LMFDB - Embedded newform 177.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,2,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, 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("177.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	177.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:	$1.41335211578$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
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			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.254102 q^{2} -1.00000 q^{3} -1.93543 q^{4} +1.68133 q^{5} +0.254102 q^{6} +2.74590 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.254102 q^{2} -1.00000 q^{3} -1.93543 q^{4} +1.68133 q^{5} +0.254102 q^{6} +2.74590 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.427229 q^{10} +2.18953 q^{11} +1.93543 q^{12} +4.18953 q^{13} -0.697737 q^{14} -1.68133 q^{15} +3.61676 q^{16} -6.29809 q^{17} -0.254102 q^{18} +4.93543 q^{19} -3.25410 q^{20} -2.74590 q^{21} -0.556364 q^{22} +3.44364 q^{23} -1.00000 q^{24} -2.17313 q^{25} -1.06457 q^{26} -1.00000 q^{27} -5.31450 q^{28} -9.12497 q^{29} +0.427229 q^{30} -0.616763 q^{31} -2.91903 q^{32} -2.18953 q^{33} +1.60036 q^{34} +4.61676 q^{35} -1.93543 q^{36} +1.44364 q^{37} -1.25410 q^{38} -4.18953 q^{39} +1.68133 q^{40} +4.10856 q^{41} +0.697737 q^{42} +8.53579 q^{43} -4.23769 q^{44} +1.68133 q^{45} -0.875034 q^{46} -6.48763 q^{47} -3.61676 q^{48} +0.539958 q^{49} +0.552195 q^{50} +6.29809 q^{51} -8.10856 q^{52} -0.664924 q^{53} +0.254102 q^{54} +3.68133 q^{55} +2.74590 q^{56} -4.93543 q^{57} +2.31867 q^{58} +1.00000 q^{59} +3.25410 q^{60} -10.9958 q^{61} +0.156721 q^{62} +2.74590 q^{63} -6.49180 q^{64} +7.04399 q^{65} +0.556364 q^{66} -9.93126 q^{67} +12.1895 q^{68} -3.44364 q^{69} -1.17313 q^{70} +6.82687 q^{71} +1.00000 q^{72} +13.5040 q^{73} -0.366830 q^{74} +2.17313 q^{75} -9.55220 q^{76} +6.01224 q^{77} +1.06457 q^{78} -1.17313 q^{79} +6.08097 q^{80} +1.00000 q^{81} -1.04399 q^{82} -15.3421 q^{83} +5.31450 q^{84} -10.5892 q^{85} -2.16896 q^{86} +9.12497 q^{87} +2.18953 q^{88} +1.97942 q^{89} -0.427229 q^{90} +11.5040 q^{91} -6.66492 q^{92} +0.616763 q^{93} +1.64852 q^{94} +8.29809 q^{95} +2.91903 q^{96} +3.33508 q^{97} -0.137204 q^{98} +2.18953 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{4} - 2 q^{5} + 9 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^4 - 2 * q^5 + 9 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{4} - 2 q^{5} + 9 q^{7} + 3 q^{8} + 3 q^{9} + 5 q^{10} - 2 q^{11} - 2 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{15} - 4 q^{16} + 3 q^{17} + 7 q^{19} - 9 q^{20} - 9 q^{21} - 11 q^{22} + q^{23} - 3 q^{24} - q^{25} - 11 q^{26} - 3 q^{27} + 9 q^{28} - 11 q^{29} - 5 q^{30} + 13 q^{31} - 4 q^{32} + 2 q^{33} - 7 q^{34} - q^{35} + 2 q^{36} - 5 q^{37} - 3 q^{38} - 4 q^{39} - 2 q^{40} - q^{41} - 8 q^{42} + 6 q^{43} - 15 q^{44} - 2 q^{45} - 19 q^{46} + 11 q^{47} + 4 q^{48} + 14 q^{49} - 21 q^{50} - 3 q^{51} - 11 q^{52} + 2 q^{53} + 4 q^{55} + 9 q^{56} - 7 q^{57} + 14 q^{58} + 3 q^{59} + 9 q^{60} - q^{61} - 2 q^{62} + 9 q^{63} - 21 q^{64} + 11 q^{66} + 10 q^{67} + 28 q^{68} - q^{69} + 2 q^{70} + 26 q^{71} + 3 q^{72} + 7 q^{73} - 19 q^{74} + q^{75} - 6 q^{76} - 17 q^{77} + 11 q^{78} + 2 q^{79} + 23 q^{80} + 3 q^{81} + 18 q^{82} - 3 q^{83} - 9 q^{84} - 35 q^{85} + 31 q^{86} + 11 q^{87} - 2 q^{88} - 23 q^{89} + 5 q^{90} + q^{91} - 16 q^{92} - 13 q^{93} + 4 q^{94} + 3 q^{95} + 4 q^{96} + 14 q^{97} + 51 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^4 - 2 * q^5 + 9 * q^7 + 3 * q^8 + 3 * q^9 + 5 * q^10 - 2 * q^11 - 2 * q^12 + 4 * q^13 + 8 * q^14 + 2 * q^15 - 4 * q^16 + 3 * q^17 + 7 * q^19 - 9 * q^20 - 9 * q^21 - 11 * q^22 + q^23 - 3 * q^24 - q^25 - 11 * q^26 - 3 * q^27 + 9 * q^28 - 11 * q^29 - 5 * q^30 + 13 * q^31 - 4 * q^32 + 2 * q^33 - 7 * q^34 - q^35 + 2 * q^36 - 5 * q^37 - 3 * q^38 - 4 * q^39 - 2 * q^40 - q^41 - 8 * q^42 + 6 * q^43 - 15 * q^44 - 2 * q^45 - 19 * q^46 + 11 * q^47 + 4 * q^48 + 14 * q^49 - 21 * q^50 - 3 * q^51 - 11 * q^52 + 2 * q^53 + 4 * q^55 + 9 * q^56 - 7 * q^57 + 14 * q^58 + 3 * q^59 + 9 * q^60 - q^61 - 2 * q^62 + 9 * q^63 - 21 * q^64 + 11 * q^66 + 10 * q^67 + 28 * q^68 - q^69 + 2 * q^70 + 26 * q^71 + 3 * q^72 + 7 * q^73 - 19 * q^74 + q^75 - 6 * q^76 - 17 * q^77 + 11 * q^78 + 2 * q^79 + 23 * q^80 + 3 * q^81 + 18 * q^82 - 3 * q^83 - 9 * q^84 - 35 * q^85 + 31 * q^86 + 11 * q^87 - 2 * q^88 - 23 * q^89 + 5 * q^90 + q^91 - 16 * q^92 - 13 * q^93 + 4 * q^94 + 3 * q^95 + 4 * q^96 + 14 * q^97 + 51 * q^98 - 2 * q^99

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 $

$2$	−0.254102	−0.179677	−0.0898385	−	0.995956i	$-0.528635\pi$
$2$	−0.254102	−0.179677	−0.0898385	+	0.995956i	$0.528635\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.93543	−0.967716
$5$	1.68133	0.751914	0.375957	−	0.926637i	$-0.377314\pi$
$5$	1.68133	0.751914	0.375957	+	0.926637i	$0.377314\pi$
$6$	0.254102	0.103737
$7$	2.74590	1.03785	0.518926	−	0.854819i	$-0.326332\pi$
$7$	2.74590	1.03785	0.518926	+	0.854819i	$0.326332\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.427229	−0.135102
$11$	2.18953	0.660169	0.330085	−	0.943951i	$-0.392923\pi$
$11$	2.18953	0.660169	0.330085	+	0.943951i	$0.392923\pi$
$12$	1.93543	0.558711
$13$	4.18953	1.16197	0.580984	−	0.813915i	$-0.302668\pi$
$13$	4.18953	1.16197	0.580984	+	0.813915i	$0.302668\pi$
$14$	−0.697737	−0.186478
$15$	−1.68133	−0.434118
$16$	3.61676	0.904191
$17$	−6.29809	−1.52751	−0.763756	−	0.645505i	$-0.776647\pi$
$17$	−6.29809	−1.52751	−0.763756	+	0.645505i	$0.776647\pi$
$18$	−0.254102	−0.0598923
$19$	4.93543	1.13227	0.566133	−	0.824314i	$-0.308439\pi$
$19$	4.93543	1.13227	0.566133	+	0.824314i	$0.308439\pi$
$20$	−3.25410	−0.727639
$21$	−2.74590	−0.599204
$22$	−0.556364	−0.118617
$23$	3.44364	0.718048	0.359024	−	0.933328i	$-0.383110\pi$
$23$	3.44364	0.718048	0.359024	+	0.933328i	$0.383110\pi$
$24$	−1.00000	−0.204124
$25$	−2.17313	−0.434625
$26$	−1.06457	−0.208779
$27$	−1.00000	−0.192450
$28$	−5.31450	−1.00435
$29$	−9.12497	−1.69446	−0.847232	−	0.531223i	$-0.821733\pi$
$29$	−9.12497	−1.69446	−0.847232	+	0.531223i	$0.821733\pi$
$30$	0.427229	0.0780010
$31$	−0.616763	−0.110774	−0.0553870	−	0.998465i	$-0.517639\pi$
