LMFDB - Embedded newform 8047.2.a.b.1.10

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:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
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)

$f(q)$	$=$	$q-2.52587 q^{2} -3.25300 q^{3} +4.38004 q^{4} -2.46689 q^{5} +8.21667 q^{6} -3.38874 q^{7} -6.01169 q^{8} +7.58202 q^{9} +O(q^{10})$	\(q-2.52587 q^{2} -3.25300 q^{3} +4.38004 q^{4} -2.46689 q^{5} +8.21667 q^{6} -3.38874 q^{7} -6.01169 q^{8} +7.58202 q^{9} +6.23106 q^{10} +4.40988 q^{11} -14.2483 q^{12} +1.00000 q^{13} +8.55953 q^{14} +8.02480 q^{15} +6.42469 q^{16} +1.63656 q^{17} -19.1512 q^{18} +4.16852 q^{19} -10.8051 q^{20} +11.0236 q^{21} -11.1388 q^{22} +1.46454 q^{23} +19.5560 q^{24} +1.08555 q^{25} -2.52587 q^{26} -14.9053 q^{27} -14.8428 q^{28} +2.21107 q^{29} -20.2696 q^{30} +0.120759 q^{31} -4.20458 q^{32} -14.3453 q^{33} -4.13376 q^{34} +8.35965 q^{35} +33.2096 q^{36} +9.09197 q^{37} -10.5292 q^{38} -3.25300 q^{39} +14.8302 q^{40} +2.20916 q^{41} -27.8442 q^{42} -2.05769 q^{43} +19.3154 q^{44} -18.7040 q^{45} -3.69923 q^{46} +2.03892 q^{47} -20.8995 q^{48} +4.48356 q^{49} -2.74197 q^{50} -5.32375 q^{51} +4.38004 q^{52} -9.70810 q^{53} +37.6490 q^{54} -10.8787 q^{55} +20.3720 q^{56} -13.5602 q^{57} -5.58490 q^{58} -4.84207 q^{59} +35.1490 q^{60} -5.70646 q^{61} -0.305022 q^{62} -25.6935 q^{63} -2.22914 q^{64} -2.46689 q^{65} +36.2345 q^{66} +5.60774 q^{67} +7.16822 q^{68} -4.76414 q^{69} -21.1154 q^{70} -8.23489 q^{71} -45.5808 q^{72} -11.0604 q^{73} -22.9652 q^{74} -3.53130 q^{75} +18.2583 q^{76} -14.9439 q^{77} +8.21667 q^{78} +7.65557 q^{79} -15.8490 q^{80} +25.7410 q^{81} -5.58006 q^{82} +1.67921 q^{83} +48.2837 q^{84} -4.03723 q^{85} +5.19746 q^{86} -7.19263 q^{87} -26.5108 q^{88} -16.1435 q^{89} +47.2440 q^{90} -3.38874 q^{91} +6.41473 q^{92} -0.392829 q^{93} -5.15005 q^{94} -10.2833 q^{95} +13.6775 q^{96} +8.90888 q^{97} -11.3249 q^{98} +33.4358 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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 $

$2$	−2.52587	−1.78606	−0.893032	−	0.449994i	$-0.851426\pi$
$2$	−2.52587	−1.78606	−0.893032	+	0.449994i	$0.851426\pi$
$3$	−3.25300	−1.87812	−0.939061	−	0.343751i	$-0.888302\pi$
$3$	−3.25300	−1.87812	−0.939061	+	0.343751i	$0.888302\pi$
$4$	4.38004	2.19002
$5$	−2.46689	−1.10323	−0.551614	−	0.834100i	$-0.685988\pi$
$5$	−2.46689	−1.10323	−0.551614	+	0.834100i	$0.685988\pi$
$6$	8.21667	3.35444
$7$	−3.38874	−1.28082	−0.640412	−	0.768032i	$-0.721236\pi$
$7$	−3.38874	−1.28082	−0.640412	+	0.768032i	$0.721236\pi$
$8$	−6.01169	−2.12545
$9$	7.58202	2.52734
$10$	6.23106	1.97043
$11$	4.40988	1.32963	0.664814	−	0.747009i	$-0.268511\pi$
$11$	4.40988	1.32963	0.664814	+	0.747009i	$0.268511\pi$
$12$	−14.2483	−4.11313
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	8.55953	2.28763
$15$	8.02480	2.07199
$16$	6.42469	1.60617
$17$	1.63656	0.396925	0.198463	−	0.980108i	$-0.436405\pi$
$17$	1.63656	0.396925	0.198463	+	0.980108i	$0.436405\pi$
$18$	−19.1512	−4.51399
$19$	4.16852	0.956325	0.478162	−	0.878271i	$-0.341303\pi$
$19$	4.16852	0.956325	0.478162	+	0.878271i	$0.341303\pi$
$20$	−10.8051	−2.41609
$21$	11.0236	2.40554
$22$	−11.1388	−2.37480
$23$	1.46454	0.305377	0.152688	−	0.988274i	$-0.451207\pi$
$23$	1.46454	0.305377	0.152688	+	0.988274i	$0.451207\pi$
$24$	19.5560	3.99186
$25$	1.08555	0.217110
$26$	−2.52587	−0.495365
$27$	−14.9053	−2.86853
$28$	−14.8428	−2.80503
$29$	2.21107	0.410586	0.205293	−	0.978701i	$-0.434185\pi$
$29$	2.21107	0.410586	0.205293	+	0.978701i	$0.434185\pi$
$30$	−20.2696	−3.70071
$31$	0.120759	0.0216889	0.0108445	−	0.999941i	$-0.496548\pi$
$31$	0.120759	0.0216889	0.0108445	+	0.999941i	$0.496548\pi$
$32$	−4.20458	−0.743271
$33$	−14.3453	−2.49720
$34$	−4.13376	−0.708933
$35$	8.35965	1.41304
$36$	33.2096	5.53493
$37$	9.09197	1.49471	0.747356	−	0.664424i	$-0.231323\pi$
$37$	9.09197	1.49471	0.747356	+	0.664424i	$0.231323\pi$
$38$	−10.5292	−1.70806
$39$	−3.25300	−0.520897
$40$	14.8302	2.34486
$41$	2.20916	0.345013	0.172506	−	0.985008i	$-0.444813\pi$
$41$	2.20916	0.345013	0.172506	+	0.985008i	$0.444813\pi$
$42$	−27.8442	−4.29645
$43$	−2.05769	−0.313794	−0.156897	−	0.987615i	$-0.550149\pi$
$43$	−2.05769	−0.313794	−0.156897	+	0.987615i	$0.550149\pi$
$44$	19.3154	2.91191
$45$	−18.7040	−2.78823
$46$	−3.69923	−0.545422
$47$	2.03892	0.297407	0.148703	−	0.988882i	$-0.452490\pi$
$47$	2.03892	0.297407	0.148703	+	0.988882i	$0.452490\pi$
$48$	−20.8995	−3.01659
$49$	4.48356	0.640508
$50$	−2.74197	−0.387773
$51$	−5.32375	−0.745474
$52$	4.38004	0.607403
$53$	−9.70810	−1.33351	−0.666755	−	0.745277i	$-0.732317\pi$
$53$	−9.70810	−1.33351	−0.666755	+	0.745277i	$0.732317\pi$
$54$	37.6490	5.12338
$55$	−10.8787	−1.46688
$56$	20.3720	2.72233
$57$	−13.5602	−1.79609
$58$	−5.58490	−0.733333
$59$	−4.84207	−0.630384	−0.315192	−	0.949028i	$-0.602069\pi$
$59$	−4.84207	−0.630384	−0.315192	+	0.949028i	$0.602069\pi$
$60$	35.1490	4.53771
$61$	−5.70646	−0.730638	−0.365319	−	0.930882i	$-0.619040\pi$
$61$	−5.70646	−0.730638	−0.365319	+	0.930882i	$0.619040\pi$
$62$	−0.305022	−0.0387378
$63$	−25.6935	−3.23708
$64$	−2.22914	−0.278643
$65$	−2.46689	−0.305980
$66$	36.2345	4.46016
$67$	5.60774	0.685095	0.342547	−	0.939501i	$-0.388710\pi$
$67$	5.60774	0.685095	0.342547	+	0.939501i	$0.388710\pi$
$68$	7.16822	0.869275
$69$	−4.76414	−0.573535
$70$	−21.1154	−2.52378
$71$	−8.23489	−0.977301	−0.488651	−	0.872479i	$-0.662511\pi$
$71$	−8.23489	−0.977301	−0.488651	+	0.872479i	$0.662511\pi$
$72$	−45.5808	−5.37174
