LMFDB - Embedded newform 8047.2.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.73051 q^{2} +1.50429 q^{3} +5.45570 q^{4} -3.60561 q^{5} -4.10747 q^{6} +2.19221 q^{7} -9.43585 q^{8} -0.737125 q^{9} +O(q^{10})$	\(q-2.73051 q^{2} +1.50429 q^{3} +5.45570 q^{4} -3.60561 q^{5} -4.10747 q^{6} +2.19221 q^{7} -9.43585 q^{8} -0.737125 q^{9} +9.84515 q^{10} +3.45904 q^{11} +8.20694 q^{12} +1.00000 q^{13} -5.98585 q^{14} -5.42386 q^{15} +14.8533 q^{16} +0.177317 q^{17} +2.01273 q^{18} +2.34233 q^{19} -19.6711 q^{20} +3.29770 q^{21} -9.44496 q^{22} -6.25076 q^{23} -14.1942 q^{24} +8.00039 q^{25} -2.73051 q^{26} -5.62170 q^{27} +11.9600 q^{28} -6.36962 q^{29} +14.8099 q^{30} -0.718381 q^{31} -21.6854 q^{32} +5.20339 q^{33} -0.484166 q^{34} -7.90423 q^{35} -4.02154 q^{36} +7.99296 q^{37} -6.39577 q^{38} +1.50429 q^{39} +34.0219 q^{40} +9.73260 q^{41} -9.00443 q^{42} +4.13139 q^{43} +18.8715 q^{44} +2.65778 q^{45} +17.0678 q^{46} -2.61233 q^{47} +22.3436 q^{48} -2.19423 q^{49} -21.8452 q^{50} +0.266735 q^{51} +5.45570 q^{52} -9.07443 q^{53} +15.3501 q^{54} -12.4719 q^{55} -20.6853 q^{56} +3.52353 q^{57} +17.3923 q^{58} +4.03636 q^{59} -29.5910 q^{60} -0.809959 q^{61} +1.96155 q^{62} -1.61593 q^{63} +29.5058 q^{64} -3.60561 q^{65} -14.2079 q^{66} -13.0123 q^{67} +0.967388 q^{68} -9.40293 q^{69} +21.5826 q^{70} +10.3060 q^{71} +6.95540 q^{72} -14.3226 q^{73} -21.8249 q^{74} +12.0349 q^{75} +12.7791 q^{76} +7.58293 q^{77} -4.10747 q^{78} -6.47622 q^{79} -53.5552 q^{80} -6.24527 q^{81} -26.5750 q^{82} +5.29002 q^{83} +17.9913 q^{84} -0.639335 q^{85} -11.2808 q^{86} -9.58173 q^{87} -32.6390 q^{88} +8.97009 q^{89} -7.25711 q^{90} +2.19221 q^{91} -34.1023 q^{92} -1.08065 q^{93} +7.13300 q^{94} -8.44552 q^{95} -32.6211 q^{96} -5.00537 q^{97} +5.99138 q^{98} -2.54975 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.73051	−1.93076	−0.965382	−	0.260839i	$-0.916001\pi$
$2$	−2.73051	−1.93076	−0.965382	+	0.260839i	$0.916001\pi$
$3$	1.50429	0.868500	0.434250	−	0.900792i	$-0.357014\pi$
$3$	1.50429	0.868500	0.434250	+	0.900792i	$0.357014\pi$
$4$	5.45570	2.72785
$5$	−3.60561	−1.61248	−0.806238	−	0.591591i	$-0.798500\pi$
$5$	−3.60561	−1.61248	−0.806238	+	0.591591i	$0.798500\pi$
$6$	−4.10747	−1.67687
$7$	2.19221	0.828576	0.414288	−	0.910146i	$-0.364031\pi$
$7$	2.19221	0.828576	0.414288	+	0.910146i	$0.364031\pi$
$8$	−9.43585	−3.33608
$9$	−0.737125	−0.245708
$10$	9.84515	3.11331
$11$	3.45904	1.04294	0.521470	−	0.853270i	$-0.325384\pi$
$11$	3.45904	1.04294	0.521470	+	0.853270i	$0.325384\pi$
$12$	8.20694	2.36914
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−5.98585	−1.59979
$15$	−5.42386	−1.40043
$16$	14.8533	3.71333
$17$	0.177317	0.0430057	0.0215028	−	0.999769i	$-0.493155\pi$
