LMFDB - Embedded newform 1764.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(1,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1764.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.a (trivial)

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:	yes
Analytic conductor:	$14.0856109166$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 9 $ x^4 - 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.57794$ of defining polynomial
Character	$\chi$	$=$	1764.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.74166 q^{5} +O(q^{10})$	$q+3.74166 q^{5} -5.29150 q^{11} -4.24264 q^{13} +3.74166 q^{17} +2.82843 q^{19} +5.29150 q^{23} +9.00000 q^{25} +5.29150 q^{29} +8.48528 q^{31} +4.00000 q^{37} +3.74166 q^{41} +8.00000 q^{43} -7.48331 q^{47} +10.5830 q^{53} -19.7990 q^{55} -7.48331 q^{59} +9.89949 q^{61} -15.8745 q^{65} +12.0000 q^{67} -15.8745 q^{71} +1.41421 q^{73} -4.00000 q^{79} -14.9666 q^{83} +14.0000 q^{85} -3.74166 q^{89} +10.5830 q^{95} -9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 36 q^{25} + 16 q^{37} + 32 q^{43} + 48 q^{67} - 16 q^{79} + 56 q^{85}+O(q^{100}) $ 4 * q + 36 * q^25 + 16 * q^37 + 32 * q^43 + 48 * q^67 - 16 * q^79 + 56 * q^85

Display 100 coefficients 10 coefficients

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
$5$	3.74166	1.67332	0.836660	−	0.547723i	$-0.184505\pi$
$5$	3.74166	1.67332	0.836660	+	0.547723i	$0.184505\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.29150	−1.59545	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	−5.29150	−1.59545	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.74166	0.907485	0.453743	−	0.891133i	$-0.350089\pi$
$17$	3.74166	0.907485	0.453743	+	0.891133i	$0.350089\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.29150	1.10335	0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150	1.10335	0.551677	+	0.834058i	$0.313988\pi$
$24$	0	0
$25$	9.00000	1.80000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.29150	0.982607	0.491304	−	0.870988i	$-0.336521\pi$
$29$	5.29150	0.982607	0.491304	+	0.870988i	$0.336521\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.74166	0.584349	0.292174	−	0.956365i	$-0.405621\pi$
$41$	3.74166	0.584349	0.292174	+	0.956365i	$0.405621\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.48331	−1.09155	−0.545777	−	0.837931i	$-0.683765\pi$
$47$	−7.48331	−1.09155	−0.545777	+	0.837931i	$0.683765\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.5830	1.45369	0.726844	−	0.686803i	$-0.240986\pi$
$53$	10.5830	1.45369	0.726844	+	0.686803i	$0.240986\pi$
$54$	0	0
$55$	−19.7990	−2.66970
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.48331	−0.974245	−0.487122	−	0.873334i	$-0.661953\pi$
$59$	−7.48331	−0.974245	−0.487122	+	0.873334i	$0.661953\pi$
$60$	0	0
$61$	9.89949	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$61$	9.89949	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−15.8745	−1.96899
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.8745	−1.88396	−0.941979	−	0.335673i	$-0.891036\pi$
$71$	−15.8745	−1.88396	−0.941979	+	0.335673i	$0.891036\pi$
$72$	0	0
$73$	1.41421	0.165521	0.0827606	−	0.996569i	$-0.473626\pi$
$73$	1.41421	0.165521	0.0827606	+	0.996569i	$0.473626\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.9666	−1.64280	−0.821401	−	0.570352i	$-0.806807\pi$
$83$	−14.9666	−1.64280	−0.821401	+	0.570352i	$0.806807\pi$
$84$	0	0
$85$	14.0000	1.51851
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.74166	−0.396615	−0.198307	−	0.980140i	$-0.563544\pi$