$73$	−11.0604	−1.29452	−0.647261	−	0.762269i	$-0.724086\pi$
$73$	−11.0604	−1.29452	−0.647261	+	0.762269i	$0.724086\pi$
$74$	−22.9652	−2.66965
$75$	−3.53130	−0.407760
$76$	18.2583	2.09437
$77$	−14.9439	−1.70302
$78$	8.21667	0.930355
$79$	7.65557	0.861319	0.430660	−	0.902514i	$-0.358281\pi$
$79$	7.65557	0.861319	0.430660	+	0.902514i	$0.358281\pi$
$80$	−15.8490	−1.77197
$81$	25.7410	2.86011
$82$	−5.58006	−0.616215
$83$	1.67921	0.184317	0.0921586	−	0.995744i	$-0.470623\pi$
$83$	1.67921	0.184317	0.0921586	+	0.995744i	$0.470623\pi$
$84$	48.2837	5.26819
$85$	−4.03723	−0.437899
$86$	5.19746	0.560456
$87$	−7.19263	−0.771131
$88$	−26.5108	−2.82606
$89$	−16.1435	−1.71121	−0.855605	−	0.517629i	$-0.826815\pi$
$89$	−16.1435	−1.71121	−0.855605	+	0.517629i	$0.826815\pi$
$90$	47.2440	4.97996
$91$	−3.38874	−0.355236
$92$	6.41473	0.668782
$93$	−0.392829	−0.0407344
$94$	−5.15005	−0.531187
$95$	−10.2833	−1.05504
$96$	13.6775	1.39595
$97$	8.90888	0.904559	0.452280	−	0.891876i	$-0.350611\pi$
$97$	8.90888	0.904559	0.452280	+	0.891876i	$0.350611\pi$
$98$	−11.3249	−1.14399
$99$	33.4358	3.36042
$100$	4.75477	0.475477
$101$	5.66423	0.563612	0.281806	−	0.959471i	$-0.409067\pi$
$101$	5.66423	0.563612	0.281806	+	0.959471i	$0.409067\pi$
$102$	13.4471	1.33146
$103$	3.22548	0.317816	0.158908	−	0.987293i	$-0.449203\pi$
$103$	3.22548	0.317816	0.158908	+	0.987293i	$0.449203\pi$
$104$	−6.01169	−0.589495
$105$	−27.1940	−2.65386
$106$	24.5215	2.38173
$107$	−8.48853	−0.820617	−0.410309	−	0.911947i	$-0.634579\pi$
$107$	−8.48853	−0.820617	−0.410309	+	0.911947i	$0.634579\pi$
$108$	−65.2860	−6.28214
$109$	−1.83986	−0.176226	−0.0881131	−	0.996110i	$-0.528084\pi$
$109$	−1.83986	−0.176226	−0.0881131	+	0.996110i	$0.528084\pi$
$110$	27.4782	2.61994
$111$	−29.5762	−2.80725
$112$	−21.7716	−2.05722
$113$	−1.91102	−0.179774	−0.0898868	−	0.995952i	$-0.528651\pi$
$113$	−1.91102	−0.179774	−0.0898868	+	0.995952i	$0.528651\pi$
$114$	34.2514	3.20794
$115$	−3.61285	−0.336900
$116$	9.68460	0.899192
$117$	7.58202	0.700958
$118$	12.2305	1.12591
$119$	−5.54589	−0.508391
$120$	−48.2426	−4.40393
$121$	8.44700	0.767909
$122$	14.4138	1.30497
$123$	−7.18640	−0.647976
$124$	0.528929	0.0474992
$125$	9.65652	0.863705
$126$	64.8985	5.78162
$127$	16.1250	1.43087	0.715433	−	0.698681i	$-0.246229\pi$
$127$	16.1250	1.43087	0.715433	+	0.698681i	$0.246229\pi$
$128$	14.0397	1.24094
$129$	6.69366	0.589343
$130$	6.23106	0.546500
$131$	−13.2142	−1.15453	−0.577265	−	0.816557i	$-0.695880\pi$
$131$	−13.2142	−1.15453	−0.577265	+	0.816557i	$0.695880\pi$
$132$	−62.8332	−5.46893
$133$	−14.1260	−1.22488
$134$	−14.1644	−1.22362
$135$	36.7698	3.16464
$136$	−9.83852	−0.843646
$137$	10.5547	0.901745	0.450872	−	0.892588i	$-0.351113\pi$
$137$	10.5547	0.901745	0.450872	+	0.892588i	$0.351113\pi$
$138$	12.0336	1.02437
$139$	13.8873	1.17791	0.588954	−	0.808166i	$-0.299540\pi$
$139$	13.8873	1.17791	0.588954	+	0.808166i	$0.299540\pi$
$140$	36.6156	3.09459
$141$	−6.63260	−0.558566
$142$	20.8003	1.74552
$143$	4.40988	0.368772
$144$	48.7121	4.05934
$145$	−5.45448	−0.452970
$146$	27.9372	2.31210
$147$	−14.5850	−1.20295
$148$	39.8232	3.27345
$149$	−13.1052	−1.07362	−0.536811	−	0.843703i	$-0.680371\pi$
$149$	−13.1052	−1.07362	−0.536811	+	0.843703i	$0.680371\pi$
$150$	8.91963	0.728285
$151$	−4.58018	−0.372730	−0.186365	−	0.982481i	$-0.559671\pi$
$151$	−4.58018	−0.372730	−0.186365	+	0.982481i	$0.559671\pi$
$152$	−25.0599	−2.03262
$153$	12.4085	1.00317
$154$	37.7465	3.04170
$155$	−0.297899	−0.0239278
$156$	−14.2483	−1.14078
$157$	−14.3771	−1.14741	−0.573707	−	0.819060i	$-0.694495\pi$
$157$	−14.3771	−1.14741	−0.573707	+	0.819060i	$0.694495\pi$
$158$	−19.3370	−1.53837
$159$	31.5805	2.50450
$160$	10.3722	0.819997
$161$	−4.96293	−0.391134
$162$	−65.0185	−5.10834
$163$	−14.2884	−1.11916	−0.559578	−	0.828778i	$-0.689037\pi$
$163$	−14.2884	−1.11916	−0.559578	+	0.828778i	$0.689037\pi$
$164$	9.67622	0.755586
$165$	35.3884	2.75498
$166$	−4.24147	−0.329202
$167$	3.15708	0.244302	0.122151	−	0.992512i	$-0.461021\pi$
$167$	3.15708	0.244302	0.122151	+	0.992512i	$0.461021\pi$
$168$	−66.2703	−5.11287
$169$	1.00000	0.0769231
$170$	10.1975	0.782115
$171$	31.6058	2.41696
$172$	−9.01275	−0.687216
$173$	−18.3857	−1.39784	−0.698918	−	0.715201i	$-0.746335\pi$
$173$	−18.3857	−1.39784	−0.698918	+	0.715201i	$0.746335\pi$
$174$	18.1677	1.37729
$175$	−3.67865	−0.278080
$176$	28.3321	2.13561
$177$	15.7513	1.18394
$178$	40.7765	3.05633
$179$	15.6377	1.16882	0.584409	−	0.811459i	$-0.301326\pi$
$179$	15.6377	1.16882	0.584409	+	0.811459i	$0.301326\pi$
$180$	−81.9244	−6.10629
$181$	−16.2267	−1.20612	−0.603061	−	0.797695i	$-0.706053\pi$
$181$	−16.2267	−1.20612	−0.603061	+	0.797695i	$0.706053\pi$
$182$	8.55953	0.634475
$183$	18.5631	1.37223
$184$	−8.80433	−0.649064
$185$	−22.4289	−1.64901
$186$	0.992236	0.0727543
$187$	7.21705	0.527763
$188$	8.93054	0.651327
$189$	50.5103	3.67408
$190$	25.9743	1.88437
$191$	13.1338	0.950332	0.475166	−	0.879896i	$-0.342388\pi$
$191$	13.1338	0.950332	0.475166	+	0.879896i	$0.342388\pi$
$192$	7.25140	0.523325
$193$	−5.92724	−0.426652	−0.213326	−	0.976981i	$-0.568430\pi$
$193$	−5.92724	−0.426652	−0.213326	+	0.976981i	$0.568430\pi$
$194$	−22.5027	−1.61560
$195$	8.02480	0.574668
$196$	19.6382	1.40273
$197$	−15.4973	−1.10413	−0.552067	−	0.833800i	$-0.686161\pi$
$197$	−15.4973	−1.10413	−0.552067	+	0.833800i	$0.686161\pi$
$198$	−84.4546	−6.00192
$199$	16.7125	1.18472	0.592359	−	0.805674i	$-0.298197\pi$
$199$	16.7125	1.18472	0.592359	+	0.805674i	$0.298197\pi$