$17$	0.177317	0.0430057	0.0215028	+	0.999769i	$0.493155\pi$
$18$	2.01273	0.474405
$19$	2.34233	0.537367	0.268684	−	0.963228i	$-0.413411\pi$
$19$	2.34233	0.537367	0.268684	+	0.963228i	$0.413411\pi$
$20$	−19.6711	−4.39860
$21$	3.29770	0.719618
$22$	−9.44496	−2.01367
$23$	−6.25076	−1.30337	−0.651687	−	0.758488i	$-0.725939\pi$
$23$	−6.25076	−1.30337	−0.651687	+	0.758488i	$0.725939\pi$
$24$	−14.1942	−2.89738
$25$	8.00039	1.60008
$26$	−2.73051	−0.535498
$27$	−5.62170	−1.08190
$28$	11.9600	2.26023
$29$	−6.36962	−1.18281	−0.591405	−	0.806375i	$-0.701426\pi$
$29$	−6.36962	−1.18281	−0.591405	+	0.806375i	$0.701426\pi$
$30$	14.8099	2.70391
$31$	−0.718381	−0.129025	−0.0645125	−	0.997917i	$-0.520549\pi$
$31$	−0.718381	−0.129025	−0.0645125	+	0.997917i	$0.520549\pi$
$32$	−21.6854	−3.83348
$33$	5.20339	0.905793
$34$	−0.484166	−0.0830338
$35$	−7.90423	−1.33606
$36$	−4.02154	−0.670256
$37$	7.99296	1.31403	0.657017	−	0.753875i	$-0.271818\pi$
$37$	7.99296	1.31403	0.657017	+	0.753875i	$0.271818\pi$
$38$	−6.39577	−1.03753
$39$	1.50429	0.240878
$40$	34.0219	5.37934
$41$	9.73260	1.51998	0.759989	−	0.649936i	$-0.225204\pi$
$41$	9.73260	1.51998	0.759989	+	0.649936i	$0.225204\pi$
$42$	−9.00443	−1.38941
$43$	4.13139	0.630031	0.315016	−	0.949086i	$-0.397990\pi$
$43$	4.13139	0.630031	0.315016	+	0.949086i	$0.397990\pi$
$44$	18.8715	2.84499
$45$	2.65778	0.396199
$46$	17.0678	2.51651
$47$	−2.61233	−0.381047	−0.190524	−	0.981683i	$-0.561019\pi$
$47$	−2.61233	−0.381047	−0.190524	+	0.981683i	$0.561019\pi$
$48$	22.3436	3.22502
$49$	−2.19423	−0.313462
$50$	−21.8452	−3.08937
$51$	0.266735	0.0373504
$52$	5.45570	0.756570
$53$	−9.07443	−1.24647	−0.623235	−	0.782035i	$-0.714182\pi$
$53$	−9.07443	−1.24647	−0.623235	+	0.782035i	$0.714182\pi$
$54$	15.3501	2.08889
$55$	−12.4719	−1.68172
$56$	−20.6853	−2.76419
$57$	3.52353	0.466703
$58$	17.3923	2.28373
$59$	4.03636	0.525489	0.262744	−	0.964865i	$-0.415372\pi$
$59$	4.03636	0.525489	0.262744	+	0.964865i	$0.415372\pi$
$60$	−29.5910	−3.82018
$61$	−0.809959	−0.103705	−0.0518523	−	0.998655i	$-0.516513\pi$
$61$	−0.809959	−0.103705	−0.0518523	+	0.998655i	$0.516513\pi$
$62$	1.96155	0.249117
$63$	−1.61593	−0.203588
$64$	29.5058	3.68823
$65$	−3.60561	−0.447220
$66$	−14.2079	−1.74887
$67$	−13.0123	−1.58970	−0.794852	−	0.606804i	$-0.792451\pi$
$67$	−13.0123	−1.58970	−0.794852	+	0.606804i	$0.792451\pi$
$68$	0.967388	0.117313
$69$	−9.40293	−1.13198
$70$	21.5826	2.57962
$71$	10.3060	1.22310	0.611550	−	0.791206i	$-0.290547\pi$
$71$	10.3060	1.22310	0.611550	+	0.791206i	$0.290547\pi$
$72$	6.95540	0.819702
$73$	−14.3226	−1.67634	−0.838169	−	0.545410i	$-0.816374\pi$
$73$	−14.3226	−1.67634	−0.838169	+	0.545410i	$0.816374\pi$
$74$	−21.8249	−2.53709
$75$	12.0349	1.38967
$76$	12.7791	1.46586
$77$	7.58293	0.864155
