LMFDB - Embedded newform 1932.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.11903$ of defining polynomial
Character	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.11903 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.11903 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.72833 q^{13} -3.11903 q^{15} +1.39070 q^{17} +3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +4.72833 q^{25} +1.00000 q^{27} +8.06596 q^{29} -5.62875 q^{31} +1.00000 q^{33} +3.11903 q^{35} +1.82790 q^{37} -3.72833 q^{39} +8.01945 q^{41} +8.50973 q^{43} -3.11903 q^{45} +4.39070 q^{47} +1.00000 q^{49} +1.39070 q^{51} +2.72833 q^{53} -3.11903 q^{55} +3.00000 q^{57} -5.74778 q^{59} +2.90043 q^{61} -1.00000 q^{63} +11.6288 q^{65} -0.337628 q^{67} +1.00000 q^{69} +7.29112 q^{71} -2.60930 q^{73} +4.72833 q^{75} -1.00000 q^{77} -10.8474 q^{79} +1.00000 q^{81} +12.0660 q^{83} -4.33763 q^{85} +8.06596 q^{87} +13.9664 q^{89} +3.72833 q^{91} -5.62875 q^{93} -9.35708 q^{95} -3.39070 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * 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
$5$	−3.11903	−1.39487	−0.697436	−	0.716647i	$-0.745676\pi$
$5$	−3.11903	−1.39487	−0.697436	+	0.716647i	$0.745676\pi$
$6$	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	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	−3.72833	−1.03405	−0.517026	−	0.855970i	$-0.672961\pi$
$13$	−3.72833	−1.03405	−0.517026	+	0.855970i	$0.672961\pi$
$14$	0	0
$15$	−3.11903	−0.805329
$16$	0	0
$17$	1.39070	0.337294	0.168647	−	0.985677i	$-0.446060\pi$
$17$	1.39070	0.337294	0.168647	+	0.985677i	$0.446060\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	4.72833	0.945665
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.06596	1.49781	0.748905	−	0.662677i	$-0.230580\pi$
$29$	8.06596	1.49781	0.748905	+	0.662677i	$0.230580\pi$
$30$	0	0
$31$	−5.62875	−1.01095	−0.505477	−	0.862840i	$-0.668683\pi$
$31$	−5.62875	−1.01095	−0.505477	+	0.862840i	$0.668683\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	3.11903	0.527212
$36$	0	0
$37$	1.82790	0.300505	0.150253	−	0.988648i	$-0.451991\pi$
$37$	1.82790	0.300505	0.150253	+	0.988648i	$0.451991\pi$
$38$	0	0
$39$	−3.72833	−0.597010
$40$	0	0
$41$	8.01945	1.25243	0.626214	−	0.779651i	$-0.284604\pi$
$41$	8.01945	1.25243	0.626214	+	0.779651i	$0.284604\pi$
$42$	0	0
$43$	8.50973	1.29772	0.648861	−	0.760907i	$-0.275246\pi$
$43$	8.50973	1.29772	0.648861	+	0.760907i	$0.275246\pi$
$44$	0	0
$45$	−3.11903	−0.464957
$46$	0	0
$47$	4.39070	0.640449	0.320225	−	0.947342i	$-0.396242\pi$
$47$	4.39070	0.640449	0.320225	+	0.947342i	$0.396242\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.39070	0.194737
$52$	0	0
$53$	2.72833	0.374765	0.187382	−	0.982287i	$-0.440000\pi$
$53$	2.72833	0.374765	0.187382	+	0.982287i	$0.440000\pi$
$54$	0	0
$55$	−3.11903	−0.420569
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	−5.74778	−0.748297	−0.374149	−	0.927369i	$-0.622065\pi$
$59$	−5.74778	−0.748297	−0.374149	+	0.927369i	$0.622065\pi$
$60$	0	0
$61$	2.90043	0.371361	0.185681	−	0.982610i	$-0.440551\pi$
$61$	2.90043	0.371361	0.185681	+	0.982610i	$0.440551\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	11.6288	1.44237
$66$	0	0
$67$	−0.337628	−0.0412478	−0.0206239	−	0.999787i	$-0.506565\pi$
$67$	−0.337628	−0.0412478	−0.0206239	+	0.999787i	$0.506565\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	7.29112	0.865297	0.432649	−	0.901563i	$-0.357579\pi$
$71$	7.29112	0.865297	0.432649	+	0.901563i	$0.357579\pi$
$72$	0	0
$73$	−2.60930	−0.305396	−0.152698	−	0.988273i	$-0.548796\pi$
$73$	−2.60930	−0.305396	−0.152698	+	0.988273i	$0.548796\pi$
$74$	0	0
$75$	4.72833	0.545980
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−10.8474	−1.22042	−0.610211	−	0.792239i	$-0.708915\pi$
$79$	−10.8474	−1.22042	−0.610211	+	0.792239i	$0.708915\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0660	1.32441	0.662205	−	0.749322i	$-0.269621\pi$
$83$	12.0660	1.32441	0.662205	+	0.749322i	$0.269621\pi$
$84$	0	0
$85$	−4.33763	−0.470482
$86$	0	0
$87$	8.06596	0.864761
$88$	0	0
$89$	13.9664	1.48043	0.740217	−	0.672368i	$-0.234723\pi$
$89$	13.9664	1.48043	0.740217	+	0.672368i	$0.234723\pi$
$90$	0	0
$91$	3.72833	0.390835
$92$	0	0
$93$	−5.62875	−0.583674
$94$	0	0
$95$	−9.35708	−0.960016
$96$	0	0
$97$	−3.39070	−0.344273	−0.172137	−	0.985073i	$-0.555067\pi$
$97$	−3.39070	−0.344273	−0.172137	+	0.985073i	$0.555067\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.1850	−1.01344	−0.506722	−	0.862110i	$-0.669143\pi$
