LMFDB - Embedded newform 847.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$6.76332905120$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.7674048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.82356$ 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)

$f(q)$	$=$	$q-0.823556 q^{2} -0.960649 q^{3} -1.32176 q^{4} +2.98565 q^{5} +0.791148 q^{6} +1.00000 q^{7} +2.73565 q^{8} -2.07715 q^{9} +O(q^{10})$	\(q-0.823556 q^{2} -0.960649 q^{3} -1.32176 q^{4} +2.98565 q^{5} +0.791148 q^{6} +1.00000 q^{7} +2.73565 q^{8} -2.07715 q^{9} -2.45885 q^{10} +1.26974 q^{12} +2.20144 q^{13} -0.823556 q^{14} -2.86816 q^{15} +0.390549 q^{16} +4.40669 q^{17} +1.71065 q^{18} -1.72563 q^{19} -3.94630 q^{20} -0.960649 q^{21} -8.39774 q^{23} -2.62800 q^{24} +3.91410 q^{25} -1.81301 q^{26} +4.87736 q^{27} -1.32176 q^{28} +3.29295 q^{29} +2.36209 q^{30} +7.47573 q^{31} -5.79294 q^{32} -3.62916 q^{34} +2.98565 q^{35} +2.74549 q^{36} -8.78071 q^{37} +1.42115 q^{38} -2.11481 q^{39} +8.16769 q^{40} +5.39351 q^{41} +0.791148 q^{42} +9.44629 q^{43} -6.20165 q^{45} +6.91601 q^{46} -5.39667 q^{47} -0.375180 q^{48} +1.00000 q^{49} -3.22348 q^{50} -4.23329 q^{51} -2.90977 q^{52} +9.39774 q^{53} -4.01678 q^{54} +2.73565 q^{56} +1.65772 q^{57} -2.71193 q^{58} +3.47462 q^{59} +3.79101 q^{60} +12.9942 q^{61} -6.15668 q^{62} -2.07715 q^{63} +3.98971 q^{64} +6.57274 q^{65} +4.32138 q^{67} -5.82457 q^{68} +8.06728 q^{69} -2.45885 q^{70} +4.40046 q^{71} -5.68237 q^{72} +14.7509 q^{73} +7.23140 q^{74} -3.76007 q^{75} +2.28086 q^{76} +1.74167 q^{78} -7.18768 q^{79} +1.16604 q^{80} +1.54603 q^{81} -4.44186 q^{82} +7.63445 q^{83} +1.26974 q^{84} +13.1568 q^{85} -7.77955 q^{86} -3.16337 q^{87} +10.8428 q^{89} +5.10741 q^{90} +2.20144 q^{91} +11.0998 q^{92} -7.18156 q^{93} +4.44446 q^{94} -5.15211 q^{95} +5.56498 q^{96} +2.85498 q^{97} -0.823556 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) $ 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9} - 8 q^{10} - 14 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} + 22 q^{17} + 24 q^{18} + 6 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{23} - 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} + 4 q^{28} + 12 q^{29} + 20 q^{30} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} + 20 q^{39} + 18 q^{40} + 26 q^{41} - 6 q^{42} - 4 q^{43} - 36 q^{45} + 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} - 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} - 32 q^{54} + 12 q^{56} + 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} - 8 q^{61} + 20 q^{62} + 8 q^{63} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} + 16 q^{72} + 14 q^{73} + 44 q^{74} - 20 q^{75} - 30 q^{76} + 32 q^{78} - 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} + 22 q^{83} - 14 q^{84} - 24 q^{85} - 30 q^{86} + 22 q^{87} - 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} - 38 q^{94} - 24 q^{95} - 62 q^{96} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9 - 8 * q^10 - 14 * q^12 + 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 + 22 * q^17 + 24 * q^18 + 6 * q^19 + 2 * q^20 - 2 * q^21 + 2 * q^23 - 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 + 4 * q^28 + 12 * q^29 + 20 * q^30 - 2 * q^31 + 8 * q^32 + 24 * q^34 - 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 + 20 * q^39 + 18 * q^40 + 26 * q^41 - 6 * q^42 - 4 * q^43 - 36 * q^45 + 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 - 4 * q^50 - 4 * q^51 + 12 * q^52 + 4 * q^53 - 32 * q^54 + 12 * q^56 + 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 - 8 * q^61 + 20 * q^62 + 8 * q^63 + 26 * q^64 + 24 * q^65 + 6 * q^67 + 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 + 16 * q^72 + 14 * q^73 + 44 * q^74 - 20 * q^75 - 30 * q^76 + 32 * q^78 - 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 + 22 * q^83 - 14 * q^84 - 24 * q^85 - 30 * q^86 + 22 * q^87 - 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 - 38 * q^94 - 24 * q^95 - 62 * q^96 - 4 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.823556	−0.582342	−0.291171	−	0.956671i	$-0.594045\pi$
$2$	−0.823556	−0.582342	−0.291171	+	0.956671i	$0.594045\pi$
$3$	−0.960649	−0.554631	−0.277315	−	0.960779i	$-0.589445\pi$
$3$	−0.960649	−0.554631	−0.277315	+	0.960779i	$0.589445\pi$
$4$	−1.32176	−0.660878
$5$	2.98565	1.33522	0.667611	−	0.744510i	$-0.267317\pi$
$5$	2.98565	1.33522	0.667611	+	0.744510i	$0.267317\pi$
$6$	0.791148	0.322985
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.73565	0.967199
$9$	−2.07715	−0.692385
$10$	−2.45885	−0.777556
$11$	0	0
$11$	0	0
$12$	1.26974	0.366543
$13$	2.20144	0.610571	0.305285	−	0.952261i	$-0.401248\pi$
$13$	2.20144	0.610571	0.305285	+	0.952261i	$0.401248\pi$
$14$	−0.823556	−0.220105
$15$	−2.86816	−0.740556
$16$	0.390549	0.0976372
$17$	4.40669	1.06878	0.534390	−	0.845238i	$-0.320541\pi$
$17$	4.40669	1.06878	0.534390	+	0.845238i	$0.320541\pi$
$18$	1.71065	0.403205
$19$	−1.72563	−0.395886	−0.197943	−	0.980214i	$-0.563426\pi$
$19$	−1.72563	−0.395886	−0.197943	+	0.980214i	$0.563426\pi$
$20$	−3.94630	−0.882419
$21$	−0.960649	−0.209631
$22$	0	0
$23$	−8.39774	−1.75105	−0.875525	−	0.483173i	$-0.839484\pi$
$23$	−8.39774	−1.75105	−0.875525	+	0.483173i	$0.839484\pi$
$24$	−2.62800	−0.536438
$25$	3.91410	0.782819
$26$	−1.81301	−0.355561
$27$	4.87736	0.938649
$28$	−1.32176	−0.249788
$29$	3.29295	0.611485	0.305743	−	0.952114i	$-0.401095\pi$
$29$	3.29295	0.611485	0.305743	+	0.952114i	$0.401095\pi$
$30$	2.36209	0.431257
$31$	7.47573	1.34268	0.671341	−	0.741149i	$-0.265719\pi$
$31$	7.47573	1.34268	0.671341	+	0.741149i	$0.265719\pi$
$32$	−5.79294	−1.02406
$33$	0	0
$34$	−3.62916	−0.622396
$35$	2.98565	0.504667
$36$	2.74549	0.457582
$37$	−8.78071	−1.44354	−0.721770	−	0.692133i	$-0.756671\pi$
$37$	−8.78071	−1.44354	−0.721770	+	0.692133i	$0.756671\pi$
$38$	1.42115	0.230541
$39$	−2.11481	−0.338641