$78$	−4.10747	−0.465080
$79$	−6.47622	−0.728632	−0.364316	−	0.931275i	$-0.618697\pi$
$79$	−6.47622	−0.728632	−0.364316	+	0.931275i	$0.618697\pi$
$80$	−53.5552	−5.98765
$81$	−6.24527	−0.693919
$82$	−26.5750	−2.93472
$83$	5.29002	0.580656	0.290328	−	0.956927i	$-0.406236\pi$
$83$	5.29002	0.580656	0.290328	+	0.956927i	$0.406236\pi$
$84$	17.9913	1.96301
$85$	−0.639335	−0.0693456
$86$	−11.2808	−1.21644
$87$	−9.58173	−1.02727
$88$	−32.6390	−3.47933
$89$	8.97009	0.950827	0.475414	−	0.879762i	$-0.342298\pi$
$89$	8.97009	0.950827	0.475414	+	0.879762i	$0.342298\pi$
$90$	−7.25711	−0.764967
$91$	2.19221	0.229806
$92$	−34.1023	−3.55541
$93$	−1.08065	−0.112058
$94$	7.13300	0.735713
$95$	−8.44552	−0.866492
$96$	−32.6211	−3.32938
$97$	−5.00537	−0.508219	−0.254109	−	0.967176i	$-0.581782\pi$
$97$	−5.00537	−0.508219	−0.254109	+	0.967176i	$0.581782\pi$
$98$	5.99138	0.605221
$99$	−2.54975	−0.256259
$100$	43.6478	4.36478
$101$	5.68847	0.566023	0.283012	−	0.959116i	$-0.408666\pi$
$101$	5.68847	0.566023	0.283012	+	0.959116i	$0.408666\pi$
$102$	−0.728324	−0.0721148
$103$	18.8027	1.85268	0.926342	−	0.376683i	$-0.122935\pi$
$103$	18.8027	1.85268	0.926342	+	0.376683i	$0.122935\pi$
$104$	−9.43585	−0.925261
$105$	−11.8902	−1.16037
$106$	24.7779	2.40664
$107$	−0.517006	−0.0499809	−0.0249905	−	0.999688i	$-0.507956\pi$
$107$	−0.517006	−0.0499809	−0.0249905	+	0.999688i	$0.507956\pi$
$108$	−30.6704	−2.95126
$109$	−14.8345	−1.42089	−0.710445	−	0.703753i	$-0.751506\pi$
$109$	−14.8345	−1.42089	−0.710445	+	0.703753i	$0.751506\pi$
$110$	34.0548	3.24700
$111$	12.0237	1.14124
$112$	32.5615	3.07677
$113$	−13.7771	−1.29604	−0.648022	−	0.761621i	$-0.724404\pi$
$113$	−13.7771	−1.29604	−0.648022	+	0.761621i	$0.724404\pi$
$114$	−9.62106	−0.901095
$115$	22.5378	2.10166
$116$	−34.7508	−3.22653
$117$	−0.737125	−0.0681472
$118$	−11.0213	−1.01460
$119$	0.388715	0.0356335
$120$	51.1787	4.67196
$121$	0.964964	0.0877240
$122$	2.21160	0.200229
$123$	14.6406	1.32010
$124$	−3.91927	−0.351961
$125$	−10.8182	−0.967612
$126$	4.41232	0.393081
$127$	12.8063	1.13637	0.568186	−	0.822900i	$-0.307646\pi$
$127$	12.8063	1.13637	0.568186	+	0.822900i	$0.307646\pi$
$128$	−37.1951	−3.28761
$129$	6.21479	0.547182
$130$	9.84515	0.863477
$131$	−17.8791	−1.56211	−0.781054	−	0.624463i	$-0.785318\pi$
$131$	−17.8791	−1.56211	−0.781054	+	0.624463i	$0.785318\pi$
$132$	28.3881	2.47087
$133$	5.13487	0.445250
$134$	35.5302	3.06934
$135$	20.2696	1.74453
$136$	−1.67314	−0.143470
$137$	−6.40781	−0.547456	−0.273728	−	0.961807i	$-0.588257\pi$
$137$	−6.40781	−0.547456	−0.273728	+	0.961807i	$0.588257\pi$
$138$	25.6748	2.18559
$139$	−16.4046	−1.39142	−0.695712	−	0.718321i	$-0.744911\pi$
$139$	−16.4046	−1.39142	−0.695712	+	0.718321i	$0.744911\pi$
$140$	−43.1231	−3.64457
