LMFDB - Embedded newform 8047.2.a.d.1.12

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			1.12
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)

$f(q)$	$=$	$q-2.42534 q^{2} -2.30091 q^{3} +3.88226 q^{4} -2.29855 q^{5} +5.58047 q^{6} -0.451041 q^{7} -4.56512 q^{8} +2.29416 q^{9} +O(q^{10})$	\(q-2.42534 q^{2} -2.30091 q^{3} +3.88226 q^{4} -2.29855 q^{5} +5.58047 q^{6} -0.451041 q^{7} -4.56512 q^{8} +2.29416 q^{9} +5.57476 q^{10} -2.08238 q^{11} -8.93271 q^{12} -1.00000 q^{13} +1.09393 q^{14} +5.28875 q^{15} +3.30743 q^{16} +2.47489 q^{17} -5.56412 q^{18} +5.13675 q^{19} -8.92358 q^{20} +1.03780 q^{21} +5.05048 q^{22} +4.90214 q^{23} +10.5039 q^{24} +0.283339 q^{25} +2.42534 q^{26} +1.62406 q^{27} -1.75106 q^{28} +4.37379 q^{29} -12.8270 q^{30} +2.22987 q^{31} +1.10861 q^{32} +4.79136 q^{33} -6.00243 q^{34} +1.03674 q^{35} +8.90655 q^{36} +5.64268 q^{37} -12.4584 q^{38} +2.30091 q^{39} +10.4932 q^{40} +8.08800 q^{41} -2.51702 q^{42} -6.52324 q^{43} -8.08435 q^{44} -5.27326 q^{45} -11.8893 q^{46} +7.00856 q^{47} -7.61008 q^{48} -6.79656 q^{49} -0.687191 q^{50} -5.69448 q^{51} -3.88226 q^{52} +14.4542 q^{53} -3.93889 q^{54} +4.78646 q^{55} +2.05905 q^{56} -11.8192 q^{57} -10.6079 q^{58} +9.70555 q^{59} +20.5323 q^{60} +0.710221 q^{61} -5.40818 q^{62} -1.03476 q^{63} -9.30360 q^{64} +2.29855 q^{65} -11.6207 q^{66} -1.11232 q^{67} +9.60815 q^{68} -11.2793 q^{69} -2.51445 q^{70} +2.91506 q^{71} -10.4731 q^{72} +9.70350 q^{73} -13.6854 q^{74} -0.651935 q^{75} +19.9422 q^{76} +0.939239 q^{77} -5.58047 q^{78} +10.9080 q^{79} -7.60229 q^{80} -10.6193 q^{81} -19.6161 q^{82} -3.11734 q^{83} +4.02902 q^{84} -5.68865 q^{85} +15.8211 q^{86} -10.0637 q^{87} +9.50631 q^{88} -15.7859 q^{89} +12.7894 q^{90} +0.451041 q^{91} +19.0314 q^{92} -5.13071 q^{93} -16.9981 q^{94} -11.8071 q^{95} -2.55080 q^{96} -0.674960 q^{97} +16.4840 q^{98} -4.77733 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.42534	−1.71497	−0.857486	−	0.514507i	$-0.827975\pi$
$2$	−2.42534	−1.71497	−0.857486	+	0.514507i	$0.827975\pi$
$3$	−2.30091	−1.32843	−0.664214	−	0.747542i	$-0.731234\pi$
$3$	−2.30091	−1.32843	−0.664214	+	0.747542i	$0.731234\pi$
$4$	3.88226	1.94113
$5$	−2.29855	−1.02794	−0.513972	−	0.857807i	$-0.671826\pi$
$5$	−2.29855	−1.02794	−0.513972	+	0.857807i	$0.671826\pi$
$6$	5.58047	2.27822
$7$	−0.451041	−0.170477	−0.0852387	−	0.996361i	$-0.527165\pi$
$7$	−0.451041	−0.170477	−0.0852387	+	0.996361i	$0.527165\pi$
$8$	−4.56512	−1.61401
$9$	2.29416	0.764722
$10$	5.57476	1.76289
$11$	−2.08238	−0.627862	−0.313931	−	0.949446i	$-0.601646\pi$
$11$	−2.08238	−0.627862	−0.313931	+	0.949446i	$0.601646\pi$
$12$	−8.93271	−2.57865
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.09393	0.292364
$15$	5.28875	1.36555
$16$	3.30743	0.826857
$17$	2.47489	0.600248	0.300124	−	0.953900i	$-0.402972\pi$
$17$	2.47489	0.600248	0.300124	+	0.953900i	$0.402972\pi$
$18$	−5.56412	−1.31148
$19$	5.13675	1.17845	0.589226	−	0.807968i	$-0.299433\pi$
$19$	5.13675	1.17845	0.589226	+	0.807968i	$0.299433\pi$
$20$	−8.92358	−1.99537
$21$	1.03780	0.226467
$22$	5.05048	1.07677
$23$	4.90214	1.02217	0.511083	−	0.859531i	$-0.329244\pi$
$23$	4.90214	1.02217	0.511083	+	0.859531i	$0.329244\pi$
$24$	10.5039	2.14410
$25$	0.283339	0.0566677
$26$	2.42534	0.475648
$27$	1.62406	0.312550
$28$	−1.75106	−0.330919
$29$	4.37379	0.812192	0.406096	−	0.913830i	$-0.366890\pi$
$29$	4.37379	0.812192	0.406096	+	0.913830i	$0.366890\pi$
$30$	−12.8270	−2.34188
$31$	2.22987	0.400496	0.200248	−	0.979745i	$-0.435825\pi$
$31$	2.22987	0.400496	0.200248	+	0.979745i	$0.435825\pi$
$32$	1.10861	0.195976
$33$	4.79136	0.834069
$34$	−6.00243	−1.02941
$35$	1.03674	0.175241
$36$	8.90655	1.48442
$37$	5.64268	0.927650	0.463825	−	0.885927i	$-0.346477\pi$
$37$	5.64268	0.927650	0.463825	+	0.885927i	$0.346477\pi$
$38$	−12.4584	−2.02101
$39$	2.30091	0.368440
$40$	10.4932	1.65911
$41$	8.08800	1.26313	0.631567	−	0.775321i	$-0.282412\pi$
$41$	8.08800	1.26313	0.631567	+	0.775321i	$0.282412\pi$
$42$	−2.51702	−0.388385
$43$	−6.52324	−0.994785	−0.497393	−	0.867526i	$-0.665709\pi$
$43$	−6.52324	−0.994785	−0.497393	+	0.867526i	$0.665709\pi$
$44$	−8.08435	−1.21876
$45$	−5.27326	−0.786091
$46$	−11.8893	−1.75299
$47$	7.00856	1.02230	0.511152	−	0.859490i	$-0.329219\pi$
$47$	7.00856	1.02230	0.511152	+	0.859490i	$0.329219\pi$
$48$	−7.61008	−1.09842
$49$	−6.79656	−0.970937
$50$	−0.687191	−0.0971835
$51$	−5.69448	−0.797387
$52$	−3.88226	−0.538373
$53$	14.4542	1.98544	0.992718	−	0.120461i	$-0.0384372\pi$
$53$	14.4542	1.98544	0.992718	+	0.120461i	$0.0384372\pi$
$54$	−3.93889	−0.536015
$55$	4.78646	0.645406
$56$	2.05905	0.275153
$57$	−11.8192	−1.56549
$58$	−10.6079	−1.39289
$59$	9.70555	1.26355	0.631777	−	0.775150i	$-0.282326\pi$
$59$	9.70555	1.26355	0.631777	+	0.775150i	$0.282326\pi$
$60$	20.5323	2.65071
$61$	0.710221	0.0909345	0.0454673	−	0.998966i	$-0.485522\pi$
$61$	0.710221	0.0909345	0.0454673	+	0.998966i	$0.485522\pi$
$62$	−5.40818	−0.686839
$63$	−1.03476	−0.130368
$64$	−9.30360	−1.16295
$65$	2.29855	0.285100
$66$	−11.6207	−1.43041
$67$	−1.11232	−0.135892	−0.0679458	−	0.997689i	$-0.521644\pi$
$67$	−1.11232	−0.135892	−0.0679458	+	0.997689i	$0.521644\pi$
$68$	9.60815	1.16516
$69$	−11.2793	−1.35787
$70$	−2.51445	−0.300534
$71$	2.91506	0.345954	0.172977	−	0.984926i	$-0.444661\pi$
$71$	2.91506	0.345954	0.172977	+	0.984926i	$0.444661\pi$
$72$	−10.4731	−1.23427
$73$	9.70350	1.13571	0.567854	−	0.823129i	$-0.307774\pi$
$73$	9.70350	1.13571	0.567854	+	0.823129i	$0.307774\pi$
