LMFDB - Embedded newform 8047.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, 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("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.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:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$168$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76928 q^{2} -1.56618 q^{3} +5.66891 q^{4} -1.99910 q^{5} +4.33718 q^{6} -3.75052 q^{7} -10.1602 q^{8} -0.547092 q^{9} +O(q^{10})$	\(q-2.76928 q^{2} -1.56618 q^{3} +5.66891 q^{4} -1.99910 q^{5} +4.33718 q^{6} -3.75052 q^{7} -10.1602 q^{8} -0.547092 q^{9} +5.53606 q^{10} -3.03819 q^{11} -8.87851 q^{12} +1.00000 q^{13} +10.3862 q^{14} +3.13094 q^{15} +16.7987 q^{16} +5.26179 q^{17} +1.51505 q^{18} -6.05437 q^{19} -11.3327 q^{20} +5.87397 q^{21} +8.41360 q^{22} +0.405316 q^{23} +15.9127 q^{24} -1.00361 q^{25} -2.76928 q^{26} +5.55537 q^{27} -21.2613 q^{28} +5.47504 q^{29} -8.67045 q^{30} -0.333037 q^{31} -26.1998 q^{32} +4.75834 q^{33} -14.5714 q^{34} +7.49766 q^{35} -3.10142 q^{36} +6.82980 q^{37} +16.7662 q^{38} -1.56618 q^{39} +20.3113 q^{40} -6.31197 q^{41} -16.2667 q^{42} -1.69926 q^{43} -17.2232 q^{44} +1.09369 q^{45} -1.12243 q^{46} +13.4745 q^{47} -26.3097 q^{48} +7.06640 q^{49} +2.77927 q^{50} -8.24088 q^{51} +5.66891 q^{52} +0.259255 q^{53} -15.3844 q^{54} +6.07364 q^{55} +38.1061 q^{56} +9.48221 q^{57} -15.1619 q^{58} -0.263315 q^{59} +17.7490 q^{60} -4.43926 q^{61} +0.922272 q^{62} +2.05188 q^{63} +38.9572 q^{64} -1.99910 q^{65} -13.1772 q^{66} -8.60130 q^{67} +29.8286 q^{68} -0.634797 q^{69} -20.7631 q^{70} +15.4790 q^{71} +5.55858 q^{72} -15.3658 q^{73} -18.9136 q^{74} +1.57182 q^{75} -34.3217 q^{76} +11.3948 q^{77} +4.33718 q^{78} -16.3183 q^{79} -33.5822 q^{80} -7.05941 q^{81} +17.4796 q^{82} -14.9428 q^{83} +33.2990 q^{84} -10.5188 q^{85} +4.70571 q^{86} -8.57487 q^{87} +30.8687 q^{88} -4.54265 q^{89} -3.02874 q^{90} -3.75052 q^{91} +2.29770 q^{92} +0.521594 q^{93} -37.3146 q^{94} +12.1033 q^{95} +41.0335 q^{96} -11.1332 q^{97} -19.5688 q^{98} +1.66217 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) $ 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.76928	−1.95818	−0.979088	−	0.203437i	$-0.934789\pi$
$2$	−2.76928	−1.95818	−0.979088	+	0.203437i	$0.934789\pi$
$3$	−1.56618	−0.904232	−0.452116	−	0.891959i	$-0.649331\pi$
$3$	−1.56618	−0.904232	−0.452116	+	0.891959i	$0.649331\pi$
$4$	5.66891	2.83445
$5$	−1.99910	−0.894024	−0.447012	−	0.894528i	$-0.647512\pi$
$5$	−1.99910	−0.894024	−0.447012	+	0.894528i	$0.647512\pi$
$6$	4.33718	1.77065
$7$	−3.75052	−1.41756	−0.708782	−	0.705428i	$-0.750755\pi$
$7$	−3.75052	−1.41756	−0.708782	+	0.705428i	$0.750755\pi$
$8$	−10.1602	−3.59218
$9$	−0.547092	−0.182364
$10$	5.53606	1.75066
$11$	−3.03819	−0.916049	−0.458025	−	0.888939i	$-0.651443\pi$
$11$	−3.03819	−0.916049	−0.458025	+	0.888939i	$0.651443\pi$
$12$	−8.87851	−2.56300
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	10.3862	2.77584
$15$	3.13094	0.808405
$16$	16.7987	4.19967
$17$	5.26179	1.27617	0.638085	−	0.769966i	$-0.279727\pi$
$17$	5.26179	1.27617	0.638085	+	0.769966i	$0.279727\pi$
$18$	1.51505	0.357101
$19$	−6.05437	−1.38897	−0.694484	−	0.719508i	$-0.744367\pi$
$19$	−6.05437	−1.38897	−0.694484	+	0.719508i	$0.744367\pi$
$20$	−11.3327	−2.53407
$21$	5.87397	1.28181
$22$	8.41360	1.79379
$23$	0.405316	0.0845143	0.0422571	−	0.999107i	$-0.486545\pi$
$23$	0.405316	0.0845143	0.0422571	+	0.999107i	$0.486545\pi$
$24$	15.9127	3.24817
$25$	−1.00361	−0.200721
$26$	−2.76928	−0.543100
$27$	5.55537	1.06913
$28$	−21.2613	−4.01802
$29$	5.47504	1.01669	0.508344	−	0.861154i	$-0.330258\pi$
$29$	5.47504	1.01669	0.508344	+	0.861154i	$0.330258\pi$
$30$	−8.67045	−1.58300
$31$	−0.333037	−0.0598152	−0.0299076	−	0.999553i	$-0.509521\pi$
$31$	−0.333037	−0.0598152	−0.0299076	+	0.999553i	$0.509521\pi$
$32$	−26.1998	−4.63152
$33$	4.75834	0.828321
$34$	−14.5714	−2.49897
$35$	7.49766	1.26734
$36$	−3.10142	−0.516903
$37$	6.82980	1.12281	0.561406	−	0.827540i	$-0.310260\pi$
$37$	6.82980	1.12281	0.561406	+	0.827540i	$0.310260\pi$
$38$	16.7662	2.71984
$39$	−1.56618	−0.250789
$40$	20.3113	3.21150
$41$	−6.31197	−0.985765	−0.492882	−	0.870096i	$-0.664057\pi$
$41$	−6.31197	−0.985765	−0.492882	+	0.870096i	$0.664057\pi$
$42$	−16.2667	−2.51000
$43$	−1.69926	−0.259134	−0.129567	−	0.991571i	$-0.541359\pi$
$43$	−1.69926	−0.259134	−0.129567	+	0.991571i	$0.541359\pi$
$44$	−17.2232	−2.59650
$45$	1.09369	0.163038
$46$	−1.12243	−0.165494
$47$	13.4745	1.96546	0.982728	−	0.185054i	$-0.0592460\pi$
$47$	13.4745	1.96546	0.982728	+	0.185054i	$0.0592460\pi$
$48$	−26.3097	−3.79748
$49$	7.06640	1.00949
$50$	2.77927	0.393047
$51$	−8.24088	−1.15395
$52$	5.66891	0.786136
$53$	0.259255	0.0356115	0.0178057	−	0.999841i	$-0.494332\pi$
$53$	0.259255	0.0356115	0.0178057	+	0.999841i	$0.494332\pi$
$54$	−15.3844	−2.09355
$55$	6.07364	0.818970
$56$	38.1061	5.09215
$57$	9.48221	1.25595
$58$	−15.1619	−1.99086
$59$	−0.263315	−0.0342807	−0.0171404	−	0.999853i	$-0.505456\pi$
$59$	−0.263315	−0.0342807	−0.0171404	+	0.999853i	$0.505456\pi$
$60$	17.7490	2.29139
$61$	−4.43926	−0.568390	−0.284195	−	0.958767i	$-0.591726\pi$
$61$	−4.43926	−0.568390	−0.284195	+	0.958767i	$0.591726\pi$
$62$	0.922272	0.117129
$63$	2.05188	0.258513
$64$	38.9572	4.86965
$65$	−1.99910	−0.247958
$66$	−13.1772	−1.62200
$67$	−8.60130	−1.05082	−0.525408	−	0.850851i	$-0.676087\pi$
$67$	−8.60130	−1.05082	−0.525408	+	0.850851i	$0.676087\pi$
$68$	29.8286	3.61725
$69$	−0.634797	−0.0764205
$70$	−20.7631	−2.48167
$71$	15.4790	1.83701	0.918507	−	0.395404i	$-0.129395\pi$
$71$	15.4790	1.83701	0.918507	+	0.395404i	$0.129395\pi$
$72$	5.55858	0.655085
