LMFDB - Embedded newform 41.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [41,2,Mod(1,41)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(41, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("41.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	41.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:	$0.327386648287$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	41.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70928 q^{2} +0.539189 q^{3} +5.34017 q^{4} +1.70928 q^{5} -1.46081 q^{6} +1.46081 q^{7} -9.04945 q^{8} -2.70928 q^{9} +O(q^{10})$	\(q-2.70928 q^{2} +0.539189 q^{3} +5.34017 q^{4} +1.70928 q^{5} -1.46081 q^{6} +1.46081 q^{7} -9.04945 q^{8} -2.70928 q^{9} -4.63090 q^{10} +3.80098 q^{11} +2.87936 q^{12} -4.34017 q^{13} -3.95774 q^{14} +0.921622 q^{15} +13.8371 q^{16} -2.00000 q^{17} +7.34017 q^{18} -1.80098 q^{19} +9.12783 q^{20} +0.787653 q^{21} -10.2979 q^{22} -1.26180 q^{23} -4.87936 q^{24} -2.07838 q^{25} +11.7587 q^{26} -3.07838 q^{27} +7.80098 q^{28} -3.07838 q^{29} -2.49693 q^{30} +0.581449 q^{31} -19.3896 q^{32} +2.04945 q^{33} +5.41855 q^{34} +2.49693 q^{35} -14.4680 q^{36} +5.12783 q^{37} +4.87936 q^{38} -2.34017 q^{39} -15.4680 q^{40} +1.00000 q^{41} -2.13397 q^{42} +2.34017 q^{43} +20.2979 q^{44} -4.63090 q^{45} +3.41855 q^{46} +12.6381 q^{47} +7.46081 q^{48} -4.86603 q^{49} +5.63090 q^{50} -1.07838 q^{51} -23.1773 q^{52} +0.921622 q^{53} +8.34017 q^{54} +6.49693 q^{55} -13.2195 q^{56} -0.971071 q^{57} +8.34017 q^{58} -5.26180 q^{59} +4.92162 q^{60} -7.75872 q^{61} -1.57531 q^{62} -3.95774 q^{63} +24.8576 q^{64} -7.41855 q^{65} -5.55252 q^{66} -3.80098 q^{67} -10.6803 q^{68} -0.680346 q^{69} -6.76487 q^{70} -1.21953 q^{71} +24.5174 q^{72} +11.6514 q^{73} -13.8927 q^{74} -1.12064 q^{75} -9.61757 q^{76} +5.55252 q^{77} +6.34017 q^{78} +14.8794 q^{79} +23.6514 q^{80} +6.46800 q^{81} -2.70928 q^{82} -2.15676 q^{83} +4.20620 q^{84} -3.41855 q^{85} -6.34017 q^{86} -1.65983 q^{87} -34.3968 q^{88} -11.9421 q^{89} +12.5464 q^{90} -6.34017 q^{91} -6.73820 q^{92} +0.313511 q^{93} -34.2401 q^{94} -3.07838 q^{95} -10.4547 q^{96} +4.15676 q^{97} +13.1834 q^{98} -10.2979 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 2 q^{5} - 6 q^{6} + 6 q^{7} - 9 q^{8} - q^{9}+O(q^{10}) $ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - 6 * q^6 + 6 * q^7 - 9 * q^8 - q^9	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - 6 q^{6} + 6 q^{7} - 9 q^{8} - q^{9} - 10 q^{10} + 2 q^{11} - 4 q^{12} - 2 q^{13} + 4 q^{14} + 6 q^{15} + 13 q^{16} - 6 q^{17} + 11 q^{18} + 4 q^{19} + 6 q^{20} - 8 q^{21} - 4 q^{22} + 4 q^{23} - 2 q^{24} - 3 q^{25} + 10 q^{26} - 6 q^{27} + 14 q^{28} - 6 q^{29} + 10 q^{30} + 16 q^{31} - 29 q^{32} - 12 q^{33} + 2 q^{34} - 10 q^{35} - 11 q^{36} - 6 q^{37} + 2 q^{38} + 4 q^{39} - 14 q^{40} + 3 q^{41} - 20 q^{42} - 4 q^{43} + 34 q^{44} - 10 q^{45} - 4 q^{46} + 24 q^{48} - q^{49} + 13 q^{50} - 30 q^{52} + 6 q^{53} + 14 q^{54} + 2 q^{55} - 16 q^{56} + 12 q^{57} + 14 q^{58} - 8 q^{59} + 18 q^{60} + 2 q^{61} + 16 q^{62} + 4 q^{63} + 13 q^{64} - 8 q^{65} - 16 q^{66} - 2 q^{67} - 10 q^{68} + 20 q^{69} - 30 q^{70} + 20 q^{71} + 23 q^{72} - 2 q^{73} - 30 q^{74} - 16 q^{75} - 24 q^{76} + 16 q^{77} + 8 q^{78} + 32 q^{79} + 34 q^{80} - 13 q^{81} - q^{82} - 12 q^{84} + 4 q^{85} - 8 q^{86} - 16 q^{87} - 40 q^{88} - 6 q^{89} + 2 q^{90} - 8 q^{91} - 28 q^{92} - 12 q^{93} - 46 q^{94} - 6 q^{95} + 2 q^{96} + 6 q^{97} + 35 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - 6 * q^6 + 6 * q^7 - 9 * q^8 - q^9 - 10 * q^10 + 2 * q^11 - 4 * q^12 - 2 * q^13 + 4 * q^14 + 6 * q^15 + 13 * q^16 - 6 * q^17 + 11 * q^18 + 4 * q^19 + 6 * q^20 - 8 * q^21 - 4 * q^22 + 4 * q^23 - 2 * q^24 - 3 * q^25 + 10 * q^26 - 6 * q^27 + 14 * q^28 - 6 * q^29 + 10 * q^30 + 16 * q^31 - 29 * q^32 - 12 * q^33 + 2 * q^34 - 10 * q^35 - 11 * q^36 - 6 * q^37 + 2 * q^38 + 4 * q^39 - 14 * q^40 + 3 * q^41 - 20 * q^42 - 4 * q^43 + 34 * q^44 - 10 * q^45 - 4 * q^46 + 24 * q^48 - q^49 + 13 * q^50 - 30 * q^52 + 6 * q^53 + 14 * q^54 + 2 * q^55 - 16 * q^56 + 12 * q^57 + 14 * q^58 - 8 * q^59 + 18 * q^60 + 2 * q^61 + 16 * q^62 + 4 * q^63 + 13 * q^64 - 8 * q^65 - 16 * q^66 - 2 * q^67 - 10 * q^68 + 20 * q^69 - 30 * q^70 + 20 * q^71 + 23 * q^72 - 2 * q^73 - 30 * q^74 - 16 * q^75 - 24 * q^76 + 16 * q^77 + 8 * q^78 + 32 * q^79 + 34 * q^80 - 13 * q^81 - q^82 - 12 * q^84 + 4 * q^85 - 8 * q^86 - 16 * q^87 - 40 * q^88 - 6 * q^89 + 2 * q^90 - 8 * q^91 - 28 * q^92 - 12 * q^93 - 46 * q^94 - 6 * q^95 + 2 * q^96 + 6 * q^97 + 35 * q^98 - 4 * 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 $

$2$	−2.70928	−1.91575	−0.957873	−	0.287190i	$-0.907279\pi$
$2$	−2.70928	−1.91575	−0.957873	+	0.287190i	$0.907279\pi$
$3$	0.539189	0.311301	0.155650	−	0.987812i	$-0.450253\pi$
$3$	0.539189	0.311301	0.155650	+	0.987812i	$0.450253\pi$
$4$	5.34017	2.67009
$5$	1.70928	0.764411	0.382206	−	0.924077i	$-0.375165\pi$
$5$	1.70928	0.764411	0.382206	+	0.924077i	$0.375165\pi$
$6$	−1.46081	−0.596374
$7$	1.46081	0.552135	0.276067	−	0.961138i	$-0.410969\pi$
$7$	1.46081	0.552135	0.276067	+	0.961138i	$0.410969\pi$
$8$	−9.04945	−3.19946
$9$	−2.70928	−0.903092
$10$	−4.63090	−1.46442
$11$	3.80098	1.14604	0.573020	−	0.819541i	$-0.305772\pi$
$11$	3.80098	1.14604	0.573020	+	0.819541i	$0.305772\pi$
$12$	2.87936	0.831200
$13$	−4.34017	−1.20375	−0.601874	−	0.798591i	$-0.705579\pi$
$13$	−4.34017	−1.20375	−0.601874	+	0.798591i	$0.705579\pi$
$14$	−3.95774	−1.05775
$15$	0.921622	0.237962
$16$	13.8371	3.45928
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	7.34017	1.73010
$19$	−1.80098	−0.413174	−0.206587	−	0.978428i	$-0.566236\pi$
$19$	−1.80098	−0.413174	−0.206587	+	0.978428i	$0.566236\pi$
$20$	9.12783	2.04104
$21$	0.787653	0.171880
$22$	−10.2979	−2.19552
$23$	−1.26180	−0.263102	−0.131551	−	0.991309i	$-0.541996\pi$
$23$	−1.26180	−0.263102	−0.131551	+	0.991309i	$0.541996\pi$
$24$	−4.87936	−0.995996
$25$	−2.07838	−0.415676
$26$	11.7587	2.30608
$27$	−3.07838	−0.592434
$28$	7.80098	1.47425
$29$	−3.07838	−0.571640	−0.285820	−	0.958283i	$-0.592266\pi$
$29$	−3.07838	−0.571640	−0.285820	+	0.958283i	$0.592266\pi$
$30$	−2.49693	−0.455875
$31$	0.581449	0.104431	0.0522157	−	0.998636i	$-0.483372\pi$
