LMFDB - Embedded newform 4620.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -3.26795 q^{13} -1.00000 q^{15} -7.92820 q^{17} -7.92820 q^{19} -1.00000 q^{21} +2.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.73205 q^{29} +0.732051 q^{31} +1.00000 q^{33} +1.00000 q^{35} +8.73205 q^{37} +3.26795 q^{39} +4.19615 q^{41} +9.19615 q^{43} +1.00000 q^{45} -3.66025 q^{47} +1.00000 q^{49} +7.92820 q^{51} +8.46410 q^{53} -1.00000 q^{55} +7.92820 q^{57} +14.6603 q^{59} -11.9282 q^{61} +1.00000 q^{63} -3.26795 q^{65} +14.0000 q^{67} -2.46410 q^{69} +4.19615 q^{71} -1.46410 q^{73} -1.00000 q^{75} -1.00000 q^{77} +2.73205 q^{79} +1.00000 q^{81} +11.0000 q^{83} -7.92820 q^{85} -1.73205 q^{87} +2.26795 q^{89} -3.26795 q^{91} -0.732051 q^{93} -7.92820 q^{95} -3.73205 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 10 q^{13} - 2 q^{15} - 2 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} + 10 q^{39} - 2 q^{41} + 8 q^{43} + 2 q^{45} + 10 q^{47} + 2 q^{49} + 2 q^{51} + 10 q^{53} - 2 q^{55} + 2 q^{57} + 12 q^{59} - 10 q^{61} + 2 q^{63} - 10 q^{65} + 28 q^{67} + 2 q^{69} - 2 q^{71} + 4 q^{73} - 2 q^{75} - 2 q^{77} + 2 q^{79} + 2 q^{81} + 22 q^{83} - 2 q^{85} + 8 q^{89} - 10 q^{91} + 2 q^{93} - 2 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 10 * q^13 - 2 * q^15 - 2 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 2 * q^33 + 2 * q^35 + 14 * q^37 + 10 * q^39 - 2 * q^41 + 8 * q^43 + 2 * q^45 + 10 * q^47 + 2 * q^49 + 2 * q^51 + 10 * q^53 - 2 * q^55 + 2 * q^57 + 12 * q^59 - 10 * q^61 + 2 * q^63 - 10 * q^65 + 28 * q^67 + 2 * q^69 - 2 * q^71 + 4 * q^73 - 2 * q^75 - 2 * q^77 + 2 * q^79 + 2 * q^81 + 22 * q^83 - 2 * q^85 + 8 * q^89 - 10 * q^91 + 2 * q^93 - 2 * q^95 - 4 * q^97 - 2 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.26795	−0.906366	−0.453183	−	0.891417i	$-0.649712\pi$
$13$	−3.26795	−0.906366	−0.453183	+	0.891417i	$0.649712\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−7.92820	−1.92287	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−7.92820	−1.92287	−0.961436	+	0.275029i	$0.911312\pi$
$18$	0	0
$19$	−7.92820	−1.81885	−0.909427	−	0.415863i	$-0.863480\pi$
$19$	−7.92820	−1.81885	−0.909427	+	0.415863i	$0.863480\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.46410	0.513801	0.256900	−	0.966438i	$-0.417299\pi$
$23$	2.46410	0.513801	0.256900	+	0.966438i	$0.417299\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	0.732051	0.131480	0.0657401	−	0.997837i	$-0.479059\pi$
$31$	0.732051	0.131480	0.0657401	+	0.997837i	$0.479059\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	8.73205	1.43554	0.717770	−	0.696280i	$-0.245163\pi$
$37$	8.73205	1.43554	0.717770	+	0.696280i	$0.245163\pi$
$38$	0	0
$39$	3.26795	0.523291
$40$	0	0
$41$	4.19615	0.655329	0.327664	−	0.944794i	$-0.393738\pi$
$41$	4.19615	0.655329	0.327664	+	0.944794i	$0.393738\pi$
$42$	0	0
$43$	9.19615	1.40240	0.701200	−	0.712965i	$-0.252648\pi$
$43$	9.19615	1.40240	0.701200	+	0.712965i	$0.252648\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−3.66025	−0.533903	−0.266951	−	0.963710i	$-0.586016\pi$
$47$	−3.66025	−0.533903	−0.266951	+	0.963710i	$0.586016\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.92820	1.11017
$52$	0	0
$53$	8.46410	1.16263	0.581317	−	0.813677i	$-0.302538\pi$
$53$	8.46410	1.16263	0.581317	+	0.813677i	$0.302538\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	7.92820	1.05012
$58$	0	0
$59$	14.6603	1.90860	0.954301	−	0.298846i	$-0.0966018\pi$
$59$	14.6603	1.90860	0.954301	+	0.298846i	$0.0966018\pi$
$60$	0	0
$61$	−11.9282	−1.52725	−0.763625	−	0.645660i	$-0.776582\pi$
$61$	−11.9282	−1.52725	−0.763625	+	0.645660i	$0.776582\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.26795	−0.405339
$66$	0	0
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	0	0
$69$	−2.46410	−0.296643
$70$	0	0
$71$	4.19615	0.497992	0.248996	−	0.968505i	$-0.419899\pi$
$71$	4.19615	0.497992	0.248996	+	0.968505i	$0.419899\pi$
$72$	0	0
$73$	−1.46410	−0.171360	−0.0856801	−	0.996323i	$-0.527306\pi$
$73$	−1.46410	−0.171360	−0.0856801	+	0.996323i	$0.527306\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.73205	0.307380	0.153690	−	0.988119i	$-0.450884\pi$
$79$	2.73205	0.307380	0.153690	+	0.988119i	$0.450884\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.0000	1.20741	0.603703	−	0.797209i	$-0.293691\pi$
$83$	11.0000	1.20741	0.603703	+	0.797209i	$0.293691\pi$
$84$	0	0
$85$	−7.92820	−0.859934
$86$	0	0
$87$	−1.73205	−0.185695
$88$	0	0
