LMFDB - Embedded newform 8047.2.a.d.1.8

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:	$156$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-2.51520 q^{2} +3.16825 q^{3} +4.32624 q^{4} +1.81160 q^{5} -7.96880 q^{6} +1.29182 q^{7} -5.85097 q^{8} +7.03782 q^{9} +O(q^{10})$	\(q-2.51520 q^{2} +3.16825 q^{3} +4.32624 q^{4} +1.81160 q^{5} -7.96880 q^{6} +1.29182 q^{7} -5.85097 q^{8} +7.03782 q^{9} -4.55654 q^{10} +1.52442 q^{11} +13.7066 q^{12} -1.00000 q^{13} -3.24919 q^{14} +5.73961 q^{15} +6.06389 q^{16} -5.34451 q^{17} -17.7016 q^{18} -0.154094 q^{19} +7.83742 q^{20} +4.09281 q^{21} -3.83422 q^{22} +2.71756 q^{23} -18.5373 q^{24} -1.71810 q^{25} +2.51520 q^{26} +12.7928 q^{27} +5.58873 q^{28} +3.59950 q^{29} -14.4363 q^{30} +5.80825 q^{31} -3.54996 q^{32} +4.82974 q^{33} +13.4425 q^{34} +2.34026 q^{35} +30.4473 q^{36} +8.44083 q^{37} +0.387577 q^{38} -3.16825 q^{39} -10.5996 q^{40} -5.21842 q^{41} -10.2943 q^{42} -3.99011 q^{43} +6.59499 q^{44} +12.7497 q^{45} -6.83520 q^{46} +2.88958 q^{47} +19.2119 q^{48} -5.33120 q^{49} +4.32138 q^{50} -16.9328 q^{51} -4.32624 q^{52} +8.74376 q^{53} -32.1766 q^{54} +2.76163 q^{55} -7.55840 q^{56} -0.488209 q^{57} -9.05347 q^{58} -1.18271 q^{59} +24.8309 q^{60} +4.63924 q^{61} -14.6089 q^{62} +9.09160 q^{63} -3.19890 q^{64} -1.81160 q^{65} -12.1478 q^{66} +1.26948 q^{67} -23.1216 q^{68} +8.60990 q^{69} -5.88623 q^{70} +10.4525 q^{71} -41.1781 q^{72} -16.2062 q^{73} -21.2304 q^{74} -5.44339 q^{75} -0.666648 q^{76} +1.96927 q^{77} +7.96880 q^{78} +9.18208 q^{79} +10.9853 q^{80} +19.4175 q^{81} +13.1254 q^{82} +3.11669 q^{83} +17.7065 q^{84} -9.68211 q^{85} +10.0359 q^{86} +11.4041 q^{87} -8.91931 q^{88} +14.9351 q^{89} -32.0681 q^{90} -1.29182 q^{91} +11.7568 q^{92} +18.4020 q^{93} -7.26787 q^{94} -0.279157 q^{95} -11.2472 q^{96} +6.06605 q^{97} +13.4090 q^{98} +10.7286 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * 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 $

$2$	−2.51520	−1.77852	−0.889258	−	0.457405i	$-0.848779\pi$
$2$	−2.51520	−1.77852	−0.889258	+	0.457405i	$0.848779\pi$
$3$	3.16825	1.82919	0.914596	−	0.404369i	$-0.132509\pi$
$3$	3.16825	1.82919	0.914596	+	0.404369i	$0.132509\pi$
$4$	4.32624	2.16312
$5$	1.81160	0.810172	0.405086	−	0.914279i	$-0.367242\pi$
$5$	1.81160	0.810172	0.405086	+	0.914279i	$0.367242\pi$
$6$	−7.96880	−3.25325
$7$	1.29182	0.488262	0.244131	−	0.969742i	$-0.421497\pi$
$7$	1.29182	0.488262	0.244131	+	0.969742i	$0.421497\pi$
$8$	−5.85097	−2.06863
$9$	7.03782	2.34594
$10$	−4.55654	−1.44090
$11$	1.52442	0.459629	0.229814	−	0.973234i	$-0.426188\pi$
$11$	1.52442	0.459629	0.229814	+	0.973234i	$0.426188\pi$
$12$	13.7066	3.95676
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.24919	−0.868382
$15$	5.73961	1.48196
$16$	6.06389	1.51597
$17$	−5.34451	−1.29623	−0.648117	−	0.761541i	$-0.724443\pi$
$17$	−5.34451	−1.29623	−0.648117	+	0.761541i	$0.724443\pi$
$18$	−17.7016	−4.17230
$19$	−0.154094	−0.0353516	−0.0176758	−	0.999844i	$-0.505627\pi$
$19$	−0.154094	−0.0353516	−0.0176758	+	0.999844i	$0.505627\pi$
$20$	7.83742	1.75250
$21$	4.09281	0.893125
$22$	−3.83422	−0.817457
$23$	2.71756	0.566649	0.283325	−	0.959024i	$-0.408563\pi$
$23$	2.71756	0.566649	0.283325	+	0.959024i	$0.408563\pi$
$24$	−18.5373	−3.78392
$25$	−1.71810	−0.343621
$26$	2.51520	0.493272
$27$	12.7928	2.46198
$28$	5.58873	1.05617
$29$	3.59950	0.668410	0.334205	−	0.942500i	$-0.391532\pi$
$29$	3.59950	0.668410	0.334205	+	0.942500i	$0.391532\pi$
$30$	−14.4363	−2.63569
$31$	5.80825	1.04319	0.521596	−	0.853193i	$-0.325337\pi$
$31$	5.80825	1.04319	0.521596	+	0.853193i	$0.325337\pi$
$32$	−3.54996	−0.627551
$33$	4.82974	0.840749
$34$	13.4425	2.30537
$35$	2.34026	0.395577
$36$	30.4473	5.07456
$37$	8.44083	1.38766	0.693832	−	0.720137i	$-0.255921\pi$
$37$	8.44083	1.38766	0.693832	+	0.720137i	$0.255921\pi$
$38$	0.387577	0.0628734
$39$	−3.16825	−0.507326
$40$	−10.5996	−1.67595
$41$	−5.21842	−0.814981	−0.407490	−	0.913210i	$-0.633596\pi$
$41$	−5.21842	−0.814981	−0.407490	+	0.913210i	$0.633596\pi$
$42$	−10.2943	−1.58844
$43$	−3.99011	−0.608486	−0.304243	−	0.952595i	$-0.598403\pi$
$43$	−3.99011	−0.608486	−0.304243	+	0.952595i	$0.598403\pi$
$44$	6.59499	0.994233
$45$	12.7497	1.90062
$46$	−6.83520	−1.00780
$47$	2.88958	0.421488	0.210744	−	0.977541i	$-0.432411\pi$
$47$	2.88958	0.421488	0.210744	+	0.977541i	$0.432411\pi$
$48$	19.2119	2.77300
$49$	−5.33120	−0.761600
$50$	4.32138	0.611135
$51$	−16.9328	−2.37106
$52$	−4.32624	−0.599942
$53$	8.74376	1.20105	0.600524	−	0.799607i	$-0.294959\pi$
$53$	8.74376	1.20105	0.600524	+	0.799607i	$0.294959\pi$
$54$	−32.1766	−4.37868
$55$	2.76163	0.372379
$56$	−7.55840	−1.01003
$57$	−0.488209	−0.0646648
$58$	−9.05347	−1.18878
$59$	−1.18271	−0.153975	−0.0769876	−	0.997032i	$-0.524530\pi$
$59$	−1.18271	−0.153975	−0.0769876	+	0.997032i	$0.524530\pi$
$60$	24.8309	3.20566
$61$	4.63924	0.593994	0.296997	−	0.954878i	$-0.404015\pi$
$61$	4.63924	0.593994	0.296997	+	0.954878i	$0.404015\pi$
$62$	−14.6089	−1.85533
$63$	9.09160	1.14543
$64$	−3.19890	−0.399863
$65$	−1.81160	−0.224701
$66$	−12.1478	−1.49529
$67$	1.26948	0.155092	0.0775458	−	0.996989i	$-0.475292\pi$
$67$	1.26948	0.155092	0.0775458	+	0.996989i	$0.475292\pi$
$68$	−23.1216	−2.80391
$69$	8.60990	1.03651
$70$	−5.88623	−0.703539
$71$	10.4525	1.24049	0.620243	−	0.784410i	$-0.287034\pi$
$71$	10.4525	1.24049	0.620243	+	0.784410i	$0.287034\pi$
$72$	−41.1781	−4.85289
$73$	−16.2062	−1.89680	−0.948398	−	0.317082i	$-0.897297\pi$
