500	+	0.772101i	$0.719206\pi$
$462$		0			0
$463$	−125.351	+	217.114i	−0.270736	+	0.468928i	−0.969050	−	0.246863i	$-0.920600\pi$
$463$	−125.351	+	217.114i	−0.270736	+	0.468928i	0.698315	+	0.715791i	$0.253934\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			263.024i			0.563220i	0.959529	+	0.281610i	$0.0908685\pi$
$467$			263.024i			0.563220i	−0.959529	+	0.281610i	$0.909132\pi$
$468$		0			0
$469$	211.400	−	366.156i	0.450747	−	0.780717i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.00319i			0.0169201i
$474$		0			0
$475$	−91.7842	−	225.372i	−0.193230	−	0.474468i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	638.279i		−	1.33252i	−0.745718	−	0.666262i	$-0.767893\pi$
$479$		−	638.279i		−	1.33252i	0.745718	−	0.666262i	$-0.232107\pi$
$480$		0			0
$481$	−263.786	−	456.890i	−0.548411	−	0.949876i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	273.809	+	158.084i	0.564556	+	0.325946i
$486$		0			0
$487$	−780.042			−1.60173			−0.800864	−	0.598846i	$-0.795626\pi$
$487$	−780.042			−1.60173			−0.800864	+	0.598846i	$0.795626\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	580.181i		−	1.18163i	−0.806806	−	0.590816i	$-0.798806\pi$
$491$		−	580.181i		−	1.18163i	0.806806	−	0.590816i	$-0.201194\pi$
$492$		0			0
$493$	−240.905	−	417.259i	−0.488651	−	0.846368i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1391.70	+	803.498i	2.80020	+	1.61670i
$498$		0			0
$499$	56.5208			0.113268			0.0566341	−	0.998395i	$-0.481963\pi$
$499$	56.5208			0.113268			0.0566341	+	0.998395i	$0.481963\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−144.369	−	83.3516i	−0.287016	−	0.165709i	0.349579	−	0.936907i	$-0.386325\pi$
$503$	−144.369	−	83.3516i	−0.287016	−	0.165709i	−0.636596	+	0.771198i	$0.719658\pi$
$504$		0			0
$505$	−18.1960			−0.0360317
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−259.974	+	150.096i	−0.510755	+	0.294885i	−0.733144	−	0.680073i	$-0.761948\pi$
$509$	−259.974	+	150.096i	−0.510755	+	0.294885i	0.222389	+	0.974958i	$0.428615\pi$
$510$		0			0
$511$	121.425	−	210.314i	0.237622	−	0.411574i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−58.1786	−	33.5894i	−0.112968	−	0.0652222i
$516$		0			0
$517$	−12.3301	−	21.3564i	−0.0238493	−	0.0413082i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			209.742i			0.402575i	0.979532	+	0.201288i	$0.0645126\pi$
$521$			209.742i			0.402575i	−0.979532	+	0.201288i	$0.935487\pi$
$522$		0			0
$523$	−492.016	+	852.197i	−0.940758	+	1.62944i	−0.176727	+	0.984260i	$0.556551\pi$
$523$	−492.016	+	852.197i	−0.940758	+	1.62944i	−0.764031	+	0.645180i	$0.776782\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			498.820i			0.946528i
$528$		0			0
$529$	339.562	−	588.138i	0.641893	−	1.11179i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.0342	−	10.9894i	0.0357114	−	0.0206180i
$534$		0			0
$535$	90.7489			0.169624
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	87.6930i		−	0.162696i
$540$		0			0
$541$	−486.250	+	842.210i	−0.898799	+	1.55677i	−0.0697678	+	0.997563i	$0.522226\pi$
$541$	−486.250	+	842.210i	−0.898799	+	1.55677i	−0.829031	+	0.559202i	$0.811107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	171.991	−	99.2993i	0.315580	−	0.182200i
$546$		0			0
$547$	−1062.63			−1.94265			−0.971326	−	0.237753i	$-0.923589\pi$
$547$	−1062.63			−1.94265			−0.971326	+	0.237753i	$0.923589\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−670.999	+	273.268i	−1.21778	+	0.495950i