$89$	2.26795	0.240402	0.120201	−	0.992750i	$-0.461646\pi$
$89$	2.26795	0.240402	0.120201	+	0.992750i	$0.461646\pi$
$90$	0	0
$91$	−3.26795	−0.342574
$92$	0	0
$93$	−0.732051	−0.0759101
$94$	0	0
$95$	−7.92820	−0.813416
$96$	0	0
$97$	−3.73205	−0.378932	−0.189466	−	0.981887i	$-0.560676\pi$
$97$	−3.73205	−0.378932	−0.189466	+	0.981887i	$0.560676\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	0.928203	0.0923597	0.0461798	−	0.998933i	$-0.485295\pi$
$101$	0.928203	0.0923597	0.0461798	+	0.998933i	$0.485295\pi$
$102$	0	0
$103$	10.6603	1.05039	0.525193	−	0.850983i	$-0.323993\pi$
$103$	10.6603	1.05039	0.525193	+	0.850983i	$0.323993\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−18.5885	−1.79701	−0.898507	−	0.438959i	$-0.855347\pi$
$107$	−18.5885	−1.79701	−0.898507	+	0.438959i	$0.855347\pi$
$108$	0	0
$109$	10.1962	0.976614	0.488307	−	0.872672i	$-0.337615\pi$
$109$	10.1962	0.976614	0.488307	+	0.872672i	$0.337615\pi$
$110$	0	0
$111$	−8.73205	−0.828810
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	2.46410	0.229779
$116$	0	0
$117$	−3.26795	−0.302122
$118$	0	0
$119$	−7.92820	−0.726777
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.19615	−0.378354
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.1962	0.993498	0.496749	−	0.867894i	$-0.334527\pi$
$127$	11.1962	0.993498	0.496749	+	0.867894i	$0.334527\pi$
$128$	0	0
$129$	−9.19615	−0.809676
$130$	0	0
$131$	−9.12436	−0.797199	−0.398599	−	0.917125i	$-0.630504\pi$
$131$	−9.12436	−0.797199	−0.398599	+	0.917125i	$0.630504\pi$
$132$	0	0
$133$	−7.92820	−0.687462
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	13.4641	1.15032	0.575158	−	0.818042i	$-0.304941\pi$
$137$	13.4641	1.15032	0.575158	+	0.818042i	$0.304941\pi$
$138$	0	0
$139$	−15.4641	−1.31165	−0.655824	−	0.754914i	$-0.727679\pi$
$139$	−15.4641	−1.31165	−0.655824	+	0.754914i	$0.727679\pi$
$140$	0	0
$141$	3.66025	0.308249
$142$	0	0
$143$	3.26795	0.273280
$144$	0	0
$145$	1.73205	0.143839
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−0.535898	−0.0439025	−0.0219513	−	0.999759i	$-0.506988\pi$
$149$	−0.535898	−0.0439025	−0.0219513	+	0.999759i	$0.506988\pi$
$150$	0	0
$151$	−7.85641	−0.639345	−0.319673	−	0.947528i	$-0.603573\pi$
$151$	−7.85641	−0.639345	−0.319673	+	0.947528i	$0.603573\pi$
$152$	0	0
$153$	−7.92820	−0.640957
$154$	0	0
$155$	0.732051	0.0587997
$156$	0	0
$157$	12.1244	0.967629	0.483814	−	0.875171i	$-0.339251\pi$
$157$	12.1244	0.967629	0.483814	+	0.875171i	$0.339251\pi$
$158$	0	0
$159$	−8.46410	−0.671247
$160$	0	0
$161$	2.46410	0.194198
$162$	0	0
$163$	−18.1962	−1.42523	−0.712616	−	0.701554i	$-0.752490\pi$
$163$	−18.1962	−1.42523	−0.712616	+	0.701554i	$0.752490\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	22.3923	1.73277	0.866384	−	0.499378i	$-0.166438\pi$
$167$	22.3923	1.73277	0.866384	+	0.499378i	$0.166438\pi$
$168$	0	0
$169$	−2.32051	−0.178501
$170$	0	0
$171$	−7.92820	−0.606285
$172$	0	0
$173$	16.3923	1.24628	0.623142	−	0.782109i	$-0.285856\pi$
$173$	16.3923	1.24628	0.623142	+	0.782109i	$0.285856\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−14.6603	−1.10193
$178$	0	0
$179$	16.1962	1.21056	0.605279	−	0.796014i	$-0.293062\pi$
$179$	16.1962	1.21056	0.605279	+	0.796014i	$0.293062\pi$
$180$	0	0
$181$	−15.8564	−1.17860	−0.589299	−	0.807915i	$-0.700596\pi$
$181$	−15.8564	−1.17860	−0.589299	+	0.807915i	$0.700596\pi$
$182$	0	0
$183$	11.9282	0.881758
$184$	0	0
$185$	8.73205	0.641993
$186$	0	0
$187$	7.92820	0.579768
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−18.3923	−1.32391	−0.661954	−	0.749545i	$-0.730272\pi$
$193$	−18.3923	−1.32391	−0.661954	+	0.749545i	$0.730272\pi$
$194$	0	0
$195$	3.26795	0.234023
$196$	0	0
$197$	−12.3923	−0.882915	−0.441458	−	0.897282i	$-0.645538\pi$
$197$	−12.3923	−0.882915	−0.441458	+	0.897282i	$0.645538\pi$
$198$	0	0
$199$	25.8564	1.83291	0.916456	−	0.400135i	$-0.131037\pi$
$199$	25.8564	1.83291	0.916456	+	0.400135i	$0.131037\pi$
$200$	0	0
$201$	−14.0000	−0.987484
$202$	0	0
$203$	1.73205	0.121566
$204$	0	0
$205$	4.19615	0.293072
$206$	0	0
$207$	2.46410	0.171267
$208$	0	0
$209$	7.92820	0.548405
$210$	0	0
$211$	−1.60770	−0.110678	−0.0553391	−	0.998468i	$-0.517624\pi$
$211$	−1.60770	−0.110678	−0.0553391	+	0.998468i	$0.517624\pi$
$212$	0	0
$213$	−4.19615	−0.287516
$214$	0	0
$215$	9.19615	0.627172
$216$	0	0
$217$	0.732051	0.0496948
$218$	0	0
$219$	1.46410	0.0989348
$220$	0	0
$221$	25.9090	1.74283
$222$	0	0
$223$	20.6603	1.38351	0.691756	−	0.722131i	$-0.256837\pi$
