LMFDB - Embedded newform 920.2.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.34623698596$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46260 q^{3} +1.00000 q^{5} +1.53740 q^{7} -0.860806 q^{9} -0.860806 q^{11} +0.139194 q^{13} +1.46260 q^{15} +5.50761 q^{17} +5.25901 q^{19} +2.24860 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.64681 q^{27} +9.76663 q^{29} +6.78600 q^{31} -1.25901 q^{33} +1.53740 q^{35} -12.0900 q^{37} +0.203585 q^{39} -9.98062 q^{41} +11.4432 q^{43} -0.860806 q^{45} -2.32340 q^{47} -4.63640 q^{49} +8.05543 q^{51} +0.149606 q^{53} -0.860806 q^{55} +7.69182 q^{57} -11.0152 q^{59} +4.43281 q^{61} -1.32340 q^{63} +0.139194 q^{65} -10.4972 q^{67} -1.46260 q^{69} +7.31299 q^{71} +7.11982 q^{73} +1.46260 q^{75} -1.32340 q^{77} +6.79641 q^{79} -5.67660 q^{81} +10.6468 q^{83} +5.50761 q^{85} +14.2847 q^{87} -17.2936 q^{89} +0.213997 q^{91} +9.92520 q^{93} +5.25901 q^{95} +1.93561 q^{97} +0.740987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 3 q^{5} + 7 q^{7} + 3 q^{9} + 3 q^{11} + 6 q^{13} + 2 q^{15} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - q^{27} - q^{29} + 10 q^{31} + 5 q^{33} + 7 q^{35} + 2 q^{37} + 7 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 3 * q^5 + 7 * q^7 + 3 * q^9 + 3 * q^11 + 6 * q^13 + 2 * q^15 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - q^27 - q^29 + 10 * q^31 + 5 * q^33 + 7 * q^35 + 2 * q^37 + 7 * q^39 - 10 * q^41 + 12 * q^43 + 3 * q^45 + q^47 + 6 * q^49 + 9 * q^51 + 10 * q^53 + 3 * q^55 - 12 * q^57 + 10 * q^59 - 13 * q^61 + 4 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 + 10 * q^71 + 7 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 - 25 * q^81 + 16 * q^83 - 5 * q^85 - 5 * q^87 - 20 * q^89 + 11 * q^91 + 25 * q^93 + 7 * q^95 + 5 * q^97 + 11 * q^99

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
$3$	1.46260	0.844432	0.422216	−	0.906495i	$-0.361252\pi$
$3$	1.46260	0.844432	0.422216	+	0.906495i	$0.361252\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.53740	0.581083	0.290542	−	0.956862i	$-0.406165\pi$
$7$	1.53740	0.581083	0.290542	+	0.956862i	$0.406165\pi$
$8$	0	0
$9$	−0.860806	−0.286935
$10$	0	0
$11$	−0.860806	−0.259543	−0.129771	−	0.991544i	$-0.541424\pi$
$11$	−0.860806	−0.259543	−0.129771	+	0.991544i	$0.541424\pi$
$12$	0	0
$13$	0.139194	0.0386055	0.0193028	−	0.999814i	$-0.493855\pi$
$13$	0.139194	0.0386055	0.0193028	+	0.999814i	$0.493855\pi$
$14$	0	0
$15$	1.46260	0.377641
$16$	0	0
$17$	5.50761	1.33579	0.667896	−	0.744254i	$-0.267195\pi$
$17$	5.50761	1.33579	0.667896	+	0.744254i	$0.267195\pi$
$18$	0	0
$19$	5.25901	1.20650	0.603250	−	0.797552i	$-0.293872\pi$
$19$	5.25901	1.20650	0.603250	+	0.797552i	$0.293872\pi$
$20$	0	0
$21$	2.24860	0.490685
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.64681	−1.08673
$28$	0	0
$29$	9.76663	1.81362	0.906809	−	0.421543i	$-0.138511\pi$
$29$	9.76663	1.81362	0.906809	+	0.421543i	$0.138511\pi$
$30$	0	0
$31$	6.78600	1.21880	0.609401	−	0.792862i	$-0.291410\pi$
$31$	6.78600	1.21880	0.609401	+	0.792862i	$0.291410\pi$
$32$	0	0
$33$	−1.25901	−0.219166
$34$	0	0
$35$	1.53740	0.259868
$36$	0	0
$37$	−12.0900	−1.98759	−0.993795	−	0.111232i	$-0.964520\pi$
$37$	−12.0900	−1.98759	−0.993795	+	0.111232i	$0.964520\pi$
$38$	0	0
$39$	0.203585	0.0325997
$40$	0	0
$41$	−9.98062	−1.55871	−0.779356	−	0.626582i	$-0.784454\pi$
$41$	−9.98062	−1.55871	−0.779356	+	0.626582i	$0.784454\pi$
$42$	0	0
$43$	11.4432	1.74508	0.872538	−	0.488547i	$-0.162473\pi$
$43$	11.4432	1.74508	0.872538	+	0.488547i	$0.162473\pi$
$44$	0	0
$45$	−0.860806	−0.128321
$46$	0	0
$47$	−2.32340	−0.338903	−0.169452	−	0.985538i	$-0.554200\pi$
$47$	−2.32340	−0.338903	−0.169452	+	0.985538i	$0.554200\pi$
$48$	0	0
$49$	−4.63640	−0.662342
$50$	0	0
$51$	8.05543	1.12799
$52$	0	0
$53$	0.149606	0.0205500	0.0102750	−	0.999947i	$-0.496729\pi$
$53$	0.149606	0.0205500	0.0102750	+	0.999947i	$0.496729\pi$
$54$	0	0
$55$	−0.860806	−0.116071
$56$	0	0
$57$	7.69182	1.01881
$58$	0	0
$59$	−11.0152	−1.43406	−0.717030	−	0.697042i	$-0.754499\pi$
$59$	−11.0152	−1.43406	−0.717030	+	0.697042i	$0.754499\pi$
$60$	0	0
$61$	4.43281	0.567563	0.283782	−	0.958889i	$-0.408411\pi$
$61$	4.43281	0.567563	0.283782	+	0.958889i	$0.408411\pi$
$62$	0	0
$63$	−1.32340	−0.166733
$64$	0	0
$65$	0.139194	0.0172649
$66$	0	0
$67$	−10.4972	−1.28244	−0.641219	−	0.767358i	$-0.721571\pi$
$67$	−10.4972	−1.28244	−0.641219	+	0.767358i	$0.721571\pi$
$68$	0	0
$69$	−1.46260	−0.176076
$70$	0	0
$71$	7.31299	0.867892	0.433946	−	0.900939i	$-0.357121\pi$
$71$	7.31299	0.867892	0.433946	+	0.900939i	$0.357121\pi$
$72$	0	0
$73$	7.11982	0.833312	0.416656	−	0.909064i	$-0.363202\pi$