$31$	0.581449	0.104431	0.0522157	+	0.998636i	$0.483372\pi$
$32$	−19.3896	−3.42763
$33$	2.04945	0.356763
$34$	5.41855	0.929274
$35$	2.49693	0.422058
$36$	−14.4680	−2.41133
$37$	5.12783	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$37$	5.12783	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$38$	4.87936	0.791537
$39$	−2.34017	−0.374728
$40$	−15.4680	−2.44571
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	−2.13397	−0.329279
$43$	2.34017	0.356873	0.178437	−	0.983951i	$-0.442896\pi$
$43$	2.34017	0.356873	0.178437	+	0.983951i	$0.442896\pi$
$44$	20.2979	3.06003
$45$	−4.63090	−0.690333
$46$	3.41855	0.504038
$47$	12.6381	1.84345	0.921727	−	0.387839i	$-0.126778\pi$
$47$	12.6381	1.84345	0.921727	+	0.387839i	$0.126778\pi$
$48$	7.46081	1.07688
$49$	−4.86603	−0.695147
$50$	5.63090	0.796329
$51$	−1.07838	−0.151003
$52$	−23.1773	−3.21411
$53$	0.921622	0.126595	0.0632973	−	0.997995i	$-0.479838\pi$
$53$	0.921622	0.126595	0.0632973	+	0.997995i	$0.479838\pi$
$54$	8.34017	1.13495
$55$	6.49693	0.876046
$56$	−13.2195	−1.76653
$57$	−0.971071	−0.128621
$58$	8.34017	1.09512
$59$	−5.26180	−0.685027	−0.342514	−	0.939513i	$-0.611278\pi$
$59$	−5.26180	−0.685027	−0.342514	+	0.939513i	$0.611278\pi$
$60$	4.92162	0.635379
$61$	−7.75872	−0.993403	−0.496701	−	0.867922i	$-0.665456\pi$
$61$	−7.75872	−0.993403	−0.496701	+	0.867922i	$0.665456\pi$
$62$	−1.57531	−0.200064
$63$	−3.95774	−0.498628
$64$	24.8576	3.10720
$65$	−7.41855	−0.920158
$66$	−5.55252	−0.683468
$67$	−3.80098	−0.464364	−0.232182	−	0.972672i	$-0.574587\pi$
$67$	−3.80098	−0.464364	−0.232182	+	0.972672i	$0.574587\pi$
$68$	−10.6803	−1.29518
$69$	−0.680346	−0.0819040
$70$	−6.76487	−0.808556
$71$	−1.21953	−0.144732	−0.0723661	−	0.997378i	$-0.523055\pi$
$71$	−1.21953	−0.144732	−0.0723661	+	0.997378i	$0.523055\pi$
$72$	24.5174	2.88941
$73$	11.6514	1.36370	0.681848	−	0.731494i	$-0.261177\pi$
$73$	11.6514	1.36370	0.681848	+	0.731494i	$0.261177\pi$
$74$	−13.8927	−1.61499
$75$	−1.12064	−0.129400
$76$	−9.61757	−1.10321
$77$	5.55252	0.632768
$78$	6.34017	0.717883
$79$	14.8794	1.67406	0.837029	−	0.547158i	$-0.184290\pi$
$79$	14.8794	1.67406	0.837029	+	0.547158i	$0.184290\pi$
$80$	23.6514	2.64431
$81$	6.46800	0.718667
$82$	−2.70928	−0.299189
$83$	−2.15676	−0.236735	−0.118367	−	0.992970i	$-0.537766\pi$
$83$	−2.15676	−0.236735	−0.118367	+	0.992970i	$0.537766\pi$
$84$	4.20620	0.458934
$85$	−3.41855	−0.370794
$86$	−6.34017	−0.683678
$87$	−1.65983	−0.177952
$88$	−34.3968	−3.66671
$89$	−11.9421	−1.26586	−0.632932	−	0.774207i	$-0.718149\pi$
$89$	−11.9421	−1.26586	−0.632932	+	0.774207i	$0.718149\pi$
$90$	12.5464	1.32250
$91$	−6.34017	−0.664631
$92$	−6.73820	−0.702506
$93$	0.313511	0.0325096
$94$	−34.2401	−3.53159
$95$	−3.07838	−0.315835
$96$	−10.4547	−1.06703
$97$	4.15676	0.422055	0.211027	−	0.977480i	$-0.432319\pi$
$97$	4.15676	0.422055	0.211027	+	0.977480i	$0.432319\pi$
$98$	13.1834	1.33173
$99$	−10.2979	−1.03498
$100$	−11.0989	−1.10989
$101$	17.0205	1.69360	0.846802	−	0.531907i	$-0.178525\pi$
$101$	17.0205	1.69360	0.846802	+	0.531907i	$0.178525\pi$
$102$	2.92162	0.289284
$103$	16.5958	1.63524	0.817618	−	0.575762i	$-0.195294\pi$
$103$	16.5958	1.63524	0.817618	+	0.575762i	$0.195294\pi$
$104$	39.2762	3.85135
$105$	1.34632	0.131387
$106$	−2.49693	−0.242523
$107$	−14.8371	−1.43436	−0.717178	−	0.696890i	$-0.754567\pi$
$107$	−14.8371	−1.43436	−0.717178	+	0.696890i	$0.754567\pi$
$108$	−16.4391	−1.58185
$109$	11.1773	1.07059	0.535294	−	0.844666i	$-0.320201\pi$
$109$	11.1773	1.07059	0.535294	+	0.844666i	$0.320201\pi$
$110$	−17.6020	−1.67828
$111$	2.76487	0.262430
$112$	20.2134	1.90999
$113$	−0.814315	−0.0766043	−0.0383022	−	0.999266i	$-0.512195\pi$
$113$	−0.814315	−0.0766043	−0.0383022	+	0.999266i	$0.512195\pi$
$114$	2.63090	0.246406
$115$	−2.15676	−0.201118
$116$	−16.4391	−1.52633
$117$	11.7587	1.08709
$118$	14.2557	1.31234
$119$	−2.92162	−0.267825
$120$	−8.34017	−0.761350
$121$	3.44748	0.313407
$122$	21.0205	1.90311
$123$	0.539189	0.0486170
$124$	3.10504	0.278841
$125$	−12.0989	−1.08216
$126$	10.7226	0.955246
$127$	2.73820	0.242976	0.121488	−	0.992593i	$-0.461233\pi$
$127$	2.73820	0.242976	0.121488	+	0.992593i	$0.461233\pi$
$128$	−28.5669	−2.52498
$129$	1.26180	0.111095
$130$	20.0989	1.76279
$131$	1.75872	0.153660	0.0768302	−	0.997044i	$-0.475520\pi$
$131$	1.75872	0.153660	0.0768302	+	0.997044i	$0.475520\pi$
$132$	10.9444	0.952588
$133$	−2.63090	−0.228128
$134$	10.2979	0.889604
$135$	−5.26180	−0.452863
$136$	18.0989	1.55197
$137$	10.3135	0.881143	0.440571	−	0.897718i	$-0.354776\pi$
$137$	10.3135	0.881143	0.440571	+	0.897718i	$0.354776\pi$
$138$	1.84324	0.156907
$139$	−20.4124	−1.73136	−0.865679	−	0.500600i	$-0.833113\pi$
$139$	−20.4124	−1.73136	−0.865679	+	0.500600i	$0.833113\pi$
$140$	13.3340	1.12693
$141$	6.81432	0.573869
$142$	3.30406	0.277270
$143$	−16.4969	−1.37954
$144$	−37.4885	−3.12404
$145$	−5.26180	−0.436968
$146$	−31.5669	−2.61249
$147$	−2.62371	−0.216400
$148$	27.3835	2.25091
$149$	7.44521	0.609936	0.304968	−	0.952363i	$-0.401354\pi$
$149$	7.44521	0.609936	0.304968	+	0.952363i	$0.401354\pi$
$150$	3.03612	0.247898
$151$	18.2979	1.48906	0.744532	−	0.667587i	$-0.232673\pi$
$151$	18.2979	1.48906	0.744532	+	0.667587i	$0.232673\pi$
$152$	16.2979	1.32194
$153$	5.41855	0.438064
$154$	−15.0433	−1.21222
$155$	0.993857	0.0798285
$156$	−12.4969	−1.00056
$157$	−12.6537	−1.00987	−0.504937	−	0.863156i	$-0.668484\pi$
$157$	−12.6537	−1.00987	−0.504937	+	0.863156i	$0.668484\pi$
$158$	−40.3123	−3.20707
$159$	0.496928	0.0394090
$160$	−33.1422	−2.62012
$161$	−1.84324	−0.145268
$162$	−17.5236	−1.37678
$163$	−12.6803	−0.993201	−0.496601	−	0.867979i	$-0.665419\pi$
$163$	−12.6803	−0.993201	−0.496601	+	0.867979i	$0.665419\pi$
$164$	5.34017	0.416997
$165$	3.50307	0.272714
$166$	5.84324	0.453524
$167$	−9.95774	−0.770553	−0.385277	−	0.922801i	$-0.625894\pi$
$167$	−9.95774	−0.770553	−0.385277	+	0.922801i	$0.625894\pi$
$168$	−7.12783	−0.549924
$169$	5.83710	0.449008
$170$	9.26180	0.710347
$171$	4.87936	0.373134
$172$	12.4969	0.952882
$173$	−12.1568	−0.924261	−0.462131	−	0.886812i	$-0.652915\pi$
$173$	−12.1568	−0.924261	−0.462131	+	0.886812i	$0.652915\pi$
$174$	4.49693	0.340911
$175$	−3.03612	−0.229509
$176$	52.5946	3.96447
$177$	−2.83710	−0.213250
$178$	32.3545	2.42508
