LMFDB - Embedded newform 847.2.a.n.1.5

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			1.5
Root			$-1.10939$ 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)

$f(q)$	$=$	$q+2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} +3.58282 q^{6} +1.00000 q^{7} +0.948212 q^{8} -0.115054 q^{9} +O(q^{10})$	\(q+2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} +3.58282 q^{6} +1.00000 q^{7} +0.948212 q^{8} -0.115054 q^{9} +1.03864 q^{10} +4.16054 q^{12} +5.30029 q^{13} +2.10939 q^{14} +0.836333 q^{15} -2.89889 q^{16} +3.03721 q^{17} -0.242693 q^{18} -4.66622 q^{19} +1.20612 q^{20} +1.69851 q^{21} -5.63835 q^{23} +1.61055 q^{24} -4.75755 q^{25} +11.1804 q^{26} -5.29096 q^{27} +2.44952 q^{28} +6.92295 q^{29} +1.76415 q^{30} -1.26565 q^{31} -8.01131 q^{32} +6.40665 q^{34} +0.492391 q^{35} -0.281826 q^{36} +10.8759 q^{37} -9.84288 q^{38} +9.00262 q^{39} +0.466891 q^{40} -1.44322 q^{41} +3.58282 q^{42} +2.88224 q^{43} -0.0566513 q^{45} -11.8935 q^{46} -8.75522 q^{47} -4.92381 q^{48} +1.00000 q^{49} -10.0355 q^{50} +5.15873 q^{51} +12.9832 q^{52} +6.63835 q^{53} -11.1607 q^{54} +0.948212 q^{56} -7.92564 q^{57} +14.6032 q^{58} -8.35733 q^{59} +2.04861 q^{60} -13.8953 q^{61} -2.66975 q^{62} -0.115054 q^{63} -11.1012 q^{64} +2.60982 q^{65} -9.70431 q^{67} +7.43970 q^{68} -9.57681 q^{69} +1.03864 q^{70} +5.94751 q^{71} -0.109095 q^{72} -3.77421 q^{73} +22.9414 q^{74} -8.08076 q^{75} -11.4300 q^{76} +18.9900 q^{78} -8.80383 q^{79} -1.42739 q^{80} -8.64160 q^{81} -3.04431 q^{82} +11.0898 q^{83} +4.16054 q^{84} +1.49549 q^{85} +6.07976 q^{86} +11.7587 q^{87} +3.10324 q^{89} -0.119500 q^{90} +5.30029 q^{91} -13.8113 q^{92} -2.14972 q^{93} -18.4682 q^{94} -2.29761 q^{95} -13.6073 q^{96} -6.31676 q^{97} +2.10939 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.10939	1.49156	0.745781	−	0.666191i	$-0.232076\pi$
$2$	2.10939	1.49156	0.745781	+	0.666191i	$0.232076\pi$
$3$	1.69851	0.980637	0.490318	−	0.871543i	$-0.336880\pi$
$3$	1.69851	0.980637	0.490318	+	0.871543i	$0.336880\pi$
$4$	2.44952	1.22476
$5$	0.492391	0.220204	0.110102	−	0.993920i	$-0.464882\pi$
$5$	0.492391	0.220204	0.110102	+	0.993920i	$0.464882\pi$
$6$	3.58282	1.46268
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.948212	0.335243
$9$	−0.115054	−0.0383512
$10$	1.03864	0.328448
$11$	0	0
$11$	0	0
$12$	4.16054	1.20104
$13$	5.30029	1.47004	0.735018	−	0.678047i	$-0.237174\pi$
$13$	5.30029	1.47004	0.735018	+	0.678047i	$0.237174\pi$
$14$	2.10939	0.563758
$15$	0.836333	0.215940
$16$	−2.89889	−0.724723
$17$	3.03721	0.736631	0.368315	−	0.929701i	$-0.379935\pi$
$17$	3.03721	0.736631	0.368315	+	0.929701i	$0.379935\pi$
$18$	−0.242693	−0.0572032
$19$	−4.66622	−1.07050	−0.535252	−	0.844692i	$-0.679784\pi$
$19$	−4.66622	−1.07050	−0.535252	+	0.844692i	$0.679784\pi$
$20$	1.20612	0.269697
$21$	1.69851	0.370646
$22$	0	0
$23$	−5.63835	−1.17568	−0.587839	−	0.808978i	$-0.700021\pi$
$23$	−5.63835	−1.17568	−0.587839	+	0.808978i	$0.700021\pi$
$24$	1.61055	0.328752
$25$	−4.75755	−0.951510
$26$	11.1804	2.19265
$27$	−5.29096	−1.01825
$28$	2.44952	0.462916
$29$	6.92295	1.28556	0.642780	−	0.766051i	$-0.277781\pi$
$29$	6.92295	1.28556	0.642780	+	0.766051i	$0.277781\pi$
$30$	1.76415	0.322088
$31$	−1.26565	−0.227317	−0.113659	−	0.993520i	$-0.536257\pi$
$31$	−1.26565	−0.227317	−0.113659	+	0.993520i	$0.536257\pi$
$32$	−8.01131	−1.41621
$33$	0	0
$34$	6.40665	1.09873
$35$	0.492391	0.0832293
$36$	−0.281826	−0.0469710
$37$	10.8759	1.78798	0.893990	−	0.448087i	$-0.147895\pi$
$37$	10.8759	1.78798	0.893990	+	0.448087i	$0.147895\pi$
$38$	−9.84288	−1.59673
$39$	9.00262	1.44157
$40$	0.466891	0.0738220
$41$	−1.44322	−0.225393	−0.112696	−	0.993629i	$-0.535949\pi$
$41$	−1.44322	−0.225393	−0.112696	+	0.993629i	$0.535949\pi$
$42$	3.58282	0.552842
$43$	2.88224	0.439537	0.219769	−	0.975552i	$-0.429470\pi$
$43$	2.88224	0.439537	0.219769	+	0.975552i	$0.429470\pi$
$44$	0	0
$45$	−0.0566513	−0.00844508
$46$	−11.8935	−1.75360
$47$	−8.75522	−1.27708	−0.638540	−	0.769589i	$-0.720461\pi$
$47$	−8.75522	−1.27708	−0.638540	+	0.769589i	$0.720461\pi$
$48$	−4.92381	−0.710690
$49$	1.00000	0.142857
$50$	−10.0355	−1.41924
$51$	5.15873	0.722367
$52$	12.9832	1.80044
$53$	6.63835	0.911848	0.455924	−	0.890019i	$-0.349309\pi$
$53$	6.63835	0.911848	0.455924	+	0.890019i	$0.349309\pi$
$54$	−11.1607	−1.51878
$55$	0	0
$56$	0.948212	0.126710
$57$	−7.92564	−1.04978
$58$	14.6032	1.91749
$59$	−8.35733	−1.08803	−0.544016	−	0.839075i	$-0.683097\pi$
$59$	−8.35733	−1.08803	−0.544016	+	0.839075i	$0.683097\pi$
$60$	2.04861	0.264475
$61$	−13.8953	−1.77911	−0.889554	−	0.456829i	$-0.848985\pi$
$61$	−13.8953	−1.77911	−0.889554	+	0.456829i	$0.848985\pi$
$62$	−2.66975	−0.339058
$63$	−0.115054	−0.0144954
$64$	−11.1012	−1.38765
$65$	2.60982	0.323708
$66$	0	0
$67$	−9.70431	−1.18557	−0.592785	−	0.805361i	$-0.701972\pi$
$67$	−9.70431	−1.18557	−0.592785	+	0.805361i	$0.701972\pi$
$68$	7.43970	0.902196
$69$	−9.57681	−1.15291
$70$	1.03864	0.124142
$71$	5.94751	0.705839	0.352920	−	0.935654i	$-0.385189\pi$
$71$	5.94751	0.705839	0.352920	+	0.935654i	$0.385189\pi$
$72$	−0.109095	−0.0128570
$73$	−3.77421	−0.441737	−0.220869	−	0.975304i	$-0.570889\pi$
$73$	−3.77421	−0.441737	−0.220869	+	0.975304i	$0.570889\pi$
$74$	22.9414	2.66688
$75$	−8.08076	−0.933086
$76$	−11.4300	−1.31111
$77$	0	0
$78$	18.9900	2.15020
$79$	−8.80383	−0.990508	−0.495254	−	0.868748i	$-0.664925\pi$
$79$	−8.80383	−0.990508	−0.495254	+	0.868748i	$0.664925\pi$
