LMFDB - Embedded newform 8031.2.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, 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("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$92$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.42805 q^{2} +1.00000 q^{3} +3.89545 q^{4} -1.43452 q^{5} -2.42805 q^{6} -1.67888 q^{7} -4.60225 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.42805 q^{2} +1.00000 q^{3} +3.89545 q^{4} -1.43452 q^{5} -2.42805 q^{6} -1.67888 q^{7} -4.60225 q^{8} +1.00000 q^{9} +3.48309 q^{10} +3.29626 q^{11} +3.89545 q^{12} +3.93906 q^{13} +4.07641 q^{14} -1.43452 q^{15} +3.38362 q^{16} +4.28727 q^{17} -2.42805 q^{18} -3.65077 q^{19} -5.58809 q^{20} -1.67888 q^{21} -8.00350 q^{22} +3.58725 q^{23} -4.60225 q^{24} -2.94215 q^{25} -9.56425 q^{26} +1.00000 q^{27} -6.53999 q^{28} -7.85512 q^{29} +3.48309 q^{30} +7.99854 q^{31} +0.988892 q^{32} +3.29626 q^{33} -10.4097 q^{34} +2.40838 q^{35} +3.89545 q^{36} -10.1617 q^{37} +8.86428 q^{38} +3.93906 q^{39} +6.60202 q^{40} +0.398644 q^{41} +4.07641 q^{42} +5.01834 q^{43} +12.8404 q^{44} -1.43452 q^{45} -8.71004 q^{46} -10.5977 q^{47} +3.38362 q^{48} -4.18136 q^{49} +7.14371 q^{50} +4.28727 q^{51} +15.3444 q^{52} +2.63647 q^{53} -2.42805 q^{54} -4.72855 q^{55} +7.72662 q^{56} -3.65077 q^{57} +19.0727 q^{58} -6.35259 q^{59} -5.58809 q^{60} -0.261087 q^{61} -19.4209 q^{62} -1.67888 q^{63} -9.16832 q^{64} -5.65065 q^{65} -8.00350 q^{66} -14.2583 q^{67} +16.7008 q^{68} +3.58725 q^{69} -5.84769 q^{70} +1.21995 q^{71} -4.60225 q^{72} -11.8903 q^{73} +24.6732 q^{74} -2.94215 q^{75} -14.2214 q^{76} -5.53403 q^{77} -9.56425 q^{78} +6.29755 q^{79} -4.85387 q^{80} +1.00000 q^{81} -0.967930 q^{82} +12.6255 q^{83} -6.53999 q^{84} -6.15017 q^{85} -12.1848 q^{86} -7.85512 q^{87} -15.1702 q^{88} +2.96264 q^{89} +3.48309 q^{90} -6.61320 q^{91} +13.9740 q^{92} +7.99854 q^{93} +25.7319 q^{94} +5.23710 q^{95} +0.988892 q^{96} +2.19808 q^{97} +10.1526 q^{98} +3.29626 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9}+O(q^{10}) $ 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9	\( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9} - 44 q^{10} - 24 q^{11} + 70 q^{12} - 48 q^{13} - 29 q^{14} - 18 q^{15} + 26 q^{16} - 69 q^{17} - 6 q^{18} - 74 q^{19} - 42 q^{20} - 42 q^{21} - 62 q^{22} - 19 q^{23} - 15 q^{24} + 16 q^{25} - 27 q^{26} + 92 q^{27} - 101 q^{28} - 54 q^{29} - 44 q^{30} - 67 q^{31} - 36 q^{32} - 24 q^{33} - 63 q^{34} - 31 q^{35} + 70 q^{36} - 70 q^{37} - 18 q^{38} - 48 q^{39} - 125 q^{40} - 98 q^{41} - 29 q^{42} - 159 q^{43} - 52 q^{44} - 18 q^{45} - 68 q^{46} - 15 q^{47} + 26 q^{48} - 28 q^{49} - 7 q^{50} - 69 q^{51} - 98 q^{52} - 23 q^{53} - 6 q^{54} - 93 q^{55} - 48 q^{56} - 74 q^{57} - 37 q^{58} - 36 q^{59} - 42 q^{60} - 172 q^{61} - 26 q^{62} - 42 q^{63} - 23 q^{64} - 66 q^{65} - 62 q^{66} - 143 q^{67} - 74 q^{68} - 19 q^{69} - 30 q^{70} - 9 q^{71} - 15 q^{72} - 134 q^{73} - 19 q^{74} + 16 q^{75} - 157 q^{76} - 25 q^{77} - 27 q^{78} - 138 q^{79} - 29 q^{80} + 92 q^{81} - 61 q^{82} - 24 q^{83} - 101 q^{84} - 84 q^{85} + 14 q^{86} - 54 q^{87} - 140 q^{88} - 148 q^{89} - 44 q^{90} - 115 q^{91} - 12 q^{92} - 67 q^{93} - 79 q^{94} - 10 q^{95} - 36 q^{96} - 165 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9 - 44 * q^10 - 24 * q^11 + 70 * q^12 - 48 * q^13 - 29 * q^14 - 18 * q^15 + 26 * q^16 - 69 * q^17 - 6 * q^18 - 74 * q^19 - 42 * q^20 - 42 * q^21 - 62 * q^22 - 19 * q^23 - 15 * q^24 + 16 * q^25 - 27 * q^26 + 92 * q^27 - 101 * q^28 - 54 * q^29 - 44 * q^30 - 67 * q^31 - 36 * q^32 - 24 * q^33 - 63 * q^34 - 31 * q^35 + 70 * q^36 - 70 * q^37 - 18 * q^38 - 48 * q^39 - 125 * q^40 - 98 * q^41 - 29 * q^42 - 159 * q^43 - 52 * q^44 - 18 * q^45 - 68 * q^46 - 15 * q^47 + 26 * q^48 - 28 * q^49 - 7 * q^50 - 69 * q^51 - 98 * q^52 - 23 * q^53 - 6 * q^54 - 93 * q^55 - 48 * q^56 - 74 * q^57 - 37 * q^58 - 36 * q^59 - 42 * q^60 - 172 * q^61 - 26 * q^62 - 42 * q^63 - 23 * q^64 - 66 * q^65 - 62 * q^66 - 143 * q^67 - 74 * q^68 - 19 * q^69 - 30 * q^70 - 9 * q^71 - 15 * q^72 - 134 * q^73 - 19 * q^74 + 16 * q^75 - 157 * q^76 - 25 * q^77 - 27 * q^78 - 138 * q^79 - 29 * q^80 + 92 * q^81 - 61 * q^82 - 24 * q^83 - 101 * q^84 - 84 * q^85 + 14 * q^86 - 54 * q^87 - 140 * q^88 - 148 * q^89 - 44 * q^90 - 115 * q^91 - 12 * q^92 - 67 * q^93 - 79 * q^94 - 10 * q^95 - 36 * q^96 - 165 * q^97 + 36 * q^98 - 24 * 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 $

$2$	−2.42805	−1.71689	−0.858447	−	0.512903i	$-0.828570\pi$
$2$	−2.42805	−1.71689	−0.858447	+	0.512903i	$0.828570\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.89545	1.94772
$5$	−1.43452	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$5$	−1.43452	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$6$	−2.42805	−0.991249
$7$	−1.67888	−0.634557	−0.317278	−	0.948332i	$-0.602769\pi$
$7$	−1.67888	−0.634557	−0.317278	+	0.948332i	$0.602769\pi$
$8$	−4.60225	−1.62714
$9$	1.00000	0.333333
$10$	3.48309	1.10145
$11$	3.29626	0.993860	0.496930	−	0.867790i	$-0.334460\pi$
$11$	3.29626	0.993860	0.496930	+	0.867790i	$0.334460\pi$
$12$	3.89545	1.12452
$13$	3.93906	1.09250	0.546249	−	0.837623i	$-0.316055\pi$
$13$	3.93906	1.09250	0.546249	+	0.837623i	$0.316055\pi$
$14$	4.07641	1.08947
$15$	−1.43452	−0.370391
$16$	3.38362	0.845905
$17$	4.28727	1.03982	0.519908	−	0.854223i	$-0.325966\pi$
$17$	4.28727	1.03982	0.519908	+	0.854223i	$0.325966\pi$
$18$	−2.42805	−0.572298
$19$	−3.65077	−0.837545	−0.418772	−	0.908091i	$-0.637539\pi$
$19$	−3.65077	−0.837545	−0.418772	+	0.908091i	$0.637539\pi$
$20$	−5.58809	−1.24954
$21$	−1.67888	−0.366362
$22$	−8.00350	−1.70635
$23$	3.58725	0.747994	0.373997	−	0.927430i	$-0.377987\pi$
$23$	3.58725	0.747994	0.373997	+	0.927430i	$0.377987\pi$
$24$	−4.60225	−0.939430
$25$	−2.94215	−0.588431
$26$	−9.56425	−1.87570
$27$	1.00000	0.192450
$28$	−6.53999	−1.23594
$29$	−7.85512	−1.45866	−0.729330	−	0.684162i	$-0.760168\pi$
$29$	−7.85512	−1.45866	−0.729330	+	0.684162i	$0.760168\pi$
$30$	3.48309	0.635922
$31$	7.99854	1.43658	0.718290	−	0.695744i	$-0.244925\pi$
$31$	7.99854	1.43658	0.718290	+	0.695744i	$0.244925\pi$
$32$	0.988892	0.174813