$223$	20.6603	1.38351	0.691756	+	0.722131i	$0.256837\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	5.00000	0.331862	0.165931	−	0.986137i	$-0.446937\pi$
$227$	5.00000	0.331862	0.165931	+	0.986137i	$0.446937\pi$
$228$	0	0
$229$	6.58846	0.435378	0.217689	−	0.976018i	$-0.430148\pi$
$229$	6.58846	0.435378	0.217689	+	0.976018i	$0.430148\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−15.5167	−1.01653	−0.508265	−	0.861201i	$-0.669713\pi$
$233$	−15.5167	−1.01653	−0.508265	+	0.861201i	$0.669713\pi$
$234$	0	0
$235$	−3.66025	−0.238769
$236$	0	0
$237$	−2.73205	−0.177466
$238$	0	0
$239$	26.6603	1.72451	0.862254	−	0.506476i	$-0.169052\pi$
$239$	26.6603	1.72451	0.862254	+	0.506476i	$0.169052\pi$
$240$	0	0
$241$	−16.0000	−1.03065	−0.515325	−	0.856995i	$-0.672329\pi$
$241$	−16.0000	−1.03065	−0.515325	+	0.856995i	$0.672329\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	25.9090	1.64855
$248$	0	0
$249$	−11.0000	−0.697097
$250$	0	0
$251$	−20.7846	−1.31191	−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	−2.46410	−0.154917
$254$	0	0
$255$	7.92820	0.496483
$256$	0	0
$257$	24.1962	1.50931	0.754657	−	0.656119i	$-0.227803\pi$
$257$	24.1962	1.50931	0.754657	+	0.656119i	$0.227803\pi$
$258$	0	0
$259$	8.73205	0.542583
$260$	0	0
$261$	1.73205	0.107211
$262$	0	0
$263$	18.5359	1.14297	0.571486	−	0.820612i	$-0.306367\pi$
$263$	18.5359	1.14297	0.571486	+	0.820612i	$0.306367\pi$
$264$	0	0
$265$	8.46410	0.519946
$266$	0	0
$267$	−2.26795	−0.138796
$268$	0	0
$269$	−10.6603	−0.649967	−0.324984	−	0.945720i	$-0.605359\pi$
$269$	−10.6603	−0.649967	−0.324984	+	0.945720i	$0.605359\pi$
$270$	0	0
$271$	21.5359	1.30821	0.654106	−	0.756403i	$-0.273045\pi$
$271$	21.5359	1.30821	0.654106	+	0.756403i	$0.273045\pi$
$272$	0	0
$273$	3.26795	0.197785
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−14.9282	−0.896949	−0.448474	−	0.893796i	$-0.648032\pi$
$277$	−14.9282	−0.896949	−0.448474	+	0.893796i	$0.648032\pi$
$278$	0	0
$279$	0.732051	0.0438267
$280$	0	0
$281$	−12.7846	−0.762666	−0.381333	−	0.924438i	$-0.624535\pi$
$281$	−12.7846	−0.762666	−0.381333	+	0.924438i	$0.624535\pi$
$282$	0	0
$283$	−5.80385	−0.345003	−0.172501	−	0.985009i	$-0.555185\pi$
$283$	−5.80385	−0.345003	−0.172501	+	0.985009i	$0.555185\pi$
$284$	0	0
$285$	7.92820	0.469626
$286$	0	0
$287$	4.19615	0.247691
$288$	0	0
$289$	45.8564	2.69744
$290$	0	0
$291$	3.73205	0.218777
$292$	0	0
$293$	19.2487	1.12452	0.562261	−	0.826960i	$-0.309932\pi$
$293$	19.2487	1.12452	0.562261	+	0.826960i	$0.309932\pi$
$294$	0	0
$295$	14.6603	0.853553
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−8.05256	−0.465692
$300$	0	0
$301$	9.19615	0.530057
$302$	0	0
$303$	−0.928203	−0.0533239
$304$	0	0
$305$	−11.9282	−0.683007
$306$	0	0
$307$	12.5359	0.715462	0.357731	−	0.933825i	$-0.383551\pi$
$307$	12.5359	0.715462	0.357731	+	0.933825i	$0.383551\pi$
$308$	0	0
$309$	−10.6603	−0.606441
$310$	0	0
$311$	−11.3205	−0.641927	−0.320964	−	0.947092i	$-0.604007\pi$
$311$	−11.3205	−0.641927	−0.320964	+	0.947092i	$0.604007\pi$
$312$	0	0
$313$	−12.8038	−0.723716	−0.361858	−	0.932233i	$-0.617858\pi$
$313$	−12.8038	−0.723716	−0.361858	+	0.932233i	$0.617858\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−24.0000	−1.34797	−0.673987	−	0.738743i	$-0.735420\pi$
$317$	−24.0000	−1.34797	−0.673987	+	0.738743i	$0.735420\pi$
$318$	0	0
$319$	−1.73205	−0.0969762
$320$	0	0
$321$	18.5885	1.03751
$322$	0	0
$323$	62.8564	3.49742
$324$	0	0
$325$	−3.26795	−0.181273
$326$	0	0
$327$	−10.1962	−0.563849
$328$	0	0
$329$	−3.66025	−0.201796
$330$	0	0
$331$	−35.2487	−1.93744	−0.968722	−	0.248148i	$-0.920178\pi$
$331$	−35.2487	−1.93744	−0.968722	+	0.248148i	$0.920178\pi$
$332$	0	0
$333$	8.73205	0.478513
$334$	0	0
$335$	14.0000	0.764902
$336$	0	0
$337$	−13.5885	−0.740210	−0.370105	−	0.928990i	$-0.620678\pi$
$337$	−13.5885	−0.740210	−0.370105	+	0.928990i	$0.620678\pi$
$338$	0	0
$339$	−1.00000	−0.0543125
$340$	0	0
$341$	−0.732051	−0.0396428
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.46410	−0.132663
$346$	0	0
$347$	33.4641	1.79645	0.898224	−	0.439539i	$-0.144858\pi$
$347$	33.4641	1.79645	0.898224	+	0.439539i	$0.144858\pi$
$348$	0	0
$349$	−20.4641	−1.09542	−0.547709	−	0.836669i	$-0.684500\pi$
$349$	−20.4641	−1.09542	−0.547709	+	0.836669i	$0.684500\pi$
$350$	0	0
$351$	3.26795	0.174430
$352$	0	0
$353$	−24.2487	−1.29063	−0.645314	−	0.763917i	$-0.723274\pi$
$353$	−24.2487	−1.29063	−0.645314	+	0.763917i	$0.723274\pi$
$354$	0	0
$355$	4.19615	0.222709
