LMFDB - Embedded newform 847.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.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:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.11447$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.11447 q^{2} -2.85882 q^{3} -0.757964 q^{4} +3.45608 q^{5} +3.18606 q^{6} -1.00000 q^{7} +3.07366 q^{8} +5.17284 q^{9} +O(q^{10})$	\(q-1.11447 q^{2} -2.85882 q^{3} -0.757964 q^{4} +3.45608 q^{5} +3.18606 q^{6} -1.00000 q^{7} +3.07366 q^{8} +5.17284 q^{9} -3.85168 q^{10} +2.16688 q^{12} +2.05965 q^{13} +1.11447 q^{14} -9.88030 q^{15} -1.90956 q^{16} -1.93373 q^{17} -5.76496 q^{18} -1.62296 q^{19} -2.61958 q^{20} +2.85882 q^{21} -0.807136 q^{23} -8.78703 q^{24} +6.94447 q^{25} -2.29541 q^{26} -6.21176 q^{27} +0.757964 q^{28} -7.97368 q^{29} +11.0113 q^{30} +0.788420 q^{31} -4.01918 q^{32} +2.15508 q^{34} -3.45608 q^{35} -3.92083 q^{36} +10.0618 q^{37} +1.80873 q^{38} -5.88817 q^{39} +10.6228 q^{40} +2.12613 q^{41} -3.18606 q^{42} -3.08043 q^{43} +17.8777 q^{45} +0.899526 q^{46} +7.56632 q^{47} +5.45909 q^{48} +1.00000 q^{49} -7.73938 q^{50} +5.52818 q^{51} -1.56114 q^{52} +10.8224 q^{53} +6.92280 q^{54} -3.07366 q^{56} +4.63975 q^{57} +8.88640 q^{58} -3.29664 q^{59} +7.48891 q^{60} +1.07663 q^{61} -0.878667 q^{62} -5.17284 q^{63} +8.29836 q^{64} +7.11832 q^{65} +2.40314 q^{67} +1.46570 q^{68} +2.30745 q^{69} +3.85168 q^{70} -3.18859 q^{71} +15.8995 q^{72} -1.22628 q^{73} -11.2135 q^{74} -19.8530 q^{75} +1.23015 q^{76} +6.56217 q^{78} +9.48182 q^{79} -6.59959 q^{80} +2.23976 q^{81} -2.36950 q^{82} +16.0694 q^{83} -2.16688 q^{84} -6.68312 q^{85} +3.43303 q^{86} +22.7953 q^{87} -4.43830 q^{89} -19.9241 q^{90} -2.05965 q^{91} +0.611780 q^{92} -2.25395 q^{93} -8.43241 q^{94} -5.60908 q^{95} +11.4901 q^{96} +6.46807 q^{97} -1.11447 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9}+O(q^{10}) $ 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9	$ 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} + q^{14} + 11 q^{15} + q^{16} + 5 q^{17} - 8 q^{18} + 13 q^{19} + 23 q^{20} - 4 q^{21} + 16 q^{23} + 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} - 7 q^{28} - 9 q^{29} + 36 q^{30} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} - 13 q^{39} - 5 q^{40} + 10 q^{41} - q^{42} + 4 q^{43} + 35 q^{45} - 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} - 6 q^{50} - 13 q^{51} + 41 q^{52} + 37 q^{53} - 30 q^{54} - 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} - 19 q^{61} + 18 q^{62} - 14 q^{63} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} + 35 q^{72} + 25 q^{73} - 33 q^{74} - 13 q^{75} - 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} + 25 q^{83} - 9 q^{84} - 3 q^{85} + 4 q^{86} + 36 q^{87} + 37 q^{89} + 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} + 42 q^{94} - 21 q^{95} + 6 q^{96} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9 - 6 * q^10 + 9 * q^12 + 6 * q^13 + q^14 + 11 * q^15 + q^16 + 5 * q^17 - 8 * q^18 + 13 * q^19 + 23 * q^20 - 4 * q^21 + 16 * q^23 + 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 - 7 * q^28 - 9 * q^29 + 36 * q^30 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 - 13 * q^39 - 5 * q^40 + 10 * q^41 - q^42 + 4 * q^43 + 35 * q^45 - 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 - 6 * q^50 - 13 * q^51 + 41 * q^52 + 37 * q^53 - 30 * q^54 - 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 - 19 * q^61 + 18 * q^62 - 14 * q^63 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 + 35 * q^72 + 25 * q^73 - 33 * q^74 - 13 * q^75 - 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 + 25 * q^83 - 9 * q^84 - 3 * q^85 + 4 * q^86 + 36 * q^87 + 37 * q^89 + 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 + 42 * q^94 - 21 * q^95 + 6 * q^96 + 15 * q^97 - q^98

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 $

$2$	−1.11447	−0.788047	−0.394023	−	0.919100i	$-0.628917\pi$
$2$	−1.11447	−0.788047	−0.394023	+	0.919100i	$0.628917\pi$
$3$	−2.85882	−1.65054	−0.825270	−	0.564739i	$-0.808977\pi$
$3$	−2.85882	−1.65054	−0.825270	+	0.564739i	$0.808977\pi$
$4$	−0.757964	−0.378982
$5$	3.45608	1.54561	0.772803	−	0.634647i	$-0.218854\pi$
$5$	3.45608	1.54561	0.772803	+	0.634647i	$0.218854\pi$
$6$	3.18606	1.30070
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.07366	1.08670
$9$	5.17284	1.72428
$10$	−3.85168	−1.21801
$11$	0	0
$11$	0	0
$12$	2.16688	0.625525
$13$	2.05965	0.571245	0.285622	−	0.958342i	$-0.407800\pi$
$13$	2.05965	0.571245	0.285622	+	0.958342i	$0.407800\pi$
$14$	1.11447	0.297854
$15$	−9.88030	−2.55108
$16$	−1.90956	−0.477390
$17$	−1.93373	−0.468998	−0.234499	−	0.972116i	$-0.575345\pi$
$17$	−1.93373	−0.468998	−0.234499	+	0.972116i	$0.575345\pi$
$18$	−5.76496	−1.35881
$19$	−1.62296	−0.372333	−0.186166	−	0.982518i	$-0.559606\pi$
$19$	−1.62296	−0.372333	−0.186166	+	0.982518i	$0.559606\pi$
$20$	−2.61958	−0.585757
$21$	2.85882	0.623845
$22$	0	0
$23$	−0.807136	−0.168299	−0.0841497	−	0.996453i	$-0.526817\pi$
$23$	−0.807136	−0.168299	−0.0841497	+	0.996453i	$0.526817\pi$
$24$	−8.78703	−1.79365
$25$	6.94447	1.38889
$26$	−2.29541	−0.450168
$27$	−6.21176	−1.19545
$28$	0.757964	0.143242
$29$	−7.97368	−1.48067	−0.740337	−	0.672235i	$-0.765334\pi$
$29$	−7.97368	−1.48067	−0.740337	+	0.672235i	$0.765334\pi$
$30$	11.0113	2.01037
$31$	0.788420	0.141604	0.0708022	−	0.997490i	$-0.477444\pi$
$31$	0.788420	0.141604	0.0708022	+	0.997490i	$0.477444\pi$
$32$	−4.01918	−0.710497
$33$	0	0
$34$	2.15508	0.369592
$35$	−3.45608	−0.584184
$36$	−3.92083	−0.653471
$37$	10.0618	1.65414	0.827072	−	0.562096i	$-0.190005\pi$
$37$	10.0618	1.65414	0.827072	+	0.562096i	$0.190005\pi$
$38$	1.80873	0.293415
$39$	−5.88817	−0.942862
$40$	10.6228	1.67961
$41$	2.12613	0.332046	0.166023	−	0.986122i	$-0.446907\pi$
$41$	2.12613	0.332046	0.166023	+	0.986122i	$0.446907\pi$
$42$	−3.18606	−0.491619
$43$	−3.08043	−0.469761	−0.234880	−	0.972024i	$-0.575470\pi$
$43$	−3.08043	−0.469761	−0.234880	+	0.972024i	$0.575470\pi$
$44$	0	0
$45$	17.8777	2.66506
$46$	0.899526	0.132628
