LMFDB - Embedded newform 8023.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8023,2,Mod(1,8023)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8023, 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("8023.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8023 = 71 \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8023.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:	$64.0639775417$
Analytic rank:	$1$
Dimension:	$155$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8023.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57220 q^{2} -0.502356 q^{3} +4.61621 q^{4} -1.32474 q^{5} +1.29216 q^{6} +1.13423 q^{7} -6.72941 q^{8} -2.74764 q^{9} +O(q^{10})$	\(q-2.57220 q^{2} -0.502356 q^{3} +4.61621 q^{4} -1.32474 q^{5} +1.29216 q^{6} +1.13423 q^{7} -6.72941 q^{8} -2.74764 q^{9} +3.40751 q^{10} -5.43416 q^{11} -2.31898 q^{12} +1.77923 q^{13} -2.91747 q^{14} +0.665494 q^{15} +8.07697 q^{16} +5.98895 q^{17} +7.06747 q^{18} +2.46587 q^{19} -6.11530 q^{20} -0.569788 q^{21} +13.9777 q^{22} +3.70947 q^{23} +3.38056 q^{24} -3.24505 q^{25} -4.57653 q^{26} +2.88736 q^{27} +5.23585 q^{28} -8.87421 q^{29} -1.71178 q^{30} -1.99570 q^{31} -7.31676 q^{32} +2.72988 q^{33} -15.4048 q^{34} -1.50257 q^{35} -12.6837 q^{36} +8.64243 q^{37} -6.34270 q^{38} -0.893806 q^{39} +8.91475 q^{40} -1.47716 q^{41} +1.46561 q^{42} -1.41721 q^{43} -25.0852 q^{44} +3.63992 q^{45} -9.54150 q^{46} +3.64961 q^{47} -4.05752 q^{48} -5.71352 q^{49} +8.34692 q^{50} -3.00859 q^{51} +8.21329 q^{52} -3.36594 q^{53} -7.42687 q^{54} +7.19887 q^{55} -7.63271 q^{56} -1.23874 q^{57} +22.8262 q^{58} -9.62178 q^{59} +3.07206 q^{60} -3.95227 q^{61} +5.13333 q^{62} -3.11646 q^{63} +2.66622 q^{64} -2.35702 q^{65} -7.02180 q^{66} +7.32489 q^{67} +27.6463 q^{68} -1.86348 q^{69} +3.86490 q^{70} +1.00000 q^{71} +18.4900 q^{72} +8.43554 q^{73} -22.2301 q^{74} +1.63017 q^{75} +11.3830 q^{76} -6.16359 q^{77} +2.29905 q^{78} +5.25624 q^{79} -10.6999 q^{80} +6.79243 q^{81} +3.79955 q^{82} -4.18982 q^{83} -2.63026 q^{84} -7.93383 q^{85} +3.64534 q^{86} +4.45801 q^{87} +36.5687 q^{88} -12.6479 q^{89} -9.36260 q^{90} +2.01805 q^{91} +17.1237 q^{92} +1.00255 q^{93} -9.38752 q^{94} -3.26664 q^{95} +3.67562 q^{96} +4.07442 q^{97} +14.6963 q^{98} +14.9311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9}+O(q^{10}) $ 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9	\( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9} - 2 q^{10} - 24 q^{11} - 32 q^{12} - 62 q^{13} - 18 q^{14} - 12 q^{15} + 155 q^{16} - 129 q^{17} - 42 q^{18} - 18 q^{19} - 59 q^{20} - 45 q^{21} - 17 q^{22} - 38 q^{23} - 27 q^{24} + 129 q^{25} - 44 q^{26} - 43 q^{27} - 100 q^{28} - 52 q^{29} - 39 q^{30} - 56 q^{31} - 145 q^{32} - 126 q^{33} - q^{34} - 49 q^{35} + 131 q^{36} - 30 q^{37} - 91 q^{38} - 29 q^{39} - 5 q^{40} - 163 q^{41} - 80 q^{42} - 15 q^{43} - 118 q^{44} - 66 q^{45} + 2 q^{46} - 111 q^{47} - 89 q^{48} + 101 q^{49} - 121 q^{50} + 5 q^{51} - 111 q^{52} - 93 q^{53} - 68 q^{54} - 60 q^{55} - 27 q^{56} - 106 q^{57} + 16 q^{58} - 79 q^{59} - 103 q^{60} - 74 q^{61} - 102 q^{62} - 118 q^{63} + 175 q^{64} - 109 q^{65} + 65 q^{66} - 18 q^{67} - 346 q^{68} - 39 q^{69} + 32 q^{70} + 155 q^{71} - 203 q^{72} - 108 q^{73} - 87 q^{74} - 22 q^{75} - 16 q^{76} - 121 q^{77} - 75 q^{78} - 6 q^{79} - 136 q^{80} + 107 q^{81} - 30 q^{82} - 116 q^{83} - 5 q^{84} - 53 q^{85} + 8 q^{86} - 100 q^{87} - 43 q^{88} - 189 q^{89} - 76 q^{90} + 14 q^{91} - 99 q^{92} - 72 q^{93} + 17 q^{94} - 18 q^{95} - 50 q^{96} - 184 q^{97} - 249 q^{98} - 114 q^{99}+O(q^{100}) \) 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9 - 2 * q^10 - 24 * q^11 - 32 * q^12 - 62 * q^13 - 18 * q^14 - 12 * q^15 + 155 * q^16 - 129 * q^17 - 42 * q^18 - 18 * q^19 - 59 * q^20 - 45 * q^21 - 17 * q^22 - 38 * q^23 - 27 * q^24 + 129 * q^25 - 44 * q^26 - 43 * q^27 - 100 * q^28 - 52 * q^29 - 39 * q^30 - 56 * q^31 - 145 * q^32 - 126 * q^33 - q^34 - 49 * q^35 + 131 * q^36 - 30 * q^37 - 91 * q^38 - 29 * q^39 - 5 * q^40 - 163 * q^41 - 80 * q^42 - 15 * q^43 - 118 * q^44 - 66 * q^45 + 2 * q^46 - 111 * q^47 - 89 * q^48 + 101 * q^49 - 121 * q^50 + 5 * q^51 - 111 * q^52 - 93 * q^53 - 68 * q^54 - 60 * q^55 - 27 * q^56 - 106 * q^57 + 16 * q^58 - 79 * q^59 - 103 * q^60 - 74 * q^61 - 102 * q^62 - 118 * q^63 + 175 * q^64 - 109 * q^65 + 65 * q^66 - 18 * q^67 - 346 * q^68 - 39 * q^69 + 32 * q^70 + 155 * q^71 - 203 * q^72 - 108 * q^73 - 87 * q^74 - 22 * q^75 - 16 * q^76 - 121 * q^77 - 75 * q^78 - 6 * q^79 - 136 * q^80 + 107 * q^81 - 30 * q^82 - 116 * q^83 - 5 * q^84 - 53 * q^85 + 8 * q^86 - 100 * q^87 - 43 * q^88 - 189 * q^89 - 76 * q^90 + 14 * q^91 - 99 * q^92 - 72 * q^93 + 17 * q^94 - 18 * q^95 - 50 * q^96 - 184 * q^97 - 249 * q^98 - 114 * 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 $

$2$	−2.57220	−1.81882	−0.909410	−	0.415901i	$-0.863466\pi$
$2$	−2.57220	−1.81882	−0.909410	+	0.415901i	$0.863466\pi$
$3$	−0.502356	−0.290035	−0.145018	−	0.989429i	$-0.546324\pi$
$3$	−0.502356	−0.290035	−0.145018	+	0.989429i	$0.546324\pi$
$4$	4.61621	2.30810
$5$	−1.32474	−0.592444	−0.296222	−	0.955119i	$-0.595727\pi$
$5$	−1.32474	−0.592444	−0.296222	+	0.955119i	$0.595727\pi$
$6$	1.29216	0.527522
$7$	1.13423	0.428699	0.214349	−	0.976757i	$-0.431237\pi$
$7$	1.13423	0.428699	0.214349	+	0.976757i	$0.431237\pi$
$8$	−6.72941	−2.37921
$9$	−2.74764	−0.915879
$10$	3.40751	1.07755
$11$	−5.43416	−1.63846	−0.819230	−	0.573466i	$-0.805599\pi$
$11$	−5.43416	−1.63846	−0.819230	+	0.573466i	$0.805599\pi$
$12$	−2.31898	−0.669432
$13$	1.77923	0.493469	0.246734	−	0.969083i	$-0.420642\pi$
$13$	1.77923	0.493469	0.246734	+	0.969083i	$0.420642\pi$
$14$	−2.91747	−0.779726
$15$	0.665494	0.171830
$16$	8.07697	2.01924
$17$	5.98895	1.45253	0.726267	−	0.687413i	$-0.241254\pi$
$17$	5.98895	1.45253	0.726267	+	0.687413i	$0.241254\pi$
$18$	7.06747	1.66582
$19$	2.46587	0.565708	0.282854	−	0.959163i	$-0.408719\pi$
$19$	2.46587	0.565708	0.282854	+	0.959163i	$0.408719\pi$
$20$	−6.11530	−1.36742
$21$	−0.569788	−0.124338
$22$	13.9777	2.98006
$23$	3.70947	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$23$	3.70947	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$24$	3.38056	0.690054
