LMFDB - Embedded newform 2704.2.f.a.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2704,2,Mod(337,2704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2704, 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("2704.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.f (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:	$21.5915487066$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2704.337
Dual form			2704.2.f.a.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -2.00000i q^{5} -1.00000i q^{7} +6.00000 q^{9} +5.00000i q^{11} +6.00000i q^{15} -3.00000 q^{17} -3.00000i q^{19} +3.00000i q^{21} -1.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -1.00000 q^{29} -8.00000i q^{31} -15.0000i q^{33} -2.00000 q^{35} +3.00000i q^{37} -3.00000i q^{41} +1.00000 q^{43} -12.0000i q^{45} -4.00000i q^{47} +6.00000 q^{49} +9.00000 q^{51} -6.00000 q^{53} +10.0000 q^{55} +9.00000i q^{57} -5.00000i q^{59} -5.00000 q^{61} -6.00000i q^{63} +7.00000i q^{67} +3.00000 q^{69} -11.0000i q^{71} +14.0000i q^{73} -3.00000 q^{75} +5.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +12.0000i q^{83} +6.00000i q^{85} +3.00000 q^{87} -9.00000i q^{89} +24.0000i q^{93} -6.00000 q^{95} +1.00000i q^{97} +30.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 12 q^{9} - 6 q^{17} - 2 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 4 q^{35} + 2 q^{43} + 12 q^{49} + 18 q^{51} - 12 q^{53} + 20 q^{55} - 10 q^{61} + 6 q^{69} - 6 q^{75} + 10 q^{77} + 8 q^{79}+ \cdots - 12 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 + 12 * q^9 - 6 * q^17 - 2 * q^23 + 2 * q^25 - 18 * q^27 - 2 * q^29 - 4 * q^35 + 2 * q^43 + 12 * q^49 + 18 * q^51 - 12 * q^53 + 20 * q^55 - 10 * q^61 + 6 * q^69 - 6 * q^75 + 10 * q^77 + 8 * q^79 + 18 * q^81 + 6 * q^87 - 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	5.00000i	1.50756i	0.657129	+	0.753778i	$0.271771\pi$
$11$	5.00000i	1.50756i	−0.657129	+	0.753778i	$0.728229\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	6.00000i	1.54919i
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 3.00000i	− 0.688247i	−0.938924	−	0.344124i	$-0.888176\pi$
$19$	− 3.00000i	− 0.688247i	0.938924	−	0.344124i	$-0.111824\pi$
$20$	0	0
$21$	3.00000i	0.654654i
$22$	0	0
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	− 8.00000i	− 1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$	− 8.00000i	− 1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$	0	0
$33$	− 15.0000i	− 2.61116i
$34$	0	0
$35$	−2.00000	−0.338062
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.00000i	− 0.468521i	−0.972174	−	0.234261i	$-0.924733\pi$
$41$	− 3.00000i	− 0.468521i	0.972174	−	0.234261i	$-0.0752669\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	− 12.0000i	− 1.78885i
$46$	0	0
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	9.00000	1.26025
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	10.0000	1.34840
$56$	0	0
$57$	9.00000i	1.19208i
$58$	0	0
$59$	− 5.00000i	− 0.650945i	−0.945552	−	0.325472i	$-0.894477\pi$
$59$	− 5.00000i	− 0.650945i	0.945552	−	0.325472i	$-0.105523\pi$
$60$	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
$63$	− 6.00000i	− 0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.00000i	0.855186i	0.903971	+	0.427593i	$0.140638\pi$
$67$	7.00000i	0.855186i	−0.903971	+	0.427593i	$0.859362\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	− 11.0000i	− 1.30546i	−0.757591	−	0.652730i	$-0.773624\pi$
$71$	− 11.0000i	− 1.30546i	0.757591	−	0.652730i	$-0.226376\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	5.00000	0.569803
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	0	0
$85$	6.00000i	0.650791i
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	− 9.00000i	− 0.953998i	−0.878904	−	0.476999i	$-0.841725\pi$
$89$	− 9.00000i	− 0.953998i	0.878904	−	0.476999i	$-0.158275\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24.0000i	2.48868i
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	1.00000i	0.101535i	0.998711	+	0.0507673i	$0.0161667\pi$
$97$	1.00000i	0.101535i	−0.998711	+	0.0507673i	$0.983833\pi$
$98$	0	0
$99$	30.0000i	3.01511i
$100$	0	0
$101$	−19.0000	−1.89057	−0.945285	−	0.326245i	$-0.894217\pi$