$179$	19.1350	1.43022	0.715109	−	0.699013i	$-0.246377\pi$
$179$	19.1350	1.43022	0.715109	+	0.699013i	$0.246377\pi$
$180$	−24.7298	−1.84325
$181$	−15.4452	−1.14803	−0.574017	−	0.818844i	$-0.694616\pi$
$181$	−15.4452	−1.14803	−0.574017	+	0.818844i	$0.694616\pi$
$182$	17.1773	1.27326
$183$	−4.18342	−0.309247
$184$	11.4186	0.841787
$185$	8.76487	0.644406
$186$	−0.849388	−0.0622801
$187$	−7.60197	−0.555911
$188$	67.4896	4.92218
$189$	−4.49693	−0.327103
$190$	8.34017	0.605060
$191$	−18.5536	−1.34249	−0.671244	−	0.741236i	$-0.734240\pi$
$191$	−18.5536	−1.34249	−0.671244	+	0.741236i	$0.734240\pi$
$192$	13.4030	0.967275
$193$	−11.2618	−0.810642	−0.405321	−	0.914174i	$-0.632840\pi$
$193$	−11.2618	−0.810642	−0.405321	+	0.914174i	$0.632840\pi$
$194$	−11.2618	−0.808550
$195$	−4.00000	−0.286446
$196$	−25.9854	−1.85610
$197$	8.24128	0.587167	0.293583	−	0.955933i	$-0.405152\pi$
$197$	8.24128	0.587167	0.293583	+	0.955933i	$0.405152\pi$
$198$	27.8999	1.98276
$199$	8.96388	0.635433	0.317716	−	0.948186i	$-0.397084\pi$
$199$	8.96388	0.635433	0.317716	+	0.948186i	$0.397084\pi$
$200$	18.8082	1.32994
$201$	−2.04945	−0.144557
$202$	−46.1133	−3.24452
$203$	−4.49693	−0.315623
$204$	−5.75872	−0.403191
$205$	1.70928	0.119381
$206$	−44.9627	−3.13270
$207$	3.41855	0.237606
$208$	−60.0554	−4.16409
$209$	−6.84551	−0.473514
$210$	−3.64754	−0.251704
$211$	14.4547	0.995100	0.497550	−	0.867435i	$-0.334233\pi$
$211$	14.4547	0.995100	0.497550	+	0.867435i	$0.334233\pi$
$212$	4.92162	0.338018
$213$	−0.657560	−0.0450552
$214$	40.1978	2.74786
$215$	4.00000	0.272798
$216$	27.8576	1.89547
$217$	0.849388	0.0576602
$218$	−30.2823	−2.05098
$219$	6.28231	0.424519
$220$	34.6947	2.33912
$221$	8.68035	0.583903
$222$	−7.49079	−0.502749
$223$	6.83710	0.457846	0.228923	−	0.973445i	$-0.426480\pi$
$223$	6.83710	0.457846	0.228923	+	0.973445i	$0.426480\pi$
$224$	−28.3246	−1.89252
$225$	5.63090	0.375393
$226$	2.20620	0.146754
$227$	−5.21953	−0.346433	−0.173216	−	0.984884i	$-0.555416\pi$
$227$	−5.21953	−0.346433	−0.173216	+	0.984884i	$0.555416\pi$
$228$	−5.18568	−0.343430
$229$	−17.3340	−1.14546	−0.572732	−	0.819742i	$-0.694117\pi$
$229$	−17.3340	−1.14546	−0.572732	+	0.819742i	$0.694117\pi$
$230$	5.84324	0.385292
$231$	2.99386	0.196981
$232$	27.8576	1.82894
$233$	21.2039	1.38912	0.694558	−	0.719437i	$-0.255600\pi$
$233$	21.2039	1.38912	0.694558	+	0.719437i	$0.255600\pi$
$234$	−31.8576	−2.08260
$235$	21.6020	1.40916
$236$	−28.0989	−1.82908
$237$	8.02279	0.521136
$238$	7.91548	0.513084
$239$	4.35577	0.281751	0.140876	−	0.990027i	$-0.455008\pi$
$239$	4.35577	0.281751	0.140876	+	0.990027i	$0.455008\pi$
$240$	12.7526	0.823176
$241$	7.39189	0.476153	0.238077	−	0.971246i	$-0.423483\pi$
$241$	7.39189	0.476153	0.238077	+	0.971246i	$0.423483\pi$
$242$	−9.34017	−0.600409
$243$	12.7226	0.816156
$244$	−41.4329	−2.65247
$245$	−8.31739	−0.531378
$246$	−1.46081	−0.0931379
$247$	7.81658	0.497357
$248$	−5.26180	−0.334124
$249$	−1.16290	−0.0736957
$250$	32.7792	2.07314
$251$	−17.7587	−1.12092	−0.560460	−	0.828181i	$-0.689376\pi$
$251$	−17.7587	−1.12092	−0.560460	+	0.828181i	$0.689376\pi$
$252$	−21.1350	−1.33138
$253$	−4.79606	−0.301526
$254$	−7.41855	−0.465481
$255$	−1.84324	−0.115428
$256$	27.6803	1.73002
$257$	22.7792	1.42093	0.710465	−	0.703732i	$-0.248485\pi$
$257$	22.7792	1.42093	0.710465	+	0.703732i	$0.248485\pi$
$258$	−3.41855	−0.212830
$259$	7.49079	0.465455
$260$	−39.6163	−2.45690
$261$	8.34017	0.516244
$262$	−4.76487	−0.294374
$263$	15.4296	0.951431	0.475715	−	0.879599i	$-0.342189\pi$
$263$	15.4296	0.951431	0.475715	+	0.879599i	$0.342189\pi$
$264$	−18.5464	−1.14145
$265$	1.57531	0.0967703
$266$	7.12783	0.437035
$267$	−6.43907	−0.394065
$268$	−20.2979	−1.23989
$269$	−10.2823	−0.626924	−0.313462	−	0.949601i	$-0.601489\pi$
$269$	−10.2823	−0.626924	−0.313462	+	0.949601i	$0.601489\pi$
$270$	14.2557	0.867571
$271$	3.41855	0.207662	0.103831	−	0.994595i	$-0.466890\pi$
$271$	3.41855	0.207662	0.103831	+	0.994595i	$0.466890\pi$
$272$	−27.6742	−1.67800
$273$	−3.41855	−0.206900
$274$	−27.9421	−1.68805
$275$	−7.89988	−0.476381
$276$	−3.63317	−0.218691
$277$	9.49466	0.570479	0.285239	−	0.958456i	$-0.407927\pi$
$277$	9.49466	0.570479	0.285239	+	0.958456i	$0.407927\pi$
$278$	55.3028	3.31684
$279$	−1.57531	−0.0943111
$280$	−22.5958	−1.35036
$281$	5.31965	0.317344	0.158672	−	0.987331i	$-0.449279\pi$
$281$	5.31965	0.317344	0.158672	+	0.987331i	$0.449279\pi$
$282$	−18.4619	−1.09939
$283$	−3.05172	−0.181406	−0.0907028	−	0.995878i	$-0.528911\pi$
$283$	−3.05172	−0.181406	−0.0907028	+	0.995878i	$0.528911\pi$
$284$	−6.51253	−0.386447
$285$	−1.65983	−0.0983197
$286$	44.6947	2.64285
$287$	1.46081	0.0862290
$288$	52.5318	3.09547
$289$	−13.0000	−0.764706
$290$	14.2557	0.837121
$291$	2.24128	0.131386
$292$	62.2206	3.64118
$293$	−9.11942	−0.532762	−0.266381	−	0.963868i	$-0.585828\pi$
$293$	−9.11942	−0.532762	−0.266381	+	0.963868i	$0.585828\pi$
$294$	7.10835	0.414568
$295$	−8.99386	−0.523643
$296$	−46.4040	−2.69718
$297$	−11.7009	−0.678953
$298$	−20.1711	−1.16848
$299$	5.47641	0.316709
$300$	−5.98440	−0.345510
$301$	3.41855	0.197042
$302$	−49.5741	−2.85267
$303$	9.17727	0.527221
$304$	−24.9204	−1.42928
$305$	−13.2618	−0.759368
$306$	−14.6803	−0.839220
$307$	−21.7587	−1.24184	−0.620918	−	0.783876i	$-0.713240\pi$
$307$	−21.7587	−1.24184	−0.620918	+	0.783876i	$0.713240\pi$
$308$	29.6514	1.68955
$309$	8.94828	0.509050
$310$	−2.69263	−0.152931
$311$	−8.66475	−0.491333	−0.245666	−	0.969354i	$-0.579007\pi$
$311$	−8.66475	−0.491333	−0.245666	+	0.969354i	$0.579007\pi$
$312$	21.1773	1.19893
$313$	−30.7792	−1.73975	−0.869873	−	0.493276i	$-0.835799\pi$
$313$	−30.7792	−1.73975	−0.869873	+	0.493276i	$0.835799\pi$
$314$	34.2823	1.93466
$315$	−6.76487	−0.381157
$316$	79.4584	4.46988
$317$	−0.822726	−0.0462089	−0.0231044	−	0.999733i	$-0.507355\pi$
$317$	−0.822726	−0.0462089	−0.0231044	+	0.999733i	$0.507355\pi$
$318$	−1.34632	−0.0754977
$319$	−11.7009	−0.655123
$320$	42.4885	2.37518
$321$	−8.00000	−0.446516
$322$	4.99386	0.278297
$323$	3.60197	0.200419
$324$	34.5402	1.91890
$325$	9.02052	0.500368
$326$	34.3545	1.90272
$327$	6.02666	0.333275
$328$	−9.04945	−0.499672
$329$	18.4619	1.01784
$330$	−9.49079	−0.522451
$331$	22.4235	1.23251	0.616253	−	0.787548i	$-0.288650\pi$