$31$	−0.616763	−0.110774	−0.0553870	+	0.998465i	$0.517639\pi$
$32$	−2.91903	−0.516016
$33$	−2.18953	−0.381149
$34$	1.60036	0.274459
$35$	4.61676	0.780375
$36$	−1.93543	−0.322572
$37$	1.44364	0.237332	0.118666	−	0.992934i	$-0.462138\pi$
$37$	1.44364	0.237332	0.118666	+	0.992934i	$0.462138\pi$
$38$	−1.25410	−0.203442
$39$	−4.18953	−0.670862
$40$	1.68133	0.265842
$41$	4.10856	0.641649	0.320825	−	0.947139i	$-0.396040\pi$
$41$	4.10856	0.641649	0.320825	+	0.947139i	$0.396040\pi$
$42$	0.697737	0.107663
$43$	8.53579	1.30170	0.650848	−	0.759208i	$-0.274414\pi$
$43$	8.53579	1.30170	0.650848	+	0.759208i	$0.274414\pi$
$44$	−4.23769	−0.638857
$45$	1.68133	0.250638
$46$	−0.875034	−0.129017
$47$	−6.48763	−0.946318	−0.473159	−	0.880977i	$-0.656886\pi$
$47$	−6.48763	−0.946318	−0.473159	+	0.880977i	$0.656886\pi$
$48$	−3.61676	−0.522035
$49$	0.539958	0.0771368
$50$	0.552195	0.0780922
$51$	6.29809	0.881910
$52$	−8.10856	−1.12445
$53$	−0.664924	−0.0913343	−0.0456672	−	0.998957i	$-0.514541\pi$
$53$	−0.664924	−0.0913343	−0.0456672	+	0.998957i	$0.514541\pi$
$54$	0.254102	0.0345789
$55$	3.68133	0.496391
$56$	2.74590	0.366936
$57$	−4.93543	−0.653714
$58$	2.31867	0.304456
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	3.25410	0.420103
$61$	−10.9958	−1.40787	−0.703936	−	0.710263i	$-0.748576\pi$
$61$	−10.9958	−1.40787	−0.703936	+	0.710263i	$0.748576\pi$
$62$	0.156721	0.0199035
$63$	2.74590	0.345951
$64$	−6.49180	−0.811475
$65$	7.04399	0.873700
$66$	0.556364	0.0684837
$67$	−9.93126	−1.21330	−0.606648	−	0.794970i	$-0.707486\pi$
$67$	−9.93126	−1.21330	−0.606648	+	0.794970i	$0.707486\pi$
$68$	12.1895	1.47820
$69$	−3.44364	−0.414565
$70$	−1.17313	−0.140216
$71$	6.82687	0.810201	0.405100	−	0.914272i	$-0.367237\pi$
$71$	6.82687	0.810201	0.405100	+	0.914272i	$0.367237\pi$
$72$	1.00000	0.117851
$73$	13.5040	1.58053	0.790264	−	0.612767i	$-0.209943\pi$
$73$	13.5040	1.58053	0.790264	+	0.612767i	$0.209943\pi$
$74$	−0.366830	−0.0426432
$75$	2.17313	0.250931
$76$	−9.55220	−1.09571
$77$	6.01224	0.685158
$78$	1.06457	0.120539
$79$	−1.17313	−0.131987	−0.0659936	−	0.997820i	$-0.521022\pi$
$79$	−1.17313	−0.131987	−0.0659936	+	0.997820i	$0.521022\pi$
$80$	6.08097	0.679874
$81$	1.00000	0.111111
$82$	−1.04399	−0.115290
$83$	−15.3421	−1.68401	−0.842006	−	0.539468i	$-0.818626\pi$
$83$	−15.3421	−1.68401	−0.842006	+	0.539468i	$0.818626\pi$
$84$	5.31450	0.579860
$85$	−10.5892	−1.14856
$86$	−2.16896	−0.233885
$87$	9.12497	0.978299
$88$	2.18953	0.233405
$89$	1.97942	0.209819	0.104909	−	0.994482i	$-0.466545\pi$
$89$	1.97942	0.209819	0.104909	+	0.994482i	$0.466545\pi$
$90$	−0.427229	−0.0450339
$91$	11.5040	1.20595
$92$	−6.66492	−0.694866
$93$	0.616763	0.0639553
$94$	1.64852	0.170032
$95$	8.29809	0.851366
$96$	2.91903	0.297922
$97$	3.33508	0.338626	0.169313	−	0.985562i	$-0.445845\pi$
$97$	3.33508	0.338626	0.169313	+	0.985562i	$0.445845\pi$
$98$	−0.137204	−0.0138597
$99$	2.18953	0.220056
$100$	4.20594	0.420594
$101$	−3.49180	−0.347447	−0.173723	−	0.984794i	$-0.555580\pi$
$101$	−3.49180	−0.347447	−0.173723	+	0.984794i	$0.555580\pi$
$102$	−1.60036	−0.158459
$103$	8.21712	0.809657	0.404828	−	0.914393i	$-0.367331\pi$
$103$	8.21712	0.809657	0.404828	+	0.914393i	$0.367331\pi$
$104$	4.18953	0.410818
$105$	−4.61676	−0.450550
$106$	0.168958	0.0164107
$107$	17.7899	1.71981	0.859907	−	0.510451i	$-0.170522\pi$
$107$	17.7899	1.71981	0.859907	+	0.510451i	$0.170522\pi$
$108$	1.93543	0.186237
$109$	−18.2981	−1.75264	−0.876320	−	0.481730i	$-0.840009\pi$
$109$	−18.2981	−1.75264	−0.876320	+	0.481730i	$0.840009\pi$
$110$	−0.935432	−0.0891900
$111$	−1.44364	−0.137024
$112$	9.93126	0.938416
$113$	−7.71414	−0.725686	−0.362843	−	0.931850i	$-0.618194\pi$
$113$	−7.71414	−0.725686	−0.362843	+	0.931850i	$0.618194\pi$
$114$	1.25410	0.117457
$115$	5.78989	0.539910
$116$	17.6608	1.63976
$117$	4.18953	0.387323
$118$	−0.254102	−0.0233920
$119$	−17.2939	−1.58533
$120$	−1.68133	−0.153484
$121$	−6.20594	−0.564176
$122$	2.79406	0.252962
$123$	−4.10856	−0.370456
$124$	1.19370	0.107198
$125$	−12.0604	−1.07871
$126$	−0.697737	−0.0621594
$127$	6.18953	0.549232	0.274616	−	0.961554i	$-0.411449\pi$
$127$	6.18953	0.549232	0.274616	+	0.961554i	$0.411449\pi$
$128$	7.48763	0.661819
$129$	−8.53579	−0.751534
$130$	−1.78989	−0.156984
$131$	−15.5798	−1.36121	−0.680606	−	0.732650i	$-0.738283\pi$
$131$	−15.5798	−1.36121	−0.680606	+	0.732650i	$0.738283\pi$
$132$	4.23769	0.368844
$133$	13.5522	1.17512
$134$	2.52355	0.218002
$135$	−1.68133	−0.144706
$136$	−6.29809	−0.540057
$137$	0.664924	0.0568083	0.0284041	−	0.999597i	$-0.490957\pi$
$137$	0.664924	0.0568083	0.0284041	+	0.999597i	$0.490957\pi$
$138$	0.875034	0.0744878
$139$	−18.7857	−1.59338	−0.796692	−	0.604385i	$-0.793419\pi$
$139$	−18.7857	−1.59338	−0.796692	+	0.604385i	$0.793419\pi$
$140$	−8.93543	−0.755182
$141$	6.48763	0.546357
$142$	−1.73472	−0.145574
$143$	9.17313	0.767095
$144$	3.61676	0.301397
$145$	−15.3421	−1.27409
$146$	−3.43140	−0.283985
$147$	−0.539958	−0.0445349
$148$	−2.79406	−0.229670
$149$	−8.04816	−0.659331	−0.329666	−	0.944098i	$-0.606936\pi$
$149$	−8.04816	−0.659331	−0.329666	+	0.944098i	$0.606936\pi$
$150$	−0.552195	−0.0450866
$151$	−17.8503	−1.45264	−0.726318	−	0.687359i	$-0.758770\pi$
$151$	−17.8503	−1.45264	−0.726318	+	0.687359i	$0.758770\pi$
$152$	4.93543	0.400316
$153$	−6.29809	−0.509171
$154$	−1.52772	−0.123107
$155$	−1.03698	−0.0832924
$156$	8.10856	0.649204
$157$	10.4395	0.833160	0.416580	−	0.909099i	$-0.363229\pi$
$157$	10.4395	0.833160	0.416580	+	0.909099i	$0.363229\pi$
$158$	0.298094	0.0237151
$159$	0.664924	0.0527319
$160$	−4.90785	−0.387999
$161$	9.45587	0.745227
$162$	−0.254102	−0.0199641
$163$	−3.76231	−0.294686	−0.147343	−	0.989085i	$-0.547072\pi$
$163$	−3.76231	−0.294686	−0.147343	+	0.989085i	$0.547072\pi$
$164$	−7.95184	−0.620934
$165$	−3.68133	−0.286591
$166$	3.89845	0.302578
$167$	18.1086	1.40128	0.700641	−	0.713514i	$-0.252897\pi$
$167$	18.1086	1.40128	0.700641	+	0.713514i	$0.252897\pi$
$168$	−2.74590	−0.211851
$169$	4.55220	0.350169
$170$	2.69073	0.206369
$171$	4.93543	0.377422
$172$	−16.5204	−1.25967
$173$	−9.66598	−0.734891	−0.367446	−	0.930045i	$-0.619768\pi$
$173$	−9.66598	−0.734891	−0.367446	+	0.930045i	$0.619768\pi$