$40$	8.16769	1.29143
$41$	5.39351	0.842325	0.421163	−	0.906985i	$-0.361622\pi$
$41$	5.39351	0.842325	0.421163	+	0.906985i	$0.361622\pi$
$42$	0.791148	0.122077
$43$	9.44629	1.44055	0.720273	−	0.693691i	$-0.244017\pi$
$43$	9.44629	1.44055	0.720273	+	0.693691i	$0.244017\pi$
$44$	0	0
$45$	−6.20165	−0.924487
$46$	6.91601	1.01971
$47$	−5.39667	−0.787185	−0.393593	−	0.919285i	$-0.628768\pi$
$47$	−5.39667	−0.787185	−0.393593	+	0.919285i	$0.628768\pi$
$48$	−0.375180	−0.0541526
$49$	1.00000	0.142857
$50$	−3.22348	−0.455869
$51$	−4.23329	−0.592779
$52$	−2.90977	−0.403513
$53$	9.39774	1.29088	0.645439	−	0.763812i	$-0.276674\pi$
$53$	9.39774	1.29088	0.645439	+	0.763812i	$0.276674\pi$
$54$	−4.01678	−0.546615
$55$	0	0
$56$	2.73565	0.365567
$57$	1.65772	0.219571
$58$	−2.71193	−0.356094
$59$	3.47462	0.452357	0.226178	−	0.974086i	$-0.427377\pi$
$59$	3.47462	0.452357	0.226178	+	0.974086i	$0.427377\pi$
$60$	3.79101	0.489417
$61$	12.9942	1.66374	0.831870	−	0.554970i	$-0.187270\pi$
$61$	12.9942	1.66374	0.831870	+	0.554970i	$0.187270\pi$
$62$	−6.15668	−0.781900
$63$	−2.07715	−0.261697
$64$	3.98971	0.498714
$65$	6.57274	0.815248
$66$	0	0
$67$	4.32138	0.527940	0.263970	−	0.964531i	$-0.414968\pi$
$67$	4.32138	0.527940	0.263970	+	0.964531i	$0.414968\pi$
$68$	−5.82457	−0.706333
$69$	8.06728	0.971186
$70$	−2.45885	−0.293889
$71$	4.40046	0.522238	0.261119	−	0.965307i	$-0.415908\pi$
$71$	4.40046	0.522238	0.261119	+	0.965307i	$0.415908\pi$
$72$	−5.68237	−0.669674
$73$	14.7509	1.72647	0.863233	−	0.504805i	$-0.168436\pi$
$73$	14.7509	1.72647	0.863233	+	0.504805i	$0.168436\pi$
$74$	7.23140	0.840634
$75$	−3.76007	−0.434176
$76$	2.28086	0.261632
$77$	0	0
$78$	1.74167	0.197205
$79$	−7.18768	−0.808677	−0.404338	−	0.914609i	$-0.632498\pi$
$79$	−7.18768	−0.808677	−0.404338	+	0.914609i	$0.632498\pi$
$80$	1.16604	0.130367
$81$	1.54603	0.171781
$82$	−4.44186	−0.490521
$83$	7.63445	0.837990	0.418995	−	0.907989i	$-0.362383\pi$
$83$	7.63445	0.837990	0.418995	+	0.907989i	$0.362383\pi$
$84$	1.26974	0.138540
$85$	13.1568	1.42706
$86$	−7.77955	−0.838890
$87$	−3.16337	−0.339149
$88$	0	0
$89$	10.8428	1.14934	0.574668	−	0.818386i	$-0.305131\pi$
$89$	10.8428	1.14934	0.574668	+	0.818386i	$0.305131\pi$
$90$	5.10741	0.538368
$91$	2.20144	0.230774
$92$	11.0998	1.15723
$93$	−7.18156	−0.744693
$94$	4.44446	0.458411
$95$	−5.15211	−0.528596
$96$	5.56498	0.567974
$97$	2.85498	0.289879	0.144940	−	0.989440i	$-0.453701\pi$
$97$	2.85498	0.289879	0.144940	+	0.989440i	$0.453701\pi$
$98$	−0.823556	−0.0831917
$99$	0	0
$100$	−5.17348	−0.517348
$101$	−14.6011	−1.45287	−0.726434	−	0.687236i	$-0.758824\pi$
$101$	−14.6011	−1.45287	−0.726434	+	0.687236i	$0.758824\pi$
$102$	3.48635	0.345200
$103$	−0.107767	−0.0106186	−0.00530932	−	0.999986i	$-0.501690\pi$
$103$	−0.107767	−0.0106186	−0.00530932	+	0.999986i	$0.501690\pi$
$104$	6.02238	0.590543
$105$	−2.86816	−0.279904
$106$	−7.73956	−0.751733
$107$	−4.64902	−0.449438	−0.224719	−	0.974424i	$-0.572146\pi$
$107$	−4.64902	−0.449438	−0.224719	+	0.974424i	$0.572146\pi$
$108$	−6.44668	−0.620332
$109$	3.23140	0.309512	0.154756	−	0.987953i	$-0.450541\pi$
$109$	3.23140	0.309512	0.154756	+	0.987953i	$0.450541\pi$
$110$	0	0
$111$	8.43518	0.800632
$112$	0.390549	0.0369034
$113$	−11.3194	−1.06484	−0.532420	−	0.846480i	$-0.678717\pi$
$113$	−11.3194	−1.06484	−0.532420	+	0.846480i	$0.678717\pi$
$114$	−1.36523	−0.127865
$115$	−25.0727	−2.33804
$116$	−4.35247	−0.404117
$117$	−4.57274	−0.422750
$118$	−2.86154	−0.263426
$119$	4.40669	0.403961
$120$	−7.84629	−0.716265
$121$	0	0
$122$	−10.7015	−0.968866
$123$	−5.18127	−0.467180
$124$	−9.88109	−0.887348
$125$	−3.24213	−0.289985
$126$	1.71065	0.152397
$127$	−19.9802	−1.77295	−0.886476	−	0.462774i	$-0.846854\pi$
$127$	−19.9802	−1.77295	−0.886476	+	0.462774i	$0.846854\pi$
$128$	8.30013	0.733635
$129$	−9.07457	−0.798971
$130$	−5.41302	−0.474753
$131$	8.45523	0.738737	0.369368	−	0.929283i	$-0.379574\pi$
$131$	8.45523	0.738737	0.369368	+	0.929283i	$0.379574\pi$
$132$	0	0
$133$	−1.72563	−0.149631
$134$	−3.55890	−0.307442
$135$	14.5621	1.25331
$136$	12.0552	1.03372
$137$	8.35456	0.713779	0.356889	−	0.934147i	$-0.383837\pi$
$137$	8.35456	0.713779	0.356889	+	0.934147i	$0.383837\pi$
$138$	−6.64385	−0.565562
$139$	−14.7134	−1.24797	−0.623986	−	0.781436i	$-0.714488\pi$
$139$	−14.7134	−1.24797	−0.623986	+	0.781436i	$0.714488\pi$
$140$	−3.94630	−0.333523
$141$	5.18430	0.436597
$142$	−3.62402	−0.304121
$143$	0	0
$144$	−0.811230	−0.0676025
$145$	9.83159	0.816469
$146$	−12.1482	−1.00539
$147$	−0.960649	−0.0792330
$148$	11.6059	0.954003
$149$	8.84271	0.724423	0.362212	−	0.932096i	$-0.382022\pi$
$149$	8.84271	0.724423	0.362212	+	0.932096i	$0.382022\pi$
$150$	3.09663	0.252839
$151$	−19.9699	−1.62513	−0.812563	−	0.582873i	$-0.801929\pi$
$151$	−19.9699	−1.62513	−0.812563	+	0.582873i	$0.801929\pi$
$152$	−4.72071	−0.382900
$153$	−9.15338	−0.740007
$154$	0	0
$155$	22.3199	1.79278
$156$	2.79527	0.223801
$157$	13.0517	1.04164	0.520818	−	0.853668i	$-0.325627\pi$
$157$	13.0517	1.04164	0.520818	+	0.853668i	$0.325627\pi$
$158$	5.91945	0.470926
$159$	−9.02793	−0.715961
$160$	−17.2957	−1.36734
$161$	−8.39774	−0.661834
$162$	−1.27324	−0.100035
$163$	5.96447	0.467174	0.233587	−	0.972336i	$-0.424954\pi$
$163$	5.96447	0.467174	0.233587	+	0.972336i	$0.424954\pi$
$164$	−7.12891	−0.556674
$165$	0	0
$166$	−6.28740	−0.487997
$167$	13.2472	1.02510	0.512548	−	0.858659i	$-0.328702\pi$
$167$	13.2472	1.02510	0.512548	+	0.858659i	$0.328702\pi$
$168$	−2.62800	−0.202755
$169$	−8.15365	−0.627204
$170$	−10.8354	−0.831037
$171$	3.58439	0.274105
$172$	−12.4857	−0.952025
$173$	−0.517716	−0.0393612	−0.0196806	−	0.999806i	$-0.506265\pi$
$173$	−0.517716	−0.0393612	−0.0196806	+	0.999806i	$0.506265\pi$
$174$	2.60521	0.197500
$175$	3.91410	0.295878
$176$	0	0
$177$	−3.33789	−0.250891
$178$	−8.92967	−0.669307