$331$	22.4235	1.23251	0.616253	+	0.787548i	$0.288650\pi$
$332$	−11.5174	−0.632102
$333$	−13.8927	−0.761315
$334$	26.9783	1.47618
$335$	−6.49693	−0.354965
$336$	10.8988	0.594580
$337$	−26.7031	−1.45461	−0.727306	−	0.686313i	$-0.759228\pi$
$337$	−26.7031	−1.45461	−0.727306	+	0.686313i	$0.759228\pi$
$338$	−15.8143	−0.860185
$339$	−0.439070	−0.0238470
$340$	−18.2557	−0.990052
$341$	2.21008	0.119683
$342$	−13.2195	−0.714831
$343$	−17.3340	−0.935950
$344$	−21.1773	−1.14180
$345$	−1.16290	−0.0626084
$346$	32.9360	1.77065
$347$	13.4030	0.719508	0.359754	−	0.933047i	$-0.382861\pi$
$347$	13.4030	0.719508	0.359754	+	0.933047i	$0.382861\pi$
$348$	−8.86376	−0.475148
$349$	−14.2907	−0.764965	−0.382482	−	0.923963i	$-0.624931\pi$
$349$	−14.2907	−0.764965	−0.382482	+	0.923963i	$0.624931\pi$
$350$	8.22568	0.439681
$351$	13.3607	0.713141
$352$	−73.6996	−3.92820
$353$	−17.4947	−0.931147	−0.465573	−	0.885009i	$-0.654152\pi$
$353$	−17.4947	−0.931147	−0.465573	+	0.885009i	$0.654152\pi$
$354$	7.68649	0.408532
$355$	−2.08452	−0.110635
$356$	−63.7731	−3.37997
$357$	−1.57531	−0.0833740
$358$	−51.8420	−2.73994
$359$	16.1978	0.854887	0.427443	−	0.904042i	$-0.359414\pi$
$359$	16.1978	0.854887	0.427443	+	0.904042i	$0.359414\pi$
$360$	41.9071	2.20870
$361$	−15.7565	−0.829287
$362$	41.8453	2.19934
$363$	1.85884	0.0975640
$364$	−33.8576	−1.77462
$365$	19.9155	1.04242
$366$	11.3340	0.592439
$367$	−5.97334	−0.311806	−0.155903	−	0.987772i	$-0.549829\pi$
$367$	−5.97334	−0.311806	−0.155903	+	0.987772i	$0.549829\pi$
$368$	−17.4596	−0.910144
$369$	−2.70928	−0.141039
$370$	−23.7464	−1.23452
$371$	1.34632	0.0698972
$372$	1.67420	0.0868034
$373$	−13.6865	−0.708660	−0.354330	−	0.935121i	$-0.615291\pi$
$373$	−13.6865	−0.708660	−0.354330	+	0.935121i	$0.615291\pi$
$374$	20.5958	1.06498
$375$	−6.52359	−0.336877
$376$	−114.368	−5.89806
$377$	13.3607	0.688111
$378$	12.1834	0.626647
$379$	28.0989	1.44334	0.721672	−	0.692235i	$-0.243374\pi$
$379$	28.0989	1.44334	0.721672	+	0.692235i	$0.243374\pi$
$380$	−16.4391	−0.843306
$381$	1.47641	0.0756388
$382$	50.2667	2.57187
$383$	−18.4547	−0.942989	−0.471495	−	0.881869i	$-0.656285\pi$
$383$	−18.4547	−0.942989	−0.471495	+	0.881869i	$0.656285\pi$
$384$	−15.4030	−0.786029
$385$	9.49079	0.483695
$386$	30.5113	1.55298
$387$	−6.34017	−0.322289
$388$	22.1978	1.12692
$389$	21.8310	1.10687	0.553437	−	0.832891i	$-0.313316\pi$
$389$	21.8310	1.10687	0.553437	+	0.832891i	$0.313316\pi$
$390$	10.8371	0.548758
$391$	2.52359	0.127623
$392$	44.0349	2.22410
$393$	0.948284	0.0478346
$394$	−22.3279	−1.12486
$395$	25.4329	1.27967
$396$	−54.9926	−2.76348
$397$	5.81658	0.291926	0.145963	−	0.989290i	$-0.453372\pi$
$397$	5.81658	0.291926	0.145963	+	0.989290i	$0.453372\pi$
$398$	−24.2856	−1.21733
$399$	−1.41855	−0.0710164
$400$	−28.7587	−1.43794
$401$	−25.5402	−1.27542	−0.637709	−	0.770277i	$-0.720118\pi$
$401$	−25.5402	−1.27542	−0.637709	+	0.770277i	$0.720118\pi$
$402$	5.55252	0.276935
$403$	−2.52359	−0.125709
$404$	90.8925	4.52207
$405$	11.0556	0.549357
$406$	12.1834	0.604653
$407$	19.4908	0.966122
$408$	9.75872	0.483129
$409$	−9.44134	−0.466844	−0.233422	−	0.972376i	$-0.574992\pi$
$409$	−9.44134	−0.466844	−0.233422	+	0.972376i	$0.574992\pi$
$410$	−4.63090	−0.228704
$411$	5.56093	0.274300
$412$	88.6246	4.36622
$413$	−7.68649	−0.378227
$414$	−9.26180	−0.455192
$415$	−3.68649	−0.180963
$416$	84.1543	4.12600
$417$	−11.0061	−0.538973
$418$	18.5464	0.907133
$419$	33.0082	1.61256	0.806279	−	0.591536i	$-0.201478\pi$
$419$	33.0082	1.61256	0.806279	+	0.591536i	$0.201478\pi$
$420$	7.18956	0.350815
$421$	−10.5958	−0.516409	−0.258204	−	0.966090i	$-0.583131\pi$
$421$	−10.5958	−0.516409	−0.258204	+	0.966090i	$0.583131\pi$
$422$	−39.1617	−1.90636
$423$	−34.2401	−1.66481
$424$	−8.34017	−0.405035
$425$	4.15676	0.201632
$426$	1.78151	0.0863144
$427$	−11.3340	−0.548492
$428$	−79.2327	−3.82986
$429$	−8.89496	−0.429453
$430$	−10.8371	−0.522611
$431$	−10.7070	−0.515738	−0.257869	−	0.966180i	$-0.583020\pi$
$431$	−10.7070	−0.515738	−0.257869	+	0.966180i	$0.583020\pi$
$432$	−42.5958	−2.04939
$433$	25.5708	1.22885	0.614426	−	0.788974i	$-0.289387\pi$
$433$	25.5708	1.22885	0.614426	+	0.788974i	$0.289387\pi$
$434$	−2.30122	−0.110462
$435$	−2.83710	−0.136029
$436$	59.6886	2.85856
$437$	2.27247	0.108707
$438$	−17.0205	−0.813272
$439$	29.5064	1.40826	0.704131	−	0.710070i	$-0.251337\pi$
$439$	29.5064	1.40826	0.704131	+	0.710070i	$0.251337\pi$
$440$	−58.7936	−2.80288
$441$	13.1834	0.627782
$442$	−23.5174	−1.11861
$443$	20.3980	0.969140	0.484570	−	0.874753i	$-0.338976\pi$
$443$	20.3980	0.969140	0.484570	+	0.874753i	$0.338976\pi$
$444$	14.7649	0.700710
$445$	−20.4124	−0.967641
$446$	−18.5236	−0.877117
$447$	4.01438	0.189873
$448$	36.3123	1.71559
$449$	13.6020	0.641917	0.320958	−	0.947093i	$-0.395995\pi$
$449$	13.6020	0.641917	0.320958	+	0.947093i	$0.395995\pi$
$450$	−15.2557	−0.719158
$451$	3.80098	0.178981
$452$	−4.34858	−0.204540
$453$	9.86603	0.463547
$454$	14.1412	0.663677
$455$	−10.8371	−0.508051
$456$	8.78765	0.411520
$457$	8.52359	0.398717	0.199358	−	0.979927i	$-0.436114\pi$
$457$	8.52359	0.398717	0.199358	+	0.979927i	$0.436114\pi$
$458$	46.9627	2.19442
$459$	6.15676	0.287373
$460$	−11.5174	−0.537004
$461$	−0.348583	−0.0162352	−0.00811758	−	0.999967i	$-0.502584\pi$
$461$	−0.348583	−0.0162352	−0.00811758	+	0.999967i	$0.502584\pi$
$462$	−8.11118	−0.377366
$463$	19.2918	0.896565	0.448282	−	0.893892i	$-0.352036\pi$
$463$	19.2918	0.896565	0.448282	+	0.893892i	$0.352036\pi$
$464$	−42.5958	−1.97746
$465$	0.535877	0.0248507
$466$	−57.4473	−2.66119
$467$	−23.4186	−1.08368	−0.541841	−	0.840481i	$-0.682272\pi$
$467$	−23.4186	−1.08368	−0.541841	+	0.840481i	$0.682272\pi$
$468$	62.7936	2.90264
$469$	−5.55252	−0.256392
$470$	−58.5257	−2.69959
$471$	−6.82273	−0.314375
$472$	47.6163	2.19172
$473$	8.89496	0.408991
$474$	−21.7359	−0.998365
$475$	3.74313	0.171746
$476$	−15.6020	−0.715115
$477$	−2.49693	−0.114327
$478$	−11.8010	−0.539764
$479$	3.98440	0.182052	0.0910260	−	0.995849i	$-0.470985\pi$
$479$	3.98440	0.182052	0.0910260	+	0.995849i	$0.470985\pi$
$480$	−17.8699	−0.815646
$481$	−22.2557	−1.01477
$482$	−20.0267	−0.912189
$483$	−0.993857	−0.0452221
$484$	18.4101	0.836825
$485$	7.10504	0.322623
$486$	−34.4690	−1.56355
