LMFDB - Embedded newform 8027.2.a.f.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8027,2,Mod(1,8027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8027, 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("8027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8027 = 23 \cdot 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8027.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:	$64.0959177025$
Analytic rank:	$0$
Dimension:	$176$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30965 q^{2} +1.95797 q^{3} +3.33448 q^{4} +2.21374 q^{5} -4.52222 q^{6} +4.98325 q^{7} -3.08217 q^{8} +0.833638 q^{9} +O(q^{10})$	\(q-2.30965 q^{2} +1.95797 q^{3} +3.33448 q^{4} +2.21374 q^{5} -4.52222 q^{6} +4.98325 q^{7} -3.08217 q^{8} +0.833638 q^{9} -5.11297 q^{10} -1.08947 q^{11} +6.52879 q^{12} -5.85991 q^{13} -11.5095 q^{14} +4.33444 q^{15} +0.449774 q^{16} -3.36258 q^{17} -1.92541 q^{18} +4.22259 q^{19} +7.38167 q^{20} +9.75704 q^{21} +2.51629 q^{22} +1.00000 q^{23} -6.03479 q^{24} -0.0993443 q^{25} +13.5343 q^{26} -4.24167 q^{27} +16.6165 q^{28} +8.10322 q^{29} -10.0110 q^{30} -3.99410 q^{31} +5.12552 q^{32} -2.13315 q^{33} +7.76638 q^{34} +11.0316 q^{35} +2.77974 q^{36} +7.43590 q^{37} -9.75270 q^{38} -11.4735 q^{39} -6.82313 q^{40} +4.59774 q^{41} -22.5353 q^{42} +4.59971 q^{43} -3.63281 q^{44} +1.84546 q^{45} -2.30965 q^{46} -7.01645 q^{47} +0.880643 q^{48} +17.8328 q^{49} +0.229450 q^{50} -6.58383 q^{51} -19.5397 q^{52} +0.967875 q^{53} +9.79676 q^{54} -2.41180 q^{55} -15.3592 q^{56} +8.26770 q^{57} -18.7156 q^{58} +3.35030 q^{59} +14.4531 q^{60} -4.37700 q^{61} +9.22496 q^{62} +4.15422 q^{63} -12.7377 q^{64} -12.9723 q^{65} +4.92682 q^{66} +2.31509 q^{67} -11.2125 q^{68} +1.95797 q^{69} -25.4792 q^{70} +8.97742 q^{71} -2.56941 q^{72} +12.7859 q^{73} -17.1743 q^{74} -0.194513 q^{75} +14.0801 q^{76} -5.42910 q^{77} +26.4998 q^{78} -5.32993 q^{79} +0.995684 q^{80} -10.8060 q^{81} -10.6192 q^{82} -2.60638 q^{83} +32.5346 q^{84} -7.44389 q^{85} -10.6237 q^{86} +15.8659 q^{87} +3.35793 q^{88} -9.08077 q^{89} -4.26236 q^{90} -29.2014 q^{91} +3.33448 q^{92} -7.82032 q^{93} +16.2055 q^{94} +9.34773 q^{95} +10.0356 q^{96} +2.64450 q^{97} -41.1874 q^{98} -0.908223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) $ 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * 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 $

$2$	−2.30965	−1.63317	−0.816584	−	0.577227i	$-0.804135\pi$
$2$	−2.30965	−1.63317	−0.816584	+	0.577227i	$0.804135\pi$
$3$	1.95797	1.13043	0.565217	−	0.824943i	$-0.308793\pi$
$3$	1.95797	1.13043	0.565217	+	0.824943i	$0.308793\pi$
$4$	3.33448	1.66724
$5$	2.21374	0.990016	0.495008	−	0.868888i	$-0.335165\pi$
$5$	2.21374	0.990016	0.495008	+	0.868888i	$0.335165\pi$
$6$	−4.52222	−1.84619
$7$	4.98325	1.88349	0.941745	−	0.336327i	$-0.109185\pi$
$7$	4.98325	1.88349	0.941745	+	0.336327i	$0.109185\pi$
$8$	−3.08217	−1.08971
$9$	0.833638	0.277879
$10$	−5.11297	−1.61686
$11$	−1.08947	−0.328487	−0.164244	−	0.986420i	$-0.552518\pi$
$11$	−1.08947	−0.328487	−0.164244	+	0.986420i	$0.552518\pi$
$12$	6.52879	1.88470
$13$	−5.85991	−1.62525	−0.812623	−	0.582790i	$-0.801961\pi$
$13$	−5.85991	−1.62525	−0.812623	+	0.582790i	$0.801961\pi$
$14$	−11.5095	−3.07606
$15$	4.33444	1.11915
$16$	0.449774	0.112443
$17$	−3.36258	−0.815546	−0.407773	−	0.913083i	$-0.633695\pi$
$17$	−3.36258	−0.815546	−0.407773	+	0.913083i	$0.633695\pi$
$18$	−1.92541	−0.453824
$19$	4.22259	0.968729	0.484364	−	0.874866i	$-0.339051\pi$
$19$	4.22259	0.968729	0.484364	+	0.874866i	$0.339051\pi$
$20$	7.38167	1.65059
$21$	9.75704	2.12916
$22$	2.51629	0.536475
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−6.03479	−1.23185
$25$	−0.0993443	−0.0198689
$26$	13.5343	2.65430
$27$	−4.24167	−0.816309
$28$	16.6165	3.14023
$29$	8.10322	1.50473	0.752365	−	0.658746i	$-0.228913\pi$
$29$	8.10322	1.50473	0.752365	+	0.658746i	$0.228913\pi$
$30$	−10.0110	−1.82775
$31$	−3.99410	−0.717361	−0.358681	−	0.933460i	$-0.616773\pi$
$31$	−3.99410	−0.717361	−0.358681	+	0.933460i	$0.616773\pi$
$32$	5.12552	0.906072
$33$	−2.13315	−0.371333
$34$	7.76638	1.33192
$35$	11.0316	1.86469
$36$	2.77974	0.463291
$37$	7.43590	1.22245	0.611227	−	0.791455i	$-0.290676\pi$
$37$	7.43590	1.22245	0.611227	+	0.791455i	$0.290676\pi$
$38$	−9.75270	−1.58210
$39$	−11.4735	−1.83723
$40$	−6.82313	−1.07883
$41$	4.59774	0.718047	0.359023	−	0.933329i	$-0.383110\pi$
$41$	4.59774	0.718047	0.359023	+	0.933329i	$0.383110\pi$
$42$	−22.5353	−3.47728
$43$	4.59971	0.701448	0.350724	−	0.936479i	$-0.385935\pi$
$43$	4.59971	0.701448	0.350724	+	0.936479i	$0.385935\pi$
$44$	−3.63281	−0.547667
$45$	1.84546	0.275105
$46$	−2.30965	−0.340539
$47$	−7.01645	−1.02345	−0.511727	−	0.859148i	$-0.670994\pi$
$47$	−7.01645	−1.02345	−0.511727	+	0.859148i	$0.670994\pi$
$48$	0.880643	0.127110
$49$	17.8328	2.54754
$50$	0.229450	0.0324492
$51$	−6.58383	−0.921921
$52$	−19.5397	−2.70967
$53$	0.967875	0.132948	0.0664740	−	0.997788i	$-0.478825\pi$
$53$	0.967875	0.132948	0.0664740	+	0.997788i	$0.478825\pi$
$54$	9.79676	1.33317
$55$	−2.41180	−0.325208
$56$	−15.3592	−2.05246
$57$	8.26770	1.09508
$58$	−18.7156	−2.45748
$59$	3.35030	0.436172	0.218086	−	0.975930i	$-0.430019\pi$
$59$	3.35030	0.436172	0.218086	+	0.975930i	$0.430019\pi$
$60$	14.4531	1.86588
$61$	−4.37700	−0.560417	−0.280209	−	0.959939i	$-0.590404\pi$
$61$	−4.37700	−0.560417	−0.280209	+	0.959939i	$0.590404\pi$
$62$	9.22496	1.17157
$63$	4.15422	0.523383
$64$	−12.7377	−1.59221
$65$	−12.9723	−1.60902
$66$	4.92682	0.606449
$67$	2.31509	0.282833	0.141417	−	0.989950i	$-0.454834\pi$