$89$	−3.74166	−0.396615	−0.198307	+	0.980140i	$0.563544\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.5830	1.08579
$96$	0	0
$97$	−9.89949	−1.00514	−0.502571	−	0.864536i	$-0.667612\pi$
$97$	−9.89949	−1.00514	−0.502571	+	0.864536i	$0.667612\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.2250	−1.11693	−0.558463	−	0.829529i	$-0.688609\pi$
$101$	−11.2250	−1.11693	−0.558463	+	0.829529i	$0.688609\pi$
$102$	0	0
$103$	−2.82843	−0.278693	−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843	−0.278693	−0.139347	+	0.990244i	$0.544500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.29150	0.511549	0.255774	−	0.966736i	$-0.417670\pi$
$107$	5.29150	0.511549	0.255774	+	0.966736i	$0.417670\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	19.7990	1.84627
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.0000	1.54545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	14.9666	1.33866
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.8745	−1.35625	−0.678125	−	0.734946i	$-0.737207\pi$
$137$	−15.8745	−1.35625	−0.678125	+	0.734946i	$0.737207\pi$
$138$	0	0
$139$	−5.65685	−0.479808	−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685	−0.479808	−0.239904	+	0.970797i	$0.577116\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.4499	1.87736
$144$	0	0
$145$	19.7990	1.64422
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.5830	0.866994	0.433497	−	0.901155i	$-0.357280\pi$
$149$	10.5830	0.866994	0.433497	+	0.901155i	$0.357280\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	31.7490	2.55014
$156$	0	0
$157$	−9.89949	−0.790066	−0.395033	−	0.918667i	$-0.629267\pi$
$157$	−9.89949	−0.790066	−0.395033	+	0.918667i	$0.629267\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.48331	0.579076	0.289538	−	0.957166i	$-0.406498\pi$
$167$	7.48331	0.579076	0.289538	+	0.957166i	$0.406498\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.2250	0.853419	0.426709	−	0.904389i	$-0.359673\pi$
$173$	11.2250	0.853419	0.426709	+	0.904389i	$0.359673\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.8745	−1.18652	−0.593258	−	0.805012i	$-0.702159\pi$
$179$	−15.8745	−1.18652	−0.593258	+	0.805012i	$0.702159\pi$
$180$	0	0
$181$	4.24264	0.315353	0.157676	−	0.987491i	$-0.449600\pi$
$181$	4.24264	0.315353	0.157676	+	0.987491i	$0.449600\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.9666	1.10037
$186$	0	0
$187$	−19.7990	−1.44785
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.29150	0.382880	0.191440	−	0.981504i	$-0.438684\pi$
$191$	5.29150	0.382880	0.191440	+	0.981504i	$0.438684\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.5830	−0.754008	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	−10.5830	−0.754008	−0.377004	+	0.926212i	$0.623046\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	14.0000	0.977802
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.9666	−1.03526
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	29.9333	2.04143
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−15.8745	−1.06783
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−22.4499	−1.49006	−0.745028	−	0.667034i	$-0.767564\pi$
$227$	−22.4499	−1.49006	−0.745028	+	0.667034i	$0.767564\pi$
$228$	0	0
$229$	−12.7279	−0.841085	−0.420542	−	0.907273i	$-0.638160\pi$
$229$	−12.7279	−0.841085	−0.420542	+	0.907273i	$0.638160\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.8745	1.03997	0.519987	−	0.854174i	$-0.325937\pi$
$233$	15.8745	1.03997	0.519987	+	0.854174i	$0.325937\pi$
$234$	0	0
$235$	−28.0000	−1.82652