$25$	−3.24505	−0.649010
$26$	−4.57653	−0.897531
$27$	2.88736	0.555673
$28$	5.23585	0.989482
$29$	−8.87421	−1.64790	−0.823949	−	0.566664i	$-0.808234\pi$
$29$	−8.87421	−1.64790	−0.823949	+	0.566664i	$0.808234\pi$
$30$	−1.71178	−0.312527
$31$	−1.99570	−0.358438	−0.179219	−	0.983809i	$-0.557357\pi$
$31$	−1.99570	−0.358438	−0.179219	+	0.983809i	$0.557357\pi$
$32$	−7.31676	−1.29343
$33$	2.72988	0.475211
$34$	−15.4048	−2.64190
$35$	−1.50257	−0.253980
$36$	−12.6837	−2.11395
$37$	8.64243	1.42081	0.710403	−	0.703795i	$-0.248513\pi$
$37$	8.64243	1.42081	0.710403	+	0.703795i	$0.248513\pi$
$38$	−6.34270	−1.02892
$39$	−0.893806	−0.143123
$40$	8.91475	1.40955
$41$	−1.47716	−0.230694	−0.115347	−	0.993325i	$-0.536798\pi$
$41$	−1.47716	−0.230694	−0.115347	+	0.993325i	$0.536798\pi$
$42$	1.46561	0.226148
$43$	−1.41721	−0.216122	−0.108061	−	0.994144i	$-0.534464\pi$
$43$	−1.41721	−0.216122	−0.108061	+	0.994144i	$0.534464\pi$
$44$	−25.0852	−3.78174
$45$	3.63992	0.542607
$46$	−9.54150	−1.40682
$47$	3.64961	0.532350	0.266175	−	0.963925i	$-0.414240\pi$
$47$	3.64961	0.532350	0.266175	+	0.963925i	$0.414240\pi$
$48$	−4.05752	−0.585652
$49$	−5.71352	−0.816217
$50$	8.34692	1.18043
$51$	−3.00859	−0.421286
$52$	8.21329	1.13898
$53$	−3.36594	−0.462347	−0.231173	−	0.972913i	$-0.574257\pi$
$53$	−3.36594	−0.462347	−0.231173	+	0.972913i	$0.574257\pi$
$54$	−7.42687	−1.01067
$55$	7.19887	0.970695
$56$	−7.63271	−1.01996
$57$	−1.23874	−0.164075
$58$	22.8262	2.99723
$59$	−9.62178	−1.25265	−0.626325	−	0.779562i	$-0.715442\pi$
$59$	−9.62178	−1.25265	−0.626325	+	0.779562i	$0.715442\pi$
$60$	3.07206	0.396601
$61$	−3.95227	−0.506036	−0.253018	−	0.967462i	$-0.581423\pi$
$61$	−3.95227	−0.506036	−0.253018	+	0.967462i	$0.581423\pi$
$62$	5.13333	0.651933
$63$	−3.11646	−0.392637
$64$	2.66622	0.333277
$65$	−2.35702	−0.292353
$66$	−7.02180	−0.864324
$67$	7.32489	0.894877	0.447439	−	0.894315i	$-0.352336\pi$
$67$	7.32489	0.894877	0.447439	+	0.894315i	$0.352336\pi$
$68$	27.6463	3.35260
$69$	−1.86348	−0.224336
$70$	3.86490	0.461944
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	18.4900	2.17907
$73$	8.43554	0.987305	0.493653	−	0.869659i	$-0.335661\pi$
$73$	8.43554	0.987305	0.493653	+	0.869659i	$0.335661\pi$
$74$	−22.2301	−2.58419
$75$	1.63017	0.188236
$76$	11.3830	1.30571
$77$	−6.16359	−0.702406
$78$	2.29905	0.260316
$79$	5.25624	0.591373	0.295687	−	0.955285i	$-0.404452\pi$
$79$	5.25624	0.591373	0.295687	+	0.955285i	$0.404452\pi$
$80$	−10.6999	−1.19629
$81$	6.79243	0.754715
$82$	3.79955	0.419591
$83$	−4.18982	−0.459892	−0.229946	−	0.973203i	$-0.573855\pi$
$83$	−4.18982	−0.459892	−0.229946	+	0.973203i	$0.573855\pi$
$84$	−2.63026	−0.286985
$85$	−7.93383	−0.860545
$86$	3.64534	0.393087
$87$	4.45801	0.477949
$88$	36.5687	3.89823
$89$	−12.6479	−1.34068	−0.670339	−	0.742055i	$-0.733851\pi$
$89$	−12.6479	−1.34068	−0.670339	+	0.742055i	$0.733851\pi$
$90$	−9.36260	−0.986904
$91$	2.01805	0.211550
$92$	17.1237	1.78527
$93$	1.00255	0.103960
$94$	−9.38752	−0.968249
$95$	−3.26664	−0.335150
$96$	3.67562	0.375141
$97$	4.07442	0.413695	0.206848	−	0.978373i	$-0.433680\pi$
$97$	4.07442	0.413695	0.206848	+	0.978373i	$0.433680\pi$
$98$	14.6963	1.48455
$99$	14.9311	1.50063
$100$	−14.9798	−1.49798
$101$	18.7177	1.86248	0.931240	−	0.364406i	$-0.118728\pi$
$101$	18.7177	1.86248	0.931240	+	0.364406i	$0.118728\pi$
$102$	7.73868	0.766244
$103$	−3.92040	−0.386289	−0.193144	−	0.981170i	$-0.561869\pi$
$103$	−3.92040	−0.386289	−0.193144	+	0.981170i	$0.561869\pi$
$104$	−11.9732	−1.17406
$105$	0.754823	0.0736632
$106$	8.65786	0.840926
$107$	17.4173	1.68379	0.841895	−	0.539642i	$-0.181440\pi$
$107$	17.4173	1.68379	0.841895	+	0.539642i	$0.181440\pi$
$108$	13.3287	1.28255
$109$	8.49517	0.813690	0.406845	−	0.913497i	$-0.366629\pi$
$109$	8.49517	0.813690	0.406845	+	0.913497i	$0.366629\pi$
$110$	−18.5169	−1.76552
$111$	−4.34158	−0.412084
$112$	9.16115	0.865647
$113$	1.00000	0.0940721
$113$	1.00000	0.0940721
$114$	3.18629	0.298424
$115$	−4.91410	−0.458243
$116$	−40.9652	−3.80352
$117$	−4.88867	−0.451958
$118$	24.7491	2.27834
$119$	6.79285	0.622700
$120$	−4.47838	−0.408818
$121$	18.5300	1.68455
$122$	10.1660	0.920388
$123$	0.742061	0.0669094
$124$	−9.21255	−0.827311
$125$	10.9226	0.976946
$126$	8.01615	0.714135
$127$	−7.15698	−0.635079	−0.317540	−	0.948245i	$-0.602857\pi$
$127$	−7.15698	−0.635079	−0.317540	+	0.948245i	$0.602857\pi$
$128$	7.77547	0.687261
$129$	0.711943	0.0626831
$130$	6.06273	0.531737
$131$	−3.85411	−0.336735	−0.168367	−	0.985724i	$-0.553849\pi$
$131$	−3.85411	−0.336735	−0.168367	+	0.985724i	$0.553849\pi$
$132$	12.6017	1.09684
$133$	2.79686	0.242519
$134$	−18.8411	−1.62762
$135$	−3.82502	−0.329205
$136$	−40.3021	−3.45588
$137$	−1.44520	−0.123472	−0.0617359	−	0.998093i	$-0.519664\pi$
$137$	−1.44520	−0.123472	−0.0617359	+	0.998093i	$0.519664\pi$
$138$	4.79323	0.408027
$139$	2.21943	0.188249	0.0941246	−	0.995560i	$-0.469995\pi$
$139$	2.21943	0.188249	0.0941246	+	0.995560i	$0.469995\pi$
$140$	−6.93616	−0.586212
$141$	−1.83340	−0.154400
$142$	−2.57220	−0.215854
$143$	−9.66860	−0.808529
$144$	−22.1926	−1.84938
$145$	11.7561	0.976287
$146$	−21.6979	−1.79573
$147$	2.87022	0.236732
$148$	39.8953	3.27937
$149$	−10.0298	−0.821673	−0.410836	−	0.911709i	$-0.634763\pi$
$149$	−10.0298	−0.821673	−0.410836	+	0.911709i	$0.634763\pi$
$150$	−4.19313	−0.342367
$151$	10.3741	0.844228	0.422114	−	0.906543i	$-0.361288\pi$
$151$	10.3741	0.844228	0.422114	+	0.906543i	$0.361288\pi$
$152$	−16.5938	−1.34594
$153$	−16.4555	−1.33035
$154$	15.8540	1.27755
$155$	2.64379	0.212354
$156$	−4.12600	−0.330344
$157$	17.6741	1.41055	0.705275	−	0.708934i	$-0.250823\pi$
$157$	17.6741	1.41055	0.705275	+	0.708934i	$0.250823\pi$
$158$	−13.5201	−1.07560
$159$	1.69090	0.134097
$160$	9.69284	0.766286
$161$	4.20740	0.331589
$162$	−17.4715	−1.37269
$163$	−19.1549	−1.50033	−0.750163	−	0.661253i	$-0.770025\pi$
$163$	−19.1549	−1.50033	−0.750163	+	0.661253i	$0.770025\pi$
$164$	−6.81889	−0.532466
$165$	−3.61640	−0.281536
$166$	10.7770	0.836461
$167$	−19.3171	−1.49480	−0.747401	−	0.664373i	$-0.768699\pi$
$167$	−19.3171	−1.49480	−0.747401	+	0.664373i	$0.768699\pi$
$168$	3.83434	0.295826
$169$	−9.83435	−0.756488
$170$	20.4074	1.56518
$171$	−6.77531	−0.518121
