LMFDB - Embedded newform 4732.2.g.b.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(337,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4732.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.g (of order $2$, degree $1$, 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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4732.337
Dual form			4732.2.g.b.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} +2.00000i q^{11} +3.00000 q^{17} -6.00000i q^{19} +4.00000 q^{23} +4.00000 q^{25} -7.00000 q^{29} +4.00000i q^{31} +1.00000 q^{35} -9.00000i q^{37} +9.00000i q^{41} -10.0000 q^{43} +3.00000i q^{45} +2.00000i q^{47} -1.00000 q^{49} +9.00000 q^{53} +2.00000 q^{55} -14.0000i q^{59} -5.00000 q^{61} -3.00000i q^{63} -8.00000i q^{67} +10.0000i q^{71} +7.00000i q^{73} -2.00000 q^{77} +2.00000 q^{79} +9.00000 q^{81} -6.00000i q^{83} -3.00000i q^{85} -6.00000i q^{89} -6.00000 q^{95} +2.00000i q^{97} -6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9} + 6 q^{17} + 8 q^{23} + 8 q^{25} - 14 q^{29} + 2 q^{35} - 20 q^{43} - 2 q^{49} + 18 q^{53} + 4 q^{55} - 10 q^{61} - 4 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{95}+O(q^{100}) $ 2 * q - 6 * q^9 + 6 * q^17 + 8 * q^23 + 8 * q^25 - 14 * q^29 + 2 * q^35 - 20 * q^43 - 2 * q^49 + 18 * q^53 + 4 * q^55 - 10 * q^61 - 4 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times$.

$n$	$2367$	$2705$	$4565$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i	−0.974679	−	0.223607i	$-0.928217\pi$
$5$	− 1.00000i	− 0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	2.00000i	0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$	2.00000i	0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.00000	−1.29987	−0.649934	−	0.759991i	$-0.725203\pi$
$29$	−7.00000	−1.29987	−0.649934	+	0.759991i	$0.725203\pi$
$30$	0	0
$31$	4.00000i	0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$	4.00000i	0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	− 9.00000i	− 1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$	− 9.00000i	− 1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.00000i	1.40556i	0.711405	+	0.702782i	$0.248059\pi$
$41$	9.00000i	1.40556i	−0.711405	+	0.702782i	$0.751941\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	3.00000i	0.447214i
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 14.0000i	− 1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$	− 14.0000i	− 1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.0000i	1.18678i	0.804914	+	0.593391i	$0.202211\pi$
$71$	10.0000i	1.18678i	−0.804914	+	0.593391i	$0.797789\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	− 3.00000i	− 0.325396i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.00000i	− 0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$	− 6.00000i	− 0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	17.0000	1.69156	0.845782	−	0.533529i	$-0.179135\pi$
$101$	17.0000	1.69156	0.845782	+	0.533529i	$0.179135\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	− 14.0000i	− 1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$	− 14.0000i	− 1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	11.0000	1.03479	0.517396	−	0.855746i	$-0.326901\pi$
$113$	11.0000	1.03479	0.517396	+	0.855746i	$0.326901\pi$
$114$	0	0
$115$	− 4.00000i	− 0.373002i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.00000i	0.275010i
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.0000i	− 1.28154i	−0.767734	−	0.640768i	$-0.778616\pi$
$137$	− 15.0000i	− 1.28154i	0.767734	−	0.640768i	$-0.221384\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	7.00000i	0.581318i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.00000i	− 0.245770i	−0.992421	−	0.122885i	$-0.960785\pi$
$149$	− 3.00000i	− 0.245770i	0.992421	−	0.122885i	$-0.0392146\pi$
$150$	0	0
$151$	− 18.0000i	− 1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$	− 18.0000i	− 1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$	0	0
$153$	−9.00000	−0.727607
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.00000i	0.315244i
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	18.0000i	1.37649i