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.8745	−1.02684	−0.513418	−	0.858138i	$-0.671621\pi$
$239$	−15.8745	−1.02684	−0.513418	+	0.858138i	$0.671621\pi$
$240$	0	0
$241$	−29.6985	−1.91305	−0.956524	−	0.291654i	$-0.905794\pi$
$241$	−29.6985	−1.91305	−0.956524	+	0.291654i	$0.905794\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.0000	−0.763542
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.48331	0.472343	0.236171	−	0.971711i	$-0.424107\pi$
$251$	7.48331	0.472343	0.236171	+	0.971711i	$0.424107\pi$
$252$	0	0
$253$	−28.0000	−1.76034
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.2250	−0.700195	−0.350097	−	0.936713i	$-0.613851\pi$
$257$	−11.2250	−0.700195	−0.350097	+	0.936713i	$0.613851\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.8745	0.978864	0.489432	−	0.872041i	$-0.337204\pi$
$263$	15.8745	0.978864	0.489432	+	0.872041i	$0.337204\pi$
$264$	0	0
$265$	39.5980	2.43248
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.74166	−0.228133	−0.114066	−	0.993473i	$-0.536388\pi$
$269$	−3.74166	−0.228133	−0.114066	+	0.993473i	$0.536388\pi$
$270$	0	0
$271$	19.7990	1.20270	0.601351	−	0.798985i	$-0.294629\pi$
$271$	19.7990	1.20270	0.601351	+	0.798985i	$0.294629\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−47.6235	−2.87181
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.29150	−0.315665	−0.157832	−	0.987466i	$-0.550451\pi$
$281$	−5.29150	−0.315665	−0.157832	+	0.987466i	$0.550451\pi$
$282$	0	0
$283$	31.1127	1.84946	0.924729	−	0.380626i	$-0.124292\pi$
$283$	31.1127	1.84946	0.924729	+	0.380626i	$0.124292\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.00000	−0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−26.1916	−1.53013	−0.765065	−	0.643953i	$-0.777293\pi$
$293$	−26.1916	−1.53013	−0.765065	+	0.643953i	$0.777293\pi$
$294$	0	0
$295$	−28.0000	−1.63022
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−22.4499	−1.29831
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	37.0405	2.12093
$306$	0	0
$307$	14.1421	0.807134	0.403567	−	0.914950i	$-0.367770\pi$
$307$	14.1421	0.807134	0.403567	+	0.914950i	$0.367770\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.48331	0.424340	0.212170	−	0.977233i	$-0.431947\pi$
$311$	7.48331	0.424340	0.212170	+	0.977233i	$0.431947\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.5830	0.594401	0.297200	−	0.954815i	$-0.403947\pi$
$317$	10.5830	0.594401	0.297200	+	0.954815i	$0.403947\pi$
$318$	0	0
$319$	−28.0000	−1.56770
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.5830	0.588854
$324$	0	0
$325$	−38.1838	−2.11805
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	44.8999	2.45314
$336$	0	0
$337$	8.00000	0.435788	0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000	0.435788	0.217894	+	0.975972i	$0.430081\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−44.8999	−2.43147
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.8745	0.852188	0.426094	−	0.904679i	$-0.359889\pi$
$347$	15.8745	0.852188	0.426094	+	0.904679i	$0.359889\pi$
$348$	0	0
$349$	29.6985	1.58972	0.794862	−	0.606791i	$-0.207543\pi$
$349$	29.6985	1.58972	0.794862	+	0.606791i	$0.207543\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.7083	−0.995742	−0.497871	−	0.867251i	$-0.665885\pi$
$353$	−18.7083	−0.995742	−0.497871	+	0.867251i	$0.665885\pi$
$354$	0	0
$355$	−59.3970	−3.15246
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.29150	−0.279275	−0.139637	−	0.990203i	$-0.544594\pi$
$359$	−5.29150	−0.279275	−0.139637	+	0.990203i	$0.544594\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	5.29150	0.276970