$101$	−19.0000	−1.89057	−0.945285	+	0.326245i	$0.894217\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	6.00000	0.585540
$106$	0	0
$107$	−11.0000	−1.06341	−0.531705	−	0.846930i	$-0.678449\pi$
$107$	−11.0000	−1.06341	−0.531705	+	0.846930i	$0.678449\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	− 9.00000i	− 0.854242i
$112$	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
$115$	2.00000i	0.186501i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.00000i	0.275010i
$120$	0	0
$121$	−14.0000	−1.27273
$122$	0	0
$123$	9.00000i	0.811503i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	−5.00000	−0.443678	−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000	−0.443678	−0.221839	+	0.975083i	$0.571206\pi$
$128$	0	0
$129$	−3.00000	−0.264135
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	18.0000i	1.54919i
$136$	0	0
$137$	15.0000i	1.28154i	0.767734	+	0.640768i	$0.221384\pi$
$137$	15.0000i	1.28154i	−0.767734	+	0.640768i	$0.778616\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	12.0000i	1.01058i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.00000i	0.166091i
$146$	0	0
$147$	−18.0000	−1.48461
$148$	0	0
$149$	21.0000i	1.72039i	0.509968	+	0.860194i	$0.329657\pi$
$149$	21.0000i	1.72039i	−0.509968	+	0.860194i	$0.670343\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	−18.0000	−1.45521
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	0	0
$161$	1.00000i	0.0788110i
$162$	0	0
$163$	7.00000i	0.548282i	0.961689	+	0.274141i	$0.0883936\pi$
$163$	7.00000i	0.548282i	−0.961689	+	0.274141i	$0.911606\pi$
$164$	0	0
$165$	−30.0000	−2.33550
$166$	0	0
$167$	9.00000i	0.696441i	0.937413	+	0.348220i	$0.113214\pi$
$167$	9.00000i	0.696441i	−0.937413	+	0.348220i	$0.886786\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 18.0000i	− 1.37649i
$172$	0	0
$173$	−11.0000	−0.836315	−0.418157	−	0.908375i	$-0.637324\pi$
$173$	−11.0000	−0.836315	−0.418157	+	0.908375i	$0.637324\pi$
$174$	0	0
$175$	− 1.00000i	− 0.0755929i
$176$	0	0
$177$	15.0000i	1.12747i
$178$	0	0
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	15.0000	1.10883
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	− 15.0000i	− 1.09691i
$188$	0	0
$189$	9.00000i	0.654654i
$190$	0	0
$191$	−1.00000	−0.0723575	−0.0361787	−	0.999345i	$-0.511519\pi$
$191$	−1.00000	−0.0723575	−0.0361787	+	0.999345i	$0.511519\pi$
$192$	0	0
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.00000i	0.641223i	0.947211	+	0.320612i	$0.103888\pi$
$197$	9.00000i	0.641223i	−0.947211	+	0.320612i	$0.896112\pi$
$198$	0	0
$199$	−19.0000	−1.34687	−0.673437	−	0.739244i	$-0.735183\pi$
$199$	−19.0000	−1.34687	−0.673437	+	0.739244i	$0.735183\pi$
$200$	0	0
$201$	− 21.0000i	− 1.48123i
$202$	0	0
$203$	1.00000i	0.0701862i
$204$	0	0
$205$	−6.00000	−0.419058
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	15.0000	1.03757
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	0	0
$213$	33.0000i	2.26112i
$214$	0	0
$215$	− 2.00000i	− 0.136399i
$216$	0	0
$217$	−8.00000	−0.543075
$218$	0	0
$219$	− 42.0000i	− 2.83810i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 9.00000i	− 0.602685i	−0.953516	−	0.301342i	$-0.902565\pi$
$223$	− 9.00000i	− 0.602685i	0.953516	−	0.301342i	$-0.0974347\pi$
$224$	0	0
$225$	6.00000	0.400000
$226$	0	0
$227$	21.0000i	1.39382i	0.717159	+	0.696909i	$0.245442\pi$
$227$	21.0000i	1.39382i	−0.717159	+	0.696909i	$0.754558\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	0	0
$231$	−15.0000	−0.986928
$232$	0	0
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	24.0000i	1.55243i	0.630468	+	0.776215i	$0.282863\pi$
$239$	24.0000i	1.55243i	−0.630468	+	0.776215i	$0.717137\pi$
$240$	0	0
$241$	− 21.0000i	− 1.35273i	−0.736567	−	0.676364i	$-0.763554\pi$
$241$	− 21.0000i	− 1.35273i	0.736567	−	0.676364i	$-0.236446\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 12.0000i	− 0.766652i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 36.0000i	− 2.28141i
$250$	0	0
$251$	9.00000	0.568075	0.284037	−	0.958813i	$-0.408326\pi$
$251$	9.00000	0.568075	0.284037	+	0.958813i	$0.408326\pi$
$252$	0	0
$253$	− 5.00000i	− 0.314347i
$254$	0	0
$255$	− 18.0000i	− 1.12720i
$256$	0	0
$257$	−3.00000	−0.187135	−0.0935674	−	0.995613i	$-0.529827\pi$
$257$	−3.00000	−0.187135	−0.0935674	+	0.995613i	$0.529827\pi$
$258$	0	0