$552$		0			0
$553$	109.727			0.198421
$554$		0			0
$555$		0			0
$556$		0			0
$557$	719.310	+	415.294i	1.29140	+	0.745590i	0.978902	−	0.204329i	$-0.0655011\pi$
$557$	719.310	+	415.294i	1.29140	+	0.745590i	0.312497	+	0.949919i	$0.398834\pi$
$558$		0			0
$559$	259.640			0.464471
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−530.247	−	306.138i	−0.941824	−	0.543762i	−0.0512923	−	0.998684i	$-0.516334\pi$
$563$	−530.247	−	306.138i	−0.941824	−	0.543762i	−0.890532	+	0.454921i	$0.849667\pi$
$564$		0			0
$565$	477.710	−	827.418i	0.845505	−	1.46446i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−177.066	+	102.229i	−0.311188	+	0.179665i	−0.647458	−	0.762101i	$-0.724168\pi$
$569$	−177.066	+	102.229i	−0.311188	+	0.179665i	0.336270	+	0.941766i	$0.390835\pi$
$570$		0			0
$571$	−244.282	+	423.109i	−0.427814	+	0.740996i	−0.996679	−	0.0814355i	$-0.974050\pi$
$571$	−244.282	+	423.109i	−0.427814	+	0.740996i	0.568865	+	0.822431i	$0.307383\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	385.528	−	222.585i	0.670484	−	0.387104i
$576$		0			0
$577$	856.457			1.48433			0.742164	−	0.670219i	$-0.233800\pi$
$577$	856.457			1.48433			0.742164	+	0.670219i	$0.233800\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1857.89	−	1072.65i	−3.19775	−	1.84622i
$582$		0			0
$583$	22.9133			0.0393024
$584$		0			0
$585$		0			0
$586$		0			0
$587$	508.197	+	293.408i	0.865753	+	0.499843i	0.865935	−	0.500157i	$-0.166724\pi$
$587$	508.197	+	293.408i	0.865753	+	0.499843i	−0.000181329		1.00000i	$0.500058\pi$
$588$		0			0
$589$	743.077	+	102.332i	1.26159	+	0.173739i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−316.266	−	182.596i	−0.533331	−	0.307919i	0.209041	−	0.977907i	$-0.432966\pi$
$593$	−316.266	−	182.596i	−0.533331	−	0.307919i	−0.742372	+	0.669988i	$0.766299\pi$
$594$		0			0
$595$	518.938	−	898.826i	0.872164	−	1.51063i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	388.322	−	224.198i	0.648283	−	0.374287i	−0.139515	−	0.990220i	$-0.544554\pi$
$599$	388.322	−	224.198i	0.648283	−	0.374287i	0.787798	+	0.615933i	$0.211221\pi$
$600$		0			0
$601$	323.504	+	560.326i	0.538277	+	0.932323i	0.998997	+	0.0447773i	$0.0142578\pi$
$601$	323.504	+	560.326i	0.538277	+	0.932323i	−0.460720	+	0.887545i	$0.652409\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	641.883	+	370.591i	1.06096	+	0.612548i
$606$		0			0
$607$	413.907	−	716.909i	0.681890	−	1.18107i	−0.292513	−	0.956262i	$-0.594491\pi$
$607$	413.907	−	716.909i	0.681890	−	1.18107i	0.974403	−	0.224807i	$-0.0721753\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−692.843	+	400.013i	−1.13395	+	0.654686i
$612$		0			0
$613$	420.668	+	728.618i	0.686244	+	1.18861i	0.973044	+	0.230619i	$0.0740752\pi$
$613$	420.668	+	728.618i	0.686244	+	1.18861i	−0.286800	+	0.957991i	$0.592591\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−870.051	+	502.324i	−1.41013	+	0.814139i	−0.995400	−	0.0958048i	$-0.969458\pi$
$617$	−870.051	+	502.324i	−1.41013	+	0.814139i	−0.414731	+	0.909944i	$0.636124\pi$
$618$		0			0
$619$	9.19386	+	15.9242i	0.0148528	+	0.0257257i	0.873356	−	0.487082i	$-0.161939\pi$
$619$	9.19386	+	15.9242i	0.0148528	+	0.0257257i	−0.858503	+	0.512808i	$0.828605\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	1417.16i		−	2.27473i
$624$		0			0
$625$	−781.155			−1.24985