$73$	−15.3658	−1.79843	−0.899213	−	0.437510i	$-0.855860\pi$
$73$	−15.3658	−1.79843	−0.899213	+	0.437510i	$0.855860\pi$
$74$	−18.9136	−2.19867
$75$	1.57182	0.181499
$76$	−34.3217	−3.93697
$77$	11.3948	1.29856
$78$	4.33718	0.491089
$79$	−16.3183	−1.83596	−0.917978	−	0.396631i	$-0.870179\pi$
$79$	−16.3183	−1.83596	−0.917978	+	0.396631i	$0.870179\pi$
$80$	−33.5822	−3.75461
$81$	−7.05941	−0.784379
$82$	17.4796	1.93030
$83$	−14.9428	−1.64019	−0.820095	−	0.572227i	$-0.806080\pi$
$83$	−14.9428	−1.64019	−0.820095	+	0.572227i	$0.806080\pi$
$84$	33.2990	3.63322
$85$	−10.5188	−1.14093
$86$	4.70571	0.507430
$87$	−8.57487	−0.919323
$88$	30.8687	3.29062
$89$	−4.54265	−0.481520	−0.240760	−	0.970585i	$-0.577397\pi$
$89$	−4.54265	−0.481520	−0.240760	+	0.970585i	$0.577397\pi$
$90$	−3.02874	−0.319257
$91$	−3.75052	−0.393161
$92$	2.29770	0.239552
$93$	0.521594	0.0540868
$94$	−37.3146	−3.84871
$95$	12.1033	1.24177
$96$	41.0335	4.18797
$97$	−11.1332	−1.13040	−0.565202	−	0.824952i	$-0.691202\pi$
$97$	−11.1332	−1.13040	−0.565202	+	0.824952i	$0.691202\pi$
$98$	−19.5688	−1.97675
$99$	1.66217	0.167054
$100$	−5.68935	−0.568935
$101$	−7.31171	−0.727543	−0.363771	−	0.931488i	$-0.618511\pi$
$101$	−7.31171	−0.727543	−0.363771	+	0.931488i	$0.618511\pi$
$102$	22.8213	2.25965
$103$	2.41846	0.238298	0.119149	−	0.992876i	$-0.461983\pi$
$103$	2.41846	0.238298	0.119149	+	0.992876i	$0.461983\pi$
$104$	−10.1602	−0.996292
$105$	−11.7427	−1.14597
$106$	−0.717951	−0.0697335
$107$	−12.8883	−1.24596	−0.622981	−	0.782237i	$-0.714079\pi$
$107$	−12.8883	−1.24596	−0.622981	+	0.782237i	$0.714079\pi$
$108$	31.4929	3.03040
$109$	−7.79156	−0.746296	−0.373148	−	0.927772i	$-0.621722\pi$
$109$	−7.79156	−0.746296	−0.373148	+	0.927772i	$0.621722\pi$
$110$	−16.8196	−1.60369
$111$	−10.6967	−1.01528
$112$	−63.0038	−5.95330
$113$	−8.57832	−0.806981	−0.403490	−	0.914984i	$-0.632203\pi$
$113$	−8.57832	−0.806981	−0.403490	+	0.914984i	$0.632203\pi$
$114$	−26.2589	−2.45937
$115$	−0.810267	−0.0755578
$116$	31.0375	2.88176
$117$	−0.547092	−0.0505787
$118$	0.729194	0.0671277
$119$	−19.7344	−1.80905
$120$	−31.8111	−2.90394
$121$	−1.76939	−0.160854
$122$	12.2936	1.11301
$123$	9.88566	0.891360
$124$	−1.88796	−0.169543
$125$	12.0018	1.07347
$126$	−5.68223	−0.506213
$127$	−12.8552	−1.14072	−0.570359	−	0.821396i	$-0.693196\pi$
$127$	−12.8552	−1.14072	−0.570359	+	0.821396i	$0.693196\pi$
$128$	−55.4838	−4.90412
$129$	2.66133	0.234317
$130$	5.53606	0.485545
$131$	−6.16123	−0.538309	−0.269154	−	0.963097i	$-0.586744\pi$
$131$	−6.16123	−0.538309	−0.269154	+	0.963097i	$0.586744\pi$
$132$	26.9746	2.34784
$133$	22.7070	1.96895
$134$	23.8194	2.05768
$135$	−11.1057	−0.955829
$136$	−53.4610	−4.58424
$137$	8.72077	0.745066	0.372533	−	0.928019i	$-0.378489\pi$
$137$	8.72077	0.745066	0.372533	+	0.928019i	$0.378489\pi$
$138$	1.75793	0.149645
$139$	−4.12653	−0.350007	−0.175004	−	0.984568i	$-0.555994\pi$
$139$	−4.12653	−0.350007	−0.175004	+	0.984568i	$0.555994\pi$
$140$	42.5035	3.59220
$141$	−21.1034	−1.77723
$142$	−42.8656	−3.59720
$143$	−3.03819	−0.254066
$144$	−9.19044	−0.765870
$145$	−10.9451	−0.908944
$146$	42.5521	3.52164
$147$	−11.0672	−0.912809
$148$	38.7175	3.18256
$149$	−8.94220	−0.732573	−0.366287	−	0.930502i	$-0.619371\pi$
$149$	−8.94220	−0.732573	−0.366287	+	0.930502i	$0.619371\pi$
$150$	−4.35282	−0.355406
$151$	−12.3225	−1.00279	−0.501396	−	0.865218i	$-0.667180\pi$
$151$	−12.3225	−1.00279	−0.501396	+	0.865218i	$0.667180\pi$
$152$	61.5138	4.98943
$153$	−2.87868	−0.232728
$154$	−31.5554	−2.54280
$155$	0.665773	0.0534762
$156$	−8.87851	−0.710849
$157$	20.9076	1.66861	0.834306	−	0.551302i	$-0.185869\pi$
$157$	20.9076	1.66861	0.834306	+	0.551302i	$0.185869\pi$
$158$	45.1900	3.59513
$159$	−0.406040	−0.0322010
$160$	52.3760	4.14069
$161$	−1.52015	−0.119804
$162$	19.5495	1.53595
$163$	25.4679	1.99480	0.997400	−	0.0720651i	$-0.0229589\pi$
$163$	25.4679	1.99480	0.997400	+	0.0720651i	$0.0229589\pi$
$164$	−35.7820	−2.79410
$165$	−9.51240	−0.740539
$166$	41.3809	3.21178
$167$	10.0612	0.778557	0.389278	−	0.921120i	$-0.372724\pi$
$167$	10.0612	0.778557	0.389278	+	0.921120i	$0.372724\pi$
$168$	−59.6809	−4.60448
$169$	1.00000	0.0769231
$170$	29.1296	2.23414
$171$	3.31230	0.253298
$172$	−9.63292	−0.734503
$173$	23.0863	1.75522	0.877611	−	0.479374i	$-0.159136\pi$
$173$	23.0863	1.75522	0.877611	+	0.479374i	$0.159136\pi$
$174$	23.7462	1.80020
$175$	3.76404	0.284535
$176$	−51.0377	−3.84711
$177$	0.412398	0.0309977
$178$	12.5799	0.942900
$179$	−18.7458	−1.40113	−0.700564	−	0.713589i	$-0.747068\pi$
$179$	−18.7458	−1.40113	−0.700564	+	0.713589i	$0.747068\pi$
$180$	6.20003	0.462123
$181$	−13.1006	−0.973760	−0.486880	−	0.873469i	$-0.661865\pi$
$181$	−13.1006	−0.973760	−0.486880	+	0.873469i	$0.661865\pi$
$182$	10.3862	0.769879
$183$	6.95267	0.513956
$184$	−4.11811	−0.303591
$185$	−13.6535	−1.00382
$186$	−1.44444	−0.105912
$187$	−15.9863	−1.16904
$188$	76.3856	5.57100
$189$	−20.8355	−1.51556
$190$	−33.5174	−2.43161
$191$	−7.25376	−0.524864	−0.262432	−	0.964951i	$-0.584525\pi$
$191$	−7.25376	−0.524864	−0.262432	+	0.964951i	$0.584525\pi$
$192$	−61.0139	−4.40330
$193$	16.8021	1.20944	0.604719	−	0.796439i	$-0.293285\pi$
$193$	16.8021	1.20944	0.604719	+	0.796439i	$0.293285\pi$
$194$	30.8309	2.21353
$195$	3.13094	0.224211
$196$	40.0588	2.86134
$197$	−19.3495	−1.37859	−0.689297	−	0.724479i	$-0.742080\pi$
$197$	−19.3495	−1.37859	−0.689297	+	0.724479i	$0.742080\pi$
$198$	−4.60302	−0.327122
$199$	−15.9174	−1.12836	−0.564178	−	0.825653i	$-0.690807\pi$
$199$	−15.9174	−1.12836	−0.564178	+	0.825653i	$0.690807\pi$
$200$	10.1969	0.721027
$201$	13.4711	0.950181
$202$	20.2482	1.42466
$203$	−20.5342	−1.44122
$204$	−46.7168	−3.27083