$73$	−16.2062	−1.89680	−0.948398	+	0.317082i	$0.897297\pi$
$74$	−21.2304	−2.46798
$75$	−5.44339	−0.628548
$76$	−0.666648	−0.0764697
$77$	1.96927	0.224419
$78$	7.96880	0.902288
$79$	9.18208	1.03306	0.516532	−	0.856268i	$-0.327223\pi$
$79$	9.18208	1.03306	0.516532	+	0.856268i	$0.327223\pi$
$80$	10.9853	1.22820
$81$	19.4175	2.15750
$82$	13.1254	1.44946
$83$	3.11669	0.342101	0.171050	−	0.985262i	$-0.445284\pi$
$83$	3.11669	0.342101	0.171050	+	0.985262i	$0.445284\pi$
$84$	17.7065	1.93194
$85$	−9.68211	−1.05017
$86$	10.0359	1.08220
$87$	11.4041	1.22265
$88$	−8.91931	−0.950802
$89$	14.9351	1.58312	0.791559	−	0.611093i	$-0.209270\pi$
$89$	14.9351	1.58312	0.791559	+	0.611093i	$0.209270\pi$
$90$	−32.0681	−3.38028
$91$	−1.29182	−0.135420
$92$	11.7568	1.22573
$93$	18.4020	1.90820
$94$	−7.26787	−0.749623
$95$	−0.279157	−0.0286409
$96$	−11.2472	−1.14791
$97$	6.06605	0.615914	0.307957	−	0.951400i	$-0.400355\pi$
$97$	6.06605	0.615914	0.307957	+	0.951400i	$0.400355\pi$
$98$	13.4090	1.35452
$99$	10.7286	1.07826
$100$	−7.43293	−0.743293
$101$	3.25959	0.324342	0.162171	−	0.986763i	$-0.448150\pi$
$101$	3.25959	0.324342	0.162171	+	0.986763i	$0.448150\pi$
$102$	42.5893	4.21697
$103$	3.85654	0.379996	0.189998	−	0.981784i	$-0.439152\pi$
$103$	3.85654	0.379996	0.189998	+	0.981784i	$0.439152\pi$
$104$	5.85097	0.573735
$105$	7.41454	0.723585
$106$	−21.9923	−2.13608
$107$	−4.89182	−0.472911	−0.236455	−	0.971642i	$-0.575986\pi$
$107$	−4.89182	−0.472911	−0.236455	+	0.971642i	$0.575986\pi$
$108$	55.3450	5.32557
$109$	−9.36239	−0.896755	−0.448377	−	0.893844i	$-0.647998\pi$
$109$	−9.36239	−0.896755	−0.448377	+	0.893844i	$0.647998\pi$
$110$	−6.94607	−0.662281
$111$	26.7427	2.53830
$112$	7.83345	0.740192
$113$	8.32097	0.782771	0.391386	−	0.920227i	$-0.371996\pi$
$113$	8.32097	0.782771	0.391386	+	0.920227i	$0.371996\pi$
$114$	1.22794	0.115007
$115$	4.92312	0.459084
$116$	15.5723	1.44585
$117$	−7.03782	−0.650647
$118$	2.97474	0.273847
$119$	−6.90414	−0.632902
$120$	−33.5823	−3.06563
$121$	−8.67615	−0.788741
$122$	−11.6686	−1.05643
$123$	−16.5333	−1.49076
$124$	25.1279	2.25655
$125$	−12.1705	−1.08856
$126$	−22.8672	−2.03717
$127$	6.18414	0.548754	0.274377	−	0.961622i	$-0.411528\pi$
$127$	6.18414	0.548754	0.274377	+	0.961622i	$0.411528\pi$
$128$	15.1458	1.33871
$129$	−12.6417	−1.11304
$130$	4.55654	0.399635
$131$	−22.0939	−1.93035	−0.965175	−	0.261606i	$-0.915748\pi$
$131$	−22.0939	−1.93035	−0.965175	+	0.261606i	$0.915748\pi$
$132$	20.8946	1.81864
$133$	−0.199062	−0.0172608
$134$	−3.19300	−0.275833
$135$	23.1755	1.99463
$136$	31.2706	2.68143
$137$	−0.808644	−0.0690872	−0.0345436	−	0.999403i	$-0.510998\pi$
$137$	−0.808644	−0.0690872	−0.0345436	+	0.999403i	$0.510998\pi$
$138$	−21.6556	−1.84345
$139$	−4.66427	−0.395618	−0.197809	−	0.980241i	$-0.563383\pi$
$139$	−4.66427	−0.395618	−0.197809	+	0.980241i	$0.563383\pi$
$140$	10.1245	0.855680
$141$	9.15490	0.770982
$142$	−26.2902	−2.20623
$143$	−1.52442	−0.127478
$144$	42.6766	3.55638
$145$	6.52085	0.541528
$146$	40.7620	3.37348
$147$	−16.8906	−1.39311
$148$	36.5171	3.00169
$149$	4.21510	0.345315	0.172657	−	0.984982i	$-0.444765\pi$
$149$	4.21510	0.345315	0.172657	+	0.984982i	$0.444765\pi$
$150$	13.6912	1.11788
$151$	6.61129	0.538019	0.269010	−	0.963138i	$-0.413304\pi$
$151$	6.61129	0.538019	0.269010	+	0.963138i	$0.413304\pi$
$152$	0.901599	0.0731293
$153$	−37.6137	−3.04089
$154$	−4.95312	−0.399134
$155$	10.5222	0.845165
$156$	−13.7066	−1.09741
$157$	−4.53111	−0.361622	−0.180811	−	0.983518i	$-0.557872\pi$
$157$	−4.53111	−0.361622	−0.180811	+	0.983518i	$0.557872\pi$
$158$	−23.0948	−1.83732
$159$	27.7025	2.19695
$160$	−6.43111	−0.508424
$161$	3.51059	0.276673
$162$	−48.8389	−3.83715
$163$	−6.63172	−0.519437	−0.259718	−	0.965684i	$-0.583630\pi$
$163$	−6.63172	−0.519437	−0.259718	+	0.965684i	$0.583630\pi$
$164$	−22.5762	−1.76290
$165$	8.74955	0.681152
$166$	−7.83909	−0.608432
$167$	19.4998	1.50894	0.754471	−	0.656334i	$-0.227894\pi$
$167$	19.4998	1.50894	0.754471	+	0.656334i	$0.227894\pi$
$168$	−23.9469	−1.84755
$169$	1.00000	0.0769231
$170$	24.3525	1.86775
$171$	−1.08449	−0.0829327
$172$	−17.2622	−1.31623
$173$	1.88315	0.143173	0.0715864	−	0.997434i	$-0.477194\pi$
$173$	1.88315	0.143173	0.0715864	+	0.997434i	$0.477194\pi$
$174$	−28.6837	−2.17450
$175$	−2.21948	−0.167777
$176$	9.24389	0.696784
$177$	−3.74711	−0.281650
$178$	−37.5648	−2.81560
$179$	−21.1034	−1.57734	−0.788670	−	0.614817i	$-0.789230\pi$
$179$	−21.1034	−1.57734	−0.788670	+	0.614817i	$0.789230\pi$
$180$	55.1584	4.11126
$181$	5.66260	0.420897	0.210449	−	0.977605i	$-0.432508\pi$
$181$	5.66260	0.420897	0.210449	+	0.977605i	$0.432508\pi$
$182$	3.24919	0.240846
$183$	14.6983	1.08653
$184$	−15.9003	−1.17219
$185$	15.2914	1.12425
$186$	−46.2847	−3.39376
$187$	−8.14726	−0.595786
$188$	12.5010	0.911729
$189$	16.5261	1.20209
$190$	0.702135	0.0509383
$191$	−3.77610	−0.273229	−0.136615	−	0.990624i	$-0.543622\pi$
$191$	−3.77610	−0.273229	−0.136615	+	0.990624i	$0.543622\pi$
$192$	−10.1349	−0.731425
$193$	1.72532	0.124192	0.0620958	−	0.998070i	$-0.480222\pi$
$193$	1.72532	0.124192	0.0620958	+	0.998070i	$0.480222\pi$
$194$	−15.2574	−1.09541
$195$	−5.73961	−0.411022
$196$	−23.0641	−1.64743
$197$	16.5950	1.18235	0.591173	−	0.806545i	$-0.298665\pi$
$197$	16.5950	1.18235	0.591173	+	0.806545i	$0.298665\pi$
$198$	−26.9845	−1.91771
$199$	−8.30463	−0.588700	−0.294350	−	0.955698i	$-0.595103\pi$
$199$	−8.30463	−0.588700	−0.294350	+	0.955698i	$0.595103\pi$
$200$	10.0526	0.710824