$487$	−36.4268	−1.65066	−0.825328	−	0.564654i	$-0.809010\pi$
$487$	−36.4268	−1.65066	−0.825328	+	0.564654i	$0.809010\pi$
$488$	70.2122	3.17836
$489$	−6.83710	−0.309184
$490$	22.5341	1.01799
$491$	−11.7321	−0.529461	−0.264730	−	0.964323i	$-0.585283\pi$
$491$	−11.7321	−0.529461	−0.264730	+	0.964323i	$0.585283\pi$
$492$	2.87936	0.129812
$493$	6.15676	0.277286
$494$	−21.1773	−0.952811
$495$	−17.6020	−0.791150
$496$	8.04557	0.361257
$497$	−1.78151	−0.0799116
$498$	3.15061	0.141182
$499$	16.5392	0.740396	0.370198	−	0.928953i	$-0.379290\pi$
$499$	16.5392	0.740396	0.370198	+	0.928953i	$0.379290\pi$
$500$	−64.6102	−2.88946
$501$	−5.36910	−0.239874
$502$	48.1133	2.14740
$503$	−31.1617	−1.38943	−0.694715	−	0.719285i	$-0.744470\pi$
$503$	−31.1617	−1.38943	−0.694715	+	0.719285i	$0.744470\pi$
$504$	35.8154	1.59534
$505$	29.0928	1.29461
$506$	12.9939	0.577647
$507$	3.14730	0.139777
$508$	14.6225	0.648768
$509$	−16.0267	−0.710369	−0.355185	−	0.934796i	$-0.615582\pi$
$509$	−16.0267	−0.710369	−0.355185	+	0.934796i	$0.615582\pi$
$510$	4.99386	0.221132
$511$	17.0205	0.752943
$512$	−17.8599	−0.789303
$513$	5.54411	0.244778
$514$	−61.7152	−2.72214
$515$	28.3668	1.24999
$516$	6.73820	0.296633
$517$	48.0372	2.11267
$518$	−20.2946	−0.891694
$519$	−6.55479	−0.287723
$520$	67.1338	2.94401
$521$	2.41241	0.105689	0.0528447	−	0.998603i	$-0.483171\pi$
$521$	2.41241	0.105689	0.0528447	+	0.998603i	$0.483171\pi$
$522$	−22.5958	−0.988992
$523$	−22.7214	−0.993537	−0.496768	−	0.867883i	$-0.665480\pi$
$523$	−22.7214	−0.993537	−0.496768	+	0.867883i	$0.665480\pi$
$524$	9.39189	0.410287
$525$	−1.63704	−0.0714463
$526$	−41.8031	−1.82270
$527$	−1.16290	−0.0506567
$528$	28.3584	1.23414
$529$	−21.4079	−0.930777
$530$	−4.26794	−0.185387
$531$	14.2557	0.618643
$532$	−14.0494	−0.609121
$533$	−4.34017	−0.187994
$534$	17.4452	0.754928
$535$	−25.3607	−1.09644
$536$	34.3968	1.48572
$537$	10.3174	0.445228
$538$	27.8576	1.20103
$539$	−18.4957	−0.796666
$540$	−28.0989	−1.20918
$541$	−21.3256	−0.916860	−0.458430	−	0.888731i	$-0.651588\pi$
$541$	−21.3256	−0.916860	−0.458430	+	0.888731i	$0.651588\pi$
$542$	−9.26180	−0.397828
$543$	−8.32789	−0.357384
$544$	38.7792	1.66265
$545$	19.1050	0.818370
$546$	9.26180	0.396368
$547$	−21.4875	−0.918738	−0.459369	−	0.888246i	$-0.651924\pi$
$547$	−21.4875	−0.918738	−0.459369	+	0.888246i	$0.651924\pi$
$548$	55.0759	2.35273
$549$	21.0205	0.897134
$550$	21.4030	0.912625
$551$	5.54411	0.236187
$552$	6.15676	0.262049
$553$	21.7359	0.924306
$554$	−25.7237	−1.09289
$555$	4.72592	0.200604
$556$	−109.006	−4.62288
$557$	−6.08452	−0.257809	−0.128905	−	0.991657i	$-0.541146\pi$
$557$	−6.08452	−0.257809	−0.128905	+	0.991657i	$0.541146\pi$
$558$	4.26794	0.180676
$559$	−10.1568	−0.429585
$560$	34.5503	1.46001
$561$	−4.09890	−0.173056
$562$	−14.4124	−0.607951
$563$	41.4752	1.74797	0.873985	−	0.485952i	$-0.161527\pi$
$563$	41.4752	1.74797	0.873985	+	0.485952i	$0.161527\pi$
$564$	36.3896	1.53228
$565$	−1.39189	−0.0585572
$566$	8.26794	0.347527
$567$	9.44852	0.396801
$568$	11.0361	0.463065
$569$	0.447480	0.0187593	0.00937967	−	0.999956i	$-0.497014\pi$
$569$	0.447480	0.0187593	0.00937967	+	0.999956i	$0.497014\pi$
$570$	4.49693	0.188356
$571$	−33.4440	−1.39959	−0.699794	−	0.714345i	$-0.746725\pi$
$571$	−33.4440	−1.39959	−0.699794	+	0.714345i	$0.746725\pi$
$572$	−88.0965	−3.68350
$573$	−10.0039	−0.417918
$574$	−3.95774	−0.165193
$575$	2.62249	0.109365
$576$	−67.3461	−2.80609
$577$	3.67420	0.152959	0.0764795	−	0.997071i	$-0.475632\pi$
$577$	3.67420	0.152959	0.0764795	+	0.997071i	$0.475632\pi$
$578$	35.2206	1.46498
$579$	−6.07223	−0.252353
$580$	−28.0989	−1.16674
$581$	−3.15061	−0.130709
$582$	−6.07223	−0.251702
$583$	3.50307	0.145082
$584$	−105.439	−4.36309
$585$	20.0989	0.830987
$586$	24.7070	1.02064
$587$	−5.46081	−0.225392	−0.112696	−	0.993630i	$-0.535949\pi$
$587$	−5.46081	−0.225392	−0.112696	+	0.993630i	$0.535949\pi$
$588$	−14.0111	−0.577807
$589$	−1.04718	−0.0431483
$590$	24.3668	1.00317
$591$	4.44360	0.182785
$592$	70.9542	2.91620
$593$	−45.9299	−1.88611	−0.943057	−	0.332632i	$-0.892063\pi$
$593$	−45.9299	−1.88611	−0.943057	+	0.332632i	$0.892063\pi$
$594$	31.7009	1.30070
$595$	−4.99386	−0.204728
$596$	39.7587	1.62858
$597$	4.83323	0.197811
$598$	−14.8371	−0.606734
$599$	9.14608	0.373699	0.186849	−	0.982389i	$-0.440172\pi$
$599$	9.14608	0.373699	0.186849	+	0.982389i	$0.440172\pi$
$600$	10.1412	0.414011
$601$	27.6742	1.12885	0.564427	−	0.825483i	$-0.309097\pi$
$601$	27.6742	1.12885	0.564427	+	0.825483i	$0.309097\pi$
$602$	−9.26180	−0.377483
$603$	10.2979	0.419363
$604$	97.7140	3.97593
$605$	5.89269	0.239572
$606$	−24.8638	−1.01002
$607$	14.8515	0.602803	0.301401	−	0.953497i	$-0.402546\pi$
$607$	14.8515	0.602803	0.301401	+	0.953497i	$0.402546\pi$
$608$	34.9204	1.41621
$609$	−2.42469	−0.0982536
$610$	35.9299	1.45476
$611$	−54.8515	−2.21905
$612$	28.9360	1.16967
$613$	22.0228	0.889492	0.444746	−	0.895657i	$-0.353294\pi$
$613$	22.0228	0.889492	0.444746	+	0.895657i	$0.353294\pi$
$614$	58.9504	2.37904
$615$	0.921622	0.0371634
$616$	−50.2472	−2.02452
$617$	40.8371	1.64404	0.822020	−	0.569459i	$-0.192847\pi$
$617$	40.8371	1.64404	0.822020	+	0.569459i	$0.192847\pi$
$618$	−24.2434	−0.975211
$619$	41.4596	1.66640	0.833201	−	0.552971i	$-0.186506\pi$
$619$	41.4596	1.66640	0.833201	+	0.552971i	$0.186506\pi$
$620$	5.30737	0.213149
$621$	3.88428	0.155871
$622$	23.4752	0.941269
$623$	−17.4452	−0.698928
$624$	−32.3812	−1.29629
$625$	−10.2885	−0.411538
$626$	83.3894	3.33291
$627$	−3.69102	−0.147405
$628$	−67.5729	−2.69645
$629$	−10.2557	−0.408920
$630$	18.3279	0.730200
$631$	22.5236	0.896650	0.448325	−	0.893871i	$-0.352021\pi$
$631$	22.5236	0.896650	0.448325	+	0.893871i	$0.352021\pi$
$632$	−134.650	−5.35609
$633$	7.79380	0.309776
$634$	2.22899	0.0885245
$635$	4.68035	0.185734
$636$	2.65368	0.105225
$637$	21.1194	0.836782
$638$	31.7009	1.25505
$639$	3.30406	0.130706
$640$	−48.8287	−1.93012
$641$	−3.26180	−0.128833	−0.0644166	−	0.997923i	$-0.520519\pi$
$641$	−3.26180	−0.128833	−0.0644166	+	0.997923i	$0.520519\pi$
$642$	21.6742	0.855413
$643$	−24.6114	−0.970580	−0.485290	−	0.874353i	$-0.661286\pi$
$643$	−24.6114	−0.970580	−0.485290	+	0.874353i	$0.661286\pi$
$644$	−9.84324	−0.387878
$645$	2.15676	0.0849222