$172$	−6.54213	−0.498832
$173$	−10.8548	−0.825278	−0.412639	−	0.910895i	$-0.635393\pi$
$173$	−10.8548	−0.825278	−0.412639	+	0.910895i	$0.635393\pi$
$174$	−11.4669	−0.869303
$175$	−3.68064	−0.278230
$176$	−43.8915	−3.30845
$177$	4.83356	0.363313
$178$	32.5330	2.43845
$179$	4.66495	0.348675	0.174338	−	0.984686i	$-0.444222\pi$
$179$	4.66495	0.348675	0.174338	+	0.984686i	$0.444222\pi$
$180$	16.8026	1.25239
$181$	10.9535	0.814169	0.407085	−	0.913391i	$-0.366545\pi$
$181$	10.9535	0.814169	0.407085	+	0.913391i	$0.366545\pi$
$182$	−5.19084	−0.384771
$183$	1.98544	0.146768
$184$	−24.9626	−1.84027
$185$	−11.4490	−0.841748
$186$	−2.57876	−0.189084
$187$	−32.5449	−2.37992
$188$	16.8474	1.22872
$189$	3.27493	0.238216
$190$	8.40245	0.609578
$191$	−10.6773	−0.772581	−0.386290	−	0.922377i	$-0.626244\pi$
$191$	−10.6773	−0.772581	−0.386290	+	0.922377i	$0.626244\pi$
$192$	−1.33939	−0.0966622
$193$	16.1203	1.16036	0.580181	−	0.814488i	$-0.302982\pi$
$193$	16.1203	1.16036	0.580181	+	0.814488i	$0.302982\pi$
$194$	−10.4802	−0.752437
$195$	1.18406	0.0847926
$196$	−26.3748	−1.88392
$197$	11.7119	0.834439	0.417219	−	0.908806i	$-0.363005\pi$
$197$	11.7119	0.834439	0.417219	+	0.908806i	$0.363005\pi$
$198$	−38.4057	−2.72938
$199$	−8.79059	−0.623148	−0.311574	−	0.950222i	$-0.600856\pi$
$199$	−8.79059	−0.623148	−0.311574	+	0.950222i	$0.600856\pi$
$200$	21.8373	1.54413
$201$	−3.67970	−0.259546
$202$	−48.1456	−3.38752
$203$	−10.0654	−0.706452
$204$	−13.8883	−0.972373
$205$	1.95686	0.136673
$206$	10.0841	0.702590
$207$	−10.1923	−0.708413
$208$	14.3708	0.996434
$209$	−13.3999	−0.926890
$210$	−1.94156	−0.133980
$211$	21.9509	1.51116	0.755581	−	0.655055i	$-0.227355\pi$
$211$	21.9509	1.51116	0.755581	+	0.655055i	$0.227355\pi$
$212$	−15.5379	−1.06715
$213$	−0.502356	−0.0344209
$214$	−44.8006	−3.06251
$215$	1.87744	0.128040
$216$	−19.4302	−1.32206
$217$	−2.26358	−0.153662
$218$	−21.8513	−1.47996
$219$	−4.23764	−0.286353
$220$	33.2315	2.24047
$221$	10.6557	0.716780
$222$	11.1674	0.749507
$223$	0.959852	0.0642764	0.0321382	−	0.999483i	$-0.489768\pi$
$223$	0.959852	0.0642764	0.0321382	+	0.999483i	$0.489768\pi$
$224$	−8.29889	−0.554493
$225$	8.91623	0.594415
$226$	−2.57220	−0.171100
$227$	−6.15490	−0.408515	−0.204257	−	0.978917i	$-0.565478\pi$
$227$	−6.15490	−0.408515	−0.204257	+	0.978917i	$0.565478\pi$
$228$	−5.71830	−0.378703
$229$	−12.0989	−0.799520	−0.399760	−	0.916620i	$-0.630907\pi$
$229$	−12.0989	−0.799520	−0.399760	+	0.916620i	$0.630907\pi$
$230$	12.6401	0.833461
$231$	3.09632	0.203723
$232$	59.7182	3.92069
$233$	9.92537	0.650232	0.325116	−	0.945674i	$-0.394597\pi$
$233$	9.92537	0.650232	0.325116	+	0.945674i	$0.394597\pi$
$234$	12.5746	0.822030
$235$	−4.83480	−0.315388
$236$	−44.4162	−2.89125
$237$	−2.64050	−0.171519
$238$	−17.4726	−1.13258
$239$	−1.90249	−0.123062	−0.0615310	−	0.998105i	$-0.519598\pi$
$239$	−1.90249	−0.123062	−0.0615310	+	0.998105i	$0.519598\pi$
$240$	5.37517	0.346966
$241$	−2.84847	−0.183486	−0.0917431	−	0.995783i	$-0.529244\pi$
$241$	−2.84847	−0.183486	−0.0917431	+	0.995783i	$0.529244\pi$
$242$	−47.6630	−3.06389
$243$	−12.0743	−0.774567
$244$	−18.2445	−1.16798
$245$	7.56896	0.483563
$246$	−1.90873	−0.121696
$247$	4.38734	0.279159
$248$	13.4299	0.852797
$249$	2.10478	0.133385
$250$	−28.0951	−1.77689
$251$	22.5693	1.42456	0.712280	−	0.701895i	$-0.247662\pi$
$251$	22.5693	1.42456	0.712280	+	0.701895i	$0.247662\pi$
$252$	−14.3862	−0.906246
$253$	−20.1579	−1.26731
$254$	18.4092	1.15509
$255$	3.98561	0.249588
$256$	−25.3325	−1.58328
$257$	10.0815	0.628865	0.314432	−	0.949280i	$-0.398186\pi$
$257$	10.0815	0.628865	0.314432	+	0.949280i	$0.398186\pi$
$258$	−1.83126	−0.114009
$259$	9.80251	0.609098
$260$	−10.8805	−0.674780
$261$	24.3831	1.50928
$262$	9.91353	0.612460
$263$	4.91625	0.303149	0.151574	−	0.988446i	$-0.451566\pi$
$263$	4.91625	0.303149	0.151574	+	0.988446i	$0.451566\pi$
$264$	−18.3705	−1.13063
$265$	4.45901	0.273915
$266$	−7.19408	−0.441097
$267$	6.35376	0.388844
$268$	33.8132	2.06547
$269$	−11.8263	−0.721061	−0.360531	−	0.932747i	$-0.617404\pi$
$269$	−11.8263	−0.721061	−0.360531	+	0.932747i	$0.617404\pi$
$270$	9.83870	0.598764
$271$	5.01245	0.304484	0.152242	−	0.988343i	$-0.451351\pi$
$271$	5.01245	0.304484	0.152242	+	0.988343i	$0.451351\pi$
$272$	48.3726	2.93302
$273$	−1.01378	−0.0613569
$274$	3.71734	0.224573
$275$	17.6341	1.06338
$276$	−8.60220	−0.517792
$277$	−17.9644	−1.07938	−0.539689	−	0.841865i	$-0.681458\pi$
$277$	−17.9644	−1.07938	−0.539689	+	0.841865i	$0.681458\pi$
$278$	−5.70881	−0.342391
$279$	5.48345	0.328286
$280$	10.1114	0.604271
$281$	−15.9656	−0.952427	−0.476214	−	0.879330i	$-0.657991\pi$
$281$	−15.9656	−0.952427	−0.476214	+	0.879330i	$0.657991\pi$
$282$	4.71588	0.280827
$283$	−7.20470	−0.428275	−0.214137	−	0.976804i	$-0.568694\pi$
$283$	−7.20470	−0.428275	−0.214137	+	0.976804i	$0.568694\pi$
$284$	4.61621	0.273922
$285$	1.64102	0.0972055
$286$	24.8696	1.47057
$287$	−1.67544	−0.0988982
$288$	20.1038	1.18463
$289$	18.8675	1.10986
$290$	−30.2389	−1.77569
$291$	−2.04681	−0.119986
$292$	38.9402	2.27880
$293$	−5.29849	−0.309541	−0.154771	−	0.987950i	$-0.549464\pi$
$293$	−5.29849	−0.309541	−0.154771	+	0.987950i	$0.549464\pi$
$294$	−7.38278	−0.430573
$295$	12.7464	0.742124
$296$	−58.1585	−3.38039
$297$	−15.6904	−0.910448
$298$	25.7986	1.49447
$299$	6.60000	0.381688
$300$	7.52521	0.434468
$301$	−1.60744	−0.0926513
$302$	−26.6841	−1.53550
$303$	−9.40295	−0.540185
$304$	19.9167	1.14230
$305$	5.23574	0.299798
$306$	42.3268	2.41966
$307$	29.2598	1.66995	0.834973	−	0.550291i	$-0.185483\pi$
$307$	29.2598	1.66995	0.834973	+	0.550291i	$0.185483\pi$
$308$	−28.4524	−1.62123
$309$	1.96944	0.112037
$310$	−6.80035	−0.386234
$311$	−26.1072	−1.48040	−0.740201	−	0.672385i	$-0.765270\pi$
$311$	−26.1072	−1.48040	−0.740201	+	0.672385i	$0.765270\pi$
$312$	6.01479	0.340520
$313$	31.7549	1.79489	0.897446	−	0.441125i	$-0.145420\pi$
$313$	31.7549	1.79489	0.897446	+	0.441125i	$0.145420\pi$
$314$	−45.4614	−2.56554
$315$	4.12851	0.232615
$316$	24.2639	1.36495
$317$	14.4681	0.812611	0.406305	−	0.913737i	$-0.366817\pi$
$317$	14.4681	0.812611	0.406305	+	0.913737i	$0.366817\pi$
$318$	−4.34933	−0.243898
$319$	48.2238	2.70001
$320$	−3.53206	−0.197448