$73$	7.11982	0.833312	0.416656	+	0.909064i	$0.363202\pi$
$74$	0	0
$75$	1.46260	0.168886
$76$	0	0
$77$	−1.32340	−0.150816
$78$	0	0
$79$	6.79641	0.764656	0.382328	−	0.924027i	$-0.375122\pi$
$79$	6.79641	0.764656	0.382328	+	0.924027i	$0.375122\pi$
$80$	0	0
$81$	−5.67660	−0.630733
$82$	0	0
$83$	10.6468	1.16864	0.584320	−	0.811524i	$-0.301361\pi$
$83$	10.6468	1.16864	0.584320	+	0.811524i	$0.301361\pi$
$84$	0	0
$85$	5.50761	0.597385
$86$	0	0
$87$	14.2847	1.53148
$88$	0	0
$89$	−17.2936	−1.83312	−0.916560	−	0.399897i	$-0.869046\pi$
$89$	−17.2936	−1.83312	−0.916560	+	0.399897i	$0.869046\pi$
$90$	0	0
$91$	0.213997	0.0224330
$92$	0	0
$93$	9.92520	1.02919
$94$	0	0
$95$	5.25901	0.539563
$96$	0	0
$97$	1.93561	0.196531	0.0982657	−	0.995160i	$-0.468671\pi$
$97$	1.93561	0.196531	0.0982657	+	0.995160i	$0.468671\pi$
$98$	0	0
$99$	0.740987	0.0744720
$100$	0	0
$101$	−15.4224	−1.53459	−0.767293	−	0.641297i	$-0.778397\pi$
$101$	−15.4224	−1.53459	−0.767293	+	0.641297i	$0.778397\pi$
$102$	0	0
$103$	7.91478	0.779867	0.389933	−	0.920843i	$-0.372498\pi$
$103$	7.91478	0.779867	0.389933	+	0.920843i	$0.372498\pi$
$104$	0	0
$105$	2.24860	0.219441
$106$	0	0
$107$	−4.27839	−0.413607	−0.206804	−	0.978382i	$-0.566306\pi$
$107$	−4.27839	−0.413607	−0.206804	+	0.978382i	$0.566306\pi$
$108$	0	0
$109$	−17.7770	−1.70273	−0.851366	−	0.524572i	$-0.824225\pi$
$109$	−17.7770	−1.70273	−0.851366	+	0.524572i	$0.824225\pi$
$110$	0	0
$111$	−17.6829	−1.67838
$112$	0	0
$113$	−3.20359	−0.301368	−0.150684	−	0.988582i	$-0.548148\pi$
$113$	−3.20359	−0.301368	−0.150684	+	0.988582i	$0.548148\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−0.119819	−0.0110773
$118$	0	0
$119$	8.46742	0.776207
$120$	0	0
$121$	−10.2590	−0.932638
$122$	0	0
$123$	−14.5976	−1.31623
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.2099	−1.34966	−0.674828	−	0.737975i	$-0.735782\pi$
$127$	−15.2099	−1.34966	−0.674828	+	0.737975i	$0.735782\pi$
$128$	0	0
$129$	16.7368	1.47360
$130$	0	0
$131$	12.6918	1.10889	0.554445	−	0.832220i	$-0.312931\pi$
$131$	12.6918	1.10889	0.554445	+	0.832220i	$0.312931\pi$
$132$	0	0
$133$	8.08522	0.701077
$134$	0	0
$135$	−5.64681	−0.486000
$136$	0	0
$137$	−14.0346	−1.19906	−0.599529	−	0.800353i	$-0.704645\pi$
$137$	−14.0346	−1.19906	−0.599529	+	0.800353i	$0.704645\pi$
$138$	0	0
$139$	14.2638	1.20984	0.604921	−	0.796285i	$-0.293205\pi$
$139$	14.2638	1.20984	0.604921	+	0.796285i	$0.293205\pi$
$140$	0	0
$141$	−3.39821	−0.286181
$142$	0	0
$143$	−0.119819	−0.0100198
$144$	0	0
$145$	9.76663	0.811074
$146$	0	0
$147$	−6.78119	−0.559303
$148$	0	0
$149$	−4.46260	−0.365590	−0.182795	−	0.983151i	$-0.558515\pi$
$149$	−4.46260	−0.365590	−0.182795	+	0.983151i	$0.558515\pi$
$150$	0	0
$151$	10.7562	0.875328	0.437664	−	0.899139i	$-0.355806\pi$
$151$	10.7562	0.875328	0.437664	+	0.899139i	$0.355806\pi$
$152$	0	0
$153$	−4.74099	−0.383286
$154$	0	0
$155$	6.78600	0.545065
$156$	0	0
$157$	14.5180	1.15866	0.579332	−	0.815091i	$-0.303313\pi$
$157$	14.5180	1.15866	0.579332	+	0.815091i	$0.303313\pi$
$158$	0	0
$159$	0.218814	0.0173531
$160$	0	0
$161$	−1.53740	−0.121164
$162$	0	0
$163$	−1.30403	−0.102139	−0.0510697	−	0.998695i	$-0.516263\pi$
$163$	−1.30403	−0.102139	−0.0510697	+	0.998695i	$0.516263\pi$
$164$	0	0
$165$	−1.25901	−0.0980141
$166$	0	0
$167$	3.44322	0.266445	0.133222	−	0.991086i	$-0.457468\pi$
$167$	3.44322	0.266445	0.133222	+	0.991086i	$0.457468\pi$
$168$	0	0
$169$	−12.9806	−0.998510
$170$	0	0
$171$	−4.52699	−0.346188
$172$	0	0
$173$	9.35801	0.711476	0.355738	−	0.934586i	$-0.384230\pi$
$173$	9.35801	0.711476	0.355738	+	0.934586i	$0.384230\pi$
$174$	0	0
$175$	1.53740	0.116217
$176$	0	0
$177$	−16.1109	−1.21097
$178$	0	0
$179$	−18.4134	−1.37628	−0.688142	−	0.725576i	$-0.741574\pi$
$179$	−18.4134	−1.37628	−0.688142	+	0.725576i	$0.741574\pi$
$180$	0	0
$181$	−1.94457	−0.144539	−0.0722694	−	0.997385i	$-0.523024\pi$
$181$	−1.94457	−0.144539	−0.0722694	+	0.997385i	$0.523024\pi$
$182$	0	0
$183$	6.48342	0.479268
$184$	0	0
$185$	−12.0900	−0.888877
$186$	0	0
$187$	−4.74099	−0.346695
$188$	0	0
$189$	−8.68141	−0.631480
$190$	0	0
$191$	−0.775591	−0.0561198	−0.0280599	−	0.999606i	$-0.508933\pi$
$191$	−0.775591	−0.0561198	−0.0280599	+	0.999606i	$0.508933\pi$
$192$	0	0
$193$	−11.7666	−0.846980	−0.423490	−	0.905901i	$-0.639195\pi$
$193$	−11.7666	−0.846980	−0.423490	+	0.905901i	$0.639195\pi$
$194$	0	0
$195$	0.203585	0.0145790
$196$	0	0
$197$	9.68141	0.689772	0.344886	−	0.938645i	$-0.387918\pi$
$197$	9.68141	0.689772	0.344886	+	0.938645i	$0.387918\pi$
$198$	0	0
$199$	−3.29362	−0.233478	−0.116739	−	0.993163i	$-0.537244\pi$
$199$	−3.29362	−0.233478	−0.116739	+	0.993163i	$0.537244\pi$
$200$	0	0
$201$	−15.3532	−1.08293
$202$	0	0
