LMFDB - Embedded newform 847.2.a.n.1.1

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:	$6$
Coefficient field:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.38595$ 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)

$f(q)$	$=$	$q-1.38595 q^{2} -0.122479 q^{3} -0.0791355 q^{4} -0.133004 q^{5} +0.169750 q^{6} +1.00000 q^{7} +2.88158 q^{8} -2.98500 q^{9} +O(q^{10})$	\(q-1.38595 q^{2} -0.122479 q^{3} -0.0791355 q^{4} -0.133004 q^{5} +0.169750 q^{6} +1.00000 q^{7} +2.88158 q^{8} -2.98500 q^{9} +0.184338 q^{10} +0.00969245 q^{12} -0.641436 q^{13} -1.38595 q^{14} +0.0162903 q^{15} -3.83547 q^{16} -1.42482 q^{17} +4.13707 q^{18} +7.18297 q^{19} +0.0105254 q^{20} -0.122479 q^{21} +1.66655 q^{23} -0.352934 q^{24} -4.98231 q^{25} +0.889000 q^{26} +0.733037 q^{27} -0.0791355 q^{28} +4.47657 q^{29} -0.0225775 q^{30} -6.83182 q^{31} -0.447392 q^{32} +1.97473 q^{34} -0.133004 q^{35} +0.236220 q^{36} +3.76864 q^{37} -9.95526 q^{38} +0.0785625 q^{39} -0.383263 q^{40} +6.17286 q^{41} +0.169750 q^{42} -1.03970 q^{43} +0.397018 q^{45} -2.30976 q^{46} +9.48937 q^{47} +0.469764 q^{48} +1.00000 q^{49} +6.90524 q^{50} +0.174511 q^{51} +0.0507604 q^{52} -0.666549 q^{53} -1.01595 q^{54} +2.88158 q^{56} -0.879764 q^{57} -6.20431 q^{58} +8.18695 q^{59} -0.00128914 q^{60} +9.49807 q^{61} +9.46858 q^{62} -2.98500 q^{63} +8.29100 q^{64} +0.0853139 q^{65} +12.0398 q^{67} +0.112754 q^{68} -0.204117 q^{69} +0.184338 q^{70} +4.83418 q^{71} -8.60152 q^{72} -8.85748 q^{73} -5.22316 q^{74} +0.610229 q^{75} -0.568428 q^{76} -0.108884 q^{78} +11.4907 q^{79} +0.510134 q^{80} +8.86521 q^{81} -8.55529 q^{82} +9.57998 q^{83} +0.00969245 q^{84} +0.189507 q^{85} +1.44098 q^{86} -0.548286 q^{87} -17.7001 q^{89} -0.550248 q^{90} -0.641436 q^{91} -0.131883 q^{92} +0.836756 q^{93} -13.1518 q^{94} -0.955367 q^{95} +0.0547962 q^{96} +6.58139 q^{97} -1.38595 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) $ 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9} - 8 q^{10} - 14 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} + 22 q^{17} + 24 q^{18} + 6 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{23} - 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} + 4 q^{28} + 12 q^{29} + 20 q^{30} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} + 20 q^{39} + 18 q^{40} + 26 q^{41} - 6 q^{42} - 4 q^{43} - 36 q^{45} + 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} - 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} - 32 q^{54} + 12 q^{56} + 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} - 8 q^{61} + 20 q^{62} + 8 q^{63} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} + 16 q^{72} + 14 q^{73} + 44 q^{74} - 20 q^{75} - 30 q^{76} + 32 q^{78} - 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} + 22 q^{83} - 14 q^{84} - 24 q^{85} - 30 q^{86} + 22 q^{87} - 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} - 38 q^{94} - 24 q^{95} - 62 q^{96} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9 - 8 * q^10 - 14 * q^12 + 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 + 22 * q^17 + 24 * q^18 + 6 * q^19 + 2 * q^20 - 2 * q^21 + 2 * q^23 - 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 + 4 * q^28 + 12 * q^29 + 20 * q^30 - 2 * q^31 + 8 * q^32 + 24 * q^34 - 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 + 20 * q^39 + 18 * q^40 + 26 * q^41 - 6 * q^42 - 4 * q^43 - 36 * q^45 + 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 - 4 * q^50 - 4 * q^51 + 12 * q^52 + 4 * q^53 - 32 * q^54 + 12 * q^56 + 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 - 8 * q^61 + 20 * q^62 + 8 * q^63 + 26 * q^64 + 24 * q^65 + 6 * q^67 + 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 + 16 * q^72 + 14 * q^73 + 44 * q^74 - 20 * q^75 - 30 * q^76 + 32 * q^78 - 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 + 22 * q^83 - 14 * q^84 - 24 * q^85 - 30 * q^86 + 22 * q^87 - 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 - 38 * q^94 - 24 * q^95 - 62 * q^96 - 4 * q^97 + 4 * 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 $

$2$	−1.38595	−0.980016	−0.490008	−	0.871718i	$-0.663006\pi$
$2$	−1.38595	−0.980016	−0.490008	+	0.871718i	$0.663006\pi$
$3$	−0.122479	−0.0707133	−0.0353567	−	0.999375i	$-0.511257\pi$
$3$	−0.122479	−0.0707133	−0.0353567	+	0.999375i	$0.511257\pi$
$4$	−0.0791355	−0.0395678
$5$	−0.133004	−0.0594814	−0.0297407	−	0.999558i	$-0.509468\pi$
$5$	−0.133004	−0.0594814	−0.0297407	+	0.999558i	$0.509468\pi$
$6$	0.169750	0.0693002
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.88158	1.01879
$9$	−2.98500	−0.995000
$10$	0.184338	0.0582928
$11$	0	0
$11$	0	0
$12$	0.00969245	0.00279797
$13$	−0.641436	−0.177902	−0.0889512	−	0.996036i	$-0.528352\pi$
$13$	−0.641436	−0.177902	−0.0889512	+	0.996036i	$0.528352\pi$
$14$	−1.38595	−0.370411
$15$	0.0162903	0.00420613
$16$	−3.83547	−0.958867
$17$	−1.42482	−0.345570	−0.172785	−	0.984960i	$-0.555277\pi$
$17$	−1.42482	−0.345570	−0.172785	+	0.984960i	$0.555277\pi$
$18$	4.13707	0.975116
$19$	7.18297	1.64789	0.823943	−	0.566672i	$-0.191769\pi$
$19$	7.18297	1.64789	0.823943	+	0.566672i	$0.191769\pi$
$20$	0.0105254	0.00235355
$21$	−0.122479	−0.0267271
$22$	0	0
$23$	1.66655	0.347500	0.173750	−	0.984790i	$-0.444412\pi$
$23$	1.66655	0.347500	0.173750	+	0.984790i	$0.444412\pi$
$24$	−0.352934	−0.0720423
$25$	−4.98231	−0.996462
$26$	0.889000	0.174347
$27$	0.733037	0.141073
$28$	−0.0791355	−0.0149552
$29$	4.47657	0.831278	0.415639	−	0.909530i	$-0.363558\pi$
$29$	4.47657	0.831278	0.415639	+	0.909530i	$0.363558\pi$
$30$	−0.0225775	−0.00412208
$31$	−6.83182	−1.22703	−0.613516	−	0.789682i	$-0.710245\pi$
$31$	−6.83182	−1.22703	−0.613516	+	0.789682i	$0.710245\pi$
$32$	−0.447392	−0.0790885
$33$	0	0
$34$	1.97473	0.338664
$35$	−0.133004	−0.0224819
$36$	0.236220	0.0393699
$37$	3.76864	0.619561	0.309781	−	0.950808i	$-0.399744\pi$
$37$	3.76864	0.619561	0.309781	+	0.950808i	$0.399744\pi$
$38$	−9.95526	−1.61496
$39$	0.0785625	0.0125801
$40$	−0.383263	−0.0605993
$41$	6.17286	0.964039	0.482019	−	0.876161i	$-0.339904\pi$
$41$	6.17286	0.964039	0.482019	+	0.876161i	$0.339904\pi$