$74$	−13.6854	−1.59089
$75$	−0.651935	−0.0752790
$76$	19.9422	2.28753
$77$	0.939239	0.107036
$78$	−5.58047	−0.631864
$79$	10.9080	1.22724	0.613621	−	0.789601i	$-0.289712\pi$
$79$	10.9080	1.22724	0.613621	+	0.789601i	$0.289712\pi$
$80$	−7.60229	−0.849962
$81$	−10.6193	−1.17992
$82$	−19.6161	−2.16624
$83$	−3.11734	−0.342173	−0.171087	−	0.985256i	$-0.554728\pi$
$83$	−3.11734	−0.342173	−0.171087	+	0.985256i	$0.554728\pi$
$84$	4.02902	0.439602
$85$	−5.68865	−0.617021
$86$	15.8211	1.70603
$87$	−10.0637	−1.07894
$88$	9.50631	1.01338
$89$	−15.7859	−1.67330	−0.836651	−	0.547736i	$-0.815490\pi$
$89$	−15.7859	−1.67330	−0.836651	+	0.547736i	$0.815490\pi$
$90$	12.7894	1.34812
$91$	0.451041	0.0472819
$92$	19.0314	1.98416
$93$	−5.13071	−0.532030
$94$	−16.9981	−1.75322
$95$	−11.8071	−1.21138
$96$	−2.55080	−0.260340
$97$	−0.674960	−0.0685318	−0.0342659	−	0.999413i	$-0.510909\pi$
$97$	−0.674960	−0.0685318	−0.0342659	+	0.999413i	$0.510909\pi$
$98$	16.4840	1.66513
$99$	−4.77733	−0.480139
$100$	1.09999	0.109999
$101$	−2.23560	−0.222451	−0.111225	−	0.993795i	$-0.535478\pi$
$101$	−2.23560	−0.222451	−0.111225	+	0.993795i	$0.535478\pi$
$102$	13.8110	1.36750
$103$	19.4826	1.91968	0.959839	−	0.280552i	$-0.0905175\pi$
$103$	19.4826	1.91968	0.959839	+	0.280552i	$0.0905175\pi$
$104$	4.56512	0.447647
$105$	−2.38544	−0.232795
$106$	−35.0563	−3.40497
$107$	14.1200	1.36504	0.682518	−	0.730868i	$-0.260885\pi$
$107$	14.1200	1.36504	0.682518	+	0.730868i	$0.260885\pi$
$108$	6.30502	0.606701
$109$	−1.19472	−0.114433	−0.0572167	−	0.998362i	$-0.518223\pi$
$109$	−1.19472	−0.114433	−0.0572167	+	0.998362i	$0.518223\pi$
$110$	−11.6088	−1.10685
$111$	−12.9833	−1.23232
$112$	−1.49179	−0.140960
$113$	6.02357	0.566650	0.283325	−	0.959024i	$-0.408562\pi$
$113$	6.02357	0.566650	0.283325	+	0.959024i	$0.408562\pi$
$114$	28.6655	2.68477
$115$	−11.2678	−1.05073
$116$	16.9802	1.57657
$117$	−2.29416	−0.212096
$118$	−23.5392	−2.16696
$119$	−1.11628	−0.102329
$120$	−24.1438	−2.20401
$121$	−6.66369	−0.605790
$122$	−1.72253	−0.155950
$123$	−18.6097	−1.67798
$124$	8.65692	0.777414
$125$	10.8415	0.969692
$126$	2.50965	0.223577
$127$	−2.05910	−0.182716	−0.0913578	−	0.995818i	$-0.529121\pi$
$127$	−2.05910	−0.182716	−0.0913578	+	0.995818i	$0.529121\pi$
$128$	20.3472	1.79845
$129$	15.0094	1.32150
$130$	−5.57476	−0.488939
$131$	7.33747	0.641078	0.320539	−	0.947235i	$-0.396136\pi$
$131$	7.33747	0.641078	0.320539	+	0.947235i	$0.396136\pi$
$132$	18.6013	1.61904
$133$	−2.31689	−0.200900
$134$	2.69775	0.233050
$135$	−3.73298	−0.321284
$136$	−11.2981	−0.968808
$137$	14.4225	1.23220	0.616099	−	0.787669i	$-0.288712\pi$
$137$	14.4225	1.23220	0.616099	+	0.787669i	$0.288712\pi$
$138$	27.3562	2.32872
$139$	−15.5965	−1.32288	−0.661440	−	0.749998i	$-0.730054\pi$
$139$	−15.5965	−1.32288	−0.661440	+	0.749998i	$0.730054\pi$
$140$	4.02490	0.340166
$141$	−16.1260	−1.35806
$142$	−7.06999	−0.593301
$143$	2.08238	0.174137
$144$	7.58778	0.632315
$145$	−10.0534	−0.834887
$146$	−23.5343	−1.94771
$147$	15.6382	1.28982
$148$	21.9063	1.80069
$149$	7.60660	0.623157	0.311579	−	0.950220i	$-0.399142\pi$
$149$	7.60660	0.623157	0.311579	+	0.950220i	$0.399142\pi$
$150$	1.58116	0.129101
$151$	15.8496	1.28982	0.644910	−	0.764258i	$-0.276895\pi$
$151$	15.8496	1.28982	0.644910	+	0.764258i	$0.276895\pi$
$152$	−23.4499	−1.90204
$153$	5.67780	0.459023
$154$	−2.27797	−0.183564
$155$	−5.12546	−0.411687
$156$	8.93271	0.715190
$157$	17.0702	1.36235	0.681177	−	0.732119i	$-0.261468\pi$
$157$	17.0702	1.36235	0.681177	+	0.732119i	$0.261468\pi$
$158$	−26.4555	−2.10468
$159$	−33.2577	−2.63751
$160$	−2.54819	−0.201452
$161$	−2.21106	−0.174256
$162$	25.7554	2.02353
$163$	−23.9542	−1.87624	−0.938118	−	0.346315i	$-0.887433\pi$
$163$	−23.9542	−1.87624	−0.938118	+	0.346315i	$0.887433\pi$
$164$	31.3997	2.45191
$165$	−11.0132	−0.857376
$166$	7.56061	0.586817
$167$	8.81911	0.682443	0.341222	−	0.939983i	$-0.389159\pi$
$167$	8.81911	0.682443	0.341222	+	0.939983i	$0.389159\pi$
$168$	−4.73769	−0.365521
$169$	1.00000	0.0769231
$170$	13.7969	1.05817
$171$	11.7846	0.901188
$172$	−25.3249	−1.93101
$173$	2.45032	0.186294	0.0931472	−	0.995652i	$-0.470307\pi$
$173$	2.45032	0.186294	0.0931472	+	0.995652i	$0.470307\pi$
$174$	24.4078	1.85035
$175$	−0.127797	−0.00966057
$176$	−6.88732	−0.519152
$177$	−22.3315	−1.67854
$178$	38.2861	2.86967
$179$	1.04275	0.0779388	0.0389694	−	0.999240i	$-0.487593\pi$
$179$	1.04275	0.0779388	0.0389694	+	0.999240i	$0.487593\pi$
$180$	−20.4722	−1.52590
$181$	6.12240	0.455074	0.227537	−	0.973769i	$-0.426933\pi$
$181$	6.12240	0.455074	0.227537	+	0.973769i	$0.426933\pi$
$182$	−1.09393	−0.0810872
$183$	−1.63415	−0.120800
$184$	−22.3788	−1.64979
$185$	−12.9700	−0.953572
$186$	12.4437	0.912416
$187$	−5.15366	−0.376873
$188$	27.2091	1.98443
$189$	−0.732518	−0.0532828
$190$	28.6362	2.07749
$191$	−4.11154	−0.297501	−0.148750	−	0.988875i	$-0.547525\pi$
$191$	−4.11154	−0.297501	−0.148750	+	0.988875i	$0.547525\pi$
$192$	21.4067	1.54490
$193$	18.4960	1.33137	0.665686	−	0.746232i	$-0.268139\pi$
$193$	18.4960	1.33137	0.665686	+	0.746232i	$0.268139\pi$
$194$	1.63700	0.117530
$195$	−5.28875	−0.378735
$196$	−26.3860	−1.88472
$197$	0.899871	0.0641131	0.0320566	−	0.999486i	$-0.489794\pi$
$197$	0.899871	0.0641131	0.0320566	+	0.999486i	$0.489794\pi$
$198$	11.5866	0.823426
$199$	−27.3909	−1.94169	−0.970845	−	0.239710i	$-0.922948\pi$
$199$	−27.3909	−1.94169	−0.970845	+	0.239710i	$0.922948\pi$
$200$	−1.29347	−0.0914624
$201$	2.55934	0.180522