$179$	1.69615	0.126776	0.0633881	−	0.997989i	$-0.479809\pi$
$179$	1.69615	0.126776	0.0633881	+	0.997989i	$0.479809\pi$
$180$	8.19707	0.610973
$181$	−9.88599	−0.734820	−0.367410	−	0.930059i	$-0.619755\pi$
$181$	−9.88599	−0.734820	−0.367410	+	0.930059i	$0.619755\pi$
$182$	−1.81301	−0.134389
$183$	−12.4829	−0.922762
$184$	−22.9733	−1.69361
$185$	−26.2161	−1.92745
$186$	5.91441	0.433666
$187$	0	0
$188$	7.13308	0.520233
$189$	4.87736	0.354776
$190$	4.24305	0.307823
$191$	23.1329	1.67384	0.836918	−	0.547328i	$-0.184355\pi$
$191$	23.1329	1.67384	0.836918	+	0.547328i	$0.184355\pi$
$192$	−3.83271	−0.276602
$193$	−23.0888	−1.66197	−0.830983	−	0.556297i	$-0.812222\pi$
$193$	−23.0888	−1.66197	−0.830983	+	0.556297i	$0.812222\pi$
$194$	−2.35124	−0.168809
$195$	−6.31409	−0.452162
$196$	−1.32176	−0.0944111
$197$	6.68989	0.476635	0.238318	−	0.971187i	$-0.423404\pi$
$197$	6.68989	0.476635	0.238318	+	0.971187i	$0.423404\pi$
$198$	0	0
$199$	−15.8233	−1.12169	−0.560844	−	0.827922i	$-0.689523\pi$
$199$	−15.8233	−1.12169	−0.560844	+	0.827922i	$0.689523\pi$
$200$	10.7076	0.757142
$201$	−4.15133	−0.292812
$202$	12.0249	0.846066
$203$	3.29295	0.231120
$204$	5.59537	0.391754
$205$	16.1031	1.12469
$206$	0.0887525	0.00618368
$207$	17.4434	1.21240
$208$	0.859771	0.0596144
$209$	0	0
$210$	2.36209	0.163000
$211$	11.3525	0.781538	0.390769	−	0.920489i	$-0.372209\pi$
$211$	11.3525	0.781538	0.390769	+	0.920489i	$0.372209\pi$
$212$	−12.4215	−0.853113
$213$	−4.22730	−0.289649
$214$	3.82873	0.261727
$215$	28.2033	1.92345
$216$	13.3428	0.907860
$217$	7.47573	0.507486
$218$	−2.66124	−0.180242
$219$	−14.1705	−0.957552
$220$	0	0
$221$	9.70109	0.652566
$222$	−6.94684	−0.466241
$223$	−10.6899	−0.715846	−0.357923	−	0.933751i	$-0.616515\pi$
$223$	−10.6899	−0.715846	−0.357923	+	0.933751i	$0.616515\pi$
$224$	−5.79294	−0.387057
$225$	−8.13018	−0.542012
$226$	9.32217	0.620102
$227$	−3.73783	−0.248089	−0.124044	−	0.992277i	$-0.539586\pi$
$227$	−3.73783	−0.248089	−0.124044	+	0.992277i	$0.539586\pi$
$228$	−2.19110	−0.145109
$229$	−23.6585	−1.56340	−0.781699	−	0.623656i	$-0.785647\pi$
$229$	−23.6585	−1.56340	−0.781699	+	0.623656i	$0.785647\pi$
$230$	20.6488	1.36154
$231$	0	0
$232$	9.00836	0.591428
$233$	3.86675	0.253319	0.126660	−	0.991946i	$-0.459574\pi$
$233$	3.86675	0.253319	0.126660	+	0.991946i	$0.459574\pi$
$234$	3.76590	0.246185
$235$	−16.1126	−1.05107
$236$	−4.59259	−0.298952
$237$	6.90483	0.448517
$238$	−3.62916	−0.235243
$239$	−10.0717	−0.651482	−0.325741	−	0.945459i	$-0.605614\pi$
$239$	−10.0717	−0.651482	−0.325741	+	0.945459i	$0.605614\pi$
$240$	−1.12016	−0.0723058
$241$	13.4265	0.864878	0.432439	−	0.901663i	$-0.357653\pi$
$241$	13.4265	0.864878	0.432439	+	0.901663i	$0.357653\pi$
$242$	0	0
$243$	−16.1173	−1.03392
$244$	−17.1752	−1.09953
$245$	2.98565	0.190746
$246$	4.26707	0.272058
$247$	−3.79887	−0.241716
$248$	20.4510	1.29864
$249$	−7.33403	−0.464775
$250$	2.67007	0.168870
$251$	0.0764471	0.00482530	0.00241265	−	0.999997i	$-0.499232\pi$
$251$	0.0764471	0.00482530	0.00241265	+	0.999997i	$0.499232\pi$
$252$	2.74549	0.172950
$253$	0	0
$254$	16.4548	1.03246
$255$	−12.6391	−0.791491
$256$	−14.8151	−0.925941
$257$	9.51194	0.593338	0.296669	−	0.954980i	$-0.404124\pi$
$257$	9.51194	0.593338	0.296669	+	0.954980i	$0.404124\pi$
$258$	7.47342	0.465275
$259$	−8.78071	−0.545607
$260$	−8.68755	−0.538779
$261$	−6.83996	−0.423383
$262$	−6.96335	−0.430197
$263$	14.7919	0.912107	0.456053	−	0.889952i	$-0.349263\pi$
$263$	14.7919	0.912107	0.456053	+	0.889952i	$0.349263\pi$
$264$	0	0
$265$	28.0583	1.72361
$266$	1.42115	0.0871363
$267$	−10.4161	−0.637458
$268$	−5.71181	−0.348904
$269$	−30.0556	−1.83252	−0.916262	−	0.400580i	$-0.868809\pi$
$269$	−30.0556	−1.83252	−0.916262	+	0.400580i	$0.868809\pi$
$270$	−11.9927	−0.729852
$271$	−10.3918	−0.631260	−0.315630	−	0.948882i	$-0.602216\pi$
$271$	−10.3918	−0.631260	−0.315630	+	0.948882i	$0.602216\pi$
$272$	1.72103	0.104353
$273$	−2.11481	−0.127994
$274$	−6.88045	−0.415663
$275$	0	0
$276$	−10.6630	−0.641835
$277$	28.1138	1.68919	0.844597	−	0.535403i	$-0.179840\pi$
$277$	28.1138	1.68919	0.844597	+	0.535403i	$0.179840\pi$
$278$	12.1173	0.726747
$279$	−15.5282	−0.929652
$280$	8.16769	0.488113
$281$	9.57010	0.570904	0.285452	−	0.958393i	$-0.407856\pi$
$281$	9.57010	0.570904	0.285452	+	0.958393i	$0.407856\pi$
$282$	−4.26956	−0.254249
$283$	−1.65526	−0.0983947	−0.0491974	−	0.998789i	$-0.515666\pi$
$283$	−1.65526	−0.0983947	−0.0491974	+	0.998789i	$0.515666\pi$
$284$	−5.81633	−0.345136
$285$	4.94937	0.293176
$286$	0	0
$287$	5.39351	0.318369
$288$	12.0328	0.709041
$289$	2.41896	0.142292
$290$	−8.09686	−0.475464
$291$	−2.74263	−0.160776
$292$	−19.4971	−1.14098
$293$	4.68188	0.273519	0.136759	−	0.990604i	$-0.456331\pi$
$293$	4.68188	0.273519	0.136759	+	0.990604i	$0.456331\pi$
$294$	0.791148	0.0461407
$295$	10.3740	0.603997
$296$	−24.0210	−1.39619
$297$	0	0
$298$	−7.28247	−0.421862
$299$	−18.4871	−1.06914
$300$	4.96990	0.286937
$301$	9.44629	0.544475
$302$	16.4463	0.946379
$303$	14.0266	0.805806
$304$	−0.673941	−0.0386532
$305$	38.7962	2.22146
$306$	7.53832	0.430937
$307$	−16.9829	−0.969266	−0.484633	−	0.874718i	$-0.661047\pi$
$307$	−16.9829	−0.969266	−0.484633	+	0.874718i	$0.661047\pi$
$308$	0	0
$309$	0.103527	0.00588942
$310$	−18.3817	−1.04401
$311$	−21.9257	−1.24329	−0.621645	−	0.783299i	$-0.713535\pi$
$311$	−21.9257	−1.24329	−0.621645	+	0.783299i	$0.713535\pi$
$312$	−5.78540	−0.327534
$313$	10.0555	0.568368	0.284184	−	0.958770i	$-0.408277\pi$
$313$	10.0555	0.568368	0.284184	+	0.958770i	$0.408277\pi$
$314$	−10.7488	−0.606589
$315$	−6.20165	−0.349423
$316$	9.50035	0.534436
$317$	23.5602	1.32327	0.661635	−	0.749826i	$-0.269863\pi$
$317$	23.5602	1.32327	0.661635	+	0.749826i	$0.269863\pi$
$318$	7.43500	0.416934
$319$	0	0
$320$	11.9119	0.665895
$321$	4.46608	0.249272
$322$	6.91601	0.385414
$323$	−7.60431	−0.423115
$324$	−2.04347	−0.113526
