LMFDB - Embedded newform 8039.2.a.a.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$1$
Dimension:	$279$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55424 q^{2} +2.06857 q^{3} +4.52414 q^{4} -3.35787 q^{5} -5.28363 q^{6} +0.600266 q^{7} -6.44725 q^{8} +1.27899 q^{9} +O(q^{10})$	\(q-2.55424 q^{2} +2.06857 q^{3} +4.52414 q^{4} -3.35787 q^{5} -5.28363 q^{6} +0.600266 q^{7} -6.44725 q^{8} +1.27899 q^{9} +8.57679 q^{10} +3.31920 q^{11} +9.35850 q^{12} -1.75205 q^{13} -1.53322 q^{14} -6.94599 q^{15} +7.41953 q^{16} +1.55971 q^{17} -3.26684 q^{18} +3.04253 q^{19} -15.1914 q^{20} +1.24169 q^{21} -8.47803 q^{22} +7.04520 q^{23} -13.3366 q^{24} +6.27527 q^{25} +4.47516 q^{26} -3.56004 q^{27} +2.71569 q^{28} -1.63683 q^{29} +17.7417 q^{30} -5.98075 q^{31} -6.05677 q^{32} +6.86600 q^{33} -3.98388 q^{34} -2.01561 q^{35} +5.78631 q^{36} -6.86992 q^{37} -7.77134 q^{38} -3.62425 q^{39} +21.6490 q^{40} +0.128893 q^{41} -3.17158 q^{42} -7.50398 q^{43} +15.0165 q^{44} -4.29467 q^{45} -17.9951 q^{46} +5.90022 q^{47} +15.3478 q^{48} -6.63968 q^{49} -16.0285 q^{50} +3.22638 q^{51} -7.92652 q^{52} -0.0214415 q^{53} +9.09318 q^{54} -11.1454 q^{55} -3.87006 q^{56} +6.29368 q^{57} +4.18086 q^{58} +4.69874 q^{59} -31.4246 q^{60} +2.91083 q^{61} +15.2763 q^{62} +0.767733 q^{63} +0.631367 q^{64} +5.88316 q^{65} -17.5374 q^{66} -14.2052 q^{67} +7.05636 q^{68} +14.5735 q^{69} +5.14836 q^{70} -3.68423 q^{71} -8.24595 q^{72} +2.07249 q^{73} +17.5474 q^{74} +12.9808 q^{75} +13.7648 q^{76} +1.99240 q^{77} +9.25719 q^{78} +5.72121 q^{79} -24.9138 q^{80} -11.2012 q^{81} -0.329223 q^{82} -9.11021 q^{83} +5.61759 q^{84} -5.23731 q^{85} +19.1669 q^{86} -3.38591 q^{87} -21.3997 q^{88} -13.6643 q^{89} +10.9696 q^{90} -1.05170 q^{91} +31.8734 q^{92} -12.3716 q^{93} -15.0706 q^{94} -10.2164 q^{95} -12.5289 q^{96} -1.46464 q^{97} +16.9593 q^{98} +4.24522 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) $ 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * 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 $

$2$	−2.55424	−1.80612	−0.903060	−	0.429515i	$-0.858685\pi$
$2$	−2.55424	−1.80612	−0.903060	+	0.429515i	$0.858685\pi$
$3$	2.06857	1.19429	0.597145	−	0.802133i	$-0.296302\pi$
$3$	2.06857	1.19429	0.597145	+	0.802133i	$0.296302\pi$
$4$	4.52414	2.26207
$5$	−3.35787	−1.50168	−0.750842	−	0.660482i	$-0.770352\pi$
$5$	−3.35787	−1.50168	−0.750842	+	0.660482i	$0.770352\pi$
$6$	−5.28363	−2.15703
$7$	0.600266	0.226879	0.113440	−	0.993545i	$-0.463813\pi$
$7$	0.600266	0.226879	0.113440	+	0.993545i	$0.463813\pi$
$8$	−6.44725	−2.27945
$9$	1.27899	0.426329
$10$	8.57679	2.71222
$11$	3.31920	1.00078	0.500388	−	0.865801i	$-0.333191\pi$
$11$	3.31920	1.00078	0.500388	+	0.865801i	$0.333191\pi$
$12$	9.35850	2.70157
$13$	−1.75205	−0.485932	−0.242966	−	0.970035i	$-0.578120\pi$
$13$	−1.75205	−0.485932	−0.242966	+	0.970035i	$0.578120\pi$
$14$	−1.53322	−0.409771
$15$	−6.94599	−1.79345
$16$	7.41953	1.85488
$17$	1.55971	0.378286	0.189143	−	0.981950i	$-0.439429\pi$
$17$	1.55971	0.378286	0.189143	+	0.981950i	$0.439429\pi$
$18$	−3.26684	−0.770002
$19$	3.04253	0.698003	0.349002	−	0.937122i	$-0.386521\pi$
$19$	3.04253	0.698003	0.349002	+	0.937122i	$0.386521\pi$
$20$	−15.1914	−3.39691
$21$	1.24169	0.270960
$22$	−8.47803	−1.80752
$23$	7.04520	1.46903	0.734513	−	0.678595i	$-0.237411\pi$
$23$	7.04520	1.46903	0.734513	+	0.678595i	$0.237411\pi$
$24$	−13.3366	−2.72232
$25$	6.27527	1.25505
$26$	4.47516	0.877651
$27$	−3.56004	−0.685129
$28$	2.71569	0.513216
$29$	−1.63683	−0.303952	−0.151976	−	0.988384i	$-0.548564\pi$
$29$	−1.63683	−0.303952	−0.151976	+	0.988384i	$0.548564\pi$
$30$	17.7417	3.23918
$31$	−5.98075	−1.07417	−0.537087	−	0.843527i	$-0.680475\pi$
$31$	−5.98075	−1.07417	−0.537087	+	0.843527i	$0.680475\pi$
$32$	−6.05677	−1.07070
$33$	6.86600	1.19522
$34$	−3.98388	−0.683230
$35$	−2.01561	−0.340701
$36$	5.78631	0.964386
$37$	−6.86992	−1.12941	−0.564704	−	0.825294i	$-0.691010\pi$
$37$	−6.86992	−1.12941	−0.564704	+	0.825294i	$0.691010\pi$
$38$	−7.77134	−1.26068
$39$	−3.62425	−0.580344
$40$	21.6490	3.42301
$41$	0.128893	0.0201297	0.0100648	−	0.999949i	$-0.496796\pi$
$41$	0.128893	0.0201297	0.0100648	+	0.999949i	$0.496796\pi$
$42$	−3.17158	−0.489386
$43$	−7.50398	−1.14435	−0.572173	−	0.820133i	$-0.693899\pi$
$43$	−7.50398	−1.14435	−0.572173	+	0.820133i	$0.693899\pi$
$44$	15.0165	2.26382
$45$	−4.29467	−0.640212
$46$	−17.9951	−2.65324
$47$	5.90022	0.860635	0.430317	−	0.902678i	$-0.358402\pi$
$47$	5.90022	0.860635	0.430317	+	0.902678i	$0.358402\pi$
$48$	15.3478	2.21527
$49$	−6.63968	−0.948526
$50$	−16.0285	−2.26678
$51$	3.22638	0.451784
$52$	−7.92652	−1.09921
$53$	−0.0214415	−0.00294522	−0.00147261	−	0.999999i	$-0.500469\pi$
$53$	−0.0214415	−0.00294522	−0.00147261	+	0.999999i	$0.500469\pi$
$54$	9.09318	1.23743
$55$	−11.1454	−1.50285
$56$	−3.87006	−0.517159
$57$	6.29368	0.833619
$58$	4.18086	0.548974
$59$	4.69874	0.611723	0.305862	−	0.952076i	$-0.401056\pi$
$59$	4.69874	0.611723	0.305862	+	0.952076i	$0.401056\pi$
$60$	−31.4246	−4.05690
$61$	2.91083	0.372693	0.186347	−	0.982484i	$-0.440335\pi$
$61$	2.91083	0.372693	0.186347	+	0.982484i	$0.440335\pi$
$62$	15.2763	1.94009
$63$	0.767733	0.0967253
$64$	0.631367	0.0789209
$65$	5.88316	0.729716
$66$	−17.5374	−2.15871
$67$	−14.2052	−1.73544	−0.867722	−	0.497051i	$-0.834416\pi$
$67$	−14.2052	−1.73544	−0.867722	+	0.497051i	$0.834416\pi$
$68$	7.05636	0.855709
$69$	14.5735	1.75444
$70$	5.14836	0.615347
$71$	−3.68423	−0.437238	−0.218619	−	0.975810i	$-0.570155\pi$
$71$	−3.68423	−0.437238	−0.218619	+	0.975810i	$0.570155\pi$