$33$	3.29626	0.573806
$34$	−10.4097	−1.78525
$35$	2.40838	0.407091
$36$	3.89545	0.649241
$37$	−10.1617	−1.67057	−0.835287	−	0.549814i	$-0.814698\pi$
$37$	−10.1617	−1.67057	−0.835287	+	0.549814i	$0.814698\pi$
$38$	8.86428	1.43798
$39$	3.93906	0.630754
$40$	6.60202	1.04387
$41$	0.398644	0.0622577	0.0311289	−	0.999515i	$-0.490090\pi$
$41$	0.398644	0.0622577	0.0311289	+	0.999515i	$0.490090\pi$
$42$	4.07641	0.629004
$43$	5.01834	0.765289	0.382644	−	0.923896i	$-0.375013\pi$
$43$	5.01834	0.765289	0.382644	+	0.923896i	$0.375013\pi$
$44$	12.8404	1.93577
$45$	−1.43452	−0.213845
$46$	−8.71004	−1.28423
$47$	−10.5977	−1.54584	−0.772919	−	0.634504i	$-0.781204\pi$
$47$	−10.5977	−1.54584	−0.772919	+	0.634504i	$0.781204\pi$
$48$	3.38362	0.488383
$49$	−4.18136	−0.597338
$50$	7.14371	1.01027
$51$	4.28727	0.600338
$52$	15.3444	2.12788
$53$	2.63647	0.362147	0.181073	−	0.983470i	$-0.442043\pi$
$53$	2.63647	0.362147	0.181073	+	0.983470i	$0.442043\pi$
$54$	−2.42805	−0.330416
$55$	−4.72855	−0.637598
$56$	7.72662	1.03251
$57$	−3.65077	−0.483557
$58$	19.0727	2.50436
$59$	−6.35259	−0.827037	−0.413519	−	0.910496i	$-0.635700\pi$
$59$	−6.35259	−0.827037	−0.413519	+	0.910496i	$0.635700\pi$
$60$	−5.58809	−0.721420
$61$	−0.261087	−0.0334288	−0.0167144	−	0.999860i	$-0.505321\pi$
$61$	−0.261087	−0.0334288	−0.0167144	+	0.999860i	$0.505321\pi$
$62$	−19.4209	−2.46646
$63$	−1.67888	−0.211519
$64$	−9.16832	−1.14604
$65$	−5.65065	−0.700877
$66$	−8.00350	−0.985163
$67$	−14.2583	−1.74193	−0.870963	−	0.491349i	$-0.836504\pi$
$67$	−14.2583	−1.74193	−0.870963	+	0.491349i	$0.836504\pi$
$68$	16.7008	2.02527
$69$	3.58725	0.431854
$70$	−5.84769	−0.698933
$71$	1.21995	0.144782	0.0723910	−	0.997376i	$-0.476937\pi$
$71$	1.21995	0.144782	0.0723910	+	0.997376i	$0.476937\pi$
$72$	−4.60225	−0.542380
$73$	−11.8903	−1.39166	−0.695830	−	0.718207i	$-0.744963\pi$
$73$	−11.8903	−1.39166	−0.695830	+	0.718207i	$0.744963\pi$
$74$	24.6732	2.86820
$75$	−2.94215	−0.339731
$76$	−14.2214	−1.63131
$77$	−5.53403	−0.630661
$78$	−9.56425	−1.08294
$79$	6.29755	0.708530	0.354265	−	0.935145i	$-0.384731\pi$
$79$	6.29755	0.708530	0.354265	+	0.935145i	$0.384731\pi$
$80$	−4.85387	−0.542679
$81$	1.00000	0.111111
$82$	−0.967930	−0.106890
$83$	12.6255	1.38583	0.692916	−	0.721018i	$-0.256326\pi$
$83$	12.6255	1.38583	0.692916	+	0.721018i	$0.256326\pi$
$84$	−6.53999	−0.713571
$85$	−6.15017	−0.667079
$86$	−12.1848	−1.31392
$87$	−7.85512	−0.842158
$88$	−15.1702	−1.61715
$89$	2.96264	0.314039	0.157020	−	0.987595i	$-0.449811\pi$
$89$	2.96264	0.314039	0.157020	+	0.987595i	$0.449811\pi$
$90$	3.48309	0.367150
$91$	−6.61320	−0.693252
$92$	13.9740	1.45689
$93$	7.99854	0.829410
$94$	25.7319	2.65404
$95$	5.23710	0.537315
$96$	0.988892	0.100928
$97$	2.19808	0.223181	0.111590	−	0.993754i	$-0.464406\pi$
$97$	2.19808	0.223181	0.111590	+	0.993754i	$0.464406\pi$
$98$	10.1526	1.02557
$99$	3.29626	0.331287
$100$	−11.4610	−1.14610
$101$	−12.0500	−1.19902	−0.599510	−	0.800367i	$-0.704638\pi$
$101$	−12.0500	−1.19902	−0.599510	+	0.800367i	$0.704638\pi$
$102$	−10.4097	−1.03072
$103$	−17.9811	−1.77173	−0.885864	−	0.463945i	$-0.846434\pi$
$103$	−17.9811	−1.77173	−0.885864	+	0.463945i	$0.846434\pi$
$104$	−18.1285	−1.77765
$105$	2.40838	0.235034
$106$	−6.40149	−0.621767
$107$	−16.6077	−1.60553	−0.802766	−	0.596295i	$-0.796639\pi$
$107$	−16.6077	−1.60553	−0.802766	+	0.596295i	$0.796639\pi$
$108$	3.89545	0.374840
$109$	3.74628	0.358829	0.179414	−	0.983774i	$-0.442580\pi$
$109$	3.74628	0.358829	0.179414	+	0.983774i	$0.442580\pi$
$110$	11.4812	1.09469
$111$	−10.1617	−0.964506
$112$	−5.68069	−0.536775
$113$	5.54109	0.521262	0.260631	−	0.965439i	$-0.416069\pi$
$113$	5.54109	0.521262	0.260631	+	0.965439i	$0.416069\pi$
$114$	8.86428	0.830215
$115$	−5.14598	−0.479865
$116$	−30.5992	−2.84107
$117$	3.93906	0.364166
$118$	15.4244	1.41993
$119$	−7.19781	−0.659822
$120$	6.60202	0.602679
$121$	−0.134655	−0.0122414
$122$	0.633934	0.0573937
$123$	0.398644	0.0359445
$124$	31.1579	2.79806
$125$	11.3932	1.01904
$126$	4.07641	0.363156
$127$	−5.98650	−0.531216	−0.265608	−	0.964081i	$-0.585573\pi$
$127$	−5.98650	−0.531216	−0.265608	+	0.964081i	$0.585573\pi$
$128$	20.2834	1.79282
$129$	5.01834	0.441840
$130$	13.7201	1.20333
$131$	7.71010	0.673635	0.336817	−	0.941570i	$-0.390649\pi$
$131$	7.71010	0.673635	0.336817	+	0.941570i	$0.390649\pi$
$132$	12.8404	1.11761
$133$	6.12921	0.531470
$134$	34.6199	2.99070
$135$	−1.43452	−0.123464
$136$	−19.7311	−1.69193
$137$	20.4994	1.75138	0.875691	−	0.482873i	$-0.160407\pi$
$137$	20.4994	1.75138	0.875691	+	0.482873i	$0.160407\pi$
$138$	−8.71004	−0.741448
$139$	−11.5766	−0.981915	−0.490957	−	0.871184i	$-0.663353\pi$
$139$	−11.5766	−0.981915	−0.490957	+	0.871184i	$0.663353\pi$
$140$	9.38174	0.792902
$141$	−10.5977	−0.892490
$142$	−2.96212	−0.248575
$143$	12.9842	1.08579
$144$	3.38362	0.281968
$145$	11.2683	0.935783
$146$	28.8704	2.38933
$147$	−4.18136	−0.344873
$148$	−39.5844	−3.25382
$149$	19.5615	1.60254	0.801270	−	0.598303i	$-0.204158\pi$
$149$	19.5615	1.60254	0.801270	+	0.598303i	$0.204158\pi$
$150$	7.14371	0.583282
$151$	19.4865	1.58579	0.792893	−	0.609361i	$-0.208574\pi$
$151$	19.4865	1.58579	0.792893	+	0.609361i	$0.208574\pi$
$152$	16.8018	1.36280
$153$	4.28727	0.346605
$154$	13.4369	1.08278
$155$	−11.4741	−0.921619
$156$	15.3444	1.22853
$157$	0.123383	0.00984705	0.00492352	−	0.999988i	$-0.498433\pi$
$157$	0.123383	0.00984705	0.00492352	+	0.999988i	$0.498433\pi$
$158$	−15.2908	−1.21647
$159$	2.63647	0.209085
$160$	−1.41858	−0.112149
$161$	−6.02256	−0.474644
$162$	−2.42805	−0.190766
$163$	−9.46538	−0.741386	−0.370693	−	0.928755i	$-0.620880\pi$
$163$	−9.46538	−0.741386	−0.370693	+	0.928755i	$0.620880\pi$
$164$	1.55290	0.121261
$165$	−4.72855	−0.368117
$166$	−30.6555	−2.37933
$167$	14.6509	1.13372	0.566859	−	0.823815i	$-0.308158\pi$
$167$	14.6509	1.13372	0.566859	+	0.823815i	$0.308158\pi$
$168$	7.72662	0.596122
$169$	2.51618	0.193552
$170$	14.9329	1.14530
$171$	−3.65077	−0.279182
$172$	19.5487	1.49057
$173$	−7.37283	−0.560546	−0.280273	−	0.959920i	$-0.590425\pi$
$173$	−7.37283	−0.560546	−0.280273	+	0.959920i	$0.590425\pi$
$174$	19.0727	1.44590