$101$	−10.1850	−1.01344	−0.506722	+	0.862110i	$0.669143\pi$
$102$	0	0
$103$	7.06596	0.696229	0.348115	−	0.937452i	$-0.386822\pi$
$103$	7.06596	0.696229	0.348115	+	0.937452i	$0.386822\pi$
$104$	0	0
$105$	3.11903	0.304386
$106$	0	0
$107$	5.72833	0.553778	0.276889	−	0.960902i	$-0.410697\pi$
$107$	5.72833	0.553778	0.276889	+	0.960902i	$0.410697\pi$
$108$	0	0
$109$	15.5951	1.49374	0.746871	−	0.664968i	$-0.231555\pi$
$109$	15.5951	1.49374	0.746871	+	0.664968i	$0.231555\pi$
$110$	0	0
$111$	1.82790	0.173497
$112$	0	0
$113$	10.3376	0.972482	0.486241	−	0.873825i	$-0.338368\pi$
$113$	10.3376	0.972482	0.486241	+	0.873825i	$0.338368\pi$
$114$	0	0
$115$	−3.11903	−0.290851
$116$	0	0
$117$	−3.72833	−0.344684
$118$	0	0
$119$	−1.39070	−0.127485
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	8.01945	0.723090
$124$	0	0
$125$	0.847354	0.0757897
$126$	0	0
$127$	16.3571	1.45146	0.725728	−	0.687982i	$-0.241503\pi$
$127$	16.3571	1.45146	0.725728	+	0.687982i	$0.241503\pi$
$128$	0	0
$129$	8.50973	0.749240
$130$	0	0
$131$	−4.45665	−0.389380	−0.194690	−	0.980865i	$-0.562370\pi$
$131$	−4.45665	−0.389380	−0.194690	+	0.980865i	$0.562370\pi$
$132$	0	0
$133$	−3.00000	−0.260133
$134$	0	0
$135$	−3.11903	−0.268443
$136$	0	0
$137$	20.2575	1.73072	0.865358	−	0.501155i	$-0.167091\pi$
$137$	20.2575	1.73072	0.865358	+	0.501155i	$0.167091\pi$
$138$	0	0
$139$	−6.77483	−0.574634	−0.287317	−	0.957836i	$-0.592763\pi$
$139$	−6.77483	−0.574634	−0.287317	+	0.957836i	$0.592763\pi$
$140$	0	0
$141$	4.39070	0.369764
$142$	0	0
$143$	−3.72833	−0.311778
$144$	0	0
$145$	−25.1579	−2.08925
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−2.04650	−0.167656	−0.0838281	−	0.996480i	$-0.526715\pi$
$149$	−2.04650	−0.167656	−0.0838281	+	0.996480i	$0.526715\pi$
$150$	0	0
$151$	−1.17210	−0.0953840	−0.0476920	−	0.998862i	$-0.515187\pi$
$151$	−1.17210	−0.0953840	−0.0476920	+	0.998862i	$0.515187\pi$
$152$	0	0
$153$	1.39070	0.112431
$154$	0	0
$155$	17.5562	1.41015
$156$	0	0
$157$	−18.0854	−1.44337	−0.721686	−	0.692220i	$-0.756633\pi$
$157$	−18.0854	−1.44337	−0.721686	+	0.692220i	$0.756633\pi$
$158$	0	0
$159$	2.72833	0.216370
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−12.2911	−0.962715	−0.481358	−	0.876524i	$-0.659856\pi$
$163$	−12.2911	−0.962715	−0.481358	+	0.876524i	$0.659856\pi$
$164$	0	0
$165$	−3.11903	−0.242816
$166$	0	0
$167$	1.23805	0.0958034	0.0479017	−	0.998852i	$-0.484747\pi$
$167$	1.23805	0.0958034	0.0479017	+	0.998852i	$0.484747\pi$
$168$	0	0
$169$	0.900425	0.0692635
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	−11.8668	−0.902217	−0.451108	−	0.892469i	$-0.648971\pi$
$173$	−11.8668	−0.902217	−0.451108	+	0.892469i	$0.648971\pi$
$174$	0	0
$175$	−4.72833	−0.357428
$176$	0	0
$177$	−5.74778	−0.432030
$178$	0	0
$179$	−8.68182	−0.648910	−0.324455	−	0.945901i	$-0.605181\pi$
$179$	−8.68182	−0.648910	−0.324455	+	0.945901i	$0.605181\pi$
$180$	0	0
$181$	−19.8668	−1.47669	−0.738344	−	0.674424i	$-0.764392\pi$
$181$	−19.8668	−1.47669	−0.738344	+	0.674424i	$0.764392\pi$
$182$	0	0
$183$	2.90043	0.214406
$184$	0	0
$185$	−5.70128	−0.419166
$186$	0	0
$187$	1.39070	0.101698
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−16.0854	−1.16390	−0.581950	−	0.813225i	$-0.697710\pi$
$191$	−16.0854	−1.16390	−0.581950	+	0.813225i	$0.697710\pi$
$192$	0	0
$193$	9.30401	0.669717	0.334859	−	0.942268i	$-0.391311\pi$
$193$	9.30401	0.669717	0.334859	+	0.942268i	$0.391311\pi$
$194$	0	0
$195$	11.6288	0.832752
$196$	0	0
$197$	−4.74778	−0.338265	−0.169133	−	0.985593i	$-0.554097\pi$
$197$	−4.74778	−0.338265	−0.169133	+	0.985593i	$0.554097\pi$
$198$	0	0
$199$	7.31058	0.518233	0.259117	−	0.965846i	$-0.416569\pi$
$199$	7.31058	0.518233	0.259117	+	0.965846i	$0.416569\pi$
$200$	0	0
$201$	−0.337628	−0.0238145
$202$	0	0
$203$	−8.06596	−0.566119
$204$	0	0
$205$	−25.0129	−1.74698
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	−0.800850	−0.0551328	−0.0275664	−	0.999620i	$-0.508776\pi$
$211$	−0.800850	−0.0551328	−0.0275664	+	0.999620i	$0.508776\pi$
$212$	0	0
$213$	7.29112	0.499580
$214$	0	0
$215$	−26.5421	−1.81015
$216$	0	0
$217$	5.62875	0.382105
$218$	0	0
$219$	−2.60930	−0.176320
$220$	0	0
$221$	−5.18498	−0.348780
$222$	0	0
$223$	22.2704	1.49134	0.745668	−	0.666318i	$-0.232131\pi$
$223$	22.2704	1.49134	0.745668	+	0.666318i	$0.232131\pi$
$224$	0	0
$225$	4.72833	0.315222
$226$	0	0
$227$	19.5562	1.29799	0.648996	−	0.760792i	$-0.275189\pi$
$227$	19.5562	1.29799	0.648996	+	0.760792i	$0.275189\pi$
$228$	0	0
$229$	−2.83447	−0.187307	−0.0936535	−	0.995605i	$-0.529855\pi$