$174$	−2.31867	−0.175778
$175$	−5.96719	−0.451077
$176$	7.91903	0.596919
$177$	−1.00000	−0.0751646
$178$	−0.502975	−0.0376996
$179$	−8.95601	−0.669403	−0.334702	−	0.942324i	$-0.608636\pi$
$179$	−8.95601	−0.669403	−0.334702	+	0.942324i	$0.608636\pi$
$180$	−3.25410	−0.242546
$181$	−0.487628	−0.0362451	−0.0181225	−	0.999836i	$-0.505769\pi$
$181$	−0.487628	−0.0362451	−0.0181225	+	0.999836i	$0.505769\pi$
$182$	−2.92319	−0.216682
$183$	10.9958	0.812835
$184$	3.44364	0.253868
$185$	2.42723	0.178453
$186$	−0.156721	−0.0114913
$187$	−13.7899	−1.00842
$188$	12.5564	0.915767
$189$	−2.74590	−0.199735
$190$	−2.10856	−0.152971
$191$	5.07681	0.367345	0.183672	−	0.982988i	$-0.441201\pi$
$191$	5.07681	0.367345	0.183672	+	0.982988i	$0.441201\pi$
$192$	6.49180	0.468505
$193$	9.36266	0.673939	0.336970	−	0.941516i	$-0.390598\pi$
$193$	9.36266	0.673939	0.336970	+	0.941516i	$0.390598\pi$
$194$	−0.847448	−0.0608433
$195$	−7.04399	−0.504431
$196$	−1.04505	−0.0746465
$197$	−16.2499	−1.15776	−0.578880	−	0.815413i	$-0.696510\pi$
$197$	−16.2499	−1.15776	−0.578880	+	0.815413i	$0.696510\pi$
$198$	−0.556364	−0.0395391
$199$	6.64958	0.471376	0.235688	−	0.971829i	$-0.424266\pi$
$199$	6.64958	0.471376	0.235688	+	0.971829i	$0.424266\pi$
$200$	−2.17313	−0.153663
$201$	9.93126	0.700497
$202$	0.887271	0.0624282
$203$	−25.0562	−1.75860
$204$	−12.1895	−0.853438
$205$	6.90785	0.482465
$206$	−2.08798	−0.145477
$207$	3.44364	0.239349
$208$	15.1526	1.05064
$209$	10.8063	0.747487
$210$	1.17313	0.0809535
$211$	13.1526	0.905459	0.452729	−	0.891648i	$-0.350450\pi$
$211$	13.1526	0.905459	0.452729	+	0.891648i	$0.350450\pi$
$212$	1.28692	0.0883857
$213$	−6.82687	−0.467770
$214$	−4.52044	−0.309011
$215$	14.3515	0.978763
$216$	−1.00000	−0.0680414
$217$	−1.69357	−0.114967
$218$	4.64958	0.314909
$219$	−13.5040	−0.912518
$220$	−7.12497	−0.480365
$221$	−26.3861	−1.77492
$222$	0.366830	0.0246200
$223$	−9.70892	−0.650157	−0.325079	−	0.945687i	$-0.605391\pi$
$223$	−9.70892	−0.650157	−0.325079	+	0.945687i	$0.605391\pi$
$224$	−8.01535	−0.535548
$225$	−2.17313	−0.144875
$226$	1.96018	0.130389
$227$	6.16896	0.409448	0.204724	−	0.978820i	$-0.434370\pi$
$227$	6.16896	0.409448	0.204724	+	0.978820i	$0.434370\pi$
$228$	9.55220	0.632610
$229$	25.8175	1.70607	0.853033	−	0.521856i	$-0.174760\pi$
$229$	25.8175	1.70607	0.853033	+	0.521856i	$0.174760\pi$
$230$	−1.47122	−0.0970094
$231$	−6.01224	−0.395576
$232$	−9.12497	−0.599083
$233$	12.2499	0.802520	0.401260	−	0.915964i	$-0.368572\pi$
$233$	12.2499	0.802520	0.401260	+	0.915964i	$0.368572\pi$
$234$	−1.06457	−0.0695930
$235$	−10.9078	−0.711549
$236$	−1.93543	−0.125986
$237$	1.17313	0.0762028
$238$	4.39442	0.284848
$239$	−22.9477	−1.48436	−0.742181	−	0.670200i	$-0.766208\pi$
$239$	−22.9477	−1.48436	−0.742181	+	0.670200i	$0.766208\pi$
$240$	−6.08097	−0.392525
$241$	4.67015	0.300831	0.150415	−	0.988623i	$-0.451939\pi$
$241$	4.67015	0.300831	0.150415	+	0.988623i	$0.451939\pi$
$242$	1.57694	0.101370
$243$	−1.00000	−0.0641500
$244$	21.2817	1.36242
$245$	0.907847	0.0580002
$246$	1.04399	0.0665625
$247$	20.6772	1.31566
$248$	−0.616763	−0.0391645
$249$	15.3421	0.972265
$250$	3.06457	0.193820
$251$	22.4067	1.41430	0.707148	−	0.707066i	$-0.249982\pi$
$251$	22.4067	1.41430	0.707148	+	0.707066i	$0.249982\pi$
$252$	−5.31450	−0.334782
$253$	7.53996	0.474033
$254$	−1.57277	−0.0986844
$255$	10.5892	0.663120
$256$	11.0810	0.692561
$257$	31.1372	1.94229	0.971143	−	0.238499i	$-0.0766554\pi$
$257$	31.1372	1.94229	0.971143	+	0.238499i	$0.0766554\pi$
$258$	2.16896	0.135033
$259$	3.96408	0.246316
$260$	−13.6332	−0.845493
$261$	−9.12497	−0.564821
$262$	3.95885	0.244578
$263$	24.9547	1.53877	0.769386	−	0.638784i	$-0.220562\pi$
$263$	24.9547	1.53877	0.769386	+	0.638784i	$0.220562\pi$
$264$	−2.18953	−0.134757
$265$	−1.11796	−0.0686755
$266$	−3.44364	−0.211143
$267$	−1.97942	−0.121139
$268$	19.2213	1.17413
$269$	5.99477	0.365508	0.182754	−	0.983159i	$-0.441499\pi$
$269$	5.99477	0.365508	0.182754	+	0.983159i	$0.441499\pi$
$270$	0.427229	0.0260003
$271$	−1.97241	−0.119816	−0.0599078	−	0.998204i	$-0.519081\pi$
$271$	−1.97241	−0.119816	−0.0599078	+	0.998204i	$0.519081\pi$
$272$	−22.7787	−1.38116
$273$	−11.5040	−0.696256
$274$	−0.168958	−0.0102071
$275$	−4.75814	−0.286926
$276$	6.66492	0.401181
$277$	−2.53579	−0.152361	−0.0761804	−	0.997094i	$-0.524272\pi$
$277$	−2.53579	−0.152361	−0.0761804	+	0.997094i	$0.524272\pi$
$278$	4.77348	0.286295
$279$	−0.616763	−0.0369246
$280$	4.61676	0.275904
$281$	−18.8545	−1.12476	−0.562381	−	0.826878i	$-0.690115\pi$
$281$	−18.8545	−1.12476	−0.562381	+	0.826878i	$0.690115\pi$
$282$	−1.64852	−0.0981678
$283$	−5.78989	−0.344173	−0.172087	−	0.985082i	$-0.555051\pi$
$283$	−5.78989	−0.344173	−0.172087	+	0.985082i	$0.555051\pi$
$284$	−13.2130	−0.784044
$285$	−8.29809	−0.491537
$286$	−2.33091	−0.137829
$287$	11.2817	0.665937
$288$	−2.91903	−0.172005
$289$	22.6660	1.33329
$290$	3.89845	0.228925
$291$	−3.33508	−0.195506
$292$	−26.1361	−1.52950
$293$	7.73472	0.451867	0.225934	−	0.974143i	$-0.427457\pi$
$293$	7.73472	0.451867	0.225934	+	0.974143i	$0.427457\pi$
$294$	0.137204	0.00800191
$295$	1.68133	0.0978909
$296$	1.44364	0.0839096
$297$	−2.18953	−0.127050
$298$	2.04505	0.118467
$299$	14.4272	0.834348
$300$	−4.20594	−0.242830
$301$	23.4384	1.35097
$302$	4.53579	0.261005
$303$	3.49180	0.200598
$304$	17.8503	1.02378
$305$	−18.4876	−1.05860
$306$	1.60036	0.0914863
$307$	16.4999	0.941697	0.470849	−	0.882214i	$-0.343948\pi$
$307$	16.4999	0.941697	0.470849	+	0.882214i	$0.343948\pi$
$308$	−11.6363	−0.663039
$309$	−8.21712	−0.467456
$310$	0.263499	0.0149657
$311$	24.9354	1.41396	0.706979	−	0.707234i	$-0.250057\pi$
$311$	24.9354	1.41396	0.706979	+	0.707234i	$0.250057\pi$
$312$	−4.18953	−0.237186
$313$	−24.1690	−1.36611	−0.683055	−	0.730367i	$-0.739349\pi$
$313$	−24.1690	−1.36611	−0.683055	+	0.730367i	$0.739349\pi$
$314$	−2.65269	−0.149700
$315$	4.61676	0.260125
$316$	2.27051	0.127726
$317$	−18.1812	−1.02116	−0.510579	−	0.859831i	$-0.670569\pi$
$317$	−18.1812	−1.02116	−0.510579	+	0.859831i	$0.670569\pi$
$318$	−0.168958	−0.00947471
$319$	−19.9794	−1.11863
$320$	−10.9149	−0.610159
$321$	−17.7899	−0.992935
$322$	−2.40275	−0.133900
$323$	−31.0838	−1.72955