$175$	4.93952	0.373393
$176$	11.1533	0.840711
$177$	−6.35259	−0.477490
$178$	−7.19345	−0.539172
$179$	−8.08265	−0.604126	−0.302063	−	0.953288i	$-0.597675\pi$
$179$	−8.08265	−0.604126	−0.302063	+	0.953288i	$0.597675\pi$
$180$	−5.58809	−0.416512
$181$	−7.92586	−0.589125	−0.294562	−	0.955632i	$-0.595174\pi$
$181$	−7.92586	−0.589125	−0.294562	+	0.955632i	$0.595174\pi$
$182$	16.0572	1.19024
$183$	−0.261087	−0.0193001
$184$	−16.5094	−1.21709
$185$	14.5772	1.07173
$186$	−19.4209	−1.42401
$187$	14.1320	1.03343
$188$	−41.2829	−3.01087
$189$	−1.67888	−0.122121
$190$	−12.7160	−0.922514
$191$	13.3533	0.966211	0.483105	−	0.875562i	$-0.339509\pi$
$191$	13.3533	0.966211	0.483105	+	0.875562i	$0.339509\pi$
$192$	−9.16832	−0.661666
$193$	16.0914	1.15828	0.579142	−	0.815226i	$-0.303388\pi$
$193$	16.0914	1.15828	0.579142	+	0.815226i	$0.303388\pi$
$194$	−5.33705	−0.383178
$195$	−5.65065	−0.404652
$196$	−16.2883	−1.16345
$197$	12.2083	0.869809	0.434904	−	0.900477i	$-0.356782\pi$
$197$	12.2083	0.869809	0.434904	+	0.900477i	$0.356782\pi$
$198$	−8.00350	−0.568784
$199$	0.377078	0.0267304	0.0133652	−	0.999911i	$-0.495746\pi$
$199$	0.377078	0.0267304	0.0133652	+	0.999911i	$0.495746\pi$
$200$	13.5405	0.957460
$201$	−14.2583	−1.00570
$202$	29.2581	2.05859
$203$	13.1878	0.925602
$204$	16.7008	1.16929
$205$	−0.571863	−0.0399406
$206$	43.6590	3.04187
$207$	3.58725	0.249331
$208$	13.3283	0.924149
$209$	−12.0339	−0.832403
$210$	−5.84769	−0.403529
$211$	−18.3256	−1.26158	−0.630792	−	0.775952i	$-0.717270\pi$
$211$	−18.3256	−1.26158	−0.630792	+	0.775952i	$0.717270\pi$
$212$	10.2702	0.705362
$213$	1.21995	0.0835899
$214$	40.3245	2.75653
$215$	−7.19890	−0.490961
$216$	−4.60225	−0.313143
$217$	−13.4286	−0.911592
$218$	−9.09618	−0.616071
$219$	−11.8903	−0.803475
$220$	−18.4198	−1.24186
$221$	16.8878	1.13600
$222$	24.6732	1.65595
$223$	−15.5332	−1.04018	−0.520089	−	0.854112i	$-0.674101\pi$
$223$	−15.5332	−1.04018	−0.520089	+	0.854112i	$0.674101\pi$
$224$	−1.66023	−0.110929
$225$	−2.94215	−0.196144
$226$	−13.4541	−0.894951
$227$	21.6939	1.43988	0.719939	−	0.694037i	$-0.244170\pi$
$227$	21.6939	1.43988	0.719939	+	0.694037i	$0.244170\pi$
$228$	−14.2214	−0.941835
$229$	−3.39287	−0.224207	−0.112104	−	0.993697i	$-0.535759\pi$
$229$	−3.39287	−0.224207	−0.112104	+	0.993697i	$0.535759\pi$
$230$	12.4947	0.823878
$231$	−5.53403	−0.364112
$232$	36.1512	2.37345
$233$	−25.2858	−1.65652	−0.828262	−	0.560341i	$-0.810670\pi$
$233$	−25.2858	−1.65652	−0.828262	+	0.560341i	$0.810670\pi$
$234$	−9.56425	−0.625234
$235$	15.2027	0.991712
$236$	−24.7462	−1.61084
$237$	6.29755	0.409070
$238$	17.4767	1.13284
$239$	−21.9949	−1.42273	−0.711367	−	0.702821i	$-0.751924\pi$
$239$	−21.9949	−1.42273	−0.711367	+	0.702821i	$0.751924\pi$
$240$	−4.85387	−0.313316
$241$	4.07314	0.262374	0.131187	−	0.991358i	$-0.458121\pi$
$241$	4.07314	0.262374	0.131187	+	0.991358i	$0.458121\pi$
$242$	0.326950	0.0210171
$243$	1.00000	0.0641500
$244$	−1.01705	−0.0651101
$245$	5.99825	0.383214
$246$	−0.967930	−0.0617129
$247$	−14.3806	−0.915016
$248$	−36.8113	−2.33752
$249$	12.6255	0.800111
$250$	−27.6632	−1.74958
$251$	7.06185	0.445740	0.222870	−	0.974848i	$-0.428457\pi$
$251$	7.06185	0.445740	0.222870	+	0.974848i	$0.428457\pi$
$252$	−6.53999	−0.411980
$253$	11.8245	0.743401
$254$	14.5355	0.912041
$255$	−6.15017	−0.385138
$256$	−30.9125	−1.93203
$257$	−9.71113	−0.605763	−0.302882	−	0.953028i	$-0.597949\pi$
$257$	−9.71113	−0.605763	−0.302882	+	0.953028i	$0.597949\pi$
$258$	−12.1848	−0.758592
$259$	17.0603	1.06007
$260$	−22.0118	−1.36512
$261$	−7.85512	−0.486220
$262$	−18.7206	−1.15656
$263$	−21.4144	−1.32047	−0.660233	−	0.751061i	$-0.729543\pi$
$263$	−21.4144	−1.32047	−0.660233	+	0.751061i	$0.729543\pi$
$264$	−15.1702	−0.933663
$265$	−3.78206	−0.232330
$266$	−14.8820	−0.912477
$267$	2.96264	0.181311
$268$	−55.5424	−3.39279
$269$	17.9418	1.09393	0.546966	−	0.837155i	$-0.315783\pi$
$269$	17.9418	1.09393	0.546966	+	0.837155i	$0.315783\pi$
$270$	3.48309	0.211974
$271$	−20.5214	−1.24659	−0.623293	−	0.781989i	$-0.714205\pi$
$271$	−20.5214	−1.24659	−0.623293	+	0.781989i	$0.714205\pi$
$272$	14.5065	0.879584
$273$	−6.61320	−0.400249
$274$	−49.7736	−3.00694
$275$	−9.69811	−0.584818
$276$	13.9740	0.841133
$277$	10.3492	0.621826	0.310913	−	0.950438i	$-0.399365\pi$
$277$	10.3492	0.621826	0.310913	+	0.950438i	$0.399365\pi$
$278$	28.1086	1.68584
$279$	7.99854	0.478860
$280$	−11.0840	−0.662395
$281$	−19.4444	−1.15995	−0.579977	−	0.814633i	$-0.696938\pi$
$281$	−19.4444	−1.15995	−0.579977	+	0.814633i	$0.696938\pi$
$282$	25.7319	1.53231
$283$	−30.6653	−1.82286	−0.911432	−	0.411451i	$-0.865022\pi$
$283$	−30.6653	−1.82286	−0.911432	+	0.411451i	$0.865022\pi$
$284$	4.75227	0.281995
$285$	5.23710	0.310219
$286$	−31.5263	−1.86419
$287$	−0.669275	−0.0395061
$288$	0.988892	0.0582710
$289$	1.38067	0.0812157
$290$	−27.3601	−1.60664
$291$	2.19808	0.128854
$292$	−46.3182	−2.71057
$293$	4.33643	0.253337	0.126669	−	0.991945i	$-0.459572\pi$
$293$	4.33643	0.253337	0.126669	+	0.991945i	$0.459572\pi$
$294$	10.1526	0.592110
$295$	9.11291	0.530574
$296$	46.7667	2.71826
$297$	3.29626	0.191269
$298$	−47.4964	−2.75139
$299$	14.1304	0.817182
$300$	−11.4610	−0.661702
$301$	−8.42518	−0.485619
$302$	−47.3142	−2.72263
$303$	−12.0500	−0.692254
$304$	−12.3528	−0.708483
$305$	0.374535	0.0214458
$306$	−10.4097	−0.595084
$307$	−22.0953	−1.26105	−0.630524	−	0.776170i	$-0.717160\pi$
$307$	−22.0953	−1.26105	−0.630524	+	0.776170i	$0.717160\pi$
$308$	−21.5575	−1.22835
$309$	−17.9811	−1.02291
$310$	27.8596	1.58232
$311$	−29.1760	−1.65442	−0.827210	−	0.561892i	$-0.810074\pi$
$311$	−29.1760	−1.65442	−0.827210	+	0.561892i	$0.810074\pi$
$312$	−18.1285	−1.02633
$313$	9.60694	0.543016	0.271508	−	0.962436i	$-0.412478\pi$
$313$	9.60694	0.543016	0.271508	+	0.962436i	$0.412478\pi$
$314$	−0.299581	−0.0169063
$315$	2.40838	0.135697
$316$	24.5318	1.38002
$317$	8.62765	0.484577	0.242289	−	0.970204i	$-0.422102\pi$
$317$	8.62765	0.484577	0.242289	+	0.970204i	$0.422102\pi$
$318$	−6.40149	−0.358978
$319$	−25.8925	−1.44970
$320$	13.1521	0.735226
$321$	−16.6077	−0.926954
$322$	14.6231	0.814914
$323$	−15.6518	−0.870892
