LMFDB - Embedded newform 8020.2.a.f.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.869740 q^{3} +1.00000 q^{5} -1.85510 q^{7} -2.24355 q^{9} +O(q^{10})$	$q-0.869740 q^{3} +1.00000 q^{5} -1.85510 q^{7} -2.24355 q^{9} -5.49756 q^{11} -0.851215 q^{13} -0.869740 q^{15} -7.59031 q^{17} -6.07960 q^{19} +1.61346 q^{21} -3.96786 q^{23} +1.00000 q^{25} +4.56053 q^{27} +2.47518 q^{29} -3.46187 q^{31} +4.78144 q^{33} -1.85510 q^{35} +6.01143 q^{37} +0.740336 q^{39} -6.31094 q^{41} -1.43182 q^{43} -2.24355 q^{45} -10.4068 q^{47} -3.55859 q^{49} +6.60160 q^{51} +0.324647 q^{53} -5.49756 q^{55} +5.28767 q^{57} -3.54379 q^{59} +6.61958 q^{61} +4.16202 q^{63} -0.851215 q^{65} -10.3649 q^{67} +3.45101 q^{69} +1.96610 q^{71} -12.3709 q^{73} -0.869740 q^{75} +10.1985 q^{77} +1.12043 q^{79} +2.76419 q^{81} +8.15985 q^{83} -7.59031 q^{85} -2.15276 q^{87} +9.14798 q^{89} +1.57909 q^{91} +3.01093 q^{93} -6.07960 q^{95} +0.182801 q^{97} +12.3341 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * 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
$3$	−0.869740	−0.502144	−0.251072	−	0.967968i	$-0.580783\pi$
$3$	−0.869740	−0.502144	−0.251072	+	0.967968i	$0.580783\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.85510	−0.701163	−0.350582	−	0.936532i	$-0.614016\pi$
$7$	−1.85510	−0.701163	−0.350582	+	0.936532i	$0.614016\pi$
$8$	0	0
$9$	−2.24355	−0.747851
$10$	0	0
$11$	−5.49756	−1.65758	−0.828788	−	0.559563i	$-0.810969\pi$
$11$	−5.49756	−1.65758	−0.828788	+	0.559563i	$0.810969\pi$
$12$	0	0
$13$	−0.851215	−0.236085	−0.118042	−	0.993009i	$-0.537662\pi$
$13$	−0.851215	−0.236085	−0.118042	+	0.993009i	$0.537662\pi$
$14$	0	0
$15$	−0.869740	−0.224566
$16$	0	0
$17$	−7.59031	−1.84092	−0.920461	−	0.390835i	$-0.872186\pi$
$17$	−7.59031	−1.84092	−0.920461	+	0.390835i	$0.872186\pi$
$18$	0	0
$19$	−6.07960	−1.39476	−0.697378	−	0.716704i	$-0.745650\pi$
$19$	−6.07960	−1.39476	−0.697378	+	0.716704i	$0.745650\pi$
$20$	0	0
$21$	1.61346	0.352085
$22$	0	0
$23$	−3.96786	−0.827357	−0.413678	−	0.910423i	$-0.635756\pi$
$23$	−3.96786	−0.827357	−0.413678	+	0.910423i	$0.635756\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.56053	0.877674
$28$	0	0
$29$	2.47518	0.459629	0.229814	−	0.973234i	$-0.426188\pi$
$29$	2.47518	0.459629	0.229814	+	0.973234i	$0.426188\pi$
$30$	0	0
$31$	−3.46187	−0.621771	−0.310885	−	0.950447i	$-0.600626\pi$
$31$	−3.46187	−0.621771	−0.310885	+	0.950447i	$0.600626\pi$
$32$	0	0
$33$	4.78144	0.832342
$34$	0	0
$35$	−1.85510	−0.313570
$36$	0	0
$37$	6.01143	0.988273	0.494136	−	0.869384i	$-0.335484\pi$
$37$	6.01143	0.988273	0.494136	+	0.869384i	$0.335484\pi$
$38$	0	0
$39$	0.740336	0.118549
$40$	0	0
$41$	−6.31094	−0.985603	−0.492802	−	0.870142i	$-0.664027\pi$
$41$	−6.31094	−0.985603	−0.492802	+	0.870142i	$0.664027\pi$
$42$	0	0
$43$	−1.43182	−0.218350	−0.109175	−	0.994023i	$-0.534821\pi$
$43$	−1.43182	−0.218350	−0.109175	+	0.994023i	$0.534821\pi$
$44$	0	0
$45$	−2.24355	−0.334449
$46$	0	0
$47$	−10.4068	−1.51799	−0.758995	−	0.651096i	$-0.774310\pi$
$47$	−10.4068	−1.51799	−0.758995	+	0.651096i	$0.774310\pi$
$48$	0	0
$49$	−3.55859	−0.508370
$50$	0	0
$51$	6.60160	0.924409
$52$	0	0
$53$	0.324647	0.0445938	0.0222969	−	0.999751i	$-0.492902\pi$
$53$	0.324647	0.0445938	0.0222969	+	0.999751i	$0.492902\pi$
$54$	0	0
$55$	−5.49756	−0.741290
$56$	0	0
$57$	5.28767	0.700369
$58$	0	0
$59$	−3.54379	−0.461362	−0.230681	−	0.973029i	$-0.574095\pi$
$59$	−3.54379	−0.461362	−0.230681	+	0.973029i	$0.574095\pi$
$60$	0	0
$61$	6.61958	0.847551	0.423775	−	0.905767i	$-0.360705\pi$
$61$	6.61958	0.847551	0.423775	+	0.905767i	$0.360705\pi$
$62$	0	0
$63$	4.16202	0.524365
$64$	0	0
$65$	−0.851215	−0.105580
$66$	0	0
$67$	−10.3649	−1.26628	−0.633139	−	0.774038i	$-0.718234\pi$
$67$	−10.3649	−1.26628	−0.633139	+	0.774038i	$0.718234\pi$
$68$	0	0
$69$	3.45101	0.415453
$70$	0	0
$71$	1.96610	0.233333	0.116666	−	0.993171i	$-0.462779\pi$
$71$	1.96610	0.233333	0.116666	+	0.993171i	$0.462779\pi$
$72$	0	0
$73$	−12.3709	−1.44791	−0.723953	−	0.689849i	$-0.757677\pi$
$73$	−12.3709	−1.44791	−0.723953	+	0.689849i	$0.757677\pi$
$74$	0	0
$75$	−0.869740	−0.100429
$76$	0	0
$77$	10.1985	1.16223
$78$	0	0
$79$	1.12043	0.126059	0.0630294	−	0.998012i	$-0.479924\pi$
$79$	1.12043	0.126059	0.0630294	+	0.998012i	$0.479924\pi$
$80$	0	0
$81$	2.76419	0.307132
$82$	0	0
$83$	8.15985	0.895660	0.447830	−	0.894119i	$-0.352197\pi$
$83$	8.15985	0.895660	0.447830	+	0.894119i	$0.352197\pi$
