LMFDB - Embedded newform 847.2.a.o.1.7

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			1.7
Root			$-1.70716$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70716 q^{2} +2.29155 q^{3} +0.914391 q^{4} +4.06637 q^{5} +3.91205 q^{6} -1.00000 q^{7} -1.85331 q^{8} +2.25122 q^{9} +O(q^{10})$	\(q+1.70716 q^{2} +2.29155 q^{3} +0.914391 q^{4} +4.06637 q^{5} +3.91205 q^{6} -1.00000 q^{7} -1.85331 q^{8} +2.25122 q^{9} +6.94194 q^{10} +2.09538 q^{12} -3.27792 q^{13} -1.70716 q^{14} +9.31831 q^{15} -4.99267 q^{16} +1.32088 q^{17} +3.84319 q^{18} -2.16175 q^{19} +3.71825 q^{20} -2.29155 q^{21} -1.86611 q^{23} -4.24695 q^{24} +11.5354 q^{25} -5.59593 q^{26} -1.71587 q^{27} -0.914391 q^{28} +0.244102 q^{29} +15.9078 q^{30} -6.86988 q^{31} -4.81667 q^{32} +2.25495 q^{34} -4.06637 q^{35} +2.05850 q^{36} +0.255619 q^{37} -3.69045 q^{38} -7.51153 q^{39} -7.53623 q^{40} +5.73233 q^{41} -3.91205 q^{42} +8.01781 q^{43} +9.15429 q^{45} -3.18575 q^{46} -4.06768 q^{47} -11.4410 q^{48} +1.00000 q^{49} +19.6927 q^{50} +3.02687 q^{51} -2.99730 q^{52} -5.00585 q^{53} -2.92926 q^{54} +1.85331 q^{56} -4.95376 q^{57} +0.416720 q^{58} -0.983592 q^{59} +8.52058 q^{60} -1.84671 q^{61} -11.7280 q^{62} -2.25122 q^{63} +1.76252 q^{64} -13.3292 q^{65} -3.00700 q^{67} +1.20780 q^{68} -4.27630 q^{69} -6.94194 q^{70} +6.47642 q^{71} -4.17220 q^{72} +9.65164 q^{73} +0.436383 q^{74} +26.4339 q^{75} -1.97668 q^{76} -12.8234 q^{78} +5.53450 q^{79} -20.3020 q^{80} -10.6857 q^{81} +9.78600 q^{82} -2.07023 q^{83} -2.09538 q^{84} +5.37119 q^{85} +13.6877 q^{86} +0.559372 q^{87} +16.6306 q^{89} +15.6278 q^{90} +3.27792 q^{91} -1.70636 q^{92} -15.7427 q^{93} -6.94417 q^{94} -8.79046 q^{95} -11.0377 q^{96} -2.58559 q^{97} +1.70716 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9}+O(q^{10}) $ 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9	$ 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} + q^{14} + 11 q^{15} + q^{16} + 5 q^{17} - 8 q^{18} + 13 q^{19} + 23 q^{20} - 4 q^{21} + 16 q^{23} + 10 q^{24} + 16 q^{25} - 6 q^{26} + 10 q^{27} - 7 q^{28} - 9 q^{29} + 36 q^{30} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} - 14 q^{36} + 7 q^{37} - 10 q^{38} - 13 q^{39} - 5 q^{40} + 10 q^{41} - q^{42} + 4 q^{43} + 35 q^{45} - 4 q^{46} + 16 q^{47} - 3 q^{48} + 8 q^{49} - 6 q^{50} - 13 q^{51} + 41 q^{52} + 37 q^{53} - 30 q^{54} - 2 q^{57} - 15 q^{58} + q^{59} + 5 q^{60} - 19 q^{61} + 18 q^{62} - 14 q^{63} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 20 q^{69} + 6 q^{70} + 13 q^{71} + 35 q^{72} + 25 q^{73} - 33 q^{74} - 13 q^{75} - 26 q^{76} - 29 q^{78} + 4 q^{80} + 8 q^{81} - 13 q^{82} + 25 q^{83} - 9 q^{84} - 3 q^{85} + 4 q^{86} + 36 q^{87} + 37 q^{89} + 2 q^{90} - 6 q^{91} + 35 q^{92} + 21 q^{93} + 42 q^{94} - 21 q^{95} + 6 q^{96} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 4 * q^3 + 7 * q^4 + 10 * q^5 + q^6 - 8 * q^7 + 14 * q^9 - 6 * q^10 + 9 * q^12 + 6 * q^13 + q^14 + 11 * q^15 + q^16 + 5 * q^17 - 8 * q^18 + 13 * q^19 + 23 * q^20 - 4 * q^21 + 16 * q^23 + 10 * q^24 + 16 * q^25 - 6 * q^26 + 10 * q^27 - 7 * q^28 - 9 * q^29 + 36 * q^30 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 - 14 * q^36 + 7 * q^37 - 10 * q^38 - 13 * q^39 - 5 * q^40 + 10 * q^41 - q^42 + 4 * q^43 + 35 * q^45 - 4 * q^46 + 16 * q^47 - 3 * q^48 + 8 * q^49 - 6 * q^50 - 13 * q^51 + 41 * q^52 + 37 * q^53 - 30 * q^54 - 2 * q^57 - 15 * q^58 + q^59 + 5 * q^60 - 19 * q^61 + 18 * q^62 - 14 * q^63 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 20 * q^69 + 6 * q^70 + 13 * q^71 + 35 * q^72 + 25 * q^73 - 33 * q^74 - 13 * q^75 - 26 * q^76 - 29 * q^78 + 4 * q^80 + 8 * q^81 - 13 * q^82 + 25 * q^83 - 9 * q^84 - 3 * q^85 + 4 * q^86 + 36 * q^87 + 37 * q^89 + 2 * q^90 - 6 * q^91 + 35 * q^92 + 21 * q^93 + 42 * q^94 - 21 * q^95 + 6 * q^96 + 15 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.70716	1.20714	0.603572	−	0.797309i	$-0.293744\pi$
$2$	1.70716	1.20714	0.603572	+	0.797309i	$0.293744\pi$
$3$	2.29155	1.32303	0.661515	−	0.749932i	$-0.269914\pi$
$3$	2.29155	1.32303	0.661515	+	0.749932i	$0.269914\pi$
$4$	0.914391	0.457196
$5$	4.06637	1.81854	0.909268	−	0.416211i	$-0.136642\pi$
$5$	4.06637	1.81854	0.909268	+	0.416211i	$0.136642\pi$
$6$	3.91205	1.59709
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.85331	−0.655243
$9$	2.25122	0.750407
$10$	6.94194	2.19523
$11$	0	0
$11$	0	0
$12$	2.09538	0.604883
$13$	−3.27792	−0.909132	−0.454566	−	0.890713i	$-0.650206\pi$
$13$	−3.27792	−0.909132	−0.454566	+	0.890713i	$0.650206\pi$
$14$	−1.70716	−0.456257
$15$	9.31831	2.40598
$16$	−4.99267	−1.24817
$17$	1.32088	0.320361	0.160180	−	0.987088i	$-0.448792\pi$
$17$	1.32088	0.320361	0.160180	+	0.987088i	$0.448792\pi$
$18$	3.84319	0.905849
$19$	−2.16175	−0.495939	−0.247969	−	0.968768i	$-0.579763\pi$
$19$	−2.16175	−0.495939	−0.247969	+	0.968768i	$0.579763\pi$
$20$	3.71825	0.831427
$21$	−2.29155	−0.500058
$22$	0	0
$23$	−1.86611	−0.389112	−0.194556	−	0.980891i	$-0.562327\pi$
$23$	−1.86611	−0.389112	−0.194556	+	0.980891i	$0.562327\pi$
$24$	−4.24695	−0.866905
$25$	11.5354	2.30707
$26$	−5.59593	−1.09745
$27$	−1.71587	−0.330219
$28$	−0.914391	−0.172804
$29$	0.244102	0.0453286	0.0226643	−	0.999743i	$-0.492785\pi$
$29$	0.244102	0.0453286	0.0226643	+	0.999743i	$0.492785\pi$
$30$	15.9078	2.90436
$31$	−6.86988	−1.23387	−0.616933	−	0.787015i	$-0.711625\pi$
$31$	−6.86988	−1.23387	−0.616933	+	0.787015i	$0.711625\pi$
$32$	−4.81667	−0.851475
$33$	0	0
$34$	2.25495	0.386721
$35$	−4.06637	−0.687342
$36$	2.05850	0.343083
$37$	0.255619	0.0420236	0.0210118	−	0.999779i	$-0.493311\pi$
$37$	0.255619	0.0420236	0.0210118	+	0.999779i	$0.493311\pi$
$38$	−3.69045	−0.598669
$39$	−7.51153	−1.20281
$40$	−7.53623	−1.19158
$41$	5.73233	0.895239	0.447620	−	0.894224i	$-0.352272\pi$
$41$	5.73233	0.895239	0.447620	+	0.894224i	$0.352272\pi$