$141$	−3.92969	−0.330940
$142$	−28.1407	−2.36152
$143$	3.45904	0.289260
$144$	−10.9487	−0.912395
$145$	22.9663	1.90725
$146$	39.1082	3.23662
$147$	−3.30075	−0.272241
$148$	43.6072	3.58449
$149$	7.34029	0.601340	0.300670	−	0.953728i	$-0.402790\pi$
$149$	7.34029	0.601340	0.300670	+	0.953728i	$0.402790\pi$
$150$	−32.8614	−2.68312
$151$	0.130754	0.0106406	0.00532032	−	0.999986i	$-0.498306\pi$
$151$	0.130754	0.0106406	0.00532032	+	0.999986i	$0.498306\pi$
$152$	−22.1019	−1.79270
$153$	−0.130705	−0.0105668
$154$	−20.7053	−1.66848
$155$	2.59020	0.208050
$156$	8.20694	0.657081
$157$	0.163004	0.0130091	0.00650457	−	0.999979i	$-0.497930\pi$
$157$	0.163004	0.0130091	0.00650457	+	0.999979i	$0.497930\pi$
$158$	17.6834	1.40682
$159$	−13.6505	−1.08256
$160$	78.1892	6.18140
$161$	−13.7030	−1.07994
$162$	17.0528	1.33979
$163$	6.00292	0.470185	0.235093	−	0.971973i	$-0.424461\pi$
$163$	6.00292	0.470185	0.235093	+	0.971973i	$0.424461\pi$
$164$	53.0982	4.14627
$165$	−18.7614	−1.46057
$166$	−14.4445	−1.12111
$167$	−1.03941	−0.0804318	−0.0402159	−	0.999191i	$-0.512805\pi$
$167$	−1.03941	−0.0804318	−0.0402159	+	0.999191i	$0.512805\pi$
$168$	−31.1166	−2.40070
$169$	1.00000	0.0769231
$170$	1.74571	0.133890
$171$	−1.72659	−0.132036
$172$	22.5397	1.71863
$173$	−2.31995	−0.176383	−0.0881914	−	0.996104i	$-0.528109\pi$
$173$	−2.31995	−0.176383	−0.0881914	+	0.996104i	$0.528109\pi$
$174$	26.1630	1.98342
$175$	17.5385	1.32579
$176$	51.3782	3.87278
$177$	6.07183	0.456387
$178$	−24.4929	−1.83582
$179$	12.6170	0.943037	0.471519	−	0.881856i	$-0.343706\pi$
$179$	12.6170	0.943037	0.471519	+	0.881856i	$0.343706\pi$
$180$	14.5001	1.08077
$181$	24.9892	1.85743	0.928716	−	0.370793i	$-0.120914\pi$
$181$	24.9892	1.85743	0.928716	+	0.370793i	$0.120914\pi$
$182$	−5.98585	−0.443701
$183$	−1.21841	−0.0900675
$184$	58.9813	4.34816
$185$	−28.8195	−2.11885
$186$	2.95073	0.216358
$187$	0.613346	0.0448523
$188$	−14.2521	−1.03944
$189$	−12.3239	−0.896434
$190$	23.0606	1.67299
$191$	−3.35168	−0.242519	−0.121259	−	0.992621i	$-0.538693\pi$
$191$	−3.35168	−0.242519	−0.121259	+	0.992621i	$0.538693\pi$
$192$	44.3852	3.20322
$193$	10.8865	0.783625	0.391813	−	0.920045i	$-0.371848\pi$
$193$	10.8865	0.783625	0.391813	+	0.920045i	$0.371848\pi$
$194$	13.6672	0.981250
$195$	−5.42386	−0.388411
$196$	−11.9711	−0.855077
$197$	−10.7451	−0.765556	−0.382778	−	0.923840i	$-0.625033\pi$
$197$	−10.7451	−0.765556	−0.382778	+	0.923840i	$0.625033\pi$
$198$	6.96212	0.494776
$199$	−17.1247	−1.21394	−0.606970	−	0.794725i	$-0.707615\pi$
$199$	−17.1247	−1.21394	−0.606970	+	0.794725i	$0.707615\pi$
$200$	−75.4905	−5.33798
$201$	−19.5742	−1.38066
$202$	−15.5324	−1.09286
$203$	−13.9635	−0.980047
$204$	1.45523	0.101886
$205$	−35.0919	−2.45093
$206$	−51.3410	−3.57710