$325$	8.61666	0.477966
$326$	−4.91208	−0.272055
$327$	−3.10425	−0.171665
$328$	14.7548	0.814696
$329$	−5.39667	−0.297528
$330$	0	0
$331$	−20.8607	−1.14661	−0.573304	−	0.819343i	$-0.694339\pi$
$331$	−20.8607	−1.14661	−0.573304	+	0.819343i	$0.694339\pi$
$332$	−10.0909	−0.553809
$333$	18.2389	0.999484
$334$	−10.9098	−0.596956
$335$	12.9021	0.704918
$336$	−0.375180	−0.0204678
$337$	13.3544	0.727458	0.363729	−	0.931505i	$-0.381503\pi$
$337$	13.3544	0.727458	0.363729	+	0.931505i	$0.381503\pi$
$338$	6.71498	0.365247
$339$	10.8740	0.590594
$340$	−17.3901	−0.943112
$341$	0	0
$342$	−2.95195	−0.159623
$343$	1.00000	0.0539949
$344$	25.8418	1.39329
$345$	24.0861	1.29675
$346$	0.426368	0.0229217
$347$	−1.59256	−0.0854933	−0.0427466	−	0.999086i	$-0.513611\pi$
$347$	−1.59256	−0.0854933	−0.0427466	+	0.999086i	$0.513611\pi$
$348$	4.18120	0.224136
$349$	6.63475	0.355150	0.177575	−	0.984107i	$-0.443175\pi$
$349$	6.63475	0.355150	0.177575	+	0.984107i	$0.443175\pi$
$350$	−3.22348	−0.172302
$351$	10.7372	0.573111
$352$	0	0
$353$	−25.4141	−1.35265	−0.676327	−	0.736601i	$-0.736430\pi$
$353$	−25.4141	−1.35265	−0.676327	+	0.736601i	$0.736430\pi$
$354$	2.74894	0.146104
$355$	13.1382	0.697304
$356$	−14.3316	−0.759571
$357$	−4.23329	−0.224049
$358$	−1.39688	−0.0738271
$359$	16.0792	0.848629	0.424314	−	0.905515i	$-0.360515\pi$
$359$	16.0792	0.848629	0.424314	+	0.905515i	$0.360515\pi$
$360$	−16.9656	−0.894163
$361$	−16.0222	−0.843274
$362$	8.14167	0.427916
$363$	0	0
$364$	−2.90977	−0.152513
$365$	44.0411	2.30522
$366$	10.2804	0.537363
$367$	−3.02606	−0.157959	−0.0789796	−	0.996876i	$-0.525166\pi$
$367$	−3.02606	−0.157959	−0.0789796	+	0.996876i	$0.525166\pi$
$368$	−3.27973	−0.170968
$369$	−11.2032	−0.583213
$370$	21.5904	1.12243
$371$	9.39774	0.487906
$372$	9.49226	0.492151
$373$	1.73856	0.0900192	0.0450096	−	0.998987i	$-0.485668\pi$
$373$	1.73856	0.0900192	0.0450096	+	0.998987i	$0.485668\pi$
$374$	0	0
$375$	3.11455	0.160834
$376$	−14.7634	−0.761365
$377$	7.24924	0.373355
$378$	−4.01678	−0.206601
$379$	19.5728	1.00538	0.502692	−	0.864465i	$-0.332343\pi$
$379$	19.5728	1.00538	0.502692	+	0.864465i	$0.332343\pi$
$380$	6.80983	0.349337
$381$	19.1939	0.983334
$382$	−19.0512	−0.974746
$383$	4.76472	0.243466	0.121733	−	0.992563i	$-0.461155\pi$
$383$	4.76472	0.243466	0.121733	+	0.992563i	$0.461155\pi$
$384$	−7.97351	−0.406897
$385$	0	0
$386$	19.0149	0.967833
$387$	−19.6214	−0.997412
$388$	−3.77359	−0.191575
$389$	−1.42208	−0.0721024	−0.0360512	−	0.999350i	$-0.511478\pi$
$389$	−1.42208	−0.0721024	−0.0360512	+	0.999350i	$0.511478\pi$
$390$	5.20001	0.263313
$391$	−37.0063	−1.87149
$392$	2.73565	0.138171
$393$	−8.12251	−0.409726
$394$	−5.50950	−0.277565
$395$	−21.4599	−1.07976
$396$	0	0
$397$	−18.3969	−0.923314	−0.461657	−	0.887059i	$-0.652745\pi$
$397$	−18.3969	−0.923314	−0.461657	+	0.887059i	$0.652745\pi$
$398$	13.0314	0.653206
$399$	1.65772	0.0829899
$400$	1.52865	0.0764323
$401$	−1.65268	−0.0825308	−0.0412654	−	0.999148i	$-0.513139\pi$
$401$	−1.65268	−0.0825308	−0.0412654	+	0.999148i	$0.513139\pi$
$402$	3.41885	0.170517
$403$	16.4574	0.819802
$404$	19.2991	0.960168
$405$	4.61589	0.229366
$406$	−2.71193	−0.134591
$407$	0	0
$408$	−11.5808	−0.573335
$409$	−36.6858	−1.81400	−0.906999	−	0.421133i	$-0.861632\pi$
$409$	−36.6858	−1.81400	−0.906999	+	0.421133i	$0.861632\pi$
$410$	−13.2618	−0.654955
$411$	−8.02580	−0.395884
$412$	0.142442	0.00701762
$413$	3.47462	0.170975
$414$	−14.3656	−0.706031
$415$	22.7938	1.11890
$416$	−12.7528	−0.625259
$417$	14.1344	0.692164
$418$	0	0
$419$	−17.3452	−0.847366	−0.423683	−	0.905810i	$-0.639263\pi$
$419$	−17.3452	−0.847366	−0.423683	+	0.905810i	$0.639263\pi$
$420$	3.79101	0.184982
$421$	5.32683	0.259614	0.129807	−	0.991539i	$-0.458564\pi$
$421$	5.32683	0.259614	0.129807	+	0.991539i	$0.458564\pi$
$422$	−9.34941	−0.455122
$423$	11.2097	0.545035
$424$	25.7089	1.24854
$425$	17.2482	0.836662
$426$	3.48141	0.168675
$427$	12.9942	0.628835
$428$	6.14487	0.297024
$429$	0	0
$430$	−23.2270	−1.12011
$431$	−28.2081	−1.35873	−0.679367	−	0.733798i	$-0.737746\pi$
$431$	−28.2081	−1.35873	−0.679367	+	0.733798i	$0.737746\pi$
$432$	1.90485	0.0916471
$433$	17.7060	0.850895	0.425447	−	0.904983i	$-0.360117\pi$
$433$	17.7060	0.850895	0.425447	+	0.904983i	$0.360117\pi$
$434$	−6.15668	−0.295530
$435$	−9.44470	−0.452839
$436$	−4.27113	−0.204550
$437$	14.4914	0.693216
$438$	11.6702	0.557623
$439$	−25.1189	−1.19886	−0.599430	−	0.800427i	$-0.704606\pi$
$439$	−25.1189	−1.19886	−0.599430	+	0.800427i	$0.704606\pi$
$440$	0	0
$441$	−2.07715	−0.0989121
$442$	−7.98939	−0.380017
$443$	−26.9567	−1.28075	−0.640376	−	0.768061i	$-0.721222\pi$
$443$	−26.9567	−1.28075	−0.640376	+	0.768061i	$0.721222\pi$
$444$	−11.1492	−0.529120
$445$	32.3728	1.53462
$446$	8.80369	0.416867
$447$	−8.49474	−0.401788
$448$	3.98971	0.188496
$449$	−21.0964	−0.995599	−0.497799	−	0.867292i	$-0.665858\pi$
$449$	−21.0964	−0.995599	−0.497799	+	0.867292i	$0.665858\pi$
$450$	6.69566	0.315636
$451$	0	0
$452$	14.9615	0.703730
$453$	19.1841	0.901345
$454$	3.07831	0.144472
$455$	6.57274	0.308135
$456$	4.53495	0.212368
$457$	−1.90356	−0.0890448	−0.0445224	−	0.999008i	$-0.514177\pi$
$457$	−1.90356	−0.0890448	−0.0445224	+	0.999008i	$0.514177\pi$
$458$	19.4841	0.910432
$459$	21.4930	1.00321
$460$	33.1400	1.54516
$461$	25.0440	1.16641	0.583207	−	0.812324i	$-0.301798\pi$
$461$	25.0440	1.16641	0.583207	+	0.812324i	$0.301798\pi$
$462$	0	0
$463$	−21.1721	−0.983952	−0.491976	−	0.870609i	$-0.663725\pi$
$463$	−21.1721	−0.983952	−0.491976	+	0.870609i	$0.663725\pi$
$464$	1.28606	0.0597037
$465$	−21.4416	−0.994330
$466$	−3.18449	−0.147519
$467$	16.6484	0.770398	0.385199	−	0.922834i	$-0.374133\pi$
$467$	16.6484	0.770398	0.385199	+	0.922834i	$0.374133\pi$
$468$	6.04404	0.279386
$469$	4.32138	0.199543
$470$	13.2696	0.612081
$471$	−12.5381	−0.577724
$472$	9.50534	0.437519