$84$	0	0
$85$	−7.59031	−0.823285
$86$	0	0
$87$	−2.15276	−0.230800
$88$	0	0
$89$	9.14798	0.969684	0.484842	−	0.874602i	$-0.338877\pi$
$89$	9.14798	0.969684	0.484842	+	0.874602i	$0.338877\pi$
$90$	0	0
$91$	1.57909	0.165534
$92$	0	0
$93$	3.01093	0.312219
$94$	0	0
$95$	−6.07960	−0.623754
$96$	0	0
$97$	0.182801	0.0185606	0.00928030	−	0.999957i	$-0.497046\pi$
$97$	0.182801	0.0185606	0.00928030	+	0.999957i	$0.497046\pi$
$98$	0	0
$99$	12.3341	1.23962
$100$	0	0
$101$	4.10830	0.408791	0.204395	−	0.978888i	$-0.434477\pi$
$101$	4.10830	0.408791	0.204395	+	0.978888i	$0.434477\pi$
$102$	0	0
$103$	−2.20411	−0.217178	−0.108589	−	0.994087i	$-0.534633\pi$
$103$	−2.20411	−0.217178	−0.108589	+	0.994087i	$0.534633\pi$
$104$	0	0
$105$	1.61346	0.157457
$106$	0	0
$107$	19.6419	1.89885	0.949425	−	0.313994i	$-0.101667\pi$
$107$	19.6419	1.89885	0.949425	+	0.313994i	$0.101667\pi$
$108$	0	0
$109$	9.25996	0.886943	0.443472	−	0.896288i	$-0.353747\pi$
$109$	9.25996	0.886943	0.443472	+	0.896288i	$0.353747\pi$
$110$	0	0
$111$	−5.22838	−0.496256
$112$	0	0
$113$	6.10969	0.574751	0.287376	−	0.957818i	$-0.407217\pi$
$113$	6.10969	0.574751	0.287376	+	0.957818i	$0.407217\pi$
$114$	0	0
$115$	−3.96786	−0.370005
$116$	0	0
$117$	1.90975	0.176556
$118$	0	0
$119$	14.0808	1.29079
$120$	0	0
$121$	19.2231	1.74756
$122$	0	0
$123$	5.48888	0.494915
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−10.0335	−0.890329	−0.445164	−	0.895449i	$-0.646855\pi$
$127$	−10.0335	−0.890329	−0.445164	+	0.895449i	$0.646855\pi$
$128$	0	0
$129$	1.24531	0.109643
$130$	0	0
$131$	9.44696	0.825384	0.412692	−	0.910871i	$-0.364589\pi$
$131$	9.44696	0.825384	0.412692	+	0.910871i	$0.364589\pi$
$132$	0	0
$133$	11.2783	0.977951
$134$	0	0
$135$	4.56053	0.392508
$136$	0	0
$137$	−0.552511	−0.0472042	−0.0236021	−	0.999721i	$-0.507513\pi$
$137$	−0.552511	−0.0472042	−0.0236021	+	0.999721i	$0.507513\pi$
$138$	0	0
$139$	−0.622098	−0.0527657	−0.0263828	−	0.999652i	$-0.508399\pi$
$139$	−0.622098	−0.0527657	−0.0263828	+	0.999652i	$0.508399\pi$
$140$	0	0
$141$	9.05123	0.762251
$142$	0	0
$143$	4.67961	0.391328
$144$	0	0
$145$	2.47518	0.205552
$146$	0	0
$147$	3.09505	0.255275
$148$	0	0
$149$	−9.60901	−0.787201	−0.393601	−	0.919282i	$-0.628771\pi$
$149$	−9.60901	−0.787201	−0.393601	+	0.919282i	$0.628771\pi$
$150$	0	0
$151$	−9.33515	−0.759684	−0.379842	−	0.925051i	$-0.624022\pi$
$151$	−9.33515	−0.759684	−0.379842	+	0.925051i	$0.624022\pi$
$152$	0	0
$153$	17.0293	1.37673
$154$	0	0
$155$	−3.46187	−0.278064
$156$	0	0
$157$	−15.8346	−1.26374	−0.631870	−	0.775074i	$-0.717713\pi$
$157$	−15.8346	−1.26374	−0.631870	+	0.775074i	$0.717713\pi$
$158$	0	0
$159$	−0.282359	−0.0223925
$160$	0	0
$161$	7.36080	0.580112
$162$	0	0
$163$	10.5649	0.827509	0.413755	−	0.910388i	$-0.364217\pi$
$163$	10.5649	0.827509	0.413755	+	0.910388i	$0.364217\pi$
$164$	0	0
$165$	4.78144	0.372235
$166$	0	0
$167$	−13.5600	−1.04930	−0.524651	−	0.851317i	$-0.675804\pi$
$167$	−13.5600	−1.04930	−0.524651	+	0.851317i	$0.675804\pi$
$168$	0	0
$169$	−12.2754	−0.944264
$170$	0	0
$171$	13.6399	1.04307
$172$	0	0
$173$	12.8989	0.980686	0.490343	−	0.871529i	$-0.336871\pi$
$173$	12.8989	0.980686	0.490343	+	0.871529i	$0.336871\pi$
$174$	0	0
$175$	−1.85510	−0.140233
$176$	0	0
$177$	3.08217	0.231670
$178$	0	0
$179$	−12.6988	−0.949154	−0.474577	−	0.880214i	$-0.657399\pi$
$179$	−12.6988	−0.949154	−0.474577	+	0.880214i	$0.657399\pi$
$180$	0	0
$181$	−16.0213	−1.19086	−0.595428	−	0.803408i	$-0.703018\pi$
$181$	−16.0213	−1.19086	−0.595428	+	0.803408i	$0.703018\pi$
$182$	0	0
$183$	−5.75731	−0.425593
$184$	0	0
$185$	6.01143	0.441969
$186$	0	0
$187$	41.7282	3.05147
$188$	0	0
$189$	−8.46025	−0.615392
$190$	0	0
$191$	3.12727	0.226281	0.113141	−	0.993579i	$-0.463909\pi$
$191$	3.12727	0.226281	0.113141	+	0.993579i	$0.463909\pi$
$192$	0	0
$193$	−4.13102	−0.297357	−0.148679	−	0.988886i	$-0.547502\pi$
$193$	−4.13102	−0.297357	−0.148679	+	0.988886i	$0.547502\pi$
$194$	0	0
$195$	0.740336	0.0530166
$196$	0	0
$197$	20.4762	1.45887	0.729435	−	0.684050i	$-0.239783\pi$
$197$	20.4762	1.45887	0.729435	+	0.684050i	$0.239783\pi$
$198$	0	0
$199$	0.335680	0.0237957	0.0118979	−	0.999929i	$-0.496213\pi$
$199$	0.335680	0.0237957	0.0118979	+	0.999929i	$0.496213\pi$
$200$	0	0
$201$	9.01479	0.635854
$202$	0	0
$203$	−4.59171	−0.322275
$204$	0	0
$205$	−6.31094	−0.440775
$206$	0	0
$207$	8.90211	0.618740
$208$	0	0
$209$	33.4229	2.31191
$210$	0	0