$366$	0	0
$367$	−11.3137	−0.590571	−0.295285	−	0.955409i	$-0.595415\pi$
$367$	−11.3137	−0.590571	−0.295285	+	0.955409i	$0.595415\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−22.4499	−1.15623
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.9666	−0.764759	−0.382380	−	0.924005i	$-0.624895\pi$
$383$	−14.9666	−0.764759	−0.382380	+	0.924005i	$0.624895\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.29150	0.268290	0.134145	−	0.990962i	$-0.457171\pi$
$389$	5.29150	0.268290	0.134145	+	0.990962i	$0.457171\pi$
$390$	0	0
$391$	19.7990	1.00128
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−14.9666	−0.753053
$396$	0	0
$397$	32.5269	1.63248	0.816239	−	0.577714i	$-0.196055\pi$
$397$	32.5269	1.63248	0.816239	+	0.577714i	$0.196055\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.8745	0.792735	0.396368	−	0.918092i	$-0.370271\pi$
$401$	15.8745	0.792735	0.396368	+	0.918092i	$0.370271\pi$
$402$	0	0
$403$	−36.0000	−1.79329
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−21.1660	−1.04916
$408$	0	0
$409$	−21.2132	−1.04893	−0.524463	−	0.851433i	$-0.675734\pi$
$409$	−21.2132	−1.04893	−0.524463	+	0.851433i	$0.675734\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−56.0000	−2.74893
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−7.48331	−0.365584	−0.182792	−	0.983152i	$-0.558513\pi$
$419$	−7.48331	−0.365584	−0.182792	+	0.983152i	$0.558513\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	33.6749	1.63347
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.8745	−0.764648	−0.382324	−	0.924028i	$-0.624876\pi$
$431$	−15.8745	−0.764648	−0.382324	+	0.924028i	$0.624876\pi$
$432$	0	0
$433$	−4.24264	−0.203888	−0.101944	−	0.994790i	$-0.532506\pi$
$433$	−4.24264	−0.203888	−0.101944	+	0.994790i	$0.532506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14.9666	0.715951
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.29150	0.251407	0.125703	−	0.992068i	$-0.459881\pi$
$443$	5.29150	0.251407	0.125703	+	0.992068i	$0.459881\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−19.7990	−0.932298
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.74166	−0.174266	−0.0871332	−	0.996197i	$-0.527771\pi$
$461$	−3.74166	−0.174266	−0.0871332	+	0.996197i	$0.527771\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.4166	1.73143	0.865716	−	0.500535i	$-0.166863\pi$
$467$	37.4166	1.73143	0.865716	+	0.500535i	$0.166863\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−42.3320	−1.94643
$474$	0	0
$475$	25.4558	1.16799
$476$	0	0
$477$	0	0
$478$	0	0
$479$	37.4166	1.70961	0.854803	−	0.518952i	$-0.173678\pi$
$479$	37.4166	1.70961	0.854803	+	0.518952i	$0.173678\pi$
$480$	0	0
$481$	−16.9706	−0.773791
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−37.0405	−1.68192
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.8745	0.716407	0.358203	−	0.933644i	$-0.383389\pi$
$491$	15.8745	0.716407	0.358203	+	0.933644i	$0.383389\pi$
$492$	0	0
$493$	19.7990	0.891702
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−14.9666	−0.667329	−0.333665	−	0.942692i	$-0.608285\pi$
$503$	−14.9666	−0.667329	−0.333665	+	0.942692i	$0.608285\pi$
$504$	0	0
$505$	−42.0000	−1.86898
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.1582	1.82431	0.912153	−	0.409849i	$-0.134419\pi$
$509$	41.1582	1.82431	0.912153	+	0.409849i	$0.134419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.5830	−0.466343
$516$	0	0
$517$	39.5980	1.74152