$67$	2.31509	0.282833	0.141417	+	0.989950i	$0.454834\pi$
$68$	−11.2125	−1.35971
$69$	1.95797	0.235712
$70$	−25.4792	−3.04534
$71$	8.97742	1.06542	0.532712	−	0.846297i	$-0.321173\pi$
$71$	8.97742	1.06542	0.532712	+	0.846297i	$0.321173\pi$
$72$	−2.56941	−0.302808
$73$	12.7859	1.49647	0.748236	−	0.663432i	$-0.230901\pi$
$73$	12.7859	1.49647	0.748236	+	0.663432i	$0.230901\pi$
$74$	−17.1743	−1.99647
$75$	−0.194513	−0.0224604
$76$	14.0801	1.61510
$77$	−5.42910	−0.618703
$78$	26.4998	3.00051
$79$	−5.32993	−0.599663	−0.299832	−	0.953992i	$-0.596931\pi$
$79$	−5.32993	−0.599663	−0.299832	+	0.953992i	$0.596931\pi$
$80$	0.995684	0.111321
$81$	−10.8060	−1.20066
$82$	−10.6192	−1.17269
$83$	−2.60638	−0.286087	−0.143044	−	0.989716i	$-0.545689\pi$
$83$	−2.60638	−0.286087	−0.143044	+	0.989716i	$0.545689\pi$
$84$	32.5346	3.54982
$85$	−7.44389	−0.807404
$86$	−10.6237	−1.14558
$87$	15.8659	1.70100
$88$	3.35793	0.357956
$89$	−9.08077	−0.962559	−0.481280	−	0.876567i	$-0.659828\pi$
$89$	−9.08077	−0.962559	−0.481280	+	0.876567i	$0.659828\pi$
$90$	−4.26236	−0.449292
$91$	−29.2014	−3.06113
$92$	3.33448	0.347643
$93$	−7.82032	−0.810929
$94$	16.2055	1.67147
$95$	9.34773	0.959057
$96$	10.0356	1.02425
$97$	2.64450	0.268508	0.134254	−	0.990947i	$-0.457136\pi$
$97$	2.64450	0.268508	0.134254	+	0.990947i	$0.457136\pi$
$98$	−41.1874	−4.16055
$99$	−0.908223	−0.0912799
$100$	−0.331261	−0.0331261
$101$	3.00941	0.299447	0.149724	−	0.988728i	$-0.452162\pi$
$101$	3.00941	0.299447	0.149724	+	0.988728i	$0.452162\pi$
$102$	15.2063	1.50565
$103$	13.7650	1.35631	0.678154	−	0.734919i	$-0.262780\pi$
$103$	13.7650	1.35631	0.678154	+	0.734919i	$0.262780\pi$
$104$	18.0612	1.77105
$105$	21.5996	2.10790
$106$	−2.23545	−0.217126
$107$	−7.67796	−0.742257	−0.371128	−	0.928582i	$-0.621029\pi$
$107$	−7.67796	−0.742257	−0.371128	+	0.928582i	$0.621029\pi$
$108$	−14.1437	−1.36098
$109$	11.9832	1.14779	0.573893	−	0.818930i	$-0.305432\pi$
$109$	11.9832	1.14779	0.573893	+	0.818930i	$0.305432\pi$
$110$	5.57042	0.531119
$111$	14.5593	1.38190
$112$	2.24133	0.211786
$113$	20.3727	1.91650	0.958249	−	0.285935i	$-0.0923040\pi$
$113$	20.3727	1.91650	0.958249	+	0.285935i	$0.0923040\pi$
$114$	−19.0955	−1.78845
$115$	2.21374	0.206433
$116$	27.0200	2.50874
$117$	−4.88504	−0.451622
$118$	−7.73801	−0.712341
$119$	−16.7566	−1.53607
$120$	−13.3595	−1.21955
$121$	−9.81306	−0.892096
$122$	10.1093	0.915255
$123$	9.00223	0.811704
$124$	−13.3182	−1.19601
$125$	−11.2886	−1.00969
$126$	−9.59480	−0.854772
$127$	12.8181	1.13742	0.568711	−	0.822537i	$-0.307442\pi$
$127$	12.8181	1.13742	0.568711	+	0.822537i	$0.307442\pi$
$128$	19.1685	1.69428
$129$	9.00607	0.792941
$130$	29.9615	2.62780
$131$	1.35102	0.118039	0.0590197	−	0.998257i	$-0.481203\pi$
$131$	1.35102	0.118039	0.0590197	+	0.998257i	$0.481203\pi$
$132$	−7.11292	−0.619100
$133$	21.0422	1.82459
$134$	−5.34705	−0.461915
$135$	−9.38996	−0.808159
$136$	10.3640	0.888710
$137$	−2.48101	−0.211967	−0.105983	−	0.994368i	$-0.533799\pi$
$137$	−2.48101	−0.211967	−0.105983	+	0.994368i	$0.533799\pi$
$138$	−4.52222	−0.384957
$139$	6.35037	0.538631	0.269316	−	0.963052i	$-0.413203\pi$
$139$	6.35037	0.538631	0.269316	+	0.963052i	$0.413203\pi$
$140$	36.7847	3.10887
$141$	−13.7380	−1.15695
$142$	−20.7347	−1.74002
$143$	6.38419	0.533873
$144$	0.374949	0.0312457
$145$	17.9384	1.48971
$146$	−29.5309	−2.44399
$147$	34.9160	2.87982
$148$	24.7948	2.03812
$149$	20.4077	1.67186	0.835930	−	0.548836i	$-0.184929\pi$
$149$	20.4077	1.67186	0.835930	+	0.548836i	$0.184929\pi$
$150$	0.449257	0.0366816
$151$	15.4653	1.25855	0.629274	−	0.777183i	$-0.283352\pi$
$151$	15.4653	1.25855	0.629274	+	0.777183i	$0.283352\pi$
$152$	−13.0147	−1.05563
$153$	−2.80318	−0.226623
$154$	12.5393	1.01045
$155$	−8.84191	−0.710199
$156$	−38.2581	−3.06310
$157$	22.8772	1.82580	0.912901	−	0.408180i	$-0.133836\pi$
$157$	22.8772	1.82580	0.912901	+	0.408180i	$0.133836\pi$
$158$	12.3103	0.979351
$159$	1.89507	0.150289
$160$	11.3466	0.897025
$161$	4.98325	0.392735
$162$	24.9580	1.96088
$163$	17.1911	1.34651	0.673257	−	0.739409i	$-0.264895\pi$
$163$	17.1911	1.34651	0.673257	+	0.739409i	$0.264895\pi$
$164$	15.3311	1.19715
$165$	−4.72224	−0.367626
$166$	6.01981	0.467228
$167$	−13.4907	−1.04394	−0.521971	−	0.852963i	$-0.674803\pi$
$167$	−13.4907	−1.04394	−0.521971	+	0.852963i	$0.674803\pi$
$168$	−30.0728	−2.32017
$169$	21.3385	1.64142
$170$	17.1928	1.31863
$171$	3.52011	0.269190
$172$	15.3376	1.16948
$173$	−22.0370	−1.67544	−0.837719	−	0.546101i	$-0.816112\pi$
$173$	−22.0370	−1.67544	−0.837719	+	0.546101i	$0.816112\pi$
$174$	−36.6445	−2.77801
$175$	−0.495057	−0.0374228
$176$	−0.490015	−0.0369363
$177$	6.55977	0.493063
$178$	20.9734	1.57202
$179$	−16.9790	−1.26907	−0.634535	−	0.772894i	$-0.718808\pi$
$179$	−16.9790	−1.26907	−0.634535	+	0.772894i	$0.718808\pi$
$180$	6.15364	0.458665
$181$	−2.88728	−0.214610	−0.107305	−	0.994226i	$-0.534222\pi$
$181$	−2.88728	−0.214610	−0.107305	+	0.994226i	$0.534222\pi$
$182$	67.4449	4.99935
$183$	−8.57002	−0.633514
$184$	−3.08217	−0.227220
$185$	16.4612	1.21025
$186$	18.0622	1.32438
$187$	3.66343	0.267897
$188$	−23.3962	−1.70634
$189$	−21.1373	−1.53751
$190$	−21.5900	−1.56630
$191$	25.3296	1.83278	0.916392	−	0.400283i	$-0.131088\pi$
$191$	25.3296	1.83278	0.916392	+	0.400283i	$0.131088\pi$
$192$	−24.9400	−1.79989
$193$	−9.75953	−0.702506	−0.351253	−	0.936281i	$-0.614244\pi$
$193$	−9.75953	−0.702506	−0.351253	+	0.936281i	$0.614244\pi$
$194$	−6.10785	−0.438518
$195$	−25.3994	−1.81889
$196$	59.4629	4.24735
$197$	3.71391	0.264605	0.132302	−	0.991209i	$-0.457763\pi$