$205$	12.6183	0.881297
$206$	−6.69740	−0.466630
$207$	−0.221745	−0.0154124
$208$	16.7987	1.16478
$209$	18.3943	1.27236
$210$	32.5187	2.24400
$211$	−5.62191	−0.387028	−0.193514	−	0.981098i	$-0.561988\pi$
$211$	−5.62191	−0.387028	−0.193514	+	0.981098i	$0.561988\pi$
$212$	1.46969	0.100939
$213$	−24.2428	−1.66109
$214$	35.6914	2.43981
$215$	3.39698	0.231672
$216$	−56.4438	−3.84052
$217$	1.24906	0.0847918
$218$	21.5770	1.46138
$219$	24.0655	1.62620
$220$	34.4309	2.32133
$221$	5.26179	0.353946
$222$	29.6221	1.98810
$223$	2.09184	0.140080	0.0700399	−	0.997544i	$-0.477687\pi$
$223$	2.09184	0.140080	0.0700399	+	0.997544i	$0.477687\pi$
$224$	98.2629	6.56547
$225$	0.549065	0.0366043
$226$	23.7558	1.58021
$227$	−10.2502	−0.680328	−0.340164	−	0.940366i	$-0.610483\pi$
$227$	−10.2502	−0.680328	−0.340164	+	0.940366i	$0.610483\pi$
$228$	53.7538	3.55993
$229$	−28.8449	−1.90612	−0.953062	−	0.302776i	$-0.902087\pi$
$229$	−28.8449	−1.90612	−0.953062	+	0.302776i	$0.902087\pi$
$230$	2.24386	0.147955
$231$	−17.8463	−1.17420
$232$	−55.6276	−3.65213
$233$	18.7155	1.22609	0.613047	−	0.790047i	$-0.289944\pi$
$233$	18.7155	1.22609	0.613047	+	0.790047i	$0.289944\pi$
$234$	1.51505	0.0990420
$235$	−26.9368	−1.75717
$236$	−1.49271	−0.0971671
$237$	25.5574	1.66013
$238$	54.6502	3.54244
$239$	3.50725	0.226865	0.113433	−	0.993546i	$-0.463815\pi$
$239$	3.50725	0.226865	0.113433	+	0.993546i	$0.463815\pi$
$240$	52.5957	3.39504
$241$	−21.5021	−1.38507	−0.692534	−	0.721385i	$-0.743506\pi$
$241$	−21.5021	−1.38507	−0.692534	+	0.721385i	$0.743506\pi$
$242$	4.89993	0.314980
$243$	−5.60983	−0.359871
$244$	−25.1658	−1.61107
$245$	−14.1264	−0.902504
$246$	−27.3762	−1.74544
$247$	−6.05437	−0.385230
$248$	3.38373	0.214867
$249$	23.4031	1.48311
$250$	−33.2363	−2.10205
$251$	−15.5937	−0.984266	−0.492133	−	0.870520i	$-0.663783\pi$
$251$	−15.5937	−0.984266	−0.492133	+	0.870520i	$0.663783\pi$
$252$	11.6319	0.732742
$253$	−1.23143	−0.0774192
$254$	35.5997	2.23373
$255$	16.4743	1.03166
$256$	75.7356	4.73348
$257$	14.5758	0.909216	0.454608	−	0.890692i	$-0.349779\pi$
$257$	14.5758	0.909216	0.454608	+	0.890692i	$0.349779\pi$
$258$	−7.36998	−0.458835
$259$	−25.6153	−1.59166
$260$	−11.3327	−0.702824
$261$	−2.99535	−0.185408
$262$	17.0622	1.05410
$263$	0.583150	0.0359586	0.0179793	−	0.999838i	$-0.494277\pi$
$263$	0.583150	0.0359586	0.0179793	+	0.999838i	$0.494277\pi$
$264$	−48.3459	−2.97548
$265$	−0.518277	−0.0318375
$266$	−62.8821	−3.85555
$267$	7.11459	0.435406
$268$	−48.7600	−2.97849
$269$	28.7749	1.75444	0.877218	−	0.480092i	$-0.159397\pi$
$269$	28.7749	1.75444	0.877218	+	0.480092i	$0.159397\pi$
$270$	30.7549	1.87168
$271$	23.5735	1.43199	0.715993	−	0.698107i	$-0.245974\pi$
$271$	23.5735	1.43199	0.715993	+	0.698107i	$0.245974\pi$
$272$	88.3911	5.35950
$273$	5.87397	0.355509
$274$	−24.1503	−1.45897
$275$	3.04915	0.183871
$276$	−3.59860	−0.216610
$277$	−8.50281	−0.510884	−0.255442	−	0.966824i	$-0.582221\pi$
$277$	−8.50281	−0.510884	−0.255442	+	0.966824i	$0.582221\pi$
$278$	11.4275	0.685376
$279$	0.182202	0.0109081
$280$	−76.1779	−4.55250
$281$	15.4924	0.924197	0.462098	−	0.886829i	$-0.347097\pi$
$281$	15.4924	0.924197	0.462098	+	0.886829i	$0.347097\pi$
$282$	58.4413	3.48013
$283$	−7.59842	−0.451679	−0.225839	−	0.974165i	$-0.572512\pi$
$283$	−7.59842	−0.451679	−0.225839	+	0.974165i	$0.572512\pi$
$284$	87.7488	5.20693
$285$	−18.9559	−1.12285
$286$	8.41360	0.497507
$287$	23.6732	1.39738
$288$	14.3337	0.844622
$289$	10.6864	0.628612
$290$	30.3101	1.77987
$291$	17.4365	1.02215
$292$	−87.1071	−5.09756
$293$	−10.7759	−0.629534	−0.314767	−	0.949169i	$-0.601926\pi$
$293$	−10.7759	−0.629534	−0.314767	+	0.949169i	$0.601926\pi$
$294$	30.6482	1.78744
$295$	0.526393	0.0306478
$296$	−69.3924	−4.03335
$297$	−16.8783	−0.979377
$298$	24.7634	1.43451
$299$	0.405316	0.0234400
$300$	8.91052	0.514449
$301$	6.37309	0.367339
$302$	34.1244	1.96364
$303$	11.4514	0.657868
$304$	−101.706	−5.83321
$305$	8.87452	0.508154
$306$	7.97188	0.455722
$307$	−15.4845	−0.883746	−0.441873	−	0.897078i	$-0.645686\pi$
$307$	−15.4845	−0.883746	−0.441873	+	0.897078i	$0.645686\pi$
$308$	64.5961	3.68070
$309$	−3.78774	−0.215477
$310$	−1.84371	−0.104716
$311$	−23.3635	−1.32482	−0.662411	−	0.749140i	$-0.730467\pi$
$311$	−23.3635	−1.32482	−0.662411	+	0.749140i	$0.730467\pi$
$312$	15.9127	0.900880
$313$	−3.26197	−0.184378	−0.0921888	−	0.995742i	$-0.529386\pi$
$313$	−3.26197	−0.184378	−0.0921888	+	0.995742i	$0.529386\pi$
$314$	−57.8991	−3.26744
$315$	−4.10191	−0.231116
$316$	−92.5071	−5.20393
$317$	−4.72783	−0.265541	−0.132771	−	0.991147i	$-0.542387\pi$
$317$	−4.72783	−0.265541	−0.132771	+	0.991147i	$0.542387\pi$
$318$	1.12444	0.0630553
$319$	−16.6342	−0.931337
$320$	−77.8793	−4.35359
$321$	20.1854	1.12664
$322$	4.20971	0.234598
$323$	−31.8568	−1.77256
$324$	−40.0192	−2.22329
$325$	−1.00361	−0.0556700
$326$	−70.5277	−3.90617
$327$	12.2030	0.674825
$328$	64.1311	3.54105
$329$	−50.5364	−2.78616
$330$	26.3425	1.45011
$331$	29.0431	1.59635	0.798177	−	0.602424i	$-0.205798\pi$
$331$	29.0431	1.59635	0.798177	+	0.602424i	$0.205798\pi$
$332$	−84.7096	−4.64904
$333$	−3.73653	−0.204761
$334$	−27.8622	−1.52455
$335$	17.1948	0.939454
$336$	98.6751	5.38317
$337$	3.94939	0.215137	0.107568	−	0.994198i	$-0.465694\pi$
$337$	3.94939	0.215137	0.107568	+	0.994198i	$0.465694\pi$
$338$	−2.76928	−0.150629
$339$	13.4352	0.729698
$340$	−59.6303	−3.23390
$341$	1.01183	0.0547937
$342$	−9.17268	−0.496002
$343$	−0.249031	−0.0134464
$344$	17.2648	0.930857
$345$	1.26902	0.0683218
$346$	−63.9325	−3.43703
$347$	−22.9206	−1.23044	−0.615221	−	0.788355i	$-0.710933\pi$
$347$	−22.9206	−1.23044	−0.615221	+	0.788355i	$0.710933\pi$
$348$	−48.6102	−2.60578