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.7083	−0.819625	−0.409812	−	0.912170i	$-0.634406\pi$
$521$	−18.7083	−0.819625	−0.409812	+	0.912170i	$0.634406\pi$
$522$	0	0
$523$	−16.9706	−0.742071	−0.371035	−	0.928619i	$-0.620997\pi$
$523$	−16.9706	−0.742071	−0.371035	+	0.928619i	$0.620997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.7490	1.38301
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.8745	−0.687601
$534$	0	0
$535$	19.7990	0.855985
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	59.8665	2.56440
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.9666	0.637600
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.7490	−1.34525	−0.672624	−	0.739984i	$-0.734833\pi$
$557$	−31.7490	−1.34525	−0.672624	+	0.739984i	$0.734833\pi$
$558$	0	0
$559$	−33.9411	−1.43556
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.48331	−0.315384	−0.157692	−	0.987488i	$-0.550405\pi$
$563$	−7.48331	−0.315384	−0.157692	+	0.987488i	$0.550405\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	26.4575	1.10916	0.554578	−	0.832132i	$-0.312880\pi$
$569$	26.4575	1.10916	0.554578	+	0.832132i	$0.312880\pi$
$570$	0	0
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	47.6235	1.98604
$576$	0	0
$577$	−21.2132	−0.883117	−0.441559	−	0.897232i	$-0.645574\pi$
$577$	−21.2132	−0.883117	−0.441559	+	0.897232i	$0.645574\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−56.0000	−2.31928
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.4499	−0.926608	−0.463304	−	0.886199i	$-0.653336\pi$
$587$	−22.4499	−0.926608	−0.463304	+	0.886199i	$0.653336\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.1916	−1.07556	−0.537780	−	0.843085i	$-0.680737\pi$
$593$	−26.1916	−1.07556	−0.537780	+	0.843085i	$0.680737\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.29150	0.216205	0.108102	−	0.994140i	$-0.465523\pi$
$599$	5.29150	0.216205	0.108102	+	0.994140i	$0.465523\pi$
$600$	0	0
$601$	4.24264	0.173061	0.0865305	−	0.996249i	$-0.472422\pi$
$601$	4.24264	0.173061	0.0865305	+	0.996249i	$0.472422\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	63.6082	2.58604
$606$	0	0
$607$	16.9706	0.688814	0.344407	−	0.938820i	$-0.388080\pi$
$607$	16.9706	0.688814	0.344407	+	0.938820i	$0.388080\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.7490	1.28443
$612$	0	0
$613$	−24.0000	−0.969351	−0.484675	−	0.874694i	$-0.661062\pi$
$613$	−24.0000	−0.969351	−0.484675	+	0.874694i	$0.661062\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.4575	1.06514	0.532570	−	0.846386i	$-0.321226\pi$
$617$	26.4575	1.06514	0.532570	+	0.846386i	$0.321226\pi$
$618$	0	0
$619$	11.3137	0.454736	0.227368	−	0.973809i	$-0.426988\pi$
$619$	11.3137	0.454736	0.227368	+	0.973809i	$0.426988\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.9666	0.596759
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.9333	−1.18787
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	37.0405	1.46301	0.731506	−	0.681835i	$-0.238818\pi$
$641$	37.0405	1.46301	0.731506	+	0.681835i	$0.238818\pi$
$642$	0	0
$643$	−14.1421	−0.557711	−0.278856	−	0.960333i	$-0.589955\pi$
$643$	−14.1421	−0.557711	−0.278856	+	0.960333i	$0.589955\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.4499	−0.882598	−0.441299	−	0.897360i	$-0.645482\pi$
$647$	−22.4499	−0.882598	−0.441299	+	0.897360i	$0.645482\pi$
$648$	0	0
$649$	39.5980	1.55436
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.8745	−0.621217	−0.310609	−	0.950538i	$-0.600533\pi$
$653$	−15.8745	−0.621217	−0.310609	+	0.950538i	$0.600533\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−47.6235	−1.85515	−0.927575	−	0.373638i	$-0.878110\pi$