$259$	3.00000	0.186411
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−15.0000	−0.924940	−0.462470	−	0.886635i	$-0.653037\pi$
$263$	−15.0000	−0.924940	−0.462470	+	0.886635i	$0.653037\pi$
$264$	0	0
$265$	12.0000i	0.737154i
$266$	0	0
$267$	27.0000i	1.65237i
$268$	0	0
$269$	15.0000	0.914566	0.457283	−	0.889321i	$-0.348823\pi$
$269$	15.0000	0.914566	0.457283	+	0.889321i	$0.348823\pi$
$270$	0	0
$271$	17.0000i	1.03268i	0.856385	+	0.516338i	$0.172705\pi$
$271$	17.0000i	1.03268i	−0.856385	+	0.516338i	$0.827295\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.00000i	0.301511i
$276$	0	0
$277$	9.00000	0.540758	0.270379	−	0.962754i	$-0.412851\pi$
$277$	9.00000	0.540758	0.270379	+	0.962754i	$0.412851\pi$
$278$	0	0
$279$	− 48.0000i	− 2.87368i
$280$	0	0
$281$	− 2.00000i	− 0.119310i	−0.998219	−	0.0596550i	$-0.981000\pi$
$281$	− 2.00000i	− 0.119310i	0.998219	−	0.0596550i	$-0.0190001\pi$
$282$	0	0
$283$	19.0000	1.12943	0.564716	−	0.825285i	$-0.308986\pi$
$283$	19.0000	1.12943	0.564716	+	0.825285i	$0.308986\pi$
$284$	0	0
$285$	18.0000	1.06623
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	− 3.00000i	− 0.175863i
$292$	0	0
$293$	19.0000i	1.10999i	0.831853	+	0.554996i	$0.187280\pi$
$293$	19.0000i	1.10999i	−0.831853	+	0.554996i	$0.812720\pi$
$294$	0	0
$295$	−10.0000	−0.582223
$296$	0	0
$297$	− 45.0000i	− 2.61116i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	57.0000	3.27456
$304$	0	0
$305$	10.0000i	0.572598i
$306$	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
$309$	24.0000	1.36531
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−12.0000	−0.676123
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	0	0
$319$	− 5.00000i	− 0.279946i
$320$	0	0
$321$	33.0000	1.84188
$322$	0	0
$323$	9.00000i	0.500773i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	30.0000i	1.65900i
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	17.0000i	0.934405i	0.884150	+	0.467202i	$0.154738\pi$
$331$	17.0000i	0.934405i	−0.884150	+	0.467202i	$0.845262\pi$
$332$	0	0
$333$	18.0000i	0.986394i
$334$	0	0
$335$	14.0000	0.764902
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	−33.0000	−1.79231
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	− 6.00000i	− 0.323029i
$346$	0	0
$347$	−33.0000	−1.77153	−0.885766	−	0.464131i	$-0.846367\pi$
$347$	−33.0000	−1.77153	−0.885766	+	0.464131i	$0.846367\pi$
$348$	0	0
$349$	19.0000i	1.01705i	0.861048	+	0.508523i	$0.169808\pi$
$349$	19.0000i	1.01705i	−0.861048	+	0.508523i	$0.830192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7.00000i	− 0.372572i	−0.982496	−	0.186286i	$-0.940355\pi$
$353$	− 7.00000i	− 0.372572i	0.982496	−	0.186286i	$-0.0596452\pi$
$354$	0	0
$355$	−22.0000	−1.16764
$356$	0	0
$357$	− 9.00000i	− 0.476331i
$358$	0	0
$359$	24.0000i	1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$	24.0000i	1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	0	0
$361$	10.0000	0.526316
$362$	0	0
$363$	42.0000	2.20443
$364$	0	0
$365$	28.0000	1.46559
$366$	0	0
$367$	−23.0000	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$367$	−23.0000	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$368$	0	0
$369$	− 18.0000i	− 0.937043i
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	23.0000	1.19089	0.595447	−	0.803394i	$-0.296975\pi$
$373$	23.0000	1.19089	0.595447	+	0.803394i	$0.296975\pi$
$374$	0	0
$375$	36.0000i	1.85903i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 5.00000i	− 0.256833i	−0.991720	−	0.128416i	$-0.959011\pi$
$379$	− 5.00000i	− 0.256833i	0.991720	−	0.128416i	$-0.0409894\pi$
$380$	0	0
$381$	15.0000	0.768473
$382$	0	0
$383$	9.00000i	0.459879i	0.973205	+	0.229939i	$0.0738528\pi$
$383$	9.00000i	0.459879i	−0.973205	+	0.229939i	$0.926147\pi$
$384$	0	0
$385$	− 10.0000i	− 0.509647i
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 8.00000i	− 0.402524i
$396$	0	0
$397$	− 25.0000i	− 1.25471i	−0.778732	−	0.627357i	$-0.784137\pi$
$397$	− 25.0000i	− 1.25471i	0.778732	−	0.627357i	$-0.215863\pi$
$398$	0	0
$399$	9.00000	0.450564
$400$	0	0
$401$	27.0000i	1.34832i	0.738587	+	0.674158i	$0.235493\pi$
$401$	27.0000i	1.34832i	−0.738587	+	0.674158i	$0.764507\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 18.0000i	− 0.894427i
$406$	0	0
$407$	−15.0000	−0.743522