$42$	0.169750	0.0261930
$43$	−1.03970	−0.158553	−0.0792764	−	0.996853i	$-0.525261\pi$
$43$	−1.03970	−0.158553	−0.0792764	+	0.996853i	$0.525261\pi$
$44$	0	0
$45$	0.397018	0.0591840
$46$	−2.30976	−0.340555
$47$	9.48937	1.38417	0.692084	−	0.721817i	$-0.256693\pi$
$47$	9.48937	1.38417	0.692084	+	0.721817i	$0.256693\pi$
$48$	0.469764	0.0678047
$49$	1.00000	0.142857
$50$	6.90524	0.976549
$51$	0.174511	0.0244364
$52$	0.0507604	0.00703920
$53$	−0.666549	−0.0915576	−0.0457788	−	0.998952i	$-0.514577\pi$
$53$	−0.666549	−0.0915576	−0.0457788	+	0.998952i	$0.514577\pi$
$54$	−1.01595	−0.138254
$55$	0	0
$56$	2.88158	0.385068
$57$	−0.879764	−0.116528
$58$	−6.20431	−0.814666
$59$	8.18695	1.06585	0.532925	−	0.846162i	$-0.321093\pi$
$59$	8.18695	1.06585	0.532925	+	0.846162i	$0.321093\pi$
$60$	−0.00128914	−0.000166427 0
$61$	9.49807	1.21610	0.608051	−	0.793898i	$-0.291951\pi$
$61$	9.49807	1.21610	0.608051	+	0.793898i	$0.291951\pi$
$62$	9.46858	1.20251
$63$	−2.98500	−0.376075
$64$	8.29100	1.03637
$65$	0.0853139	0.0105819
$66$	0	0
$67$	12.0398	1.47089	0.735446	−	0.677583i	$-0.236973\pi$
$67$	12.0398	1.47089	0.735446	+	0.677583i	$0.236973\pi$
$68$	0.112754	0.0136734
$69$	−0.204117	−0.0245729
$70$	0.184338	0.0220326
$71$	4.83418	0.573711	0.286856	−	0.957974i	$-0.407390\pi$
$71$	4.83418	0.573711	0.286856	+	0.957974i	$0.407390\pi$
$72$	−8.60152	−1.01370
$73$	−8.85748	−1.03669	−0.518345	−	0.855172i	$-0.673452\pi$
$73$	−8.85748	−1.03669	−0.518345	+	0.855172i	$0.673452\pi$
$74$	−5.22316	−0.607180
$75$	0.610229	0.0704632
$76$	−0.568428	−0.0652032
$77$	0	0
$78$	−0.108884	−0.0123287
$79$	11.4907	1.29280	0.646400	−	0.762998i	$-0.276274\pi$
$79$	11.4907	1.29280	0.646400	+	0.762998i	$0.276274\pi$
$80$	0.510134	0.0570347
$81$	8.86521	0.985024
$82$	−8.55529	−0.944774
$83$	9.57998	1.05154	0.525770	−	0.850627i	$-0.323777\pi$
$83$	9.57998	1.05154	0.525770	+	0.850627i	$0.323777\pi$
$84$	0.00969245	0.00105753
$85$	0.189507	0.0205550
$86$	1.44098	0.155384
$87$	−0.548286	−0.0587824
$88$	0	0
$89$	−17.7001	−1.87621	−0.938104	−	0.346353i	$-0.887420\pi$
$89$	−17.7001	−1.87621	−0.938104	+	0.346353i	$0.887420\pi$
$90$	−0.550248	−0.0580013
$91$	−0.641436	−0.0672408
$92$	−0.131883	−0.0137498
$93$	0.836756	0.0867675
$94$	−13.1518	−1.35651
$95$	−0.955367	−0.0980186
$96$	0.0547962	0.00559261
$97$	6.58139	0.668239	0.334119	−	0.942531i	$-0.391561\pi$
$97$	6.58139	0.668239	0.334119	+	0.942531i	$0.391561\pi$
$98$	−1.38595	−0.140002
$99$	0	0
$100$	0.394278	0.0394278
$101$	18.6204	1.85280	0.926400	−	0.376541i	$-0.122887\pi$
$101$	18.6204	1.85280	0.926400	+	0.376541i	$0.122887\pi$
$102$	−0.241864	−0.0239481
$103$	−6.92356	−0.682199	−0.341099	−	0.940027i	$-0.610799\pi$
$103$	−6.92356	−0.682199	−0.341099	+	0.940027i	$0.610799\pi$
$104$	−1.84835	−0.181246
$105$	0.0162903	0.00158977
$106$	0.923806	0.0897279
$107$	−5.44446	−0.526336	−0.263168	−	0.964750i	$-0.584767\pi$
$107$	−5.44446	−0.526336	−0.263168	+	0.964750i	$0.584767\pi$
$108$	−0.0580093	−0.00558195
$109$	−9.22316	−0.883419	−0.441709	−	0.897158i	$-0.645628\pi$
$109$	−9.22316	−0.883419	−0.441709	+	0.897158i	$0.645628\pi$
$110$	0	0
$111$	−0.461580	−0.0438112
$112$	−3.83547	−0.362418
$113$	9.38223	0.882606	0.441303	−	0.897358i	$-0.354516\pi$
$113$	9.38223	0.882606	0.441303	+	0.897358i	$0.354516\pi$
$114$	1.21931	0.114199
$115$	−0.221659	−0.0206698
$116$	−0.354256	−0.0328918
$117$	1.91469	0.177013
$118$	−11.3467	−1.04455
$119$	−1.42482	−0.130613
$120$	0.0469418	0.00428518
$121$	0	0
$122$	−13.1639	−1.19180
$123$	−0.756046	−0.0681704
$124$	0.540640	0.0485509
$125$	1.32769	0.118752
$126$	4.13707	0.368559
$127$	15.4560	1.37150	0.685749	−	0.727838i	$-0.259475\pi$
$127$	15.4560	1.37150	0.685749	+	0.727838i	$0.259475\pi$
$128$	−10.5961	−0.936576
$129$	0.127342	0.0112118
$130$	−0.118241	−0.0103704
$131$	17.6675	1.54362	0.771810	−	0.635854i	$-0.219352\pi$
$131$	17.6675	1.54362	0.771810	+	0.635854i	$0.219352\pi$
$132$	0	0
$133$	7.18297	0.622843
$134$	−16.6866	−1.44150
$135$	−0.0974972	−0.00839123
$136$	−4.10574	−0.352064
$137$	19.0323	1.62604	0.813021	−	0.582235i	$-0.197822\pi$
$137$	19.0323	1.62604	0.813021	+	0.582235i	$0.197822\pi$
$138$	0.282897	0.0240818
$139$	−14.0665	−1.19310	−0.596552	−	0.802574i	$-0.703463\pi$
$139$	−14.0665	−1.19310	−0.596552	+	0.802574i	$0.703463\pi$
$140$	0.0105254	0.000889557 0
$141$	−1.16225	−0.0978791
$142$	−6.69994	−0.562247
$143$	0	0
$144$	11.4489	0.954072
$145$	−0.595404	−0.0494456
$146$	12.2760	1.01597
$147$	−0.122479	−0.0101019
$148$	−0.298234	−0.0245147
$149$	−11.0367	−0.904160	−0.452080	−	0.891978i	$-0.649318\pi$
$149$	−11.0367	−0.904160	−0.452080	+	0.891978i	$0.649318\pi$
$150$	−0.845748	−0.0690551
$151$	−16.1350	−1.31305	−0.656523	−	0.754306i	$-0.727974\pi$
$151$	−16.1350	−1.31305	−0.656523	+	0.754306i	$0.727974\pi$
$152$	20.6983	1.67886
$153$	4.25309	0.343842
$154$	0	0
$155$	0.908663	0.0729856
$156$	−0.00621709	−0.000497765 0
$157$	−14.8699	−1.18674	−0.593372	−	0.804928i	$-0.702204\pi$
$157$	−14.8699	−1.18674	−0.593372	+	0.804928i	$0.702204\pi$
$158$	−15.9255	−1.26697
$159$	0.0816383	0.00647434
$160$	0.0595051	0.00470429
$161$	1.66655	0.131342
$162$	−12.2868	−0.965340
$163$	−16.4539	−1.28877	−0.644385	−	0.764701i	$-0.722887\pi$
$163$	−16.4539	−1.28877	−0.644385	+	0.764701i	$0.722887\pi$
$164$	−0.488493	−0.0381449
$165$	0	0
$166$	−13.2774	−1.03053
$167$	−19.1519	−1.48201	−0.741007	−	0.671497i	$-0.765652\pi$
$167$	−19.1519	−1.48201	−0.741007	+	0.671497i	$0.765652\pi$
$168$	−0.352934	−0.0272294
$169$	−12.5886	−0.968351
$170$	−0.262648	−0.0201442
$171$	−21.4412	−1.63965
$172$	0.0822773	0.00627358
$173$	15.2246	1.15750	0.578752	−	0.815504i	$-0.303540\pi$
$173$	15.2246	1.15750	0.578752	+	0.815504i	$0.303540\pi$
$174$	0.759898	0.0576077
$175$	−4.98231	−0.376627
$176$	0	0
$177$	−1.00273	−0.0753698
$178$	24.5315	1.83872
$179$	1.12862	0.0843574	0.0421787	−	0.999110i	$-0.486570\pi$
$179$	1.12862	0.0843574	0.0421787	+	0.999110i	$0.486570\pi$
$180$	−0.0314183	−0.00234178
$181$	18.4775	1.37342	0.686711	−	0.726931i	$-0.259054\pi$
$181$	18.4775	1.37342	0.686711	+	0.726931i	$0.259054\pi$
$182$	0.889000	0.0658971