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	4.00000i	0.302372i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−15.0000	−1.11494	−0.557471	−	0.830197i	$-0.688228\pi$
$181$	−15.0000	−1.11494	−0.557471	+	0.830197i	$0.688228\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.00000	−0.661693
$186$	0	0
$187$	6.00000i	0.438763i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	− 11.0000i	− 0.791797i	−0.918294	−	0.395899i	$-0.870433\pi$
$193$	− 11.0000i	− 0.791797i	0.918294	−	0.395899i	$-0.129567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.00000i	− 0.491304i
$204$	0	0
$205$	9.00000	0.628587
$206$	0	0
$207$	−12.0000	−0.834058
$208$	0	0
$209$	12.0000	0.830057
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.0000i	0.681994i
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.00000i	0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$	6.00000i	0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$	0	0
$225$	−12.0000	−0.800000
$226$	0	0
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	6.00000i	0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$	6.00000i	0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	2.00000	0.130466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 6.00000i	− 0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$	− 6.00000i	− 0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$	0	0
$241$	3.00000i	0.193247i	0.995321	+	0.0966235i	$0.0308043\pi$
$241$	3.00000i	0.193247i	−0.995321	+	0.0966235i	$0.969196\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000i	0.0638877i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−30.0000	−1.89358	−0.946792	−	0.321847i	$-0.895696\pi$
$251$	−30.0000	−1.89358	−0.946792	+	0.321847i	$0.895696\pi$
$252$	0	0
$253$	8.00000i	0.502956i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	0	0
$261$	21.0000	1.29987
$262$	0	0
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	− 9.00000i	− 0.552866i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	− 10.0000i	− 0.607457i	−0.952759	−	0.303728i	$-0.901768\pi$
$271$	− 10.0000i	− 0.607457i	0.952759	−	0.303728i	$-0.0982315\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8.00000i	0.482418i
$276$	0	0
$277$	3.00000	0.180253	0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000	0.180253	0.0901263	+	0.995930i	$0.471273\pi$
$278$	0	0
$279$	− 12.0000i	− 0.718421i
$280$	0	0
$281$	− 7.00000i	− 0.417585i	−0.977960	−	0.208792i	$-0.933047\pi$
$281$	− 7.00000i	− 0.417585i	0.977960	−	0.208792i	$-0.0669533\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 31.0000i	− 1.81104i	−0.424304	−	0.905520i	$-0.639481\pi$
$293$	− 31.0000i	− 1.81104i	0.424304	−	0.905520i	$-0.360519\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 10.0000i	− 0.576390i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.00000i	0.286299i
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	0	0
$315$	−3.00000	−0.169031
$316$	0	0
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	0	0
$319$	− 14.0000i	− 0.783850i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 18.0000i	− 1.00155i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2.00000	−0.110264
$330$	0	0
$331$	14.0000i	0.769510i	0.923019	+	0.384755i	$0.125714\pi$
$331$	14.0000i	0.769510i	−0.923019	+	0.384755i	$0.874286\pi$
$332$	0	0
$333$	27.0000i	1.47959i
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	25.0000	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$337$	25.0000	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	14.0000i	0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$	14.0000i	0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.00000i	0.0532246i	0.999646	+	0.0266123i	$0.00847196\pi$
$353$	1.00000i	0.0532246i	−0.999646	+	0.0266123i	$0.991528\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.00000	0.366397
$366$	0	0
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	− 27.0000i	− 1.40556i
$370$	0	0
$371$	9.00000i	0.467257i
$372$	0	0
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	16.0000i	0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$	16.0000i	0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.0000i	1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$	30.0000i	1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$	0	0