$201$	4.02203	0.283692
$202$	−8.19854	−0.576847
$203$	4.64991	0.326359
$204$	−73.2552	−5.12889
$205$	−9.45370	−0.660275
$206$	−9.69997	−0.675829
$207$	19.1257	1.32933
$208$	−6.06389	−0.420455
$209$	−0.234903	−0.0162486
$210$	−18.6491	−1.28691
$211$	7.80322	0.537196	0.268598	−	0.963252i	$-0.413440\pi$
$211$	7.80322	0.537196	0.268598	+	0.963252i	$0.413440\pi$
$212$	37.8276	2.59801
$213$	33.1162	2.26909
$214$	12.3039	0.841079
$215$	−7.22848	−0.492978
$216$	−74.8506	−5.09294
$217$	7.50321	0.509351
$218$	23.5483	1.59489
$219$	−51.3454	−3.46960
$220$	11.9475	0.805500
$221$	5.34451	0.359511
$222$	−67.2633	−4.51442
$223$	16.1102	1.07882	0.539411	−	0.842043i	$-0.318647\pi$
$223$	16.1102	1.07882	0.539411	+	0.842043i	$0.318647\pi$
$224$	−4.58591	−0.306409
$225$	−12.0917	−0.806114
$226$	−20.9289	−1.39217
$227$	20.8647	1.38484	0.692420	−	0.721495i	$-0.256545\pi$
$227$	20.8647	1.38484	0.692420	+	0.721495i	$0.256545\pi$
$228$	−2.11211	−0.139878
$229$	−8.68362	−0.573830	−0.286915	−	0.957956i	$-0.592630\pi$
$229$	−8.68362	−0.573830	−0.286915	+	0.957956i	$0.592630\pi$
$230$	−12.3827	−0.816488
$231$	6.23915	0.410506
$232$	−21.0606	−1.38269
$233$	10.4478	0.684459	0.342230	−	0.939616i	$-0.388818\pi$
$233$	10.4478	0.684459	0.342230	+	0.939616i	$0.388818\pi$
$234$	17.7016	1.15719
$235$	5.23476	0.341478
$236$	−5.11667	−0.333067
$237$	29.0911	1.88967
$238$	17.3653	1.12563
$239$	27.5854	1.78435	0.892177	−	0.451686i	$-0.149177\pi$
$239$	27.5854	1.78435	0.892177	+	0.451686i	$0.149177\pi$
$240$	34.8043	2.24661
$241$	30.4299	1.96016	0.980081	−	0.198597i	$-0.0636386\pi$
$241$	30.4299	1.96016	0.980081	+	0.198597i	$0.0636386\pi$
$242$	21.8223	1.40279
$243$	23.1410	1.48450
$244$	20.0705	1.28488
$245$	−9.65800	−0.617027
$246$	41.5845	2.65133
$247$	0.154094	0.00980476
$248$	−33.9839	−2.15798
$249$	9.87445	0.625768
$250$	30.6113	1.93603
$251$	11.6483	0.735236	0.367618	−	0.929977i	$-0.380173\pi$
$251$	11.6483	0.735236	0.367618	+	0.929977i	$0.380173\pi$
$252$	39.3325	2.47771
$253$	4.14269	0.260448
$254$	−15.5544	−0.975968
$255$	−30.6754	−1.92097
$256$	−31.6970	−1.98106
$257$	−9.34468	−0.582905	−0.291453	−	0.956585i	$-0.594139\pi$
$257$	−9.34468	−0.582905	−0.291453	+	0.956585i	$0.594139\pi$
$258$	31.7963	1.97955
$259$	10.9040	0.677544
$260$	−7.83742	−0.486056
$261$	25.3326	1.56805
$262$	55.5705	3.43316
$263$	21.9842	1.35560	0.677801	−	0.735246i	$-0.262933\pi$
$263$	21.9842	1.35560	0.677801	+	0.735246i	$0.262933\pi$
$264$	−28.2586	−1.73920
$265$	15.8402	0.973056
$266$	0.500680	0.0306987
$267$	47.3182	2.89583
$268$	5.49207	0.335482
$269$	−32.0310	−1.95296	−0.976481	−	0.215603i	$-0.930828\pi$
$269$	−32.0310	−1.95296	−0.976481	+	0.215603i	$0.930828\pi$
$270$	−58.2911	−3.54749
$271$	11.7164	0.711724	0.355862	−	0.934539i	$-0.384187\pi$
$271$	11.7164	0.711724	0.355862	+	0.934539i	$0.384187\pi$
$272$	−32.4085	−1.96505
$273$	−4.09281	−0.247708
$274$	2.03390	0.122873
$275$	−2.61911	−0.157938
$276$	37.2485	2.24210
$277$	−29.5894	−1.77785	−0.888927	−	0.458049i	$-0.848548\pi$
$277$	−29.5894	−1.77785	−0.888927	+	0.458049i	$0.848548\pi$
$278$	11.7316	0.703614
$279$	40.8774	2.44727
$280$	−13.6928	−0.818301
$281$	−0.626570	−0.0373780	−0.0186890	−	0.999825i	$-0.505949\pi$
$281$	−0.626570	−0.0373780	−0.0186890	+	0.999825i	$0.505949\pi$
$282$	−23.0264	−1.37120
$283$	−17.1781	−1.02113	−0.510565	−	0.859839i	$-0.670564\pi$
$283$	−17.1781	−1.02113	−0.510565	+	0.859839i	$0.670564\pi$
$284$	45.2202	2.68332
$285$	−0.884439	−0.0523896
$286$	3.83422	0.226722
$287$	−6.74126	−0.397924
$288$	−24.9840	−1.47220
$289$	11.5638	0.680222
$290$	−16.4013	−0.963116
$291$	19.2188	1.12663
$292$	−70.1121	−4.10300
$293$	−17.6481	−1.03101	−0.515505	−	0.856886i	$-0.672396\pi$
$293$	−17.6481	−1.03101	−0.515505	+	0.856886i	$0.672396\pi$
$294$	42.4832	2.47767
$295$	−2.14259	−0.124746
$296$	−49.3871	−2.87056
$297$	19.5016	1.13160
$298$	−10.6018	−0.614148
$299$	−2.71756	−0.157160
$300$	−23.5494	−1.35963
$301$	−5.15450	−0.297101
$302$	−16.6287	−0.956876
$303$	10.3272	0.593283
$304$	−0.934408	−0.0535920
$305$	8.40446	0.481238
$306$	94.6061	5.40827
$307$	−18.5188	−1.05692	−0.528462	−	0.848957i	$-0.677231\pi$
$307$	−18.5188	−1.05692	−0.528462	+	0.848957i	$0.677231\pi$
$308$	8.51955	0.485446
$309$	12.2185	0.695085
$310$	−26.4655	−1.50314
$311$	−12.2395	−0.694037	−0.347019	−	0.937858i	$-0.612806\pi$
$311$	−12.2395	−0.694037	−0.347019	+	0.937858i	$0.612806\pi$
$312$	18.5373	1.04947
$313$	17.5378	0.991295	0.495647	−	0.868524i	$-0.334931\pi$
$313$	17.5378	0.991295	0.495647	+	0.868524i	$0.334931\pi$
$314$	11.3967	0.643150
$315$	16.4704	0.927999
$316$	39.7239	2.23464
$317$	−3.77029	−0.211760	−0.105880	−	0.994379i	$-0.533766\pi$
$317$	−3.77029	−0.211760	−0.105880	+	0.994379i	$0.533766\pi$
$318$	−69.6773	−3.90731
$319$	5.48714	0.307221
$320$	−5.79513	−0.323958
$321$	−15.4985	−0.865044
$322$	−8.82985	−0.492068
$323$	0.823556	0.0458239
$324$	84.0048	4.66693
$325$	1.71810	0.0953033
$326$	16.6801	0.923827
$327$	−29.6624	−1.64034
$328$	30.5328	1.68589
$329$	3.73281	0.205797
$330$	−22.0069	−1.21144
$331$	−2.58474	−0.142070	−0.0710352	−	0.997474i	$-0.522630\pi$
$331$	−2.58474	−0.142070	−0.0710352	+	0.997474i	$0.522630\pi$
$332$	13.4835	0.740005
$333$	59.4051	3.25538
$334$	−49.0460	−2.68368
$335$	2.29979	0.125651
$336$	24.8184	1.35395
$337$	12.4844	0.680069	0.340034	−	0.940413i	$-0.389561\pi$
$337$	12.4844	0.680069	0.340034	+	0.940413i	$0.389561\pi$
$338$	−2.51520	−0.136809
$339$	26.3629	1.43184
$340$	−41.8872	−2.27165
$341$	8.85419	0.479481