$321$	−8.74966	−0.488359
$322$	−10.8223	−0.603101
$323$	14.7679	0.821711
$324$	31.3553	1.74196
$325$	−5.77369	−0.320266
$326$	49.2701	2.72882
$327$	−4.26760	−0.235999
$328$	9.94043	0.548869
$329$	4.13950	0.228218
$330$	9.30209	0.512063
$331$	18.1153	0.995707	0.497853	−	0.867261i	$-0.334122\pi$
$331$	18.1153	0.995707	0.497853	+	0.867261i	$0.334122\pi$
$332$	−19.3411	−1.06148
$333$	−23.7463	−1.30129
$334$	49.6875	2.71878
$335$	−9.70360	−0.530164
$336$	−4.60216	−0.251068
$337$	−26.1097	−1.42229	−0.711143	−	0.703047i	$-0.751822\pi$
$337$	−26.1097	−1.42229	−0.711143	+	0.703047i	$0.751822\pi$
$338$	25.2959	1.37592
$339$	−0.502356	−0.0272842
$340$	−36.6242	−1.98623
$341$	10.8449	0.587285
$342$	17.4274	0.942368
$343$	−14.4201	−0.778610
$344$	9.53697	0.514199
$345$	2.46863	0.132907
$346$	27.9208	1.50103
$347$	−13.1558	−0.706239	−0.353120	−	0.935578i	$-0.614879\pi$
$347$	−13.1558	−0.706239	−0.353120	+	0.935578i	$0.614879\pi$
$348$	20.5791	1.10316
$349$	−5.44438	−0.291431	−0.145716	−	0.989327i	$-0.546548\pi$
$349$	−5.44438	−0.291431	−0.145716	+	0.989327i	$0.546548\pi$
$350$	9.46733	0.506050
$351$	5.13727	0.274207
$352$	39.7604	2.11924
$353$	−34.3923	−1.83052	−0.915258	−	0.402869i	$-0.868013\pi$
$353$	−34.3923	−1.83052	−0.915258	+	0.402869i	$0.868013\pi$
$354$	−12.4329	−0.660800
$355$	−1.32474	−0.0703101
$356$	−58.3855	−3.09443
$357$	−3.41243	−0.180605
$358$	−11.9992	−0.634177
$359$	15.5739	0.821961	0.410980	−	0.911644i	$-0.365186\pi$
$359$	15.5739	0.821961	0.410980	+	0.911644i	$0.365186\pi$
$360$	−24.4945	−1.29097
$361$	−12.9195	−0.679974
$362$	−28.1746	−1.48083
$363$	−9.30868	−0.488579
$364$	9.31576	0.488279
$365$	−11.1749	−0.584923
$366$	−5.10696	−0.266945
$367$	17.1160	0.893448	0.446724	−	0.894672i	$-0.352591\pi$
$367$	17.1160	0.893448	0.446724	+	0.894672i	$0.352591\pi$
$368$	29.9613	1.56184
$369$	4.05871	0.211288
$370$	29.4491	1.53099
$371$	−3.81775	−0.198208
$372$	4.62798	0.239950
$373$	13.5081	0.699425	0.349712	−	0.936857i	$-0.386279\pi$
$373$	13.5081	0.699425	0.349712	+	0.936857i	$0.386279\pi$
$374$	83.7119	4.32864
$375$	−5.48703	−0.283349
$376$	−24.5597	−1.26657
$377$	−15.7892	−0.813187
$378$	−8.42378	−0.433273
$379$	−27.4081	−1.40786	−0.703929	−	0.710271i	$-0.748572\pi$
$379$	−27.4081	−1.40786	−0.703929	+	0.710271i	$0.748572\pi$
$380$	−15.0795	−0.773562
$381$	3.59535	0.184195
$382$	27.4641	1.40519
$383$	13.5965	0.694749	0.347374	−	0.937727i	$-0.387073\pi$
$383$	13.5965	0.694749	0.347374	+	0.937727i	$0.387073\pi$
$384$	−3.90606	−0.199330
$385$	8.16518	0.416136
$386$	−41.4645	−2.11049
$387$	3.89397	0.197942
$388$	18.8084	0.954851
$389$	4.31447	0.218752	0.109376	−	0.994000i	$-0.465115\pi$
$389$	4.31447	0.218752	0.109376	+	0.994000i	$0.465115\pi$
$390$	−3.04565	−0.154222
$391$	22.2159	1.12350
$392$	38.4486	1.94195
$393$	1.93613	0.0976650
$394$	−30.1254	−1.51769
$395$	−6.96318	−0.350355
$396$	68.9251	3.46361
$397$	−0.688316	−0.0345456	−0.0172728	−	0.999851i	$-0.505498\pi$
$397$	−0.688316	−0.0345456	−0.0172728	+	0.999851i	$0.505498\pi$
$398$	22.6111	1.13339
$399$	−1.40502	−0.0703390
$400$	−26.2102	−1.31051
$401$	0.379993	0.0189759	0.00948797	−	0.999955i	$-0.496980\pi$
$401$	0.379993	0.0189759	0.00948797	+	0.999955i	$0.496980\pi$
$402$	9.46493	0.472068
$403$	−3.55080	−0.176878
$404$	86.4048	4.29880
$405$	−8.99824	−0.447126
$406$	25.8902	1.28491
$407$	−46.9643	−2.32793
$408$	20.2460	1.00233
$409$	−19.2154	−0.950142	−0.475071	−	0.879948i	$-0.657578\pi$
$409$	−19.2154	−0.950142	−0.475071	+	0.879948i	$0.657578\pi$
$410$	−5.03344	−0.248584
$411$	0.726005	0.0358112
$412$	−18.0974	−0.891595
$413$	−10.9133	−0.537009
$414$	26.2166	1.28848
$415$	5.55044	0.272460
$416$	−13.0182	−0.638269
$417$	−1.11494	−0.0545989
$418$	34.4672	1.68585
$419$	37.5519	1.83453	0.917266	−	0.398274i	$-0.130391\pi$
$419$	37.5519	1.83453	0.917266	+	0.398274i	$0.130391\pi$
$420$	3.48442	0.170022
$421$	−8.31510	−0.405253	−0.202627	−	0.979256i	$-0.564948\pi$
$421$	−8.31510	−0.405253	−0.202627	+	0.979256i	$0.564948\pi$
$422$	−56.4621	−2.74853
$423$	−10.0278	−0.487569
$424$	22.6508	1.10002
$425$	−19.4345	−0.942710
$426$	1.29216	0.0626054
$427$	−4.48278	−0.216937
$428$	80.4017	3.88636
$429$	4.85708	0.234502
$430$	−4.82914	−0.232882
$431$	−5.56995	−0.268295	−0.134147	−	0.990961i	$-0.542830\pi$
$431$	−5.56995	−0.268295	−0.134147	+	0.990961i	$0.542830\pi$
$432$	23.3211	1.12204
$433$	6.24912	0.300314	0.150157	−	0.988662i	$-0.452022\pi$
$433$	6.24912	0.300314	0.150157	+	0.988662i	$0.452022\pi$
$434$	5.82238	0.279483
$435$	−5.90573	−0.283158
$436$	39.2155	1.87808
$437$	9.14706	0.437563
$438$	10.9001	0.520825
$439$	1.90182	0.0907690	0.0453845	−	0.998970i	$-0.485549\pi$
$439$	1.90182	0.0907690	0.0453845	+	0.998970i	$0.485549\pi$
$440$	−48.4442	−2.30948
$441$	15.6987	0.747557
$442$	−27.4086	−1.30369
$443$	−9.25639	−0.439785	−0.219892	−	0.975524i	$-0.570571\pi$
$443$	−9.25639	−0.439785	−0.219892	+	0.975524i	$0.570571\pi$
$444$	−20.0416	−0.951134
$445$	16.7553	0.794276
$446$	−2.46893	−0.116907
$447$	5.03853	0.238314
$448$	3.02411	0.142876
$449$	−29.9526	−1.41355	−0.706774	−	0.707439i	$-0.749850\pi$
$449$	−29.9526	−1.41355	−0.706774	+	0.707439i	$0.749850\pi$
$450$	−22.9343	−1.08113
$451$	8.02713	0.377983
$452$	4.61621	0.217128
$453$	−5.21147	−0.244856
$454$	15.8316	0.743015
$455$	−2.67341	−0.125331
$456$	8.33601	0.390369
$457$	−11.1683	−0.522430	−0.261215	−	0.965281i	$-0.584123\pi$
$457$	−11.1683	−0.522430	−0.261215	+	0.965281i	$0.584123\pi$
$458$	31.1209	1.45418
$459$	17.2923	0.807134
$460$	−22.6845	−1.05767
$461$	−4.54094	−0.211493	−0.105746	−	0.994393i	$-0.533723\pi$
$461$	−4.54094	−0.211493	−0.105746	+	0.994393i	$0.533723\pi$
$462$	−7.96434	−0.370535
$463$	16.9377	0.787161	0.393580	−	0.919290i	$-0.371236\pi$
$463$	16.9377	0.787161	0.393580	+	0.919290i	$0.371236\pi$
$464$	−71.6767	−3.32751
$465$	−1.32812	−0.0615902
$466$	−25.5300	−1.18266
$467$	−12.8452	−0.594406	−0.297203	−	0.954814i	$-0.596054\pi$
$467$	−12.8452	−0.594406	−0.297203	+	0.954814i	$0.596054\pi$
$468$	−22.5671	−1.04317
$469$	8.30811	0.383633
$470$	12.4361	0.573633
$471$	−8.87872	−0.409110
$472$	64.7489	2.98031
$473$	7.70132	0.354107
$474$	6.79190	0.311963
$475$	−8.00186	−0.367151
$476$	31.3572	1.43726