$72$	−8.24595	−0.971794
$73$	2.07249	0.242567	0.121283	−	0.992618i	$-0.461299\pi$
$73$	2.07249	0.242567	0.121283	+	0.992618i	$0.461299\pi$
$74$	17.5474	2.03984
$75$	12.9808	1.49890
$76$	13.7648	1.57893
$77$	1.99240	0.227055
$78$	9.25719	1.04817
$79$	5.72121	0.643686	0.321843	−	0.946793i	$-0.395698\pi$
$79$	5.72121	0.643686	0.321843	+	0.946793i	$0.395698\pi$
$80$	−24.9138	−2.78545
$81$	−11.2012	−1.24457
$82$	−0.329223	−0.0363566
$83$	−9.11021	−0.999976	−0.499988	−	0.866032i	$-0.666662\pi$
$83$	−9.11021	−0.999976	−0.499988	+	0.866032i	$0.666662\pi$
$84$	5.61759	0.612929
$85$	−5.23731	−0.568066
$86$	19.1669	2.06683
$87$	−3.38591	−0.363007
$88$	−21.3997	−2.28122
$89$	−13.6643	−1.44841	−0.724205	−	0.689585i	$-0.757793\pi$
$89$	−13.6643	−1.44841	−0.724205	+	0.689585i	$0.757793\pi$
$90$	10.9696	1.15630
$91$	−1.05170	−0.110248
$92$	31.8734	3.32304
$93$	−12.3716	−1.28288
$94$	−15.0706	−1.55441
$95$	−10.2164	−1.04818
$96$	−12.5289	−1.27872
$97$	−1.46464	−0.148712	−0.0743560	−	0.997232i	$-0.523690\pi$
$97$	−1.46464	−0.148712	−0.0743560	+	0.997232i	$0.523690\pi$
$98$	16.9593	1.71315
$99$	4.24522	0.426660
$100$	28.3902	2.83902
$101$	−2.60326	−0.259034	−0.129517	−	0.991577i	$-0.541343\pi$
$101$	−2.60326	−0.259034	−0.129517	+	0.991577i	$0.541343\pi$
$102$	−8.24095	−0.815975
$103$	9.62466	0.948346	0.474173	−	0.880432i	$-0.342747\pi$
$103$	9.62466	0.948346	0.474173	+	0.880432i	$0.342747\pi$
$104$	11.2959	1.10766
$105$	−4.16944	−0.406896
$106$	0.0547667	0.00531941
$107$	11.0751	1.07067	0.535336	−	0.844639i	$-0.320185\pi$
$107$	11.0751	1.07067	0.535336	+	0.844639i	$0.320185\pi$
$108$	−16.1061	−1.54981
$109$	−3.01846	−0.289116	−0.144558	−	0.989496i	$-0.546176\pi$
$109$	−3.01846	−0.289116	−0.144558	+	0.989496i	$0.546176\pi$
$110$	28.4681	2.71433
$111$	−14.2109	−1.34884
$112$	4.45369	0.420835
$113$	0.255392	0.0240253	0.0120126	−	0.999928i	$-0.496176\pi$
$113$	0.255392	0.0240253	0.0120126	+	0.999928i	$0.496176\pi$
$114$	−16.0756	−1.50562
$115$	−23.6568	−2.20601
$116$	−7.40525	−0.687560
$117$	−2.24085	−0.207167
$118$	−12.0017	−1.10485
$119$	0.936244	0.0858253
$120$	44.7825	4.08806
$121$	0.0170900	0.00155364
$122$	−7.43495	−0.673128
$123$	0.266624	0.0240407
$124$	−27.0577	−2.42986
$125$	−4.28220	−0.383011
$126$	−1.96097	−0.174697
$127$	9.56933	0.849141	0.424570	−	0.905395i	$-0.360425\pi$
$127$	9.56933	0.849141	0.424570	+	0.905395i	$0.360425\pi$
$128$	10.5009	0.928155
$129$	−15.5225	−1.36668
$130$	−15.0270	−1.31795
$131$	16.4809	1.43994	0.719972	−	0.694003i	$-0.244154\pi$
$131$	16.4809	1.43994	0.719972	+	0.694003i	$0.244154\pi$
$132$	31.0627	2.70366
$133$	1.82633	0.158363
$134$	36.2835	3.13442
$135$	11.9541	1.02885
$136$	−10.0559	−0.862283
$137$	5.56153	0.475154	0.237577	−	0.971369i	$-0.423647\pi$
$137$	5.56153	0.475154	0.237577	+	0.971369i	$0.423647\pi$
$138$	−37.2242	−3.16873
$139$	−6.94421	−0.589000	−0.294500	−	0.955651i	$-0.595153\pi$
$139$	−6.94421	−0.589000	−0.294500	+	0.955651i	$0.595153\pi$
$140$	−9.11891	−0.770689
$141$	12.2050	1.02785
$142$	9.41041	0.789704
$143$	−5.81541	−0.486309
$144$	9.48949	0.790791
$145$	5.49627	0.456440
$146$	−5.29364	−0.438105
$147$	−13.7347	−1.13282
$148$	−31.0804	−2.55480
$149$	−2.73880	−0.224372	−0.112186	−	0.993687i	$-0.535785\pi$
$149$	−2.73880	−0.224372	−0.112186	+	0.993687i	$0.535785\pi$
$150$	−33.1562	−2.70719
$151$	23.8234	1.93872	0.969362	−	0.245635i	$-0.0789965\pi$
$151$	23.8234	1.93872	0.969362	+	0.245635i	$0.0789965\pi$
$152$	−19.6159	−1.59106
$153$	1.99486	0.161275
$154$	−5.08907	−0.410089
$155$	20.0826	1.61307
$156$	−16.3966	−1.31278
$157$	−0.343690	−0.0274294	−0.0137147	−	0.999906i	$-0.504366\pi$
$157$	−0.343690	−0.0274294	−0.0137147	+	0.999906i	$0.504366\pi$
$158$	−14.6133	−1.16257
$159$	−0.0443533	−0.00351744
$160$	20.3378	1.60785
$161$	4.22899	0.333291
$162$	28.6104	2.24785
$163$	12.2461	0.959190	0.479595	−	0.877490i	$-0.340784\pi$
$163$	12.2461	0.959190	0.479595	+	0.877490i	$0.340784\pi$
$164$	0.583128	0.0455347
$165$	−23.0551	−1.79484
$166$	23.2697	1.80608
$167$	19.8326	1.53470	0.767348	−	0.641231i	$-0.221576\pi$
$167$	19.8326	1.53470	0.767348	+	0.641231i	$0.221576\pi$
$168$	−8.00550	−0.617638
$169$	−9.93031	−0.763870
$170$	13.3773	1.02600
$171$	3.89135	0.297579
$172$	−33.9490	−2.58859
$173$	−11.1944	−0.851093	−0.425547	−	0.904937i	$-0.639918\pi$
$173$	−11.1944	−0.851093	−0.425547	+	0.904937i	$0.639918\pi$
$174$	8.64841	0.655634
$175$	3.76683	0.284746
$176$	24.6269	1.85632
$177$	9.71967	0.730575
$178$	34.9018	2.61600
$179$	8.27486	0.618492	0.309246	−	0.950982i	$-0.399923\pi$
$179$	8.27486	0.618492	0.309246	+	0.950982i	$0.399923\pi$
$180$	−19.4297	−1.44820
$181$	8.76385	0.651412	0.325706	−	0.945471i	$-0.394398\pi$
$181$	8.76385	0.651412	0.325706	+	0.945471i	$0.394398\pi$
$182$	2.68629	0.199121
$183$	6.02125	0.445104
$184$	−45.4221	−3.34856
$185$	23.0683	1.69601
$186$	31.6000	2.31703
$187$	5.17700	0.378580
$188$	26.6934	1.94681
$189$	−2.13697	−0.155442
$190$	26.0951	1.89314
$191$	−17.7188	−1.28209	−0.641043	−	0.767505i	$-0.721498\pi$
$191$	−17.7188	−1.28209	−0.641043	+	0.767505i	$0.721498\pi$
$192$	1.30603	0.0942544
$193$	−19.7292	−1.42014	−0.710070	−	0.704131i	$-0.751337\pi$
$193$	−19.7292	−1.42014	−0.710070	+	0.704131i	$0.751337\pi$
$194$	3.74105	0.268592
$195$	12.1697	0.871493
$196$	−30.0388	−2.14563
$197$	2.78304	0.198283	0.0991416	−	0.995073i	$-0.468390\pi$
$197$	2.78304	0.198283	0.0991416	+	0.995073i	$0.468390\pi$
$198$	−10.8433	−0.770600
$199$	−9.32156	−0.660788	−0.330394	−	0.943843i	$-0.607182\pi$