$202$	5.42209	0.381497
$203$	−1.97276	−0.138460
$204$	−22.1075	−1.54783
$205$	−18.5907	−1.29843
$206$	−47.2519	−3.29219
$207$	11.2463	0.781672
$208$	−3.30743	−0.229329
$209$	−10.6967	−0.739905
$210$	5.78550	0.399238
$211$	10.3373	0.711648	0.355824	−	0.934553i	$-0.384200\pi$
$211$	10.3373	0.711648	0.355824	+	0.934553i	$0.384200\pi$
$212$	56.1149	3.85399
$213$	−6.70727	−0.459574
$214$	−34.2459	−2.34100
$215$	14.9940	1.02258
$216$	−7.41402	−0.504460
$217$	−1.00576	−0.0682755
$218$	2.89760	0.196250
$219$	−22.3268	−1.50871
$220$	18.5823	1.25282
$221$	−2.47489	−0.166479
$222$	31.4888	2.11339
$223$	11.6279	0.778660	0.389330	−	0.921098i	$-0.372706\pi$
$223$	11.6279	0.778660	0.389330	+	0.921098i	$0.372706\pi$
$224$	−0.500028	−0.0334095
$225$	0.650025	0.0433350
$226$	−14.6092	−0.971789
$227$	−12.2681	−0.814265	−0.407133	−	0.913369i	$-0.633471\pi$
$227$	−12.2681	−0.814265	−0.407133	+	0.913369i	$0.633471\pi$
$228$	−45.8852	−3.03882
$229$	−5.70180	−0.376785	−0.188393	−	0.982094i	$-0.560328\pi$
$229$	−5.70180	−0.376785	−0.188393	+	0.982094i	$0.560328\pi$
$230$	27.3282	1.80197
$231$	−2.16110	−0.142190
$232$	−19.9668	−1.31089
$233$	−26.0787	−1.70847	−0.854235	−	0.519887i	$-0.825974\pi$
$233$	−26.0787	−1.70847	−0.854235	+	0.519887i	$0.825974\pi$
$234$	5.56412	0.363738
$235$	−16.1095	−1.05087
$236$	37.6795	2.45272
$237$	−25.0982	−1.63030
$238$	2.70734	0.175491
$239$	23.8773	1.54449	0.772247	−	0.635323i	$-0.219133\pi$
$239$	23.8773	1.54449	0.772247	+	0.635323i	$0.219133\pi$
$240$	17.4922	1.12911
$241$	−10.9257	−0.703783	−0.351892	−	0.936041i	$-0.614461\pi$
$241$	−10.9257	−0.703783	−0.351892	+	0.936041i	$0.614461\pi$
$242$	16.1617	1.03891
$243$	19.5618	1.25489
$244$	2.75726	0.176516
$245$	15.6222	0.998069
$246$	45.1349	2.87769
$247$	−5.13675	−0.326844
$248$	−10.1796	−0.646405
$249$	7.17271	0.454552
$250$	−26.2943	−1.66300
$251$	−5.98128	−0.377535	−0.188768	−	0.982022i	$-0.560449\pi$
$251$	−5.98128	−0.377535	−0.188768	+	0.982022i	$0.560449\pi$
$252$	−4.01722	−0.253061
$253$	−10.2081	−0.641779
$254$	4.99401	0.313352
$255$	13.0891	0.819668
$256$	−30.7415	−1.92134
$257$	−10.8549	−0.677111	−0.338556	−	0.940946i	$-0.609938\pi$
$257$	−10.8549	−0.677111	−0.338556	+	0.940946i	$0.609938\pi$
$258$	−36.4028	−2.26634
$259$	−2.54508	−0.158143
$260$	8.92358	0.553417
$261$	10.0342	0.621101
$262$	−17.7958	−1.09943
$263$	−8.94369	−0.551492	−0.275746	−	0.961231i	$-0.588925\pi$
$263$	−8.94369	−0.551492	−0.275746	+	0.961231i	$0.588925\pi$
$264$	−21.8731	−1.34620
$265$	−33.2237	−2.04092
$266$	5.61923	0.344537
$267$	36.3219	2.22286
$268$	−4.31832	−0.263783
$269$	−3.70161	−0.225691	−0.112846	−	0.993613i	$-0.535997\pi$
$269$	−3.70161	−0.225691	−0.112846	+	0.993613i	$0.535997\pi$
$270$	9.05375	0.550993
$271$	−24.7141	−1.50128	−0.750638	−	0.660714i	$-0.770254\pi$
$271$	−24.7141	−1.50128	−0.750638	+	0.660714i	$0.770254\pi$
$272$	8.18551	0.496319
$273$	−1.03780	−0.0628107
$274$	−34.9795	−2.11319
$275$	−0.590019	−0.0355795
$276$	−43.7894	−2.63581
$277$	−9.24287	−0.555350	−0.277675	−	0.960675i	$-0.589564\pi$
$277$	−9.24287	−0.555350	−0.277675	+	0.960675i	$0.589564\pi$
$278$	37.8268	2.26870
$279$	5.11568	0.306268
$280$	−4.73284	−0.282842
$281$	24.7622	1.47719	0.738594	−	0.674151i	$-0.235490\pi$
$281$	24.7622	1.47719	0.738594	+	0.674151i	$0.235490\pi$
$282$	39.1111	2.32903
$283$	−1.03417	−0.0614750	−0.0307375	−	0.999527i	$-0.509786\pi$
$283$	−1.03417	−0.0614750	−0.0307375	+	0.999527i	$0.509786\pi$
$284$	11.3170	0.671541
$285$	27.1670	1.60923
$286$	−5.05048	−0.298641
$287$	−3.64802	−0.215336
$288$	2.54333	0.149867
$289$	−10.8749	−0.639702
$290$	24.3828	1.43181
$291$	1.55302	0.0910395
$292$	37.6715	2.20456
$293$	−17.0229	−0.994487	−0.497244	−	0.867611i	$-0.665654\pi$
$293$	−17.0229	−0.994487	−0.497244	+	0.867611i	$0.665654\pi$
$294$	−37.9280	−2.21201
$295$	−22.3087	−1.29886
$296$	−25.7595	−1.49724
$297$	−3.38191	−0.196238
$298$	−18.4486	−1.06870
$299$	−4.90214	−0.283498
$300$	−2.53098	−0.146126
$301$	2.94225	0.169588
$302$	−38.4406	−2.21201
$303$	5.14391	0.295510
$304$	16.9894	0.974411
$305$	−1.63248	−0.0934755
$306$	−13.7706	−0.787211
$307$	−20.3450	−1.16115	−0.580575	−	0.814207i	$-0.697172\pi$
$307$	−20.3450	−1.16115	−0.580575	+	0.814207i	$0.697172\pi$
$308$	3.64637	0.207771
$309$	−44.8276	−2.55015
$310$	12.4310	0.706032
$311$	−29.9655	−1.69919	−0.849593	−	0.527438i	$-0.823153\pi$
$311$	−29.9655	−1.69919	−0.849593	+	0.527438i	$0.823153\pi$
$312$	−10.5039	−0.594666
$313$	15.1820	0.858138	0.429069	−	0.903272i	$-0.358842\pi$
$313$	15.1820	0.858138	0.429069	+	0.903272i	$0.358842\pi$
$314$	−41.4011	−2.33640
$315$	2.37845	0.134011
$316$	42.3475	2.38224
$317$	32.0465	1.79991	0.899956	−	0.435981i	$-0.143598\pi$
$317$	32.0465	1.79991	0.899956	+	0.435981i	$0.143598\pi$
$318$	80.6612	4.52326
$319$	−9.10789	−0.509944
$320$	21.3848	1.19545
$321$	−32.4889	−1.81335
$322$	5.36258	0.298845
$323$	12.7129	0.707364
$324$	−41.2269	−2.29038
$325$	−0.283339	−0.0157168
$326$	58.0970	3.21769
$327$	2.74894	0.152017
$328$	−36.9227	−2.03871
$329$	−3.16115	−0.174280
$330$	26.7107	1.47038
$331$	−2.55593	−0.140487	−0.0702433	−	0.997530i	$-0.522378\pi$
$331$	−2.55593	−0.140487	−0.0702433	+	0.997530i	$0.522378\pi$
$332$	−12.1023	−0.664202
$333$	12.9452	0.709394
$334$	−21.3893	−1.17037
$335$	2.55673	0.139689
$336$	3.43246	0.187256
$337$	−26.4484	−1.44074	−0.720369	−	0.693591i	$-0.756027\pi$
$337$	−26.4484	−1.44074	−0.720369	+	0.693591i	$0.756027\pi$
$338$	−2.42534	−0.131921
$339$	−13.8597	−0.752754