$473$	0	0
$474$	−5.68652	−0.261190
$475$	−6.75427	−0.309907
$476$	−5.82457	−0.266969
$477$	−19.5205	−0.893784
$478$	8.29459	0.379386
$479$	21.6613	0.989730	0.494865	−	0.868970i	$-0.335217\pi$
$479$	21.6613	0.989730	0.494865	+	0.868970i	$0.335217\pi$
$480$	16.6151	0.758371
$481$	−19.3302	−0.881383
$482$	−11.0575	−0.503655
$483$	8.06728	0.367074
$484$	0	0
$485$	8.52397	0.387053
$486$	13.2735	0.602097
$487$	−2.81335	−0.127485	−0.0637426	−	0.997966i	$-0.520304\pi$
$487$	−2.81335	−0.127485	−0.0637426	+	0.997966i	$0.520304\pi$
$488$	35.5477	1.60917
$489$	−5.72976	−0.259109
$490$	−2.45885	−0.111079
$491$	3.20580	0.144676	0.0723379	−	0.997380i	$-0.476954\pi$
$491$	3.20580	0.144676	0.0723379	+	0.997380i	$0.476954\pi$
$492$	6.84838	0.308749
$493$	14.5110	0.653543
$494$	3.12858	0.140762
$495$	0	0
$496$	2.91964	0.131096
$497$	4.40046	0.197388
$498$	6.03998	0.270658
$499$	−20.0512	−0.897616	−0.448808	−	0.893628i	$-0.648151\pi$
$499$	−20.0512	−0.897616	−0.448808	+	0.893628i	$0.648151\pi$
$500$	4.28530	0.191644
$501$	−12.7259	−0.568550
$502$	−0.0629585	−0.00280997
$503$	35.5089	1.58326	0.791631	−	0.610999i	$-0.209232\pi$
$503$	35.5089	1.58326	0.791631	+	0.610999i	$0.209232\pi$
$504$	−5.68237	−0.253113
$505$	−43.5939	−1.93990
$506$	0	0
$507$	7.83279	0.347867
$508$	26.4089	1.17170
$509$	9.18980	0.407331	0.203665	−	0.979041i	$-0.434715\pi$
$509$	9.18980	0.407331	0.203665	+	0.979041i	$0.434715\pi$
$510$	10.4090	0.460919
$511$	14.7509	0.652543
$512$	−4.39924	−0.194421
$513$	−8.41650	−0.371598
$514$	−7.83361	−0.345526
$515$	−0.321756	−0.0141782
$516$	11.9944	0.528022
$517$	0	0
$518$	7.23140	0.317730
$519$	0.497343	0.0218310
$520$	17.9807	0.788507
$521$	12.3998	0.543246	0.271623	−	0.962404i	$-0.412440\pi$
$521$	12.3998	0.543246	0.271623	+	0.962404i	$0.412440\pi$
$522$	5.63309	0.246554
$523$	22.7105	0.993062	0.496531	−	0.868019i	$-0.334607\pi$
$523$	22.7105	0.993062	0.496531	+	0.868019i	$0.334607\pi$
$524$	−11.1757	−0.488215
$525$	−3.76007	−0.164103
$526$	−12.1819	−0.531158
$527$	32.9433	1.43503
$528$	0	0
$529$	47.5220	2.06617
$530$	−23.1076	−1.00373
$531$	−7.21731	−0.313205
$532$	2.28086	0.0988877
$533$	11.8735	0.514299
$534$	8.57828	0.371218
$535$	−13.8803	−0.600100
$536$	11.8218	0.510623
$537$	−1.62941	−0.0703140
$538$	24.7525	1.06716
$539$	0	0
$540$	−19.2475	−0.828281
$541$	17.2748	0.742703	0.371351	−	0.928492i	$-0.378895\pi$
$541$	17.2748	0.742703	0.371351	+	0.928492i	$0.378895\pi$
$542$	8.55827	0.367609
$543$	9.49697	0.407554
$544$	−25.5277	−1.09449
$545$	9.64784	0.413268
$546$	1.74167	0.0745365
$547$	−13.1740	−0.563278	−0.281639	−	0.959520i	$-0.590878\pi$
$547$	−13.1740	−0.563278	−0.281639	+	0.959520i	$0.590878\pi$
$548$	−11.0427	−0.471720
$549$	−26.9910	−1.15195
$550$	0	0
$551$	−5.68240	−0.242078
$552$	22.0693	0.939330
$553$	−7.18768	−0.305651
$554$	−23.1533	−0.983689
$555$	25.1845	1.06902
$556$	19.4475	0.824757
$557$	−12.6216	−0.534793	−0.267396	−	0.963587i	$-0.586163\pi$
$557$	−12.6216	−0.534793	−0.267396	+	0.963587i	$0.586163\pi$
$558$	12.7884	0.541375
$559$	20.7955	0.879555
$560$	1.16604	0.0492743
$561$	0	0
$562$	−7.88151	−0.332462
$563$	9.81924	0.413832	0.206916	−	0.978359i	$-0.433657\pi$
$563$	9.81924	0.413832	0.206916	+	0.978359i	$0.433657\pi$
$564$	−6.85238	−0.288537
$565$	−33.7958	−1.42180
$566$	1.36320	0.0572994
$567$	1.54603	0.0649271
$568$	12.0381	0.505108
$569$	20.3544	0.853301	0.426650	−	0.904417i	$-0.359693\pi$
$569$	20.3544	0.853301	0.426650	+	0.904417i	$0.359693\pi$
$570$	−4.07608	−0.170728
$571$	−32.3174	−1.35244	−0.676221	−	0.736699i	$-0.736384\pi$
$571$	−32.3174	−1.35244	−0.676221	+	0.736699i	$0.736384\pi$
$572$	0	0
$573$	−22.2226	−0.928362
$574$	−4.44186	−0.185400
$575$	−32.8696	−1.37076
$576$	−8.28725	−0.345302
$577$	−11.3181	−0.471179	−0.235590	−	0.971853i	$-0.575702\pi$
$577$	−11.3181	−0.471179	−0.235590	+	0.971853i	$0.575702\pi$
$578$	−1.99215	−0.0828623
$579$	22.1802	0.921778
$580$	−12.9950	−0.539586
$581$	7.63445	0.316730
$582$	2.25871	0.0936266
$583$	0	0
$584$	40.3534	1.66984
$585$	−13.6526	−0.564465
$586$	−3.85579	−0.159281
$587$	−38.2388	−1.57828	−0.789142	−	0.614211i	$-0.789474\pi$
$587$	−38.2388	−1.57828	−0.789142	+	0.614211i	$0.789474\pi$
$588$	1.26974	0.0523633
$589$	−12.9003	−0.531548
$590$	−8.54356	−0.351733
$591$	−6.42664	−0.264357
$592$	−3.42930	−0.140943
$593$	−26.4263	−1.08520	−0.542598	−	0.839992i	$-0.682559\pi$
$593$	−26.4263	−1.08520	−0.542598	+	0.839992i	$0.682559\pi$
$594$	0	0
$595$	13.1568	0.539378
$596$	−11.6879	−0.478755
$597$	15.2007	0.622123
$598$	15.2252	0.622605
$599$	−20.2994	−0.829413	−0.414706	−	0.909955i	$-0.636116\pi$
$599$	−20.2994	−0.829413	−0.414706	+	0.909955i	$0.636116\pi$
$600$	−10.2862	−0.419934
$601$	−5.09410	−0.207793	−0.103896	−	0.994588i	$-0.533131\pi$
$601$	−5.09410	−0.207793	−0.103896	+	0.994588i	$0.533131\pi$
$602$	−7.77955	−0.317071
$603$	−8.97617	−0.365538
$604$	26.3953	1.07401
$605$	0	0
$606$	−11.5517	−0.469254
$607$	30.6407	1.24367	0.621833	−	0.783150i	$-0.286388\pi$
$607$	30.6407	1.24367	0.621833	+	0.783150i	$0.286388\pi$
$608$	9.99645	0.405410
$609$	−3.16337	−0.128186
$610$	−31.9508	−1.29365
$611$	−11.8805	−0.480632
$612$	12.0985	0.489054
$613$	−24.8391	−1.00324	−0.501620	−	0.865088i	$-0.667263\pi$
$613$	−24.8391	−1.00324	−0.501620	+	0.865088i	$0.667263\pi$
$614$	13.9864	0.564444
$615$	−15.4695	−0.623789
$616$	0	0
$617$	0.521714	0.0210034	0.0105017	−	0.999945i	$-0.496657\pi$
$617$	0.521714	0.0210034	0.0105017	+	0.999945i	$0.496657\pi$
$618$	−0.0852600	−0.00342966
$619$	31.4859	1.26552	0.632762	−	0.774346i	$-0.281921\pi$
$619$	31.4859	1.26552	0.632762	+	0.774346i	$0.281921\pi$
$620$	−29.5015	−1.18481
$621$	−40.9588	−1.64362
$622$	18.0570	0.724020
$623$	10.8428	0.434408
$624$	−0.825938	−0.0330640
$625$	−29.2503	−1.17001
$626$	−8.28124	−0.330985
$627$	0	0
$628$	−17.2511	−0.688395
$629$	−38.6939	−1.54283