$47$	7.56632	1.10366	0.551831	−	0.833956i	$-0.313930\pi$
$47$	7.56632	1.10366	0.551831	+	0.833956i	$0.313930\pi$
$48$	5.45909	0.787952
$49$	1.00000	0.142857
$50$	−7.73938	−1.09451
$51$	5.52818	0.774100
$52$	−1.56114	−0.216492
$53$	10.8224	1.48658	0.743289	−	0.668971i	$-0.233265\pi$
$53$	10.8224	1.48658	0.743289	+	0.668971i	$0.233265\pi$
$54$	6.92280	0.942073
$55$	0	0
$56$	−3.07366	−0.410735
$57$	4.63975	0.614550
$58$	8.88640	1.16684
$59$	−3.29664	−0.429186	−0.214593	−	0.976704i	$-0.568842\pi$
$59$	−3.29664	−0.429186	−0.214593	+	0.976704i	$0.568842\pi$
$60$	7.48891	0.966814
$61$	1.07663	0.137848	0.0689240	−	0.997622i	$-0.478043\pi$
$61$	1.07663	0.137848	0.0689240	+	0.997622i	$0.478043\pi$
$62$	−0.878667	−0.111591
$63$	−5.17284	−0.651717
$64$	8.29836	1.03729
$65$	7.11832	0.882919
$66$	0	0
$67$	2.40314	0.293590	0.146795	−	0.989167i	$-0.453104\pi$
$67$	2.40314	0.293590	0.146795	+	0.989167i	$0.453104\pi$
$68$	1.46570	0.177742
$69$	2.30745	0.277785
$70$	3.85168	0.460364
$71$	−3.18859	−0.378417	−0.189208	−	0.981937i	$-0.560592\pi$
$71$	−3.18859	−0.378417	−0.189208	+	0.981937i	$0.560592\pi$
$72$	15.8995	1.87378
$73$	−1.22628	−0.143525	−0.0717624	−	0.997422i	$-0.522862\pi$
$73$	−1.22628	−0.143525	−0.0717624	+	0.997422i	$0.522862\pi$
$74$	−11.2135	−1.30354
$75$	−19.8530	−2.29243
$76$	1.23015	0.141107
$77$	0	0
$78$	6.56217	0.743020
$79$	9.48182	1.06679	0.533394	−	0.845867i	$-0.320916\pi$
$79$	9.48182	1.06679	0.533394	+	0.845867i	$0.320916\pi$
$80$	−6.59959	−0.737857
$81$	2.23976	0.248862
$82$	−2.36950	−0.261668
$83$	16.0694	1.76385	0.881923	−	0.471394i	$-0.156249\pi$
$83$	16.0694	1.76385	0.881923	+	0.471394i	$0.156249\pi$
$84$	−2.16688	−0.236426
$85$	−6.68312	−0.724886
$86$	3.43303	0.370193
$87$	22.7953	2.44391
$88$	0	0
$89$	−4.43830	−0.470459	−0.235230	−	0.971940i	$-0.575584\pi$
$89$	−4.43830	−0.470459	−0.235230	+	0.971940i	$0.575584\pi$
$90$	−19.9241	−2.10019
$91$	−2.05965	−0.215910
$92$	0.611780	0.0637825
$93$	−2.25395	−0.233724
$94$	−8.43241	−0.869737
$95$	−5.60908	−0.575479
$96$	11.4901	1.17270
$97$	6.46807	0.656733	0.328366	−	0.944550i	$-0.393502\pi$
$97$	6.46807	0.656733	0.328366	+	0.944550i	$0.393502\pi$
$98$	−1.11447	−0.112578
$99$	0	0
$100$	−5.26366	−0.526366
$101$	15.4449	1.53682	0.768412	−	0.639955i	$-0.221047\pi$
$101$	15.4449	1.53682	0.768412	+	0.639955i	$0.221047\pi$
$102$	−6.16097	−0.610027
$103$	8.90812	0.877743	0.438872	−	0.898550i	$-0.355378\pi$
$103$	8.90812	0.877743	0.438872	+	0.898550i	$0.355378\pi$
$104$	6.33067	0.620773
$105$	9.88030	0.964218
$106$	−12.0613	−1.17149
$107$	−3.51219	−0.339536	−0.169768	−	0.985484i	$-0.554302\pi$
$107$	−3.51219	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$108$	4.70829	0.453055
$109$	−3.87655	−0.371306	−0.185653	−	0.982615i	$-0.559440\pi$
$109$	−3.87655	−0.371306	−0.185653	+	0.982615i	$0.559440\pi$
$110$	0	0
$111$	−28.7648	−2.73023
$112$	1.90956	0.180437
$113$	10.6539	1.00224	0.501118	−	0.865379i	$-0.332922\pi$
$113$	10.6539	1.00224	0.501118	+	0.865379i	$0.332922\pi$
$114$	−5.17084	−0.484294
$115$	−2.78952	−0.260124
$116$	6.04376	0.561149
$117$	10.6543	0.984986
$118$	3.67399	0.338218
$119$	1.93373	0.177265
$120$	−30.3687	−2.77227
$121$	0	0
$122$	−1.19987	−0.108631
$123$	−6.07823	−0.548056
$124$	−0.597594	−0.0536655
$125$	6.72025	0.601078
$126$	5.76496	0.513583
$127$	−19.4509	−1.72599	−0.862994	−	0.505214i	$-0.831414\pi$
$127$	−19.4509	−1.72599	−0.862994	+	0.505214i	$0.831414\pi$
$128$	−1.20989	−0.106941
$129$	8.80638	0.775359
$130$	−7.93313	−0.695782
$131$	−5.11284	−0.446711	−0.223355	−	0.974737i	$-0.571701\pi$
$131$	−5.11284	−0.446711	−0.223355	+	0.974737i	$0.571701\pi$
$132$	0	0
$133$	1.62296	0.140728
$134$	−2.67822	−0.231363
$135$	−21.4683	−1.84770
$136$	−5.94362	−0.509661
$137$	9.10052	0.777510	0.388755	−	0.921341i	$-0.372905\pi$
$137$	9.10052	0.777510	0.388755	+	0.921341i	$0.372905\pi$
$138$	−2.57158	−0.218907
$139$	13.0166	1.10405	0.552026	−	0.833827i	$-0.313855\pi$
$139$	13.0166	1.10405	0.552026	+	0.833827i	$0.313855\pi$
$140$	2.61958	0.221395
$141$	−21.6307	−1.82164
$142$	3.55358	0.298210
$143$	0	0
$144$	−9.87786	−0.823155
$145$	−27.5577	−2.28854
$146$	1.36664	0.113104
$147$	−2.85882	−0.235791
$148$	−7.62646	−0.626891
$149$	3.14650	0.257771	0.128886	−	0.991659i	$-0.458860\pi$
$149$	3.14650	0.257771	0.128886	+	0.991659i	$0.458860\pi$
$150$	22.1255	1.80654
$151$	−2.86696	−0.233310	−0.116655	−	0.993172i	$-0.537217\pi$
$151$	−2.86696	−0.233310	−0.116655	+	0.993172i	$0.537217\pi$
$152$	−4.98843	−0.404615
$153$	−10.0029	−0.808684
$154$	0	0
$155$	2.72484	0.218864
$156$	4.46302	0.357328
$157$	−21.4895	−1.71505	−0.857524	−	0.514443i	$-0.827999\pi$
$157$	−21.4895	−1.71505	−0.857524	+	0.514443i	$0.827999\pi$
$158$	−10.5672	−0.840679
$159$	−30.9394	−2.45365
$160$	−13.8906	−1.09815
$161$	0.807136	0.0636112
$162$	−2.49614	−0.196115
$163$	−8.22245	−0.644032	−0.322016	−	0.946734i	$-0.604361\pi$
$163$	−8.22245	−0.644032	−0.322016	+	0.946734i	$0.604361\pi$
$164$	−1.61153	−0.125840
$165$	0	0
$166$	−17.9088	−1.38999
$167$	21.7086	1.67986	0.839930	−	0.542695i	$-0.182596\pi$
$167$	21.7086	1.67986	0.839930	+	0.542695i	$0.182596\pi$
$168$	8.78703	0.677934
$169$	−8.75783	−0.673679
$170$	7.44811	0.571244
$171$	−8.39531	−0.642006
$172$	2.33485	0.178031
$173$	8.04563	0.611698	0.305849	−	0.952080i	$-0.401060\pi$
$173$	8.04563	0.611698	0.305849	+	0.952080i	$0.401060\pi$
$174$	−25.4046	−1.92592
$175$	−6.94447	−0.524953
$176$	0	0
$177$	9.42449	0.708388
$178$	4.94634	0.370744
$179$	−3.62091	−0.270640	−0.135320	−	0.990802i	$-0.543206\pi$
$179$	−3.62091	−0.270640	−0.135320	+	0.990802i	$0.543206\pi$
$180$	−13.5507	−1.01001
$181$	15.8179	1.17574	0.587868	−	0.808957i	$-0.299967\pi$
$181$	15.8179	1.17574	0.587868	+	0.808957i	$0.299967\pi$
$182$	2.29541	0.170147
$183$	−3.07788	−0.227524
$184$	−2.48086	−0.182891
$185$	34.7743	2.55665
$186$	2.51195	0.184185
$187$	0	0
$188$	−5.73500	−0.418268
$189$	6.21176	0.451839
$190$	6.25113	0.453504
$191$	0.429081	0.0310472	0.0155236	−	0.999880i	$-0.495058\pi$