$324$	−1.93543	−0.107524
$325$	−9.10439	−0.505021
$326$	0.956008	0.0529484
$327$	18.2981	1.01189
$328$	4.10856	0.226857
$329$	−17.8144	−0.982138
$330$	0.935432	0.0514939
$331$	−14.6290	−0.804083	−0.402041	−	0.915622i	$-0.631699\pi$
$331$	−14.6290	−0.804083	−0.402041	+	0.915622i	$0.631699\pi$
$332$	29.6936	1.62965
$333$	1.44364	0.0791107
$334$	−4.60142	−0.251778
$335$	−16.6977	−0.912295
$336$	−9.93126	−0.541795
$337$	−19.1526	−1.04331	−0.521653	−	0.853158i	$-0.674684\pi$
$337$	−19.1526	−1.04331	−0.521653	+	0.853158i	$0.674684\pi$
$338$	−1.15672	−0.0629173
$339$	7.71414	0.418975
$340$	20.4946	1.11148
$341$	−1.35042	−0.0731295
$342$	−1.25410	−0.0678141
$343$	−17.7386	−0.957795
$344$	8.53579	0.460219
$345$	−5.78989	−0.311717
$346$	2.45614	0.132043
$347$	29.4301	1.57989	0.789944	−	0.613178i	$-0.210109\pi$
$347$	29.4301	1.57989	0.789944	+	0.613178i	$0.210109\pi$
$348$	−17.6608	−0.946716
$349$	28.9065	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$349$	28.9065	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$350$	1.51627	0.0810482
$351$	−4.18953	−0.223621
$352$	−6.39131	−0.340658
$353$	0.0481609	0.00256335	0.00128167	−	0.999999i	$-0.499592\pi$
$353$	0.0481609	0.00256335	0.00128167	+	0.999999i	$0.499592\pi$
$354$	0.254102	0.0135054
$355$	11.4782	0.609201
$356$	−3.83104	−0.203045
$357$	17.2939	0.915292
$358$	2.27574	0.120276
$359$	11.7417	0.619705	0.309852	−	0.950785i	$-0.399720\pi$
$359$	11.7417	0.619705	0.309852	+	0.950785i	$0.399720\pi$
$360$	1.68133	0.0886139
$361$	5.35849	0.282026
$362$	0.123907	0.00651241
$363$	6.20594	0.325727
$364$	−22.2653	−1.16702
$365$	22.7047	1.18842
$366$	−2.79406	−0.146048
$367$	−7.04399	−0.367693	−0.183847	−	0.982955i	$-0.558855\pi$
$367$	−7.04399	−0.367693	−0.183847	+	0.982955i	$0.558855\pi$
$368$	12.4548	0.649252
$369$	4.10856	0.213883
$370$	−0.616763	−0.0320640
$371$	−1.82581	−0.0947915
$372$	−1.19370	−0.0618906
$373$	31.5316	1.63265	0.816323	−	0.577596i	$-0.196009\pi$
$373$	31.5316	1.63265	0.816323	+	0.577596i	$0.196009\pi$
$374$	3.50403	0.181189
$375$	12.0604	0.622796
$376$	−6.48763	−0.334574
$377$	−38.2294	−1.96891
$378$	0.697737	0.0358877
$379$	−10.5030	−0.539502	−0.269751	−	0.962930i	$-0.586941\pi$
$379$	−10.5030	−0.539502	−0.269751	+	0.962930i	$0.586941\pi$
$380$	−16.0604	−0.823881
$381$	−6.18953	−0.317099
$382$	−1.29002	−0.0660034
$383$	23.3023	1.19069	0.595345	−	0.803470i	$-0.297015\pi$
$383$	23.3023	1.19069	0.595345	+	0.803470i	$0.297015\pi$
$384$	−7.48763	−0.382101
$385$	10.1086	0.515180
$386$	−2.37907	−0.121091
$387$	8.53579	0.433899
$388$	−6.45481	−0.327694
$389$	−33.4835	−1.69768	−0.848839	−	0.528651i	$-0.822698\pi$
$389$	−33.4835	−1.69768	−0.848839	+	0.528651i	$0.822698\pi$
$390$	1.78989	0.0906346
$391$	−21.6883	−1.09683
$392$	0.539958	0.0272720
$393$	15.5798	0.785896
$394$	4.12914	0.208023
$395$	−1.97241	−0.0992430
$396$	−4.23769	−0.212952
$397$	−10.6977	−0.536904	−0.268452	−	0.963293i	$-0.586512\pi$
$397$	−10.6977	−0.536904	−0.268452	+	0.963293i	$0.586512\pi$
$398$	−1.68967	−0.0846954
$399$	−13.5522	−0.678458
$400$	−7.85969	−0.392984
$401$	−23.3145	−1.16427	−0.582135	−	0.813092i	$-0.697783\pi$
$401$	−23.3145	−1.16427	−0.582135	+	0.813092i	$0.697783\pi$
$402$	−2.52355	−0.125863
$403$	−2.58395	−0.128716
$404$	6.75814	0.336230
$405$	1.68133	0.0835460
$406$	6.36683	0.315980
$407$	3.16089	0.156679
$408$	6.29809	0.311802
$409$	−8.65970	−0.428194	−0.214097	−	0.976812i	$-0.568681\pi$
$409$	−8.65970	−0.428194	−0.214097	+	0.976812i	$0.568681\pi$
$410$	−1.75530	−0.0866879
$411$	−0.664924	−0.0327983
$412$	−15.9037	−0.783518
$413$	2.74590	0.135117
$414$	−0.875034	−0.0430056
$415$	−25.7951	−1.26623
$416$	−12.2294	−0.599594
$417$	18.7857	0.919941
$418$	−2.74590	−0.134306
$419$	4.63734	0.226549	0.113274	−	0.993564i	$-0.463866\pi$
$419$	4.63734	0.226549	0.113274	+	0.993564i	$0.463866\pi$
$420$	8.93543	0.436004
$421$	1.68133	0.0819430	0.0409715	−	0.999160i	$-0.486955\pi$
$421$	1.68133	0.0819430	0.0409715	+	0.999160i	$0.486955\pi$
$422$	−3.34209	−0.162690
$423$	−6.48763	−0.315439
$424$	−0.664924	−0.0322916
$425$	13.6866	0.663896
$426$	1.73472	0.0840475
$427$	−30.1934	−1.46116
$428$	−34.4311	−1.66429
$429$	−9.17313	−0.442883
$430$	−3.64674	−0.175861
$431$	11.5040	0.554130	0.277065	−	0.960851i	$-0.410638\pi$
$431$	11.5040	0.554130	0.277065	+	0.960851i	$0.410638\pi$
$432$	−3.61676	−0.174012
$433$	24.5069	1.17773	0.588863	−	0.808233i	$-0.299576\pi$
$433$	24.5069	1.17773	0.588863	+	0.808233i	$0.299576\pi$
$434$	0.430339	0.0206569
$435$	15.3421	0.735597
$436$	35.4147	1.69606
$437$	16.9958	0.813021
$438$	3.43140	0.163959
$439$	−5.63840	−0.269106	−0.134553	−	0.990906i	$-0.542960\pi$
$439$	−5.63840	−0.269106	−0.134553	+	0.990906i	$0.542960\pi$
$440$	3.68133	0.175501
$441$	0.539958	0.0257123
$442$	6.70475	0.318912
$443$	−38.0070	−1.80577	−0.902884	−	0.429885i	$-0.858554\pi$
$443$	−38.0070	−1.80577	−0.902884	+	0.429885i	$0.858554\pi$
$444$	2.79406	0.132600
$445$	3.32807	0.157765
$446$	2.46705	0.116818
$447$	8.04816	0.380665
$448$	−17.8258	−0.842191
$449$	−11.0357	−0.520805	−0.260402	−	0.965500i	$-0.583855\pi$
$449$	−11.0357	−0.520805	−0.260402	+	0.965500i	$0.583855\pi$
$450$	0.552195	0.0260307
$451$	8.99583	0.423597
$452$	14.9302	0.702258
$453$	17.8503	0.838680
$454$	−1.56754	−0.0735684
$455$	19.3421	0.906771
$456$	−4.93543	−0.231123
$457$	−32.3913	−1.51520	−0.757601	−	0.652718i	$-0.773629\pi$
$457$	−32.3913	−1.51520	−0.757601	+	0.652718i	$0.773629\pi$
$458$	−6.56026	−0.306541
$459$	6.29809	0.293970
$460$	−11.2059	−0.522480
$461$	34.9599	1.62825	0.814123	−	0.580693i	$-0.197218\pi$
$461$	34.9599	1.62825	0.814123	+	0.580693i	$0.197218\pi$
$462$	1.52772	0.0710760
$463$	−22.8063	−1.05990	−0.529949	−	0.848029i	$-0.677789\pi$
$463$	−22.8063	−1.05990	−0.529949	+	0.848029i	$0.677789\pi$
$464$	−33.0028	−1.53212
$465$	1.03698	0.0480889
$466$	−3.11273	−0.144194
$467$	−36.2416	−1.67706	−0.838530	−	0.544855i	$-0.816585\pi$
$467$	−36.2416	−1.67706	−0.838530	+	0.544855i	$0.816585\pi$
$468$	−8.10856	−0.374818
$469$	−27.2702	−1.25922
$470$	2.77170	0.127849
$471$	−10.4395	−0.481025
$472$	1.00000	0.0460287
$473$	18.6894	0.859340
$474$	−0.298094	−0.0136919
$475$	−10.7253	−0.492112
$476$	33.4712	1.53415
$477$	−0.664924	−0.0304448