$200$	−6.52600	−0.461458
$201$	−18.2420	−1.28669
$202$	−14.3071	−1.00665
$203$	−7.49275	−0.525888
$204$	−23.3182	−1.63260
$205$	−5.44976	−0.380628
$206$	−8.14716	−0.567640
$207$	11.1041	0.771791
$208$	6.42469	0.445472
$209$	18.3827	1.27156
$210$	68.6885	4.73996
$211$	−5.56321	−0.382988	−0.191494	−	0.981494i	$-0.561333\pi$
$211$	−5.56321	−0.382988	−0.191494	+	0.981494i	$0.561333\pi$
$212$	−42.5219	−2.92042
$213$	26.7881	1.83549
$214$	21.4410	1.46567
$215$	5.07609	0.346186
$216$	89.6062	6.09693
$217$	−0.409220	−0.0277797
$218$	4.64724	0.314751
$219$	35.9795	2.43127
$220$	−47.6491	−3.21250
$221$	1.63656	0.110087
$222$	74.7058	5.01392
$223$	−21.8320	−1.46198	−0.730989	−	0.682389i	$-0.760941\pi$
$223$	−21.8320	−1.46198	−0.730989	+	0.682389i	$0.760941\pi$
$224$	14.2482	0.951999
$225$	8.23068	0.548712
$226$	4.82700	0.321087
$227$	1.71020	0.113510	0.0567551	−	0.998388i	$-0.481925\pi$
$227$	1.71020	0.113510	0.0567551	+	0.998388i	$0.481925\pi$
$228$	−59.3943	−3.93348
$229$	20.9060	1.38151	0.690755	−	0.723089i	$-0.257279\pi$
$229$	20.9060	1.38151	0.690755	+	0.723089i	$0.257279\pi$
$230$	9.12560	0.601725
$231$	48.6126	3.19847
$232$	−13.2923	−0.872682
$233$	−21.9669	−1.43910	−0.719549	−	0.694442i	$-0.755651\pi$
$233$	−21.9669	−1.43910	−0.719549	+	0.694442i	$0.755651\pi$
$234$	−19.1512	−1.25196
$235$	−5.02979	−0.328107
$236$	−21.2085	−1.38055
$237$	−24.9036	−1.61766
$238$	14.0082	0.908018
$239$	0.839606	0.0543096	0.0271548	−	0.999631i	$-0.491355\pi$
$239$	0.839606	0.0543096	0.0271548	+	0.999631i	$0.491355\pi$
$240$	51.5568	3.32798
$241$	19.0933	1.22991	0.614953	−	0.788563i	$-0.289175\pi$
$241$	19.0933	1.22991	0.614953	+	0.788563i	$0.289175\pi$
$242$	−21.3361	−1.37153
$243$	−39.0195	−2.50310
$244$	−24.9946	−1.60011
$245$	−11.0604	−0.706626
$246$	18.1520	1.15733
$247$	4.16852	0.265237
$248$	−0.725965	−0.0460988
$249$	−5.46247	−0.346170
$250$	−24.3911	−1.54263
$251$	−7.66731	−0.483956	−0.241978	−	0.970282i	$-0.577796\pi$
$251$	−7.66731	−0.483956	−0.241978	+	0.970282i	$0.577796\pi$
$252$	−112.539	−7.08927
$253$	6.45842	0.406037
$254$	−40.7298	−2.55562
$255$	13.1331	0.822427
$256$	−31.0042	−1.93776
$257$	−14.0903	−0.878926	−0.439463	−	0.898261i	$-0.644831\pi$
$257$	−14.0903	−0.878926	−0.439463	+	0.898261i	$0.644831\pi$
$258$	−16.9073	−1.05260
$259$	−30.8103	−1.91446
$260$	−10.8051	−0.670103
$261$	16.7644	1.03769
$262$	33.3774	2.06206
$263$	−2.01515	−0.124260	−0.0621299	−	0.998068i	$-0.519789\pi$
$263$	−2.01515	−0.124260	−0.0621299	+	0.998068i	$0.519789\pi$
$264$	86.2397	5.30769
$265$	23.9488	1.47117
$266$	35.6806	2.18772
$267$	52.5149	3.21386
$268$	24.5621	1.50037
$269$	−4.15091	−0.253085	−0.126543	−	0.991961i	$-0.540388\pi$
$269$	−4.15091	−0.253085	−0.126543	+	0.991961i	$0.540388\pi$
$270$	−92.8759	−5.65225
$271$	18.1385	1.10183	0.550916	−	0.834560i	$-0.314278\pi$
$271$	18.1385	1.10183	0.550916	+	0.834560i	$0.314278\pi$
$272$	10.5144	0.637530
$273$	11.0236	0.667177
$274$	−26.6597	−1.61057
$275$	4.78715	0.288676
$276$	−20.8671	−1.25605
$277$	−0.0231764	−0.00139254	−0.000696269	−	1.00000i	$-0.500222\pi$
$277$	−0.0231764	−0.00139254	−0.000696269		1.00000i	$0.500222\pi$
$278$	−35.0777	−2.10382
$279$	0.915596	0.0548153
$280$	−50.2556	−3.00335
$281$	−0.321557	−0.0191825	−0.00959126	−	0.999954i	$-0.503053\pi$
$281$	−0.321557	−0.0191825	−0.00959126	+	0.999954i	$0.503053\pi$
$282$	16.7531	0.997634
$283$	14.7800	0.878580	0.439290	−	0.898345i	$-0.355230\pi$
$283$	14.7800	0.878580	0.439290	+	0.898345i	$0.355230\pi$
$284$	−36.0692	−2.14031
$285$	33.4516	1.98150
$286$	−11.1388	−0.658651
$287$	−7.48627	−0.441900
$288$	−31.8792	−1.87850
$289$	−14.3217	−0.842450
$290$	13.7773	0.809033
$291$	−28.9806	−1.69887
$292$	−48.4450	−2.83503
$293$	−6.33053	−0.369833	−0.184917	−	0.982754i	$-0.559202\pi$
$293$	−6.33053	−0.369833	−0.184917	+	0.982754i	$0.559202\pi$
$294$	36.8399	2.14855
$295$	11.9449	0.695457
$296$	−54.6581	−3.17694
$297$	−65.7306	−3.81408
$298$	33.1021	1.91756
$299$	1.46454	0.0846963
$300$	−15.4673	−0.893003
$301$	6.97296	0.401915
$302$	11.5690	0.665719
$303$	−18.4258	−1.05853
$304$	26.7815	1.53602
$305$	14.0772	0.806060
$306$	−31.3422	−1.79172
$307$	9.48140	0.541132	0.270566	−	0.962701i	$-0.412789\pi$
$307$	9.48140	0.541132	0.270566	+	0.962701i	$0.412789\pi$
$308$	−65.4550	−3.72965
$309$	−10.4925	−0.596897
$310$	0.752455	0.0427366
$311$	21.7050	1.23078	0.615390	−	0.788223i	$-0.288999\pi$
$311$	21.7050	1.23078	0.615390	+	0.788223i	$0.288999\pi$
$312$	19.5560	1.10714
$313$	33.7685	1.90871	0.954355	−	0.298674i	$-0.0965444\pi$
$313$	33.7685	1.90871	0.954355	+	0.298674i	$0.0965444\pi$
$314$	36.3147	2.04935
$315$	63.3831	3.57123
$316$	33.5317	1.88631
$317$	15.7820	0.886406	0.443203	−	0.896421i	$-0.353842\pi$
$317$	15.7820	0.886406	0.443203	+	0.896421i	$0.353842\pi$
$318$	−79.7683	−4.47319
$319$	9.75056	0.545927
$320$	5.49905	0.307406
$321$	27.6132	1.54122
$322$	12.5357	0.698589
$323$	6.82206	0.379589
$324$	112.747	6.26370
$325$	1.08555	0.0602156
$326$	36.0908	1.99888
$327$	5.98505	0.330974
$328$	−13.2808	−0.733309
$329$	−6.90936	−0.380925
$330$	−89.3866	−4.92057
$331$	23.7160	1.30355	0.651774	−	0.758414i	$-0.274025\pi$
$331$	23.7160	1.30355	0.651774	+	0.758414i	$0.274025\pi$
$332$	7.35501	0.403659
$333$	68.9355	3.77764
$334$	−7.97439	−0.436339
$335$	−13.8337	−0.755815
$336$	70.8230	3.86371
$337$	27.2889	1.48652	0.743260	−	0.669003i	$-0.233279\pi$
$337$	27.2889	1.48652	0.743260	+	0.669003i	$0.233279\pi$
$338$	−2.52587	−0.137389