$207$	4.60759	0.320250
$208$	14.8533	1.02989
$209$	8.10222	0.560442
$210$	32.4664	2.24039
$211$	7.75204	0.533673	0.266836	−	0.963742i	$-0.414022\pi$
$211$	7.75204	0.533673	0.266836	+	0.963742i	$0.414022\pi$
$212$	−49.5074	−3.40018
$213$	15.5032	1.06226
$214$	1.41169	0.0965014
$215$	−14.8962	−1.01591
$216$	53.0455	3.60929
$217$	−1.57484	−0.106907
$218$	40.5059	2.74340
$219$	−21.5453	−1.45590
$220$	−68.0432	−4.58747
$221$	0.177317	0.0119276
$222$	−32.8309	−2.20346
$223$	6.86594	0.459778	0.229889	−	0.973217i	$-0.426164\pi$
$223$	6.86594	0.459778	0.229889	+	0.973217i	$0.426164\pi$
$224$	−47.5390	−3.17633
$225$	−5.89729	−0.393153
$226$	37.6187	2.50236
$227$	−18.3671	−1.21906	−0.609532	−	0.792761i	$-0.708643\pi$
$227$	−18.3671	−1.21906	−0.609532	+	0.792761i	$0.708643\pi$
$228$	19.2234	1.27310
$229$	22.0981	1.46029	0.730143	−	0.683295i	$-0.239454\pi$
$229$	22.0981	1.46029	0.730143	+	0.683295i	$0.239454\pi$
$230$	−61.5397	−4.05781
$231$	11.4069	0.750518
$232$	60.1028	3.94594
$233$	0.429269	0.0281223	0.0140612	−	0.999901i	$-0.495524\pi$
$233$	0.429269	0.0281223	0.0140612	+	0.999901i	$0.495524\pi$
$234$	2.01273	0.131576
$235$	9.41903	0.614430
$236$	22.0212	1.43346
$237$	−9.74209	−0.632817
$238$	−1.06139	−0.0687998
$239$	27.2498	1.76265	0.881323	−	0.472515i	$-0.156654\pi$
$239$	27.2498	1.76265	0.881323	+	0.472515i	$0.156654\pi$
$240$	−80.5622	−5.20027
$241$	−25.1285	−1.61867	−0.809335	−	0.587347i	$-0.800172\pi$
$241$	−25.1285	−1.61867	−0.809335	+	0.587347i	$0.800172\pi$
$242$	−2.63485	−0.169374
$243$	7.47044	0.479229
$244$	−4.41890	−0.282891
$245$	7.91153	0.505449
$246$	−39.9764	−2.54880
$247$	2.34233	0.149039
$248$	6.77853	0.430437
$249$	7.95771	0.504299
$250$	29.5393	1.86823
$251$	5.41024	0.341491	0.170746	−	0.985315i	$-0.445382\pi$
$251$	5.41024	0.341491	0.170746	+	0.985315i	$0.445382\pi$
$252$	−8.81604	−0.555358
$253$	−21.6216	−1.35934
$254$	−34.9677	−2.19407
$255$	−0.961742	−0.0602266
$256$	42.5501	2.65938
$257$	−28.3502	−1.76844	−0.884218	−	0.467075i	$-0.845308\pi$
$257$	−28.3502	−1.76844	−0.884218	+	0.467075i	$0.845308\pi$
$258$	−16.9696	−1.05648
$259$	17.5222	1.08878
$260$	−19.6711	−1.21995
$261$	4.69521	0.290626
$262$	48.8193	3.01606
$263$	14.8441	0.915325	0.457663	−	0.889126i	$-0.348687\pi$
$263$	14.8441	0.915325	0.457663	+	0.889126i	$0.348687\pi$
$264$	−49.0984	−3.02179
$265$	32.7188	2.00990
$266$	−14.0208	−0.859673
$267$	13.4936	0.825793
$268$	−70.9912	−4.33648
$269$	11.2264	0.684484	0.342242	−	0.939612i	$-0.388814\pi$
$269$	11.2264	0.684484	0.342242	+	0.939612i	$0.388814\pi$
$270$	−55.3465	−3.36828
$271$	−4.31255	−0.261969	−0.130984	−	0.991384i	$-0.541814\pi$
$271$	−4.31255	−0.261969	−0.130984	+	0.991384i	$0.541814\pi$
$272$	2.63374	0.159694
$273$	3.29770	0.199586