$478$	5.83104	0.266706
$479$	17.0562	0.779319	0.389660	−	0.920959i	$-0.372593\pi$
$479$	17.0562	0.779319	0.389660	+	0.920959i	$0.372593\pi$
$480$	4.90785	0.224012
$481$	6.04816	0.275772
$482$	−1.18669	−0.0540524
$483$	−9.45587	−0.430257
$484$	12.0112	0.545963
$485$	5.60737	0.254617
$486$	0.254102	0.0115263
$487$	15.3473	0.695453	0.347727	−	0.937596i	$-0.386954\pi$
$487$	15.3473	0.695453	0.347727	+	0.937596i	$0.386954\pi$
$488$	−10.9958	−0.497758
$489$	3.76231	0.170137
$490$	−0.230685	−0.0104213
$491$	18.0122	0.812881	0.406440	−	0.913677i	$-0.366770\pi$
$491$	18.0122	0.812881	0.406440	+	0.913677i	$0.366770\pi$
$492$	7.95184	0.358497
$493$	57.4699	2.58831
$494$	−5.25410	−0.236393
$495$	3.68133	0.165464
$496$	−2.23069	−0.100161
$497$	18.7459	0.840868
$498$	−3.89845	−0.174694
$499$	−21.6074	−0.967279	−0.483639	−	0.875267i	$-0.660685\pi$
$499$	−21.6074	−0.967279	−0.483639	+	0.875267i	$0.660685\pi$
$500$	23.3421	1.04389
$501$	−18.1086	−0.809031
$502$	−5.69357	−0.254116
$503$	7.76231	0.346104	0.173052	−	0.984913i	$-0.444637\pi$
$503$	7.76231	0.346104	0.173052	+	0.984913i	$0.444637\pi$
$504$	2.74590	0.122312
$505$	−5.87086	−0.261250
$506$	−1.91592	−0.0851729
$507$	−4.55220	−0.202170
$508$	−11.9794	−0.531501
$509$	40.7704	1.80712	0.903558	−	0.428467i	$-0.140946\pi$
$509$	40.7704	1.80712	0.903558	+	0.428467i	$0.140946\pi$
$510$	−2.69073	−0.119147
$511$	37.0807	1.64035
$512$	−17.7909	−0.786256
$513$	−4.93543	−0.217905
$514$	−7.91202	−0.348984
$515$	13.8157	0.608792
$516$	16.5204	0.727272
$517$	−14.2049	−0.624730
$518$	−1.00728	−0.0442573
$519$	9.66598	0.424290
$520$	7.04399	0.308899
$521$	41.0427	1.79811	0.899056	−	0.437834i	$-0.144254\pi$
$521$	41.0427	1.79811	0.899056	+	0.437834i	$0.144254\pi$
$522$	2.31867	0.101485
$523$	−22.5686	−0.986856	−0.493428	−	0.869787i	$-0.664256\pi$
$523$	−22.5686	−0.986856	−0.493428	+	0.869787i	$0.664256\pi$
$524$	30.1536	1.31727
$525$	5.96719	0.260429
$526$	−6.34103	−0.276482
$527$	3.88443	0.169208
$528$	−7.91903	−0.344631
$529$	−11.1414	−0.484408
$530$	0.284075	0.0123394
$531$	1.00000	0.0433963
$532$	−26.2294	−1.13719
$533$	17.2130	0.745576
$534$	0.502975	0.0217659
$535$	29.9107	1.29315
$536$	−9.93126	−0.428965
$537$	8.95601	0.386480
$538$	−1.52328	−0.0656733
$539$	1.18226	0.0509233
$540$	3.25410	0.140034
$541$	35.0357	1.50630	0.753150	−	0.657849i	$-0.228533\pi$
$541$	35.0357	1.50630	0.753150	+	0.657849i	$0.228533\pi$
$542$	0.501194	0.0215281
$543$	0.487628	0.0209261
$544$	18.3843	0.788220
$545$	−30.7651	−1.31783
$546$	2.92319	0.125101
$547$	3.70191	0.158282	0.0791410	−	0.996863i	$-0.474782\pi$
$547$	3.70191	0.158282	0.0791410	+	0.996863i	$0.474782\pi$
$548$	−1.28692	−0.0549743
$549$	−10.9958	−0.469291
$550$	1.20905	0.0515541
$551$	−45.0357	−1.91858
$552$	−3.44364	−0.146571
$553$	−3.22129	−0.136983
$554$	0.644348	0.0273757
$555$	−2.42723	−0.103030
$556$	36.3585	1.54194
$557$	−16.7253	−0.708675	−0.354337	−	0.935118i	$-0.615294\pi$
$557$	−16.7253	−0.708675	−0.354337	+	0.935118i	$0.615294\pi$
$558$	0.156721	0.00663451
$559$	35.7610	1.51253
$560$	16.6977	0.705608
$561$	13.7899	0.582210
$562$	4.79095	0.202094
$563$	24.9149	1.05004	0.525018	−	0.851091i	$-0.324059\pi$
$563$	24.9149	1.05004	0.525018	+	0.851091i	$0.324059\pi$
$564$	−12.5564	−0.528718
$565$	−12.9700	−0.545653
$566$	1.47122	0.0618400
$567$	2.74590	0.115317
$568$	6.82687	0.286449
$569$	42.4548	1.77980	0.889899	−	0.456157i	$-0.150775\pi$
$569$	42.4548	1.77980	0.889899	+	0.456157i	$0.150775\pi$
$570$	2.10856	0.0883178
$571$	−32.6977	−1.36836	−0.684179	−	0.729314i	$-0.739839\pi$
$571$	−32.6977	−1.36836	−0.684179	+	0.729314i	$0.739839\pi$
$572$	−17.7540	−0.742331
$573$	−5.07681	−0.212087
$574$	−2.86670	−0.119654
$575$	−7.48346	−0.312082
$576$	−6.49180	−0.270492
$577$	−13.2716	−0.552503	−0.276251	−	0.961085i	$-0.589092\pi$
$577$	−13.2716	−0.552503	−0.276251	+	0.961085i	$0.589092\pi$
$578$	−5.75946	−0.239562
$579$	−9.36266	−0.389099
$580$	29.6936	1.23296
$581$	−42.1278	−1.74776
$582$	0.847448	0.0351279
$583$	−1.45587	−0.0602961
$584$	13.5040	0.558801
$585$	7.04399	0.291233
$586$	−1.96541	−0.0811901
$587$	−14.4067	−0.594626	−0.297313	−	0.954780i	$-0.596091\pi$
$587$	−14.4067	−0.594626	−0.297313	+	0.954780i	$0.596091\pi$
$588$	1.04505	0.0430972
$589$	−3.04399	−0.125426
$590$	−0.427229	−0.0175887
$591$	16.2499	0.668433
$592$	5.22129	0.214594
$593$	−1.85863	−0.0763247	−0.0381623	−	0.999272i	$-0.512150\pi$
$593$	−1.85863	−0.0763247	−0.0381623	+	0.999272i	$0.512150\pi$
$594$	0.556364	0.0228279
$595$	−29.0768	−1.19203
$596$	15.5767	0.638045
$597$	−6.64958	−0.272149
$598$	−3.66598	−0.149913
$599$	−20.2171	−0.826049	−0.413025	−	0.910720i	$-0.635528\pi$
$599$	−20.2171	−0.826049	−0.413025	+	0.910720i	$0.635528\pi$
$600$	2.17313	0.0887175
$601$	−29.6608	−1.20989	−0.604944	−	0.796268i	$-0.706804\pi$
$601$	−29.6608	−1.20989	−0.604944	+	0.796268i	$0.706804\pi$
$602$	−5.95574	−0.242738
$603$	−9.93126	−0.404432
$604$	34.5480	1.40574
$605$	−10.4342	−0.424212
$606$	−0.887271	−0.0360429
$607$	8.06040	0.327161	0.163581	−	0.986530i	$-0.447696\pi$
$607$	8.06040	0.327161	0.163581	+	0.986530i	$0.447696\pi$
$608$	−14.4067	−0.584267
$609$	25.0562	1.01533
$610$	4.69774	0.190206
$611$	−27.1801	−1.09959
$612$	12.1895	0.492733
$613$	−35.7886	−1.44549	−0.722743	−	0.691117i	$-0.757119\pi$
$613$	−35.7886	−1.44549	−0.722743	+	0.691117i	$0.757119\pi$
$614$	−4.19264	−0.169201
$615$	−6.90785	−0.278551
$616$	6.01224	0.242240
$617$	21.7159	0.874250	0.437125	−	0.899401i	$-0.355997\pi$
$617$	21.7159	0.874250	0.437125	+	0.899401i	$0.355997\pi$
$618$	2.08798	0.0839910
$619$	−15.3679	−0.617688	−0.308844	−	0.951113i	$-0.599942\pi$
$619$	−15.3679	−0.617688	−0.308844	+	0.951113i	$0.599942\pi$
$620$	2.00701	0.0806034
$621$	−3.44364	−0.138188
$622$	−6.33614	−0.254056
$623$	5.43530	0.217761
$624$	−15.1526	−0.606588
$625$	−9.41188	−0.376475
$626$	6.14137	0.245459
$627$	−10.8063	−0.431562
$628$	−20.2049	−0.806263
$629$	−9.09215	−0.362528
$630$	−1.17313	−0.0467385
$631$	−24.5358	−0.976754	−0.488377	−	0.872633i	$-0.662411\pi$
$631$	−24.5358	−0.976754	−0.488377	+	0.872633i	$0.662411\pi$
$632$	−1.17313	−0.0466645
$633$	−13.1526	−0.522767
$634$	4.61987	0.183479
$635$	10.4067	0.412975
