LMFDB - Embedded newform 847.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

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

Embedding invariants

Embedding label			1.4
Root			$0.226211$ 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)

$f(q)$	$=$	$q+0.226211 q^{2} -0.219130 q^{3} -1.94883 q^{4} -2.49552 q^{5} -0.0495696 q^{6} +1.00000 q^{7} -0.893270 q^{8} -2.95198 q^{9} +O(q^{10})$	\(q+0.226211 q^{2} -0.219130 q^{3} -1.94883 q^{4} -2.49552 q^{5} -0.0495696 q^{6} +1.00000 q^{7} -0.893270 q^{8} -2.95198 q^{9} -0.564516 q^{10} +0.427046 q^{12} +5.13499 q^{13} +0.226211 q^{14} +0.546843 q^{15} +3.69559 q^{16} -1.43752 q^{17} -0.667772 q^{18} -6.06848 q^{19} +4.86335 q^{20} -0.219130 q^{21} +7.08292 q^{23} +0.195742 q^{24} +1.22764 q^{25} +1.16159 q^{26} +1.30426 q^{27} -1.94883 q^{28} +6.51769 q^{29} +0.123702 q^{30} +7.68895 q^{31} +2.62252 q^{32} -0.325184 q^{34} -2.49552 q^{35} +5.75291 q^{36} -3.98432 q^{37} -1.37276 q^{38} -1.12523 q^{39} +2.22918 q^{40} +6.74900 q^{41} -0.0495696 q^{42} -0.802299 q^{43} +7.36674 q^{45} +1.60224 q^{46} +6.75222 q^{47} -0.809813 q^{48} +1.00000 q^{49} +0.277706 q^{50} +0.315004 q^{51} -10.0072 q^{52} +6.58167 q^{53} +0.295037 q^{54} -0.893270 q^{56} +1.32978 q^{57} +1.47437 q^{58} +2.87625 q^{59} -1.06570 q^{60} -0.855342 q^{61} +1.73933 q^{62} -2.95198 q^{63} -6.79793 q^{64} -12.8145 q^{65} -1.64668 q^{67} +2.80149 q^{68} -1.55208 q^{69} -0.564516 q^{70} +4.52077 q^{71} +2.63692 q^{72} -14.8479 q^{73} -0.901299 q^{74} -0.269012 q^{75} +11.8264 q^{76} -0.254539 q^{78} +2.45291 q^{79} -9.22243 q^{80} +8.57015 q^{81} +1.52670 q^{82} +2.24780 q^{83} +0.427046 q^{84} +3.58738 q^{85} -0.181489 q^{86} -1.42822 q^{87} +1.73566 q^{89} +1.66644 q^{90} +5.13499 q^{91} -13.8034 q^{92} -1.68488 q^{93} +1.52743 q^{94} +15.1440 q^{95} -0.574672 q^{96} -12.0776 q^{97} +0.226211 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9}+O(q^{10}) $ 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9	$ 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + 6 q^{10} + 9 q^{12} - 6 q^{13} + q^{14} + 11 q^{15} + q^{16} - 5 q^{17} + 8 q^{18} - 13 q^{19} + 23 q^{20} + 4 q^{21} + 16 q^{23} - 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} + 7 q^{28} + 9 q^{29} - 36 q^{30} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} + 13 q^{39} + 5 q^{40} - 10 q^{41} - q^{42} - 4 q^{43} + 35 q^{45} + 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} + 6 q^{50} + 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} + 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} + 14 q^{63} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} - 35 q^{72} - 25 q^{73} + 33 q^{74} - 13 q^{75} + 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} - 25 q^{83} + 9 q^{84} + 3 q^{85} + 4 q^{86} - 36 q^{87} + 37 q^{89} - 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} - 42 q^{94} + 21 q^{95} - 6 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) $ 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9 + 6 * q^10 + 9 * q^12 - 6 * q^13 + q^14 + 11 * q^15 + q^16 - 5 * q^17 + 8 * q^18 - 13 * q^19 + 23 * q^20 + 4 * q^21 + 16 * q^23 - 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 + 7 * q^28 + 9 * q^29 - 36 * q^30 + 9 * q^31 + 16 * q^32 - 12 * q^34 + 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 + 13 * q^39 + 5 * q^40 - 10 * q^41 - q^42 - 4 * q^43 + 35 * q^45 + 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 + 6 * q^50 + 13 * q^51 - 41 * q^52 + 37 * q^53 + 30 * q^54 + 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 + 19 * q^61 - 18 * q^62 + 14 * q^63 - 4 * q^64 - 4 * q^65 - 19 * q^67 + 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 - 35 * q^72 - 25 * q^73 + 33 * q^74 - 13 * q^75 + 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 - 25 * q^83 + 9 * q^84 + 3 * q^85 + 4 * q^86 - 36 * q^87 + 37 * q^89 - 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 - 42 * q^94 + 21 * q^95 - 6 * q^96 + 15 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.226211	0.159956	0.0799778	−	0.996797i	$-0.474515\pi$
$2$	0.226211	0.159956	0.0799778	+	0.996797i	$0.474515\pi$
$3$	−0.219130	−0.126514	−0.0632572	−	0.997997i	$-0.520149\pi$
$3$	−0.219130	−0.126514	−0.0632572	+	0.997997i	$0.520149\pi$
$4$	−1.94883	−0.974414
$5$	−2.49552	−1.11603	−0.558016	−	0.829830i	$-0.688437\pi$
$5$	−2.49552	−1.11603	−0.558016	+	0.829830i	$0.688437\pi$
$6$	−0.0495696	−0.0202367
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−0.893270	−0.315819
$9$	−2.95198	−0.983994
$10$	−0.564516	−0.178516
$11$	0	0
$11$	0	0
$12$	0.427046	0.123278
$13$	5.13499	1.42419	0.712094	−	0.702084i	$-0.247747\pi$
$13$	5.13499	1.42419	0.712094	+	0.702084i	$0.247747\pi$
$14$	0.226211	0.0604575
$15$	0.546843	0.141194
$16$	3.69559	0.923897
$17$	−1.43752	−0.348651	−0.174325	−	0.984688i	$-0.555774\pi$
$17$	−1.43752	−0.348651	−0.174325	+	0.984688i	$0.555774\pi$
$18$	−0.667772	−0.157395
$19$	−6.06848	−1.39220	−0.696102	−	0.717943i	$-0.745084\pi$
$19$	−6.06848	−1.39220	−0.696102	+	0.717943i	$0.745084\pi$
$20$	4.86335	1.08748
$21$	−0.219130	−0.0478180
$22$	0	0
$23$	7.08292	1.47689	0.738446	−	0.674313i	$-0.235560\pi$
$23$	7.08292	1.47689	0.738446	+	0.674313i	$0.235560\pi$
$24$	0.195742	0.0399556
$25$	1.22764	0.245528
$26$	1.16159	0.227807
$27$	1.30426	0.251004
$28$	−1.94883	−0.368294
$29$	6.51769	1.21030	0.605152	−	0.796110i	$-0.293112\pi$
$29$	6.51769	1.21030	0.605152	+	0.796110i	$0.293112\pi$
$30$	0.123702	0.0225848
$31$	7.68895	1.38098	0.690488	−	0.723344i	$-0.257396\pi$
$31$	7.68895	1.38098	0.690488	+	0.723344i	$0.257396\pi$
$32$	2.62252	0.463601
$33$	0	0
$34$	−0.325184	−0.0557687
$35$	−2.49552	−0.421820
$36$	5.75291	0.958818
$37$	−3.98432	−0.655019	−0.327509	−	0.944848i	$-0.606209\pi$
$37$	−3.98432	−0.655019	−0.327509	+	0.944848i	$0.606209\pi$
$38$	−1.37276	−0.222691
$39$	−1.12523	−0.180181
$40$	2.22918	0.352464
$41$	6.74900	1.05402	0.527008	−	0.849860i	$-0.323314\pi$
$41$	6.74900	1.05402	0.527008	+	0.849860i	$0.323314\pi$
$42$	−0.0495696	−0.00764875
$43$	−0.802299	−0.122349	−0.0611747	−	0.998127i	$-0.519485\pi$
$43$	−0.802299	−0.122349	−0.0611747	+	0.998127i	$0.519485\pi$