$659$	−47.6235	−1.85515	−0.927575	+	0.373638i	$0.878110\pi$
$660$	0	0
$661$	−21.2132	−0.825098	−0.412549	−	0.910935i	$-0.635361\pi$
$661$	−21.2132	−0.825098	−0.412549	+	0.910935i	$0.635361\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.0000	1.08416
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−52.3832	−2.02223
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−33.6749	−1.29423	−0.647116	−	0.762391i	$-0.724025\pi$
$677$	−33.6749	−1.29423	−0.647116	+	0.762391i	$0.724025\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.8745	−0.607421	−0.303711	−	0.952764i	$-0.598226\pi$
$683$	−15.8745	−0.607421	−0.303711	+	0.952764i	$0.598226\pi$
$684$	0	0
$685$	−59.3970	−2.26944
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−44.8999	−1.71055
$690$	0	0
$691$	−16.9706	−0.645591	−0.322795	−	0.946469i	$-0.604623\pi$
$691$	−16.9706	−0.645591	−0.322795	+	0.946469i	$0.604623\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−21.1660	−0.802873
$696$	0	0
$697$	14.0000	0.530288
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15.8745	0.599572	0.299786	−	0.954006i	$-0.403085\pi$
$701$	15.8745	0.599572	0.299786	+	0.954006i	$0.403085\pi$
$702$	0	0
$703$	11.3137	0.426705
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	44.8999	1.68151
$714$	0	0
$715$	84.0000	3.14142
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−29.9333	−1.11632	−0.558161	−	0.829733i	$-0.688493\pi$
$719$	−29.9333	−1.11632	−0.558161	+	0.829733i	$0.688493\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	47.6235	1.76869
$726$	0	0
$727$	−14.1421	−0.524503	−0.262251	−	0.965000i	$-0.584465\pi$
$727$	−14.1421	−0.524503	−0.262251	+	0.965000i	$0.584465\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.9333	1.10712
$732$	0	0
$733$	−46.6690	−1.72376	−0.861880	−	0.507112i	$-0.830713\pi$
$733$	−46.6690	−1.72376	−0.861880	+	0.507112i	$0.830713\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−63.4980	−2.33898
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.0405	−1.35888	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	−37.0405	−1.35888	−0.679442	+	0.733729i	$0.737778\pi$
$744$	0	0
$745$	39.5980	1.45076
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14.9666	−0.544691
$756$	0	0
$757$	−16.0000	−0.581530	−0.290765	−	0.956795i	$-0.593910\pi$
$757$	−16.0000	−0.581530	−0.290765	+	0.956795i	$0.593910\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.1582	1.49198	0.745992	−	0.665955i	$-0.231976\pi$
$761$	41.1582	1.49198	0.745992	+	0.665955i	$0.231976\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	31.7490	1.14639
$768$	0	0
$769$	−4.24264	−0.152994	−0.0764968	−	0.997070i	$-0.524373\pi$
$769$	−4.24264	−0.152994	−0.0764968	+	0.997070i	$0.524373\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−48.6415	−1.74951	−0.874757	−	0.484561i	$-0.838979\pi$
$773$	−48.6415	−1.74951	−0.874757	+	0.484561i	$0.838979\pi$
$774$	0	0
$775$	76.3675	2.74320
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.5830	0.379176
$780$	0	0
$781$	84.0000	3.00576
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−37.0405	−1.32203
$786$	0	0
$787$	39.5980	1.41152	0.705758	−	0.708453i	$-0.250607\pi$
$787$	39.5980	1.41152	0.705758	+	0.708453i	$0.250607\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−42.0000	−1.49146
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.2250	−0.397609	−0.198804	−	0.980039i	$-0.563706\pi$
$797$	−11.2250	−0.397609	−0.198804	+	0.980039i	$0.563706\pi$
