LMFDB - Embedded newform 4031.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53428 q^{2} -2.04027 q^{3} +4.42257 q^{4} -2.47983 q^{5} +5.17061 q^{6} -0.629929 q^{7} -6.13948 q^{8} +1.16269 q^{9} +O(q^{10})$	\(q-2.53428 q^{2} -2.04027 q^{3} +4.42257 q^{4} -2.47983 q^{5} +5.17061 q^{6} -0.629929 q^{7} -6.13948 q^{8} +1.16269 q^{9} +6.28458 q^{10} +0.0387158 q^{11} -9.02323 q^{12} -2.09282 q^{13} +1.59642 q^{14} +5.05951 q^{15} +6.71401 q^{16} +1.11384 q^{17} -2.94657 q^{18} -4.51218 q^{19} -10.9672 q^{20} +1.28522 q^{21} -0.0981165 q^{22} +4.29752 q^{23} +12.5262 q^{24} +1.14956 q^{25} +5.30378 q^{26} +3.74861 q^{27} -2.78591 q^{28} -1.00000 q^{29} -12.8222 q^{30} -9.00294 q^{31} -4.73622 q^{32} -0.0789904 q^{33} -2.82278 q^{34} +1.56212 q^{35} +5.14207 q^{36} +0.632125 q^{37} +11.4351 q^{38} +4.26990 q^{39} +15.2249 q^{40} +6.59466 q^{41} -3.25712 q^{42} -0.733209 q^{43} +0.171223 q^{44} -2.88327 q^{45} -10.8911 q^{46} +2.09354 q^{47} -13.6984 q^{48} -6.60319 q^{49} -2.91330 q^{50} -2.27253 q^{51} -9.25563 q^{52} -5.00250 q^{53} -9.50002 q^{54} -0.0960085 q^{55} +3.86744 q^{56} +9.20606 q^{57} +2.53428 q^{58} +7.01828 q^{59} +22.3761 q^{60} +8.02767 q^{61} +22.8160 q^{62} -0.732410 q^{63} -1.42512 q^{64} +5.18983 q^{65} +0.200184 q^{66} +9.08801 q^{67} +4.92603 q^{68} -8.76810 q^{69} -3.95884 q^{70} -7.30583 q^{71} -7.13829 q^{72} +1.15403 q^{73} -1.60198 q^{74} -2.34540 q^{75} -19.9555 q^{76} -0.0243882 q^{77} -10.8211 q^{78} -8.00526 q^{79} -16.6496 q^{80} -11.1362 q^{81} -16.7127 q^{82} +7.30793 q^{83} +5.68399 q^{84} -2.76213 q^{85} +1.85816 q^{86} +2.04027 q^{87} -0.237695 q^{88} -0.395426 q^{89} +7.30700 q^{90} +1.31833 q^{91} +19.0061 q^{92} +18.3684 q^{93} -5.30562 q^{94} +11.1895 q^{95} +9.66315 q^{96} +15.2066 q^{97} +16.7343 q^{98} +0.0450143 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	−2.53428	−1.79201	−0.896003	−	0.444048i	$-0.853542\pi$
$2$	−2.53428	−1.79201	−0.896003	+	0.444048i	$0.853542\pi$
$3$	−2.04027	−1.17795	−0.588974	−	0.808152i	$-0.700468\pi$
$3$	−2.04027	−1.17795	−0.588974	+	0.808152i	$0.700468\pi$
$4$	4.42257	2.21129
$5$	−2.47983	−1.10901	−0.554507	−	0.832179i	$-0.687093\pi$
$5$	−2.47983	−1.10901	−0.554507	+	0.832179i	$0.687093\pi$
$6$	5.17061	2.11089
$7$	−0.629929	−0.238091	−0.119045	−	0.992889i	$-0.537983\pi$
$7$	−0.629929	−0.238091	−0.119045	+	0.992889i	$0.537983\pi$
$8$	−6.13948	−2.17063
$9$	1.16269	0.387562
$10$	6.28458	1.98736
$11$	0.0387158	0.0116732	0.00583662	−	0.999983i	$-0.498142\pi$
$11$	0.0387158	0.0116732	0.00583662	+	0.999983i	$0.498142\pi$
$12$	−9.02323	−2.60478
$13$	−2.09282	−0.580443	−0.290221	−	0.956960i	$-0.593729\pi$
$13$	−2.09282	−0.580443	−0.290221	+	0.956960i	$0.593729\pi$
$14$	1.59642	0.426660
$15$	5.05951	1.30636
$16$	6.71401	1.67850
$17$	1.11384	0.270145	0.135073	−	0.990836i	$-0.456873\pi$
$17$	1.11384	0.270145	0.135073	+	0.990836i	$0.456873\pi$
$18$	−2.94657	−0.694514
$19$	−4.51218	−1.03517	−0.517583	−	0.855633i	$-0.673168\pi$
$19$	−4.51218	−1.03517	−0.517583	+	0.855633i	$0.673168\pi$
$20$	−10.9672	−2.45235
$21$	1.28522	0.280459
$22$	−0.0981165	−0.0209185
$23$	4.29752	0.896096	0.448048	−	0.894010i	$-0.352119\pi$
$23$	4.29752	0.896096	0.448048	+	0.894010i	$0.352119\pi$
$24$	12.5262	2.55689
$25$	1.14956	0.229912
$26$	5.30378	1.04016
$27$	3.74861	0.721420
$28$	−2.78591	−0.526487
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−12.8222	−2.34101
$31$	−9.00294	−1.61698	−0.808488	−	0.588513i	$-0.799714\pi$
$31$	−9.00294	−1.61698	−0.808488	+	0.588513i	$0.799714\pi$
$32$	−4.73622	−0.837253
$33$	−0.0789904	−0.0137505
$34$	−2.82278	−0.484102
$35$	1.56212	0.264046
$36$	5.14207	0.857011
$37$	0.632125	0.103921	0.0519603	−	0.998649i	$-0.483453\pi$
$37$	0.632125	0.103921	0.0519603	+	0.998649i	$0.483453\pi$
$38$	11.4351	1.85502
$39$	4.26990	0.683732
$40$	15.2249	2.40726
$41$	6.59466	1.02991	0.514956	−	0.857217i	$-0.327808\pi$
$41$	6.59466	1.02991	0.514956	+	0.857217i	$0.327808\pi$
$42$	−3.25712	−0.502584
$43$	−0.733209	−0.111813	−0.0559067	−	0.998436i	$-0.517805\pi$
$43$	−0.733209	−0.111813	−0.0559067	+	0.998436i	$0.517805\pi$
$44$	0.171223	0.0258129
$45$	−2.88327	−0.429812
$46$	−10.8911	−1.60581
$47$	2.09354	0.305374	0.152687	−	0.988275i	$-0.451207\pi$
$47$	2.09354	0.305374	0.152687	+	0.988275i	$0.451207\pi$
$48$	−13.6984	−1.97719
$49$	−6.60319	−0.943313
$50$	−2.91330	−0.412003
$51$	−2.27253	−0.318217
$52$	−9.25563	−1.28353
$53$	−5.00250	−0.687147	−0.343573	−	0.939126i	$-0.611637\pi$
$53$	−5.00250	−0.687147	−0.343573	+	0.939126i	$0.611637\pi$
$54$	−9.50002	−1.29279
$55$	−0.0960085	−0.0129458
$56$	3.86744	0.516808
$57$	9.20606	1.21937
$58$	2.53428	0.332767
$59$	7.01828	0.913702	0.456851	−	0.889543i	$-0.348977\pi$
$59$	7.01828	0.913702	0.456851	+	0.889543i	$0.348977\pi$
$60$	22.3761	2.88874
$61$	8.02767	1.02784	0.513919	−	0.857839i	$-0.328193\pi$
$61$	8.02767	1.02784	0.513919	+	0.857839i	$0.328193\pi$
$62$	22.8160	2.89763
$63$	−0.732410	−0.0922750
$64$	−1.42512	−0.178140
$65$	5.18983	0.643719
$66$	0.200184	0.0246409
$67$	9.08801	1.11028	0.555138	−	0.831758i	$-0.312665\pi$
$67$	9.08801	1.11028	0.555138	+	0.831758i	$0.312665\pi$
$68$	4.92603	0.597369
$69$	−8.76810	−1.05555
$70$	−3.95884	−0.473172
$71$	−7.30583	−0.867042	−0.433521	−	0.901143i	$-0.642729\pi$
$71$	−7.30583	−0.867042	−0.433521	+	0.901143i	$0.642729\pi$
$72$	−7.13829	−0.841256
$73$	1.15403	0.135069	0.0675347	−	0.997717i	$-0.478487\pi$
$73$	1.15403	0.135069	0.0675347	+	0.997717i	$0.478487\pi$
$74$	−1.60198	−0.186227
$75$	−2.34540	−0.270824