$229$	−2.83447	−0.187307	−0.0936535	+	0.995605i	$0.529855\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	2.77483	0.181785	0.0908926	−	0.995861i	$-0.471028\pi$
$233$	2.77483	0.181785	0.0908926	+	0.995861i	$0.471028\pi$
$234$	0	0
$235$	−13.6947	−0.893344
$236$	0	0
$237$	−10.8474	−0.704611
$238$	0	0
$239$	−0.536778	−0.0347213	−0.0173606	−	0.999849i	$-0.505526\pi$
$239$	−0.536778	−0.0347213	−0.0173606	+	0.999849i	$0.505526\pi$
$240$	0	0
$241$	14.9328	0.961904	0.480952	−	0.876747i	$-0.340291\pi$
$241$	14.9328	0.961904	0.480952	+	0.876747i	$0.340291\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.11903	−0.199267
$246$	0	0
$247$	−11.1850	−0.711683
$248$	0	0
$249$	12.0660	0.764649
$250$	0	0
$251$	−21.9328	−1.38438	−0.692192	−	0.721714i	$-0.743355\pi$
$251$	−21.9328	−1.38438	−0.692192	+	0.721714i	$0.743355\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	0	0
$255$	−4.33763	−0.271633
$256$	0	0
$257$	−12.0854	−0.753867	−0.376934	−	0.926240i	$-0.623021\pi$
$257$	−12.0854	−0.753867	−0.376934	+	0.926240i	$0.623021\pi$
$258$	0	0
$259$	−1.82790	−0.113580
$260$	0	0
$261$	8.06596	0.499270
$262$	0	0
$263$	2.01945	0.124525	0.0622624	−	0.998060i	$-0.480168\pi$
$263$	2.01945	0.124525	0.0622624	+	0.998060i	$0.480168\pi$
$264$	0	0
$265$	−8.50973	−0.522748
$266$	0	0
$267$	13.9664	0.854729
$268$	0	0
$269$	−19.5292	−1.19071	−0.595357	−	0.803461i	$-0.702990\pi$
$269$	−19.5292	−1.19071	−0.595357	+	0.803461i	$0.702990\pi$
$270$	0	0
$271$	15.5615	0.945295	0.472647	−	0.881252i	$-0.343298\pi$
$271$	15.5615	0.945295	0.472647	+	0.881252i	$0.343298\pi$
$272$	0	0
$273$	3.72833	0.225649
$274$	0	0
$275$	4.72833	0.285129
$276$	0	0
$277$	8.35708	0.502128	0.251064	−	0.967970i	$-0.419219\pi$
$277$	8.35708	0.502128	0.251064	+	0.967970i	$0.419219\pi$
$278$	0	0
$279$	−5.62875	−0.336985
$280$	0	0
$281$	19.0195	1.13461	0.567303	−	0.823509i	$-0.307987\pi$
$281$	19.0195	1.13461	0.567303	+	0.823509i	$0.307987\pi$
$282$	0	0
$283$	−9.35708	−0.556221	−0.278110	−	0.960549i	$-0.589708\pi$
$283$	−9.35708	−0.556221	−0.278110	+	0.960549i	$0.589708\pi$
$284$	0	0
$285$	−9.35708	−0.554266
$286$	0	0
$287$	−8.01945	−0.473373
$288$	0	0
$289$	−15.0660	−0.886233
$290$	0	0
$291$	−3.39070	−0.198766
$292$	0	0
$293$	12.4761	0.728862	0.364431	−	0.931230i	$-0.381264\pi$
$293$	12.4761	0.728862	0.364431	+	0.931230i	$0.381264\pi$
$294$	0	0
$295$	17.9275	1.04378
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−3.72833	−0.215615
$300$	0	0
$301$	−8.50973	−0.490492
$302$	0	0
$303$	−10.1850	−0.585112
$304$	0	0
$305$	−9.04650	−0.518001
$306$	0	0
$307$	27.7607	1.58438	0.792192	−	0.610271i	$-0.208940\pi$
$307$	27.7607	1.58438	0.792192	+	0.610271i	$0.208940\pi$
$308$	0	0
$309$	7.06596	0.401968
$310$	0	0
$311$	27.5757	1.56367	0.781837	−	0.623483i	$-0.214283\pi$
$311$	27.5757	1.56367	0.781837	+	0.623483i	$0.214283\pi$
$312$	0	0
$313$	7.75435	0.438302	0.219151	−	0.975691i	$-0.429671\pi$
$313$	7.75435	0.438302	0.219151	+	0.975691i	$0.429671\pi$
$314$	0	0
$315$	3.11903	0.175737
$316$	0	0
$317$	21.3960	1.20172	0.600859	−	0.799355i	$-0.294825\pi$
$317$	21.3960	1.20172	0.600859	+	0.799355i	$0.294825\pi$
$318$	0	0
$319$	8.06596	0.451607
$320$	0	0
$321$	5.72833	0.319724
$322$	0	0
$323$	4.17210	0.232142
$324$	0	0
$325$	−17.6288	−0.977867
$326$	0	0
$327$	15.5951	0.862413
$328$	0	0
$329$	−4.39070	−0.242067
$330$	0	0
$331$	26.0182	1.43009	0.715044	−	0.699080i	$-0.246407\pi$
$331$	26.0182	1.43009	0.715044	+	0.699080i	$0.246407\pi$
$332$	0	0
$333$	1.82790	0.100168
$334$	0	0
$335$	1.05307	0.0575354
$336$	0	0
$337$	−5.09197	−0.277378	−0.138689	−	0.990336i	$-0.544289\pi$
$337$	−5.09197	−0.277378	−0.138689	+	0.990336i	$0.544289\pi$
$338$	0	0
$339$	10.3376	0.561463
$340$	0	0
$341$	−5.62875	−0.304814
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.11903	−0.167923
$346$	0	0
$347$	−6.30401	−0.338417	−0.169208	−	0.985580i	$-0.554121\pi$
$347$	−6.30401	−0.338417	−0.169208	+	0.985580i	$0.554121\pi$
$348$	0	0
$349$	−32.9199	−1.76216	−0.881080	−	0.472967i	$-0.843183\pi$
$349$	−32.9199	−1.76216	−0.881080	+	0.472967i	$0.843183\pi$
$350$	0	0
$351$	−3.72833	−0.199003
$352$	0	0
$353$	−11.5226	−0.613287	−0.306643	−	0.951824i	$-0.599206\pi$
$353$	−11.5226	−0.613287	−0.306643	+	0.951824i	$0.599206\pi$
$354$	0	0
$355$	−22.7412	−1.20698
$356$	0	0
$357$	−1.39070	−0.0736036
$358$	0	0
$359$	7.05179	0.372179	0.186090	−	0.982533i	$-0.440419\pi$
$359$	7.05179	0.372179	0.186090	+	0.982533i	$0.440419\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	−10.0000	−0.524864
$364$	0	0
$365$	8.13848	0.425987