$324$	3.89545	0.216414
$325$	−11.5893	−0.642860
$326$	22.9825	1.27288
$327$	3.74628	0.207170
$328$	−1.83466	−0.101302
$329$	17.7923	0.980922
$330$	11.4812	0.632018
$331$	11.3872	0.625898	0.312949	−	0.949770i	$-0.398683\pi$
$331$	11.3872	0.625898	0.312949	+	0.949770i	$0.398683\pi$
$332$	49.1821	2.69922
$333$	−10.1617	−0.556858
$334$	−35.5731	−1.94647
$335$	20.4538	1.11751
$336$	−5.68069	−0.309907
$337$	−32.0520	−1.74598	−0.872992	−	0.487735i	$-0.837823\pi$
$337$	−32.0520	−1.74598	−0.872992	+	0.487735i	$0.837823\pi$
$338$	−6.10941	−0.332308
$339$	5.54109	0.300951
$340$	−23.9577	−1.29929
$341$	26.3653	1.42776
$342$	8.86428	0.479325
$343$	18.7722	1.01360
$344$	−23.0956	−1.24523
$345$	−5.14598	−0.277050
$346$	17.9016	0.962397
$347$	13.6915	0.734998	0.367499	−	0.930024i	$-0.380214\pi$
$347$	13.6915	0.734998	0.367499	+	0.930024i	$0.380214\pi$
$348$	−30.5992	−1.64029
$349$	2.62357	0.140436	0.0702182	−	0.997532i	$-0.477630\pi$
$349$	2.62357	0.140436	0.0702182	+	0.997532i	$0.477630\pi$
$350$	−11.9934	−0.641076
$351$	3.93906	0.210251
$352$	3.25965	0.173740
$353$	7.37537	0.392551	0.196276	−	0.980549i	$-0.437115\pi$
$353$	7.37537	0.392551	0.196276	+	0.980549i	$0.437115\pi$
$354$	15.4244	0.819800
$355$	−1.75005	−0.0928829
$356$	11.5408	0.611662
$357$	−7.19781	−0.380948
$358$	19.6251	1.03722
$359$	1.61520	0.0852470	0.0426235	−	0.999091i	$-0.486428\pi$
$359$	1.61520	0.0852470	0.0426235	+	0.999091i	$0.486428\pi$
$360$	6.60202	0.347957
$361$	−5.67186	−0.298519
$362$	19.2444	1.01146
$363$	−0.134655	−0.00706755
$364$	−25.7614	−1.35026
$365$	17.0569	0.892800
$366$	0.633934	0.0331363
$367$	−14.8727	−0.776350	−0.388175	−	0.921586i	$-0.626894\pi$
$367$	−14.8727	−0.776350	−0.388175	+	0.921586i	$0.626894\pi$
$368$	12.1379	0.632731
$369$	0.398644	0.0207526
$370$	−35.3941	−1.84005
$371$	−4.42631	−0.229803
$372$	31.1579	1.61546
$373$	22.8214	1.18165	0.590823	−	0.806801i	$-0.298803\pi$
$373$	22.8214	1.18165	0.590823	+	0.806801i	$0.298803\pi$
$374$	−34.3132	−1.77429
$375$	11.3932	0.588341
$376$	48.7734	2.51530
$377$	−30.9418	−1.59358
$378$	4.07641	0.209668
$379$	9.40349	0.483025	0.241512	−	0.970398i	$-0.422357\pi$
$379$	9.40349	0.483025	0.241512	+	0.970398i	$0.422357\pi$
$380$	20.4009	1.04654
$381$	−5.98650	−0.306698
$382$	−32.4225	−1.65888
$383$	13.5650	0.693140	0.346570	−	0.938024i	$-0.387346\pi$
$383$	13.5650	0.693140	0.346570	+	0.938024i	$0.387346\pi$
$384$	20.2834	1.03508
$385$	7.93867	0.404592
$386$	−39.0708	−1.98865
$387$	5.01834	0.255096
$388$	8.56249	0.434695
$389$	9.93174	0.503559	0.251780	−	0.967785i	$-0.418984\pi$
$389$	9.93174	0.503559	0.251780	+	0.967785i	$0.418984\pi$
$390$	13.7201	0.694744
$391$	15.3795	0.777775
$392$	19.2437	0.971953
$393$	7.71010	0.388923
$394$	−29.6425	−1.49337
$395$	−9.03395	−0.454548
$396$	12.8404	0.645255
$397$	−2.37657	−0.119276	−0.0596382	−	0.998220i	$-0.518995\pi$
$397$	−2.37657	−0.119276	−0.0596382	+	0.998220i	$0.518995\pi$
$398$	−0.915567	−0.0458932
$399$	6.12921	0.306844
$400$	−9.95513	−0.497756
$401$	−24.8791	−1.24240	−0.621202	−	0.783650i	$-0.713356\pi$
$401$	−24.8791	−1.24240	−0.621202	+	0.783650i	$0.713356\pi$
$402$	34.6199	1.72668
$403$	31.5067	1.56946
$404$	−46.9401	−2.33536
$405$	−1.43452	−0.0712818
$406$	−32.0207	−1.58916
$407$	−33.4956	−1.66032
$408$	−19.7311	−0.976834
$409$	26.4400	1.30737	0.653687	−	0.756765i	$-0.273221\pi$
$409$	26.4400	1.30737	0.653687	+	0.756765i	$0.273221\pi$
$410$	1.38851	0.0685738
$411$	20.4994	1.01116
$412$	−70.0443	−3.45084
$413$	10.6652	0.524802
$414$	−8.71004	−0.428075
$415$	−18.1116	−0.889062
$416$	3.89530	0.190983
$417$	−11.5766	−0.566909
$418$	29.2190	1.42915
$419$	−6.37365	−0.311373	−0.155687	−	0.987806i	$-0.549759\pi$
$419$	−6.37365	−0.311373	−0.155687	+	0.987806i	$0.549759\pi$
$420$	9.38174	0.457782
$421$	7.92173	0.386081	0.193041	−	0.981191i	$-0.438165\pi$
$421$	7.92173	0.386081	0.193041	+	0.981191i	$0.438165\pi$
$422$	44.4955	2.16601
$423$	−10.5977	−0.515279
$424$	−12.1337	−0.589264
$425$	−12.6138	−0.611859
$426$	−2.96212	−0.143515
$427$	0.438334	0.0212125
$428$	−64.6946	−3.12713
$429$	12.9842	0.626882
$430$	17.4793	0.842927
$431$	−17.8289	−0.858790	−0.429395	−	0.903117i	$-0.641273\pi$
$431$	−17.8289	−0.858790	−0.429395	+	0.903117i	$0.641273\pi$
$432$	3.38362	0.162794
$433$	24.1973	1.16285	0.581424	−	0.813600i	$-0.302496\pi$
$433$	24.1973	1.16285	0.581424	+	0.813600i	$0.302496\pi$
$434$	32.6053	1.56511
$435$	11.2683	0.540275
$436$	14.5935	0.698900
$437$	−13.0962	−0.626478
$438$	28.8704	1.37948
$439$	−3.14704	−0.150200	−0.0750999	−	0.997176i	$-0.523928\pi$
$439$	−3.14704	−0.150200	−0.0750999	+	0.997176i	$0.523928\pi$
$440$	21.7620	1.03746
$441$	−4.18136	−0.199113
$442$	−41.0045	−1.95038
$443$	15.0077	0.713037	0.356519	−	0.934288i	$-0.383964\pi$
$443$	15.0077	0.713037	0.356519	+	0.934288i	$0.383964\pi$
$444$	−39.5844	−1.87859
$445$	−4.24996	−0.201468
$446$	37.7154	1.78587
$447$	19.5615	0.925227
$448$	15.3925	0.727227
$449$	−8.64650	−0.408054	−0.204027	−	0.978965i	$-0.565403\pi$
$449$	−8.64650	−0.408054	−0.204027	+	0.978965i	$0.565403\pi$
$450$	7.14371	0.336758
$451$	1.31404	0.0618755
$452$	21.5850	1.01527
$453$	19.4865	0.915554
$454$	−52.6741	−2.47212
$455$	9.48677	0.444746
$456$	16.8018	0.786815
$457$	0.228603	0.0106936	0.00534679	−	0.999986i	$-0.498298\pi$
$457$	0.228603	0.0106936	0.00534679	+	0.999986i	$0.498298\pi$
$458$	8.23808	0.384940
$459$	4.28727	0.200113
$460$	−20.0459	−0.934645
$461$	−16.8223	−0.783493	−0.391746	−	0.920073i	$-0.628129\pi$
$461$	−16.8223	−0.783493	−0.391746	+	0.920073i	$0.628129\pi$
$462$	13.4369	0.625142
$463$	30.9174	1.43685	0.718426	−	0.695603i	$-0.244863\pi$
$463$	30.9174	1.43685	0.718426	+	0.695603i	$0.244863\pi$
$464$	−26.5787	−1.23389
$465$	−11.4741	−0.532097
$466$	61.3952	2.84408
$467$	20.2173	0.935547	0.467774	−	0.883848i	$-0.345056\pi$
$467$	20.2173	0.935547	0.467774	+	0.883848i	$0.345056\pi$
$468$	15.3444	0.709295
$469$	23.9379	1.10535
$470$	−36.9129	−1.70266
$471$	0.123383	0.00568520
$472$	29.2362	1.34571
$473$	16.5417	0.760590
$474$	−15.2908	−0.702329
$475$	10.7411	0.492837
$476$	−28.0387	−1.28515
$477$	2.63647	0.120716
$478$	53.4049	2.44268