$76$	−19.9555	−2.28905
$77$	−0.0243882	−0.00277929
$78$	−10.8211	−1.22525
$79$	−8.00526	−0.900662	−0.450331	−	0.892862i	$-0.648694\pi$
$79$	−8.00526	−0.900662	−0.450331	+	0.892862i	$0.648694\pi$
$80$	−16.6496	−1.86148
$81$	−11.1362	−1.23736
$82$	−16.7127	−1.84561
$83$	7.30793	0.802150	0.401075	−	0.916045i	$-0.368637\pi$
$83$	7.30793	0.802150	0.401075	+	0.916045i	$0.368637\pi$
$84$	5.68399	0.620175
$85$	−2.76213	−0.299595
$86$	1.85816	0.200370
$87$	2.04027	0.218740
$88$	−0.237695	−0.0253383
$89$	−0.395426	−0.0419150	−0.0209575	−	0.999780i	$-0.506671\pi$
$89$	−0.395426	−0.0419150	−0.0209575	+	0.999780i	$0.506671\pi$
$90$	7.30700	0.770226
$91$	1.31833	0.138198
$92$	19.0061	1.98152
$93$	18.3684	1.90471
$94$	−5.30562	−0.547233
$95$	11.1895	1.14801
$96$	9.66315	0.986241
$97$	15.2066	1.54399	0.771997	−	0.635627i	$-0.219258\pi$
$97$	15.2066	1.54399	0.771997	+	0.635627i	$0.219258\pi$
$98$	16.7343	1.69042
$99$	0.0450143	0.00452411
$100$	5.08401	0.508401
$101$	−6.72285	−0.668948	−0.334474	−	0.942405i	$-0.608559\pi$
$101$	−6.72285	−0.668948	−0.334474	+	0.942405i	$0.608559\pi$
$102$	5.75922	0.570248
$103$	−1.70519	−0.168018	−0.0840088	−	0.996465i	$-0.526772\pi$
$103$	−1.70519	−0.168018	−0.0840088	+	0.996465i	$0.526772\pi$
$104$	12.8488	1.25993
$105$	−3.18714	−0.311033
$106$	12.6777	1.23137
$107$	−2.03038	−0.196284	−0.0981420	−	0.995172i	$-0.531290\pi$
$107$	−2.03038	−0.196284	−0.0981420	+	0.995172i	$0.531290\pi$
$108$	16.5785	1.59527
$109$	−0.0254372	−0.00243644	−0.00121822	−	0.999999i	$-0.500388\pi$
$109$	−0.0254372	−0.00243644	−0.00121822	+	0.999999i	$0.500388\pi$
$110$	0.243312	0.0231989
$111$	−1.28970	−0.122413
$112$	−4.22935	−0.399636
$113$	10.3511	0.973753	0.486876	−	0.873471i	$-0.338136\pi$
$113$	10.3511	0.973753	0.486876	+	0.873471i	$0.338136\pi$
$114$	−23.3307	−2.18512
$115$	−10.6571	−0.993783
$116$	−4.42257	−0.410626
$117$	−2.43329	−0.224958
$118$	−17.7863	−1.63736
$119$	−0.701639	−0.0643192
$120$	−31.0628	−2.83563
$121$	−10.9985	−0.999864
$122$	−20.3444	−1.84189
$123$	−13.4549	−1.21318
$124$	−39.8162	−3.57560
$125$	9.54844	0.854039
$126$	1.85613	0.165357
$127$	15.8767	1.40883	0.704413	−	0.709790i	$-0.251210\pi$
$127$	15.8767	1.40883	0.704413	+	0.709790i	$0.251210\pi$
$128$	13.0841	1.15648
$129$	1.49594	0.131710
$130$	−13.1525	−1.15355
$131$	15.7505	1.37613	0.688064	−	0.725650i	$-0.258461\pi$
$131$	15.7505	1.37613	0.688064	+	0.725650i	$0.258461\pi$
$132$	−0.349341	−0.0304062
$133$	2.84236	0.246464
$134$	−23.0316	−1.98962
$135$	−9.29591	−0.800065
$136$	−6.83839	−0.586387
$137$	9.04429	0.772706	0.386353	−	0.922351i	$-0.373735\pi$
$137$	9.04429	0.772706	0.386353	+	0.922351i	$0.373735\pi$
$138$	22.2208	1.89156
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	6.90858	0.583881
$141$	−4.27138	−0.359715
$142$	18.5150	1.55375
$143$	−0.0810249	−0.00677565
$144$	7.80629	0.650524
$145$	2.47983	0.205939
$146$	−2.92464	−0.242045
$147$	13.4723	1.11117
$148$	2.79562	0.229798
$149$	12.6443	1.03586	0.517932	−	0.855422i	$-0.326702\pi$
$149$	12.6443	1.03586	0.517932	+	0.855422i	$0.326702\pi$
$150$	5.94391	0.485318
$151$	11.5787	0.942262	0.471131	−	0.882063i	$-0.343846\pi$
$151$	11.5787	0.942262	0.471131	+	0.882063i	$0.343846\pi$
$152$	27.7025	2.24697
$153$	1.29505	0.104698
$154$	0.0618065	0.00498051
$155$	22.3258	1.79325
$156$	18.8840	1.51193
$157$	−5.42291	−0.432795	−0.216398	−	0.976305i	$-0.569431\pi$
$157$	−5.42291	−0.432795	−0.216398	+	0.976305i	$0.569431\pi$
$158$	20.2876	1.61399
$159$	10.2064	0.809423
$160$	11.7450	0.928525
$161$	−2.70714	−0.213352
$162$	28.2223	2.21735
$163$	−8.88009	−0.695543	−0.347771	−	0.937579i	$-0.613061\pi$
$163$	−8.88009	−0.695543	−0.347771	+	0.937579i	$0.613061\pi$
$164$	29.1653	2.27743
$165$	0.195883	0.0152495
$166$	−18.5203	−1.43746
$167$	−13.1561	−1.01805	−0.509024	−	0.860752i	$-0.669994\pi$
$167$	−13.1561	−1.01805	−0.509024	+	0.860752i	$0.669994\pi$
$168$	−7.89060	−0.608773
$169$	−8.62012	−0.663086
$170$	7.00001	0.536876
$171$	−5.24626	−0.401191
$172$	−3.24267	−0.247251
$173$	19.2022	1.45992	0.729960	−	0.683490i	$-0.239539\pi$
$173$	19.2022	1.45992	0.729960	+	0.683490i	$0.239539\pi$
$174$	−5.17061	−0.391983
$175$	−0.724140	−0.0547399
$176$	0.259938	0.0195936
$177$	−14.3192	−1.07629
$178$	1.00212	0.0751120
$179$	−2.91749	−0.218064	−0.109032	−	0.994038i	$-0.534775\pi$
$179$	−2.91749	−0.218064	−0.109032	+	0.994038i	$0.534775\pi$
$180$	−12.7515	−0.950437
$181$	15.8412	1.17747	0.588733	−	0.808327i	$-0.299627\pi$
$181$	15.8412	1.17747	0.588733	+	0.808327i	$0.299627\pi$
$182$	−3.34101	−0.247652
$183$	−16.3786	−1.21074
$184$	−26.3846	−1.94510
$185$	−1.56756	−0.115249
$186$	−46.5507	−3.41326
$187$	0.0431231	0.00315347
$188$	9.25884	0.675270
$189$	−2.36136	−0.171763
$190$	−28.3572	−2.05725
$191$	−0.0385705	−0.00279086	−0.00139543	−	0.999999i	$-0.500444\pi$
$191$	−0.0385705	−0.00279086	−0.00139543	+	0.999999i	$0.500444\pi$
$192$	2.90762	0.209839
$193$	−11.6336	−0.837403	−0.418701	−	0.908124i	$-0.637515\pi$
$193$	−11.6336	−0.837403	−0.418701	+	0.908124i	$0.637515\pi$
$194$	−38.5377	−2.76685
$195$	−10.5886	−0.758268
$196$	−29.2031	−2.08593
$197$	17.5139	1.24781	0.623906	−	0.781499i	$-0.285545\pi$
$197$	17.5139	1.24781	0.623906	+	0.781499i	$0.285545\pi$
$198$	−0.114079	−0.00810723
$199$	20.0932	1.42437	0.712186	−	0.701991i	$-0.247705\pi$
$199$	20.0932	1.42437	0.712186	+	0.701991i	$0.247705\pi$
$200$	−7.05769	−0.499054
$201$	−18.5420	−1.30785
$202$	17.0376	1.19876
$203$	0.629929	0.0442124
$204$	−10.0504	−0.703670
$205$	−16.3536	−1.14219
$206$	4.32144	0.301089