$366$	0	0
$367$	15.1385	0.790222	0.395111	−	0.918633i	$-0.370706\pi$
$367$	15.1385	0.790222	0.395111	+	0.918633i	$0.370706\pi$
$368$	0	0
$369$	8.01945	0.417476
$370$	0	0
$371$	−2.72833	−0.141648
$372$	0	0
$373$	−6.17210	−0.319579	−0.159790	−	0.987151i	$-0.551082\pi$
$373$	−6.17210	−0.319579	−0.159790	+	0.987151i	$0.551082\pi$
$374$	0	0
$375$	0.847354	0.0437572
$376$	0	0
$377$	−30.0725	−1.54881
$378$	0	0
$379$	10.3442	0.531346	0.265673	−	0.964063i	$-0.414406\pi$
$379$	10.3442	0.531346	0.265673	+	0.964063i	$0.414406\pi$
$380$	0	0
$381$	16.3571	0.837999
$382$	0	0
$383$	24.1979	1.23645	0.618227	−	0.786000i	$-0.287851\pi$
$383$	24.1979	1.23645	0.618227	+	0.786000i	$0.287851\pi$
$384$	0	0
$385$	3.11903	0.158960
$386$	0	0
$387$	8.50973	0.432574
$388$	0	0
$389$	−35.1243	−1.78087	−0.890437	−	0.455107i	$-0.849601\pi$
$389$	−35.1243	−1.78087	−0.890437	+	0.455107i	$0.849601\pi$
$390$	0	0
$391$	1.39070	0.0703307
$392$	0	0
$393$	−4.45665	−0.224808
$394$	0	0
$395$	33.8332	1.70233
$396$	0	0
$397$	−22.6753	−1.13804	−0.569019	−	0.822324i	$-0.692677\pi$
$397$	−22.6753	−1.13804	−0.569019	+	0.822324i	$0.692677\pi$
$398$	0	0
$399$	−3.00000	−0.150188
$400$	0	0
$401$	32.6275	1.62934	0.814669	−	0.579926i	$-0.196919\pi$
$401$	32.6275	1.62934	0.814669	+	0.579926i	$0.196919\pi$
$402$	0	0
$403$	20.9858	1.04538
$404$	0	0
$405$	−3.11903	−0.154986
$406$	0	0
$407$	1.82790	0.0906057
$408$	0	0
$409$	−37.6288	−1.86062	−0.930311	−	0.366772i	$-0.880463\pi$
$409$	−37.6288	−1.86062	−0.930311	+	0.366772i	$0.880463\pi$
$410$	0	0
$411$	20.2575	0.999229
$412$	0	0
$413$	5.74778	0.282830
$414$	0	0
$415$	−37.6340	−1.84738
$416$	0	0
$417$	−6.77483	−0.331765
$418$	0	0
$419$	−25.5951	−1.25040	−0.625202	−	0.780463i	$-0.714983\pi$
$419$	−25.5951	−1.25040	−0.625202	+	0.780463i	$0.714983\pi$
$420$	0	0
$421$	−38.3363	−1.86840	−0.934200	−	0.356751i	$-0.883885\pi$
$421$	−38.3363	−1.86840	−0.934200	+	0.356751i	$0.883885\pi$
$422$	0	0
$423$	4.39070	0.213483
$424$	0	0
$425$	6.57568	0.318967
$426$	0	0
$427$	−2.90043	−0.140361
$428$	0	0
$429$	−3.72833	−0.180005
$430$	0	0
$431$	−11.1385	−0.536522	−0.268261	−	0.963346i	$-0.586449\pi$
$431$	−11.1385	−0.536522	−0.268261	+	0.963346i	$0.586449\pi$
$432$	0	0
$433$	−1.70231	−0.0818077	−0.0409039	−	0.999163i	$-0.513024\pi$
$433$	−1.70231	−0.0818077	−0.0409039	+	0.999163i	$0.513024\pi$
$434$	0	0
$435$	−25.1579	−1.20623
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−1.45794	−0.0695836	−0.0347918	−	0.999395i	$-0.511077\pi$
$439$	−1.45794	−0.0695836	−0.0347918	+	0.999395i	$0.511077\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−15.1256	−0.718639	−0.359319	−	0.933215i	$-0.616991\pi$
$443$	−15.1256	−0.718639	−0.359319	+	0.933215i	$0.616991\pi$
$444$	0	0
$445$	−43.5615	−2.06501
$446$	0	0
$447$	−2.04650	−0.0967963
$448$	0	0
$449$	−12.0323	−0.567841	−0.283921	−	0.958848i	$-0.591635\pi$
$449$	−12.0323	−0.567841	−0.283921	+	0.958848i	$0.591635\pi$
$450$	0	0
$451$	8.01945	0.377621
$452$	0	0
$453$	−1.17210	−0.0550700
$454$	0	0
$455$	−11.6288	−0.545164
$456$	0	0
$457$	−37.7270	−1.76480	−0.882398	−	0.470503i	$-0.844072\pi$
$457$	−37.7270	−1.76480	−0.882398	+	0.470503i	$0.844072\pi$
$458$	0	0
$459$	1.39070	0.0649123
$460$	0	0
$461$	13.0518	0.607882	0.303941	−	0.952691i	$-0.401697\pi$
$461$	13.0518	0.607882	0.303941	+	0.952691i	$0.401697\pi$
$462$	0	0
$463$	5.30529	0.246558	0.123279	−	0.992372i	$-0.460659\pi$
$463$	5.30529	0.246558	0.123279	+	0.992372i	$0.460659\pi$
$464$	0	0
$465$	17.5562	0.814151
$466$	0	0
$467$	−3.58985	−0.166118	−0.0830592	−	0.996545i	$-0.526469\pi$
$467$	−3.58985	−0.166118	−0.0830592	+	0.996545i	$0.526469\pi$
$468$	0	0
$469$	0.337628	0.0155902
$470$	0	0
$471$	−18.0854	−0.833332
$472$	0	0
$473$	8.50973	0.391278
$474$	0	0
$475$	14.1850	0.650852
$476$	0	0
$477$	2.72833	0.124922
$478$	0	0
$479$	3.69599	0.168874	0.0844371	−	0.996429i	$-0.473091\pi$
$479$	3.69599	0.168874	0.0844371	+	0.996429i	$0.473091\pi$
$480$	0	0
$481$	−6.81502	−0.310738
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	10.5757	0.480217
$486$	0	0
$487$	−9.32346	−0.422486	−0.211243	−	0.977434i	$-0.567751\pi$
$487$	−9.32346	−0.422486	−0.211243	+	0.977434i	$0.567751\pi$
$488$	0	0
$489$	−12.2911	−0.555824
$490$	0	0
$491$	28.2044	1.27285	0.636424	−	0.771339i	$-0.280413\pi$
$491$	28.2044	1.27285	0.636424	+	0.771339i	$0.280413\pi$
$492$	0	0
$493$	11.2173	0.505203
$494$	0	0
$495$	−3.11903	−0.140190
$496$	0	0
$497$	−7.29112	−0.327052
$498$	0	0
$499$	−4.88097	−0.218502	−0.109251	−	0.994014i	$-0.534845\pi$