$199$	−9.32156	−0.660788	−0.330394	+	0.943843i	$0.607182\pi$
$200$	−40.4582	−2.86083
$201$	−29.3845	−2.07262
$202$	6.64935	0.467847
$203$	−0.982535	−0.0689604
$204$	14.5966	1.02197
$205$	−0.432805	−0.0302284
$206$	−24.5837	−1.71283
$207$	9.01072	0.626289
$208$	−12.9994	−0.901347
$209$	10.0988	0.698545
$210$	10.6497	0.734902
$211$	10.0067	0.688888	0.344444	−	0.938807i	$-0.388067\pi$
$211$	10.0067	0.688888	0.344444	+	0.938807i	$0.388067\pi$
$212$	−0.0970043	−0.00666228
$213$	−7.62110	−0.522189
$214$	−28.2885	−1.93376
$215$	25.1974	1.71845
$216$	22.9524	1.56172
$217$	−3.59004	−0.243708
$218$	7.70986	0.522177
$219$	4.28710	0.289695
$220$	−50.4235	−3.39955
$221$	−2.73270	−0.183821
$222$	36.2981	2.43617
$223$	−5.49438	−0.367931	−0.183966	−	0.982933i	$-0.558894\pi$
$223$	−5.49438	−0.367931	−0.183966	+	0.982933i	$0.558894\pi$
$224$	−3.63567	−0.242919
$225$	8.02600	0.535066
$226$	−0.652332	−0.0433925
$227$	−7.90549	−0.524706	−0.262353	−	0.964972i	$-0.584498\pi$
$227$	−7.90549	−0.524706	−0.262353	+	0.964972i	$0.584498\pi$
$228$	28.4735	1.88570
$229$	7.32223	0.483866	0.241933	−	0.970293i	$-0.422219\pi$
$229$	7.32223	0.483866	0.241933	+	0.970293i	$0.422219\pi$
$230$	60.4252	3.98432
$231$	4.12143	0.271170
$232$	10.5531	0.692842
$233$	22.6079	1.48109	0.740547	−	0.672004i	$-0.234566\pi$
$233$	22.6079	1.48109	0.740547	+	0.672004i	$0.234566\pi$
$234$	5.72368	0.374168
$235$	−19.8121	−1.29240
$236$	21.2577	1.38376
$237$	11.8347	0.768748
$238$	−2.39139	−0.155011
$239$	−10.4997	−0.679167	−0.339583	−	0.940576i	$-0.610286\pi$
$239$	−10.4997	−0.679167	−0.339583	+	0.940576i	$0.610286\pi$
$240$	−51.5360	−3.32663
$241$	−22.3641	−1.44060	−0.720299	−	0.693664i	$-0.755995\pi$
$241$	−22.3641	−1.44060	−0.720299	+	0.693664i	$0.755995\pi$
$242$	−0.0436520	−0.00280605
$243$	−12.4903	−0.801252
$244$	13.1690	0.843057
$245$	22.2952	1.42439
$246$	−0.681021	−0.0434203
$247$	−5.33067	−0.339182
$248$	38.5594	2.44852
$249$	−18.8451	−1.19426
$250$	10.9378	0.691764
$251$	−15.5331	−0.980439	−0.490219	−	0.871599i	$-0.663083\pi$
$251$	−15.5331	−0.980439	−0.490219	+	0.871599i	$0.663083\pi$
$252$	3.47333	0.218799
$253$	23.3844	1.47017
$254$	−24.4423	−1.53365
$255$	−10.8338	−0.678436
$256$	−28.0845	−1.75528
$257$	−3.96399	−0.247267	−0.123633	−	0.992328i	$-0.539455\pi$
$257$	−3.96399	−0.247267	−0.123633	+	0.992328i	$0.539455\pi$
$258$	39.6482	2.46839
$259$	−4.12378	−0.256239
$260$	26.6162	1.65067
$261$	−2.09349	−0.129584
$262$	−42.0962	−2.60071
$263$	−12.5981	−0.776835	−0.388418	−	0.921483i	$-0.626978\pi$
$263$	−12.5981	−0.776835	−0.388418	+	0.921483i	$0.626978\pi$
$264$	−44.2668	−2.72443
$265$	0.0719977	0.00442278
$266$	−4.66487	−0.286022
$267$	−28.2655	−1.72982
$268$	−64.2663	−3.92569
$269$	28.1036	1.71351	0.856754	−	0.515725i	$-0.172477\pi$
$269$	28.1036	1.71351	0.856754	+	0.515725i	$0.172477\pi$
$270$	−30.5337	−1.85822
$271$	−12.2538	−0.744368	−0.372184	−	0.928159i	$-0.621391\pi$
$271$	−12.2538	−0.744368	−0.372184	+	0.928159i	$0.621391\pi$
$272$	11.5724	0.701677
$273$	−2.17551	−0.131668
$274$	−14.2055	−0.858185
$275$	20.8289	1.25603
$276$	65.9325	3.96867
$277$	12.3403	0.741457	0.370729	−	0.928741i	$-0.379108\pi$
$277$	12.3403	0.741457	0.370729	+	0.928741i	$0.379108\pi$
$278$	17.7372	1.06380
$279$	−7.64931	−0.457952
$280$	12.9952	0.776609
$281$	−15.2135	−0.907562	−0.453781	−	0.891113i	$-0.649925\pi$
$281$	−15.2135	−0.907562	−0.453781	+	0.891113i	$0.649925\pi$
$282$	−31.1745	−1.85642
$283$	7.52407	0.447260	0.223630	−	0.974674i	$-0.428209\pi$
$283$	7.52407	0.447260	0.223630	+	0.974674i	$0.428209\pi$
$284$	−16.6680	−0.989062
$285$	−21.1334	−1.25183
$286$	14.8540	0.878333
$287$	0.0773699	0.00456700
$288$	−7.74653	−0.456469
$289$	−14.5673	−0.856899
$290$	−14.0388	−0.824385
$291$	−3.02972	−0.177605
$292$	9.37624	0.548703
$293$	−18.8155	−1.09921	−0.549605	−	0.835424i	$-0.685222\pi$
$293$	−18.8155	−1.09921	−0.549605	+	0.835424i	$0.685222\pi$
$294$	35.0816	2.04600
$295$	−15.7777	−0.918615
$296$	44.2920	2.57442
$297$	−11.8165	−0.685661
$298$	6.99556	0.405242
$299$	−12.3436	−0.713847
$300$	58.7271	3.39061
$301$	−4.50438	−0.259628
$302$	−60.8508	−3.50157
$303$	−5.38503	−0.309362
$304$	22.5741	1.29472
$305$	−9.77417	−0.559667
$306$	−5.09534	−0.291281
$307$	−18.8315	−1.07477	−0.537384	−	0.843338i	$-0.680587\pi$
$307$	−18.8315	−1.07477	−0.537384	+	0.843338i	$0.680587\pi$
$308$	9.01390	0.513615
$309$	19.9093	1.13260
$310$	−51.2957	−2.91340
$311$	11.3735	0.644931	0.322466	−	0.946581i	$-0.395488\pi$
$311$	11.3735	0.644931	0.322466	+	0.946581i	$0.395488\pi$
$312$	23.3664	1.32286
$313$	−17.1557	−0.969695	−0.484847	−	0.874599i	$-0.661125\pi$
$313$	−17.1557	−0.969695	−0.484847	+	0.874599i	$0.661125\pi$
$314$	0.877866	0.0495408
$315$	−2.57795	−0.145251
$316$	25.8835	1.45606
$317$	−0.361731	−0.0203169	−0.0101584	−	0.999948i	$-0.503234\pi$
$317$	−0.361731	−0.0203169	−0.0101584	+	0.999948i	$0.503234\pi$
$318$	0.113289	0.00635292
$319$	−5.43298	−0.304188
$320$	−2.12005	−0.118514
$321$	22.9097	1.27869
$322$	−10.8019	−0.601964
$323$	4.74547	0.264045
$324$	−50.6755	−2.81531
$325$	−10.9946	−0.609871
$326$	−31.2795	−1.73241
$327$	−6.24389	−0.345288
$328$	−0.831003	−0.0458845
$329$	3.54170	0.195260
$330$	58.8883	3.24169
$331$	−25.5360	−1.40358	−0.701792	−	0.712381i	$-0.747617\pi$
$331$	−25.5360	−1.40358	−0.701792	+	0.712381i	$0.747617\pi$
$332$	−41.2158	−2.26201
$333$	−8.78654	−0.481499
$334$	−50.6573	−2.77185
$335$	47.6992	2.60609
$336$	9.21278	0.502599
$337$	−33.4370	−1.82143	−0.910716	−	0.413034i	$-0.864469\pi$
$337$	−33.4370	−1.82143	−0.910716	+	0.413034i	$0.864469\pi$