$191$	0.429081	0.0310472	0.0155236	+	0.999880i	$0.495058\pi$
$192$	−23.7235	−1.71210
$193$	15.1748	1.09231	0.546154	−	0.837685i	$-0.316091\pi$
$193$	15.1748	1.09231	0.546154	+	0.837685i	$0.316091\pi$
$194$	−7.20845	−0.517536
$195$	−20.3500	−1.45729
$196$	−0.757964	−0.0541403
$197$	20.8082	1.48252	0.741262	−	0.671216i	$-0.234228\pi$
$197$	20.8082	1.48252	0.741262	+	0.671216i	$0.234228\pi$
$198$	0	0
$199$	8.44567	0.598698	0.299349	−	0.954144i	$-0.403231\pi$
$199$	8.44567	0.598698	0.299349	+	0.954144i	$0.403231\pi$
$200$	21.3449	1.50932
$201$	−6.87014	−0.484582
$202$	−17.2128	−1.21109
$203$	7.97368	0.559642
$204$	−4.19016	−0.293370
$205$	7.34808	0.513212
$206$	−9.92781	−0.691703
$207$	−4.17518	−0.290195
$208$	−3.93303	−0.272707
$209$	0	0
$210$	−11.0113	−0.759849
$211$	9.85927	0.678740	0.339370	−	0.940653i	$-0.389786\pi$
$211$	9.85927	0.678740	0.339370	+	0.940653i	$0.389786\pi$
$212$	−8.20303	−0.563386
$213$	9.11561	0.624592
$214$	3.91422	0.267570
$215$	−10.6462	−0.726065
$216$	−19.0928	−1.29910
$217$	−0.788420	−0.0535214
$218$	4.32028	0.292606
$219$	3.50570	0.236893
$220$	0	0
$221$	−3.98281	−0.267913
$222$	32.0574	2.15155
$223$	17.3959	1.16491	0.582457	−	0.812861i	$-0.302091\pi$
$223$	17.3959	1.16491	0.582457	+	0.812861i	$0.302091\pi$
$224$	4.01918	0.268542
$225$	35.9227	2.39484
$226$	−11.8734	−0.789809
$227$	12.6315	0.838381	0.419190	−	0.907898i	$-0.362314\pi$
$227$	12.6315	0.838381	0.419190	+	0.907898i	$0.362314\pi$
$228$	−3.51676	−0.232903
$229$	4.57341	0.302220	0.151110	−	0.988517i	$-0.451715\pi$
$229$	4.57341	0.302220	0.151110	+	0.988517i	$0.451715\pi$
$230$	3.10883	0.204990
$231$	0	0
$232$	−24.5084	−1.60905
$233$	−23.8569	−1.56292	−0.781458	−	0.623957i	$-0.785524\pi$
$233$	−23.8569	−1.56292	−0.781458	+	0.623957i	$0.785524\pi$
$234$	−11.8738	−0.776215
$235$	26.1498	1.70582
$236$	2.49873	0.162654
$237$	−27.1068	−1.76078
$238$	−2.15508	−0.139693
$239$	−8.83814	−0.571692	−0.285846	−	0.958276i	$-0.592275\pi$
$239$	−8.83814	−0.571692	−0.285846	+	0.958276i	$0.592275\pi$
$240$	18.8670	1.21786
$241$	−18.9464	−1.22045	−0.610224	−	0.792229i	$-0.708921\pi$
$241$	−18.9464	−1.22045	−0.610224	+	0.792229i	$0.708921\pi$
$242$	0	0
$243$	12.2322	0.784696
$244$	−0.816045	−0.0522419
$245$	3.45608	0.220801
$246$	6.77398	0.431893
$247$	−3.34273	−0.212693
$248$	2.42333	0.153882
$249$	−45.9395	−2.91130
$250$	−7.48950	−0.473678
$251$	−2.86691	−0.180958	−0.0904790	−	0.995898i	$-0.528840\pi$
$251$	−2.86691	−0.180958	−0.0904790	+	0.995898i	$0.528840\pi$
$252$	3.92083	0.246989
$253$	0	0
$254$	21.6774	1.36016
$255$	19.1058	1.19645
$256$	−15.2483	−0.953021
$257$	22.4159	1.39826	0.699132	−	0.714993i	$-0.253570\pi$
$257$	22.4159	1.39826	0.699132	+	0.714993i	$0.253570\pi$
$258$	−9.81442	−0.611019
$259$	−10.0618	−0.625208
$260$	−5.39543	−0.334611
$261$	−41.2466	−2.55310
$262$	5.69808	0.352029
$263$	−0.990706	−0.0610895	−0.0305448	−	0.999533i	$-0.509724\pi$
$263$	−0.990706	−0.0610895	−0.0305448	+	0.999533i	$0.509724\pi$
$264$	0	0
$265$	37.4032	2.29766
$266$	−1.80873	−0.110901
$267$	12.6883	0.776511
$268$	−1.82149	−0.111265
$269$	−7.19036	−0.438404	−0.219202	−	0.975679i	$-0.570345\pi$
$269$	−7.19036	−0.438404	−0.219202	+	0.975679i	$0.570345\pi$
$270$	23.9257	1.45607
$271$	27.1643	1.65011	0.825056	−	0.565050i	$-0.191143\pi$
$271$	27.1643	1.65011	0.825056	+	0.565050i	$0.191143\pi$
$272$	3.69257	0.223895
$273$	5.88817	0.356368
$274$	−10.1422	−0.612714
$275$	0	0
$276$	−1.74897	−0.105275
$277$	−20.9856	−1.26090	−0.630451	−	0.776229i	$-0.717130\pi$
$277$	−20.9856	−1.26090	−0.630451	+	0.776229i	$0.717130\pi$
$278$	−14.5065	−0.870045
$279$	4.07837	0.244166
$280$	−10.6228	−0.634834
$281$	28.1580	1.67977	0.839883	−	0.542768i	$-0.182624\pi$
$281$	28.1580	1.67977	0.839883	+	0.542768i	$0.182624\pi$
$282$	24.1067	1.43553
$283$	26.1917	1.55694	0.778469	−	0.627684i	$-0.215997\pi$
$283$	26.1917	1.55694	0.778469	+	0.627684i	$0.215997\pi$
$284$	2.41684	0.143413
$285$	16.0353	0.949851
$286$	0	0
$287$	−2.12613	−0.125502
$288$	−20.7906	−1.22510
$289$	−13.2607	−0.780041
$290$	30.7121	1.80348
$291$	−18.4910	−1.08396
$292$	0.929473	0.0543933
$293$	4.46385	0.260781	0.130391	−	0.991463i	$-0.458377\pi$
$293$	4.46385	0.260781	0.130391	+	0.991463i	$0.458377\pi$
$294$	3.18606	0.185815
$295$	−11.3934	−0.663351
$296$	30.9264	1.79756
$297$	0	0
$298$	−3.50667	−0.203136
$299$	−1.66242	−0.0961402
$300$	15.0479	0.868788
$301$	3.08043	0.177553
$302$	3.19513	0.183859
$303$	−44.1542	−2.53659
$304$	3.09914	0.177748
$305$	3.72091	0.213059
$306$	11.1479	0.637281
$307$	−12.8841	−0.735334	−0.367667	−	0.929957i	$-0.619843\pi$
$307$	−12.8841	−0.735334	−0.367667	+	0.929957i	$0.619843\pi$
$308$	0	0
$309$	−25.4667	−1.44875
$310$	−3.03674	−0.172475
$311$	26.8199	1.52081	0.760407	−	0.649447i	$-0.224999\pi$
$311$	26.8199	1.52081	0.760407	+	0.649447i	$0.224999\pi$
$312$	−18.0982	−1.02461
$313$	3.58869	0.202845	0.101422	−	0.994843i	$-0.467661\pi$
$313$	3.58869	0.202845	0.101422	+	0.994843i	$0.467661\pi$
$314$	23.9493	1.35154
$315$	−17.8777	−1.00730
$316$	−7.18688	−0.404294
$317$	−16.9181	−0.950213	−0.475106	−	0.879928i	$-0.657590\pi$
$317$	−16.9181	−0.950213	−0.475106	+	0.879928i	$0.657590\pi$
$318$	34.4809	1.93359
$319$	0	0
$320$	28.6798	1.60325
$321$	10.0407	0.560418
$322$	−0.899526	−0.0501286
$323$	3.13836	0.174623
$324$	−1.69766	−0.0943144
$325$	14.3032	0.793399
$326$	9.16365	0.507528
$327$	11.0823	0.612855
$328$	6.53501	0.360836
$329$	−7.56632	−0.417145
$330$	0	0
$331$	−1.23826	−0.0680610	−0.0340305	−	0.999421i	$-0.510834\pi$
$331$	−1.23826	−0.0680610	−0.0340305	+	0.999421i	$0.510834\pi$
$332$	−12.1800	−0.668466
$333$	52.0479	2.85221
$334$	−24.1935	−1.32381
$335$	8.30544	0.453774
$336$	−5.45909	−0.297818
$337$	20.4806	1.11565	0.557824	−	0.829959i	$-0.311636\pi$
$337$	20.4806	1.11565	0.557824	+	0.829959i	$0.311636\pi$
$338$	9.76031	0.530891
$339$	−30.4576	−1.65423
$340$	5.06556	0.274719
$341$	0	0
$342$	9.35630	0.505931
$343$	−1.00000	−0.0539949
$344$	−9.46818	−0.510490
$345$	7.97474	0.429346
$346$	−8.96659	−0.482047
$347$	−27.2699	−1.46392	−0.731961	−	0.681346i	$-0.761395\pi$