$274$	17.4966	1.05701
$275$	27.6737	1.66879
$276$	−51.2996	−3.08787
$277$	−12.7004	−0.763095	−0.381547	−	0.924349i	$-0.624609\pi$
$277$	−12.7004	−0.763095	−0.381547	+	0.924349i	$0.624609\pi$
$278$	44.7931	2.68651
$279$	0.529537	0.0317025
$280$	74.5831	4.45719
$281$	−22.8079	−1.36060	−0.680302	−	0.732932i	$-0.738151\pi$
$281$	−22.8079	−1.36060	−0.680302	+	0.732932i	$0.738151\pi$
$282$	10.7301	0.638966
$283$	−11.7465	−0.698258	−0.349129	−	0.937075i	$-0.613522\pi$
$283$	−11.7465	−0.698258	−0.349129	+	0.937075i	$0.613522\pi$
$284$	56.2266	3.33643
$285$	−12.7045	−0.752548
$286$	−9.44496	−0.558492
$287$	21.3359	1.25942
$288$	15.9849	0.941919
$289$	−16.9686	−0.998151
$290$	−62.7099	−3.68245
$291$	−7.52951	−0.441388
$292$	−78.1401	−4.57280
$293$	4.23620	0.247481	0.123741	−	0.992315i	$-0.460511\pi$
$293$	4.23620	0.247481	0.123741	+	0.992315i	$0.460511\pi$
$294$	9.01275	0.525634
$295$	−14.5535	−0.847338
$296$	−75.4204	−4.38372
$297$	−19.4457	−1.12835
$298$	−20.0428	−1.16105
$299$	−6.25076	−0.361491
$300$	65.6587	3.79081
$301$	9.05686	0.522029
$302$	−0.357027	−0.0205446
$303$	8.55708	0.491591
$304$	34.7913	1.99542
$305$	2.92039	0.167221
$306$	0.356891	0.0204021
$307$	−13.7975	−0.787463	−0.393731	−	0.919226i	$-0.628816\pi$
$307$	−13.7975	−0.787463	−0.393731	+	0.919226i	$0.628816\pi$
$308$	41.3702	2.35729
$309$	28.2846	1.60906
$310$	−7.07257	−0.401695
$311$	2.50210	0.141881	0.0709404	−	0.997481i	$-0.477400\pi$
$311$	2.50210	0.141881	0.0709404	+	0.997481i	$0.477400\pi$
$312$	−14.1942	−0.803589
$313$	4.99839	0.282526	0.141263	−	0.989972i	$-0.454884\pi$
$313$	4.99839	0.282526	0.141263	+	0.989972i	$0.454884\pi$
$314$	−0.445085	−0.0251176
$315$	5.82641	0.328281
$316$	−35.3324	−1.98760
$317$	−17.6102	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$317$	−17.6102	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$318$	37.2730	2.09017
$319$	−22.0328	−1.23360
$320$	−106.386	−5.94718
$321$	−0.777725	−0.0434084
$322$	37.4161	2.08512
$323$	0.415335	0.0231098
$324$	−34.0724	−1.89291
$325$	8.00039	0.443782
$326$	−16.3911	−0.907817
$327$	−22.3154	−1.23404
$328$	−91.8354	−5.07076
$329$	−5.72676	−0.315727
$330$	51.2281	2.82002
$331$	−33.4265	−1.83729	−0.918644	−	0.395086i	$-0.870715\pi$
$331$	−33.4265	−1.83729	−0.918644	+	0.395086i	$0.870715\pi$
$332$	28.8608	1.58394
$333$	−5.89181	−0.322869
$334$	2.83812	0.155295
$335$	46.9171	2.56336
$336$	48.9818	2.67218
$337$	−27.3565	−1.49020	−0.745102	−	0.666951i	$-0.767599\pi$
$337$	−27.3565	−1.49020	−0.745102	+	0.666951i	$0.767599\pi$
$338$	−2.73051	−0.148520
$339$	−20.7248	−1.12561
$340$	−3.48802	−0.189165
$341$	−2.48491	−0.134565
$342$	4.71448	0.254930
$343$	−20.1557	−1.08830
$344$	−38.9832	−2.10183
$345$	33.9033	1.82529
$346$	6.33467	0.340554