$207$	4.99668	0.347293
$208$	−14.0512	−0.974274
$209$	−0.174693	−0.0120837
$210$	8.07709	0.557372
$211$	1.68594	0.116065	0.0580325	−	0.998315i	$-0.481517\pi$
$211$	1.68594	0.116065	0.0580325	+	0.998315i	$0.481517\pi$
$212$	−22.1239	−1.51948
$213$	14.9058	1.02133
$214$	5.14555	0.351742
$215$	1.81823	0.124002
$216$	−23.0145	−1.56594
$217$	5.67122	0.384987
$218$	0.0644650	0.00436612
$219$	−2.35453	−0.159105
$220$	−0.424605	−0.0286268
$221$	−2.33106	−0.156804
$222$	3.26847	0.219365
$223$	−6.69856	−0.448569	−0.224284	−	0.974524i	$-0.572004\pi$
$223$	−6.69856	−0.448569	−0.224284	+	0.974524i	$0.572004\pi$
$224$	2.98348	0.199342
$225$	1.33658	0.0891051
$226$	−26.2327	−1.74497
$227$	0.666704	0.0442507	0.0221254	−	0.999755i	$-0.492957\pi$
$227$	0.666704	0.0442507	0.0221254	+	0.999755i	$0.492957\pi$
$228$	40.7145	2.69638
$229$	24.7669	1.63665	0.818323	−	0.574759i	$-0.194904\pi$
$229$	24.7669	1.63665	0.818323	+	0.574759i	$0.194904\pi$
$230$	27.0082	1.78086
$231$	0.0497584	0.00327386
$232$	6.13948	0.403076
$233$	−7.46473	−0.489031	−0.244515	−	0.969645i	$-0.578629\pi$
$233$	−7.46473	−0.489031	−0.244515	+	0.969645i	$0.578629\pi$
$234$	6.16664	0.403126
$235$	−5.19163	−0.338664
$236$	31.0388	2.02046
$237$	16.3329	1.06093
$238$	1.77815	0.115260
$239$	15.1859	0.982297	0.491148	−	0.871076i	$-0.336577\pi$
$239$	15.1859	0.982297	0.491148	+	0.871076i	$0.336577\pi$
$240$	33.9696	2.19273
$241$	−25.0731	−1.61510	−0.807549	−	0.589801i	$-0.799206\pi$
$241$	−25.0731	−1.61510	−0.807549	+	0.589801i	$0.799206\pi$
$242$	27.8733	1.79176
$243$	11.4750	0.736124
$244$	35.5030	2.27284
$245$	16.3748	1.04615
$246$	34.0984	2.17403
$247$	9.44317	0.600855
$248$	55.2734	3.50986
$249$	−14.9101	−0.944891
$250$	−24.1984	−1.53044
$251$	−2.13311	−0.134640	−0.0673202	−	0.997731i	$-0.521445\pi$
$251$	−2.13311	−0.134640	−0.0673202	+	0.997731i	$0.521445\pi$
$252$	−3.23914	−0.204047
$253$	0.166382	0.0104603
$254$	−40.2359	−2.52463
$255$	5.63548	0.352907
$256$	−30.3085	−1.89428
$257$	−0.832643	−0.0519388	−0.0259694	−	0.999663i	$-0.508267\pi$
$257$	−0.832643	−0.0519388	−0.0259694	+	0.999663i	$0.508267\pi$
$258$	−3.79113	−0.236026
$259$	−0.398194	−0.0247426
$260$	22.9524	1.42345
$261$	−1.16269	−0.0719685
$262$	−39.9162	−2.46603
$263$	19.2457	1.18674	0.593371	−	0.804929i	$-0.297797\pi$
$263$	19.2457	1.18674	0.593371	+	0.804929i	$0.297797\pi$
$264$	0.484960	0.0298472
$265$	12.4054	0.762055
$266$	−7.20333	−0.441664
$267$	0.806774	0.0493738
$268$	40.1924	2.45514
$269$	−5.86704	−0.357719	−0.178860	−	0.983875i	$-0.557241\pi$
$269$	−5.86704	−0.357719	−0.178860	+	0.983875i	$0.557241\pi$
$270$	23.5584	1.43372
$271$	4.41421	0.268144	0.134072	−	0.990972i	$-0.457195\pi$
$271$	4.41421	0.268144	0.134072	+	0.990972i	$0.457195\pi$
$272$	7.47832	0.453440
$273$	−2.68974	−0.162790
$274$	−22.9208	−1.38469
$275$	0.0445060	0.00268381
$276$	−38.7775	−2.33413
$277$	−8.35250	−0.501853	−0.250927	−	0.968006i	$-0.580735\pi$
$277$	−8.35250	−0.501853	−0.250927	+	0.968006i	$0.580735\pi$
$278$	2.53428	0.151996
$279$	−10.4676	−0.626679
$280$	−9.59059	−0.573147
$281$	0.482945	0.0288101	0.0144050	−	0.999896i	$-0.495415\pi$
$281$	0.482945	0.0288101	0.0144050	+	0.999896i	$0.495415\pi$
$282$	10.8249	0.644612
$283$	−9.14175	−0.543421	−0.271710	−	0.962379i	$-0.587589\pi$
$283$	−9.14175	−0.543421	−0.271710	+	0.962379i	$0.587589\pi$
$284$	−32.3106	−1.91728
$285$	−22.8295	−1.35230
$286$	0.205340	0.0121420
$287$	−4.15417	−0.245213
$288$	−5.50674	−0.324488
$289$	−15.7594	−0.927021
$290$	−6.28458	−0.369043
$291$	−31.0255	−1.81874
$292$	5.10379	0.298677
$293$	9.06814	0.529767	0.264883	−	0.964280i	$-0.414667\pi$
$293$	9.06814	0.529767	0.264883	+	0.964280i	$0.414667\pi$
$294$	−34.1425	−1.99123
$295$	−17.4041	−1.01331
$296$	−3.88092	−0.225574
$297$	0.145130	0.00842131
$298$	−32.0443	−1.85627
$299$	−8.99393	−0.520132
$300$	−10.3727	−0.598870
$301$	0.461870	0.0266217
$302$	−29.3437	−1.68854
$303$	13.7164	0.787986
$304$	−30.2948	−1.73753
$305$	−19.9073	−1.13989
$306$	−3.28201	−0.187620
$307$	22.1343	1.26327	0.631635	−	0.775266i	$-0.282384\pi$
$307$	22.1343	1.26327	0.631635	+	0.775266i	$0.282384\pi$
$308$	−0.107859	−0.00614581
$309$	3.47905	0.197916
$310$	−56.5797	−3.21351
$311$	−29.3636	−1.66505	−0.832527	−	0.553984i	$-0.813107\pi$
$311$	−29.3636	−1.66505	−0.832527	+	0.553984i	$0.813107\pi$
$312$	−26.2150	−1.48413
$313$	−17.3492	−0.980636	−0.490318	−	0.871544i	$-0.663119\pi$
$313$	−17.3492	−0.980636	−0.490318	+	0.871544i	$0.663119\pi$
$314$	13.7432	0.775572
$315$	1.81625	0.102334
$316$	−35.4039	−1.99162
$317$	3.38474	0.190106	0.0950531	−	0.995472i	$-0.469698\pi$
$317$	3.38474	0.190106	0.0950531	+	0.995472i	$0.469698\pi$
$318$	−25.8660	−1.45049
$319$	−0.0387158	−0.00216767
$320$	3.53405	0.197559
$321$	4.14251	0.231212
$322$	6.86064	0.382329
$323$	−5.02584	−0.279645
$324$	−49.2507	−2.73615
$325$	−2.40581	−0.133451
$326$	22.5046	1.24642
$327$	0.0518987	0.00287000
$328$	−40.4877	−2.23556
$329$	−1.31878	−0.0727068
$330$	−0.496422	−0.0273271
$331$	18.9850	1.04351	0.521756	−	0.853095i	$-0.325277\pi$
$331$	18.9850	1.04351	0.521756	+	0.853095i	$0.325277\pi$
$332$	32.3199	1.77378
$333$	0.734963	0.0402757
$334$	33.3412	1.82435
$335$	−22.5367	−1.23131
$336$	8.62900	0.470751
$337$	2.41583	0.131599	0.0657993	−	0.997833i	$-0.479040\pi$
$337$	2.41583	0.131599	0.0657993	+	0.997833i	$0.479040\pi$
$338$	21.8458	1.18825
$339$	−21.1191	−1.14703
$340$	−12.2157	−0.662491
$341$	−0.348556	−0.0188753
$342$	13.2955	0.718937
$343$	8.56905	0.462685
$344$	4.50152	0.242706
$345$	21.7434	1.17062
$346$	−48.6638	−2.61618