$80$	−1.42739	−0.159587
$81$	−8.64160	−0.960178
$82$	−3.04431	−0.336188
$83$	11.0898	1.21726	0.608632	−	0.793453i	$-0.291719\pi$
$83$	11.0898	1.21726	0.608632	+	0.793453i	$0.291719\pi$
$84$	4.16054	0.453952
$85$	1.49549	0.162209
$86$	6.07976	0.655597
$87$	11.7587	1.26067
$88$	0	0
$89$	3.10324	0.328943	0.164472	−	0.986382i	$-0.447408\pi$
$89$	3.10324	0.328943	0.164472	+	0.986382i	$0.447408\pi$
$90$	−0.119500	−0.0125964
$91$	5.30029	0.555622
$92$	−13.8113	−1.43992
$93$	−2.14972	−0.222916
$94$	−18.4682	−1.90484
$95$	−2.29761	−0.235730
$96$	−13.6073	−1.38879
$97$	−6.31676	−0.641370	−0.320685	−	0.947186i	$-0.603913\pi$
$97$	−6.31676	−0.641370	−0.320685	+	0.947186i	$0.603913\pi$
$98$	2.10939	0.213080
$99$	0	0
$100$	−11.6537	−1.16537
$101$	11.7984	1.17399	0.586993	−	0.809592i	$-0.300312\pi$
$101$	11.7984	1.17399	0.586993	+	0.809592i	$0.300312\pi$
$102$	10.8818	1.07746
$103$	7.00565	0.690287	0.345144	−	0.938550i	$-0.387830\pi$
$103$	7.00565	0.690287	0.345144	+	0.938550i	$0.387830\pi$
$104$	5.02580	0.492820
$105$	0.836333	0.0816177
$106$	14.0029	1.36008
$107$	11.3547	1.09770	0.548850	−	0.835921i	$-0.315066\pi$
$107$	11.3547	1.09770	0.548850	+	0.835921i	$0.315066\pi$
$108$	−12.9603	−1.24711
$109$	18.9414	1.81426	0.907129	−	0.420853i	$-0.138269\pi$
$109$	18.9414	1.81426	0.907129	+	0.420853i	$0.138269\pi$
$110$	0	0
$111$	18.4728	1.75336
$112$	−2.89889	−0.273920
$113$	−13.4961	−1.26961	−0.634804	−	0.772673i	$-0.718919\pi$
$113$	−13.4961	−1.26961	−0.634804	+	0.772673i	$0.718919\pi$
$114$	−16.7183	−1.56581
$115$	−2.77627	−0.258889
$116$	16.9579	1.57450
$117$	−0.609817	−0.0563776
$118$	−17.6289	−1.62287
$119$	3.03721	0.278420
$120$	0.793021	0.0723925
$121$	0	0
$122$	−29.3106	−2.65365
$123$	−2.45133	−0.221029
$124$	−3.10023	−0.278409
$125$	−4.80453	−0.429730
$126$	−0.242693	−0.0216208
$127$	−4.66064	−0.413565	−0.206782	−	0.978387i	$-0.566299\pi$
$127$	−4.66064	−0.413565	−0.206782	+	0.978387i	$0.566299\pi$
$128$	−7.39409	−0.653551
$129$	4.89552	0.431026
$130$	5.50512	0.482831
$131$	9.03676	0.789545	0.394773	−	0.918779i	$-0.370823\pi$
$131$	9.03676	0.789545	0.394773	+	0.918779i	$0.370823\pi$
$132$	0	0
$133$	−4.66622	−0.404613
$134$	−20.4702	−1.76835
$135$	−2.60522	−0.224222
$136$	2.87991	0.246951
$137$	−1.63772	−0.139920	−0.0699600	−	0.997550i	$-0.522287\pi$
$137$	−1.63772	−0.139920	−0.0699600	+	0.997550i	$0.522287\pi$
$138$	−20.2012	−1.71964
$139$	−1.53472	−0.130173	−0.0650866	−	0.997880i	$-0.520732\pi$
$139$	−1.53472	−0.130173	−0.0650866	+	0.997880i	$0.520732\pi$
$140$	1.20612	0.101936
$141$	−14.8709	−1.25235
$142$	12.5456	1.05280
$143$	0	0
$144$	0.333528	0.0277940
$145$	3.40880	0.283086
$146$	−7.96127	−0.658879
$147$	1.69851	0.140091
$148$	26.6406	2.18985
$149$	13.4909	1.10522	0.552610	−	0.833440i	$-0.313632\pi$
$149$	13.4909	1.10522	0.552610	+	0.833440i	$0.313632\pi$
$150$	−17.0455	−1.39176
$151$	−12.2370	−0.995835	−0.497917	−	0.867225i	$-0.665902\pi$
$151$	−12.2370	−0.995835	−0.497917	+	0.867225i	$0.665902\pi$
$152$	−4.42457	−0.358880
$153$	−0.349441	−0.0282507
$154$	0	0
$155$	−0.623194	−0.0500562
$156$	22.0521	1.76558
$157$	2.52042	0.201152	0.100576	−	0.994929i	$-0.467932\pi$
$157$	2.52042	0.201152	0.100576	+	0.994929i	$0.467932\pi$
$158$	−18.5707	−1.47741
$159$	11.2753	0.894191
$160$	−3.94470	−0.311856
$161$	−5.63835	−0.444364
$162$	−18.2285	−1.43217
$163$	7.87905	0.617135	0.308567	−	0.951203i	$-0.400151\pi$
$163$	7.87905	0.617135	0.308567	+	0.951203i	$0.400151\pi$
$164$	−3.53519	−0.276052
$165$	0	0
$166$	23.3927	1.81562
$167$	−2.05485	−0.159009	−0.0795047	−	0.996834i	$-0.525334\pi$
$167$	−2.05485	−0.159009	−0.0795047	+	0.996834i	$0.525334\pi$
$168$	1.61055	0.124257
$169$	15.0931	1.16101
$170$	3.15458	0.241945
$171$	0.536865	0.0410551
$172$	7.06010	0.538327
$173$	−23.2707	−1.76923	−0.884617	−	0.466318i	$-0.845580\pi$
$173$	−23.2707	−1.76923	−0.884617	+	0.466318i	$0.845580\pi$
$174$	24.8037	1.88037
$175$	−4.75755	−0.359637
$176$	0	0
$177$	−14.1950	−1.06696
$178$	6.54595	0.490640
$179$	17.6596	1.31994	0.659969	−	0.751293i	$-0.270569\pi$
$179$	17.6596	1.31994	0.659969	+	0.751293i	$0.270569\pi$
$180$	−0.138769	−0.0103432
$181$	15.4701	1.14988	0.574941	−	0.818195i	$-0.305025\pi$
$181$	15.4701	1.14988	0.574941	+	0.818195i	$0.305025\pi$
$182$	11.1804	0.828745
$183$	−23.6013	−1.74466
$184$	−5.34635	−0.394138
$185$	5.35518	0.393720
$186$	−4.53460	−0.332493
$187$	0	0
$188$	−21.4461	−1.56412
$189$	−5.29096	−0.384861
$190$	−4.84655	−0.351605
$191$	15.9385	1.15327	0.576635	−	0.817002i	$-0.304366\pi$
$191$	15.9385	1.15327	0.576635	+	0.817002i	$0.304366\pi$
$192$	−18.8555	−1.36078
$193$	8.45386	0.608522	0.304261	−	0.952589i	$-0.401590\pi$
$193$	8.45386	0.608522	0.304261	+	0.952589i	$0.401590\pi$
$194$	−13.3245	−0.956643
$195$	4.43281	0.317440
$196$	2.44952	0.174966
$197$	−14.3384	−1.02157	−0.510785	−	0.859708i	$-0.670645\pi$
$197$	−14.3384	−1.02157	−0.510785	+	0.859708i	$0.670645\pi$
$198$	0	0
$199$	−22.1343	−1.56906	−0.784528	−	0.620094i	$-0.787095\pi$
$199$	−22.1343	−1.56906	−0.784528	+	0.620094i	$0.787095\pi$
$200$	−4.51117	−0.318988
$201$	−16.4829	−1.16261
$202$	24.8874	1.75107
$203$	6.92295	0.485896
$204$	12.6364	0.884727
$205$	−0.710628	−0.0496324
$206$	14.7776	1.02961
$207$	0.648712	0.0450886
$208$	−15.3650	−1.06537
$209$	0	0
$210$	1.76415	0.121738
$211$	2.18302	0.150286	0.0751428	−	0.997173i	$-0.476059\pi$
$211$	2.18302	0.150286	0.0751428	+	0.997173i	$0.476059\pi$
$212$	16.2608	1.11679