$203$	15.0152	1.05386
$204$	0	0
$205$	−9.98062	−0.697077
$206$	0	0
$207$	0.860806	0.0598301
$208$	0	0
$209$	−4.52699	−0.313138
$210$	0	0
$211$	−16.2396	−1.11798	−0.558991	−	0.829173i	$-0.688812\pi$
$211$	−16.2396	−1.11798	−0.558991	+	0.829173i	$0.688812\pi$
$212$	0	0
$213$	10.6960	0.732876
$214$	0	0
$215$	11.4432	0.780421
$216$	0	0
$217$	10.4328	0.708225
$218$	0	0
$219$	10.4134	0.703675
$220$	0	0
$221$	0.766628	0.0515690
$222$	0	0
$223$	21.2936	1.42593	0.712963	−	0.701202i	$-0.247353\pi$
$223$	21.2936	1.42593	0.712963	+	0.701202i	$0.247353\pi$
$224$	0	0
$225$	−0.860806	−0.0573871
$226$	0	0
$227$	−29.6620	−1.96874	−0.984369	−	0.176117i	$-0.943646\pi$
$227$	−29.6620	−1.96874	−0.984369	+	0.176117i	$0.943646\pi$
$228$	0	0
$229$	19.6829	1.30068	0.650340	−	0.759643i	$-0.274626\pi$
$229$	19.6829	1.30068	0.650340	+	0.759643i	$0.274626\pi$
$230$	0	0
$231$	−1.93561	−0.127354
$232$	0	0
$233$	−5.52699	−0.362085	−0.181043	−	0.983475i	$-0.557947\pi$
$233$	−5.52699	−0.362085	−0.181043	+	0.983475i	$0.557947\pi$
$234$	0	0
$235$	−2.32340	−0.151562
$236$	0	0
$237$	9.94043	0.645700
$238$	0	0
$239$	−17.3982	−1.12540	−0.562698	−	0.826662i	$-0.690237\pi$
$239$	−17.3982	−1.12540	−0.562698	+	0.826662i	$0.690237\pi$
$240$	0	0
$241$	−16.0900	−1.03645	−0.518225	−	0.855244i	$-0.673407\pi$
$241$	−16.0900	−1.03645	−0.518225	+	0.855244i	$0.673407\pi$
$242$	0	0
$243$	8.63785	0.554118
$244$	0	0
$245$	−4.63640	−0.296209
$246$	0	0
$247$	0.732024	0.0465776
$248$	0	0
$249$	15.5720	0.986836
$250$	0	0
$251$	−29.0498	−1.83361	−0.916805	−	0.399336i	$-0.869241\pi$
$251$	−29.0498	−1.83361	−0.916805	+	0.399336i	$0.869241\pi$
$252$	0	0
$253$	0.860806	0.0541184
$254$	0	0
$255$	8.05543	0.504450
$256$	0	0
$257$	−6.56304	−0.409391	−0.204696	−	0.978826i	$-0.565620\pi$
$257$	−6.56304	−0.409391	−0.204696	+	0.978826i	$0.565620\pi$
$258$	0	0
$259$	−18.5872	−1.15495
$260$	0	0
$261$	−8.40717	−0.520391
$262$	0	0
$263$	−0.561593	−0.0346293	−0.0173147	−	0.999850i	$-0.505512\pi$
$263$	−0.561593	−0.0346293	−0.0173147	+	0.999850i	$0.505512\pi$
$264$	0	0
$265$	0.149606	0.00919024
$266$	0	0
$267$	−25.2936	−1.54794
$268$	0	0
$269$	−7.26943	−0.443225	−0.221612	−	0.975135i	$-0.571132\pi$
$269$	−7.26943	−0.443225	−0.221612	+	0.975135i	$0.571132\pi$
$270$	0	0
$271$	0.184210	0.0111900	0.00559498	−	0.999984i	$-0.498219\pi$
$271$	0.184210	0.0111900	0.00559498	+	0.999984i	$0.498219\pi$
$272$	0	0
$273$	0.312992	0.0189431
$274$	0	0
$275$	−0.860806	−0.0519085
$276$	0	0
$277$	25.8954	1.55590	0.777952	−	0.628323i	$-0.216259\pi$
$277$	25.8954	1.55590	0.777952	+	0.628323i	$0.216259\pi$
$278$	0	0
$279$	−5.84143	−0.349717
$280$	0	0
$281$	−26.7756	−1.59730	−0.798649	−	0.601797i	$-0.794452\pi$
$281$	−26.7756	−1.59730	−0.798649	+	0.601797i	$0.794452\pi$
$282$	0	0
$283$	5.70079	0.338877	0.169438	−	0.985541i	$-0.445805\pi$
$283$	5.70079	0.338877	0.169438	+	0.985541i	$0.445805\pi$
$284$	0	0
$285$	7.69182	0.455624
$286$	0	0
$287$	−15.3442	−0.905741
$288$	0	0
$289$	13.3338	0.784342
$290$	0	0
$291$	2.83102	0.165957
$292$	0	0
$293$	1.05398	0.0615741	0.0307871	−	0.999526i	$-0.490199\pi$
$293$	1.05398	0.0615741	0.0307871	+	0.999526i	$0.490199\pi$
$294$	0	0
$295$	−11.0152	−0.641331
$296$	0	0
$297$	4.86081	0.282053
$298$	0	0
$299$	−0.139194	−0.00804981
$300$	0	0
$301$	17.5928	1.01403
$302$	0	0
$303$	−22.5568	−1.29585
$304$	0	0
$305$	4.43281	0.253822
$306$	0	0
$307$	17.8760	1.02024	0.510120	−	0.860103i	$-0.329601\pi$
$307$	17.8760	1.02024	0.510120	+	0.860103i	$0.329601\pi$
$308$	0	0
$309$	11.5762	0.658544
$310$	0	0
$311$	16.9315	0.960095	0.480048	−	0.877242i	$-0.340619\pi$
$311$	16.9315	0.960095	0.480048	+	0.877242i	$0.340619\pi$
$312$	0	0
$313$	−19.9959	−1.13023	−0.565116	−	0.825011i	$-0.691169\pi$
$313$	−19.9959	−1.13023	−0.565116	+	0.825011i	$0.691169\pi$
$314$	0	0
$315$	−1.32340	−0.0745654
$316$	0	0
$317$	3.29776	0.185221	0.0926104	−	0.995702i	$-0.470479\pi$
$317$	3.29776	0.185221	0.0926104	+	0.995702i	$0.470479\pi$
$318$	0	0
$319$	−8.40717	−0.470711
$320$	0	0
$321$	−6.25756	−0.349263
$322$	0	0
$323$	28.9646	1.61163
$324$	0	0
$325$	0.139194	0.00772110
$326$	0	0
$327$	−26.0007	−1.43784
$328$	0	0
$329$	−3.57201	−0.196931
$330$	0	0
$331$	−12.8802	−0.707959	−0.353979	−	0.935253i	$-0.615172\pi$
$331$	−12.8802	−0.707959	−0.353979	+	0.935253i	$0.615172\pi$
$332$	0	0
$333$	10.4072	0.570309
$334$	0	0
$335$	−10.4972	−0.573523
$336$	0	0
$337$	−9.22026	−0.502260	−0.251130	−	0.967953i	$-0.580802\pi$
$337$	−9.22026	−0.502260	−0.251130	+	0.967953i	$0.580802\pi$
$338$	0	0
$339$	−4.68556	−0.254485
$340$	0	0
$341$	−5.84143	−0.316331
$342$	0	0
$343$	−17.8898	−0.965959
$344$	0	0
$345$	−1.46260	−0.0787437
$346$	0	0
$347$	31.2984	1.68019	0.840094	−	0.542441i	$-0.182500\pi$