$646$	−9.75872	−0.383952
$647$	−5.56093	−0.218623	−0.109311	−	0.994008i	$-0.534865\pi$
$647$	−5.56093	−0.218623	−0.109311	+	0.994008i	$0.534865\pi$
$648$	−58.5318	−2.29935
$649$	−20.0000	−0.785069
$650$	−24.4391	−0.958579
$651$	0.457980	0.0179497
$652$	−67.7152	−2.65193
$653$	−19.4908	−0.762733	−0.381367	−	0.924424i	$-0.624546\pi$
$653$	−19.4908	−0.762733	−0.381367	+	0.924424i	$0.624546\pi$
$654$	−16.3279	−0.638471
$655$	3.00614	0.117460
$656$	13.8371	0.540248
$657$	−31.5669	−1.23154
$658$	−50.0183	−1.94991
$659$	−30.5946	−1.19180	−0.595898	−	0.803060i	$-0.703204\pi$
$659$	−30.5946	−1.19180	−0.595898	+	0.803060i	$0.703204\pi$
$660$	18.7070	0.728169
$661$	9.76260	0.379721	0.189861	−	0.981811i	$-0.439196\pi$
$661$	9.76260	0.379721	0.189861	+	0.981811i	$0.439196\pi$
$662$	−60.7514	−2.36117
$663$	4.68035	0.181770
$664$	19.5174	0.757424
$665$	−4.49693	−0.174383
$666$	37.6391	1.45849
$667$	3.88428	0.150400
$668$	−53.1761	−2.05744
$669$	3.68649	0.142528
$670$	17.6020	0.680023
$671$	−29.4908	−1.13848
$672$	−15.2723	−0.589142
$673$	6.94828	0.267837	0.133918	−	0.990992i	$-0.457244\pi$
$673$	6.94828	0.267837	0.133918	+	0.990992i	$0.457244\pi$
$674$	72.3461	2.78667
$675$	6.39803	0.246260
$676$	31.1711	1.19889
$677$	−22.0060	−0.845758	−0.422879	−	0.906186i	$-0.638980\pi$
$677$	−22.0060	−0.845758	−0.422879	+	0.906186i	$0.638980\pi$
$678$	1.18956	0.0456848
$679$	6.07223	0.233031
$680$	30.9360	1.18634
$681$	−2.81432	−0.107845
$682$	−5.98771	−0.229281
$683$	21.2339	0.812493	0.406247	−	0.913764i	$-0.366837\pi$
$683$	21.2339	0.812493	0.406247	+	0.913764i	$0.366837\pi$
$684$	26.0566	0.996300
$685$	17.6286	0.673555
$686$	46.9627	1.79304
$687$	−9.34632	−0.356584
$688$	32.3812	1.23452
$689$	−4.00000	−0.152388
$690$	3.15061	0.119942
$691$	−8.27125	−0.314653	−0.157327	−	0.987547i	$-0.550288\pi$
$691$	−8.27125	−0.314653	−0.157327	+	0.987547i	$0.550288\pi$
$692$	−64.9192	−2.46786
$693$	−15.0433	−0.571448
$694$	−36.3123	−1.37840
$695$	−34.8904	−1.32347
$696$	15.0205	0.569351
$697$	−2.00000	−0.0757554
$698$	38.7175	1.46548
$699$	11.4329	0.432433
$700$	−16.2134	−0.612809
$701$	−23.1857	−0.875711	−0.437856	−	0.899045i	$-0.644262\pi$
$701$	−23.1857	−0.875711	−0.437856	+	0.899045i	$0.644262\pi$
$702$	−36.1978	−1.36620
$703$	−9.23513	−0.348310
$704$	94.4834	3.56098
$705$	11.6475	0.438672
$706$	47.3979	1.78384
$707$	24.8638	0.935098
$708$	−15.1506	−0.569395
$709$	34.0144	1.27744	0.638718	−	0.769441i	$-0.279465\pi$
$709$	34.0144	1.27744	0.638718	+	0.769441i	$0.279465\pi$
$710$	5.64754	0.211948
$711$	−40.3123	−1.51183
$712$	108.070	4.05009
$713$	−0.733670	−0.0274762
$714$	4.26794	0.159724
$715$	−28.1978	−1.05454
$716$	102.184	3.81881
$717$	2.34858	0.0877095
$718$	−43.8843	−1.63775
$719$	−29.2872	−1.09223	−0.546115	−	0.837710i	$-0.683894\pi$
$719$	−29.2872	−1.09223	−0.546115	+	0.837710i	$0.683894\pi$
$720$	−64.0782	−2.38805
$721$	24.2434	0.902870
$722$	42.6886	1.58870
$723$	3.98562	0.148227
$724$	−82.4801	−3.06535
$725$	6.39803	0.237617
$726$	−5.03612	−0.186908
$727$	−26.6791	−0.989474	−0.494737	−	0.869043i	$-0.664736\pi$
$727$	−26.6791	−0.989474	−0.494737	+	0.869043i	$0.664736\pi$
$728$	57.3751	2.12646
$729$	−12.5441	−0.464597
$730$	−53.9565	−1.99702
$731$	−4.68035	−0.173109
$732$	−22.3402	−0.825717
$733$	38.6491	1.42754	0.713769	−	0.700381i	$-0.246986\pi$
$733$	38.6491	1.42754	0.713769	+	0.700381i	$0.246986\pi$
$734$	16.1834	0.597341
$735$	−4.48464	−0.165419
$736$	24.4657	0.901819
$737$	−14.4475	−0.532180
$738$	7.34017	0.270196
$739$	26.1568	0.962192	0.481096	−	0.876668i	$-0.340239\pi$
$739$	26.1568	0.962192	0.481096	+	0.876668i	$0.340239\pi$
$740$	46.8059	1.72062
$741$	4.21461	0.154828
$742$	−3.64754	−0.133905
$743$	−18.5548	−0.680709	−0.340355	−	0.940297i	$-0.610547\pi$
$743$	−18.5548	−0.680709	−0.340355	+	0.940297i	$0.610547\pi$
$744$	−2.83710	−0.104013
$745$	12.7259	0.466242
$746$	37.0805	1.35761
$747$	5.84324	0.213793
$748$	−40.5958	−1.48433
$749$	−21.6742	−0.791958
$750$	17.6742	0.645371
$751$	23.0361	0.840600	0.420300	−	0.907385i	$-0.361925\pi$
$751$	23.0361	0.840600	0.420300	+	0.907385i	$0.361925\pi$
$752$	174.874	6.37702
$753$	−9.57531	−0.348944
$754$	−36.1978	−1.31825
$755$	31.2762	1.13826
$756$	−24.0144	−0.873394
$757$	−37.5851	−1.36606	−0.683028	−	0.730392i	$-0.739337\pi$
$757$	−37.5851	−1.36606	−0.683028	+	0.730392i	$0.739337\pi$
$758$	−76.1276	−2.76508
$759$	−2.58598	−0.0938653
$760$	27.8576	1.01050
$761$	5.39576	0.195596	0.0977982	−	0.995206i	$-0.468820\pi$
$761$	5.39576	0.195596	0.0977982	+	0.995206i	$0.468820\pi$
$762$	−4.00000	−0.144905
$763$	16.3279	0.591109
$764$	−99.0792	−3.58456
$765$	9.26180	0.334861
$766$	49.9988	1.80653
$767$	22.8371	0.824600
$768$	14.9249	0.538557
$769$	−6.19779	−0.223498	−0.111749	−	0.993736i	$-0.535645\pi$
$769$	−6.19779	−0.223498	−0.111749	+	0.993736i	$0.535645\pi$
$770$	−25.7132	−0.926638
$771$	12.2823	0.442337
$772$	−60.1399	−2.16448
$773$	17.1194	0.615743	0.307871	−	0.951428i	$-0.400383\pi$
$773$	17.1194	0.615743	0.307871	+	0.951428i	$0.400383\pi$
$774$	17.1773	0.617424
$775$	−1.20847	−0.0434096
$776$	−37.6163	−1.35035
$777$	4.03895	0.144896
$778$	−59.1461	−2.12049
$779$	−1.80098	−0.0645270
$780$	−21.3607	−0.764835
$781$	−4.63543	−0.165869
$782$	−6.83710	−0.244494
$783$	9.47641	0.338659
$784$	−67.3318	−2.40471
$785$	−21.6286	−0.771959
$786$	−2.56916	−0.0916390
$787$	0.595825	0.0212389	0.0106194	−	0.999944i	$-0.496620\pi$
$787$	0.595825	0.0212389	0.0106194	+	0.999944i	$0.496620\pi$
$788$	44.0098	1.56779
$789$	8.31948	0.296181
$790$	−68.9048	−2.45152
$791$	−1.18956	−0.0422959
$792$	93.1904	3.31138
$793$	33.6742	1.19581
$794$	−15.7587	−0.559256
$795$	0.849388	0.0301247
$796$	47.8687	1.69666
$797$	13.9155	0.492912	0.246456	−	0.969154i	$-0.420734\pi$
$797$	13.9155	0.492912	0.246456	+	0.969154i	$0.420734\pi$
$798$	3.84324	0.136049
$799$	−25.2762	−0.894207
$800$	40.2990	1.42478
$801$	32.3545	1.14319
$802$	69.1955	2.44338
$803$	44.2868	1.56285
$804$	−10.9444	−0.385980
$805$	−3.15061	−0.111044
$806$	6.83710	0.240827
$807$	−5.54411	−0.195162
$808$	−154.026	−5.41863
$809$	−36.1568	−1.27120	−0.635602	−	0.772017i	$-0.719248\pi$
$809$	−36.1568	−1.27120	−0.635602	+	0.772017i	$0.719248\pi$
$810$	−29.9526	−1.05243
$811$	−8.76487	−0.307776	−0.153888	−	0.988088i	$-0.549180\pi$