$630$	5.10741	0.203484
$631$	−0.114230	−0.00454742	−0.00227371	−	0.999997i	$-0.500724\pi$
$631$	−0.114230	−0.00454742	−0.00227371	+	0.999997i	$0.500724\pi$
$632$	−19.6630	−0.782151
$633$	−10.9058	−0.433465
$634$	−19.4031	−0.770596
$635$	−59.6537	−2.36729
$636$	11.9327	0.473163
$637$	2.20144	0.0872244
$638$	0	0
$639$	−9.14043	−0.361590
$640$	24.7813	0.979566
$641$	13.2141	0.521927	0.260963	−	0.965349i	$-0.415960\pi$
$641$	13.2141	0.521927	0.260963	+	0.965349i	$0.415960\pi$
$642$	−3.67807	−0.145162
$643$	2.45599	0.0968550	0.0484275	−	0.998827i	$-0.484579\pi$
$643$	2.45599	0.0968550	0.0484275	+	0.998827i	$0.484579\pi$
$644$	11.0998	0.437392
$645$	−27.0935	−1.06680
$646$	6.26257	0.246398
$647$	−18.6116	−0.731698	−0.365849	−	0.930674i	$-0.619221\pi$
$647$	−18.6116	−0.731698	−0.365849	+	0.930674i	$0.619221\pi$
$648$	4.22939	0.166146
$649$	0	0
$650$	−7.09630	−0.278340
$651$	−7.18156	−0.281467
$652$	−7.88357	−0.308745
$653$	19.4481	0.761065	0.380532	−	0.924768i	$-0.375741\pi$
$653$	19.4481	0.761065	0.380532	+	0.924768i	$0.375741\pi$
$654$	2.55652	0.0999678
$655$	25.2443	0.986378
$656$	2.10643	0.0822423
$657$	−30.6400	−1.19538
$658$	4.44446	0.173263
$659$	4.71629	0.183721	0.0918603	−	0.995772i	$-0.470719\pi$
$659$	4.71629	0.183721	0.0918603	+	0.995772i	$0.470719\pi$
$660$	0	0
$661$	24.9330	0.969782	0.484891	−	0.874575i	$-0.338859\pi$
$661$	24.9330	0.969782	0.484891	+	0.874575i	$0.338859\pi$
$662$	17.1800	0.667718
$663$	−9.31934	−0.361933
$664$	20.8852	0.810503
$665$	−5.15211	−0.199790
$666$	−15.0207	−0.582042
$667$	−27.6533	−1.07074
$668$	−17.5095	−0.677463
$669$	10.2692	0.397030
$670$	−10.6256	−0.410503
$671$	0	0
$672$	5.56498	0.214674
$673$	−5.12972	−0.197736	−0.0988681	−	0.995101i	$-0.531522\pi$
$673$	−5.12972	−0.197736	−0.0988681	+	0.995101i	$0.531522\pi$
$674$	−10.9981	−0.423629
$675$	19.0905	0.734792
$676$	10.7771	0.414505
$677$	−51.2610	−1.97012	−0.985060	−	0.172213i	$-0.944908\pi$
$677$	−51.2610	−1.97012	−0.985060	+	0.172213i	$0.944908\pi$
$678$	−8.95533	−0.343927
$679$	2.85498	0.109564
$680$	35.9925	1.38025
$681$	3.59074	0.137598
$682$	0	0
$683$	−24.9448	−0.954485	−0.477242	−	0.878772i	$-0.658364\pi$
$683$	−24.9448	−0.954485	−0.477242	+	0.878772i	$0.658364\pi$
$684$	−4.73769	−0.181150
$685$	24.9438	0.953053
$686$	−0.823556	−0.0314435
$687$	22.7275	0.867109
$688$	3.68924	0.140651
$689$	20.6886	0.788172
$690$	−19.8362	−0.755152
$691$	11.7847	0.448312	0.224156	−	0.974553i	$-0.428038\pi$
$691$	11.7847	0.448312	0.224156	+	0.974553i	$0.428038\pi$
$692$	0.684294	0.0260130
$693$	0	0
$694$	1.31156	0.0497863
$695$	−43.9290	−1.66632
$696$	−8.65387	−0.328024
$697$	23.7676	0.900261
$698$	−5.46409	−0.206819
$699$	−3.71459	−0.140499
$700$	−5.17348	−0.195539
$701$	−31.3172	−1.18283	−0.591417	−	0.806366i	$-0.701431\pi$
$701$	−31.3172	−1.18283	−0.591417	+	0.806366i	$0.701431\pi$
$702$	−8.84272	−0.333747
$703$	15.1522	0.571477
$704$	0	0
$705$	15.4785	0.582954
$706$	20.9299	0.787708
$707$	−14.6011	−0.549133
$708$	4.41187	0.165808
$709$	−41.4751	−1.55763	−0.778815	−	0.627254i	$-0.784179\pi$
$709$	−41.4751	−1.55763	−0.778815	+	0.627254i	$0.784179\pi$
$710$	−10.8201	−0.406070
$711$	14.9299	0.559915
$712$	29.6622	1.11164
$713$	−62.7792	−2.35110
$714$	3.48635	0.130473
$715$	0	0
$716$	−2.24190	−0.0837836
$717$	9.67534	0.361332
$718$	−13.2421	−0.494192
$719$	−7.89027	−0.294258	−0.147129	−	0.989117i	$-0.547003\pi$
$719$	−7.89027	−0.294258	−0.147129	+	0.989117i	$0.547003\pi$
$720$	−2.42205	−0.0902644
$721$	−0.107767	−0.00401347
$722$	13.1952	0.491074
$723$	−12.8982	−0.479688
$724$	13.0669	0.485626
$725$	12.8889	0.478682
$726$	0	0
$727$	5.12729	0.190161	0.0950803	−	0.995470i	$-0.469689\pi$
$727$	5.12729	0.190161	0.0950803	+	0.995470i	$0.469689\pi$
$728$	6.02238	0.223204
$729$	10.8450	0.401665
$730$	−36.2703	−1.34242
$731$	41.6269	1.53963
$732$	16.4993	0.609833
$733$	−33.7736	−1.24746	−0.623728	−	0.781642i	$-0.714383\pi$
$733$	−33.7736	−1.24746	−0.623728	+	0.781642i	$0.714383\pi$
$734$	2.49213	0.0919863
$735$	−2.86816	−0.105794
$736$	48.6476	1.79317
$737$	0	0
$738$	9.22643	0.339629
$739$	−42.7431	−1.57233	−0.786164	−	0.618017i	$-0.787936\pi$
$739$	−42.7431	−1.57233	−0.786164	+	0.618017i	$0.787936\pi$
$740$	34.6513	1.27381
$741$	3.64938	0.134063
$742$	−7.73956	−0.284128
$743$	22.9565	0.842192	0.421096	−	0.907016i	$-0.361646\pi$
$743$	22.9565	0.842192	0.421096	+	0.907016i	$0.361646\pi$
$744$	−19.6462	−0.720266
$745$	26.4012	0.967266
$746$	−1.43180	−0.0524220
$747$	−15.8579	−0.580211
$748$	0	0
$749$	−4.64902	−0.169872
$750$	−2.56500	−0.0936607
$751$	−0.930719	−0.0339624	−0.0169812	−	0.999856i	$-0.505406\pi$
$751$	−0.930719	−0.0339624	−0.0169812	+	0.999856i	$0.505406\pi$
$752$	−2.10766	−0.0768586
$753$	−0.0734388	−0.00267626
$754$	−5.97016	−0.217420
$755$	−59.6231	−2.16991
$756$	−6.44668	−0.234464
$757$	−11.4824	−0.417334	−0.208667	−	0.977987i	$-0.566913\pi$
$757$	−11.4824	−0.417334	−0.208667	+	0.977987i	$0.566913\pi$
$758$	−16.1193	−0.585478
$759$	0	0
$760$	−14.0944	−0.511257
$761$	38.8640	1.40882	0.704410	−	0.709794i	$-0.251212\pi$
$761$	38.8640	1.40882	0.704410	+	0.709794i	$0.251212\pi$
$762$	−15.8073	−0.572637
$763$	3.23140	0.116985
$764$	−30.5760	−1.10620
$765$	−27.3288	−0.988074
$766$	−3.92402	−0.141780
$767$	7.64917	0.276196
$768$	14.2321	0.513555
$769$	−35.6509	−1.28560	−0.642802	−	0.766032i	$-0.722228\pi$
$769$	−35.6509	−1.28560	−0.642802	+	0.766032i	$0.722228\pi$
$770$	0	0
$771$	−9.13763	−0.329084
$772$	30.5177	1.09836
$773$	−1.77045	−0.0636788	−0.0318394	−	0.999493i	$-0.510137\pi$
$773$	−1.77045	−0.0636788	−0.0318394	+	0.999493i	$0.510137\pi$
$774$	16.1593	0.580835
$775$	29.2607	1.05108
$776$	7.81023	0.280371
$777$	8.43518	0.302610
$778$	1.17116	0.0419883
$779$	−9.30719	−0.333465
$780$	8.34569	0.298824
$781$	0	0
$782$	30.4767	1.08985
$783$	16.0609	0.573970
$784$	0.390549	0.0139482
$785$	38.9677	1.39082