$213$	10.1019	0.692172
$214$	23.9515	1.63729
$215$	1.41919	0.0967878
$216$	−5.01695	−0.341360
$217$	−1.26565	−0.0859178
$218$	39.9548	2.70608
$219$	−6.41054	−0.433184
$220$	0	0
$221$	16.0981	1.08287
$222$	38.9663	2.61525
$223$	27.2603	1.82549	0.912744	−	0.408533i	$-0.133960\pi$
$223$	27.2603	1.82549	0.912744	+	0.408533i	$0.133960\pi$
$224$	−8.01131	−0.535278
$225$	0.547373	0.0364915
$226$	−28.4685	−1.89370
$227$	−13.5892	−0.901945	−0.450973	−	0.892538i	$-0.648923\pi$
$227$	−13.5892	−0.901945	−0.450973	+	0.892538i	$0.648923\pi$
$228$	−19.4140	−1.28572
$229$	−8.30141	−0.548573	−0.274286	−	0.961648i	$-0.588442\pi$
$229$	−8.30141	−0.548573	−0.274286	+	0.961648i	$0.588442\pi$
$230$	−5.85624	−0.386149
$231$	0	0
$232$	6.56443	0.430976
$233$	12.9476	0.848229	0.424114	−	0.905609i	$-0.360585\pi$
$233$	12.9476	0.848229	0.424114	+	0.905609i	$0.360585\pi$
$234$	−1.28634	−0.0840908
$235$	−4.31099	−0.281218
$236$	−20.4715	−1.33258
$237$	−14.9534	−0.971329
$238$	6.40665	0.415281
$239$	1.89342	0.122475	0.0612375	−	0.998123i	$-0.480495\pi$
$239$	1.89342	0.122475	0.0612375	+	0.998123i	$0.480495\pi$
$240$	−2.42444	−0.156497
$241$	−11.6983	−0.753557	−0.376778	−	0.926303i	$-0.622968\pi$
$241$	−11.6983	−0.753557	−0.376778	+	0.926303i	$0.622968\pi$
$242$	0	0
$243$	1.19500	0.0766595
$244$	−34.0368	−2.17898
$245$	0.492391	0.0314577
$246$	−5.17080	−0.329678
$247$	−24.7323	−1.57368
$248$	−1.20010	−0.0762066
$249$	18.8362	1.19369
$250$	−10.1346	−0.640970
$251$	−13.1860	−0.832291	−0.416146	−	0.909298i	$-0.636619\pi$
$251$	−13.1860	−0.832291	−0.416146	+	0.909298i	$0.636619\pi$
$252$	−0.281826	−0.0177534
$253$	0	0
$254$	−9.83110	−0.616858
$255$	2.54012	0.159068
$256$	6.60537	0.412836
$257$	−15.3445	−0.957165	−0.478582	−	0.878043i	$-0.658849\pi$
$257$	−15.3445	−0.957165	−0.478582	+	0.878043i	$0.658849\pi$
$258$	10.3265	0.642903
$259$	10.8759	0.675793
$260$	6.39280	0.396465
$261$	−0.796510	−0.0493027
$262$	19.0620	1.17766
$263$	10.4197	0.642507	0.321254	−	0.946993i	$-0.395896\pi$
$263$	10.4197	0.642507	0.321254	+	0.946993i	$0.395896\pi$
$264$	0	0
$265$	3.26867	0.200792
$266$	−9.84288	−0.603506
$267$	5.27090	0.322574
$268$	−23.7709	−1.45204
$269$	16.2712	0.992074	0.496037	−	0.868301i	$-0.334788\pi$
$269$	16.2712	0.992074	0.496037	+	0.868301i	$0.334788\pi$
$270$	−5.49542	−0.334441
$271$	−5.83922	−0.354707	−0.177354	−	0.984147i	$-0.556754\pi$
$271$	−5.83922	−0.354707	−0.177354	+	0.984147i	$0.556754\pi$
$272$	−8.80454	−0.533853
$273$	9.00262	0.544863
$274$	−3.45459	−0.208700
$275$	0	0
$276$	−23.4586	−1.41204
$277$	15.9255	0.956868	0.478434	−	0.878123i	$-0.341205\pi$
$277$	15.9255	0.956868	0.478434	+	0.878123i	$0.341205\pi$
$278$	−3.23732	−0.194162
$279$	0.145617	0.00871788
$280$	0.466891	0.0279021
$281$	10.2004	0.608504	0.304252	−	0.952592i	$-0.401594\pi$
$281$	10.2004	0.608504	0.304252	+	0.952592i	$0.401594\pi$
$282$	−31.3684	−1.86796
$283$	−16.1634	−0.960812	−0.480406	−	0.877046i	$-0.659511\pi$
$283$	−16.1634	−0.960812	−0.480406	+	0.877046i	$0.659511\pi$
$284$	14.5685	0.864483
$285$	−3.90252	−0.231165
$286$	0	0
$287$	−1.44322	−0.0851905
$288$	0.921730	0.0543135
$289$	−7.77538	−0.457375
$290$	7.19049	0.422240
$291$	−10.7291	−0.628951
$292$	−9.24499	−0.541022
$293$	27.0517	1.58038	0.790188	−	0.612864i	$-0.209983\pi$
$293$	27.0517	1.58038	0.790188	+	0.612864i	$0.209983\pi$
$294$	3.58282	0.208955
$295$	−4.11508	−0.239589
$296$	10.3126	0.599408
$297$	0	0
$298$	28.4576	1.64851
$299$	−29.8849	−1.72829
$300$	−19.7940	−1.14281
$301$	2.88224	0.166129
$302$	−25.8126	−1.48535
$303$	20.0398	1.15125
$304$	13.5269	0.775820
$305$	−6.84192	−0.391767
$306$	−0.737107	−0.0421376
$307$	29.7251	1.69650	0.848250	−	0.529596i	$-0.177657\pi$
$307$	29.7251	1.69650	0.848250	+	0.529596i	$0.177657\pi$
$308$	0	0
$309$	11.8992	0.676921
$310$	−1.31456	−0.0746619
$311$	22.4029	1.27035	0.635176	−	0.772367i	$-0.280928\pi$
$311$	22.4029	1.27035	0.635176	+	0.772367i	$0.280928\pi$
$312$	8.53639	0.483278
$313$	9.94833	0.562313	0.281157	−	0.959662i	$-0.409282\pi$
$313$	9.94833	0.562313	0.281157	+	0.959662i	$0.409282\pi$
$314$	5.31655	0.300030
$315$	−0.0566513	−0.00319194
$316$	−21.5652	−1.21313
$317$	−11.1420	−0.625796	−0.312898	−	0.949787i	$-0.601300\pi$
$317$	−11.1420	−0.625796	−0.312898	+	0.949787i	$0.601300\pi$
$318$	23.7840	1.33374
$319$	0	0
$320$	−5.46613	−0.305566
$321$	19.2861	1.07645
$322$	−11.8935	−0.662797
$323$	−14.1723	−0.788567
$324$	−21.1678	−1.17599
$325$	−25.2164	−1.39875
$326$	16.6200	0.920495
$327$	32.1722	1.77913
$328$	−1.36848	−0.0755615
$329$	−8.75522	−0.482691
$330$	0	0
$331$	14.5950	0.802214	0.401107	−	0.916031i	$-0.368626\pi$
$331$	14.5950	0.802214	0.401107	+	0.916031i	$0.368626\pi$
$332$	27.1647	1.49086
$333$	−1.25131	−0.0685711
$334$	−4.33449	−0.237172
$335$	−4.77832	−0.261067
$336$	−4.92381	−0.268616
$337$	12.6059	0.686688	0.343344	−	0.939210i	$-0.388440\pi$
$337$	12.6059	0.686688	0.343344	+	0.939210i	$0.388440\pi$
$338$	31.8372	1.73172
$339$	−22.9233	−1.24502
$340$	3.66324	0.198667
$341$	0	0
$342$	1.13246	0.0612363
$343$	1.00000	0.0539949
$344$	2.73297	0.147352
$345$	−4.71554	−0.253876
$346$	−49.0868	−2.63893
$347$	0.410734	0.0220494	0.0110247	−	0.999939i	$-0.496491\pi$
$347$	0.410734	0.0220494	0.0110247	+	0.999939i	$0.496491\pi$
$348$	28.8032	1.54402
$349$	13.4025	0.717422	0.358711	−	0.933449i	$-0.383216\pi$
$349$	13.4025	0.717422	0.358711	+	0.933449i	$0.383216\pi$
$350$	−10.0355	−0.536421
$351$	−28.0436	−1.49686
$352$	0	0