$338$	25.3644	1.37964
$339$	0.528296	0.0286931
$340$	−23.6943	−1.28500
$341$	−19.8513	−1.07501
$342$	−9.93945	−0.537464
$343$	−8.18744	−0.442080
$344$	48.3800	2.60847
$345$	−48.9359	−2.63462
$346$	28.5931	1.53718
$347$	−34.6713	−1.86125	−0.930627	−	0.365968i	$-0.880738\pi$
$347$	−34.6713	−1.86125	−0.930627	+	0.365968i	$0.880738\pi$
$348$	−15.3183	−0.821147
$349$	−24.4503	−1.30879	−0.654396	−	0.756152i	$-0.727077\pi$
$349$	−24.4503	−1.30879	−0.654396	+	0.756152i	$0.727077\pi$
$350$	−9.62139	−0.514285
$351$	6.23737	0.332926
$352$	−20.1036	−1.07153
$353$	0.529052	0.0281586	0.0140793	−	0.999901i	$-0.495518\pi$
$353$	0.529052	0.0281586	0.0140793	+	0.999901i	$0.495518\pi$
$354$	−24.8264	−1.31951
$355$	12.3712	0.656593
$356$	−61.8190	−3.27640
$357$	1.93669	0.102500
$358$	−21.1360	−1.11707
$359$	−24.7373	−1.30558	−0.652791	−	0.757538i	$-0.726402\pi$
$359$	−24.7373	−1.30558	−0.652791	+	0.757538i	$0.726402\pi$
$360$	27.6888	1.45933
$361$	−9.74303	−0.512791
$362$	−22.3850	−1.17653
$363$	0.0353519	0.00185549
$364$	−4.75802	−0.249388
$365$	−6.95916	−0.364259
$366$	−15.3797	−0.803911
$367$	10.3645	0.541021	0.270510	−	0.962717i	$-0.412808\pi$
$367$	10.3645	0.541021	0.270510	+	0.962717i	$0.412808\pi$
$368$	52.2721	2.72487
$369$	0.164852	0.00858186
$370$	−58.9219	−3.06320
$371$	−0.0128706	−0.000668209 0
$372$	−55.9708	−2.90195
$373$	−24.3783	−1.26226	−0.631131	−	0.775676i	$-0.717409\pi$
$373$	−24.3783	−1.26226	−0.631131	+	0.775676i	$0.717409\pi$
$374$	−13.2233	−0.683761
$375$	−8.85803	−0.457427
$376$	−38.0401	−1.96177
$377$	2.86782	0.147700
$378$	5.45833	0.280746
$379$	−15.1734	−0.779407	−0.389703	−	0.920940i	$-0.627422\pi$
$379$	−15.1734	−0.779407	−0.389703	+	0.920940i	$0.627422\pi$
$380$	−46.2204	−2.37106
$381$	19.7948	1.01412
$382$	45.2580	2.31560
$383$	1.96762	0.100540	0.0502702	−	0.998736i	$-0.483992\pi$
$383$	1.96762	0.100540	0.0502702	+	0.998736i	$0.483992\pi$
$384$	21.7218	1.10849
$385$	−6.69023	−0.340965
$386$	50.3931	2.56494
$387$	−9.59749	−0.487868
$388$	−6.62624	−0.336397
$389$	26.1395	1.32533	0.662663	−	0.748918i	$-0.269426\pi$
$389$	26.1395	1.32533	0.662663	+	0.748918i	$0.269426\pi$
$390$	−31.0844	−1.57402
$391$	10.9885	0.555712
$392$	42.8077	2.16211
$393$	34.0920	1.71971
$394$	−7.10854	−0.358123
$395$	−19.2111	−0.966613
$396$	19.2059	0.965135
$397$	−6.69314	−0.335919	−0.167960	−	0.985794i	$-0.553718\pi$
$397$	−6.69314	−0.335919	−0.167960	+	0.985794i	$0.553718\pi$
$398$	23.8095	1.19346
$399$	3.77788	0.189131
$400$	46.5596	2.32798
$401$	−27.8506	−1.39079	−0.695396	−	0.718626i	$-0.744771\pi$
$401$	−27.8506	−1.39079	−0.695396	+	0.718626i	$0.744771\pi$
$402$	75.0550	3.74340
$403$	10.4786	0.521976
$404$	−11.7775	−0.585953
$405$	37.6120	1.86895
$406$	2.50963	0.124551
$407$	−22.8026	−1.13028
$408$	−20.8013	−1.02982
$409$	−16.4291	−0.812369	−0.406184	−	0.913791i	$-0.633141\pi$
$409$	−16.4291	−0.812369	−0.406184	+	0.913791i	$0.633141\pi$
$410$	1.10549	0.0545961
$411$	11.5044	0.567472
$412$	43.5433	2.14522
$413$	2.82049	0.138787
$414$	−23.0155	−1.13115
$415$	30.5909	1.50165
$416$	10.6118	0.520285
$417$	−14.3646	−0.703437
$418$	−25.7946	−1.26166
$419$	−11.5773	−0.565587	−0.282793	−	0.959181i	$-0.591261\pi$
$419$	−11.5773	−0.565587	−0.282793	+	0.959181i	$0.591261\pi$
$420$	−18.8631	−0.920426
$421$	29.9946	1.46185	0.730923	−	0.682460i	$-0.239090\pi$
$421$	29.9946	1.46185	0.730923	+	0.682460i	$0.239090\pi$
$422$	−25.5595	−1.24421
$423$	7.54630	0.366914
$424$	0.138239	0.00671346
$425$	9.78763	0.474770
$426$	19.4661	0.943136
$427$	1.74727	0.0845563
$428$	50.1053	2.42193
$429$	−12.0296	−0.580794
$430$	−64.3601	−3.10372
$431$	−23.8094	−1.14686	−0.573428	−	0.819256i	$-0.694387\pi$
$431$	−23.8094	−1.14686	−0.573428	+	0.819256i	$0.694387\pi$
$432$	−26.4138	−1.27084
$433$	−15.9433	−0.766188	−0.383094	−	0.923709i	$-0.625142\pi$
$433$	−15.9433	−0.766188	−0.383094	+	0.923709i	$0.625142\pi$
$434$	9.16982	0.440166
$435$	11.3694	0.545122
$436$	−13.6559	−0.653999
$437$	21.4352	1.02539
$438$	−10.9503	−0.523225
$439$	19.0959	0.911396	0.455698	−	0.890135i	$-0.349390\pi$
$439$	19.0959	0.911396	0.455698	+	0.890135i	$0.349390\pi$
$440$	71.8573	3.42566
$441$	−8.49207	−0.404384
$442$	6.97997	0.332003
$443$	−5.05195	−0.240025	−0.120013	−	0.992772i	$-0.538293\pi$
$443$	−5.05195	−0.240025	−0.120013	+	0.992772i	$0.538293\pi$
$444$	−64.2921	−3.05117
$445$	45.8828	2.17505
$446$	14.0340	0.664528
$447$	−5.66541	−0.267965
$448$	0.378988	0.0179055
$449$	−1.32425	−0.0624954	−0.0312477	−	0.999512i	$-0.509948\pi$
$449$	−1.32425	−0.0624954	−0.0312477	+	0.999512i	$0.509948\pi$
$450$	−20.5003	−0.966394
$451$	0.427821	0.0201453
$452$	1.15543	0.0543468
$453$	49.2805	2.31540
$454$	20.1925	0.947682
$455$	3.53146	0.165557
$456$	−40.5769	−1.90019
$457$	18.7138	0.875397	0.437698	−	0.899122i	$-0.355794\pi$
$457$	18.7138	0.875397	0.437698	+	0.899122i	$0.355794\pi$
$458$	−18.7027	−0.873921
$459$	−5.55264	−0.259175
$460$	−107.027	−4.99015
$461$	−1.94379	−0.0905313	−0.0452657	−	0.998975i	$-0.514413\pi$
$461$	−1.94379	−0.0905313	−0.0452657	+	0.998975i	$0.514413\pi$
$462$	−10.5271	−0.489766
$463$	14.2751	0.663418	0.331709	−	0.943382i	$-0.392375\pi$
$463$	14.2751	0.663418	0.331709	+	0.943382i	$0.392375\pi$
$464$	−12.1445	−0.563796
$465$	41.5422	1.92647
$466$	−57.7460	−2.67503
$467$	5.30336	0.245410	0.122705	−	0.992443i	$-0.460843\pi$
$467$	5.30336	0.245410	0.122705	+	0.992443i	$0.460843\pi$
$468$	−10.1379	−0.468626
$469$	−8.52691	−0.393736
$470$	50.6049	2.33423
$471$	−0.710947	−0.0327587
$472$	−30.2939	−1.39439
$473$	−24.9072	−1.14523