$408$	0	0
$409$	5.00000i	0.247234i	0.992330	+	0.123617i	$0.0394494\pi$
$409$	5.00000i	0.247234i	−0.992330	+	0.123617i	$0.960551\pi$
$410$	0	0
$411$	− 45.0000i	− 2.21969i
$412$	0	0
$413$	−5.00000	−0.246034
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−33.0000	−1.61602
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	− 10.0000i	− 0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$	− 10.0000i	− 0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$	0	0
$423$	− 24.0000i	− 1.16692i
$424$	0	0
$425$	−3.00000	−0.145521
$426$	0	0
$427$	5.00000i	0.241967i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.00000i	0.144505i	0.997386	+	0.0722525i	$0.0230187\pi$
$431$	3.00000i	0.144505i	−0.997386	+	0.0722525i	$0.976981\pi$
$432$	0	0
$433$	−19.0000	−0.913082	−0.456541	−	0.889702i	$-0.650912\pi$
$433$	−19.0000	−0.913082	−0.456541	+	0.889702i	$0.650912\pi$
$434$	0	0
$435$	− 6.00000i	− 0.287678i
$436$	0	0
$437$	3.00000i	0.143509i
$438$	0	0
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	0	0
$441$	36.0000	1.71429
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	− 63.0000i	− 2.97980i
$448$	0	0
$449$	− 5.00000i	− 0.235965i	−0.993016	−	0.117982i	$-0.962357\pi$
$449$	− 5.00000i	− 0.235965i	0.993016	−	0.117982i	$-0.0376426\pi$
$450$	0	0
$451$	15.0000	0.706322
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17.0000i	0.795226i	0.917553	+	0.397613i	$0.130161\pi$
$457$	17.0000i	0.795226i	−0.917553	+	0.397613i	$0.869839\pi$
$458$	0	0
$459$	27.0000	1.26025
$460$	0	0
$461$	9.00000i	0.419172i	0.977790	+	0.209586i	$0.0672116\pi$
$461$	9.00000i	0.419172i	−0.977790	+	0.209586i	$0.932788\pi$
$462$	0	0
$463$	− 40.0000i	− 1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$	− 40.0000i	− 1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$	0	0
$465$	48.0000	2.22595
$466$	0	0
$467$	−16.0000	−0.740392	−0.370196	−	0.928954i	$-0.620709\pi$
$467$	−16.0000	−0.740392	−0.370196	+	0.928954i	$0.620709\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	66.0000	3.04112
$472$	0	0
$473$	5.00000i	0.229900i
$474$	0	0
$475$	− 3.00000i	− 0.137649i
$476$	0	0
$477$	−36.0000	−1.64833
$478$	0	0
$479$	− 3.00000i	− 0.137073i	−0.997649	−	0.0685367i	$-0.978167\pi$
$479$	− 3.00000i	− 0.137073i	0.997649	−	0.0685367i	$-0.0218330\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 3.00000i	− 0.136505i
$484$	0	0
$485$	2.00000	0.0908153
$486$	0	0
$487$	29.0000i	1.31412i	0.753840	+	0.657058i	$0.228199\pi$
$487$	29.0000i	1.31412i	−0.753840	+	0.657058i	$0.771801\pi$
$488$	0	0
$489$	− 21.0000i	− 0.949653i
$490$	0	0
$491$	31.0000	1.39901	0.699505	−	0.714628i	$-0.253404\pi$
$491$	31.0000	1.39901	0.699505	+	0.714628i	$0.253404\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	0	0
$495$	60.0000	2.69680
$496$	0	0
$497$	−11.0000	−0.493417
$498$	0	0
$499$	4.00000i	0.179065i	0.995984	+	0.0895323i	$0.0285372\pi$
$499$	4.00000i	0.179065i	−0.995984	+	0.0895323i	$0.971463\pi$
$500$	0	0
$501$	− 27.0000i	− 1.20627i
$502$	0	0
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	0	0
$505$	38.0000i	1.69098i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.00000i	0.221621i	0.993842	+	0.110811i	$0.0353447\pi$
$509$	5.00000i	0.221621i	−0.993842	+	0.110811i	$0.964655\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	0	0
$513$	27.0000i	1.19208i
$514$	0	0
$515$	16.0000i	0.705044i
$516$	0	0
$517$	20.0000	0.879599
$518$	0	0
$519$	33.0000	1.44854
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−15.0000	−0.655904	−0.327952	−	0.944694i	$-0.606358\pi$
$523$	−15.0000	−0.655904	−0.327952	+	0.944694i	$0.606358\pi$
$524$	0	0
$525$	3.00000i	0.130931i
$526$	0	0
$527$	24.0000i	1.04546i
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	− 30.0000i	− 1.30189i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	22.0000i	0.951143i
$536$	0	0
$537$	63.0000	2.71865
$538$	0	0
$539$	30.0000i	1.29219i
$540$	0	0
$541$	− 34.0000i	− 1.46177i	−0.682498	−	0.730887i	$-0.739107\pi$
$541$	− 34.0000i	− 1.46177i	0.682498	−	0.730887i	$-0.260893\pi$
$542$	0	0
$543$	54.0000	2.31736
$544$	0	0
$545$	−20.0000	−0.856706
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	−30.0000	−1.28037
$550$	0	0
$551$	3.00000i	0.127804i
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	0	0