$183$	−1.16331	−0.0859947
$184$	4.80230	0.354030
$185$	−0.501246	−0.0368524
$186$	−1.15970	−0.0850336
$187$	0	0
$188$	−0.750947	−0.0547684
$189$	0.733037	0.0533206
$190$	1.32409	0.0960599
$191$	−2.42981	−0.175815	−0.0879076	−	0.996129i	$-0.528018\pi$
$191$	−2.42981	−0.175815	−0.0879076	+	0.996129i	$0.528018\pi$
$192$	−1.01547	−0.0732855
$193$	−0.263324	−0.0189545	−0.00947725	−	0.999955i	$-0.503017\pi$
$193$	−0.263324	−0.0189545	−0.00947725	+	0.999955i	$0.503017\pi$
$194$	−9.12149	−0.654885
$195$	−0.0104492	−0.000748280 0
$196$	−0.0791355	−0.00565254
$197$	17.4681	1.24455	0.622276	−	0.782798i	$-0.286208\pi$
$197$	17.4681	1.24455	0.622276	+	0.782798i	$0.286208\pi$
$198$	0	0
$199$	−20.1415	−1.42780	−0.713898	−	0.700250i	$-0.753072\pi$
$199$	−20.1415	−1.42780	−0.713898	+	0.700250i	$0.753072\pi$
$200$	−14.3569	−1.01519
$201$	−1.47462	−0.104012
$202$	−25.8070	−1.81577
$203$	4.47657	0.314193
$204$	−0.0138100	−0.000966893 0
$205$	−0.821018	−0.0573424
$206$	9.59572	0.668566
$207$	−4.97465	−0.345762
$208$	2.46021	0.170585
$209$	0	0
$210$	−0.0225775	−0.00155800
$211$	−14.0802	−0.969320	−0.484660	−	0.874702i	$-0.661057\pi$
$211$	−14.0802	−0.969320	−0.484660	+	0.874702i	$0.661057\pi$
$212$	0.0527477	0.00362273
$213$	−0.592086	−0.0405690
$214$	7.54576	0.515817
$215$	0.138285	0.00943095
$216$	2.11231	0.143724
$217$	−6.83182	−0.463774
$218$	12.7829	0.865765
$219$	1.08486	0.0733078
$220$	0	0
$221$	0.913931	0.0614777
$222$	0.639728	0.0429357
$223$	−9.63364	−0.645116	−0.322558	−	0.946550i	$-0.604543\pi$
$223$	−9.63364	−0.645116	−0.322558	+	0.946550i	$0.604543\pi$
$224$	−0.447392	−0.0298926
$225$	14.8722	0.991479
$226$	−13.0033	−0.864969
$227$	−10.4092	−0.690880	−0.345440	−	0.938441i	$-0.612270\pi$
$227$	−10.4092	−0.690880	−0.345440	+	0.938441i	$0.612270\pi$
$228$	0.0696206	0.00461074
$229$	−9.32529	−0.616232	−0.308116	−	0.951349i	$-0.599699\pi$
$229$	−9.32529	−0.616232	−0.308116	+	0.951349i	$0.599699\pi$
$230$	0.307208	0.0202567
$231$	0	0
$232$	12.8996	0.846900
$233$	16.9560	1.11082	0.555412	−	0.831575i	$-0.312560\pi$
$233$	16.9560	1.11082	0.555412	+	0.831575i	$0.312560\pi$
$234$	−2.65366	−0.173475
$235$	−1.26213	−0.0823322
$236$	−0.647879	−0.0421733
$237$	−1.40737	−0.0914183
$238$	1.97473	0.128003
$239$	8.27964	0.535565	0.267783	−	0.963479i	$-0.413709\pi$
$239$	8.27964	0.535565	0.267783	+	0.963479i	$0.413709\pi$
$240$	−0.0624808	−0.00403312
$241$	9.31212	0.599846	0.299923	−	0.953963i	$-0.403039\pi$
$241$	9.31212	0.599846	0.299923	+	0.953963i	$0.403039\pi$
$242$	0	0
$243$	−3.28492	−0.210727
$244$	−0.751635	−0.0481185
$245$	−0.133004	−0.00849734
$246$	1.04784	0.0668081
$247$	−4.60742	−0.293163
$248$	−19.6865	−1.25009
$249$	−1.17335	−0.0743579
$250$	−1.84012	−0.116379
$251$	−4.93532	−0.311514	−0.155757	−	0.987795i	$-0.549782\pi$
$251$	−4.93532	−0.311514	−0.155757	+	0.987795i	$0.549782\pi$
$252$	0.236220	0.0148804
$253$	0	0
$254$	−21.4213	−1.34409
$255$	−0.0232107	−0.00145351
$256$	−1.89624	−0.118515
$257$	−16.8159	−1.04895	−0.524474	−	0.851426i	$-0.675738\pi$
$257$	−16.8159	−1.04895	−0.524474	+	0.851426i	$0.675738\pi$
$258$	−0.176489	−0.0109878
$259$	3.76864	0.234172
$260$	−0.00675136	−0.000418702 0
$261$	−13.3625	−0.827121
$262$	−24.4864	−1.51277
$263$	4.11162	0.253533	0.126767	−	0.991933i	$-0.459540\pi$
$263$	4.11162	0.253533	0.126767	+	0.991933i	$0.459540\pi$
$264$	0	0
$265$	0.0886540	0.00544597
$266$	−9.95526	−0.610396
$267$	2.16789	0.132673
$268$	−0.952774	−0.0581999
$269$	−23.9684	−1.46138	−0.730690	−	0.682710i	$-0.760801\pi$
$269$	−23.9684	−1.46138	−0.730690	+	0.682710i	$0.760801\pi$
$270$	0.135127	0.00822354
$271$	−6.00791	−0.364954	−0.182477	−	0.983210i	$-0.558412\pi$
$271$	−6.00791	−0.364954	−0.182477	+	0.983210i	$0.558412\pi$
$272$	5.46485	0.331355
$273$	0.0785625	0.00475482
$274$	−26.3779	−1.59355
$275$	0	0
$276$	0.0161529	0.000972293 0
$277$	−9.16206	−0.550495	−0.275247	−	0.961373i	$-0.588760\pi$
$277$	−9.16206	−0.550495	−0.275247	+	0.961373i	$0.588760\pi$
$278$	19.4955	1.16926
$279$	20.3930	1.22090
$280$	−0.383263	−0.0229044
$281$	12.9579	0.773006	0.386503	−	0.922288i	$-0.373683\pi$
$281$	12.9579	0.773006	0.386503	+	0.922288i	$0.373683\pi$
$282$	1.61082	0.0959231
$283$	−3.78781	−0.225162	−0.112581	−	0.993643i	$-0.535912\pi$
$283$	−3.78781	−0.225162	−0.112581	+	0.993643i	$0.535912\pi$
$284$	−0.382555	−0.0227005
$285$	0.117013	0.00693122
$286$	0	0
$287$	6.17286	0.364372
$288$	1.33546	0.0786930
$289$	−14.9699	−0.880582
$290$	0.825201	0.0484575
$291$	−0.806083	−0.0472534
$292$	0.700941	0.0410195
$293$	21.3690	1.24839	0.624195	−	0.781269i	$-0.285427\pi$
$293$	21.3690	1.24839	0.624195	+	0.781269i	$0.285427\pi$
$294$	0.169750	0.00990003
$295$	−1.08890	−0.0633983
$296$	10.8597	0.631205
$297$	0	0
$298$	15.2963	0.886091
$299$	−1.06898	−0.0618210
$300$	−0.0482908	−0.00278807
$301$	−1.03970	−0.0599274
$302$	22.3623	1.28681
$303$	−2.28061	−0.131018
$304$	−27.5500	−1.58010
$305$	−1.26329	−0.0723355
$306$	−5.89458	−0.336971
$307$	−8.44677	−0.482082	−0.241041	−	0.970515i	$-0.577489\pi$
$307$	−8.44677	−0.482082	−0.241041	+	0.970515i	$0.577489\pi$
$308$	0	0
$309$	0.847991	0.0482405
$310$	−1.25936	−0.0715271
$311$	18.0190	1.02176	0.510882	−	0.859651i	$-0.329319\pi$
$311$	18.0190	1.02176	0.510882	+	0.859651i	$0.329319\pi$
$312$	0.226384	0.0128165
$313$	0.236340	0.0133587	0.00667937	−	0.999978i	$-0.497874\pi$
$313$	0.236340	0.0133587	0.00667937	+	0.999978i	$0.497874\pi$
$314$	20.6089	1.16303
$315$	0.397018	0.0223694
$316$	−0.909320	−0.0511532
$317$	17.0931	0.960044	0.480022	−	0.877256i	$-0.340629\pi$
$317$	17.0931	0.960044	0.480022	+	0.877256i	$0.340629\pi$
$318$	−0.113147	−0.00634496
$319$	0	0
$320$	−1.10274	−0.0616450
$321$	0.666832	0.0372189
$322$	−2.30976	−0.128718
$323$	−10.2344	−0.569460
$324$	−0.701554	−0.0389752
$325$	3.19583	0.177273
$326$	22.8043	1.26302
$327$	1.12964	0.0624695
$328$	17.7876	0.982156
$329$	9.48937	0.523166
$330$	0	0
$331$	4.41186	0.242498	0.121249	−	0.992622i	$-0.461310\pi$
$331$	4.41186	0.242498	0.121249	+	0.992622i	$0.461310\pi$
$332$	−0.758117	−0.0416071
$333$	−11.2494	−0.616463
$334$	26.5436	1.45240