$340$	−22.0848	−1.19772
$341$	−4.64343	−0.251456
$342$	−28.5815	−1.54551
$343$	6.22282	0.336000
$344$	29.7794	1.60560
$345$	25.9262	1.39582
$346$	−5.94285	−0.319490
$347$	−19.4670	−1.04504	−0.522521	−	0.852627i	$-0.675008\pi$
$347$	−19.4670	−1.04504	−0.522521	+	0.852627i	$0.675008\pi$
$348$	−39.0698	−2.09436
$349$	31.9276	1.70905	0.854523	−	0.519414i	$-0.173850\pi$
$349$	31.9276	1.70905	0.854523	+	0.519414i	$0.173850\pi$
$350$	0.309952	0.0165676
$351$	−1.62406	−0.0866859
$352$	−2.30854	−0.123046
$353$	28.0092	1.49078	0.745389	−	0.666629i	$-0.232264\pi$
$353$	28.0092	1.49078	0.745389	+	0.666629i	$0.232264\pi$
$354$	54.1615	2.87865
$355$	−6.70041	−0.355621
$356$	−61.2850	−3.24810
$357$	2.56844	0.135936
$358$	−2.52902	−0.133663
$359$	37.1174	1.95898	0.979490	−	0.201491i	$-0.0645786\pi$
$359$	37.1174	1.95898	0.979490	+	0.201491i	$0.0645786\pi$
$360$	24.0730	1.26876
$361$	7.38625	0.388750
$362$	−14.8489	−0.780440
$363$	15.3325	0.804748
$364$	1.75106	0.0917804
$365$	−22.3040	−1.16744
$366$	3.96337	0.207169
$367$	4.65238	0.242852	0.121426	−	0.992600i	$-0.461253\pi$
$367$	4.65238	0.242852	0.121426	+	0.992600i	$0.461253\pi$
$368$	16.2135	0.845185
$369$	18.5552	0.965946
$370$	31.4566	1.63535
$371$	−6.51943	−0.338472
$372$	−19.9187	−1.03274
$373$	9.44275	0.488927	0.244463	−	0.969659i	$-0.421388\pi$
$373$	9.44275	0.488927	0.244463	+	0.969659i	$0.421388\pi$
$374$	12.4994	0.646326
$375$	−24.9452	−1.28817
$376$	−31.9949	−1.65001
$377$	−4.37379	−0.225261
$378$	1.77660	0.0913785
$379$	−7.72772	−0.396946	−0.198473	−	0.980106i	$-0.563598\pi$
$379$	−7.72772	−0.396946	−0.198473	+	0.980106i	$0.563598\pi$
$380$	−45.8382	−2.35145
$381$	4.73779	0.242724
$382$	9.97187	0.510205
$383$	30.8563	1.57668	0.788341	−	0.615238i	$-0.210940\pi$
$383$	30.8563	1.57668	0.788341	+	0.615238i	$0.210940\pi$
$384$	−46.8169	−2.38911
$385$	−2.15889	−0.110027
$386$	−44.8591	−2.28327
$387$	−14.9654	−0.760734
$388$	−2.62037	−0.133029
$389$	20.9430	1.06185	0.530926	−	0.847418i	$-0.321844\pi$
$389$	20.9430	1.06185	0.530926	+	0.847418i	$0.321844\pi$
$390$	12.8270	0.649520
$391$	12.1322	0.613553
$392$	31.0271	1.56711
$393$	−16.8828	−0.851626
$394$	−2.18249	−0.109952
$395$	−25.0725	−1.26153
$396$	−18.5468	−0.932013
$397$	−34.9791	−1.75555	−0.877776	−	0.479071i	$-0.840974\pi$
$397$	−34.9791	−1.75555	−0.877776	+	0.479071i	$0.840974\pi$
$398$	66.4322	3.32994
$399$	5.33094	0.266881
$400$	0.937121	0.0468561
$401$	−16.5880	−0.828365	−0.414182	−	0.910194i	$-0.635932\pi$
$401$	−16.5880	−0.828365	−0.414182	+	0.910194i	$0.635932\pi$
$402$	−6.20727	−0.309591
$403$	−2.22987	−0.111077
$404$	−8.67919	−0.431806
$405$	24.4090	1.21289
$406$	4.78460	0.237456
$407$	−11.7502	−0.582436
$408$	25.9960	1.28699
$409$	−7.00570	−0.346410	−0.173205	−	0.984886i	$-0.555412\pi$
$409$	−7.00570	−0.346410	−0.173205	+	0.984886i	$0.555412\pi$
$410$	45.0887	2.22677
$411$	−33.1848	−1.63689
$412$	75.6365	3.72634
$413$	−4.37760	−0.215408
$414$	−27.2761	−1.34055
$415$	7.16538	0.351735
$416$	−1.10861	−0.0543540
$417$	35.8861	1.75735
$418$	25.9431	1.26892
$419$	18.1142	0.884936	0.442468	−	0.896784i	$-0.354103\pi$
$419$	18.1142	0.884936	0.442468	+	0.896784i	$0.354103\pi$
$420$	−9.26091	−0.451886
$421$	−7.95641	−0.387772	−0.193886	−	0.981024i	$-0.562109\pi$
$421$	−7.95641	−0.387772	−0.193886	+	0.981024i	$0.562109\pi$
$422$	−25.0714	−1.22046
$423$	16.0788	0.781778
$424$	−65.9851	−3.20452
$425$	0.701231	0.0340147
$426$	16.2674	0.788157
$427$	−0.320339	−0.0155023
$428$	54.8177	2.64971
$429$	−4.79136	−0.231329
$430$	−36.3655	−1.75370
$431$	−13.2812	−0.639734	−0.319867	−	0.947462i	$-0.603638\pi$
$431$	−13.2812	−0.639734	−0.319867	+	0.947462i	$0.603638\pi$
$432$	5.37146	0.258434
$433$	11.6852	0.561554	0.280777	−	0.959773i	$-0.409408\pi$
$433$	11.6852	0.561554	0.280777	+	0.959773i	$0.409408\pi$
$434$	2.43931	0.117091
$435$	23.1319	1.10909
$436$	−4.63822	−0.222130
$437$	25.1811	1.20457
$438$	54.1501	2.58739
$439$	13.5799	0.648134	0.324067	−	0.946034i	$-0.394950\pi$
$439$	13.5799	0.648134	0.324067	+	0.946034i	$0.394950\pi$
$440$	−21.8508	−1.04169
$441$	−15.5924	−0.742497
$442$	6.00243	0.285507
$443$	−39.9923	−1.90009	−0.950045	−	0.312112i	$-0.898964\pi$
$443$	−39.9923	−1.90009	−0.950045	+	0.312112i	$0.898964\pi$
$444$	−50.4044	−2.39209
$445$	36.2847	1.72006
$446$	−28.2015	−1.33538
$447$	−17.5021	−0.827820
$448$	4.19631	0.198257
$449$	7.05890	0.333130	0.166565	−	0.986030i	$-0.446732\pi$
$449$	7.05890	0.333130	0.166565	+	0.986030i	$0.446732\pi$
$450$	−1.57653	−0.0743184
$451$	−16.8423	−0.793073
$452$	23.3851	1.09994
$453$	−36.4684	−1.71343
$454$	29.7544	1.39644
$455$	−1.03674	−0.0486032
$456$	53.9560	2.52672
$457$	21.2536	0.994202	0.497101	−	0.867693i	$-0.334398\pi$
$457$	21.2536	0.994202	0.497101	+	0.867693i	$0.334398\pi$
$458$	13.8288	0.646176
$459$	4.01936	0.187608
$460$	−43.7446	−2.03960
$461$	−22.4400	−1.04513	−0.522567	−	0.852598i	$-0.675025\pi$
$461$	−22.4400	−1.04513	−0.522567	+	0.852598i	$0.675025\pi$
$462$	5.24140	0.243852
$463$	34.9955	1.62638	0.813190	−	0.581998i	$-0.197729\pi$
$463$	34.9955	1.62638	0.813190	+	0.581998i	$0.197729\pi$
$464$	14.4660	0.671566
$465$	11.7932	0.546896
$466$	63.2495	2.92998
$467$	−31.6424	−1.46423	−0.732117	−	0.681179i	$-0.761468\pi$
$467$	−31.6424	−1.46423	−0.732117	+	0.681179i	$0.761468\pi$
$468$	−8.90655	−0.411705
$469$	0.501702	0.0231665
$470$	39.0711	1.80221
$471$	−39.2770	−1.80979
$472$	−44.3070	−2.03939
$473$	13.5839	0.624587
$474$	60.8716	2.79592
$475$	1.45544	0.0667802