$197$	3.71391	0.264605	0.132302	+	0.991209i	$0.457763\pi$
$198$	2.09768	0.149075
$199$	−11.8445	−0.839636	−0.419818	−	0.907608i	$-0.637906\pi$
$199$	−11.8445	−0.839636	−0.419818	+	0.907608i	$0.637906\pi$
$200$	0.306196	0.0216513
$201$	4.53287	0.319724
$202$	−6.95067	−0.489047
$203$	40.3804	2.83415
$204$	−21.9536	−1.53706
$205$	10.1782	0.710878
$206$	−31.7924	−2.21508
$207$	0.833638	0.0579418
$208$	−2.63563	−0.182748
$209$	−4.60038	−0.318215
$210$	−49.8874	−3.44256
$211$	2.96372	0.204031	0.102015	−	0.994783i	$-0.467471\pi$
$211$	2.96372	0.204031	0.102015	+	0.994783i	$0.467471\pi$
$212$	3.22736	0.221656
$213$	17.5775	1.20439
$214$	17.7334	1.21223
$215$	10.1826	0.694445
$216$	13.0735	0.889541
$217$	−19.9036	−1.35114
$218$	−27.6771	−1.87453
$219$	25.0343	1.69166
$220$	−8.04210	−0.542198
$221$	19.7044	1.32546
$222$	−33.6268	−2.25688
$223$	−20.5574	−1.37662	−0.688312	−	0.725414i	$-0.741648\pi$
$223$	−20.5574	−1.37662	−0.688312	+	0.725414i	$0.741648\pi$
$224$	25.5417	1.70658
$225$	−0.0828172	−0.00552115
$226$	−47.0537	−3.12996
$227$	12.0617	0.800563	0.400281	−	0.916392i	$-0.368912\pi$
$227$	12.0617	0.800563	0.400281	+	0.916392i	$0.368912\pi$
$228$	27.5684	1.82576
$229$	−6.00492	−0.396816	−0.198408	−	0.980119i	$-0.563577\pi$
$229$	−6.00492	−0.396816	−0.198408	+	0.980119i	$0.563577\pi$
$230$	−5.11297	−0.337139
$231$	−10.6300	−0.699402
$232$	−24.9755	−1.63972
$233$	20.7440	1.35899	0.679493	−	0.733682i	$-0.262200\pi$
$233$	20.7440	1.35899	0.679493	+	0.733682i	$0.262200\pi$
$234$	11.2827	0.737575
$235$	−15.5326	−1.01324
$236$	11.1715	0.727202
$237$	−10.4358	−0.677879
$238$	38.7018	2.50867
$239$	−7.00811	−0.453317	−0.226658	−	0.973974i	$-0.572780\pi$
$239$	−7.00811	−0.453317	−0.226658	+	0.973974i	$0.572780\pi$
$240$	1.94952	0.125841
$241$	4.95966	0.319480	0.159740	−	0.987159i	$-0.448934\pi$
$241$	4.95966	0.319480	0.159740	+	0.987159i	$0.448934\pi$
$242$	22.6647	1.45694
$243$	−8.43272	−0.540959
$244$	−14.5950	−0.934348
$245$	39.4771	2.52210
$246$	−20.7920	−1.32565
$247$	−24.7440	−1.57442
$248$	12.3105	0.781717
$249$	−5.10320	−0.323402
$250$	26.0728	1.64899
$251$	4.75309	0.300012	0.150006	−	0.988685i	$-0.452071\pi$
$251$	4.75309	0.300012	0.150006	+	0.988685i	$0.452071\pi$
$252$	13.8522	0.872604
$253$	−1.08947	−0.0684944
$254$	−29.6053	−1.85760
$255$	−14.5749	−0.912716
$256$	−18.7972	−1.17483
$257$	−12.5137	−0.780584	−0.390292	−	0.920691i	$-0.627626\pi$
$257$	−12.5137	−0.780584	−0.390292	+	0.920691i	$0.627626\pi$
$258$	−20.8009	−1.29501
$259$	37.0549	2.30248
$260$	−43.2559	−2.68262
$261$	6.75515	0.418134
$262$	−3.12039	−0.192778
$263$	−3.96093	−0.244242	−0.122121	−	0.992515i	$-0.538970\pi$
$263$	−3.96093	−0.244242	−0.122121	+	0.992515i	$0.538970\pi$
$264$	6.57472	0.404646
$265$	2.14263	0.131621
$266$	−48.6001	−2.97986
$267$	−17.7798	−1.08811
$268$	7.71962	0.471551
$269$	−12.5826	−0.767172	−0.383586	−	0.923505i	$-0.625311\pi$
$269$	−12.5826	−0.767172	−0.383586	+	0.923505i	$0.625311\pi$
$270$	21.6875	1.31986
$271$	19.3596	1.17601	0.588006	−	0.808857i	$-0.299913\pi$
$271$	19.3596	1.17601	0.588006	+	0.808857i	$0.299913\pi$
$272$	−1.51240	−0.0917029
$273$	−57.1753	−3.46041
$274$	5.73025	0.346177
$275$	0.108233	0.00652667
$276$	6.52879	0.392987
$277$	29.2090	1.75500	0.877500	−	0.479577i	$-0.159210\pi$
$277$	29.2090	1.75500	0.877500	+	0.479577i	$0.159210\pi$
$278$	−14.6671	−0.879675
$279$	−3.32963	−0.199340
$280$	−34.0013	−2.03197
$281$	−26.7947	−1.59844	−0.799218	−	0.601042i	$-0.794753\pi$
$281$	−26.7947	−1.59844	−0.799218	+	0.601042i	$0.794753\pi$
$282$	31.7299	1.88949
$283$	−5.70733	−0.339266	−0.169633	−	0.985507i	$-0.554258\pi$
$283$	−5.70733	−0.339266	−0.169633	+	0.985507i	$0.554258\pi$
$284$	29.9350	1.77631
$285$	18.3026	1.08415
$286$	−14.7452	−0.871904
$287$	22.9117	1.35243
$288$	4.27283	0.251779
$289$	−5.69303	−0.334884
$290$	−41.4315	−2.43294
$291$	5.17784	0.303530
$292$	42.6342	2.49498
$293$	−10.2729	−0.600148	−0.300074	−	0.953916i	$-0.597011\pi$
$293$	−10.2729	−0.600148	−0.300074	+	0.953916i	$0.597011\pi$
$294$	−80.6436	−4.70323
$295$	7.41670	0.431817
$296$	−22.9187	−1.33212
$297$	4.62117	0.268147
$298$	−47.1345	−2.73043
$299$	−5.85991	−0.338887
$300$	−0.648599	−0.0374469
$301$	22.9215	1.32117
$302$	−35.7194	−2.05542
$303$	5.89232	0.338505
$304$	1.89921	0.108927
$305$	−9.68954	−0.554822
$306$	6.47435	0.370114
$307$	−17.6042	−1.00473	−0.502364	−	0.864656i	$-0.667536\pi$
$307$	−17.6042	−1.00473	−0.502364	+	0.864656i	$0.667536\pi$
$308$	−18.1032	−1.03152
$309$	26.9515	1.53322
$310$	20.4217	1.15987
$311$	8.61767	0.488663	0.244332	−	0.969692i	$-0.421431\pi$
$311$	8.61767	0.488663	0.244332	+	0.969692i	$0.421431\pi$
$312$	35.3633	2.00205
$313$	25.1918	1.42393	0.711964	−	0.702216i	$-0.247806\pi$
$313$	25.1918	1.42393	0.711964	+	0.702216i	$0.247806\pi$
$314$	−52.8384	−2.98184
$315$	9.19638	0.518157
$316$	−17.7725	−0.999781
$317$	−12.0992	−0.679558	−0.339779	−	0.940505i	$-0.610352\pi$
$317$	−12.0992	−0.679558	−0.339779	+	0.940505i	$0.610352\pi$
$318$	−4.37694	−0.245447
$319$	−8.82821	−0.494285
$320$	−28.1980	−1.57631
$321$	−15.0332	−0.839072
$322$	−11.5095	−0.641402
$323$	−14.1988	−0.790043
$324$	−36.0322	−2.00179
$325$	0.582149	0.0322918
$326$	−39.7055	−2.19908
$327$	23.4628	1.29750
$328$	−14.1710	−0.782463
$329$	−34.9647	−1.92767
$330$	10.9067	0.600394
$331$	30.0755	1.65310	0.826550	−	0.562864i	$-0.190300\pi$
$331$	30.0755	1.65310	0.826550	+	0.562864i	$0.190300\pi$
$332$	−8.69090	−0.476975
$333$	6.19885	0.339695
$334$	31.1588	1.70493
$335$	5.12502	0.280010
$336$	4.38846	0.239410