$335$	−1.60134	−0.0874908
$336$	0.469764	0.0256278
$337$	27.7879	1.51370	0.756852	−	0.653586i	$-0.226736\pi$
$337$	27.7879	1.51370	0.756852	+	0.653586i	$0.226736\pi$
$338$	17.4471	0.949000
$339$	−1.14913	−0.0624120
$340$	−0.0149968	−0.000813315 0
$341$	0	0
$342$	29.7164	1.60688
$343$	1.00000	0.0539949
$344$	−2.99598	−0.161533
$345$	0.0271485	0.00146163
$346$	−21.1006	−1.13437
$347$	31.2515	1.67767	0.838834	−	0.544388i	$-0.183238\pi$
$347$	31.2515	1.67767	0.838834	+	0.544388i	$0.183238\pi$
$348$	0.0433889	0.00232589
$349$	−0.543438	−0.0290896	−0.0145448	−	0.999894i	$-0.504630\pi$
$349$	−0.543438	−0.0290896	−0.0145448	+	0.999894i	$0.504630\pi$
$350$	6.90524	0.369101
$351$	−0.470197	−0.0250972
$352$	0	0
$353$	−17.8517	−0.950148	−0.475074	−	0.879946i	$-0.657579\pi$
$353$	−17.8517	−0.950148	−0.475074	+	0.879946i	$0.657579\pi$
$354$	1.38974	0.0738637
$355$	−0.642967	−0.0341252
$356$	1.40071	0.0742374
$357$	0.174511	0.00923609
$358$	−1.56422	−0.0826716
$359$	−11.9909	−0.632854	−0.316427	−	0.948617i	$-0.602483\pi$
$359$	−11.9909	−0.632854	−0.316427	+	0.948617i	$0.602483\pi$
$360$	1.14404	0.0602963
$361$	32.5951	1.71553
$362$	−25.6089	−1.34598
$363$	0	0
$364$	0.0507604	0.00266057
$365$	1.17808	0.0616637
$366$	1.61230	0.0842762
$367$	29.5958	1.54489	0.772444	−	0.635083i	$-0.219034\pi$
$367$	29.5958	1.54489	0.772444	+	0.635083i	$0.219034\pi$
$368$	−6.39199	−0.333206
$369$	−18.4260	−0.959218
$370$	0.694704	0.0361159
$371$	−0.666549	−0.0346055
$372$	−0.0662171	−0.00343320
$373$	10.5209	0.544753	0.272377	−	0.962191i	$-0.412190\pi$
$373$	10.5209	0.544753	0.272377	+	0.962191i	$0.412190\pi$
$374$	0	0
$375$	−0.162615	−0.00839738
$376$	27.3444	1.41018
$377$	−2.87143	−0.147886
$378$	−1.01595	−0.0522551
$379$	−8.34913	−0.428866	−0.214433	−	0.976739i	$-0.568790\pi$
$379$	−8.34913	−0.428866	−0.214433	+	0.976739i	$0.568790\pi$
$380$	0.0756035	0.00387838
$381$	−1.89304	−0.0969832
$382$	3.36761	0.172302
$383$	−18.4737	−0.943965	−0.471982	−	0.881608i	$-0.656461\pi$
$383$	−18.4737	−0.943965	−0.471982	+	0.881608i	$0.656461\pi$
$384$	1.29781	0.0662284
$385$	0	0
$386$	0.364955	0.0185757
$387$	3.10351	0.157760
$388$	−0.520822	−0.0264407
$389$	23.2560	1.17913	0.589564	−	0.807722i	$-0.299300\pi$
$389$	23.2560	1.17913	0.589564	+	0.807722i	$0.299300\pi$
$390$	0.0144821	0.000733327 0
$391$	−2.37453	−0.120085
$392$	2.88158	0.145542
$393$	−2.16390	−0.109154
$394$	−24.2100	−1.21968
$395$	−1.52831	−0.0768976
$396$	0	0
$397$	21.6794	1.08806	0.544029	−	0.839066i	$-0.316898\pi$
$397$	21.6794	1.08806	0.544029	+	0.839066i	$0.316898\pi$
$398$	27.9152	1.39926
$399$	−0.879764	−0.0440433
$400$	19.1095	0.955474
$401$	−34.8075	−1.73821	−0.869103	−	0.494631i	$-0.835303\pi$
$401$	−34.8075	−1.73821	−0.869103	+	0.494631i	$0.835303\pi$
$402$	2.04375	0.101933
$403$	4.38218	0.218292
$404$	−1.47354	−0.0733112
$405$	−1.17911	−0.0585906
$406$	−6.20431	−0.307915
$407$	0	0
$408$	0.502867	0.0248956
$409$	16.2927	0.805620	0.402810	−	0.915284i	$-0.368033\pi$
$409$	16.2927	0.805620	0.402810	+	0.915284i	$0.368033\pi$
$410$	1.13789	0.0561965
$411$	−2.33106	−0.114983
$412$	0.547900	0.0269931
$413$	8.18695	0.402853
$414$	6.89463	0.338852
$415$	−1.27418	−0.0625471
$416$	0.286973	0.0140700
$417$	1.72285	0.0843684
$418$	0	0
$419$	−22.6536	−1.10670	−0.553350	−	0.832949i	$-0.686651\pi$
$419$	−22.6536	−1.10670	−0.553350	+	0.832949i	$0.686651\pi$
$420$	−0.00128914	−6.29036e−5 0
$421$	−15.2061	−0.741102	−0.370551	−	0.928812i	$-0.620831\pi$
$421$	−15.2061	−0.741102	−0.370551	+	0.928812i	$0.620831\pi$
$422$	19.5145	0.949950
$423$	−28.3258	−1.37725
$424$	−1.92072	−0.0932783
$425$	7.09890	0.344347
$426$	0.820603	0.0397583
$427$	9.49807	0.459644
$428$	0.430850	0.0208259
$429$	0	0
$430$	−0.191656	−0.00924249
$431$	−3.20691	−0.154471	−0.0772356	−	0.997013i	$-0.524609\pi$
$431$	−3.20691	−0.154471	−0.0772356	+	0.997013i	$0.524609\pi$
$432$	−2.81154	−0.135270
$433$	5.99246	0.287979	0.143990	−	0.989579i	$-0.454007\pi$
$433$	5.99246	0.287979	0.143990	+	0.989579i	$0.454007\pi$
$434$	9.46858	0.454507
$435$	0.0729245	0.00349646
$436$	0.729880	0.0349549
$437$	11.9708	0.572640
$438$	−1.50356	−0.0718428
$439$	−7.68712	−0.366886	−0.183443	−	0.983030i	$-0.558724\pi$
$439$	−7.68712	−0.366886	−0.183443	+	0.983030i	$0.558724\pi$
$440$	0	0
$441$	−2.98500	−0.142143
$442$	−1.26667	−0.0602491
$443$	37.2499	1.76980	0.884898	−	0.465784i	$-0.154228\pi$
$443$	37.2499	1.76980	0.884898	+	0.465784i	$0.154228\pi$
$444$	0.0365274	0.00173351
$445$	2.35419	0.111600
$446$	13.3518	0.632224
$447$	1.35176	0.0639361
$448$	8.29100	0.391713
$449$	11.0724	0.522539	0.261269	−	0.965266i	$-0.415859\pi$
$449$	11.0724	0.522539	0.261269	+	0.965266i	$0.415859\pi$
$450$	−20.6121	−0.971666
$451$	0	0
$452$	−0.742468	−0.0349228
$453$	1.97620	0.0928499
$454$	14.4266	0.677074
$455$	0.0853139	0.00399958
$456$	−2.53511	−0.118718
$457$	28.3961	1.32831	0.664157	−	0.747593i	$-0.268791\pi$
$457$	28.3961	1.32831	0.664157	+	0.747593i	$0.268791\pi$
$458$	12.9244	0.603918
$459$	−1.04445	−0.0487506
$460$	0.0175411	0.000817856 0
$461$	−25.7420	−1.19892	−0.599462	−	0.800403i	$-0.704619\pi$
$461$	−25.7420	−1.19892	−0.599462	+	0.800403i	$0.704619\pi$
$462$	0	0
$463$	37.6543	1.74995	0.874973	−	0.484172i	$-0.160879\pi$
$463$	37.6543	1.74995	0.874973	+	0.484172i	$0.160879\pi$
$464$	−17.1697	−0.797084
$465$	−0.111292	−0.00516105
$466$	−23.5002	−1.08863
$467$	−15.3875	−0.712046	−0.356023	−	0.934477i	$-0.615868\pi$
$467$	−15.3875	−0.712046	−0.356023	+	0.934477i	$0.615868\pi$
$468$	−0.151520	−0.00700400
$469$	12.0398	0.555945
$470$	1.74925	0.0806869
$471$	1.82125	0.0839187
$472$	23.5914	1.08588
$473$	0	0
$474$	1.95054	0.0895914
$475$	−35.7878	−1.64206
$476$	0.112754	0.00516807
$477$	1.98965	0.0910998
$478$	−11.4752	−0.524863
$479$	−0.600178	−0.0274228	−0.0137114	−	0.999906i	$-0.504365\pi$
$479$	−0.600178	−0.0274228	−0.0137114	+	0.999906i	$0.504365\pi$
$480$	−0.00728813	−0.000332656 0
$481$	−2.41734	−0.110221
$482$	−12.9062	−0.587859
$483$	−0.204117	−0.00928767
$484$	0	0
$485$	−0.875354	−0.0397478
$486$	4.55274	0.206516
$487$	10.2991	0.466698	0.233349	−	0.972393i	$-0.425032\pi$