$353$	12.3419	0.656892	0.328446	−	0.944523i	$-0.393475\pi$
$353$	12.3419	0.656892	0.328446	+	0.944523i	$0.393475\pi$
$354$	−29.9429	−1.59144
$355$	2.92850	0.155429
$356$	7.60146	0.402877
$357$	5.15873	0.273029
$358$	37.2509	1.96877
$359$	−25.0097	−1.31996	−0.659981	−	0.751283i	$-0.729435\pi$
$359$	−25.0097	−1.31996	−0.659981	+	0.751283i	$0.729435\pi$
$360$	−0.0537175	−0.00283116
$361$	2.77364	0.145981
$362$	32.6324	1.71512
$363$	0	0
$364$	12.9832	0.680503
$365$	−1.85839	−0.0972724
$366$	−49.7844	−2.60227
$367$	−2.06915	−0.108009	−0.0540043	−	0.998541i	$-0.517198\pi$
$367$	−2.06915	−0.108009	−0.0540043	+	0.998541i	$0.517198\pi$
$368$	16.3450	0.852041
$369$	0.166047	0.00864408
$370$	11.2961	0.587259
$371$	6.63835	0.344646
$372$	−5.26578	−0.273018
$373$	−14.7623	−0.764365	−0.382183	−	0.924087i	$-0.624828\pi$
$373$	−14.7623	−0.764365	−0.382183	+	0.924087i	$0.624828\pi$
$374$	0	0
$375$	−8.16056	−0.421410
$376$	−8.30180	−0.428133
$377$	36.6937	1.88982
$378$	−11.1607	−0.574044
$379$	27.7508	1.42546	0.712730	−	0.701438i	$-0.247458\pi$
$379$	27.7508	1.42546	0.712730	+	0.701438i	$0.247458\pi$
$380$	−5.62803	−0.288712
$381$	−7.91615	−0.405557
$382$	33.6205	1.72017
$383$	−18.0334	−0.921464	−0.460732	−	0.887539i	$-0.652413\pi$
$383$	−18.0334	−0.921464	−0.460732	+	0.887539i	$0.652413\pi$
$384$	−12.5590	−0.640897
$385$	0	0
$386$	17.8325	0.907649
$387$	−0.331612	−0.0168568
$388$	−15.4730	−0.785524
$389$	13.5412	0.686564	0.343282	−	0.939232i	$-0.388461\pi$
$389$	13.5412	0.686564	0.343282	+	0.939232i	$0.388461\pi$
$390$	9.35052	0.473482
$391$	−17.1248	−0.866040
$392$	0.948212	0.0478919
$393$	15.3491	0.774257
$394$	−30.2453	−1.52374
$395$	−4.33493	−0.218114
$396$	0	0
$397$	−24.6525	−1.23727	−0.618637	−	0.785677i	$-0.712315\pi$
$397$	−24.6525	−1.23727	−0.618637	+	0.785677i	$0.712315\pi$
$398$	−46.6897	−2.34035
$399$	−7.92564	−0.396778
$400$	13.7916	0.689581
$401$	18.7389	0.935778	0.467889	−	0.883787i	$-0.345015\pi$
$401$	18.7389	0.935778	0.467889	+	0.883787i	$0.345015\pi$
$402$	−34.7688	−1.73411
$403$	−6.70831	−0.334165
$404$	28.9004	1.43785
$405$	−4.25505	−0.211435
$406$	14.6032	0.724745
$407$	0	0
$408$	4.89157	0.242169
$409$	15.7098	0.776801	0.388401	−	0.921491i	$-0.373028\pi$
$409$	15.7098	0.776801	0.388401	+	0.921491i	$0.373028\pi$
$410$	−1.49899	−0.0740299
$411$	−2.78169	−0.137211
$412$	17.1605	0.845436
$413$	−8.35733	−0.411238
$414$	1.36839	0.0672525
$415$	5.46052	0.268046
$416$	−42.4623	−2.08189
$417$	−2.60674	−0.127653
$418$	0	0
$419$	22.6034	1.10425	0.552125	−	0.833761i	$-0.313817\pi$
$419$	22.6034	1.10425	0.552125	+	0.833761i	$0.313817\pi$
$420$	2.04861	0.0999621
$421$	−23.3311	−1.13709	−0.568544	−	0.822653i	$-0.692493\pi$
$421$	−23.3311	−1.13709	−0.568544	+	0.822653i	$0.692493\pi$
$422$	4.60485	0.224160
$423$	1.00732	0.0489775
$424$	6.29456	0.305691
$425$	−14.4497	−0.700912
$426$	21.3089	1.03242
$427$	−13.8953	−0.672440
$428$	27.8136	1.34442
$429$	0	0
$430$	2.99362	0.144365
$431$	−9.53898	−0.459476	−0.229738	−	0.973252i	$-0.573787\pi$
$431$	−9.53898	−0.459476	−0.229738	+	0.973252i	$0.573787\pi$
$432$	15.3379	0.737946
$433$	−23.0105	−1.10581	−0.552907	−	0.833243i	$-0.686482\pi$
$433$	−23.0105	−1.10581	−0.552907	+	0.833243i	$0.686482\pi$
$434$	−2.66975	−0.128152
$435$	5.78989	0.277604
$436$	46.3973	2.22203
$437$	26.3098	1.25857
$438$	−13.5223	−0.646121
$439$	−27.6434	−1.31935	−0.659673	−	0.751553i	$-0.729305\pi$
$439$	−27.6434	−1.31935	−0.659673	+	0.751553i	$0.729305\pi$
$440$	0	0
$441$	−0.115054	−0.00547874
$442$	33.9571	1.61518
$443$	14.7713	0.701807	0.350904	−	0.936412i	$-0.385875\pi$
$443$	14.7713	0.701807	0.350904	+	0.936412i	$0.385875\pi$
$444$	45.2494	2.14744
$445$	1.52801	0.0724346
$446$	57.5026	2.72283
$447$	22.9145	1.08382
$448$	−11.1012	−0.524482
$449$	30.2669	1.42838	0.714192	−	0.699950i	$-0.246795\pi$
$449$	30.2669	1.42838	0.714192	+	0.699950i	$0.246795\pi$
$450$	1.15462	0.0544294
$451$	0	0
$452$	−33.0590	−1.55496
$453$	−20.7847	−0.976552
$454$	−28.6648	−1.34531
$455$	2.60982	0.122350
$456$	−7.51518	−0.351931
$457$	−20.2565	−0.947558	−0.473779	−	0.880644i	$-0.657111\pi$
$457$	−20.2565	−0.947558	−0.473779	+	0.880644i	$0.657111\pi$
$458$	−17.5109	−0.818231
$459$	−16.0697	−0.750071
$460$	−6.80054	−0.317077
$461$	8.51184	0.396436	0.198218	−	0.980158i	$-0.436485\pi$
$461$	8.51184	0.396436	0.198218	+	0.980158i	$0.436485\pi$
$462$	0	0
$463$	−0.591469	−0.0274879	−0.0137440	−	0.999906i	$-0.504375\pi$
$463$	−0.591469	−0.0274879	−0.0137440	+	0.999906i	$0.504375\pi$
$464$	−20.0689	−0.931675
$465$	−1.05850	−0.0490869
$466$	27.3116	1.26519
$467$	−41.0347	−1.89886	−0.949430	−	0.313979i	$-0.898338\pi$
$467$	−41.0347	−1.89886	−0.949430	+	0.313979i	$0.898338\pi$
$468$	−1.49376	−0.0690491
$469$	−9.70431	−0.448103
$470$	−9.09356	−0.419454
$471$	4.28097	0.197257
$472$	−7.92452	−0.364756
$473$	0	0
$474$	−31.5426	−1.44880
$475$	22.1998	1.01860
$476$	7.43970	0.340998
$477$	−0.763766	−0.0349704
$478$	3.99395	0.182679
$479$	20.3437	0.929527	0.464763	−	0.885435i	$-0.346139\pi$
$479$	20.3437	0.929527	0.464763	+	0.885435i	$0.346139\pi$
$480$	−6.70012	−0.305817
$481$	57.6452	2.62840
$482$	−24.6764	−1.12398
$483$	−9.57681	−0.435760
$484$	0	0
$485$	−3.11032	−0.141232
$486$	2.52073	0.114342
$487$	−28.8165	−1.30580	−0.652900	−	0.757444i	$-0.726448\pi$
$487$	−28.8165	−1.30580	−0.652900	+	0.757444i	$0.726448\pi$
$488$	−13.1757	−0.596435
$489$	13.3827	0.605185
$490$	1.03864	0.0469212
$491$	2.68045	0.120967	0.0604834	−	0.998169i	$-0.480736\pi$