$476$	−4.33367	−0.198634
$477$	33.1603	1.51831
$478$	−57.9105	−2.64876
$479$	−30.3300	−1.38581	−0.692907	−	0.721027i	$-0.743670\pi$
$479$	−30.3300	−1.38581	−0.692907	+	0.721027i	$0.743670\pi$
$480$	5.86315	0.267615
$481$	−5.64268	−0.257284
$482$	26.4984	1.20697
$483$	5.08745	0.231487
$484$	−25.8702	−1.17592
$485$	1.55143	0.0704468
$486$	−47.4440	−2.15210
$487$	2.93985	0.133217	0.0666085	−	0.997779i	$-0.478782\pi$
$487$	2.93985	0.133217	0.0666085	+	0.997779i	$0.478782\pi$
$488$	−3.24224	−0.146769
$489$	55.1163	2.49245
$490$	−37.8892	−1.71166
$491$	−2.77109	−0.125057	−0.0625287	−	0.998043i	$-0.519916\pi$
$491$	−2.77109	−0.125057	−0.0625287	+	0.998043i	$0.519916\pi$
$492$	−72.2478	−3.25718
$493$	10.8246	0.487517
$494$	12.4584	0.560528
$495$	10.9809	0.493556
$496$	7.37512	0.331153
$497$	−1.31481	−0.0589773
$498$	−17.3963	−0.779545
$499$	−11.6580	−0.521885	−0.260942	−	0.965354i	$-0.584033\pi$
$499$	−11.6580	−0.521885	−0.260942	+	0.965354i	$0.584033\pi$
$500$	42.0895	1.88230
$501$	−20.2919	−0.906577
$502$	14.5066	0.647462
$503$	11.9351	0.532160	0.266080	−	0.963951i	$-0.414271\pi$
$503$	11.9351	0.532160	0.266080	+	0.963951i	$0.414271\pi$
$504$	4.72381	0.210415
$505$	5.13865	0.228667
$506$	24.7581	1.10063
$507$	−2.30091	−0.102187
$508$	−7.99396	−0.354675
$509$	−29.5611	−1.31027	−0.655136	−	0.755511i	$-0.727389\pi$
$509$	−29.5611	−1.31027	−0.655136	+	0.755511i	$0.727389\pi$
$510$	−31.7454	−1.40571
$511$	−4.37668	−0.193613
$512$	33.8642	1.49660
$513$	8.34240	0.368326
$514$	26.3268	1.16123
$515$	−44.7818	−1.97332
$516$	58.2703	2.56520
$517$	−14.5945	−0.641865
$518$	6.17267	0.271212
$519$	−5.63795	−0.247479
$520$	−10.4932	−0.460155
$521$	−15.6602	−0.686085	−0.343043	−	0.939320i	$-0.611458\pi$
$521$	−15.6602	−0.686085	−0.343043	+	0.939320i	$0.611458\pi$
$522$	−24.3363	−1.06517
$523$	24.0283	1.05068	0.525342	−	0.850891i	$-0.323937\pi$
$523$	24.0283	1.05068	0.525342	+	0.850891i	$0.323937\pi$
$524$	28.4860	1.24442
$525$	0.294049	0.0128334
$526$	21.6915	0.945793
$527$	5.51866	0.240397
$528$	15.8471	0.689656
$529$	1.03093	0.0448232
$530$	80.5787	3.50011
$531$	22.2661	0.966268
$532$	−8.99476	−0.389972
$533$	−8.08800	−0.350330
$534$	−88.0928	−3.81215
$535$	−32.4557	−1.40318
$536$	5.07788	0.219331
$537$	−2.39927	−0.103536
$538$	8.97766	0.387054
$539$	14.1530	0.609614
$540$	−14.4924	−0.623654
$541$	−37.3339	−1.60511	−0.802554	−	0.596579i	$-0.796526\pi$
$541$	−37.3339	−1.60511	−0.802554	+	0.596579i	$0.796526\pi$
$542$	59.9401	2.57465
$543$	−14.0871	−0.604534
$544$	2.74368	0.117634
$545$	2.74613	0.117631
$546$	2.51702	0.107719
$547$	−39.3729	−1.68347	−0.841733	−	0.539894i	$-0.818464\pi$
$547$	−39.3729	−1.68347	−0.841733	+	0.539894i	$0.818464\pi$
$548$	55.9919	2.39186
$549$	1.62936	0.0695396
$550$	1.43099	0.0610178
$551$	22.4671	0.957129
$552$	51.4916	2.19163
$553$	−4.91994	−0.209217
$554$	22.4171	0.952410
$555$	29.8427	1.26675
$556$	−60.5498	−2.56788
$557$	−46.3980	−1.96595	−0.982973	−	0.183753i	$-0.941175\pi$
$557$	−46.3980	−1.96595	−0.982973	+	0.183753i	$0.941175\pi$
$558$	−12.4072	−0.525241
$559$	6.52324	0.275904
$560$	3.42894	0.144899
$561$	11.8581	0.500648
$562$	−60.0566	−2.53334
$563$	−7.08093	−0.298426	−0.149213	−	0.988805i	$-0.547674\pi$
$563$	−7.08093	−0.298426	−0.149213	+	0.988805i	$0.547674\pi$
$564$	−62.6055	−2.63617
$565$	−13.8455	−0.582484
$566$	2.50821	0.105428
$567$	4.78974	0.201150
$568$	−13.3076	−0.558373
$569$	10.0267	0.420342	0.210171	−	0.977665i	$-0.432598\pi$
$569$	10.0267	0.420342	0.210171	+	0.977665i	$0.432598\pi$
$570$	−65.8892	−2.75979
$571$	30.2314	1.26514	0.632571	−	0.774502i	$-0.281999\pi$
$571$	30.2314	1.26514	0.632571	+	0.774502i	$0.281999\pi$
$572$	8.08435	0.338024
$573$	9.46026	0.395208
$574$	8.84768	0.369295
$575$	1.38896	0.0579238
$576$	−21.3440	−0.889333
$577$	44.6404	1.85840	0.929201	−	0.369574i	$-0.120496\pi$
$577$	44.6404	1.85840	0.929201	+	0.369574i	$0.120496\pi$
$578$	26.3754	1.09707
$579$	−42.5576	−1.76863
$580$	−39.0298	−1.62062
$581$	1.40605	0.0583328
$582$	−3.76659	−0.156130
$583$	−30.0991	−1.24658
$584$	−44.2976	−1.83305
$585$	5.27326	0.218022
$586$	41.2862	1.70552
$587$	28.1540	1.16204	0.581020	−	0.813889i	$-0.302654\pi$
$587$	28.1540	1.16204	0.581020	+	0.813889i	$0.302654\pi$
$588$	60.7117	2.50371
$589$	11.4543	0.471965
$590$	54.1061	2.22751
$591$	−2.07052	−0.0851697
$592$	18.6627	0.767034
$593$	14.9685	0.614681	0.307341	−	0.951600i	$-0.400561\pi$
$593$	14.9685	0.614681	0.307341	+	0.951600i	$0.400561\pi$
$594$	8.20228	0.336543
$595$	2.56582	0.105188
$596$	29.5308	1.20963
$597$	63.0239	2.57939
$598$	11.8893	0.486191
$599$	30.1242	1.23084	0.615421	−	0.788198i	$-0.288986\pi$
$599$	30.1242	1.23084	0.615421	+	0.788198i	$0.288986\pi$
$600$	2.97616	0.121501
$601$	10.0455	0.409766	0.204883	−	0.978786i	$-0.434319\pi$
$601$	10.0455	0.409766	0.204883	+	0.978786i	$0.434319\pi$
$602$	−7.13595	−0.290839
$603$	−2.55185	−0.103919
$604$	61.5322	2.50371
$605$	15.3168	0.622718
$606$	−12.4757	−0.506791
$607$	−0.396457	−0.0160917	−0.00804584	−	0.999968i	$-0.502561\pi$
$607$	−0.396457	−0.0160917	−0.00804584	+	0.999968i	$0.502561\pi$
$608$	5.69465	0.230948
$609$	4.53913	0.183935
$610$	3.95931	0.160308
$611$	−7.00856	−0.283536
$612$	22.0427	0.891023
$613$	0.838115	0.0338511	0.0169256	−	0.999857i	$-0.494612\pi$
$613$	0.838115	0.0338511	0.0169256	+	0.999857i	$0.494612\pi$
$614$	49.3434	1.99134
$615$	42.7754	1.72487
$616$	−4.28774	−0.172758
$617$	38.7794	1.56120	0.780599	−	0.625032i	$-0.214914\pi$
$617$	38.7794	1.56120	0.780599	+	0.625032i	$0.214914\pi$