$342$	2.72770	0.147497
$343$	−15.9297	−0.860123
$344$	23.3460	1.25873
$345$	15.5977	0.839752
$346$	−4.73649	−0.254635
$347$	−9.30942	−0.499756	−0.249878	−	0.968277i	$-0.580391\pi$
$347$	−9.30942	−0.499756	−0.249878	+	0.968277i	$0.580391\pi$
$348$	49.3370	2.64474
$349$	−0.885935	−0.0474230	−0.0237115	−	0.999719i	$-0.507548\pi$
$349$	−0.885935	−0.0474230	−0.0237115	+	0.999719i	$0.507548\pi$
$350$	5.58244	0.298394
$351$	−12.7928	−0.682832
$352$	−5.41162	−0.288440
$353$	8.16467	0.434561	0.217281	−	0.976109i	$-0.430281\pi$
$353$	8.16467	0.434561	0.217281	+	0.976109i	$0.430281\pi$
$354$	9.42474	0.500919
$355$	18.9358	1.00501
$356$	64.6129	3.42448
$357$	−21.8741	−1.15770
$358$	53.0792	2.80532
$359$	33.4590	1.76590	0.882949	−	0.469469i	$-0.155555\pi$
$359$	33.4590	1.76590	0.882949	+	0.469469i	$0.155555\pi$
$360$	−74.5983	−3.93167
$361$	−18.9763	−0.998750
$362$	−14.2426	−0.748573
$363$	−27.4882	−1.44276
$364$	−5.58873	−0.292929
$365$	−29.3592	−1.53673
$366$	−36.9692	−1.93241
$367$	0.467043	0.0243794	0.0121897	−	0.999926i	$-0.496120\pi$
$367$	0.467043	0.0243794	0.0121897	+	0.999926i	$0.496120\pi$
$368$	16.4789	0.859024
$369$	−36.7263	−1.91190
$370$	−38.4610	−1.99949
$371$	11.2954	0.586426
$372$	79.6115	4.12766
$373$	−12.8782	−0.666809	−0.333405	−	0.942784i	$-0.608198\pi$
$373$	−12.8782	−0.666809	−0.333405	+	0.942784i	$0.608198\pi$
$374$	20.4920	1.05962
$375$	−38.5593	−1.99119
$376$	−16.9068	−0.871902
$377$	−3.59950	−0.185384
$378$	−41.5664	−2.13794
$379$	−9.49097	−0.487518	−0.243759	−	0.969836i	$-0.578381\pi$
$379$	−9.49097	−0.487518	−0.243759	+	0.969836i	$0.578381\pi$
$380$	−1.20770	−0.0619537
$381$	19.5929	1.00378
$382$	9.49766	0.485942
$383$	26.9402	1.37658	0.688291	−	0.725435i	$-0.258361\pi$
$383$	26.9402	1.37658	0.688291	+	0.725435i	$0.258361\pi$
$384$	47.9857	2.44876
$385$	3.56753	0.181818
$386$	−4.33954	−0.220877
$387$	−28.0817	−1.42747
$388$	26.2432	1.33230
$389$	−22.5183	−1.14172	−0.570862	−	0.821046i	$-0.693391\pi$
$389$	−22.5183	−1.14172	−0.570862	+	0.821046i	$0.693391\pi$
$390$	14.4363	0.731009
$391$	−14.5240	−0.734510
$392$	31.1927	1.57547
$393$	−69.9989	−3.53098
$394$	−41.7398	−2.10282
$395$	16.6343	0.836960
$396$	46.4144	2.33241
$397$	2.69215	0.135115	0.0675577	−	0.997715i	$-0.478479\pi$
$397$	2.69215	0.135115	0.0675577	+	0.997715i	$0.478479\pi$
$398$	20.8878	1.04701
$399$	−0.630678	−0.0315734
$400$	−10.4184	−0.520919
$401$	−20.7771	−1.03756	−0.518778	−	0.854909i	$-0.673613\pi$
$401$	−20.7771	−1.03756	−0.518778	+	0.854909i	$0.673613\pi$
$402$	−10.1162	−0.504551
$403$	−5.80825	−0.289329
$404$	14.1018	0.701590
$405$	35.1768	1.74795
$406$	−11.6955	−0.580436
$407$	12.8673	0.637811
$408$	99.0730	4.90485
$409$	−23.5214	−1.16306	−0.581529	−	0.813526i	$-0.697545\pi$
$409$	−23.5214	−1.16306	−0.581529	+	0.813526i	$0.697545\pi$
$410$	23.7780	1.17431
$411$	−2.56199	−0.126374
$412$	16.6843	0.821977
$413$	−1.52784	−0.0751802
$414$	−48.1049	−2.36423
$415$	5.64619	0.277161
$416$	3.54996	0.174051
$417$	−14.7776	−0.723662
$418$	0.590829	0.0288984
$419$	22.5009	1.09924	0.549621	−	0.835414i	$-0.314772\pi$
$419$	22.5009	1.09924	0.549621	+	0.835414i	$0.314772\pi$
$420$	32.0771	1.56520
$421$	−11.0320	−0.537666	−0.268833	−	0.963187i	$-0.586638\pi$
$421$	−11.0320	−0.537666	−0.268833	+	0.963187i	$0.586638\pi$
$422$	−19.6267	−0.955412
$423$	20.3363	0.988786
$424$	−51.1595	−2.48452
$425$	9.18242	0.445413
$426$	−83.2940	−4.03561
$427$	5.99307	0.290025
$428$	−21.1632	−1.02296
$429$	−4.82974	−0.233182
$430$	18.1811	0.876770
$431$	7.05294	0.339728	0.169864	−	0.985467i	$-0.445667\pi$
$431$	7.05294	0.339728	0.169864	+	0.985467i	$0.445667\pi$
$432$	77.5744	3.73230
$433$	28.8346	1.38570	0.692850	−	0.721081i	$-0.256355\pi$
$433$	28.8346	1.38570	0.692850	+	0.721081i	$0.256355\pi$
$434$	−18.8721	−0.905890
$435$	20.6597	0.990558
$436$	−40.5040	−1.93979
$437$	−0.418759	−0.0200319
$438$	129.144	6.17075
$439$	−5.17222	−0.246857	−0.123428	−	0.992353i	$-0.539389\pi$
$439$	−5.17222	−0.246857	−0.123428	+	0.992353i	$0.539389\pi$
$440$	−16.1582	−0.770313
$441$	−37.5201	−1.78667
$442$	−13.4425	−0.639395
$443$	8.28359	0.393565	0.196783	−	0.980447i	$-0.436951\pi$
$443$	8.28359	0.393565	0.196783	+	0.980447i	$0.436951\pi$
$444$	115.695	5.49066
$445$	27.0564	1.28260
$446$	−40.5205	−1.91870
$447$	13.3545	0.631647
$448$	−4.13240	−0.195238
$449$	−13.6724	−0.645240	−0.322620	−	0.946529i	$-0.604564\pi$
$449$	−13.6724	−0.645240	−0.322620	+	0.946529i	$0.604564\pi$
$450$	30.4131	1.43369
$451$	−7.95505	−0.374589
$452$	35.9986	1.69323
$453$	20.9462	0.984140
$454$	−52.4790	−2.46296
$455$	−2.34026	−0.109713
$456$	2.85649	0.133768
$457$	−21.6195	−1.01132	−0.505659	−	0.862734i	$-0.668751\pi$
$457$	−21.6195	−1.01132	−0.505659	+	0.862734i	$0.668751\pi$
$458$	21.8411	1.02057
$459$	−68.3715	−3.19131
$460$	21.2986	0.993054
$461$	−6.07529	−0.282954	−0.141477	−	0.989942i	$-0.545185\pi$
$461$	−6.07529	−0.282954	−0.141477	+	0.989942i	$0.545185\pi$
$462$	−15.6927	−0.730092
$463$	−16.1244	−0.749365	−0.374683	−	0.927153i	$-0.622248\pi$
$463$	−16.1244	−0.749365	−0.374683	+	0.927153i	$0.622248\pi$
$464$	21.8270	1.01329
$465$	33.3371	1.54597
$466$	−26.2784	−1.21732
$467$	−1.15849	−0.0536085	−0.0268043	−	0.999641i	$-0.508533\pi$
$467$	−1.15849	−0.0536085	−0.0268043	+	0.999641i	$0.508533\pi$
$468$	−30.4473	−1.40743
$469$	1.63994	0.0757253
$470$	−13.1665	−0.607324
$471$	−14.3557	−0.661476
$472$	6.91997	0.318518
$473$	−6.08258	−0.279678
$474$	−73.1701	−3.36081
$475$	0.264749	0.0121475
$476$	−29.8690	−1.36904