$211$	7.57457	0.521455	0.260728	−	0.965412i	$-0.416038\pi$
$211$	7.57457	0.521455	0.260728	+	0.965412i	$0.416038\pi$
$212$	0	0
$213$	−1.70999	−0.117167
$214$	0	0
$215$	−1.43182	−0.0976492
$216$	0	0
$217$	6.42213	0.435963
$218$	0	0
$219$	10.7595	0.727058
$220$	0	0
$221$	6.46099	0.434613
$222$	0	0
$223$	−24.4166	−1.63506	−0.817528	−	0.575888i	$-0.804656\pi$
$223$	−24.4166	−1.63506	−0.817528	+	0.575888i	$0.804656\pi$
$224$	0	0
$225$	−2.24355	−0.149570
$226$	0	0
$227$	−21.0011	−1.39389	−0.696947	−	0.717123i	$-0.745459\pi$
$227$	−21.0011	−1.39389	−0.696947	+	0.717123i	$0.745459\pi$
$228$	0	0
$229$	−29.2561	−1.93330	−0.966649	−	0.256106i	$-0.917560\pi$
$229$	−29.2561	−1.93330	−0.966649	+	0.256106i	$0.917560\pi$
$230$	0	0
$231$	−8.87007	−0.583608
$232$	0	0
$233$	8.70405	0.570222	0.285111	−	0.958495i	$-0.407970\pi$
$233$	8.70405	0.570222	0.285111	+	0.958495i	$0.407970\pi$
$234$	0	0
$235$	−10.4068	−0.678866
$236$	0	0
$237$	−0.974487	−0.0632997
$238$	0	0
$239$	1.04778	0.0677755	0.0338878	−	0.999426i	$-0.489211\pi$
$239$	1.04778	0.0677755	0.0338878	+	0.999426i	$0.489211\pi$
$240$	0	0
$241$	−21.5632	−1.38901	−0.694505	−	0.719488i	$-0.744376\pi$
$241$	−21.5632	−1.38901	−0.694505	+	0.719488i	$0.744376\pi$
$242$	0	0
$243$	−16.0857	−1.03190
$244$	0	0
$245$	−3.55859	−0.227350
$246$	0	0
$247$	5.17505	0.329280
$248$	0	0
$249$	−7.09695	−0.449751
$250$	0	0
$251$	−14.2953	−0.902309	−0.451155	−	0.892446i	$-0.648988\pi$
$251$	−14.2953	−0.902309	−0.451155	+	0.892446i	$0.648988\pi$
$252$	0	0
$253$	21.8136	1.37141
$254$	0	0
$255$	6.60160	0.413408
$256$	0	0
$257$	−3.36011	−0.209598	−0.104799	−	0.994493i	$-0.533420\pi$
$257$	−3.36011	−0.209598	−0.104799	+	0.994493i	$0.533420\pi$
$258$	0	0
$259$	−11.1518	−0.692940
$260$	0	0
$261$	−5.55319	−0.343734
$262$	0	0
$263$	1.54432	0.0952271	0.0476135	−	0.998866i	$-0.484838\pi$
$263$	1.54432	0.0952271	0.0476135	+	0.998866i	$0.484838\pi$
$264$	0	0
$265$	0.324647	0.0199429
$266$	0	0
$267$	−7.95637	−0.486922
$268$	0	0
$269$	−11.7385	−0.715709	−0.357854	−	0.933777i	$-0.616492\pi$
$269$	−11.7385	−0.715709	−0.357854	+	0.933777i	$0.616492\pi$
$270$	0	0
$271$	−3.45734	−0.210018	−0.105009	−	0.994471i	$-0.533487\pi$
$271$	−3.45734	−0.210018	−0.105009	+	0.994471i	$0.533487\pi$
$272$	0	0
$273$	−1.37340	−0.0831219
$274$	0	0
$275$	−5.49756	−0.331515
$276$	0	0
$277$	16.4748	0.989874	0.494937	−	0.868929i	$-0.335191\pi$
$277$	16.4748	0.989874	0.494937	+	0.868929i	$0.335191\pi$
$278$	0	0
$279$	7.76690	0.464992
$280$	0	0
$281$	−22.5560	−1.34558	−0.672788	−	0.739835i	$-0.734904\pi$
$281$	−22.5560	−1.34558	−0.672788	+	0.739835i	$0.734904\pi$
$282$	0	0
$283$	22.8171	1.35634	0.678169	−	0.734906i	$-0.262774\pi$
$283$	22.8171	1.35634	0.678169	+	0.734906i	$0.262774\pi$
$284$	0	0
$285$	5.28767	0.313214
$286$	0	0
$287$	11.7074	0.691069
$288$	0	0
$289$	40.6129	2.38899
$290$	0	0
$291$	−0.158989	−0.00932010
$292$	0	0
$293$	−12.4313	−0.726245	−0.363122	−	0.931741i	$-0.618289\pi$
$293$	−12.4313	−0.726245	−0.363122	+	0.931741i	$0.618289\pi$
$294$	0	0
$295$	−3.54379	−0.206327
$296$	0	0
$297$	−25.0718	−1.45481
$298$	0	0
$299$	3.37751	0.195326
$300$	0	0
$301$	2.65617	0.153099
$302$	0	0
$303$	−3.57315	−0.205272
$304$	0	0
$305$	6.61958	0.379036
$306$	0	0
$307$	−13.9198	−0.794447	−0.397224	−	0.917722i	$-0.630026\pi$
$307$	−13.9198	−0.794447	−0.397224	+	0.917722i	$0.630026\pi$
$308$	0	0
$309$	1.91700	0.109054
$310$	0	0
$311$	0.813892	0.0461516	0.0230758	−	0.999734i	$-0.492654\pi$
$311$	0.813892	0.0461516	0.0230758	+	0.999734i	$0.492654\pi$
$312$	0	0
$313$	−6.30521	−0.356392	−0.178196	−	0.983995i	$-0.557026\pi$
$313$	−6.30521	−0.356392	−0.178196	+	0.983995i	$0.557026\pi$
$314$	0	0
$315$	4.16202	0.234503
$316$	0	0
$317$	20.7290	1.16425	0.582127	−	0.813098i	$-0.302221\pi$
$317$	20.7290	1.16425	0.582127	+	0.813098i	$0.302221\pi$
$318$	0	0
$319$	−13.6074	−0.761869
$320$	0	0
$321$	−17.0833	−0.953497
$322$	0	0
$323$	46.1461	2.56764
$324$	0	0
$325$	−0.851215	−0.0472169
$326$	0	0
$327$	−8.05375	−0.445374
$328$	0	0
$329$	19.3057	1.06436
$330$	0	0
$331$	−5.09455	−0.280022	−0.140011	−	0.990150i	$-0.544714\pi$
$331$	−5.09455	−0.280022	−0.140011	+	0.990150i	$0.544714\pi$
$332$	0	0
$333$	−13.4870	−0.739081
$334$	0	0
$335$	−10.3649	−0.566297
$336$	0	0
$337$	18.8928	1.02916	0.514578	−	0.857443i	$-0.327949\pi$
$337$	18.8928	1.02916	0.514578	+	0.857443i	$0.327949\pi$
$338$	0	0
$339$	−5.31384	−0.288608
$340$	0	0
$341$	19.0318	1.03063