$337$	34.1753	1.86164	0.930822	−	0.365472i	$-0.119092\pi$
$337$	34.1753	1.86164	0.930822	+	0.365472i	$0.119092\pi$
$338$	−49.2845	−2.68072
$339$	39.8890	2.16647
$340$	−24.8215	−1.34613
$341$	4.35145	0.235644
$342$	−8.13022	−0.439632
$343$	53.9823	2.91477
$344$	−14.1771	−0.764376
$345$	4.33444	0.233358
$346$	50.8976	2.73627
$347$	−32.3387	−1.73603	−0.868017	−	0.496535i	$-0.834606\pi$
$347$	−32.3387	−1.73603	−0.868017	+	0.496535i	$0.834606\pi$
$348$	52.9043	2.83597
$349$	−1.00000	−0.0535288
$349$	−1.00000	−0.0535288
$350$	1.14341	0.0611177
$351$	24.8558	1.32670
$352$	−5.58409	−0.297633
$353$	14.9319	0.794743	0.397372	−	0.917658i	$-0.369922\pi$
$353$	14.9319	0.794743	0.397372	+	0.917658i	$0.369922\pi$
$354$	−15.1508	−0.805254
$355$	19.8737	1.05479
$356$	−30.2796	−1.60481
$357$	−32.8089	−1.73643
$358$	39.2155	2.07260
$359$	−28.9110	−1.52586	−0.762931	−	0.646480i	$-0.776241\pi$
$359$	−28.9110	−1.52586	−0.762931	+	0.646480i	$0.776241\pi$
$360$	−5.68802	−0.299785
$361$	−1.16972	−0.0615644
$362$	6.66861	0.350494
$363$	−19.2136	−1.00845
$364$	−97.3712	−5.10364
$365$	28.3046	1.48153
$366$	19.7937	1.03463
$367$	−10.1043	−0.527439	−0.263719	−	0.964599i	$-0.584949\pi$
$367$	−10.1043	−0.527439	−0.263719	+	0.964599i	$0.584949\pi$
$368$	0.449774	0.0234461
$369$	3.83285	0.199530
$370$	−38.0195	−1.97654
$371$	4.82316	0.250406
$372$	−26.0767	−1.35201
$373$	3.28564	0.170124	0.0850619	−	0.996376i	$-0.472891\pi$
$373$	3.28564	0.170124	0.0850619	+	0.996376i	$0.472891\pi$
$374$	−8.46124	−0.437520
$375$	−22.1028	−1.14138
$376$	21.6259	1.11527
$377$	−47.4841	−2.44556
$378$	48.8197	2.51101
$379$	−12.0431	−0.618612	−0.309306	−	0.950963i	$-0.600097\pi$
$379$	−12.0431	−0.618612	−0.309306	+	0.950963i	$0.600097\pi$
$380$	31.1698	1.59898
$381$	25.0974	1.28578
$382$	−58.5024	−2.99324
$383$	−5.39916	−0.275884	−0.137942	−	0.990440i	$-0.544049\pi$
$383$	−5.39916	−0.275884	−0.137942	+	0.990440i	$0.544049\pi$
$384$	37.5314	1.91527
$385$	−12.0186	−0.612526
$386$	22.5411	1.14731
$387$	3.83449	0.194918
$388$	8.81800	0.447666
$389$	−15.6298	−0.792460	−0.396230	−	0.918151i	$-0.629682\pi$
$389$	−15.6298	−0.792460	−0.396230	+	0.918151i	$0.629682\pi$
$390$	58.6637	2.97055
$391$	−3.36258	−0.170053
$392$	−54.9635	−2.77608
$393$	2.64526	0.133436
$394$	−8.57782	−0.432144
$395$	−11.7991	−0.593676
$396$	−3.02845	−0.152185
$397$	8.83680	0.443506	0.221753	−	0.975103i	$-0.428822\pi$
$397$	8.83680	0.443506	0.221753	+	0.975103i	$0.428822\pi$
$398$	27.3567	1.37127
$399$	41.2000	2.06258
$400$	−0.0446825	−0.00223412
$401$	9.63522	0.481160	0.240580	−	0.970629i	$-0.422662\pi$
$401$	9.63522	0.481160	0.240580	+	0.970629i	$0.422662\pi$
$402$	−10.4693	−0.522164
$403$	23.4050	1.16589
$404$	10.0348	0.499249
$405$	−23.9216	−1.18867
$406$	−93.2644	−4.62864
$407$	−8.10119	−0.401561
$408$	20.2925	1.00463
$409$	12.5750	0.621796	0.310898	−	0.950443i	$-0.399370\pi$
$409$	12.5750	0.621796	0.310898	+	0.950443i	$0.399370\pi$
$410$	−23.5081	−1.16098
$411$	−4.85773	−0.239614
$412$	45.8992	2.26129
$413$	16.6954	0.821525
$414$	−1.92541	−0.0946288
$415$	−5.76985	−0.283231
$416$	−30.0351	−1.47259
$417$	12.4338	0.608887
$418$	10.6253	0.519699
$419$	10.8668	0.530879	0.265440	−	0.964127i	$-0.414483\pi$
$419$	10.8668	0.530879	0.265440	+	0.964127i	$0.414483\pi$
$420$	72.0232	3.51437
$421$	22.4441	1.09386	0.546929	−	0.837179i	$-0.315797\pi$
$421$	22.4441	1.09386	0.546929	+	0.837179i	$0.315797\pi$
$422$	−6.84515	−0.333217
$423$	−5.84918	−0.284397
$424$	−2.98315	−0.144875
$425$	0.334054	0.0162040
$426$	−40.5978	−1.96697
$427$	−21.8117	−1.05554
$428$	−25.6020	−1.23752
$429$	12.5000	0.603508
$430$	−23.5181	−1.13415
$431$	−4.56255	−0.219770	−0.109885	−	0.993944i	$-0.535048\pi$
$431$	−4.56255	−0.219770	−0.109885	+	0.993944i	$0.535048\pi$
$432$	−1.90779	−0.0917886
$433$	6.23755	0.299758	0.149879	−	0.988704i	$-0.452112\pi$
$433$	6.23755	0.299758	0.149879	+	0.988704i	$0.452112\pi$
$434$	45.9703	2.20664
$435$	35.1229	1.68401
$436$	39.9578	1.91363
$437$	4.22259	0.201994
$438$	−57.8205	−2.76277
$439$	6.16399	0.294191	0.147096	−	0.989122i	$-0.453007\pi$
$439$	6.16399	0.294191	0.147096	+	0.989122i	$0.453007\pi$
$440$	7.43359	0.354382
$441$	14.8661	0.707908
$442$	−45.5103	−2.16470
$443$	−4.13367	−0.196397	−0.0981983	−	0.995167i	$-0.531308\pi$
$443$	−4.13367	−0.196397	−0.0981983	+	0.995167i	$0.531308\pi$
$444$	48.5475	2.30396
$445$	−20.1025	−0.952949
$446$	47.4803	2.24826
$447$	39.9575	1.88993
$448$	−63.4750	−2.99891
$449$	1.92458	0.0908263	0.0454132	−	0.998968i	$-0.485540\pi$
$449$	1.92458	0.0908263	0.0454132	+	0.998968i	$0.485540\pi$
$450$	0.191279	0.00901696
$451$	−5.00910	−0.235869
$452$	67.9321	3.19526
$453$	30.2806	1.42271
$454$	−27.8583	−1.30745
$455$	−64.6443	−3.03057
$456$	−25.4824	−1.19332
$457$	−41.5873	−1.94537	−0.972686	−	0.232123i	$-0.925433\pi$
$457$	−41.5873	−1.94537	−0.972686	+	0.232123i	$0.925433\pi$
$458$	13.8693	0.648068
$459$	14.2630	0.665738
$460$	7.38167	0.344172
$461$	−14.2515	−0.663758	−0.331879	−	0.943322i	$-0.607683\pi$
$461$	−14.2515	−0.663758	−0.331879	+	0.943322i	$0.607683\pi$
$462$	24.5515	1.14224
$463$	−8.40781	−0.390744	−0.195372	−	0.980729i	$-0.562591\pi$
$463$	−8.40781	−0.390744	−0.195372	+	0.980729i	$0.562591\pi$
$464$	3.64462	0.169197
$465$	−17.3122	−0.802833
$466$	−47.9114	−2.21945
$467$	−7.21854	−0.334034	−0.167017	−	0.985954i	$-0.553413\pi$
$467$	−7.21854	−0.334034	−0.167017	+	0.985954i	$0.553413\pi$
$468$	−16.2890	−0.752961
$469$	11.5367	0.532714
$470$	35.8749	1.65478
$471$	44.7929	2.06395
$472$	−10.3262	−0.475301
$473$	−5.01124	−0.230417
$474$	24.1031	1.10709