$555$	−18.0000	−0.764057
$556$	0	0
$557$	− 33.0000i	− 1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$	− 33.0000i	− 1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	45.0000i	1.89990i
$562$	0	0
$563$	−19.0000	−0.800755	−0.400377	−	0.916350i	$-0.631121\pi$
$563$	−19.0000	−0.800755	−0.400377	+	0.916350i	$0.631121\pi$
$564$	0	0
$565$	− 22.0000i	− 0.925547i
$566$	0	0
$567$	− 9.00000i	− 0.377964i
$568$	0	0
$569$	−7.00000	−0.293455	−0.146728	−	0.989177i	$-0.546874\pi$
$569$	−7.00000	−0.293455	−0.146728	+	0.989177i	$0.546874\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	3.00000	0.125327
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	− 6.00000i	− 0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$	− 6.00000i	− 0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$	0	0
$579$	− 33.0000i	− 1.37143i
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 30.0000i	− 1.24247i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 25.0000i	− 1.03186i	−0.856631	−	0.515930i	$-0.827446\pi$
$587$	− 25.0000i	− 1.03186i	0.856631	−	0.515930i	$-0.172554\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	− 27.0000i	− 1.11063i
$592$	0	0
$593$	− 34.0000i	− 1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$	− 34.0000i	− 1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	0	0
$597$	57.0000	2.33285
$598$	0	0
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	42.0000i	1.71037i
$604$	0	0
$605$	28.0000i	1.13836i
$606$	0	0
$607$	27.0000	1.09590	0.547948	−	0.836512i	$-0.315409\pi$
$607$	27.0000	1.09590	0.547948	+	0.836512i	$0.315409\pi$
$608$	0	0
$609$	− 3.00000i	− 0.121566i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 11.0000i	− 0.444286i	−0.975014	−	0.222143i	$-0.928695\pi$
$613$	− 11.0000i	− 0.444286i	0.975014	−	0.222143i	$-0.0713052\pi$
$614$	0	0
$615$	18.0000	0.725830
$616$	0	0
$617$	− 7.00000i	− 0.281809i	−0.990023	−	0.140905i	$-0.954999\pi$
$617$	− 7.00000i	− 0.281809i	0.990023	−	0.140905i	$-0.0450011\pi$
$618$	0	0
$619$	20.0000i	0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$	20.0000i	0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$	0	0
$621$	9.00000	0.361158
$622$	0	0
$623$	−9.00000	−0.360577
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	−45.0000	−1.79713
$628$	0	0
$629$	− 9.00000i	− 0.358854i
$630$	0	0
$631$	− 37.0000i	− 1.47295i	−0.676467	−	0.736473i	$-0.736490\pi$
$631$	− 37.0000i	− 1.47295i	0.676467	−	0.736473i	$-0.263510\pi$
$632$	0	0
$633$	−39.0000	−1.55011
$634$	0	0
$635$	10.0000i	0.396838i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 66.0000i	− 2.61092i
$640$	0	0
$641$	5.00000	0.197488	0.0987441	−	0.995113i	$-0.468517\pi$
$641$	5.00000	0.197488	0.0987441	+	0.995113i	$0.468517\pi$
$642$	0	0
$643$	− 31.0000i	− 1.22252i	−0.791430	−	0.611260i	$-0.790663\pi$
$643$	− 31.0000i	− 1.22252i	0.791430	−	0.611260i	$-0.209337\pi$
$644$	0	0
$645$	6.00000i	0.236250i
$646$	0	0
$647$	−29.0000	−1.14011	−0.570054	−	0.821607i	$-0.693078\pi$
$647$	−29.0000	−1.14011	−0.570054	+	0.821607i	$0.693078\pi$
$648$	0	0
$649$	25.0000	0.981336
$650$	0	0
$651$	24.0000	0.940634
$652$	0	0
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	0	0
$655$	− 8.00000i	− 0.312586i
$656$	0	0
$657$	84.0000i	3.27715i
$658$	0	0
$659$	−33.0000	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$659$	−33.0000	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$660$	0	0
$661$	− 37.0000i	− 1.43913i	−0.694423	−	0.719567i	$-0.744340\pi$
$661$	− 37.0000i	− 1.43913i	0.694423	−	0.719567i	$-0.255660\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.00000i	0.232670i
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	27.0000i	1.04388i
$670$	0	0
$671$	− 25.0000i	− 0.965114i
$672$	0	0
$673$	−7.00000	−0.269830	−0.134915	−	0.990857i	$-0.543076\pi$
$673$	−7.00000	−0.269830	−0.134915	+	0.990857i	$0.543076\pi$
$674$	0	0
$675$	−9.00000	−0.346410
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	− 63.0000i	− 2.41417i
$682$	0	0
$683$	11.0000i	0.420903i	0.977604	+	0.210452i	$0.0674935\pi$
$683$	11.0000i	0.420903i	−0.977604	+	0.210452i	$0.932507\pi$
$684$	0	0
$685$	30.0000	1.14624
$686$	0	0
$687$	18.0000i	0.686743i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 5.00000i	− 0.190209i	−0.995467	−	0.0951045i	$-0.969681\pi$