$42$	−3.91205	−0.603642
$43$	8.01781	1.22270	0.611352	−	0.791358i	$-0.290626\pi$
$43$	8.01781	1.22270	0.611352	+	0.791358i	$0.290626\pi$
$44$	0	0
$45$	9.15429	1.36464
$46$	−3.18575	−0.469714
$47$	−4.06768	−0.593332	−0.296666	−	0.954981i	$-0.595875\pi$
$47$	−4.06768	−0.593332	−0.296666	+	0.954981i	$0.595875\pi$
$48$	−11.4410	−1.65136
$49$	1.00000	0.142857
$50$	19.6927	2.78497
$51$	3.02687	0.423847
$52$	−2.99730	−0.415651
$53$	−5.00585	−0.687607	−0.343803	−	0.939042i	$-0.611715\pi$
$53$	−5.00585	−0.687607	−0.343803	+	0.939042i	$0.611715\pi$
$54$	−2.92926	−0.398622
$55$	0	0
$56$	1.85331	0.247658
$57$	−4.95376	−0.656142
$58$	0.416720	0.0547181
$59$	−0.983592	−0.128053	−0.0640264	−	0.997948i	$-0.520394\pi$
$59$	−0.983592	−0.128053	−0.0640264	+	0.997948i	$0.520394\pi$
$60$	8.52058	1.10000
$61$	−1.84671	−0.236447	−0.118223	−	0.992987i	$-0.537720\pi$
$61$	−1.84671	−0.236447	−0.118223	+	0.992987i	$0.537720\pi$
$62$	−11.7280	−1.48945
$63$	−2.25122	−0.283627
$64$	1.76252	0.220315
$65$	−13.3292	−1.65329
$66$	0	0
$67$	−3.00700	−0.367364	−0.183682	−	0.982986i	$-0.558802\pi$
$67$	−3.00700	−0.367364	−0.183682	+	0.982986i	$0.558802\pi$
$68$	1.20780	0.146468
$69$	−4.27630	−0.514806
$70$	−6.94194	−0.829720
$71$	6.47642	0.768609	0.384305	−	0.923206i	$-0.374441\pi$
$71$	6.47642	0.768609	0.384305	+	0.923206i	$0.374441\pi$
$72$	−4.17220	−0.491699
$73$	9.65164	1.12964	0.564820	−	0.825214i	$-0.308946\pi$
$73$	9.65164	1.12964	0.564820	+	0.825214i	$0.308946\pi$
$74$	0.436383	0.0507285
$75$	26.4339	3.05233
$76$	−1.97668	−0.226741
$77$	0	0
$78$	−12.8234	−1.45196
$79$	5.53450	0.622680	0.311340	−	0.950299i	$-0.399222\pi$
$79$	5.53450	0.622680	0.311340	+	0.950299i	$0.399222\pi$
$80$	−20.3020	−2.26984
$81$	−10.6857	−1.18730
$82$	9.78600	1.08068
$83$	−2.07023	−0.227237	−0.113618	−	0.993524i	$-0.536244\pi$
$83$	−2.07023	−0.227237	−0.113618	+	0.993524i	$0.536244\pi$
$84$	−2.09538	−0.228624
$85$	5.37119	0.582588
$86$	13.6877	1.47598
$87$	0.559372	0.0599710
$88$	0	0
$89$	16.6306	1.76284	0.881421	−	0.472331i	$-0.156587\pi$
$89$	16.6306	1.76284	0.881421	+	0.472331i	$0.156587\pi$
$90$	15.6278	1.64732
$91$	3.27792	0.343620
$92$	−1.70636	−0.177900
$93$	−15.7427	−1.63244
$94$	−6.94417	−0.716237
$95$	−8.79046	−0.901883
$96$	−11.0377	−1.12653
$97$	−2.58559	−0.262527	−0.131264	−	0.991348i	$-0.541903\pi$
$97$	−2.58559	−0.262527	−0.131264	+	0.991348i	$0.541903\pi$
$98$	1.70716	0.172449
$99$	0	0
$100$	10.5478	1.05478
$101$	−10.5686	−1.05162	−0.525808	−	0.850604i	$-0.676237\pi$
$101$	−10.5686	−1.05162	−0.525808	+	0.850604i	$0.676237\pi$
$102$	5.16735	0.511644
$103$	13.8489	1.36457	0.682286	−	0.731086i	$-0.260986\pi$
$103$	13.8489	1.36457	0.682286	+	0.731086i	$0.260986\pi$
$104$	6.07499	0.595702
$105$	−9.31831	−0.909374
$106$	−8.54578	−0.830040
$107$	−12.4286	−1.20152	−0.600761	−	0.799428i	$-0.705136\pi$
$107$	−12.4286	−1.20152	−0.600761	+	0.799428i	$0.705136\pi$
$108$	−1.56898	−0.150975
$109$	7.36748	0.705676	0.352838	−	0.935684i	$-0.385217\pi$
$109$	7.36748	0.705676	0.352838	+	0.935684i	$0.385217\pi$
$110$	0	0
$111$	0.585766	0.0555984
$112$	4.99267	0.471763
$113$	11.4083	1.07320	0.536600	−	0.843837i	$-0.319708\pi$
$113$	11.4083	1.07320	0.536600	+	0.843837i	$0.319708\pi$
$114$	−8.45686	−0.792057
$115$	−7.58831	−0.707614
$116$	0.223205	0.0207240
$117$	−7.37932	−0.682219
$118$	−1.67915	−0.154578
$119$	−1.32088	−0.121085
$120$	−17.2697	−1.57650
$121$	0	0
$122$	−3.15263	−0.285425
$123$	13.1359	1.18443
$124$	−6.28176	−0.564119
$125$	26.5752	2.37696
$126$	−3.84319	−0.342379
$127$	8.74845	0.776300	0.388150	−	0.921596i	$-0.373114\pi$
$127$	8.74845	0.776300	0.388150	+	0.921596i	$0.373114\pi$
$128$	12.6422	1.11743
$129$	18.3732	1.61767
$130$	−22.7551	−1.99576
$131$	12.5516	1.09664	0.548319	−	0.836269i	$-0.315268\pi$
$131$	12.5516	1.09664	0.548319	+	0.836269i	$0.315268\pi$
$132$	0	0
$133$	2.16175	0.187447
$134$	−5.13343	−0.443461
$135$	−6.97736	−0.600516
$136$	−2.44800	−0.209914
$137$	−6.32819	−0.540654	−0.270327	−	0.962769i	$-0.587132\pi$
$137$	−6.32819	−0.540654	−0.270327	+	0.962769i	$0.587132\pi$
$138$	−7.30033	−0.621445
$139$	−14.4429	−1.22503	−0.612517	−	0.790457i	$-0.709843\pi$
$139$	−14.4429	−1.22503	−0.612517	+	0.790457i	$0.709843\pi$
$140$	−3.71825	−0.314250
$141$	−9.32130	−0.784995
$142$	11.0563	0.927822
$143$	0	0
$144$	−11.2396	−0.936633
$145$	0.992608	0.0824316
$146$	16.4769	1.36364
$147$	2.29155	0.189004
$148$	0.233736	0.0192130
$149$	−24.0592	−1.97101	−0.985504	−	0.169653i	$-0.945735\pi$
$149$	−24.0592	−1.97101	−0.985504	+	0.169653i	$0.945735\pi$
$150$	45.1269	3.68460
$151$	1.32418	0.107760	0.0538802	−	0.998547i	$-0.482841\pi$
$151$	1.32418	0.107760	0.0538802	+	0.998547i	$0.482841\pi$
$152$	4.00638	0.324960
$153$	2.97359	0.240401
$154$	0	0
$155$	−27.9355	−2.24383
$156$	−6.86848	−0.549919
$157$	14.6141	1.16633	0.583165	−	0.812354i	$-0.301814\pi$
$157$	14.6141	1.16633	0.583165	+	0.812354i	$0.301814\pi$
$158$	9.44827	0.751664
$159$	−11.4712	−0.909724
$160$	−19.5864	−1.54844
$161$	1.86611	0.147070
$162$	−18.2421	−1.43324
$163$	−20.9215	−1.63870	−0.819349	−	0.573296i	$-0.805665\pi$
$163$	−20.9215	−1.63870	−0.819349	+	0.573296i	$0.805665\pi$
$164$	5.24159	0.409300
$165$	0	0
$166$	−3.53421	−0.274308
$167$	−4.91335	−0.380206	−0.190103	−	0.981764i	$-0.560882\pi$
$167$	−4.91335	−0.380206	−0.190103	+	0.981764i	$0.560882\pi$
$168$	4.24695	0.327659
$169$	−2.25523	−0.173479
$170$	9.16948	0.703267
$171$	−4.86657	−0.372156
$172$	7.33142	0.559015
$173$	−19.6578	−1.49456	−0.747279	−	0.664511i	$-0.768640\pi$
$173$	−19.6578	−1.49456	−0.747279	+	0.664511i	$0.768640\pi$
$174$	0.954937	0.0723936
$175$	−11.5354	−0.871992
$176$	0	0
$177$	−2.25395	−0.169418
$178$	28.3911	2.12800
$179$	−25.3588	−1.89541	−0.947705	−	0.319149i	$-0.896603\pi$