$475$	−0.419491	−0.0192475
$476$	−55.8744	−2.56100
$477$	0.806858	0.0369435
$478$	16.1863	0.740343
$479$	2.29995	0.105088	0.0525438	−	0.998619i	$-0.483267\pi$
$479$	2.29995	0.105088	0.0525438	+	0.998619i	$0.483267\pi$
$480$	22.2162	1.01403
$481$	−43.5737	−1.98679
$482$	−11.4551	−0.521764
$483$	9.75704	0.443961
$484$	−32.7214	−1.48734
$485$	5.85423	0.265827
$486$	19.4766	0.883478
$487$	5.42000	0.245603	0.122802	−	0.992431i	$-0.460812\pi$
$487$	5.42000	0.245603	0.122802	+	0.992431i	$0.460812\pi$
$488$	13.4906	0.610693
$489$	33.6597	1.52214
$490$	−91.1783	−4.11901
$491$	3.66652	0.165468	0.0827338	−	0.996572i	$-0.473635\pi$
$491$	3.66652	0.165468	0.0827338	+	0.996572i	$0.473635\pi$
$492$	30.0177	1.35330
$493$	−27.2478	−1.22718
$494$	57.1499	2.57130
$495$	−2.01057	−0.0903685
$496$	−1.79644	−0.0806626
$497$	44.7367	2.00671
$498$	11.7866	0.528170
$499$	−12.8305	−0.574371	−0.287186	−	0.957875i	$-0.592720\pi$
$499$	−12.8305	−0.574371	−0.287186	+	0.957875i	$0.592720\pi$
$500$	−37.6417	−1.68339
$501$	−26.4144	−1.18011
$502$	−10.9780	−0.489971
$503$	−2.25672	−0.100622	−0.0503110	−	0.998734i	$-0.516021\pi$
$503$	−2.25672	−0.100622	−0.0503110	+	0.998734i	$0.516021\pi$
$504$	−12.8040	−0.570336
$505$	6.66205	0.296457
$506$	2.51629	0.111863
$507$	41.7801	1.85552
$508$	42.7416	1.89635
$509$	−23.0317	−1.02086	−0.510431	−	0.859919i	$-0.670514\pi$
$509$	−23.0317	−1.02086	−0.510431	+	0.859919i	$0.670514\pi$
$510$	33.6629	1.49062
$511$	63.7151	2.81859
$512$	5.07788	0.224413
$513$	−17.9108	−0.790782
$514$	28.9023	1.27482
$515$	30.4722	1.34277
$516$	30.0305	1.32202
$517$	7.64421	0.336192
$518$	−85.5839	−3.76034
$519$	−43.1477	−1.89397
$520$	39.9829	1.75337
$521$	−28.4736	−1.24745	−0.623726	−	0.781643i	$-0.714382\pi$
$521$	−28.4736	−1.24745	−0.623726	+	0.781643i	$0.714382\pi$
$522$	−15.6020	−0.682882
$523$	0.652547	0.0285339	0.0142670	−	0.999898i	$-0.495459\pi$
$523$	0.652547	0.0285339	0.0142670	+	0.999898i	$0.495459\pi$
$524$	4.50495	0.196800
$525$	−0.969306	−0.0423040
$526$	9.14837	0.398888
$527$	13.4305	0.585041
$528$	−0.959433	−0.0417540
$529$	1.00000	0.0434783
$530$	−4.94871	−0.214958
$531$	2.79293	0.121203
$532$	70.1647	3.04203
$533$	−26.9423	−1.16700
$534$	41.0652	1.77706
$535$	−16.9970	−0.734846
$536$	−7.13550	−0.308207
$537$	−33.2443	−1.43460
$538$	29.0613	1.25292
$539$	−19.4282	−0.836834
$540$	−31.3106	−1.34739
$541$	−23.5573	−1.01281	−0.506404	−	0.862296i	$-0.669025\pi$
$541$	−23.5573	−1.01281	−0.506404	+	0.862296i	$0.669025\pi$
$542$	−44.7139	−1.92062
$543$	−5.65321	−0.242602
$544$	−17.2350	−0.738944
$545$	26.5278	1.13633
$546$	132.055	5.65143
$547$	−27.1115	−1.15920	−0.579602	−	0.814899i	$-0.696792\pi$
$547$	−27.1115	−1.15920	−0.579602	+	0.814899i	$0.696792\pi$
$548$	−8.27286	−0.353399
$549$	−3.64883	−0.155728
$550$	−0.249979	−0.0106592
$551$	34.2166	1.45768
$552$	−6.03479	−0.256858
$553$	−26.5603	−1.12946
$554$	−67.4626	−2.86621
$555$	32.2304	1.36811
$556$	21.1751	0.898026
$557$	−43.3501	−1.83680	−0.918401	−	0.395652i	$-0.870519\pi$
$557$	−43.3501	−1.83680	−0.918401	+	0.395652i	$0.870519\pi$
$558$	7.69028	0.325556
$559$	−26.9538	−1.14003
$560$	4.96174	0.209672
$561$	7.17288	0.302839
$562$	61.8862	2.61051
$563$	40.2540	1.69650	0.848251	−	0.529594i	$-0.177656\pi$
$563$	40.2540	1.69650	0.848251	+	0.529594i	$0.177656\pi$
$564$	−45.8090	−1.92891
$565$	45.0998	1.89736
$566$	13.1819	0.554078
$567$	−53.8488	−2.26144
$568$	−27.6699	−1.16100
$569$	−16.2379	−0.680730	−0.340365	−	0.940293i	$-0.610551\pi$
$569$	−16.2379	−0.680730	−0.340365	+	0.940293i	$0.610551\pi$
$570$	−42.2725	−1.77060
$571$	−10.6855	−0.447173	−0.223586	−	0.974684i	$-0.571777\pi$
$571$	−10.6855	−0.447173	−0.223586	+	0.974684i	$0.571777\pi$
$572$	21.2879	0.890093
$573$	49.5945	2.07184
$574$	−52.9179	−2.20875
$575$	−0.0993443	−0.00414294
$576$	−10.6186	−0.442443
$577$	−5.04145	−0.209878	−0.104939	−	0.994479i	$-0.533465\pi$
$577$	−5.04145	−0.209878	−0.104939	+	0.994479i	$0.533465\pi$
$578$	13.1489	0.546922
$579$	−19.1088	−0.794136
$580$	59.8153	2.48370
$581$	−12.9882	−0.538842
$582$	−11.9590	−0.495716
$583$	−1.05447	−0.0436717
$584$	−39.4082	−1.63072
$585$	−10.8142	−0.447113
$586$	23.7267	0.980142
$587$	25.6438	1.05843	0.529217	−	0.848486i	$-0.322486\pi$
$587$	25.6438	1.05843	0.529217	+	0.848486i	$0.322486\pi$
$588$	116.426	4.80134
$589$	−16.8654	−0.694929
$590$	−17.1300	−0.705229
$591$	7.27171	0.299118
$592$	3.34447	0.137457
$593$	−3.48479	−0.143103	−0.0715516	−	0.997437i	$-0.522795\pi$
$593$	−3.48479	−0.143103	−0.0715516	+	0.997437i	$0.522795\pi$
$594$	−10.6733	−0.437930
$595$	−37.0948	−1.52074
$596$	68.0488	2.78739
$597$	−23.1912	−0.949153
$598$	13.5343	0.553460
$599$	39.6790	1.62124	0.810619	−	0.585574i	$-0.199131\pi$
$599$	39.6790	1.62124	0.810619	+	0.585574i	$0.199131\pi$
$600$	0.599522	0.0244754
$601$	−37.8911	−1.54561	−0.772806	−	0.634643i	$-0.781147\pi$
$601$	−37.8911	−1.54561	−0.772806	+	0.634643i	$0.781147\pi$
$602$	−52.9405	−2.15769
$603$	1.92995	0.0785936
$604$	51.5687	2.09830
$605$	−21.7236	−0.883189
$606$	−13.6092	−0.552835
$607$	−42.5259	−1.72607	−0.863036	−	0.505142i	$-0.831440\pi$
$607$	−42.5259	−1.72607	−0.863036	+	0.505142i	$0.831440\pi$
$608$	21.6430	0.877738
$609$	79.0635	3.20381
$610$	22.3794	0.906117
$611$	41.1157	1.66336
$612$	−9.34712	−0.377835
$613$	−5.58104	−0.225416	−0.112708	−	0.993628i	$-0.535952\pi$
$613$	−5.58104	−0.225416	−0.112708	+	0.993628i	$0.535952\pi$
$614$	40.6596	1.64089
$615$	19.9286	0.803600
$616$	16.7334	0.674207
$617$	−16.5443	−0.666050	−0.333025	−	0.942918i	$-0.608069\pi$
$617$	−16.5443	−0.666050	−0.333025	+	0.942918i	$0.608069\pi$