$356$	0	0
$357$	7.92820	0.419605
$358$	0	0
$359$	−14.2679	−0.753034	−0.376517	−	0.926410i	$-0.622878\pi$
$359$	−14.2679	−0.753034	−0.376517	+	0.926410i	$0.622878\pi$
$360$	0	0
$361$	43.8564	2.30823
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−1.46410	−0.0766346
$366$	0	0
$367$	11.8756	0.619904	0.309952	−	0.950752i	$-0.399687\pi$
$367$	11.8756	0.619904	0.309952	+	0.950752i	$0.399687\pi$
$368$	0	0
$369$	4.19615	0.218443
$370$	0	0
$371$	8.46410	0.439434
$372$	0	0
$373$	10.2679	0.531654	0.265827	−	0.964021i	$-0.414355\pi$
$373$	10.2679	0.531654	0.265827	+	0.964021i	$0.414355\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.66025	−0.291518
$378$	0	0
$379$	−33.6410	−1.72802	−0.864011	−	0.503472i	$-0.832056\pi$
$379$	−33.6410	−1.72802	−0.864011	+	0.503472i	$0.832056\pi$
$380$	0	0
$381$	−11.1962	−0.573596
$382$	0	0
$383$	8.87564	0.453524	0.226762	−	0.973950i	$-0.427186\pi$
$383$	8.87564	0.453524	0.226762	+	0.973950i	$0.427186\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	9.19615	0.467467
$388$	0	0
$389$	−7.66025	−0.388390	−0.194195	−	0.980963i	$-0.562210\pi$
$389$	−7.66025	−0.388390	−0.194195	+	0.980963i	$0.562210\pi$
$390$	0	0
$391$	−19.5359	−0.987973
$392$	0	0
$393$	9.12436	0.460263
$394$	0	0
$395$	2.73205	0.137464
$396$	0	0
$397$	−18.9282	−0.949979	−0.474990	−	0.879991i	$-0.657548\pi$
$397$	−18.9282	−0.949979	−0.474990	+	0.879991i	$0.657548\pi$
$398$	0	0
$399$	7.92820	0.396907
$400$	0	0
$401$	5.66025	0.282660	0.141330	−	0.989963i	$-0.454862\pi$
$401$	5.66025	0.282660	0.141330	+	0.989963i	$0.454862\pi$
$402$	0	0
$403$	−2.39230	−0.119169
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.73205	−0.432832
$408$	0	0
$409$	14.9282	0.738152	0.369076	−	0.929399i	$-0.379674\pi$
$409$	14.9282	0.738152	0.369076	+	0.929399i	$0.379674\pi$
$410$	0	0
$411$	−13.4641	−0.664135
$412$	0	0
$413$	14.6603	0.721384
$414$	0	0
$415$	11.0000	0.539969
$416$	0	0
$417$	15.4641	0.757280
$418$	0	0
$419$	−1.33975	−0.0654509	−0.0327254	−	0.999464i	$-0.510419\pi$
$419$	−1.33975	−0.0654509	−0.0327254	+	0.999464i	$0.510419\pi$
$420$	0	0
$421$	−12.8564	−0.626583	−0.313291	−	0.949657i	$-0.601432\pi$
$421$	−12.8564	−0.626583	−0.313291	+	0.949657i	$0.601432\pi$
$422$	0	0
$423$	−3.66025	−0.177968
$424$	0	0
$425$	−7.92820	−0.384574
$426$	0	0
$427$	−11.9282	−0.577246
$428$	0	0
$429$	−3.26795	−0.157778
$430$	0	0
$431$	−17.4641	−0.841216	−0.420608	−	0.907242i	$-0.638183\pi$
$431$	−17.4641	−0.841216	−0.420608	+	0.907242i	$0.638183\pi$
$432$	0	0
$433$	−8.24871	−0.396408	−0.198204	−	0.980161i	$-0.563511\pi$
$433$	−8.24871	−0.396408	−0.198204	+	0.980161i	$0.563511\pi$
$434$	0	0
$435$	−1.73205	−0.0830455
$436$	0	0
$437$	−19.5359	−0.934529
$438$	0	0
$439$	39.1051	1.86639	0.933193	−	0.359376i	$-0.117011\pi$
$439$	39.1051	1.86639	0.933193	+	0.359376i	$0.117011\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−24.2487	−1.15209	−0.576046	−	0.817418i	$-0.695405\pi$
$443$	−24.2487	−1.15209	−0.576046	+	0.817418i	$0.695405\pi$
$444$	0	0
$445$	2.26795	0.107511
$446$	0	0
$447$	0.535898	0.0253471
$448$	0	0
$449$	34.0526	1.60704	0.803520	−	0.595278i	$-0.202958\pi$
$449$	34.0526	1.60704	0.803520	+	0.595278i	$0.202958\pi$
$450$	0	0
$451$	−4.19615	−0.197589
$452$	0	0
$453$	7.85641	0.369126
$454$	0	0
$455$	−3.26795	−0.153204
$456$	0	0
$457$	−39.7321	−1.85859	−0.929293	−	0.369342i	$-0.879583\pi$
$457$	−39.7321	−1.85859	−0.929293	+	0.369342i	$0.879583\pi$
$458$	0	0
$459$	7.92820	0.370057
$460$	0	0
$461$	11.4641	0.533936	0.266968	−	0.963705i	$-0.413978\pi$
$461$	11.4641	0.533936	0.266968	+	0.963705i	$0.413978\pi$
$462$	0	0
$463$	19.4641	0.904574	0.452287	−	0.891873i	$-0.350608\pi$
$463$	19.4641	0.904574	0.452287	+	0.891873i	$0.350608\pi$
$464$	0	0
$465$	−0.732051	−0.0339480
$466$	0	0
$467$	−0.679492	−0.0314431	−0.0157216	−	0.999876i	$-0.505005\pi$
$467$	−0.679492	−0.0314431	−0.0157216	+	0.999876i	$0.505005\pi$
$468$	0	0
$469$	14.0000	0.646460
$470$	0	0
$471$	−12.1244	−0.558661
$472$	0	0
$473$	−9.19615	−0.422840
$474$	0	0
$475$	−7.92820	−0.363771
$476$	0	0
$477$	8.46410	0.387545
$478$	0	0
$479$	37.5167	1.71418	0.857090	−	0.515167i	$-0.172270\pi$
$479$	37.5167	1.71418	0.857090	+	0.515167i	$0.172270\pi$
$480$	0	0
$481$	−28.5359	−1.30112
$482$	0	0
$483$	−2.46410	−0.112121
$484$	0	0
$485$	−3.73205	−0.169464
$486$	0	0
$487$	−17.3205	−0.784867	−0.392434	−	0.919780i	$-0.628367\pi$
$487$	−17.3205	−0.784867	−0.392434	+	0.919780i	$0.628367\pi$
$488$	0	0
$489$	18.1962	0.822858
$490$	0	0