$618$	108.722	4.37344
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−19.8984	−0.799138
$621$	7.96136	0.319478
$622$	72.6764	2.91406
$623$	7.12009	0.285260
$624$	7.61008	0.304647
$625$	−26.3364	−1.05346
$626$	−36.8215	−1.47168
$627$	24.6120	0.982911
$628$	66.2711	2.64451
$629$	13.9650	0.556820
$630$	−5.76855	−0.229825
$631$	24.1091	0.959766	0.479883	−	0.877332i	$-0.340679\pi$
$631$	24.1091	0.959766	0.479883	+	0.877332i	$0.340679\pi$
$632$	−49.7961	−1.98078
$633$	−23.7851	−0.945373
$634$	−77.7236	−3.08680
$635$	4.73295	0.187821
$636$	−129.115	−5.11975
$637$	6.79656	0.269290
$638$	22.0897	0.874540
$639$	6.68762	0.264558
$640$	−46.7690	−1.84871
$641$	18.6769	0.737691	0.368846	−	0.929491i	$-0.379753\pi$
$641$	18.6769	0.737691	0.368846	+	0.929491i	$0.379753\pi$
$642$	78.7965	3.10985
$643$	40.8688	1.61171	0.805854	−	0.592114i	$-0.201706\pi$
$643$	40.8688	1.61171	0.805854	+	0.592114i	$0.201706\pi$
$644$	−8.58393	−0.338254
$645$	−34.4998	−1.35843
$646$	−30.8330	−1.21311
$647$	−2.56629	−0.100891	−0.0504455	−	0.998727i	$-0.516064\pi$
$647$	−2.56629	−0.100891	−0.0504455	+	0.998727i	$0.516064\pi$
$648$	48.4784	1.90441
$649$	−20.2106	−0.793337
$650$	0.687191	0.0269539
$651$	2.31416	0.0906991
$652$	−92.9964	−3.64202
$653$	0.255229	0.00998788	0.00499394	−	0.999988i	$-0.498410\pi$
$653$	0.255229	0.00998788	0.00499394	+	0.999988i	$0.498410\pi$
$654$	−6.66710	−0.260704
$655$	−16.8655	−0.658991
$656$	26.7505	1.04443
$657$	22.2614	0.868501
$658$	7.66685	0.298885
$659$	−10.9191	−0.425346	−0.212673	−	0.977123i	$-0.568217\pi$
$659$	−10.9191	−0.425346	−0.212673	+	0.977123i	$0.568217\pi$
$660$	−42.7561	−1.66428
$661$	41.0055	1.59493	0.797465	−	0.603366i	$-0.206174\pi$
$661$	41.0055	1.59493	0.797465	+	0.603366i	$0.206174\pi$
$662$	6.19900	0.240931
$663$	5.69448	0.221155
$664$	14.2310	0.552272
$665$	5.32548	0.206513
$666$	−31.3965	−1.21659
$667$	21.4409	0.830195
$668$	34.2381	1.32471
$669$	−26.7547	−1.03439
$670$	−6.20092	−0.239563
$671$	−1.47895	−0.0570943
$672$	1.15052	0.0443821
$673$	1.93481	0.0745815	0.0372908	−	0.999304i	$-0.488127\pi$
$673$	1.93481	0.0745815	0.0372908	+	0.999304i	$0.488127\pi$
$674$	64.1463	2.47082
$675$	0.460159	0.0177115
$676$	3.88226	0.149318
$677$	25.4931	0.979778	0.489889	−	0.871785i	$-0.337037\pi$
$677$	25.4931	0.979778	0.489889	+	0.871785i	$0.337037\pi$
$678$	33.6144	1.29095
$679$	0.304434	0.0116831
$680$	25.9694	0.995880
$681$	28.2278	1.08169
$682$	11.2619	0.431240
$683$	−20.7080	−0.792370	−0.396185	−	0.918171i	$-0.629666\pi$
$683$	−20.7080	−0.792370	−0.396185	+	0.918171i	$0.629666\pi$
$684$	45.7507	1.74932
$685$	−33.1509	−1.26663
$686$	−15.0924	−0.576231
$687$	13.1193	0.500532
$688$	−21.5751	−0.822545
$689$	−14.4542	−0.550661
$690$	−62.8797	−2.39379
$691$	−32.1100	−1.22152	−0.610760	−	0.791816i	$-0.709136\pi$
$691$	−32.1100	−1.22152	−0.610760	+	0.791816i	$0.709136\pi$
$692$	9.51278	0.361622
$693$	2.15477	0.0818529
$694$	47.2139	1.79222
$695$	35.8494	1.35985
$696$	45.9418	1.74142
$697$	20.0169	0.758194
$698$	−77.4352	−2.93097
$699$	60.0045	2.26958
$700$	−0.496142	−0.0187524
$701$	27.5006	1.03868	0.519342	−	0.854567i	$-0.326177\pi$
$701$	27.5006	1.03868	0.519342	+	0.854567i	$0.326177\pi$
$702$	3.93889	0.148664
$703$	28.9850	1.09319
$704$	19.3736	0.730172
$705$	37.0665	1.39601
$706$	−67.9317	−2.55664
$707$	1.00835	0.0379229
$708$	−86.6969	−3.25827
$709$	51.9559	1.95125	0.975623	−	0.219454i	$-0.0704277\pi$
$709$	51.9559	1.95125	0.975623	+	0.219454i	$0.0704277\pi$
$710$	16.2507	0.609880
$711$	25.0247	0.938498
$712$	72.0645	2.70073
$713$	10.9311	0.409373
$714$	−6.22934	−0.233127
$715$	−4.78646	−0.179003
$716$	4.04823	0.151289
$717$	−54.9394	−2.05175
$718$	−90.0222	−3.35960
$719$	−33.6056	−1.25328	−0.626638	−	0.779310i	$-0.715570\pi$
$719$	−33.6056	−1.25328	−0.626638	+	0.779310i	$0.715570\pi$
$720$	−17.4409	−0.649984
$721$	−8.78745	−0.327262
$722$	−17.9141	−0.666696
$723$	25.1389	0.934926
$724$	23.7687	0.883359
$725$	1.23926	0.0460250
$726$	−37.1865	−1.38012
$727$	10.6977	0.396757	0.198379	−	0.980125i	$-0.436432\pi$
$727$	10.6977	0.396757	0.198379	+	0.980125i	$0.436432\pi$
$728$	−2.05905	−0.0763137
$729$	−13.1520	−0.487111
$730$	54.0947	2.00213
$731$	−16.1443	−0.597118
$732$	−6.34420	−0.234488
$733$	−35.3165	−1.30444	−0.652222	−	0.758028i	$-0.726163\pi$
$733$	−35.3165	−1.30444	−0.652222	+	0.758028i	$0.726163\pi$
$734$	−11.2836	−0.416485
$735$	−35.9453	−1.32586
$736$	5.43455	0.200320
$737$	2.31628	0.0853211
$738$	−45.0026	−1.65657
$739$	−29.3756	−1.08060	−0.540300	−	0.841473i	$-0.681689\pi$
$739$	−29.3756	−1.08060	−0.540300	+	0.841473i	$0.681689\pi$
$740$	−50.3528	−1.85101
$741$	11.8192	0.434189
$742$	15.8118	0.580470
$743$	42.6201	1.56358	0.781789	−	0.623543i	$-0.214307\pi$
$743$	42.6201	1.56358	0.781789	+	0.623543i	$0.214307\pi$
$744$	23.4223	0.858703
$745$	−17.4842	−0.640570
$746$	−22.9018	−0.838496
$747$	−7.15170	−0.261667
$748$	−20.0078	−0.731559
$749$	−6.36872	−0.232708
$750$	60.5006	2.20917
$751$	41.4519	1.51260	0.756301	−	0.654224i	$-0.227004\pi$
$751$	41.4519	1.51260	0.756301	+	0.654224i	$0.227004\pi$
$752$	23.1803	0.845299
$753$	13.7624	0.501528
$754$	10.6079	0.386317
$755$	−36.4311	−1.32586
$756$	−2.84382	−0.103429
$757$	−20.5729	−0.747735	−0.373868	−	0.927482i	$-0.621969\pi$
$757$	−20.5729	−0.747735	−0.373868	+	0.927482i	$0.621969\pi$
$758$	18.7423	0.680752
$759$	23.4879	0.852557
$760$	53.9008	1.95519
$761$	8.53345	0.309337	0.154669	−	0.987966i	$-0.450569\pi$