$626$		0			0
$627$		0			0
$628$		0			0
$629$	262.700	−	151.670i	0.417648	−	0.241129i
$630$		0			0
$631$	511.523	−	885.984i	0.810655	−	1.40410i	−0.101752	−	0.994810i	$-0.532445\pi$
$631$	511.523	−	885.984i	0.810655	−	1.40410i	0.912406	−	0.409285i	$-0.134222\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−64.1038	−	37.0104i	−0.100951	−	0.0582841i
$636$		0			0
$637$	−2844.94			−4.46615
$638$		0			0
$639$		0			0
$640$		0			0
$641$	475.462	+	274.508i	0.741750	+	0.428250i	0.822705	−	0.568468i	$-0.192464\pi$
$641$	475.462	+	274.508i	0.741750	+	0.428250i	−0.0809552	+	0.996718i	$0.525797\pi$
$642$		0			0
$643$	−550.274			−0.855791			−0.427896	−	0.903828i	$-0.640745\pi$
$643$	−550.274			−0.855791			−0.427896	+	0.903828i	$0.640745\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			246.104i			0.380376i	0.981748	+	0.190188i	$0.0609098\pi$
$647$			246.104i			0.380376i	−0.981748	+	0.190188i	$0.939090\pi$
$648$		0			0
$649$	20.9653	+	36.3129i	0.0323039	+	0.0559521i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−70.4998	−	40.7031i	−0.107963	−	0.0623324i	0.445046	−	0.895508i	$-0.353187\pi$
$653$	−70.4998	−	40.7031i	−0.107963	−	0.0623324i	−0.553009	+	0.833175i	$0.686521\pi$
$654$		0			0
$655$	65.1347			0.0994422
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	569.031i		−	0.863477i	−0.901999	−	0.431738i	$-0.857900\pi$
$659$		−	569.031i		−	0.863477i	0.901999	−	0.431738i	$-0.142100\pi$
$660$		0			0
$661$	−91.5566	−	158.581i	−0.138512	−	0.239910i	0.788421	−	0.615136i	$-0.210899\pi$
$661$	−91.5566	−	158.581i	−0.138512	−	0.239910i	−0.926934	+	0.375225i	$0.877565\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1232.49	−	957.437i	−1.85337	−	1.43976i
$666$		0			0
$667$	−662.699	−	1147.83i	−0.993552	−	1.72088i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−58.6253	+	33.8474i	−0.0873701	+	0.0504431i
$672$		0			0
$673$	120.295	+	208.356i	0.178744	+	0.309593i	0.941451	−	0.337151i	$-0.109463\pi$
$673$	120.295	+	208.356i	0.178744	+	0.309593i	−0.762707	+	0.646744i	$0.776130\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−497.193	+	287.055i	−0.734407	+	0.424010i	−0.820032	−	0.572317i	$-0.806045\pi$
$677$	−497.193	+	287.055i	−0.734407	+	0.424010i	0.0856253	+	0.996327i	$0.472711\pi$
$678$		0			0
$679$	686.908			1.01165
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1104.28i			1.61681i	0.588628	+	0.808404i	$0.299668\pi$
$683$			1104.28i			1.61681i	−0.588628	+	0.808404i	$0.700332\pi$
$684$		0			0
$685$	−113.669			−0.165941
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	743.354i		−	1.07889i
$690$		0			0
$691$	−182.004	−	315.241i	−0.263393	−	0.456210i	0.703748	−	0.710449i	$-0.251508\pi$
$691$	−182.004	−	315.241i	−0.263393	−	0.456210i	−0.967141	+	0.254239i	$0.918175\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	677.516	−	391.164i	0.974844	−	0.562826i
$696$		0			0
$697$	6.31862	+	10.9442i	0.00906546	+	0.0157018i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−629.268	+	363.308i	−0.897672	+	0.518271i	−0.876444	−	0.481503i	$-0.840091\pi$
$701$	−629.268	+	363.308i	−0.897672	+	0.518271i	−0.0212278	+	0.999775i	$0.506758\pi$
$702$		0			0
$703$	−172.046	−	422.451i	−0.244731	−	0.600927i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−34.2363	+	19.7664i	−0.0484248	+	0.0279581i
$708$		0			0
$709$	−32.0041			−0.0451398			−0.0225699	−	0.999745i	$-0.507185\pi$