$798$	0	0
$799$	−28.0000	−0.990569
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.48331	−0.264080
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.1660	0.744157	0.372079	−	0.928201i	$-0.378645\pi$
$809$	21.1660	0.744157	0.372079	+	0.928201i	$0.378645\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−74.8331	−2.62129
$816$	0	0
$817$	22.6274	0.791633
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31.7490	−1.10805	−0.554024	−	0.832501i	$-0.686908\pi$
$821$	−31.7490	−1.10805	−0.554024	+	0.832501i	$0.686908\pi$
$822$	0	0
$823$	4.00000	0.139431	0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000	0.139431	0.0697156	+	0.997567i	$0.477791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.8745	−0.552011	−0.276005	−	0.961156i	$-0.589011\pi$
$827$	−15.8745	−0.552011	−0.276005	+	0.961156i	$0.589011\pi$
$828$	0	0
$829$	18.3848	0.638530	0.319265	−	0.947666i	$-0.396564\pi$
$829$	18.3848	0.638530	0.319265	+	0.947666i	$0.396564\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	28.0000	0.968980
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.48331	−0.258353	−0.129176	−	0.991622i	$-0.541233\pi$
$839$	−7.48331	−0.258353	−0.129176	+	0.991622i	$0.541233\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	18.7083	0.643585
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	21.1660	0.725561
$852$	0	0
$853$	−55.1543	−1.88845	−0.944224	−	0.329304i	$-0.893186\pi$
$853$	−55.1543	−1.88845	−0.944224	+	0.329304i	$0.893186\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	41.1582	1.40594	0.702969	−	0.711220i	$-0.251857\pi$
$857$	41.1582	1.40594	0.702969	+	0.711220i	$0.251857\pi$
$858$	0	0
$859$	−19.7990	−0.675533	−0.337766	−	0.941230i	$-0.609671\pi$
$859$	−19.7990	−0.675533	−0.337766	+	0.941230i	$0.609671\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.29150	0.180125	0.0900624	−	0.995936i	$-0.471293\pi$
$863$	5.29150	0.180125	0.0900624	+	0.995936i	$0.471293\pi$
$864$	0	0
$865$	42.0000	1.42804
$866$	0	0
$867$	0	0
$868$	0	0
$869$	21.1660	0.718008
$870$	0	0
$871$	−50.9117	−1.72508
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−56.1249	−1.89089	−0.945447	−	0.325775i	$-0.894375\pi$
$881$	−56.1249	−1.89089	−0.945447	+	0.325775i	$0.894375\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.4166	−1.25633	−0.628163	−	0.778082i	$-0.716193\pi$
$887$	−37.4166	−1.25633	−0.628163	+	0.778082i	$0.716193\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.1660	−0.708294
$894$	0	0
$895$	−59.3970	−1.98542
$896$	0	0
$897$	0	0
$898$	0	0
$899$	44.8999	1.49750
$900$	0	0
$901$	39.5980	1.31920
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.8745	0.527686
$906$	0	0
$907$	16.0000	0.531271	0.265636	−	0.964073i	$-0.414418\pi$
$907$	16.0000	0.531271	0.265636	+	0.964073i	$0.414418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5.29150	0.175315	0.0876577	−	0.996151i	$-0.472062\pi$
$911$	5.29150	0.175315	0.0876577	+	0.996151i	$0.472062\pi$
$912$	0	0
$913$	79.1960	2.62100
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	67.3498	2.21685
$924$	0	0
$925$	36.0000	1.18367
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−18.7083	−0.613799	−0.306899	−	0.951742i	$-0.599292\pi$
$929$	−18.7083	−0.613799	−0.306899	+	0.951742i	$0.599292\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−74.0810	−2.42271
$936$	0	0
$937$	−35.3553	−1.15501	−0.577504	−	0.816388i	$-0.695973\pi$
$937$	−35.3553	−1.15501	−0.577504	+	0.816388i	$0.695973\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.1916	0.853822	0.426911	−	0.904294i	$-0.359602\pi$