$618$	−62.2485	−2.50400
$619$	−26.3776	−1.06020	−0.530102	−	0.847934i	$-0.677846\pi$
$619$	−26.3776	−1.06020	−0.530102	+	0.847934i	$0.677846\pi$
$620$	−29.4831	−1.18407
$621$	−4.24167	−0.170212
$622$	−19.9038	−0.798069
$623$	−45.2517	−1.81297
$624$	−5.16049	−0.206585
$625$	−24.4934	−0.979736
$626$	−58.1843	−2.32551
$627$	−9.00740	−0.359721
$628$	76.2836	3.04405
$629$	−25.0038	−0.996968
$630$	−21.2404	−0.846238
$631$	−4.24167	−0.168858	−0.0844291	−	0.996429i	$-0.526907\pi$
$631$	−4.24167	−0.168858	−0.0844291	+	0.996429i	$0.526907\pi$
$632$	16.4277	0.653460
$633$	5.80287	0.230643
$634$	27.9449	1.10983
$635$	28.3760	1.12607
$636$	6.31906	0.250567
$637$	−104.498	−4.14037
$638$	20.3901	0.807251
$639$	7.48391	0.296059
$640$	42.4342	1.67736
$641$	−4.60379	−0.181839	−0.0909194	−	0.995858i	$-0.528981\pi$
$641$	−4.60379	−0.181839	−0.0909194	+	0.995858i	$0.528981\pi$
$642$	34.7214	1.37034
$643$	2.06716	0.0815210	0.0407605	−	0.999169i	$-0.487022\pi$
$643$	2.06716	0.0815210	0.0407605	+	0.999169i	$0.487022\pi$
$644$	16.6165	0.654782
$645$	19.9371	0.785024
$646$	32.7943	1.29027
$647$	5.36556	0.210942	0.105471	−	0.994422i	$-0.466365\pi$
$647$	5.36556	0.210942	0.105471	+	0.994422i	$0.466365\pi$
$648$	33.3058	1.30837
$649$	−3.65005	−0.143277
$650$	−1.34456	−0.0527379
$651$	−38.9706	−1.52738
$652$	57.3234	2.24496
$653$	−26.8599	−1.05111	−0.525554	−	0.850760i	$-0.676142\pi$
$653$	−26.8599	−1.05111	−0.525554	+	0.850760i	$0.676142\pi$
$654$	−54.1908	−2.11903
$655$	2.99082	0.116861
$656$	2.06794	0.0807397
$657$	10.6588	0.415839
$658$	80.7562	3.14820
$659$	−0.517386	−0.0201545	−0.0100773	−	0.999949i	$-0.503208\pi$
$659$	−0.517386	−0.0201545	−0.0100773	+	0.999949i	$0.503208\pi$
$660$	−15.7462	−0.612919
$661$	28.2255	1.09784	0.548921	−	0.835874i	$-0.315039\pi$
$661$	28.2255	1.09784	0.548921	+	0.835874i	$0.315039\pi$
$662$	−69.4638	−2.69979
$663$	38.5806	1.49835
$664$	8.03329	0.311752
$665$	46.5820	1.80637
$666$	−14.3172	−0.554779
$667$	8.10322	0.313758
$668$	−44.9844	−1.74050
$669$	−40.2507	−1.55618
$670$	−11.8370	−0.457303
$671$	4.76860	0.184090
$672$	50.0099	1.92917
$673$	−6.37316	−0.245667	−0.122834	−	0.992427i	$-0.539198\pi$
$673$	−6.37316	−0.245667	−0.122834	+	0.992427i	$0.539198\pi$
$674$	−78.9328	−3.04038
$675$	0.421386	0.0162191
$676$	71.1527	2.73664
$677$	22.3795	0.860113	0.430056	−	0.902802i	$-0.358494\pi$
$677$	22.3795	0.860113	0.430056	+	0.902802i	$0.358494\pi$
$678$	−92.1296	−3.53821
$679$	13.1782	0.505732
$680$	22.9433	0.879837
$681$	23.6164	0.904983
$682$	−10.0503	−0.384847
$683$	−33.0860	−1.26600	−0.633000	−	0.774152i	$-0.718177\pi$
$683$	−33.0860	−1.26600	−0.633000	+	0.774152i	$0.718177\pi$
$684$	11.7377	0.448803
$685$	−5.49231	−0.209850
$686$	−124.680	−4.76031
$687$	−11.7574	−0.448574
$688$	2.06883	0.0788733
$689$	−5.67166	−0.216073
$690$	−10.0110	−0.381113
$691$	14.3979	0.547722	0.273861	−	0.961769i	$-0.411699\pi$
$691$	14.3979	0.547722	0.273861	+	0.961769i	$0.411699\pi$
$692$	−73.4817	−2.79335
$693$	−4.52590	−0.171925
$694$	74.6911	2.83523
$695$	14.0581	0.533253
$696$	−48.9012	−1.85360
$697$	−15.4603	−0.585600
$698$	2.30965	0.0874215
$699$	40.6161	1.53624
$700$	−1.65076	−0.0623927
$701$	−13.6789	−0.516643	−0.258322	−	0.966059i	$-0.583169\pi$
$701$	−13.6789	−0.516643	−0.258322	+	0.966059i	$0.583169\pi$
$702$	−57.4081	−2.16673
$703$	31.3988	1.18423
$704$	13.8773	0.523021
$705$	−30.4124	−1.14540
$706$	−34.4874	−1.29795
$707$	14.9966	0.564006
$708$	21.8734	0.822053
$709$	24.0575	0.903500	0.451750	−	0.892145i	$-0.350800\pi$
$709$	24.0575	0.903500	0.451750	+	0.892145i	$0.350800\pi$
$710$	−45.9012	−1.72264
$711$	−4.44323	−0.166634
$712$	27.9884	1.04891
$713$	−3.99410	−0.149580
$714$	75.7769	2.83588
$715$	14.1330	0.528542
$716$	−56.6161	−2.11584
$717$	−13.7217	−0.512445
$718$	66.7742	2.49199
$719$	4.67069	0.174187	0.0870936	−	0.996200i	$-0.472242\pi$
$719$	4.67069	0.174187	0.0870936	+	0.996200i	$0.472242\pi$
$720$	0.830040	0.0309337
$721$	68.5945	2.55459
$722$	2.70165	0.100545
$723$	9.71085	0.361150
$724$	−9.62758	−0.357806
$725$	−0.805009	−0.0298973
$726$	44.3768	1.64698
$727$	11.9978	0.444975	0.222487	−	0.974936i	$-0.428582\pi$
$727$	11.9978	0.444975	0.222487	+	0.974936i	$0.428582\pi$
$728$	90.0035	3.33575
$729$	15.9069	0.589144
$730$	−65.3737	−2.41959
$731$	−15.4669	−0.572064
$732$	−28.5765	−1.05622
$733$	−22.0100	−0.812956	−0.406478	−	0.913661i	$-0.633243\pi$
$733$	−22.0100	−0.812956	−0.406478	+	0.913661i	$0.633243\pi$
$734$	23.3373	0.861396
$735$	77.2949	2.85107
$736$	5.12552	0.188929
$737$	−2.52222	−0.0929072
$738$	−8.85254	−0.325867
$739$	15.8025	0.581306	0.290653	−	0.956829i	$-0.406127\pi$
$739$	15.8025	0.581306	0.290653	+	0.956829i	$0.406127\pi$
$740$	54.8894	2.01777
$741$	−48.4479	−1.77978
$742$	−11.1398	−0.408955
$743$	0.948804	0.0348082	0.0174041	−	0.999849i	$-0.494460\pi$
$743$	0.948804	0.0348082	0.0174041	+	0.999849i	$0.494460\pi$
$744$	24.1035	0.883678
$745$	45.1773	1.65517
$746$	−7.58867	−0.277841
$747$	−2.17277	−0.0794977
$748$	12.2156	0.446647
$749$	−38.2612	−1.39803
$750$	51.0497	1.86407
$751$	27.3398	0.997644	0.498822	−	0.866704i	$-0.333766\pi$
$751$	27.3398	0.997644	0.498822	+	0.866704i	$0.333766\pi$
$752$	−3.15582	−0.115081
$753$	9.30640	0.339144
$754$	109.672	3.99401
$755$	34.2362	1.24598
$756$	−70.4817	−2.56340
$757$	11.4806	0.417268	0.208634	−	0.977994i	$-0.433098\pi$
$757$	11.4806	0.417268	0.208634	+	0.977994i	$0.433098\pi$
$758$	27.8153	1.01030
$759$	−2.13315	−0.0774283
$760$	−28.8113	−1.04509
$761$	9.14340	0.331448	0.165724	−	0.986172i	$-0.447004\pi$