$761$	8.53345	0.309337	0.154669	+	0.987966i	$0.450569\pi$
$762$	−11.4907	−0.416266
$763$	0.538868	0.0195083
$764$	−15.9621	−0.577487
$765$	−13.0507	−0.471849
$766$	−74.8369	−2.70397
$767$	−9.70555	−0.350447
$768$	70.7333	2.55237
$769$	48.4473	1.74705	0.873527	−	0.486775i	$-0.161827\pi$
$769$	48.4473	1.74705	0.873527	+	0.486775i	$0.161827\pi$
$770$	5.23604	0.188694
$771$	24.9761	0.899494
$772$	71.8064	2.58437
$773$	−35.2092	−1.26639	−0.633193	−	0.773994i	$-0.718256\pi$
$773$	−35.2092	−1.26639	−0.633193	+	0.773994i	$0.718256\pi$
$774$	36.2961	1.30464
$775$	0.631807	0.0226952
$776$	3.08127	0.110611
$777$	5.85598	0.210082
$778$	−50.7938	−1.82105
$779$	41.5461	1.48854
$780$	−20.5323	−0.735174
$781$	−6.07026	−0.217211
$782$	−29.4247	−1.05223
$783$	7.10329	0.253851
$784$	−22.4791	−0.802826
$785$	−39.2368	−1.40042
$786$	40.9465	1.46051
$787$	42.6638	1.52080	0.760401	−	0.649454i	$-0.225002\pi$
$787$	42.6638	1.52080	0.760401	+	0.649454i	$0.225002\pi$
$788$	3.49353	0.124452
$789$	20.5786	0.732617
$790$	60.8093	2.16350
$791$	−2.71688	−0.0966011
$792$	21.8091	0.774951
$793$	−0.710221	−0.0252207
$794$	84.8362	3.01072
$795$	76.4446	2.71121
$796$	−106.339	−3.76907
$797$	30.6025	1.08400	0.541999	−	0.840379i	$-0.317668\pi$
$797$	30.6025	1.08400	0.541999	+	0.840379i	$0.317668\pi$
$798$	−12.9293	−0.457693
$799$	17.3454	0.613636
$800$	0.314111	0.0111055
$801$	−36.2155	−1.27961
$802$	40.2315	1.42062
$803$	−20.2064	−0.713068
$804$	9.93604	0.350417
$805$	5.08224	0.179126
$806$	5.40818	0.190495
$807$	8.51706	0.299815
$808$	10.2058	0.359038
$809$	−40.7031	−1.43104	−0.715522	−	0.698590i	$-0.753811\pi$
$809$	−40.7031	−1.43104	−0.715522	+	0.698590i	$0.753811\pi$
$810$	−59.2001	−2.08008
$811$	−40.0560	−1.40656	−0.703279	−	0.710914i	$-0.748281\pi$
$811$	−40.0560	−1.40656	−0.703279	+	0.710914i	$0.748281\pi$
$812$	−7.65876	−0.268770
$813$	56.8649	1.99434
$814$	28.4982	0.998861
$815$	55.0599	1.92866
$816$	−18.8341	−0.659324
$817$	−33.5083	−1.17231
$818$	16.9912	0.594083
$819$	1.03476	0.0361575
$820$	−72.1739	−2.52042
$821$	20.8112	0.726314	0.363157	−	0.931728i	$-0.381699\pi$
$821$	20.8112	0.726314	0.363157	+	0.931728i	$0.381699\pi$
$822$	80.4844	2.80722
$823$	−9.01669	−0.314302	−0.157151	−	0.987575i	$-0.550231\pi$
$823$	−9.01669	−0.314302	−0.157151	+	0.987575i	$0.550231\pi$
$824$	−88.9404	−3.09838
$825$	1.35758	0.0472648
$826$	10.6172	0.369418
$827$	−44.1433	−1.53501	−0.767506	−	0.641042i	$-0.778503\pi$
$827$	−44.1433	−1.53501	−0.767506	+	0.641042i	$0.778503\pi$
$828$	43.6611	1.51733
$829$	14.1263	0.490627	0.245314	−	0.969444i	$-0.421109\pi$
$829$	14.1263	0.490627	0.245314	+	0.969444i	$0.421109\pi$
$830$	−17.3785	−0.603215
$831$	21.2670	0.737743
$832$	9.30360	0.322544
$833$	−16.8207	−0.582803
$834$	−87.0360	−3.01381
$835$	−20.2712	−0.701513
$836$	−41.5273	−1.43625
$837$	3.62143	0.125175
$838$	−43.9330	−1.51764
$839$	37.4692	1.29358	0.646791	−	0.762668i	$-0.276111\pi$
$839$	37.4692	1.29358	0.646791	+	0.762668i	$0.276111\pi$
$840$	10.8898	0.375735
$841$	−9.87000	−0.340345
$842$	19.2970	0.665018
$843$	−56.9754	−1.96234
$844$	40.1320	1.38140
$845$	−2.29855	−0.0790726
$846$	−38.9965	−1.34073
$847$	3.00560	0.103274
$848$	47.8062	1.64167
$849$	2.37953	0.0816652
$850$	−1.70072	−0.0583342
$851$	27.6612	0.948212
$852$	−26.0394	−0.892094
$853$	3.68034	0.126012	0.0630062	−	0.998013i	$-0.479931\pi$
$853$	3.68034	0.126012	0.0630062	+	0.998013i	$0.479931\pi$
$854$	0.776930	0.0265860
$855$	−27.0874	−0.926370
$856$	−64.4597	−2.20319
$857$	−4.35440	−0.148743	−0.0743717	−	0.997231i	$-0.523695\pi$
$857$	−4.35440	−0.148743	−0.0743717	+	0.997231i	$0.523695\pi$
$858$	11.6207	0.396723
$859$	−13.9457	−0.475820	−0.237910	−	0.971287i	$-0.576462\pi$
$859$	−13.9457	−0.475820	−0.237910	+	0.971287i	$0.576462\pi$
$860$	58.2106	1.98497
$861$	8.39375	0.286058
$862$	32.2114	1.09713
$863$	−0.191609	−0.00652243	−0.00326122	−	0.999995i	$-0.501038\pi$
$863$	−0.191609	−0.00652243	−0.00326122	+	0.999995i	$0.501038\pi$
$864$	1.80045	0.0612524
$865$	−5.63219	−0.191500
$866$	−28.3405	−0.963049
$867$	25.0222	0.849798
$868$	−3.90462	−0.132532
$869$	−22.7145	−0.770538
$870$	−56.1026	−1.90205
$871$	1.11232	0.0376896
$872$	5.45404	0.184697
$873$	−1.54847	−0.0524077
$874$	−61.0726	−2.06581
$875$	−4.88996	−0.165311
$876$	−86.6786	−2.92860
$877$	45.7352	1.54437	0.772183	−	0.635400i	$-0.219165\pi$
$877$	45.7352	1.54437	0.772183	+	0.635400i	$0.219165\pi$
$878$	−32.9359	−1.11153
$879$	39.1680	1.32110
$880$	15.8309	0.533658
$881$	−1.82831	−0.0615973	−0.0307986	−	0.999526i	$-0.509805\pi$
$881$	−1.82831	−0.0615973	−0.0307986	+	0.999526i	$0.509805\pi$
$882$	37.8169	1.27336
$883$	43.1166	1.45099	0.725494	−	0.688228i	$-0.241611\pi$
$883$	43.1166	1.45099	0.725494	+	0.688228i	$0.241611\pi$
$884$	−9.60815	−0.323157
$885$	51.3302	1.72545
$886$	96.9948	3.25860
$887$	−19.3689	−0.650343	−0.325172	−	0.945655i	$-0.605422\pi$
$887$	−19.3689	−0.650343	−0.325172	+	0.945655i	$0.605422\pi$
$888$	59.2701	1.98897
$889$	0.928738	0.0311489
$890$	−88.0027	−2.94986
$891$	22.1134	0.740828
$892$	45.1425	1.51148
$893$	36.0013	1.20474
$894$	42.4484	1.41969
$895$	−2.39681	−0.0801166
$896$	−9.17740	−0.306595
$897$	11.2793	0.376607
$898$	−17.1202	−0.571309
$899$	9.75295	0.325279
$900$	2.52357	0.0841189
$901$	35.7725	1.19175
$902$	40.8483	1.36010
$903$	−6.76984	−0.225286
$904$	−27.4983	−0.914580
$905$	−14.0726	−0.467791
$906$	88.4481	2.93849
$907$	−0.0206324	−0.000685087 0	−0.000342544	−	1.00000i	$-0.500109\pi$
$907$	−0.0206324	−0.000685087 0	−0.000342544		1.00000i	$0.500109\pi$