$44$	0	0
$45$	7.36674	1.09817
$46$	1.60224	0.236237
$47$	6.75222	0.984912	0.492456	−	0.870337i	$-0.336099\pi$
$47$	6.75222	0.984912	0.492456	+	0.870337i	$0.336099\pi$
$48$	−0.809813	−0.116886
$49$	1.00000	0.142857
$50$	0.277706	0.0392735
$51$	0.315004	0.0441094
$52$	−10.0072	−1.38775
$53$	6.58167	0.904062	0.452031	−	0.892002i	$-0.350700\pi$
$53$	6.58167	0.904062	0.452031	+	0.892002i	$0.350700\pi$
$54$	0.295037	0.0401495
$55$	0	0
$56$	−0.893270	−0.119368
$57$	1.32978	0.176134
$58$	1.47437	0.193595
$59$	2.87625	0.374456	0.187228	−	0.982316i	$-0.440050\pi$
$59$	2.87625	0.374456	0.187228	+	0.982316i	$0.440050\pi$
$60$	−1.06570	−0.137582
$61$	−0.855342	−0.109515	−0.0547576	−	0.998500i	$-0.517439\pi$
$61$	−0.855342	−0.109515	−0.0547576	+	0.998500i	$0.517439\pi$
$62$	1.73933	0.220895
$63$	−2.95198	−0.371915
$64$	−6.79793	−0.849742
$65$	−12.8145	−1.58944
$66$	0	0
$67$	−1.64668	−0.201174	−0.100587	−	0.994928i	$-0.532072\pi$
$67$	−1.64668	−0.201174	−0.100587	+	0.994928i	$0.532072\pi$
$68$	2.80149	0.339730
$69$	−1.55208	−0.186848
$70$	−0.564516	−0.0674725
$71$	4.52077	0.536517	0.268258	−	0.963347i	$-0.413552\pi$
$71$	4.52077	0.536517	0.268258	+	0.963347i	$0.413552\pi$
$72$	2.63692	0.310764
$73$	−14.8479	−1.73782	−0.868910	−	0.494970i	$-0.835179\pi$
$73$	−14.8479	−1.73782	−0.868910	+	0.494970i	$0.835179\pi$
$74$	−0.901299	−0.104774
$75$	−0.269012	−0.0310628
$76$	11.8264	1.35658
$77$	0	0
$78$	−0.254539	−0.0288209
$79$	2.45291	0.275973	0.137987	−	0.990434i	$-0.455937\pi$
$79$	2.45291	0.275973	0.137987	+	0.990434i	$0.455937\pi$
$80$	−9.22243	−1.03110
$81$	8.57015	0.952238
$82$	1.52670	0.168596
$83$	2.24780	0.246728	0.123364	−	0.992361i	$-0.460632\pi$
$83$	2.24780	0.246728	0.123364	+	0.992361i	$0.460632\pi$
$84$	0.427046	0.0465945
$85$	3.58738	0.389106
$86$	−0.181489	−0.0195705
$87$	−1.42822	−0.153121
$88$	0	0
$89$	1.73566	0.183980	0.0919898	−	0.995760i	$-0.470677\pi$
$89$	1.73566	0.183980	0.0919898	+	0.995760i	$0.470677\pi$
$90$	1.66644	0.175658
$91$	5.13499	0.538293
$92$	−13.8034	−1.43910
$93$	−1.68488	−0.174714
$94$	1.52743	0.157542
$95$	15.1440	1.55374
$96$	−0.574672	−0.0586523
$97$	−12.0776	−1.22629	−0.613145	−	0.789970i	$-0.710096\pi$
$97$	−12.0776	−1.22629	−0.613145	+	0.789970i	$0.710096\pi$
$98$	0.226211	0.0228508
$99$	0	0
$100$	−2.39246	−0.239246
$101$	−3.69338	−0.367505	−0.183753	−	0.982973i	$-0.558825\pi$
$101$	−3.69338	−0.367505	−0.183753	+	0.982973i	$0.558825\pi$
$102$	0.0712575	0.00705554
$103$	−1.15156	−0.113467	−0.0567334	−	0.998389i	$-0.518069\pi$
$103$	−1.15156	−0.113467	−0.0567334	+	0.998389i	$0.518069\pi$
$104$	−4.58693	−0.449785
$105$	0.546843	0.0533664
$106$	1.48885	0.144610
$107$	1.16714	0.112831	0.0564157	−	0.998407i	$-0.482033\pi$
$107$	1.16714	0.112831	0.0564157	+	0.998407i	$0.482033\pi$
$108$	−2.54177	−0.244582
$109$	9.30234	0.891003	0.445501	−	0.895281i	$-0.353025\pi$
$109$	9.30234	0.891003	0.445501	+	0.895281i	$0.353025\pi$
$110$	0	0
$111$	0.873083	0.0828694
$112$	3.69559	0.349200
$113$	3.29733	0.310187	0.155093	−	0.987900i	$-0.450432\pi$
$113$	3.29733	0.310187	0.155093	+	0.987900i	$0.450432\pi$
$114$	0.300812	0.0281736
$115$	−17.6756	−1.64826
$116$	−12.7019	−1.17934
$117$	−15.1584	−1.40139
$118$	0.650640	0.0598963
$119$	−1.43752	−0.131778
$120$	−0.488478	−0.0445918
$121$	0	0
$122$	−0.193488	−0.0175176
$123$	−1.47891	−0.133348
$124$	−14.9844	−1.34564
$125$	9.41402	0.842015
$126$	−0.667772	−0.0594898
$127$	−0.289205	−0.0256628	−0.0128314	−	0.999918i	$-0.504084\pi$
$127$	−0.289205	−0.0256628	−0.0128314	+	0.999918i	$0.504084\pi$
$128$	−6.78282	−0.599522
$129$	0.175807	0.0154790
$130$	−2.89878	−0.254240
$131$	−16.5059	−1.44212	−0.721062	−	0.692871i	$-0.756346\pi$
$131$	−16.5059	−1.44212	−0.721062	+	0.692871i	$0.756346\pi$
$132$	0	0
$133$	−6.06848	−0.526204
$134$	−0.372498	−0.0321789
$135$	−3.25480	−0.280129
$136$	1.28410	0.110110
$137$	9.32588	0.796763	0.398382	−	0.917220i	$-0.369572\pi$
$137$	9.32588	0.796763	0.398382	+	0.917220i	$0.369572\pi$
$138$	−0.351097	−0.0298874
$139$	4.82649	0.409377	0.204689	−	0.978827i	$-0.434382\pi$
$139$	4.82649	0.409377	0.204689	+	0.978827i	$0.434382\pi$
$140$	4.86335	0.411028
$141$	−1.47961	−0.124606
$142$	1.02265	0.0858189
$143$	0	0
$144$	−10.9093	−0.909109
$145$	−16.2650	−1.35074
$146$	−3.35877	−0.277974
$147$	−0.219130	−0.0180735
$148$	7.76476	0.638260
$149$	0.921915	0.0755262	0.0377631	−	0.999287i	$-0.487977\pi$
$149$	0.921915	0.0755262	0.0377631	+	0.999287i	$0.487977\pi$
$150$	−0.0608535	−0.00496867
$151$	18.0964	1.47267	0.736333	−	0.676620i	$-0.236556\pi$
$151$	18.0964	1.47267	0.736333	+	0.676620i	$0.236556\pi$
$152$	5.42079	0.439684
$153$	4.24355	0.343070
$154$	0	0
$155$	−19.1880	−1.54121
$156$	2.19287	0.175570
$157$	−12.2733	−0.979518	−0.489759	−	0.871858i	$-0.662915\pi$
$157$	−12.2733	−0.979518	−0.489759	+	0.871858i	$0.662915\pi$
$158$	0.554875	0.0441435
$159$	−1.44224	−0.114377
$160$	−6.54457	−0.517394
$161$	7.08292	0.558212
$162$	1.93866	0.152316
$163$	8.09913	0.634373	0.317186	−	0.948363i	$-0.397262\pi$
$163$	8.09913	0.634373	0.317186	+	0.948363i	$0.397262\pi$
$164$	−13.1526	−1.02705
$165$	0	0
$166$	0.508478	0.0394655
$167$	−13.0516	−1.00996	−0.504982	−	0.863130i	$-0.668501\pi$
$167$	−13.0516	−1.00996	−0.504982	+	0.863130i	$0.668501\pi$
$168$	0.195742	0.0151018
$169$	13.3681	1.02831
$170$	0.811505	0.0622396
$171$	17.9140	1.36992
$172$	1.56354	0.119219
$173$	5.91219	0.449495	0.224748	−	0.974417i	$-0.427844\pi$
$173$	5.91219	0.449495	0.224748	+	0.974417i	$0.427844\pi$
$174$	−0.323079	−0.0244926
$175$	1.22764	0.0928007
$176$	0	0
$177$	−0.630271	−0.0473741
$178$	0.392626	0.0294286
$179$	4.33508	0.324019	0.162009	−	0.986789i	$-0.448202\pi$
$179$	4.33508	0.324019	0.162009	+	0.986789i	$0.448202\pi$
$180$	−14.3565	−1.07007
$181$	10.8307	0.805040	0.402520	−	0.915411i	$-0.368134\pi$
$181$	10.8307	0.805040	0.402520	+	0.915411i	$0.368134\pi$
$182$	1.16159	0.0861029
$183$	0.187431	0.0138553
$184$	−6.32696	−0.466430
$185$	9.94297	0.731022
$186$	−0.381138	−0.0279464
$187$	0	0
$188$	−13.1589	−0.959713
$189$	1.30426	0.0948706
$190$	3.42575	0.248530
$191$	11.6556	0.843370	0.421685	−	0.906742i	$-0.361439\pi$