$474$	−30.2287	−1.38845
$475$	19.0927	0.876032
$476$	4.23569	0.194143
$477$	−0.0274234	−0.00125563
$478$	26.8186	1.22666
$479$	−25.5530	−1.16755	−0.583774	−	0.811916i	$-0.698425\pi$
$479$	−25.5530	−1.16755	−0.583774	+	0.811916i	$0.698425\pi$
$480$	42.0702	1.92024
$481$	12.0365	0.548815
$482$	57.1232	2.60189
$483$	8.74798	0.398047
$484$	0.0773175	0.00351443
$485$	4.91808	0.223318
$486$	31.9031	1.44716
$487$	−6.41911	−0.290878	−0.145439	−	0.989367i	$-0.546459\pi$
$487$	−6.41911	−0.290878	−0.145439	+	0.989367i	$0.546459\pi$
$488$	−18.7668	−0.849534
$489$	25.3320	1.14555
$490$	−56.9472	−2.57261
$491$	−22.9683	−1.03654	−0.518272	−	0.855216i	$-0.673425\pi$
$491$	−22.9683	−1.03654	−0.518272	+	0.855216i	$0.673425\pi$
$492$	1.20624	0.0543816
$493$	−2.55299	−0.114981
$494$	13.6158	0.612604
$495$	−14.2549	−0.640709
$496$	−44.3744	−1.99247
$497$	−2.21152	−0.0992003
$498$	48.1349	2.15698
$499$	0.335176	0.0150045	0.00750226	−	0.999972i	$-0.497612\pi$
$499$	0.335176	0.0150045	0.00750226	+	0.999972i	$0.497612\pi$
$500$	−19.3732	−0.866398
$501$	41.0252	1.83287
$502$	39.6752	1.77079
$503$	−42.3030	−1.88620	−0.943100	−	0.332510i	$-0.892105\pi$
$503$	−42.3030	−1.88620	−0.943100	+	0.332510i	$0.892105\pi$
$504$	−4.94976	−0.220480
$505$	8.74140	0.388987
$506$	−59.7294	−2.65530
$507$	−20.5416	−0.912283
$508$	43.2929	1.92081
$509$	24.8874	1.10311	0.551557	−	0.834137i	$-0.314034\pi$
$509$	24.8874	1.10311	0.551557	+	0.834137i	$0.314034\pi$
$510$	27.6720	1.22534
$511$	1.24405	0.0550334
$512$	50.7327	2.24209
$513$	−10.8315	−0.478223
$514$	10.1250	0.446593
$515$	−32.3183	−1.42412
$516$	−70.2260	−3.09153
$517$	19.5840	0.861303
$518$	10.5331	0.462798
$519$	−23.1564	−1.01645
$520$	−37.9302	−1.66335
$521$	8.79725	0.385415	0.192707	−	0.981256i	$-0.438273\pi$
$521$	8.79725	0.385415	0.192707	+	0.981256i	$0.438273\pi$
$522$	5.34727	0.234044
$523$	12.3253	0.538947	0.269474	−	0.963008i	$-0.413150\pi$
$523$	12.3253	0.538947	0.269474	+	0.963008i	$0.413150\pi$
$524$	74.5619	3.25725
$525$	7.79196	0.340069
$526$	32.1787	1.40306
$527$	−9.32826	−0.406345
$528$	50.9425	2.21699
$529$	26.6348	1.15804
$530$	−0.183899	−0.00798808
$531$	6.00963	0.260796
$532$	8.26254	0.358227
$533$	−0.225827	−0.00978164
$534$	72.1969	3.12426
$535$	−37.1888	−1.60781
$536$	91.5845	3.95585
$537$	17.1171	0.738659
$538$	−71.7834	−3.09480
$539$	−22.0384	−0.949262
$540$	54.0821	2.32732
$541$	13.8872	0.597059	0.298529	−	0.954400i	$-0.403504\pi$
$541$	13.8872	0.597059	0.298529	+	0.954400i	$0.403504\pi$
$542$	31.2992	1.34442
$543$	18.1286	0.777975
$544$	−9.44683	−0.405029
$545$	10.1356	0.434160
$546$	5.55678	0.237808
$547$	32.3056	1.38129	0.690645	−	0.723194i	$-0.257327\pi$
$547$	32.3056	1.38129	0.690645	+	0.723194i	$0.257327\pi$
$548$	25.1611	1.07483
$549$	3.72291	0.158890
$550$	−53.2019	−2.26854
$551$	−4.98011	−0.212160
$552$	−93.9589	−3.99916
$553$	3.43425	0.146039
$554$	−31.5201	−1.33916
$555$	47.7184	2.02553
$556$	−31.4166	−1.33236
$557$	6.71032	0.284325	0.142163	−	0.989843i	$-0.454594\pi$
$557$	6.71032	0.284325	0.142163	+	0.989843i	$0.454594\pi$
$558$	19.5382	0.827116
$559$	13.1474	0.556074
$560$	−14.9549	−0.631960
$561$	10.7090	0.452134
$562$	38.8589	1.63917
$563$	−25.0949	−1.05762	−0.528812	−	0.848739i	$-0.677362\pi$
$563$	−25.0949	−1.05762	−0.528812	+	0.848739i	$0.677362\pi$
$564$	55.2172	2.32506
$565$	−0.857572	−0.0360783
$566$	−19.2183	−0.807805
$567$	−6.72367	−0.282368
$568$	23.7532	0.996661
$569$	38.0939	1.59698	0.798490	−	0.602008i	$-0.205632\pi$
$569$	38.0939	1.59698	0.798490	+	0.602008i	$0.205632\pi$
$570$	53.9796	2.26096
$571$	−33.5722	−1.40495	−0.702476	−	0.711707i	$-0.747922\pi$
$571$	−33.5722	−1.40495	−0.702476	+	0.711707i	$0.747922\pi$
$572$	−26.3097	−1.10006
$573$	−36.6526	−1.53118
$574$	−0.197621	−0.00824855
$575$	44.2106	1.84371
$576$	0.807510	0.0336463
$577$	4.56595	0.190083	0.0950416	−	0.995473i	$-0.469702\pi$
$577$	4.56595	0.190083	0.0950416	+	0.995473i	$0.469702\pi$
$578$	37.2083	1.54766
$579$	−40.8113	−1.69606
$580$	24.8659	1.03250
$581$	−5.46855	−0.226874
$582$	7.73863	0.320776
$583$	−0.0711686	−0.00294750
$584$	−13.3619	−0.552918
$585$	7.52449	0.311099
$586$	48.0592	1.98531
$587$	18.5816	0.766943	0.383471	−	0.923553i	$-0.374728\pi$
$587$	18.5816	0.766943	0.383471	+	0.923553i	$0.374728\pi$
$588$	−62.1374	−2.56250
$589$	−18.1966	−0.749777
$590$	40.3001	1.65913
$591$	5.75691	0.236808
$592$	−50.9716	−2.09492
$593$	3.03764	0.124741	0.0623704	−	0.998053i	$-0.480134\pi$
$593$	3.03764	0.124741	0.0623704	+	0.998053i	$0.480134\pi$
$594$	30.1821	1.23839
$595$	−3.14378	−0.128882
$596$	−12.3907	−0.507544
$597$	−19.2823	−0.789173
$598$	31.5284	1.28929
$599$	27.0249	1.10421	0.552104	−	0.833775i	$-0.313825\pi$
$599$	27.0249	1.10421	0.552104	+	0.833775i	$0.313825\pi$
$600$	−83.6907	−3.41666
$601$	−4.42499	−0.180499	−0.0902495	−	0.995919i	$-0.528766\pi$
$601$	−4.42499	−0.180499	−0.0902495	+	0.995919i	$0.528766\pi$
$602$	11.5053	0.468920
$603$	−18.1683	−0.739870
$604$	107.781	4.38553
$605$	−0.0573860	−0.00233307
$606$	13.7547	0.558745
$607$	−1.93810	−0.0786651	−0.0393325	−	0.999226i	$-0.512523\pi$
$607$	−1.93810	−0.0786651	−0.0393325	+	0.999226i	$0.512523\pi$
$608$	−18.4279	−0.747349
$609$	−2.03244	−0.0823588
$610$	24.9656	1.01083
$611$	−10.3375	−0.418210
$612$	9.02500	0.364814
$613$	4.17939	0.168804	0.0844020	−	0.996432i	$-0.473102\pi$
$613$	4.17939	0.168804	0.0844020	+	0.996432i	$0.473102\pi$
$614$	48.1001	1.94116
$615$	−0.895287	−0.0361015
$616$	−12.8455	−0.517560
$617$	−25.6874	−1.03414	−0.517069	−	0.855944i	$-0.672977\pi$