$811$	−8.76487	−0.307776	−0.153888	+	0.988088i	$0.549180\pi$
$812$	−24.0144	−0.842739
$813$	1.84324	0.0646454
$814$	−52.8059	−1.85085
$815$	−21.6742	−0.759214
$816$	−14.9216	−0.522361
$817$	−4.21461	−0.147451
$818$	25.5792	0.894355
$819$	17.1773	0.600223
$820$	9.12783	0.318758
$821$	−28.8599	−1.00722	−0.503609	−	0.863932i	$-0.667995\pi$
$821$	−28.8599	−1.00722	−0.503609	+	0.863932i	$0.667995\pi$
$822$	−15.0661	−0.525490
$823$	21.7165	0.756988	0.378494	−	0.925604i	$-0.376442\pi$
$823$	21.7165	0.756988	0.378494	+	0.925604i	$0.376442\pi$
$824$	−150.183	−5.23187
$825$	−4.25953	−0.148298
$826$	20.8248	0.724588
$827$	24.0156	0.835104	0.417552	−	0.908653i	$-0.362888\pi$
$827$	24.0156	0.835104	0.417552	+	0.908653i	$0.362888\pi$
$828$	18.2557	0.634428
$829$	−0.187293	−0.00650496	−0.00325248	−	0.999995i	$-0.501035\pi$
$829$	−0.187293	−0.00650496	−0.00325248	+	0.999995i	$0.501035\pi$
$830$	9.98771	0.346679
$831$	5.11942	0.177591
$832$	−107.886	−3.74029
$833$	9.73206	0.337196
$834$	29.8187	1.03254
$835$	−17.0205	−0.589019
$836$	−36.5562	−1.26432
$837$	−1.78992	−0.0618687
$838$	−89.4284	−3.08925
$839$	−9.65860	−0.333452	−0.166726	−	0.986003i	$-0.553320\pi$
$839$	−9.65860	−0.333452	−0.166726	+	0.986003i	$0.553320\pi$
$840$	−12.1834	−0.420368
$841$	−19.5236	−0.673227
$842$	28.7070	0.989309
$843$	2.86830	0.0987894
$844$	77.1904	2.65700
$845$	9.97721	0.343227
$846$	92.7657	3.18935
$847$	5.03612	0.173043
$848$	12.7526	0.437925
$849$	−1.64545	−0.0564717
$850$	−11.2618	−0.386276
$851$	−6.47027	−0.221798
$852$	−3.51148	−0.120301
$853$	25.0882	0.859004	0.429502	−	0.903066i	$-0.358689\pi$
$853$	25.0882	0.859004	0.429502	+	0.903066i	$0.358689\pi$
$854$	30.7070	1.05077
$855$	8.34017	0.285228
$856$	134.268	4.58917
$857$	41.9604	1.43334	0.716670	−	0.697413i	$-0.245665\pi$
$857$	41.9604	1.43334	0.716670	+	0.697413i	$0.245665\pi$
$858$	24.0989	0.822723
$859$	14.1256	0.481958	0.240979	−	0.970530i	$-0.422532\pi$
$859$	14.1256	0.481958	0.240979	+	0.970530i	$0.422532\pi$
$860$	21.3607	0.728394
$861$	0.787653	0.0268431
$862$	29.0082	0.988024
$863$	38.5380	1.31185	0.655924	−	0.754827i	$-0.272279\pi$
$863$	38.5380	1.31185	0.655924	+	0.754827i	$0.272279\pi$
$864$	59.6886	2.03065
$865$	−20.7792	−0.706515
$866$	−69.2783	−2.35417
$867$	−7.00946	−0.238054
$868$	4.53588	0.153958
$869$	56.5562	1.91854
$870$	7.68649	0.260596
$871$	16.4969	0.558977
$872$	−101.148	−3.42531
$873$	−11.2618	−0.381154
$874$	−6.15676	−0.208255
$875$	−17.6742	−0.597497
$876$	33.5486	1.13350
$877$	−30.7936	−1.03983	−0.519913	−	0.854219i	$-0.674036\pi$
$877$	−30.7936	−1.03983	−0.519913	+	0.854219i	$0.674036\pi$
$878$	−79.9409	−2.69788
$879$	−4.91709	−0.165849
$880$	89.8987	3.03048
$881$	42.8248	1.44280	0.721402	−	0.692516i	$-0.243498\pi$
$881$	42.8248	1.44280	0.721402	+	0.692516i	$0.243498\pi$
$882$	−35.7175	−1.20267
$883$	48.8503	1.64394	0.821971	−	0.569529i	$-0.192875\pi$
$883$	48.8503	1.64394	0.821971	+	0.569529i	$0.192875\pi$
$884$	46.3545	1.55907
$885$	−4.84939	−0.163010
$886$	−55.2639	−1.85663
$887$	−30.2401	−1.01536	−0.507681	−	0.861545i	$-0.669497\pi$
$887$	−30.2401	−1.01536	−0.507681	+	0.861545i	$0.669497\pi$
$888$	−25.0205	−0.839634
$889$	4.00000	0.134156
$890$	55.3028	1.85376
$891$	24.5848	0.823621
$892$	36.5113	1.22249
$893$	−22.7610	−0.761668
$894$	−10.8760	−0.363750
$895$	32.7070	1.09327
$896$	−41.7308	−1.39413
$897$	2.95282	0.0985918
$898$	−36.8515	−1.22975
$899$	−1.78992	−0.0596972
$900$	30.0700	1.00233
$901$	−1.84324	−0.0614074
$902$	−10.2979	−0.342883
$903$	1.84324	0.0613393
$904$	7.36910	0.245093
$905$	−26.4001	−0.877570
$906$	−26.7298	−0.888038
$907$	2.86830	0.0952403	0.0476201	−	0.998866i	$-0.484836\pi$
$907$	2.86830	0.0952403	0.0476201	+	0.998866i	$0.484836\pi$
$908$	−27.8732	−0.925005
$909$	−46.1133	−1.52948
$910$	29.3607	0.973297
$911$	10.8227	0.358573	0.179286	−	0.983797i	$-0.442621\pi$
$911$	10.8227	0.358573	0.179286	+	0.983797i	$0.442621\pi$
$912$	−13.4368	−0.444937
$913$	−8.19779	−0.271307
$914$	−23.0928	−0.763840
$915$	−7.15061	−0.236392
$916$	−92.5667	−3.05849
$917$	2.56916	0.0848412
$918$	−16.6803	−0.550533
$919$	50.4235	1.66332	0.831658	−	0.555288i	$-0.187392\pi$
$919$	50.4235	1.66332	0.831658	+	0.555288i	$0.187392\pi$
$920$	19.5174	0.643471
$921$	−11.7321	−0.386585
$922$	0.944409	0.0311024
$923$	5.29299	0.174221
$924$	15.9877	0.525957
$925$	−10.6576	−0.350418
$926$	−52.2667	−1.71759
$927$	−44.9627	−1.47677
$928$	59.6886	1.95937
$929$	−36.5692	−1.19980	−0.599898	−	0.800077i	$-0.704792\pi$
$929$	−36.5692	−1.19980	−0.599898	+	0.800077i	$0.704792\pi$
$930$	−1.45184	−0.0476076
$931$	8.76364	0.287217
$932$	113.233	3.70906
$933$	−4.67194	−0.152952
$934$	63.4473	2.07606
$935$	−12.9939	−0.424945
$936$	−106.410	−3.47812
$937$	−10.0989	−0.329917	−0.164958	−	0.986301i	$-0.552749\pi$
$937$	−10.0989	−0.329917	−0.164958	+	0.986301i	$0.552749\pi$
$938$	15.0433	0.491181
$939$	−16.5958	−0.541584
$940$	115.358	3.76257
$941$	9.94668	0.324252	0.162126	−	0.986770i	$-0.448165\pi$
$941$	9.94668	0.324252	0.162126	+	0.986770i	$0.448165\pi$
$942$	18.4846	0.602262
$943$	−1.26180	−0.0410897
$944$	−72.8080	−2.36970
$945$	−7.68649	−0.250042
$946$	−24.0989	−0.783523
$947$	−9.31512	−0.302701	−0.151350	−	0.988480i	$-0.548362\pi$
$947$	−9.31512	−0.302701	−0.151350	+	0.988480i	$0.548362\pi$
$948$	42.8431	1.39148
$949$	−50.5692	−1.64154
$950$	−10.1412	−0.329023
$951$	−0.443604	−0.0143849
$952$	26.4391	0.856895
$953$	8.74417	0.283251	0.141626	−	0.989920i	$-0.454767\pi$
$953$	8.74417	0.283251	0.141626	+	0.989920i	$0.454767\pi$
$954$	6.76487	0.219021
$955$	−31.7132	−1.02621
$956$	23.2606	0.752301
$957$	−6.30898	−0.203940
$958$	−10.7948	−0.348765
$959$	15.0661	0.486509
$960$	22.9093	0.739396
$961$	−30.6619	−0.989094
$962$	60.2967	1.94404
$963$	40.1978	1.29536
$964$	39.4740	1.27137
$965$	−19.2495	−0.619664
$966$	2.69263	0.0866340
$967$	4.65037	0.149546	0.0747729	−	0.997201i	$-0.476177\pi$
$967$	4.65037	0.149546	0.0747729	+	0.997201i	$0.476177\pi$
$968$	−31.1978	−1.00274
$969$	1.94214	0.0623906
$970$	−19.2495	−0.618064
$971$	−47.6586	−1.52944	−0.764719	−	0.644364i	$-0.777122\pi$
$971$	−47.6586	−1.52944	−0.764719	+	0.644364i	$0.777122\pi$
$972$	67.9409	2.17921
$973$	−29.8187	−0.955943
$974$	98.6902	3.16224