$191$	11.6556	0.843370	0.421685	+	0.906742i	$0.361439\pi$
$192$	1.48963	0.107505
$193$	22.4454	1.61566	0.807829	−	0.589418i	$-0.200643\pi$
$193$	22.4454	1.61566	0.807829	+	0.589418i	$0.200643\pi$
$194$	−2.73208	−0.196152
$195$	2.80803	0.201087
$196$	−1.94883	−0.139202
$197$	24.1022	1.71721	0.858604	−	0.512639i	$-0.171332\pi$
$197$	24.1022	1.71721	0.858604	+	0.512639i	$0.171332\pi$
$198$	0	0
$199$	18.7205	1.32706	0.663531	−	0.748148i	$-0.269057\pi$
$199$	18.7205	1.32706	0.663531	+	0.748148i	$0.269057\pi$
$200$	−1.09661	−0.0775422
$201$	0.360836	0.0254514
$202$	−0.835485	−0.0587845
$203$	6.51769	0.457452
$204$	−0.613889	−0.0429808
$205$	−16.8423	−1.17632
$206$	−0.260497	−0.0181497
$207$	−20.9087	−1.45325
$208$	18.9768	1.31580
$209$	0	0
$210$	0.123702	0.00853625
$211$	−7.56636	−0.520890	−0.260445	−	0.965489i	$-0.583869\pi$
$211$	−7.56636	−0.520890	−0.260445	+	0.965489i	$0.583869\pi$
$212$	−12.8266	−0.880931
$213$	−0.990634	−0.0678771
$214$	0.264019	0.0180480
$215$	2.00216	0.136546
$216$	−1.16505	−0.0792717
$217$	7.68895	0.521960
$218$	2.10430	0.142521
$219$	3.25362	0.219859
$220$	0	0
$221$	−7.38167	−0.496545
$222$	0.197501	0.0132554
$223$	17.5244	1.17352	0.586760	−	0.809761i	$-0.300403\pi$
$223$	17.5244	1.17352	0.586760	+	0.809761i	$0.300403\pi$
$224$	2.62252	0.175225
$225$	−3.62397	−0.241598
$226$	0.745893	0.0496161
$227$	−25.6773	−1.70426	−0.852130	−	0.523330i	$-0.824689\pi$
$227$	−25.6773	−1.70426	−0.852130	+	0.523330i	$0.824689\pi$
$228$	−2.59152	−0.171627
$229$	−19.8369	−1.31086	−0.655429	−	0.755257i	$-0.727512\pi$
$229$	−19.8369	−1.31086	−0.655429	+	0.755257i	$0.727512\pi$
$230$	−3.99842	−0.263648
$231$	0	0
$232$	−5.82205	−0.382236
$233$	−20.2146	−1.32430	−0.662151	−	0.749371i	$-0.730356\pi$
$233$	−20.2146	−1.32430	−0.662151	+	0.749371i	$0.730356\pi$
$234$	−3.42900	−0.224161
$235$	−16.8503	−1.09919
$236$	−5.60532	−0.364875
$237$	−0.537504	−0.0349146
$238$	−0.325184	−0.0210786
$239$	17.1004	1.10613	0.553066	−	0.833137i	$-0.313458\pi$
$239$	17.1004	1.10613	0.553066	+	0.833137i	$0.313458\pi$
$240$	2.02091	0.130449
$241$	−24.1529	−1.55582	−0.777912	−	0.628373i	$-0.783721\pi$
$241$	−24.1529	−1.55582	−0.777912	+	0.628373i	$0.783721\pi$
$242$	0	0
$243$	−5.79074	−0.371476
$244$	1.66691	0.106713
$245$	−2.49552	−0.159433
$246$	−0.334545	−0.0213298
$247$	−31.1615	−1.98276
$248$	−6.86831	−0.436138
$249$	−0.492559	−0.0312147
$250$	2.12956	0.134685
$251$	11.8947	0.750790	0.375395	−	0.926865i	$-0.377507\pi$
$251$	11.8947	0.750790	0.375395	+	0.926865i	$0.377507\pi$
$252$	5.75291	0.362399
$253$	0	0
$254$	−0.0654215	−0.00410491
$255$	−0.786100	−0.0492275
$256$	12.0615	0.753845
$257$	22.9609	1.43226	0.716131	−	0.697966i	$-0.245911\pi$
$257$	22.9609	1.43226	0.716131	+	0.697966i	$0.245911\pi$
$258$	0.0397696	0.00247595
$259$	−3.98432	−0.247574
$260$	24.9732	1.54877
$261$	−19.2401	−1.19093
$262$	−3.73381	−0.230676
$263$	1.93774	0.119486	0.0597432	−	0.998214i	$-0.480972\pi$
$263$	1.93774	0.119486	0.0597432	+	0.998214i	$0.480972\pi$
$264$	0	0
$265$	−16.4247	−1.00896
$266$	−1.37276	−0.0841692
$267$	−0.380335	−0.0232761
$268$	3.20910	0.196027
$269$	7.18676	0.438184	0.219092	−	0.975704i	$-0.429690\pi$
$269$	7.18676	0.438184	0.219092	+	0.975704i	$0.429690\pi$
$270$	−0.736273	−0.0448081
$271$	1.19110	0.0723543	0.0361772	−	0.999345i	$-0.488482\pi$
$271$	1.19110	0.0723543	0.0361772	+	0.999345i	$0.488482\pi$
$272$	−5.31250	−0.322118
$273$	−1.12523	−0.0681018
$274$	2.10962	0.127447
$275$	0	0
$276$	3.02473	0.182067
$277$	10.3402	0.621285	0.310642	−	0.950527i	$-0.399456\pi$
$277$	10.3402	0.621285	0.310642	+	0.950527i	$0.399456\pi$
$278$	1.09181	0.0654822
$279$	−22.6977	−1.35887
$280$	2.22918	0.133219
$281$	−13.1513	−0.784541	−0.392271	−	0.919850i	$-0.628310\pi$
$281$	−13.1513	−0.784541	−0.392271	+	0.919850i	$0.628310\pi$
$282$	−0.334705	−0.0199314
$283$	0.300031	0.0178350	0.00891748	−	0.999960i	$-0.497161\pi$
$283$	0.300031	0.0178350	0.00891748	+	0.999960i	$0.497161\pi$
$284$	−8.81021	−0.522790
$285$	−3.31850	−0.196571
$286$	0	0
$287$	6.74900	0.398381
$288$	−7.74164	−0.456181
$289$	−14.9335	−0.878443
$290$	−3.67934	−0.216058
$291$	2.64655	0.155143
$292$	28.9361	1.69336
$293$	16.1441	0.943148	0.471574	−	0.881826i	$-0.343686\pi$
$293$	16.1441	0.943148	0.471574	+	0.881826i	$0.343686\pi$
$294$	−0.0495696	−0.00289096
$295$	−7.17775	−0.417905
$296$	3.55908	0.206867
$297$	0	0
$298$	0.208548	0.0120808
$299$	36.3707	2.10337
$300$	0.524258	0.0302680
$301$	−0.802299	−0.0462437
$302$	4.09361	0.235561
$303$	0.809329	0.0464947
$304$	−22.4266	−1.28625
$305$	2.13453	0.122223
$306$	0.959939	0.0548760
$307$	−28.6376	−1.63443	−0.817217	−	0.576330i	$-0.804484\pi$
$307$	−28.6376	−1.63443	−0.817217	+	0.576330i	$0.804484\pi$
$308$	0	0
$309$	0.252341	0.0143552
$310$	−4.34053	−0.246526
$311$	−31.8228	−1.80450	−0.902252	−	0.431210i	$-0.858087\pi$
$311$	−31.8228	−1.80450	−0.902252	+	0.431210i	$0.858087\pi$
$312$	1.00513	0.0569044
$313$	0.0342232	0.00193441	0.000967206	−	1.00000i	$-0.499692\pi$
$313$	0.0342232	0.00193441	0.000967206		1.00000i	$0.499692\pi$
$314$	−2.77636	−0.156679
$315$	7.36674	0.415069
$316$	−4.78029	−0.268912
$317$	22.3894	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$317$	22.3894	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$318$	−0.326251	−0.0182952
$319$	0	0
$320$	16.9644	0.948339
$321$	−0.255754	−0.0142748
$322$	1.60224	0.0892892
$323$	8.72359	0.485393
$324$	−16.7017	−0.927875
$325$	6.30391	0.349678
$326$	1.83211	0.101471
$327$	−2.03842	−0.112725
$328$	−6.02868	−0.332878
$329$	6.75222	0.372262
$330$	0	0
$331$	10.7577	0.591297	0.295648	−	0.955297i	$-0.404464\pi$
$331$	10.7577	0.591297	0.295648	+	0.955297i	$0.404464\pi$
$332$	−4.38057	−0.240415
$333$	11.7617	0.644535
$334$	−2.95242	−0.161549
$335$	4.10933	0.224517
$336$	−0.809813	−0.0441789
$337$	7.50492	0.408819	0.204410	−	0.978885i	$-0.434473\pi$
$337$	7.50492	0.408819	0.204410	+	0.978885i	$0.434473\pi$
$338$	3.02401	0.164485
$339$	−0.722542	−0.0392431
$340$	−6.99118	−0.379150
$341$	0	0
$342$	4.05236	0.219126
$343$	1.00000	0.0539949
$344$	0.716669	0.0386402
$345$	3.87325	0.208529
$346$	1.33740	0.0718993