$908$	−47.6281	−1.58059
$909$	−5.12884	−0.170113
$910$	2.51445	0.0833531
$911$	20.6880	0.685425	0.342712	−	0.939440i	$-0.388654\pi$
$911$	20.6880	0.685425	0.342712	+	0.939440i	$0.388654\pi$
$912$	−39.0911	−1.29444
$913$	6.49150	0.214837
$914$	−51.5471	−1.70503
$915$	3.75618	0.124176
$916$	−22.1359	−0.731389
$917$	−3.30950	−0.109289
$918$	−9.74831	−0.321742
$919$	−12.7927	−0.421993	−0.210997	−	0.977487i	$-0.567671\pi$
$919$	−12.7927	−0.421993	−0.210997	+	0.977487i	$0.567671\pi$
$920$	51.4389	1.69589
$921$	46.8119	1.54250
$922$	54.4245	1.79237
$923$	−2.91506	−0.0959502
$924$	−8.38996	−0.276009
$925$	1.59879	0.0525678
$926$	−84.8760	−2.78920
$927$	44.6963	1.46802
$928$	4.84882	0.159170
$929$	−39.6607	−1.30123	−0.650613	−	0.759410i	$-0.725488\pi$
$929$	−39.6607	−1.30123	−0.650613	+	0.759410i	$0.725488\pi$
$930$	−28.6025	−0.937912
$931$	−34.9123	−1.14420
$932$	−101.244	−3.31636
$933$	68.9477	2.25725
$934$	76.7434	2.51112
$935$	11.8459	0.387404
$936$	10.4731	0.342325
$937$	0.804325	0.0262761	0.0131381	−	0.999914i	$-0.495818\pi$
$937$	0.804325	0.0262761	0.0131381	+	0.999914i	$0.495818\pi$
$938$	−1.21680	−0.0397298
$939$	−34.9324	−1.13998
$940$	−62.5414	−2.03988
$941$	−6.67048	−0.217451	−0.108726	−	0.994072i	$-0.534677\pi$
$941$	−6.67048	−0.217451	−0.108726	+	0.994072i	$0.534677\pi$
$942$	95.2600	3.10374
$943$	39.6485	1.29113
$944$	32.1004	1.04478
$945$	1.68373	0.0547717
$946$	−32.9455	−1.07115
$947$	−2.18228	−0.0709145	−0.0354573	−	0.999371i	$-0.511289\pi$
$947$	−2.18228	−0.0709145	−0.0354573	+	0.999371i	$0.511289\pi$
$948$	−97.4377	−3.16463
$949$	−9.70350	−0.314989
$950$	−3.52993	−0.114526
$951$	−73.7360	−2.39105
$952$	5.09593	0.165160
$953$	−7.21188	−0.233616	−0.116808	−	0.993155i	$-0.537266\pi$
$953$	−7.21188	−0.233616	−0.116808	+	0.993155i	$0.537266\pi$
$954$	−80.4249	−2.60385
$955$	9.45059	0.305814
$956$	92.6979	2.99806
$957$	20.9564	0.677424
$958$	73.5605	2.37663
$959$	−6.50514	−0.210062
$960$	−49.2044	−1.58807
$961$	−26.0277	−0.839603
$962$	13.6854	0.441235
$963$	32.3937	1.04387
$964$	−42.4162	−1.36614
$965$	−42.5141	−1.36858
$966$	−12.3388	−0.396994
$967$	24.1350	0.776130	0.388065	−	0.921632i	$-0.373144\pi$
$967$	24.1350	0.776130	0.388065	+	0.921632i	$0.373144\pi$
$968$	30.4205	0.977753
$969$	−29.2511	−0.939682
$970$	−3.76274	−0.120814
$971$	34.8091	1.11708	0.558539	−	0.829478i	$-0.311362\pi$
$971$	34.8091	1.11708	0.558539	+	0.829478i	$0.311362\pi$
$972$	75.9441	2.43591
$973$	7.03467	0.225521
$974$	−7.13012	−0.228464
$975$	0.651935	0.0208786
$976$	2.34900	0.0751898
$977$	−47.6428	−1.52423	−0.762114	−	0.647443i	$-0.775838\pi$
$977$	−47.6428	−1.52423	−0.762114	+	0.647443i	$0.775838\pi$
$978$	−133.676	−4.27448
$979$	32.8723	1.05060
$980$	60.6496	1.93738
$981$	−2.74089	−0.0875098
$982$	6.72082	0.214470
$983$	−46.8279	−1.49358	−0.746788	−	0.665062i	$-0.768405\pi$
$983$	−46.8279	−1.49358	−0.746788	+	0.665062i	$0.768405\pi$
$984$	84.9556	2.70828
$985$	−2.06840	−0.0659047
$986$	−26.2534	−0.836077
$987$	7.27350	0.231518
$988$	−19.9422	−0.634447
$989$	−31.9778	−1.01684
$990$	−26.6325	−0.846435
$991$	14.3044	0.454394	0.227197	−	0.973849i	$-0.427044\pi$
$991$	14.3044	0.454394	0.227197	+	0.973849i	$0.427044\pi$
$992$	2.47205	0.0784876
$993$	5.88096	0.186626
$994$	3.18886	0.101144
$995$	62.9594	1.99595
$996$	27.8463	0.882345
$997$	33.0823	1.04773	0.523863	−	0.851802i	$-0.324490\pi$
$997$	33.0823	1.04773	0.523863	+	0.851802i	$0.324490\pi$
$998$	28.2746	0.895018
$999$	9.16404	0.289937

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.12	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-2.42534 q^{2} -2.30091 q^{3} +3.88226 q^{4} -2.29855 q^{5} +5.58047 q^{6} -0.451041 q^{7} -4.56512 q^{8} +2.29416 q^{9} +O(q^{10})\)	\(q-2.42534 q^{2} -2.30091 q^{3} +3.88226 q^{4} -2.29855 q^{5} +5.58047 q^{6} -0.451041 q^{7} -4.56512 q^{8} +2.29416 q^{9} +5.57476 q^{10} -2.08238 q^{11} -8.93271 q^{12} -1.00000 q^{13} +1.09393 q^{14} +5.28875 q^{15} +3.30743 q^{16} +2.47489 q^{17} -5.56412 q^{18} +5.13675 q^{19} -8.92358 q^{20} +1.03780 q^{21} +5.05048 q^{22} +4.90214 q^{23} +10.5039 q^{24} +0.283339 q^{25} +2.42534 q^{26} +1.62406 q^{27} -1.75106 q^{28} +4.37379 q^{29} -12.8270 q^{30} +2.22987 q^{31} +1.10861 q^{32} +4.79136 q^{33} -6.00243 q^{34} +1.03674 q^{35} +8.90655 q^{36} +5.64268 q^{37} -12.4584 q^{38} +2.30091 q^{39} +10.4932 q^{40} +8.08800 q^{41} -2.51702 q^{42} -6.52324 q^{43} -8.08435 q^{44} -5.27326 q^{45} -11.8893 q^{46} +7.00856 q^{47} -7.61008 q^{48} -6.79656 q^{49} -0.687191 q^{50} -5.69448 q^{51} -3.88226 q^{52} +14.4542 q^{53} -3.93889 q^{54} +4.78646 q^{55} +2.05905 q^{56} -11.8192 q^{57} -10.6079 q^{58} +9.70555 q^{59} +20.5323 q^{60} +0.710221 q^{61} -5.40818 q^{62} -1.03476 q^{63} -9.30360 q^{64} +2.29855 q^{65} -11.6207 q^{66} -1.11232 q^{67} +9.60815 q^{68} -11.2793 q^{69} -2.51445 q^{70} +2.91506 q^{71} -10.4731 q^{72} +9.70350 q^{73} -13.6854 q^{74} -0.651935 q^{75} +19.9422 q^{76} +0.939239 q^{77} -5.58047 q^{78} +10.9080 q^{79} -7.60229 q^{80} -10.6193 q^{81} -19.6161 q^{82} -3.11734 q^{83} +4.02902 q^{84} -5.68865 q^{85} +15.8211 q^{86} -10.0637 q^{87} +9.50631 q^{88} -15.7859 q^{89} +12.7894 q^{90} +0.451041 q^{91} +19.0314 q^{92} -5.13071 q^{93} -16.9981 q^{94} -11.8071 q^{95} -2.55080 q^{96} -0.674960 q^{97} +16.4840 q^{98} -4.77733 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 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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