$347$	31.2984	1.68019	0.840094	+	0.542441i	$0.182500\pi$
$348$	0	0
$349$	9.89541	0.529689	0.264845	−	0.964291i	$-0.414679\pi$
$349$	9.89541	0.529689	0.264845	+	0.964291i	$0.414679\pi$
$350$	0	0
$351$	−0.786003	−0.0419537
$352$	0	0
$353$	19.4287	1.03408	0.517042	−	0.855960i	$-0.327033\pi$
$353$	19.4287	1.03408	0.517042	+	0.855960i	$0.327033\pi$
$354$	0	0
$355$	7.31299	0.388133
$356$	0	0
$357$	12.3844	0.655453
$358$	0	0
$359$	−13.1440	−0.693714	−0.346857	−	0.937918i	$-0.612751\pi$
$359$	−13.1440	−0.693714	−0.346857	+	0.937918i	$0.612751\pi$
$360$	0	0
$361$	8.65722	0.455643
$362$	0	0
$363$	−15.0048	−0.787549
$364$	0	0
$365$	7.11982	0.372668
$366$	0	0
$367$	−15.7424	−0.821748	−0.410874	−	0.911692i	$-0.634776\pi$
$367$	−15.7424	−0.821748	−0.410874	+	0.911692i	$0.634776\pi$
$368$	0	0
$369$	8.59138	0.447249
$370$	0	0
$371$	0.230005	0.0119413
$372$	0	0
$373$	27.6620	1.43229	0.716143	−	0.697954i	$-0.245906\pi$
$373$	27.6620	1.43229	0.716143	+	0.697954i	$0.245906\pi$
$374$	0	0
$375$	1.46260	0.0755283
$376$	0	0
$377$	1.35946	0.0700156
$378$	0	0
$379$	28.2140	1.44926	0.724628	−	0.689140i	$-0.242012\pi$
$379$	28.2140	1.44926	0.724628	+	0.689140i	$0.242012\pi$
$380$	0	0
$381$	−22.2459	−1.13969
$382$	0	0
$383$	−21.5928	−1.10334	−0.551671	−	0.834062i	$-0.686010\pi$
$383$	−21.5928	−1.10334	−0.551671	+	0.834062i	$0.686010\pi$
$384$	0	0
$385$	−1.32340	−0.0674469
$386$	0	0
$387$	−9.85039	−0.500724
$388$	0	0
$389$	17.2501	0.874612	0.437306	−	0.899313i	$-0.355933\pi$
$389$	17.2501	0.874612	0.437306	+	0.899313i	$0.355933\pi$
$390$	0	0
$391$	−5.50761	−0.278532
$392$	0	0
$393$	18.5630	0.936382
$394$	0	0
$395$	6.79641	0.341965
$396$	0	0
$397$	34.8850	1.75083	0.875414	−	0.483374i	$-0.160589\pi$
$397$	34.8850	1.75083	0.875414	+	0.483374i	$0.160589\pi$
$398$	0	0
$399$	11.8254	0.592012
$400$	0	0
$401$	4.21881	0.210678	0.105339	−	0.994436i	$-0.466407\pi$
$401$	4.21881	0.210678	0.105339	+	0.994436i	$0.466407\pi$
$402$	0	0
$403$	0.944572	0.0470525
$404$	0	0
$405$	−5.67660	−0.282072
$406$	0	0
$407$	10.4072	0.515864
$408$	0	0
$409$	38.8165	1.91935	0.959675	−	0.281111i	$-0.0907030\pi$
$409$	38.8165	1.91935	0.959675	+	0.281111i	$0.0907030\pi$
$410$	0	0
$411$	−20.5270	−1.01252
$412$	0	0
$413$	−16.9348	−0.833309
$414$	0	0
$415$	10.6468	0.522631
$416$	0	0
$417$	20.8623	1.02163
$418$	0	0
$419$	22.0305	1.07626	0.538129	−	0.842862i	$-0.319131\pi$
$419$	22.0305	1.07626	0.538129	+	0.842862i	$0.319131\pi$
$420$	0	0
$421$	−17.2203	−0.839264	−0.419632	−	0.907694i	$-0.637841\pi$
$421$	−17.2203	−0.839264	−0.419632	+	0.907694i	$0.637841\pi$
$422$	0	0
$423$	2.00000	0.0972433
$424$	0	0
$425$	5.50761	0.267159
$426$	0	0
$427$	6.81501	0.329801
$428$	0	0
$429$	−0.175247	−0.00846102
$430$	0	0
$431$	−11.8116	−0.568947	−0.284473	−	0.958684i	$-0.591819\pi$
$431$	−11.8116	−0.568947	−0.284473	+	0.958684i	$0.591819\pi$
$432$	0	0
$433$	19.8642	0.954611	0.477306	−	0.878737i	$-0.341613\pi$
$433$	19.8642	0.954611	0.477306	+	0.878737i	$0.341613\pi$
$434$	0	0
$435$	14.2847	0.684897
$436$	0	0
$437$	−5.25901	−0.251573
$438$	0	0
$439$	−5.04647	−0.240855	−0.120427	−	0.992722i	$-0.538426\pi$
$439$	−5.04647	−0.240855	−0.120427	+	0.992722i	$0.538426\pi$
$440$	0	0
$441$	3.99104	0.190049
$442$	0	0
$443$	−14.3193	−0.680328	−0.340164	−	0.940366i	$-0.610483\pi$
$443$	−14.3193	−0.680328	−0.340164	+	0.940366i	$0.610483\pi$
$444$	0	0
$445$	−17.2936	−0.819796
$446$	0	0
$447$	−6.52699	−0.308716
$448$	0	0
$449$	−13.3699	−0.630963	−0.315482	−	0.948932i	$-0.602166\pi$
$449$	−13.3699	−0.630963	−0.315482	+	0.948932i	$0.602166\pi$
$450$	0	0
$451$	8.59138	0.404552
$452$	0	0
$453$	15.7320	0.739155
$454$	0	0
$455$	0.213997	0.0100323
$456$	0	0
$457$	2.38924	0.111764	0.0558821	−	0.998437i	$-0.482203\pi$
$457$	2.38924	0.111764	0.0558821	+	0.998437i	$0.482203\pi$
$458$	0	0
$459$	−31.1004	−1.45164
$460$	0	0
$461$	21.2278	0.988676	0.494338	−	0.869270i	$-0.335410\pi$
$461$	21.2278	0.988676	0.494338	+	0.869270i	$0.335410\pi$
$462$	0	0
$463$	35.1261	1.63245	0.816224	−	0.577736i	$-0.196064\pi$
$463$	35.1261	1.63245	0.816224	+	0.577736i	$0.196064\pi$
$464$	0	0
$465$	9.92520	0.460270
$466$	0	0
$467$	16.7756	0.776282	0.388141	−	0.921600i	$-0.373117\pi$
$467$	16.7756	0.776282	0.388141	+	0.921600i	$0.373117\pi$
$468$	0	0
$469$	−16.1384	−0.745203
$470$	0	0
$471$	21.2340	0.978413
$472$	0	0
$473$	−9.85039	−0.452922
$474$	0	0
$475$	5.25901	0.241300
$476$	0	0
$477$	−0.128782	−0.00589652
$478$	0	0
$479$	−5.61076	−0.256362	−0.128181	−	0.991751i	$-0.540914\pi$
$479$	−5.61076	−0.256362	−0.128181	+	0.991751i	$0.540914\pi$
$480$	0	0
$481$	−1.68286	−0.0767319
$482$	0	0
$483$	−2.24860	−0.102315
$484$	0	0
$485$	1.93561	0.0878915