$179$	−25.3588	−1.89541	−0.947705	+	0.319149i	$0.896603\pi$
$180$	8.37061	0.623908
$181$	−1.97569	−0.146852	−0.0734258	−	0.997301i	$-0.523393\pi$
$181$	−1.97569	−0.146852	−0.0734258	+	0.997301i	$0.523393\pi$
$182$	5.59593	0.414798
$183$	−4.23184	−0.312826
$184$	3.45848	0.254963
$185$	1.03944	0.0764214
$186$	−26.8753	−1.97059
$187$	0	0
$188$	−3.71945	−0.271269
$189$	1.71587	0.124811
$190$	−15.0067	−1.08870
$191$	0.972995	0.0704035	0.0352017	−	0.999380i	$-0.488793\pi$
$191$	0.972995	0.0704035	0.0352017	+	0.999380i	$0.488793\pi$
$192$	4.03891	0.291484
$193$	23.1326	1.66512	0.832562	−	0.553932i	$-0.186873\pi$
$193$	23.1326	1.66512	0.832562	+	0.553932i	$0.186873\pi$
$194$	−4.41402	−0.316908
$195$	−30.5447	−2.18735
$196$	0.914391	0.0653137
$197$	9.91237	0.706227	0.353114	−	0.935580i	$-0.385123\pi$
$197$	9.91237	0.706227	0.353114	+	0.935580i	$0.385123\pi$
$198$	0	0
$199$	10.3847	0.736154	0.368077	−	0.929795i	$-0.380016\pi$
$199$	10.3847	0.736154	0.368077	+	0.929795i	$0.380016\pi$
$200$	−21.3786	−1.51169
$201$	−6.89070	−0.486033
$202$	−18.0423	−1.26945
$203$	−0.244102	−0.0171326
$204$	2.76774	0.193781
$205$	23.3098	1.62803
$206$	23.6422	1.64723
$207$	−4.20103	−0.291992
$208$	16.3656	1.13475
$209$	0	0
$210$	−15.9078	−1.09774
$211$	12.9916	0.894378	0.447189	−	0.894439i	$-0.352425\pi$
$211$	12.9916	0.894378	0.447189	+	0.894439i	$0.352425\pi$
$212$	−4.57731	−0.314371
$213$	14.8411	1.01689
$214$	−21.2177	−1.45041
$215$	32.6034	2.22353
$216$	3.18003	0.216374
$217$	6.86988	0.466358
$218$	12.5775	0.851853
$219$	22.1173	1.49455
$220$	0	0
$221$	−4.32975	−0.291250
$222$	0.999995	0.0671153
$223$	−14.3527	−0.961127	−0.480563	−	0.876960i	$-0.659568\pi$
$223$	−14.3527	−0.961127	−0.480563	+	0.876960i	$0.659568\pi$
$224$	4.81667	0.321827
$225$	25.9686	1.73124
$226$	19.4757	1.29551
$227$	−6.37479	−0.423110	−0.211555	−	0.977366i	$-0.567853\pi$
$227$	−6.37479	−0.423110	−0.211555	+	0.977366i	$0.567853\pi$
$228$	−4.52968	−0.299985
$229$	−8.00237	−0.528812	−0.264406	−	0.964412i	$-0.585176\pi$
$229$	−8.00237	−0.528812	−0.264406	+	0.964412i	$0.585176\pi$
$230$	−12.9545	−0.854191
$231$	0	0
$232$	−0.452395	−0.0297012
$233$	10.8022	0.707675	0.353837	−	0.935307i	$-0.384877\pi$
$233$	10.8022	0.707675	0.353837	+	0.935307i	$0.384877\pi$
$234$	−12.5977	−0.823536
$235$	−16.5407	−1.07900
$236$	−0.899388	−0.0585452
$237$	12.6826	0.823824
$238$	−2.25495	−0.146167
$239$	−25.1583	−1.62735	−0.813676	−	0.581318i	$-0.802537\pi$
$239$	−25.1583	−1.62735	−0.813676	+	0.581318i	$0.802537\pi$
$240$	−46.5232	−3.00306
$241$	18.2462	1.17534	0.587669	−	0.809101i	$-0.300046\pi$
$241$	18.2462	1.17534	0.587669	+	0.809101i	$0.300046\pi$
$242$	0	0
$243$	−19.3392	−1.24061
$244$	−1.68862	−0.108103
$245$	4.06637	0.259791
$246$	22.4251	1.42977
$247$	7.08604	0.450874
$248$	12.7320	0.808482
$249$	−4.74404	−0.300641
$250$	45.3681	2.86933
$251$	15.3284	0.967520	0.483760	−	0.875201i	$-0.339271\pi$
$251$	15.3284	0.967520	0.483760	+	0.875201i	$0.339271\pi$
$252$	−2.05850	−0.129673
$253$	0	0
$254$	14.9350	0.937105
$255$	12.3084	0.770780
$256$	18.0573	1.12858
$257$	2.67980	0.167161	0.0835805	−	0.996501i	$-0.473364\pi$
$257$	2.67980	0.167161	0.0835805	+	0.996501i	$0.473364\pi$
$258$	31.3661	1.95277
$259$	−0.255619	−0.0158834
$260$	−12.1881	−0.755877
$261$	0.549527	0.0340149
$262$	21.4276	1.32380
$263$	9.97733	0.615229	0.307614	−	0.951511i	$-0.400469\pi$
$263$	9.97733	0.615229	0.307614	+	0.951511i	$0.400469\pi$
$264$	0	0
$265$	−20.3556	−1.25044
$266$	3.69045	0.226276
$267$	38.1100	2.33229
$268$	−2.74958	−0.167957
$269$	−21.5774	−1.31560	−0.657798	−	0.753194i	$-0.728512\pi$
$269$	−21.5774	−1.31560	−0.657798	+	0.753194i	$0.728512\pi$
$270$	−11.9115	−0.724909
$271$	14.9862	0.910349	0.455175	−	0.890402i	$-0.349577\pi$
$271$	14.9862	0.910349	0.455175	+	0.890402i	$0.349577\pi$
$272$	−6.59473	−0.399864
$273$	7.51153	0.454619
$274$	−10.8032	−0.652647
$275$	0	0
$276$	−3.91021	−0.235367
$277$	−5.20640	−0.312822	−0.156411	−	0.987692i	$-0.549992\pi$
$277$	−5.20640	−0.312822	−0.156411	+	0.987692i	$0.549992\pi$
$278$	−24.6564	−1.47879
$279$	−15.4656	−0.925902
$280$	7.53623	0.450376
$281$	−15.1601	−0.904374	−0.452187	−	0.891923i	$-0.649356\pi$
$281$	−15.1601	−0.904374	−0.452187	+	0.891923i	$0.649356\pi$
$282$	−15.9129	−0.947602
$283$	5.66559	0.336784	0.168392	−	0.985720i	$-0.446143\pi$
$283$	5.66559	0.336784	0.168392	+	0.985720i	$0.446143\pi$
$284$	5.92198	0.351405
$285$	−20.1438	−1.19322
$286$	0	0
$287$	−5.73233	−0.338369
$288$	−10.8434	−0.638952
$289$	−15.2553	−0.897369
$290$	1.69454	0.0995068
$291$	−5.92502	−0.347331
$292$	8.82538	0.516466
$293$	26.9403	1.57387	0.786935	−	0.617035i	$-0.211666\pi$
$293$	26.9403	1.57387	0.786935	+	0.617035i	$0.211666\pi$
$294$	3.91205	0.228155
$295$	−3.99965	−0.232869
$296$	−0.473741	−0.0275356
$297$	0	0
$298$	−41.0729	−2.37929
$299$	6.11698	0.353754
$300$	24.1709	1.39551
$301$	−8.01781	−0.462139
$302$	2.26059	0.130082
$303$	−24.2185	−1.39132
$304$	10.7929	0.619015
$305$	−7.50941	−0.429987
$306$	5.07640	0.290198
$307$	−24.6157	−1.40489	−0.702447	−	0.711736i	$-0.747909\pi$
$307$	−24.6157	−1.40489	−0.702447	+	0.711736i	$0.747909\pi$
$308$	0	0
$309$	31.7355	1.80537
$310$	−47.6903	−2.70863
$311$	−10.2131	−0.579132	−0.289566	−	0.957158i	$-0.593511\pi$
$311$	−10.2131	−0.579132	−0.289566	+	0.957158i	$0.593511\pi$
$312$	13.9212	0.788131
$313$	22.4003	1.26614	0.633069	−	0.774096i	$-0.281795\pi$
$313$	22.4003	1.26614	0.633069	+	0.774096i	$0.281795\pi$
$314$	24.9485	1.40793
$315$	−9.15429	−0.515786
$316$	5.06070	0.284687
$317$	23.3595	1.31200	0.656000	−	0.754761i	$-0.272247\pi$
$317$	23.3595	1.31200	0.656000	+	0.754761i	$0.272247\pi$
$318$	−19.5831	−1.09817
$319$	0	0
$320$	7.16707	0.400651
$321$	−28.4809	−1.58965
$322$	3.18575	0.177535
$323$	−2.85541	−0.158879
$324$	−9.77088	−0.542827
$325$	−37.8120	−2.09743