$347$	−3.23651	−0.173745	−0.0868725	−	0.996219i	$-0.527687\pi$
$347$	−3.23651	−0.173745	−0.0868725	+	0.996219i	$0.527687\pi$
$348$	−52.2751	−2.80224
$349$	−5.88567	−0.315052	−0.157526	−	0.987515i	$-0.550352\pi$
$349$	−5.88567	−0.315052	−0.157526	+	0.987515i	$0.550352\pi$
$350$	−47.8891	−2.55978
$351$	−5.62170	−0.300064
$352$	−75.0109	−3.99809
$353$	−5.87941	−0.312930	−0.156465	−	0.987684i	$-0.550010\pi$
$353$	−5.87941	−0.312930	−0.156465	+	0.987684i	$0.550010\pi$
$354$	−16.5792	−0.881176
$355$	−37.1594	−1.97222
$356$	48.9381	2.59372
$357$	0.584739	0.0309476
$358$	−34.4508	−1.82078
$359$	20.2716	1.06989	0.534946	−	0.844886i	$-0.320332\pi$
$359$	20.2716	1.06989	0.534946	+	0.844886i	$0.320332\pi$
$360$	−25.0784	−1.32175
$361$	−13.5135	−0.711236
$362$	−68.2333	−3.58626
$363$	1.45158	0.0761883
$364$	11.9600	0.626876
$365$	51.6418	2.70306
$366$	3.32689	0.173899
$367$	−7.08194	−0.369674	−0.184837	−	0.982769i	$-0.559176\pi$
$367$	−7.08194	−0.369674	−0.184837	+	0.982769i	$0.559176\pi$
$368$	−92.8445	−4.83985
$369$	−7.17415	−0.373471
$370$	78.6919	4.09100
$371$	−19.8930	−1.03279
$372$	−5.89571	−0.305678
$373$	3.30909	0.171338	0.0856691	−	0.996324i	$-0.472697\pi$
$373$	3.30909	0.171338	0.0856691	+	0.996324i	$0.472697\pi$
$374$	−1.67475	−0.0865993
$375$	−16.2737	−0.840370
$376$	24.6495	1.27120
$377$	−6.36962	−0.328052
$378$	33.6507	1.73080
$379$	31.5820	1.62226	0.811129	−	0.584868i	$-0.198854\pi$
$379$	31.5820	1.62226	0.811129	+	0.584868i	$0.198854\pi$
$380$	−46.0763	−2.36366
$381$	19.2643	0.986938
$382$	9.15180	0.468247
$383$	12.0317	0.614792	0.307396	−	0.951582i	$-0.400542\pi$
$383$	12.0317	0.614792	0.307396	+	0.951582i	$0.400542\pi$
$384$	−55.9521	−2.85529
$385$	−27.3411	−1.39343
$386$	−29.7256	−1.51300
$387$	−3.04535	−0.154804
$388$	−27.3078	−1.38635
$389$	25.1996	1.27767	0.638834	−	0.769344i	$-0.279417\pi$
$389$	25.1996	1.27767	0.638834	+	0.769344i	$0.279417\pi$
$390$	14.8099	0.749930
$391$	−1.10837	−0.0560525
$392$	20.7044	1.04573
$393$	−26.8953	−1.35669
$394$	29.3396	1.47811
$395$	23.3507	1.17490
$396$	−13.9107	−0.699037
$397$	21.2739	1.06771	0.533854	−	0.845576i	$-0.320743\pi$
$397$	21.2739	1.06771	0.533854	+	0.845576i	$0.320743\pi$
$398$	46.7593	2.34383
$399$	7.72431	0.386699
$400$	118.832	5.94161
$401$	18.1041	0.904076	0.452038	−	0.891999i	$-0.350697\pi$
$401$	18.1041	0.904076	0.452038	+	0.891999i	$0.350697\pi$
$402$	53.4476	2.66572
$403$	−0.718381	−0.0357851
$404$	31.0346	1.54403
$405$	22.5180	1.11893
$406$	38.1276	1.89224
$407$	27.6480	1.37046
$408$	−2.51687	−0.124604
$409$	−11.1956	−0.553586	−0.276793	−	0.960930i	$-0.589272\pi$
$409$	−11.1956	−0.553586	−0.276793	+	0.960930i	$0.589272\pi$
$410$	95.8190	4.73216
$411$	−9.63917	−0.475465
$412$	102.582	5.05385
$413$	8.84853	0.435408