$486$	0	0
$487$	10.2638	0.465099	0.232549	−	0.972585i	$-0.425293\pi$
$487$	10.2638	0.465099	0.232549	+	0.972585i	$0.425293\pi$
$488$	0	0
$489$	−1.90727	−0.0862498
$490$	0	0
$491$	9.28735	0.419132	0.209566	−	0.977794i	$-0.432795\pi$
$491$	9.28735	0.419132	0.209566	+	0.977794i	$0.432795\pi$
$492$	0	0
$493$	53.7908	2.42262
$494$	0	0
$495$	0.740987	0.0333049
$496$	0	0
$497$	11.2430	0.504318
$498$	0	0
$499$	13.3082	0.595756	0.297878	−	0.954604i	$-0.403721\pi$
$499$	13.3082	0.595756	0.297878	+	0.954604i	$0.403721\pi$
$500$	0	0
$501$	5.03605	0.224994
$502$	0	0
$503$	−37.7383	−1.68267	−0.841334	−	0.540516i	$-0.818229\pi$
$503$	−37.7383	−1.68267	−0.841334	+	0.540516i	$0.818229\pi$
$504$	0	0
$505$	−15.4224	−0.686288
$506$	0	0
$507$	−18.9854	−0.843173
$508$	0	0
$509$	−1.11982	−0.0496351	−0.0248176	−	0.999692i	$-0.507900\pi$
$509$	−1.11982	−0.0496351	−0.0248176	+	0.999692i	$0.507900\pi$
$510$	0	0
$511$	10.9460	0.484223
$512$	0	0
$513$	−29.6966	−1.31114
$514$	0	0
$515$	7.91478	0.348767
$516$	0	0
$517$	2.00000	0.0879599
$518$	0	0
$519$	13.6870	0.600793
$520$	0	0
$521$	−13.9100	−0.609407	−0.304703	−	0.952447i	$-0.598557\pi$
$521$	−13.9100	−0.609407	−0.304703	+	0.952447i	$0.598557\pi$
$522$	0	0
$523$	8.19799	0.358473	0.179237	−	0.983806i	$-0.442637\pi$
$523$	8.19799	0.358473	0.179237	+	0.983806i	$0.442637\pi$
$524$	0	0
$525$	2.24860	0.0981370
$526$	0	0
$527$	37.3747	1.62807
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.48197	0.411483
$532$	0	0
$533$	−1.38924	−0.0601749
$534$	0	0
$535$	−4.27839	−0.184971
$536$	0	0
$537$	−26.9315	−1.16218
$538$	0	0
$539$	3.99104	0.171906
$540$	0	0
$541$	23.5478	1.01240	0.506200	−	0.862416i	$-0.331050\pi$
$541$	23.5478	1.01240	0.506200	+	0.862416i	$0.331050\pi$
$542$	0	0
$543$	−2.84413	−0.122053
$544$	0	0
$545$	−17.7770	−0.761485
$546$	0	0
$547$	1.77077	0.0757128	0.0378564	−	0.999283i	$-0.487947\pi$
$547$	1.77077	0.0757128	0.0378564	+	0.999283i	$0.487947\pi$
$548$	0	0
$549$	−3.81579	−0.162854
$550$	0	0
$551$	51.3628	2.18813
$552$	0	0
$553$	10.4488	0.444329
$554$	0	0
$555$	−17.6829	−0.750596
$556$	0	0
$557$	−22.5872	−0.957052	−0.478526	−	0.878073i	$-0.658829\pi$
$557$	−22.5872	−0.957052	−0.478526	+	0.878073i	$0.658829\pi$
$558$	0	0
$559$	1.59283	0.0673695
$560$	0	0
$561$	−6.93416	−0.292760
$562$	0	0
$563$	21.8504	0.920884	0.460442	−	0.887690i	$-0.347691\pi$
$563$	21.8504	0.920884	0.460442	+	0.887690i	$0.347691\pi$
$564$	0	0
$565$	−3.20359	−0.134776
$566$	0	0
$567$	−8.72721	−0.366508
$568$	0	0
$569$	11.4737	0.481002	0.240501	−	0.970649i	$-0.422688\pi$
$569$	11.4737	0.481002	0.240501	+	0.970649i	$0.422688\pi$
$570$	0	0
$571$	27.7085	1.15956	0.579782	−	0.814771i	$-0.303138\pi$
$571$	27.7085	1.15956	0.579782	+	0.814771i	$0.303138\pi$
$572$	0	0
$573$	−1.13438	−0.0473893
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	20.3442	0.846941	0.423471	−	0.905910i	$-0.360812\pi$
$577$	20.3442	0.846941	0.423471	+	0.905910i	$0.360812\pi$
$578$	0	0
$579$	−17.2099	−0.715217
$580$	0	0
$581$	16.3684	0.679076
$582$	0	0
$583$	−0.128782	−0.00533360
$584$	0	0
$585$	−0.119819	−0.00495391
$586$	0	0
$587$	12.1690	0.502268	0.251134	−	0.967952i	$-0.419197\pi$
$587$	12.1690	0.502268	0.251134	+	0.967952i	$0.419197\pi$
$588$	0	0
$589$	35.6877	1.47049
$590$	0	0
$591$	14.1600	0.582465
$592$	0	0
$593$	−40.5180	−1.66388	−0.831938	−	0.554869i	$-0.812769\pi$
$593$	−40.5180	−1.66388	−0.831938	+	0.554869i	$0.812769\pi$
$594$	0	0
$595$	8.46742	0.347130
$596$	0	0
$597$	−4.81724	−0.197156
$598$	0	0
$599$	−28.8913	−1.18047	−0.590233	−	0.807233i	$-0.700964\pi$
$599$	−28.8913	−1.18047	−0.590233	+	0.807233i	$0.700964\pi$
$600$	0	0
$601$	5.34278	0.217937	0.108968	−	0.994045i	$-0.465245\pi$
$601$	5.34278	0.217937	0.108968	+	0.994045i	$0.465245\pi$
$602$	0	0
$603$	9.03605	0.367977
$604$	0	0
$605$	−10.2590	−0.417088
$606$	0	0
$607$	−40.4376	−1.64131	−0.820656	−	0.571422i	$-0.806392\pi$
$607$	−40.4376	−1.64131	−0.820656	+	0.571422i	$0.806392\pi$
$608$	0	0
$609$	21.9612	0.889915
$610$	0	0
$611$	−0.323404	−0.0130835
$612$	0	0
$613$	−5.31154	−0.214531	−0.107266	−	0.994230i	$-0.534210\pi$
$613$	−5.31154	−0.214531	−0.107266	+	0.994230i	$0.534210\pi$
$614$	0	0
$615$	−14.5976	−0.588634
$616$	0	0
$617$	9.71680	0.391183	0.195592	−	0.980685i	$-0.437337\pi$
$617$	9.71680	0.391183	0.195592	+	0.980685i	$0.437337\pi$
$618$	0	0
$619$	−23.7562	−0.954843	−0.477421	−	0.878674i	$-0.658428\pi$
$619$	−23.7562	−0.954843	−0.477421	+	0.878674i	$0.658428\pi$
$620$	0	0
$621$	5.64681	0.226599
$622$	0	0
$623$	−26.5872	−1.06520
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.62117	−0.264424
$628$	0	0
$629$	−66.5872	−2.65501
$630$	0	0