$349$	20.3803	1.09093	0.545466	−	0.838133i	$-0.316353\pi$
$349$	20.3803	1.09093	0.545466	+	0.838133i	$0.316353\pi$
$350$	−10.4237	−0.557170
$351$	5.55537	0.296524
$352$	79.6001	4.24270
$353$	−14.5220	−0.772927	−0.386463	−	0.922305i	$-0.626303\pi$
$353$	−14.5220	−0.772927	−0.386463	+	0.922305i	$0.626303\pi$
$354$	−1.14205	−0.0606990
$355$	−30.9440	−1.64234
$356$	−25.7519	−1.36485
$357$	30.9076	1.63580
$358$	51.9124	2.74366
$359$	−22.3070	−1.17732	−0.588660	−	0.808380i	$-0.700344\pi$
$359$	−22.3070	−1.17732	−0.588660	+	0.808380i	$0.700344\pi$
$360$	−11.1122	−0.585662
$361$	17.6554	0.929233
$362$	36.2792	1.90679
$363$	2.77118	0.145449
$364$	−21.2613	−1.11440
$365$	30.7177	1.60784
$366$	−19.2539	−1.00642
$367$	−0.565156	−0.0295009	−0.0147504	−	0.999891i	$-0.504695\pi$
$367$	−0.565156	−0.0295009	−0.0147504	+	0.999891i	$0.504695\pi$
$368$	6.80878	0.354932
$369$	3.45323	0.179768
$370$	37.8102	1.96566
$371$	−0.972343	−0.0504815
$372$	2.95687	0.153307
$373$	−4.04459	−0.209421	−0.104710	−	0.994503i	$-0.533392\pi$
$373$	−4.04459	−0.209421	−0.104710	+	0.994503i	$0.533392\pi$
$374$	44.2706	2.28918
$375$	−18.7969	−0.970669
$376$	−136.904	−7.06028
$377$	5.47504	0.281979
$378$	57.6994	2.96774
$379$	−17.4894	−0.898368	−0.449184	−	0.893439i	$-0.648285\pi$
$379$	−17.4894	−0.898368	−0.449184	+	0.893439i	$0.648285\pi$
$380$	68.6124	3.51974
$381$	20.1336	1.03147
$382$	20.0877	1.02778
$383$	17.3032	0.884152	0.442076	−	0.896978i	$-0.354242\pi$
$383$	17.3032	0.884152	0.442076	+	0.896978i	$0.354242\pi$
$384$	86.8974	4.43446
$385$	−22.7793	−1.16094
$386$	−46.5296	−2.36829
$387$	0.929649	0.0472567
$388$	−63.1130	−3.20408
$389$	−2.10430	−0.106692	−0.0533461	−	0.998576i	$-0.516989\pi$
$389$	−2.10430	−0.106692	−0.0533461	+	0.998576i	$0.516989\pi$
$390$	−8.67045	−0.439045
$391$	2.13269	0.107855
$392$	−71.7962	−3.62626
$393$	9.64957	0.486756
$394$	53.5841	2.69953
$395$	32.6220	1.64139
$396$	9.42269	0.473508
$397$	32.4235	1.62729	0.813645	−	0.581362i	$-0.197480\pi$
$397$	32.4235	1.62729	0.813645	+	0.581362i	$0.197480\pi$
$398$	44.0798	2.20952
$399$	−35.5632	−1.78039
$400$	−16.8593	−0.842964
$401$	38.2455	1.90989	0.954946	−	0.296781i	$-0.0959131\pi$
$401$	38.2455	1.90989	0.954946	+	0.296781i	$0.0959131\pi$
$402$	−37.3054	−1.86062
$403$	−0.333037	−0.0165897
$404$	−41.4494	−2.06219
$405$	14.1125	0.701254
$406$	56.8650	2.82216
$407$	−20.7503	−1.02855
$408$	83.7293	4.14522
$409$	−6.00486	−0.296921	−0.148461	−	0.988918i	$-0.547432\pi$
$409$	−6.00486	−0.296921	−0.148461	+	0.988918i	$0.547432\pi$
$410$	−34.9435	−1.72573
$411$	−13.6583	−0.673713
$412$	13.7100	0.675445
$413$	0.987569	0.0485951
$414$	0.614075	0.0301801
$415$	29.8722	1.46637
$416$	−26.1998	−1.28455
$417$	6.46287	0.316488
$418$	−50.9391	−2.49151
$419$	−12.7214	−0.621482	−0.310741	−	0.950495i	$-0.600577\pi$
$419$	−12.7214	−0.621482	−0.310741	+	0.950495i	$0.600577\pi$
$420$	−66.5680	−3.24819
$421$	27.1149	1.32150	0.660749	−	0.750607i	$-0.270239\pi$
$421$	27.1149	1.32150	0.660749	+	0.750607i	$0.270239\pi$
$422$	15.5686	0.757869
$423$	−7.37179	−0.358429
$424$	−2.63409	−0.127923
$425$	−5.28076	−0.256155
$426$	67.1350	3.25270
$427$	16.6495	0.805728
$428$	−73.0628	−3.53162
$429$	4.75834	0.229735
$430$	−9.40718	−0.453655
$431$	−16.7185	−0.805303	−0.402652	−	0.915353i	$-0.631911\pi$
$431$	−16.7185	−0.805303	−0.402652	+	0.915353i	$0.631911\pi$
$432$	93.3230	4.49000
$433$	−19.6959	−0.946526	−0.473263	−	0.880921i	$-0.656924\pi$
$433$	−19.6959	−0.946526	−0.473263	+	0.880921i	$0.656924\pi$
$434$	−3.45900	−0.166037
$435$	17.1420	0.821897
$436$	−44.1696	−2.11534
$437$	−2.45394	−0.117388
$438$	−66.6441	−3.18438
$439$	4.45477	0.212614	0.106307	−	0.994333i	$-0.466097\pi$
$439$	4.45477	0.212614	0.106307	+	0.994333i	$0.466097\pi$
$440$	−61.7096	−2.94189
$441$	−3.86597	−0.184094
$442$	−14.5714	−0.693089
$443$	−32.8694	−1.56167	−0.780836	−	0.624737i	$-0.785206\pi$
$443$	−32.8694	−1.56167	−0.780836	+	0.624737i	$0.785206\pi$
$444$	−60.6385	−2.87777
$445$	9.08120	0.430490
$446$	−5.79288	−0.274301
$447$	14.0051	0.662417
$448$	−146.110	−6.90304
$449$	−7.69062	−0.362943	−0.181471	−	0.983396i	$-0.558086\pi$
$449$	−7.69062	−0.362943	−0.181471	+	0.983396i	$0.558086\pi$
$450$	−1.52051	−0.0716777
$451$	19.1770	0.903009
$452$	−48.6297	−2.28735
$453$	19.2992	0.906756
$454$	28.3856	1.33220
$455$	7.49766	0.351496
$456$	−96.3415	−4.51160
$457$	−36.9144	−1.72678	−0.863392	−	0.504534i	$-0.831664\pi$
$457$	−36.9144	−1.72678	−0.863392	+	0.504534i	$0.831664\pi$
$458$	79.8795	3.73253
$459$	29.2312	1.36439
$460$	−4.59333	−0.214165
$461$	−20.6606	−0.962262	−0.481131	−	0.876649i	$-0.659774\pi$
$461$	−20.6606	−0.962262	−0.481131	+	0.876649i	$0.659774\pi$
$462$	49.4213	2.29929
$463$	−29.1663	−1.35547	−0.677736	−	0.735305i	$-0.737039\pi$
$463$	−29.1663	−1.35547	−0.677736	+	0.735305i	$0.737039\pi$
$464$	91.9735	4.26976
$465$	−1.04272	−0.0483549
$466$	−51.8285	−2.40091
$467$	−1.68816	−0.0781188	−0.0390594	−	0.999237i	$-0.512436\pi$
$467$	−1.68816	−0.0781188	−0.0390594	+	0.999237i	$0.512436\pi$
$468$	−3.10142	−0.143363
$469$	32.2593	1.48960
$470$	74.5956	3.44084
$471$	−32.7451	−1.50881
$472$	2.67534	0.123143
$473$	5.16266	0.237380
$474$	−70.7756	−3.25083
$475$	6.07620	0.278795
$476$	−111.873	−5.12768
$477$	−0.141837	−0.00649425
$478$	−9.71256	−0.444242
$479$	−0.479364	−0.0219027	−0.0109514	−	0.999940i	$-0.503486\pi$
$479$	−0.479364	−0.0219027	−0.0109514	+	0.999940i	$0.503486\pi$
$480$	−82.0301	−3.74414
$481$	6.82980	0.311412
$482$	59.5452	2.71221
$483$	2.38082	0.108331
$484$	−10.0305	−0.455932
$485$	22.2563	1.01061
$486$	15.5352	0.704690
$487$	−28.2587	−1.28052	−0.640261	−	0.768157i	$-0.721174\pi$
$487$	−28.2587	−1.28052	−0.640261	+	0.768157i	$0.721174\pi$