$709$	−32.0041			−0.0451398			−0.0225699	+	0.999745i	$0.507185\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1372.19i			1.92454i
$714$		0			0
$715$	−45.7640	+	79.2655i	−0.0640055	+	0.110861i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−737.472	+	425.780i	−1.02569	+	0.592183i	−0.915747	−	0.401755i	$-0.868400\pi$
$719$	−737.472	+	425.780i	−1.02569	+	0.592183i	−0.109944	+	0.993938i	$0.535067\pi$
$720$		0			0
$721$	−145.953			−0.202432
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	488.383i		−	0.673632i
$726$		0			0
$727$	−286.128	+	495.588i	−0.393574	+	0.681690i	−0.992918	−	0.118802i	$-0.962095\pi$
$727$	−286.128	+	495.588i	−0.393574	+	0.681690i	0.599344	+	0.800491i	$0.295428\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			149.286i			0.204222i
$732$		0			0
$733$	−388.675	+	673.206i	−0.530253	+	0.918425i	0.469124	+	0.883132i	$0.344570\pi$
$733$	−388.675	+	673.206i	−0.530253	+	0.918425i	−0.999377	+	0.0352927i	$0.988764\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−18.5662	−	10.7192i	−0.0251916	−	0.0145444i
$738$		0			0
$739$	186.632	+	323.256i	0.252546	+	0.437423i	0.964226	−	0.265081i	$-0.0853987\pi$
$739$	186.632	+	323.256i	0.252546	+	0.437423i	−0.711680	+	0.702504i	$0.752065\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	648.046i		−	0.872202i	−0.899898	−	0.436101i	$-0.856359\pi$
$743$		−	648.046i		−	0.872202i	0.899898	−	0.436101i	$-0.143641\pi$
$744$		0			0
$745$	−1226.97			−1.64693
$746$		0			0
$747$		0			0
$748$		0			0
$749$	170.747	−	98.5808i	0.227967	−	0.131617i
$750$		0			0
$751$	202.582	+	350.883i	0.269750	+	0.467221i	0.968797	−	0.247855i	$-0.0797255\pi$
$751$	202.582	+	350.883i	0.269750	+	0.467221i	−0.699047	+	0.715076i	$0.746392\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−916.652	+	529.229i	−1.21411	+	0.700966i
$756$		0			0
$757$	473.100	+	819.433i	0.624966	+	1.08247i	0.988547	+	0.150911i	$0.0482207\pi$
$757$	473.100	+	819.433i	0.624966	+	1.08247i	−0.363581	+	0.931563i	$0.618446\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	646.639	+	373.337i	0.849722	+	0.490588i	0.860557	−	0.509354i	$-0.170116\pi$
$761$	646.639	+	373.337i	0.849722	+	0.490588i	−0.0108347	+	0.999941i	$0.503449\pi$
$762$		0			0
$763$	215.738	−	373.670i	0.282750	−	0.489737i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1178.06	−	680.155i	1.53594	−	0.886773i
$768$		0			0
$769$	340.080	+	589.035i	0.442236	+	0.765975i	0.997855	−	0.0654616i	$-0.0208520\pi$
$769$	340.080	+	589.035i	0.442236	+	0.765975i	−0.555619	+	0.831437i	$0.687519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	334.968	+	193.394i	0.433335	+	0.250186i	0.700766	−	0.713391i	$-0.252842\pi$
$773$	334.968	+	193.394i	0.433335	+	0.250186i	−0.267431	+	0.963577i	$0.586175\pi$
$774$		0			0
$775$	−252.813	+	437.885i	−0.326210	+	0.565013i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17.5994	−	7.16748i	0.0225924	−	0.00920087i
$780$		0			0
$781$	40.7420	−	70.5672i	0.0521664	−	0.0903549i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			775.446i			0.987830i
$786$		0			0
$787$	150.489	−	260.655i	0.191219	−	0.331201i	−0.754436	−	0.656374i	$-0.772089\pi$
$787$	150.489	−	260.655i	0.191219	−	0.331201i	0.945654	+	0.325173i	$0.105423\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2075.75i		−	2.62421i
$792$		0			0