$347$	−25.9390	−1.39248	−0.696239	−	0.717810i	$-0.745145\pi$
$347$	−25.9390	−1.39248	−0.696239	+	0.717810i	$0.745145\pi$
$348$	9.02323	0.483696
$349$	−17.1162	−0.916207	−0.458103	−	0.888899i	$-0.651471\pi$
$349$	−17.1162	−0.916207	−0.458103	+	0.888899i	$0.651471\pi$
$350$	1.83517	0.0980942
$351$	−7.84515	−0.418743
$352$	−0.183366	−0.00977345
$353$	−34.2932	−1.82524	−0.912620	−	0.408808i	$-0.865944\pi$
$353$	−34.2932	−1.82524	−0.912620	+	0.408808i	$0.865944\pi$
$354$	36.2887	1.92872
$355$	18.1172	0.961562
$356$	−1.74880	−0.0926862
$357$	1.43153	0.0757647
$358$	7.39375	0.390772
$359$	−15.3319	−0.809188	−0.404594	−	0.914496i	$-0.632587\pi$
$359$	−15.3319	−0.809188	−0.404594	+	0.914496i	$0.632587\pi$
$360$	17.7017	0.932964
$361$	1.35981	0.0715690
$362$	−40.1460	−2.11003
$363$	22.4399	1.17779
$364$	5.83039	0.305596
$365$	−2.86181	−0.149794
$366$	41.5079	2.16965
$367$	12.8028	0.668300	0.334150	−	0.942520i	$-0.391551\pi$
$367$	12.8028	0.668300	0.334150	+	0.942520i	$0.391551\pi$
$368$	28.8536	1.50410
$369$	7.66752	0.399155
$370$	3.97264	0.206528
$371$	3.15122	0.163603
$372$	81.2356	4.21187
$373$	−7.55022	−0.390935	−0.195468	−	0.980710i	$-0.562622\pi$
$373$	−7.55022	−0.390935	−0.195468	+	0.980710i	$0.562622\pi$
$374$	−0.109286	−0.00565104
$375$	−19.4814	−1.00601
$376$	−12.8532	−0.662856
$377$	2.09282	0.107786
$378$	5.98434	0.307801
$379$	−19.0391	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$379$	−19.0391	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$380$	49.4862	2.53859
$381$	−32.3926	−1.65953
$382$	0.0977485	0.00500125
$383$	21.0226	1.07420	0.537102	−	0.843517i	$-0.319519\pi$
$383$	21.0226	1.07420	0.537102	+	0.843517i	$0.319519\pi$
$384$	−26.6950	−1.36227
$385$	0.0604785	0.00308227
$386$	29.4827	1.50063
$387$	−0.852492	−0.0433346
$388$	67.2522	3.41421
$389$	14.8273	0.751773	0.375886	−	0.926666i	$-0.377338\pi$
$389$	14.8273	0.751773	0.375886	+	0.926666i	$0.377338\pi$
$390$	26.8346	1.35882
$391$	4.78675	0.242076
$392$	40.5401	2.04759
$393$	−32.1352	−1.62101
$394$	−44.3851	−2.23609
$395$	19.8517	0.998847
$396$	0.199079	0.0100041
$397$	−2.20756	−0.110794	−0.0553972	−	0.998464i	$-0.517643\pi$
$397$	−2.20756	−0.110794	−0.0553972	+	0.998464i	$0.517643\pi$
$398$	−50.9219	−2.55248
$399$	−5.79917	−0.290321
$400$	7.71814	0.385907
$401$	−16.5629	−0.827112	−0.413556	−	0.910479i	$-0.635713\pi$
$401$	−16.5629	−0.827112	−0.413556	+	0.910479i	$0.635713\pi$
$402$	46.9905	2.34367
$403$	18.8415	0.938562
$404$	−29.7323	−1.47924
$405$	27.6159	1.37225
$406$	−1.59642	−0.0792288
$407$	0.0244732	0.00121309
$408$	13.9521	0.690733
$409$	−13.5897	−0.671968	−0.335984	−	0.941868i	$-0.609069\pi$
$409$	−13.5897	−0.671968	−0.335984	+	0.941868i	$0.609069\pi$
$410$	41.4447	2.04681
$411$	−18.4528	−0.910208
$412$	−7.54134	−0.371535
$413$	−4.42102	−0.217544
$414$	−12.6630	−0.622351
$415$	−18.1224	−0.889595
$416$	9.91203	0.485977
$417$	2.04027	0.0999123
$418$	0.442720	0.0216541
$419$	−17.9232	−0.875604	−0.437802	−	0.899071i	$-0.644243\pi$
$419$	−17.9232	−0.875604	−0.437802	+	0.899071i	$0.644243\pi$
$420$	−14.0953	−0.687782
$421$	7.35827	0.358620	0.179310	−	0.983793i	$-0.442613\pi$
$421$	7.35827	0.358620	0.179310	+	0.983793i	$0.442613\pi$
$422$	−4.27265	−0.207989
$423$	2.43413	0.118352
$424$	30.7128	1.49154
$425$	1.28042	0.0621096
$426$	−37.7756	−1.83023
$427$	−5.05686	−0.244719
$428$	−8.97950	−0.434040
$429$	0.165312	0.00798136
$430$	−4.60791	−0.222213
$431$	−13.8655	−0.667880	−0.333940	−	0.942594i	$-0.608378\pi$
$431$	−13.8655	−0.667880	−0.333940	+	0.942594i	$0.608378\pi$
$432$	25.1682	1.21090
$433$	0.389905	0.0187377	0.00936883	−	0.999956i	$-0.497018\pi$
$433$	0.389905	0.0187377	0.00936883	+	0.999956i	$0.497018\pi$
$434$	−14.3724	−0.689899
$435$	−5.05951	−0.242585
$436$	−0.112498	−0.00538767
$437$	−19.3912	−0.927608
$438$	5.96705	0.285117
$439$	−23.8276	−1.13723	−0.568615	−	0.822604i	$-0.692521\pi$
$439$	−23.8276	−1.13723	−0.568615	+	0.822604i	$0.692521\pi$
$440$	0.589442	0.0281005
$441$	−7.67744	−0.365592
$442$	5.90755	0.280994
$443$	−28.7945	−1.36807	−0.684035	−	0.729450i	$-0.739776\pi$
$443$	−28.7945	−1.36807	−0.684035	+	0.729450i	$0.739776\pi$
$444$	−5.70381	−0.270691
$445$	0.980589	0.0464844
$446$	16.9760	0.803838
$447$	−25.7978	−1.22019
$448$	0.897722	0.0424134
$449$	4.27392	0.201699	0.100849	−	0.994902i	$-0.467844\pi$
$449$	4.27392	0.201699	0.100849	+	0.994902i	$0.467844\pi$
$450$	−3.38726	−0.159677
$451$	0.255317	0.0120224
$452$	45.7786	2.15325
$453$	−23.6236	−1.10994
$454$	−1.68962	−0.0792976
$455$	−3.26922	−0.153264
$456$	−56.5204	−2.64681
$457$	−36.6346	−1.71369	−0.856847	−	0.515571i	$-0.827580\pi$
$457$	−36.6346	−1.71369	−0.856847	+	0.515571i	$0.827580\pi$
$458$	−62.7664	−2.93288
$459$	4.17534	0.194888
$460$	−47.1319	−2.19754
$461$	23.5352	1.09614	0.548072	−	0.836431i	$-0.315362\pi$
$461$	23.5352	1.09614	0.548072	+	0.836431i	$0.315362\pi$
$462$	−0.126102	−0.00586678
$463$	−18.5748	−0.863242	−0.431621	−	0.902055i	$-0.642058\pi$
$463$	−18.5748	−0.863242	−0.431621	+	0.902055i	$0.642058\pi$
$464$	−6.71401	−0.311690
$465$	−45.5505	−2.11235
$466$	18.9177	0.876347
$467$	−0.0296519	−0.00137212	−0.000686062	−	1.00000i	$-0.500218\pi$
$467$	−0.0296519	−0.00137212	−0.000686062		1.00000i	$0.500218\pi$
$468$	−10.7614	−0.497446
$469$	−5.72480	−0.264347
$470$	13.1570	0.606889
$471$	11.0642	0.509811
$472$	−43.0885	−1.98331
$473$	−0.0283867	−0.00130522
$474$	−41.3921	−1.90120
$475$	−5.18702	−0.237997
$476$	−3.10305	−0.142228
$477$	−5.81634	−0.266312
$478$	−38.4854	−1.76028