$339$	6.21655	0.337637
$340$	−17.6832	−0.959008
$341$	0.532531	0.0288382
$342$	−79.8324	−4.31684
$343$	8.52757	0.460446
$344$	12.3702	0.666955
$345$	11.7526	0.632739
$346$	46.4399	2.49662
$347$	−18.3794	−0.986657	−0.493329	−	0.869843i	$-0.664220\pi$
$347$	−18.3794	−0.986657	−0.493329	+	0.869843i	$0.664220\pi$
$348$	−31.5040	−1.68879
$349$	−6.34447	−0.339612	−0.169806	−	0.985478i	$-0.554314\pi$
$349$	−6.34447	−0.339612	−0.169806	+	0.985478i	$0.554314\pi$
$350$	9.29182	0.496669
$351$	−14.9053	−0.795587
$352$	−18.5417	−0.988274
$353$	−13.6413	−0.726053	−0.363027	−	0.931779i	$-0.618257\pi$
$353$	−13.6413	−0.726053	−0.363027	+	0.931779i	$0.618257\pi$
$354$	−39.7857	−2.11459
$355$	20.3146	1.07819
$356$	−70.7093	−3.74759
$357$	18.0408	0.954820
$358$	−39.4989	−2.08758
$359$	22.4081	1.18265	0.591327	−	0.806432i	$-0.298604\pi$
$359$	22.4081	1.18265	0.591327	+	0.806432i	$0.298604\pi$
$360$	112.443	5.92625
$361$	−1.62342	−0.0854430
$362$	40.9867	2.15421
$363$	−27.4781	−1.44223
$364$	−14.8428	−0.777975
$365$	27.2848	1.42815
$366$	−46.8882	−2.45088
$367$	−31.9021	−1.66528	−0.832638	−	0.553818i	$-0.813170\pi$
$367$	−31.9021	−1.66528	−0.832638	+	0.553818i	$0.813170\pi$
$368$	9.40918	0.490488
$369$	16.7499	0.871965
$370$	56.6526	2.94523
$371$	32.8982	1.70799
$372$	−1.72061	−0.0892093
$373$	31.8156	1.64735	0.823674	−	0.567063i	$-0.191920\pi$
$373$	31.8156	1.64735	0.823674	+	0.567063i	$0.191920\pi$
$374$	−18.2294	−0.942617
$375$	−31.4127	−1.62214
$376$	−12.2573	−0.632124
$377$	2.21107	0.113876
$378$	−127.583	−6.56214
$379$	12.0375	0.618327	0.309163	−	0.951009i	$-0.399951\pi$
$379$	12.0375	0.618327	0.309163	+	0.951009i	$0.399951\pi$
$380$	−45.0413	−2.31057
$381$	−52.4548	−2.68734
$382$	−33.1744	−1.69735
$383$	−1.29214	−0.0660252	−0.0330126	−	0.999455i	$-0.510510\pi$
$383$	−1.29214	−0.0660252	−0.0330126	+	0.999455i	$0.510510\pi$
$384$	−45.6711	−2.33064
$385$	36.8650	1.87882
$386$	14.9715	0.762027
$387$	−15.6014	−0.793065
$388$	39.0213	1.98100
$389$	10.9433	0.554845	0.277423	−	0.960748i	$-0.410520\pi$
$389$	10.9433	0.554845	0.277423	+	0.960748i	$0.410520\pi$
$390$	−20.2696	−1.02639
$391$	2.39681	0.121212
$392$	−26.9537	−1.36137
$393$	42.9858	2.16835
$394$	39.1441	1.97205
$395$	−18.8855	−0.950231
$396$	146.450	7.35939
$397$	37.9222	1.90326	0.951631	−	0.307243i	$-0.0994064\pi$
$397$	37.9222	1.90326	0.951631	+	0.307243i	$0.0994064\pi$
$398$	−42.2137	−2.11598
$399$	45.9520	2.30048
$400$	6.97433	0.348717
$401$	−29.8711	−1.49169	−0.745846	−	0.666118i	$-0.767955\pi$
$401$	−29.8711	−1.49169	−0.745846	+	0.666118i	$0.767955\pi$
$402$	46.0770	2.29811
$403$	0.120759	0.00601543
$404$	24.8096	1.23432
$405$	−63.5002	−3.15535
$406$	18.9258	0.939270
$407$	40.0945	1.98741
$408$	32.0047	1.58447
$409$	−4.53690	−0.224335	−0.112168	−	0.993689i	$-0.535779\pi$
$409$	−4.53690	−0.224335	−0.112168	+	0.993689i	$0.535779\pi$
$410$	13.7654	0.679825
$411$	−34.3343	−1.69359
$412$	14.1277	0.696024
$413$	16.4085	0.807410
$414$	−28.0477	−1.37847
$415$	−4.14243	−0.203344
$416$	−4.20458	−0.206146
$417$	−45.1755	−2.21226
$418$	−46.4323	−2.27108
$419$	−34.8034	−1.70026	−0.850128	−	0.526577i	$-0.823475\pi$
$419$	−34.8034	−1.70026	−0.850128	+	0.526577i	$0.823475\pi$
$420$	−119.111	−5.81201
$421$	8.40464	0.409617	0.204808	−	0.978802i	$-0.434343\pi$
$421$	8.40464	0.409617	0.204808	+	0.978802i	$0.434343\pi$
$422$	14.0520	0.684040
$423$	15.4591	0.751648
$424$	58.3621	2.83431
$425$	1.77658	0.0861766
$426$	−67.6634	−3.27830
$427$	19.3377	0.935818
$428$	−37.1801	−1.79717
$429$	−14.3453	−0.692599
$430$	−12.8216	−0.618310
$431$	−0.953929	−0.0459491	−0.0229746	−	0.999736i	$-0.507314\pi$
$431$	−0.953929	−0.0459491	−0.0229746	+	0.999736i	$0.507314\pi$
$432$	−95.7621	−4.60735
$433$	−17.7329	−0.852189	−0.426094	−	0.904679i	$-0.640111\pi$
$433$	−17.7329	−0.852189	−0.426094	+	0.904679i	$0.640111\pi$
$434$	1.03364	0.0496163
$435$	17.7434	0.850732
$436$	−8.05865	−0.385939
$437$	6.10495	0.292039
$438$	−90.8797	−4.34240
$439$	−34.5804	−1.65043	−0.825216	−	0.564817i	$-0.808947\pi$
$439$	−34.5804	−1.65043	−0.825216	+	0.564817i	$0.808947\pi$
$440$	65.3993	3.11779
$441$	33.9944	1.61878
$442$	−4.13376	−0.196623
$443$	−5.88598	−0.279651	−0.139826	−	0.990176i	$-0.544654\pi$
$443$	−5.88598	−0.279651	−0.139826	+	0.990176i	$0.544654\pi$
$444$	−129.545	−6.14794
$445$	39.8243	1.88785
$446$	55.1449	2.61119
$447$	42.6313	2.01639
$448$	7.55398	0.356892
$449$	31.1042	1.46790	0.733949	−	0.679205i	$-0.237675\pi$
$449$	31.1042	1.46790	0.733949	+	0.679205i	$0.237675\pi$
$450$	−20.7897	−0.980034
$451$	9.74212	0.458739
$452$	−8.37035	−0.393708
$453$	14.8993	0.700032
$454$	−4.31976	−0.202736
$455$	8.35965	0.391907
$456$	81.5198	3.81751
$457$	−36.5753	−1.71092	−0.855461	−	0.517867i	$-0.826726\pi$
$457$	−36.5753	−1.71092	−0.855461	+	0.517867i	$0.826726\pi$
$458$	−52.8060	−2.46746
$459$	−24.3935	−1.13859
$460$	−15.8244	−0.737818
$461$	−10.5570	−0.491686	−0.245843	−	0.969310i	$-0.579065\pi$
$461$	−10.5570	−0.491686	−0.245843	+	0.969310i	$0.579065\pi$
$462$	−122.789	−5.71268
$463$	2.59293	0.120504	0.0602519	−	0.998183i	$-0.480810\pi$
$463$	2.59293	0.120504	0.0602519	+	0.998183i	$0.480810\pi$
$464$	14.2055	0.659472
$465$	0.969066	0.0449393
$466$	55.4856	2.57032
$467$	−7.75694	−0.358948	−0.179474	−	0.983763i	$-0.557440\pi$
$467$	−7.75694	−0.358948	−0.179474	+	0.983763i	$0.557440\pi$
$468$	33.2096	1.53511
$469$	−19.0032	−0.877485
$470$	12.7046	0.586020
$471$	46.7686	2.15498
$472$	29.1090	1.33985
$473$	−9.07414	−0.417229
$474$	62.9034	2.88925