$793$	1098.08	+	1901.92i	1.38471	+	2.39839i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−252.290	−	145.660i	−0.316550	−	0.182760i	0.333304	−	0.942820i	$-0.391837\pi$
$797$	−252.290	−	145.660i	−0.316550	−	0.182760i	−0.649854	+	0.760059i	$0.725170\pi$
$798$		0			0
$799$	−229.997	−	398.367i	−0.287856	−	0.498582i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.6641	−	6.15695i	−0.0132804	−	0.00766743i
$804$		0			0
$805$	1427.53	−	2472.56i	1.77333	−	3.07150i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1049.66i		−	1.29748i	−0.761011	−	0.648739i	$-0.775297\pi$
$809$		−	1049.66i		−	1.29748i	0.761011	−	0.648739i	$-0.224703\pi$
$810$		0			0
$811$	208.027	−	360.313i	0.256506	−	0.444282i	−0.708797	−	0.705412i	$-0.750762\pi$
$811$	208.027	−	360.313i	0.256506	−	0.444282i	0.965304	+	0.261130i	$0.0840952\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			799.459i			0.980931i
$816$		0			0
$817$	222.387	+	30.6258i	0.272199	+	0.0374856i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			83.9316i			0.102231i	0.998693	+	0.0511154i	$0.0162776\pi$
$821$			83.9316i			0.102231i	−0.998693	+	0.0511154i	$0.983722\pi$
$822$		0			0
$823$	−382.995	−	663.366i	−0.465364	−	0.806034i	0.533854	−	0.845577i	$-0.320743\pi$
$823$	−382.995	−	663.366i	−0.465364	−	0.806034i	−0.999218	+	0.0395426i	$0.987410\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1372.99	−	792.695i	−1.66020	−	0.958518i	−0.972617	−	0.232415i	$-0.925337\pi$
$827$	−1372.99	−	792.695i	−1.66020	−	0.958518i	−0.687586	−	0.726103i	$-0.741329\pi$
$828$		0			0
$829$	−543.934			−0.656133			−0.328067	−	0.944655i	$-0.606397\pi$
$829$	−543.934			−0.656133			−0.328067	+	0.944655i	$0.606397\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	1635.77i		−	1.96370i
$834$		0			0
$835$	−53.8264	−	93.2301i	−0.0644628	−	0.111653i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	297.919	+	172.003i	0.355088	+	0.205010i	0.666924	−	0.745126i	$-0.267611\pi$
$839$	297.919	+	172.003i	0.355088	+	0.205010i	−0.311836	+	0.950136i	$0.600944\pi$
$840$		0			0
$841$	−613.058			−0.728964
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1671.60	+	965.101i	1.97823	+	1.14213i
$846$		0			0
$847$	1610.30			1.90118
$848$		0			0
$849$		0			0
$850$		0			0
$851$	722.657	−	417.226i	0.849185	−	0.490277i
$852$		0			0
$853$	291.512	−	504.914i	0.341749	−	0.591927i	−0.643008	−	0.765859i	$-0.722314\pi$
$853$	291.512	−	504.914i	0.341749	−	0.591927i	0.984758	+	0.173932i	$0.0556473\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1216.36	−	702.265i	−1.41932	−	0.819446i	−0.423083	−	0.906091i	$-0.639052\pi$
$857$	−1216.36	−	702.265i	−1.41932	−	0.819446i	−0.996239	+	0.0866446i	$0.972386\pi$
$858$		0			0
$859$	543.718	+	941.748i	0.632967	+	1.09633i	0.986942	+	0.161076i	$0.0514963\pi$
$859$	543.718	+	941.748i	0.632967	+	1.09633i	−0.353975	+	0.935255i	$0.615170\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1009.86i			1.17018i	0.810970	+	0.585088i	$0.198940\pi$
$863$			1009.86i			1.17018i	−0.810970	+	0.585088i	$0.801060\pi$
$864$		0			0
$865$	−798.806	+	1383.57i	−0.923475	+	1.59951i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	5.56378i		−	0.00640251i
$870$		0			0
$871$	−347.753	+	602.325i	−0.399257	+	0.691533i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−867.318	+	500.746i	−0.991221	+	0.572282i
$876$		0			0