$385$	2.00000i	0.101929i
$386$	0	0
$387$	30.0000	1.52499
$388$	0	0
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 2.00000i	− 0.100631i
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 15.0000i	− 0.749064i	−0.927214	−	0.374532i	$-0.877803\pi$
$401$	− 15.0000i	− 0.749064i	0.927214	−	0.374532i	$-0.122197\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 9.00000i	− 0.447214i
$406$	0	0
$407$	18.0000	0.892227
$408$	0	0
$409$	1.00000i	0.0494468i	0.999694	+	0.0247234i	$0.00787051\pi$
$409$	1.00000i	0.0494468i	−0.999694	+	0.0247234i	$0.992129\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	14.0000	0.688895
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	25.0000i	1.21843i	0.793007	+	0.609213i	$0.208514\pi$
$421$	25.0000i	1.21843i	−0.793007	+	0.609213i	$0.791486\pi$
$422$	0	0
$423$	− 6.00000i	− 0.291730i
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	− 5.00000i	− 0.241967i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 24.0000i	− 1.14808i
$438$	0	0
$439$	34.0000	1.62273	0.811366	−	0.584539i	$-0.198725\pi$
$439$	34.0000	1.62273	0.811366	+	0.584539i	$0.198725\pi$
$440$	0	0
$441$	3.00000	0.142857
$442$	0	0
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.0000i	1.22702i	0.789689	+	0.613508i	$0.210242\pi$
$449$	26.0000i	1.22702i	−0.789689	+	0.613508i	$0.789758\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 21.0000i	− 0.978068i	−0.872265	−	0.489034i	$-0.837349\pi$
$461$	− 21.0000i	− 0.978068i	0.872265	−	0.489034i	$-0.162651\pi$
$462$	0	0
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.0000	−0.647843	−0.323921	−	0.946084i	$-0.605001\pi$
$467$	−14.0000	−0.647843	−0.323921	+	0.946084i	$0.605001\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 20.0000i	− 0.919601i
$474$	0	0
$475$	− 24.0000i	− 1.10120i
$476$	0	0
$477$	−27.0000	−1.23625
$478$	0	0
$479$	− 12.0000i	− 0.548294i	−0.961688	−	0.274147i	$-0.911605\pi$
$479$	− 12.0000i	− 0.548294i	0.961688	−	0.274147i	$-0.0883955\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	0	0
$493$	−21.0000	−0.945792
$494$	0	0
$495$	−6.00000	−0.269680
$496$	0	0
$497$	−10.0000	−0.448561
$498$	0	0
$499$	40.0000i	1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$	40.0000i	1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.00000	−0.0891756	−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000	−0.0891756	−0.0445878	+	0.999005i	$0.514197\pi$
$504$	0	0
$505$	− 17.0000i	− 0.756490i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 29.0000i	− 1.28540i	−0.766117	−	0.642701i	$-0.777814\pi$
$509$	− 29.0000i	− 1.28540i	0.766117	−	0.642701i	$-0.222186\pi$
$510$	0	0
$511$	−7.00000	−0.309662
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14.0000i	− 0.616914i
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.0000	0.920027	0.460013	−	0.887912i	$-0.347845\pi$
$521$	21.0000	0.920027	0.460013	+	0.887912i	$0.347845\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	42.0000i	1.82264i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 2.00000i	− 0.0864675i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.00000i	− 0.0861461i
$540$	0	0
$541$	7.00000i	0.300954i	0.988614	+	0.150477i	$0.0480809\pi$
$541$	7.00000i	0.300954i	−0.988614	+	0.150477i	$0.951919\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	15.0000	0.640184
$550$	0	0
$551$	42.0000i	1.78926i
$552$	0	0
$553$	2.00000i	0.0850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.00000i	0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$	3.00000i	0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10.0000	0.421450	0.210725	−	0.977545i	$-0.432418\pi$
$563$	10.0000	0.421450	0.210725	+	0.977545i	$0.432418\pi$
$564$	0	0
$565$	− 11.0000i	− 0.462773i
$566$	0	0
$567$	9.00000i	0.377964i
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	16.0000	0.667246
$576$	0	0
$577$	− 27.0000i	− 1.12402i	−0.827129	−	0.562012i	$-0.810027\pi$
$577$	− 27.0000i	− 1.12402i	0.827129	−	0.562012i	$-0.189973\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	18.0000i	0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.0000i	1.81607i	0.418890	+	0.908037i	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 29.0000i	− 1.19089i	−0.803397	−	0.595444i	$-0.796976\pi$