$786$	6.68934	0.238601
$787$	3.35315	0.119527	0.0597635	−	0.998213i	$-0.480965\pi$
$787$	3.35315	0.119527	0.0597635	+	0.998213i	$0.480965\pi$
$788$	−8.84240	−0.314998
$789$	−14.2098	−0.505883
$790$	17.6734	0.628792
$791$	−11.3194	−0.402472
$792$	0	0
$793$	28.6061	1.01583
$794$	15.1509	0.537684
$795$	−26.9542	−0.955968
$796$	20.9146	0.741298
$797$	4.92542	0.174467	0.0872337	−	0.996188i	$-0.472197\pi$
$797$	4.92542	0.174467	0.0872337	+	0.996188i	$0.472197\pi$
$798$	−1.36523	−0.0483285
$799$	−23.7815	−0.841328
$800$	−22.6741	−0.801652
$801$	−22.5222	−0.795783
$802$	1.36107	0.0480612
$803$	0	0
$804$	5.48704	0.193513
$805$	−25.0727	−0.883696
$806$	−13.5536	−0.477405
$807$	28.8729	1.01637
$808$	−39.9436	−1.40521
$809$	−6.97322	−0.245165	−0.122583	−	0.992458i	$-0.539118\pi$
$809$	−6.97322	−0.245165	−0.122583	+	0.992458i	$0.539118\pi$
$810$	−3.80145	−0.133569
$811$	5.93377	0.208363	0.104181	−	0.994558i	$-0.466778\pi$
$811$	5.93377	0.208363	0.104181	+	0.994558i	$0.466778\pi$
$812$	−4.35247	−0.152742
$813$	9.98292	0.350116
$814$	0	0
$815$	17.8078	0.623781
$816$	−1.65331	−0.0578773
$817$	−16.3008	−0.570292
$818$	30.2128	1.05637
$819$	−4.57274	−0.159784
$820$	−21.2844	−0.743284
$821$	41.7468	1.45697	0.728487	−	0.685060i	$-0.240224\pi$
$821$	41.7468	1.45697	0.728487	+	0.685060i	$0.240224\pi$
$822$	6.60970	0.230540
$823$	−23.2004	−0.808714	−0.404357	−	0.914601i	$-0.632505\pi$
$823$	−23.2004	−0.808714	−0.404357	+	0.914601i	$0.632505\pi$
$824$	−0.294814	−0.0102703
$825$	0	0
$826$	−2.86154	−0.0995658
$827$	−13.1779	−0.458240	−0.229120	−	0.973398i	$-0.573585\pi$
$827$	−13.1779	−0.458240	−0.229120	+	0.973398i	$0.573585\pi$
$828$	−23.0559	−0.801248
$829$	4.29090	0.149029	0.0745146	−	0.997220i	$-0.476259\pi$
$829$	4.29090	0.149029	0.0745146	+	0.997220i	$0.476259\pi$
$830$	−18.7720	−0.651584
$831$	−27.0075	−0.936879
$832$	8.78313	0.304500
$833$	4.40669	0.152683
$834$	−11.6405	−0.403076
$835$	39.5513	1.36873
$836$	0	0
$837$	36.4619	1.26031
$838$	14.2847	0.493457
$839$	−15.7453	−0.543588	−0.271794	−	0.962355i	$-0.587617\pi$
$839$	−15.7453	−0.543588	−0.271794	+	0.962355i	$0.587617\pi$
$840$	−7.84629	−0.270723
$841$	−18.1565	−0.626086
$842$	−4.38694	−0.151184
$843$	−9.19351	−0.316641
$844$	−15.0052	−0.516501
$845$	−24.3439	−0.837456
$846$	−9.23182	−0.317397
$847$	0	0
$848$	3.67028	0.126038
$849$	1.59012	0.0545728
$850$	−14.2049	−0.487223
$851$	73.7381	2.52771
$852$	5.58745	0.191423
$853$	6.14015	0.210235	0.105117	−	0.994460i	$-0.466478\pi$
$853$	6.14015	0.210235	0.105117	+	0.994460i	$0.466478\pi$
$854$	−10.7015	−0.366197
$855$	10.7017	0.365991
$856$	−12.7181	−0.434696
$857$	34.4740	1.17761	0.588804	−	0.808276i	$-0.299599\pi$
$857$	34.4740	1.17761	0.588804	+	0.808276i	$0.299599\pi$
$858$	0	0
$859$	8.21553	0.280310	0.140155	−	0.990130i	$-0.455240\pi$
$859$	8.21553	0.280310	0.140155	+	0.990130i	$0.455240\pi$
$860$	−37.2779	−1.27116
$861$	−5.18127	−0.176577
$862$	23.2309	0.791248
$863$	13.0390	0.443852	0.221926	−	0.975063i	$-0.428766\pi$
$863$	13.0390	0.443852	0.221926	+	0.975063i	$0.428766\pi$
$864$	−28.2543	−0.961230
$865$	−1.54572	−0.0525560
$866$	−14.5819	−0.495512
$867$	−2.32377	−0.0789193
$868$	−9.88109	−0.335386
$869$	0	0
$870$	7.77824	0.263707
$871$	9.51327	0.322345
$872$	8.84000	0.299360
$873$	−5.93023	−0.200708
$874$	−11.9344	−0.403689
$875$	−3.24213	−0.109604
$876$	18.7299	0.632825
$877$	17.9259	0.605316	0.302658	−	0.953099i	$-0.402126\pi$
$877$	17.9259	0.605316	0.302658	+	0.953099i	$0.402126\pi$
$878$	20.6868	0.698147
$879$	−4.49765	−0.151702
$880$	0	0
$881$	−17.3276	−0.583780	−0.291890	−	0.956452i	$-0.594284\pi$
$881$	−17.3276	−0.583780	−0.291890	+	0.956452i	$0.594284\pi$
$882$	1.71065	0.0576007
$883$	21.8740	0.736120	0.368060	−	0.929802i	$-0.380022\pi$
$883$	21.8740	0.736120	0.368060	+	0.929802i	$0.380022\pi$
$884$	−12.8225	−0.431266
$885$	−9.96576	−0.334995
$886$	22.2004	0.745836
$887$	25.6098	0.859893	0.429946	−	0.902854i	$-0.358532\pi$
$887$	25.6098	0.859893	0.429946	+	0.902854i	$0.358532\pi$
$888$	23.0757	0.774370
$889$	−19.9802	−0.670113
$890$	−26.6608	−0.893674
$891$	0	0
$892$	14.1294	0.473087
$893$	9.31263	0.311635
$894$	6.99590	0.233978
$895$	5.06411	0.169275
$896$	8.30013	0.277288
$897$	17.7597	0.592978
$898$	17.3740	0.579779
$899$	24.6172	0.821030
$900$	10.7461	0.358204
$901$	41.4130	1.37967
$902$	0	0
$903$	−9.07457	−0.301983
$904$	−30.9660	−1.02991
$905$	−29.5161	−0.981148
$906$	−15.7991	−0.524891
$907$	−14.4388	−0.479431	−0.239716	−	0.970843i	$-0.577054\pi$
$907$	−14.4388	−0.479431	−0.239716	+	0.970843i	$0.577054\pi$
$908$	4.94050	0.163956
$909$	30.3288	1.00594
$910$	−5.41302	−0.179440
$911$	37.1858	1.23202	0.616010	−	0.787739i	$-0.288748\pi$
$911$	37.1858	1.23202	0.616010	+	0.787739i	$0.288748\pi$
$912$	0.647421	0.0214383
$913$	0	0
$914$	1.56769	0.0518545
$915$	−37.2695	−1.23209
$916$	31.2708	1.03321
$917$	8.45523	0.279216
$918$	−17.7007	−0.584211
$919$	−16.0496	−0.529426	−0.264713	−	0.964327i	$-0.585277\pi$
$919$	−16.0496	−0.529426	−0.264713	+	0.964327i	$0.585277\pi$
$920$	−68.5901	−2.26135
$921$	16.3146	0.537585
$922$	−20.6251	−0.679252
$923$	9.68736	0.318863
$924$	0	0
$925$	−34.3685	−1.13003
$926$	17.4364	0.572997
$927$	0.223849	0.00735218
$928$	−19.0759	−0.626196
$929$	33.8914	1.11194	0.555970	−	0.831202i	$-0.312347\pi$
$929$	33.8914	1.11194	0.555970	+	0.831202i	$0.312347\pi$
$930$	17.6584	0.579040
$931$	−1.72563	−0.0565551
$932$	−5.11090	−0.167413
$933$	21.0629	0.689567
$934$	−13.7109	−0.448635
$935$	0	0
$936$	−12.5094	−0.408883
$937$	−9.22624	−0.301408	−0.150704	−	0.988579i	$-0.548154\pi$
$937$	−9.22624	−0.301408	−0.150704	+	0.988579i	$0.548154\pi$
$938$	−3.55890	−0.116202
$939$	−9.65977	−0.315235
$940$	21.2969	0.694627
$941$	−53.3626	−1.73957	−0.869786	−	0.493430i	$-0.835743\pi$
$941$	−53.3626	−1.73957	−0.869786	+	0.493430i	$0.835743\pi$
$942$	10.3258	0.336433