$636$	−1.28692	−0.0510295
$637$	2.26217	0.0896305
$638$	5.07681	0.200993
$639$	6.82687	0.270067
$640$	12.5892	0.497631
$641$	−16.0521	−0.634018	−0.317009	−	0.948422i	$-0.602679\pi$
$641$	−16.0521	−0.634018	−0.317009	+	0.948422i	$0.602679\pi$
$642$	4.52044	0.178408
$643$	−9.47645	−0.373715	−0.186857	−	0.982387i	$-0.559830\pi$
$643$	−9.47645	−0.373715	−0.186857	+	0.982387i	$0.559830\pi$
$644$	−18.3012	−0.721168
$645$	−14.3515	−0.565089
$646$	7.89845	0.310760
$647$	41.1924	1.61944	0.809720	−	0.586817i	$-0.199619\pi$
$647$	41.1924	1.61944	0.809720	+	0.586817i	$0.199619\pi$
$648$	1.00000	0.0392837
$649$	2.18953	0.0859467
$650$	2.31344	0.0907406
$651$	1.69357	0.0663762
$652$	7.28169	0.285173
$653$	14.0757	0.550827	0.275413	−	0.961326i	$-0.411185\pi$
$653$	14.0757	0.550827	0.275413	+	0.961326i	$0.411185\pi$
$654$	−4.64958	−0.181813
$655$	−26.1948	−1.02351
$656$	14.8597	0.580173
$657$	13.5040	0.526843
$658$	4.52666	0.176468
$659$	−20.6220	−0.803319	−0.401659	−	0.915789i	$-0.631566\pi$
$659$	−20.6220	−0.803319	−0.401659	+	0.915789i	$0.631566\pi$
$660$	7.12497	0.277339
$661$	−30.1414	−1.17236	−0.586182	−	0.810180i	$-0.699370\pi$
$661$	−30.1414	−1.17236	−0.586182	+	0.810180i	$0.699370\pi$
$662$	3.71725	0.144475
$663$	26.3861	1.02475
$664$	−15.3421	−0.595388
$665$	22.7857	0.883592
$666$	−0.366830	−0.0142144
$667$	−31.4231	−1.21671
$668$	−35.0479	−1.35604
$669$	9.70892	0.375368
$670$	4.24292	0.163918
$671$	−24.0757	−0.929434
$672$	8.01535	0.309199
$673$	3.27468	0.126230	0.0631148	−	0.998006i	$-0.479897\pi$
$673$	3.27468	0.126230	0.0631148	+	0.998006i	$0.479897\pi$
$674$	4.86670	0.187458
$675$	2.17313	0.0836437
$676$	−8.81047	−0.338864
$677$	19.6126	0.753773	0.376887	−	0.926259i	$-0.376995\pi$
$677$	19.6126	0.753773	0.376887	+	0.926259i	$0.376995\pi$
$678$	−1.96018	−0.0752801
$679$	9.15778	0.351443
$680$	−10.5892	−0.406076
$681$	−6.16896	−0.236395
$682$	0.343145	0.0131397
$683$	12.1812	0.466101	0.233050	−	0.972465i	$-0.425129\pi$
$683$	12.1812	0.466101	0.233050	+	0.972465i	$0.425129\pi$
$684$	−9.55220	−0.365237
$685$	1.11796	0.0427149
$686$	4.50741	0.172094
$687$	−25.8175	−0.984998
$688$	30.8719	1.17698
$689$	−2.78572	−0.106128
$690$	1.47122	0.0560084
$691$	−13.8052	−0.525176	−0.262588	−	0.964908i	$-0.584576\pi$
$691$	−13.8052	−0.525176	−0.262588	+	0.964908i	$0.584576\pi$
$692$	18.7079	0.711166
$693$	6.01224	0.228386
$694$	−7.47823	−0.283870
$695$	−31.5850	−1.19809
$696$	9.12497	0.345881
$697$	−25.8761	−0.980127
$698$	−7.34520	−0.278020
$699$	−12.2499	−0.463335
$700$	11.5491	0.436514
$701$	−9.06980	−0.342561	−0.171281	−	0.985222i	$-0.554791\pi$
$701$	−9.06980	−0.342561	−0.171281	+	0.985222i	$0.554791\pi$
$702$	1.06457	0.0401795
$703$	7.12497	0.268723
$704$	−14.2140	−0.535711
$705$	10.9078	0.410813
$706$	−0.0122378	−0.000460575 0
$707$	−9.58812	−0.360598
$708$	1.93543	0.0727380
$709$	26.2171	0.984605	0.492302	−	0.870424i	$-0.336155\pi$
$709$	26.2171	0.984605	0.492302	+	0.870424i	$0.336155\pi$
$710$	−2.91664	−0.109459
$711$	−1.17313	−0.0439957
$712$	1.97942	0.0741821
$713$	−2.12391	−0.0795409
$714$	−4.39442	−0.164457
$715$	15.4231	0.576790
$716$	17.3337	0.647793
$717$	22.9477	0.856996
$718$	−2.98359	−0.111347
$719$	−37.5972	−1.40214	−0.701070	−	0.713092i	$-0.747294\pi$
$719$	−37.5972	−1.40214	−0.701070	+	0.713092i	$0.747294\pi$
$720$	6.08097	0.226625
$721$	22.5634	0.840304
$722$	−1.36160	−0.0506736
$723$	−4.67015	−0.173685
$724$	0.943770	0.0350749
$725$	19.8297	0.736457
$726$	−1.57694	−0.0585257
$727$	42.3463	1.57054	0.785268	−	0.619156i	$-0.212525\pi$
$727$	42.3463	1.57054	0.785268	+	0.619156i	$0.212525\pi$
$728$	11.5040	0.426368
$729$	1.00000	0.0370370
$730$	−5.76931	−0.213532
$731$	−53.7592	−1.98836
$732$	−21.2817	−0.786594
$733$	−5.96408	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$733$	−5.96408	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$734$	1.78989	0.0660661
$735$	−0.907847	−0.0334864
$736$	−10.0521	−0.370524
$737$	−21.7448	−0.800981
$738$	−1.04399	−0.0384299
$739$	39.9435	1.46935	0.734673	−	0.678422i	$-0.237336\pi$
$739$	39.9435	1.46935	0.734673	+	0.678422i	$0.237336\pi$
$740$	−4.69774	−0.172692
$741$	−20.6772	−0.759595
$742$	0.463942	0.0170319
$743$	22.7407	0.834274	0.417137	−	0.908844i	$-0.363033\pi$
$743$	22.7407	0.834274	0.417137	+	0.908844i	$0.363033\pi$
$744$	0.616763	0.0226116
$745$	−13.5316	−0.495760
$746$	−8.01224	−0.293349
$747$	−15.3421	−0.561338
$748$	26.6894	0.975861
$749$	48.8492	1.78491
$750$	−3.06457	−0.111902
$751$	17.8052	0.649722	0.324861	−	0.945762i	$-0.394682\pi$
$751$	17.8052	0.649722	0.324861	+	0.945762i	$0.394682\pi$
$752$	−23.4642	−0.855652
$753$	−22.4067	−0.816544
$754$	9.71414	0.353768
$755$	−30.0122	−1.09226
$756$	5.31450	0.193287
$757$	27.4353	0.997153	0.498576	−	0.866846i	$-0.333856\pi$
$757$	27.4353	0.997153	0.498576	+	0.866846i	$0.333856\pi$
$758$	2.66882	0.0969360
$759$	−7.53996	−0.273683
$760$	8.29809	0.301003
$761$	10.1812	0.369068	0.184534	−	0.982826i	$-0.440922\pi$
$761$	10.1812	0.369068	0.184534	+	0.982826i	$0.440922\pi$
$762$	1.57277	0.0569755
$763$	−50.2447	−1.81898
$764$	−9.82581	−0.355485
$765$	−10.5892	−0.382853
$766$	−5.92114	−0.213940
$767$	4.18953	0.151275
$768$	−11.0810	−0.399850
$769$	38.0398	1.37175	0.685876	−	0.727719i	$-0.259419\pi$
$769$	38.0398	1.37175	0.685876	+	0.727719i	$0.259419\pi$
$770$	−2.56860	−0.0925660
$771$	−31.1372	−1.12138
$772$	−18.1208	−0.652182
$773$	−44.1676	−1.58860	−0.794300	−	0.607526i	$-0.792162\pi$
$773$	−44.1676	−1.58860	−0.794300	+	0.607526i	$0.792162\pi$
$774$	−2.16896	−0.0779616
$775$	1.34030	0.0481452
$776$	3.33508	0.119722
$777$	−3.96408	−0.142210
$778$	8.50820	0.305034
$779$	20.2775	0.726517
$780$	13.6332	0.488146
$781$	14.9477	0.534870
$782$	5.51104	0.197075
$783$	9.12497	0.326100
$784$	1.95290	0.0697464
$785$	17.5522	0.626465
$786$	−3.95885	−0.141207
$787$	−16.3379	−0.582384	−0.291192	−	0.956665i	$-0.594052\pi$
$787$	−16.3379	−0.582384	−0.291192	+	0.956665i	$0.594052\pi$
$788$	31.4506	1.12038
$789$	−24.9547	−0.888410
$790$	0.501194	0.0178317
$791$	−21.1823	−0.753154
$792$	2.18953	0.0778017
$793$	−46.0674	−1.63590
$794$	2.71831	0.0964693
$795$	1.11796	0.0396498
$796$	−12.8698	−0.456158
$797$	−32.9424	−1.16688	−0.583441	−	0.812156i	$-0.698294\pi$