$491$	−11.0526	−0.498795	−0.249397	−	0.968401i	$-0.580233\pi$
$491$	−11.0526	−0.498795	−0.249397	+	0.968401i	$0.580233\pi$
$492$	0	0
$493$	−13.7321	−0.618460
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	4.19615	0.188223
$498$	0	0
$499$	−35.2487	−1.57795	−0.788975	−	0.614426i	$-0.789388\pi$
$499$	−35.2487	−1.57795	−0.788975	+	0.614426i	$0.789388\pi$
$500$	0	0
$501$	−22.3923	−1.00041
$502$	0	0
$503$	26.3205	1.17357	0.586787	−	0.809742i	$-0.300393\pi$
$503$	26.3205	1.17357	0.586787	+	0.809742i	$0.300393\pi$
$504$	0	0
$505$	0.928203	0.0413045
$506$	0	0
$507$	2.32051	0.103057
$508$	0	0
$509$	6.12436	0.271457	0.135729	−	0.990746i	$-0.456662\pi$
$509$	6.12436	0.271457	0.135729	+	0.990746i	$0.456662\pi$
$510$	0	0
$511$	−1.46410	−0.0647680
$512$	0	0
$513$	7.92820	0.350039
$514$	0	0
$515$	10.6603	0.469747
$516$	0	0
$517$	3.66025	0.160978
$518$	0	0
$519$	−16.3923	−0.719542
$520$	0	0
$521$	17.9808	0.787751	0.393876	−	0.919164i	$-0.371134\pi$
$521$	17.9808	0.787751	0.393876	+	0.919164i	$0.371134\pi$
$522$	0	0
$523$	11.3205	0.495011	0.247506	−	0.968886i	$-0.420389\pi$
$523$	11.3205	0.495011	0.247506	+	0.968886i	$0.420389\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−5.80385	−0.252820
$528$	0	0
$529$	−16.9282	−0.736009
$530$	0	0
$531$	14.6603	0.636201
$532$	0	0
$533$	−13.7128	−0.593968
$534$	0	0
$535$	−18.5885	−0.803649
$536$	0	0
$537$	−16.1962	−0.698916
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−5.80385	−0.249527	−0.124763	−	0.992187i	$-0.539817\pi$
$541$	−5.80385	−0.249527	−0.124763	+	0.992187i	$0.539817\pi$
$542$	0	0
$543$	15.8564	0.680464
$544$	0	0
$545$	10.1962	0.436755
$546$	0	0
$547$	28.1244	1.20251	0.601255	−	0.799057i	$-0.294667\pi$
$547$	28.1244	1.20251	0.601255	+	0.799057i	$0.294667\pi$
$548$	0	0
$549$	−11.9282	−0.509083
$550$	0	0
$551$	−13.7321	−0.585005
$552$	0	0
$553$	2.73205	0.116179
$554$	0	0
$555$	−8.73205	−0.370655
$556$	0	0
$557$	7.46410	0.316264	0.158132	−	0.987418i	$-0.449453\pi$
$557$	7.46410	0.316264	0.158132	+	0.987418i	$0.449453\pi$
$558$	0	0
$559$	−30.0526	−1.27109
$560$	0	0
$561$	−7.92820	−0.334729
$562$	0	0
$563$	43.8564	1.84833	0.924164	−	0.381997i	$-0.124764\pi$
$563$	43.8564	1.84833	0.924164	+	0.381997i	$0.124764\pi$
$564$	0	0
$565$	1.00000	0.0420703
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	4.12436	0.172902	0.0864510	−	0.996256i	$-0.472447\pi$
$569$	4.12436	0.172902	0.0864510	+	0.996256i	$0.472447\pi$
$570$	0	0
$571$	23.9090	1.00056	0.500280	−	0.865864i	$-0.333231\pi$
$571$	23.9090	1.00056	0.500280	+	0.865864i	$0.333231\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	2.46410	0.102760
$576$	0	0
$577$	7.85641	0.327066	0.163533	−	0.986538i	$-0.447711\pi$
$577$	7.85641	0.327066	0.163533	+	0.986538i	$0.447711\pi$
$578$	0	0
$579$	18.3923	0.764358
$580$	0	0
$581$	11.0000	0.456357
$582$	0	0
$583$	−8.46410	−0.350547
$584$	0	0
$585$	−3.26795	−0.135113
$586$	0	0
$587$	34.0526	1.40550	0.702750	−	0.711437i	$-0.251955\pi$
$587$	34.0526	1.40550	0.702750	+	0.711437i	$0.251955\pi$
$588$	0	0
$589$	−5.80385	−0.239143
$590$	0	0
$591$	12.3923	0.509751
$592$	0	0
$593$	24.7846	1.01778	0.508891	−	0.860831i	$-0.330056\pi$
$593$	24.7846	1.01778	0.508891	+	0.860831i	$0.330056\pi$
$594$	0	0
$595$	−7.92820	−0.325025
$596$	0	0
$597$	−25.8564	−1.05823
$598$	0	0
$599$	−1.60770	−0.0656886	−0.0328443	−	0.999460i	$-0.510457\pi$
$599$	−1.60770	−0.0656886	−0.0328443	+	0.999460i	$0.510457\pi$
$600$	0	0
$601$	16.3205	0.665727	0.332864	−	0.942975i	$-0.391985\pi$
$601$	16.3205	0.665727	0.332864	+	0.942975i	$0.391985\pi$
$602$	0	0
$603$	14.0000	0.570124
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−35.1769	−1.42779	−0.713893	−	0.700254i	$-0.753070\pi$
$607$	−35.1769	−1.42779	−0.713893	+	0.700254i	$0.753070\pi$
$608$	0	0
$609$	−1.73205	−0.0701862
$610$	0	0
$611$	11.9615	0.483911
$612$	0	0
$613$	28.6410	1.15680	0.578400	−	0.815753i	$-0.303677\pi$
$613$	28.6410	1.15680	0.578400	+	0.815753i	$0.303677\pi$
$614$	0	0
$615$	−4.19615	−0.169205
$616$	0	0
$617$	−0.784610	−0.0315872	−0.0157936	−	0.999875i	$-0.505027\pi$
$617$	−0.784610	−0.0315872	−0.0157936	+	0.999875i	$0.505027\pi$
$618$	0	0
$619$	13.7128	0.551164	0.275582	−	0.961277i	$-0.411129\pi$
$619$	13.7128	0.551164	0.275582	+	0.961277i	$0.411129\pi$
$620$	0	0
$621$	−2.46410	−0.0988810
$622$	0	0
$623$	2.26795	0.0908635
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.92820	−0.316622
$628$	0	0
$629$	−69.2295	−2.76036
$630$	0	0