$475$	4.52515	0.207628
$476$	−24.2912	−1.11339
$477$	−73.6071	−3.37024
$478$	−2.12074	−0.0970003
$479$	−9.12367	−0.416871	−0.208436	−	0.978036i	$-0.566837\pi$
$479$	−9.12367	−0.416871	−0.208436	+	0.978036i	$0.566837\pi$
$480$	−33.7409	−1.54005
$481$	9.09197	0.414558
$482$	−48.2273	−2.19669
$483$	16.1444	0.734596
$484$	36.9982	1.68174
$485$	−21.9772	−0.997934
$486$	98.5584	4.47070
$487$	28.5242	1.29256	0.646279	−	0.763102i	$-0.276324\pi$
$487$	28.5242	1.29256	0.646279	+	0.763102i	$0.276324\pi$
$488$	34.3055	1.55294
$489$	46.4803	2.10191
$490$	27.9373	1.26208
$491$	12.0146	0.542211	0.271106	−	0.962550i	$-0.412611\pi$
$491$	12.0146	0.542211	0.271106	+	0.962550i	$0.412611\pi$
$492$	−31.4767	−1.41908
$493$	3.61857	0.162972
$494$	−10.5292	−0.473730
$495$	−82.4824	−3.70731
$496$	0.775838	0.0348361
$497$	27.9059	1.25175
$498$	13.7975	0.618282
$499$	18.5350	0.829739	0.414869	−	0.909881i	$-0.363827\pi$
$499$	18.5350	0.829739	0.414869	+	0.909881i	$0.363827\pi$
$500$	42.2960	1.89153
$501$	−10.2700	−0.458829
$502$	19.3667	0.864376
$503$	−33.2696	−1.48342	−0.741708	−	0.670723i	$-0.765984\pi$
$503$	−33.2696	−1.48342	−0.741708	+	0.670723i	$0.765984\pi$
$504$	154.461	6.88025
$505$	−13.9730	−0.621792
$506$	−16.3132	−0.725208
$507$	−3.25300	−0.144471
$508$	70.6284	3.13363
$509$	2.99173	0.132606	0.0663029	−	0.997800i	$-0.478880\pi$
$509$	2.99173	0.132606	0.0663029	+	0.997800i	$0.478880\pi$
$510$	−33.1726	−1.46891
$511$	37.4808	1.65805
$512$	50.2334	2.22002
$513$	−62.1332	−2.74325
$514$	35.5902	1.56982
$515$	−7.95691	−0.350623
$516$	29.3185	1.29067
$517$	8.99137	0.395440
$518$	77.8230	3.41935
$519$	59.8086	2.62531
$520$	14.8302	0.650347
$521$	−39.1306	−1.71434	−0.857171	−	0.515032i	$-0.827780\pi$
$521$	−39.1306	−1.71434	−0.857171	+	0.515032i	$0.827780\pi$
$522$	−42.3448	−1.85338
$523$	22.2910	0.974718	0.487359	−	0.873202i	$-0.337960\pi$
$523$	22.2910	0.974718	0.487359	+	0.873202i	$0.337960\pi$
$524$	−57.8788	−2.52845
$525$	11.9667	0.522268
$526$	5.09003	0.221936
$527$	0.197630	0.00860888
$528$	−92.1643	−4.01094
$529$	−20.8551	−0.906745
$530$	−60.4918	−2.62759
$531$	−36.7127	−1.59319
$532$	−61.8726	−2.68252
$533$	2.20916	0.0956894
$534$	−132.646	−5.74016
$535$	20.9403	0.905327
$536$	−33.7120	−1.45614
$537$	−50.8695	−2.19518
$538$	10.4847	0.452026
$539$	19.7719	0.851637
$540$	161.053	6.93063
$541$	2.82447	0.121433	0.0607167	−	0.998155i	$-0.480661\pi$
$541$	2.82447	0.121433	0.0607167	+	0.998155i	$0.480661\pi$
$542$	−45.8155	−1.96794
$543$	52.7856	2.26525
$544$	−6.88106	−0.295023
$545$	4.53872	0.194418
$546$	−27.8442	−1.19162
$547$	−45.5705	−1.94846	−0.974228	−	0.225567i	$-0.927577\pi$
$547$	−45.5705	−1.94846	−0.974228	+	0.225567i	$0.927577\pi$
$548$	46.2298	1.97484
$549$	−43.2665	−1.84657
$550$	−12.0917	−0.515594
$551$	9.21691	0.392654
$552$	28.6405	1.21902
$553$	−25.9427	−1.10320
$554$	0.0585408	0.00248716
$555$	72.9613	3.09703
$556$	60.8271	2.57965
$557$	45.1243	1.91198	0.955989	−	0.293404i	$-0.0947880\pi$
$557$	45.1243	1.91198	0.955989	+	0.293404i	$0.0947880\pi$
$558$	−2.31268	−0.0979036
$559$	−2.05769	−0.0870308
$560$	53.7081	2.26958
$561$	−23.4771	−0.991203
$562$	0.812214	0.0342612
$563$	34.7076	1.46275	0.731376	−	0.681975i	$-0.238879\pi$
$563$	34.7076	1.46275	0.731376	+	0.681975i	$0.238879\pi$
$564$	−29.0511	−1.22327
$565$	4.71428	0.198331
$566$	−37.3324	−1.56920
$567$	−87.2295	−3.66329
$568$	49.5056	2.07721
$569$	−31.9039	−1.33748	−0.668741	−	0.743495i	$-0.733166\pi$
$569$	−31.9039	−1.33748	−0.668741	+	0.743495i	$0.733166\pi$
$570$	−84.4945	−3.53908
$571$	−23.3277	−0.976235	−0.488118	−	0.872778i	$-0.662316\pi$
$571$	−23.3277	−0.976235	−0.488118	+	0.872778i	$0.662316\pi$
$572$	19.3154	0.807619
$573$	−42.7244	−1.78484
$574$	18.9094	0.789262
$575$	1.58983	0.0663005
$576$	−16.9014	−0.704225
$577$	42.3485	1.76299	0.881495	−	0.472193i	$-0.156537\pi$
$577$	42.3485	1.76299	0.881495	+	0.472193i	$0.156537\pi$
$578$	36.1747	1.50467
$579$	19.2813	0.801304
$580$	−23.8909	−0.992014
$581$	−5.69040	−0.236078
$582$	73.2013	3.03429
$583$	−42.8115	−1.77307
$584$	66.4917	2.75145
$585$	−18.7040	−0.773316
$586$	15.9901	0.660546
$587$	−42.5550	−1.75643	−0.878217	−	0.478262i	$-0.841267\pi$
$587$	−42.5550	−1.75643	−0.878217	+	0.478262i	$0.841267\pi$
$588$	−63.8830	−2.63449
$589$	0.503386	0.0207417
$590$	−30.1712	−1.24213
$591$	50.4126	2.07370
$592$	58.4131	2.40076
$593$	−43.6727	−1.79342	−0.896711	−	0.442617i	$-0.854050\pi$
$593$	−43.6727	−1.79342	−0.896711	+	0.442617i	$0.854050\pi$
$594$	166.027	6.81218
$595$	13.6811	0.560871
$596$	−57.4014	−2.35125
$597$	−54.3658	−2.22504
$598$	−3.69923	−0.151273
$599$	−17.4443	−0.712755	−0.356377	−	0.934342i	$-0.615988\pi$
$599$	−17.4443	−0.712755	−0.356377	+	0.934342i	$0.615988\pi$
$600$	21.2291	0.866674
$601$	−34.3766	−1.40225	−0.701125	−	0.713039i	$-0.747318\pi$
$601$	−34.3766	−1.40225	−0.701125	+	0.713039i	$0.747318\pi$
$602$	−17.6128	−0.717845
$603$	42.5180	1.73147
$604$	−20.0614	−0.816286
$605$	−20.8378	−0.847179
$606$	46.5412	1.89061
$607$	−39.9912	−1.62319	−0.811596	−	0.584220i	$-0.801401\pi$
$607$	−39.9912	−1.62319	−0.811596	+	0.584220i	$0.801401\pi$
$608$	−17.5269	−0.710809
$609$	24.3739	0.987682
$610$	−35.5573	−1.43967
$611$	2.03892	0.0824857
$612$	54.3496	2.19695
$613$	33.3161	1.34563	0.672813	−	0.739813i	$-0.265086\pi$
$613$	33.3161	1.34563	0.672813	+	0.739813i	$0.265086\pi$
$614$	−23.9488	−0.966496
$615$	17.7281	0.714865
$616$	89.8382	3.61968
$617$	−5.20072	−0.209373	−0.104687	−	0.994505i	$-0.533384\pi$