$477$	9.24838	0.423454
$478$	4.89359	0.223828
$479$	17.9977	0.822338	0.411169	−	0.911559i	$-0.365121\pi$
$479$	17.9977	0.822338	0.411169	+	0.911559i	$0.365121\pi$
$480$	−4.86926	−0.222250
$481$	15.3768	0.701124
$482$	7.32683	0.333728
$483$	−2.11361	−0.0961727
$484$	85.5386	3.88812
$485$	−5.39757	−0.245091
$486$	31.0575	1.40880
$487$	−21.6256	−0.979949	−0.489975	−	0.871737i	$-0.662994\pi$
$487$	−21.6256	−0.979949	−0.489975	+	0.871737i	$0.662994\pi$
$488$	26.5964	1.20396
$489$	9.62257	0.435147
$490$	−19.4689	−0.879514
$491$	18.1951	0.821131	0.410566	−	0.911831i	$-0.365331\pi$
$491$	18.1951	0.821131	0.410566	+	0.911831i	$0.365331\pi$
$492$	3.42551	0.154434
$493$	−53.1472	−2.39363
$494$	−11.2851	−0.507741
$495$	−19.7799	−0.889040
$496$	−16.1192	−0.723773
$497$	1.13423	0.0508772
$498$	−5.41392	−0.242603
$499$	−22.6955	−1.01599	−0.507994	−	0.861361i	$-0.669613\pi$
$499$	−22.6955	−1.01599	−0.507994	+	0.861361i	$0.669613\pi$
$500$	50.4210	2.25489
$501$	9.70407	0.433546
$502$	−58.0527	−2.59102
$503$	17.0495	0.760199	0.380099	−	0.924946i	$-0.375890\pi$
$503$	17.0495	0.760199	0.380099	+	0.924946i	$0.375890\pi$
$504$	20.9719	0.934164
$505$	−24.7962	−1.10341
$506$	51.8500	2.30501
$507$	4.94035	0.219408
$508$	−33.0381	−1.46583
$509$	−8.81302	−0.390630	−0.195315	−	0.980741i	$-0.562573\pi$
$509$	−8.81302	−0.390630	−0.195315	+	0.980741i	$0.562573\pi$
$510$	−10.2518	−0.453956
$511$	9.56785	0.423257
$512$	49.6093	2.19244
$513$	7.11984	0.314349
$514$	−25.9316	−1.14379
$515$	5.19353	0.228854
$516$	3.28648	0.144679
$517$	−19.8325	−0.872234
$518$	−25.2140	−1.10784
$519$	5.45299	0.239360
$520$	15.8614	0.695567
$521$	−18.1058	−0.793229	−0.396615	−	0.917985i	$-0.629815\pi$
$521$	−18.1058	−0.793229	−0.396615	+	0.917985i	$0.629815\pi$
$522$	−62.7182	−2.74510
$523$	−19.5603	−0.855311	−0.427656	−	0.903942i	$-0.640660\pi$
$523$	−19.5603	−0.855311	−0.427656	+	0.903942i	$0.640660\pi$
$524$	−17.7914	−0.777219
$525$	1.84899	0.0806966
$526$	−12.6456	−0.551373
$527$	−11.9521	−0.520643
$528$	22.0492	0.959567
$529$	−9.23981	−0.401731
$530$	−11.4695	−0.498201
$531$	26.4372	1.14728
$532$	12.9109	0.559758
$533$	−2.62821	−0.113840
$534$	−16.3432	−0.707237
$535$	−23.0734	−0.997550
$536$	−49.2922	−2.12910
$537$	−2.34347	−0.101128
$538$	30.4196	1.31148
$539$	31.0482	1.33734
$540$	−17.6571	−0.759840
$541$	−1.67235	−0.0718998	−0.0359499	−	0.999354i	$-0.511446\pi$
$541$	−1.67235	−0.0718998	−0.0359499	+	0.999354i	$0.511446\pi$
$542$	−12.8930	−0.553802
$543$	−5.50257	−0.236138
$544$	−43.8197	−1.87875
$545$	−11.2539	−0.482066
$546$	2.60765	0.111597
$547$	16.4930	0.705190	0.352595	−	0.935776i	$-0.385299\pi$
$547$	16.4930	0.705190	0.352595	+	0.935776i	$0.385299\pi$
$548$	−6.67135	−0.284986
$549$	10.8594	0.463468
$550$	−45.3585	−1.93409
$551$	−21.8826	−0.932230
$552$	12.5401	0.533742
$553$	5.96179	0.253521
$554$	46.2081	1.96319
$555$	5.75148	0.244137
$556$	10.2453	0.434499
$557$	−12.7256	−0.539202	−0.269601	−	0.962972i	$-0.586892\pi$
$557$	−12.7256	−0.539202	−0.269601	+	0.962972i	$0.586892\pi$
$558$	−14.1045	−0.597092
$559$	−2.52153	−0.106650
$560$	−12.1362	−0.512847
$561$	16.3491	0.690261
$562$	41.0667	1.73229
$563$	−4.34108	−0.182955	−0.0914774	−	0.995807i	$-0.529159\pi$
$563$	−4.34108	−0.182955	−0.0914774	+	0.995807i	$0.529159\pi$
$564$	−8.46338	−0.356372
$565$	−1.32474	−0.0557324
$566$	18.5319	0.778955
$567$	7.70418	0.323545
$568$	−6.72941	−0.282360
$569$	−26.1979	−1.09827	−0.549137	−	0.835732i	$-0.685044\pi$
$569$	−26.1979	−1.09827	−0.549137	+	0.835732i	$0.685044\pi$
$570$	−4.22102	−0.176799
$571$	−1.88802	−0.0790111	−0.0395056	−	0.999219i	$-0.512578\pi$
$571$	−1.88802	−0.0790111	−0.0395056	+	0.999219i	$0.512578\pi$
$572$	−44.6323	−1.86617
$573$	5.36380	0.224076
$574$	4.30957	0.179878
$575$	−12.0374	−0.501996
$576$	−7.32580	−0.305242
$577$	17.7994	0.740997	0.370498	−	0.928833i	$-0.379187\pi$
$577$	17.7994	0.740997	0.370498	+	0.928833i	$0.379187\pi$
$578$	−48.5311	−2.01863
$579$	−8.09811	−0.336546
$580$	54.2684	2.25337
$581$	−4.75222	−0.197155
$582$	5.26481	0.218233
$583$	18.2910	0.757537
$584$	−56.7662	−2.34900
$585$	6.47624	0.267760
$586$	13.6288	0.563000
$587$	17.8730	0.737700	0.368850	−	0.929489i	$-0.379752\pi$
$587$	17.8730	0.737700	0.368850	+	0.929489i	$0.379752\pi$
$588$	13.2495	0.546402
$589$	−4.92112	−0.202771
$590$	−32.7863	−1.34979
$591$	−5.88355	−0.242017
$592$	69.8047	2.86895
$593$	−11.2676	−0.462703	−0.231352	−	0.972870i	$-0.574315\pi$
$593$	−11.2676	−0.462703	−0.231352	+	0.972870i	$0.574315\pi$
$594$	40.3588	1.65594
$595$	−8.99879	−0.368915
$596$	−46.2996	−1.89651
$597$	4.41601	0.180735
$598$	−16.9765	−0.694221
$599$	−23.4485	−0.958080	−0.479040	−	0.877793i	$-0.659015\pi$
$599$	−23.4485	−0.958080	−0.479040	+	0.877793i	$0.659015\pi$
$600$	−10.9701	−0.447852
$601$	−11.0544	−0.450920	−0.225460	−	0.974252i	$-0.572389\pi$
$601$	−11.0544	−0.450920	−0.225460	+	0.974252i	$0.572389\pi$
$602$	4.13466	0.168516
$603$	−20.1261	−0.819600
$604$	47.8888	1.94857
$605$	−24.5476	−0.998001
$606$	24.1863	0.982500
$607$	−2.19933	−0.0892680	−0.0446340	−	0.999003i	$-0.514212\pi$
$607$	−2.19933	−0.0892680	−0.0446340	+	0.999003i	$0.514212\pi$
$608$	−18.0421	−0.731706
$609$	5.05641	0.204896
$610$	−13.4674	−0.545278
$611$	6.49349	0.262698
$612$	−75.9619	−3.07058
$613$	13.9809	0.564682	0.282341	−	0.959314i	$-0.408889\pi$
$613$	13.9809	0.564682	0.282341	+	0.959314i	$0.408889\pi$
$614$	−75.2621	−3.03733
$615$	−0.983042	−0.0396401
$616$	41.4773	1.67117
$617$	44.5063	1.79176	0.895879	−	0.444299i	$-0.146547\pi$
$617$	44.5063	1.79176	0.895879	+	0.444299i	$0.146547\pi$
$618$	−5.06579	−0.203776
$619$	−34.6387	−1.39225	−0.696123	−	0.717922i	$-0.745093\pi$
$619$	−34.6387	−1.39225	−0.696123	+	0.717922i	$0.745093\pi$
$620$	12.2043	0.490136
$621$	10.7106	0.429801
$622$	67.1529	2.69259
$623$	−14.3457	−0.574747
$624$	−7.21925	−0.289001
$625$	1.75562	0.0702249
$626$	−81.6799	−3.26458
$627$	6.73152	0.268831
$628$	81.5876	3.25570
$629$	51.7591	2.06377
$630$	−10.6193	−0.423085
$631$	−37.7492	−1.50277	−0.751387	−	0.659862i	$-0.770615\pi$
$631$	−37.7492	−1.50277	−0.751387	+	0.659862i	$0.770615\pi$
$632$	−35.3714	−1.40700
$633$	−11.0272	−0.438290
$634$	−37.2149	−1.47799
$635$	9.48117	0.376249