$499$	−4.88097	−0.218502	−0.109251	+	0.994014i	$0.534845\pi$
$500$	0	0
$501$	1.23805	0.0553121
$502$	0	0
$503$	11.9393	0.532348	0.266174	−	0.963925i	$-0.414240\pi$
$503$	11.9393	0.532348	0.266174	+	0.963925i	$0.414240\pi$
$504$	0	0
$505$	31.7672	1.41362
$506$	0	0
$507$	0.900425	0.0399893
$508$	0	0
$509$	−7.60930	−0.337276	−0.168638	−	0.985678i	$-0.553937\pi$
$509$	−7.60930	−0.337276	−0.168638	+	0.985678i	$0.553937\pi$
$510$	0	0
$511$	2.60930	0.115429
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	−22.0389	−0.971150
$516$	0	0
$517$	4.39070	0.193103
$518$	0	0
$519$	−11.8668	−0.520895
$520$	0	0
$521$	25.6017	1.12163	0.560815	−	0.827941i	$-0.310488\pi$
$521$	25.6017	1.12163	0.560815	+	0.827941i	$0.310488\pi$
$522$	0	0
$523$	−6.93404	−0.303205	−0.151602	−	0.988442i	$-0.548443\pi$
$523$	−6.93404	−0.303205	−0.151602	+	0.988442i	$0.548443\pi$
$524$	0	0
$525$	−4.72833	−0.206361
$526$	0	0
$527$	−7.82790	−0.340989
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.74778	−0.249432
$532$	0	0
$533$	−29.8991	−1.29508
$534$	0	0
$535$	−17.8668	−0.772449
$536$	0	0
$537$	−8.68182	−0.374648
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	31.7801	1.36633	0.683167	−	0.730262i	$-0.260602\pi$
$541$	31.7801	1.36633	0.683167	+	0.730262i	$0.260602\pi$
$542$	0	0
$543$	−19.8668	−0.852566
$544$	0	0
$545$	−48.6416	−2.08358
$546$	0	0
$547$	19.5951	0.837827	0.418914	−	0.908026i	$-0.362411\pi$
$547$	19.5951	0.837827	0.418914	+	0.908026i	$0.362411\pi$
$548$	0	0
$549$	2.90043	0.123787
$550$	0	0
$551$	24.1979	1.03086
$552$	0	0
$553$	10.8474	0.461276
$554$	0	0
$555$	−5.70128	−0.242006
$556$	0	0
$557$	−8.58225	−0.363642	−0.181821	−	0.983332i	$-0.558199\pi$
$557$	−8.58225	−0.363642	−0.181821	+	0.983332i	$0.558199\pi$
$558$	0	0
$559$	−31.7270	−1.34191
$560$	0	0
$561$	1.39070	0.0587154
$562$	0	0
$563$	−5.18498	−0.218521	−0.109260	−	0.994013i	$-0.534848\pi$
$563$	−5.18498	−0.218521	−0.109260	+	0.994013i	$0.534848\pi$
$564$	0	0
$565$	−32.2433	−1.35649
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	9.41015	0.394494	0.197247	−	0.980354i	$-0.436800\pi$
$569$	9.41015	0.394494	0.197247	+	0.980354i	$0.436800\pi$
$570$	0	0
$571$	27.1526	1.13630	0.568151	−	0.822924i	$-0.307659\pi$
$571$	27.1526	1.13630	0.568151	+	0.822924i	$0.307659\pi$
$572$	0	0
$573$	−16.0854	−0.671978
$574$	0	0
$575$	4.72833	0.197185
$576$	0	0
$577$	18.9522	0.788991	0.394495	−	0.918898i	$-0.370919\pi$
$577$	18.9522	0.788991	0.394495	+	0.918898i	$0.370919\pi$
$578$	0	0
$579$	9.30401	0.386661
$580$	0	0
$581$	−12.0660	−0.500580
$582$	0	0
$583$	2.72833	0.112996
$584$	0	0
$585$	11.6288	0.480790
$586$	0	0
$587$	22.6624	0.935376	0.467688	−	0.883894i	$-0.345087\pi$
$587$	22.6624	0.935376	0.467688	+	0.883894i	$0.345087\pi$
$588$	0	0
$589$	−16.8863	−0.695786
$590$	0	0
$591$	−4.74778	−0.195298
$592$	0	0
$593$	−17.4567	−0.716859	−0.358429	−	0.933557i	$-0.616688\pi$
$593$	−17.4567	−0.716859	−0.358429	+	0.933557i	$0.616688\pi$
$594$	0	0
$595$	4.33763	0.177825
$596$	0	0
$597$	7.31058	0.299202
$598$	0	0
$599$	42.0311	1.71734	0.858671	−	0.512527i	$-0.171291\pi$
$599$	42.0311	1.71734	0.858671	+	0.512527i	$0.171291\pi$
$600$	0	0
$601$	−26.7207	−1.08996	−0.544981	−	0.838449i	$-0.683463\pi$
$601$	−26.7207	−1.08996	−0.544981	+	0.838449i	$0.683463\pi$
$602$	0	0
$603$	−0.337628	−0.0137493
$604$	0	0
$605$	31.1903	1.26806
$606$	0	0
$607$	0.310576	0.0126059	0.00630295	−	0.999980i	$-0.497994\pi$
$607$	0.310576	0.0126059	0.00630295	+	0.999980i	$0.497994\pi$
$608$	0	0
$609$	−8.06596	−0.326849
$610$	0	0
$611$	−16.3700	−0.662258
$612$	0	0
$613$	−5.28456	−0.213441	−0.106721	−	0.994289i	$-0.534035\pi$
$613$	−5.28456	−0.213441	−0.106721	+	0.994289i	$0.534035\pi$
$614$	0	0
$615$	−25.0129	−1.00862
$616$	0	0
$617$	4.47082	0.179989	0.0899943	−	0.995942i	$-0.471315\pi$
$617$	4.47082	0.179989	0.0899943	+	0.995942i	$0.471315\pi$
$618$	0	0
$619$	−15.8345	−0.636441	−0.318221	−	0.948017i	$-0.603085\pi$
$619$	−15.8345	−0.636441	−0.318221	+	0.948017i	$0.603085\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−13.9664	−0.559551
$624$	0	0
$625$	−26.2846	−1.05138
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	2.54206	0.101359
$630$	0	0
$631$	29.8279	1.18743	0.593715	−	0.804675i	$-0.297661\pi$
$631$	29.8279	1.18743	0.593715	+	0.804675i	$0.297661\pi$
$632$	0	0
$633$	−0.800850	−0.0318309
$634$	0	0
$635$	−51.0182	−2.02459
$636$	0	0
$637$	−3.72833	−0.147722
$638$	0	0
$639$	7.29112	0.288432
$640$	0	0
$641$	−13.0712	−0.516283	−0.258141	−	0.966107i	$-0.583110\pi$
$641$	−13.0712	−0.516283	−0.258141	+	0.966107i	$0.583110\pi$