$479$	−35.4362	−1.61912	−0.809561	−	0.587036i	$-0.800295\pi$
$479$	−35.4362	−1.61912	−0.809561	+	0.587036i	$0.800295\pi$
$480$	−1.41858	−0.0647492
$481$	−40.0275	−1.82510
$482$	−9.88981	−0.450469
$483$	−6.02256	−0.274036
$484$	−0.524541	−0.0238428
$485$	−3.15318	−0.143179
$486$	−2.42805	−0.110139
$487$	2.64650	0.119924	0.0599622	−	0.998201i	$-0.480902\pi$
$487$	2.64650	0.119924	0.0599622	+	0.998201i	$0.480902\pi$
$488$	1.20159	0.0543934
$489$	−9.46538	−0.428039
$490$	−14.5641	−0.657937
$491$	−32.7673	−1.47877	−0.739383	−	0.673285i	$-0.764883\pi$
$491$	−32.7673	−1.47877	−0.739383	+	0.673285i	$0.764883\pi$
$492$	1.55290	0.0700100
$493$	−33.6770	−1.51674
$494$	34.9169	1.57099
$495$	−4.72855	−0.212533
$496$	27.0640	1.21521
$497$	−2.04816	−0.0918724
$498$	−30.6555	−1.37371
$499$	−12.8705	−0.576163	−0.288082	−	0.957606i	$-0.593017\pi$
$499$	−12.8705	−0.576163	−0.288082	+	0.957606i	$0.593017\pi$
$500$	44.3815	1.98480
$501$	14.6509	0.654552
$502$	−17.1466	−0.765288
$503$	16.6476	0.742281	0.371140	−	0.928577i	$-0.378967\pi$
$503$	16.6476	0.742281	0.371140	+	0.928577i	$0.378967\pi$
$504$	7.72662	0.344171
$505$	17.2860	0.769215
$506$	−28.7106	−1.27634
$507$	2.51618	0.111747
$508$	−23.3201	−1.03466
$509$	−32.3750	−1.43500	−0.717499	−	0.696559i	$-0.754713\pi$
$509$	−32.3750	−1.43500	−0.717499	+	0.696559i	$0.754713\pi$
$510$	14.9329	0.661242
$511$	19.9624	0.883087
$512$	34.4905	1.52428
$513$	−3.65077	−0.161186
$514$	23.5791	1.04003
$515$	25.7942	1.13663
$516$	19.5487	0.860582
$517$	−34.9329	−1.53635
$518$	−41.4233	−1.82003
$519$	−7.37283	−0.323631
$520$	26.0057	1.14043
$521$	17.2923	0.757588	0.378794	−	0.925481i	$-0.376339\pi$
$521$	17.2923	0.757588	0.378794	+	0.925481i	$0.376339\pi$
$522$	19.0727	0.834788
$523$	7.52378	0.328992	0.164496	−	0.986378i	$-0.447400\pi$
$523$	7.52378	0.328992	0.164496	+	0.986378i	$0.447400\pi$
$524$	30.0343	1.31205
$525$	4.93952	0.215578
$526$	51.9953	2.26710
$527$	34.2919	1.49378
$528$	11.1533	0.485385
$529$	−10.1316	−0.440505
$530$	9.18305	0.398886
$531$	−6.35259	−0.275679
$532$	23.8760	1.03516
$533$	1.57028	0.0680165
$534$	−7.19345	−0.311291
$535$	23.8241	1.03001
$536$	65.6202	2.83436
$537$	−8.08265	−0.348792
$538$	−43.5637	−1.87816
$539$	−13.7829	−0.593670
$540$	−5.58809	−0.240473
$541$	41.2341	1.77279	0.886397	−	0.462926i	$-0.153201\pi$
$541$	41.2341	1.77279	0.886397	+	0.462926i	$0.153201\pi$
$542$	49.8270	2.14025
$543$	−7.92586	−0.340131
$544$	4.23964	0.181773
$545$	−5.37412	−0.230202
$546$	16.0572	0.687185
$547$	−15.9501	−0.681977	−0.340989	−	0.940067i	$-0.610762\pi$
$547$	−15.9501	−0.681977	−0.340989	+	0.940067i	$0.610762\pi$
$548$	79.8543	3.41121
$549$	−0.261087	−0.0111429
$550$	23.5475	1.00407
$551$	28.6773	1.22169
$552$	−16.5094	−0.702688
$553$	−10.5728	−0.449602
$554$	−25.1285	−1.06761
$555$	14.5772	0.618766
$556$	−45.0961	−1.91250
$557$	−6.18087	−0.261892	−0.130946	−	0.991390i	$-0.541801\pi$
$557$	−6.18087	−0.261892	−0.130946	+	0.991390i	$0.541801\pi$
$558$	−19.4209	−0.822152
$559$	19.7675	0.836077
$560$	8.14905	0.344360
$561$	14.1320	0.596652
$562$	47.2120	1.99152
$563$	12.7917	0.539106	0.269553	−	0.962986i	$-0.413124\pi$
$563$	12.7917	0.539106	0.269553	+	0.962986i	$0.413124\pi$
$564$	−41.2829	−1.73832
$565$	−7.94879	−0.334408
$566$	74.4571	3.12966
$567$	−1.67888	−0.0705063
$568$	−5.61454	−0.235581
$569$	−36.9812	−1.55033	−0.775167	−	0.631756i	$-0.782334\pi$
$569$	−36.9812	−1.55033	−0.775167	+	0.631756i	$0.782334\pi$
$570$	−12.7160	−0.532613
$571$	−1.52524	−0.0638294	−0.0319147	−	0.999491i	$-0.510160\pi$
$571$	−1.52524	−0.0638294	−0.0319147	+	0.999491i	$0.510160\pi$
$572$	50.5791	2.11482
$573$	13.3533	0.557842
$574$	1.62504	0.0678277
$575$	−10.5543	−0.440143
$576$	−9.16832	−0.382013
$577$	−24.6633	−1.02675	−0.513374	−	0.858165i	$-0.671605\pi$
$577$	−24.6633	−1.02675	−0.513374	+	0.858165i	$0.671605\pi$
$578$	−3.35233	−0.139439
$579$	16.0914	0.668736
$580$	43.8952	1.82265
$581$	−21.1968	−0.879389
$582$	−5.33705	−0.221228
$583$	8.69049	0.359923
$584$	54.7223	2.26443
$585$	−5.65065	−0.233626
$586$	−10.5291	−0.434953
$587$	−33.6557	−1.38912	−0.694559	−	0.719435i	$-0.744401\pi$
$587$	−33.6557	−1.38912	−0.694559	+	0.719435i	$0.744401\pi$
$588$	−16.2883	−0.671718
$589$	−29.2009	−1.20320
$590$	−22.1267	−0.910940
$591$	12.2083	0.502184
$592$	−34.3833	−1.41315
$593$	−8.64424	−0.354976	−0.177488	−	0.984123i	$-0.556797\pi$
$593$	−8.64424	−0.354976	−0.177488	+	0.984123i	$0.556797\pi$
$594$	−8.00350	−0.328388
$595$	10.3254	0.423300
$596$	76.2008	3.12131
$597$	0.377078	0.0154328
$598$	−34.3094	−1.40301
$599$	−21.7468	−0.888549	−0.444275	−	0.895891i	$-0.646539\pi$
$599$	−21.7468	−0.888549	−0.444275	+	0.895891i	$0.646539\pi$
$600$	13.5405	0.552790
$601$	−15.2539	−0.622219	−0.311110	−	0.950374i	$-0.600701\pi$
$601$	−15.2539	−0.622219	−0.311110	+	0.950374i	$0.600701\pi$
$602$	20.4568	0.833757
$603$	−14.2583	−0.580642
$604$	75.9085	3.08867
$605$	0.193165	0.00785328
$606$	29.2581	1.18853
$607$	−13.7823	−0.559405	−0.279703	−	0.960087i	$-0.590236\pi$
$607$	−13.7823	−0.559405	−0.279703	+	0.960087i	$0.590236\pi$
$608$	−3.61022	−0.146414
$609$	13.1878	0.534397
$610$	−0.909391	−0.0368202
$611$	−41.7451	−1.68883
$612$	16.7008	0.675091
$613$	34.0501	1.37527	0.687636	−	0.726056i	$-0.258649\pi$
$613$	34.0501	1.37527	0.687636	+	0.726056i	$0.258649\pi$
$614$	53.6487	2.16508
$615$	−0.571863	−0.0230597
$616$	25.4690	1.02617
$617$	−20.3824	−0.820566	−0.410283	−	0.911958i	$-0.634570\pi$
$617$	−20.3824	−0.820566	−0.410283	+	0.911958i	$0.634570\pi$
$618$	43.6590	1.75622
$619$	−18.6411	−0.749247	−0.374624	−	0.927177i	$-0.622228\pi$
$619$	−18.6411	−0.749247	−0.374624	+	0.927177i	$0.622228\pi$
$620$	−44.6966	−1.79506
$621$	3.58725	0.143951
$622$	70.8410	2.84046
$623$	−4.97391	−0.199276
$624$	13.3283	0.533558
$625$	−1.63295	−0.0653180
$626$	−23.3262	−0.932301
$627$	−12.0339	−0.480588
$628$	0.480633	0.0191793
$629$	−43.5659	−1.73709
$630$	−5.84769	−0.232978
$631$	−49.1391	−1.95620	−0.978098	−	0.208144i	$-0.933258\pi$
$631$	−49.1391	−1.95620	−0.978098	+	0.208144i	$0.933258\pi$
$632$	−28.9829	−1.15288
$633$	−18.3256	−0.728376
$634$	−20.9484	−0.831968
$635$	8.58774	0.340794