$479$	−11.5808	−0.529142	−0.264571	−	0.964366i	$-0.585230\pi$
$479$	−11.5808	−0.529142	−0.264571	+	0.964366i	$0.585230\pi$
$480$	−23.9630	−1.09375
$481$	−1.32292	−0.0603200
$482$	63.5421	2.89426
$483$	5.52328	0.251318
$484$	−48.6417	−2.21099
$485$	−37.7097	−1.71231
$486$	−29.0809	−1.31914
$487$	8.42949	0.381977	0.190988	−	0.981592i	$-0.438831\pi$
$487$	8.42949	0.381977	0.190988	+	0.981592i	$0.438831\pi$
$488$	−49.2857	−2.23106
$489$	18.1178	0.819313
$490$	−41.4983	−1.87470
$491$	−20.8335	−0.940204	−0.470102	−	0.882612i	$-0.655783\pi$
$491$	−20.8335	−0.940204	−0.470102	+	0.882612i	$0.655783\pi$
$492$	−59.5051	−2.68270
$493$	−1.11384	−0.0501648
$494$	−23.9316	−1.07674
$495$	−0.111628	−0.00501730
$496$	−60.4458	−2.71410
$497$	4.60216	0.206435
$498$	37.7864	1.69325
$499$	27.9109	1.24947	0.624733	−	0.780839i	$-0.285208\pi$
$499$	27.9109	1.24947	0.624733	+	0.780839i	$0.285208\pi$
$500$	42.2287	1.88852
$501$	26.8419	1.19921
$502$	5.40589	0.241277
$503$	−6.43258	−0.286815	−0.143407	−	0.989664i	$-0.545806\pi$
$503$	−6.43258	−0.286815	−0.143407	+	0.989664i	$0.545806\pi$
$504$	4.49662	0.200295
$505$	16.6715	0.741873
$506$	−0.421658	−0.0187450
$507$	17.5873	0.781081
$508$	70.2157	3.11532
$509$	−28.7881	−1.27601	−0.638005	−	0.770033i	$-0.720240\pi$
$509$	−28.7881	−1.27601	−0.638005	+	0.770033i	$0.720240\pi$
$510$	−14.2819	−0.632412
$511$	−0.726959	−0.0321588
$512$	50.6420	2.23808
$513$	−16.9144	−0.746790
$514$	2.11015	0.0930747
$515$	4.22859	0.186334
$516$	6.61591	0.291249
$517$	0.0810530	0.00356471
$518$	1.00913	0.0443388
$519$	−39.1777	−1.71971
$520$	−31.8628	−1.39728
$521$	−25.5383	−1.11885	−0.559426	−	0.828880i	$-0.688979\pi$
$521$	−25.5383	−1.11885	−0.559426	+	0.828880i	$0.688979\pi$
$522$	2.94657	0.128968
$523$	19.6507	0.859264	0.429632	−	0.903004i	$-0.358643\pi$
$523$	19.6507	0.859264	0.429632	+	0.903004i	$0.358643\pi$
$524$	69.6578	3.04301
$525$	1.47744	0.0644807
$526$	−48.7741	−2.12665
$527$	−10.0278	−0.436819
$528$	−0.530342	−0.0230802
$529$	−4.53128	−0.197012
$530$	−31.4386	−1.36561
$531$	8.16006	0.354116
$532$	12.5705	0.545002
$533$	−13.8014	−0.597805
$534$	−2.04459	−0.0884781
$535$	5.03499	0.217682
$536$	−55.7956	−2.41000
$537$	5.95247	0.256868
$538$	14.8687	0.641036
$539$	−0.255647	−0.0110115
$540$	−41.1119	−1.76917
$541$	−31.3091	−1.34608	−0.673042	−	0.739604i	$-0.735013\pi$
$541$	−31.3091	−1.34608	−0.673042	+	0.739604i	$0.735013\pi$
$542$	−11.1868	−0.480516
$543$	−32.3202	−1.38699
$544$	−5.27538	−0.226180
$545$	0.0630800	0.00270205
$546$	6.81654	0.291721
$547$	−42.1175	−1.80082	−0.900408	−	0.435046i	$-0.856732\pi$
$547$	−42.1175	−1.80082	−0.900408	+	0.435046i	$0.856732\pi$
$548$	39.9991	1.70867
$549$	9.33367	0.398351
$550$	−0.112791	−0.00480941
$551$	4.51218	0.192226
$552$	53.8315	2.29122
$553$	5.04275	0.214439
$554$	21.1676	0.899324
$555$	3.19824	0.135758
$556$	−4.42257	−0.187559
$557$	−14.1445	−0.599324	−0.299662	−	0.954045i	$-0.596874\pi$
$557$	−14.1445	−0.599324	−0.299662	+	0.954045i	$0.596874\pi$
$558$	26.5278	1.12301
$559$	1.53447	0.0649012
$560$	10.4881	0.443202
$561$	−0.0879826	−0.00371463
$562$	−1.22392	−0.0516279
$563$	−40.2579	−1.69667	−0.848333	−	0.529463i	$-0.822393\pi$
$563$	−40.2579	−1.69667	−0.848333	+	0.529463i	$0.822393\pi$
$564$	−18.8905	−0.795433
$565$	−25.6691	−1.07991
$566$	23.1677	0.973813
$567$	7.01503	0.294604
$568$	44.8540	1.88203
$569$	25.9081	1.08613	0.543063	−	0.839692i	$-0.317265\pi$
$569$	25.9081	1.08613	0.543063	+	0.839692i	$0.317265\pi$
$570$	57.8562	2.42333
$571$	18.2387	0.763267	0.381633	−	0.924314i	$-0.375362\pi$
$571$	18.2387	0.763267	0.381633	+	0.924314i	$0.375362\pi$
$572$	−0.358339	−0.0149829
$573$	0.0786941	0.00328749
$574$	10.5278	0.439423
$575$	4.94025	0.206023
$576$	−1.65696	−0.0690402
$577$	21.0699	0.877152	0.438576	−	0.898694i	$-0.355483\pi$
$577$	21.0699	0.877152	0.438576	+	0.898694i	$0.355483\pi$
$578$	39.9386	1.66123
$579$	23.7356	0.986417
$580$	10.9672	0.455389
$581$	−4.60348	−0.190984
$582$	78.6272	3.25920
$583$	−0.193676	−0.00802123
$584$	−7.08516	−0.293186
$585$	6.03415	0.249481
$586$	−22.9812	−0.949345
$587$	−10.7856	−0.445170	−0.222585	−	0.974913i	$-0.571449\pi$
$587$	−10.7856	−0.445170	−0.222585	+	0.974913i	$0.571449\pi$
$588$	59.5821	2.45712
$589$	40.6229	1.67384
$590$	44.1069	1.81585
$591$	−35.7330	−1.46986
$592$	4.24409	0.174431
$593$	−1.13587	−0.0466445	−0.0233223	−	0.999728i	$-0.507424\pi$
$593$	−1.13587	−0.0466445	−0.0233223	+	0.999728i	$0.507424\pi$
$594$	−0.367800	−0.0150910
$595$	1.73995	0.0713308
$596$	55.9205	2.29059
$597$	−40.9955	−1.67784
$598$	22.7931	0.932080
$599$	0.647277	0.0264470	0.0132235	−	0.999913i	$-0.495791\pi$
$599$	0.647277	0.0264470	0.0132235	+	0.999913i	$0.495791\pi$
$600$	14.3996	0.587860
$601$	−1.18958	−0.0485239	−0.0242619	−	0.999706i	$-0.507724\pi$
$601$	−1.18958	−0.0485239	−0.0242619	+	0.999706i	$0.507724\pi$
$602$	−1.17051	−0.0477063
$603$	10.5665	0.430301
$604$	51.2077	2.08361
$605$	27.2744	1.10886
$606$	−34.7612	−1.41208
$607$	7.41191	0.300840	0.150420	−	0.988622i	$-0.451937\pi$
$607$	7.41191	0.300840	0.150420	+	0.988622i	$0.451937\pi$
$608$	21.3707	0.866696
$609$	−1.28522	−0.0520799
$610$	50.4506	2.04268
$611$	−4.38140	−0.177252
$612$	5.72743	0.231518
$613$	−18.7227	−0.756203	−0.378102	−	0.925764i	$-0.623423\pi$
$613$	−18.7227	−0.756203	−0.378102	+	0.925764i	$0.623423\pi$
$614$	−56.0945	−2.26379
$615$	33.3658	1.34544
$616$	0.149731	0.00603282
$617$	−24.1954	−0.974071	−0.487035	−	0.873382i	$-0.661922\pi$
$617$	−24.1954	−0.974071	−0.487035	+	0.873382i	$0.661922\pi$
$618$	−8.81688	−0.354667