$347$	−27.2699	−1.46392	−0.731961	+	0.681346i	$0.761395\pi$
$348$	−17.2780	−0.926199
$349$	−7.98434	−0.427392	−0.213696	−	0.976900i	$-0.568550\pi$
$349$	−7.98434	−0.427392	−0.213696	+	0.976900i	$0.568550\pi$
$350$	7.73938	0.413688
$351$	−12.7941	−0.682897
$352$	0	0
$353$	5.93472	0.315873	0.157937	−	0.987449i	$-0.449516\pi$
$353$	5.93472	0.315873	0.157937	+	0.987449i	$0.449516\pi$
$354$	−10.5033	−0.558243
$355$	−11.0200	−0.584883
$356$	3.36407	0.178296
$357$	−5.52818	−0.292582
$358$	4.03538	0.213277
$359$	−28.4203	−1.49996	−0.749982	−	0.661458i	$-0.769938\pi$
$359$	−28.4203	−1.49996	−0.749982	+	0.661458i	$0.769938\pi$
$360$	54.9501	2.89612
$361$	−16.3660	−0.861368
$362$	−17.6285	−0.926535
$363$	0	0
$364$	1.56114	0.0818261
$365$	−4.23810	−0.221832
$366$	3.43020	0.179299
$367$	−30.4014	−1.58694	−0.793471	−	0.608608i	$-0.791728\pi$
$367$	−30.4014	−1.58694	−0.793471	+	0.608608i	$0.791728\pi$
$368$	1.54128	0.0803445
$369$	10.9982	0.572541
$370$	−38.7547	−2.01476
$371$	−10.8224	−0.561873
$372$	1.70841	0.0885771
$373$	−14.4226	−0.746772	−0.373386	−	0.927676i	$-0.621803\pi$
$373$	−14.4226	−0.746772	−0.373386	+	0.927676i	$0.621803\pi$
$374$	0	0
$375$	−19.2120	−0.992103
$376$	23.2563	1.19935
$377$	−16.4230	−0.845828
$378$	−6.92280	−0.356070
$379$	−22.4072	−1.15098	−0.575490	−	0.817809i	$-0.695189\pi$
$379$	−22.4072	−1.15098	−0.575490	+	0.817809i	$0.695189\pi$
$380$	4.25148	0.218096
$381$	55.6066	2.84881
$382$	−0.478196	−0.0244667
$383$	33.6785	1.72089	0.860446	−	0.509541i	$-0.170185\pi$
$383$	33.6785	1.72089	0.860446	+	0.509541i	$0.170185\pi$
$384$	3.45887	0.176510
$385$	0	0
$386$	−16.9118	−0.860790
$387$	−15.9346	−0.809999
$388$	−4.90256	−0.248890
$389$	2.42783	0.123096	0.0615479	−	0.998104i	$-0.480396\pi$
$389$	2.42783	0.123096	0.0615479	+	0.998104i	$0.480396\pi$
$390$	22.6794	1.14841
$391$	1.56078	0.0789321
$392$	3.07366	0.155243
$393$	14.6167	0.737313
$394$	−23.1900	−1.16830
$395$	32.7699	1.64883
$396$	0	0
$397$	−5.89696	−0.295960	−0.147980	−	0.988990i	$-0.547277\pi$
$397$	−5.89696	−0.295960	−0.147980	+	0.988990i	$0.547277\pi$
$398$	−9.41242	−0.471802
$399$	−4.63975	−0.232278
$400$	−13.2609	−0.663045
$401$	11.2396	0.561278	0.280639	−	0.959813i	$-0.409454\pi$
$401$	11.2396	0.561278	0.280639	+	0.959813i	$0.409454\pi$
$402$	7.65654	0.381873
$403$	1.62387	0.0808908
$404$	−11.7067	−0.582429
$405$	7.74079	0.384643
$406$	−8.88640	−0.441025
$407$	0	0
$408$	16.9917	0.841216
$409$	−29.3344	−1.45049	−0.725246	−	0.688490i	$-0.758274\pi$
$409$	−29.3344	−1.45049	−0.725246	+	0.688490i	$0.758274\pi$
$410$	−8.18919	−0.404435
$411$	−26.0167	−1.28331
$412$	−6.75204	−0.332649
$413$	3.29664	0.162217
$414$	4.65310	0.228688
$415$	55.5371	2.72621
$416$	−8.27811	−0.405868
$417$	−37.2120	−1.82228
$418$	0	0
$419$	−20.2858	−0.991027	−0.495514	−	0.868600i	$-0.665020\pi$
$419$	−20.2858	−0.991027	−0.495514	+	0.868600i	$0.665020\pi$
$420$	−7.48891	−0.365422
$421$	−3.06003	−0.149137	−0.0745683	−	0.997216i	$-0.523758\pi$
$421$	−3.06003	−0.149137	−0.0745683	+	0.997216i	$0.523758\pi$
$422$	−10.9878	−0.534879
$423$	39.1394	1.90302
$424$	33.2645	1.61547
$425$	−13.4287	−0.651389
$426$	−10.1590	−0.492207
$427$	−1.07663	−0.0521017
$428$	2.66211	0.128678
$429$	0	0
$430$	11.8648	0.572173
$431$	−7.54197	−0.363284	−0.181642	−	0.983365i	$-0.558141\pi$
$431$	−7.54197	−0.363284	−0.181642	+	0.983365i	$0.558141\pi$
$432$	11.8617	0.570698
$433$	−32.1616	−1.54559	−0.772794	−	0.634657i	$-0.781141\pi$
$433$	−32.1616	−1.54559	−0.772794	+	0.634657i	$0.781141\pi$
$434$	0.878667	0.0421774
$435$	78.7823	3.77732
$436$	2.93828	0.140718
$437$	1.30995	0.0626633
$438$	−3.90698	−0.186683
$439$	4.66725	0.222756	0.111378	−	0.993778i	$-0.464474\pi$
$439$	4.66725	0.222756	0.111378	+	0.993778i	$0.464474\pi$
$440$	0	0
$441$	5.17284	0.246326
$442$	4.43871	0.211128
$443$	17.2772	0.820862	0.410431	−	0.911892i	$-0.365378\pi$
$443$	17.2772	0.820862	0.410431	+	0.911892i	$0.365378\pi$
$444$	21.8027	1.03471
$445$	−15.3391	−0.727144
$446$	−19.3871	−0.918007
$447$	−8.99527	−0.425461
$448$	−8.29836	−0.392061
$449$	−16.7401	−0.790013	−0.395007	−	0.918678i	$-0.629258\pi$
$449$	−16.7401	−0.790013	−0.395007	+	0.918678i	$0.629258\pi$
$450$	−40.0346	−1.88725
$451$	0	0
$452$	−8.07529	−0.379829
$453$	8.19612	0.385088
$454$	−14.0774	−0.660683
$455$	−7.11832	−0.333712
$456$	14.2610	0.667833
$457$	21.6974	1.01496	0.507481	−	0.861663i	$-0.330577\pi$
$457$	21.6974	1.01496	0.507481	+	0.861663i	$0.330577\pi$
$458$	−5.09692	−0.238163
$459$	12.0119	0.560665
$460$	2.11436	0.0985825
$461$	6.07778	0.283070	0.141535	−	0.989933i	$-0.454796\pi$
$461$	6.07778	0.283070	0.141535	+	0.989933i	$0.454796\pi$
$462$	0	0
$463$	−5.14719	−0.239210	−0.119605	−	0.992822i	$-0.538163\pi$
$463$	−5.14719	−0.239210	−0.119605	+	0.992822i	$0.538163\pi$
$464$	15.2262	0.706860
$465$	−7.78982	−0.361244
$466$	26.5877	1.23165
$467$	−3.91927	−0.181362	−0.0906812	−	0.995880i	$-0.528904\pi$
$467$	−3.91927	−0.181362	−0.0906812	+	0.995880i	$0.528904\pi$
$468$	−8.07555	−0.373292
$469$	−2.40314	−0.110967
$470$	−29.1431	−1.34427
$471$	61.4346	2.83076
$472$	−10.1327	−0.466397
$473$	0	0
$474$	30.2096	1.38757
$475$	−11.2706	−0.517131
$476$	−1.46570	−0.0671801
$477$	55.9828	2.56328
$478$	9.84982	0.450520
$479$	15.1129	0.690527	0.345263	−	0.938506i	$-0.387790\pi$
$479$	15.1129	0.690527	0.345263	+	0.938506i	$0.387790\pi$
$480$	39.7106	1.81253
$481$	20.7237	0.944922
$482$	21.1152	0.961770
$483$	−2.30745	−0.104993
$484$	0	0
$485$	22.3541	1.01505
$486$	−13.6324	−0.618377
$487$	25.0768	1.13634	0.568169	−	0.822912i	$-0.307652\pi$
$487$	25.0768	1.13634	0.568169	+	0.822912i	$0.307652\pi$
$488$	3.30919	0.149800
$489$	23.5065	1.06300
$490$	−3.85168	−0.174001
$491$	−37.2208	−1.67975	−0.839875	−	0.542780i	$-0.817372\pi$
$491$	−37.2208	−1.67975	−0.839875	+	0.542780i	$0.817372\pi$
$492$	4.60708	0.207703
$493$	15.4189	0.694434
$494$	3.72536	0.167612
$495$	0	0
$496$	−1.50554	−0.0676006
$497$	3.18859	0.143028
$498$	51.1980	2.29424
$499$	−31.8134	−1.42416	−0.712081	−	0.702098i	$-0.752247\pi$