$636$	7.80554	0.309510
$637$	−10.1657	−0.402778
$638$	−124.041	−4.91084
$639$	−2.74764	−0.108695
$640$	−10.3005	−0.407164
$641$	−9.96961	−0.393776	−0.196888	−	0.980426i	$-0.563084\pi$
$641$	−9.96961	−0.393776	−0.196888	+	0.980426i	$0.563084\pi$
$642$	22.5059	0.888236
$643$	6.38171	0.251670	0.125835	−	0.992051i	$-0.459839\pi$
$643$	6.38171	0.251670	0.125835	+	0.992051i	$0.459839\pi$
$644$	19.4222	0.765343
$645$	−0.943142	−0.0371362
$646$	−37.9861	−1.49454
$647$	−28.3292	−1.11374	−0.556868	−	0.830601i	$-0.687997\pi$
$647$	−28.3292	−1.11374	−0.556868	+	0.830601i	$0.687997\pi$
$648$	−45.7091	−1.79562
$649$	52.2863	2.05241
$650$	14.8511	0.582507
$651$	1.13712	0.0445674
$652$	−88.4229	−3.46291
$653$	−1.92362	−0.0752773	−0.0376386	−	0.999291i	$-0.511984\pi$
$653$	−1.92362	−0.0752773	−0.0376386	+	0.999291i	$0.511984\pi$
$654$	10.9771	0.429240
$655$	5.10570	0.199496
$656$	−11.9310	−0.465827
$657$	−23.1778	−0.904252
$658$	−10.6476	−0.415087
$659$	35.5539	1.38498	0.692491	−	0.721427i	$-0.256513\pi$
$659$	35.5539	1.38498	0.692491	+	0.721427i	$0.256513\pi$
$660$	−16.6940	−0.649815
$661$	10.4814	0.407679	0.203840	−	0.979004i	$-0.434658\pi$
$661$	10.4814	0.407679	0.203840	+	0.979004i	$0.434658\pi$
$662$	−46.5962	−1.81101
$663$	−5.35296	−0.207892
$664$	28.1950	1.09418
$665$	−3.70513	−0.143679
$666$	61.0801	2.36681
$667$	−32.9186	−1.27461
$668$	−89.1718	−3.45016
$669$	−0.482187	−0.0186424
$670$	24.9596	0.964273
$671$	21.4772	0.829119
$672$	4.16900	0.160823
$673$	−43.2801	−1.66832	−0.834162	−	0.551519i	$-0.814048\pi$
$673$	−43.2801	−1.66832	−0.834162	+	0.551519i	$0.814048\pi$
$674$	67.1594	2.58688
$675$	−9.36964	−0.360638
$676$	−45.3974	−1.74605
$677$	−43.9072	−1.68749	−0.843746	−	0.536743i	$-0.819655\pi$
$677$	−43.9072	−1.68749	−0.843746	+	0.536743i	$0.819655\pi$
$678$	1.29216	0.0496251
$679$	4.62134	0.177351
$680$	53.3900	2.04741
$681$	3.09195	0.118484
$682$	−27.8953	−1.06817
$683$	−35.0390	−1.34073	−0.670366	−	0.742031i	$-0.733863\pi$
$683$	−35.0390	−1.34073	−0.670366	+	0.742031i	$0.733863\pi$
$684$	−31.2762	−1.19588
$685$	1.91452	0.0731501
$686$	37.0913	1.41615
$687$	6.07797	0.231889
$688$	−11.4467	−0.436403
$689$	−5.98877	−0.228154
$690$	−6.34981	−0.241733
$691$	5.83488	0.221969	0.110985	−	0.993822i	$-0.464600\pi$
$691$	5.83488	0.221969	0.110985	+	0.993822i	$0.464600\pi$
$692$	−50.1082	−1.90483
$693$	16.9353	0.643319
$694$	33.8393	1.28452
$695$	−2.94017	−0.111527
$696$	−29.9998	−1.13714
$697$	−8.84665	−0.335091
$698$	14.0040	0.530061
$699$	−4.98607	−0.188590
$700$	−16.9906	−0.642184
$701$	−0.0120259	−0.000454214 0	−0.000227107	−	1.00000i	$-0.500072\pi$
$701$	−0.0120259	−0.000454214 0	−0.000227107		1.00000i	$0.500072\pi$
$702$	−13.2141	−0.498734
$703$	21.3111	0.803762
$704$	−14.4886	−0.546061
$705$	2.42879	0.0914736
$706$	88.4638	3.32938
$707$	21.2302	0.798443
$708$	22.3127	0.838564
$709$	−43.7112	−1.64161	−0.820805	−	0.571208i	$-0.806475\pi$
$709$	−43.7112	−1.64161	−0.820805	+	0.571208i	$0.806475\pi$
$710$	3.40751	0.127881
$711$	−14.4422	−0.541627
$712$	85.1131	3.18975
$713$	−7.40298	−0.277244
$714$	8.77745	0.328488
$715$	12.8084	0.479008
$716$	21.5344	0.804779
$717$	0.955728	0.0356923
$718$	−40.0593	−1.49500
$719$	−10.1606	−0.378925	−0.189463	−	0.981888i	$-0.560675\pi$
$719$	−10.1606	−0.378925	−0.189463	+	0.981888i	$0.560675\pi$
$720$	29.3995	1.09566
$721$	−4.44664	−0.165602
$722$	33.2316	1.23675
$723$	1.43095	0.0532175
$724$	50.5638	1.87919
$725$	28.7973	1.06950
$726$	23.9438	0.888637
$727$	−35.1559	−1.30386	−0.651929	−	0.758280i	$-0.726040\pi$
$727$	−35.1559	−1.30386	−0.651929	+	0.758280i	$0.726040\pi$
$728$	−13.5803	−0.503320
$729$	−14.3117	−0.530063
$730$	28.7442	1.06387
$731$	−8.48758	−0.313925
$732$	9.16523	0.338757
$733$	17.6102	0.650449	0.325224	−	0.945637i	$-0.394560\pi$
$733$	17.6102	0.650449	0.325224	+	0.945637i	$0.394560\pi$
$734$	−44.0258	−1.62502
$735$	−3.80231	−0.140250
$736$	−27.1413	−1.00044
$737$	−39.8046	−1.46622
$738$	−10.4398	−0.384294
$739$	−15.5642	−0.572540	−0.286270	−	0.958149i	$-0.592415\pi$
$739$	−15.5642	−0.572540	−0.286270	+	0.958149i	$0.592415\pi$
$740$	−52.8510	−1.94284
$741$	−2.20401	−0.0809661
$742$	9.82001	0.360504
$743$	7.54561	0.276822	0.138411	−	0.990375i	$-0.455801\pi$
$743$	7.54561	0.276822	0.138411	+	0.990375i	$0.455801\pi$
$744$	−6.74657	−0.247341
$745$	13.2869	0.486795
$746$	−34.7456	−1.27213
$747$	11.5121	0.421206
$748$	−150.234	−5.49310
$749$	19.7552	0.721839
$750$	14.1137	0.515361
$751$	−29.6148	−1.08066	−0.540329	−	0.841454i	$-0.681700\pi$
$751$	−29.6148	−1.08066	−0.540329	+	0.841454i	$0.681700\pi$
$752$	29.4778	1.07494
$753$	−11.3378	−0.413173
$754$	40.6131	1.47904
$755$	−13.7430	−0.500158
$756$	15.1178	0.549828
$757$	10.9691	0.398678	0.199339	−	0.979931i	$-0.436121\pi$
$757$	10.9691	0.398678	0.199339	+	0.979931i	$0.436121\pi$
$758$	70.4990	2.56064
$759$	10.1264	0.367566
$760$	21.9826	0.797392
$761$	29.3096	1.06247	0.531236	−	0.847224i	$-0.321728\pi$
$761$	29.3096	1.06247	0.531236	+	0.847224i	$0.321728\pi$
$762$	−9.24796	−0.335018
$763$	9.63549	0.348828
$764$	−49.2886	−1.78320
$765$	21.7993	0.788155
$766$	−34.9729	−1.26362
$767$	−17.1193	−0.618143
$768$	12.7259	0.459208
$769$	15.8247	0.570654	0.285327	−	0.958430i	$-0.407898\pi$
$769$	15.8247	0.570654	0.285327	+	0.958430i	$0.407898\pi$
$770$	−21.0025	−0.756876
$771$	−5.06449	−0.182393
$772$	74.4145	2.67824
$773$	54.1301	1.94692	0.973462	−	0.228848i	$-0.0734960\pi$
$773$	54.1301	1.94692	0.973462	+	0.228848i	$0.0734960\pi$
$774$	−10.0161	−0.360020
$775$	6.47614	0.232630
$776$	−27.4185	−0.984266
$777$	−4.92435	−0.176660
$778$	−11.0977	−0.397871
$779$	−3.64248	−0.130505
$780$	5.46589	0.195710
$781$	−5.43416	−0.194449
$782$	−57.1436	−2.04345
$783$	−25.6230	−0.915693
$784$	−46.1480	−1.64814
$785$	−23.4137	−0.835672
$786$	−4.98012	−0.177635
$787$	19.2985	0.687916	0.343958	−	0.938985i	$-0.388232\pi$
$787$	19.2985	0.687916	0.343958	+	0.938985i	$0.388232\pi$
$788$	54.0646	1.92597
$789$	−2.46971	−0.0879239
$790$	17.9107	0.637233
$791$	1.13423	0.0403286
$792$	−100.477	−3.57031
$793$	−7.03198	−0.249713
$794$	1.77049	0.0628322
$795$	−2.24001	−0.0794449
$796$	−40.5792	−1.43829
$797$	−31.7830	−1.12581	−0.562905	−	0.826522i	$-0.690316\pi$