$631$	−5.14359	−0.204763	−0.102382	−	0.994745i	$-0.532646\pi$
$631$	−5.14359	−0.204763	−0.102382	+	0.994745i	$0.532646\pi$
$632$	0	0
$633$	1.60770	0.0639001
$634$	0	0
$635$	11.1962	0.444306
$636$	0	0
$637$	−3.26795	−0.129481
$638$	0	0
$639$	4.19615	0.165997
$640$	0	0
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	0	0
$643$	−24.8038	−0.978168	−0.489084	−	0.872237i	$-0.662669\pi$
$643$	−24.8038	−0.978168	−0.489084	+	0.872237i	$0.662669\pi$
$644$	0	0
$645$	−9.19615	−0.362098
$646$	0	0
$647$	2.53590	0.0996965	0.0498482	−	0.998757i	$-0.484126\pi$
$647$	2.53590	0.0996965	0.0498482	+	0.998757i	$0.484126\pi$
$648$	0	0
$649$	−14.6603	−0.575465
$650$	0	0
$651$	−0.732051	−0.0286913
$652$	0	0
$653$	38.4641	1.50522	0.752608	−	0.658468i	$-0.228795\pi$
$653$	38.4641	1.50522	0.752608	+	0.658468i	$0.228795\pi$
$654$	0	0
$655$	−9.12436	−0.356518
$656$	0	0
$657$	−1.46410	−0.0571200
$658$	0	0
$659$	21.3397	0.831278	0.415639	−	0.909530i	$-0.363558\pi$
$659$	21.3397	0.831278	0.415639	+	0.909530i	$0.363558\pi$
$660$	0	0
$661$	−46.1962	−1.79682	−0.898411	−	0.439156i	$-0.855278\pi$
$661$	−46.1962	−1.79682	−0.898411	+	0.439156i	$0.855278\pi$
$662$	0	0
$663$	−25.9090	−1.00622
$664$	0	0
$665$	−7.92820	−0.307443
$666$	0	0
$667$	4.26795	0.165256
$668$	0	0
$669$	−20.6603	−0.798772
$670$	0	0
$671$	11.9282	0.460483
$672$	0	0
$673$	42.9090	1.65402	0.827010	−	0.562188i	$-0.190040\pi$
$673$	42.9090	1.65402	0.827010	+	0.562188i	$0.190040\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−17.0000	−0.653363	−0.326682	−	0.945134i	$-0.605930\pi$
$677$	−17.0000	−0.653363	−0.326682	+	0.945134i	$0.605930\pi$
$678$	0	0
$679$	−3.73205	−0.143223
$680$	0	0
$681$	−5.00000	−0.191600
$682$	0	0
$683$	20.7846	0.795301	0.397650	−	0.917537i	$-0.369826\pi$
$683$	20.7846	0.795301	0.397650	+	0.917537i	$0.369826\pi$
$684$	0	0
$685$	13.4641	0.514437
$686$	0	0
$687$	−6.58846	−0.251365
$688$	0	0
$689$	−27.6603	−1.05377
$690$	0	0
$691$	−2.00000	−0.0760836	−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000	−0.0760836	−0.0380418	+	0.999276i	$0.512112\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−15.4641	−0.586587
$696$	0	0
$697$	−33.2679	−1.26011
$698$	0	0
$699$	15.5167	0.586894
$700$	0	0
$701$	37.1962	1.40488	0.702440	−	0.711743i	$-0.252094\pi$
$701$	37.1962	1.40488	0.702440	+	0.711743i	$0.252094\pi$
$702$	0	0
$703$	−69.2295	−2.61104
$704$	0	0
$705$	3.66025	0.137853
$706$	0	0
$707$	0.928203	0.0349087
$708$	0	0
$709$	−15.2487	−0.572677	−0.286339	−	0.958128i	$-0.592438\pi$
$709$	−15.2487	−0.572677	−0.286339	+	0.958128i	$0.592438\pi$
$710$	0	0
$711$	2.73205	0.102460
$712$	0	0
$713$	1.80385	0.0675546
$714$	0	0
$715$	3.26795	0.122214
$716$	0	0
$717$	−26.6603	−0.995645
$718$	0	0
$719$	−38.9090	−1.45106	−0.725530	−	0.688191i	$-0.758405\pi$
$719$	−38.9090	−1.45106	−0.725530	+	0.688191i	$0.758405\pi$
$720$	0	0
$721$	10.6603	0.397009
$722$	0	0
$723$	16.0000	0.595046
$724$	0	0
$725$	1.73205	0.0643268
$726$	0	0
$727$	−24.3731	−0.903947	−0.451974	−	0.892031i	$-0.649280\pi$
$727$	−24.3731	−0.903947	−0.451974	+	0.892031i	$0.649280\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−72.9090	−2.69664
$732$	0	0
$733$	38.3923	1.41805	0.709026	−	0.705182i	$-0.249135\pi$
$733$	38.3923	1.41805	0.709026	+	0.705182i	$0.249135\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−14.0000	−0.515697
$738$	0	0
$739$	−37.3205	−1.37286	−0.686429	−	0.727197i	$-0.740823\pi$
$739$	−37.3205	−1.37286	−0.686429	+	0.727197i	$0.740823\pi$
$740$	0	0
$741$	−25.9090	−0.951790
$742$	0	0
$743$	−2.67949	−0.0983010	−0.0491505	−	0.998791i	$-0.515651\pi$
$743$	−2.67949	−0.0983010	−0.0491505	+	0.998791i	$0.515651\pi$
$744$	0	0
$745$	−0.535898	−0.0196338
$746$	0	0
$747$	11.0000	0.402469
$748$	0	0
$749$	−18.5885	−0.679207
$750$	0	0
$751$	15.3923	0.561673	0.280837	−	0.959756i	$-0.409388\pi$
$751$	15.3923	0.561673	0.280837	+	0.959756i	$0.409388\pi$
$752$	0	0
$753$	20.7846	0.757433
$754$	0	0
$755$	−7.85641	−0.285924
$756$	0	0
$757$	13.3205	0.484142	0.242071	−	0.970259i	$-0.422173\pi$
$757$	13.3205	0.484142	0.242071	+	0.970259i	$0.422173\pi$
$758$	0	0
$759$	2.46410	0.0894412
$760$	0	0
$761$	−40.6410	−1.47324	−0.736618	−	0.676309i	$-0.763578\pi$
$761$	−40.6410	−1.47324	−0.736618	+	0.676309i	$0.763578\pi$
$762$	0	0
$763$	10.1962	0.369126
$764$	0	0
$765$	−7.92820	−0.286645
$766$	0	0
$767$	−47.9090	−1.72989
$768$	0	0
$769$	29.3923	1.05991	0.529957	−	0.848025i	$-0.322208\pi$
$769$	29.3923	1.05991	0.529957	+	0.848025i	$0.322208\pi$
$770$	0	0