$347$	−27.6818	−1.48604	−0.743019	−	0.669271i	$-0.766607\pi$
$347$	−27.6818	−1.48604	−0.743019	+	0.669271i	$0.766607\pi$
$348$	2.78335	0.149203
$349$	−11.0605	−0.592055	−0.296028	−	0.955179i	$-0.595662\pi$
$349$	−11.0605	−0.592055	−0.296028	+	0.955179i	$0.595662\pi$
$350$	0.277706	0.0148440
$351$	6.69733	0.357477
$352$	0	0
$353$	31.9202	1.69894	0.849469	−	0.527638i	$-0.176922\pi$
$353$	31.9202	1.69894	0.849469	+	0.527638i	$0.176922\pi$
$354$	−0.142575	−0.00757775
$355$	−11.2817	−0.598770
$356$	−3.38251	−0.179272
$357$	0.315004	0.0166718
$358$	0.980644	0.0518286
$359$	3.57826	0.188854	0.0944268	−	0.995532i	$-0.469898\pi$
$359$	3.57826	0.188854	0.0944268	+	0.995532i	$0.469898\pi$
$360$	−6.58049	−0.346822
$361$	17.8264	0.938233
$362$	2.45003	0.128771
$363$	0	0
$364$	−10.0072	−0.524520
$365$	37.0534	1.93946
$366$	0.0423989	0.00221623
$367$	2.29397	0.119744	0.0598720	−	0.998206i	$-0.480931\pi$
$367$	2.29397	0.119744	0.0598720	+	0.998206i	$0.480931\pi$
$368$	26.1756	1.36450
$369$	−19.9229	−1.03715
$370$	2.24921	0.116931
$371$	6.58167	0.341703
$372$	3.28354	0.170243
$373$	7.96856	0.412596	0.206298	−	0.978489i	$-0.433858\pi$
$373$	7.96856	0.412596	0.206298	+	0.978489i	$0.433858\pi$
$374$	0	0
$375$	−2.06289	−0.106527
$376$	−6.03155	−0.311054
$377$	33.4682	1.72370
$378$	0.295037	0.0151751
$379$	−11.6212	−0.596941	−0.298470	−	0.954419i	$-0.596476\pi$
$379$	−11.6212	−0.596941	−0.298470	+	0.954419i	$0.596476\pi$
$380$	−29.5131	−1.51399
$381$	0.0633734	0.00324672
$382$	2.63663	0.134902
$383$	12.5785	0.642729	0.321364	−	0.946956i	$-0.395859\pi$
$383$	12.5785	0.642729	0.321364	+	0.946956i	$0.395859\pi$
$384$	1.48632	0.0758482
$385$	0	0
$386$	5.07741	0.258433
$387$	2.36837	0.120391
$388$	23.5371	1.19491
$389$	−0.438294	−0.0222224	−0.0111112	−	0.999938i	$-0.503537\pi$
$389$	−0.438294	−0.0222224	−0.0111112	+	0.999938i	$0.503537\pi$
$390$	0.635208	0.0321650
$391$	−10.1819	−0.514920
$392$	−0.893270	−0.0451169
$393$	3.61692	0.182450
$394$	5.45218	0.274677
$395$	−6.12128	−0.307995
$396$	0	0
$397$	16.8147	0.843905	0.421952	−	0.906618i	$-0.361345\pi$
$397$	16.8147	0.843905	0.421952	+	0.906618i	$0.361345\pi$
$398$	4.23480	0.212271
$399$	1.32978	0.0665724
$400$	4.53685	0.226842
$401$	36.8609	1.84074	0.920372	−	0.391043i	$-0.127886\pi$
$401$	36.8609	1.84074	0.920372	+	0.391043i	$0.127886\pi$
$402$	0.0816253	0.00407110
$403$	39.4827	1.96677
$404$	7.19776	0.358102
$405$	−21.3870	−1.06273
$406$	1.47437	0.0731720
$407$	0	0
$408$	−0.281384	−0.0139306
$409$	16.2739	0.804692	0.402346	−	0.915488i	$-0.368195\pi$
$409$	16.2739	0.804692	0.402346	+	0.915488i	$0.368195\pi$
$410$	−3.80992	−0.188158
$411$	−2.04357	−0.100802
$412$	2.24420	0.110564
$413$	2.87625	0.141531
$414$	−4.72978	−0.232456
$415$	−5.60944	−0.275356
$416$	13.4666	0.660255
$417$	−1.05763	−0.0517922
$418$	0	0
$419$	5.56352	0.271796	0.135898	−	0.990723i	$-0.456608\pi$
$419$	5.56352	0.271796	0.135898	+	0.990723i	$0.456608\pi$
$420$	−1.06570	−0.0520010
$421$	21.4914	1.04743	0.523713	−	0.851895i	$-0.324546\pi$
$421$	21.4914	1.04743	0.523713	+	0.851895i	$0.324546\pi$
$422$	−1.71160	−0.0833192
$423$	−19.9324	−0.969148
$424$	−5.87921	−0.285520
$425$	−1.76476	−0.0856035
$426$	−0.224093	−0.0108573
$427$	−0.855342	−0.0413929
$428$	−2.27455	−0.109944
$429$	0	0
$430$	0.452910	0.0218413
$431$	−27.7188	−1.33517	−0.667583	−	0.744536i	$-0.732671\pi$
$431$	−27.7188	−1.33517	−0.667583	+	0.744536i	$0.732671\pi$
$432$	4.81999	0.231902
$433$	9.28812	0.446358	0.223179	−	0.974777i	$-0.428356\pi$
$433$	9.28812	0.446358	0.223179	+	0.974777i	$0.428356\pi$
$434$	1.73933	0.0834904
$435$	3.56415	0.170888
$436$	−18.1287	−0.868206
$437$	−42.9826	−2.05613
$438$	0.736006	0.0351678
$439$	−14.7118	−0.702156	−0.351078	−	0.936346i	$-0.614185\pi$
$439$	−14.7118	−0.702156	−0.351078	+	0.936346i	$0.614185\pi$
$440$	0	0
$441$	−2.95198	−0.140571
$442$	−1.66982	−0.0794251
$443$	2.44345	0.116092	0.0580459	−	0.998314i	$-0.481513\pi$
$443$	2.44345	0.116092	0.0580459	+	0.998314i	$0.481513\pi$
$444$	−1.70149	−0.0807491
$445$	−4.33138	−0.205327
$446$	3.96421	0.187711
$447$	−0.202019	−0.00955516
$448$	−6.79793	−0.321172
$449$	4.76935	0.225080	0.112540	−	0.993647i	$-0.464101\pi$
$449$	4.76935	0.225080	0.112540	+	0.993647i	$0.464101\pi$
$450$	−0.819782	−0.0386449
$451$	0	0
$452$	−6.42593	−0.302250
$453$	−3.96546	−0.186313
$454$	−5.80849	−0.272606
$455$	−12.8145	−0.600752
$456$	−1.18785	−0.0556264
$457$	31.0875	1.45421	0.727106	−	0.686525i	$-0.240865\pi$
$457$	31.0875	1.45421	0.727106	+	0.686525i	$0.240865\pi$
$458$	−4.48733	−0.209679
$459$	−1.87490	−0.0875128
$460$	34.4467	1.60609
$461$	29.7215	1.38427	0.692134	−	0.721769i	$-0.256671\pi$
$461$	29.7215	1.38427	0.692134	+	0.721769i	$0.256671\pi$
$462$	0	0
$463$	−25.4553	−1.18301	−0.591505	−	0.806302i	$-0.701466\pi$
$463$	−25.4553	−1.18301	−0.591505	+	0.806302i	$0.701466\pi$
$464$	24.0867	1.11820
$465$	4.20465	0.194986
$466$	−4.57277	−0.211829
$467$	−3.18491	−0.147380	−0.0736901	−	0.997281i	$-0.523478\pi$
$467$	−3.18491	−0.147380	−0.0736901	+	0.997281i	$0.523478\pi$
$468$	29.5411	1.36554
$469$	−1.64668	−0.0760366
$470$	−3.81173	−0.175822
$471$	2.68945	0.123923
$472$	−2.56927	−0.118260
$473$	0	0
$474$	−0.121590	−0.00558479
$475$	−7.44990	−0.341825
$476$	2.80149	0.128406
$477$	−19.4290	−0.889592
$478$	3.86830	0.176932
$479$	3.23354	0.147744	0.0738720	−	0.997268i	$-0.476464\pi$
$479$	3.23354	0.147744	0.0738720	+	0.997268i	$0.476464\pi$
$480$	1.43411	0.0654578
$481$	−20.4594	−0.932870
$482$	−5.46366	−0.248863
$483$	−1.55208	−0.0706220
$484$	0	0
$485$	30.1398	1.36858
$486$	−1.30993	−0.0594197
$487$	9.87096	0.447296	0.223648	−	0.974670i	$-0.428203\pi$
$487$	9.87096	0.447296	0.223648	+	0.974670i	$0.428203\pi$
$488$	0.764051	0.0345870
$489$	−1.77476	−0.0802573
$490$	−0.564516	−0.0255022
$491$	4.22600	0.190717	0.0953584	−	0.995443i	$-0.469600\pi$
$491$	4.22600	0.190717	0.0953584	+	0.995443i	$0.469600\pi$
$492$	2.88213	0.129937
$493$	−9.36933	−0.421974
$494$	−7.04910	−0.317154
$495$	0	0
$496$	28.4152	1.27588
$497$	4.52077	0.202784
$498$	−0.111422	−0.00499296
$499$	−20.3953	−0.913018	−0.456509	−	0.889719i	$-0.650900\pi$