$593$	− 29.0000i	− 1.19089i	0.803397	−	0.595444i	$-0.203024\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	0	0
$597$	0	0
$598$	0	0
$599$	34.0000	1.38920	0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000	1.38920	0.694601	+	0.719395i	$0.255581\pi$
$600$	0	0
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	0	0
$603$	24.0000i	0.977356i
$604$	0	0
$605$	− 7.00000i	− 0.284590i
$606$	0	0
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	17.0000i	0.686624i	0.939222	+	0.343312i	$0.111549\pi$
$613$	17.0000i	0.686624i	−0.939222	+	0.343312i	$0.888451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 41.0000i	− 1.65060i	−0.564696	−	0.825299i	$-0.691007\pi$
$617$	− 41.0000i	− 1.65060i	0.564696	−	0.825299i	$-0.308993\pi$
$618$	0	0
$619$	2.00000i	0.0803868i	0.999192	+	0.0401934i	$0.0127974\pi$
$619$	2.00000i	0.0803868i	−0.999192	+	0.0401934i	$0.987203\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 27.0000i	− 1.07656i
$630$	0	0
$631$	32.0000i	1.27390i	0.770905	+	0.636950i	$0.219804\pi$
$631$	32.0000i	1.27390i	−0.770905	+	0.636950i	$0.780196\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 14.0000i	− 0.555573i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 30.0000i	− 1.18678i
$640$	0	0
$641$	41.0000	1.61940	0.809701	−	0.586842i	$-0.199629\pi$
$641$	41.0000	1.61940	0.809701	+	0.586842i	$0.199629\pi$
$642$	0	0
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−46.0000	−1.80845	−0.904223	−	0.427060i	$-0.859549\pi$
$647$	−46.0000	−1.80845	−0.904223	+	0.427060i	$0.859549\pi$
$648$	0	0
$649$	28.0000	1.09910
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	− 14.0000i	− 0.547025i
$656$	0	0
$657$	− 21.0000i	− 0.819288i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	13.0000i	0.505641i	0.967513	+	0.252821i	$0.0813583\pi$
$661$	13.0000i	0.505641i	−0.967513	+	0.252821i	$0.918642\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 6.00000i	− 0.232670i
$666$	0	0
$667$	−28.0000	−1.08416
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 10.0000i	− 0.386046i
$672$	0	0
$673$	41.0000	1.58043	0.790217	−	0.612827i	$-0.209968\pi$
$673$	41.0000	1.58043	0.790217	+	0.612827i	$0.209968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	−2.00000	−0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.00000i	0.0765279i	0.999268	+	0.0382639i	$0.0121828\pi$
$683$	2.00000i	0.0765279i	−0.999268	+	0.0382639i	$0.987817\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 20.0000i	− 0.760836i	−0.924815	−	0.380418i	$-0.875780\pi$
$691$	− 20.0000i	− 0.760836i	0.924815	−	0.380418i	$-0.124220\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	14.0000i	0.531050i
$696$	0	0
$697$	27.0000i	1.02270i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−54.0000	−2.03665
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.0000i	0.639351i
$708$	0	0
$709$	43.0000i	1.61490i	0.589937	+	0.807449i	$0.299153\pi$
$709$	43.0000i	1.61490i	−0.589937	+	0.807449i	$0.700847\pi$
$710$	0	0
$711$	−6.00000	−0.225018
$712$	0	0
$713$	16.0000i	0.599205i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−14.0000	−0.522112	−0.261056	−	0.965324i	$-0.584071\pi$
$719$	−14.0000	−0.522112	−0.261056	+	0.965324i	$0.584071\pi$
$720$	0	0
$721$	14.0000i	0.521387i
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−28.0000	−1.03989
$726$	0	0
$727$	−52.0000	−1.92857	−0.964287	−	0.264861i	$-0.914674\pi$
$727$	−52.0000	−1.92857	−0.964287	+	0.264861i	$0.914674\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−30.0000	−1.10959
$732$	0	0
$733$	− 1.00000i	− 0.0369358i	−0.999829	−	0.0184679i	$-0.994121\pi$
$733$	− 1.00000i	− 0.0369358i	0.999829	−	0.0184679i	$-0.00587886\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	36.0000i	1.32428i	0.749380	+	0.662141i	$0.230352\pi$
$739$	36.0000i	1.32428i	−0.749380	+	0.662141i	$0.769648\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0000i	0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$	16.0000i	0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	0	0
$747$	18.0000i	0.658586i