$499$	−31.8134	−1.42416	−0.712081	+	0.702098i	$0.752247\pi$
$500$	−5.09371	−0.227798
$501$	−62.0609	−2.77267
$502$	3.19508	0.142603
$503$	19.2058	0.856346	0.428173	−	0.903697i	$-0.359157\pi$
$503$	19.2058	0.856346	0.428173	+	0.903697i	$0.359157\pi$
$504$	−15.8995	−0.708222
$505$	53.3788	2.37532
$506$	0	0
$507$	25.0370	1.11193
$508$	14.7431	0.654119
$509$	2.51549	0.111497	0.0557485	−	0.998445i	$-0.482246\pi$
$509$	2.51549	0.111497	0.0557485	+	0.998445i	$0.482246\pi$
$510$	−21.2928	−0.942861
$511$	1.22628	0.0542472
$512$	19.4135	0.857966
$513$	10.0814	0.445106
$514$	−24.9817	−1.10190
$515$	30.7872	1.35664
$516$	−6.67492	−0.293847
$517$	0	0
$518$	11.2135	0.492693
$519$	−23.0010	−1.00963
$520$	21.8793	0.959470
$521$	14.8968	0.652639	0.326320	−	0.945260i	$-0.394191\pi$
$521$	14.8968	0.652639	0.326320	+	0.945260i	$0.394191\pi$
$522$	45.9679	2.01196
$523$	9.90502	0.433116	0.216558	−	0.976270i	$-0.430517\pi$
$523$	9.90502	0.433116	0.216558	+	0.976270i	$0.430517\pi$
$524$	3.87535	0.169295
$525$	19.8530	0.866455
$526$	1.10411	0.0481414
$527$	−1.52459	−0.0664122
$528$	0	0
$529$	−22.3485	−0.971675
$530$	−41.6846	−1.81066
$531$	−17.0530	−0.740036
$532$	−1.23015	−0.0533336
$533$	4.37910	0.189680
$534$	−14.1407	−0.611927
$535$	−12.1384	−0.524789
$536$	7.38643	0.319045
$537$	10.3515	0.446701
$538$	8.01342	0.345483
$539$	0	0
$540$	16.2722	0.700245
$541$	22.0084	0.946214	0.473107	−	0.881005i	$-0.343132\pi$
$541$	22.0084	0.946214	0.473107	+	0.881005i	$0.343132\pi$
$542$	−30.2737	−1.30037
$543$	−45.2205	−1.94060
$544$	7.77199	0.333222
$545$	−13.3977	−0.573892
$546$	−6.56217	−0.280835
$547$	10.8643	0.464523	0.232261	−	0.972653i	$-0.425388\pi$
$547$	10.8643	0.464523	0.232261	+	0.972653i	$0.425388\pi$
$548$	−6.89787	−0.294662
$549$	5.56922	0.237689
$550$	0	0
$551$	12.9410	0.551303
$552$	7.09233	0.301869
$553$	−9.48182	−0.403208
$554$	23.3878	0.993650
$555$	−99.4133	−4.21986
$556$	−9.86610	−0.418416
$557$	−33.6126	−1.42421	−0.712106	−	0.702072i	$-0.752258\pi$
$557$	−33.6126	−1.42421	−0.712106	+	0.702072i	$0.752258\pi$
$558$	−4.54521	−0.192414
$559$	−6.34461	−0.268348
$560$	6.59959	0.278884
$561$	0	0
$562$	−31.3811	−1.32373
$563$	2.05348	0.0865440	0.0432720	−	0.999063i	$-0.486222\pi$
$563$	2.05348	0.0865440	0.0432720	+	0.999063i	$0.486222\pi$
$564$	16.3953	0.690367
$565$	36.8208	1.54906
$566$	−29.1898	−1.22694
$567$	−2.23976	−0.0940611
$568$	−9.80065	−0.411226
$569$	−3.27416	−0.137260	−0.0686300	−	0.997642i	$-0.521863\pi$
$569$	−3.27416	−0.137260	−0.0686300	+	0.997642i	$0.521863\pi$
$570$	−17.8708	−0.748527
$571$	43.8897	1.83673	0.918363	−	0.395738i	$-0.129511\pi$
$571$	43.8897	1.83673	0.918363	+	0.395738i	$0.129511\pi$
$572$	0	0
$573$	−1.22666	−0.0512446
$574$	2.36950	0.0989012
$575$	−5.60513	−0.233750
$576$	42.9261	1.78859
$577$	−43.8904	−1.82718	−0.913591	−	0.406635i	$-0.866702\pi$
$577$	−43.8904	−1.82718	−0.913591	+	0.406635i	$0.866702\pi$
$578$	14.7786	0.614709
$579$	−43.3821	−1.80290
$580$	20.8877	0.867315
$581$	−16.0694	−0.666671
$582$	20.6076	0.854214
$583$	0	0
$584$	−3.76915	−0.155969
$585$	36.8219	1.52240
$586$	−4.97481	−0.205508
$587$	2.79166	0.115224	0.0576121	−	0.998339i	$-0.481651\pi$
$587$	2.79166	0.115224	0.0576121	+	0.998339i	$0.481651\pi$
$588$	2.16688	0.0893607
$589$	−1.27957	−0.0527239
$590$	12.6976	0.522752
$591$	−59.4869	−2.44696
$592$	−19.2136	−0.789673
$593$	−23.2526	−0.954871	−0.477435	−	0.878667i	$-0.658434\pi$
$593$	−23.2526	−0.954871	−0.477435	+	0.878667i	$0.658434\pi$
$594$	0	0
$595$	6.68312	0.273981
$596$	−2.38493	−0.0976907
$597$	−24.1446	−0.988175
$598$	1.85271	0.0757630
$599$	10.4595	0.427362	0.213681	−	0.976904i	$-0.431455\pi$
$599$	10.4595	0.427362	0.213681	+	0.976904i	$0.431455\pi$
$600$	−61.0213	−2.49118
$601$	−22.3096	−0.910029	−0.455015	−	0.890484i	$-0.650366\pi$
$601$	−22.3096	−0.910029	−0.455015	+	0.890484i	$0.650366\pi$
$602$	−3.43303	−0.139920
$603$	12.4311	0.506232
$604$	2.17306	0.0884204
$605$	0	0
$606$	49.2083	1.99895
$607$	−18.9884	−0.770716	−0.385358	−	0.922767i	$-0.625922\pi$
$607$	−18.9884	−0.770716	−0.385358	+	0.922767i	$0.625922\pi$
$608$	6.52296	0.264541
$609$	−22.7953	−0.923712
$610$	−4.14683	−0.167900
$611$	15.5840	0.630461
$612$	7.58182	0.306477
$613$	−20.1617	−0.814323	−0.407161	−	0.913356i	$-0.633481\pi$
$613$	−20.1617	−0.814323	−0.407161	+	0.913356i	$0.633481\pi$
$614$	14.3589	0.579478
$615$	−21.0068	−0.847077
$616$	0	0
$617$	7.03919	0.283387	0.141694	−	0.989911i	$-0.454745\pi$
$617$	7.03919	0.283387	0.141694	+	0.989911i	$0.454745\pi$
$618$	28.3818	1.14168
$619$	−31.0831	−1.24933	−0.624667	−	0.780891i	$-0.714765\pi$
$619$	−31.0831	−1.24933	−0.624667	+	0.780891i	$0.714765\pi$
$620$	−2.06533	−0.0829457
$621$	5.01373	0.201194
$622$	−29.8898	−1.19847
$623$	4.43830	0.177817
$624$	11.2438	0.450113
$625$	−11.4966	−0.459866
$626$	−3.99947	−0.159851
$627$	0	0
$628$	16.2883	0.649973
$629$	−19.4567	−0.775791
$630$	19.9241	0.793797
$631$	21.9720	0.874690	0.437345	−	0.899294i	$-0.355919\pi$
$631$	21.9720	0.874690	0.437345	+	0.899294i	$0.355919\pi$
$632$	29.1439	1.15928
$633$	−28.1859	−1.12029
$634$	18.8546	0.748812
$635$	−67.2238	−2.66770
$636$	23.4510	0.929891
$637$	2.05965	0.0816064
$638$	0	0
$639$	−16.4941	−0.652496
$640$	−4.18149	−0.165288
$641$	−13.2909	−0.524961	−0.262480	−	0.964937i	$-0.584540\pi$
$641$	−13.2909	−0.524961	−0.262480	+	0.964937i	$0.584540\pi$
$642$	−11.1900	−0.441636
$643$	14.6904	0.579333	0.289666	−	0.957128i	$-0.406456\pi$
$643$	14.6904	0.579333	0.289666	+	0.957128i	$0.406456\pi$
$644$	−0.611780	−0.0241075
$645$	30.4355	1.19840
$646$	−3.49760	−0.137611
$647$	−6.14523	−0.241594	−0.120797	−	0.992677i	$-0.538545\pi$
$647$	−6.14523	−0.241594	−0.120797	+	0.992677i	$0.538545\pi$
$648$	6.88426	0.270439
$649$	0	0
$650$	−15.9404	−0.625236
$651$	2.25395	0.0883392
$652$	6.23232	0.244077
$653$	−40.6691	−1.59151	−0.795753	−	0.605622i	$-0.792924\pi$
$653$	−40.6691	−1.59151	−0.795753	+	0.605622i	$0.792924\pi$
$654$	−12.3509	−0.482959
$655$	−17.6704	−0.690438
$656$	−4.05998	−0.158516
$657$	−6.34333	−0.247477