$617$	−25.6874	−1.03414	−0.517069	+	0.855944i	$0.672977\pi$
$618$	−50.8531	−2.04561
$619$	3.61165	0.145164	0.0725821	−	0.997362i	$-0.476876\pi$
$619$	3.61165	0.145164	0.0725821	+	0.997362i	$0.476876\pi$
$620$	90.8562	3.64887
$621$	−25.0812	−1.00647
$622$	−29.0506	−1.16482
$623$	−8.20220	−0.328614
$624$	−26.8902	−1.07647
$625$	−16.9973	−0.679893
$626$	43.8196	1.75138
$627$	20.8900	0.834266
$628$	−1.55490	−0.0620472
$629$	−10.7151	−0.427239
$630$	6.58469	0.262340
$631$	−5.91452	−0.235453	−0.117727	−	0.993046i	$-0.537561\pi$
$631$	−5.91452	−0.235453	−0.117727	+	0.993046i	$0.537561\pi$
$632$	−36.8860	−1.46725
$633$	20.6995	0.822733
$634$	0.923948	0.0366947
$635$	−32.1325	−1.27514
$636$	−0.200660	−0.00795670
$637$	11.6331	0.460919
$638$	13.8771	0.549400
$639$	−4.71209	−0.186407
$640$	−35.2605	−1.39380
$641$	−10.8446	−0.428336	−0.214168	−	0.976797i	$-0.568704\pi$
$641$	−10.8446	−0.428336	−0.214168	+	0.976797i	$0.568704\pi$
$642$	−58.5168	−2.30947
$643$	−2.91258	−0.114861	−0.0574304	−	0.998350i	$-0.518291\pi$
$643$	−2.91258	−0.114861	−0.0574304	+	0.998350i	$0.518291\pi$
$644$	19.1325	0.753928
$645$	52.1225	2.05232
$646$	−12.1211	−0.476897
$647$	0.983498	0.0386653	0.0193327	−	0.999813i	$-0.493846\pi$
$647$	0.983498	0.0386653	0.0193327	+	0.999813i	$0.493846\pi$
$648$	72.2166	2.83694
$649$	15.5960	0.612198
$650$	28.0829	1.10150
$651$	−7.42626	−0.291058
$652$	55.4031	2.16975
$653$	15.9471	0.624059	0.312030	−	0.950072i	$-0.398991\pi$
$653$	15.9471	0.624059	0.312030	+	0.950072i	$0.398991\pi$
$654$	15.9484	0.623631
$655$	−55.3407	−2.16234
$656$	0.956324	0.0373382
$657$	2.65069	0.103413
$658$	−9.04635	−0.352663
$659$	−13.7651	−0.536214	−0.268107	−	0.963389i	$-0.586398\pi$
$659$	−13.7651	−0.536214	−0.268107	+	0.963389i	$0.586398\pi$
$660$	−104.305	−4.06005
$661$	−13.0731	−0.508486	−0.254243	−	0.967140i	$-0.581826\pi$
$661$	−13.0731	−0.508486	−0.254243	+	0.967140i	$0.581826\pi$
$662$	65.2250	2.53504
$663$	−5.65279	−0.219536
$664$	58.7358	2.27939
$665$	−6.13256	−0.237810
$666$	22.4429	0.869645
$667$	−11.5318	−0.446514
$668$	89.7256	3.47159
$669$	−11.3655	−0.439417
$670$	−121.835	−4.70690
$671$	9.66162	0.372983
$672$	−7.52065	−0.290115
$673$	7.48433	0.288500	0.144250	−	0.989541i	$-0.453923\pi$
$673$	7.48433	0.288500	0.144250	+	0.989541i	$0.453923\pi$
$674$	85.4062	3.28972
$675$	−22.3402	−0.859875
$676$	−44.9261	−1.72793
$677$	26.4949	1.01828	0.509140	−	0.860683i	$-0.329963\pi$
$677$	26.4949	1.01828	0.509140	+	0.860683i	$0.329963\pi$
$678$	−1.34940	−0.0518232
$679$	−0.879176	−0.0337397
$680$	33.7662	1.29488
$681$	−16.3531	−0.626651
$682$	50.7050	1.94159
$683$	20.6973	0.791960	0.395980	−	0.918259i	$-0.370405\pi$
$683$	20.6973	0.791960	0.395980	+	0.918259i	$0.370405\pi$
$684$	17.6050	0.673145
$685$	−18.6749	−0.713531
$686$	20.9127	0.798449
$687$	15.1465	0.577877
$688$	−55.6760	−2.12263
$689$	0.0375666	0.00143118
$690$	124.994	4.75844
$691$	1.85949	0.0707384	0.0353692	−	0.999374i	$-0.488739\pi$
$691$	1.85949	0.0707384	0.0353692	+	0.999374i	$0.488739\pi$
$692$	−50.6449	−1.92523
$693$	2.54826	0.0968004
$694$	88.5589	3.36165
$695$	23.3177	0.884492
$696$	21.8298	0.827455
$697$	0.201036	0.00761477
$698$	62.4518	2.36384
$699$	46.7661	1.76886
$700$	17.0417	0.644114
$701$	15.3334	0.579133	0.289567	−	0.957158i	$-0.406489\pi$
$701$	15.3334	0.579133	0.289567	+	0.957158i	$0.406489\pi$
$702$	−15.9317	−0.601305
$703$	−20.9019	−0.788330
$704$	2.09563	0.0789821
$705$	−40.9828	−1.54350
$706$	−1.35133	−0.0508578
$707$	−1.56265	−0.0587695
$708$	43.9731	1.65261
$709$	−19.7781	−0.742781	−0.371390	−	0.928477i	$-0.621119\pi$
$709$	−19.7781	−0.742781	−0.371390	+	0.928477i	$0.621119\pi$
$710$	−31.5989	−1.18589
$711$	7.31736	0.274422
$712$	88.0969	3.30157
$713$	−42.1356	−1.57799
$714$	−4.94676	−0.185128
$715$	19.5274	0.730283
$716$	37.4366	1.39907
$717$	−21.7193	−0.811122
$718$	63.1849	2.35804
$719$	−32.7625	−1.22184	−0.610918	−	0.791694i	$-0.709200\pi$
$719$	−32.7625	−1.22184	−0.610918	+	0.791694i	$0.709200\pi$
$720$	−31.8645	−1.18752
$721$	5.77735	0.215160
$722$	24.8860	0.926162
$723$	−46.2617	−1.72049
$724$	39.6488	1.47354
$725$	−10.2716	−0.381477
$726$	−0.0902972	−0.00335124
$727$	27.3053	1.01270	0.506348	−	0.862329i	$-0.330995\pi$
$727$	27.3053	1.01270	0.506348	+	0.862329i	$0.330995\pi$
$728$	6.78055	0.251304
$729$	7.76643	0.287646
$730$	17.7753	0.657895
$731$	−11.7041	−0.432890
$732$	27.2410	1.00686
$733$	−38.3155	−1.41522	−0.707608	−	0.706606i	$-0.750226\pi$
$733$	−38.3155	−1.41522	−0.707608	+	0.706606i	$0.750226\pi$
$734$	−26.4733	−0.977149
$735$	46.1191	1.70113
$736$	−42.6711	−1.57288
$737$	−47.1499	−1.73679
$738$	−0.421072	−0.0154999
$739$	24.5174	0.901886	0.450943	−	0.892553i	$-0.351088\pi$
$739$	24.5174	0.901886	0.450943	+	0.892553i	$0.351088\pi$
$740$	104.364	3.83650
$741$	−11.0269	−0.405082
$742$	0.0328746	0.00120686
$743$	18.1547	0.666032	0.333016	−	0.942921i	$-0.391934\pi$
$743$	18.1547	0.666032	0.333016	+	0.942921i	$0.391934\pi$
$744$	79.7628	2.92425
$745$	9.19654	0.336935
$746$	62.2680	2.27980
$747$	−11.6518	−0.426319
$748$	23.4215	0.856374
$749$	6.64802	0.242913
$750$	22.6255	0.826167
$751$	−29.5776	−1.07930	−0.539651	−	0.841889i	$-0.681444\pi$
$751$	−29.5776	−1.07930	−0.539651	+	0.841889i	$0.681444\pi$
$752$	43.7768	1.59638
$753$	−32.1313	−1.17093
$754$	−7.32509	−0.266764
$755$	−79.9960	−2.91135
$756$	−9.66794	−0.351620
$757$	−6.24912	−0.227128	−0.113564	−	0.993531i	$-0.536227\pi$
$757$	−6.24912	−0.227128	−0.113564	+	0.993531i	$0.536227\pi$
$758$	38.7566	1.40770
$759$	48.3724	1.75581
$760$	65.8676	2.38927