$491$	2.68045	0.120967	0.0604834	+	0.998169i	$0.480736\pi$
$492$	−6.00457	−0.270707
$493$	21.0264	0.946983
$494$	−52.1701	−2.34725
$495$	0	0
$496$	3.66898	0.164742
$497$	5.94751	0.266782
$498$	39.7328	1.78047
$499$	22.3425	1.00019	0.500095	−	0.865971i	$-0.333298\pi$
$499$	22.3425	1.00019	0.500095	+	0.865971i	$0.333298\pi$
$500$	−11.7688	−0.526317
$501$	−3.49020	−0.155930
$502$	−27.8143	−1.24141
$503$	−4.47599	−0.199575	−0.0997873	−	0.995009i	$-0.531816\pi$
$503$	−4.47599	−0.199575	−0.0997873	+	0.995009i	$0.531816\pi$
$504$	−0.109095	−0.00485948
$505$	5.80943	0.258516
$506$	0	0
$507$	25.6358	1.13853
$508$	−11.4163	−0.506518
$509$	−17.1547	−0.760368	−0.380184	−	0.924911i	$-0.624139\pi$
$509$	−17.1547	−0.760368	−0.380184	+	0.924911i	$0.624139\pi$
$510$	5.35809	0.237260
$511$	−3.77421	−0.166961
$512$	28.7215	1.26932
$513$	24.6888	1.09004
$514$	−32.3676	−1.42767
$515$	3.44952	0.152004
$516$	11.9917	0.527904
$517$	0	0
$518$	22.9414	1.00799
$519$	−39.5255	−1.73498
$520$	2.47466	0.108521
$521$	1.00957	0.0442300	0.0221150	−	0.999755i	$-0.492960\pi$
$521$	1.00957	0.0442300	0.0221150	+	0.999755i	$0.492960\pi$
$522$	−1.68015	−0.0735382
$523$	−13.6433	−0.596578	−0.298289	−	0.954476i	$-0.596416\pi$
$523$	−13.6433	−0.596578	−0.298289	+	0.954476i	$0.596416\pi$
$524$	22.1357	0.967004
$525$	−8.08076	−0.352673
$526$	21.9792	0.958340
$527$	−3.84404	−0.167449
$528$	0	0
$529$	8.79099	0.382217
$530$	6.89488	0.299495
$531$	0.961541	0.0417273
$532$	−11.4300	−0.495554
$533$	−7.64948	−0.331336
$534$	11.1184	0.481139
$535$	5.59095	0.241718
$536$	−9.20174	−0.397455
$537$	29.9950	1.29438
$538$	34.3223	1.47974
$539$	0	0
$540$	−6.38154	−0.274618
$541$	−2.76335	−0.118806	−0.0594028	−	0.998234i	$-0.518920\pi$
$541$	−2.76335	−0.118806	−0.0594028	+	0.998234i	$0.518920\pi$
$542$	−12.3172	−0.529068
$543$	26.2761	1.12762
$544$	−24.3320	−1.04323
$545$	9.32658	0.399507
$546$	18.9900	0.812698
$547$	15.2417	0.651687	0.325843	−	0.945424i	$-0.394352\pi$
$547$	15.2417	0.651687	0.325843	+	0.945424i	$0.394352\pi$
$548$	−4.01163	−0.171368
$549$	1.59870	0.0682309
$550$	0	0
$551$	−32.3041	−1.37620
$552$	−9.08084	−0.386506
$553$	−8.80383	−0.374377
$554$	33.5930	1.42723
$555$	9.09584	0.386097
$556$	−3.75933	−0.159431
$557$	3.56730	0.151151	0.0755757	−	0.997140i	$-0.475921\pi$
$557$	3.56730	0.151151	0.0755757	+	0.997140i	$0.475921\pi$
$558$	0.307164	0.0130033
$559$	15.2767	0.646136
$560$	−1.42739	−0.0603182
$561$	0	0
$562$	21.5166	0.907621
$563$	36.7500	1.54883	0.774414	−	0.632679i	$-0.218045\pi$
$563$	36.7500	1.54883	0.774414	+	0.632679i	$0.218045\pi$
$564$	−36.4264	−1.53383
$565$	−6.64537	−0.279573
$566$	−34.0948	−1.43311
$567$	−8.64160	−0.362913
$568$	5.63949	0.236628
$569$	−34.0802	−1.42872	−0.714359	−	0.699779i	$-0.753282\pi$
$569$	−34.0802	−1.42872	−0.714359	+	0.699779i	$0.753282\pi$
$570$	−8.23192	−0.344797
$571$	−5.79312	−0.242434	−0.121217	−	0.992626i	$-0.538680\pi$
$571$	−5.79312	−0.242434	−0.121217	+	0.992626i	$0.538680\pi$
$572$	0	0
$573$	27.0718	1.13094
$574$	−3.04431	−0.127067
$575$	26.8247	1.11867
$576$	1.27723	0.0532179
$577$	−31.1231	−1.29567	−0.647837	−	0.761779i	$-0.724326\pi$
$577$	−31.1231	−1.29567	−0.647837	+	0.761779i	$0.724326\pi$
$578$	−16.4013	−0.682204
$579$	14.3590	0.596739
$580$	8.34993	0.346712
$581$	11.0898	0.460082
$582$	−22.6318	−0.938120
$583$	0	0
$584$	−3.57875	−0.148090
$585$	−0.300269	−0.0124146
$586$	57.0625	2.35723
$587$	−47.6946	−1.96857	−0.984283	−	0.176601i	$-0.943490\pi$
$587$	−47.6946	−1.96857	−0.984283	+	0.176601i	$0.943490\pi$
$588$	4.16054	0.171578
$589$	5.90580	0.243344
$590$	−8.68030	−0.357362
$591$	−24.3540	−1.00179
$592$	−31.5279	−1.29579
$593$	44.5859	1.83092	0.915462	−	0.402404i	$-0.131825\pi$
$593$	44.5859	1.83092	0.915462	+	0.402404i	$0.131825\pi$
$594$	0	0
$595$	1.49549	0.0613093
$596$	33.0463	1.35363
$597$	−37.5953	−1.53867
$598$	−63.0389	−2.57785
$599$	−8.06937	−0.329706	−0.164853	−	0.986318i	$-0.552715\pi$
$599$	−8.06937	−0.329706	−0.164853	+	0.986318i	$0.552715\pi$
$600$	−7.66227	−0.312811
$601$	−28.9086	−1.17921	−0.589603	−	0.807693i	$-0.700716\pi$
$601$	−28.9086	−1.17921	−0.589603	+	0.807693i	$0.700716\pi$
$602$	6.07976	0.247792
$603$	1.11651	0.0454680
$604$	−29.9748	−1.21966
$605$	0	0
$606$	42.2716	1.71717
$607$	−9.85310	−0.399925	−0.199962	−	0.979804i	$-0.564082\pi$
$607$	−9.85310	−0.399925	−0.199962	+	0.979804i	$0.564082\pi$
$608$	37.3826	1.51606
$609$	11.7587	0.476488
$610$	−14.4323	−0.584345
$611$	−46.4052	−1.87735
$612$	−0.855963	−0.0346003
$613$	−35.8329	−1.44728	−0.723638	−	0.690180i	$-0.757531\pi$
$613$	−35.8329	−1.44728	−0.723638	+	0.690180i	$0.757531\pi$
$614$	62.7017	2.53044
$615$	−1.20701	−0.0486714
$616$	0	0
$617$	−38.4398	−1.54753	−0.773764	−	0.633474i	$-0.781628\pi$
$617$	−38.4398	−1.54753	−0.773764	+	0.633474i	$0.781628\pi$
$618$	25.1000	1.00967
$619$	4.19552	0.168632	0.0843162	−	0.996439i	$-0.473129\pi$
$619$	4.19552	0.168632	0.0843162	+	0.996439i	$0.473129\pi$
$620$	−1.52653	−0.0613068
$621$	29.8323	1.19713
$622$	47.2564	1.89481
$623$	3.10324	0.124329
$624$	−26.0976	−1.04474
$625$	21.4220	0.856882
$626$	20.9849	0.838725
$627$	0	0
$628$	6.17382	0.246362
$629$	33.0322	1.31708
$630$	−0.119500	−0.00476098
$631$	−3.51798	−0.140049	−0.0700243	−	0.997545i	$-0.522308\pi$
$631$	−3.51798	−0.140049	−0.0700243	+	0.997545i	$0.522308\pi$
$632$	−8.34790	−0.332061
$633$	3.70789	0.147376
$634$	−23.5028	−0.933414
$635$	−2.29486	−0.0910686