$642$	0	0
$643$	−29.9845	−1.18248	−0.591238	−	0.806497i	$-0.701360\pi$
$643$	−29.9845	−1.18248	−0.591238	+	0.806497i	$0.701360\pi$
$644$	0	0
$645$	−26.5421	−1.04509
$646$	0	0
$647$	−18.0323	−0.708924	−0.354462	−	0.935070i	$-0.615336\pi$
$647$	−18.0323	−0.708924	−0.354462	+	0.935070i	$0.615336\pi$
$648$	0	0
$649$	−5.74778	−0.225620
$650$	0	0
$651$	5.62875	0.220608
$652$	0	0
$653$	−0.749062	−0.0293131	−0.0146565	−	0.999893i	$-0.504665\pi$
$653$	−0.749062	−0.0293131	−0.0146565	+	0.999893i	$0.504665\pi$
$654$	0	0
$655$	13.9004	0.543135
$656$	0	0
$657$	−2.60930	−0.101799
$658$	0	0
$659$	24.8474	0.967915	0.483958	−	0.875091i	$-0.339199\pi$
$659$	24.8474	0.967915	0.483958	+	0.875091i	$0.339199\pi$
$660$	0	0
$661$	23.8461	0.927505	0.463752	−	0.885965i	$-0.346503\pi$
$661$	23.8461	0.927505	0.463752	+	0.885965i	$0.346503\pi$
$662$	0	0
$663$	−5.18498	−0.201368
$664$	0	0
$665$	9.35708	0.362852
$666$	0	0
$667$	8.06596	0.312315
$668$	0	0
$669$	22.2704	0.861023
$670$	0	0
$671$	2.90043	0.111970
$672$	0	0
$673$	8.31161	0.320389	0.160194	−	0.987085i	$-0.448788\pi$
$673$	8.31161	0.320389	0.160194	+	0.987085i	$0.448788\pi$
$674$	0	0
$675$	4.72833	0.181993
$676$	0	0
$677$	−13.9275	−0.535276	−0.267638	−	0.963519i	$-0.586243\pi$
$677$	−13.9275	−0.535276	−0.267638	+	0.963519i	$0.586243\pi$
$678$	0	0
$679$	3.39070	0.130123
$680$	0	0
$681$	19.5562	0.749396
$682$	0	0
$683$	−14.2651	−0.545839	−0.272920	−	0.962037i	$-0.587989\pi$
$683$	−14.2651	−0.545839	−0.272920	+	0.962037i	$0.587989\pi$
$684$	0	0
$685$	−63.1837	−2.41413
$686$	0	0
$687$	−2.83447	−0.108142
$688$	0	0
$689$	−10.1721	−0.387526
$690$	0	0
$691$	−25.1850	−0.958082	−0.479041	−	0.877793i	$-0.659015\pi$
$691$	−25.1850	−0.958082	−0.479041	+	0.877793i	$0.659015\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	21.1309	0.801540
$696$	0	0
$697$	11.1526	0.422437
$698$	0	0
$699$	2.77483	0.104954
$700$	0	0
$701$	−19.1114	−0.721829	−0.360914	−	0.932599i	$-0.617535\pi$
$701$	−19.1114	−0.721829	−0.360914	+	0.932599i	$0.617535\pi$
$702$	0	0
$703$	5.48371	0.206822
$704$	0	0
$705$	−13.6947	−0.515773
$706$	0	0
$707$	10.1850	0.383046
$708$	0	0
$709$	30.7735	1.15572	0.577862	−	0.816134i	$-0.303887\pi$
$709$	30.7735	1.15572	0.577862	+	0.816134i	$0.303887\pi$
$710$	0	0
$711$	−10.8474	−0.406808
$712$	0	0
$713$	−5.62875	−0.210798
$714$	0	0
$715$	11.6288	0.434891
$716$	0	0
$717$	−0.536778	−0.0200463
$718$	0	0
$719$	−2.67654	−0.0998181	−0.0499090	−	0.998754i	$-0.515893\pi$
$719$	−2.67654	−0.0998181	−0.0499090	+	0.998754i	$0.515893\pi$
$720$	0	0
$721$	−7.06596	−0.263150
$722$	0	0
$723$	14.9328	0.555355
$724$	0	0
$725$	38.1385	1.41643
$726$	0	0
$727$	30.3505	1.12564	0.562819	−	0.826580i	$-0.309717\pi$
$727$	30.3505	1.12564	0.562819	+	0.826580i	$0.309717\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.8345	0.437714
$732$	0	0
$733$	7.01945	0.259269	0.129635	−	0.991562i	$-0.458620\pi$
$733$	7.01945	0.259269	0.129635	+	0.991562i	$0.458620\pi$
$734$	0	0
$735$	−3.11903	−0.115047
$736$	0	0
$737$	−0.337628	−0.0124367
$738$	0	0
$739$	−50.9509	−1.87426	−0.937130	−	0.348980i	$-0.886528\pi$
$739$	−50.9509	−1.87426	−0.937130	+	0.348980i	$0.886528\pi$
$740$	0	0
$741$	−11.1850	−0.410891
$742$	0	0
$743$	−8.39727	−0.308066	−0.154033	−	0.988066i	$-0.549226\pi$
$743$	−8.39727	−0.308066	−0.154033	+	0.988066i	$0.549226\pi$
$744$	0	0
$745$	6.38310	0.233859
$746$	0	0
$747$	12.0660	0.441470
$748$	0	0
$749$	−5.72833	−0.209309
$750$	0	0
$751$	20.2057	0.737317	0.368659	−	0.929565i	$-0.379817\pi$
$751$	20.2057	0.737317	0.368659	+	0.929565i	$0.379817\pi$
$752$	0	0
$753$	−21.9328	−0.799274
$754$	0	0
$755$	3.65580	0.133048
$756$	0	0
$757$	51.2951	1.86435	0.932177	−	0.362004i	$-0.117907\pi$
$757$	51.2951	1.86435	0.932177	+	0.362004i	$0.117907\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	25.5032	0.924489	0.462244	−	0.886753i	$-0.347044\pi$
$761$	25.5032	0.924489	0.462244	+	0.886753i	$0.347044\pi$
$762$	0	0
$763$	−15.5951	−0.564582
$764$	0	0
$765$	−4.33763	−0.156827
$766$	0	0
$767$	21.4296	0.773778
$768$	0	0
$769$	12.3700	0.446072	0.223036	−	0.974810i	$-0.428403\pi$
$769$	12.3700	0.446072	0.223036	+	0.974810i	$0.428403\pi$
$770$	0	0
$771$	−12.0854	−0.435245
$772$	0	0
$773$	−0.503158	−0.0180974	−0.00904868	−	0.999959i	$-0.502880\pi$
$773$	−0.503158	−0.0180974	−0.00904868	+	0.999959i	$0.502880\pi$
$774$	0	0
$775$	−26.6146	−0.956024
$776$	0	0
$777$	−1.82790	−0.0655756
$778$	0	0
$779$	24.0584	0.861980
$780$	0	0
$781$	7.29112	0.260897
$782$	0	0
$783$	8.06596	0.288254
$784$	0	0