$877$	−606.932			−0.692055			−0.346028	−	0.938224i	$-0.612470\pi$
$877$	−606.932			−0.692055			−0.346028	+	0.938224i	$0.612470\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1319.04i		−	1.49721i	−0.663015	−	0.748606i	$-0.730723\pi$
$881$		−	1319.04i		−	1.49721i	0.663015	−	0.748606i	$-0.269277\pi$
$882$		0			0
$883$	632.552	−	1095.61i	0.716367	−	1.24078i	−0.246063	−	0.969254i	$-0.579137\pi$
$883$	632.552	−	1095.61i	0.716367	−	1.24078i	0.962430	−	0.271530i	$-0.0875295\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−420.192	+	242.598i	−0.473723	+	0.273504i	−0.717797	−	0.696252i	$-0.754849\pi$
$887$	−420.192	+	242.598i	−0.473723	+	0.273504i	0.244074	+	0.969757i	$0.421516\pi$
$888$		0			0
$889$	−160.818			−0.180898
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−640.618	+	260.896i	−0.717378	+	0.292156i
$894$		0			0
$895$	743.376			0.830588
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1303.71	+	752.698i	1.45018	+	0.837261i
$900$		0			0
$901$	427.410			0.474372
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1173.08	+	677.280i	1.29622	+	0.748375i
$906$		0			0
$907$	−221.770	+	384.116i	−0.244509	+	0.423502i	−0.961993	−	0.273072i	$-0.911960\pi$
$907$	−221.770	+	384.116i	−0.244509	+	0.423502i	0.717484	+	0.696575i	$0.245293\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	162.124	−	93.6021i	0.177962	−	0.102747i	−0.408373	−	0.912815i	$-0.633904\pi$
$911$	162.124	−	93.6021i	0.177962	−	0.102747i	0.586335	+	0.810069i	$0.300570\pi$
$912$		0			0
$913$	−54.3897	+	94.2058i	−0.0595725	+	0.103183i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	122.553	−	70.7560i	0.133646	−	0.0771603i
$918$		0			0
$919$	−829.412			−0.902516			−0.451258	−	0.892394i	$-0.649025\pi$
$919$	−829.412			−0.902516			−0.451258	+	0.892394i	$0.649025\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2289.34	−	1321.75i	−2.48032	−	1.43202i
$924$		0			0
$925$	307.479			0.332410
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.9987	−	17.8971i	−0.0333678	−	0.0192649i	0.483223	−	0.875497i	$-0.339466\pi$
$929$	−30.9987	−	17.8971i	−0.0333678	−	0.0192649i	−0.516591	+	0.856232i	$0.672799\pi$
$930$		0			0
$931$	−2436.75	−	335.574i	−2.61735	−	0.360445i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−45.5757	−	26.3131i	−0.0487440	−	0.0281424i
$936$		0			0
$937$	−678.876	+	1175.85i	−0.724521	+	1.25491i	0.234650	+	0.972080i	$0.424606\pi$
$937$	−678.876	+	1175.85i	−0.724521	+	1.25491i	−0.959171	+	0.282827i	$0.908728\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1013.23	−	584.991i	1.07676	−	0.621670i	0.146742	−	0.989175i	$-0.453121\pi$
$941$	1013.23	−	584.991i	1.07676	−	0.621670i	0.930022	+	0.367505i	$0.119788\pi$
$942$		0			0
$943$	17.3818	+	30.1061i	0.0184324	+	0.0319259i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	897.336	+	518.077i	0.947557	+	0.547072i	0.892321	−	0.451401i	$-0.149076\pi$
$947$	897.336	+	518.077i	0.947557	+	0.547072i	0.0552355	+	0.998473i	$0.482409\pi$
$948$		0			0
$949$	−199.744	+	345.966i	−0.210478	+	0.364559i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.9587	+	12.6779i	−0.0230416	+	0.0133031i	−0.511477	−	0.859297i	$-0.670901\pi$
$953$	−21.9587	+	12.6779i	−0.0230416	+	0.0133031i	0.488435	+	0.872600i	$0.337568\pi$
$954$		0			0
$955$	−311.583	−	539.677i	−0.326265	−	0.565107i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−213.873	+	123.479i	−0.223016	+	0.128758i