$975$	4.86376	0.155765
$976$	−107.358	−3.43645
$977$	12.2557	0.392093	0.196047	−	0.980595i	$-0.437190\pi$
$977$	12.2557	0.392093	0.196047	+	0.980595i	$0.437190\pi$
$978$	18.5236	0.592319
$979$	−45.3919	−1.45073
$980$	−44.4163	−1.41883
$981$	−30.2823	−0.966840
$982$	31.7854	1.01431
$983$	54.9360	1.75219	0.876093	−	0.482142i	$-0.160141\pi$
$983$	54.9360	1.75219	0.876093	+	0.482142i	$0.160141\pi$
$984$	−4.87936	−0.155548
$985$	14.0866	0.448837
$986$	−16.6803	−0.531210
$987$	9.95443	0.316853
$988$	41.7419	1.32799
$989$	−2.95282	−0.0938942
$990$	47.6886	1.51564
$991$	14.4136	0.457864	0.228932	−	0.973442i	$-0.426477\pi$
$991$	14.4136	0.457864	0.228932	+	0.973442i	$0.426477\pi$
$992$	−11.2741	−0.357952
$993$	12.0905	0.383680
$994$	4.82660	0.153090
$995$	15.3217	0.485732
$996$	−6.21008	−0.196774
$997$	−6.64915	−0.210581	−0.105290	−	0.994442i	$-0.533577\pi$
$997$	−6.64915	−0.210581	−0.105290	+	0.994442i	$0.533577\pi$
$998$	−44.8092	−1.41841
$999$	−15.7854	−0.499428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	41.2.a.a.1.1	✓	3
3.2	odd	2		369.2.a.f.1.3		3
4.3	odd	2		656.2.a.f.1.2		3
5.2	odd	4		1025.2.b.h.124.1		6
5.3	odd	4		1025.2.b.h.124.6		6
5.4	even	2		1025.2.a.j.1.3		3
7.6	odd	2		2009.2.a.g.1.1		3
8.3	odd	2		2624.2.a.q.1.2		3
8.5	even	2		2624.2.a.r.1.2		3
11.10	odd	2		4961.2.a.d.1.3		3
12.11	even	2		5904.2.a.bk.1.1		3
13.12	even	2		6929.2.a.b.1.3		3
15.14	odd	2		9225.2.a.bv.1.1		3
41.40	even	2		1681.2.a.d.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.2.a.a.1.1	✓	3	1.1	even	1	trivial
369.2.a.f.1.3		3	3.2	odd	2
656.2.a.f.1.2		3	4.3	odd	2
1025.2.a.j.1.3		3	5.4	even	2
1025.2.b.h.124.1		6	5.2	odd	4
1025.2.b.h.124.6		6	5.3	odd	4
1681.2.a.d.1.1		3	41.40	even	2
2009.2.a.g.1.1		3	7.6	odd	2
2624.2.a.q.1.2		3	8.3	odd	2
2624.2.a.r.1.2		3	8.5	even	2
4961.2.a.d.1.3		3	11.10	odd	2
5904.2.a.bk.1.1		3	12.11	even	2
6929.2.a.b.1.3		3	13.12	even	2
9225.2.a.bv.1.1		3	15.14	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:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	41.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70928 q^{2} +0.539189 q^{3} +5.34017 q^{4} +1.70928 q^{5} -1.46081 q^{6} +1.46081 q^{7} -9.04945 q^{8} -2.70928 q^{9} +O(q^{10})\)	\(q-2.70928 q^{2} +0.539189 q^{3} +5.34017 q^{4} +1.70928 q^{5} -1.46081 q^{6} +1.46081 q^{7} -9.04945 q^{8} -2.70928 q^{9} -4.63090 q^{10} +3.80098 q^{11} +2.87936 q^{12} -4.34017 q^{13} -3.95774 q^{14} +0.921622 q^{15} +13.8371 q^{16} -2.00000 q^{17} +7.34017 q^{18} -1.80098 q^{19} +9.12783 q^{20} +0.787653 q^{21} -10.2979 q^{22} -1.26180 q^{23} -4.87936 q^{24} -2.07838 q^{25} +11.7587 q^{26} -3.07838 q^{27} +7.80098 q^{28} -3.07838 q^{29} -2.49693 q^{30} +0.581449 q^{31} -19.3896 q^{32} +2.04945 q^{33} +5.41855 q^{34} +2.49693 q^{35} -14.4680 q^{36} +5.12783 q^{37} +4.87936 q^{38} -2.34017 q^{39} -15.4680 q^{40} +1.00000 q^{41} -2.13397 q^{42} +2.34017 q^{43} +20.2979 q^{44} -4.63090 q^{45} +3.41855 q^{46} +12.6381 q^{47} +7.46081 q^{48} -4.86603 q^{49} +5.63090 q^{50} -1.07838 q^{51} -23.1773 q^{52} +0.921622 q^{53} +8.34017 q^{54} +6.49693 q^{55} -13.2195 q^{56} -0.971071 q^{57} +8.34017 q^{58} -5.26180 q^{59} +4.92162 q^{60} -7.75872 q^{61} -1.57531 q^{62} -3.95774 q^{63} +24.8576 q^{64} -7.41855 q^{65} -5.55252 q^{66} -3.80098 q^{67} -10.6803 q^{68} -0.680346 q^{69} -6.76487 q^{70} -1.21953 q^{71} +24.5174 q^{72} +11.6514 q^{73} -13.8927 q^{74} -1.12064 q^{75} -9.61757 q^{76} +5.55252 q^{77} +6.34017 q^{78} +14.8794 q^{79} +23.6514 q^{80} +6.46800 q^{81} -2.70928 q^{82} -2.15676 q^{83} +4.20620 q^{84} -3.41855 q^{85} -6.34017 q^{86} -1.65983 q^{87} -34.3968 q^{88} -11.9421 q^{89} +12.5464 q^{90} -6.34017 q^{91} -6.73820 q^{92} +0.313511 q^{93} -34.2401 q^{94} -3.07838 q^{95} -10.4547 q^{96} +4.15676 q^{97} +13.1834 q^{98} -10.2979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - 6 q^{6} + 6 q^{7} - 9 q^{8} - q^{9}+O(q^{10}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - 6 * q^6 + 6 * q^7 - 9 * q^8 - q^9	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - 6 q^{6} + 6 q^{7} - 9 q^{8} - q^{9} - 10 q^{10} + 2 q^{11} - 4 q^{12} - 2 q^{13} + 4 q^{14} + 6 q^{15} + 13 q^{16} - 6 q^{17} + 11 q^{18} + 4 q^{19} + 6 q^{20} - 8 q^{21} - 4 q^{22} + 4 q^{23} - 2 q^{24} - 3 q^{25} + 10 q^{26} - 6 q^{27} + 14 q^{28} - 6 q^{29} + 10 q^{30} + 16 q^{31} - 29 q^{32} - 12 q^{33} + 2 q^{34} - 10 q^{35} - 11 q^{36} - 6 q^{37} + 2 q^{38} + 4 q^{39} - 14 q^{40} + 3 q^{41} - 20 q^{42} - 4 q^{43} + 34 q^{44} - 10 q^{45} - 4 q^{46} + 24 q^{48} - q^{49} + 13 q^{50} - 30 q^{52} + 6 q^{53} + 14 q^{54} + 2 q^{55} - 16 q^{56} + 12 q^{57} + 14 q^{58} - 8 q^{59} + 18 q^{60} + 2 q^{61} + 16 q^{62} + 4 q^{63} + 13 q^{64} - 8 q^{65} - 16 q^{66} - 2 q^{67} - 10 q^{68} + 20 q^{69} - 30 q^{70} + 20 q^{71} + 23 q^{72} - 2 q^{73} - 30 q^{74} - 16 q^{75} - 24 q^{76} + 16 q^{77} + 8 q^{78} + 32 q^{79} + 34 q^{80} - 13 q^{81} - q^{82} - 12 q^{84} + 4 q^{85} - 8 q^{86} - 16 q^{87} - 40 q^{88} - 6 q^{89} + 2 q^{90} - 8 q^{91} - 28 q^{92} - 12 q^{93} - 46 q^{94} - 6 q^{95} + 2 q^{96} + 6 q^{97} + 35 q^{98} - 4 q^{99}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - 6 * q^6 + 6 * q^7 - 9 * q^8 - q^9 - 10 * q^10 + 2 * q^11 - 4 * q^12 - 2 * q^13 + 4 * q^14 + 6 * q^15 + 13 * q^16 - 6 * q^17 + 11 * q^18 + 4 * q^19 + 6 * q^20 - 8 * q^21 - 4 * q^22 + 4 * q^23 - 2 * q^24 - 3 * q^25 + 10 * q^26 - 6 * q^27 + 14 * q^28 - 6 * q^29 + 10 * q^30 + 16 * q^31 - 29 * q^32 - 12 * q^33 + 2 * q^34 - 10 * q^35 - 11 * q^36 - 6 * q^37 + 2 * q^38 + 4 * q^39 - 14 * q^40 + 3 * q^41 - 20 * q^42 - 4 * q^43 + 34 * q^44 - 10 * q^45 - 4 * q^46 + 24 * q^48 - q^49 + 13 * q^50 - 30 * q^52 + 6 * q^53 + 14 * q^54 + 2 * q^55 - 16 * q^56 + 12 * q^57 + 14 * q^58 - 8 * q^59 + 18 * q^60 + 2 * q^61 + 16 * q^62 + 4 * q^63 + 13 * q^64 - 8 * q^65 - 16 * q^66 - 2 * q^67 - 10 * q^68 + 20 * q^69 - 30 * q^70 + 20 * q^71 + 23 * q^72 - 2 * q^73 - 30 * q^74 - 16 * q^75 - 24 * q^76 + 16 * q^77 + 8 * q^78 + 32 * q^79 + 34 * q^80 - 13 * q^81 - q^82 - 12 * q^84 + 4 * q^85 - 8 * q^86 - 16 * q^87 - 40 * q^88 - 6 * q^89 + 2 * q^90 - 8 * q^91 - 28 * q^92 - 12 * q^93 - 46 * q^94 - 6 * q^95 + 2 * q^96 + 6 * q^97 + 35 * q^98 - 4 * q^99

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