$631$	1.24234	0.0494566	0.0247283	−	0.999694i	$-0.492128\pi$
$631$	1.24234	0.0494566	0.0247283	+	0.999694i	$0.492128\pi$
$632$	0	0
$633$	−23.7521	−0.944060
$634$	0	0
$635$	−15.2099	−0.603585
$636$	0	0
$637$	−0.645359	−0.0255701
$638$	0	0
$639$	−6.29507	−0.249029
$640$	0	0
$641$	−34.8864	−1.37793	−0.688966	−	0.724794i	$-0.741935\pi$
$641$	−34.8864	−1.37793	−0.688966	+	0.724794i	$0.741935\pi$
$642$	0	0
$643$	32.1592	1.26824	0.634118	−	0.773236i	$-0.281363\pi$
$643$	32.1592	1.26824	0.634118	+	0.773236i	$0.281363\pi$
$644$	0	0
$645$	16.7368	0.659012
$646$	0	0
$647$	23.9550	0.941768	0.470884	−	0.882195i	$-0.343935\pi$
$647$	23.9550	0.941768	0.470884	+	0.882195i	$0.343935\pi$
$648$	0	0
$649$	9.48197	0.372200
$650$	0	0
$651$	15.2590	0.598048
$652$	0	0
$653$	4.31926	0.169026	0.0845128	−	0.996422i	$-0.473067\pi$
$653$	4.31926	0.169026	0.0845128	+	0.996422i	$0.473067\pi$
$654$	0	0
$655$	12.6918	0.495911
$656$	0	0
$657$	−6.12878	−0.239107
$658$	0	0
$659$	−6.14961	−0.239555	−0.119777	−	0.992801i	$-0.538218\pi$
$659$	−6.14961	−0.239555	−0.119777	+	0.992801i	$0.538218\pi$
$660$	0	0
$661$	−15.7473	−0.612497	−0.306249	−	0.951952i	$-0.599074\pi$
$661$	−15.7473	−0.612497	−0.306249	+	0.951952i	$0.599074\pi$
$662$	0	0
$663$	1.12127	0.0435465
$664$	0	0
$665$	8.08522	0.313531
$666$	0	0
$667$	−9.76663	−0.378165
$668$	0	0
$669$	31.1440	1.20410
$670$	0	0
$671$	−3.81579	−0.147307
$672$	0	0
$673$	43.5783	1.67982	0.839909	−	0.542727i	$-0.182608\pi$
$673$	43.5783	1.67982	0.839909	+	0.542727i	$0.182608\pi$
$674$	0	0
$675$	−5.64681	−0.217346
$676$	0	0
$677$	30.1801	1.15991	0.579957	−	0.814647i	$-0.303069\pi$
$677$	30.1801	1.15991	0.579957	+	0.814647i	$0.303069\pi$
$678$	0	0
$679$	2.97581	0.114201
$680$	0	0
$681$	−43.3836	−1.66247
$682$	0	0
$683$	29.2832	1.12049	0.560245	−	0.828327i	$-0.310707\pi$
$683$	29.2832	1.12049	0.560245	+	0.828327i	$0.310707\pi$
$684$	0	0
$685$	−14.0346	−0.536235
$686$	0	0
$687$	28.7881	1.09834
$688$	0	0
$689$	0.0208243	0.000793344 0
$690$	0	0
$691$	−14.9557	−0.568940	−0.284470	−	0.958685i	$-0.591818\pi$
$691$	−14.9557	−0.568940	−0.284470	+	0.958685i	$0.591818\pi$
$692$	0	0
$693$	1.13919	0.0432744
$694$	0	0
$695$	14.2638	0.541058
$696$	0	0
$697$	−54.9694	−2.08212
$698$	0	0
$699$	−8.08377	−0.305756
$700$	0	0
$701$	24.9627	0.942828	0.471414	−	0.881912i	$-0.343744\pi$
$701$	24.9627	0.942828	0.471414	+	0.881912i	$0.343744\pi$
$702$	0	0
$703$	−63.5816	−2.39803
$704$	0	0
$705$	−3.39821	−0.127984
$706$	0	0
$707$	−23.7104	−0.891722
$708$	0	0
$709$	−38.9717	−1.46361	−0.731806	−	0.681513i	$-0.761322\pi$
$709$	−38.9717	−1.46361	−0.731806	+	0.681513i	$0.761322\pi$
$710$	0	0
$711$	−5.85039	−0.219407
$712$	0	0
$713$	−6.78600	−0.254138
$714$	0	0
$715$	−0.119819	−0.00448098
$716$	0	0
$717$	−25.4466	−0.950320
$718$	0	0
$719$	−25.2978	−0.943447	−0.471724	−	0.881746i	$-0.656368\pi$
$719$	−25.2978	−0.943447	−0.471724	+	0.881746i	$0.656368\pi$
$720$	0	0
$721$	12.1682	0.453168
$722$	0	0
$723$	−23.5333	−0.875211
$724$	0	0
$725$	9.76663	0.362723
$726$	0	0
$727$	12.7417	0.472562	0.236281	−	0.971685i	$-0.424071\pi$
$727$	12.7417	0.472562	0.236281	+	0.971685i	$0.424071\pi$
$728$	0	0
$729$	29.6635	1.09865
$730$	0	0
$731$	63.0249	2.33106
$732$	0	0
$733$	−9.83247	−0.363170	−0.181585	−	0.983375i	$-0.558123\pi$
$733$	−9.83247	−0.363170	−0.181585	+	0.983375i	$0.558123\pi$
$734$	0	0
$735$	−6.78119	−0.250128
$736$	0	0
$737$	9.03605	0.332847
$738$	0	0
$739$	22.6918	0.834732	0.417366	−	0.908738i	$-0.362953\pi$
$739$	22.6918	0.834732	0.417366	+	0.908738i	$0.362953\pi$
$740$	0	0
$741$	1.07066	0.0393316
$742$	0	0
$743$	3.08858	0.113309	0.0566546	−	0.998394i	$-0.481957\pi$
$743$	3.08858	0.113309	0.0566546	+	0.998394i	$0.481957\pi$
$744$	0	0
$745$	−4.46260	−0.163497
$746$	0	0
$747$	−9.16484	−0.335324
$748$	0	0
$749$	−6.57760	−0.240340
$750$	0	0
$751$	−10.7577	−0.392553	−0.196276	−	0.980549i	$-0.562885\pi$
$751$	−10.7577	−0.392553	−0.196276	+	0.980549i	$0.562885\pi$
$752$	0	0
$753$	−42.4882	−1.54836
$754$	0	0
$755$	10.7562	0.391459
$756$	0	0
$757$	−19.5333	−0.709948	−0.354974	−	0.934876i	$-0.615510\pi$
$757$	−19.5333	−0.709948	−0.354974	+	0.934876i	$0.615510\pi$
$758$	0	0
$759$	1.25901	0.0456993
$760$	0	0
$761$	−15.6933	−0.568881	−0.284440	−	0.958694i	$-0.591808\pi$
$761$	−15.6933	−0.568881	−0.284440	+	0.958694i	$0.591808\pi$
$762$	0	0
$763$	−27.3304	−0.989429
$764$	0	0
$765$	−4.74099	−0.171411
$766$	0	0
$767$	−1.53326	−0.0553626
$768$	0	0
$769$	11.4128	0.411555	0.205777	−	0.978599i	$-0.434028\pi$
$769$	11.4128	0.411555	0.205777	+	0.978599i	$0.434028\pi$
$770$	0	0
$771$	−9.59910	−0.345703
$772$	0	0
$773$	−22.5872	−0.812406	−0.406203	−	0.913783i	$-0.633147\pi$
$773$	−22.5872	−0.812406	−0.406203	+	0.913783i	$0.633147\pi$