$499$	−20.3953	−0.913018	−0.456509	+	0.889719i	$0.650900\pi$
$500$	−18.3463	−0.820472
$501$	2.85999	0.127775
$502$	2.69073	0.120093
$503$	−23.9593	−1.06829	−0.534145	−	0.845393i	$-0.679366\pi$
$503$	−23.9593	−1.06829	−0.534145	+	0.845393i	$0.679366\pi$
$504$	2.63692	0.117458
$505$	9.21692	0.410147
$506$	0	0
$507$	−2.92934	−0.130097
$508$	0.563611	0.0250062
$509$	3.69341	0.163708	0.0818538	−	0.996644i	$-0.473916\pi$
$509$	3.69341	0.163708	0.0818538	+	0.996644i	$0.473916\pi$
$510$	−0.177825	−0.00787421
$511$	−14.8479	−0.656834
$512$	16.2941	0.720104
$513$	−7.91484	−0.349449
$514$	5.19402	0.229098
$515$	2.87375	0.126633
$516$	−0.342618	−0.0150829
$517$	0	0
$518$	−0.901299	−0.0396008
$519$	−1.29554	−0.0568677
$520$	11.4468	0.501975
$521$	−0.238270	−0.0104388	−0.00521939	−	0.999986i	$-0.501661\pi$
$521$	−0.238270	−0.0104388	−0.00521939	+	0.999986i	$0.501661\pi$
$522$	−4.35233	−0.190496
$523$	−21.9914	−0.961618	−0.480809	−	0.876825i	$-0.659657\pi$
$523$	−21.9914	−0.961618	−0.480809	+	0.876825i	$0.659657\pi$
$524$	32.1671	1.40523
$525$	−0.269012	−0.0117406
$526$	0.438339	0.0191125
$527$	−11.0531	−0.481479
$528$	0	0
$529$	27.1678	1.18121
$530$	−3.71546	−0.161389
$531$	−8.49064	−0.368462
$532$	11.8264	0.512740
$533$	34.6560	1.50112
$534$	−0.0860360	−0.00372314
$535$	−2.91262	−0.125923
$536$	1.47093	0.0635345
$537$	−0.949944	−0.0409931
$538$	1.62573	0.0700900
$539$	0	0
$540$	6.34305	0.272961
$541$	28.6309	1.23094	0.615470	−	0.788161i	$-0.288966\pi$
$541$	28.6309	1.23094	0.615470	+	0.788161i	$0.288966\pi$
$542$	0.269441	0.0115735
$543$	−2.37333	−0.101849
$544$	−3.76994	−0.161635
$545$	−23.2142	−0.994388
$546$	−0.254539	−0.0108933
$547$	6.40847	0.274007	0.137003	−	0.990571i	$-0.456253\pi$
$547$	6.40847	0.274007	0.137003	+	0.990571i	$0.456253\pi$
$548$	−18.1745	−0.776378
$549$	2.52495	0.107762
$550$	0	0
$551$	−39.5524	−1.68499
$552$	1.38642	0.0590101
$553$	2.45291	0.104308
$554$	2.33908	0.0993780
$555$	−2.17880	−0.0924849
$556$	−9.40600	−0.398903
$557$	−22.5193	−0.954172	−0.477086	−	0.878857i	$-0.658307\pi$
$557$	−22.5193	−0.954172	−0.477086	+	0.878857i	$0.658307\pi$
$558$	−5.13447	−0.217359
$559$	−4.11979	−0.174249
$560$	−9.22243	−0.389719
$561$	0	0
$562$	−2.97498	−0.125492
$563$	−29.8267	−1.25705	−0.628523	−	0.777791i	$-0.716340\pi$
$563$	−29.8267	−1.25705	−0.628523	+	0.777791i	$0.716340\pi$
$564$	2.88351	0.121418
$565$	−8.22856	−0.346178
$566$	0.0678703	0.00285280
$567$	8.57015	0.359912
$568$	−4.03827	−0.169442
$569$	29.2764	1.22733	0.613665	−	0.789567i	$-0.289695\pi$
$569$	29.2764	1.22733	0.613665	+	0.789567i	$0.289695\pi$
$570$	−0.750683	−0.0314427
$571$	37.9252	1.58712	0.793559	−	0.608493i	$-0.208226\pi$
$571$	37.9252	1.58712	0.793559	+	0.608493i	$0.208226\pi$
$572$	0	0
$573$	−2.55409	−0.106699
$574$	1.52670	0.0637232
$575$	8.69527	0.362618
$576$	20.0674	0.836141
$577$	8.77263	0.365209	0.182605	−	0.983186i	$-0.441547\pi$
$577$	8.77263	0.365209	0.182605	+	0.983186i	$0.441547\pi$
$578$	−3.37813	−0.140512
$579$	−4.91846	−0.204404
$580$	31.6978	1.31618
$581$	2.24780	0.0932544
$582$	0.598679	0.0248161
$583$	0	0
$584$	13.2632	0.548836
$585$	37.8281	1.56400
$586$	3.65198	0.150862
$587$	9.17011	0.378491	0.189246	−	0.981930i	$-0.439396\pi$
$587$	9.17011	0.378491	0.189246	+	0.981930i	$0.439396\pi$
$588$	0.427046	0.0176111
$589$	−46.6602	−1.92260
$590$	−1.62369	−0.0668462
$591$	−5.28150	−0.217252
$592$	−14.7244	−0.605170
$593$	7.25596	0.297967	0.148983	−	0.988840i	$-0.452400\pi$
$593$	7.25596	0.297967	0.148983	+	0.988840i	$0.452400\pi$
$594$	0	0
$595$	3.58738	0.147068
$596$	−1.79665	−0.0735938
$597$	−4.10222	−0.167893
$598$	8.22747	0.336446
$599$	−20.0181	−0.817916	−0.408958	−	0.912553i	$-0.634108\pi$
$599$	−20.0181	−0.817916	−0.408958	+	0.912553i	$0.634108\pi$
$600$	0.240300	0.00981021
$601$	−11.2706	−0.459736	−0.229868	−	0.973222i	$-0.573829\pi$
$601$	−11.2706	−0.459736	−0.229868	+	0.973222i	$0.573829\pi$
$602$	−0.181489	−0.00739694
$603$	4.86097	0.197954
$604$	−35.2668	−1.43499
$605$	0	0
$606$	0.183079	0.00743709
$607$	−13.5260	−0.549005	−0.274502	−	0.961586i	$-0.588513\pi$
$607$	−13.5260	−0.549005	−0.274502	+	0.961586i	$0.588513\pi$
$608$	−15.9147	−0.645427
$609$	−1.42822	−0.0578743
$610$	0.482854	0.0195502
$611$	34.6725	1.40270
$612$	−8.26995	−0.334293
$613$	−30.7968	−1.24387	−0.621935	−	0.783069i	$-0.713653\pi$
$613$	−30.7968	−1.24387	−0.621935	+	0.783069i	$0.713653\pi$
$614$	−6.47815	−0.261437
$615$	3.69064	0.148821
$616$	0	0
$617$	23.6896	0.953707	0.476853	−	0.878983i	$-0.341777\pi$
$617$	23.6896	0.953707	0.476853	+	0.878983i	$0.341777\pi$
$618$	0.0570825	0.00229619
$619$	−32.4878	−1.30579	−0.652897	−	0.757446i	$-0.726447\pi$
$619$	−32.4878	−1.30579	−0.652897	+	0.757446i	$0.726447\pi$
$620$	37.3940	1.50178
$621$	9.23794	0.370706
$622$	−7.19867	−0.288640
$623$	1.73566	0.0695378
$624$	−4.15838	−0.166468
$625$	−29.6311	−1.18524
$626$	0.00774169	0.000309420 0
$627$	0	0
$628$	23.9186	0.954456
$629$	5.72756	0.228373
$630$	1.66644	0.0663926
$631$	−15.1333	−0.602448	−0.301224	−	0.953553i	$-0.597395\pi$
$631$	−15.1333	−0.602448	−0.301224	+	0.953553i	$0.597395\pi$
$632$	−2.19111	−0.0871575
$633$	1.65801	0.0659001
$634$	5.06474	0.201147
$635$	0.721718	0.0286405
$636$	2.81068	0.111451
$637$	5.13499	0.203456
$638$	0	0
$639$	−13.3452	−0.527929
$640$	16.9267	0.669086
$641$	−16.5502	−0.653695	−0.326848	−	0.945077i	$-0.605986\pi$
$641$	−16.5502	−0.653695	−0.326848	+	0.945077i	$0.605986\pi$
$642$	−0.0578545	−0.00228333
$643$	1.99506	0.0786773	0.0393387	−	0.999226i	$-0.487475\pi$
$643$	1.99506	0.0786773	0.0393387	+	0.999226i	$0.487475\pi$
$644$	−13.8034	−0.543930
$645$	−0.438731	−0.0172750
$646$	1.97337	0.0776414
$647$	40.4517	1.59032	0.795160	−	0.606399i	$-0.207387\pi$
$647$	40.4517	1.59032	0.795160	+	0.606399i	$0.207387\pi$
$648$	−7.65545	−0.300735
$649$	0	0
$650$	1.42602	0.0559329
$651$	−1.68488	−0.0660355
$652$	−15.7838	−0.618142
$653$	−41.4856	−1.62346	−0.811729	−	0.584034i	$-0.801473\pi$
$653$	−41.4856	−1.62346	−0.811729	+	0.584034i	$0.801473\pi$
$654$	−0.461113	−0.0180310
$655$	41.1908	1.60946
$656$	24.9415	0.973803
$657$	43.8309	1.71001
$658$	1.52743	0.0595454
$659$	−51.1359	−1.99197	−0.995985	−	0.0895158i	$-0.971468\pi$