$326$	−35.7163	−1.97814
$327$	16.8830	0.933631
$328$	−10.6238	−0.586599
$329$	4.06768	0.224258
$330$	0	0
$331$	15.6444	0.859892	0.429946	−	0.902855i	$-0.358533\pi$
$331$	15.6444	0.859892	0.429946	+	0.902855i	$0.358533\pi$
$332$	−1.89300	−0.103892
$333$	0.575455	0.0315348
$334$	−8.38787	−0.458964
$335$	−12.2276	−0.668064
$336$	11.4410	0.624156
$337$	−32.5910	−1.77534	−0.887672	−	0.460477i	$-0.847678\pi$
$337$	−32.5910	−1.77534	−0.887672	+	0.460477i	$0.847678\pi$
$338$	−3.85004	−0.209414
$339$	26.1427	1.41988
$340$	4.91137	0.266356
$341$	0	0
$342$	−8.30801	−0.449245
$343$	−1.00000	−0.0539949
$344$	−14.8595	−0.801169
$345$	−17.3890	−0.936194
$346$	−33.5590	−1.80415
$347$	−5.75284	−0.308829	−0.154414	−	0.988006i	$-0.549349\pi$
$347$	−5.75284	−0.308829	−0.154414	+	0.988006i	$0.549349\pi$
$348$	0.511485	0.0274185
$349$	0.183939	0.00984606	0.00492303	−	0.999988i	$-0.498433\pi$
$349$	0.183939	0.00984606	0.00492303	+	0.999988i	$0.498433\pi$
$350$	−19.6927	−1.05262
$351$	5.62449	0.300213
$352$	0	0
$353$	19.0211	1.01239	0.506195	−	0.862419i	$-0.331052\pi$
$353$	19.0211	1.01239	0.506195	+	0.862419i	$0.331052\pi$
$354$	−3.84786	−0.204511
$355$	26.3355	1.39774
$356$	15.2069	0.805964
$357$	−3.02687	−0.160199
$358$	−43.2916	−2.28803
$359$	0.718928	0.0379435	0.0189718	−	0.999820i	$-0.493961\pi$
$359$	0.718928	0.0379435	0.0189718	+	0.999820i	$0.493961\pi$
$360$	−16.9657	−0.894172
$361$	−14.3268	−0.754045
$362$	−3.37281	−0.177271
$363$	0	0
$364$	2.99730	0.157101
$365$	39.2471	2.05429
$366$	−7.22442	−0.377626
$367$	−7.89295	−0.412009	−0.206004	−	0.978551i	$-0.566046\pi$
$367$	−7.89295	−0.412009	−0.206004	+	0.978551i	$0.566046\pi$
$368$	9.31689	0.485677
$369$	12.9047	0.671794
$370$	1.77449	0.0922516
$371$	5.00585	0.259891
$372$	−14.3950	−0.746345
$373$	−15.6686	−0.811292	−0.405646	−	0.914030i	$-0.632953\pi$
$373$	−15.6686	−0.811292	−0.405646	+	0.914030i	$0.632953\pi$
$374$	0	0
$375$	60.8985	3.14479
$376$	7.53865	0.388776
$377$	−0.800146	−0.0412096
$378$	2.92926	0.150665
$379$	−13.3144	−0.683914	−0.341957	−	0.939716i	$-0.611090\pi$
$379$	−13.3144	−0.683914	−0.341957	+	0.939716i	$0.611090\pi$
$380$	−8.03792	−0.412337
$381$	20.0476	1.02707
$382$	1.66106	0.0849871
$383$	11.8484	0.605426	0.302713	−	0.953082i	$-0.402108\pi$
$383$	11.8484	0.605426	0.302713	+	0.953082i	$0.402108\pi$
$384$	28.9704	1.47839
$385$	0	0
$386$	39.4911	2.01004
$387$	18.0499	0.917526
$388$	−2.36424	−0.120026
$389$	−6.04972	−0.306733	−0.153366	−	0.988169i	$-0.549011\pi$
$389$	−6.04972	−0.306733	−0.153366	+	0.988169i	$0.549011\pi$
$390$	−52.1446	−2.64045
$391$	−2.46491	−0.124656
$392$	−1.85331	−0.0936061
$393$	28.7627	1.45089
$394$	16.9220	0.852518
$395$	22.5053	1.13237
$396$	0	0
$397$	−11.5763	−0.580996	−0.290498	−	0.956876i	$-0.593821\pi$
$397$	−11.5763	−0.580996	−0.290498	+	0.956876i	$0.593821\pi$
$398$	17.7284	0.888643
$399$	4.95376	0.247998
$400$	−57.5923	−2.87961
$401$	33.7924	1.68751	0.843755	−	0.536728i	$-0.180340\pi$
$401$	33.7924	1.68751	0.843755	+	0.536728i	$0.180340\pi$
$402$	−11.7635	−0.586711
$403$	22.5189	1.12175
$404$	−9.66384	−0.480794
$405$	−43.4519	−2.15914
$406$	−0.416720	−0.0206815
$407$	0	0
$408$	−5.60972	−0.277722
$409$	−18.0520	−0.892613	−0.446306	−	0.894880i	$-0.647261\pi$
$409$	−18.0520	−0.892613	−0.446306	+	0.894880i	$0.647261\pi$
$410$	39.7935	1.96526
$411$	−14.5014	−0.715301
$412$	12.6633	0.623876
$413$	0.983592	0.0483994
$414$	−7.17183	−0.352476
$415$	−8.41831	−0.413239
$416$	15.7887	0.774103
$417$	−33.0968	−1.62076
$418$	0	0
$419$	2.58559	0.126314	0.0631571	−	0.998004i	$-0.479883\pi$
$419$	2.58559	0.126314	0.0631571	+	0.998004i	$0.479883\pi$
$420$	−8.52058	−0.415762
$421$	13.9948	0.682064	0.341032	−	0.940052i	$-0.389224\pi$
$421$	13.9948	0.682064	0.341032	+	0.940052i	$0.389224\pi$
$422$	22.1787	1.07964
$423$	−9.15724	−0.445240
$424$	9.27738	0.450549
$425$	15.2368	0.739096
$426$	25.3360	1.22754
$427$	1.84671	0.0893686
$428$	−11.3646	−0.549331
$429$	0	0
$430$	55.6592	2.68412
$431$	33.6682	1.62174	0.810870	−	0.585227i	$-0.198995\pi$
$431$	33.6682	1.62174	0.810870	+	0.585227i	$0.198995\pi$
$432$	8.56677	0.412169
$433$	7.52292	0.361529	0.180764	−	0.983526i	$-0.442143\pi$
$433$	7.52292	0.361529	0.180764	+	0.983526i	$0.442143\pi$
$434$	11.7280	0.562961
$435$	2.27461	0.109059
$436$	6.73676	0.322632
$437$	4.03407	0.192976
$438$	37.7577	1.80413
$439$	22.0123	1.05059	0.525294	−	0.850921i	$-0.323955\pi$
$439$	22.0123	1.05059	0.525294	+	0.850921i	$0.323955\pi$
$440$	0	0
$441$	2.25122	0.107201
$442$	−7.39156	−0.351581
$443$	23.6893	1.12551	0.562756	−	0.826623i	$-0.309741\pi$
$443$	23.6893	1.12551	0.562756	+	0.826623i	$0.309741\pi$
$444$	0.535619	0.0254194
$445$	67.6263	3.20579
$446$	−24.5023	−1.16022
$447$	−55.1330	−2.60770
$448$	−1.76252	−0.0832713
$449$	2.27513	0.107370	0.0536851	−	0.998558i	$-0.482903\pi$
$449$	2.27513	0.107370	0.0536851	+	0.998558i	$0.482903\pi$
$450$	44.3326	2.08986
$451$	0	0
$452$	10.4316	0.490663
$453$	3.03443	0.142570
$454$	−10.8828	−0.510754
$455$	13.3292	0.624885
$456$	9.18084	0.429932
$457$	27.3134	1.27767	0.638833	−	0.769345i	$-0.279417\pi$
$457$	27.3134	1.27767	0.638833	+	0.769345i	$0.279417\pi$
$458$	−13.6613	−0.638351
$459$	−2.26646	−0.105789
$460$	−6.93868	−0.323518
$461$	8.21908	0.382801	0.191400	−	0.981512i	$-0.438697\pi$
$461$	8.21908	0.382801	0.191400	+	0.981512i	$0.438697\pi$
$462$	0	0
$463$	−27.9839	−1.30052	−0.650261	−	0.759711i	$-0.725340\pi$
$463$	−27.9839	−1.30052	−0.650261	+	0.759711i	$0.725340\pi$
$464$	−1.21872	−0.0565776
$465$	−64.0156	−2.96865
$466$	18.4411	0.854265
$467$	26.6745	1.23435	0.617175	−	0.786826i	$-0.288277\pi$
$467$	26.6745	1.23435	0.617175	+	0.786826i	$0.288277\pi$
$468$	−6.74759	−0.311907
$469$	3.00700	0.138850
$470$	−28.2376	−1.30250
$471$	33.4889	1.54309
$472$	1.82290	0.0839057
$473$	0	0