$774$	0	0
$775$	6.78600	0.243760
$776$	0	0
$777$	−27.1857	−0.975280
$778$	0	0
$779$	−52.4882	−1.88059
$780$	0	0
$781$	−6.29507	−0.225255
$782$	0	0
$783$	−55.1503	−1.97091
$784$	0	0
$785$	14.5180	0.518171
$786$	0	0
$787$	−4.42799	−0.157841	−0.0789205	−	0.996881i	$-0.525147\pi$
$787$	−4.42799	−0.157841	−0.0789205	+	0.996881i	$0.525147\pi$
$788$	0	0
$789$	−0.821385	−0.0292421
$790$	0	0
$791$	−4.92520	−0.175120
$792$	0	0
$793$	0.617021	0.0219111
$794$	0	0
$795$	0.218814	0.00776053
$796$	0	0
$797$	21.5124	0.762009	0.381005	−	0.924573i	$-0.375578\pi$
$797$	21.5124	0.762009	0.381005	+	0.924573i	$0.375578\pi$
$798$	0	0
$799$	−12.7964	−0.452705
$800$	0	0
$801$	14.8864	0.525987
$802$	0	0
$803$	−6.12878	−0.216280
$804$	0	0
$805$	−1.53740	−0.0541863
$806$	0	0
$807$	−10.6323	−0.374273
$808$	0	0
$809$	7.37883	0.259426	0.129713	−	0.991552i	$-0.458594\pi$
$809$	7.37883	0.259426	0.129713	+	0.991552i	$0.458594\pi$
$810$	0	0
$811$	29.0423	1.01981	0.509907	−	0.860230i	$-0.329680\pi$
$811$	29.0423	1.01981	0.509907	+	0.860230i	$0.329680\pi$
$812$	0	0
$813$	0.269425	0.00944916
$814$	0	0
$815$	−1.30403	−0.0456782
$816$	0	0
$817$	60.1801	2.10543
$818$	0	0
$819$	−0.184210	−0.00643682
$820$	0	0
$821$	5.70079	0.198959	0.0994794	−	0.995040i	$-0.468282\pi$
$821$	5.70079	0.198959	0.0994794	+	0.995040i	$0.468282\pi$
$822$	0	0
$823$	−16.0034	−0.557842	−0.278921	−	0.960314i	$-0.589977\pi$
$823$	−16.0034	−0.557842	−0.278921	+	0.960314i	$0.589977\pi$
$824$	0	0
$825$	−1.25901	−0.0438332
$826$	0	0
$827$	39.7937	1.38376	0.691882	−	0.722011i	$-0.256782\pi$
$827$	39.7937	1.38376	0.691882	+	0.722011i	$0.256782\pi$
$828$	0	0
$829$	−23.3353	−0.810467	−0.405234	−	0.914213i	$-0.632810\pi$
$829$	−23.3353	−0.810467	−0.405234	+	0.914213i	$0.632810\pi$
$830$	0	0
$831$	37.8746	1.31385
$832$	0	0
$833$	−25.5355	−0.884752
$834$	0	0
$835$	3.44322	0.119158
$836$	0	0
$837$	−38.3193	−1.32451
$838$	0	0
$839$	37.1469	1.28245	0.641227	−	0.767351i	$-0.278426\pi$
$839$	37.1469	1.28245	0.641227	+	0.767351i	$0.278426\pi$
$840$	0	0
$841$	66.3870	2.28921
$842$	0	0
$843$	−39.1619	−1.34881
$844$	0	0
$845$	−12.9806	−0.446547
$846$	0	0
$847$	−15.7722	−0.541940
$848$	0	0
$849$	8.33796	0.286158
$850$	0	0
$851$	12.0900	0.414441
$852$	0	0
$853$	−53.7658	−1.84091	−0.920454	−	0.390851i	$-0.872181\pi$
$853$	−53.7658	−1.84091	−0.920454	+	0.390851i	$0.872181\pi$
$854$	0	0
$855$	−4.52699	−0.154820
$856$	0	0
$857$	−23.9675	−0.818715	−0.409357	−	0.912374i	$-0.634247\pi$
$857$	−23.9675	−0.818715	−0.409357	+	0.912374i	$0.634247\pi$
$858$	0	0
$859$	3.70705	0.126483	0.0632415	−	0.997998i	$-0.479856\pi$
$859$	3.70705	0.126483	0.0632415	+	0.997998i	$0.479856\pi$
$860$	0	0
$861$	−22.4424	−0.764836
$862$	0	0
$863$	49.6296	1.68941	0.844705	−	0.535232i	$-0.179776\pi$
$863$	49.6296	1.68941	0.844705	+	0.535232i	$0.179776\pi$
$864$	0	0
$865$	9.35801	0.318182
$866$	0	0
$867$	19.5020	0.662323
$868$	0	0
$869$	−5.85039	−0.198461
$870$	0	0
$871$	−1.46115	−0.0495091
$872$	0	0
$873$	−1.66618	−0.0563918
$874$	0	0
$875$	1.53740	0.0519737
$876$	0	0
$877$	−13.1004	−0.442371	−0.221185	−	0.975232i	$-0.570993\pi$
$877$	−13.1004	−0.442371	−0.221185	+	0.975232i	$0.570993\pi$
$878$	0	0
$879$	1.54155	0.0519951
$880$	0	0
$881$	3.88085	0.130749	0.0653746	−	0.997861i	$-0.479176\pi$
$881$	3.88085	0.130749	0.0653746	+	0.997861i	$0.479176\pi$
$882$	0	0
$883$	−10.8131	−0.363890	−0.181945	−	0.983309i	$-0.558239\pi$
$883$	−10.8131	−0.363890	−0.181945	+	0.983309i	$0.558239\pi$
$884$	0	0
$885$	−16.1109	−0.541561
$886$	0	0
$887$	−8.58387	−0.288218	−0.144109	−	0.989562i	$-0.546032\pi$
$887$	−8.58387	−0.288218	−0.144109	+	0.989562i	$0.546032\pi$
$888$	0	0
$889$	−23.3836	−0.784262
$890$	0	0
$891$	4.88645	0.163702
$892$	0	0
$893$	−12.2188	−0.408887
$894$	0	0
$895$	−18.4134	−0.615493
$896$	0	0
$897$	−0.203585	−0.00679751
$898$	0	0
$899$	66.2764	2.21044
$900$	0	0
$901$	0.823974	0.0274505
$902$	0	0
$903$	25.7312	0.856282
$904$	0	0
$905$	−1.94457	−0.0646398
$906$	0	0
$907$	−21.2728	−0.706351	−0.353176	−	0.935557i	$-0.614898\pi$
$907$	−21.2728	−0.706351	−0.353176	+	0.935557i	$0.614898\pi$
$908$	0	0
$909$	13.2757	0.440327
$910$	0	0
$911$	−27.6441	−0.915890	−0.457945	−	0.888980i	$-0.651414\pi$
$911$	−27.6441	−0.915890	−0.457945	+	0.888980i	$0.651414\pi$
$912$	0	0
$913$	−9.16484	−0.303312
$914$	0	0
$915$	6.48342	0.214335
$916$	0	0
$917$	19.5124	0.644357
$918$	0	0
$919$	−0.994404	−0.0328024	−0.0164012	−	0.999865i	$-0.505221\pi$
$919$	−0.994404	−0.0328024	−0.0164012	+	0.999865i	$0.505221\pi$
$920$	0	0
$921$	26.1455	0.861522
$922$	0	0
$923$	1.01793	0.0335054
$924$	0	0
$925$	−12.0900	−0.397518
$926$	0	0
$927$	−6.81309	−0.223771
$928$	0	0