$691$	− 5.00000i	− 0.190209i	0.995467	−	0.0951045i	$-0.0303185\pi$
$692$	0	0
$693$	30.0000	1.13961
$694$	0	0
$695$	− 22.0000i	− 0.834508i
$696$	0	0
$697$	9.00000i	0.340899i
$698$	0	0
$699$	66.0000	2.49635
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	9.00000	0.339441
$704$	0	0
$705$	24.0000	0.903892
$706$	0	0
$707$	19.0000i	0.714569i
$708$	0	0
$709$	− 1.00000i	− 0.0375558i	−0.999824	−	0.0187779i	$-0.994022\pi$
$709$	− 1.00000i	− 0.0375558i	0.999824	−	0.0187779i	$-0.00597754\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 72.0000i	− 2.68889i
$718$	0	0
$719$	17.0000	0.633993	0.316997	−	0.948427i	$-0.397326\pi$
$719$	17.0000	0.633993	0.316997	+	0.948427i	$0.397326\pi$
$720$	0	0
$721$	8.00000i	0.297936i
$722$	0	0
$723$	63.0000i	2.34300i
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	0	0
$735$	36.0000i	1.32788i
$736$	0	0
$737$	−35.0000	−1.28924
$738$	0	0
$739$	− 15.0000i	− 0.551784i	−0.961189	−	0.275892i	$-0.911027\pi$
$739$	− 15.0000i	− 0.551784i	0.961189	−	0.275892i	$-0.0889732\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 29.0000i	− 1.06391i	−0.846774	−	0.531953i	$-0.821458\pi$
$743$	− 29.0000i	− 1.06391i	0.846774	−	0.531953i	$-0.178542\pi$
$744$	0	0
$745$	42.0000	1.53876
$746$	0	0
$747$	72.0000i	2.63434i
$748$	0	0
$749$	11.0000i	0.401931i
$750$	0	0
$751$	−9.00000	−0.328415	−0.164207	−	0.986426i	$-0.552507\pi$
$751$	−9.00000	−0.328415	−0.164207	+	0.986426i	$0.552507\pi$
$752$	0	0
$753$	−27.0000	−0.983935
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.00000	0.109037	0.0545184	−	0.998513i	$-0.482638\pi$
$757$	3.00000	0.109037	0.0545184	+	0.998513i	$0.482638\pi$
$758$	0	0
$759$	15.0000i	0.544466i
$760$	0	0
$761$	27.0000i	0.978749i	0.872074	+	0.489375i	$0.162775\pi$
$761$	27.0000i	0.978749i	−0.872074	+	0.489375i	$0.837225\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	0	0
$765$	36.0000i	1.30158i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	21.0000i	0.757279i	0.925544	+	0.378640i	$0.123608\pi$
$769$	21.0000i	0.757279i	−0.925544	+	0.378640i	$0.876392\pi$
$770$	0	0
$771$	9.00000	0.324127
$772$	0	0
$773$	1.00000i	0.0359675i	0.999838	+	0.0179838i	$0.00572471\pi$
$773$	1.00000i	0.0359675i	−0.999838	+	0.0179838i	$0.994275\pi$
$774$	0	0
$775$	− 8.00000i	− 0.287368i
$776$	0	0
$777$	−9.00000	−0.322873
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	55.0000	1.96805
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	44.0000i	1.57043i
$786$	0	0
$787$	7.00000i	0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$	7.00000i	0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$	0	0
$789$	45.0000	1.60204
$790$	0	0
$791$	− 11.0000i	− 0.391115i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 36.0000i	− 1.27679i
$796$	0	0
$797$	−27.0000	−0.956389	−0.478195	−	0.878254i	$-0.658709\pi$
$797$	−27.0000	−0.956389	−0.478195	+	0.878254i	$0.658709\pi$
$798$	0	0
$799$	12.0000i	0.424529i
$800$	0	0
$801$	− 54.0000i	− 1.90800i
$802$	0	0
$803$	−70.0000	−2.47025
$804$	0	0
$805$	2.00000	0.0704907
$806$	0	0
$807$	−45.0000	−1.58408
$808$	0	0
$809$	−45.0000	−1.58212	−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000	−1.58212	−0.791058	+	0.611741i	$0.790469\pi$
$810$	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
$813$	− 51.0000i	− 1.78865i
$814$	0	0
$815$	14.0000	0.490399
$816$	0	0
$817$	− 3.00000i	− 0.104957i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 47.0000i	− 1.64031i	−0.572140	−	0.820156i	$-0.693887\pi$
$821$	− 47.0000i	− 1.64031i	0.572140	−	0.820156i	$-0.306113\pi$
$822$	0	0
$823$	−3.00000	−0.104573	−0.0522867	−	0.998632i	$-0.516651\pi$
$823$	−3.00000	−0.104573	−0.0522867	+	0.998632i	$0.516651\pi$
$824$	0	0
$825$	− 15.0000i	− 0.522233i
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	−27.0000	−0.936620
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	18.0000	0.622916
$836$	0	0
$837$	72.0000i	2.48868i
$838$	0	0
$839$	43.0000i	1.48452i	0.670109	+	0.742262i	$0.266247\pi$
$839$	43.0000i	1.48452i	−0.670109	+	0.742262i	$0.733753\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	6.00000i	0.206651i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	0	0
$849$	−57.0000	−1.95623
$850$	0	0