$477$	61.5371	2.81759
$478$	−69.3830	−3.17350
$479$	−15.7352	−0.718960	−0.359480	−	0.933153i	$-0.617046\pi$
$479$	−15.7352	−0.718960	−0.359480	+	0.933153i	$0.617046\pi$
$480$	−20.3754	−0.930005
$481$	−8.44083	−0.384869
$482$	−76.5374	−3.48618
$483$	11.1224	0.506089
$484$	−37.5351	−1.70614
$485$	10.9893	0.498997
$486$	−58.2043	−2.64020
$487$	20.9871	0.951016	0.475508	−	0.879711i	$-0.342264\pi$
$487$	20.9871	0.951016	0.475508	+	0.879711i	$0.342264\pi$
$488$	−27.1441	−1.22875
$489$	−21.0110	−0.950149
$490$	24.2918	1.09739
$491$	−17.0231	−0.768240	−0.384120	−	0.923283i	$-0.625495\pi$
$491$	−17.0231	−0.768240	−0.384120	+	0.923283i	$0.625495\pi$
$492$	−71.5270	−3.22468
$493$	−19.2376	−0.866416
$494$	−0.387577	−0.0174379
$495$	19.4359	0.873578
$496$	35.2206	1.58145
$497$	13.5028	0.605683
$498$	−24.8362	−1.11294
$499$	3.61170	0.161682	0.0808408	−	0.996727i	$-0.474239\pi$
$499$	3.61170	0.161682	0.0808408	+	0.996727i	$0.474239\pi$
$500$	−52.6526	−2.35470
$501$	61.7804	2.76014
$502$	−29.2979	−1.30763
$503$	26.9491	1.20160	0.600800	−	0.799399i	$-0.294849\pi$
$503$	26.9491	1.20160	0.600800	+	0.799399i	$0.294849\pi$
$504$	−53.1947	−2.36948
$505$	5.90508	0.262773
$506$	−10.4197	−0.463212
$507$	3.16825	0.140707
$508$	26.7541	1.18702
$509$	−19.4193	−0.860747	−0.430373	−	0.902651i	$-0.641618\pi$
$509$	−19.4193	−0.860747	−0.430373	+	0.902651i	$0.641618\pi$
$510$	77.1548	3.41647
$511$	−20.9355	−0.926134
$512$	49.4327	2.18464
$513$	−1.97130	−0.0870350
$514$	23.5038	1.03671
$515$	6.98650	0.307862
$516$	−54.6909	−2.40763
$517$	4.40492	0.193728
$518$	−27.4259	−1.20502
$519$	5.96628	0.261891
$520$	10.5996	0.464824
$521$	−16.6332	−0.728713	−0.364357	−	0.931259i	$-0.618711\pi$
$521$	−16.6332	−0.728713	−0.364357	+	0.931259i	$0.618711\pi$
$522$	−63.7167	−2.78881
$523$	−29.6498	−1.29650	−0.648248	−	0.761429i	$-0.724498\pi$
$523$	−29.6498	−1.29650	−0.648248	+	0.761429i	$0.724498\pi$
$524$	−95.5834	−4.17558
$525$	−7.03188	−0.306896
$526$	−55.2946	−2.41096
$527$	−31.0422	−1.35222
$528$	29.2870	1.27455
$529$	−15.6149	−0.678908
$530$	−39.8413	−1.73060
$531$	−8.32367	−0.361217
$532$	−0.861189	−0.0373373
$533$	5.21842	0.226035
$534$	−119.015	−5.15028
$535$	−8.86203	−0.383139
$536$	−7.42768	−0.320827
$537$	−66.8608	−2.88526
$538$	80.5644	3.47338
$539$	−8.12697	−0.350053
$540$	100.263	4.31463
$541$	−30.5356	−1.31283	−0.656414	−	0.754401i	$-0.727927\pi$
$541$	−30.5356	−1.31283	−0.656414	+	0.754401i	$0.727927\pi$
$542$	−29.4692	−1.26581
$543$	17.9405	0.769902
$544$	18.9728	0.813452
$545$	−16.9609	−0.726526
$546$	10.2943	0.440553
$547$	−11.9059	−0.509060	−0.254530	−	0.967065i	$-0.581921\pi$
$547$	−11.9059	−0.509060	−0.254530	+	0.967065i	$0.581921\pi$
$548$	−3.49839	−0.149444
$549$	32.6502	1.39348
$550$	6.58758	0.280895
$551$	−0.554661	−0.0236294
$552$	−50.3763	−2.14416
$553$	11.8616	0.504406
$554$	74.4233	3.16194
$555$	48.4471	2.05646
$556$	−20.1788	−0.855770
$557$	−0.111954	−0.00474364	−0.00237182	−	0.999997i	$-0.500755\pi$
$557$	−0.111954	−0.00474364	−0.00237182	+	0.999997i	$0.500755\pi$
$558$	−102.815	−4.35251
$559$	3.99011	0.168764
$560$	14.1911	0.599683
$561$	−25.8126	−1.08981
$562$	1.57595	0.0664774
$563$	21.8132	0.919315	0.459658	−	0.888096i	$-0.347972\pi$
$563$	21.8132	0.919315	0.459658	+	0.888096i	$0.347972\pi$
$564$	39.6063	1.66773
$565$	15.0743	0.634180
$566$	43.2063	1.81610
$567$	25.0839	1.05343
$568$	−61.1574	−2.56611
$569$	0.243757	0.0102188	0.00510941	−	0.999987i	$-0.498374\pi$
$569$	0.243757	0.0102188	0.00510941	+	0.999987i	$0.498374\pi$
$570$	2.22454	0.0931758
$571$	35.9084	1.50272	0.751360	−	0.659892i	$-0.229398\pi$
$571$	35.9084	1.50272	0.751360	+	0.659892i	$0.229398\pi$
$572$	−6.59499	−0.275751
$573$	−11.9636	−0.499788
$574$	16.9556	0.707715
$575$	−4.66904	−0.194713
$576$	−22.5133	−0.938054
$577$	12.8710	0.535827	0.267914	−	0.963443i	$-0.413666\pi$
$577$	12.8710	0.535827	0.267914	+	0.963443i	$0.413666\pi$
$578$	−29.0852	−1.20979
$579$	5.46626	0.227170
$580$	28.2108	1.17139
$581$	4.02620	0.167035
$582$	−48.3391	−2.00372
$583$	13.3291	0.552036
$584$	94.8222	3.92377
$585$	−12.7497	−0.527136
$586$	44.3884	1.83367
$587$	−38.9782	−1.60880	−0.804401	−	0.594087i	$-0.797513\pi$
$587$	−38.9782	−1.60880	−0.804401	+	0.594087i	$0.797513\pi$
$588$	−73.0728	−3.01347
$589$	−0.895016	−0.0368785
$590$	5.38905	0.221863
$591$	52.5772	2.16274
$592$	51.1843	2.10366
$593$	28.8651	1.18535	0.592674	−	0.805442i	$-0.298072\pi$
$593$	28.8651	1.18535	0.592674	+	0.805442i	$0.298072\pi$
$594$	−49.0505	−2.01257
$595$	−12.5076	−0.512760
$596$	18.2356	0.746957
$597$	−26.3112	−1.07684
$598$	6.83520	0.279512
$599$	−27.4911	−1.12326	−0.561628	−	0.827390i	$-0.689825\pi$
$599$	−27.4911	−1.12326	−0.561628	+	0.827390i	$0.689825\pi$
$600$	31.8491	1.30023
$601$	−43.2111	−1.76262	−0.881309	−	0.472541i	$-0.843337\pi$
$601$	−43.2111	−1.76262	−0.881309	+	0.472541i	$0.843337\pi$
$602$	12.9646	0.528398
$603$	8.93437	0.363836
$604$	28.6020	1.16380
$605$	−15.7177	−0.639016
$606$	−25.9750	−1.05516
$607$	13.7527	0.558206	0.279103	−	0.960261i	$-0.409963\pi$
$607$	13.7527	0.558206	0.279103	+	0.960261i	$0.409963\pi$
$608$	0.547028	0.0221849
$609$	14.7321	0.596974
$610$	−21.1389	−0.855889
$611$	−2.88958	−0.116900
$612$	−162.726	−6.57781
$613$	−5.82538	−0.235285	−0.117642	−	0.993056i	$-0.537534\pi$
$613$	−5.82538	−0.235285	−0.117642	+	0.993056i	$0.537534\pi$
$614$	46.5786	1.87976
$615$	−29.9517	−1.20777
$616$	−11.5221	−0.464241
$617$	−45.1072	−1.81595	−0.907974	−	0.419026i	$-0.862372\pi$