$619$	27.6196	1.11012	0.555062	−	0.831809i	$-0.312694\pi$
$619$	27.6196	1.11012	0.555062	+	0.831809i	$0.312694\pi$
$620$	98.7373	3.96539
$621$	16.1097	0.646461
$622$	74.4155	2.98379
$623$	0.249090	0.00997959
$624$	28.6682	1.14764
$625$	−29.4263	−1.17705
$626$	43.9678	1.75731
$627$	0.356419	0.0142340
$628$	−23.9832	−0.957035
$629$	0.704085	0.0280737
$630$	−4.60289	−0.183384
$631$	22.2756	0.886777	0.443389	−	0.896329i	$-0.353776\pi$
$631$	22.2756	0.886777	0.443389	+	0.896329i	$0.353776\pi$
$632$	49.1481	1.95501
$633$	−3.43977	−0.136719
$634$	−8.57789	−0.340671
$635$	−39.3714	−1.56241
$636$	45.1387	1.78987
$637$	13.8193	0.547539
$638$	0.0981165	0.00388447
$639$	−8.49439	−0.336033
$640$	−32.4463	−1.28255
$641$	11.9095	0.470397	0.235199	−	0.971947i	$-0.424426\pi$
$641$	11.9095	0.470397	0.235199	+	0.971947i	$0.424426\pi$
$642$	−10.4983	−0.414334
$643$	25.5380	1.00712	0.503561	−	0.863960i	$-0.332023\pi$
$643$	25.5380	1.00712	0.503561	+	0.863960i	$0.332023\pi$
$644$	−11.9725	−0.471783
$645$	−3.70968	−0.146069
$646$	12.7369	0.501126
$647$	19.0558	0.749162	0.374581	−	0.927194i	$-0.377787\pi$
$647$	19.0558	0.749162	0.374581	+	0.927194i	$0.377787\pi$
$648$	68.3706	2.68585
$649$	0.271718	0.0106659
$650$	6.09700	0.239144
$651$	−11.5708	−0.453495
$652$	−39.2729	−1.53804
$653$	−14.2224	−0.556564	−0.278282	−	0.960499i	$-0.589765\pi$
$653$	−14.2224	−0.556564	−0.278282	+	0.960499i	$0.589765\pi$
$654$	−0.131526	−0.00514306
$655$	−39.0586	−1.52614
$656$	44.2766	1.72871
$657$	1.34178	0.0523478
$658$	3.34216	0.130291
$659$	0.462001	0.0179970	0.00899851	−	0.999960i	$-0.497136\pi$
$659$	0.462001	0.0179970	0.00899851	+	0.999960i	$0.497136\pi$
$660$	0.866306	0.0337209
$661$	0.698548	0.0271703	0.0135852	−	0.999908i	$-0.495676\pi$
$661$	0.698548	0.0271703	0.0135852	+	0.999908i	$0.495676\pi$
$662$	−48.1134	−1.86998
$663$	4.75598	0.184707
$664$	−44.8669	−1.74117
$665$	−7.04856	−0.273332
$666$	−1.86260	−0.0721744
$667$	−4.29752	−0.166401
$668$	−58.1837	−2.25120
$669$	13.6668	0.528391
$670$	57.1143	2.20652
$671$	0.310797	0.0119982
$672$	−6.08710	−0.234815
$673$	−26.7501	−1.03114	−0.515571	−	0.856847i	$-0.672420\pi$
$673$	−26.7501	−1.03114	−0.515571	+	0.856847i	$0.672420\pi$
$674$	−6.12239	−0.235826
$675$	4.30924	0.165863
$676$	−38.1231	−1.46627
$677$	−15.5018	−0.595784	−0.297892	−	0.954600i	$-0.596284\pi$
$677$	−15.5018	−0.595784	−0.297892	+	0.954600i	$0.596284\pi$
$678$	53.5216	2.05549
$679$	−9.57906	−0.367611
$680$	16.9580	0.650311
$681$	−1.36025	−0.0521251
$682$	0.883337	0.0338247
$683$	45.6589	1.74709	0.873545	−	0.486743i	$-0.161815\pi$
$683$	45.6589	1.74709	0.873545	+	0.486743i	$0.161815\pi$
$684$	−23.2020	−0.887149
$685$	−22.4283	−0.856942
$686$	−21.7164	−0.829134
$687$	−50.5312	−1.92788
$688$	−4.92277	−0.187679
$689$	10.4693	0.398849
$690$	−55.1038	−2.09777
$691$	11.0375	0.419888	0.209944	−	0.977713i	$-0.432672\pi$
$691$	11.0375	0.419888	0.209944	+	0.977713i	$0.432672\pi$
$692$	84.9233	3.22830
$693$	−0.0283558	−0.00107715
$694$	65.7367	2.49533
$695$	2.47983	0.0940653
$696$	−12.5262	−0.474803
$697$	7.34538	0.278226
$698$	43.3771	1.64185
$699$	15.2300	0.576053
$700$	−3.20256	−0.121046
$701$	47.3794	1.78950	0.894748	−	0.446571i	$-0.147355\pi$
$701$	47.3794	1.78950	0.894748	+	0.446571i	$0.147355\pi$
$702$	19.8818	0.750390
$703$	−2.85226	−0.107575
$704$	−0.0551745	−0.00207947
$705$	10.5923	0.398929
$706$	86.9084	3.27084
$707$	4.23492	0.159270
$708$	−63.3275	−2.37999
$709$	−29.2500	−1.09851	−0.549253	−	0.835656i	$-0.685088\pi$
$709$	−29.2500	−1.09851	−0.549253	+	0.835656i	$0.685088\pi$
$710$	−45.9141	−1.72313
$711$	−9.30761	−0.349063
$712$	2.42771	0.0909822
$713$	−38.6904	−1.44897
$714$	−3.62790	−0.135771
$715$	0.200928	0.00751428
$716$	−12.9028	−0.482202
$717$	−30.9834	−1.15709
$718$	38.8554	1.45007
$719$	−19.4417	−0.725052	−0.362526	−	0.931974i	$-0.618086\pi$
$719$	−19.4417	−0.725052	−0.362526	+	0.931974i	$0.618086\pi$
$720$	−19.3583	−0.721440
$721$	1.07415	0.0400035
$722$	−3.44614	−0.128252
$723$	51.1557	1.90250
$724$	70.0588	2.60372
$725$	−1.14956	−0.0426935
$726$	−56.8689	−2.11060
$727$	16.1524	0.599059	0.299529	−	0.954087i	$-0.403170\pi$
$727$	16.1524	0.599059	0.299529	+	0.954087i	$0.403170\pi$
$728$	−8.09383	−0.299977
$729$	9.99654	0.370242
$730$	7.25262	0.268431
$731$	−0.816676	−0.0302059
$732$	−72.4355	−2.67729
$733$	11.2349	0.414972	0.207486	−	0.978238i	$-0.433472\pi$
$733$	11.2349	0.414972	0.207486	+	0.978238i	$0.433472\pi$
$734$	−32.4458	−1.19760
$735$	−33.4089	−1.23231
$736$	−20.3540	−0.750259
$737$	0.351849	0.0129605
$738$	−19.4316	−0.715289
$739$	1.20092	0.0441764	0.0220882	−	0.999756i	$-0.492969\pi$
$739$	1.20092	0.0441764	0.0220882	+	0.999756i	$0.492969\pi$
$740$	−6.93266	−0.254850
$741$	−19.2666	−0.707776
$742$	−7.98608	−0.293178
$743$	25.1822	0.923845	0.461922	−	0.886920i	$-0.347160\pi$
$743$	25.1822	0.923845	0.461922	+	0.886920i	$0.347160\pi$
$744$	−112.772	−4.13444
$745$	−31.3558	−1.14879
$746$	19.1344	0.700559
$747$	8.49684	0.310883
$748$	0.190715	0.00697323
$749$	1.27899	0.0467334
$750$	49.3712	1.80278
$751$	−0.808800	−0.0295135	−0.0147568	−	0.999891i	$-0.504697\pi$
$751$	−0.808800	−0.0295135	−0.0147568	+	0.999891i	$0.504697\pi$
$752$	14.0561	0.512571
$753$	4.35210	0.158599
$754$	−5.30378	−0.193152
$755$	−28.7132	−1.04498
$756$	−10.4433	−0.379818
$757$	39.3734	1.43105	0.715525	−	0.698587i	$-0.246187\pi$
$757$	39.3734	1.43105	0.715525	+	0.698587i	$0.246187\pi$
$758$	48.2503	1.75253
$759$	−0.339463	−0.0123217
$760$	−68.6974	−2.49192
$761$	−10.7107	−0.388264	−0.194132	−	0.980975i	$-0.562189\pi$