$636$	27.6191	1.09517
$637$	5.30029	0.210005
$638$	0	0
$639$	−0.684281	−0.0270698
$640$	−3.64078	−0.143915
$641$	16.0952	0.635723	0.317862	−	0.948137i	$-0.397035\pi$
$641$	16.0952	0.635723	0.317862	+	0.948137i	$0.397035\pi$
$642$	40.6819	1.60559
$643$	−34.8261	−1.37341	−0.686704	−	0.726938i	$-0.740943\pi$
$643$	−34.8261	−1.37341	−0.686704	+	0.726938i	$0.740943\pi$
$644$	−13.8113	−0.544239
$645$	2.41051	0.0949137
$646$	−29.8948	−1.17620
$647$	16.9051	0.664607	0.332304	−	0.943172i	$-0.392174\pi$
$647$	16.9051	0.664607	0.332304	+	0.943172i	$0.392174\pi$
$648$	−8.19407	−0.321893
$649$	0	0
$650$	−53.1912	−2.08633
$651$	−2.14972	−0.0842542
$652$	19.2999	0.755842
$653$	3.52799	0.138061	0.0690304	−	0.997615i	$-0.478009\pi$
$653$	3.52799	0.138061	0.0690304	+	0.997615i	$0.478009\pi$
$654$	67.8637	2.65368
$655$	4.44962	0.173861
$656$	4.18374	0.163347
$657$	0.434236	0.0169411
$658$	−18.4682	−0.719964
$659$	29.4409	1.14686	0.573428	−	0.819256i	$-0.305613\pi$
$659$	29.4409	1.14686	0.573428	+	0.819256i	$0.305613\pi$
$660$	0	0
$661$	15.2989	0.595059	0.297529	−	0.954713i	$-0.403837\pi$
$661$	15.2989	0.595059	0.297529	+	0.954713i	$0.403837\pi$
$662$	30.7865	1.19655
$663$	27.3428	1.06191
$664$	10.5155	0.408080
$665$	−2.29761	−0.0890974
$666$	−2.63949	−0.102278
$667$	−39.0340	−1.51140
$668$	−5.03341	−0.194748
$669$	46.3020	1.79014
$670$	−10.0793	−0.389398
$671$	0	0
$672$	−13.6073	−0.524914
$673$	12.1652	0.468936	0.234468	−	0.972124i	$-0.424665\pi$
$673$	12.1652	0.468936	0.234468	+	0.972124i	$0.424665\pi$
$674$	26.5908	1.02424
$675$	25.1720	0.968871
$676$	36.9709	1.42196
$677$	−0.313619	−0.0120533	−0.00602667	−	0.999982i	$-0.501918\pi$
$677$	−0.313619	−0.0120533	−0.00602667	+	0.999982i	$0.501918\pi$
$678$	−48.3542	−1.85703
$679$	−6.31676	−0.242415
$680$	1.41804	0.0543795
$681$	−23.0814	−0.884481
$682$	0	0
$683$	38.2419	1.46328	0.731642	−	0.681689i	$-0.238754\pi$
$683$	38.2419	1.46328	0.731642	+	0.681689i	$0.238754\pi$
$684$	1.31506	0.0502827
$685$	−0.806400	−0.0308110
$686$	2.10939	0.0805368
$687$	−14.1001	−0.537951
$688$	−8.35530	−0.318543
$689$	35.1852	1.34045
$690$	−9.94690	−0.378672
$691$	−10.2754	−0.390895	−0.195448	−	0.980714i	$-0.562616\pi$
$691$	−10.2754	−0.390895	−0.195448	+	0.980714i	$0.562616\pi$
$692$	−57.0019	−2.16689
$693$	0	0
$694$	0.866398	0.0328880
$695$	−0.755683	−0.0286647
$696$	11.1498	0.422631
$697$	−4.38335	−0.166031
$698$	28.2712	1.07008
$699$	21.9917	0.831804
$700$	−11.6537	−0.440469
$701$	18.9188	0.714552	0.357276	−	0.933999i	$-0.383706\pi$
$701$	18.9188	0.714552	0.357276	+	0.933999i	$0.383706\pi$
$702$	−59.1549	−2.23266
$703$	−50.7492	−1.91404
$704$	0	0
$705$	−7.32228	−0.275773
$706$	26.0338	0.979795
$707$	11.7984	0.443725
$708$	−34.7710	−1.30678
$709$	−14.1498	−0.531409	−0.265704	−	0.964055i	$-0.585605\pi$
$709$	−14.1498	−0.531409	−0.265704	+	0.964055i	$0.585605\pi$
$710$	6.17734	0.231832
$711$	1.01291	0.0379872
$712$	2.94253	0.110276
$713$	7.13617	0.267252
$714$	10.8818	0.407240
$715$	0	0
$716$	43.2575	1.61661
$717$	3.21599	0.120104
$718$	−52.7552	−1.96880
$719$	−15.9330	−0.594201	−0.297101	−	0.954846i	$-0.596020\pi$
$719$	−15.9330	−0.594201	−0.297101	+	0.954846i	$0.596020\pi$
$720$	0.164226	0.00612035
$721$	7.00565	0.260904
$722$	5.85068	0.217740
$723$	−19.8698	−0.738965
$724$	37.8943	1.40833
$725$	−32.9363	−1.22322
$726$	0	0
$727$	−4.20455	−0.155938	−0.0779691	−	0.996956i	$-0.524844\pi$
$727$	−4.20455	−0.155938	−0.0779691	+	0.996956i	$0.524844\pi$
$728$	5.02580	0.186269
$729$	27.9545	1.03535
$730$	−3.92006	−0.145088
$731$	8.75395	0.323777
$732$	−57.8119	−2.13679
$733$	34.0777	1.25869	0.629345	−	0.777126i	$-0.283323\pi$
$733$	34.0777	1.25869	0.629345	+	0.777126i	$0.283323\pi$
$734$	−4.36464	−0.161102
$735$	0.836333	0.0308486
$736$	45.1706	1.66501
$737$	0	0
$738$	0.350258	0.0128932
$739$	26.1306	0.961230	0.480615	−	0.876932i	$-0.340413\pi$
$739$	26.1306	0.961230	0.480615	+	0.876932i	$0.340413\pi$
$740$	13.1176	0.482213
$741$	−42.0082	−1.54321
$742$	14.0029	0.514061
$743$	−23.6263	−0.866765	−0.433383	−	0.901210i	$-0.642680\pi$
$743$	−23.6263	−0.866765	−0.433383	+	0.901210i	$0.642680\pi$
$744$	−2.03839	−0.0747310
$745$	6.64282	0.243374
$746$	−31.1395	−1.14010
$747$	−1.27592	−0.0466835
$748$	0	0
$749$	11.3547	0.414892
$750$	−17.2138	−0.628559
$751$	−2.32359	−0.0847889	−0.0423945	−	0.999101i	$-0.513499\pi$
$751$	−2.32359	−0.0847889	−0.0423945	+	0.999101i	$0.513499\pi$
$752$	25.3804	0.925529
$753$	−22.3965	−0.816175
$754$	77.4012	2.81879
$755$	−6.02540	−0.219287
$756$	−12.9603	−0.471362
$757$	−14.7651	−0.536647	−0.268324	−	0.963329i	$-0.586470\pi$
$757$	−14.7651	−0.536647	−0.268324	+	0.963329i	$0.586470\pi$
$758$	58.5371	2.12616
$759$	0	0
$760$	−2.17862	−0.0790268
$761$	−47.1876	−1.71055	−0.855274	−	0.518175i	$-0.826611\pi$
$761$	−47.1876	−1.71055	−0.855274	+	0.518175i	$0.826611\pi$
$762$	−16.6982	−0.604914
$763$	18.9414	0.685725
$764$	39.0417	1.41248
$765$	−0.172062	−0.00622091
$766$	−38.0395	−1.37442
$767$	−44.2963	−1.59945
$768$	11.2193	0.404842
$769$	−12.4418	−0.448662	−0.224331	−	0.974513i	$-0.572020\pi$
$769$	−12.4418	−0.448662	−0.224331	+	0.974513i	$0.572020\pi$
$770$	0	0
$771$	−26.0629	−0.938631
$772$	20.7079	0.745294
$773$	−35.2698	−1.26857	−0.634284	−	0.773101i	$-0.718705\pi$
$773$	−35.2698	−1.26857	−0.634284	+	0.773101i	$0.718705\pi$
$774$	−0.699497	−0.0251429
$775$	6.02139	0.216295
$776$	−5.98962	−0.215015
$777$	18.4728	0.662707
$778$	28.5636	1.02405
$779$	6.73438	0.241284