$488$	45.1039	2.04176
$489$	−39.8872	−1.80376
$490$	39.1200	1.76726
$491$	11.6329	0.524983	0.262492	−	0.964934i	$-0.415456\pi$
$491$	11.6329	0.524983	0.262492	+	0.964934i	$0.415456\pi$
$492$	56.0409	2.52652
$493$	28.8085	1.29747
$494$	16.7662	0.754349
$495$	−3.32284	−0.149351
$496$	−5.59458	−0.251204
$497$	−58.0542	−2.60408
$498$	−64.8098	−2.90420
$499$	−28.8486	−1.29144	−0.645720	−	0.763575i	$-0.723442\pi$
$499$	−28.8486	−1.29144	−0.645720	+	0.763575i	$0.723442\pi$
$500$	68.0371	3.04271
$501$	−15.7576	−0.703996
$502$	43.1833	1.92737
$503$	2.76865	0.123448	0.0617240	−	0.998093i	$-0.480340\pi$
$503$	2.76865	0.123448	0.0617240	+	0.998093i	$0.480340\pi$
$504$	−20.8476	−0.928625
$505$	14.6168	0.650441
$506$	3.41017	0.151601
$507$	−1.56618	−0.0695563
$508$	−72.8751	−3.23331
$509$	23.6829	1.04973	0.524863	−	0.851187i	$-0.324117\pi$
$509$	23.6829	1.04973	0.524863	+	0.851187i	$0.324117\pi$
$510$	−45.6220	−2.02018
$511$	57.6296	2.54938
$512$	−98.7656	−4.36486
$513$	−33.6343	−1.48499
$514$	−40.3646	−1.78041
$515$	−4.83474	−0.213044
$516$	15.0869	0.664162
$517$	−40.9381	−1.80046
$518$	70.9360	3.11675
$519$	−36.1573	−1.58713
$520$	20.3113	0.890709
$521$	0.00484218	0.000212140 0	0.000106070	−	1.00000i	$-0.499966\pi$
$521$	0.00484218	0.000212140 0	0.000106070		1.00000i	$0.499966\pi$
$522$	8.29496	0.363061
$523$	7.13921	0.312176	0.156088	−	0.987743i	$-0.450112\pi$
$523$	7.13921	0.312176	0.156088	+	0.987743i	$0.450112\pi$
$524$	−34.9274	−1.52581
$525$	−5.89516	−0.257286
$526$	−1.61491	−0.0704132
$527$	−1.75237	−0.0763344
$528$	79.9340	3.47868
$529$	−22.8357	−0.992857
$530$	1.43525	0.0623434
$531$	0.144058	0.00625157
$532$	128.724	5.58090
$533$	−6.31197	−0.273402
$534$	−19.7023	−0.852601
$535$	25.7651	1.11392
$536$	87.3911	3.77472
$537$	29.3593	1.26695
$538$	−79.6857	−3.43549
$539$	−21.4691	−0.924739
$540$	−62.9574	−2.70925
$541$	16.9613	0.729221	0.364611	−	0.931160i	$-0.381202\pi$
$541$	16.9613	0.729221	0.364611	+	0.931160i	$0.381202\pi$
$542$	−65.2815	−2.80408
$543$	20.5178	0.880505
$544$	−137.858	−5.91061
$545$	15.5761	0.667207
$546$	−16.2667	−0.696149
$547$	−21.3376	−0.912329	−0.456164	−	0.889896i	$-0.650777\pi$
$547$	−21.3376	−0.912329	−0.456164	+	0.889896i	$0.650777\pi$
$548$	49.4373	2.11185
$549$	2.42869	0.103654
$550$	−8.44394	−0.360051
$551$	−33.1479	−1.41215
$552$	6.44968	0.274517
$553$	61.2023	2.60258
$554$	23.5466	1.00040
$555$	21.3837	0.907688
$556$	−23.3929	−0.992080
$557$	24.5024	1.03820	0.519100	−	0.854713i	$-0.326267\pi$
$557$	24.5024	1.03820	0.519100	+	0.854713i	$0.326267\pi$
$558$	−0.504568	−0.0213601
$559$	−1.69926	−0.0718708
$560$	125.951	5.32240
$561$	25.0374	1.05708
$562$	−42.9027	−1.80974
$563$	−32.0456	−1.35056	−0.675281	−	0.737561i	$-0.735977\pi$
$563$	−32.0456	−1.35056	−0.675281	+	0.737561i	$0.735977\pi$
$564$	−119.633	−5.03747
$565$	17.1489	0.721460
$566$	21.0421	0.884467
$567$	26.4765	1.11191
$568$	−157.270	−6.59889
$569$	1.73558	0.0727594	0.0363797	−	0.999338i	$-0.488417\pi$
$569$	1.73558	0.0727594	0.0363797	+	0.999338i	$0.488417\pi$
$570$	52.4941	2.19874
$571$	−7.69119	−0.321866	−0.160933	−	0.986965i	$-0.551450\pi$
$571$	−7.69119	−0.321866	−0.160933	+	0.986965i	$0.551450\pi$
$572$	−17.2232	−0.720139
$573$	11.3607	0.474599
$574$	−65.5576	−2.73632
$575$	−0.406778	−0.0169638
$576$	−21.3132	−0.888050
$577$	−2.41700	−0.100621	−0.0503105	−	0.998734i	$-0.516021\pi$
$577$	−2.41700	−0.100621	−0.0503105	+	0.998734i	$0.516021\pi$
$578$	−29.5936	−1.23093
$579$	−26.3150	−1.09361
$580$	−62.0470	−2.57636
$581$	56.0434	2.32507
$582$	−48.2867	−2.00155
$583$	−0.787668	−0.0326219
$584$	156.120	6.46028
$585$	1.09369	0.0452186
$586$	29.8414	1.23274
$587$	22.8254	0.942106	0.471053	−	0.882105i	$-0.343874\pi$
$587$	22.8254	0.942106	0.471053	+	0.882105i	$0.343874\pi$
$588$	−62.7391	−2.58732
$589$	2.01633	0.0830814
$590$	−1.45773	−0.0600138
$591$	30.3047	1.24657
$592$	114.732	4.71545
$593$	24.5718	1.00904	0.504522	−	0.863399i	$-0.331669\pi$
$593$	24.5718	1.00904	0.504522	+	0.863399i	$0.331669\pi$
$594$	46.7407	1.91779
$595$	39.4511	1.61734
$596$	−50.6925	−2.07645
$597$	24.9295	1.02030
$598$	−1.12243	−0.0458997
$599$	2.55855	0.104540	0.0522698	−	0.998633i	$-0.483354\pi$
$599$	2.55855	0.104540	0.0522698	+	0.998633i	$0.483354\pi$
$600$	−15.9701	−0.651976
$601$	−38.7497	−1.58063	−0.790317	−	0.612698i	$-0.790084\pi$
$601$	−38.7497	−1.58063	−0.790317	+	0.612698i	$0.790084\pi$
$602$	−17.6489	−0.719314
$603$	4.70570	0.191631
$604$	−69.8551	−2.84236
$605$	3.53718	0.143807
$606$	−31.7122	−1.28822
$607$	30.0665	1.22036	0.610180	−	0.792263i	$-0.291097\pi$
$607$	30.0665	1.22036	0.610180	+	0.792263i	$0.291097\pi$
$608$	158.623	6.43303
$609$	32.1602	1.30320
$610$	−24.5760	−0.995055
$611$	13.4745	0.545120
$612$	−16.3190	−0.659656
$613$	27.4603	1.10911	0.554556	−	0.832146i	$-0.312888\pi$
$613$	27.4603	1.10911	0.554556	+	0.832146i	$0.312888\pi$
$614$	42.8809	1.73053
$615$	−19.7624	−0.796897
$616$	−115.774	−4.66466
$617$	−33.8132	−1.36127	−0.680634	−	0.732624i	$-0.738295\pi$
$617$	−33.8132	−1.36127	−0.680634	+	0.732624i	$0.738295\pi$
$618$	10.4893	0.421942
$619$	−1.00000	−0.0401934
$619$	−1.00000	−0.0401934
$620$	3.77421	0.151576
$621$	2.25168	0.0903569
$622$	64.7001	2.59424
$623$	17.0373	0.682585
$624$	−26.3097	−1.05323
$625$	−18.9747	−0.758990
$626$	9.03331	0.361044
$627$	−28.8088	−1.15051
$628$	118.523	4.72960
$629$	35.9370	1.43290
$630$	11.3593	0.452567
$631$	16.3193	0.649661	0.324831	−	0.945772i	$-0.394693\pi$
$631$	16.3193	0.649661	0.324831	+	0.945772i	$0.394693\pi$
$632$	165.798	6.59509
$633$	8.80490	0.349963
$634$	13.0927	0.519977
$635$	25.6989	1.01983
$636$	−2.30180	−0.0912724
$637$	7.06640	0.279981
$638$	46.0648	1.82372