$929$	−18.3747	−0.602854	−0.301427	−	0.953489i	$-0.597463\pi$
$929$	−18.3747	−0.602854	−0.301427	+	0.953489i	$0.597463\pi$
$930$	0	0
$931$	−24.3829	−0.799116
$932$	0	0
$933$	24.7639	0.810735
$934$	0	0
$935$	−4.74099	−0.155047
$936$	0	0
$937$	−24.0942	−0.787122	−0.393561	−	0.919298i	$-0.628757\pi$
$937$	−24.0942	−0.787122	−0.393561	+	0.919298i	$0.628757\pi$
$938$	0	0
$939$	−29.2459	−0.954404
$940$	0	0
$941$	51.9244	1.69269	0.846344	−	0.532637i	$-0.178799\pi$
$941$	51.9244	1.69269	0.846344	+	0.532637i	$0.178799\pi$
$942$	0	0
$943$	9.98062	0.325014
$944$	0	0
$945$	−8.68141	−0.282406
$946$	0	0
$947$	−4.72643	−0.153588	−0.0767941	−	0.997047i	$-0.524468\pi$
$947$	−4.72643	−0.153588	−0.0767941	+	0.997047i	$0.524468\pi$
$948$	0	0
$949$	0.991037	0.0321704
$950$	0	0
$951$	4.82330	0.156406
$952$	0	0
$953$	3.57682	0.115865	0.0579323	−	0.998321i	$-0.481549\pi$
$953$	3.57682	0.115865	0.0579323	+	0.998321i	$0.481549\pi$
$954$	0	0
$955$	−0.775591	−0.0250975
$956$	0	0
$957$	−12.2963	−0.397483
$958$	0	0
$959$	−21.5768	−0.696752
$960$	0	0
$961$	15.0498	0.485478
$962$	0	0
$963$	3.68286	0.118679
$964$	0	0
$965$	−11.7666	−0.378781
$966$	0	0
$967$	33.9646	1.09223	0.546114	−	0.837711i	$-0.316106\pi$
$967$	33.9646	1.09223	0.546114	+	0.837711i	$0.316106\pi$
$968$	0	0
$969$	42.3636	1.36092
$970$	0	0
$971$	−56.0561	−1.79893	−0.899463	−	0.436997i	$-0.856042\pi$
$971$	−56.0561	−1.79893	−0.899463	+	0.436997i	$0.856042\pi$
$972$	0	0
$973$	21.9292	0.703019
$974$	0	0
$975$	0.203585	0.00651994
$976$	0	0
$977$	−53.1219	−1.69952	−0.849761	−	0.527169i	$-0.823254\pi$
$977$	−53.1219	−1.69952	−0.849761	+	0.527169i	$0.823254\pi$
$978$	0	0
$979$	14.8864	0.475773
$980$	0	0
$981$	15.3026	0.488574
$982$	0	0
$983$	28.7833	0.918045	0.459022	−	0.888425i	$-0.348200\pi$
$983$	28.7833	0.918045	0.459022	+	0.888425i	$0.348200\pi$
$984$	0	0
$985$	9.68141	0.308475
$986$	0	0
$987$	−5.22441	−0.166295
$988$	0	0
$989$	−11.4432	−0.363873
$990$	0	0
$991$	48.1641	1.52998	0.764991	−	0.644041i	$-0.222743\pi$
$991$	48.1641	1.52998	0.764991	+	0.644041i	$0.222743\pi$
$992$	0	0
$993$	−18.8385	−0.597823
$994$	0	0
$995$	−3.29362	−0.104415
$996$	0	0
$997$	43.5153	1.37814	0.689072	−	0.724693i	$-0.258018\pi$
$997$	43.5153	1.37814	0.689072	+	0.724693i	$0.258018\pi$
$998$	0	0
$999$	68.2701	2.15997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.i.1.2	✓	3
3.2	odd	2		8280.2.a.bl.1.2		3
4.3	odd	2		1840.2.a.q.1.2		3
5.2	odd	4		4600.2.e.q.4049.3		6
5.3	odd	4		4600.2.e.q.4049.4		6
5.4	even	2		4600.2.a.v.1.2		3
8.3	odd	2		7360.2.a.cf.1.2		3
8.5	even	2		7360.2.a.bw.1.2		3
20.19	odd	2		9200.2.a.ci.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.i.1.2	✓	3	1.1	even	1	trivial
1840.2.a.q.1.2		3	4.3	odd	2
4600.2.a.v.1.2		3	5.4	even	2
4600.2.e.q.4049.3		6	5.2	odd	4
4600.2.e.q.4049.4		6	5.3	odd	4
7360.2.a.bw.1.2		3	8.5	even	2
7360.2.a.cf.1.2		3	8.3	odd	2
8280.2.a.bl.1.2		3	3.2	odd	2
9200.2.a.ci.1.2		3	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46260 q^{3} +1.00000 q^{5} +1.53740 q^{7} -0.860806 q^{9} -0.860806 q^{11} +0.139194 q^{13} +1.46260 q^{15} +5.50761 q^{17} +5.25901 q^{19} +2.24860 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.64681 q^{27} +9.76663 q^{29} +6.78600 q^{31} -1.25901 q^{33} +1.53740 q^{35} -12.0900 q^{37} +0.203585 q^{39} -9.98062 q^{41} +11.4432 q^{43} -0.860806 q^{45} -2.32340 q^{47} -4.63640 q^{49} +8.05543 q^{51} +0.149606 q^{53} -0.860806 q^{55} +7.69182 q^{57} -11.0152 q^{59} +4.43281 q^{61} -1.32340 q^{63} +0.139194 q^{65} -10.4972 q^{67} -1.46260 q^{69} +7.31299 q^{71} +7.11982 q^{73} +1.46260 q^{75} -1.32340 q^{77} +6.79641 q^{79} -5.67660 q^{81} +10.6468 q^{83} +5.50761 q^{85} +14.2847 q^{87} -17.2936 q^{89} +0.213997 q^{91} +9.92520 q^{93} +5.25901 q^{95} +1.93561 q^{97} +0.740987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 3 q^{5} + 7 q^{7} + 3 q^{9} + 3 q^{11} + 6 q^{13} + 2 q^{15} - 5 q^{17} + 7 q^{19} - 6 q^{21} - 3 q^{23} + 3 q^{25} - q^{27} - q^{29} + 10 q^{31} + 5 q^{33} + 7 q^{35} + 2 q^{37} + 7 q^{39}+ \cdots + 11 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 3 * q^5 + 7 * q^7 + 3 * q^9 + 3 * q^11 + 6 * q^13 + 2 * q^15 - 5 * q^17 + 7 * q^19 - 6 * q^21 - 3 * q^23 + 3 * q^25 - q^27 - q^29 + 10 * q^31 + 5 * q^33 + 7 * q^35 + 2 * q^37 + 7 * q^39 - 10 * q^41 + 12 * q^43 + 3 * q^45 + q^47 + 6 * q^49 + 9 * q^51 + 10 * q^53 + 3 * q^55 - 12 * q^57 + 10 * q^59 - 13 * q^61 + 4 * q^63 + 6 * q^65 - 6 * q^67 - 2 * q^69 + 10 * q^71 + 7 * q^73 + 2 * q^75 + 4 * q^77 + 14 * q^79 - 25 * q^81 + 16 * q^83 - 5 * q^85 - 5 * q^87 - 20 * q^89 + 11 * q^91 + 25 * q^93 + 7 * q^95 + 5 * q^97 + 11 * q^99

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