$780$	10.8583	0.388788
$781$	0	0
$782$	−36.1229	−1.29175
$783$	−36.6291	−1.30902
$784$	−2.89889	−0.103532
$785$	1.24103	0.0442944
$786$	32.3771	1.15485
$787$	−16.3383	−0.582397	−0.291198	−	0.956663i	$-0.594054\pi$
$787$	−16.3383	−0.582397	−0.291198	+	0.956663i	$0.594054\pi$
$788$	−35.1223	−1.25118
$789$	17.6980	0.630067
$790$	−9.14405	−0.325331
$791$	−13.4961	−0.479867
$792$	0	0
$793$	−73.6491	−2.61536
$794$	−52.0017	−1.84547
$795$	5.55187	0.196905
$796$	−54.2183	−1.92172
$797$	−0.292902	−0.0103751	−0.00518755	−	0.999987i	$-0.501651\pi$
$797$	−0.292902	−0.0103751	−0.00518755	+	0.999987i	$0.501651\pi$
$798$	−16.7183	−0.591820
$799$	−26.5914	−0.940736
$800$	38.1142	1.34754
$801$	−0.357039	−0.0126154
$802$	39.5277	1.39577
$803$	0	0
$804$	−40.3752	−1.42392
$805$	−2.77627	−0.0978508
$806$	−14.1504	−0.498428
$807$	27.6369	0.972865
$808$	11.1874	0.393571
$809$	−11.0373	−0.388050	−0.194025	−	0.980997i	$-0.562154\pi$
$809$	−11.0373	−0.388050	−0.194025	+	0.980997i	$0.562154\pi$
$810$	−8.97555	−0.315369
$811$	−5.21763	−0.183216	−0.0916078	−	0.995795i	$-0.529201\pi$
$811$	−5.21763	−0.183216	−0.0916078	+	0.995795i	$0.529201\pi$
$812$	16.9579	0.595106
$813$	−9.91799	−0.347839
$814$	0	0
$815$	3.87957	0.135896
$816$	−14.9546	−0.523516
$817$	−13.4492	−0.470527
$818$	33.1381	1.15865
$819$	−0.609817	−0.0213087
$820$	−1.74070	−0.0607878
$821$	26.4468	0.923000	0.461500	−	0.887140i	$-0.347311\pi$
$821$	26.4468	0.923000	0.461500	+	0.887140i	$0.347311\pi$
$822$	−5.86767	−0.204658
$823$	−49.1895	−1.71464	−0.857319	−	0.514786i	$-0.827871\pi$
$823$	−49.1895	−1.71464	−0.857319	+	0.514786i	$0.827871\pi$
$824$	6.64284	0.231414
$825$	0	0
$826$	−17.6289	−0.613387
$827$	−41.8006	−1.45355	−0.726774	−	0.686876i	$-0.758981\pi$
$827$	−41.8006	−1.45355	−0.726774	+	0.686876i	$0.758981\pi$
$828$	1.58903	0.0552227
$829$	−24.8988	−0.864771	−0.432385	−	0.901689i	$-0.642328\pi$
$829$	−24.8988	−0.864771	−0.432385	+	0.901689i	$0.642328\pi$
$830$	11.5184	0.399808
$831$	27.0496	0.938340
$832$	−58.8395	−2.03989
$833$	3.03721	0.105233
$834$	−5.49863	−0.190402
$835$	−1.01179	−0.0350145
$836$	0	0
$837$	6.69650	0.231465
$838$	47.6794	1.64706
$839$	−53.2912	−1.83982	−0.919908	−	0.392135i	$-0.871737\pi$
$839$	−53.2912	−1.83982	−0.919908	+	0.392135i	$0.871737\pi$
$840$	0.793021	0.0273618
$841$	18.9273	0.652666
$842$	−49.2143	−1.69604
$843$	17.3255	0.596721
$844$	5.34736	0.184064
$845$	7.43171	0.255659
$846$	2.12483	0.0730530
$847$	0	0
$848$	−19.2439	−0.660837
$849$	−27.4537	−0.942207
$850$	−30.4800	−1.04545
$851$	−61.3219	−2.10209
$852$	24.7448	0.847744
$853$	13.3706	0.457800	0.228900	−	0.973450i	$-0.426487\pi$
$853$	13.3706	0.457800	0.228900	+	0.973450i	$0.426487\pi$
$854$	−29.3106	−1.00299
$855$	0.264348	0.00904050
$856$	10.7667	0.367997
$857$	43.4419	1.48395	0.741974	−	0.670429i	$-0.233890\pi$
$857$	43.4419	1.48395	0.741974	+	0.670429i	$0.233890\pi$
$858$	0	0
$859$	−7.96898	−0.271898	−0.135949	−	0.990716i	$-0.543408\pi$
$859$	−7.96898	−0.271898	−0.135949	+	0.990716i	$0.543408\pi$
$860$	3.47633	0.118542
$861$	−2.45133	−0.0835410
$862$	−20.1214	−0.685338
$863$	50.3471	1.71384	0.856918	−	0.515452i	$-0.172376\pi$
$863$	50.3471	1.71384	0.856918	+	0.515452i	$0.172376\pi$
$864$	42.3875	1.44205
$865$	−11.4583	−0.389593
$866$	−48.5381	−1.64939
$867$	−13.2066	−0.448519
$868$	−3.10023	−0.105229
$869$	0	0
$870$	12.2131	0.414064
$871$	−51.4357	−1.74283
$872$	17.9605	0.608218
$873$	0.726765	0.0245973
$874$	55.4976	1.87723
$875$	−4.80453	−0.162423
$876$	−15.7027	−0.530546
$877$	−55.1801	−1.86330	−0.931649	−	0.363359i	$-0.881630\pi$
$877$	−55.1801	−1.86330	−0.931649	+	0.363359i	$0.881630\pi$
$878$	−58.3106	−1.96789
$879$	45.9476	1.54978
$880$	0	0
$881$	−22.6475	−0.763014	−0.381507	−	0.924366i	$-0.624595\pi$
$881$	−22.6475	−0.763014	−0.381507	+	0.924366i	$0.624595\pi$
$882$	−0.242693	−0.00817188
$883$	−12.7050	−0.427557	−0.213779	−	0.976882i	$-0.568577\pi$
$883$	−12.7050	−0.427557	−0.213779	+	0.976882i	$0.568577\pi$
$884$	39.4326	1.32626
$885$	−6.98951	−0.234950
$886$	31.1585	1.04679
$887$	6.70004	0.224965	0.112483	−	0.993654i	$-0.464120\pi$
$887$	6.70004	0.224965	0.112483	+	0.993654i	$0.464120\pi$
$888$	17.5161	0.587802
$889$	−4.66064	−0.156313
$890$	3.22317	0.108041
$891$	0	0
$892$	66.7747	2.23578
$893$	40.8538	1.36712
$894$	48.3357	1.61659
$895$	8.69542	0.290656
$896$	−7.39409	−0.247019
$897$	−50.7599	−1.69482
$898$	63.8446	2.13052
$899$	−8.76203	−0.292230
$900$	1.34080	0.0446934
$901$	20.1620	0.671695
$902$	0	0
$903$	4.89552	0.162913
$904$	−12.7972	−0.425628
$905$	7.61733	0.253209
$906$	−43.8431	−1.45659
$907$	−57.7184	−1.91651	−0.958254	−	0.285918i	$-0.907701\pi$
$907$	−57.7184	−1.91651	−0.958254	+	0.285918i	$0.907701\pi$
$908$	−33.2869	−1.10467
$909$	−1.35745	−0.0450237
$910$	5.50512	0.182493
$911$	−2.84362	−0.0942134	−0.0471067	−	0.998890i	$-0.515000\pi$
$911$	−2.84362	−0.0942134	−0.0471067	+	0.998890i	$0.515000\pi$
$912$	22.9756	0.760798
$913$	0	0
$914$	−42.7288	−1.41334
$915$	−11.6211	−0.384181
$916$	−20.3345	−0.671870
$917$	9.03676	0.298420
$918$	−33.8973	−1.11878
$919$	−9.36824	−0.309030	−0.154515	−	0.987990i	$-0.549381\pi$
$919$	−9.36824	−0.309030	−0.154515	+	0.987990i	$0.549381\pi$
$920$	−2.63250	−0.0867908
$921$	50.4884	1.66365
$922$	17.9548	0.591309
$923$	31.5235	1.03761
$924$	0	0
$925$	−51.7424	−1.70128
$926$	−1.24764	−0.0410000
$927$	−0.806025	−0.0264733
$928$	−55.4620	−1.82063
$929$	−43.8437	−1.43847	−0.719233	−	0.694769i	$-0.755506\pi$