$342$	0	0
$343$	19.5873	1.05761
$344$	0	0
$345$	3.45101	0.185796
$346$	0	0
$347$	−25.9352	−1.39228	−0.696138	−	0.717908i	$-0.745100\pi$
$347$	−25.9352	−1.39228	−0.696138	+	0.717908i	$0.745100\pi$
$348$	0	0
$349$	−28.6816	−1.53529	−0.767645	−	0.640876i	$-0.778571\pi$
$349$	−28.6816	−1.53529	−0.767645	+	0.640876i	$0.778571\pi$
$350$	0	0
$351$	−3.88199	−0.207205
$352$	0	0
$353$	31.1211	1.65641	0.828205	−	0.560426i	$-0.189362\pi$
$353$	31.1211	1.65641	0.828205	+	0.560426i	$0.189362\pi$
$354$	0	0
$355$	1.96610	0.104350
$356$	0	0
$357$	−12.2466	−0.648161
$358$	0	0
$359$	−29.7264	−1.56890	−0.784449	−	0.620193i	$-0.787054\pi$
$359$	−29.7264	−1.56890	−0.784449	+	0.620193i	$0.787054\pi$
$360$	0	0
$361$	17.9615	0.945343
$362$	0	0
$363$	−16.7191	−0.877526
$364$	0	0
$365$	−12.3709	−0.647524
$366$	0	0
$367$	4.07965	0.212956	0.106478	−	0.994315i	$-0.466043\pi$
$367$	4.07965	0.212956	0.106478	+	0.994315i	$0.466043\pi$
$368$	0	0
$369$	14.1589	0.737084
$370$	0	0
$371$	−0.602254	−0.0312675
$372$	0	0
$373$	7.32519	0.379284	0.189642	−	0.981853i	$-0.439267\pi$
$373$	7.32519	0.379284	0.189642	+	0.981853i	$0.439267\pi$
$374$	0	0
$375$	−0.869740	−0.0449132
$376$	0	0
$377$	−2.10691	−0.108511
$378$	0	0
$379$	24.4065	1.25368	0.626840	−	0.779148i	$-0.284348\pi$
$379$	24.4065	1.25368	0.626840	+	0.779148i	$0.284348\pi$
$380$	0	0
$381$	8.72653	0.447074
$382$	0	0
$383$	12.7241	0.650169	0.325085	−	0.945685i	$-0.394607\pi$
$383$	12.7241	0.650169	0.325085	+	0.945685i	$0.394607\pi$
$384$	0	0
$385$	10.1985	0.519765
$386$	0	0
$387$	3.21236	0.163293
$388$	0	0
$389$	28.0555	1.42247	0.711236	−	0.702953i	$-0.248136\pi$
$389$	28.0555	1.42247	0.711236	+	0.702953i	$0.248136\pi$
$390$	0	0
$391$	30.1173	1.52310
$392$	0	0
$393$	−8.21639	−0.414462
$394$	0	0
$395$	1.12043	0.0563752
$396$	0	0
$397$	16.0007	0.803053	0.401527	−	0.915847i	$-0.368480\pi$
$397$	16.0007	0.803053	0.401527	+	0.915847i	$0.368480\pi$
$398$	0	0
$399$	−9.80917	−0.491073
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	2.94680	0.146791
$404$	0	0
$405$	2.76419	0.137354
$406$	0	0
$407$	−33.0482	−1.63814
$408$	0	0
$409$	14.1633	0.700328	0.350164	−	0.936688i	$-0.386126\pi$
$409$	14.1633	0.700328	0.350164	+	0.936688i	$0.386126\pi$
$410$	0	0
$411$	0.480541	0.0237033
$412$	0	0
$413$	6.57409	0.323490
$414$	0	0
$415$	8.15985	0.400551
$416$	0	0
$417$	0.541063	0.0264960
$418$	0	0
$419$	22.4760	1.09802	0.549011	−	0.835815i	$-0.315004\pi$
$419$	22.4760	1.09802	0.549011	+	0.835815i	$0.315004\pi$
$420$	0	0
$421$	−14.2981	−0.696845	−0.348422	−	0.937338i	$-0.613282\pi$
$421$	−14.2981	−0.696845	−0.348422	+	0.937338i	$0.613282\pi$
$422$	0	0
$423$	23.3483	1.13523
$424$	0	0
$425$	−7.59031	−0.368184
$426$	0	0
$427$	−12.2800	−0.594271
$428$	0	0
$429$	−4.07004	−0.196503
$430$	0	0
$431$	0.516789	0.0248928	0.0124464	−	0.999923i	$-0.496038\pi$
$431$	0.516789	0.0248928	0.0124464	+	0.999923i	$0.496038\pi$
$432$	0	0
$433$	18.4398	0.886161	0.443080	−	0.896482i	$-0.353886\pi$
$433$	18.4398	0.886161	0.443080	+	0.896482i	$0.353886\pi$
$434$	0	0
$435$	−2.15276	−0.103217
$436$	0	0
$437$	24.1230	1.15396
$438$	0	0
$439$	−20.1411	−0.961284	−0.480642	−	0.876917i	$-0.659596\pi$
$439$	−20.1411	−0.961284	−0.480642	+	0.876917i	$0.659596\pi$
$440$	0	0
$441$	7.98389	0.380185
$442$	0	0
$443$	14.1434	0.671973	0.335986	−	0.941867i	$-0.390930\pi$
$443$	14.1434	0.671973	0.335986	+	0.941867i	$0.390930\pi$
$444$	0	0
$445$	9.14798	0.433656
$446$	0	0
$447$	8.35734	0.395289
$448$	0	0
$449$	−4.41670	−0.208437	−0.104219	−	0.994554i	$-0.533234\pi$
$449$	−4.41670	−0.208437	−0.104219	+	0.994554i	$0.533234\pi$
$450$	0	0
$451$	34.6948	1.63371
$452$	0	0
$453$	8.11915	0.381471
$454$	0	0
$455$	1.57909	0.0740290
$456$	0	0
$457$	4.65249	0.217634	0.108817	−	0.994062i	$-0.465294\pi$
$457$	4.65249	0.217634	0.108817	+	0.994062i	$0.465294\pi$
$458$	0	0
$459$	−34.6158	−1.61573
$460$	0	0
$461$	−28.8247	−1.34250	−0.671250	−	0.741231i	$-0.734242\pi$
$461$	−28.8247	−1.34250	−0.671250	+	0.741231i	$0.734242\pi$
$462$	0	0
$463$	−13.9732	−0.649388	−0.324694	−	0.945819i	$-0.605261\pi$
$463$	−13.9732	−0.649388	−0.324694	+	0.945819i	$0.605261\pi$
$464$	0	0
$465$	3.01093	0.139628
$466$	0	0
$467$	−16.5975	−0.768040	−0.384020	−	0.923325i	$-0.625461\pi$
$467$	−16.5975	−0.768040	−0.384020	+	0.923325i	$0.625461\pi$
$468$	0	0
$469$	19.2280	0.887867
$470$	0	0
$471$	13.7720	0.634580
$472$	0	0
$473$	7.87150	0.361932
$474$	0	0
$475$	−6.07960	−0.278951