$487$	10.2991	0.466698	0.233349	+	0.972393i	$0.425032\pi$
$488$	27.3695	1.23896
$489$	2.01526	0.0911333
$490$	0.184338	0.00832754
$491$	−10.0925	−0.455466	−0.227733	−	0.973724i	$-0.573131\pi$
$491$	−10.0925	−0.455466	−0.227733	+	0.973724i	$0.573131\pi$
$492$	0.0598301	0.00269735
$493$	−6.37830	−0.287264
$494$	6.38566	0.287304
$495$	0	0
$496$	26.2032	1.17656
$497$	4.83418	0.216842
$498$	1.62620	0.0728720
$499$	19.7955	0.886170	0.443085	−	0.896480i	$-0.353884\pi$
$499$	19.7955	0.886170	0.443085	+	0.896480i	$0.353884\pi$
$500$	−0.105068	−0.00469877
$501$	2.34570	0.104798
$502$	6.84012	0.305289
$503$	−14.3034	−0.637757	−0.318879	−	0.947796i	$-0.603306\pi$
$503$	−14.3034	−0.637757	−0.318879	+	0.947796i	$0.603306\pi$
$504$	−8.60152	−0.383142
$505$	−2.47660	−0.110207
$506$	0	0
$507$	1.54184	0.0684753
$508$	−1.22312	−0.0542671
$509$	−42.0975	−1.86594	−0.932970	−	0.359953i	$-0.882793\pi$
$509$	−42.0975	−1.86594	−0.932970	+	0.359953i	$0.882793\pi$
$510$	0.0321689	0.00142446
$511$	−8.85748	−0.391832
$512$	23.8204	1.05272
$513$	5.26539	0.232472
$514$	23.3061	1.02799
$515$	0.920864	0.0405781
$516$	−0.0100772	−0.000443626 0
$517$	0	0
$518$	−5.22316	−0.229492
$519$	−1.86469	−0.0818510
$520$	0.245839	0.0107808
$521$	−21.4472	−0.939619	−0.469810	−	0.882768i	$-0.655677\pi$
$521$	−21.4472	−0.939619	−0.469810	+	0.882768i	$0.655677\pi$
$522$	18.5199	0.810592
$523$	−35.5946	−1.55644	−0.778222	−	0.627989i	$-0.783878\pi$
$523$	−35.5946	−1.55644	−0.778222	+	0.627989i	$0.783878\pi$
$524$	−1.39813	−0.0610776
$525$	0.610229	0.0266326
$526$	−5.69851	−0.248467
$527$	9.73412	0.424025
$528$	0	0
$529$	−20.2226	−0.879244
$530$	−0.122870	−0.00533714
$531$	−24.4380	−1.06052
$532$	−0.568428	−0.0246445
$533$	−3.95949	−0.171505
$534$	−3.00460	−0.130022
$535$	0.724137	0.0313072
$536$	34.6936	1.49854
$537$	−0.138233	−0.00596519
$538$	33.2191	1.43218
$539$	0	0
$540$	0.00771550	0.000332022 0
$541$	16.9350	0.728092	0.364046	−	0.931381i	$-0.381395\pi$
$541$	16.9350	0.728092	0.364046	+	0.931381i	$0.381395\pi$
$542$	8.32667	0.357661
$543$	−2.26311	−0.0971192
$544$	0.637453	0.0273306
$545$	1.22672	0.0525470
$546$	−0.108884	−0.00465980
$547$	−8.79082	−0.375868	−0.187934	−	0.982182i	$-0.560179\pi$
$547$	−8.79082	−0.375868	−0.187934	+	0.982182i	$0.560179\pi$
$548$	−1.50613	−0.0643388
$549$	−28.3517	−1.21002
$550$	0	0
$551$	32.1551	1.36985
$552$	−0.588181	−0.0250347
$553$	11.4907	0.488633
$554$	12.6982	0.539494
$555$	0.0613922	0.00260595
$556$	1.11316	0.0472085
$557$	21.7276	0.920629	0.460315	−	0.887756i	$-0.347737\pi$
$557$	21.7276	0.920629	0.460315	+	0.887756i	$0.347737\pi$
$558$	−28.2637	−1.19650
$559$	0.666902	0.0282069
$560$	0.510134	0.0215571
$561$	0	0
$562$	−17.9591	−0.757559
$563$	−1.82134	−0.0767605	−0.0383803	−	0.999263i	$-0.512220\pi$
$563$	−1.82134	−0.0767605	−0.0383803	+	0.999263i	$0.512220\pi$
$564$	0.0919753	0.00387286
$565$	−1.24788	−0.0524987
$566$	5.24972	0.220662
$567$	8.86521	0.372304
$568$	13.9301	0.584493
$569$	32.9418	1.38099	0.690496	−	0.723336i	$-0.257392\pi$
$569$	32.9418	1.38099	0.690496	+	0.723336i	$0.257392\pi$
$570$	−0.162174	−0.00679271
$571$	−23.4394	−0.980908	−0.490454	−	0.871467i	$-0.663169\pi$
$571$	−23.4394	−0.980908	−0.490454	+	0.871467i	$0.663169\pi$
$572$	0	0
$573$	0.297601	0.0124325
$574$	−8.55529	−0.357091
$575$	−8.30326	−0.346270
$576$	−24.7486	−1.03119
$577$	34.4539	1.43433	0.717166	−	0.696902i	$-0.245439\pi$
$577$	34.4539	1.43433	0.717166	+	0.696902i	$0.245439\pi$
$578$	20.7476	0.862984
$579$	0.0322517	0.00134034
$580$	0.0471176	0.00195645
$581$	9.57998	0.397445
$582$	1.11719	0.0463091
$583$	0	0
$584$	−25.5236	−1.05617
$585$	−0.254662	−0.0105290
$586$	−29.6164	−1.22344
$587$	−20.8670	−0.861274	−0.430637	−	0.902525i	$-0.641711\pi$
$587$	−20.8670	−0.861274	−0.430637	+	0.902525i	$0.641711\pi$
$588$	0.00969245	0.000399710 0
$589$	−49.0728	−2.02201
$590$	1.50916	0.0621313
$591$	−2.13948	−0.0880064
$592$	−14.4545	−0.594076
$593$	26.6381	1.09389	0.546947	−	0.837167i	$-0.315790\pi$
$593$	26.6381	1.09389	0.546947	+	0.837167i	$0.315790\pi$
$594$	0	0
$595$	0.189507	0.00776905
$596$	0.873393	0.0357756
$597$	2.46692	0.100964
$598$	1.48156	0.0605856
$599$	−20.6712	−0.844602	−0.422301	−	0.906456i	$-0.638777\pi$
$599$	−20.6712	−0.844602	−0.422301	+	0.906456i	$0.638777\pi$
$600$	1.75843	0.0717874
$601$	−4.94812	−0.201838	−0.100919	−	0.994895i	$-0.532178\pi$
$601$	−4.94812	−0.201838	−0.100919	+	0.994895i	$0.532178\pi$
$602$	1.44098	0.0587298
$603$	−35.9387	−1.46354
$604$	1.27685	0.0519543
$605$	0	0
$606$	3.16082	0.128399
$607$	−27.6086	−1.12060	−0.560300	−	0.828290i	$-0.689314\pi$
$607$	−27.6086	−1.12060	−0.560300	+	0.828290i	$0.689314\pi$
$608$	−3.21360	−0.130329
$609$	−0.548286	−0.0222177
$610$	1.75085	0.0708900
$611$	−6.08683	−0.246247
$612$	−0.336570	−0.0136051
$613$	7.18919	0.290368	0.145184	−	0.989405i	$-0.453623\pi$
$613$	7.18919	0.290368	0.145184	+	0.989405i	$0.453623\pi$
$614$	11.7068	0.472449
$615$	0.100558	0.00405487
$616$	0	0
$617$	21.1215	0.850320	0.425160	−	0.905118i	$-0.360218\pi$
$617$	21.1215	0.850320	0.425160	+	0.905118i	$0.360218\pi$
$618$	−1.17528	−0.0472765
$619$	2.77412	0.111501	0.0557506	−	0.998445i	$-0.482245\pi$
$619$	2.77412	0.111501	0.0557506	+	0.998445i	$0.482245\pi$
$620$	−0.0719076	−0.00288788
$621$	1.22164	0.0490228
$622$	−24.9735	−1.00134
$623$	−17.7001	−0.709140
$624$	−0.301324	−0.0120626
$625$	24.7350	0.989398
$626$	−0.327556	−0.0130918
$627$	0	0
$628$	1.17673	0.0469568
$629$	−5.36964	−0.214102
$630$	−0.550248	−0.0219224
$631$	−24.2090	−0.963743	−0.481872	−	0.876242i	$-0.660043\pi$
$631$	−24.2090	−0.963743	−0.481872	+	0.876242i	$0.660043\pi$
$632$	33.1113	1.31710
$633$	1.72453	0.0685439
$634$	−23.6902	−0.940859
$635$	−2.05572	−0.0815786
$636$	−0.00646049	−0.000256175 0
$637$	−0.641436	−0.0254146
$638$	0	0
$639$	−14.4300	−0.570843
$640$	1.40933	0.0557088
$641$	43.4897	1.71774	0.858871	−	0.512192i	$-0.171166\pi$
$641$	43.4897	1.71774	0.858871	+	0.512192i	$0.171166\pi$
$642$	−0.924198	−0.0364752
$643$	−17.1517	−0.676397	−0.338199	−	0.941075i	$-0.609818\pi$
$643$	−17.1517	−0.676397	−0.338199	+	0.941075i	$0.609818\pi$