$797$	−31.7830	−1.12581	−0.562905	+	0.826522i	$0.690316\pi$
$798$	3.61399	0.127934
$799$	21.8573	0.773257
$800$	23.7433	0.839451
$801$	34.7519	1.22790
$802$	−0.977418	−0.0345138
$803$	−45.8400	−1.61766
$804$	−16.9863	−0.599060
$805$	−5.57373	−0.196448
$806$	9.13336	0.321709
$807$	5.94100	0.209133
$808$	−125.959	−4.43123
$809$	−3.39697	−0.119431	−0.0597156	−	0.998215i	$-0.519019\pi$
$809$	−3.39697	−0.119431	−0.0597156	+	0.998215i	$0.519019\pi$
$810$	23.1453	0.813241
$811$	−1.49719	−0.0525736	−0.0262868	−	0.999654i	$-0.508368\pi$
$811$	−1.49719	−0.0525736	−0.0262868	+	0.999654i	$0.508368\pi$
$812$	−46.4640	−1.63057
$813$	−2.51803	−0.0883113
$814$	120.802	4.23409
$815$	25.3753	0.888858
$816$	−24.3003	−0.850680
$817$	−3.49464	−0.122262
$818$	49.4259	1.72814
$819$	−5.54488	−0.193754
$820$	9.03329	0.315456
$821$	−38.0695	−1.32863	−0.664317	−	0.747451i	$-0.731278\pi$
$821$	−38.0695	−1.32863	−0.664317	+	0.747451i	$0.731278\pi$
$822$	−1.86743	−0.0651341
$823$	−56.3705	−1.96495	−0.982477	−	0.186385i	$-0.940323\pi$
$823$	−56.3705	−1.96495	−0.982477	+	0.186385i	$0.940323\pi$
$824$	26.3820	0.919061
$825$	−8.85861	−0.308417
$826$	28.0712	0.976723
$827$	24.4817	0.851311	0.425656	−	0.904885i	$-0.360043\pi$
$827$	24.4817	0.851311	0.425656	+	0.904885i	$0.360043\pi$
$828$	−47.0498	−1.63509
$829$	−40.0081	−1.38954	−0.694770	−	0.719232i	$-0.744494\pi$
$829$	−40.0081	−1.38954	−0.694770	+	0.719232i	$0.744494\pi$
$830$	−14.2768	−0.495556
$831$	9.02454	0.313058
$832$	4.74381	0.164462
$833$	−34.2180	−1.18558
$834$	2.86785	0.0993056
$835$	25.5902	0.885587
$836$	−61.8567	−2.13936
$837$	−5.76230	−0.199174
$838$	−96.5911	−3.33668
$839$	−34.9156	−1.20542	−0.602709	−	0.797961i	$-0.705912\pi$
$839$	−34.9156	−1.20542	−0.602709	+	0.797961i	$0.705912\pi$
$840$	−5.07952	−0.175260
$841$	49.7515	1.71557
$842$	21.3881	0.737083
$843$	8.02041	0.276238
$844$	101.330	3.48792
$845$	13.0280	0.448177
$846$	25.7935	0.886799
$847$	21.0173	0.722164
$848$	−27.1866	−0.933591
$849$	3.61932	0.124215
$850$	49.9893	1.71462
$851$	32.0589	1.09896
$852$	−2.31898	−0.0794470
$853$	15.4819	0.530091	0.265045	−	0.964236i	$-0.414613\pi$
$853$	15.4819	0.530091	0.265045	+	0.964236i	$0.414613\pi$
$854$	11.5306	0.394569
$855$	8.97555	0.306957
$856$	−117.208	−4.00608
$857$	28.7546	0.982237	0.491118	−	0.871093i	$-0.336588\pi$
$857$	28.7546	0.982237	0.491118	+	0.871093i	$0.336588\pi$
$858$	−12.4934	−0.426517
$859$	−4.87779	−0.166428	−0.0832141	−	0.996532i	$-0.526519\pi$
$859$	−4.87779	−0.166428	−0.0832141	+	0.996532i	$0.526519\pi$
$860$	8.66664	0.295530
$861$	0.841669	0.0286840
$862$	14.3270	0.487980
$863$	−42.6990	−1.45349	−0.726745	−	0.686907i	$-0.758968\pi$
$863$	−42.6990	−1.45349	−0.726745	+	0.686907i	$0.758968\pi$
$864$	−21.1261	−0.718726
$865$	14.3799	0.488931
$866$	−16.0740	−0.546216
$867$	−9.47822	−0.321897
$868$	−10.4492	−0.354668
$869$	−28.5632	−0.968941
$870$	15.1907	0.515013
$871$	13.0326	0.441594
$872$	−57.1675	−1.93594
$873$	−11.1950	−0.378895
$874$	−23.5281	−0.795849
$875$	12.3887	0.418816
$876$	−19.5619	−0.660934
$877$	0.271415	0.00916503	0.00458252	−	0.999990i	$-0.498541\pi$
$877$	0.271415	0.00916503	0.00458252	+	0.999990i	$0.498541\pi$
$878$	−4.89186	−0.165092
$879$	2.66173	0.0897780
$880$	58.1451	1.96007
$881$	−10.8261	−0.364740	−0.182370	−	0.983230i	$-0.558377\pi$
$881$	−10.8261	−0.364740	−0.182370	+	0.983230i	$0.558377\pi$
$882$	−40.3802	−1.35967
$883$	2.46180	0.0828461	0.0414231	−	0.999142i	$-0.486811\pi$
$883$	2.46180	0.0828461	0.0414231	+	0.999142i	$0.486811\pi$
$884$	49.1890	1.65440
$885$	−6.40323	−0.215242
$886$	23.8093	0.799889
$887$	10.4973	0.352463	0.176232	−	0.984349i	$-0.443609\pi$
$887$	10.4973	0.352463	0.176232	+	0.984349i	$0.443609\pi$
$888$	29.2163	0.980434
$889$	−8.11766	−0.272258
$890$	−43.0979	−1.44465
$891$	−36.9111	−1.23657
$892$	4.43088	0.148357
$893$	8.99945	0.301155
$894$	−12.9601	−0.433451
$895$	−6.17987	−0.206570
$896$	8.81918	0.294628
$897$	−3.31555	−0.110703
$898$	77.0439	2.57099
$899$	17.7102	0.590669
$900$	41.1592	1.37197
$901$	−20.1584	−0.671575
$902$	−20.6474	−0.687482
$903$	0.807507	0.0268722
$904$	−6.72941	−0.223817
$905$	−14.5106	−0.482349
$906$	13.4049	0.445349
$907$	−6.05706	−0.201121	−0.100561	−	0.994931i	$-0.532064\pi$
$907$	−6.05706	−0.201121	−0.100561	+	0.994931i	$0.532064\pi$
$908$	−28.4123	−0.942895
$909$	−51.4295	−1.70581
$910$	6.87654	0.227955
$911$	9.79446	0.324505	0.162252	−	0.986749i	$-0.448124\pi$
$911$	9.79446	0.324505	0.162252	+	0.986749i	$0.448124\pi$
$912$	−10.0053	−0.331308
$913$	22.7681	0.753515
$914$	28.7270	0.950205
$915$	−2.63021	−0.0869520
$916$	−55.8512	−1.84538
$917$	−4.37144	−0.144358
$918$	−44.4792	−1.46803
$919$	−26.7780	−0.883325	−0.441663	−	0.897181i	$-0.645611\pi$
$919$	−26.7780	−0.883325	−0.441663	+	0.897181i	$0.645611\pi$
$920$	33.0690	1.09025
$921$	−14.6989	−0.484344
$922$	11.6802	0.384667
$923$	1.77923	0.0585640
$924$	14.2932	0.470213
$925$	−28.0451	−0.922118
$926$	−43.5671	−1.43170
$927$	10.7718	0.353794
$928$	64.9304	2.13145
$929$	−3.86898	−0.126937	−0.0634686	−	0.997984i	$-0.520216\pi$
$929$	−3.86898	−0.126937	−0.0634686	+	0.997984i	$0.520216\pi$
$930$	3.41620	0.112021
$931$	−14.0888	−0.461741
$932$	45.8176	1.50080
$933$	13.1151	0.429369
$934$	33.0404	1.08112
$935$	43.1137	1.40997
$936$	32.8979	1.07530
$937$	−41.8117	−1.36593	−0.682964	−	0.730452i	$-0.739309\pi$
$937$	−41.8117	−1.36593	−0.682964	+	0.730452i	$0.739309\pi$
$938$	−21.3701	−0.697759
$939$	−15.9523	−0.520582
$940$	−22.3185	−0.727948
$941$	6.90703	0.225163	0.112581	−	0.993643i	$-0.464088\pi$
$941$	6.90703	0.225163	0.112581	+	0.993643i	$0.464088\pi$
$942$	22.8378	0.744097
$943$	−5.47949	−0.178437
$944$	−77.7149	−2.52940
$945$	−4.33845	−0.141130
$946$	−19.8093	−0.644057
$947$	−7.27619	−0.236444	−0.118222	−	0.992987i	$-0.537719\pi$
$947$	−7.27619	−0.236444	−0.118222	+	0.992987i	$0.537719\pi$
$948$	−12.1891	−0.395884
$949$	15.0087	0.487204
$950$	20.5824	0.667781
$951$	−7.26815	−0.235686
$952$	−45.7119	−1.48153
$953$	46.0390	1.49135	0.745675	−	0.666309i	$-0.232127\pi$
$953$	46.0390	1.49135	0.745675	+	0.666309i	$0.232127\pi$
$954$	−23.7887	−0.770187
$955$	14.1447	0.457711
$956$	−8.78230	−0.284040
$957$	−24.2255	−0.783100
$958$	−46.2938	−1.49568