$636$	10.2702	0.407241
$637$	−16.4706	−0.652590
$638$	62.8685	2.48899
$639$	1.21995	0.0482606
$640$	−29.0969	−1.15016
$641$	8.66520	0.342255	0.171127	−	0.985249i	$-0.445259\pi$
$641$	8.66520	0.342255	0.171127	+	0.985249i	$0.445259\pi$
$642$	40.3245	1.59148
$643$	35.3302	1.39329	0.696643	−	0.717418i	$-0.254676\pi$
$643$	35.3302	1.39329	0.696643	+	0.717418i	$0.254676\pi$
$644$	−23.4606	−0.924476
$645$	−7.19890	−0.283456
$646$	38.0035	1.49523
$647$	39.5133	1.55343	0.776714	−	0.629854i	$-0.216885\pi$
$647$	39.5133	1.55343	0.776714	+	0.629854i	$0.216885\pi$
$648$	−4.60225	−0.180793
$649$	−20.9398	−0.821959
$650$	28.1395	1.10372
$651$	−13.4286	−0.526308
$652$	−36.8719	−1.44402
$653$	6.82684	0.267155	0.133578	−	0.991038i	$-0.457353\pi$
$653$	6.82684	0.267155	0.133578	+	0.991038i	$0.457353\pi$
$654$	−9.09618	−0.355689
$655$	−11.0603	−0.432161
$656$	1.34886	0.0526641
$657$	−11.8903	−0.463886
$658$	−43.2007	−1.68414
$659$	−4.96039	−0.193229	−0.0966147	−	0.995322i	$-0.530801\pi$
$659$	−4.96039	−0.193229	−0.0966147	+	0.995322i	$0.530801\pi$
$660$	−18.4198	−0.716991
$661$	−14.1606	−0.550783	−0.275391	−	0.961332i	$-0.588807\pi$
$661$	−14.1606	−0.550783	−0.275391	+	0.961332i	$0.588807\pi$
$662$	−27.6488	−1.07460
$663$	16.8878	0.655868
$664$	−58.1059	−2.25495
$665$	−8.79247	−0.340957
$666$	24.6732	0.956066
$667$	−28.1783	−1.09107
$668$	57.0717	2.20817
$669$	−15.5332	−0.600547
$670$	−49.6629	−1.91864
$671$	−0.860612	−0.0332236
$672$	−1.66023	−0.0640448
$673$	−33.6329	−1.29645	−0.648226	−	0.761448i	$-0.724489\pi$
$673$	−33.6329	−1.29645	−0.648226	+	0.761448i	$0.724489\pi$
$674$	77.8240	2.99767
$675$	−2.94215	−0.113244
$676$	9.80163	0.376986
$677$	−37.2108	−1.43013	−0.715063	−	0.699060i	$-0.753602\pi$
$677$	−37.2108	−1.43013	−0.715063	+	0.699060i	$0.753602\pi$
$678$	−13.4541	−0.516700
$679$	−3.69031	−0.141621
$680$	28.3046	1.08543
$681$	21.6939	0.831314
$682$	−64.0164	−2.45131
$683$	−1.72078	−0.0658440	−0.0329220	−	0.999458i	$-0.510481\pi$
$683$	−1.72078	−0.0658440	−0.0329220	+	0.999458i	$0.510481\pi$
$684$	−14.2214	−0.543769
$685$	−29.4068	−1.12357
$686$	−45.5798	−1.74025
$687$	−3.39287	−0.129446
$688$	16.9801	0.647361
$689$	10.3852	0.395645
$690$	12.4947	0.475666
$691$	26.7045	1.01589	0.507944	−	0.861390i	$-0.330406\pi$
$691$	26.7045	1.01589	0.507944	+	0.861390i	$0.330406\pi$
$692$	−28.7205	−1.09179
$693$	−5.53403	−0.210220
$694$	−33.2437	−1.26191
$695$	16.6069	0.629934
$696$	36.1512	1.37031
$697$	1.70909	0.0647366
$698$	−6.37017	−0.241114
$699$	−25.2858	−0.956395
$700$	19.2417	0.727266
$701$	−27.4666	−1.03740	−0.518700	−	0.854956i	$-0.673584\pi$
$701$	−27.4666	−1.03740	−0.518700	+	0.854956i	$0.673584\pi$
$702$	−9.56425	−0.360979
$703$	37.0981	1.39918
$704$	−30.2212	−1.13900
$705$	15.2027	0.572565
$706$	−17.9078	−0.673969
$707$	20.2305	0.760846
$708$	−24.7462	−0.930019
$709$	15.0012	0.563383	0.281692	−	0.959505i	$-0.409104\pi$
$709$	15.0012	0.563383	0.281692	+	0.959505i	$0.409104\pi$
$710$	4.24921	0.159470
$711$	6.29755	0.236177
$712$	−13.6348	−0.510986
$713$	28.6928	1.07455
$714$	17.4767	0.654048
$715$	−18.6260	−0.696574
$716$	−31.4855	−1.17667
$717$	−21.9949	−0.821416
$718$	−3.92179	−0.146360
$719$	30.5196	1.13819	0.569094	−	0.822272i	$-0.307294\pi$
$719$	30.5196	1.13819	0.569094	+	0.822272i	$0.307294\pi$
$720$	−4.85387	−0.180893
$721$	30.1881	1.12426
$722$	13.7716	0.512525
$723$	4.07314	0.151482
$724$	−30.8748	−1.14745
$725$	23.1110	0.858321
$726$	0.326950	0.0121342
$727$	32.1511	1.19242	0.596210	−	0.802829i	$-0.296673\pi$
$727$	32.1511	1.19242	0.596210	+	0.802829i	$0.296673\pi$
$728$	30.4356	1.12802
$729$	1.00000	0.0370370
$730$	−41.4151	−1.53284
$731$	21.5149	0.795759
$732$	−1.01705	−0.0375913
$733$	−21.6279	−0.798844	−0.399422	−	0.916767i	$-0.630789\pi$
$733$	−21.6279	−0.798844	−0.399422	+	0.916767i	$0.630789\pi$
$734$	36.1118	1.33291
$735$	5.99825	0.221249
$736$	3.54740	0.130759
$737$	−46.9990	−1.73123
$738$	−0.967930	−0.0356300
$739$	11.6013	0.426761	0.213380	−	0.976969i	$-0.431553\pi$
$739$	11.6013	0.426761	0.213380	+	0.976969i	$0.431553\pi$
$740$	56.7846	2.08744
$741$	−14.3806	−0.528285
$742$	10.7473	0.394547
$743$	51.4457	1.88736	0.943680	−	0.330861i	$-0.107339\pi$
$743$	51.4457	1.88736	0.943680	+	0.330861i	$0.107339\pi$
$744$	−36.8113	−1.34957
$745$	−28.0613	−1.02809
$746$	−55.4116	−2.02876
$747$	12.6255	0.461944
$748$	55.0503	2.01284
$749$	27.8824	1.01880
$750$	−27.6632	−1.01012
$751$	43.0134	1.56958	0.784790	−	0.619762i	$-0.212771\pi$
$751$	43.0134	1.56958	0.784790	+	0.619762i	$0.212771\pi$
$752$	−35.8587	−1.30763
$753$	7.06185	0.257348
$754$	75.1283	2.73601
$755$	−27.9537	−1.01734
$756$	−6.53999	−0.237857
$757$	−5.03133	−0.182867	−0.0914334	−	0.995811i	$-0.529145\pi$
$757$	−5.03133	−0.182867	−0.0914334	+	0.995811i	$0.529145\pi$
$758$	−22.8322	−0.829303
$759$	11.8245	0.429203
$760$	−24.1025	−0.874288
$761$	−13.3095	−0.482468	−0.241234	−	0.970467i	$-0.577552\pi$
$761$	−13.3095	−0.482468	−0.241234	+	0.970467i	$0.577552\pi$
$762$	14.5355	0.526567
$763$	−6.28956	−0.227697
$764$	52.0171	1.88191
$765$	−6.15017	−0.222360
$766$	−32.9366	−1.19005
$767$	−25.0232	−0.903536
$768$	−30.9125	−1.11546
$769$	18.9003	0.681564	0.340782	−	0.940142i	$-0.389308\pi$
$769$	18.9003	0.681564	0.340782	+	0.940142i	$0.389308\pi$
$770$	−19.2755	−0.694641
$771$	−9.71113	−0.349738
$772$	62.6832	2.25602
$773$	23.0607	0.829435	0.414717	−	0.909950i	$-0.363880\pi$
$773$	23.0607	0.829435	0.414717	+	0.909950i	$0.363880\pi$
$774$	−12.1848	−0.437973
$775$	−23.5329	−0.845328
$776$	−10.1161	−0.363147
$777$	17.0603	0.612034
$778$	−24.1148	−0.864558
$779$	−1.45536	−0.0521437
$780$	−22.0118	−0.788150
$781$	4.02129	0.143893
$782$	−37.3423	−1.33536
$783$	−7.85512	−0.280719
$784$	−14.1481	−0.505291
$785$	−0.176995	−0.00631724
$786$	−18.7206	−0.667740
$787$	−5.69413	−0.202974	−0.101487	−	0.994837i	$-0.532360\pi$
$787$	−5.69413	−0.202974	−0.101487	+	0.994837i	$0.532360\pi$
$788$	47.5570	1.69415
$789$	−21.4144	−0.762372
$790$	21.9349	0.780410
$791$	−9.30282	−0.330770
$792$	−15.1702	−0.539050
$793$	−1.02844	−0.0365209
$794$	5.77043	0.204785
$795$	−3.78206	−0.134136
$796$	1.46889	0.0520634