$659$	−51.1359	−1.99197	−0.995985	+	0.0895158i	$0.971468\pi$
$660$	0	0
$661$	−42.8840	−1.66800	−0.833998	−	0.551768i	$-0.813954\pi$
$661$	−42.8840	−1.66800	−0.833998	+	0.551768i	$0.813954\pi$
$662$	2.43351	0.0945812
$663$	1.61754	0.0628201
$664$	−2.00789	−0.0779213
$665$	15.1440	0.587260
$666$	2.66062	0.103097
$667$	46.1643	1.78749
$668$	25.4353	0.984122
$669$	−3.84011	−0.148467
$670$	0.929577	0.0359127
$671$	0	0
$672$	−0.574672	−0.0221685
$673$	−25.0072	−0.963958	−0.481979	−	0.876183i	$-0.660082\pi$
$673$	−25.0072	−0.963958	−0.481979	+	0.876183i	$0.660082\pi$
$674$	1.69770	0.0653929
$675$	1.60115	0.0616284
$676$	−26.0521	−1.00200
$677$	−9.91890	−0.381214	−0.190607	−	0.981666i	$-0.561046\pi$
$677$	−9.91890	−0.381214	−0.190607	+	0.981666i	$0.561046\pi$
$678$	−0.163447	−0.00627715
$679$	−12.0776	−0.463494
$680$	−3.20450	−0.122887
$681$	5.62665	0.215614
$682$	0	0
$683$	−39.8980	−1.52666	−0.763328	−	0.646011i	$-0.776436\pi$
$683$	−39.8980	−1.52666	−0.763328	+	0.646011i	$0.776436\pi$
$684$	−34.9114	−1.33487
$685$	−23.2729	−0.889214
$686$	0.226211	0.00863679
$687$	4.34685	0.165843
$688$	−2.96497	−0.113038
$689$	33.7968	1.28756
$690$	0.876172	0.0333553
$691$	−6.61401	−0.251609	−0.125804	−	0.992055i	$-0.540151\pi$
$691$	−6.61401	−0.251609	−0.125804	+	0.992055i	$0.540151\pi$
$692$	−11.5218	−0.437995
$693$	0	0
$694$	−6.26194	−0.237700
$695$	−12.0446	−0.456878
$696$	1.27578	0.0483585
$697$	−9.70185	−0.367484
$698$	−2.50201	−0.0947025
$699$	4.42961	0.167543
$700$	−2.39246	−0.0904264
$701$	14.6016	0.551495	0.275748	−	0.961230i	$-0.411075\pi$
$701$	14.6016	0.551495	0.275748	+	0.961230i	$0.411075\pi$
$702$	1.51501	0.0571805
$703$	24.1788	0.911920
$704$	0	0
$705$	3.69240	0.139064
$706$	7.22070	0.271755
$707$	−3.69338	−0.138904
$708$	1.22829	0.0461620
$709$	−4.14406	−0.155633	−0.0778167	−	0.996968i	$-0.524795\pi$
$709$	−4.14406	−0.155633	−0.0778167	+	0.996968i	$0.524795\pi$
$710$	−2.55205	−0.0957766
$711$	−7.24093	−0.271556
$712$	−1.55041	−0.0581042
$713$	54.4602	2.03955
$714$	0.0712575	0.00266674
$715$	0	0
$716$	−8.44832	−0.315729
$717$	−3.74720	−0.139942
$718$	0.809444	0.0302082
$719$	17.0223	0.634823	0.317412	−	0.948288i	$-0.397186\pi$
$719$	17.0223	0.634823	0.317412	+	0.948288i	$0.397186\pi$
$720$	27.2245	1.01460
$721$	−1.15156	−0.0428864
$722$	4.03254	0.150076
$723$	5.29261	0.196834
$724$	−21.1072	−0.784442
$725$	8.00136	0.297163
$726$	0	0
$727$	−21.6199	−0.801837	−0.400918	−	0.916114i	$-0.631309\pi$
$727$	−21.6199	−0.801837	−0.400918	+	0.916114i	$0.631309\pi$
$728$	−4.58693	−0.170003
$729$	−24.4415	−0.905241
$730$	8.38190	0.310228
$731$	1.15332	0.0426572
$732$	−0.365270	−0.0135008
$733$	−48.2326	−1.78151	−0.890755	−	0.454484i	$-0.849824\pi$
$733$	−48.2326	−1.78151	−0.890755	+	0.454484i	$0.849824\pi$
$734$	0.518921	0.0191537
$735$	0.546843	0.0201706
$736$	18.5751	0.684688
$737$	0	0
$738$	−4.50679	−0.165897
$739$	−8.16347	−0.300298	−0.150149	−	0.988663i	$-0.547975\pi$
$739$	−8.16347	−0.300298	−0.150149	+	0.988663i	$0.547975\pi$
$740$	−19.3772	−0.712318
$741$	6.82842	0.250848
$742$	1.48885	0.0546574
$743$	19.5612	0.717633	0.358816	−	0.933408i	$-0.383180\pi$
$743$	19.5612	0.717633	0.358816	+	0.933408i	$0.383180\pi$
$744$	1.50505	0.0551778
$745$	−2.30066	−0.0842897
$746$	1.80258	0.0659971
$747$	−6.63546	−0.242779
$748$	0	0
$749$	1.16714	0.0426462
$750$	−0.466649	−0.0170396
$751$	1.11630	0.0407343	0.0203671	−	0.999793i	$-0.493516\pi$
$751$	1.11630	0.0407343	0.0203671	+	0.999793i	$0.493516\pi$
$752$	24.9534	0.909958
$753$	−2.60649	−0.0949858
$754$	7.57089	0.275716
$755$	−45.1600	−1.64354
$756$	−2.54177	−0.0924433
$757$	−26.5773	−0.965969	−0.482984	−	0.875629i	$-0.660447\pi$
$757$	−26.5773	−0.965969	−0.482984	+	0.875629i	$0.660447\pi$
$758$	−2.62885	−0.0954840
$759$	0	0
$760$	−13.5277	−0.490701
$761$	6.30003	0.228376	0.114188	−	0.993459i	$-0.463573\pi$
$761$	6.30003	0.228376	0.114188	+	0.993459i	$0.463573\pi$
$762$	0.0143358	0.000519330 0
$763$	9.30234	0.336767
$764$	−22.7148	−0.821792
$765$	−10.5899	−0.382878
$766$	2.84539	0.102808
$767$	14.7695	0.533296
$768$	−2.64303	−0.0953723
$769$	13.1916	0.475700	0.237850	−	0.971302i	$-0.423557\pi$
$769$	13.1916	0.475700	0.237850	+	0.971302i	$0.423557\pi$
$770$	0	0
$771$	−5.03141	−0.181202
$772$	−43.7423	−1.57432
$773$	45.8178	1.64795	0.823976	−	0.566625i	$-0.191751\pi$
$773$	45.8178	1.64795	0.823976	+	0.566625i	$0.191751\pi$
$774$	0.535752	0.0192572
$775$	9.43925	0.339068
$776$	10.7885	0.387285
$777$	0.873083	0.0313217
$778$	−0.0991470	−0.00355459
$779$	−40.9562	−1.46741
$780$	−5.47237	−0.195942
$781$	0	0
$782$	−2.30326	−0.0823643
$783$	8.50073	0.303791
$784$	3.69559	0.131985
$785$	30.6284	1.09317
$786$	0.818189	0.0291838
$787$	−18.8479	−0.671856	−0.335928	−	0.941888i	$-0.609050\pi$
$787$	−18.8479	−0.671856	−0.335928	+	0.941888i	$0.609050\pi$
$788$	−46.9710	−1.67327
$789$	−0.424617	−0.0151168
$790$	−1.38470	−0.0492656
$791$	3.29733	0.117240
$792$	0	0
$793$	−4.39217	−0.155970
$794$	3.80367	0.134987
$795$	3.59914	0.127648
$796$	−36.4831	−1.29311
$797$	−28.1613	−0.997526	−0.498763	−	0.866738i	$-0.666212\pi$
$797$	−28.1613	−0.997526	−0.498763	+	0.866738i	$0.666212\pi$
$798$	0.300812	0.0106486
$799$	−9.70648	−0.343391
$800$	3.21951	0.113827
$801$	−5.12364	−0.181035
$802$	8.33835	0.294437
$803$	0	0
$804$	−0.703208	−0.0248002
$805$	−17.6756	−0.622983
$806$	8.93143	0.314596
$807$	−1.57483	−0.0554367
$808$	3.29919	0.116065
$809$	−6.54552	−0.230128	−0.115064	−	0.993358i	$-0.536707\pi$
$809$	−6.54552	−0.230128	−0.115064	+	0.993358i	$0.536707\pi$
$810$	−4.83798	−0.169989
$811$	18.8046	0.660320	0.330160	−	0.943925i	$-0.392897\pi$
$811$	18.8046	0.660320	0.330160	+	0.943925i	$0.392897\pi$
$812$	−12.7019	−0.445748
$813$	−0.261006	−0.00915387
$814$	0	0
$815$	−20.2116	−0.707980
$816$	1.16413	0.0407526
$817$	4.86873	0.170335
$818$	3.68134	0.128715
$819$	−15.1584	−0.529677
$820$	32.8227	1.14622
$821$	−11.5880	−0.404424	−0.202212	−	0.979342i	$-0.564813\pi$
$821$	−11.5880	−0.404424	−0.202212	+	0.979342i	$0.564813\pi$
$822$	−0.462280	−0.0161239
$823$	12.3464	0.430368	0.215184	−	0.976574i	$-0.430965\pi$