$761$	−10.7107	−0.388264	−0.194132	+	0.980975i	$0.562189\pi$
$762$	82.0920	2.97388
$763$	0.0160236	0.000580095 0
$764$	−0.170581	−0.00617140
$765$	−3.21149	−0.116112
$766$	−53.2772	−1.92498
$767$	−14.6880	−0.530351
$768$	61.8374	2.23136
$769$	46.2157	1.66658	0.833291	−	0.552834i	$-0.186454\pi$
$769$	46.2157	1.66658	0.833291	+	0.552834i	$0.186454\pi$
$770$	−0.153270	−0.00552345
$771$	1.69881	0.0611812
$772$	−51.4503	−1.85174
$773$	40.0454	1.44033	0.720166	−	0.693802i	$-0.244066\pi$
$773$	40.0454	1.44033	0.720166	+	0.693802i	$0.244066\pi$
$774$	2.16045	0.0776559
$775$	−10.3494	−0.371762
$776$	−93.3604	−3.35144
$777$	0.812422	0.0291455
$778$	−37.5765	−1.34718
$779$	−29.7563	−1.06613
$780$	−46.8290	−1.67675
$781$	−0.282851	−0.0101212
$782$	−12.1310	−0.433802
$783$	−3.74861	−0.133964
$784$	−44.3339	−1.58335
$785$	13.4479	0.479976
$786$	81.4396	2.90486
$787$	47.6121	1.69719	0.848593	−	0.529046i	$-0.177450\pi$
$787$	47.6121	1.69719	0.848593	+	0.529046i	$0.177450\pi$
$788$	77.4564	2.75927
$789$	−39.2664	−1.39792
$790$	−50.3097	−1.78994
$791$	−6.52048	−0.231842
$792$	−0.276364	−0.00982018
$793$	−16.8004	−0.596601
$794$	5.59458	0.198544
$795$	−25.3102	−0.897661
$796$	88.8638	3.14969
$797$	33.2123	1.17644	0.588220	−	0.808701i	$-0.299829\pi$
$797$	33.2123	1.17644	0.588220	+	0.808701i	$0.299829\pi$
$798$	14.6967	0.520258
$799$	2.33187	0.0824955
$800$	−5.44456	−0.192494
$801$	−0.459756	−0.0162447
$802$	41.9751	1.48219
$803$	0.0446792	0.00157670
$804$	−82.0032	−2.89203
$805$	6.71324	0.236611
$806$	−47.7496	−1.68191
$807$	11.9703	0.421375
$808$	41.2748	1.45204
$809$	22.2372	0.781819	0.390910	−	0.920429i	$-0.372161\pi$
$809$	22.2372	0.781819	0.390910	+	0.920429i	$0.372161\pi$
$810$	−69.9865	−2.45908
$811$	7.63656	0.268156	0.134078	−	0.990971i	$-0.457193\pi$
$811$	7.63656	0.268156	0.134078	+	0.990971i	$0.457193\pi$
$812$	2.78591	0.0977662
$813$	−9.00617	−0.315860
$814$	−0.0620219	−0.00217387
$815$	22.0211	0.771366
$816$	−15.2578	−0.534129
$817$	3.30837	0.115745
$818$	34.4401	1.20417
$819$	1.53280	0.0535604
$820$	−72.3251	−2.52570
$821$	−36.3186	−1.26753	−0.633765	−	0.773526i	$-0.718491\pi$
$821$	−36.3186	−1.26753	−0.633765	+	0.773526i	$0.718491\pi$
$822$	46.7645	1.63110
$823$	−45.5195	−1.58671	−0.793356	−	0.608758i	$-0.791668\pi$
$823$	−45.5195	−1.58671	−0.793356	+	0.608758i	$0.791668\pi$
$824$	10.4690	0.364705
$825$	−0.0908041	−0.00316139
$826$	11.2041	0.389840
$827$	−39.0261	−1.35707	−0.678535	−	0.734568i	$-0.737385\pi$
$827$	−39.0261	−1.35707	−0.678535	+	0.734568i	$0.737385\pi$
$828$	22.0982	0.767964
$829$	−33.5883	−1.16657	−0.583286	−	0.812267i	$-0.698233\pi$
$829$	−33.5883	−1.16657	−0.583286	+	0.812267i	$0.698233\pi$
$830$	45.9273	1.59416
$831$	17.0413	0.591157
$832$	2.98251	0.103400
$833$	−7.35489	−0.254832
$834$	−5.17061	−0.179043
$835$	32.6248	1.12903
$836$	−0.772591	−0.0267206
$837$	−33.7485	−1.16652
$838$	45.4223	1.56909
$839$	47.7165	1.64736	0.823679	−	0.567057i	$-0.191918\pi$
$839$	47.7165	1.64736	0.823679	+	0.567057i	$0.191918\pi$
$840$	19.5674	0.675138
$841$	1.00000	0.0344828
$842$	−18.6479	−0.642649
$843$	−0.985336	−0.0339368
$844$	7.45620	0.256653
$845$	21.3764	0.735372
$846$	−6.16877	−0.212087
$847$	6.92828	0.238058
$848$	−33.5868	−1.15338
$849$	18.6516	0.640121
$850$	−3.24495	−0.111301
$851$	2.71657	0.0931229
$852$	65.9222	2.25846
$853$	−38.0888	−1.30414	−0.652068	−	0.758161i	$-0.726098\pi$
$853$	−38.0888	−1.30414	−0.652068	+	0.758161i	$0.726098\pi$
$854$	12.8155	0.438538
$855$	13.0098	0.444927
$856$	12.4655	0.426061
$857$	4.47298	0.152794	0.0763970	−	0.997077i	$-0.475658\pi$
$857$	4.47298	0.152794	0.0763970	+	0.997077i	$0.475658\pi$
$858$	−0.418948	−0.0143026
$859$	−32.3534	−1.10388	−0.551942	−	0.833883i	$-0.686113\pi$
$859$	−32.3534	−1.10388	−0.551942	+	0.833883i	$0.686113\pi$
$860$	8.04127	0.274205
$861$	8.47561	0.288848
$862$	35.1392	1.19684
$863$	7.89144	0.268628	0.134314	−	0.990939i	$-0.457117\pi$
$863$	7.89144	0.268628	0.134314	+	0.990939i	$0.457117\pi$
$864$	−17.7542	−0.604011
$865$	−47.6183	−1.61907
$866$	−0.988129	−0.0335780
$867$	32.1533	1.09198
$868$	25.0814	0.851317
$869$	−0.309930	−0.0105136
$870$	12.8222	0.434714
$871$	−19.0195	−0.644452
$872$	0.156171	0.00528862
$873$	17.6805	0.598394
$874$	49.1428	1.66228
$875$	−6.01484	−0.203339
$876$	−10.4131	−0.351826
$877$	−51.8770	−1.75176	−0.875881	−	0.482527i	$-0.839719\pi$
$877$	−51.8770	−1.75176	−0.875881	+	0.482527i	$0.839719\pi$
$878$	60.3858	2.03792
$879$	−18.5014	−0.624038
$880$	−0.644602	−0.0217295
$881$	−31.5281	−1.06221	−0.531105	−	0.847306i	$-0.678223\pi$
$881$	−31.5281	−1.06221	−0.531105	+	0.847306i	$0.678223\pi$
$882$	19.4568	0.655144
$883$	58.7926	1.97853	0.989265	−	0.146135i	$-0.0466833\pi$
$883$	58.7926	1.97853	0.989265	+	0.146135i	$0.0466833\pi$
$884$	−10.3093	−0.346739
$885$	35.5091	1.19362
$886$	72.9734	2.45159
$887$	1.69885	0.0570417	0.0285208	−	0.999593i	$-0.490920\pi$
$887$	1.69885	0.0570417	0.0285208	+	0.999593i	$0.490920\pi$
$888$	7.91810	0.265714
$889$	−10.0012	−0.335429
$890$	−2.48509	−0.0833003
$891$	−0.431147	−0.0144440
$892$	−29.6249	−0.991914
$893$	−9.44644	−0.316113
$894$	65.3788	2.18660
$895$	7.23489	0.241836
$896$	−8.24204	−0.275347
$897$	18.3500	0.612689
$898$	−10.8313	−0.361446
$899$	9.00294	0.300265
$900$	5.91111	0.197037
$901$	−5.57198	−0.185630
$902$	−0.647045	−0.0215442
$903$	−0.942337	−0.0313590
$904$	−63.5506	−2.11366
$905$	−39.2835	−1.30583
$906$	59.8689	1.98901
$907$	−40.5913	−1.34781	−0.673906	−	0.738817i	$-0.735385\pi$