$785$	56.4089	2.01332
$786$	0	0
$787$	20.4903	0.730399	0.365200	−	0.930929i	$-0.381001\pi$
$787$	20.4903	0.730399	0.365200	+	0.930929i	$0.381001\pi$
$788$	0	0
$789$	2.01945	0.0718944
$790$	0	0
$791$	−10.3376	−0.367564
$792$	0	0
$793$	−10.8137	−0.384007
$794$	0	0
$795$	−8.50973	−0.301809
$796$	0	0
$797$	−11.3636	−0.402521	−0.201261	−	0.979538i	$-0.564504\pi$
$797$	−11.3636	−0.402521	−0.201261	+	0.979538i	$0.564504\pi$
$798$	0	0
$799$	6.10614	0.216020
$800$	0	0
$801$	13.9664	0.493478
$802$	0	0
$803$	−2.60930	−0.0920802
$804$	0	0
$805$	3.11903	0.109931
$806$	0	0
$807$	−19.5292	−0.687460
$808$	0	0
$809$	18.4425	0.648403	0.324202	−	0.945988i	$-0.394904\pi$
$809$	18.4425	0.648403	0.324202	+	0.945988i	$0.394904\pi$
$810$	0	0
$811$	23.1243	0.812004	0.406002	−	0.913872i	$-0.366923\pi$
$811$	23.1243	0.812004	0.406002	+	0.913872i	$0.366923\pi$
$812$	0	0
$813$	15.5615	0.545766
$814$	0	0
$815$	38.3363	1.34286
$816$	0	0
$817$	25.5292	0.893153
$818$	0	0
$819$	3.72833	0.130278
$820$	0	0
$821$	−32.4089	−1.13108	−0.565539	−	0.824722i	$-0.691332\pi$
$821$	−32.4089	−1.13108	−0.565539	+	0.824722i	$0.691332\pi$
$822$	0	0
$823$	−4.92748	−0.171761	−0.0858805	−	0.996305i	$-0.527370\pi$
$823$	−4.92748	−0.171761	−0.0858805	+	0.996305i	$0.527370\pi$
$824$	0	0
$825$	4.72833	0.164619
$826$	0	0
$827$	0.635320	0.0220922	0.0110461	−	0.999939i	$-0.496484\pi$
$827$	0.635320	0.0220922	0.0110461	+	0.999939i	$0.496484\pi$
$828$	0	0
$829$	24.1438	0.838548	0.419274	−	0.907860i	$-0.362285\pi$
$829$	24.1438	0.838548	0.419274	+	0.907860i	$0.362285\pi$
$830$	0	0
$831$	8.35708	0.289904
$832$	0	0
$833$	1.39070	0.0481849
$834$	0	0
$835$	−3.86152	−0.133633
$836$	0	0
$837$	−5.62875	−0.194558
$838$	0	0
$839$	43.5939	1.50503	0.752513	−	0.658577i	$-0.228841\pi$
$839$	43.5939	1.50503	0.752513	+	0.658577i	$0.228841\pi$
$840$	0	0
$841$	36.0596	1.24344
$842$	0	0
$843$	19.0195	0.655065
$844$	0	0
$845$	−2.80845	−0.0966136
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	−9.35708	−0.321134
$850$	0	0
$851$	1.82790	0.0626597
$852$	0	0
$853$	44.7531	1.53232	0.766158	−	0.642652i	$-0.222166\pi$
$853$	44.7531	1.53232	0.766158	+	0.642652i	$0.222166\pi$
$854$	0	0
$855$	−9.35708	−0.320005
$856$	0	0
$857$	11.5704	0.395237	0.197619	−	0.980279i	$-0.436679\pi$
$857$	11.5704	0.395237	0.197619	+	0.980279i	$0.436679\pi$
$858$	0	0
$859$	−43.2575	−1.47593	−0.737964	−	0.674841i	$-0.764212\pi$
$859$	−43.2575	−1.47593	−0.737964	+	0.674841i	$0.764212\pi$
$860$	0	0
$861$	−8.01945	−0.273302
$862$	0	0
$863$	16.9805	0.578025	0.289012	−	0.957325i	$-0.406673\pi$
$863$	16.9805	0.578025	0.289012	+	0.957325i	$0.406673\pi$
$864$	0	0
$865$	37.0129	1.25848
$866$	0	0
$867$	−15.0660	−0.511667
$868$	0	0
$869$	−10.8474	−0.367971
$870$	0	0
$871$	1.25879	0.0426524
$872$	0	0
$873$	−3.39070	−0.114758
$874$	0	0
$875$	−0.847354	−0.0286458
$876$	0	0
$877$	11.4491	0.386607	0.193304	−	0.981139i	$-0.438080\pi$
$877$	11.4491	0.386607	0.193304	+	0.981139i	$0.438080\pi$
$878$	0	0
$879$	12.4761	0.420809
$880$	0	0
$881$	−21.0584	−0.709474	−0.354737	−	0.934966i	$-0.615430\pi$
$881$	−21.0584	−0.709474	−0.354737	+	0.934966i	$0.615430\pi$
$882$	0	0
$883$	12.4966	0.420544	0.210272	−	0.977643i	$-0.432565\pi$
$883$	12.4966	0.420544	0.210272	+	0.977643i	$0.432565\pi$
$884$	0	0
$885$	17.9275	0.602626
$886$	0	0
$887$	−57.3169	−1.92451	−0.962256	−	0.272144i	$-0.912267\pi$
$887$	−57.3169	−1.92451	−0.962256	+	0.272144i	$0.912267\pi$
$888$	0	0
$889$	−16.3571	−0.548599
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	13.1721	0.440787
$894$	0	0
$895$	27.0788	0.905146
$896$	0	0
$897$	−3.72833	−0.124485
$898$	0	0
$899$	−45.4013	−1.51422
$900$	0	0
$901$	3.79428	0.126406
$902$	0	0
$903$	−8.50973	−0.283186
$904$	0	0
$905$	61.9651	2.05979
$906$	0	0
$907$	45.5562	1.51267	0.756335	−	0.654185i	$-0.226988\pi$
$907$	45.5562	1.51267	0.756335	+	0.654185i	$0.226988\pi$
$908$	0	0
$909$	−10.1850	−0.337815
$910$	0	0
$911$	15.9328	0.527876	0.263938	−	0.964540i	$-0.414979\pi$
$911$	15.9328	0.527876	0.263938	+	0.964540i	$0.414979\pi$
$912$	0	0
$913$	12.0660	0.399325
$914$	0	0
$915$	−9.04650	−0.299068
$916$	0	0
$917$	4.45665	0.147172
$918$	0	0
$919$	−35.1243	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$919$	−35.1243	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$920$	0	0
$921$	27.7607	0.914745
$922$	0	0
$923$	−27.1837	−0.894762
$924$	0	0
$925$	8.64292	0.284177
$926$	0	0
$927$	7.06596	0.232076
$928$	0	0
$929$	−30.2704	−0.993139	−0.496570	−	0.867997i	$-0.665407\pi$