$476$	0	0
$477$	−0.728364	−0.0333495
$478$	0	0
$479$	−11.3457	−0.518398	−0.259199	−	0.965824i	$-0.583458\pi$
$479$	−11.3457	−0.518398	−0.259199	+	0.965824i	$0.583458\pi$
$480$	0	0
$481$	−5.11702	−0.233316
$482$	0	0
$483$	−6.40198	−0.291300
$484$	0	0
$485$	0.182801	0.00830055
$486$	0	0
$487$	−2.44531	−0.110807	−0.0554037	−	0.998464i	$-0.517645\pi$
$487$	−2.44531	−0.110807	−0.0554037	+	0.998464i	$0.517645\pi$
$488$	0	0
$489$	−9.18874	−0.415529
$490$	0	0
$491$	−33.7070	−1.52118	−0.760589	−	0.649234i	$-0.775090\pi$
$491$	−33.7070	−1.52118	−0.760589	+	0.649234i	$0.775090\pi$
$492$	0	0
$493$	−18.7874	−0.846140
$494$	0	0
$495$	12.3341	0.554375
$496$	0	0
$497$	−3.64731	−0.163604
$498$	0	0
$499$	−12.7408	−0.570356	−0.285178	−	0.958475i	$-0.592053\pi$
$499$	−12.7408	−0.570356	−0.285178	+	0.958475i	$0.592053\pi$
$500$	0	0
$501$	11.7937	0.526902
$502$	0	0
$503$	−4.71589	−0.210271	−0.105136	−	0.994458i	$-0.533528\pi$
$503$	−4.71589	−0.210271	−0.105136	+	0.994458i	$0.533528\pi$
$504$	0	0
$505$	4.10830	0.182817
$506$	0	0
$507$	10.6764	0.474157
$508$	0	0
$509$	13.2669	0.588046	0.294023	−	0.955798i	$-0.405006\pi$
$509$	13.2669	0.588046	0.294023	+	0.955798i	$0.405006\pi$
$510$	0	0
$511$	22.9493	1.01522
$512$	0	0
$513$	−27.7262	−1.22414
$514$	0	0
$515$	−2.20411	−0.0971247
$516$	0	0
$517$	57.2121	2.51618
$518$	0	0
$519$	−11.2187	−0.492446
$520$	0	0
$521$	−25.2628	−1.10678	−0.553392	−	0.832921i	$-0.686667\pi$
$521$	−25.2628	−1.10678	−0.553392	+	0.832921i	$0.686667\pi$
$522$	0	0
$523$	0.524563	0.0229376	0.0114688	−	0.999934i	$-0.496349\pi$
$523$	0.524563	0.0229376	0.0114688	+	0.999934i	$0.496349\pi$
$524$	0	0
$525$	1.61346	0.0704170
$526$	0	0
$527$	26.2767	1.14463
$528$	0	0
$529$	−7.25605	−0.315480
$530$	0	0
$531$	7.95068	0.345030
$532$	0	0
$533$	5.37197	0.232686
$534$	0	0
$535$	19.6419	0.849191
$536$	0	0
$537$	11.0447	0.476612
$538$	0	0
$539$	19.5636	0.842662
$540$	0	0
$541$	−36.2909	−1.56027	−0.780134	−	0.625612i	$-0.784849\pi$
$541$	−36.2909	−1.56027	−0.780134	+	0.625612i	$0.784849\pi$
$542$	0	0
$543$	13.9344	0.597982
$544$	0	0
$545$	9.25996	0.396653
$546$	0	0
$547$	−29.3566	−1.25520	−0.627599	−	0.778537i	$-0.715962\pi$
$547$	−29.3566	−1.25520	−0.627599	+	0.778537i	$0.715962\pi$
$548$	0	0
$549$	−14.8514	−0.633842
$550$	0	0
$551$	−15.0481	−0.641070
$552$	0	0
$553$	−2.07852	−0.0883878
$554$	0	0
$555$	−5.22838	−0.221932
$556$	0	0
$557$	5.65359	0.239550	0.119775	−	0.992801i	$-0.461783\pi$
$557$	5.65359	0.239550	0.119775	+	0.992801i	$0.461783\pi$
$558$	0	0
$559$	1.21879	0.0515492
$560$	0	0
$561$	−36.2927	−1.53228
$562$	0	0
$563$	−15.4269	−0.650165	−0.325083	−	0.945686i	$-0.605392\pi$
$563$	−15.4269	−0.650165	−0.325083	+	0.945686i	$0.605392\pi$
$564$	0	0
$565$	6.10969	0.257037
$566$	0	0
$567$	−5.12785	−0.215350
$568$	0	0
$569$	39.1423	1.64093	0.820464	−	0.571698i	$-0.193715\pi$
$569$	39.1423	1.64093	0.820464	+	0.571698i	$0.193715\pi$
$570$	0	0
$571$	−4.70250	−0.196794	−0.0983969	−	0.995147i	$-0.531371\pi$
$571$	−4.70250	−0.196794	−0.0983969	+	0.995147i	$0.531371\pi$
$572$	0	0
$573$	−2.71991	−0.113626
$574$	0	0
$575$	−3.96786	−0.165471
$576$	0	0
$577$	−4.86115	−0.202372	−0.101186	−	0.994868i	$-0.532264\pi$
$577$	−4.86115	−0.202372	−0.101186	+	0.994868i	$0.532264\pi$
$578$	0	0
$579$	3.59291	0.149316
$580$	0	0
$581$	−15.1374	−0.628004
$582$	0	0
$583$	−1.78477	−0.0739175
$584$	0	0
$585$	1.90975	0.0789583
$586$	0	0
$587$	−6.30225	−0.260122	−0.130061	−	0.991506i	$-0.541517\pi$
$587$	−6.30225	−0.260122	−0.130061	+	0.991506i	$0.541517\pi$
$588$	0	0
$589$	21.0468	0.867218
$590$	0	0
$591$	−17.8090	−0.732563
$592$	0	0
$593$	−47.7067	−1.95908	−0.979541	−	0.201247i	$-0.935501\pi$
$593$	−47.7067	−1.95908	−0.979541	+	0.201247i	$0.935501\pi$
$594$	0	0
$595$	14.0808	0.577257
$596$	0	0
$597$	−0.291954	−0.0119489
$598$	0	0
$599$	43.2890	1.76874	0.884370	−	0.466786i	$-0.154588\pi$
$599$	43.2890	1.76874	0.884370	+	0.466786i	$0.154588\pi$
$600$	0	0
$601$	7.91221	0.322746	0.161373	−	0.986894i	$-0.448408\pi$
$601$	7.91221	0.322746	0.161373	+	0.986894i	$0.448408\pi$
$602$	0	0
$603$	23.2543	0.946987
$604$	0	0
$605$	19.2231	0.781531
$606$	0	0
$607$	7.41151	0.300824	0.150412	−	0.988623i	$-0.451940\pi$
$607$	7.41151	0.300824	0.150412	+	0.988623i	$0.451940\pi$
$608$	0	0
$609$	3.99359	0.161828
$610$	0	0
$611$	8.85845	0.358374
$612$	0	0
$613$	−13.1374	−0.530613	−0.265307	−	0.964164i	$-0.585473\pi$
$613$	−13.1374	−0.530613	−0.265307	+	0.964164i	$0.585473\pi$