$748$	0	0
$749$	2.00000i	0.0730784i
$750$	0	0
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.0000	−0.655087
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.00000i	0.217500i	0.994069	+	0.108750i	$0.0346848\pi$
$761$	6.00000i	0.217500i	−0.994069	+	0.108750i	$0.965315\pi$
$762$	0	0
$763$	14.0000	0.506834
$764$	0	0
$765$	9.00000i	0.325396i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.0000i	0.649097i	0.945869	+	0.324548i	$0.105212\pi$
$769$	18.0000i	0.649097i	−0.945869	+	0.324548i	$0.894788\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	0	0
$775$	16.0000i	0.574737i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	54.0000	1.93475
$780$	0	0
$781$	−20.0000	−0.715656
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 11.0000i	− 0.392607i
$786$	0	0
$787$	52.0000i	1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$	52.0000i	1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.0000i	0.391115i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	0	0
$799$	6.00000i	0.212265i
$800$	0	0
$801$	18.0000i	0.635999i
$802$	0	0
$803$	−14.0000	−0.494049
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	0	0
$811$	− 40.0000i	− 1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$	− 40.0000i	− 1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	60.0000i	2.09913i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.0000i	− 0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$	− 10.0000i	− 0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	0	0
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.00000i	0.208640i	0.994544	+	0.104320i	$0.0332667\pi$
$827$	6.00000i	0.208640i	−0.994544	+	0.104320i	$0.966733\pi$
$828$	0	0
$829$	−55.0000	−1.91023	−0.955114	−	0.296237i	$-0.904268\pi$
$829$	−55.0000	−1.91023	−0.955114	+	0.296237i	$0.904268\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−18.0000	−0.622916
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 26.0000i	− 0.897620i	−0.893627	−	0.448810i	$-0.851848\pi$
$839$	− 26.0000i	− 0.897620i	0.893627	−	0.448810i	$-0.148152\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.00000i	0.240523i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 36.0000i	− 1.23406i
$852$	0	0
$853$	5.00000i	0.171197i	0.996330	+	0.0855984i	$0.0272802\pi$
$853$	5.00000i	0.171197i	−0.996330	+	0.0855984i	$0.972720\pi$
$854$	0	0
$855$	18.0000	0.615587
$856$	0	0
$857$	−37.0000	−1.26390	−0.631948	−	0.775011i	$-0.717744\pi$
$857$	−37.0000	−1.26390	−0.631948	+	0.775011i	$0.717744\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.0000i	0.340404i	0.985409	+	0.170202i	$0.0544420\pi$
$863$	10.0000i	0.340404i	−0.985409	+	0.170202i	$0.945558\pi$
$864$	0	0
$865$	2.00000i	0.0680020i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.00000i	0.135691i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 6.00000i	− 0.203069i
$874$	0	0
$875$	9.00000	0.304256
$876$	0	0
$877$	21.0000i	0.709120i	0.935033	+	0.354560i	$0.115369\pi$
$877$	21.0000i	0.709120i	−0.935033	+	0.354560i	$0.884631\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−57.0000	−1.92038	−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000	−1.92038	−0.960189	+	0.279350i	$0.909881\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.0000	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$887$	16.0000	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$888$	0	0
$889$	14.0000i	0.469545i
$890$	0	0
$891$	18.0000i	0.603023i
$892$	0	0
$893$	12.0000	0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 28.0000i	− 0.933852i
$900$	0	0
$901$	27.0000	0.899500
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.0000i	0.498617i
$906$	0	0
$907$	18.0000	0.597680	0.298840	−	0.954303i	$-0.403400\pi$
$907$	18.0000	0.597680	0.298840	+	0.954303i	$0.403400\pi$
$908$	0	0
$909$	−51.0000	−1.69156
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.0000i	0.462321i
$918$	0	0
$919$	−18.0000	−0.593765	−0.296883	−	0.954914i	$-0.595947\pi$
$919$	−18.0000	−0.593765	−0.296883	+	0.954914i	$0.595947\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 36.0000i	− 1.18367i
$926$	0	0
$927$	−42.0000	−1.37946
$928$	0	0