$644$	−0.131883	−0.00519693
$645$	−0.0169370	−0.000666894 0
$646$	14.1845	0.558080
$647$	14.2909	0.561833	0.280916	−	0.959732i	$-0.409362\pi$
$647$	14.2909	0.561833	0.280916	+	0.959732i	$0.409362\pi$
$648$	25.5459	1.00354
$649$	0	0
$650$	−4.42927	−0.173730
$651$	0.836756	0.0327950
$652$	1.30209	0.0509938
$653$	−37.8518	−1.48126	−0.740628	−	0.671915i	$-0.765472\pi$
$653$	−37.8518	−1.48126	−0.740628	+	0.671915i	$0.765472\pi$
$654$	−1.56563	−0.0612211
$655$	−2.34986	−0.0918166
$656$	−23.6758	−0.924384
$657$	26.4396	1.03151
$658$	−13.1518	−0.512711
$659$	8.13829	0.317023	0.158511	−	0.987357i	$-0.449331\pi$
$659$	8.13829	0.317023	0.158511	+	0.987357i	$0.449331\pi$
$660$	0	0
$661$	−5.33161	−0.207375	−0.103688	−	0.994610i	$-0.533064\pi$
$661$	−5.33161	−0.207375	−0.103688	+	0.994610i	$0.533064\pi$
$662$	−6.11463	−0.237652
$663$	−0.111937	−0.00434729
$664$	27.6055	1.07130
$665$	−0.955367	−0.0370476
$666$	15.5911	0.604144
$667$	7.46042	0.288869
$668$	1.51559	0.0586400
$669$	1.17992	0.0456183
$670$	2.21939	0.0857424
$671$	0	0
$672$	0.0547962	0.00211381
$673$	−19.4495	−0.749724	−0.374862	−	0.927081i	$-0.622310\pi$
$673$	−19.4495	−0.749724	−0.374862	+	0.927081i	$0.622310\pi$
$674$	−38.5127	−1.48345
$675$	−3.65222	−0.140574
$676$	0.996203	0.0383155
$677$	−36.3042	−1.39528	−0.697642	−	0.716446i	$-0.745768\pi$
$677$	−36.3042	−1.39528	−0.697642	+	0.716446i	$0.745768\pi$
$678$	1.59264	0.0611648
$679$	6.58139	0.252571
$680$	0.546082	0.0209413
$681$	1.27490	0.0488545
$682$	0	0
$683$	−20.7805	−0.795142	−0.397571	−	0.917571i	$-0.630147\pi$
$683$	−20.7805	−0.795142	−0.397571	+	0.917571i	$0.630147\pi$
$684$	1.69676	0.0648772
$685$	−2.53138	−0.0967192
$686$	−1.38595	−0.0529159
$687$	1.14215	0.0435758
$688$	3.98774	0.152031
$689$	0.427549	0.0162883
$690$	−0.0376266	−0.00143242
$691$	−50.1050	−1.90608	−0.953042	−	0.302838i	$-0.902066\pi$
$691$	−50.1050	−1.90608	−0.953042	+	0.302838i	$0.902066\pi$
$692$	−1.20481	−0.0457998
$693$	0	0
$694$	−43.3131	−1.64414
$695$	1.87091	0.0709675
$696$	−1.57993	−0.0598871
$697$	−8.79521	−0.333143
$698$	0.753179	0.0285083
$699$	−2.07676	−0.0785501
$700$	0.394278	0.0149023
$701$	−16.3229	−0.616508	−0.308254	−	0.951304i	$-0.599745\pi$
$701$	−16.3229	−0.616508	−0.308254	+	0.951304i	$0.599745\pi$
$702$	0.651670	0.0245957
$703$	27.0701	1.02097
$704$	0	0
$705$	0.154584	0.00582199
$706$	24.7416	0.931161
$707$	18.6204	0.700292
$708$	0.0793516	0.00298222
$709$	33.2299	1.24798	0.623988	−	0.781434i	$-0.285511\pi$
$709$	33.2299	1.24798	0.623988	+	0.781434i	$0.285511\pi$
$710$	0.891122	0.0334432
$711$	−34.2996	−1.28634
$712$	−51.0044	−1.91147
$713$	−11.3856	−0.426393
$714$	−0.241864	−0.00905152
$715$	0	0
$716$	−0.0893143	−0.00333783
$717$	−1.01408	−0.0378716
$718$	16.6188	0.620207
$719$	−25.2455	−0.941498	−0.470749	−	0.882267i	$-0.656016\pi$
$719$	−25.2455	−0.941498	−0.470749	+	0.882267i	$0.656016\pi$
$720$	−1.52275	−0.0567495
$721$	−6.92356	−0.257847
$722$	−45.1752	−1.68125
$723$	−1.14054	−0.0424171
$724$	−1.46223	−0.0543432
$725$	−22.3036	−0.828337
$726$	0	0
$727$	1.86242	0.0690733	0.0345366	−	0.999403i	$-0.489004\pi$
$727$	1.86242	0.0690733	0.0345366	+	0.999403i	$0.489004\pi$
$728$	−1.84835	−0.0685045
$729$	−26.1933	−0.970123
$730$	−1.63277	−0.0604315
$731$	1.48139	0.0547911
$732$	0.0920595	0.00340262
$733$	20.0651	0.741123	0.370561	−	0.928808i	$-0.379165\pi$
$733$	20.0651	0.741123	0.370561	+	0.928808i	$0.379165\pi$
$734$	−41.0184	−1.51402
$735$	0.0162903	0.000600876 0
$736$	−0.745601	−0.0274832
$737$	0	0
$738$	25.5375	0.940049
$739$	−22.3970	−0.823886	−0.411943	−	0.911210i	$-0.635150\pi$
$739$	−22.3970	−0.823886	−0.411943	+	0.911210i	$0.635150\pi$
$740$	0.0396664	0.00145817
$741$	0.564312	0.0207305
$742$	0.923806	0.0339140
$743$	−36.2705	−1.33063	−0.665317	−	0.746561i	$-0.731704\pi$
$743$	−36.2705	−1.33063	−0.665317	+	0.746561i	$0.731704\pi$
$744$	2.41118	0.0883982
$745$	1.46793	0.0537807
$746$	−14.5815	−0.533867
$747$	−28.5962	−1.04628
$748$	0	0
$749$	−5.44446	−0.198936
$750$	0.225376	0.00822957
$751$	44.6237	1.62834	0.814170	−	0.580626i	$-0.197192\pi$
$751$	44.6237	1.62834	0.814170	+	0.580626i	$0.197192\pi$
$752$	−36.3962	−1.32723
$753$	0.604473	0.0220282
$754$	3.97967	0.144931
$755$	2.14603	0.0781019
$756$	−0.0580093	−0.00210978
$757$	8.99964	0.327097	0.163549	−	0.986535i	$-0.447706\pi$
$757$	8.99964	0.327097	0.163549	+	0.986535i	$0.447706\pi$
$758$	11.5715	0.420296
$759$	0	0
$760$	−2.75297	−0.0998607
$761$	5.15155	0.186743	0.0933717	−	0.995631i	$-0.470235\pi$
$761$	5.15155	0.186743	0.0933717	+	0.995631i	$0.470235\pi$
$762$	2.62366	0.0950451
$763$	−9.22316	−0.333901
$764$	0.192285	0.00695661
$765$	−0.565680	−0.0204522
$766$	25.6037	0.925101
$767$	−5.25140	−0.189617
$768$	0.232250	0.00838059
$769$	−38.5242	−1.38922	−0.694608	−	0.719388i	$-0.744422\pi$
$769$	−38.5242	−1.38922	−0.694608	+	0.719388i	$0.744422\pi$
$770$	0	0
$771$	2.05960	0.0741746
$772$	0.0208383	0.000749987 0
$773$	−6.99024	−0.251421	−0.125711	−	0.992067i	$-0.540121\pi$
$773$	−6.99024	−0.251421	−0.125711	+	0.992067i	$0.540121\pi$
$774$	−4.30131	−0.154607
$775$	34.0383	1.22269
$776$	18.9648	0.680797
$777$	−0.461580	−0.0165591
$778$	−32.2317	−1.15556
$779$	44.3395	1.58863
$780$	0.000826901 0	2.96078e−5 0
$781$	0	0
$782$	3.29099	0.117686
$783$	3.28149	0.117271
$784$	−3.83547	−0.136981
$785$	1.97776	0.0705892
$786$	2.99907	0.106973
$787$	31.3859	1.11879	0.559393	−	0.828903i	$-0.311034\pi$
$787$	31.3859	1.11879	0.559393	+	0.828903i	$0.311034\pi$
$788$	−1.38235	−0.0492441
$789$	−0.503587	−0.0179282
$790$	2.11816	0.0753609
$791$	9.38223	0.333594
$792$	0	0
$793$	−6.09240	−0.216348
$794$	−30.0466	−1.06631
$795$	−0.0108583	−0.000385103 0
$796$	1.59391	0.0564947
$797$	26.2972	0.931496	0.465748	−	0.884917i	$-0.345785\pi$
$797$	26.2972	0.931496	0.465748	+	0.884917i	$0.345785\pi$
$798$	1.21931	0.0431631
$799$	−13.5207	−0.478326
$800$	2.22904	0.0788086
$801$	52.8348	1.86683
$802$	48.2416	1.70347
$803$	0	0
$804$	0.116695	0.00411551
$805$	−0.221659	−0.00781244
$806$	−6.07349	−0.213930
$807$	2.93563	0.103339
$808$	53.6562	1.88762