$617$	−45.1072	−1.81595	−0.907974	+	0.419026i	$0.862372\pi$
$618$	−30.7320	−1.23622
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	45.5217	1.82820
$621$	34.7653	1.39508
$622$	30.7848	1.23436
$623$	19.2935	0.772977
$624$	−19.2119	−0.769092
$625$	−13.4576	−0.538304
$626$	−44.1111	−1.76303
$627$	−0.744233	−0.0297218
$628$	−19.6027	−0.782232
$629$	−45.1121	−1.79874
$630$	−41.4263	−1.65046
$631$	−10.1925	−0.405758	−0.202879	−	0.979204i	$-0.565030\pi$
$631$	−10.1925	−0.405758	−0.202879	+	0.979204i	$0.565030\pi$
$632$	−53.7240	−2.13703
$633$	24.7226	0.982634
$634$	9.48303	0.376619
$635$	11.2032	0.444585
$636$	119.848	4.75226
$637$	5.33120	0.211230
$638$	−13.8013	−0.546397
$639$	73.5630	2.91011
$640$	27.4381	1.08459
$641$	36.0744	1.42485	0.712426	−	0.701747i	$-0.247596\pi$
$641$	36.0744	1.42485	0.712426	+	0.701747i	$0.247596\pi$
$642$	38.9820	1.53849
$643$	−9.53401	−0.375985	−0.187992	−	0.982170i	$-0.560198\pi$
$643$	−9.53401	−0.375985	−0.187992	+	0.982170i	$0.560198\pi$
$644$	15.1877	0.598478
$645$	−22.9017	−0.901752
$646$	−2.07141	−0.0814986
$647$	−10.9390	−0.430058	−0.215029	−	0.976608i	$-0.568985\pi$
$647$	−10.9390	−0.430058	−0.215029	+	0.976608i	$0.568985\pi$
$648$	−113.611	−4.46307
$649$	−1.80294	−0.0707714
$650$	−4.32138	−0.169498
$651$	23.7721	0.931701
$652$	−28.6904	−1.12360
$653$	−26.1304	−1.02256	−0.511281	−	0.859414i	$-0.670829\pi$
$653$	−26.1304	−1.02256	−0.511281	+	0.859414i	$0.670829\pi$
$654$	74.6070	2.91736
$655$	−40.0252	−1.56392
$656$	−31.6439	−1.23549
$657$	−114.057	−4.44977
$658$	−9.38878	−0.366013
$659$	26.8823	1.04719	0.523593	−	0.851969i	$-0.324591\pi$
$659$	26.8823	1.04719	0.523593	+	0.851969i	$0.324591\pi$
$660$	37.8527	1.47341
$661$	−28.8913	−1.12374	−0.561870	−	0.827226i	$-0.689918\pi$
$661$	−28.8913	−1.12374	−0.561870	+	0.827226i	$0.689918\pi$
$662$	6.50115	0.252675
$663$	16.9328	0.657614
$664$	−18.2356	−0.707680
$665$	−0.360620	−0.0139843
$666$	−149.416	−5.78975
$667$	9.78184	0.378754
$668$	84.3610	3.26402
$669$	51.0413	1.97337
$670$	−5.78443	−0.223472
$671$	7.07214	0.273017
$672$	−14.5293	−0.560481
$673$	−28.1500	−1.08510	−0.542551	−	0.840023i	$-0.682541\pi$
$673$	−28.1500	−1.08510	−0.542551	+	0.840023i	$0.682541\pi$
$674$	−31.4008	−1.20951
$675$	−21.9794	−0.845989
$676$	4.32624	0.166394
$677$	−13.5286	−0.519945	−0.259972	−	0.965616i	$-0.583713\pi$
$677$	−13.5286	−0.519945	−0.259972	+	0.965616i	$0.583713\pi$
$678$	−66.3081	−2.54655
$679$	7.83625	0.300728
$680$	56.6498	2.17242
$681$	66.1047	2.53314
$682$	−22.2701	−0.852765
$683$	18.7037	0.715677	0.357838	−	0.933784i	$-0.383514\pi$
$683$	18.7037	0.715677	0.357838	+	0.933784i	$0.383514\pi$
$684$	−4.69175	−0.179394
$685$	−1.46494	−0.0559725
$686$	40.0664	1.52974
$687$	−27.5119	−1.04964
$688$	−24.1956	−0.922447
$689$	−8.74376	−0.333111
$690$	−39.2314	−1.49351
$691$	−47.0623	−1.79034	−0.895168	−	0.445729i	$-0.852944\pi$
$691$	−47.0623	−1.79034	−0.895168	+	0.445729i	$0.852944\pi$
$692$	8.14694	0.309700
$693$	13.8594	0.526475
$694$	23.4151	0.888824
$695$	−8.44979	−0.320519
$696$	−66.7252	−2.52921
$697$	27.8899	1.05641
$698$	2.22831	0.0843426
$699$	33.1013	1.25201
$700$	−9.60201	−0.362922
$701$	−26.6066	−1.00492	−0.502459	−	0.864601i	$-0.667571\pi$
$701$	−26.6066	−1.00492	−0.502459	+	0.864601i	$0.667571\pi$
$702$	32.1766	1.21443
$703$	−1.30068	−0.0490561
$704$	−4.87646	−0.183788
$705$	16.5850	0.624628
$706$	−20.5358	−0.772875
$707$	4.21081	0.158364
$708$	−16.2109	−0.609243
$709$	−15.1224	−0.567935	−0.283967	−	0.958834i	$-0.591651\pi$
$709$	−15.1224	−0.567935	−0.283967	+	0.958834i	$0.591651\pi$
$710$	−47.6274	−1.78742
$711$	64.6218	2.42351
$712$	−87.3849	−3.27489
$713$	15.7842	0.591124
$714$	55.0177	2.05899
$715$	−2.76163	−0.103279
$716$	−91.2983	−3.41198
$717$	87.3977	3.26393
$718$	−84.1561	−3.14068
$719$	6.01264	0.224234	0.112117	−	0.993695i	$-0.464237\pi$
$719$	6.01264	0.224234	0.112117	+	0.993695i	$0.464237\pi$
$720$	77.3129	2.88128
$721$	4.98195	0.185538
$722$	47.7291	1.77629
$723$	96.4096	3.58551
$724$	24.4978	0.910452
$725$	−6.18431	−0.229680
$726$	69.1385	2.56597
$727$	−28.3834	−1.05268	−0.526341	−	0.850273i	$-0.676437\pi$
$727$	−28.3834	−1.05268	−0.526341	+	0.850273i	$0.676437\pi$
$728$	7.55840	0.280133
$729$	15.0641	0.557928
$730$	73.8444	2.73310
$731$	21.3252	0.788740
$732$	63.5884	2.35029
$733$	21.9308	0.810032	0.405016	−	0.914310i	$-0.367266\pi$
$733$	21.9308	0.810032	0.405016	+	0.914310i	$0.367266\pi$
$734$	−1.17471	−0.0433592
$735$	−30.5990	−1.12866
$736$	−9.64722	−0.355601
$737$	1.93521	0.0712846
$738$	92.3742	3.40034
$739$	25.2878	0.930227	0.465114	−	0.885251i	$-0.346013\pi$
$739$	25.2878	0.930227	0.465114	+	0.885251i	$0.346013\pi$
$740$	66.1544	2.43188
$741$	0.488209	0.0179348
$742$	−28.4101	−1.04297
$743$	6.33037	0.232239	0.116119	−	0.993235i	$-0.462954\pi$
$743$	6.33037	0.232239	0.116119	+	0.993235i	$0.462954\pi$
$744$	−107.670	−3.94736
$745$	7.63608	0.279764
$746$	32.3913	1.18593
$747$	21.9347	0.802548
$748$	−35.2470	−1.28876
$749$	−6.31936	−0.230904
$750$	96.9844	3.54137
$751$	−50.2341	−1.83307	−0.916534	−	0.399957i	$-0.869025\pi$
$751$	−50.2341	−1.83307	−0.916534	+	0.399957i	$0.869025\pi$
$752$	17.5221	0.638964
$753$	36.9049	1.34489
$754$	9.05347	0.329708
$755$	11.9770	0.435888
$756$	71.4957	2.60028
$757$	−35.0953	−1.27556	−0.637779	−	0.770219i	$-0.720147\pi$
$757$	−35.0953	−1.27556	−0.637779	+	0.770219i	$0.720147\pi$
$758$	23.8717	0.867060
$759$	13.1251	0.476410
$760$	1.63334	0.0592474
$761$	−28.1134	−1.01911	−0.509556	−	0.860438i	$-0.670190\pi$