$474$	21.6512	0.994474
$475$	−24.9365	−1.14417
$476$	−1.20780	−0.0553595
$477$	−11.2693	−0.515985
$478$	−42.9491	−1.96445
$479$	23.2240	1.06113	0.530566	−	0.847644i	$-0.321980\pi$
$479$	23.2240	1.06113	0.530566	+	0.847644i	$0.321980\pi$
$480$	−44.8832	−2.04863
$481$	−0.837900	−0.0382050
$482$	31.1491	1.41880
$483$	4.27630	0.194578
$484$	0	0
$485$	−10.5140	−0.477415
$486$	−33.0150	−1.49759
$487$	−18.7920	−0.851548	−0.425774	−	0.904830i	$-0.639998\pi$
$487$	−18.7920	−0.851548	−0.425774	+	0.904830i	$0.639998\pi$
$488$	3.42252	0.154930
$489$	−47.9427	−2.16804
$490$	6.94194	0.313605
$491$	−7.40573	−0.334216	−0.167108	−	0.985939i	$-0.553443\pi$
$491$	−7.40573	−0.334216	−0.167108	+	0.985939i	$0.553443\pi$
$492$	12.0114	0.541515
$493$	0.322429	0.0145215
$494$	12.0970	0.544269
$495$	0	0
$496$	34.2991	1.54007
$497$	−6.47642	−0.290507
$498$	−8.09882	−0.362917
$499$	23.6703	1.05963	0.529815	−	0.848113i	$-0.322261\pi$
$499$	23.6703	1.05963	0.529815	+	0.848113i	$0.322261\pi$
$500$	24.3001	1.08674
$501$	−11.2592	−0.503024
$502$	26.1680	1.16794
$503$	8.39731	0.374418	0.187209	−	0.982320i	$-0.440056\pi$
$503$	8.39731	0.374418	0.187209	+	0.982320i	$0.440056\pi$
$504$	4.17220	0.185845
$505$	−42.9758	−1.91240
$506$	0	0
$507$	−5.16798	−0.229518
$508$	7.99951	0.354921
$509$	14.3065	0.634123	0.317061	−	0.948405i	$-0.397304\pi$
$509$	14.3065	0.634123	0.317061	+	0.948405i	$0.397304\pi$
$510$	21.0124	0.930443
$511$	−9.65164	−0.426963
$512$	5.54215	0.244931
$513$	3.70928	0.163769
$514$	4.57484	0.201787
$515$	56.3147	2.48152
$516$	16.8003	0.739594
$517$	0	0
$518$	−0.436383	−0.0191736
$519$	−45.0470	−1.97734
$520$	24.7032	1.08331
$521$	−33.3684	−1.46189	−0.730947	−	0.682434i	$-0.760921\pi$
$521$	−33.3684	−1.46189	−0.730947	+	0.682434i	$0.760921\pi$
$522$	0.938129	0.0410608
$523$	−13.3817	−0.585143	−0.292571	−	0.956244i	$-0.594511\pi$
$523$	−13.3817	−0.585143	−0.292571	+	0.956244i	$0.594511\pi$
$524$	11.4771	0.501378
$525$	−26.4339	−1.15367
$526$	17.0329	0.742669
$527$	−9.07430	−0.395283
$528$	0	0
$529$	−19.5176	−0.848592
$530$	−34.7503	−1.50946
$531$	−2.21428	−0.0960917
$532$	1.97668	0.0857001
$533$	−18.7901	−0.813891
$534$	65.0598	2.81541
$535$	−50.5395	−2.18501
$536$	5.57289	0.240712
$537$	−58.1112	−2.50768
$538$	−36.8360	−1.58811
$539$	0	0
$540$	−6.38004	−0.274553
$541$	34.0639	1.46452	0.732262	−	0.681023i	$-0.238465\pi$
$541$	34.0639	1.46452	0.732262	+	0.681023i	$0.238465\pi$
$542$	25.5839	1.09892
$543$	−4.52739	−0.194289
$544$	−6.36225	−0.272779
$545$	29.9589	1.28330
$546$	12.8234	0.548790
$547$	15.7614	0.673909	0.336954	−	0.941521i	$-0.390603\pi$
$547$	15.7614	0.673909	0.336954	+	0.941521i	$0.390603\pi$
$548$	−5.78644	−0.247185
$549$	−4.15735	−0.177431
$550$	0	0
$551$	−0.527686	−0.0224802
$552$	7.92530	0.337323
$553$	−5.53450	−0.235351
$554$	−8.88815	−0.377621
$555$	2.38194	0.101108
$556$	−13.2065	−0.560080
$557$	−16.1024	−0.682282	−0.341141	−	0.940012i	$-0.610813\pi$
$557$	−16.1024	−0.682282	−0.341141	+	0.940012i	$0.610813\pi$
$558$	−26.4023	−1.11770
$559$	−26.2818	−1.11160
$560$	20.3020	0.857918
$561$	0	0
$562$	−25.8806	−1.09171
$563$	17.2305	0.726177	0.363089	−	0.931755i	$-0.381722\pi$
$563$	17.2305	0.726177	0.363089	+	0.931755i	$0.381722\pi$
$564$	−8.52332	−0.358896
$565$	46.3903	1.95165
$566$	9.67206	0.406547
$567$	10.6857	0.448756
$568$	−12.0028	−0.503626
$569$	35.6143	1.49303	0.746514	−	0.665370i	$-0.231726\pi$
$569$	35.6143	1.49303	0.746514	+	0.665370i	$0.231726\pi$
$570$	−34.3887	−1.44038
$571$	−26.6026	−1.11329	−0.556643	−	0.830752i	$-0.687911\pi$
$571$	−26.6026	−1.11329	−0.556643	+	0.830752i	$0.687911\pi$
$572$	0	0
$573$	2.22967	0.0931459
$574$	−9.78600	−0.408460
$575$	−21.5263	−0.897709
$576$	3.96783	0.165326
$577$	28.8725	1.20198	0.600989	−	0.799257i	$-0.294773\pi$
$577$	28.8725	1.20198	0.600989	+	0.799257i	$0.294773\pi$
$578$	−26.0432	−1.08325
$579$	53.0097	2.20301
$580$	0.907632	0.0376874
$581$	2.07023	0.0858875
$582$	−10.1150	−0.419278
$583$	0	0
$584$	−17.8874	−0.740188
$585$	−30.0071	−1.24064
$586$	45.9914	1.89989
$587$	2.43414	0.100468	0.0502339	−	0.998737i	$-0.484003\pi$
$587$	2.43414	0.100468	0.0502339	+	0.998737i	$0.484003\pi$
$588$	2.09538	0.0864119
$589$	14.8509	0.611922
$590$	−6.82804	−0.281106
$591$	22.7147	0.934359
$592$	−1.27622	−0.0524525
$593$	−15.6870	−0.644187	−0.322093	−	0.946708i	$-0.604387\pi$
$593$	−15.6870	−0.644187	−0.322093	+	0.946708i	$0.604387\pi$
$594$	0	0
$595$	−5.37119	−0.220197
$596$	−21.9995	−0.901136
$597$	23.7972	0.973953
$598$	10.4426	0.427032
$599$	−13.2218	−0.540229	−0.270115	−	0.962828i	$-0.587062\pi$
$599$	−13.2218	−0.540229	−0.270115	+	0.962828i	$0.587062\pi$
$600$	−48.9901	−2.00001
$601$	11.2303	0.458093	0.229047	−	0.973415i	$-0.426439\pi$
$601$	11.2303	0.458093	0.229047	+	0.973415i	$0.426439\pi$
$602$	−13.6877	−0.557868
$603$	−6.76942	−0.275672
$604$	1.21082	0.0492676
$605$	0	0
$606$	−41.3449	−1.67952
$607$	−19.3477	−0.785298	−0.392649	−	0.919688i	$-0.628441\pi$
$607$	−19.3477	−0.785298	−0.392649	+	0.919688i	$0.628441\pi$
$608$	10.4124	0.422279
$609$	−0.559372	−0.0226669
$610$	−12.8197	−0.519056
$611$	13.3335	0.539417
$612$	2.71903	0.109910
$613$	−21.0549	−0.850398	−0.425199	−	0.905100i	$-0.639796\pi$
$613$	−21.0549	−0.850398	−0.425199	+	0.905100i	$0.639796\pi$
$614$	−42.0230	−1.69591
$615$	53.4156	2.15392
$616$	0	0
$617$	−24.1496	−0.972228	−0.486114	−	0.873895i	$-0.661586\pi$
$617$	−24.1496	−0.972228	−0.486114	+	0.873895i	$0.661586\pi$
$618$	54.1775	2.17934
$619$	2.62475	0.105497	0.0527487	−	0.998608i	$-0.483202\pi$
$619$	2.62475	0.105497	0.0527487	+	0.998608i	$0.483202\pi$
$620$	−25.5440	−1.02587
$621$	3.20201	0.128492
$622$	−17.4354	−0.699095
$623$	−16.6306	−0.666292
$624$	37.5026	1.50131
$625$	50.3878	2.01551
$626$	38.2408	1.52841
$627$	0	0
$628$	13.3630	0.533241
$629$	0.337643	0.0134627