$614$	0	0
$615$	5.48888	0.221333
$616$	0	0
$617$	36.0685	1.45206	0.726032	−	0.687661i	$-0.241362\pi$
$617$	36.0685	1.45206	0.726032	+	0.687661i	$0.241362\pi$
$618$	0	0
$619$	−2.79664	−0.112407	−0.0562033	−	0.998419i	$-0.517899\pi$
$619$	−2.79664	−0.112407	−0.0562033	+	0.998419i	$0.517899\pi$
$620$	0	0
$621$	−18.0956	−0.726149
$622$	0	0
$623$	−16.9705	−0.679907
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−29.0693	−1.16091
$628$	0	0
$629$	−45.6286	−1.81933
$630$	0	0
$631$	−24.9049	−0.991448	−0.495724	−	0.868480i	$-0.665097\pi$
$631$	−24.9049	−0.991448	−0.495724	+	0.868480i	$0.665097\pi$
$632$	0	0
$633$	−6.58791	−0.261846
$634$	0	0
$635$	−10.0335	−0.398167
$636$	0	0
$637$	3.02913	0.120018
$638$	0	0
$639$	−4.41104	−0.174498
$640$	0	0
$641$	0.454652	0.0179577	0.00897883	−	0.999960i	$-0.497142\pi$
$641$	0.454652	0.0179577	0.00897883	+	0.999960i	$0.497142\pi$
$642$	0	0
$643$	−28.2701	−1.11486	−0.557432	−	0.830222i	$-0.688213\pi$
$643$	−28.2701	−1.11486	−0.557432	+	0.830222i	$0.688213\pi$
$644$	0	0
$645$	1.24531	0.0490340
$646$	0	0
$647$	−26.7854	−1.05304	−0.526522	−	0.850162i	$-0.676504\pi$
$647$	−26.7854	−1.05304	−0.526522	+	0.850162i	$0.676504\pi$
$648$	0	0
$649$	19.4822	0.764742
$650$	0	0
$651$	−5.58558	−0.218916
$652$	0	0
$653$	12.6301	0.494253	0.247127	−	0.968983i	$-0.420514\pi$
$653$	12.6301	0.494253	0.247127	+	0.968983i	$0.420514\pi$
$654$	0	0
$655$	9.44696	0.369123
$656$	0	0
$657$	27.7548	1.08282
$658$	0	0
$659$	30.9227	1.20458	0.602288	−	0.798279i	$-0.294256\pi$
$659$	30.9227	1.20458	0.602288	+	0.798279i	$0.294256\pi$
$660$	0	0
$661$	28.2113	1.09729	0.548646	−	0.836055i	$-0.315143\pi$
$661$	28.2113	1.09729	0.548646	+	0.836055i	$0.315143\pi$
$662$	0	0
$663$	−5.61938	−0.218239
$664$	0	0
$665$	11.2783	0.437353
$666$	0	0
$667$	−9.82117	−0.380277
$668$	0	0
$669$	21.2361	0.821035
$670$	0	0
$671$	−36.3915	−1.40488
$672$	0	0
$673$	40.7866	1.57221	0.786103	−	0.618095i	$-0.212095\pi$
$673$	40.7866	1.57221	0.786103	+	0.618095i	$0.212095\pi$
$674$	0	0
$675$	4.56053	0.175535
$676$	0	0
$677$	18.3628	0.705738	0.352869	−	0.935673i	$-0.385206\pi$
$677$	18.3628	0.705738	0.352869	+	0.935673i	$0.385206\pi$
$678$	0	0
$679$	−0.339114	−0.0130140
$680$	0	0
$681$	18.2655	0.699936
$682$	0	0
$683$	40.2636	1.54064	0.770322	−	0.637656i	$-0.220096\pi$
$683$	40.2636	1.54064	0.770322	+	0.637656i	$0.220096\pi$
$684$	0	0
$685$	−0.552511	−0.0211104
$686$	0	0
$687$	25.4452	0.970795
$688$	0	0
$689$	−0.276345	−0.0105279
$690$	0	0
$691$	45.6678	1.73728	0.868642	−	0.495440i	$-0.164993\pi$
$691$	45.6678	1.73728	0.868642	+	0.495440i	$0.164993\pi$
$692$	0	0
$693$	−22.8810	−0.869176
$694$	0	0
$695$	−0.622098	−0.0235975
$696$	0	0
$697$	47.9020	1.81442
$698$	0	0
$699$	−7.57026	−0.286334
$700$	0	0
$701$	−8.68135	−0.327890	−0.163945	−	0.986469i	$-0.552422\pi$
$701$	−8.68135	−0.327890	−0.163945	+	0.986469i	$0.552422\pi$
$702$	0	0
$703$	−36.5471	−1.37840
$704$	0	0
$705$	9.05123	0.340889
$706$	0	0
$707$	−7.62132	−0.286629
$708$	0	0
$709$	−19.8339	−0.744879	−0.372439	−	0.928057i	$-0.621479\pi$
$709$	−19.8339	−0.744879	−0.372439	+	0.928057i	$0.621479\pi$
$710$	0	0
$711$	−2.51375	−0.0942732
$712$	0	0
$713$	13.7362	0.514426
$714$	0	0
$715$	4.67961	0.175007
$716$	0	0
$717$	−0.911299	−0.0340331
$718$	0	0
$719$	22.8267	0.851293	0.425646	−	0.904890i	$-0.360047\pi$
$719$	22.8267	0.851293	0.425646	+	0.904890i	$0.360047\pi$
$720$	0	0
$721$	4.08885	0.152277
$722$	0	0
$723$	18.7544	0.697483
$724$	0	0
$725$	2.47518	0.0919257
$726$	0	0
$727$	−46.9125	−1.73989	−0.869944	−	0.493150i	$-0.835845\pi$
$727$	−46.9125	−1.73989	−0.869944	+	0.493150i	$0.835845\pi$
$728$	0	0
$729$	5.69781	0.211030
$730$	0	0
$731$	10.8680	0.401966
$732$	0	0
$733$	10.3128	0.380912	0.190456	−	0.981696i	$-0.439003\pi$
$733$	10.3128	0.380912	0.190456	+	0.981696i	$0.439003\pi$
$734$	0	0
$735$	3.09505	0.114163
$736$	0	0
$737$	56.9818	2.09895
$738$	0	0
$739$	8.49621	0.312538	0.156269	−	0.987715i	$-0.450053\pi$
$739$	8.49621	0.312538	0.156269	+	0.987715i	$0.450053\pi$
$740$	0	0
$741$	−4.50094	−0.165346
$742$	0	0
$743$	3.40892	0.125061	0.0625306	−	0.998043i	$-0.480083\pi$
$743$	3.40892	0.125061	0.0625306	+	0.998043i	$0.480083\pi$
$744$	0	0
$745$	−9.60901	−0.352047
$746$	0	0
$747$	−18.3071	−0.669820
$748$	0	0
$749$	−36.4377	−1.33140
$750$	0	0
$751$	−34.9485	−1.27529	−0.637645	−	0.770330i	$-0.720091\pi$
$751$	−34.9485	−1.27529	−0.637645	+	0.770330i	$0.720091\pi$