$639$	−8.46842	−0.335006
$640$	110.918	4.38440
$641$	−38.5714	−1.52348	−0.761739	−	0.647885i	$-0.775654\pi$
$641$	−38.5714	−1.52348	−0.761739	+	0.647885i	$0.775654\pi$
$642$	−55.8990	−2.20616
$643$	−10.7615	−0.424394	−0.212197	−	0.977227i	$-0.568062\pi$
$643$	−10.7615	−0.424394	−0.212197	+	0.977227i	$0.568062\pi$
$644$	−8.61757	−0.339580
$645$	−5.32027	−0.209485
$646$	88.2204	3.47099
$647$	−12.0791	−0.474879	−0.237440	−	0.971402i	$-0.576308\pi$
$647$	−12.0791	−0.474879	−0.237440	+	0.971402i	$0.576308\pi$
$648$	71.7253	2.81763
$649$	0.800002	0.0314028
$650$	2.77927	0.109012
$651$	−1.95625	−0.0766715
$652$	144.375	5.65417
$653$	33.5927	1.31459	0.657293	−	0.753635i	$-0.271702\pi$
$653$	33.5927	1.31459	0.657293	+	0.753635i	$0.271702\pi$
$654$	−33.7934	−1.32143
$655$	12.3169	0.481261
$656$	−106.033	−4.13989
$657$	8.40649	0.327968
$658$	139.949	5.45579
$659$	−30.8831	−1.20304	−0.601518	−	0.798859i	$-0.705437\pi$
$659$	−30.8831	−1.20304	−0.601518	+	0.798859i	$0.705437\pi$
$660$	−53.9249	−2.09902
$661$	34.1412	1.32794	0.663969	−	0.747760i	$-0.268870\pi$
$661$	34.1412	1.32794	0.663969	+	0.747760i	$0.268870\pi$
$662$	−80.4285	−3.12594
$663$	−8.24088	−0.320049
$664$	151.823	5.89186
$665$	−45.3936	−1.76029
$666$	10.3475	0.400958
$667$	2.21912	0.0859247
$668$	57.0359	2.20678
$669$	−3.27619	−0.126665
$670$	−47.6173	−1.83962
$671$	13.4873	0.520673
$672$	−153.897	−5.93671
$673$	25.5766	0.985904	0.492952	−	0.870057i	$-0.335918\pi$
$673$	25.5766	0.985904	0.492952	+	0.870057i	$0.335918\pi$
$674$	−10.9370	−0.421276
$675$	−5.57540	−0.214597
$676$	5.66891	0.218035
$677$	−24.7127	−0.949787	−0.474894	−	0.880043i	$-0.657513\pi$
$677$	−24.7127	−0.949787	−0.474894	+	0.880043i	$0.657513\pi$
$678$	−37.2057	−1.42888
$679$	41.7553	1.60242
$680$	106.874	4.09842
$681$	16.0536	0.615175
$682$	−2.80204	−0.107296
$683$	−11.5202	−0.440807	−0.220403	−	0.975409i	$-0.570737\pi$
$683$	−11.5202	−0.440807	−0.220403	+	0.975409i	$0.570737\pi$
$684$	18.7771	0.717961
$685$	−17.4337	−0.666107
$686$	0.689637	0.0263304
$687$	45.1762	1.72358
$688$	−28.5453	−1.08828
$689$	0.259255	0.00987684
$690$	−3.51427	−0.133786
$691$	25.9305	0.986443	0.493221	−	0.869904i	$-0.335819\pi$
$691$	25.9305	0.986443	0.493221	+	0.869904i	$0.335819\pi$
$692$	130.874	4.97509
$693$	−6.23401	−0.236810
$694$	63.4735	2.40942
$695$	8.24933	0.312915
$696$	87.1227	3.30238
$697$	−33.2123	−1.25800
$698$	−56.4387	−2.13624
$699$	−29.3118	−1.10867
$700$	21.3380	0.806501
$701$	−18.7878	−0.709605	−0.354803	−	0.934941i	$-0.615452\pi$
$701$	−18.7878	−0.709605	−0.354803	+	0.934941i	$0.615452\pi$
$702$	−15.3844	−0.580646
$703$	−41.3502	−1.55955
$704$	−118.360	−4.46084
$705$	42.1878	1.58889
$706$	40.2154	1.51353
$707$	27.4227	1.03134
$708$	2.33785	0.0878617
$709$	−38.5089	−1.44623	−0.723117	−	0.690726i	$-0.757291\pi$
$709$	−38.5089	−1.44623	−0.723117	+	0.690726i	$0.757291\pi$
$710$	85.6925	3.21598
$711$	8.92764	0.334813
$712$	46.1543	1.72971
$713$	−0.134985	−0.00505524
$714$	−85.5918	−3.20319
$715$	6.07364	0.227141
$716$	−106.268	−3.97143
$717$	−5.49297	−0.205139
$718$	61.7744	2.30540
$719$	−45.5765	−1.69972	−0.849858	−	0.527012i	$-0.823312\pi$
$719$	−45.5765	−1.69972	−0.849858	+	0.527012i	$0.823312\pi$
$720$	18.3726	0.684706
$721$	−9.07049	−0.337803
$722$	−48.8928	−1.81960
$723$	33.6760	1.25242
$724$	−74.2661	−2.76008
$725$	−5.49478	−0.204071
$726$	−7.67416	−0.284815
$727$	−22.8089	−0.845934	−0.422967	−	0.906145i	$-0.639011\pi$
$727$	−22.8089	−0.845934	−0.422967	+	0.906145i	$0.639011\pi$
$728$	38.1061	1.41231
$729$	29.9642	1.10979
$730$	−85.0658	−3.14843
$731$	−8.94112	−0.330699
$732$	39.4140	1.45678
$733$	−29.3481	−1.08400	−0.541998	−	0.840380i	$-0.682332\pi$
$733$	−29.3481	−1.08400	−0.541998	+	0.840380i	$0.682332\pi$
$734$	1.56507	0.0577679
$735$	22.1245	0.816073
$736$	−10.6192	−0.391429
$737$	26.1324	0.962599
$738$	−9.56296	−0.352017
$739$	4.87612	0.179371	0.0896854	−	0.995970i	$-0.471414\pi$
$739$	4.87612	0.179371	0.0896854	+	0.995970i	$0.471414\pi$
$740$	−77.4001	−2.84529
$741$	9.48221	0.348338
$742$	2.69269	0.0988517
$743$	15.5096	0.568992	0.284496	−	0.958677i	$-0.408174\pi$
$743$	15.5096	0.568992	0.284496	+	0.958677i	$0.408174\pi$
$744$	−5.29952	−0.194290
$745$	17.8763	0.654938
$746$	11.2006	0.410083
$747$	8.17512	0.299112
$748$	−90.6249	−3.31358
$749$	48.3380	1.76623
$750$	52.0539	1.90074
$751$	8.42073	0.307277	0.153638	−	0.988127i	$-0.450901\pi$
$751$	8.42073	0.307277	0.153638	+	0.988127i	$0.450901\pi$
$752$	226.354	8.25428
$753$	24.4225	0.890005
$754$	−15.1619	−0.552164
$755$	24.6339	0.896519
$756$	−118.115	−4.29579
$757$	20.8671	0.758426	0.379213	−	0.925309i	$-0.376195\pi$
$757$	20.8671	0.758426	0.379213	+	0.925309i	$0.376195\pi$
$758$	48.4329	1.75916
$759$	1.92863	0.0700050
$760$	−122.972	−4.46067
$761$	0.981649	0.0355847	0.0177924	−	0.999842i	$-0.494336\pi$
$761$	0.981649	0.0355847	0.0177924	+	0.999842i	$0.494336\pi$
$762$	−55.7554	−2.01981
$763$	29.2224	1.05792
$764$	−41.1209	−1.48770
$765$	5.75477	0.208064
$766$	−47.9174	−1.73133
$767$	−0.263315	−0.00950776
$768$	−118.615	−4.28016
$769$	31.4350	1.13358	0.566788	−	0.823863i	$-0.308186\pi$
$769$	31.4350	1.13358	0.566788	+	0.823863i	$0.308186\pi$
$770$	63.0823	2.27333
$771$	−22.8283	−0.822143
$772$	95.2493	3.42810
$773$	−11.7794	−0.423677	−0.211839	−	0.977305i	$-0.567945\pi$
$773$	−11.7794	−0.423677	−0.211839	+	0.977305i	$0.567945\pi$
$774$	−2.57446	−0.0925370
$775$	0.334238	0.0120062
$776$	113.116	4.06062
$777$	40.1181	1.43923
$778$	5.82740	0.208922
$779$	38.2150	1.36920
$780$	17.7490	0.635516
$781$	−47.0281	−1.68280
$782$	−5.90601	−0.211198
$783$	30.4159	1.08697
$784$	118.706	4.23951
$785$	−41.7964	−1.49178
$786$	−26.7223	−0.953154