$797$	−32.9424	−1.16688	−0.583441	+	0.812156i	$0.698294\pi$
$798$	3.44364	0.121903
$799$	40.8597	1.44551
$800$	6.34341	0.224274
$801$	1.97942	0.0699395
$802$	5.92425	0.209193
$803$	29.5675	1.04342
$804$	−19.2213	−0.677883
$805$	15.8984	0.560347
$806$	0.656586	0.0231273
$807$	−5.99477	−0.211026
$808$	−3.49180	−0.122841
$809$	−51.9659	−1.82702	−0.913511	−	0.406814i	$-0.866640\pi$
$809$	−51.9659	−1.82702	−0.913511	+	0.406814i	$0.866640\pi$
$810$	−0.427229	−0.0150113
$811$	26.9424	0.946077	0.473039	−	0.881042i	$-0.343157\pi$
$811$	26.9424	0.946077	0.473039	+	0.881042i	$0.343157\pi$
$812$	48.4946	1.70183
$813$	1.97241	0.0691756
$814$	−0.803187	−0.0281517
$815$	−6.32568	−0.221579
$816$	22.7787	0.797414
$817$	42.1278	1.47387
$818$	2.20044	0.0769367
$819$	11.5040	0.401983
$820$	−13.3697	−0.466889
$821$	−27.8914	−0.973418	−0.486709	−	0.873564i	$-0.661803\pi$
$821$	−27.8914	−0.973418	−0.486709	+	0.873564i	$0.661803\pi$
$822$	0.168958	0.00589310
$823$	18.3103	0.638258	0.319129	−	0.947711i	$-0.396610\pi$
$823$	18.3103	0.638258	0.319129	+	0.947711i	$0.396610\pi$
$824$	8.21712	0.286257
$825$	4.75814	0.165657
$826$	−0.697737	−0.0242774
$827$	−4.37384	−0.152093	−0.0760467	−	0.997104i	$-0.524230\pi$
$827$	−4.37384	−0.152093	−0.0760467	+	0.997104i	$0.524230\pi$
$828$	−6.66492	−0.231622
$829$	30.5204	1.06002	0.530009	−	0.847992i	$-0.322188\pi$
$829$	30.5204	1.06002	0.530009	+	0.847992i	$0.322188\pi$
$830$	6.55458	0.227513
$831$	2.53579	0.0879655
$832$	−27.1976	−0.942907
$833$	−3.40070	−0.117827
$834$	−4.77348	−0.165292
$835$	30.4465	1.05364
$836$	−20.9149	−0.723355
$837$	0.616763	0.0213184
$838$	−1.17836	−0.0407056
$839$	−0.859686	−0.0296797	−0.0148398	−	0.999890i	$-0.504724\pi$
$839$	−0.859686	−0.0296797	−0.0148398	+	0.999890i	$0.504724\pi$
$840$	−4.61676	−0.159293
$841$	54.2650	1.87121
$842$	−0.427229	−0.0147233
$843$	18.8545	0.649382
$844$	−25.4559	−0.876227
$845$	7.65375	0.263297
$846$	1.64852	0.0566772
$847$	−17.0409	−0.585532
$848$	−2.40487	−0.0825836
$849$	5.78989	0.198709
$850$	−3.47778	−0.119287
$851$	4.97136	0.170416
$852$	13.2130	0.452668
$853$	50.4741	1.72820	0.864099	−	0.503321i	$-0.167889\pi$
$853$	50.4741	1.72820	0.864099	+	0.503321i	$0.167889\pi$
$854$	7.67220	0.262537
$855$	8.29809	0.283789
$856$	17.7899	0.608046
$857$	38.4946	1.31495	0.657476	−	0.753476i	$-0.271624\pi$
$857$	38.4946	1.31495	0.657476	+	0.753476i	$0.271624\pi$
$858$	2.33091	0.0795759
$859$	−27.1044	−0.924790	−0.462395	−	0.886674i	$-0.653010\pi$
$859$	−27.1044	−0.924790	−0.462395	+	0.886674i	$0.653010\pi$
$860$	−27.7763	−0.947165
$861$	−11.2817	−0.384479
$862$	−2.92319	−0.0995644
$863$	10.6443	0.362338	0.181169	−	0.983452i	$-0.442012\pi$
$863$	10.6443	0.362338	0.181169	+	0.983452i	$0.442012\pi$
$864$	2.91903	0.0993073
$865$	−16.2517	−0.552575
$866$	−6.22724	−0.211610
$867$	−22.6660	−0.769777
$868$	3.27779	0.111255
$869$	−2.56860	−0.0871339
$870$	−3.89845	−0.132170
$871$	−41.6074	−1.40981
$872$	−18.2981	−0.619652
$873$	3.33508	0.112875
$874$	−4.31867	−0.146081
$875$	−33.1166	−1.11955
$876$	26.1361	0.883058
$877$	40.0328	1.35181	0.675906	−	0.736988i	$-0.263753\pi$
$877$	40.0328	1.35181	0.675906	+	0.736988i	$0.263753\pi$
$878$	1.43273	0.0483522
$879$	−7.73472	−0.260886
$880$	13.3145	0.448832
$881$	29.9588	1.00934	0.504670	−	0.863313i	$-0.331614\pi$
$881$	29.9588	1.00934	0.504670	+	0.863313i	$0.331614\pi$
$882$	−0.137204	−0.00461990
$883$	−12.3239	−0.414732	−0.207366	−	0.978263i	$-0.566489\pi$
$883$	−12.3239	−0.414732	−0.207366	+	0.978263i	$0.566489\pi$
$884$	51.0685	1.71762
$885$	−1.68133	−0.0565173
$886$	9.65765	0.324455
$887$	25.4559	0.854725	0.427362	−	0.904080i	$-0.359443\pi$
$887$	25.4559	0.854725	0.427362	+	0.904080i	$0.359443\pi$
$888$	−1.44364	−0.0484452
$889$	16.9958	0.570022
$890$	−0.845667	−0.0283468
$891$	2.18953	0.0733521
$892$	18.7909	0.629168
$893$	−32.0192	−1.07148
$894$	−2.04505	−0.0683967
$895$	−15.0580	−0.503334
$896$	20.5603	0.686870
$897$	−14.4272	−0.481711
$898$	2.80418	0.0935766
$899$	5.62794	0.187702
$900$	4.20594	0.140198
$901$	4.18775	0.139514
$902$	−2.28586	−0.0761107
$903$	−23.4384	−0.779981
$904$	−7.71414	−0.256569
$905$	−0.819863	−0.0272532
$906$	−4.53579	−0.150691
$907$	38.4863	1.27792	0.638958	−	0.769241i	$-0.279366\pi$
$907$	38.4863	1.27792	0.638958	+	0.769241i	$0.279366\pi$
$908$	−11.9396	−0.396230
$909$	−3.49180	−0.115816
$910$	−4.91486	−0.162926
$911$	21.2234	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$911$	21.2234	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$912$	−17.8503	−0.591082
$913$	−33.5920	−1.11173
$914$	8.23069	0.272247
$915$	18.4876	0.611182
$916$	−49.9680	−1.65099
$917$	−42.7805	−1.41274
$918$	−1.60036	−0.0528196
$919$	46.3051	1.52746	0.763732	−	0.645533i	$-0.223365\pi$
$919$	46.3051	1.52746	0.763732	+	0.645533i	$0.223365\pi$
$920$	5.78989	0.190887
$921$	−16.4999	−0.543689
$922$	−8.88337	−0.292558
$923$	28.6014	0.941427
$924$	11.6363	0.382805
$925$	−3.13720	−0.103151
$926$	5.79512	0.190439
$927$	8.21712	0.269886
$928$	26.6360	0.874370
$929$	31.9313	1.04763	0.523815	−	0.851832i	$-0.324508\pi$
$929$	31.9313	1.04763	0.523815	+	0.851832i	$0.324508\pi$
$930$	−0.263499	−0.00864047
$931$	2.66492	0.0873394
$932$	−23.7089	−0.776611
$933$	−24.9354	−0.816349
$934$	9.20905	0.301329
$935$	−23.1854	−0.758243
$936$	4.18953	0.136939
$937$	−33.6813	−1.10032	−0.550161	−	0.835059i	$-0.685433\pi$
$937$	−33.6813	−1.10032	−0.550161	+	0.835059i	$0.685433\pi$
$938$	6.92941	0.226253
$939$	24.1690	0.788724
$940$	21.1114	0.688578
$941$	−22.6977	−0.739925	−0.369963	−	0.929047i	$-0.620629\pi$
$941$	−22.6977	−0.739925	−0.369963	+	0.929047i	$0.620629\pi$
$942$	2.65269	0.0864292
$943$	14.1484	0.460735
$944$	3.61676	0.117716
$945$	−4.61676	−0.150183
$946$	−4.74901	−0.154404
$947$	22.5533	0.732882	0.366441	−	0.930441i	$-0.380576\pi$
$947$	22.5533	0.732882	0.366441	+	0.930441i	$0.380576\pi$
$948$	−2.27051	−0.0737427
$949$	56.5756	1.83652
$950$	2.72532	0.0884211
$951$	18.1812	0.589566
$952$	−17.2939	−0.560499
$953$	−10.3791	−0.336211	−0.168105	−	0.985769i	$-0.553765\pi$
$953$	−10.3791	−0.336211	−0.168105	+	0.985769i	$0.553765\pi$
$954$	0.168958	0.00547023
$955$	8.53579	0.276212
$956$	44.4137	1.43644