$959$	−1.63919	−0.0529322
$960$	1.77435	0.0572669
$961$	−27.0172	−0.871523
$962$	−39.5523	−1.27522
$963$	−47.8563	−1.54215
$964$	−13.1491	−0.423505
$965$	−21.3552	−0.687449
$966$	5.43663	0.174921
$967$	−50.8310	−1.63461	−0.817307	−	0.576203i	$-0.804534\pi$
$967$	−50.8310	−1.63461	−0.817307	+	0.576203i	$0.804534\pi$
$968$	−124.696	−4.00789
$969$	−7.41877	−0.238325
$970$	13.8836	0.445776
$971$	49.8096	1.59847	0.799233	−	0.601021i	$-0.205239\pi$
$971$	49.8096	1.59847	0.799233	+	0.601021i	$0.205239\pi$
$972$	−55.7375	−1.78778
$973$	2.51734	0.0807022
$974$	55.6253	1.78235
$975$	2.90045	0.0928886
$976$	−31.9223	−1.02181
$977$	−50.5109	−1.61599	−0.807994	−	0.589191i	$-0.799446\pi$
$977$	−50.5109	−1.61599	−0.807994	+	0.589191i	$0.799446\pi$
$978$	−24.7512	−0.791455
$979$	68.7308	2.19665
$980$	34.9399	1.11611
$981$	−23.3417	−0.745242
$982$	−46.8013	−1.49349
$983$	−25.1050	−0.800725	−0.400362	−	0.916357i	$-0.631116\pi$
$983$	−25.1050	−0.800725	−0.400362	+	0.916357i	$0.631116\pi$
$984$	−4.99364	−0.159191
$985$	−15.5153	−0.494358
$986$	136.705	4.35358
$987$	−2.07950	−0.0661913
$988$	20.2529	0.644329
$989$	−5.25709	−0.167166
$990$	50.8778	1.61700
$991$	−52.7113	−1.67443	−0.837215	−	0.546874i	$-0.815818\pi$
$991$	−52.7113	−1.67443	−0.837215	+	0.546874i	$0.815818\pi$
$992$	14.6020	0.463615
$993$	−9.10033	−0.288790
$994$	−2.91747	−0.0925365
$995$	11.6453	0.369180
$996$	9.71611	0.307867
$997$	58.0038	1.83700	0.918499	−	0.395423i	$-0.129402\pi$
$997$	58.0038	1.83700	0.918499	+	0.395423i	$0.129402\pi$
$998$	58.3772	1.84790
$999$	24.9538	0.789504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8023.2.a.b.1.12	✓	155

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8023.2.a.b.1.12	✓	155	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:	\( N \)	\(=\)	\( 8023 = 71 \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8023.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57220 q^{2} -0.502356 q^{3} +4.61621 q^{4} -1.32474 q^{5} +1.29216 q^{6} +1.13423 q^{7} -6.72941 q^{8} -2.74764 q^{9} +O(q^{10})\)	\(q-2.57220 q^{2} -0.502356 q^{3} +4.61621 q^{4} -1.32474 q^{5} +1.29216 q^{6} +1.13423 q^{7} -6.72941 q^{8} -2.74764 q^{9} +3.40751 q^{10} -5.43416 q^{11} -2.31898 q^{12} +1.77923 q^{13} -2.91747 q^{14} +0.665494 q^{15} +8.07697 q^{16} +5.98895 q^{17} +7.06747 q^{18} +2.46587 q^{19} -6.11530 q^{20} -0.569788 q^{21} +13.9777 q^{22} +3.70947 q^{23} +3.38056 q^{24} -3.24505 q^{25} -4.57653 q^{26} +2.88736 q^{27} +5.23585 q^{28} -8.87421 q^{29} -1.71178 q^{30} -1.99570 q^{31} -7.31676 q^{32} +2.72988 q^{33} -15.4048 q^{34} -1.50257 q^{35} -12.6837 q^{36} +8.64243 q^{37} -6.34270 q^{38} -0.893806 q^{39} +8.91475 q^{40} -1.47716 q^{41} +1.46561 q^{42} -1.41721 q^{43} -25.0852 q^{44} +3.63992 q^{45} -9.54150 q^{46} +3.64961 q^{47} -4.05752 q^{48} -5.71352 q^{49} +8.34692 q^{50} -3.00859 q^{51} +8.21329 q^{52} -3.36594 q^{53} -7.42687 q^{54} +7.19887 q^{55} -7.63271 q^{56} -1.23874 q^{57} +22.8262 q^{58} -9.62178 q^{59} +3.07206 q^{60} -3.95227 q^{61} +5.13333 q^{62} -3.11646 q^{63} +2.66622 q^{64} -2.35702 q^{65} -7.02180 q^{66} +7.32489 q^{67} +27.6463 q^{68} -1.86348 q^{69} +3.86490 q^{70} +1.00000 q^{71} +18.4900 q^{72} +8.43554 q^{73} -22.2301 q^{74} +1.63017 q^{75} +11.3830 q^{76} -6.16359 q^{77} +2.29905 q^{78} +5.25624 q^{79} -10.6999 q^{80} +6.79243 q^{81} +3.79955 q^{82} -4.18982 q^{83} -2.63026 q^{84} -7.93383 q^{85} +3.64534 q^{86} +4.45801 q^{87} +36.5687 q^{88} -12.6479 q^{89} -9.36260 q^{90} +2.01805 q^{91} +17.1237 q^{92} +1.00255 q^{93} -9.38752 q^{94} -3.26664 q^{95} +3.67562 q^{96} +4.07442 q^{97} +14.6963 q^{98} +14.9311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9}+O(q^{10}) \) 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9	\( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9} - 2 q^{10} - 24 q^{11} - 32 q^{12} - 62 q^{13} - 18 q^{14} - 12 q^{15} + 155 q^{16} - 129 q^{17} - 42 q^{18} - 18 q^{19} - 59 q^{20} - 45 q^{21} - 17 q^{22} - 38 q^{23} - 27 q^{24} + 129 q^{25} - 44 q^{26} - 43 q^{27} - 100 q^{28} - 52 q^{29} - 39 q^{30} - 56 q^{31} - 145 q^{32} - 126 q^{33} - q^{34} - 49 q^{35} + 131 q^{36} - 30 q^{37} - 91 q^{38} - 29 q^{39} - 5 q^{40} - 163 q^{41} - 80 q^{42} - 15 q^{43} - 118 q^{44} - 66 q^{45} + 2 q^{46} - 111 q^{47} - 89 q^{48} + 101 q^{49} - 121 q^{50} + 5 q^{51} - 111 q^{52} - 93 q^{53} - 68 q^{54} - 60 q^{55} - 27 q^{56} - 106 q^{57} + 16 q^{58} - 79 q^{59} - 103 q^{60} - 74 q^{61} - 102 q^{62} - 118 q^{63} + 175 q^{64} - 109 q^{65} + 65 q^{66} - 18 q^{67} - 346 q^{68} - 39 q^{69} + 32 q^{70} + 155 q^{71} - 203 q^{72} - 108 q^{73} - 87 q^{74} - 22 q^{75} - 16 q^{76} - 121 q^{77} - 75 q^{78} - 6 q^{79} - 136 q^{80} + 107 q^{81} - 30 q^{82} - 116 q^{83} - 5 q^{84} - 53 q^{85} + 8 q^{86} - 100 q^{87} - 43 q^{88} - 189 q^{89} - 76 q^{90} + 14 q^{91} - 99 q^{92} - 72 q^{93} + 17 q^{94} - 18 q^{95} - 50 q^{96} - 184 q^{97} - 249 q^{98} - 114 q^{99}+O(q^{100}) \) 155 * q - 21 * q^2 - 16 * q^3 + 151 * q^4 - 26 * q^5 - 10 * q^6 - 40 * q^7 - 57 * q^8 + 135 * q^9 - 2 * q^10 - 24 * q^11 - 32 * q^12 - 62 * q^13 - 18 * q^14 - 12 * q^15 + 155 * q^16 - 129 * q^17 - 42 * q^18 - 18 * q^19 - 59 * q^20 - 45 * q^21 - 17 * q^22 - 38 * q^23 - 27 * q^24 + 129 * q^25 - 44 * q^26 - 43 * q^27 - 100 * q^28 - 52 * q^29 - 39 * q^30 - 56 * q^31 - 145 * q^32 - 126 * q^33 - q^34 - 49 * q^35 + 131 * q^36 - 30 * q^37 - 91 * q^38 - 29 * q^39 - 5 * q^40 - 163 * q^41 - 80 * q^42 - 15 * q^43 - 118 * q^44 - 66 * q^45 + 2 * q^46 - 111 * q^47 - 89 * q^48 + 101 * q^49 - 121 * q^50 + 5 * q^51 - 111 * q^52 - 93 * q^53 - 68 * q^54 - 60 * q^55 - 27 * q^56 - 106 * q^57 + 16 * q^58 - 79 * q^59 - 103 * q^60 - 74 * q^61 - 102 * q^62 - 118 * q^63 + 175 * q^64 - 109 * q^65 + 65 * q^66 - 18 * q^67 - 346 * q^68 - 39 * q^69 + 32 * q^70 + 155 * q^71 - 203 * q^72 - 108 * q^73 - 87 * q^74 - 22 * q^75 - 16 * q^76 - 121 * q^77 - 75 * q^78 - 6 * q^79 - 136 * q^80 + 107 * q^81 - 30 * q^82 - 116 * q^83 - 5 * q^84 - 53 * q^85 + 8 * q^86 - 100 * q^87 - 43 * q^88 - 189 * q^89 - 76 * q^90 + 14 * q^91 - 99 * q^92 - 72 * q^93 + 17 * q^94 - 18 * q^95 - 50 * q^96 - 184 * q^97 - 249 * q^98 - 114 * q^99

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