$617$	−5.20072	−0.209373	−0.104687	+	0.994505i	$0.533384\pi$
$618$	26.5027	1.06610
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−1.30481	−0.0524024
$621$	−21.8294	−0.875983
$622$	−54.8242	−2.19825
$623$	54.7062	2.19176
$624$	−20.8995	−0.836650
$625$	−29.2493	−1.16997
$626$	−85.2951	−3.40908
$627$	−59.7989	−2.38814
$628$	−62.9721	−2.51286
$629$	14.8796	0.593289
$630$	−160.098	−6.37844
$631$	−3.51758	−0.140033	−0.0700163	−	0.997546i	$-0.522305\pi$
$631$	−3.51758	−0.140033	−0.0700163	+	0.997546i	$0.522305\pi$
$632$	−46.0229	−1.83069
$633$	18.0971	0.719297
$634$	−39.8634	−1.58318
$635$	−39.7787	−1.57857
$636$	138.324	5.48490
$637$	4.48356	0.177645
$638$	−24.6287	−0.975059
$639$	−62.4371	−2.46997
$640$	−34.6344	−1.36904
$641$	−9.75926	−0.385468	−0.192734	−	0.981251i	$-0.561735\pi$
$641$	−9.75926	−0.385468	−0.192734	+	0.981251i	$0.561735\pi$
$642$	−69.7475	−2.75271
$643$	−44.7857	−1.76618	−0.883088	−	0.469208i	$-0.844540\pi$
$643$	−44.7857	−1.76618	−0.883088	+	0.469208i	$0.844540\pi$
$644$	−21.7378	−0.856591
$645$	−16.5125	−0.650180
$646$	−17.2317	−0.677971
$647$	34.1283	1.34172	0.670860	−	0.741584i	$-0.265925\pi$
$647$	34.1283	1.34172	0.670860	+	0.741584i	$0.265925\pi$
$648$	−154.747	−6.07903
$649$	−21.3529	−0.838176
$650$	−2.74197	−0.107549
$651$	1.33119	0.0521736
$652$	−62.5839	−2.45098
$653$	−15.2619	−0.597244	−0.298622	−	0.954371i	$-0.596527\pi$
$653$	−15.2619	−0.597244	−0.298622	+	0.954371i	$0.596527\pi$
$654$	−15.1175	−0.591141
$655$	32.5980	1.27371
$656$	14.1932	0.554150
$657$	−83.8602	−3.27170
$658$	17.4522	0.680356
$659$	−23.6298	−0.920487	−0.460243	−	0.887793i	$-0.652238\pi$
$659$	−23.6298	−0.920487	−0.460243	+	0.887793i	$0.652238\pi$
$660$	155.003	6.03347
$661$	3.33121	0.129569	0.0647845	−	0.997899i	$-0.479364\pi$
$661$	3.33121	0.129569	0.0647845	+	0.997899i	$0.479364\pi$
$662$	−59.9035	−2.32822
$663$	−5.32375	−0.206757
$664$	−10.0949	−0.391757
$665$	34.8474	1.35132
$666$	−174.123	−6.74711
$667$	3.23820	0.125383
$668$	13.8281	0.535027
$669$	71.0195	2.74577
$670$	34.9422	1.34993
$671$	−25.1648	−0.971476
$672$	−46.3495	−1.78797
$673$	22.2686	0.858392	0.429196	−	0.903211i	$-0.358797\pi$
$673$	22.2686	0.858392	0.429196	+	0.903211i	$0.358797\pi$
$674$	−68.9283	−2.65502
$675$	−16.1805	−0.622788
$676$	4.38004	0.168463
$677$	12.3002	0.472734	0.236367	−	0.971664i	$-0.424043\pi$
$677$	12.3002	0.472734	0.236367	+	0.971664i	$0.424043\pi$
$678$	−15.7022	−0.603041
$679$	−30.1899	−1.15858
$680$	24.2706	0.930733
$681$	−5.56329	−0.213186
$682$	−1.34511	−0.0515068
$683$	19.3669	0.741053	0.370526	−	0.928822i	$-0.379177\pi$
$683$	19.3669	0.741053	0.370526	+	0.928822i	$0.379177\pi$
$684$	138.435	5.29319
$685$	−26.0372	−0.994829
$686$	−21.5396	−0.822385
$687$	−68.0073	−2.59464
$688$	−13.2200	−0.504007
$689$	−9.70810	−0.369849
$690$	−29.6856	−1.13011
$691$	−20.5961	−0.783513	−0.391756	−	0.920069i	$-0.628132\pi$
$691$	−20.5961	−0.783513	−0.391756	+	0.920069i	$0.628132\pi$
$692$	−80.5300	−3.06129
$693$	−113.305	−4.30411
$694$	46.4240	1.76223
$695$	−34.2586	−1.29950
$696$	43.2398	1.63900
$697$	3.61543	0.136944
$698$	16.0253	0.606568
$699$	71.4583	2.70280
$700$	−16.1127	−0.609001
$701$	−34.5345	−1.30435	−0.652175	−	0.758069i	$-0.726143\pi$
$701$	−34.5345	−1.30435	−0.652175	+	0.758069i	$0.726143\pi$
$702$	37.6490	1.42097
$703$	37.9001	1.42943
$704$	−9.83023	−0.370491
$705$	16.3619	0.616225
$706$	34.4562	1.29678
$707$	−19.1946	−0.721887
$708$	68.9912	2.59285
$709$	12.6790	0.476168	0.238084	−	0.971245i	$-0.423481\pi$
$709$	12.6790	0.476168	0.238084	+	0.971245i	$0.423481\pi$
$710$	−51.3121	−1.92571
$711$	58.0447	2.17685
$712$	97.0499	3.63710
$713$	0.176856	0.00662329
$714$	−45.5688	−1.70537
$715$	−10.8787	−0.406840
$716$	68.4939	2.55974
$717$	−2.73124	−0.102000
$718$	−56.6000	−2.11229
$719$	−39.4697	−1.47197	−0.735986	−	0.676997i	$-0.763281\pi$
$719$	−39.4697	−1.47197	−0.735986	+	0.676997i	$0.763281\pi$
$720$	−120.167	−4.47838
$721$	−10.9303	−0.407066
$722$	4.10055	0.152607
$723$	−62.1105	−2.30991
$724$	−71.0737	−2.64143
$725$	2.40024	0.0891426
$726$	69.4063	2.57591
$727$	1.45149	0.0538327	0.0269164	−	0.999638i	$-0.491431\pi$
$727$	1.45149	0.0538327	0.0269164	+	0.999638i	$0.491431\pi$
$728$	20.3720	0.755038
$729$	49.7076	1.84102
$730$	−68.9180	−2.55077
$731$	−3.36754	−0.124553
$732$	81.3073	3.00521
$733$	1.84156	0.0680196	0.0340098	−	0.999421i	$-0.489172\pi$
$733$	1.84156	0.0680196	0.0340098	+	0.999421i	$0.489172\pi$
$734$	80.5807	2.97429
$735$	35.9796	1.32713
$736$	−6.15775	−0.226978
$737$	24.7294	0.910921
$738$	−42.3081	−1.55738
$739$	15.1247	0.556371	0.278186	−	0.960527i	$-0.410267\pi$
$739$	15.1247	0.556371	0.278186	+	0.960527i	$0.410267\pi$
$740$	−98.2396	−3.61136
$741$	−13.5602	−0.498147
$742$	−83.0968	−3.05058
$743$	18.6613	0.684617	0.342309	−	0.939588i	$-0.388791\pi$
$743$	18.6613	0.684617	0.342309	+	0.939588i	$0.388791\pi$
$744$	2.36156	0.0865791
$745$	32.3291	1.18445
$746$	−80.3622	−2.94227
$747$	12.7318	0.465832
$748$	31.6110	1.15581
$749$	28.7654	1.05107
$750$	79.3445	2.89725
$751$	27.9338	1.01932	0.509659	−	0.860377i	$-0.329772\pi$
$751$	27.9338	1.01932	0.509659	+	0.860377i	$0.329772\pi$
$752$	13.0994	0.477686
$753$	24.9418	0.908929
$754$	−5.58490	−0.203390
$755$	11.2988	0.411206
$756$	221.237	8.04632
$757$	−20.7823	−0.755346	−0.377673	−	0.925939i	$-0.623276\pi$
$757$	−20.7823	−0.755346	−0.377673	+	0.925939i	$0.623276\pi$
$758$	−30.4053	−1.10437
$759$	−21.0092	−0.762587
$760$	61.8200	2.24245
$761$	33.1746	1.20258	0.601290	−	0.799031i	$-0.294654\pi$