$851$	− 3.00000i	− 0.102839i
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	0	0
$855$	−36.0000	−1.23117
$856$	0	0
$857$	2.00000	0.0683187	0.0341593	−	0.999416i	$-0.489125\pi$
$857$	2.00000	0.0683187	0.0341593	+	0.999416i	$0.489125\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	0	0
$863$	16.0000i	0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$	16.0000i	0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$	0	0
$865$	22.0000i	0.748022i
$866$	0	0
$867$	24.0000	0.815083
$868$	0	0
$869$	20.0000i	0.678454i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.00000i	0.203069i
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	45.0000i	1.51954i	0.650191	+	0.759771i	$0.274689\pi$
$877$	45.0000i	1.51954i	−0.650191	+	0.759771i	$0.725311\pi$
$878$	0	0
$879$	− 57.0000i	− 1.92256i
$880$	0	0
$881$	−51.0000	−1.71823	−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000	−1.71823	−0.859117	+	0.511780i	$0.828986\pi$
$882$	0	0
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	0	0
$885$	30.0000	1.00844
$886$	0	0
$887$	53.0000	1.77957	0.889783	−	0.456384i	$-0.150856\pi$
$887$	53.0000	1.77957	0.889783	+	0.456384i	$0.150856\pi$
$888$	0	0
$889$	5.00000i	0.167695i
$890$	0	0
$891$	45.0000i	1.50756i
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	42.0000i	1.40391i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.00000i	0.266815i
$900$	0	0
$901$	18.0000	0.599667
$902$	0	0
$903$	3.00000i	0.0998337i
$904$	0	0
$905$	36.0000i	1.19668i
$906$	0	0
$907$	3.00000	0.0996134	0.0498067	−	0.998759i	$-0.484139\pi$
$907$	3.00000	0.0996134	0.0498067	+	0.998759i	$0.484139\pi$
$908$	0	0
$909$	−114.000	−3.78114
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	− 30.0000i	− 0.991769i
$916$	0	0
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	−33.0000	−1.08857	−0.544285	−	0.838901i	$-0.683199\pi$
$919$	−33.0000	−1.08857	−0.544285	+	0.838901i	$0.683199\pi$
$920$	0	0
$921$	− 12.0000i	− 0.395413i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	3.00000i	0.0986394i
$926$	0	0
$927$	−48.0000	−1.57653
$928$	0	0
$929$	21.0000i	0.688988i	0.938789	+	0.344494i	$0.111949\pi$
$929$	21.0000i	0.688988i	−0.938789	+	0.344494i	$0.888051\pi$
$930$	0	0
$931$	− 18.0000i	− 0.589926i
$932$	0	0
$933$	72.0000	2.35717
$934$	0	0
$935$	−30.0000	−0.981105
$936$	0	0
$937$	−18.0000	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$938$	0	0
$939$	−42.0000	−1.37062
$940$	0	0
$941$	50.0000i	1.62995i	0.579494	+	0.814977i	$0.303250\pi$
$941$	50.0000i	1.62995i	−0.579494	+	0.814977i	$0.696750\pi$
$942$	0	0
$943$	3.00000i	0.0976934i
$944$	0	0
$945$	18.0000	0.585540
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	90.0000i	2.91845i
$952$	0	0
$953$	1.00000	0.0323932	0.0161966	−	0.999869i	$-0.494844\pi$
$953$	1.00000	0.0323932	0.0161966	+	0.999869i	$0.494844\pi$
$954$	0	0
$955$	2.00000i	0.0647185i
$956$	0	0
$957$	15.0000i	0.484881i
$958$	0	0
$959$	15.0000	0.484375
$960$	0	0
$961$	−33.0000	−1.06452
$962$	0	0
$963$	−66.0000	−2.12682
$964$	0	0
$965$	22.0000	0.708205
$966$	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
$969$	− 27.0000i	− 0.867365i
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	− 11.0000i	− 0.352644i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 39.0000i	− 1.24772i	−0.781536	−	0.623860i	$-0.785563\pi$
$977$	− 39.0000i	− 1.24772i	0.781536	−	0.623860i	$-0.214437\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	− 60.0000i	− 1.91565i
$982$	0	0
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	12.0000	0.381964
$988$	0	0
$989$	−1.00000	−0.0317982
$990$	0	0
$991$	−27.0000	−0.857683	−0.428842	−	0.903380i	$-0.641078\pi$
$991$	−27.0000	−0.857683	−0.428842	+	0.903380i	$0.641078\pi$
$992$	0	0
$993$	− 51.0000i	− 1.61844i
$994$	0	0
$995$	38.0000i	1.20468i
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	0	0
$999$	− 27.0000i	− 0.854242i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.f.a.337.1		2
4.3	odd	2		676.2.d.d.337.1		2
12.11	even	2		6084.2.b.l.4393.2		2
13.5	odd	4		2704.2.a.a.1.1		1
13.7	odd	12		208.2.i.d.81.1		2
13.8	odd	4		2704.2.a.b.1.1		1
13.11	odd	12		208.2.i.d.113.1		2
13.12	even	2	inner	2704.2.f.a.337.2		2
39.11	even	12		1872.2.t.f.1153.1		2
39.20	even	12		1872.2.t.f.289.1		2