$658$	8.43241	0.328730
$659$	−18.0090	−0.701531	−0.350765	−	0.936463i	$-0.614079\pi$
$659$	−18.0090	−0.701531	−0.350765	+	0.936463i	$0.614079\pi$
$660$	0	0
$661$	−17.1420	−0.666745	−0.333373	−	0.942795i	$-0.608187\pi$
$661$	−17.1420	−0.666745	−0.333373	+	0.942795i	$0.608187\pi$
$662$	1.38000	0.0536353
$663$	11.3861	0.442201
$664$	49.3919	1.91678
$665$	5.60908	0.217511
$666$	−58.0057	−2.24767
$667$	6.43584	0.249197
$668$	−16.4543	−0.636637
$669$	−49.7317	−1.92274
$670$	−9.25613	−0.357596
$671$	0	0
$672$	−11.4901	−0.443240
$673$	23.1926	0.894008	0.447004	−	0.894532i	$-0.352491\pi$
$673$	23.1926	0.894008	0.447004	+	0.894532i	$0.352491\pi$
$674$	−22.8249	−0.879183
$675$	−43.1374	−1.66036
$676$	6.63812	0.255312
$677$	27.5705	1.05962	0.529811	−	0.848116i	$-0.322263\pi$
$677$	27.5705	1.05962	0.529811	+	0.848116i	$0.322263\pi$
$678$	33.9440	1.30361
$679$	−6.46807	−0.248222
$680$	−20.5416	−0.787735
$681$	−36.1111	−1.38378
$682$	0	0
$683$	21.9351	0.839322	0.419661	−	0.907681i	$-0.362149\pi$
$683$	21.9351	0.839322	0.419661	+	0.907681i	$0.362149\pi$
$684$	6.36335	0.243309
$685$	31.4521	1.20172
$686$	1.11447	0.0425505
$687$	−13.0746	−0.498826
$688$	5.88227	0.224259
$689$	22.2905	0.849200
$690$	−8.88758	−0.338344
$691$	−25.6666	−0.976404	−0.488202	−	0.872731i	$-0.662347\pi$
$691$	−25.6666	−0.976404	−0.488202	+	0.872731i	$0.662347\pi$
$692$	−6.09830	−0.231823
$693$	0	0
$694$	30.3913	1.15364
$695$	44.9863	1.70643
$696$	70.0650	2.65581
$697$	−4.11137	−0.155729
$698$	8.89828	0.336805
$699$	68.2025	2.57966
$700$	5.26366	0.198948
$701$	18.4135	0.695467	0.347733	−	0.937593i	$-0.386951\pi$
$701$	18.4135	0.695467	0.347733	+	0.937593i	$0.386951\pi$
$702$	14.2586	0.538154
$703$	−16.3298	−0.615892
$704$	0	0
$705$	−74.7575	−2.81553
$706$	−6.61404	−0.248923
$707$	−15.4449	−0.580865
$708$	−7.14342	−0.268466
$709$	23.6621	0.888647	0.444323	−	0.895866i	$-0.353444\pi$
$709$	23.6621	0.888647	0.444323	+	0.895866i	$0.353444\pi$
$710$	12.2815	0.460915
$711$	49.0480	1.83944
$712$	−13.6418	−0.511249
$713$	−0.636362	−0.0238319
$714$	6.16097	0.230569
$715$	0	0
$716$	2.74452	0.102568
$717$	25.2666	0.943600
$718$	31.6734	1.18204
$719$	11.1222	0.414790	0.207395	−	0.978257i	$-0.433502\pi$
$719$	11.1222	0.414790	0.207395	+	0.978257i	$0.433502\pi$
$720$	−34.1387	−1.27227
$721$	−8.90812	−0.331756
$722$	18.2394	0.678799
$723$	54.1644	2.01440
$724$	−11.9894	−0.445583
$725$	−55.3730	−2.05650
$726$	0	0
$727$	42.4803	1.57551	0.787753	−	0.615991i	$-0.211244\pi$
$727$	42.4803	1.57551	0.787753	+	0.615991i	$0.211244\pi$
$728$	−6.33067	−0.234630
$729$	−41.6889	−1.54403
$730$	4.72323	0.174814
$731$	5.95671	0.220317
$732$	2.33292	0.0862274
$733$	−22.1884	−0.819548	−0.409774	−	0.912187i	$-0.634393\pi$
$733$	−22.1884	−0.819548	−0.409774	+	0.912187i	$0.634393\pi$
$734$	33.8814	1.25058
$735$	−9.88030	−0.364440
$736$	3.24402	0.119576
$737$	0	0
$738$	−12.2571	−0.451189
$739$	−29.4481	−1.08327	−0.541633	−	0.840615i	$-0.682194\pi$
$739$	−29.4481	−1.08327	−0.541633	+	0.840615i	$0.682194\pi$
$740$	−26.3576	−0.968926
$741$	9.55627	0.351058
$742$	12.0613	0.442783
$743$	−16.9059	−0.620217	−0.310109	−	0.950701i	$-0.600365\pi$
$743$	−16.9059	−0.620217	−0.310109	+	0.950701i	$0.600365\pi$
$744$	−6.92787	−0.253988
$745$	10.8745	0.398412
$746$	16.0735	0.588491
$747$	83.1244	3.04136
$748$	0	0
$749$	3.51219	0.128333
$750$	21.4111	0.781823
$751$	−1.55147	−0.0566138	−0.0283069	−	0.999599i	$-0.509012\pi$
$751$	−1.55147	−0.0566138	−0.0283069	+	0.999599i	$0.509012\pi$
$752$	−14.4484	−0.526877
$753$	8.19598	0.298678
$754$	18.3029	0.666552
$755$	−9.90845	−0.360605
$756$	−4.70829	−0.171239
$757$	12.5467	0.456016	0.228008	−	0.973659i	$-0.426779\pi$
$757$	12.5467	0.456016	0.228008	+	0.973659i	$0.426779\pi$
$758$	24.9721	0.907027
$759$	0	0
$760$	−17.2404	−0.625375
$761$	9.03436	0.327495	0.163748	−	0.986502i	$-0.447642\pi$
$761$	9.03436	0.327495	0.163748	+	0.986502i	$0.447642\pi$
$762$	−61.9717	−2.24500
$763$	3.87655	0.140340
$764$	−0.325228	−0.0117663
$765$	−34.5707	−1.24991
$766$	−37.5336	−1.35614
$767$	−6.78993	−0.245170
$768$	43.5922	1.57300
$769$	16.1383	0.581963	0.290981	−	0.956729i	$-0.406018\pi$
$769$	16.1383	0.581963	0.290981	+	0.956729i	$0.406018\pi$
$770$	0	0
$771$	−64.0829	−2.30789
$772$	−11.5020	−0.413965
$773$	−18.4134	−0.662285	−0.331143	−	0.943581i	$-0.607434\pi$
$773$	−18.4134	−0.662285	−0.331143	+	0.943581i	$0.607434\pi$
$774$	17.7585	0.638317
$775$	5.47516	0.196674
$776$	19.8806	0.713673
$777$	28.7648	1.03193
$778$	−2.70573	−0.0970053
$779$	−3.45063	−0.123632
$780$	15.4246	0.552288
$781$	0	0
$782$	−1.73944	−0.0622022
$783$	49.5306	1.77008
$784$	−1.90956	−0.0681986
$785$	−74.2694	−2.65079
$786$	−16.2898	−0.581037
$787$	−46.9870	−1.67491	−0.837453	−	0.546509i	$-0.815957\pi$
$787$	−46.9870	−1.67491	−0.837453	+	0.546509i	$0.815957\pi$
$788$	−15.7719	−0.561850
$789$	2.83225	0.100831
$790$	−36.5210	−1.29936
$791$	−10.6539	−0.378810
$792$	0	0
$793$	2.21748	0.0787450
$794$	6.57197	0.233230
$795$	−106.929	−3.79238
$796$	−6.40152	−0.226896
$797$	−3.12454	−0.110677	−0.0553385	−	0.998468i	$-0.517624\pi$
$797$	−3.12454	−0.110677	−0.0553385	+	0.998468i	$0.517624\pi$
$798$	5.17084	0.183046
$799$	−14.6312	−0.517615
$800$	−27.9111	−0.986805
$801$	−22.9586	−0.811203
$802$	−12.5261	−0.442314
$803$	0	0
$804$	5.20732	0.183648
$805$	2.78952	0.0983178
$806$	−1.80975	−0.0637457
$807$	20.5559	0.723604
$808$	47.4724	1.67007
$809$	32.7257	1.15057	0.575286	−	0.817952i	$-0.304891\pi$
$809$	32.7257	1.15057	0.575286	+	0.817952i	$0.304891\pi$
$810$	−8.62685	−0.303117
$811$	−32.6613	−1.14689	−0.573447	−	0.819243i	$-0.694394\pi$
$811$	−32.6613	−1.14689	−0.573447	+	0.819243i	$0.694394\pi$
$812$	−6.04376	−0.212095
$813$	−77.6578	−2.72358
$814$	0	0
$815$	−28.4174	−0.995419
$816$	−10.5564	−0.369548
$817$	4.99941	0.174907
$818$	32.6922	1.14306
$819$	−10.6543	−0.372290
$820$	−5.56958	−0.194498
$821$	7.43142	0.259358	0.129679	−	0.991556i	$-0.458605\pi$
$821$	7.43142	0.259358	0.129679	+	0.991556i	$0.458605\pi$
$822$	28.9948	1.01131