$761$	−28.1134	−1.01911	−0.509556	+	0.860438i	$0.670190\pi$
$762$	−49.2802	−1.78523
$763$	−12.0945	−0.437851
$764$	−16.3363	−0.591027
$765$	−68.1410	−2.46364
$766$	−67.7602	−2.44827
$767$	1.18271	0.0427050
$768$	−100.424	−3.62374
$769$	18.3039	0.660055	0.330028	−	0.943971i	$-0.392942\pi$
$769$	18.3039	0.660055	0.330028	+	0.943971i	$0.392942\pi$
$770$	−8.97307	−0.323367
$771$	−29.6063	−1.06625
$772$	7.46417	0.268641
$773$	13.1012	0.471218	0.235609	−	0.971848i	$-0.424292\pi$
$773$	13.1012	0.471218	0.235609	+	0.971848i	$0.424292\pi$
$774$	70.6311	2.53878
$775$	−9.97917	−0.358463
$776$	−35.4923	−1.27410
$777$	34.5468	1.23936
$778$	56.6381	2.03058
$779$	0.804127	0.0288108
$780$	−24.8309	−0.889090
$781$	15.9340	0.570163
$782$	36.5308	1.30634
$783$	46.0479	1.64562
$784$	−32.3278	−1.15456
$785$	−8.20856	−0.292976
$786$	176.061	6.27990
$787$	−38.2585	−1.36377	−0.681885	−	0.731460i	$-0.738839\pi$
$787$	−38.2585	−1.36377	−0.681885	+	0.731460i	$0.738839\pi$
$788$	71.7941	2.55756
$789$	69.6514	2.47965
$790$	−41.8385	−1.48855
$791$	10.7492	0.382198
$792$	−62.7726	−2.23053
$793$	−4.63924	−0.164744
$794$	−6.77131	−0.240305
$795$	50.1858	1.77991
$796$	−35.9278	−1.27343
$797$	49.2804	1.74560	0.872801	−	0.488077i	$-0.162301\pi$
$797$	49.2804	1.74560	0.872801	+	0.488077i	$0.162301\pi$
$798$	1.58628	0.0561538
$799$	−15.4434	−0.546347
$800$	6.09920	0.215639
$801$	105.111	3.71390
$802$	52.2585	1.84531
$803$	−24.7050	−0.871822
$804$	17.4003	0.613661
$805$	6.35979	0.224153
$806$	14.6089	0.514577
$807$	−101.482	−3.57234
$808$	−19.0718	−0.670943
$809$	−24.0104	−0.844161	−0.422081	−	0.906558i	$-0.638700\pi$
$809$	−24.0104	−0.844161	−0.422081	+	0.906558i	$0.638700\pi$
$810$	−88.4767	−3.10875
$811$	−0.408788	−0.0143545	−0.00717724	−	0.999974i	$-0.502285\pi$
$811$	−0.408788	−0.0143545	−0.00717724	+	0.999974i	$0.502285\pi$
$812$	20.1166	0.705955
$813$	37.1207	1.30188
$814$	−32.3640	−1.13436
$815$	−12.0140	−0.420833
$816$	−102.678	−3.59446
$817$	0.614851	0.0215109
$818$	59.1610	2.06852
$819$	−9.09160	−0.317686
$820$	−40.8990	−1.42825
$821$	5.55179	0.193759	0.0968795	−	0.995296i	$-0.469114\pi$
$821$	5.55179	0.193759	0.0968795	+	0.995296i	$0.469114\pi$
$822$	6.44392	0.224758
$823$	−41.0826	−1.43205	−0.716025	−	0.698075i	$-0.754040\pi$
$823$	−41.0826	−1.43205	−0.716025	+	0.698075i	$0.754040\pi$
$824$	−22.5645	−0.786071
$825$	−8.29799	−0.288899
$826$	3.84283	0.133709
$827$	14.5760	0.506858	0.253429	−	0.967354i	$-0.418442\pi$
$827$	14.5760	0.506858	0.253429	+	0.967354i	$0.418442\pi$
$828$	82.7423	2.87549
$829$	40.9233	1.42132	0.710662	−	0.703533i	$-0.248395\pi$
$829$	40.9233	1.42132	0.710662	+	0.703533i	$0.248395\pi$
$830$	−14.2013	−0.492935
$831$	−93.7466	−3.25204
$832$	3.19890	0.110902
$833$	28.4926	0.987212
$834$	37.1686	1.28704
$835$	35.3259	1.22250
$836$	−1.01625	−0.0351477
$837$	74.3040	2.56832
$838$	−56.5943	−1.95502
$839$	−19.0051	−0.656127	−0.328064	−	0.944656i	$-0.606396\pi$
$839$	−19.0051	−0.656127	−0.328064	+	0.944656i	$0.606396\pi$
$840$	−43.3823	−1.49683
$841$	−16.0436	−0.553228
$842$	27.7477	0.956247
$843$	−1.98513	−0.0683716
$844$	33.7586	1.16202
$845$	1.81160	0.0623209
$846$	−51.1500	−1.75857
$847$	−11.2080	−0.385113
$848$	53.0212	1.82076
$849$	−54.4244	−1.86784
$850$	−23.0956	−0.792174
$851$	22.9384	0.786319
$852$	143.269	4.90831
$853$	8.64295	0.295929	0.147964	−	0.988993i	$-0.452728\pi$
$853$	8.64295	0.295929	0.147964	+	0.988993i	$0.452728\pi$
$854$	−15.0738	−0.515814
$855$	−1.96466	−0.0671898
$856$	28.6219	0.978277
$857$	15.4885	0.529077	0.264539	−	0.964375i	$-0.414780\pi$
$857$	15.4885	0.529077	0.264539	+	0.964375i	$0.414780\pi$
$858$	12.1478	0.414718
$859$	34.7575	1.18591	0.592956	−	0.805235i	$-0.297961\pi$
$859$	34.7575	1.18591	0.592956	+	0.805235i	$0.297961\pi$
$860$	−31.2722	−1.06637
$861$	−21.3580	−0.727880
$862$	−17.7396	−0.604212
$863$	−33.6611	−1.14584	−0.572919	−	0.819612i	$-0.694189\pi$
$863$	−33.6611	−1.14584	−0.572919	+	0.819612i	$0.694189\pi$
$864$	−45.4141	−1.54502
$865$	3.41151	0.115995
$866$	−72.5248	−2.46449
$867$	36.6369	1.24426
$868$	32.4607	1.10179
$869$	13.9973	0.474826
$870$	−51.9634	−1.76172
$871$	−1.26948	−0.0430147
$872$	54.7791	1.85505
$873$	42.6918	1.44490
$874$	1.05326	0.0356272
$875$	−15.7221	−0.531505
$876$	−222.133	−7.50517
$877$	−30.0011	−1.01306	−0.506532	−	0.862221i	$-0.669073\pi$
$877$	−30.0011	−1.01306	−0.506532	+	0.862221i	$0.669073\pi$
$878$	13.0092	0.439038
$879$	−55.9135	−1.88592
$880$	16.7462	0.564515
$881$	−33.5342	−1.12979	−0.564897	−	0.825161i	$-0.691084\pi$
$881$	−33.5342	−1.12979	−0.564897	+	0.825161i	$0.691084\pi$
$882$	94.3705	3.17762
$883$	−15.3852	−0.517752	−0.258876	−	0.965911i	$-0.583352\pi$
$883$	−15.3852	−0.517752	−0.258876	+	0.965911i	$0.583352\pi$
$884$	23.1216	0.777665
$885$	−6.78826	−0.228185
$886$	−20.8349	−0.699962
$887$	57.8564	1.94263	0.971314	−	0.237800i	$-0.0764263\pi$
$887$	57.8564	1.94263	0.971314	+	0.237800i	$0.0764263\pi$
$888$	−156.471	−5.25081
$889$	7.98880	0.267936
$890$	−68.0524	−2.28112
$891$	29.6004	0.991649
$892$	69.6968	2.33362
$893$	−0.445266	−0.0149003
$894$	−33.5893	−1.12339
$895$	−38.2309	−1.27792
$896$	19.5657	0.653643
$897$	−8.60990	−0.287476
$898$	34.3888	1.14757
$899$	20.9068	0.697280
$900$	−52.3117	−1.74372
$901$	−46.7311	−1.55684
$902$	20.0086	0.666212
$903$	−16.3308	−0.543454
$904$	−48.6858	−1.61926
$905$	10.2584	0.340999
$906$	−52.6840	−1.75031
$907$	1.12126	0.0372307	0.0186154	−	0.999827i	$-0.494074\pi$
$907$	1.12126	0.0372307	0.0186154	+	0.999827i	$0.494074\pi$