$761$	−51.6154	−1.87106	−0.935529	−	0.353251i	$-0.885076\pi$
$761$	−51.6154	−1.87106	−0.935529	+	0.353251i	$0.885076\pi$
$762$	−50.5607	−1.83162
$763$	−1.81188	−0.0655943
$764$	−80.1622	−2.90016
$765$	−6.69846	−0.242183
$766$	−5.02576	−0.181588
$767$	−8.23243	−0.297256
$768$	−58.0947	−2.09631
$769$	−25.8466	−0.932054	−0.466027	−	0.884771i	$-0.654315\pi$
$769$	−25.8466	−0.932054	−0.466027	+	0.884771i	$0.654315\pi$
$770$	17.0884	0.615824
$771$	−8.19979	−0.295308
$772$	−89.2577	−3.21245
$773$	−28.6547	−1.03064	−0.515319	−	0.856998i	$-0.672327\pi$
$773$	−28.6547	−1.03064	−0.515319	+	0.856998i	$0.672327\pi$
$774$	24.5143	0.881148
$775$	−37.5308	−1.34815
$776$	9.44291	0.338981
$777$	−8.53033	−0.306024
$778$	−66.7666	−2.39370
$779$	0.392159	0.0140506
$780$	55.0575	1.97138
$781$	−12.2287	−0.437578
$782$	−28.0673	−1.00368
$783$	5.82718	0.208247
$784$	−49.2633	−1.75940
$785$	1.15406	0.0411903
$786$	−87.0790	−3.10601
$787$	36.1988	1.29035	0.645173	−	0.764036i	$-0.276785\pi$
$787$	36.1988	1.29035	0.645173	+	0.764036i	$0.276785\pi$
$788$	12.5908	0.448530
$789$	−26.0602	−0.927767
$790$	49.0696	1.74582
$791$	0.153303	0.00545083
$792$	−27.3700	−0.972549
$793$	−5.09992	−0.181104
$794$	17.0959	0.606710
$795$	0.148932	0.00528209
$796$	−42.1720	−1.49475
$797$	35.8433	1.26964	0.634818	−	0.772662i	$-0.281075\pi$
$797$	35.8433	1.26964	0.634818	+	0.772662i	$0.281075\pi$
$798$	−9.64962	−0.341593
$799$	9.20265	0.325566
$800$	−38.0079	−1.34378
$801$	−17.4764	−0.617499
$802$	71.1371	2.51194
$803$	6.87902	0.242755
$804$	−132.939	−4.68841
$805$	−14.2004	−0.500498
$806$	−26.7648	−0.942750
$807$	58.1344	2.04643
$808$	16.7839	0.590454
$809$	1.54061	0.0541651	0.0270826	−	0.999633i	$-0.491378\pi$
$809$	1.54061	0.0541651	0.0270826	+	0.999633i	$0.491378\pi$
$810$	−96.0700	−3.37556
$811$	7.76565	0.272689	0.136344	−	0.990661i	$-0.456465\pi$
$811$	7.76565	0.272689	0.136344	+	0.990661i	$0.456465\pi$
$812$	−4.44512	−0.155993
$813$	−25.3480	−0.888992
$814$	58.2434	2.04143
$815$	−41.1209	−1.44040
$816$	23.9382	0.838006
$817$	−22.8310	−0.798757
$818$	41.9640	1.46724
$819$	−1.34511	−0.0470019
$820$	−1.95807	−0.0683787
$821$	−45.2059	−1.57770	−0.788848	−	0.614588i	$-0.789322\pi$
$821$	−45.2059	−1.57770	−0.788848	+	0.614588i	$0.789322\pi$
$822$	−29.3851	−1.02492
$823$	−43.5268	−1.51725	−0.758625	−	0.651528i	$-0.774128\pi$
$823$	−43.5268	−1.51725	−0.758625	+	0.651528i	$0.774128\pi$
$824$	−62.0525	−2.16170
$825$	43.0860	1.50006
$826$	−7.20421	−0.250666
$827$	−50.0432	−1.74017	−0.870086	−	0.492901i	$-0.835937\pi$
$827$	−50.0432	−1.74017	−0.870086	+	0.492901i	$0.835937\pi$
$828$	40.7657	1.41671
$829$	−1.64272	−0.0570540	−0.0285270	−	0.999593i	$-0.509082\pi$
$829$	−1.64272	−0.0570540	−0.0285270	+	0.999593i	$0.509082\pi$
$830$	−78.1364	−2.71215
$831$	25.5268	0.885515
$832$	−1.10619	−0.0383502
$833$	−10.3560	−0.358814
$834$	36.6906	1.27049
$835$	−66.5954	−2.30463
$836$	45.6881	1.58016
$837$	21.2917	0.735948
$838$	29.5711	1.02152
$839$	−29.7373	−1.02665	−0.513323	−	0.858195i	$-0.671586\pi$
$839$	−29.7373	−1.02665	−0.513323	+	0.858195i	$0.671586\pi$
$840$	26.8814	0.927497
$841$	−26.3208	−0.907613
$842$	−76.6133	−2.64027
$843$	−31.4702	−1.08389
$844$	45.2716	1.55831
$845$	33.3447	1.14709
$846$	−19.2751	−0.662690
$847$	0.0102586	0.000352488 0
$848$	−0.159086	−0.00546303
$849$	15.5641	0.534158
$850$	−25.0000	−0.857491
$851$	−48.3999	−1.65913
$852$	−34.4789	−1.18123
$853$	−27.7354	−0.949644	−0.474822	−	0.880082i	$-0.657487\pi$
$853$	−27.7354	−0.949644	−0.474822	+	0.880082i	$0.657487\pi$
$854$	−4.46295	−0.152719
$855$	−13.0667	−0.446870
$856$	−71.4040	−2.44054
$857$	23.2875	0.795485	0.397743	−	0.917497i	$-0.369794\pi$
$857$	23.2875	0.795485	0.397743	+	0.917497i	$0.369794\pi$
$858$	30.7265	1.04898
$859$	1.57598	0.0537718	0.0268859	−	0.999639i	$-0.491441\pi$
$859$	1.57598	0.0537718	0.0268859	+	0.999639i	$0.491441\pi$
$860$	113.996	3.88724
$861$	0.160045	0.00545433
$862$	60.8148	2.07136
$863$	32.0229	1.09007	0.545036	−	0.838412i	$-0.316516\pi$
$863$	32.0229	1.09007	0.545036	+	0.838412i	$0.316516\pi$
$864$	21.5623	0.733565
$865$	37.5892	1.27807
$866$	40.7231	1.38383
$867$	−30.1335	−1.02339
$868$	−16.2418	−0.551284
$869$	18.9898	0.644186
$870$	−29.0402	−0.984555
$871$	24.8883	0.843307
$872$	19.4607	0.659023
$873$	−1.87326	−0.0634003
$874$	−54.7506	−1.85197
$875$	−2.57046	−0.0868973
$876$	19.3954	0.655311
$877$	36.0868	1.21856	0.609282	−	0.792953i	$-0.291458\pi$
$877$	36.0868	1.21856	0.609282	+	0.792953i	$0.291458\pi$
$878$	−48.7754	−1.64609
$879$	−38.9211	−1.31278
$880$	−82.6939	−2.78761
$881$	−31.3587	−1.05650	−0.528250	−	0.849089i	$-0.677152\pi$
$881$	−31.3587	−1.05650	−0.528250	+	0.849089i	$0.677152\pi$
$882$	21.6908	0.730366
$883$	41.5333	1.39771	0.698853	−	0.715265i	$-0.253694\pi$
$883$	41.5333	1.39771	0.698853	+	0.715265i	$0.253694\pi$
$884$	−12.3631	−0.415816
$885$	−32.6374	−1.09709
$886$	12.9039	0.433514
$887$	−25.0488	−0.841055	−0.420527	−	0.907280i	$-0.638155\pi$
$887$	−25.0488	−0.841055	−0.420527	+	0.907280i	$0.638155\pi$
$888$	91.6212	3.07461
$889$	5.74414	0.192652
$890$	−117.196	−3.92841
$891$	−37.1789	−1.24554
$892$	−24.8573	−0.832285
$893$	17.9516	0.600726
$894$	14.4708	0.483977
$895$	−27.7859	−0.928779
$896$	6.30332	0.210579
$897$	−25.5335	−0.852540
$898$	3.38246	0.112874
$899$	9.78949	0.326498
$900$	36.3107	1.21036
$901$	−0.0334426	−0.00111414
$902$	−1.09276	−0.0363848
$903$	−9.31764	−0.310072
$904$	−1.64657	−0.0547643
$905$	−29.4278	−0.978214
$906$	−125.874	−4.18189
$907$	31.7942	1.05571	0.527855	−	0.849334i	$-0.322996\pi$