$941$	26.1916	0.853822	0.426911	+	0.904294i	$0.359602\pi$
$942$	0	0
$943$	19.7990	0.644744
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.6235	1.54756	0.773778	−	0.633457i	$-0.218364\pi$
$947$	47.6235	1.54756	0.773778	+	0.633457i	$0.218364\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	42.3320	1.37127	0.685634	−	0.727946i	$-0.259525\pi$
$953$	42.3320	1.37127	0.685634	+	0.727946i	$0.259525\pi$
$954$	0	0
$955$	19.7990	0.640680
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.48331	0.240896
$966$	0	0
$967$	48.0000	1.54358	0.771788	−	0.635880i	$-0.219363\pi$
$967$	48.0000	1.54358	0.771788	+	0.635880i	$0.219363\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.9666	0.480302	0.240151	−	0.970736i	$-0.422803\pi$
$971$	14.9666	0.480302	0.240151	+	0.970736i	$0.422803\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.8745	−0.507871	−0.253935	−	0.967221i	$-0.581725\pi$
$977$	−15.8745	−0.507871	−0.253935	+	0.967221i	$0.581725\pi$
$978$	0	0
$979$	19.7990	0.632778
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−14.9666	−0.477361	−0.238681	−	0.971098i	$-0.576715\pi$
$983$	−14.9666	−0.477361	−0.238681	+	0.971098i	$0.576715\pi$
$984$	0	0
$985$	−39.5980	−1.26170
$986$	0	0
$987$	0	0
$988$	0	0
$989$	42.3320	1.34608
$990$	0	0
$991$	44.0000	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$991$	44.0000	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.41421	0.0447886	0.0223943	−	0.999749i	$-0.492871\pi$
$997$	1.41421	0.0447886	0.0223943	+	0.999749i	$0.492871\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.a.m.1.3	yes	4
3.2	odd	2	inner	1764.2.a.m.1.2	yes	4
4.3	odd	2		7056.2.a.cy.1.4		4
7.2	even	3		1764.2.k.m.361.2		8
7.3	odd	6		1764.2.k.m.1549.4		8
7.4	even	3		1764.2.k.m.1549.2		8
7.5	odd	6		1764.2.k.m.361.4		8
7.6	odd	2	inner	1764.2.a.m.1.1	✓	4
12.11	even	2		7056.2.a.cy.1.1		4
21.2	odd	6		1764.2.k.m.361.3		8
21.5	even	6		1764.2.k.m.361.1		8
21.11	odd	6		1764.2.k.m.1549.3		8
21.17	even	6		1764.2.k.m.1549.1		8
21.20	even	2	inner	1764.2.a.m.1.4	yes	4
28.27	even	2		7056.2.a.cy.1.2		4
84.83	odd	2		7056.2.a.cy.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.2.a.m.1.1	✓	4	7.6	odd	2	inner
1764.2.a.m.1.2	yes	4	3.2	odd	2	inner
1764.2.a.m.1.3	yes	4	1.1	even	1	trivial
1764.2.a.m.1.4	yes	4	21.20	even	2	inner
1764.2.k.m.361.1		8	21.5	even	6
1764.2.k.m.361.2		8	7.2	even	3
1764.2.k.m.361.3		8	21.2	odd	6
1764.2.k.m.361.4		8	7.5	odd	6
1764.2.k.m.1549.1		8	21.17	even	6
1764.2.k.m.1549.2		8	7.4	even	3
1764.2.k.m.1549.3		8	21.11	odd	6
1764.2.k.m.1549.4		8	7.3	odd	6
7056.2.a.cy.1.1		4	12.11	even	2
7056.2.a.cy.1.2		4	28.27	even	2
7056.2.a.cy.1.3		4	84.83	odd	2
7056.2.a.cy.1.4		4	4.3	odd	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

\(f(q)\)	\(=\)	\(q+3.74166 q^{5} +O(q^{10})\)	\(q+3.74166 q^{5} -5.29150 q^{11} -4.24264 q^{13} +3.74166 q^{17} +2.82843 q^{19} +5.29150 q^{23} +9.00000 q^{25} +5.29150 q^{29} +8.48528 q^{31} +4.00000 q^{37} +3.74166 q^{41} +8.00000 q^{43} -7.48331 q^{47} +10.5830 q^{53} -19.7990 q^{55} -7.48331 q^{59} +9.89949 q^{61} -15.8745 q^{65} +12.0000 q^{67} -15.8745 q^{71} +1.41421 q^{73} -4.00000 q^{79} -14.9666 q^{83} +14.0000 q^{85} -3.74166 q^{89} +10.5830 q^{95} -9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 36 q^{25} + 16 q^{37} + 32 q^{43} + 48 q^{67} - 16 q^{79} + 56 q^{85}+O(q^{100}) \) 4 * q + 36 * q^25 + 16 * q^37 + 32 * q^43 + 48 * q^67 - 16 * q^79 + 56 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more