$630$	−15.6278	−0.622628
$631$	−25.1176	−0.999914	−0.499957	−	0.866050i	$-0.666651\pi$
$631$	−25.1176	−0.999914	−0.499957	+	0.866050i	$0.666651\pi$
$632$	−10.2571	−0.408006
$633$	29.7709	1.18329
$634$	39.8784	1.58377
$635$	35.5744	1.41173
$636$	−10.4891	−0.415922
$637$	−3.27792	−0.129876
$638$	0	0
$639$	14.5798	0.576770
$640$	51.4080	2.03208
$641$	20.6584	0.815958	0.407979	−	0.912991i	$-0.366234\pi$
$641$	20.6584	0.815958	0.407979	+	0.912991i	$0.366234\pi$
$642$	−48.6214	−1.91894
$643$	16.4632	0.649246	0.324623	−	0.945844i	$-0.394763\pi$
$643$	16.4632	0.649246	0.324623	+	0.945844i	$0.394763\pi$
$644$	1.70636	0.0672399
$645$	74.7124	2.94180
$646$	−4.87464	−0.191790
$647$	−34.9584	−1.37435	−0.687177	−	0.726490i	$-0.741150\pi$
$647$	−34.9584	−1.37435	−0.687177	+	0.726490i	$0.741150\pi$
$648$	19.8038	0.777967
$649$	0	0
$650$	−64.5511	−2.53190
$651$	15.7427	0.617005
$652$	−19.1304	−0.749205
$653$	2.41649	0.0945645	0.0472823	−	0.998882i	$-0.484944\pi$
$653$	2.41649	0.0945645	0.0472823	+	0.998882i	$0.484944\pi$
$654$	28.8219	1.12703
$655$	51.0395	1.99428
$656$	−28.6196	−1.11741
$657$	21.7280	0.847689
$658$	6.94417	0.270712
$659$	−16.9733	−0.661186	−0.330593	−	0.943773i	$-0.607249\pi$
$659$	−16.9733	−0.661186	−0.330593	+	0.943773i	$0.607249\pi$
$660$	0	0
$661$	7.96946	0.309976	0.154988	−	0.987916i	$-0.450466\pi$
$661$	7.96946	0.309976	0.154988	+	0.987916i	$0.450466\pi$
$662$	26.7074	1.03801
$663$	−9.92184	−0.385333
$664$	3.83677	0.148895
$665$	8.79046	0.340880
$666$	0.982394	0.0380670
$667$	−0.455522	−0.0176379
$668$	−4.49272	−0.173829
$669$	−32.8899	−1.27160
$670$	−20.8744	−0.806449
$671$	0	0
$672$	11.0377	0.425787
$673$	−16.0250	−0.617717	−0.308859	−	0.951108i	$-0.599947\pi$
$673$	−16.0250	−0.617717	−0.308859	+	0.951108i	$0.599947\pi$
$674$	−55.6380	−2.14309
$675$	−19.7932	−0.761840
$676$	−2.06216	−0.0793140
$677$	−15.2366	−0.585589	−0.292794	−	0.956175i	$-0.594585\pi$
$677$	−15.2366	−0.585589	−0.292794	+	0.956175i	$0.594585\pi$
$678$	44.6297	1.71399
$679$	2.58559	0.0992259
$680$	−9.95446	−0.381736
$681$	−14.6082	−0.559787
$682$	0	0
$683$	14.3742	0.550013	0.275007	−	0.961442i	$-0.411320\pi$
$683$	14.3742	0.550013	0.275007	+	0.961442i	$0.411320\pi$
$684$	−4.44995	−0.170148
$685$	−25.7328	−0.983198
$686$	−1.70716	−0.0651796
$687$	−18.3379	−0.699633
$688$	−40.0303	−1.52614
$689$	16.4088	0.625125
$690$	−29.6858	−1.13012
$691$	−44.7978	−1.70419	−0.852095	−	0.523387i	$-0.824668\pi$
$691$	−44.7978	−1.70419	−0.852095	+	0.523387i	$0.824668\pi$
$692$	−17.9750	−0.683305
$693$	0	0
$694$	−9.82101	−0.372800
$695$	−58.7303	−2.22777
$696$	−1.03669	−0.0392956
$697$	7.57173	0.286800
$698$	0.314014	0.0118856
$699$	24.7538	0.936275
$700$	−10.5478	−0.398671
$701$	0.240330	0.00907713	0.00453857	−	0.999990i	$-0.498555\pi$
$701$	0.240330	0.00907713	0.00453857	+	0.999990i	$0.498555\pi$
$702$	9.60189	0.362400
$703$	−0.552584	−0.0208411
$704$	0	0
$705$	−37.9039	−1.42754
$706$	32.4720	1.22210
$707$	10.5686	0.397473
$708$	−2.06100	−0.0774570
$709$	10.7534	0.403852	0.201926	−	0.979401i	$-0.435280\pi$
$709$	10.7534	0.403852	0.201926	+	0.979401i	$0.435280\pi$
$710$	44.9589	1.68728
$711$	12.4594	0.467263
$712$	−30.8216	−1.15509
$713$	12.8200	0.480112
$714$	−5.16735	−0.193383
$715$	0	0
$716$	−23.1879	−0.866573
$717$	−57.6515	−2.15304
$718$	1.22732	0.0458033
$719$	−25.7748	−0.961239	−0.480620	−	0.876929i	$-0.659588\pi$
$719$	−25.7748	−0.961239	−0.480620	+	0.876929i	$0.659588\pi$
$720$	−45.7044	−1.70330
$721$	−13.8489	−0.515759
$722$	−24.4582	−0.910240
$723$	41.8120	1.55501
$724$	−1.80655	−0.0671399
$725$	2.81580	0.104576
$726$	0	0
$727$	−5.47160	−0.202930	−0.101465	−	0.994839i	$-0.532353\pi$
$727$	−5.47160	−0.202930	−0.101465	+	0.994839i	$0.532353\pi$
$728$	−6.07499	−0.225154
$729$	−12.2598	−0.454065
$730$	67.0011	2.47982
$731$	10.5906	0.391707
$732$	−3.86955	−0.143023
$733$	19.8563	0.733411	0.366705	−	0.930337i	$-0.380486\pi$
$733$	19.8563	0.733411	0.366705	+	0.930337i	$0.380486\pi$
$734$	−13.4745	−0.497354
$735$	9.31831	0.343711
$736$	8.98845	0.331319
$737$	0	0
$738$	22.0304	0.810951
$739$	−1.80077	−0.0662423	−0.0331212	−	0.999451i	$-0.510545\pi$
$739$	−1.80077	−0.0662423	−0.0331212	+	0.999451i	$0.510545\pi$
$740$	0.950458	0.0349395
$741$	16.2380	0.596519
$742$	8.54578	0.313726
$743$	−48.0579	−1.76307	−0.881537	−	0.472115i	$-0.843491\pi$
$743$	−48.0579	−1.76307	−0.881537	+	0.472115i	$0.843491\pi$
$744$	29.1760	1.06965
$745$	−97.8337	−3.58435
$746$	−26.7489	−0.979346
$747$	−4.66054	−0.170520
$748$	0	0
$749$	12.4286	0.454133
$750$	103.963	3.79621
$751$	−25.0465	−0.913961	−0.456980	−	0.889477i	$-0.651069\pi$
$751$	−25.0465	−0.913961	−0.456980	+	0.889477i	$0.651069\pi$
$752$	20.3086	0.740578
$753$	35.1259	1.28006
$754$	−1.36598	−0.0497459
$755$	5.38461	0.195966
$756$	1.56898	0.0570631
$757$	10.6487	0.387032	0.193516	−	0.981097i	$-0.438011\pi$
$757$	10.6487	0.387032	0.193516	+	0.981097i	$0.438011\pi$
$758$	−22.7298	−0.825583
$759$	0	0
$760$	16.2914	0.590952
$761$	29.8160	1.08083	0.540414	−	0.841399i	$-0.318268\pi$
$761$	29.8160	1.08083	0.540414	+	0.841399i	$0.318268\pi$
$762$	34.2244	1.23982
$763$	−7.36748	−0.266721
$764$	0.889698	0.0321882
$765$	12.0917	0.437178
$766$	20.2271	0.730836
$767$	3.22414	0.116417
$768$	41.3792	1.49314
$769$	28.3115	1.02094	0.510470	−	0.859896i	$-0.329471\pi$
$769$	28.3115	1.02094	0.510470	+	0.859896i	$0.329471\pi$
$770$	0	0
$771$	6.14090	0.221159
$772$	21.1523	0.761287
$773$	13.2564	0.476800	0.238400	−	0.971167i	$-0.423377\pi$
$773$	13.2564	0.476800	0.238400	+	0.971167i	$0.423377\pi$
$774$	30.8140	1.10759
$775$	−79.2466	−2.84662
$776$	4.79189	0.172019
$777$	−0.585766	−0.0210142
$778$	−10.3278	−0.370271
$779$	−12.3918	−0.443984
$780$	−27.9298	−1.00005
$781$	0	0
$782$	−4.20800	−0.150478
$783$	−0.418847	−0.0149684