$908$	90.2658	2.99558
$909$	22.9404	0.760887
$910$	5.88623	0.195127
$911$	−11.3938	−0.377494	−0.188747	−	0.982026i	$-0.560443\pi$
$911$	−11.3938	−0.377494	−0.188747	+	0.982026i	$0.560443\pi$
$912$	−2.96044	−0.0980300
$913$	4.75113	0.157239
$914$	54.3774	1.79864
$915$	26.6274	0.880276
$916$	−37.5674	−1.24126
$917$	−28.5413	−0.942517
$918$	171.968	5.67579
$919$	38.9525	1.28493	0.642463	−	0.766317i	$-0.277913\pi$
$919$	38.9525	1.28493	0.642463	+	0.766317i	$0.277913\pi$
$920$	−28.8050	−0.949674
$921$	−58.6723	−1.93332
$922$	15.2806	0.503239
$923$	−10.4525	−0.344049
$924$	26.9921	0.887974
$925$	−14.5022	−0.476830
$926$	40.5562	1.33276
$927$	27.1416	0.891448
$928$	−12.7781	−0.419461
$929$	28.8405	0.946228	0.473114	−	0.881001i	$-0.343130\pi$
$929$	28.8405	0.946228	0.473114	+	0.881001i	$0.343130\pi$
$930$	−83.8495	−2.74953
$931$	0.821506	0.0269238
$932$	45.1998	1.48057
$933$	−38.7778	−1.26953
$934$	2.91384	0.0953437
$935$	−14.7596	−0.482690
$936$	41.1781	1.34595
$937$	0.671854	0.0219485	0.0109743	−	0.999940i	$-0.496507\pi$
$937$	0.671854	0.0219485	0.0109743	+	0.999940i	$0.496507\pi$
$938$	−4.12478	−0.134679
$939$	55.5642	1.81327
$940$	22.6468	0.738658
$941$	−12.8587	−0.419180	−0.209590	−	0.977789i	$-0.567213\pi$
$941$	−12.8587	−0.419180	−0.209590	+	0.977789i	$0.567213\pi$
$942$	36.1075	1.17645
$943$	−14.1814	−0.461808
$944$	−7.17179	−0.233422
$945$	29.9386	0.973903
$946$	15.2989	0.497411
$947$	36.1853	1.17587	0.587933	−	0.808910i	$-0.299942\pi$
$947$	36.1853	1.17587	0.587933	+	0.808910i	$0.299942\pi$
$948$	125.855	4.08759
$949$	16.2062	0.526077
$950$	−0.665898	−0.0216046
$951$	−11.9452	−0.387350
$952$	40.3959	1.30924
$953$	26.0118	0.842606	0.421303	−	0.906920i	$-0.361573\pi$
$953$	26.0118	0.842606	0.421303	+	0.906920i	$0.361573\pi$
$954$	−154.778	−5.01113
$955$	−6.84078	−0.221363
$956$	119.341	3.85977
$957$	17.3846	0.561965
$958$	39.5772	1.27868
$959$	−1.04462	−0.0337326
$960$	−18.3604	−0.592581
$961$	2.73575	0.0882500
$962$	21.2304	0.684496
$963$	−34.4278	−1.10942
$964$	131.647	4.24007
$965$	3.12560	0.100617
$966$	−27.9752	−0.900087
$967$	−39.5850	−1.27297	−0.636484	−	0.771290i	$-0.719612\pi$
$967$	−39.5850	−1.27297	−0.636484	+	0.771290i	$0.719612\pi$
$968$	50.7639	1.63161
$969$	2.60923	0.0838207
$970$	−27.6402	−0.887474
$971$	30.0445	0.964174	0.482087	−	0.876123i	$-0.339879\pi$
$971$	30.0445	0.964174	0.482087	+	0.876123i	$0.339879\pi$
$972$	100.114	3.21115
$973$	−6.02540	−0.193165
$974$	−52.7868	−1.69140
$975$	5.44339	0.174328
$976$	28.1318	0.900479
$977$	52.0746	1.66601	0.833007	−	0.553262i	$-0.186617\pi$
$977$	52.0746	1.66601	0.833007	+	0.553262i	$0.186617\pi$
$978$	52.8469	1.68986
$979$	22.7673	0.727647
$980$	−41.7829	−1.33470
$981$	−65.8909	−2.10373
$982$	42.8165	1.36633
$983$	−44.0826	−1.40602	−0.703008	−	0.711182i	$-0.748160\pi$
$983$	−44.0826	−1.40602	−0.703008	+	0.711182i	$0.748160\pi$
$984$	96.7357	3.08382
$985$	30.0636	0.957905
$986$	48.3863	1.54094
$987$	11.8265	0.376441
$988$	0.666648	0.0212089
$989$	−10.8433	−0.344798
$990$	−48.8852	−1.55367
$991$	55.4111	1.76019	0.880095	−	0.474797i	$-0.157479\pi$
$991$	55.4111	1.76019	0.880095	+	0.474797i	$0.157479\pi$
$992$	−20.6191	−0.654656
$993$	−8.18912	−0.259874
$994$	−33.9622	−1.07722
$995$	−15.0447	−0.476948
$996$	42.7192	1.35361
$997$	30.1161	0.953785	0.476893	−	0.878962i	$-0.341763\pi$
$997$	30.1161	0.953785	0.476893	+	0.878962i	$0.341763\pi$
$998$	−9.08414	−0.287554
$999$	107.982	3.41641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.8	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.8	✓	156	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)

\(f(q)\)	\(=\)	\(q-2.51520 q^{2} +3.16825 q^{3} +4.32624 q^{4} +1.81160 q^{5} -7.96880 q^{6} +1.29182 q^{7} -5.85097 q^{8} +7.03782 q^{9} +O(q^{10})\)	\(q-2.51520 q^{2} +3.16825 q^{3} +4.32624 q^{4} +1.81160 q^{5} -7.96880 q^{6} +1.29182 q^{7} -5.85097 q^{8} +7.03782 q^{9} -4.55654 q^{10} +1.52442 q^{11} +13.7066 q^{12} -1.00000 q^{13} -3.24919 q^{14} +5.73961 q^{15} +6.06389 q^{16} -5.34451 q^{17} -17.7016 q^{18} -0.154094 q^{19} +7.83742 q^{20} +4.09281 q^{21} -3.83422 q^{22} +2.71756 q^{23} -18.5373 q^{24} -1.71810 q^{25} +2.51520 q^{26} +12.7928 q^{27} +5.58873 q^{28} +3.59950 q^{29} -14.4363 q^{30} +5.80825 q^{31} -3.54996 q^{32} +4.82974 q^{33} +13.4425 q^{34} +2.34026 q^{35} +30.4473 q^{36} +8.44083 q^{37} +0.387577 q^{38} -3.16825 q^{39} -10.5996 q^{40} -5.21842 q^{41} -10.2943 q^{42} -3.99011 q^{43} +6.59499 q^{44} +12.7497 q^{45} -6.83520 q^{46} +2.88958 q^{47} +19.2119 q^{48} -5.33120 q^{49} +4.32138 q^{50} -16.9328 q^{51} -4.32624 q^{52} +8.74376 q^{53} -32.1766 q^{54} +2.76163 q^{55} -7.55840 q^{56} -0.488209 q^{57} -9.05347 q^{58} -1.18271 q^{59} +24.8309 q^{60} +4.63924 q^{61} -14.6089 q^{62} +9.09160 q^{63} -3.19890 q^{64} -1.81160 q^{65} -12.1478 q^{66} +1.26948 q^{67} -23.1216 q^{68} +8.60990 q^{69} -5.88623 q^{70} +10.4525 q^{71} -41.1781 q^{72} -16.2062 q^{73} -21.2304 q^{74} -5.44339 q^{75} -0.666648 q^{76} +1.96927 q^{77} +7.96880 q^{78} +9.18208 q^{79} +10.9853 q^{80} +19.4175 q^{81} +13.1254 q^{82} +3.11669 q^{83} +17.7065 q^{84} -9.68211 q^{85} +10.0359 q^{86} +11.4041 q^{87} -8.91931 q^{88} +14.9351 q^{89} -32.0681 q^{90} -1.29182 q^{91} +11.7568 q^{92} +18.4020 q^{93} -7.26787 q^{94} -0.279157 q^{95} -11.2472 q^{96} +6.06605 q^{97} +13.4090 q^{98} +10.7286 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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