$787$	−49.3816	−1.76026	−0.880132	−	0.474729i	$-0.842546\pi$
$787$	−49.3816	−1.76026	−0.880132	+	0.474729i	$0.842546\pi$
$788$	−109.690	−3.90756
$789$	−0.913316	−0.0325149
$790$	−90.3393	−3.21413
$791$	32.1732	1.14395
$792$	−16.8880	−0.600090
$793$	−4.43926	−0.157643
$794$	−89.7898	−3.18652
$795$	0.811713	0.0287885
$796$	−90.2344	−3.19827
$797$	14.2049	0.503163	0.251581	−	0.967836i	$-0.419049\pi$
$797$	14.2049	0.503163	0.251581	+	0.967836i	$0.419049\pi$
$798$	98.4845	3.48631
$799$	70.8999	2.50826
$800$	26.2943	0.929644
$801$	2.48525	0.0878119
$802$	−105.913	−3.73990
$803$	46.6841	1.64745
$804$	76.3667	2.69325
$805$	3.03892	0.107108
$806$	0.922272	0.0324856
$807$	−45.0666	−1.58642
$808$	74.2887	2.61347
$809$	11.9873	0.421450	0.210725	−	0.977545i	$-0.432418\pi$
$809$	11.9873	0.421450	0.210725	+	0.977545i	$0.432418\pi$
$810$	−39.0813	−1.37318
$811$	−31.5950	−1.10945	−0.554725	−	0.832034i	$-0.687176\pi$
$811$	−31.5950	−1.10945	−0.554725	+	0.832034i	$0.687176\pi$
$812$	−116.407	−4.08507
$813$	−36.9202	−1.29485
$814$	57.4633	2.01409
$815$	−50.9128	−1.78340
$816$	−138.436	−4.84623
$817$	10.2879	0.359929
$818$	16.6291	0.581424
$819$	2.05188	0.0716985
$820$	71.5317	2.49800
$821$	6.63996	0.231736	0.115868	−	0.993265i	$-0.463035\pi$
$821$	6.63996	0.231736	0.115868	+	0.993265i	$0.463035\pi$
$822$	37.8236	1.31925
$823$	34.0727	1.18770	0.593850	−	0.804576i	$-0.297607\pi$
$823$	34.0727	1.18770	0.593850	+	0.804576i	$0.297607\pi$
$824$	−24.5721	−0.856011
$825$	−4.77550	−0.166262
$826$	−2.73485	−0.0951578
$827$	24.1497	0.839768	0.419884	−	0.907578i	$-0.362071\pi$
$827$	24.1497	0.839768	0.419884	+	0.907578i	$0.362071\pi$
$828$	−1.25705	−0.0436856
$829$	−25.6061	−0.889338	−0.444669	−	0.895695i	$-0.646679\pi$
$829$	−25.6061	−0.889338	−0.444669	+	0.895695i	$0.646679\pi$
$830$	−82.7245	−2.87141
$831$	13.3169	0.461958
$832$	38.9572	1.35060
$833$	37.1819	1.28828
$834$	−17.8975	−0.619739
$835$	−20.1133	−0.696048
$836$	104.276	3.60646
$837$	−1.85014	−0.0639503
$838$	35.2292	1.21697
$839$	−17.7582	−0.613082	−0.306541	−	0.951857i	$-0.599172\pi$
$839$	−17.7582	−0.613082	−0.306541	+	0.951857i	$0.599172\pi$
$840$	119.308	4.11652
$841$	0.976033	0.0336563
$842$	−75.0887	−2.58773
$843$	−24.2638	−0.835689
$844$	−31.8701	−1.09701
$845$	−1.99910	−0.0687711
$846$	20.4145	0.701866
$847$	6.63613	0.228020
$848$	4.35515	0.149557
$849$	11.9005	0.408423
$850$	14.6239	0.501596
$851$	2.76823	0.0948937
$852$	−137.430	−4.70828
$853$	16.8249	0.576073	0.288036	−	0.957619i	$-0.406998\pi$
$853$	16.8249	0.576073	0.288036	+	0.957619i	$0.406998\pi$
$854$	−46.1072	−1.57776
$855$	−6.62161	−0.226454
$856$	130.948	4.47573
$857$	9.67475	0.330483	0.165242	−	0.986253i	$-0.447160\pi$
$857$	9.67475	0.330483	0.165242	+	0.986253i	$0.447160\pi$
$858$	−13.1772	−0.449862
$859$	−23.1578	−0.790135	−0.395067	−	0.918652i	$-0.629279\pi$
$859$	−23.1578	−0.790135	−0.395067	+	0.918652i	$0.629279\pi$
$860$	19.2572	0.656664
$861$	−37.0764	−1.26356
$862$	46.2983	1.57693
$863$	−21.7950	−0.741911	−0.370956	−	0.928651i	$-0.620970\pi$
$863$	−21.7950	−0.741911	−0.370956	+	0.928651i	$0.620970\pi$
$864$	−145.550	−4.95170
$865$	−46.1519	−1.56921
$866$	54.5435	1.85346
$867$	−16.7368	−0.568411
$868$	7.08081	0.240338
$869$	49.5782	1.68183
$870$	−47.4710	−1.60942
$871$	−8.60130	−0.291444
$872$	79.1640	2.68083
$873$	6.09088	0.206145
$874$	6.79563	0.229866
$875$	−45.0130	−1.52172
$876$	136.425	4.60938
$877$	19.0172	0.642167	0.321083	−	0.947051i	$-0.395953\pi$
$877$	19.0172	0.642167	0.321083	+	0.947051i	$0.395953\pi$
$878$	−12.3365	−0.416337
$879$	16.8769	0.569245
$880$	102.029	3.43941
$881$	15.3891	0.518473	0.259236	−	0.965814i	$-0.416529\pi$
$881$	15.3891	0.518473	0.259236	+	0.965814i	$0.416529\pi$
$882$	10.7060	0.360488
$883$	−14.7724	−0.497132	−0.248566	−	0.968615i	$-0.579959\pi$
$883$	−14.7724	−0.497132	−0.248566	+	0.968615i	$0.579959\pi$
$884$	29.8286	1.00324
$885$	−0.824424	−0.0277127
$886$	91.0245	3.05803
$887$	58.7398	1.97229	0.986144	−	0.165891i	$-0.0530498\pi$
$887$	58.7398	1.97229	0.986144	+	0.165891i	$0.0530498\pi$
$888$	108.681	3.64708
$889$	48.2138	1.61704
$890$	−25.1484	−0.842976
$891$	21.4479	0.718530
$892$	11.8584	0.397050
$893$	−81.5796	−2.72996
$894$	−38.7839	−1.29713
$895$	37.4747	1.25264
$896$	208.093	6.95190
$897$	−0.634797	−0.0211952
$898$	21.2975	0.710706
$899$	−1.82339	−0.0608134
$900$	3.11260	0.103753
$901$	1.36415	0.0454463
$902$	−53.1064	−1.76825
$903$	−9.98138	−0.332160
$904$	87.1577	2.89882
$905$	26.1894	0.870565
$906$	−53.4449	−1.77559
$907$	−15.8922	−0.527693	−0.263847	−	0.964565i	$-0.584991\pi$
$907$	−15.8922	−0.527693	−0.263847	+	0.964565i	$0.584991\pi$
$908$	−58.1073	−1.92836
$909$	4.00018	0.132678
$910$	−20.7631	−0.688290
$911$	10.2652	0.340100	0.170050	−	0.985435i	$-0.445607\pi$
$911$	10.2652	0.340100	0.170050	+	0.985435i	$0.445607\pi$
$912$	159.289	5.27458
$913$	45.3992	1.50250
$914$	102.226	3.38135
$915$	−13.8991	−0.459489
$916$	−163.519	−5.40282
$917$	23.1078	0.763087
$918$	−80.9493	−2.67172
$919$	58.1348	1.91769	0.958845	−	0.283929i	$-0.0916381\pi$
$919$	58.1348	1.91769	0.958845	+	0.283929i	$0.0916381\pi$
$920$	8.23250	0.271417
$921$	24.2514	0.799112
$922$	57.2151	1.88428
$923$	15.4790	0.509496
$924$	−101.169	−3.32821
$925$	−6.85443	−0.225372
$926$	80.7696	2.65425
$927$	−1.32312	−0.0434570
$928$	−143.445	−4.70881
$929$	−22.5207	−0.738879	−0.369440	−	0.929255i	$-0.620450\pi$
$929$	−22.5207	−0.738879	−0.369440	+	0.929255i	$0.620450\pi$
$930$	2.88758	0.0946874
$931$	−42.7826	−1.40214
$932$	106.096	3.47531
$933$	36.5914	1.19795
$934$	4.67499	0.152970
$935$	31.9582	1.04515
$936$	5.55858	0.181688
$937$	−29.2227	−0.954662	−0.477331	−	0.878723i	$-0.658396\pi$