$752$	0	0
$753$	12.4332	0.453090
$754$	0	0
$755$	−9.33515	−0.339741
$756$	0	0
$757$	−50.8194	−1.84706	−0.923531	−	0.383524i	$-0.874710\pi$
$757$	−50.8194	−1.84706	−0.923531	+	0.383524i	$0.874710\pi$
$758$	0	0
$759$	−18.9721	−0.688644
$760$	0	0
$761$	39.1698	1.41990	0.709952	−	0.704250i	$-0.248717\pi$
$761$	39.1698	1.41990	0.709952	+	0.704250i	$0.248717\pi$
$762$	0	0
$763$	−17.1782	−0.621892
$764$	0	0
$765$	17.0293	0.615695
$766$	0	0
$767$	3.01653	0.108920
$768$	0	0
$769$	−35.2165	−1.26994	−0.634970	−	0.772536i	$-0.718988\pi$
$769$	−35.2165	−1.26994	−0.634970	+	0.772536i	$0.718988\pi$
$770$	0	0
$771$	2.92242	0.105248
$772$	0	0
$773$	14.5395	0.522950	0.261475	−	0.965210i	$-0.415791\pi$
$773$	14.5395	0.522950	0.261475	+	0.965210i	$0.415791\pi$
$774$	0	0
$775$	−3.46187	−0.124354
$776$	0	0
$777$	9.69918	0.347956
$778$	0	0
$779$	38.3680	1.37468
$780$	0	0
$781$	−10.8087	−0.386767
$782$	0	0
$783$	11.2881	0.403404
$784$	0	0
$785$	−15.8346	−0.565162
$786$	0	0
$787$	−4.24267	−0.151235	−0.0756175	−	0.997137i	$-0.524093\pi$
$787$	−4.24267	−0.151235	−0.0756175	+	0.997137i	$0.524093\pi$
$788$	0	0
$789$	−1.34316	−0.0478177
$790$	0	0
$791$	−11.3341	−0.402994
$792$	0	0
$793$	−5.63469	−0.200094
$794$	0	0
$795$	−0.282359	−0.0100142
$796$	0	0
$797$	3.78772	0.134168	0.0670839	−	0.997747i	$-0.478630\pi$
$797$	3.78772	0.134168	0.0670839	+	0.997747i	$0.478630\pi$
$798$	0	0
$799$	78.9910	2.79450
$800$	0	0
$801$	−20.5240	−0.725179
$802$	0	0
$803$	68.0098	2.40002
$804$	0	0
$805$	7.36080	0.259434
$806$	0	0
$807$	10.2094	0.359389
$808$	0	0
$809$	9.04551	0.318023	0.159012	−	0.987277i	$-0.449169\pi$
$809$	9.04551	0.318023	0.159012	+	0.987277i	$0.449169\pi$
$810$	0	0
$811$	9.96371	0.349873	0.174937	−	0.984580i	$-0.444028\pi$
$811$	9.96371	0.349873	0.174937	+	0.984580i	$0.444028\pi$
$812$	0	0
$813$	3.00699	0.105460
$814$	0	0
$815$	10.5649	0.370073
$816$	0	0
$817$	8.70488	0.304545
$818$	0	0
$819$	−3.54278	−0.123795
$820$	0	0
$821$	−10.2646	−0.358236	−0.179118	−	0.983828i	$-0.557324\pi$
$821$	−10.2646	−0.358236	−0.179118	+	0.983828i	$0.557324\pi$
$822$	0	0
$823$	−10.2214	−0.356297	−0.178148	−	0.984004i	$-0.557011\pi$
$823$	−10.2214	−0.356297	−0.178148	+	0.984004i	$0.557011\pi$
$824$	0	0
$825$	4.78144	0.166468
$826$	0	0
$827$	−26.0286	−0.905104	−0.452552	−	0.891738i	$-0.649486\pi$
$827$	−26.0286	−0.905104	−0.452552	+	0.891738i	$0.649486\pi$
$828$	0	0
$829$	16.5463	0.574676	0.287338	−	0.957829i	$-0.407230\pi$
$829$	16.5463	0.574676	0.287338	+	0.957829i	$0.407230\pi$
$830$	0	0
$831$	−14.3288	−0.497060
$832$	0	0
$833$	27.0108	0.935870
$834$	0	0
$835$	−13.5600	−0.469263
$836$	0	0
$837$	−15.7880	−0.545712
$838$	0	0
$839$	−16.1059	−0.556038	−0.278019	−	0.960576i	$-0.589678\pi$
$839$	−16.1059	−0.556038	−0.278019	+	0.960576i	$0.589678\pi$
$840$	0	0
$841$	−22.8735	−0.788741
$842$	0	0
$843$	19.6178	0.675674
$844$	0	0
$845$	−12.2754	−0.422288
$846$	0	0
$847$	−35.6609	−1.22532
$848$	0	0
$849$	−19.8450	−0.681077
$850$	0	0
$851$	−23.8525	−0.817654
$852$	0	0
$853$	47.6361	1.63103	0.815514	−	0.578737i	$-0.196454\pi$
$853$	47.6361	1.63103	0.815514	+	0.578737i	$0.196454\pi$
$854$	0	0
$855$	13.6399	0.466475
$856$	0	0
$857$	6.20282	0.211884	0.105942	−	0.994372i	$-0.466214\pi$
$857$	6.20282	0.211884	0.105942	+	0.994372i	$0.466214\pi$
$858$	0	0
$859$	−57.3081	−1.95533	−0.977663	−	0.210179i	$-0.932595\pi$
$859$	−57.3081	−1.95533	−0.977663	+	0.210179i	$0.932595\pi$
$860$	0	0
$861$	−10.1824	−0.347016
$862$	0	0
$863$	22.4085	0.762793	0.381397	−	0.924411i	$-0.375443\pi$
$863$	22.4085	0.762793	0.381397	+	0.924411i	$0.375443\pi$
$864$	0	0
$865$	12.8989	0.438576
$866$	0	0
$867$	−35.3226	−1.19962
$868$	0	0
$869$	−6.15965	−0.208952
$870$	0	0
$871$	8.82279	0.298949
$872$	0	0
$873$	−0.410123	−0.0138806
$874$	0	0
$875$	−1.85510	−0.0627139
$876$	0	0
$877$	−25.2472	−0.852538	−0.426269	−	0.904596i	$-0.640172\pi$
$877$	−25.2472	−0.852538	−0.426269	+	0.904596i	$0.640172\pi$
$878$	0	0
$879$	10.8120	0.364680
$880$	0	0
$881$	−16.5209	−0.556605	−0.278302	−	0.960494i	$-0.589772\pi$
$881$	−16.5209	−0.556605	−0.278302	+	0.960494i	$0.589772\pi$
$882$	0	0
$883$	24.3827	0.820543	0.410272	−	0.911963i	$-0.365434\pi$
$883$	24.3827	0.820543	0.410272	+	0.911963i	$0.365434\pi$
$884$	0	0
$885$	3.08217	0.103606
$886$	0	0
$887$	22.5055	0.755660	0.377830	−	0.925875i	$-0.376670\pi$
$887$	22.5055	0.755660	0.377830	+	0.925875i	$0.376670\pi$
$888$	0	0