$929$	−43.8437	−1.43847	−0.719233	+	0.694769i	$0.755506\pi$
$930$	−2.23280	−0.0732162
$931$	−4.66622	−0.152929
$932$	31.7155	1.03888
$933$	38.0516	1.24575
$934$	−86.5581	−2.83227
$935$	0	0
$936$	−0.578236	−0.0189002
$937$	3.20925	0.104842	0.0524208	−	0.998625i	$-0.483306\pi$
$937$	3.20925	0.104842	0.0524208	+	0.998625i	$0.483306\pi$
$938$	−20.4702	−0.668374
$939$	16.8974	0.551425
$940$	−10.5599	−0.344425
$941$	11.1320	0.362893	0.181446	−	0.983401i	$-0.441922\pi$
$941$	11.1320	0.362893	0.181446	+	0.983401i	$0.441922\pi$
$942$	9.03022	0.294221
$943$	8.13737	0.264989
$944$	24.2270	0.788522
$945$	−2.60522	−0.0847479
$946$	0	0
$947$	51.6934	1.67981	0.839905	−	0.542733i	$-0.182610\pi$
$947$	51.6934	1.67981	0.839905	+	0.542733i	$0.182610\pi$
$948$	−36.6287	−1.18964
$949$	−20.0044	−0.649370
$950$	46.8280	1.51930
$951$	−18.9248	−0.613679
$952$	2.87991	0.0933386
$953$	28.0305	0.907996	0.453998	−	0.891003i	$-0.349997\pi$
$953$	28.0305	0.907996	0.453998	+	0.891003i	$0.349997\pi$
$954$	−1.61108	−0.0521606
$955$	7.84798	0.253955
$956$	4.63796	0.150002
$957$	0	0
$958$	42.9127	1.38645
$959$	−1.63772	−0.0528848
$960$	−9.28429	−0.299649
$961$	−29.3981	−0.948327
$962$	121.596	3.92042
$963$	−1.30640	−0.0420981
$964$	−28.6553	−0.922926
$965$	4.16261	0.133999
$966$	−20.2012	−0.649963
$967$	−25.5912	−0.822957	−0.411479	−	0.911419i	$-0.634988\pi$
$967$	−25.5912	−0.822957	−0.411479	+	0.911419i	$0.634988\pi$
$968$	0	0
$969$	−24.0718	−0.773298
$970$	−6.56086	−0.210657
$971$	12.5737	0.403508	0.201754	−	0.979436i	$-0.435336\pi$
$971$	12.5737	0.403508	0.201754	+	0.979436i	$0.435336\pi$
$972$	2.92718	0.0938895
$973$	−1.53472	−0.0492009
$974$	−60.7852	−1.94768
$975$	−42.8304	−1.37167
$976$	40.2809	1.28936
$977$	−14.5164	−0.464420	−0.232210	−	0.972666i	$-0.574596\pi$
$977$	−14.5164	−0.464420	−0.232210	+	0.972666i	$0.574596\pi$
$978$	28.2292	0.902671
$979$	0	0
$980$	1.20612	0.0385281
$981$	−2.17928	−0.0695789
$982$	5.65410	0.180430
$983$	−12.9883	−0.414264	−0.207132	−	0.978313i	$-0.566413\pi$
$983$	−12.9883	−0.414264	−0.207132	+	0.978313i	$0.566413\pi$
$984$	−2.32438	−0.0740984
$985$	−7.06012	−0.224954
$986$	44.3529	1.41249
$987$	−14.8709	−0.473344
$988$	−60.5824	−1.92738
$989$	−16.2511	−0.516754
$990$	0	0
$991$	7.01006	0.222682	0.111341	−	0.993782i	$-0.464485\pi$
$991$	7.01006	0.222682	0.111341	+	0.993782i	$0.464485\pi$
$992$	10.1395	0.321930
$993$	24.7898	0.786681
$994$	12.5456	0.397922
$995$	−10.8987	−0.345512
$996$	46.1396	1.46199
$997$	−31.8789	−1.00961	−0.504807	−	0.863232i	$-0.668436\pi$
$997$	−31.8789	−1.00961	−0.504807	+	0.863232i	$0.668436\pi$
$998$	47.1291	1.49185
$999$	−57.5437	−1.82060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.n.1.5	yes	6
3.2	odd	2		7623.2.a.cp.1.2		6
7.6	odd	2		5929.2.a.bm.1.5		6
11.2	odd	10		847.2.f.z.323.5		24
11.3	even	5		847.2.f.y.372.5		24
11.4	even	5		847.2.f.y.148.5		24
11.5	even	5		847.2.f.y.729.2		24
11.6	odd	10		847.2.f.z.729.5		24
11.7	odd	10		847.2.f.z.148.2		24
11.8	odd	10		847.2.f.z.372.2		24
11.9	even	5		847.2.f.y.323.2		24
11.10	odd	2		847.2.a.m.1.2	✓	6
33.32	even	2		7623.2.a.cs.1.5		6
77.76	even	2		5929.2.a.bj.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.m.1.2	✓	6	11.10	odd	2
847.2.a.n.1.5	yes	6	1.1	even	1	trivial
847.2.f.y.148.5		24	11.4	even	5
847.2.f.y.323.2		24	11.9	even	5
847.2.f.y.372.5		24	11.3	even	5
847.2.f.y.729.2		24	11.5	even	5
847.2.f.z.148.2		24	11.7	odd	10
847.2.f.z.323.5		24	11.2	odd	10
847.2.f.z.372.2		24	11.8	odd	10
847.2.f.z.729.5		24	11.6	odd	10
5929.2.a.bj.1.2		6	77.76	even	2
5929.2.a.bm.1.5		6	7.6	odd	2
7623.2.a.cp.1.2		6	3.2	odd	2
7623.2.a.cs.1.5		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)

\(f(q)\)	\(=\)	\(q+2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} +3.58282 q^{6} +1.00000 q^{7} +0.948212 q^{8} -0.115054 q^{9} +O(q^{10})\)	\(q+2.10939 q^{2} +1.69851 q^{3} +2.44952 q^{4} +0.492391 q^{5} +3.58282 q^{6} +1.00000 q^{7} +0.948212 q^{8} -0.115054 q^{9} +1.03864 q^{10} +4.16054 q^{12} +5.30029 q^{13} +2.10939 q^{14} +0.836333 q^{15} -2.89889 q^{16} +3.03721 q^{17} -0.242693 q^{18} -4.66622 q^{19} +1.20612 q^{20} +1.69851 q^{21} -5.63835 q^{23} +1.61055 q^{24} -4.75755 q^{25} +11.1804 q^{26} -5.29096 q^{27} +2.44952 q^{28} +6.92295 q^{29} +1.76415 q^{30} -1.26565 q^{31} -8.01131 q^{32} +6.40665 q^{34} +0.492391 q^{35} -0.281826 q^{36} +10.8759 q^{37} -9.84288 q^{38} +9.00262 q^{39} +0.466891 q^{40} -1.44322 q^{41} +3.58282 q^{42} +2.88224 q^{43} -0.0566513 q^{45} -11.8935 q^{46} -8.75522 q^{47} -4.92381 q^{48} +1.00000 q^{49} -10.0355 q^{50} +5.15873 q^{51} +12.9832 q^{52} +6.63835 q^{53} -11.1607 q^{54} +0.948212 q^{56} -7.92564 q^{57} +14.6032 q^{58} -8.35733 q^{59} +2.04861 q^{60} -13.8953 q^{61} -2.66975 q^{62} -0.115054 q^{63} -11.1012 q^{64} +2.60982 q^{65} -9.70431 q^{67} +7.43970 q^{68} -9.57681 q^{69} +1.03864 q^{70} +5.94751 q^{71} -0.109095 q^{72} -3.77421 q^{73} +22.9414 q^{74} -8.08076 q^{75} -11.4300 q^{76} +18.9900 q^{78} -8.80383 q^{79} -1.42739 q^{80} -8.64160 q^{81} -3.04431 q^{82} +11.0898 q^{83} +4.16054 q^{84} +1.49549 q^{85} +6.07976 q^{86} +11.7587 q^{87} +3.10324 q^{89} -0.119500 q^{90} +5.30029 q^{91} -13.8113 q^{92} -2.14972 q^{93} -18.4682 q^{94} -2.29761 q^{95} -13.6073 q^{96} -6.31676 q^{97} +2.10939 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 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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