$761$	9.14340	0.331448	0.165724	+	0.986172i	$0.447004\pi$
$762$	−57.9662	−2.09989
$763$	59.7155	2.16184
$764$	84.4608	3.05569
$765$	−6.20551	−0.224361
$766$	12.4702	0.450565
$767$	−19.6324	−0.708886
$768$	−36.8044	−1.32806
$769$	−44.5901	−1.60796	−0.803981	−	0.594656i	$-0.797288\pi$
$769$	−44.5901	−1.60796	−0.803981	+	0.594656i	$0.797288\pi$
$770$	27.7588	1.00036
$771$	−24.5014	−0.882398
$772$	−32.5429	−1.17124
$773$	−46.7806	−1.68258	−0.841291	−	0.540582i	$-0.818204\pi$
$773$	−46.7806	−1.68258	−0.841291	+	0.540582i	$0.818204\pi$
$774$	−8.85632	−0.318334
$775$	0.396791	0.0142532
$776$	−8.15078	−0.292596
$777$	72.5524	2.60280
$778$	36.0992	1.29422
$779$	19.4144	0.695593
$780$	−84.6936	−3.03252
$781$	−9.78062	−0.349978
$782$	7.76638	0.277725
$783$	−34.3712	−1.22833
$784$	8.02071	0.286454
$785$	50.6443	1.80757
$786$	−6.10962	−0.217923
$787$	25.0934	0.894484	0.447242	−	0.894413i	$-0.352406\pi$
$787$	25.0934	0.894484	0.447242	+	0.894413i	$0.352406\pi$
$788$	12.3839	0.441159
$789$	−7.75538	−0.276099
$790$	27.2517	0.969573
$791$	101.522	3.60971
$792$	2.79930	0.0994687
$793$	25.6488	0.910815
$794$	−20.4099	−0.724320
$795$	4.19519	0.148788
$796$	−39.4953	−1.39987
$797$	22.0738	0.781895	0.390948	−	0.920413i	$-0.372147\pi$
$797$	22.0738	0.781895	0.390948	+	0.920413i	$0.372147\pi$
$798$	−95.1575	−3.36854
$799$	23.5934	0.834674
$800$	−0.509191	−0.0180026
$801$	−7.57007	−0.267475
$802$	−22.2540	−0.785815
$803$	−13.9298	−0.491572
$804$	15.1148	0.533056
$805$	11.0316	0.388814
$806$	−54.0574	−1.90409
$807$	−24.6362	−0.867236
$808$	−9.27549	−0.326311
$809$	−4.51267	−0.158657	−0.0793285	−	0.996849i	$-0.525278\pi$
$809$	−4.51267	−0.158657	−0.0793285	+	0.996849i	$0.525278\pi$
$810$	55.2505	1.94131
$811$	−33.9368	−1.19168	−0.595841	−	0.803103i	$-0.703181\pi$
$811$	−33.9368	−1.19168	−0.595841	+	0.803103i	$0.703181\pi$
$812$	134.647	4.72519
$813$	37.9055	1.32940
$814$	18.7109	0.655816
$815$	38.0567	1.33307
$816$	−2.96123	−0.103664
$817$	19.4227	0.679513
$818$	−29.0439	−1.01550
$819$	−24.3434	−0.850626
$820$	33.9390	1.18520
$821$	−18.6825	−0.652025	−0.326013	−	0.945365i	$-0.605705\pi$
$821$	−18.6825	−0.652025	−0.326013	+	0.945365i	$0.605705\pi$
$822$	11.2197	0.391330
$823$	−12.0677	−0.420653	−0.210327	−	0.977631i	$-0.567453\pi$
$823$	−12.0677	−0.420653	−0.210327	+	0.977631i	$0.567453\pi$
$824$	−42.4261	−1.47798
$825$	0.211916	0.00737797
$826$	−38.5604	−1.34169
$827$	2.21520	0.0770301	0.0385150	−	0.999258i	$-0.487737\pi$
$827$	2.21520	0.0770301	0.0385150	+	0.999258i	$0.487737\pi$
$828$	2.77974	0.0966028
$829$	−35.7236	−1.24073	−0.620365	−	0.784313i	$-0.713016\pi$
$829$	−35.7236	−1.24073	−0.620365	+	0.784313i	$0.713016\pi$
$830$	13.3263	0.462563
$831$	57.1903	1.98391
$832$	74.6417	2.58773
$833$	−59.9641	−2.07763
$834$	−28.7177	−0.994414
$835$	−29.8650	−1.03352
$836$	−15.3399	−0.530540
$837$	16.9416	0.585589
$838$	−25.0986	−0.867015
$839$	47.8380	1.65155	0.825775	−	0.563999i	$-0.190738\pi$
$839$	47.8380	1.65155	0.825775	+	0.563999i	$0.190738\pi$
$840$	−66.5735	−2.29700
$841$	36.6622	1.26421
$842$	−51.8379	−1.78645
$843$	−52.4631	−1.80692
$844$	9.88245	0.340168
$845$	47.2380	1.62504
$846$	13.5095	0.464468
$847$	−48.9009	−1.68025
$848$	0.435325	0.0149491
$849$	−11.1748	−0.383517
$850$	−0.771546	−0.0264638
$851$	7.43590	0.254899
$852$	58.6117	2.00800
$853$	−51.9232	−1.77782	−0.888908	−	0.458085i	$-0.848536\pi$
$853$	−51.9232	−1.77782	−0.888908	+	0.458085i	$0.848536\pi$
$854$	50.3773	1.72387
$855$	7.79262	0.266502
$856$	23.6648	0.808845
$857$	−2.82859	−0.0966228	−0.0483114	−	0.998832i	$-0.515384\pi$
$857$	−2.82859	−0.0966228	−0.0483114	+	0.998832i	$0.515384\pi$
$858$	−28.8707	−0.985629
$859$	−13.8748	−0.473403	−0.236702	−	0.971582i	$-0.576066\pi$
$859$	−13.8748	−0.473403	−0.236702	+	0.971582i	$0.576066\pi$
$860$	33.9535	1.15780
$861$	44.8603	1.52884
$862$	10.5379	0.358922
$863$	−24.0921	−0.820106	−0.410053	−	0.912062i	$-0.634490\pi$
$863$	−24.0921	−0.820106	−0.410053	+	0.912062i	$0.634490\pi$
$864$	−21.7407	−0.739635
$865$	−48.7842	−1.65871
$866$	−14.4065	−0.489554
$867$	−11.1468	−0.378564
$868$	−66.3680	−2.25268
$869$	5.80679	0.196982
$870$	−81.1216	−2.75028
$871$	−13.5662	−0.459674
$872$	−36.9344	−1.25076
$873$	2.20455	0.0746128
$874$	−9.75270	−0.329890
$875$	−56.2541	−1.90173
$876$	83.4763	2.82040
$877$	−38.3489	−1.29495	−0.647476	−	0.762086i	$-0.724175\pi$
$877$	−38.3489	−1.29495	−0.647476	+	0.762086i	$0.724175\pi$
$878$	−14.2367	−0.480464
$879$	−20.1140	−0.678427
$880$	−1.08477	−0.0365675
$881$	−4.18008	−0.140830	−0.0704152	−	0.997518i	$-0.522432\pi$
$881$	−4.18008	−0.140830	−0.0704152	+	0.997518i	$0.522432\pi$
$882$	−34.3354	−1.15613
$883$	40.3284	1.35716	0.678580	−	0.734527i	$-0.262596\pi$
$883$	40.3284	1.35716	0.678580	+	0.734527i	$0.262596\pi$
$884$	65.7039	2.20986
$885$	14.5217	0.488140
$886$	9.54732	0.320749
$887$	−25.2631	−0.848252	−0.424126	−	0.905603i	$-0.639419\pi$
$887$	−25.2631	−0.848252	−0.424126	+	0.905603i	$0.639419\pi$
$888$	−44.8741	−1.50588
$889$	63.8757	2.14232
$890$	46.4296	1.55633
$891$	11.7728	0.394402
$892$	−68.5481	−2.29516
$893$	−29.6276	−0.991450
$894$	−92.2878	−3.08657
$895$	−37.5871	−1.25640
$896$	95.5216	3.19115
$897$	−11.4735	−0.383089
$898$	−4.44509	−0.148335
$899$	−32.3651	−1.07944
$900$	−0.276152	−0.00920506
$901$	−3.25456	−0.108425
$902$	11.5693	0.385214
$903$	44.8795	1.49350
$904$	−62.7919	−2.08843
$905$	−6.39170	−0.212467
$906$	−69.9375	−2.32352
$907$	−8.94543	−0.297028	−0.148514	−	0.988910i	$-0.547449\pi$
$907$	−8.94543	−0.297028	−0.148514	+	0.988910i	$0.547449\pi$
$908$	40.2194	1.33473