$823$	12.3464	0.430368	0.215184	+	0.976574i	$0.430965\pi$
$824$	1.02866	0.0358349
$825$	0	0
$826$	0.650640	0.0226387
$827$	−4.49579	−0.156334	−0.0781669	−	0.996940i	$-0.524907\pi$
$827$	−4.49579	−0.156334	−0.0781669	+	0.996940i	$0.524907\pi$
$828$	40.7474	1.41607
$829$	−19.8629	−0.689866	−0.344933	−	0.938627i	$-0.612098\pi$
$829$	−19.8629	−0.689866	−0.344933	+	0.938627i	$0.612098\pi$
$830$	−1.26892	−0.0440448
$831$	−2.26585	−0.0786015
$832$	−34.9073	−1.21019
$833$	−1.43752	−0.0498073
$834$	−0.239247	−0.00828444
$835$	32.5706	1.12715
$836$	0	0
$837$	10.0284	0.346631
$838$	1.25853	0.0434753
$839$	−43.8528	−1.51397	−0.756984	−	0.653434i	$-0.773328\pi$
$839$	−43.8528	−1.51397	−0.756984	+	0.653434i	$0.773328\pi$
$840$	−0.488478	−0.0168541
$841$	13.4802	0.464836
$842$	4.86160	0.167542
$843$	2.88184	0.0992558
$844$	14.7455	0.507562
$845$	−33.3604	−1.14763
$846$	−4.50894	−0.155021
$847$	0	0
$848$	24.3232	0.835261
$849$	−0.0657456	−0.00225638
$850$	−0.399209	−0.0136928
$851$	−28.2207	−0.967392
$852$	1.93058	0.0661405
$853$	39.1407	1.34015	0.670076	−	0.742293i	$-0.266262\pi$
$853$	39.1407	1.34015	0.670076	+	0.742293i	$0.266262\pi$
$854$	−0.193488	−0.00662102
$855$	−44.7049	−1.52888
$856$	−1.04257	−0.0356342
$857$	35.0524	1.19737	0.598684	−	0.800986i	$-0.295691\pi$
$857$	35.0524	1.19737	0.598684	+	0.800986i	$0.295691\pi$
$858$	0	0
$859$	−32.5206	−1.10959	−0.554794	−	0.831988i	$-0.687203\pi$
$859$	−32.5206	−1.10959	−0.554794	+	0.831988i	$0.687203\pi$
$860$	−3.90186	−0.133052
$861$	−1.47891	−0.0504010
$862$	−6.27030	−0.213567
$863$	12.6940	0.432107	0.216054	−	0.976381i	$-0.430681\pi$
$863$	12.6940	0.432107	0.216054	+	0.976381i	$0.430681\pi$
$864$	3.42044	0.116366
$865$	−14.7540	−0.501651
$866$	2.10108	0.0713975
$867$	3.27238	0.111136
$868$	−14.9844	−0.508605
$869$	0	0
$870$	0.806251	0.0273345
$871$	−8.45568	−0.286510
$872$	−8.30950	−0.281395
$873$	35.6527	1.20666
$874$	−9.72314	−0.328890
$875$	9.41402	0.318252
$876$	−6.34075	−0.214234
$877$	−28.1340	−0.950017	−0.475008	−	0.879981i	$-0.657555\pi$
$877$	−28.1340	−0.950017	−0.475008	+	0.879981i	$0.657555\pi$
$878$	−3.32798	−0.112314
$879$	−3.53765	−0.119322
$880$	0	0
$881$	−36.7964	−1.23970	−0.619850	−	0.784720i	$-0.712807\pi$
$881$	−36.7964	−1.23970	−0.619850	+	0.784720i	$0.712807\pi$
$882$	−0.667772	−0.0224850
$883$	−2.28419	−0.0768692	−0.0384346	−	0.999261i	$-0.512237\pi$
$883$	−2.28419	−0.0768692	−0.0384346	+	0.999261i	$0.512237\pi$
$884$	14.3856	0.483840
$885$	1.57286	0.0528710
$886$	0.552736	0.0185695
$887$	42.2676	1.41921	0.709604	−	0.704600i	$-0.248874\pi$
$887$	42.2676	1.41921	0.709604	+	0.704600i	$0.248874\pi$
$888$	−0.779899	−0.0261717
$889$	−0.289205	−0.00969963
$890$	−0.979808	−0.0328432
$891$	0	0
$892$	−34.1520	−1.14349
$893$	−40.9757	−1.37120
$894$	−0.0456989	−0.00152840
$895$	−10.8183	−0.361616
$896$	−6.78282	−0.226598
$897$	−7.96990	−0.266107
$898$	1.07888	0.0360027
$899$	50.1142	1.67140
$900$	7.06249	0.235416
$901$	−9.46132	−0.315202
$902$	0	0
$903$	0.175807	0.00585050
$904$	−2.94540	−0.0979627
$905$	−27.0283	−0.898451
$906$	−0.897032	−0.0298019
$907$	39.5286	1.31252	0.656262	−	0.754533i	$-0.272136\pi$
$907$	39.5286	1.31252	0.656262	+	0.754533i	$0.272136\pi$
$908$	50.0406	1.66065
$909$	10.9028	0.361623
$910$	−2.89878	−0.0960936
$911$	−35.2296	−1.16721	−0.583604	−	0.812039i	$-0.698358\pi$
$911$	−35.2296	−1.16721	−0.583604	+	0.812039i	$0.698358\pi$
$912$	4.91433	0.162730
$913$	0	0
$914$	7.03234	0.232609
$915$	−0.467737	−0.0154629
$916$	38.6587	1.27732
$917$	−16.5059	−0.545072
$918$	−0.424123	−0.0139982
$919$	−39.6105	−1.30663	−0.653314	−	0.757087i	$-0.726622\pi$
$919$	−39.6105	−1.30663	−0.653314	+	0.757087i	$0.726622\pi$
$920$	15.7891	0.520551
$921$	6.27534	0.206780
$922$	6.72334	0.221421
$923$	23.2141	0.764101
$924$	0	0
$925$	−4.89131	−0.160825
$926$	−5.75828	−0.189229
$927$	3.39939	0.111651
$928$	17.0928	0.561098
$929$	2.61657	0.0858469	0.0429234	−	0.999078i	$-0.486333\pi$
$929$	2.61657	0.0858469	0.0429234	+	0.999078i	$0.486333\pi$
$930$	0.951139	0.0311891
$931$	−6.06848	−0.198886
$932$	39.3947	1.29042
$933$	6.97331	0.228296
$934$	−0.720463	−0.0235743
$935$	0	0
$936$	13.5405	0.442586
$937$	53.8401	1.75888	0.879439	−	0.476011i	$-0.157918\pi$
$937$	53.8401	1.75888	0.879439	+	0.476011i	$0.157918\pi$
$938$	−0.372498	−0.0121625
$939$	−0.00749932	−0.000244731 0
$940$	32.8384	1.07107
$941$	29.6476	0.966484	0.483242	−	0.875487i	$-0.339459\pi$
$941$	29.6476	0.966484	0.483242	+	0.875487i	$0.339459\pi$
$942$	0.608384	0.0198222
$943$	47.8026	1.55667
$944$	10.6294	0.345959
$945$	−3.25480	−0.105879
$946$	0	0
$947$	−15.7861	−0.512980	−0.256490	−	0.966547i	$-0.582566\pi$
$947$	−15.7861	−0.512980	−0.256490	+	0.966547i	$0.582566\pi$
$948$	1.04750	0.0340213
$949$	−76.2440	−2.47498
$950$	−1.68525	−0.0546768
$951$	−4.90618	−0.159094
$952$	1.28410	0.0416178
$953$	−35.1119	−1.13739	−0.568693	−	0.822550i	$-0.692551\pi$
$953$	−35.1119	−1.13739	−0.568693	+	0.822550i	$0.692551\pi$
$954$	−4.39506	−0.142295
$955$	−29.0869	−0.941229
$956$	−33.3257	−1.07783
$957$	0	0
$958$	0.731463	0.0236325
$959$	9.32588	0.301148
$960$	−3.71740	−0.119979
$961$	28.1200	0.907096
$962$	−4.62816	−0.149218
$963$	−3.44537	−0.111025
$964$	47.0698	1.51602
$965$	−56.0131	−1.80313
$966$	−0.351097	−0.0112964
$967$	−49.2820	−1.58480	−0.792401	−	0.610001i	$-0.791169\pi$
$967$	−49.2820	−1.58480	−0.792401	+	0.610001i	$0.791169\pi$
$968$	0	0
$969$	−1.91160	−0.0614093
$970$	6.81797	0.218912
$971$	37.3424	1.19838	0.599188	−	0.800608i	$-0.295490\pi$
$971$	37.3424	1.19838	0.599188	+	0.800608i	$0.295490\pi$
$972$	11.2852	0.361971
$973$	4.82649	0.154730
$974$	2.23292	0.0715475
$975$	−1.38137	−0.0442393
$976$	−3.16099	−0.101181
$977$	40.0485	1.28126	0.640632	−	0.767848i	$-0.278672\pi$
$977$	40.0485	1.28126	0.640632	+	0.767848i	$0.278672\pi$
$978$	−0.401470	−0.0128376
$979$	0	0
$980$	4.86335	0.155354
$981$	−27.4603	−0.876741
$982$	0.955970	0.0305062
$983$	−52.0414	−1.65986	−0.829932	−	0.557864i	$-0.811621\pi$
$983$	−52.0414	−1.65986	−0.829932	+	0.557864i	$0.811621\pi$
$984$	1.32106	0.0421139
$985$	−60.1475	−1.91646
$986$	−2.11945	−0.0674970
$987$	−1.47961	−0.0470965
$988$	60.7285	1.93203