$929$	−30.2704	−0.993139	−0.496570	+	0.867997i	$0.665407\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	0	0
$933$	27.5757	0.902787
$934$	0	0
$935$	−4.33763	−0.141856
$936$	0	0
$937$	24.8266	0.811050	0.405525	−	0.914084i	$-0.367089\pi$
$937$	24.8266	0.811050	0.405525	+	0.914084i	$0.367089\pi$
$938$	0	0
$939$	7.75435	0.253054
$940$	0	0
$941$	−7.60170	−0.247808	−0.123904	−	0.992294i	$-0.539542\pi$
$941$	−7.60170	−0.247808	−0.123904	+	0.992294i	$0.539542\pi$
$942$	0	0
$943$	8.01945	0.261149
$944$	0	0
$945$	3.11903	0.101462
$946$	0	0
$947$	−22.8592	−0.742824	−0.371412	−	0.928468i	$-0.621126\pi$
$947$	−22.8592	−0.742824	−0.371412	+	0.928468i	$0.621126\pi$
$948$	0	0
$949$	9.72833	0.315795
$950$	0	0
$951$	21.3960	0.693812
$952$	0	0
$953$	−34.3828	−1.11377	−0.556885	−	0.830590i	$-0.688004\pi$
$953$	−34.3828	−1.11377	−0.556885	+	0.830590i	$0.688004\pi$
$954$	0	0
$955$	50.1708	1.62349
$956$	0	0
$957$	8.06596	0.260735
$958$	0	0
$959$	−20.2575	−0.654149
$960$	0	0
$961$	0.682856	0.0220276
$962$	0	0
$963$	5.72833	0.184593
$964$	0	0
$965$	−29.0195	−0.934169
$966$	0	0
$967$	4.74249	0.152508	0.0762542	−	0.997088i	$-0.475704\pi$
$967$	4.74249	0.152508	0.0762542	+	0.997088i	$0.475704\pi$
$968$	0	0
$969$	4.17210	0.134027
$970$	0	0
$971$	52.2691	1.67740	0.838698	−	0.544597i	$-0.183317\pi$
$971$	52.2691	1.67740	0.838698	+	0.544597i	$0.183317\pi$
$972$	0	0
$973$	6.77483	0.217191
$974$	0	0
$975$	−17.6288	−0.564572
$976$	0	0
$977$	33.1385	1.06019	0.530097	−	0.847937i	$-0.322156\pi$
$977$	33.1385	1.06019	0.530097	+	0.847937i	$0.322156\pi$
$978$	0	0
$979$	13.9664	0.446367
$980$	0	0
$981$	15.5951	0.497914
$982$	0	0
$983$	23.5357	0.750674	0.375337	−	0.926888i	$-0.377527\pi$
$983$	23.5357	0.750674	0.375337	+	0.926888i	$0.377527\pi$
$984$	0	0
$985$	14.8085	0.471836
$986$	0	0
$987$	−4.39070	−0.139757
$988$	0	0
$989$	8.50973	0.270594
$990$	0	0
$991$	−10.4890	−0.333194	−0.166597	−	0.986025i	$-0.553278\pi$
$991$	−10.4890	−0.333194	−0.166597	+	0.986025i	$0.553278\pi$
$992$	0	0
$993$	26.0182	0.825662
$994$	0	0
$995$	−22.8019	−0.722868
$996$	0	0
$997$	−43.0182	−1.36240	−0.681200	−	0.732098i	$-0.738541\pi$
$997$	−43.0182	−1.36240	−0.681200	+	0.732098i	$0.738541\pi$
$998$	0	0
$999$	1.82790	0.0578323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.j.1.1	✓	3
3.2	odd	2		5796.2.a.o.1.3		3
4.3	odd	2		7728.2.a.br.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.j.1.1	✓	3	1.1	even	1	trivial
5796.2.a.o.1.3		3	3.2	odd	2
7728.2.a.br.1.1		3	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.11903 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.11903 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -3.72833 q^{13} -3.11903 q^{15} +1.39070 q^{17} +3.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +4.72833 q^{25} +1.00000 q^{27} +8.06596 q^{29} -5.62875 q^{31} +1.00000 q^{33} +3.11903 q^{35} +1.82790 q^{37} -3.72833 q^{39} +8.01945 q^{41} +8.50973 q^{43} -3.11903 q^{45} +4.39070 q^{47} +1.00000 q^{49} +1.39070 q^{51} +2.72833 q^{53} -3.11903 q^{55} +3.00000 q^{57} -5.74778 q^{59} +2.90043 q^{61} -1.00000 q^{63} +11.6288 q^{65} -0.337628 q^{67} +1.00000 q^{69} +7.29112 q^{71} -2.60930 q^{73} +4.72833 q^{75} -1.00000 q^{77} -10.8474 q^{79} +1.00000 q^{81} +12.0660 q^{83} -4.33763 q^{85} +8.06596 q^{87} +13.9664 q^{89} +3.72833 q^{91} -5.62875 q^{93} -9.35708 q^{95} -3.39070 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 3 q^{7} + 3 q^{9} + 3 q^{11} - q^{13} + q^{15} + 4 q^{17} + 9 q^{19} - 3 q^{21} + 3 q^{23} + 4 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 3 q^{33} - q^{35} + 6 q^{37} - q^{39} + 3 q^{41} + 15 q^{43} + q^{45} + 13 q^{47} + 3 q^{49} + 4 q^{51} - 2 q^{53} + q^{55} + 9 q^{57} + 14 q^{59} - 2 q^{61} - 3 q^{63} + 14 q^{65} + 9 q^{67} + 3 q^{69} + 11 q^{71} - 8 q^{73} + 4 q^{75} - 3 q^{77} - 12 q^{79} + 3 q^{81} + 16 q^{83} - 3 q^{85} + 4 q^{87} + 11 q^{89} + q^{91} + 4 q^{93} + 3 q^{95} - 10 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 3 * q^7 + 3 * q^9 + 3 * q^11 - q^13 + q^15 + 4 * q^17 + 9 * q^19 - 3 * q^21 + 3 * q^23 + 4 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 3 * q^33 - q^35 + 6 * q^37 - q^39 + 3 * q^41 + 15 * q^43 + q^45 + 13 * q^47 + 3 * q^49 + 4 * q^51 - 2 * q^53 + q^55 + 9 * q^57 + 14 * q^59 - 2 * q^61 - 3 * q^63 + 14 * q^65 + 9 * q^67 + 3 * q^69 + 11 * q^71 - 8 * q^73 + 4 * q^75 - 3 * q^77 - 12 * q^79 + 3 * q^81 + 16 * q^83 - 3 * q^85 + 4 * q^87 + 11 * q^89 + q^91 + 4 * q^93 + 3 * q^95 - 10 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

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