$761$	33.1746	1.20258	0.601290	+	0.799031i	$0.294654\pi$
$762$	132.494	4.79976
$763$	6.23479	0.225715
$764$	57.5268	2.08125
$765$	−30.6103	−1.10672
$766$	3.26378	0.117925
$767$	−4.84207	−0.174837
$768$	100.857	3.63935
$769$	11.9381	0.430497	0.215249	−	0.976559i	$-0.430944\pi$
$769$	11.9381	0.430497	0.215249	+	0.976559i	$0.430944\pi$
$770$	−93.1164	−3.35568
$771$	45.8356	1.65073
$772$	−25.9615	−0.934377
$773$	36.8151	1.32415	0.662074	−	0.749438i	$-0.269676\pi$
$773$	36.8151	1.32415	0.662074	+	0.749438i	$0.269676\pi$
$774$	39.4072	1.41646
$775$	0.131090	0.00470889
$776$	−53.5574	−1.92260
$777$	100.226	3.59559
$778$	−27.6413	−0.990989
$779$	9.20894	0.329944
$780$	35.1490	1.25854
$781$	−36.3148	−1.29945
$782$	−6.05403	−0.216492
$783$	−32.9568	−1.17778
$784$	28.8054	1.02877
$785$	35.4666	1.26586
$786$	−108.577	−3.87281
$787$	−49.7745	−1.77427	−0.887135	−	0.461510i	$-0.847308\pi$
$787$	−49.7745	−1.77427	−0.887135	+	0.461510i	$0.847308\pi$
$788$	−67.8786	−2.41808
$789$	6.55530	0.233375
$790$	47.7023	1.69717
$791$	6.47595	0.230258
$792$	−201.005	−7.14242
$793$	−5.70646	−0.202643
$794$	−95.7868	−3.39935
$795$	−77.9056	−2.76303
$796$	73.2015	2.59456
$797$	−1.29914	−0.0460177	−0.0230089	−	0.999735i	$-0.507325\pi$
$797$	−1.29914	−0.0460177	−0.0230089	+	0.999735i	$0.507325\pi$
$798$	−116.069	−4.10880
$799$	3.33682	0.118048
$800$	−4.56429	−0.161372
$801$	−122.401	−4.32481
$802$	75.4507	2.66426
$803$	−48.7750	−1.72123
$804$	−79.9007	−2.81788
$805$	12.2430	0.431509
$806$	−0.305022	−0.0107439
$807$	13.5029	0.475325
$808$	−34.0516	−1.19793
$809$	−16.1240	−0.566888	−0.283444	−	0.958989i	$-0.591477\pi$
$809$	−16.1240	−0.566888	−0.283444	+	0.958989i	$0.591477\pi$
$810$	160.394	5.63566
$811$	28.6228	1.00508	0.502541	−	0.864553i	$-0.332399\pi$
$811$	28.6228	1.00508	0.502541	+	0.864553i	$0.332399\pi$
$812$	−32.8186	−1.15171
$813$	−59.0044	−2.06938
$814$	−101.274	−3.54964
$815$	35.2480	1.23468
$816$	−34.2034	−1.19736
$817$	−8.57751	−0.300089
$818$	11.4596	0.400677
$819$	−25.6935	−0.897803
$820$	−23.8702	−0.833583
$821$	5.53525	0.193182	0.0965908	−	0.995324i	$-0.469206\pi$
$821$	5.53525	0.193182	0.0965908	+	0.995324i	$0.469206\pi$
$822$	86.7241	3.02485
$823$	37.7630	1.31634	0.658168	−	0.752871i	$-0.271331\pi$
$823$	37.7630	1.31634	0.658168	+	0.752871i	$0.271331\pi$
$824$	−19.3906	−0.675503
$825$	−15.5726	−0.542169
$826$	−41.4458	−1.44209
$827$	50.2629	1.74781	0.873906	−	0.486095i	$-0.161579\pi$
$827$	50.2629	1.74781	0.873906	+	0.486095i	$0.161579\pi$
$828$	48.6366	1.69024
$829$	−8.09939	−0.281303	−0.140652	−	0.990059i	$-0.544920\pi$
$829$	−8.09939	−0.281303	−0.140652	+	0.990059i	$0.544920\pi$
$830$	10.4633	0.363185
$831$	0.0753930	0.00261535
$832$	−2.22914	−0.0772816
$833$	7.33763	0.254234
$834$	114.108	3.95123
$835$	−7.78817	−0.269521
$836$	80.5169	2.78473
$837$	−1.79995	−0.0622154
$838$	87.9089	3.03676
$839$	14.5271	0.501530	0.250765	−	0.968048i	$-0.419318\pi$
$839$	14.5271	0.501530	0.250765	+	0.968048i	$0.419318\pi$
$840$	163.482	5.64065
$841$	−24.1112	−0.831419
$842$	−21.2291	−0.731602
$843$	1.04603	0.0360271
$844$	−24.3671	−0.838751
$845$	−2.46689	−0.0848636
$846$	−39.0478	−1.34249
$847$	−28.6247	−0.983556
$848$	−62.3715	−2.14185
$849$	−48.0794	−1.65008
$850$	−4.48741	−0.153917
$851$	13.3155	0.456450
$852$	117.333	4.01976
$853$	1.67049	0.0571966	0.0285983	−	0.999591i	$-0.490896\pi$
$853$	1.67049	0.0571966	0.0285983	+	0.999591i	$0.490896\pi$
$854$	−48.8447	−1.67143
$855$	−77.9681	−2.66645
$856$	51.0304	1.74418
$857$	−17.6379	−0.602500	−0.301250	−	0.953545i	$-0.597404\pi$
$857$	−17.6379	−0.602500	−0.301250	+	0.953545i	$0.597404\pi$
$858$	36.2345	1.23703
$859$	−17.0412	−0.581437	−0.290719	−	0.956809i	$-0.593894\pi$
$859$	−17.0412	−0.581437	−0.290719	+	0.956809i	$0.593894\pi$
$860$	22.2335	0.758155
$861$	24.3528	0.829943
$862$	2.40950	0.0820681
$863$	49.6007	1.68843	0.844214	−	0.536006i	$-0.180068\pi$
$863$	49.6007	1.68843	0.844214	+	0.536006i	$0.180068\pi$
$864$	62.6706	2.13210
$865$	45.3555	1.54213
$866$	44.7911	1.52206
$867$	46.5884	1.58222
$868$	−1.79240	−0.0608381
$869$	33.7601	1.14523
$870$	−44.8177	−1.51946
$871$	5.60774	0.190011
$872$	11.0606	0.374561
$873$	67.5473	2.28613
$874$	−15.4203	−0.521601
$875$	−32.7234	−1.10625
$876$	157.592	5.32453
$877$	−18.3764	−0.620527	−0.310264	−	0.950651i	$-0.600417\pi$
$877$	−18.3764	−0.620527	−0.310264	+	0.950651i	$0.600417\pi$
$878$	87.3457	2.94778
$879$	20.5932	0.694592
$880$	−69.8921	−2.35606
$881$	−44.8623	−1.51145	−0.755724	−	0.654890i	$-0.772715\pi$
$881$	−44.8623	−1.51145	−0.755724	+	0.654890i	$0.772715\pi$
$882$	−85.8656	−2.89125
$883$	50.0351	1.68382	0.841908	−	0.539621i	$-0.181432\pi$
$883$	50.0351	1.68382	0.841908	+	0.539621i	$0.181432\pi$
$884$	7.16822	0.241093
$885$	−38.8566	−1.30615
$886$	14.8673	0.499475
$887$	48.8566	1.64044	0.820221	−	0.572046i	$-0.193850\pi$
$887$	48.8566	1.64044	0.820221	+	0.572046i	$0.193850\pi$
$888$	177.803	5.96668
$889$	−54.6436	−1.83269
$890$	−100.591	−3.37183
$891$	113.515	3.80288
$892$	−95.6250	−3.20176
$893$	8.49927	0.284417
$894$	−107.681	−3.60140
$895$	−38.5765	−1.28947
$896$	−47.5768	−1.58943
$897$	−4.76414	−0.159070
$898$	−78.5653	−2.62176
$899$	0.267007	0.00890517
$900$	36.0507	1.20169
$901$	−15.8879	−0.529304
$902$	−24.6074	−0.819336
$903$	−22.6831	−0.754845
$904$	11.4885	0.382100
$905$	40.0296	1.33063
$906$	−37.6338	−1.25030
$907$	38.7907	1.28802	0.644012	−	0.765016i	$-0.277269\pi$
$907$	38.7907	1.28802	0.644012	+	0.765016i	$0.277269\pi$