$957$	19.9794	0.645843
$958$	−4.33402	−0.140026
$959$	1.82581	0.0589586
$960$	10.9149	0.352275
$961$	−30.6196	−0.987729
$962$	−1.53685	−0.0495500
$963$	17.7899	0.573271
$964$	−9.03876	−0.291119
$965$	15.7417	0.506744
$966$	2.40275	0.0773073
$967$	32.1913	1.03520	0.517601	−	0.855622i	$-0.326825\pi$
$967$	32.1913	1.03520	0.517601	+	0.855622i	$0.326825\pi$
$968$	−6.20594	−0.199466
$969$	31.0838	0.998556
$970$	−1.42484	−0.0457489
$971$	−33.1114	−1.06260	−0.531298	−	0.847185i	$-0.678295\pi$
$971$	−33.1114	−1.06260	−0.531298	+	0.847185i	$0.678295\pi$
$972$	1.93543	0.0620790
$973$	−51.5837	−1.65370
$974$	−3.89978	−0.124957
$975$	9.10439	0.291574
$976$	−39.7693	−1.27298
$977$	12.2307	0.391294	0.195647	−	0.980674i	$-0.437319\pi$
$977$	12.2307	0.391294	0.195647	+	0.980674i	$0.437319\pi$
$978$	−0.956008	−0.0305698
$979$	4.33402	0.138516
$980$	−1.75708	−0.0561278
$981$	−18.2981	−0.584213
$982$	−4.57694	−0.146056
$983$	−7.26111	−0.231593	−0.115797	−	0.993273i	$-0.536942\pi$
$983$	−7.26111	−0.231593	−0.115797	+	0.993273i	$0.536942\pi$
$984$	−4.10856	−0.130976
$985$	−27.3215	−0.870536
$986$	−14.6032	−0.465061
$987$	17.8144	0.567038
$988$	−40.0192	−1.27318
$989$	29.3941	0.934679
$990$	−0.935432	−0.0297300
$991$	−20.1054	−0.638671	−0.319335	−	0.947642i	$-0.603460\pi$
$991$	−20.1054	−0.638671	−0.319335	+	0.947642i	$0.603460\pi$
$992$	1.80035	0.0571611
$993$	14.6290	0.464237
$994$	−4.76336	−0.151085
$995$	11.1801	0.354434
$996$	−29.6936	−0.940877
$997$	−42.3173	−1.34020	−0.670102	−	0.742269i	$-0.733750\pi$
$997$	−42.3173	−1.34020	−0.670102	+	0.742269i	$0.733750\pi$
$998$	5.49047	0.173798
$999$	−1.44364	−0.0456746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.a.d.1.2	✓	3
3.2	odd	2		531.2.a.d.1.2		3
4.3	odd	2		2832.2.a.t.1.3		3
5.4	even	2		4425.2.a.w.1.2		3
7.6	odd	2		8673.2.a.s.1.2		3
12.11	even	2		8496.2.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.d.1.2	✓	3	1.1	even	1	trivial
531.2.a.d.1.2		3	3.2	odd	2
2832.2.a.t.1.3		3	4.3	odd	2
4425.2.a.w.1.2		3	5.4	even	2
8496.2.a.bl.1.1		3	12.11	even	2
8673.2.a.s.1.2		3	7.6	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q-0.254102 q^{2} -1.00000 q^{3} -1.93543 q^{4} +1.68133 q^{5} +0.254102 q^{6} +2.74590 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.254102 q^{2} -1.00000 q^{3} -1.93543 q^{4} +1.68133 q^{5} +0.254102 q^{6} +2.74590 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.427229 q^{10} +2.18953 q^{11} +1.93543 q^{12} +4.18953 q^{13} -0.697737 q^{14} -1.68133 q^{15} +3.61676 q^{16} -6.29809 q^{17} -0.254102 q^{18} +4.93543 q^{19} -3.25410 q^{20} -2.74590 q^{21} -0.556364 q^{22} +3.44364 q^{23} -1.00000 q^{24} -2.17313 q^{25} -1.06457 q^{26} -1.00000 q^{27} -5.31450 q^{28} -9.12497 q^{29} +0.427229 q^{30} -0.616763 q^{31} -2.91903 q^{32} -2.18953 q^{33} +1.60036 q^{34} +4.61676 q^{35} -1.93543 q^{36} +1.44364 q^{37} -1.25410 q^{38} -4.18953 q^{39} +1.68133 q^{40} +4.10856 q^{41} +0.697737 q^{42} +8.53579 q^{43} -4.23769 q^{44} +1.68133 q^{45} -0.875034 q^{46} -6.48763 q^{47} -3.61676 q^{48} +0.539958 q^{49} +0.552195 q^{50} +6.29809 q^{51} -8.10856 q^{52} -0.664924 q^{53} +0.254102 q^{54} +3.68133 q^{55} +2.74590 q^{56} -4.93543 q^{57} +2.31867 q^{58} +1.00000 q^{59} +3.25410 q^{60} -10.9958 q^{61} +0.156721 q^{62} +2.74590 q^{63} -6.49180 q^{64} +7.04399 q^{65} +0.556364 q^{66} -9.93126 q^{67} +12.1895 q^{68} -3.44364 q^{69} -1.17313 q^{70} +6.82687 q^{71} +1.00000 q^{72} +13.5040 q^{73} -0.366830 q^{74} +2.17313 q^{75} -9.55220 q^{76} +6.01224 q^{77} +1.06457 q^{78} -1.17313 q^{79} +6.08097 q^{80} +1.00000 q^{81} -1.04399 q^{82} -15.3421 q^{83} +5.31450 q^{84} -10.5892 q^{85} -2.16896 q^{86} +9.12497 q^{87} +2.18953 q^{88} +1.97942 q^{89} -0.427229 q^{90} +11.5040 q^{91} -6.66492 q^{92} +0.616763 q^{93} +1.64852 q^{94} +8.29809 q^{95} +2.91903 q^{96} +3.33508 q^{97} -0.137204 q^{98} +2.18953 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{4} - 2 q^{5} + 9 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^4 - 2 * q^5 + 9 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{4} - 2 q^{5} + 9 q^{7} + 3 q^{8} + 3 q^{9} + 5 q^{10} - 2 q^{11} - 2 q^{12} + 4 q^{13} + 8 q^{14} + 2 q^{15} - 4 q^{16} + 3 q^{17} + 7 q^{19} - 9 q^{20} - 9 q^{21} - 11 q^{22} + q^{23} - 3 q^{24} - q^{25} - 11 q^{26} - 3 q^{27} + 9 q^{28} - 11 q^{29} - 5 q^{30} + 13 q^{31} - 4 q^{32} + 2 q^{33} - 7 q^{34} - q^{35} + 2 q^{36} - 5 q^{37} - 3 q^{38} - 4 q^{39} - 2 q^{40} - q^{41} - 8 q^{42} + 6 q^{43} - 15 q^{44} - 2 q^{45} - 19 q^{46} + 11 q^{47} + 4 q^{48} + 14 q^{49} - 21 q^{50} - 3 q^{51} - 11 q^{52} + 2 q^{53} + 4 q^{55} + 9 q^{56} - 7 q^{57} + 14 q^{58} + 3 q^{59} + 9 q^{60} - q^{61} - 2 q^{62} + 9 q^{63} - 21 q^{64} + 11 q^{66} + 10 q^{67} + 28 q^{68} - q^{69} + 2 q^{70} + 26 q^{71} + 3 q^{72} + 7 q^{73} - 19 q^{74} + q^{75} - 6 q^{76} - 17 q^{77} + 11 q^{78} + 2 q^{79} + 23 q^{80} + 3 q^{81} + 18 q^{82} - 3 q^{83} - 9 q^{84} - 35 q^{85} + 31 q^{86} + 11 q^{87} - 2 q^{88} - 23 q^{89} + 5 q^{90} + q^{91} - 16 q^{92} - 13 q^{93} + 4 q^{94} + 3 q^{95} + 4 q^{96} + 14 q^{97} + 51 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^4 - 2 * q^5 + 9 * q^7 + 3 * q^8 + 3 * q^9 + 5 * q^10 - 2 * q^11 - 2 * q^12 + 4 * q^13 + 8 * q^14 + 2 * q^15 - 4 * q^16 + 3 * q^17 + 7 * q^19 - 9 * q^20 - 9 * q^21 - 11 * q^22 + q^23 - 3 * q^24 - q^25 - 11 * q^26 - 3 * q^27 + 9 * q^28 - 11 * q^29 - 5 * q^30 + 13 * q^31 - 4 * q^32 + 2 * q^33 - 7 * q^34 - q^35 + 2 * q^36 - 5 * q^37 - 3 * q^38 - 4 * q^39 - 2 * q^40 - q^41 - 8 * q^42 + 6 * q^43 - 15 * q^44 - 2 * q^45 - 19 * q^46 + 11 * q^47 + 4 * q^48 + 14 * q^49 - 21 * q^50 - 3 * q^51 - 11 * q^52 + 2 * q^53 + 4 * q^55 + 9 * q^56 - 7 * q^57 + 14 * q^58 + 3 * q^59 + 9 * q^60 - q^61 - 2 * q^62 + 9 * q^63 - 21 * q^64 + 11 * q^66 + 10 * q^67 + 28 * q^68 - q^69 + 2 * q^70 + 26 * q^71 + 3 * q^72 + 7 * q^73 - 19 * q^74 + q^75 - 6 * q^76 - 17 * q^77 + 11 * q^78 + 2 * q^79 + 23 * q^80 + 3 * q^81 + 18 * q^82 - 3 * q^83 - 9 * q^84 - 35 * q^85 + 31 * q^86 + 11 * q^87 - 2 * q^88 - 23 * q^89 + 5 * q^90 + q^91 - 16 * q^92 - 13 * q^93 + 4 * q^94 + 3 * q^95 + 4 * q^96 + 14 * q^97 + 51 * q^98 - 2 * q^99

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