$784$	−4.99267	−0.178310
$785$	59.4262	2.12101
$786$	49.1025	1.75143
$787$	−52.6771	−1.87774	−0.938868	−	0.344279i	$-0.888124\pi$
$787$	−52.6771	−1.87774	−0.938868	+	0.344279i	$0.888124\pi$
$788$	9.06379	0.322884
$789$	22.8636	0.813965
$790$	38.4202	1.36693
$791$	−11.4083	−0.405632
$792$	0	0
$793$	6.05337	0.214961
$794$	−19.7625	−0.701346
$795$	−46.6461	−1.65437
$796$	9.49570	0.336566
$797$	−7.33480	−0.259812	−0.129906	−	0.991526i	$-0.541468\pi$
$797$	−7.33480	−0.259812	−0.129906	+	0.991526i	$0.541468\pi$
$798$	8.45686	0.299369
$799$	−5.37292	−0.190080
$800$	−55.5620	−1.96442
$801$	37.4392	1.32285
$802$	57.6890	2.03707
$803$	0	0
$804$	−6.30080	−0.222212
$805$	7.58831	0.267453
$806$	38.4434	1.35411
$807$	−49.4458	−1.74057
$808$	19.5869	0.689063
$809$	−8.66809	−0.304754	−0.152377	−	0.988322i	$-0.548693\pi$
$809$	−8.66809	−0.304754	−0.152377	+	0.988322i	$0.548693\pi$
$810$	−74.1793	−2.60639
$811$	−19.9487	−0.700494	−0.350247	−	0.936657i	$-0.613902\pi$
$811$	−19.9487	−0.700494	−0.350247	+	0.936657i	$0.613902\pi$
$812$	−0.223205	−0.00783294
$813$	34.3418	1.20442
$814$	0	0
$815$	−85.0745	−2.98003
$816$	−15.1122	−0.529032
$817$	−17.3325	−0.606387
$818$	−30.8176	−1.07751
$819$	7.37932	0.257854
$820$	21.3143	0.744326
$821$	−53.5648	−1.86943	−0.934713	−	0.355404i	$-0.884343\pi$
$821$	−53.5648	−1.86943	−0.934713	+	0.355404i	$0.884343\pi$
$822$	−24.7562	−0.863471
$823$	−8.44842	−0.294493	−0.147247	−	0.989100i	$-0.547041\pi$
$823$	−8.44842	−0.294493	−0.147247	+	0.989100i	$0.547041\pi$
$824$	−25.6662	−0.894125
$825$	0	0
$826$	1.67915	0.0584250
$827$	3.18260	0.110670	0.0553349	−	0.998468i	$-0.482377\pi$
$827$	3.18260	0.110670	0.0553349	+	0.998468i	$0.482377\pi$
$828$	−3.84139	−0.133497
$829$	57.5089	1.99737	0.998684	−	0.0512831i	$-0.0163311\pi$
$829$	57.5089	1.99737	0.998684	+	0.0512831i	$0.0163311\pi$
$830$	−14.3714	−0.498838
$831$	−11.9307	−0.413873
$832$	−5.77741	−0.200296
$833$	1.32088	0.0457658
$834$	−56.5015	−1.95649
$835$	−19.9795	−0.691419
$836$	0	0
$837$	11.7878	0.407447
$838$	4.41401	0.152479
$839$	12.5594	0.433600	0.216800	−	0.976216i	$-0.430438\pi$
$839$	12.5594	0.433600	0.216800	+	0.976216i	$0.430438\pi$
$840$	17.2697	0.595861
$841$	−28.9404	−0.997945
$842$	23.8913	0.823349
$843$	−34.7401	−1.19651
$844$	11.8794	0.408906
$845$	−9.17060	−0.315478
$846$	−15.6329	−0.537469
$847$	0	0
$848$	24.9926	0.858248
$849$	12.9830	0.445575
$850$	26.0117	0.892195
$851$	−0.477015	−0.0163519
$852$	13.5705	0.464919
$853$	−33.3315	−1.14125	−0.570625	−	0.821211i	$-0.693299\pi$
$853$	−33.3315	−1.14125	−0.570625	+	0.821211i	$0.693299\pi$
$854$	3.15263	0.107881
$855$	−19.7893	−0.676779
$856$	23.0341	0.787289
$857$	−54.0291	−1.84560	−0.922800	−	0.385279i	$-0.874105\pi$
$857$	−54.0291	−1.84560	−0.922800	+	0.385279i	$0.874105\pi$
$858$	0	0
$859$	−12.8624	−0.438860	−0.219430	−	0.975628i	$-0.570420\pi$
$859$	−12.8624	−0.438860	−0.219430	+	0.975628i	$0.570420\pi$
$860$	29.8123	1.01659
$861$	−13.1359	−0.447672
$862$	57.4769	1.95767
$863$	−33.2271	−1.13106	−0.565531	−	0.824727i	$-0.691329\pi$
$863$	−33.2271	−1.13106	−0.565531	+	0.824727i	$0.691329\pi$
$864$	8.26478	0.281173
$865$	−79.9360	−2.71791
$866$	12.8428	0.436417
$867$	−34.9583	−1.18725
$868$	6.28176	0.213217
$869$	0	0
$870$	3.88313	0.131650
$871$	9.85671	0.333982
$872$	−13.6542	−0.462389
$873$	−5.82074	−0.197002
$874$	6.88679	0.232949
$875$	−26.5752	−0.898406
$876$	20.2238	0.683300
$877$	−27.6709	−0.934379	−0.467189	−	0.884157i	$-0.654733\pi$
$877$	−27.6709	−0.934379	−0.467189	+	0.884157i	$0.654733\pi$
$878$	37.5784	1.26821
$879$	61.7352	2.08228
$880$	0	0
$881$	48.9636	1.64963	0.824813	−	0.565406i	$-0.191280\pi$
$881$	48.9636	1.64963	0.824813	+	0.565406i	$0.191280\pi$
$882$	3.84319	0.129407
$883$	21.3223	0.717554	0.358777	−	0.933423i	$-0.383194\pi$
$883$	21.3223	0.717554	0.358777	+	0.933423i	$0.383194\pi$
$884$	−3.95908	−0.133158
$885$	−9.16541	−0.308092
$886$	40.4414	1.35865
$887$	45.2483	1.51929	0.759645	−	0.650338i	$-0.225373\pi$
$887$	45.2483	1.51929	0.759645	+	0.650338i	$0.225373\pi$
$888$	−1.08560	−0.0364305
$889$	−8.74845	−0.293414
$890$	115.449	3.86985
$891$	0	0
$892$	−13.1240	−0.439423
$893$	8.79329	0.294256
$894$	−94.1208	−3.14787
$895$	−103.118	−3.44687
$896$	−12.6422	−0.422348
$897$	14.0174	0.468027
$898$	3.88401	0.129611
$899$	−1.67695	−0.0559294
$900$	23.7455	0.791517
$901$	−6.61213	−0.220282
$902$	0	0
$903$	−18.3732	−0.611423
$904$	−21.1430	−0.703207
$905$	−8.03388	−0.267055
$906$	5.18026	0.172103
$907$	−45.0665	−1.49641	−0.748205	−	0.663468i	$-0.769084\pi$
$907$	−45.0665	−1.49641	−0.748205	+	0.663468i	$0.769084\pi$
$908$	−5.82906	−0.193444
$909$	−23.7922	−0.789139
$910$	22.7551	0.754325
$911$	−9.21469	−0.305296	−0.152648	−	0.988281i	$-0.548780\pi$
$911$	−9.21469	−0.305296	−0.152648	+	0.988281i	$0.548780\pi$
$912$	24.7325	0.818975
$913$	0	0
$914$	46.6283	1.54233
$915$	−17.2082	−0.568886
$916$	−7.31730	−0.241770
$917$	−12.5516	−0.414491
$918$	−3.86921	−0.127703
$919$	−49.7144	−1.63993	−0.819963	−	0.572417i	$-0.806006\pi$
$919$	−49.7144	−1.63993	−0.819963	+	0.572417i	$0.806006\pi$
$920$	14.0635	0.463659
$921$	−56.4083	−1.85872
$922$	14.0313	0.462095
$923$	−21.2292	−0.698767
$924$	0	0
$925$	2.94866	0.0969514
$926$	−47.7729	−1.56992
$927$	31.1769	1.02398
$928$	−1.17576	−0.0385961
$929$	49.5105	1.62439	0.812193	−	0.583389i	$-0.198274\pi$
$929$	49.5105	1.62439	0.812193	+	0.583389i	$0.198274\pi$
$930$	−109.285	−3.58359
$931$	−2.16175	−0.0708484
$932$	9.87743	0.323546
$933$	−23.4039	−0.766208
$934$	45.5377	1.49004
$935$	0	0
$936$	13.6761	0.447019
$937$	−6.98790	−0.228285	−0.114142	−	0.993464i	$-0.536412\pi$
$937$	−6.98790	−0.228285	−0.114142	+	0.993464i	$0.536412\pi$
$938$	5.13343	0.167612
$939$	51.3314	1.67514
$940$	−15.1247	−0.493312
$941$	−46.8241	−1.52642	−0.763211	−	0.646150i	$-0.776378\pi$