$797$	−32.0826	−1.13642	−0.568211	−	0.822883i	$-0.692364\pi$
$797$	−32.0826	−1.13642	−0.568211	+	0.822883i	$0.692364\pi$
$798$	−14.8820	−0.526819
$799$	−45.4353	−1.60739
$800$	−2.90947	−0.102865
$801$	2.96264	0.104680
$802$	60.4079	2.13308
$803$	−39.1937	−1.38312
$804$	−55.5424	−1.95883
$805$	8.63948	0.304502
$806$	−76.5000	−2.69460
$807$	17.9418	0.631582
$808$	55.4571	1.95097
$809$	−44.8711	−1.57758	−0.788791	−	0.614661i	$-0.789293\pi$
$809$	−44.8711	−1.57758	−0.788791	+	0.614661i	$0.789293\pi$
$810$	3.48309	0.122383
$811$	−21.7831	−0.764907	−0.382454	−	0.923975i	$-0.624921\pi$
$811$	−21.7831	−0.764907	−0.382454	+	0.923975i	$0.624921\pi$
$812$	51.3724	1.80282
$813$	−20.5214	−0.719716
$814$	81.3292	2.85059
$815$	13.5783	0.475626
$816$	14.5065	0.507828
$817$	−18.3208	−0.640964
$818$	−64.1977	−2.24462
$819$	−6.61320	−0.231084
$820$	−2.22766	−0.0777933
$821$	−55.2241	−1.92733	−0.963667	−	0.267106i	$-0.913933\pi$
$821$	−55.2241	−1.92733	−0.963667	+	0.267106i	$0.913933\pi$
$822$	−49.7736	−1.73605
$823$	2.65769	0.0926413	0.0463206	−	0.998927i	$-0.485250\pi$
$823$	2.65769	0.0926413	0.0463206	+	0.998927i	$0.485250\pi$
$824$	82.7534	2.88285
$825$	−9.69811	−0.337645
$826$	−25.8958	−0.901029
$827$	17.8016	0.619023	0.309511	−	0.950896i	$-0.399835\pi$
$827$	17.8016	0.619023	0.309511	+	0.950896i	$0.399835\pi$
$828$	13.9740	0.485628
$829$	10.3114	0.358129	0.179064	−	0.983837i	$-0.442693\pi$
$829$	10.3114	0.358129	0.179064	+	0.983837i	$0.442693\pi$
$830$	43.9759	1.52642
$831$	10.3492	0.359011
$832$	−36.1145	−1.25205
$833$	−17.9266	−0.621121
$834$	28.1086	0.973322
$835$	−21.0169	−0.727321
$836$	−46.8775	−1.62129
$837$	7.99854	0.276470
$838$	15.4756	0.534595
$839$	11.9487	0.412516	0.206258	−	0.978498i	$-0.433871\pi$
$839$	11.9487	0.412516	0.206258	+	0.978498i	$0.433871\pi$
$840$	−11.0840	−0.382434
$841$	32.7030	1.12769
$842$	−19.2344	−0.662861
$843$	−19.4444	−0.669700
$844$	−71.3863	−2.45722
$845$	−3.60950	−0.124171
$846$	25.7319	0.884680
$847$	0.226069	0.00776784
$848$	8.92080	0.306342
$849$	−30.6653	−1.05243
$850$	30.6270	1.05050
$851$	−36.4526	−1.24958
$852$	4.75227	0.162810
$853$	3.64104	0.124667	0.0623334	−	0.998055i	$-0.480146\pi$
$853$	3.64104	0.124667	0.0623334	+	0.998055i	$0.480146\pi$
$854$	−1.06430	−0.0364196
$855$	5.23710	0.179105
$856$	76.4330	2.61243
$857$	−16.9309	−0.578349	−0.289174	−	0.957276i	$-0.593381\pi$
$857$	−16.9309	−0.578349	−0.289174	+	0.957276i	$0.593381\pi$
$858$	−31.5263	−1.07629
$859$	−6.39988	−0.218361	−0.109181	−	0.994022i	$-0.534823\pi$
$859$	−6.39988	−0.218361	−0.109181	+	0.994022i	$0.534823\pi$
$860$	−28.0429	−0.956256
$861$	−0.669275	−0.0228088
$862$	43.2896	1.47445
$863$	−49.1496	−1.67307	−0.836535	−	0.547913i	$-0.815423\pi$
$863$	−49.1496	−1.67307	−0.836535	+	0.547913i	$0.815423\pi$
$864$	0.988892	0.0336428
$865$	10.5765	0.359610
$866$	−58.7524	−1.99649
$867$	1.38067	0.0468899
$868$	−52.3104	−1.77553
$869$	20.7584	0.704179
$870$	−27.3601	−0.927594
$871$	−56.1642	−1.90305
$872$	−17.2413	−0.583865
$873$	2.19808	0.0743936
$874$	31.7984	1.07560
$875$	−19.1278	−0.646636
$876$	−46.3182	−1.56495
$877$	−1.45743	−0.0492140	−0.0246070	−	0.999697i	$-0.507833\pi$
$877$	−1.45743	−0.0492140	−0.0246070	+	0.999697i	$0.507833\pi$
$878$	7.64118	0.257877
$879$	4.33643	0.146264
$880$	−15.9996	−0.539347
$881$	−40.9223	−1.37871	−0.689354	−	0.724425i	$-0.742105\pi$
$881$	−40.9223	−1.37871	−0.689354	+	0.724425i	$0.742105\pi$
$882$	10.1526	0.341855
$883$	8.27135	0.278353	0.139177	−	0.990268i	$-0.455554\pi$
$883$	8.27135	0.278353	0.139177	+	0.990268i	$0.455554\pi$
$884$	65.7855	2.21261
$885$	9.11291	0.306327
$886$	−36.4395	−1.22421
$887$	−25.1365	−0.844000	−0.422000	−	0.906596i	$-0.638672\pi$
$887$	−25.1365	−0.844000	−0.422000	+	0.906596i	$0.638672\pi$
$888$	46.7667	1.56939
$889$	10.0506	0.337087
$890$	10.3191	0.345898
$891$	3.29626	0.110429
$892$	−60.5086	−2.02598
$893$	38.6899	1.29471
$894$	−47.4964	−1.58852
$895$	11.5947	0.387569
$896$	−34.0534	−1.13764
$897$	14.1304	0.471800
$898$	20.9942	0.700585
$899$	−62.8295	−2.09548
$900$	−11.4610	−0.382034
$901$	11.3032	0.376566
$902$	−3.19055	−0.106234
$903$	−8.42518	−0.280372
$904$	−25.5015	−0.848166
$905$	11.3698	0.377945
$906$	−47.3142	−1.57191
$907$	22.2578	0.739057	0.369528	−	0.929219i	$-0.379519\pi$
$907$	22.2578	0.739057	0.369528	+	0.929219i	$0.379519\pi$
$908$	84.5076	2.80448
$909$	−12.0500	−0.399673
$910$	−23.0344	−0.763582
$911$	22.1199	0.732864	0.366432	−	0.930445i	$-0.380579\pi$
$911$	22.1199	0.732864	0.366432	+	0.930445i	$0.380579\pi$
$912$	−12.3528	−0.409043
$913$	41.6171	1.37732
$914$	−0.555059	−0.0183597
$915$	0.374535	0.0123817
$916$	−13.2168	−0.436694
$917$	−12.9443	−0.427460
$918$	−10.4097	−0.343572
$919$	−38.7749	−1.27907	−0.639534	−	0.768763i	$-0.720873\pi$
$919$	−38.7749	−1.27907	−0.639534	+	0.768763i	$0.720873\pi$
$920$	23.6831	0.780808
$921$	−22.0953	−0.728066
$922$	40.8455	1.34517
$923$	4.80547	0.158174
$924$	−21.5575	−0.709190
$925$	29.8973	0.983017
$926$	−75.0691	−2.46692
$927$	−17.9811	−0.590576
$928$	−7.76787	−0.254993
$929$	−49.5713	−1.62638	−0.813190	−	0.581998i	$-0.802271\pi$
$929$	−49.5713	−1.62638	−0.813190	+	0.581998i	$0.802271\pi$
$930$	27.8596	0.913554
$931$	15.2652	0.500297
$932$	−98.4993	−3.22645
$933$	−29.1760	−0.955180
$934$	−49.0888	−1.60623
$935$	−20.2726	−0.662984
$936$	−18.1285	−0.592550
$937$	30.0260	0.980908	0.490454	−	0.871467i	$-0.336831\pi$
$937$	30.0260	0.980908	0.490454	+	0.871467i	$0.336831\pi$
$938$	−58.1226	−1.89777
$939$	9.60694	0.313511
$940$	59.2211	1.93158
$941$	−9.57537	−0.312148	−0.156074	−	0.987745i	$-0.549884\pi$
$941$	−9.57537	−0.312148	−0.156074	+	0.987745i	$0.549884\pi$
$942$	−0.299581	−0.00976088
$943$	1.43004	0.0465684
$944$	−21.4947	−0.699594
$945$	2.40838	0.0783448
$946$	−40.1643	−1.30585
$947$	49.5140	1.60899	0.804494	−	0.593961i	$-0.202437\pi$
$947$	49.5140	1.60899	0.804494	+	0.593961i	$0.202437\pi$
$948$	24.5318	0.796755
$949$	−46.8367	−1.52038
$950$	−26.0801	−0.846149
$951$	8.62765	0.279771
$952$	33.1261	1.07362
$953$	7.21861	0.233834	0.116917	−	0.993142i	$-0.462699\pi$
$953$	7.21861	0.233834	0.116917	+	0.993142i	$0.462699\pi$
$954$	−6.40149	−0.207256