$809$	11.7634	0.413578	0.206789	−	0.978386i	$-0.433699\pi$
$809$	11.7634	0.413578	0.206789	+	0.978386i	$0.433699\pi$
$810$	1.63420	0.0574198
$811$	−8.27304	−0.290506	−0.145253	−	0.989395i	$-0.546400\pi$
$811$	−8.27304	−0.290506	−0.145253	+	0.989395i	$0.546400\pi$
$812$	−0.354256	−0.0124319
$813$	0.735843	0.0258071
$814$	0	0
$815$	2.18844	0.0766579
$816$	−0.669330	−0.0234312
$817$	−7.46814	−0.261277
$818$	−22.5809	−0.789521
$819$	1.91469	0.0669045
$820$	0.0649717	0.00226891
$821$	−7.38513	−0.257743	−0.128871	−	0.991661i	$-0.541135\pi$
$821$	−7.38513	−0.257743	−0.128871	+	0.991661i	$0.541135\pi$
$822$	3.23074	0.112685
$823$	49.4433	1.72349	0.861743	−	0.507345i	$-0.169373\pi$
$823$	49.4433	1.72349	0.861743	+	0.507345i	$0.169373\pi$
$824$	−19.9508	−0.695019
$825$	0	0
$826$	−11.3467	−0.394803
$827$	23.2339	0.807922	0.403961	−	0.914776i	$-0.367633\pi$
$827$	23.2339	0.807922	0.403961	+	0.914776i	$0.367633\pi$
$828$	0.393671	0.0136810
$829$	30.1938	1.04867	0.524336	−	0.851511i	$-0.324314\pi$
$829$	30.1938	1.04867	0.524336	+	0.851511i	$0.324314\pi$
$830$	1.76595	0.0612972
$831$	1.12216	0.0389273
$832$	−5.31814	−0.184374
$833$	−1.42482	−0.0493671
$834$	−2.38779	−0.0826824
$835$	2.54728	0.0881523
$836$	0	0
$837$	−5.00798	−0.173101
$838$	31.3968	1.08458
$839$	16.6125	0.573529	0.286764	−	0.958001i	$-0.407420\pi$
$839$	16.6125	0.573529	0.286764	+	0.958001i	$0.407420\pi$
$840$	0.0469418	0.00161964
$841$	−8.96035	−0.308977
$842$	21.0750	0.726292
$843$	−1.58708	−0.0546619
$844$	1.11424	0.0383539
$845$	1.67433	0.0575989
$846$	39.2582	1.34972
$847$	0	0
$848$	2.55653	0.0877915
$849$	0.463927	0.0159219
$850$	−9.83873	−0.337466
$851$	6.28063	0.215297
$852$	0.0468550	0.00160523
$853$	−40.0507	−1.37131	−0.685654	−	0.727927i	$-0.740484\pi$
$853$	−40.0507	−1.37131	−0.685654	+	0.727927i	$0.740484\pi$
$854$	−13.1639	−0.450458
$855$	2.85177	0.0975285
$856$	−15.6887	−0.536227
$857$	−31.7070	−1.08309	−0.541546	−	0.840671i	$-0.682161\pi$
$857$	−31.7070	−1.08309	−0.541546	+	0.840671i	$0.682161\pi$
$858$	0	0
$859$	1.63654	0.0558381	0.0279190	−	0.999610i	$-0.491112\pi$
$859$	1.63654	0.0558381	0.0279190	+	0.999610i	$0.491112\pi$
$860$	−0.0109432	−0.000373162 0
$861$	−0.756046	−0.0257660
$862$	4.44462	0.151384
$863$	−17.4055	−0.592489	−0.296244	−	0.955112i	$-0.595734\pi$
$863$	−17.4055	−0.592489	−0.296244	+	0.955112i	$0.595734\pi$
$864$	−0.327955	−0.0111573
$865$	−2.02494	−0.0688500
$866$	−8.30526	−0.282224
$867$	1.83350	0.0622689
$868$	0.540640	0.0183505
$869$	0	0
$870$	−0.101070	−0.00342659
$871$	−7.72275	−0.261675
$872$	−26.5773	−0.900021
$873$	−19.6454	−0.664897
$874$	−16.5909	−0.561196
$875$	1.32769	0.0448842
$876$	−0.0858507	−0.00290062
$877$	−48.1667	−1.62647	−0.813237	−	0.581933i	$-0.802297\pi$
$877$	−48.1667	−1.62647	−0.813237	+	0.581933i	$0.802297\pi$
$878$	10.6540	0.359555
$879$	−2.61725	−0.0882778
$880$	0	0
$881$	−30.0141	−1.01120	−0.505601	−	0.862767i	$-0.668729\pi$
$881$	−30.0141	−1.01120	−0.505601	+	0.862767i	$0.668729\pi$
$882$	4.13707	0.139302
$883$	56.3355	1.89584	0.947921	−	0.318506i	$-0.103181\pi$
$883$	56.3355	1.89584	0.947921	+	0.318506i	$0.103181\pi$
$884$	−0.0723244	−0.00243253
$885$	0.133368	0.00448310
$886$	−51.6266	−1.73443
$887$	−4.53095	−0.152135	−0.0760673	−	0.997103i	$-0.524236\pi$
$887$	−4.53095	−0.152135	−0.0760673	+	0.997103i	$0.524236\pi$
$888$	−1.33008	−0.0446346
$889$	15.4560	0.518377
$890$	−3.26280	−0.109369
$891$	0	0
$892$	0.762363	0.0255258
$893$	68.1619	2.28095
$894$	−1.87348	−0.0626585
$895$	−0.150112	−0.00501770
$896$	−10.5961	−0.353992
$897$	0.130928	0.00437157
$898$	−15.3458	−0.512097
$899$	−30.5831	−1.02000
$900$	−1.17692	−0.0392306
$901$	0.949713	0.0316395
$902$	0	0
$903$	0.127342	0.00423766
$904$	27.0357	0.899193
$905$	−2.45759	−0.0816930
$906$	−2.73892	−0.0909944
$907$	−20.7609	−0.689355	−0.344678	−	0.938721i	$-0.612012\pi$
$907$	−20.7609	−0.689355	−0.344678	+	0.938721i	$0.612012\pi$
$908$	0.823735	0.0273366
$909$	−55.5819	−1.84353
$910$	−0.118241	−0.00391965
$911$	−15.3952	−0.510066	−0.255033	−	0.966932i	$-0.582086\pi$
$911$	−15.3952	−0.510066	−0.255033	+	0.966932i	$0.582086\pi$
$912$	3.37430	0.111734
$913$	0	0
$914$	−39.3557	−1.30177
$915$	0.154726	0.00511509
$916$	0.737962	0.0243829
$917$	17.6675	0.583433
$918$	1.44755	0.0477764
$919$	21.3581	0.704540	0.352270	−	0.935898i	$-0.385410\pi$
$919$	21.3581	0.704540	0.352270	+	0.935898i	$0.385410\pi$
$920$	−0.638727	−0.0210582
$921$	1.03455	0.0340897
$922$	35.6772	1.17496
$923$	−3.10082	−0.102065
$924$	0	0
$925$	−18.7765	−0.617369
$926$	−52.1871	−1.71498
$927$	20.6668	0.678787
$928$	−2.00278	−0.0657445
$929$	16.1336	0.529325	0.264662	−	0.964341i	$-0.414739\pi$
$929$	16.1336	0.529325	0.264662	+	0.964341i	$0.414739\pi$
$930$	0.154246	0.00505792
$931$	7.18297	0.235412
$932$	−1.34182	−0.0439529
$933$	−2.20695	−0.0722523
$934$	21.3263	0.697817
$935$	0	0
$936$	5.51733	0.180339
$937$	−4.56306	−0.149069	−0.0745343	−	0.997218i	$-0.523747\pi$
$937$	−4.56306	−0.149069	−0.0745343	+	0.997218i	$0.523747\pi$
$938$	−16.6866	−0.544835
$939$	−0.0289468	−0.000944642 0
$940$	0.0998793	0.00325770
$941$	12.5504	0.409130	0.204565	−	0.978853i	$-0.434422\pi$
$941$	12.5504	0.409130	0.204565	+	0.978853i	$0.434422\pi$
$942$	−2.52416	−0.0822417
$943$	10.2874	0.335003
$944$	−31.4008	−1.02201
$945$	−0.0974972	−0.00317159
$946$	0	0
$947$	51.6790	1.67934	0.839672	−	0.543094i	$-0.182747\pi$
$947$	51.6790	1.67934	0.839672	+	0.543094i	$0.182747\pi$
$948$	0.111373	0.00361722
$949$	5.68151	0.184429
$950$	49.6002	1.60924
$951$	−2.09355	−0.0678879
$952$	−4.10574	−0.133068
$953$	1.32251	0.0428403	0.0214202	−	0.999771i	$-0.493181\pi$
$953$	1.32251	0.0428403	0.0214202	+	0.999771i	$0.493181\pi$
$954$	−2.75756	−0.0892793
$955$	0.323176	0.0104577
$956$	−0.655213	−0.0211911
$957$	0	0
$958$	0.831818	0.0268748
$959$	19.0323	0.614586
$960$	0.135063	0.00435913
$961$	15.6738	0.505607
$962$	3.35032	0.108019
$963$	16.2517	0.523704
$964$	−0.736919	−0.0237346
$965$	0.0350233	0.00112744
$966$	0.282897	0.00910206
$967$	−44.1214	−1.41885	−0.709425	−	0.704781i	$-0.751045\pi$
$967$	−44.1214	−1.41885	−0.709425	+	0.704781i	$0.751045\pi$