$907$	31.7942	1.05571	0.527855	+	0.849334i	$0.322996\pi$
$908$	−35.7655	−1.18692
$909$	−3.32954	−0.110434
$910$	−9.02020	−0.299017
$911$	−38.5261	−1.27643	−0.638214	−	0.769859i	$-0.720327\pi$
$911$	−38.5261	−1.27643	−0.638214	+	0.769859i	$0.720327\pi$
$912$	46.6962	1.54627
$913$	−30.2386	−1.00075
$914$	−47.7996	−1.58107
$915$	−20.2186	−0.668405
$916$	33.1267	1.09454
$917$	9.89294	0.326694
$918$	14.1828	0.468101
$919$	−31.2986	−1.03244	−0.516222	−	0.856455i	$-0.672662\pi$
$919$	−31.2986	−1.03244	−0.516222	+	0.856455i	$0.672662\pi$
$920$	152.522	5.02849
$921$	−38.9542	−1.28359
$922$	4.96490	0.163510
$923$	6.45497	0.212468
$924$	18.6459	0.613405
$925$	−43.1106	−1.41747
$926$	−36.4619	−1.19821
$927$	12.3098	0.404307
$928$	9.91392	0.325440
$929$	−42.6974	−1.40086	−0.700429	−	0.713722i	$-0.747008\pi$
$929$	−42.6974	−1.40086	−0.700429	+	0.713722i	$0.747008\pi$
$930$	−106.109	−3.47944
$931$	−20.2014	−0.662074
$932$	102.281	3.35034
$933$	23.5269	0.770235
$934$	−13.5460	−0.443240
$935$	−17.3837	−0.568508
$936$	14.4473	0.472226
$937$	45.9490	1.50109	0.750545	−	0.660820i	$-0.229791\pi$
$937$	45.9490	1.50109	0.750545	+	0.660820i	$0.229791\pi$
$938$	21.7798	0.711134
$939$	−35.4877	−1.15810
$940$	−89.6328	−2.92350
$941$	33.5157	1.09258	0.546290	−	0.837596i	$-0.316040\pi$
$941$	33.5157	1.09258	0.546290	+	0.837596i	$0.316040\pi$
$942$	1.81593	0.0591661
$943$	0.908075	0.0295710
$944$	34.8624	1.13468
$945$	7.17566	0.233424
$946$	63.6189	2.06843
$947$	40.2477	1.30787	0.653936	−	0.756549i	$-0.273116\pi$
$947$	40.2477	1.30787	0.653936	+	0.756549i	$0.273116\pi$
$948$	53.5419	1.73896
$949$	−3.63112	−0.117871
$950$	−48.7673	−1.58222
$951$	−0.748267	−0.0242642
$952$	−6.03619	−0.195634
$953$	2.49927	0.0809592	0.0404796	−	0.999180i	$-0.487111\pi$
$953$	2.49927	0.0809592	0.0404796	+	0.999180i	$0.487111\pi$
$954$	0.0700460	0.00226782
$955$	59.4973	1.92529
$956$	−47.5019	−1.53632
$957$	−11.2385	−0.363289
$958$	65.2685	2.10873
$959$	3.33840	0.107803
$960$	−4.38547	−0.141540
$961$	4.76936	0.153850
$962$	−30.7440	−0.991226
$963$	14.1649	0.456459
$964$	−101.178	−3.25873
$965$	66.2481	2.13260
$966$	−22.3444	−0.718920
$967$	12.2122	0.392716	0.196358	−	0.980532i	$-0.437088\pi$
$967$	12.2122	0.392716	0.196358	+	0.980532i	$0.437088\pi$
$968$	−0.110183	−0.00354143
$969$	9.81635	0.315347
$970$	−12.5619	−0.403340
$971$	−2.08241	−0.0668277	−0.0334139	−	0.999442i	$-0.510638\pi$
$971$	−2.08241	−0.0668277	−0.0334139	+	0.999442i	$0.510638\pi$
$972$	−56.5077	−1.81249
$973$	−4.16837	−0.133632
$974$	16.3959	0.525360
$975$	−22.7431	−0.728363
$976$	21.5970	0.691302
$977$	−18.2426	−0.583633	−0.291817	−	0.956474i	$-0.594260\pi$
$977$	−18.2426	−0.583633	−0.291817	+	0.956474i	$0.594260\pi$
$978$	−64.7039	−2.06900
$979$	−45.3544	−1.44953
$980$	100.866	3.22206
$981$	−3.86057	−0.123258
$982$	58.6665	1.87212
$983$	−41.8466	−1.33470	−0.667349	−	0.744745i	$-0.732571\pi$
$983$	−41.8466	−1.33470	−0.667349	+	0.744745i	$0.732571\pi$
$984$	−1.71899	−0.0547994
$985$	−9.34507	−0.297759
$986$	6.52095	0.207669
$987$	7.32626	0.233197
$988$	−24.1167	−0.767253
$989$	−52.8670	−1.68107
$990$	36.4104	1.15720
$991$	−13.3539	−0.424201	−0.212100	−	0.977248i	$-0.568030\pi$
$991$	−13.3539	−0.424201	−0.212100	+	0.977248i	$0.568030\pi$
$992$	36.2240	1.15011
$993$	−52.8230	−1.67629
$994$	5.64875	0.179168
$995$	31.3006	0.992295
$996$	−85.2579	−2.70150
$997$	−16.4393	−0.520637	−0.260319	−	0.965523i	$-0.583828\pi$
$997$	−16.4393	−0.520637	−0.260319	+	0.965523i	$0.583828\pi$
$998$	−0.856118	−0.0271000
$999$	24.4572	0.773790

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.a.1.16	✓	279

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.a.1.16	✓	279	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55424 q^{2} +2.06857 q^{3} +4.52414 q^{4} -3.35787 q^{5} -5.28363 q^{6} +0.600266 q^{7} -6.44725 q^{8} +1.27899 q^{9} +O(q^{10})\)	\(q-2.55424 q^{2} +2.06857 q^{3} +4.52414 q^{4} -3.35787 q^{5} -5.28363 q^{6} +0.600266 q^{7} -6.44725 q^{8} +1.27899 q^{9} +8.57679 q^{10} +3.31920 q^{11} +9.35850 q^{12} -1.75205 q^{13} -1.53322 q^{14} -6.94599 q^{15} +7.41953 q^{16} +1.55971 q^{17} -3.26684 q^{18} +3.04253 q^{19} -15.1914 q^{20} +1.24169 q^{21} -8.47803 q^{22} +7.04520 q^{23} -13.3366 q^{24} +6.27527 q^{25} +4.47516 q^{26} -3.56004 q^{27} +2.71569 q^{28} -1.63683 q^{29} +17.7417 q^{30} -5.98075 q^{31} -6.05677 q^{32} +6.86600 q^{33} -3.98388 q^{34} -2.01561 q^{35} +5.78631 q^{36} -6.86992 q^{37} -7.77134 q^{38} -3.62425 q^{39} +21.6490 q^{40} +0.128893 q^{41} -3.17158 q^{42} -7.50398 q^{43} +15.0165 q^{44} -4.29467 q^{45} -17.9951 q^{46} +5.90022 q^{47} +15.3478 q^{48} -6.63968 q^{49} -16.0285 q^{50} +3.22638 q^{51} -7.92652 q^{52} -0.0214415 q^{53} +9.09318 q^{54} -11.1454 q^{55} -3.87006 q^{56} +6.29368 q^{57} +4.18086 q^{58} +4.69874 q^{59} -31.4246 q^{60} +2.91083 q^{61} +15.2763 q^{62} +0.767733 q^{63} +0.631367 q^{64} +5.88316 q^{65} -17.5374 q^{66} -14.2052 q^{67} +7.05636 q^{68} +14.5735 q^{69} +5.14836 q^{70} -3.68423 q^{71} -8.24595 q^{72} +2.07249 q^{73} +17.5474 q^{74} +12.9808 q^{75} +13.7648 q^{76} +1.99240 q^{77} +9.25719 q^{78} +5.72121 q^{79} -24.9138 q^{80} -11.2012 q^{81} -0.329223 q^{82} -9.11021 q^{83} +5.61759 q^{84} -5.23731 q^{85} +19.1669 q^{86} -3.38591 q^{87} -21.3997 q^{88} -13.6643 q^{89} +10.9696 q^{90} -1.05170 q^{91} +31.8734 q^{92} -12.3716 q^{93} -15.0706 q^{94} -10.2164 q^{95} -12.5289 q^{96} -1.46464 q^{97} +16.9593 q^{98} +4.24522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * q^99

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