$909$	2.50875	0.0832101
$910$	149.306	4.94943
$911$	32.1930	1.06660	0.533300	−	0.845926i	$-0.320952\pi$
$911$	32.1930	1.06660	0.533300	+	0.845926i	$0.320952\pi$
$912$	3.71859	0.123135
$913$	2.83957	0.0939760
$914$	96.0521	3.17712
$915$	−18.9718	−0.627189
$916$	−20.0233	−0.661587
$917$	6.73248	0.222326
$918$	−32.9424	−1.08726
$919$	58.8113	1.94001	0.970003	−	0.243093i	$-0.0781621\pi$
$919$	58.8113	1.94001	0.970003	+	0.243093i	$0.0781621\pi$
$920$	−6.82313	−0.224952
$921$	−34.4686	−1.13578
$922$	32.9159	1.08403
$923$	−52.6068	−1.73157
$924$	−35.4455	−1.16607
$925$	−0.738715	−0.0242888
$926$	19.4191	0.638151
$927$	11.4751	0.376890
$928$	41.5332	1.36339
$929$	−25.1883	−0.826402	−0.413201	−	0.910640i	$-0.635589\pi$
$929$	−25.1883	−0.826402	−0.413201	+	0.910640i	$0.635589\pi$
$930$	39.9850	1.31116
$931$	75.3004	2.46787
$932$	69.1704	2.26575
$933$	16.8731	0.552401
$934$	16.6723	0.545534
$935$	8.10989	0.265222
$936$	15.0565	0.492138
$937$	17.6303	0.575956	0.287978	−	0.957637i	$-0.407017\pi$
$937$	17.6303	0.575956	0.287978	+	0.957637i	$0.407017\pi$
$938$	−26.6457	−0.870012
$939$	49.3248	1.60965
$940$	−51.7931	−1.68930
$941$	−37.9738	−1.23791	−0.618956	−	0.785426i	$-0.712444\pi$
$941$	−37.9738	−1.23791	−0.618956	+	0.785426i	$0.712444\pi$
$942$	−103.456	−3.37077
$943$	4.59774	0.149723
$944$	1.50688	0.0490446
$945$	−46.7925	−1.52216
$946$	11.5742	0.376310
$947$	−7.60852	−0.247244	−0.123622	−	0.992329i	$-0.539451\pi$
$947$	−7.60852	−0.247244	−0.123622	+	0.992329i	$0.539451\pi$
$948$	−34.7980	−1.13019
$949$	−74.9240	−2.43214
$950$	0.968876	0.0314345
$951$	−23.6898	−0.768195
$952$	51.6466	1.67388
$953$	33.0983	1.07216	0.536080	−	0.844167i	$-0.319905\pi$
$953$	33.0983	1.07216	0.536080	+	0.844167i	$0.319905\pi$
$954$	−1.86356	−0.0603349
$955$	56.0731	1.81448
$956$	−23.3684	−0.755787
$957$	−17.2854	−0.558756
$958$	−5.31209	−0.171626
$959$	−12.3635	−0.399237
$960$	−55.2107	−1.78192
$961$	−15.0472	−0.485393
$962$	100.640	3.24476
$963$	−6.40064	−0.206258
$964$	16.5378	0.532648
$965$	−21.6051	−0.695492
$966$	−22.5353	−0.725062
$967$	−49.4697	−1.59084	−0.795419	−	0.606060i	$-0.792749\pi$
$967$	−49.4697	−1.59084	−0.795419	+	0.606060i	$0.792749\pi$
$968$	30.2455	0.972127
$969$	−27.8008	−0.893091
$970$	−13.5212	−0.434140
$971$	9.15851	0.293911	0.146955	−	0.989143i	$-0.453053\pi$
$971$	9.15851	0.293911	0.146955	+	0.989143i	$0.453053\pi$
$972$	−28.1187	−0.901908
$973$	31.6455	1.01451
$974$	−12.5183	−0.401112
$975$	1.13983	0.0365037
$976$	−1.96866	−0.0630152
$977$	36.1889	1.15779	0.578894	−	0.815403i	$-0.303485\pi$
$977$	36.1889	1.15779	0.578894	+	0.815403i	$0.303485\pi$
$978$	−77.7420	−2.48592
$979$	9.89322	0.316189
$980$	131.635	4.20494
$981$	9.98969	0.318946
$982$	−8.46837	−0.270236
$983$	−34.6899	−1.10644	−0.553219	−	0.833036i	$-0.686601\pi$
$983$	−34.6899	−1.10644	−0.553219	+	0.833036i	$0.686601\pi$
$984$	−27.7464	−0.884523
$985$	8.22163	0.261963
$986$	62.9327	2.00419
$987$	−68.4598	−2.17910
$988$	−82.5082	−2.62494
$989$	4.59971	0.146262
$990$	4.64371	0.147587
$991$	−43.8457	−1.39280	−0.696402	−	0.717652i	$-0.745217\pi$
$991$	−43.8457	−1.39280	−0.696402	+	0.717652i	$0.745217\pi$
$992$	−20.4718	−0.649981
$993$	58.8869	1.86872
$994$	−103.326	−3.27730
$995$	−26.2207	−0.831253
$996$	−17.0165	−0.539188
$997$	−31.3040	−0.991406	−0.495703	−	0.868492i	$-0.665090\pi$
$997$	−31.3040	−0.991406	−0.495703	+	0.868492i	$0.665090\pi$
$998$	29.6339	0.938044
$999$	−31.5406	−0.997901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8027.2.a.f.1.20	✓	176

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8027.2.a.f.1.20	✓	176	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:	\( N \)	\(=\)	\( 8027 = 23 \cdot 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30965 q^{2} +1.95797 q^{3} +3.33448 q^{4} +2.21374 q^{5} -4.52222 q^{6} +4.98325 q^{7} -3.08217 q^{8} +0.833638 q^{9} +O(q^{10})\)	\(q-2.30965 q^{2} +1.95797 q^{3} +3.33448 q^{4} +2.21374 q^{5} -4.52222 q^{6} +4.98325 q^{7} -3.08217 q^{8} +0.833638 q^{9} -5.11297 q^{10} -1.08947 q^{11} +6.52879 q^{12} -5.85991 q^{13} -11.5095 q^{14} +4.33444 q^{15} +0.449774 q^{16} -3.36258 q^{17} -1.92541 q^{18} +4.22259 q^{19} +7.38167 q^{20} +9.75704 q^{21} +2.51629 q^{22} +1.00000 q^{23} -6.03479 q^{24} -0.0993443 q^{25} +13.5343 q^{26} -4.24167 q^{27} +16.6165 q^{28} +8.10322 q^{29} -10.0110 q^{30} -3.99410 q^{31} +5.12552 q^{32} -2.13315 q^{33} +7.76638 q^{34} +11.0316 q^{35} +2.77974 q^{36} +7.43590 q^{37} -9.75270 q^{38} -11.4735 q^{39} -6.82313 q^{40} +4.59774 q^{41} -22.5353 q^{42} +4.59971 q^{43} -3.63281 q^{44} +1.84546 q^{45} -2.30965 q^{46} -7.01645 q^{47} +0.880643 q^{48} +17.8328 q^{49} +0.229450 q^{50} -6.58383 q^{51} -19.5397 q^{52} +0.967875 q^{53} +9.79676 q^{54} -2.41180 q^{55} -15.3592 q^{56} +8.26770 q^{57} -18.7156 q^{58} +3.35030 q^{59} +14.4531 q^{60} -4.37700 q^{61} +9.22496 q^{62} +4.15422 q^{63} -12.7377 q^{64} -12.9723 q^{65} +4.92682 q^{66} +2.31509 q^{67} -11.2125 q^{68} +1.95797 q^{69} -25.4792 q^{70} +8.97742 q^{71} -2.56941 q^{72} +12.7859 q^{73} -17.1743 q^{74} -0.194513 q^{75} +14.0801 q^{76} -5.42910 q^{77} +26.4998 q^{78} -5.32993 q^{79} +0.995684 q^{80} -10.8060 q^{81} -10.6192 q^{82} -2.60638 q^{83} +32.5346 q^{84} -7.44389 q^{85} -10.6237 q^{86} +15.8659 q^{87} +3.35793 q^{88} -9.08077 q^{89} -4.26236 q^{90} -29.2014 q^{91} +3.33448 q^{92} -7.82032 q^{93} +16.2055 q^{94} +9.34773 q^{95} +10.0356 q^{96} +2.64450 q^{97} -41.1874 q^{98} -0.908223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

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