$960$		0			0
$961$	−298.773	−	517.490i	−0.310898	−	0.538491i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1966.08i			2.03738i
$966$		0			0
$967$	−1194.01			−1.23475			−0.617376	−	0.786668i	$-0.711804\pi$
$967$	−1194.01			−1.23475			−0.617376	+	0.786668i	$0.711804\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	903.364	−	521.557i	0.930344	−	0.537134i	0.0434236	−	0.999057i	$-0.486173\pi$
$971$	903.364	−	521.557i	0.930344	−	0.537134i	0.886920	+	0.461922i	$0.152840\pi$
$972$		0			0
$973$	849.846	−	1471.98i	0.873429	−	1.51282i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	22.5462	+	13.0171i	0.0230770	+	0.0133235i	0.511494	−	0.859287i	$-0.329092\pi$
$977$	22.5462	+	13.0171i	0.0230770	+	0.0133235i	−0.488417	+	0.872610i	$0.662426\pi$
$978$		0			0
$979$	−71.8579			−0.0733993
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−283.804	−	163.855i	−0.288713	−	0.166688i	0.348649	−	0.937254i	$-0.386641\pi$
$983$	−283.804	−	163.855i	−0.288713	−	0.166688i	−0.637361	+	0.770565i	$0.719974\pi$
$984$		0			0
$985$	2229.98			2.26394
$986$		0			0
$987$		0			0
$988$		0			0
$989$			410.668i			0.415236i
$990$		0			0
$991$	−559.358	−	968.836i	−0.564438	−	0.977635i	−0.997102	−	0.0760795i	$-0.975760\pi$
$991$	−559.358	−	968.836i	−0.564438	−	0.977635i	0.432664	−	0.901555i	$-0.357574\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	671.179	+	387.505i	0.674551	+	0.389452i
$996$		0			0
$997$	−817.886			−0.820347			−0.410174	−	0.912007i	$-0.634532\pi$
$997$	−817.886			−0.820347			−0.410174	+	0.912007i	$0.634532\pi$
$998$		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.881.9		80
3.2	odd	2		684.3.m.a.653.35	yes	80
9.2	odd	6		2052.3.be.a.197.9		80
9.7	even	3		684.3.be.a.425.9	yes	80
19.11	even	3		2052.3.be.a.125.9		80
57.11	odd	6		684.3.be.a.581.9	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.32		80
171.106	even	3		684.3.m.a.353.35	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.35	✓	80	171.106	even	3
684.3.m.a.653.35	yes	80	3.2	odd	2
684.3.be.a.425.9	yes	80	9.7	even	3
684.3.be.a.581.9	yes	80	57.11	odd	6
2052.3.m.a.881.9		80	1.1	even	1	trivial
2052.3.m.a.1493.32		80	171.11	odd	6	inner
2052.3.be.a.125.9		80	19.11	even	3
2052.3.be.a.197.9		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)

\(f(q)\)	\(=\)	\(q-6.14879i q^{5} +(-6.67945 - 11.5692i) q^{7} +O(q^{10})\)	\(q-6.14879i q^{5} +(-6.67945 - 11.5692i) q^{7} +(-0.586623 + 0.338687i) q^{11} +(10.9877 + 19.0312i) q^{13} +(-10.9425 + 6.31763i) q^{17} +(7.16635 + 17.5967i) q^{19} +(-30.1014 + 17.3790i) q^{23} -12.8077 q^{25} +38.1321i q^{29} +(19.7392 - 34.1893i) q^{31} +(-71.1363 + 41.0706i) q^{35} -24.0074 q^{37} -1.00016i q^{41} +(5.90752 - 10.2321i) q^{43} +36.4056i q^{47} +(-64.7302 + 112.116i) q^{49} +(-29.2948 - 16.9134i) q^{53} +(2.08251 + 3.60702i) q^{55} -61.9016i q^{59} +99.9371 q^{61} +(117.019 - 67.5609i) q^{65} +(15.8247 + 27.4092i) q^{67} +(-104.178 + 60.1470i) q^{71} +(9.08945 + 15.7434i) q^{73} +(7.83664 + 4.52448i) q^{77} +(-4.10688 + 7.11332i) q^{79} +(139.075 - 80.2951i) q^{83} +(38.8458 + 67.2829i) q^{85} +(91.8708 + 53.0416i) q^{89} +(146.783 - 254.236i) q^{91} +(108.198 - 44.0644i) q^{95} +(-25.7098 + 44.5306i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 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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