$771$	−24.1962	−0.871403
$772$	0	0
$773$	−7.46410	−0.268465	−0.134233	−	0.990950i	$-0.542857\pi$
$773$	−7.46410	−0.268465	−0.134233	+	0.990950i	$0.542857\pi$
$774$	0	0
$775$	0.732051	0.0262960
$776$	0	0
$777$	−8.73205	−0.313261
$778$	0	0
$779$	−33.2679	−1.19195
$780$	0	0
$781$	−4.19615	−0.150150
$782$	0	0
$783$	−1.73205	−0.0618984
$784$	0	0
$785$	12.1244	0.432737
$786$	0	0
$787$	−27.7128	−0.987855	−0.493928	−	0.869503i	$-0.664439\pi$
$787$	−27.7128	−0.987855	−0.493928	+	0.869503i	$0.664439\pi$
$788$	0	0
$789$	−18.5359	−0.659895
$790$	0	0
$791$	1.00000	0.0355559
$792$	0	0
$793$	38.9808	1.38425
$794$	0	0
$795$	−8.46410	−0.300191
$796$	0	0
$797$	5.60770	0.198635	0.0993174	−	0.995056i	$-0.468334\pi$
$797$	5.60770	0.198635	0.0993174	+	0.995056i	$0.468334\pi$
$798$	0	0
$799$	29.0192	1.02663
$800$	0	0
$801$	2.26795	0.0801340
$802$	0	0
$803$	1.46410	0.0516670
$804$	0	0
$805$	2.46410	0.0868482
$806$	0	0
$807$	10.6603	0.375259
$808$	0	0
$809$	−10.3923	−0.365374	−0.182687	−	0.983171i	$-0.558479\pi$
$809$	−10.3923	−0.365374	−0.182687	+	0.983171i	$0.558479\pi$
$810$	0	0
$811$	−14.1436	−0.496649	−0.248324	−	0.968677i	$-0.579880\pi$
$811$	−14.1436	−0.496649	−0.248324	+	0.968677i	$0.579880\pi$
$812$	0	0
$813$	−21.5359	−0.755297
$814$	0	0
$815$	−18.1962	−0.637383
$816$	0	0
$817$	−72.9090	−2.55076
$818$	0	0
$819$	−3.26795	−0.114191
$820$	0	0
$821$	8.66025	0.302245	0.151122	−	0.988515i	$-0.451711\pi$
$821$	8.66025	0.302245	0.151122	+	0.988515i	$0.451711\pi$
$822$	0	0
$823$	42.1962	1.47087	0.735433	−	0.677598i	$-0.236979\pi$
$823$	42.1962	1.47087	0.735433	+	0.677598i	$0.236979\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	29.1769	1.01458	0.507290	−	0.861775i	$-0.330647\pi$
$827$	29.1769	1.01458	0.507290	+	0.861775i	$0.330647\pi$
$828$	0	0
$829$	−30.3923	−1.05557	−0.527784	−	0.849379i	$-0.676977\pi$
$829$	−30.3923	−1.05557	−0.527784	+	0.849379i	$0.676977\pi$
$830$	0	0
$831$	14.9282	0.517854
$832$	0	0
$833$	−7.92820	−0.274696
$834$	0	0
$835$	22.3923	0.774918
$836$	0	0
$837$	−0.732051	−0.0253034
$838$	0	0
$839$	38.9090	1.34329	0.671643	−	0.740875i	$-0.265589\pi$
$839$	38.9090	1.34329	0.671643	+	0.740875i	$0.265589\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	0	0
$843$	12.7846	0.440325
$844$	0	0
$845$	−2.32051	−0.0798279
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	5.80385	0.199188
$850$	0	0
$851$	21.5167	0.737582
$852$	0	0
$853$	6.44486	0.220668	0.110334	−	0.993895i	$-0.464808\pi$
$853$	6.44486	0.220668	0.110334	+	0.993895i	$0.464808\pi$
$854$	0	0
$855$	−7.92820	−0.271139
$856$	0	0
$857$	43.8564	1.49811	0.749053	−	0.662510i	$-0.230509\pi$
$857$	43.8564	1.49811	0.749053	+	0.662510i	$0.230509\pi$
$858$	0	0
$859$	0.0525589	0.00179329	0.000896643	−	1.00000i	$-0.499715\pi$
$859$	0.0525589	0.00179329	0.000896643		1.00000i	$0.499715\pi$
$860$	0	0
$861$	−4.19615	−0.143004
$862$	0	0
$863$	−0.607695	−0.0206862	−0.0103431	−	0.999947i	$-0.503292\pi$
$863$	−0.607695	−0.0206862	−0.0103431	+	0.999947i	$0.503292\pi$
$864$	0	0
$865$	16.3923	0.557355
$866$	0	0
$867$	−45.8564	−1.55737
$868$	0	0
$869$	−2.73205	−0.0926785
$870$	0	0
$871$	−45.7513	−1.55022
$872$	0	0
$873$	−3.73205	−0.126311
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−18.4115	−0.621714	−0.310857	−	0.950457i	$-0.600616\pi$
$877$	−18.4115	−0.621714	−0.310857	+	0.950457i	$0.600616\pi$
$878$	0	0
$879$	−19.2487	−0.649243
$880$	0	0
$881$	−33.0526	−1.11357	−0.556785	−	0.830657i	$-0.687965\pi$
$881$	−33.0526	−1.11357	−0.556785	+	0.830657i	$0.687965\pi$
$882$	0	0
$883$	11.6077	0.390630	0.195315	−	0.980741i	$-0.437427\pi$
$883$	11.6077	0.390630	0.195315	+	0.980741i	$0.437427\pi$
$884$	0	0
$885$	−14.6603	−0.492799
$886$	0	0
$887$	51.9282	1.74358	0.871789	−	0.489881i	$-0.162960\pi$
$887$	51.9282	1.74358	0.871789	+	0.489881i	$0.162960\pi$
$888$	0	0
$889$	11.1962	0.375507
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	29.0192	0.971092
$894$	0	0
$895$	16.1962	0.541378
$896$	0	0
$897$	8.05256	0.268867
$898$	0	0
$899$	1.26795	0.0422885
$900$	0	0
$901$	−67.1051	−2.23560
$902$	0	0
$903$	−9.19615	−0.306029
$904$	0	0
$905$	−15.8564	−0.527085
$906$	0	0
$907$	−8.33975	−0.276917	−0.138458	−	0.990368i	$-0.544215\pi$
$907$	−8.33975	−0.276917	−0.138458	+	0.990368i	$0.544215\pi$
$908$	0	0
$909$	0.928203	0.0307866
$910$	0	0
$911$	−8.39230	−0.278049	−0.139025	−	0.990289i	$-0.544397\pi$
$911$	−8.39230	−0.278049	−0.139025	+	0.990289i	$0.544397\pi$
$912$	0	0
$913$	−11.0000	−0.364047
$914$	0	0