$941$	−46.8241	−1.52642	−0.763211	+	0.646150i	$0.776378\pi$
$942$	57.1709	1.86273
$943$	−10.6972	−0.348348
$944$	4.91075	0.159831
$945$	6.97736	0.226974
$946$	0	0
$947$	15.1286	0.491614	0.245807	−	0.969319i	$-0.420947\pi$
$947$	15.1286	0.491614	0.245807	+	0.969319i	$0.420947\pi$
$948$	11.5969	0.376649
$949$	−31.6373	−1.02699
$950$	−42.5706	−1.38117
$951$	53.5295	1.73581
$952$	2.44800	0.0793401
$953$	56.2273	1.82138	0.910690	−	0.413090i	$-0.135550\pi$
$953$	56.2273	1.82138	0.910690	+	0.413090i	$0.135550\pi$
$954$	−19.2384	−0.622867
$955$	3.95656	0.128031
$956$	−23.0045	−0.744019
$957$	0	0
$958$	39.6470	1.28094
$959$	6.32819	0.204348
$960$	16.4237	0.530073
$961$	16.1952	0.522427
$962$	−1.43043	−0.0461189
$963$	−27.9796	−0.901631
$964$	16.6841	0.537359
$965$	94.0658	3.02809
$966$	7.30033	0.234884
$967$	−46.3761	−1.49136	−0.745678	−	0.666307i	$-0.767874\pi$
$967$	−46.3761	−1.49136	−0.745678	+	0.666307i	$0.767874\pi$
$968$	0	0
$969$	−6.54333	−0.210202
$970$	−17.9490	−0.576308
$971$	15.5400	0.498701	0.249351	−	0.968413i	$-0.419783\pi$
$971$	15.5400	0.498701	0.249351	+	0.968413i	$0.419783\pi$
$972$	−17.6836	−0.567201
$973$	14.4429	0.463019
$974$	−32.0810	−1.02794
$975$	−86.6483	−2.77497
$976$	9.22002	0.295125
$977$	32.8287	1.05028	0.525141	−	0.851015i	$-0.324012\pi$
$977$	32.8287	1.05028	0.525141	+	0.851015i	$0.324012\pi$
$978$	−81.8458	−2.61714
$979$	0	0
$980$	3.71825	0.118775
$981$	16.5858	0.529544
$982$	−12.6428	−0.403447
$983$	24.3184	0.775635	0.387817	−	0.921736i	$-0.373229\pi$
$983$	24.3184	0.775635	0.387817	+	0.921736i	$0.373229\pi$
$984$	−24.3449	−0.776088
$985$	40.3074	1.28430
$986$	0.550438	0.0175295
$987$	9.32130	0.296700
$988$	6.47941	0.206138
$989$	−14.9622	−0.475769
$990$	0	0
$991$	−47.7104	−1.51557	−0.757786	−	0.652503i	$-0.773719\pi$
$991$	−47.7104	−1.51557	−0.757786	+	0.652503i	$0.773719\pi$
$992$	33.0899	1.05061
$993$	35.8499	1.13766
$994$	−11.0563	−0.350684
$995$	42.2281	1.33872
$996$	−4.33791	−0.137452
$997$	−19.8449	−0.628494	−0.314247	−	0.949341i	$-0.601752\pi$
$997$	−19.8449	−0.628494	−0.314247	+	0.949341i	$0.601752\pi$
$998$	40.4090	1.27913
$999$	−0.438610	−0.0138770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.o.1.7		8
3.2	odd	2		7623.2.a.cw.1.2		8
7.6	odd	2		5929.2.a.bs.1.7		8
11.2	odd	10		77.2.f.b.15.4	✓	16
11.3	even	5		847.2.f.v.372.4		16
11.4	even	5		847.2.f.v.148.4		16
11.5	even	5		847.2.f.x.729.1		16
11.6	odd	10		77.2.f.b.36.4	yes	16
11.7	odd	10		847.2.f.w.148.1		16
11.8	odd	10		847.2.f.w.372.1		16
11.9	even	5		847.2.f.x.323.1		16
11.10	odd	2		847.2.a.p.1.2		8
33.2	even	10		693.2.m.i.631.1		16
33.17	even	10		693.2.m.i.190.1		16
33.32	even	2		7623.2.a.ct.1.7		8
77.2	odd	30		539.2.q.g.312.4		32
77.6	even	10		539.2.f.e.344.4		16
77.13	even	10		539.2.f.e.246.4		16
77.17	even	30		539.2.q.f.520.4		32
77.24	even	30		539.2.q.f.422.1		32
77.39	odd	30		539.2.q.g.520.4		32
77.46	odd	30		539.2.q.g.422.1		32
77.61	even	30		539.2.q.f.410.1		32
77.68	even	30		539.2.q.f.312.4		32
77.72	odd	30		539.2.q.g.410.1		32
77.76	even	2		5929.2.a.bt.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.15.4	✓	16	11.2	odd	10
77.2.f.b.36.4	yes	16	11.6	odd	10
539.2.f.e.246.4		16	77.13	even	10
539.2.f.e.344.4		16	77.6	even	10
539.2.q.f.312.4		32	77.68	even	30
539.2.q.f.410.1		32	77.61	even	30
539.2.q.f.422.1		32	77.24	even	30
539.2.q.f.520.4		32	77.17	even	30
539.2.q.g.312.4		32	77.2	odd	30
539.2.q.g.410.1		32	77.72	odd	30
539.2.q.g.422.1		32	77.46	odd	30
539.2.q.g.520.4		32	77.39	odd	30
693.2.m.i.190.1		16	33.17	even	10
693.2.m.i.631.1		16	33.2	even	10
847.2.a.o.1.7		8	1.1	even	1	trivial
847.2.a.p.1.2		8	11.10	odd	2
847.2.f.v.148.4		16	11.4	even	5
847.2.f.v.372.4		16	11.3	even	5
847.2.f.w.148.1		16	11.7	odd	10
847.2.f.w.372.1		16	11.8	odd	10
847.2.f.x.323.1		16	11.9	even	5
847.2.f.x.729.1		16	11.5	even	5
5929.2.a.bs.1.7		8	7.6	odd	2
5929.2.a.bt.1.2		8	77.76	even	2
7623.2.a.ct.1.7		8	33.32	even	2
7623.2.a.cw.1.2		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.70716 q^{2} +2.29155 q^{3} +0.914391 q^{4} +4.06637 q^{5} +3.91205 q^{6} -1.00000 q^{7} -1.85331 q^{8} +2.25122 q^{9} +O(q^{10})\)	\(q+1.70716 q^{2} +2.29155 q^{3} +0.914391 q^{4} +4.06637 q^{5} +3.91205 q^{6} -1.00000 q^{7} -1.85331 q^{8} +2.25122 q^{9} +6.94194 q^{10} +2.09538 q^{12} -3.27792 q^{13} -1.70716 q^{14} +9.31831 q^{15} -4.99267 q^{16} +1.32088 q^{17} +3.84319 q^{18} -2.16175 q^{19} +3.71825 q^{20} -2.29155 q^{21} -1.86611 q^{23} -4.24695 q^{24} +11.5354 q^{25} -5.59593 q^{26} -1.71587 q^{27} -0.914391 q^{28} +0.244102 q^{29} +15.9078 q^{30} -6.86988 q^{31} -4.81667 q^{32} +2.25495 q^{34} -4.06637 q^{35} +2.05850 q^{36} +0.255619 q^{37} -3.69045 q^{38} -7.51153 q^{39} -7.53623 q^{40} +5.73233 q^{41} -3.91205 q^{42} +8.01781 q^{43} +9.15429 q^{45} -3.18575 q^{46} -4.06768 q^{47} -11.4410 q^{48} +1.00000 q^{49} +19.6927 q^{50} +3.02687 q^{51} -2.99730 q^{52} -5.00585 q^{53} -2.92926 q^{54} +1.85331 q^{56} -4.95376 q^{57} +0.416720 q^{58} -0.983592 q^{59} +8.52058 q^{60} -1.84671 q^{61} -11.7280 q^{62} -2.25122 q^{63} +1.76252 q^{64} -13.3292 q^{65} -3.00700 q^{67} +1.20780 q^{68} -4.27630 q^{69} -6.94194 q^{70} +6.47642 q^{71} -4.17220 q^{72} +9.65164 q^{73} +0.436383 q^{74} +26.4339 q^{75} -1.97668 q^{76} -12.8234 q^{78} +5.53450 q^{79} -20.3020 q^{80} -10.6857 q^{81} +9.78600 q^{82} -2.07023 q^{83} -2.09538 q^{84} +5.37119 q^{85} +13.6877 q^{86} +0.559372 q^{87} +16.6306 q^{89} +15.6278 q^{90} +3.27792 q^{91} -1.70636 q^{92} -15.7427 q^{93} -6.94417 q^{94} -8.79046 q^{95} -11.0377 q^{96} -2.58559 q^{97} +1.70716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 4 q^{3} + 7 q^{4} + 10 q^{5} + q^{6} - 8 q^{7} + 14 q^{9}+O(q^{10}) \) 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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