$823$	6.55866	0.228621	0.114310	−	0.993445i	$-0.463534\pi$
$823$	6.55866	0.228621	0.114310	+	0.993445i	$0.463534\pi$
$824$	27.3805	0.953846
$825$	0	0
$826$	−3.67399	−0.127835
$827$	−23.8538	−0.829479	−0.414739	−	0.909940i	$-0.636127\pi$
$827$	−23.8538	−0.829479	−0.414739	+	0.909940i	$0.636127\pi$
$828$	3.16464	0.109979
$829$	−26.1431	−0.907989	−0.453994	−	0.891005i	$-0.650001\pi$
$829$	−26.1431	−0.907989	−0.453994	+	0.891005i	$0.650001\pi$
$830$	−61.8942	−2.14838
$831$	59.9940	2.08117
$832$	17.0917	0.592549
$833$	−1.93373	−0.0669997
$834$	41.4716	1.43604
$835$	75.0265	2.59640
$836$	0	0
$837$	−4.89747	−0.169281
$838$	22.6079	0.780976
$839$	34.2890	1.18379	0.591893	−	0.806016i	$-0.298381\pi$
$839$	34.2890	1.18379	0.591893	+	0.806016i	$0.298381\pi$
$840$	30.3687	1.04782
$841$	34.5795	1.19240
$842$	3.41030	0.117527
$843$	−80.4986	−2.77252
$844$	−7.47297	−0.257230
$845$	−30.2677	−1.04124
$846$	−43.6195	−1.49967
$847$	0	0
$848$	−20.6661	−0.709678
$849$	−74.8774	−2.56979
$850$	14.9659	0.513325
$851$	−8.12121	−0.278392
$852$	−6.90931	−0.236709
$853$	−21.3842	−0.732181	−0.366090	−	0.930579i	$-0.619304\pi$
$853$	−21.3842	−0.732181	−0.366090	+	0.930579i	$0.619304\pi$
$854$	1.19987	0.0410585
$855$	−29.0149	−0.992287
$856$	−10.7953	−0.368975
$857$	−42.8697	−1.46440	−0.732200	−	0.681090i	$-0.761506\pi$
$857$	−42.8697	−1.46440	−0.732200	+	0.681090i	$0.761506\pi$
$858$	0	0
$859$	−30.3915	−1.03695	−0.518473	−	0.855094i	$-0.673499\pi$
$859$	−30.3915	−1.03695	−0.518473	+	0.855094i	$0.673499\pi$
$860$	8.06944	0.275165
$861$	6.07823	0.207146
$862$	8.40527	0.286285
$863$	11.8184	0.402304	0.201152	−	0.979560i	$-0.435532\pi$
$863$	11.8184	0.402304	0.201152	+	0.979560i	$0.435532\pi$
$864$	24.9661	0.849365
$865$	27.8063	0.945444
$866$	35.8430	1.21800
$867$	37.9099	1.28749
$868$	0.597594	0.0202837
$869$	0	0
$870$	−87.8003	−2.97671
$871$	4.94963	0.167712
$872$	−11.9152	−0.403499
$873$	33.4583	1.13239
$874$	−1.45989	−0.0493816
$875$	−6.72025	−0.227186
$876$	−2.65719	−0.0897783
$877$	9.20488	0.310827	0.155413	−	0.987850i	$-0.450329\pi$
$877$	9.20488	0.310827	0.155413	+	0.987850i	$0.450329\pi$
$878$	−5.20149	−0.175542
$879$	−12.7613	−0.430429
$880$	0	0
$881$	−41.9030	−1.41175	−0.705874	−	0.708338i	$-0.749445\pi$
$881$	−41.9030	−1.41175	−0.705874	+	0.708338i	$0.749445\pi$
$882$	−5.76496	−0.194116
$883$	−16.0478	−0.540053	−0.270026	−	0.962853i	$-0.587032\pi$
$883$	−16.0478	−0.540053	−0.270026	+	0.962853i	$0.587032\pi$
$884$	3.01883	0.101534
$885$	32.5718	1.09489
$886$	−19.2548	−0.646878
$887$	33.8483	1.13652	0.568258	−	0.822850i	$-0.307618\pi$
$887$	33.8483	1.13652	0.568258	+	0.822850i	$0.307618\pi$
$888$	−88.4131	−2.96695
$889$	19.4509	0.652362
$890$	17.0949	0.573024
$891$	0	0
$892$	−13.1855	−0.441482
$893$	−12.2798	−0.410929
$894$	10.0249	0.335284
$895$	−12.5141	−0.418302
$896$	1.20989	0.0404197
$897$	4.75255	0.158683
$898$	18.6563	0.622568
$899$	−6.28660	−0.209670
$900$	−27.2281	−0.907603
$901$	−20.9277	−0.697202
$902$	0	0
$903$	−8.80638	−0.293058
$904$	32.7465	1.08913
$905$	54.6679	1.81722
$906$	−9.13431	−0.303467
$907$	−44.1013	−1.46436	−0.732181	−	0.681111i	$-0.761497\pi$
$907$	−44.1013	−1.46436	−0.732181	+	0.681111i	$0.761497\pi$
$908$	−9.57421	−0.317731
$909$	79.8940	2.64992
$910$	7.93313	0.262981
$911$	49.5756	1.64251	0.821257	−	0.570558i	$-0.193273\pi$
$911$	49.5756	1.64251	0.821257	+	0.570558i	$0.193273\pi$
$912$	−8.85988	−0.293380
$913$	0	0
$914$	−24.1810	−0.799837
$915$	−10.6374	−0.351662
$916$	−3.46648	−0.114536
$917$	5.11284	0.168841
$918$	−13.3868	−0.441831
$919$	40.7926	1.34563	0.672813	−	0.739813i	$-0.265086\pi$
$919$	40.7926	1.34563	0.672813	+	0.739813i	$0.265086\pi$
$920$	−8.57404	−0.282678
$921$	36.8333	1.21370
$922$	−6.77348	−0.223073
$923$	−6.56740	−0.216169
$924$	0	0
$925$	69.8737	2.29743
$926$	5.73637	0.188509
$927$	46.0803	1.51348
$928$	32.0476	1.05201
$929$	41.3929	1.35806	0.679029	−	0.734112i	$-0.262401\pi$
$929$	41.3929	1.35806	0.679029	+	0.734112i	$0.262401\pi$
$930$	8.68150	0.284677
$931$	−1.62296	−0.0531904
$932$	18.0827	0.592318
$933$	−76.6731	−2.51016
$934$	4.36790	0.142922
$935$	0	0
$936$	32.7476	1.07039
$937$	−1.59644	−0.0521534	−0.0260767	−	0.999660i	$-0.508301\pi$
$937$	−1.59644	−0.0521534	−0.0260767	+	0.999660i	$0.508301\pi$
$938$	2.67822	0.0874469
$939$	−10.2594	−0.334803
$940$	−19.8206	−0.646477
$941$	−7.10787	−0.231710	−0.115855	−	0.993266i	$-0.536961\pi$
$941$	−7.10787	−0.231710	−0.115855	+	0.993266i	$0.536961\pi$
$942$	−68.4668	−2.23077
$943$	−1.71608	−0.0558832
$944$	6.29513	0.204889
$945$	21.4683	0.698364
$946$	0	0
$947$	−2.45986	−0.0799347	−0.0399674	−	0.999201i	$-0.512725\pi$
$947$	−2.45986	−0.0799347	−0.0399674	+	0.999201i	$0.512725\pi$
$948$	20.5460	0.667303
$949$	−2.52570	−0.0819878
$950$	12.5607	0.407523
$951$	48.3656	1.56836
$952$	5.94362	0.192634
$953$	28.6000	0.926446	0.463223	−	0.886242i	$-0.346693\pi$
$953$	28.6000	0.926446	0.463223	+	0.886242i	$0.346693\pi$
$954$	−62.3910	−2.01998
$955$	1.48294	0.0479867
$956$	6.69900	0.216661
$957$	0	0
$958$	−16.8428	−0.544168
$959$	−9.10052	−0.293871
$960$	−81.9903	−2.64622
$961$	−30.3784	−0.979948
$962$	−23.0959	−0.744643
$963$	−18.1680	−0.585456
$964$	14.3607	0.462528
$965$	52.4454	1.68828
$966$	2.57158	0.0827392
$967$	0.213338	0.00686047	0.00343024	−	0.999994i	$-0.498908\pi$
$967$	0.213338	0.00686047	0.00343024	+	0.999994i	$0.498908\pi$
$968$	0	0
$969$	−8.97201	−0.288223
$970$	−24.9129	−0.799907
$971$	−0.828199	−0.0265782	−0.0132891	−	0.999912i	$-0.504230\pi$
$971$	−0.828199	−0.0265782	−0.0132891	+	0.999912i	$0.504230\pi$
$972$	−9.27157	−0.297386
$973$	−13.0166	−0.417292
$974$	−27.9473	−0.895488
$975$	−40.8903	−1.30954
$976$	−2.05589	−0.0658073
$977$	9.75714	0.312159	0.156079	−	0.987745i	$-0.450114\pi$
$977$	9.75714	0.312159	0.156079	+	0.987745i	$0.450114\pi$
$978$	−26.1972	−0.837694
$979$	0	0
$980$	−2.61958	−0.0836795
$981$	−20.0528	−0.640236
$982$	41.4813	1.32372
$983$	45.1198	1.43910	0.719548	−	0.694442i	$-0.244349\pi$
$983$	45.1198	1.43910	0.719548	+	0.694442i	$0.244349\pi$
$984$	−18.6824	−0.595573