$915$	11.9282	0.394334
$916$	0	0
$917$	−9.12436	−0.301313
$918$	0	0
$919$	19.8564	0.655002	0.327501	−	0.944851i	$-0.393793\pi$
$919$	19.8564	0.655002	0.327501	+	0.944851i	$0.393793\pi$
$920$	0	0
$921$	−12.5359	−0.413072
$922$	0	0
$923$	−13.7128	−0.451363
$924$	0	0
$925$	8.73205	0.287108
$926$	0	0
$927$	10.6603	0.350129
$928$	0	0
$929$	34.6410	1.13653	0.568267	−	0.822844i	$-0.307614\pi$
$929$	34.6410	1.13653	0.568267	+	0.822844i	$0.307614\pi$
$930$	0	0
$931$	−7.92820	−0.259836
$932$	0	0
$933$	11.3205	0.370617
$934$	0	0
$935$	7.92820	0.259280
$936$	0	0
$937$	18.0526	0.589751	0.294876	−	0.955536i	$-0.404722\pi$
$937$	18.0526	0.589751	0.294876	+	0.955536i	$0.404722\pi$
$938$	0	0
$939$	12.8038	0.417838
$940$	0	0
$941$	38.9808	1.27074	0.635368	−	0.772209i	$-0.280848\pi$
$941$	38.9808	1.27074	0.635368	+	0.772209i	$0.280848\pi$
$942$	0	0
$943$	10.3397	0.336708
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−14.2154	−0.461938	−0.230969	−	0.972961i	$-0.574190\pi$
$947$	−14.2154	−0.461938	−0.230969	+	0.972961i	$0.574190\pi$
$948$	0	0
$949$	4.78461	0.155315
$950$	0	0
$951$	24.0000	0.778253
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	8.00000	0.258874
$956$	0	0
$957$	1.73205	0.0559893
$958$	0	0
$959$	13.4641	0.434779
$960$	0	0
$961$	−30.4641	−0.982713
$962$	0	0
$963$	−18.5885	−0.599005
$964$	0	0
$965$	−18.3923	−0.592069
$966$	0	0
$967$	30.2679	0.973352	0.486676	−	0.873583i	$-0.338209\pi$
$967$	30.2679	0.973352	0.486676	+	0.873583i	$0.338209\pi$
$968$	0	0
$969$	−62.8564	−2.01924
$970$	0	0
$971$	41.4449	1.33003	0.665014	−	0.746830i	$-0.268425\pi$
$971$	41.4449	1.33003	0.665014	+	0.746830i	$0.268425\pi$
$972$	0	0
$973$	−15.4641	−0.495756
$974$	0	0
$975$	3.26795	0.104658
$976$	0	0
$977$	46.3205	1.48192	0.740962	−	0.671547i	$-0.234370\pi$
$977$	46.3205	1.48192	0.740962	+	0.671547i	$0.234370\pi$
$978$	0	0
$979$	−2.26795	−0.0724840
$980$	0	0
$981$	10.1962	0.325538
$982$	0	0
$983$	−34.3923	−1.09694	−0.548472	−	0.836169i	$-0.684790\pi$
$983$	−34.3923	−1.09694	−0.548472	+	0.836169i	$0.684790\pi$
$984$	0	0
$985$	−12.3923	−0.394852
$986$	0	0
$987$	3.66025	0.116507
$988$	0	0
$989$	22.6603	0.720554
$990$	0	0
$991$	−29.1436	−0.925777	−0.462888	−	0.886417i	$-0.653187\pi$
$991$	−29.1436	−0.925777	−0.462888	+	0.886417i	$0.653187\pi$
$992$	0	0
$993$	35.2487	1.11858
$994$	0	0
$995$	25.8564	0.819703
$996$	0	0
$997$	−47.8564	−1.51563	−0.757814	−	0.652471i	$-0.773732\pi$
$997$	−47.8564	−1.51563	−0.757814	+	0.652471i	$0.773732\pi$
$998$	0	0
$999$	−8.73205	−0.276270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.s.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.s.1.2	✓	2	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -3.26795 q^{13} -1.00000 q^{15} -7.92820 q^{17} -7.92820 q^{19} -1.00000 q^{21} +2.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.73205 q^{29} +0.732051 q^{31} +1.00000 q^{33} +1.00000 q^{35} +8.73205 q^{37} +3.26795 q^{39} +4.19615 q^{41} +9.19615 q^{43} +1.00000 q^{45} -3.66025 q^{47} +1.00000 q^{49} +7.92820 q^{51} +8.46410 q^{53} -1.00000 q^{55} +7.92820 q^{57} +14.6603 q^{59} -11.9282 q^{61} +1.00000 q^{63} -3.26795 q^{65} +14.0000 q^{67} -2.46410 q^{69} +4.19615 q^{71} -1.46410 q^{73} -1.00000 q^{75} -1.00000 q^{77} +2.73205 q^{79} +1.00000 q^{81} +11.0000 q^{83} -7.92820 q^{85} -1.73205 q^{87} +2.26795 q^{89} -3.26795 q^{91} -0.732051 q^{93} -7.92820 q^{95} -3.73205 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} - 10 q^{13} - 2 q^{15} - 2 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 2 q^{33} + 2 q^{35} + 14 q^{37} + 10 q^{39} - 2 q^{41} + 8 q^{43} + 2 q^{45} + 10 q^{47} + 2 q^{49} + 2 q^{51} + 10 q^{53} - 2 q^{55} + 2 q^{57} + 12 q^{59} - 10 q^{61} + 2 q^{63} - 10 q^{65} + 28 q^{67} + 2 q^{69} - 2 q^{71} + 4 q^{73} - 2 q^{75} - 2 q^{77} + 2 q^{79} + 2 q^{81} + 22 q^{83} - 2 q^{85} + 8 q^{89} - 10 q^{91} + 2 q^{93} - 2 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 - 10 * q^13 - 2 * q^15 - 2 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 2 * q^33 + 2 * q^35 + 14 * q^37 + 10 * q^39 - 2 * q^41 + 8 * q^43 + 2 * q^45 + 10 * q^47 + 2 * q^49 + 2 * q^51 + 10 * q^53 - 2 * q^55 + 2 * q^57 + 12 * q^59 - 10 * q^61 + 2 * q^63 - 10 * q^65 + 28 * q^67 + 2 * q^69 - 2 * q^71 + 4 * q^73 - 2 * q^75 - 2 * q^77 + 2 * q^79 + 2 * q^81 + 22 * q^83 - 2 * q^85 + 8 * q^89 - 10 * q^91 + 2 * q^93 - 2 * q^95 - 4 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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