$937$	−29.2227	−0.954662	−0.477331	+	0.878723i	$0.658396\pi$
$938$	−89.3351	−2.91689
$939$	5.10882	0.166720
$940$	−152.702	−4.98060
$941$	44.4499	1.44903	0.724513	−	0.689261i	$-0.242065\pi$
$941$	44.4499	1.44903	0.724513	+	0.689261i	$0.242065\pi$
$942$	90.6802	2.95452
$943$	−2.55835	−0.0833112
$944$	−4.42335	−0.143968
$945$	41.6523	1.35495
$946$	−14.2969	−0.464831
$947$	29.5182	0.959214	0.479607	−	0.877483i	$-0.340779\pi$
$947$	29.5182	0.959214	0.479607	+	0.877483i	$0.340779\pi$
$948$	144.882	4.70556
$949$	−15.3658	−0.498794
$950$	−16.8267	−0.545930
$951$	7.40461	0.240111
$952$	200.506	6.49845
$953$	39.1263	1.26742	0.633712	−	0.773569i	$-0.281530\pi$
$953$	39.1263	1.26742	0.633712	+	0.773569i	$0.281530\pi$
$954$	0.392785	0.0127169
$955$	14.5010	0.469241
$956$	19.8823	0.643039
$957$	26.0521	0.842145
$958$	1.32749	0.0428893
$959$	−32.7074	−1.05618
$960$	121.973	3.93665
$961$	−30.8891	−0.996422
$962$	−18.9136	−0.609800
$963$	7.05111	0.227219
$964$	−121.893	−3.92591
$965$	−33.5890	−1.08127
$966$	−6.59315	−0.212131
$967$	43.3754	1.39486	0.697430	−	0.716653i	$-0.254327\pi$
$967$	43.3754	1.39486	0.697430	+	0.716653i	$0.254327\pi$
$968$	17.9774	0.577816
$969$	49.8934	1.60281
$970$	−61.6340	−1.97895
$971$	46.2316	1.48364	0.741821	−	0.670598i	$-0.233962\pi$
$971$	46.2316	1.48364	0.741821	+	0.670598i	$0.233962\pi$
$972$	−31.8016	−1.02004
$973$	15.4766	0.496158
$974$	78.2561	2.50749
$975$	1.57182	0.0503387
$976$	−74.5738	−2.38705
$977$	14.8875	0.476294	0.238147	−	0.971229i	$-0.423460\pi$
$977$	14.8875	0.476294	0.238147	+	0.971229i	$0.423460\pi$
$978$	110.459	3.53208
$979$	13.8014	0.441096
$980$	−80.0814	−2.55811
$981$	4.26270	0.136098
$982$	−32.2146	−1.02801
$983$	−16.7294	−0.533584	−0.266792	−	0.963754i	$-0.585964\pi$
$983$	−16.7294	−0.533584	−0.266792	+	0.963754i	$0.585964\pi$
$984$	−100.441	−3.20193
$985$	38.6815	1.23250
$986$	−79.7787	−2.54067
$987$	79.1488	2.51933
$988$	−34.3217	−1.09192
$989$	−0.688736	−0.0219005
$990$	9.20188	0.292455
$991$	−13.5856	−0.431560	−0.215780	−	0.976442i	$-0.569229\pi$
$991$	−13.5856	−0.431560	−0.215780	+	0.976442i	$0.569229\pi$
$992$	8.72551	0.277035
$993$	−45.4866	−1.44347
$994$	160.768	5.09926
$995$	31.8205	1.00878
$996$	132.670	4.20382
$997$	−28.4243	−0.900207	−0.450103	−	0.892977i	$-0.648613\pi$
$997$	−28.4243	−0.900207	−0.450103	+	0.892977i	$0.648613\pi$
$998$	79.8897	2.52887
$999$	37.9421	1.20043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.e.1.2	✓	168

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.e.1.2	✓	168	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.76928 q^{2} -1.56618 q^{3} +5.66891 q^{4} -1.99910 q^{5} +4.33718 q^{6} -3.75052 q^{7} -10.1602 q^{8} -0.547092 q^{9} +O(q^{10})\)	\(q-2.76928 q^{2} -1.56618 q^{3} +5.66891 q^{4} -1.99910 q^{5} +4.33718 q^{6} -3.75052 q^{7} -10.1602 q^{8} -0.547092 q^{9} +5.53606 q^{10} -3.03819 q^{11} -8.87851 q^{12} +1.00000 q^{13} +10.3862 q^{14} +3.13094 q^{15} +16.7987 q^{16} +5.26179 q^{17} +1.51505 q^{18} -6.05437 q^{19} -11.3327 q^{20} +5.87397 q^{21} +8.41360 q^{22} +0.405316 q^{23} +15.9127 q^{24} -1.00361 q^{25} -2.76928 q^{26} +5.55537 q^{27} -21.2613 q^{28} +5.47504 q^{29} -8.67045 q^{30} -0.333037 q^{31} -26.1998 q^{32} +4.75834 q^{33} -14.5714 q^{34} +7.49766 q^{35} -3.10142 q^{36} +6.82980 q^{37} +16.7662 q^{38} -1.56618 q^{39} +20.3113 q^{40} -6.31197 q^{41} -16.2667 q^{42} -1.69926 q^{43} -17.2232 q^{44} +1.09369 q^{45} -1.12243 q^{46} +13.4745 q^{47} -26.3097 q^{48} +7.06640 q^{49} +2.77927 q^{50} -8.24088 q^{51} +5.66891 q^{52} +0.259255 q^{53} -15.3844 q^{54} +6.07364 q^{55} +38.1061 q^{56} +9.48221 q^{57} -15.1619 q^{58} -0.263315 q^{59} +17.7490 q^{60} -4.43926 q^{61} +0.922272 q^{62} +2.05188 q^{63} +38.9572 q^{64} -1.99910 q^{65} -13.1772 q^{66} -8.60130 q^{67} +29.8286 q^{68} -0.634797 q^{69} -20.7631 q^{70} +15.4790 q^{71} +5.55858 q^{72} -15.3658 q^{73} -18.9136 q^{74} +1.57182 q^{75} -34.3217 q^{76} +11.3948 q^{77} +4.33718 q^{78} -16.3183 q^{79} -33.5822 q^{80} -7.05941 q^{81} +17.4796 q^{82} -14.9428 q^{83} +33.2990 q^{84} -10.5188 q^{85} +4.70571 q^{86} -8.57487 q^{87} +30.8687 q^{88} -4.54265 q^{89} -3.02874 q^{90} -3.75052 q^{91} +2.29770 q^{92} +0.521594 q^{93} -37.3146 q^{94} +12.1033 q^{95} +41.0335 q^{96} -11.1332 q^{97} -19.5688 q^{98} +1.66217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9	\( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 168 * q + 11 * q^2 + 26 * q^3 + 181 * q^4 + 41 * q^5 + 11 * q^6 + 12 * q^7 + 27 * q^8 + 220 * q^9 + 11 * q^10 + 23 * q^11 + 78 * q^12 + 168 * q^13 + 47 * q^14 + 10 * q^15 + 203 * q^16 + 147 * q^17 + 13 * q^18 + 17 * q^19 + 81 * q^20 + 13 * q^21 + 20 * q^22 + 85 * q^23 + 14 * q^24 + 225 * q^25 + 11 * q^26 + 89 * q^27 + 12 * q^28 + 137 * q^29 + 26 * q^30 + 13 * q^31 + 60 * q^32 + 78 * q^33 - 2 * q^34 + 77 * q^35 + 278 * q^36 + 41 * q^37 + 68 * q^38 + 26 * q^39 + 11 * q^40 + 107 * q^41 + 43 * q^42 + 27 * q^43 + 39 * q^44 + 88 * q^45 - 23 * q^46 + 112 * q^47 + 127 * q^48 + 236 * q^49 + 14 * q^50 + 55 * q^51 + 181 * q^52 + 149 * q^53 + 3 * q^54 + 40 * q^55 + 134 * q^56 + 55 * q^57 - q^58 + 44 * q^59 - 13 * q^60 + 81 * q^61 + 106 * q^62 + 34 * q^63 + 197 * q^64 + 41 * q^65 - 20 * q^66 - q^67 + 278 * q^68 + 75 * q^69 - 42 * q^70 + 48 * q^71 - 34 * q^72 + 107 * q^73 + 74 * q^74 + 93 * q^75 + 20 * q^76 + 206 * q^77 + 11 * q^78 + 14 * q^79 + 115 * q^80 + 328 * q^81 + 48 * q^82 + 62 * q^83 - 11 * q^84 + 6 * q^85 + 27 * q^86 + 51 * q^87 + 31 * q^88 + 173 * q^89 - 21 * q^90 + 12 * q^91 + 179 * q^92 + 73 * q^93 + 17 * q^94 + 90 * q^95 - 33 * q^96 + 110 * q^97 - 13 * q^98 + 24 * q^99

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