$929$	− 27.0000i	− 0.885841i	−0.896561	−	0.442921i	$-0.853942\pi$
$929$	− 27.0000i	− 0.885841i	0.896561	−	0.442921i	$-0.146058\pi$
$930$	0	0
$931$	6.00000i	0.196642i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.00000	0.196221
$936$	0	0
$937$	33.0000	1.07806	0.539032	−	0.842286i	$-0.318790\pi$
$937$	33.0000	1.07806	0.539032	+	0.842286i	$0.318790\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.0000i	0.325991i	0.986627	+	0.162995i	$0.0521156\pi$
$941$	10.0000i	0.325991i	−0.986627	+	0.162995i	$0.947884\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 60.0000i	− 1.94974i	−0.222779	−	0.974869i	$-0.571513\pi$
$947$	− 60.0000i	− 1.94974i	0.222779	−	0.974869i	$-0.428487\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.0000	0.484375
$960$	0	0
$961$	15.0000	0.483871
$962$	0	0
$963$	−6.00000	−0.193347
$964$	0	0
$965$	−11.0000	−0.354103
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	0	0
$973$	− 14.0000i	− 0.448819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.0000i	− 1.05576i	−0.849318	−	0.527882i	$-0.822986\pi$
$977$	− 33.0000i	− 1.05576i	0.849318	−	0.527882i	$-0.177014\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	42.0000i	1.34096i
$982$	0	0
$983$	− 2.00000i	− 0.0637901i	−0.999491	−	0.0318950i	$-0.989846\pi$
$983$	− 2.00000i	− 0.0637901i	0.999491	−	0.0318950i	$-0.0101542\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−40.0000	−1.27193
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14.0000i	0.443830i
$996$	0	0
$997$	59.0000	1.86855	0.934274	−	0.356555i	$-0.116049\pi$
$997$	59.0000	1.86855	0.934274	+	0.356555i	$0.116049\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.g.b.337.1		2
13.2	odd	12		364.2.k.a.113.1	yes	2
13.5	odd	4		4732.2.a.c.1.1		1
13.6	odd	12		364.2.k.a.29.1	✓	2
13.8	odd	4		4732.2.a.d.1.1		1
13.12	even	2	inner	4732.2.g.b.337.2		2
39.2	even	12		3276.2.z.c.3025.1		2
39.32	even	12		3276.2.z.c.757.1		2
52.15	even	12		1456.2.s.c.113.1		2
52.19	even	12		1456.2.s.c.1121.1		2
91.2	odd	12		2548.2.i.e.165.1		2
91.6	even	12		2548.2.k.c.393.1		2
91.19	even	12		2548.2.l.d.1537.1		2
91.32	odd	12		2548.2.i.e.1745.1		2
91.41	even	12		2548.2.k.c.1569.1		2
91.45	even	12		2548.2.i.d.1745.1		2
91.54	even	12		2548.2.i.d.165.1		2
91.58	odd	12		2548.2.l.e.1537.1		2
91.67	odd	12		2548.2.l.e.373.1		2
91.80	even	12		2548.2.l.d.373.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.k.a.29.1	✓	2	13.6	odd	12
364.2.k.a.113.1	yes	2	13.2	odd	12
1456.2.s.c.113.1		2	52.15	even	12
1456.2.s.c.1121.1		2	52.19	even	12
2548.2.i.d.165.1		2	91.54	even	12
2548.2.i.d.1745.1		2	91.45	even	12
2548.2.i.e.165.1		2	91.2	odd	12
2548.2.i.e.1745.1		2	91.32	odd	12
2548.2.k.c.393.1		2	91.6	even	12
2548.2.k.c.1569.1		2	91.41	even	12
2548.2.l.d.373.1		2	91.80	even	12
2548.2.l.d.1537.1		2	91.19	even	12
2548.2.l.e.373.1		2	91.67	odd	12
2548.2.l.e.1537.1		2	91.58	odd	12
3276.2.z.c.757.1		2	39.32	even	12
3276.2.z.c.3025.1		2	39.2	even	12
4732.2.a.c.1.1		1	13.5	odd	4
4732.2.a.d.1.1		1	13.8	odd	4
4732.2.g.b.337.1		2	1.1	even	1	trivial
4732.2.g.b.337.2		2	13.12	even	2	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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +1.00000i q^{7} -3.00000 q^{9} +2.00000i q^{11} +3.00000 q^{17} -6.00000i q^{19} +4.00000 q^{23} +4.00000 q^{25} -7.00000 q^{29} +4.00000i q^{31} +1.00000 q^{35} -9.00000i q^{37} +9.00000i q^{41} -10.0000 q^{43} +3.00000i q^{45} +2.00000i q^{47} -1.00000 q^{49} +9.00000 q^{53} +2.00000 q^{55} -14.0000i q^{59} -5.00000 q^{61} -3.00000i q^{63} -8.00000i q^{67} +10.0000i q^{71} +7.00000i q^{73} -2.00000 q^{77} +2.00000 q^{79} +9.00000 q^{81} -6.00000i q^{83} -3.00000i q^{85} -6.00000i q^{89} -6.00000 q^{95} +2.00000i q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} + 6 q^{17} + 8 q^{23} + 8 q^{25} - 14 q^{29} + 2 q^{35} - 20 q^{43} - 2 q^{49} + 18 q^{53} + 4 q^{55} - 10 q^{61} - 4 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{95}+O(q^{100}) \) 2 * q - 6 * q^9 + 6 * q^17 + 8 * q^23 + 8 * q^25 - 14 * q^29 + 2 * q^35 - 20 * q^43 - 2 * q^49 + 18 * q^53 + 4 * q^55 - 10 * q^61 - 4 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^95

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