52.3	odd	6		676.2.h.b.485.1		4
52.7	even	12		52.2.e.a.29.1	yes	2
52.11	even	12		52.2.e.a.9.1	✓	2
52.15	even	12		676.2.e.a.529.1		2
52.19	even	12		676.2.e.a.653.1		2
52.23	odd	6		676.2.h.b.485.2		4
52.31	even	4		676.2.a.d.1.1		1
52.35	odd	6		676.2.h.b.361.1		4
52.43	odd	6		676.2.h.b.361.2		4
52.47	even	4		676.2.a.e.1.1		1
52.51	odd	2		676.2.d.d.337.2		2
104.11	even	12		832.2.i.j.321.1		2
104.37	odd	12		832.2.i.a.321.1		2
104.59	even	12		832.2.i.j.705.1		2
104.85	odd	12		832.2.i.a.705.1		2
156.11	odd	12		468.2.l.a.217.1		2
156.47	odd	4		6084.2.a.f.1.1		1
156.59	odd	12		468.2.l.a.289.1		2
156.83	odd	4		6084.2.a.k.1.1		1
156.155	even	2		6084.2.b.l.4393.1		2
260.7	odd	12		1300.2.bb.f.549.1		4
260.59	even	12		1300.2.i.f.601.1		2
260.63	odd	12		1300.2.bb.f.1049.1		4
260.163	odd	12		1300.2.bb.f.549.2		4
260.167	odd	12		1300.2.bb.f.1049.2		4
260.219	even	12		1300.2.i.f.1101.1		2
364.11	even	12		2548.2.l.h.373.1		2
364.59	odd	12		2548.2.i.h.1745.1		2
364.111	odd	12		2548.2.k.d.393.1		2
364.115	odd	12		2548.2.l.a.373.1		2
364.163	even	12		2548.2.l.h.1537.1		2
364.167	odd	12		2548.2.k.d.1569.1		2
364.215	odd	12		2548.2.l.a.1537.1		2
364.219	even	12		2548.2.i.a.165.1		2
364.271	odd	12		2548.2.i.h.165.1		2
364.319	even	12		2548.2.i.a.1745.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.2.e.a.9.1	✓	2	52.11	even	12
52.2.e.a.29.1	yes	2	52.7	even	12
208.2.i.d.81.1		2	13.7	odd	12
208.2.i.d.113.1		2	13.11	odd	12
468.2.l.a.217.1		2	156.11	odd	12
468.2.l.a.289.1		2	156.59	odd	12
676.2.a.d.1.1		1	52.31	even	4
676.2.a.e.1.1		1	52.47	even	4
676.2.d.d.337.1		2	4.3	odd	2
676.2.d.d.337.2		2	52.51	odd	2
676.2.e.a.529.1		2	52.15	even	12
676.2.e.a.653.1		2	52.19	even	12
676.2.h.b.361.1		4	52.35	odd	6
676.2.h.b.361.2		4	52.43	odd	6
676.2.h.b.485.1		4	52.3	odd	6
676.2.h.b.485.2		4	52.23	odd	6
832.2.i.a.321.1		2	104.37	odd	12
832.2.i.a.705.1		2	104.85	odd	12
832.2.i.j.321.1		2	104.11	even	12
832.2.i.j.705.1		2	104.59	even	12
1300.2.i.f.601.1		2	260.59	even	12
1300.2.i.f.1101.1		2	260.219	even	12
1300.2.bb.f.549.1		4	260.7	odd	12
1300.2.bb.f.549.2		4	260.163	odd	12
1300.2.bb.f.1049.1		4	260.63	odd	12
1300.2.bb.f.1049.2		4	260.167	odd	12
1872.2.t.f.289.1		2	39.20	even	12
1872.2.t.f.1153.1		2	39.11	even	12
2548.2.i.a.165.1		2	364.219	even	12
2548.2.i.a.1745.1		2	364.319	even	12
2548.2.i.h.165.1		2	364.271	odd	12
2548.2.i.h.1745.1		2	364.59	odd	12
2548.2.k.d.393.1		2	364.111	odd	12
2548.2.k.d.1569.1		2	364.167	odd	12
2548.2.l.a.373.1		2	364.115	odd	12
2548.2.l.a.1537.1		2	364.215	odd	12
2548.2.l.h.373.1		2	364.11	even	12
2548.2.l.h.1537.1		2	364.163	even	12
2704.2.a.a.1.1		1	13.5	odd	4
2704.2.a.b.1.1		1	13.8	odd	4
2704.2.f.a.337.1		2	1.1	even	1	trivial
2704.2.f.a.337.2		2	13.12	even	2	inner
6084.2.a.f.1.1		1	156.47	odd	4
6084.2.a.k.1.1		1	156.83	odd	4
6084.2.b.l.4393.1		2	156.155	even	2
6084.2.b.l.4393.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -2.00000i q^{5} -1.00000i q^{7} +6.00000 q^{9} +5.00000i q^{11} +6.00000i q^{15} -3.00000 q^{17} -3.00000i q^{19} +3.00000i q^{21} -1.00000 q^{23} +1.00000 q^{25} -9.00000 q^{27} -1.00000 q^{29} -8.00000i q^{31} -15.0000i q^{33} -2.00000 q^{35} +3.00000i q^{37} -3.00000i q^{41} +1.00000 q^{43} -12.0000i q^{45} -4.00000i q^{47} +6.00000 q^{49} +9.00000 q^{51} -6.00000 q^{53} +10.0000 q^{55} +9.00000i q^{57} -5.00000i q^{59} -5.00000 q^{61} -6.00000i q^{63} +7.00000i q^{67} +3.00000 q^{69} -11.0000i q^{71} +14.0000i q^{73} -3.00000 q^{75} +5.00000 q^{77} +4.00000 q^{79} +9.00000 q^{81} +12.0000i q^{83} +6.00000i q^{85} +3.00000 q^{87} -9.00000i q^{89} +24.0000i q^{93} -6.00000 q^{95} +1.00000i q^{97} +30.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 12 q^{9} - 6 q^{17} - 2 q^{23} + 2 q^{25} - 18 q^{27} - 2 q^{29} - 4 q^{35} + 2 q^{43} + 12 q^{49} + 18 q^{51} - 12 q^{53} + 20 q^{55} - 10 q^{61} + 6 q^{69} - 6 q^{75} + 10 q^{77} + 8 q^{79}+ \cdots - 12 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 + 12 * q^9 - 6 * q^17 - 2 * q^23 + 2 * q^25 - 18 * q^27 - 2 * q^29 - 4 * q^35 + 2 * q^43 + 12 * q^49 + 18 * q^51 - 12 * q^53 + 20 * q^55 - 10 * q^61 + 6 * q^69 - 6 * q^75 + 10 * q^77 + 8 * q^79 + 18 * q^81 + 6 * q^87 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more