$985$	71.9148	2.29140
$986$	−17.1839	−0.547246
$987$	21.6307	0.688514
$988$	2.53367	0.0806069
$989$	2.48632	0.0790604
$990$	0	0
$991$	53.5405	1.70077	0.850384	−	0.526162i	$-0.176369\pi$
$991$	53.5405	1.70077	0.850384	+	0.526162i	$0.176369\pi$
$992$	−3.16880	−0.100609
$993$	3.53997	0.112337
$994$	−3.55358	−0.112713
$995$	29.1889	0.925351
$996$	34.8205	1.10333
$997$	31.1607	0.986869	0.493435	−	0.869783i	$-0.335741\pi$
$997$	31.1607	0.986869	0.493435	+	0.869783i	$0.335741\pi$
$998$	35.4549	1.12231
$999$	−62.5013	−1.97745

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.o.1.3		8
3.2	odd	2		7623.2.a.cw.1.6		8
7.6	odd	2		5929.2.a.bs.1.3		8
11.2	odd	10		77.2.f.b.15.2	✓	16
11.3	even	5		847.2.f.v.372.2		16
11.4	even	5		847.2.f.v.148.2		16
11.5	even	5		847.2.f.x.729.3		16
11.6	odd	10		77.2.f.b.36.2	yes	16
11.7	odd	10		847.2.f.w.148.3		16
11.8	odd	10		847.2.f.w.372.3		16
11.9	even	5		847.2.f.x.323.3		16
11.10	odd	2		847.2.a.p.1.6		8
33.2	even	10		693.2.m.i.631.3		16
33.17	even	10		693.2.m.i.190.3		16
33.32	even	2		7623.2.a.ct.1.3		8
77.2	odd	30		539.2.q.g.312.2		32
77.6	even	10		539.2.f.e.344.2		16
77.13	even	10		539.2.f.e.246.2		16
77.17	even	30		539.2.q.f.520.2		32
77.24	even	30		539.2.q.f.422.3		32
77.39	odd	30		539.2.q.g.520.2		32
77.46	odd	30		539.2.q.g.422.3		32
77.61	even	30		539.2.q.f.410.3		32
77.68	even	30		539.2.q.f.312.2		32
77.72	odd	30		539.2.q.g.410.3		32
77.76	even	2		5929.2.a.bt.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.15.2	✓	16	11.2	odd	10
77.2.f.b.36.2	yes	16	11.6	odd	10
539.2.f.e.246.2		16	77.13	even	10
539.2.f.e.344.2		16	77.6	even	10
539.2.q.f.312.2		32	77.68	even	30
539.2.q.f.410.3		32	77.61	even	30
539.2.q.f.422.3		32	77.24	even	30
539.2.q.f.520.2		32	77.17	even	30
539.2.q.g.312.2		32	77.2	odd	30
539.2.q.g.410.3		32	77.72	odd	30
539.2.q.g.422.3		32	77.46	odd	30
539.2.q.g.520.2		32	77.39	odd	30
693.2.m.i.190.3		16	33.17	even	10
693.2.m.i.631.3		16	33.2	even	10
847.2.a.o.1.3		8	1.1	even	1	trivial
847.2.a.p.1.6		8	11.10	odd	2
847.2.f.v.148.2		16	11.4	even	5
847.2.f.v.372.2		16	11.3	even	5
847.2.f.w.148.3		16	11.7	odd	10
847.2.f.w.372.3		16	11.8	odd	10
847.2.f.x.323.3		16	11.9	even	5
847.2.f.x.729.3		16	11.5	even	5
5929.2.a.bs.1.3		8	7.6	odd	2
5929.2.a.bt.1.6		8	77.76	even	2
7623.2.a.ct.1.3		8	33.32	even	2
7623.2.a.cw.1.6		8	3.2	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-1.11447 q^{2} -2.85882 q^{3} -0.757964 q^{4} +3.45608 q^{5} +3.18606 q^{6} -1.00000 q^{7} +3.07366 q^{8} +5.17284 q^{9} +O(q^{10})\)	\(q-1.11447 q^{2} -2.85882 q^{3} -0.757964 q^{4} +3.45608 q^{5} +3.18606 q^{6} -1.00000 q^{7} +3.07366 q^{8} +5.17284 q^{9} -3.85168 q^{10} +2.16688 q^{12} +2.05965 q^{13} +1.11447 q^{14} -9.88030 q^{15} -1.90956 q^{16} -1.93373 q^{17} -5.76496 q^{18} -1.62296 q^{19} -2.61958 q^{20} +2.85882 q^{21} -0.807136 q^{23} -8.78703 q^{24} +6.94447 q^{25} -2.29541 q^{26} -6.21176 q^{27} +0.757964 q^{28} -7.97368 q^{29} +11.0113 q^{30} +0.788420 q^{31} -4.01918 q^{32} +2.15508 q^{34} -3.45608 q^{35} -3.92083 q^{36} +10.0618 q^{37} +1.80873 q^{38} -5.88817 q^{39} +10.6228 q^{40} +2.12613 q^{41} -3.18606 q^{42} -3.08043 q^{43} +17.8777 q^{45} +0.899526 q^{46} +7.56632 q^{47} +5.45909 q^{48} +1.00000 q^{49} -7.73938 q^{50} +5.52818 q^{51} -1.56114 q^{52} +10.8224 q^{53} +6.92280 q^{54} -3.07366 q^{56} +4.63975 q^{57} +8.88640 q^{58} -3.29664 q^{59} +7.48891 q^{60} +1.07663 q^{61} -0.878667 q^{62} -5.17284 q^{63} +8.29836 q^{64} +7.11832 q^{65} +2.40314 q^{67} +1.46570 q^{68} +2.30745 q^{69} +3.85168 q^{70} -3.18859 q^{71} +15.8995 q^{72} -1.22628 q^{73} -11.2135 q^{74} -19.8530 q^{75} +1.23015 q^{76} +6.56217 q^{78} +9.48182 q^{79} -6.59959 q^{80} +2.23976 q^{81} -2.36950 q^{82} +16.0694 q^{83} -2.16688 q^{84} -6.68312 q^{85} +3.43303 q^{86} +22.7953 q^{87} -4.43830 q^{89} -19.9241 q^{90} -2.05965 q^{91} +0.611780 q^{92} -2.25395 q^{93} -8.43241 q^{94} -5.60908 q^{95} +11.4901 q^{96} +6.46807 q^{97} -1.11447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9}+O(q^{10}) \) 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9	\( 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} + q^{14} + 11 q^{15} + q^{16} + 5 q^{17} - 8 q^{18} + 13 q^{19} + 23 q^{20} - 4 q^{21} + 16 q^{23} + 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} - 7 q^{28} - 9 q^{29} + 36 q^{30} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} - 13 q^{39} - 5 q^{40} + 10 q^{41} - q^{42} + 4 q^{43} + 35 q^{45} - 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} - 6 q^{50} - 13 q^{51} + 41 q^{52} + 37 q^{53} - 30 q^{54} - 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} - 19 q^{61} + 18 q^{62} - 14 q^{63} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} + 35 q^{72} + 25 q^{73} - 33 q^{74} - 13 q^{75} - 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} + 25 q^{83} - 9 q^{84} - 3 q^{85} + 4 q^{86} + 36 q^{87} + 37 q^{89} + 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} + 42 q^{94} - 21 q^{95} + 6 q^{96} + 15 q^{97} - q^{98}+O(q^{100}) \) 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9 - 6 * q^10 + 9 * q^12 + 6 * q^13 + q^14 + 11 * q^15 + q^16 + 5 * q^17 - 8 * q^18 + 13 * q^19 + 23 * q^20 - 4 * q^21 + 16 * q^23 + 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 - 7 * q^28 - 9 * q^29 + 36 * q^30 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 - 13 * q^39 - 5 * q^40 + 10 * q^41 - q^42 + 4 * q^43 + 35 * q^45 - 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 - 6 * q^50 - 13 * q^51 + 41 * q^52 + 37 * q^53 - 30 * q^54 - 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 - 19 * q^61 + 18 * q^62 - 14 * q^63 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 + 35 * q^72 + 25 * q^73 - 33 * q^74 - 13 * q^75 - 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 + 25 * q^83 - 9 * q^84 - 3 * q^85 + 4 * q^86 + 36 * q^87 + 37 * q^89 + 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 + 42 * q^94 - 21 * q^95 + 6 * q^96 + 15 * q^97 - q^98

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