$908$	7.49076	0.248590
$909$	42.9463	1.42444
$910$	−21.1154	−0.699970
$911$	−12.7878	−0.423680	−0.211840	−	0.977304i	$-0.567946\pi$
$911$	−12.7878	−0.423680	−0.211840	+	0.977304i	$0.567946\pi$
$912$	−87.1201	−2.88484
$913$	7.40510	0.245073
$914$	92.3847	3.05581
$915$	−45.7932	−1.51388
$916$	91.5693	3.02553
$917$	44.7795	1.47875
$918$	61.6150	2.03360
$919$	−13.1655	−0.434290	−0.217145	−	0.976139i	$-0.569674\pi$
$919$	−13.1655	−0.434290	−0.217145	+	0.976139i	$0.569674\pi$
$920$	21.7193	0.716065
$921$	−30.8430	−1.01631
$922$	26.6655	0.878183
$923$	−8.23489	−0.271055
$924$	212.925	7.00473
$925$	9.86981	0.324517
$926$	−6.54943	−0.215228
$927$	24.4557	0.803229
$928$	−9.29663	−0.305177
$929$	−31.3068	−1.02714	−0.513571	−	0.858047i	$-0.671678\pi$
$929$	−31.3068	−1.02714	−0.513571	+	0.858047i	$0.671678\pi$
$930$	−2.44774	−0.0802645
$931$	18.6898	0.612534
$932$	−96.2158	−3.15165
$933$	−70.6065	−2.31155
$934$	19.5931	0.641105
$935$	−17.8037	−0.582242
$936$	−45.5808	−1.48985
$937$	−1.79387	−0.0586031	−0.0293015	−	0.999571i	$-0.509328\pi$
$937$	−1.79387	−0.0586031	−0.0293015	+	0.999571i	$0.509328\pi$
$938$	47.9996	1.56724
$939$	−109.849	−3.58479
$940$	−22.0307	−0.718561
$941$	−38.5341	−1.25617	−0.628087	−	0.778143i	$-0.716162\pi$
$941$	−38.5341	−1.25617	−0.628087	+	0.778143i	$0.716162\pi$
$942$	−118.132	−3.84894
$943$	3.23539	0.105359
$944$	−31.1088	−1.01250
$945$	−124.603	−4.05335
$946$	22.9201	0.745198
$947$	44.8393	1.45708	0.728541	−	0.685002i	$-0.240199\pi$
$947$	44.8393	1.45708	0.728541	+	0.685002i	$0.240199\pi$
$948$	−109.079	−3.54272
$949$	−11.0604	−0.359036
$950$	−11.4300	−0.370837
$951$	−51.3389	−1.66478
$952$	33.3402	1.08056
$953$	−17.1480	−0.555480	−0.277740	−	0.960656i	$-0.589585\pi$
$953$	−17.1480	−0.555480	−0.277740	+	0.960656i	$0.589585\pi$
$954$	185.922	6.01945
$955$	−32.3998	−1.04843
$956$	3.67751	0.118939
$957$	−31.7186	−1.02532
$958$	23.0453	0.744558
$959$	−35.7670	−1.15498
$960$	−17.8884	−0.577346
$961$	−30.9854	−0.999530
$962$	−22.9652	−0.740427
$963$	−64.3602	−2.07398
$964$	83.6294	2.69352
$965$	14.6218	0.470694
$966$	−40.7788	−1.31204
$967$	−1.81842	−0.0584764	−0.0292382	−	0.999572i	$-0.509308\pi$
$967$	−1.81842	−0.0584764	−0.0292382	+	0.999572i	$0.509308\pi$
$968$	−50.7808	−1.63216
$969$	−22.1922	−0.712915
$970$	55.5117	1.78237
$971$	52.4539	1.68333	0.841664	−	0.540002i	$-0.181577\pi$
$971$	52.4539	1.68333	0.841664	+	0.540002i	$0.181577\pi$
$972$	−170.907	−5.48185
$973$	−47.0606	−1.50869
$974$	−72.0487	−2.30859
$975$	−3.53130	−0.113092
$976$	−36.6623	−1.17353
$977$	−42.3402	−1.35458	−0.677292	−	0.735714i	$-0.736847\pi$
$977$	−42.3402	−1.35458	−0.677292	+	0.735714i	$0.736847\pi$
$978$	−117.403	−3.75415
$979$	−71.1910	−2.27527
$980$	−48.4452	−1.54753
$981$	−13.9498	−0.445384
$982$	−30.3474	−0.968424
$983$	45.4109	1.44838	0.724192	−	0.689598i	$-0.242213\pi$
$983$	45.4109	1.44838	0.724192	+	0.689598i	$0.242213\pi$
$984$	43.2024	1.37724
$985$	38.2300	1.21811
$986$	−9.14004	−0.291078
$987$	22.4762	0.715424
$988$	18.2583	0.580874
$989$	−3.01355	−0.0958254
$990$	208.340	6.62149
$991$	29.8715	0.948900	0.474450	−	0.880283i	$-0.342647\pi$
$991$	29.8715	0.948900	0.474450	+	0.880283i	$0.342647\pi$
$992$	−0.507740	−0.0161208
$993$	−77.1481	−2.44822
$994$	−70.4868	−2.23571
$995$	−41.2279	−1.30701
$996$	−23.9259	−0.758120
$997$	−38.9231	−1.23271	−0.616353	−	0.787470i	$-0.711390\pi$
$997$	−38.9231	−1.23271	−0.616353	+	0.787470i	$0.711390\pi$
$998$	−46.8170	−1.48197
$999$	−135.519	−4.28763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.10	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.10	✓	142	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)

\(f(q)\)	\(=\)	\(q-2.52587 q^{2} -3.25300 q^{3} +4.38004 q^{4} -2.46689 q^{5} +8.21667 q^{6} -3.38874 q^{7} -6.01169 q^{8} +7.58202 q^{9} +O(q^{10})\)	\(q-2.52587 q^{2} -3.25300 q^{3} +4.38004 q^{4} -2.46689 q^{5} +8.21667 q^{6} -3.38874 q^{7} -6.01169 q^{8} +7.58202 q^{9} +6.23106 q^{10} +4.40988 q^{11} -14.2483 q^{12} +1.00000 q^{13} +8.55953 q^{14} +8.02480 q^{15} +6.42469 q^{16} +1.63656 q^{17} -19.1512 q^{18} +4.16852 q^{19} -10.8051 q^{20} +11.0236 q^{21} -11.1388 q^{22} +1.46454 q^{23} +19.5560 q^{24} +1.08555 q^{25} -2.52587 q^{26} -14.9053 q^{27} -14.8428 q^{28} +2.21107 q^{29} -20.2696 q^{30} +0.120759 q^{31} -4.20458 q^{32} -14.3453 q^{33} -4.13376 q^{34} +8.35965 q^{35} +33.2096 q^{36} +9.09197 q^{37} -10.5292 q^{38} -3.25300 q^{39} +14.8302 q^{40} +2.20916 q^{41} -27.8442 q^{42} -2.05769 q^{43} +19.3154 q^{44} -18.7040 q^{45} -3.69923 q^{46} +2.03892 q^{47} -20.8995 q^{48} +4.48356 q^{49} -2.74197 q^{50} -5.32375 q^{51} +4.38004 q^{52} -9.70810 q^{53} +37.6490 q^{54} -10.8787 q^{55} +20.3720 q^{56} -13.5602 q^{57} -5.58490 q^{58} -4.84207 q^{59} +35.1490 q^{60} -5.70646 q^{61} -0.305022 q^{62} -25.6935 q^{63} -2.22914 q^{64} -2.46689 q^{65} +36.2345 q^{66} +5.60774 q^{67} +7.16822 q^{68} -4.76414 q^{69} -21.1154 q^{70} -8.23489 q^{71} -45.5808 q^{72} -11.0604 q^{73} -22.9652 q^{74} -3.53130 q^{75} +18.2583 q^{76} -14.9439 q^{77} +8.21667 q^{78} +7.65557 q^{79} -15.8490 q^{80} +25.7410 q^{81} -5.58006 q^{82} +1.67921 q^{83} +48.2837 q^{84} -4.03723 q^{85} +5.19746 q^{86} -7.19263 q^{87} -26.5108 q^{88} -16.1435 q^{89} +47.2440 q^{90} -3.38874 q^{91} +6.41473 q^{92} -0.392829 q^{93} -5.15005 q^{94} -10.2833 q^{95} +13.6775 q^{96} +8.90888 q^{97} -11.3249 q^{98} +33.4358 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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