$889$	18.6132	0.624266
$890$	0	0
$891$	−15.1963	−0.509094
$892$	0	0
$893$	63.2693	2.11723
$894$	0	0
$895$	−12.6988	−0.424475
$896$	0	0
$897$	−2.93755	−0.0980820
$898$	0	0
$899$	−8.56875	−0.285784
$900$	0	0
$901$	−2.46418	−0.0820936
$902$	0	0
$903$	−2.31018	−0.0768779
$904$	0	0
$905$	−16.0213	−0.532567
$906$	0	0
$907$	44.9439	1.49234	0.746168	−	0.665757i	$-0.231891\pi$
$907$	44.9439	1.49234	0.746168	+	0.665757i	$0.231891\pi$
$908$	0	0
$909$	−9.21718	−0.305715
$910$	0	0
$911$	−24.6918	−0.818076	−0.409038	−	0.912517i	$-0.634136\pi$
$911$	−24.6918	−0.818076	−0.409038	+	0.912517i	$0.634136\pi$
$912$	0	0
$913$	−44.8592	−1.48462
$914$	0	0
$915$	−5.75731	−0.190331
$916$	0	0
$917$	−17.5251	−0.578729
$918$	0	0
$919$	31.7204	1.04636	0.523179	−	0.852223i	$-0.324746\pi$
$919$	31.7204	1.04636	0.523179	+	0.852223i	$0.324746\pi$
$920$	0	0
$921$	12.1066	0.398927
$922$	0	0
$923$	−1.67357	−0.0550863
$924$	0	0
$925$	6.01143	0.197655
$926$	0	0
$927$	4.94504	0.162416
$928$	0	0
$929$	−0.0490739	−0.00161006	−0.000805031	−	1.00000i	$-0.500256\pi$
$929$	−0.0490739	−0.00161006	−0.000805031		1.00000i	$0.500256\pi$
$930$	0	0
$931$	21.6348	0.709052
$932$	0	0
$933$	−0.707874	−0.0231748
$934$	0	0
$935$	41.7282	1.36466
$936$	0	0
$937$	8.83717	0.288698	0.144349	−	0.989527i	$-0.453891\pi$
$937$	8.83717	0.288698	0.144349	+	0.989527i	$0.453891\pi$
$938$	0	0
$939$	5.48390	0.178960
$940$	0	0
$941$	−32.6660	−1.06488	−0.532441	−	0.846467i	$-0.678725\pi$
$941$	−32.6660	−1.06488	−0.532441	+	0.846467i	$0.678725\pi$
$942$	0	0
$943$	25.0410	0.815446
$944$	0	0
$945$	−8.46025	−0.275212
$946$	0	0
$947$	49.4646	1.60738	0.803692	−	0.595046i	$-0.202866\pi$
$947$	49.4646	1.60738	0.803692	+	0.595046i	$0.202866\pi$
$948$	0	0
$949$	10.5303	0.341829
$950$	0	0
$951$	−18.0288	−0.584624
$952$	0	0
$953$	53.6616	1.73827	0.869136	−	0.494574i	$-0.164676\pi$
$953$	53.6616	1.73827	0.869136	+	0.494574i	$0.164676\pi$
$954$	0	0
$955$	3.12727	0.101196
$956$	0	0
$957$	11.8349	0.382569
$958$	0	0
$959$	1.02497	0.0330979
$960$	0	0
$961$	−19.0154	−0.613401
$962$	0	0
$963$	−44.0675	−1.42006
$964$	0	0
$965$	−4.13102	−0.132982
$966$	0	0
$967$	−21.1003	−0.678540	−0.339270	−	0.940689i	$-0.610180\pi$
$967$	−21.1003	−0.678540	−0.339270	+	0.940689i	$0.610180\pi$
$968$	0	0
$969$	−40.1351	−1.28932
$970$	0	0
$971$	−44.5298	−1.42903	−0.714515	−	0.699620i	$-0.753353\pi$
$971$	−44.5298	−1.42903	−0.714515	+	0.699620i	$0.753353\pi$
$972$	0	0
$973$	1.15406	0.0369973
$974$	0	0
$975$	0.740336	0.0237097
$976$	0	0
$977$	3.79384	0.121376	0.0606878	−	0.998157i	$-0.480671\pi$
$977$	3.79384	0.121376	0.0606878	+	0.998157i	$0.480671\pi$
$978$	0	0
$979$	−50.2916	−1.60733
$980$	0	0
$981$	−20.7752	−0.663301
$982$	0	0
$983$	40.8708	1.30357	0.651787	−	0.758402i	$-0.274019\pi$
$983$	40.8708	1.30357	0.651787	+	0.758402i	$0.274019\pi$
$984$	0	0
$985$	20.4762	0.652426
$986$	0	0
$987$	−16.7910	−0.534462
$988$	0	0
$989$	5.68126	0.180654
$990$	0	0
$991$	−54.0799	−1.71791	−0.858953	−	0.512054i	$-0.828885\pi$
$991$	−54.0799	−1.71791	−0.858953	+	0.512054i	$0.828885\pi$
$992$	0	0
$993$	4.43093	0.140611
$994$	0	0
$995$	0.335680	0.0106418
$996$	0	0
$997$	47.3648	1.50006	0.750029	−	0.661405i	$-0.230039\pi$
$997$	47.3648	1.50006	0.750029	+	0.661405i	$0.230039\pi$
$998$	0	0
$999$	27.4153	0.867381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.14	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.14	✓	37	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-0.869740 q^{3} +1.00000 q^{5} -1.85510 q^{7} -2.24355 q^{9} +O(q^{10})\)	\(q-0.869740 q^{3} +1.00000 q^{5} -1.85510 q^{7} -2.24355 q^{9} -5.49756 q^{11} -0.851215 q^{13} -0.869740 q^{15} -7.59031 q^{17} -6.07960 q^{19} +1.61346 q^{21} -3.96786 q^{23} +1.00000 q^{25} +4.56053 q^{27} +2.47518 q^{29} -3.46187 q^{31} +4.78144 q^{33} -1.85510 q^{35} +6.01143 q^{37} +0.740336 q^{39} -6.31094 q^{41} -1.43182 q^{43} -2.24355 q^{45} -10.4068 q^{47} -3.55859 q^{49} +6.60160 q^{51} +0.324647 q^{53} -5.49756 q^{55} +5.28767 q^{57} -3.54379 q^{59} +6.61958 q^{61} +4.16202 q^{63} -0.851215 q^{65} -10.3649 q^{67} +3.45101 q^{69} +1.96610 q^{71} -12.3709 q^{73} -0.869740 q^{75} +10.1985 q^{77} +1.12043 q^{79} +2.76419 q^{81} +8.15985 q^{83} -7.59031 q^{85} -2.15276 q^{87} +9.14798 q^{89} +1.57909 q^{91} +3.01093 q^{93} -6.07960 q^{95} +0.182801 q^{97} +12.3341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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