$955$	−19.1556	−0.619859
$956$	−85.6801	−2.77109
$957$	−25.8925	−0.836987
$958$	86.0410	2.77986
$959$	−34.4160	−1.11135
$960$	13.1521	0.424483
$961$	32.9767	1.06376
$962$	97.1890	3.13350
$963$	−16.6077	−0.535177
$964$	15.8667	0.511032
$965$	−23.0834	−0.743082
$966$	14.6231	0.470491
$967$	26.3075	0.845992	0.422996	−	0.906132i	$-0.360978\pi$
$967$	26.3075	0.845992	0.422996	+	0.906132i	$0.360978\pi$
$968$	0.619716	0.0199184
$969$	−15.6518	−0.502810
$970$	7.65610	0.245823
$971$	−42.7825	−1.37295	−0.686477	−	0.727151i	$-0.740844\pi$
$971$	−42.7825	−1.37295	−0.686477	+	0.727151i	$0.740844\pi$
$972$	3.89545	0.124947
$973$	19.4357	0.623081
$974$	−6.42585	−0.205898
$975$	−11.5893	−0.371155
$976$	−0.883420	−0.0282776
$977$	9.97936	0.319268	0.159634	−	0.987176i	$-0.448969\pi$
$977$	9.97936	0.319268	0.159634	+	0.987176i	$0.448969\pi$
$978$	22.9825	0.734898
$979$	9.76564	0.312111
$980$	23.3659	0.746395
$981$	3.74628	0.119610
$982$	79.5607	2.53889
$983$	38.3220	1.22228	0.611140	−	0.791522i	$-0.290711\pi$
$983$	38.3220	1.22228	0.611140	+	0.791522i	$0.290711\pi$
$984$	−1.83466	−0.0584868
$985$	−17.5131	−0.558014
$986$	81.7696	2.60408
$987$	17.7923	0.566336
$988$	−56.0189	−1.78220
$989$	18.0020	0.572431
$990$	11.4812	0.364896
$991$	−37.4569	−1.18986	−0.594929	−	0.803778i	$-0.702820\pi$
$991$	−37.4569	−1.18986	−0.594929	+	0.803778i	$0.702820\pi$
$992$	7.90969	0.251133
$993$	11.3872	0.361362
$994$	4.97303	0.157735
$995$	−0.540926	−0.0171485
$996$	49.1821	1.55839
$997$	1.42413	0.0451026	0.0225513	−	0.999746i	$-0.492821\pi$
$997$	1.42413	0.0451026	0.0225513	+	0.999746i	$0.492821\pi$
$998$	31.2503	0.989211
$999$	−10.1617	−0.321502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.a.1.8	✓	92

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.a.1.8	✓	92	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.42805 q^{2} +1.00000 q^{3} +3.89545 q^{4} -1.43452 q^{5} -2.42805 q^{6} -1.67888 q^{7} -4.60225 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.42805 q^{2} +1.00000 q^{3} +3.89545 q^{4} -1.43452 q^{5} -2.42805 q^{6} -1.67888 q^{7} -4.60225 q^{8} +1.00000 q^{9} +3.48309 q^{10} +3.29626 q^{11} +3.89545 q^{12} +3.93906 q^{13} +4.07641 q^{14} -1.43452 q^{15} +3.38362 q^{16} +4.28727 q^{17} -2.42805 q^{18} -3.65077 q^{19} -5.58809 q^{20} -1.67888 q^{21} -8.00350 q^{22} +3.58725 q^{23} -4.60225 q^{24} -2.94215 q^{25} -9.56425 q^{26} +1.00000 q^{27} -6.53999 q^{28} -7.85512 q^{29} +3.48309 q^{30} +7.99854 q^{31} +0.988892 q^{32} +3.29626 q^{33} -10.4097 q^{34} +2.40838 q^{35} +3.89545 q^{36} -10.1617 q^{37} +8.86428 q^{38} +3.93906 q^{39} +6.60202 q^{40} +0.398644 q^{41} +4.07641 q^{42} +5.01834 q^{43} +12.8404 q^{44} -1.43452 q^{45} -8.71004 q^{46} -10.5977 q^{47} +3.38362 q^{48} -4.18136 q^{49} +7.14371 q^{50} +4.28727 q^{51} +15.3444 q^{52} +2.63647 q^{53} -2.42805 q^{54} -4.72855 q^{55} +7.72662 q^{56} -3.65077 q^{57} +19.0727 q^{58} -6.35259 q^{59} -5.58809 q^{60} -0.261087 q^{61} -19.4209 q^{62} -1.67888 q^{63} -9.16832 q^{64} -5.65065 q^{65} -8.00350 q^{66} -14.2583 q^{67} +16.7008 q^{68} +3.58725 q^{69} -5.84769 q^{70} +1.21995 q^{71} -4.60225 q^{72} -11.8903 q^{73} +24.6732 q^{74} -2.94215 q^{75} -14.2214 q^{76} -5.53403 q^{77} -9.56425 q^{78} +6.29755 q^{79} -4.85387 q^{80} +1.00000 q^{81} -0.967930 q^{82} +12.6255 q^{83} -6.53999 q^{84} -6.15017 q^{85} -12.1848 q^{86} -7.85512 q^{87} -15.1702 q^{88} +2.96264 q^{89} +3.48309 q^{90} -6.61320 q^{91} +13.9740 q^{92} +7.99854 q^{93} +25.7319 q^{94} +5.23710 q^{95} +0.988892 q^{96} +2.19808 q^{97} +10.1526 q^{98} +3.29626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9}+O(q^{10}) \) 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9	\( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9} - 44 q^{10} - 24 q^{11} + 70 q^{12} - 48 q^{13} - 29 q^{14} - 18 q^{15} + 26 q^{16} - 69 q^{17} - 6 q^{18} - 74 q^{19} - 42 q^{20} - 42 q^{21} - 62 q^{22} - 19 q^{23} - 15 q^{24} + 16 q^{25} - 27 q^{26} + 92 q^{27} - 101 q^{28} - 54 q^{29} - 44 q^{30} - 67 q^{31} - 36 q^{32} - 24 q^{33} - 63 q^{34} - 31 q^{35} + 70 q^{36} - 70 q^{37} - 18 q^{38} - 48 q^{39} - 125 q^{40} - 98 q^{41} - 29 q^{42} - 159 q^{43} - 52 q^{44} - 18 q^{45} - 68 q^{46} - 15 q^{47} + 26 q^{48} - 28 q^{49} - 7 q^{50} - 69 q^{51} - 98 q^{52} - 23 q^{53} - 6 q^{54} - 93 q^{55} - 48 q^{56} - 74 q^{57} - 37 q^{58} - 36 q^{59} - 42 q^{60} - 172 q^{61} - 26 q^{62} - 42 q^{63} - 23 q^{64} - 66 q^{65} - 62 q^{66} - 143 q^{67} - 74 q^{68} - 19 q^{69} - 30 q^{70} - 9 q^{71} - 15 q^{72} - 134 q^{73} - 19 q^{74} + 16 q^{75} - 157 q^{76} - 25 q^{77} - 27 q^{78} - 138 q^{79} - 29 q^{80} + 92 q^{81} - 61 q^{82} - 24 q^{83} - 101 q^{84} - 84 q^{85} + 14 q^{86} - 54 q^{87} - 140 q^{88} - 148 q^{89} - 44 q^{90} - 115 q^{91} - 12 q^{92} - 67 q^{93} - 79 q^{94} - 10 q^{95} - 36 q^{96} - 165 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 92 * q - 6 * q^2 + 92 * q^3 + 70 * q^4 - 18 * q^5 - 6 * q^6 - 42 * q^7 - 15 * q^8 + 92 * q^9 - 44 * q^10 - 24 * q^11 + 70 * q^12 - 48 * q^13 - 29 * q^14 - 18 * q^15 + 26 * q^16 - 69 * q^17 - 6 * q^18 - 74 * q^19 - 42 * q^20 - 42 * q^21 - 62 * q^22 - 19 * q^23 - 15 * q^24 + 16 * q^25 - 27 * q^26 + 92 * q^27 - 101 * q^28 - 54 * q^29 - 44 * q^30 - 67 * q^31 - 36 * q^32 - 24 * q^33 - 63 * q^34 - 31 * q^35 + 70 * q^36 - 70 * q^37 - 18 * q^38 - 48 * q^39 - 125 * q^40 - 98 * q^41 - 29 * q^42 - 159 * q^43 - 52 * q^44 - 18 * q^45 - 68 * q^46 - 15 * q^47 + 26 * q^48 - 28 * q^49 - 7 * q^50 - 69 * q^51 - 98 * q^52 - 23 * q^53 - 6 * q^54 - 93 * q^55 - 48 * q^56 - 74 * q^57 - 37 * q^58 - 36 * q^59 - 42 * q^60 - 172 * q^61 - 26 * q^62 - 42 * q^63 - 23 * q^64 - 66 * q^65 - 62 * q^66 - 143 * q^67 - 74 * q^68 - 19 * q^69 - 30 * q^70 - 9 * q^71 - 15 * q^72 - 134 * q^73 - 19 * q^74 + 16 * q^75 - 157 * q^76 - 25 * q^77 - 27 * q^78 - 138 * q^79 - 29 * q^80 + 92 * q^81 - 61 * q^82 - 24 * q^83 - 101 * q^84 - 84 * q^85 + 14 * q^86 - 54 * q^87 - 140 * q^88 - 148 * q^89 - 44 * q^90 - 115 * q^91 - 12 * q^92 - 67 * q^93 - 79 * q^94 - 10 * q^95 - 36 * q^96 - 165 * q^97 + 36 * q^98 - 24 * q^99

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