$989$	−5.68262	−0.180697
$990$	0	0
$991$	45.4828	1.44481	0.722404	−	0.691471i	$-0.243037\pi$
$991$	45.4828	1.44481	0.722404	+	0.691471i	$0.243037\pi$
$992$	20.1645	0.640222
$993$	−2.35733	−0.0748076
$994$	1.02265	0.0324365
$995$	−46.7175	−1.48104
$996$	0.959913	0.0304160
$997$	27.9594	0.885482	0.442741	−	0.896650i	$-0.354006\pi$
$997$	27.9594	0.885482	0.442741	+	0.896650i	$0.354006\pi$
$998$	−4.61364	−0.146042
$999$	−5.19657	−0.164412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.p.1.4		8
3.2	odd	2		7623.2.a.ct.1.5		8
7.6	odd	2		5929.2.a.bt.1.4		8
11.2	odd	10		847.2.f.x.323.2		16
11.3	even	5		847.2.f.w.372.2		16
11.4	even	5		847.2.f.w.148.2		16
11.5	even	5		77.2.f.b.36.3	yes	16
11.6	odd	10		847.2.f.x.729.2		16
11.7	odd	10		847.2.f.v.148.3		16
11.8	odd	10		847.2.f.v.372.3		16
11.9	even	5		77.2.f.b.15.3	✓	16
11.10	odd	2		847.2.a.o.1.5		8
33.5	odd	10		693.2.m.i.190.2		16
33.20	odd	10		693.2.m.i.631.2		16
33.32	even	2		7623.2.a.cw.1.4		8
77.5	odd	30		539.2.q.f.410.2		32
77.9	even	15		539.2.q.g.312.3		32
77.16	even	15		539.2.q.g.410.2		32
77.20	odd	10		539.2.f.e.246.3		16
77.27	odd	10		539.2.f.e.344.3		16
77.31	odd	30		539.2.q.f.422.2		32
77.38	odd	30		539.2.q.f.520.3		32
77.53	even	15		539.2.q.g.422.2		32
77.60	even	15		539.2.q.g.520.3		32
77.75	odd	30		539.2.q.f.312.3		32
77.76	even	2		5929.2.a.bs.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.15.3	✓	16	11.9	even	5
77.2.f.b.36.3	yes	16	11.5	even	5
539.2.f.e.246.3		16	77.20	odd	10
539.2.f.e.344.3		16	77.27	odd	10
539.2.q.f.312.3		32	77.75	odd	30
539.2.q.f.410.2		32	77.5	odd	30
539.2.q.f.422.2		32	77.31	odd	30
539.2.q.f.520.3		32	77.38	odd	30
539.2.q.g.312.3		32	77.9	even	15
539.2.q.g.410.2		32	77.16	even	15
539.2.q.g.422.2		32	77.53	even	15
539.2.q.g.520.3		32	77.60	even	15
693.2.m.i.190.2		16	33.5	odd	10
693.2.m.i.631.2		16	33.20	odd	10
847.2.a.o.1.5		8	11.10	odd	2
847.2.a.p.1.4		8	1.1	even	1	trivial
847.2.f.v.148.3		16	11.7	odd	10
847.2.f.v.372.3		16	11.8	odd	10
847.2.f.w.148.2		16	11.4	even	5
847.2.f.w.372.2		16	11.3	even	5
847.2.f.x.323.2		16	11.2	odd	10
847.2.f.x.729.2		16	11.6	odd	10
5929.2.a.bs.1.5		8	77.76	even	2
5929.2.a.bt.1.4		8	7.6	odd	2
7623.2.a.ct.1.5		8	3.2	odd	2
7623.2.a.cw.1.4		8	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)

\(f(q)\)	\(=\)	\(q+0.226211 q^{2} -0.219130 q^{3} -1.94883 q^{4} -2.49552 q^{5} -0.0495696 q^{6} +1.00000 q^{7} -0.893270 q^{8} -2.95198 q^{9} +O(q^{10})\)	\(q+0.226211 q^{2} -0.219130 q^{3} -1.94883 q^{4} -2.49552 q^{5} -0.0495696 q^{6} +1.00000 q^{7} -0.893270 q^{8} -2.95198 q^{9} -0.564516 q^{10} +0.427046 q^{12} +5.13499 q^{13} +0.226211 q^{14} +0.546843 q^{15} +3.69559 q^{16} -1.43752 q^{17} -0.667772 q^{18} -6.06848 q^{19} +4.86335 q^{20} -0.219130 q^{21} +7.08292 q^{23} +0.195742 q^{24} +1.22764 q^{25} +1.16159 q^{26} +1.30426 q^{27} -1.94883 q^{28} +6.51769 q^{29} +0.123702 q^{30} +7.68895 q^{31} +2.62252 q^{32} -0.325184 q^{34} -2.49552 q^{35} +5.75291 q^{36} -3.98432 q^{37} -1.37276 q^{38} -1.12523 q^{39} +2.22918 q^{40} +6.74900 q^{41} -0.0495696 q^{42} -0.802299 q^{43} +7.36674 q^{45} +1.60224 q^{46} +6.75222 q^{47} -0.809813 q^{48} +1.00000 q^{49} +0.277706 q^{50} +0.315004 q^{51} -10.0072 q^{52} +6.58167 q^{53} +0.295037 q^{54} -0.893270 q^{56} +1.32978 q^{57} +1.47437 q^{58} +2.87625 q^{59} -1.06570 q^{60} -0.855342 q^{61} +1.73933 q^{62} -2.95198 q^{63} -6.79793 q^{64} -12.8145 q^{65} -1.64668 q^{67} +2.80149 q^{68} -1.55208 q^{69} -0.564516 q^{70} +4.52077 q^{71} +2.63692 q^{72} -14.8479 q^{73} -0.901299 q^{74} -0.269012 q^{75} +11.8264 q^{76} -0.254539 q^{78} +2.45291 q^{79} -9.22243 q^{80} +8.57015 q^{81} +1.52670 q^{82} +2.24780 q^{83} +0.427046 q^{84} +3.58738 q^{85} -0.181489 q^{86} -1.42822 q^{87} +1.73566 q^{89} +1.66644 q^{90} +5.13499 q^{91} -13.8034 q^{92} -1.68488 q^{93} +1.52743 q^{94} +15.1440 q^{95} -0.574672 q^{96} -12.0776 q^{97} +0.226211 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9}+O(q^{10}) \) 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9	\( 8 q + q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} - q^{6} + 8 q^{7} + 14 q^{9} + 6 q^{10} + 9 q^{12} - 6 q^{13} + q^{14} + 11 q^{15} + q^{16} - 5 q^{17} + 8 q^{18} - 13 q^{19} + 23 q^{20} + 4 q^{21} + 16 q^{23} - 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} + 7 q^{28} + 9 q^{29} - 36 q^{30} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} + 13 q^{39} + 5 q^{40} - 10 q^{41} - q^{42} - 4 q^{43} + 35 q^{45} + 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} + 6 q^{50} + 13 q^{51} - 41 q^{52} + 37 q^{53} + 30 q^{54} + 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} + 19 q^{61} - 18 q^{62} + 14 q^{63} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} - 35 q^{72} - 25 q^{73} + 33 q^{74} - 13 q^{75} + 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} - 25 q^{83} + 9 q^{84} + 3 q^{85} + 4 q^{86} - 36 q^{87} + 37 q^{89} - 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} - 42 q^{94} + 21 q^{95} - 6 q^{96} + 15 q^{97} + q^{98}+O(q^{100}) \) 8 * q + q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 - q^6 + 8 * q^7 + 14 * q^9 + 6 * q^10 + 9 * q^12 - 6 * q^13 + q^14 + 11 * q^15 + q^16 - 5 * q^17 + 8 * q^18 - 13 * q^19 + 23 * q^20 + 4 * q^21 + 16 * q^23 - 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 + 7 * q^28 + 9 * q^29 - 36 * q^30 + 9 * q^31 + 16 * q^32 - 12 * q^34 + 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 + 13 * q^39 + 5 * q^40 - 10 * q^41 - q^42 - 4 * q^43 + 35 * q^45 + 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 + 6 * q^50 + 13 * q^51 - 41 * q^52 + 37 * q^53 + 30 * q^54 + 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 + 19 * q^61 - 18 * q^62 + 14 * q^63 - 4 * q^64 - 4 * q^65 - 19 * q^67 + 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 - 35 * q^72 - 25 * q^73 + 33 * q^74 - 13 * q^75 + 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 - 25 * q^83 + 9 * q^84 + 3 * q^85 + 4 * q^86 - 36 * q^87 + 37 * q^89 - 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 - 42 * q^94 + 21 * q^95 - 6 * q^96 + 15 * q^97 + q^98

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