$907$	−40.5913	−1.34781	−0.673906	+	0.738817i	$0.735385\pi$
$908$	2.94855	0.0978510
$909$	−7.81656	−0.259259
$910$	8.28513	0.274649
$911$	4.06313	0.134617	0.0673087	−	0.997732i	$-0.478559\pi$
$911$	4.06313	0.134617	0.0673087	+	0.997732i	$0.478559\pi$
$912$	61.8096	2.04672
$913$	0.282932	0.00936368
$914$	92.8423	3.07095
$915$	40.6161	1.34273
$916$	109.534	3.61909
$917$	−9.92170	−0.327643
$918$	−10.5815	−0.349241
$919$	−5.19417	−0.171340	−0.0856699	−	0.996324i	$-0.527303\pi$
$919$	−5.19417	−0.171340	−0.0856699	+	0.996324i	$0.527303\pi$
$920$	65.4292	2.15714
$921$	−45.1598	−1.48807
$922$	−59.6448	−1.96430
$923$	15.2898	0.503268
$924$	0.220060	0.00723945
$925$	0.726664	0.0238926
$926$	47.0736	1.54694
$927$	−1.98261	−0.0651173
$928$	4.73622	0.155474
$929$	14.1571	0.464478	0.232239	−	0.972659i	$-0.425395\pi$
$929$	14.1571	0.464478	0.232239	+	0.972659i	$0.425395\pi$
$930$	115.438	3.78535
$931$	29.7948	0.976485
$932$	−33.0133	−1.08139
$933$	59.9095	1.96135
$934$	0.0751461	0.00245886
$935$	−0.106938	−0.00349724
$936$	14.9391	0.488301
$937$	−6.43455	−0.210207	−0.105104	−	0.994461i	$-0.533517\pi$
$937$	−6.43455	−0.210207	−0.105104	+	0.994461i	$0.533517\pi$
$938$	14.5082	0.473711
$939$	35.3970	1.15514
$940$	−22.9603	−0.748884
$941$	7.80257	0.254357	0.127178	−	0.991880i	$-0.459408\pi$
$941$	7.80257	0.254357	0.127178	+	0.991880i	$0.459408\pi$
$942$	−28.0397	−0.913584
$943$	28.3407	0.922900
$944$	47.1208	1.53365
$945$	5.85577	0.190488
$946$	0.0719399	0.00233897
$947$	45.6451	1.48327	0.741633	−	0.670806i	$-0.234051\pi$
$947$	45.6451	1.48327	0.741633	+	0.670806i	$0.234051\pi$
$948$	72.2333	2.34603
$949$	−2.41518	−0.0784000
$950$	13.1454	0.426492
$951$	−6.90578	−0.223935
$952$	4.30770	0.139613
$953$	−41.0425	−1.32950	−0.664748	−	0.747067i	$-0.731461\pi$
$953$	−41.0425	−1.32950	−0.664748	+	0.747067i	$0.731461\pi$
$954$	14.7402	0.477233
$955$	0.0956483	0.00309511
$956$	67.1609	2.17214
$957$	0.0789904	0.00255340
$958$	29.3491	0.948227
$959$	−5.69726	−0.183974
$960$	−7.21040	−0.232715
$961$	50.0529	1.61461
$962$	3.35265	0.108094
$963$	−2.36069	−0.0760723
$964$	−110.887	−3.57144
$965$	28.8493	0.928691
$966$	−13.9975	−0.450363
$967$	2.07805	0.0668255	0.0334128	−	0.999442i	$-0.489362\pi$
$967$	2.07805	0.0668255	0.0334128	+	0.999442i	$0.489362\pi$
$968$	67.5251	2.17034
$969$	10.2541	0.329408
$970$	95.5670	3.06847
$971$	−28.5806	−0.917195	−0.458598	−	0.888644i	$-0.651648\pi$
$971$	−28.5806	−0.917195	−0.458598	+	0.888644i	$0.651648\pi$
$972$	50.7492	1.62778
$973$	0.629929	0.0201946
$974$	−21.3627	−0.684504
$975$	4.90850	0.157198
$976$	53.8978	1.72523
$977$	11.8580	0.379371	0.189685	−	0.981845i	$-0.439253\pi$
$977$	11.8580	0.379371	0.189685	+	0.981845i	$0.439253\pi$
$978$	−45.9155	−1.46821
$979$	−0.0153092	−0.000489284 0
$980$	72.4187	2.31333
$981$	−0.0295755	−0.000944273 0
$982$	52.7980	1.68485
$983$	14.4929	0.462252	0.231126	−	0.972924i	$-0.425759\pi$
$983$	14.4929	0.462252	0.231126	+	0.972924i	$0.425759\pi$
$984$	82.6058	2.63338
$985$	−43.4315	−1.38384
$986$	2.82278	0.0898956
$987$	2.69067	0.0856449
$988$	41.7631	1.32866
$989$	−3.15098	−0.100195
$990$	0.282896	0.00899103
$991$	−6.11159	−0.194141	−0.0970706	−	0.995278i	$-0.530947\pi$
$991$	−6.11159	−0.194141	−0.0970706	+	0.995278i	$0.530947\pi$
$992$	42.6399	1.35382
$993$	−38.7345	−1.22920
$994$	−11.6631	−0.369933
$995$	−49.8278	−1.57965
$996$	−65.9411	−2.08942
$997$	12.8682	0.407540	0.203770	−	0.979019i	$-0.434681\pi$
$997$	12.8682	0.407540	0.203770	+	0.979019i	$0.434681\pi$
$998$	−70.7341	−2.23905
$999$	2.36959	0.0749705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.3	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.3	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53428 q^{2} -2.04027 q^{3} +4.42257 q^{4} -2.47983 q^{5} +5.17061 q^{6} -0.629929 q^{7} -6.13948 q^{8} +1.16269 q^{9} +O(q^{10})\)	\(q-2.53428 q^{2} -2.04027 q^{3} +4.42257 q^{4} -2.47983 q^{5} +5.17061 q^{6} -0.629929 q^{7} -6.13948 q^{8} +1.16269 q^{9} +6.28458 q^{10} +0.0387158 q^{11} -9.02323 q^{12} -2.09282 q^{13} +1.59642 q^{14} +5.05951 q^{15} +6.71401 q^{16} +1.11384 q^{17} -2.94657 q^{18} -4.51218 q^{19} -10.9672 q^{20} +1.28522 q^{21} -0.0981165 q^{22} +4.29752 q^{23} +12.5262 q^{24} +1.14956 q^{25} +5.30378 q^{26} +3.74861 q^{27} -2.78591 q^{28} -1.00000 q^{29} -12.8222 q^{30} -9.00294 q^{31} -4.73622 q^{32} -0.0789904 q^{33} -2.82278 q^{34} +1.56212 q^{35} +5.14207 q^{36} +0.632125 q^{37} +11.4351 q^{38} +4.26990 q^{39} +15.2249 q^{40} +6.59466 q^{41} -3.25712 q^{42} -0.733209 q^{43} +0.171223 q^{44} -2.88327 q^{45} -10.8911 q^{46} +2.09354 q^{47} -13.6984 q^{48} -6.60319 q^{49} -2.91330 q^{50} -2.27253 q^{51} -9.25563 q^{52} -5.00250 q^{53} -9.50002 q^{54} -0.0960085 q^{55} +3.86744 q^{56} +9.20606 q^{57} +2.53428 q^{58} +7.01828 q^{59} +22.3761 q^{60} +8.02767 q^{61} +22.8160 q^{62} -0.732410 q^{63} -1.42512 q^{64} +5.18983 q^{65} +0.200184 q^{66} +9.08801 q^{67} +4.92603 q^{68} -8.76810 q^{69} -3.95884 q^{70} -7.30583 q^{71} -7.13829 q^{72} +1.15403 q^{73} -1.60198 q^{74} -2.34540 q^{75} -19.9555 q^{76} -0.0243882 q^{77} -10.8211 q^{78} -8.00526 q^{79} -16.6496 q^{80} -11.1362 q^{81} -16.7127 q^{82} +7.30793 q^{83} +5.68399 q^{84} -2.76213 q^{85} +1.85816 q^{86} +2.04027 q^{87} -0.237695 q^{88} -0.395426 q^{89} +7.30700 q^{90} +1.31833 q^{91} +19.0061 q^{92} +18.3684 q^{93} -5.30562 q^{94} +11.1895 q^{95} +9.66315 q^{96} +15.2066 q^{97} +16.7343 q^{98} +0.0450143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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