$943$	−45.2933	−1.47495
$944$	1.35701	0.0441668
$945$	14.5621	0.473705
$946$	0	0
$947$	31.0986	1.01057	0.505285	−	0.862953i	$-0.331388\pi$
$947$	31.0986	1.01057	0.505285	+	0.862953i	$0.331388\pi$
$948$	−9.12650	−0.296415
$949$	32.4734	1.05413
$950$	5.56252	0.180472
$951$	−22.6330	−0.733927
$952$	12.0552	0.390711
$953$	34.9286	1.13145	0.565724	−	0.824595i	$-0.308597\pi$
$953$	34.9286	1.13145	0.565724	+	0.824595i	$0.308597\pi$
$954$	16.0763	0.520488
$955$	69.0667	2.23494
$956$	13.3123	0.430550
$957$	0	0
$958$	−17.8393	−0.576361
$959$	8.35456	0.269783
$960$	−11.4431	−0.369326
$961$	24.8866	0.802793
$962$	15.9195	0.513266
$963$	9.65673	0.311184
$964$	−17.7466	−0.571579
$965$	−68.9350	−2.21910
$966$	−6.64385	−0.213763
$967$	−20.8029	−0.668975	−0.334488	−	0.942400i	$-0.608563\pi$
$967$	−20.8029	−0.668975	−0.334488	+	0.942400i	$0.608563\pi$
$968$	0	0
$969$	7.30507	0.234673
$970$	−7.01996	−0.225397
$971$	−40.0308	−1.28465	−0.642325	−	0.766433i	$-0.722030\pi$
$971$	−40.0308	−1.28465	−0.642325	+	0.766433i	$0.722030\pi$
$972$	21.3031	0.683297
$973$	−14.7134	−0.471689
$974$	2.31695	0.0742400
$975$	−8.27759	−0.265095
$976$	5.07488	0.162443
$977$	6.48609	0.207509	0.103754	−	0.994603i	$-0.466914\pi$
$977$	6.48609	0.207509	0.103754	+	0.994603i	$0.466914\pi$
$978$	4.71878	0.150890
$979$	0	0
$980$	−3.94630	−0.126060
$981$	−6.71212	−0.214302
$982$	−2.64016	−0.0842508
$983$	10.9782	0.350150	0.175075	−	0.984555i	$-0.443983\pi$
$983$	10.9782	0.350150	0.175075	+	0.984555i	$0.443983\pi$
$984$	−14.1742	−0.451856
$985$	19.9737	0.636414
$986$	−11.9506	−0.380586
$987$	5.18430	0.165018
$988$	5.02118	0.159745
$989$	−79.3275	−2.52247
$990$	0	0
$991$	−59.2666	−1.88267	−0.941333	−	0.337479i	$-0.890426\pi$
$991$	−59.2666	−1.88267	−0.941333	+	0.337479i	$0.890426\pi$
$992$	−43.3065	−1.37498
$993$	20.0398	0.635944
$994$	−3.62402	−0.114947
$995$	−47.2430	−1.49770
$996$	9.69379	0.307160
$997$	−43.3517	−1.37296	−0.686481	−	0.727148i	$-0.740845\pi$
$997$	−43.3517	−1.37296	−0.686481	+	0.727148i	$0.740845\pi$
$998$	16.5133	0.522720
$999$	−42.8267	−1.35498

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.823556 q^{2} -0.960649 q^{3} -1.32176 q^{4} +2.98565 q^{5} +0.791148 q^{6} +1.00000 q^{7} +2.73565 q^{8} -2.07715 q^{9} +O(q^{10})\)	\(q-0.823556 q^{2} -0.960649 q^{3} -1.32176 q^{4} +2.98565 q^{5} +0.791148 q^{6} +1.00000 q^{7} +2.73565 q^{8} -2.07715 q^{9} -2.45885 q^{10} +1.26974 q^{12} +2.20144 q^{13} -0.823556 q^{14} -2.86816 q^{15} +0.390549 q^{16} +4.40669 q^{17} +1.71065 q^{18} -1.72563 q^{19} -3.94630 q^{20} -0.960649 q^{21} -8.39774 q^{23} -2.62800 q^{24} +3.91410 q^{25} -1.81301 q^{26} +4.87736 q^{27} -1.32176 q^{28} +3.29295 q^{29} +2.36209 q^{30} +7.47573 q^{31} -5.79294 q^{32} -3.62916 q^{34} +2.98565 q^{35} +2.74549 q^{36} -8.78071 q^{37} +1.42115 q^{38} -2.11481 q^{39} +8.16769 q^{40} +5.39351 q^{41} +0.791148 q^{42} +9.44629 q^{43} -6.20165 q^{45} +6.91601 q^{46} -5.39667 q^{47} -0.375180 q^{48} +1.00000 q^{49} -3.22348 q^{50} -4.23329 q^{51} -2.90977 q^{52} +9.39774 q^{53} -4.01678 q^{54} +2.73565 q^{56} +1.65772 q^{57} -2.71193 q^{58} +3.47462 q^{59} +3.79101 q^{60} +12.9942 q^{61} -6.15668 q^{62} -2.07715 q^{63} +3.98971 q^{64} +6.57274 q^{65} +4.32138 q^{67} -5.82457 q^{68} +8.06728 q^{69} -2.45885 q^{70} +4.40046 q^{71} -5.68237 q^{72} +14.7509 q^{73} +7.23140 q^{74} -3.76007 q^{75} +2.28086 q^{76} +1.74167 q^{78} -7.18768 q^{79} +1.16604 q^{80} +1.54603 q^{81} -4.44186 q^{82} +7.63445 q^{83} +1.26974 q^{84} +13.1568 q^{85} -7.77955 q^{86} -3.16337 q^{87} +10.8428 q^{89} +5.10741 q^{90} +2.20144 q^{91} +11.0998 q^{92} -7.18156 q^{93} +4.44446 q^{94} -5.15211 q^{95} +5.56498 q^{96} +2.85498 q^{97} -0.823556 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9	\( 6 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{7} + 12 q^{8} + 8 q^{9} - 8 q^{10} - 14 q^{12} + 4 q^{13} + 4 q^{14} + 2 q^{15} + 8 q^{16} + 22 q^{17} + 24 q^{18} + 6 q^{19} + 2 q^{20} - 2 q^{21} + 2 q^{23} - 20 q^{24} + 4 q^{25} + 6 q^{26} - 2 q^{27} + 4 q^{28} + 12 q^{29} + 20 q^{30} - 2 q^{31} + 8 q^{32} + 24 q^{34} - 4 q^{35} + 18 q^{36} + 14 q^{37} - 22 q^{38} + 20 q^{39} + 18 q^{40} + 26 q^{41} - 6 q^{42} - 4 q^{43} - 36 q^{45} + 12 q^{46} - 16 q^{47} - 24 q^{48} + 6 q^{49} - 4 q^{50} - 4 q^{51} + 12 q^{52} + 4 q^{53} - 32 q^{54} + 12 q^{56} + 20 q^{57} - 2 q^{58} - 4 q^{59} + 24 q^{60} - 8 q^{61} + 20 q^{62} + 8 q^{63} + 26 q^{64} + 24 q^{65} + 6 q^{67} + 12 q^{68} - 14 q^{69} - 8 q^{70} + 22 q^{71} + 16 q^{72} + 14 q^{73} + 44 q^{74} - 20 q^{75} - 30 q^{76} + 32 q^{78} - 28 q^{79} - 4 q^{80} - 6 q^{81} - 4 q^{82} + 22 q^{83} - 14 q^{84} - 24 q^{85} - 30 q^{86} + 22 q^{87} - 22 q^{90} + 4 q^{91} + 10 q^{92} - 50 q^{93} - 38 q^{94} - 24 q^{95} - 62 q^{96} - 4 q^{97} + 4 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 - 6 * q^6 + 6 * q^7 + 12 * q^8 + 8 * q^9 - 8 * q^10 - 14 * q^12 + 4 * q^13 + 4 * q^14 + 2 * q^15 + 8 * q^16 + 22 * q^17 + 24 * q^18 + 6 * q^19 + 2 * q^20 - 2 * q^21 + 2 * q^23 - 20 * q^24 + 4 * q^25 + 6 * q^26 - 2 * q^27 + 4 * q^28 + 12 * q^29 + 20 * q^30 - 2 * q^31 + 8 * q^32 + 24 * q^34 - 4 * q^35 + 18 * q^36 + 14 * q^37 - 22 * q^38 + 20 * q^39 + 18 * q^40 + 26 * q^41 - 6 * q^42 - 4 * q^43 - 36 * q^45 + 12 * q^46 - 16 * q^47 - 24 * q^48 + 6 * q^49 - 4 * q^50 - 4 * q^51 + 12 * q^52 + 4 * q^53 - 32 * q^54 + 12 * q^56 + 20 * q^57 - 2 * q^58 - 4 * q^59 + 24 * q^60 - 8 * q^61 + 20 * q^62 + 8 * q^63 + 26 * q^64 + 24 * q^65 + 6 * q^67 + 12 * q^68 - 14 * q^69 - 8 * q^70 + 22 * q^71 + 16 * q^72 + 14 * q^73 + 44 * q^74 - 20 * q^75 - 30 * q^76 + 32 * q^78 - 28 * q^79 - 4 * q^80 - 6 * q^81 - 4 * q^82 + 22 * q^83 - 14 * q^84 - 24 * q^85 - 30 * q^86 + 22 * q^87 - 22 * q^90 + 4 * q^91 + 10 * q^92 - 50 * q^93 - 38 * q^94 - 24 * q^95 - 62 * q^96 - 4 * q^97 + 4 * q^98

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