$968$	0	0
$969$	1.25351	0.0402684
$970$	1.21320	0.0389535
$971$	−1.31170	−0.0420944	−0.0210472	−	0.999778i	$-0.506700\pi$
$971$	−1.31170	−0.0420944	−0.0210472	+	0.999778i	$0.506700\pi$
$972$	0.259954	0.00833801
$973$	−14.0665	−0.450951
$974$	−14.2741	−0.457371
$975$	−0.391423	−0.0125356
$976$	−36.4295	−1.16608
$977$	−12.4102	−0.397036	−0.198518	−	0.980097i	$-0.563613\pi$
$977$	−12.4102	−0.397036	−0.198518	+	0.980097i	$0.563613\pi$
$978$	−2.79306	−0.0893121
$979$	0	0
$980$	0.0105254	0.000336221 0
$981$	27.5311	0.879001
$982$	13.9877	0.446364
$983$	−31.9565	−1.01925	−0.509627	−	0.860395i	$-0.670217\pi$
$983$	−31.9565	−1.01925	−0.509627	+	0.860395i	$0.670217\pi$
$984$	−2.17861	−0.0694516
$985$	−2.32334	−0.0740277
$986$	8.84003	0.281524
$987$	−1.16225	−0.0369948
$988$	0.364610	0.0115998
$989$	−1.73271	−0.0550970
$990$	0	0
$991$	−9.17926	−0.291589	−0.145794	−	0.989315i	$-0.546574\pi$
$991$	−9.17926	−0.291589	−0.145794	+	0.989315i	$0.546574\pi$
$992$	3.05650	0.0970440
$993$	−0.540360	−0.0171478
$994$	−6.69994	−0.212509
$995$	2.67891	0.0849273
$996$	0.0928535	0.00294218
$997$	14.3428	0.454242	0.227121	−	0.973867i	$-0.427069\pi$
$997$	14.3428	0.454242	0.227121	+	0.973867i	$0.427069\pi$
$998$	−27.4357	−0.868461
$999$	2.76256	0.0874034

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.n.1.1	yes	6
3.2	odd	2		7623.2.a.cp.1.6		6
7.6	odd	2		5929.2.a.bm.1.1		6
11.2	odd	10		847.2.f.z.323.1		24
11.3	even	5		847.2.f.y.372.1		24
11.4	even	5		847.2.f.y.148.1		24
11.5	even	5		847.2.f.y.729.6		24
11.6	odd	10		847.2.f.z.729.1		24
11.7	odd	10		847.2.f.z.148.6		24
11.8	odd	10		847.2.f.z.372.6		24
11.9	even	5		847.2.f.y.323.6		24
11.10	odd	2		847.2.a.m.1.6	✓	6
33.32	even	2		7623.2.a.cs.1.1		6
77.76	even	2		5929.2.a.bj.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.6	✓	6	11.10	odd	2
847.2.a.n.1.1	yes	6	1.1	even	1	trivial
847.2.f.y.148.1		24	11.4	even	5
847.2.f.y.323.6		24	11.9	even	5
847.2.f.y.372.1		24	11.3	even	5
847.2.f.y.729.6		24	11.5	even	5
847.2.f.z.148.6		24	11.7	odd	10
847.2.f.z.323.1		24	11.2	odd	10
847.2.f.z.372.6		24	11.8	odd	10
847.2.f.z.729.1		24	11.6	odd	10
5929.2.a.bj.1.6		6	77.76	even	2
5929.2.a.bm.1.1		6	7.6	odd	2
7623.2.a.cp.1.6		6	3.2	odd	2
7623.2.a.cs.1.1		6	33.32	even	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)

\(f(q)\)	\(=\)	\(q-1.38595 q^{2} -0.122479 q^{3} -0.0791355 q^{4} -0.133004 q^{5} +0.169750 q^{6} +1.00000 q^{7} +2.88158 q^{8} -2.98500 q^{9} +O(q^{10})\)	\(q-1.38595 q^{2} -0.122479 q^{3} -0.0791355 q^{4} -0.133004 q^{5} +0.169750 q^{6} +1.00000 q^{7} +2.88158 q^{8} -2.98500 q^{9} +0.184338 q^{10} +0.00969245 q^{12} -0.641436 q^{13} -1.38595 q^{14} +0.0162903 q^{15} -3.83547 q^{16} -1.42482 q^{17} +4.13707 q^{18} +7.18297 q^{19} +0.0105254 q^{20} -0.122479 q^{21} +1.66655 q^{23} -0.352934 q^{24} -4.98231 q^{25} +0.889000 q^{26} +0.733037 q^{27} -0.0791355 q^{28} +4.47657 q^{29} -0.0225775 q^{30} -6.83182 q^{31} -0.447392 q^{32} +1.97473 q^{34} -0.133004 q^{35} +0.236220 q^{36} +3.76864 q^{37} -9.95526 q^{38} +0.0785625 q^{39} -0.383263 q^{40} +6.17286 q^{41} +0.169750 q^{42} -1.03970 q^{43} +0.397018 q^{45} -2.30976 q^{46} +9.48937 q^{47} +0.469764 q^{48} +1.00000 q^{49} +6.90524 q^{50} +0.174511 q^{51} +0.0507604 q^{52} -0.666549 q^{53} -1.01595 q^{54} +2.88158 q^{56} -0.879764 q^{57} -6.20431 q^{58} +8.18695 q^{59} -0.00128914 q^{60} +9.49807 q^{61} +9.46858 q^{62} -2.98500 q^{63} +8.29100 q^{64} +0.0853139 q^{65} +12.0398 q^{67} +0.112754 q^{68} -0.204117 q^{69} +0.184338 q^{70} +4.83418 q^{71} -8.60152 q^{72} -8.85748 q^{73} -5.22316 q^{74} +0.610229 q^{75} -0.568428 q^{76} -0.108884 q^{78} +11.4907 q^{79} +0.510134 q^{80} +8.86521 q^{81} -8.55529 q^{82} +9.57998 q^{83} +0.00969245 q^{84} +0.189507 q^{85} +1.44098 q^{86} -0.548286 q^{87} -17.7001 q^{89} -0.550248 q^{90} -0.641436 q^{91} -0.131883 q^{92} +0.836756 q^{93} -13.1518 q^{94} -0.955367 q^{95} +0.0547962 q^{96} +6.58139 q^{97} -1.38595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9} - 8 q^{10} - 14 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} + 22 q^{17} + 24 q^{18} + 6 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{23} - 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} + 4 q^{28} + 12 q^{29} + 20 q^{30} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} + 20 q^{39} + 18 q^{40} + 26 q^{41} - 6 q^{42} - 4 q^{43} - 36 q^{45} + 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} - 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} - 32 q^{54} + 12 q^{56} + 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} - 8 q^{61} + 20 q^{62} + 8 q^{63} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} + 16 q^{72} + 14 q^{73} + 44 q^{74} - 20 q^{75} - 30 q^{76} + 32 q^{78} - 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} + 22 q^{83} - 14 q^{84} - 24 q^{85} - 30 q^{86} + 22 q^{87} - 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} - 38 q^{94} - 24 q^{95} - 62 q^{96} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9 - 8 * q^10 - 14 * q^12 + 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 + 22 * q^17 + 24 * q^18 + 6 * q^19 + 2 * q^20 - 2 * q^21 + 2 * q^23 - 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 + 4 * q^28 + 12 * q^29 + 20 * q^30 - 2 * q^31 + 8 * q^32 + 24 * q^34 - 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 + 20 * q^39 + 18 * q^40 + 26 * q^41 - 6 * q^42 - 4 * q^43 - 36 * q^45 + 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 - 4 * q^50 - 4 * q^51 + 12 * q^52 + 4 * q^53 - 32 * q^54 + 12 * q^56 + 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 - 8 * q^61 + 20 * q^62 + 8 * q^63 + 26 * q^64 + 24 * q^65 + 6 * q^67 + 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 + 16 * q^72 + 14 * q^73 + 44 * q^74 - 20 * q^75 - 30 * q^76 + 32 * q^78 - 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 + 22 * q^83 - 14 * q^84 - 24 * q^85 - 30 * q^86 + 22 * q^87 - 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 - 38 * q^94 - 24 * q^95 - 62 * q^96 - 4 * q^97 + 4 * q^98

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