$414$	−12.5811	−0.618327
$415$	−19.0737	−0.936293
$416$	−21.6854	−1.06322
$417$	−24.6773	−1.20845
$418$	−22.1232	−1.08208
$419$	−21.3250	−1.04179	−0.520897	−	0.853619i	$-0.674403\pi$
$419$	−21.3250	−1.04179	−0.520897	+	0.853619i	$0.674403\pi$
$420$	−64.8695	−3.16531
$421$	−22.5343	−1.09825	−0.549126	−	0.835739i	$-0.685039\pi$
$421$	−22.5343	−1.09825	−0.549126	+	0.835739i	$0.685039\pi$
$422$	−21.1671	−1.03040
$423$	1.92561	0.0936265
$424$	85.6250	4.15832
$425$	1.41860	0.0688124
$426$	−42.3317	−2.05098
$427$	−1.77560	−0.0859272
$428$	−2.82063	−0.136341
$429$	5.20339	0.251222
$430$	40.6742	1.96148
$431$	−18.0048	−0.867261	−0.433630	−	0.901091i	$-0.642768\pi$
$431$	−18.0048	−0.867261	−0.433630	+	0.901091i	$0.642768\pi$
$432$	−83.5009	−4.01744
$433$	19.3407	0.929456	0.464728	−	0.885453i	$-0.346152\pi$
$433$	19.3407	0.929456	0.464728	+	0.885453i	$0.346152\pi$
$434$	4.30012	0.206412
$435$	34.5479	1.65645
$436$	−80.9328	−3.87598
$437$	−14.6414	−0.700391
$438$	58.8299	2.81100
$439$	−33.9153	−1.61869	−0.809344	−	0.587335i	$-0.800177\pi$
$439$	−33.9153	−1.61869	−0.809344	+	0.587335i	$0.800177\pi$
$440$	117.683	5.61033
$441$	1.61742	0.0770202
$442$	−0.484166	−0.0230294
$443$	3.94013	0.187201	0.0936005	−	0.995610i	$-0.470162\pi$
$443$	3.94013	0.187201	0.0936005	+	0.995610i	$0.470162\pi$
$444$	65.5977	3.11313
$445$	−32.3426	−1.53319
$446$	−18.7476	−0.887722
$447$	11.0419	0.522264
$448$	64.6828	3.05598
$449$	5.74186	0.270975	0.135488	−	0.990779i	$-0.456740\pi$
$449$	5.74186	0.270975	0.135488	+	0.990779i	$0.456740\pi$
$450$	16.1026	0.759085
$451$	33.6655	1.58525
$452$	−75.1640	−3.53542
$453$	0.196692	0.00924140
$454$	50.1515	2.35373
$455$	−7.90423	−0.370556
$456$	−33.2475	−1.55696
$457$	10.9875	0.513972	0.256986	−	0.966415i	$-0.417271\pi$
$457$	10.9875	0.513972	0.256986	+	0.966415i	$0.417271\pi$
$458$	−60.3392	−2.81947
$459$	−0.996823	−0.0465277
$460$	122.960	5.73302
$461$	8.55511	0.398451	0.199226	−	0.979954i	$-0.436157\pi$
$461$	8.55511	0.398451	0.199226	+	0.979954i	$0.436157\pi$
$462$	−31.1467	−1.44907
$463$	3.42436	0.159143	0.0795717	−	0.996829i	$-0.474645\pi$
$463$	3.42436	0.159143	0.0795717	+	0.996829i	$0.474645\pi$
$464$	−94.6099	−4.39216
$465$	3.89640	0.180691
$466$	−1.17212	−0.0542976
$467$	−21.3397	−0.987483	−0.493742	−	0.869609i	$-0.664371\pi$
$467$	−21.3397	−0.987483	−0.493742	+	0.869609i	$0.664371\pi$
$468$	−4.02154	−0.185896
$469$	−28.5256	−1.31719
$470$	−25.7188	−1.18632
$471$	0.245205	0.0112984
$472$	−38.0865	−1.75307
$473$	14.2907	0.657085
$474$	26.6009	1.22182
$475$	18.7396	0.859830
$476$	2.12071	0.0972028
$477$	6.68899	0.306268
$478$	−74.4060	−3.40325
$479$	6.83265	0.312192	0.156096	−	0.987742i	$-0.450109\pi$
$479$	6.83265	0.312192	0.156096	+	0.987742i	$0.450109\pi$
$480$	117.619	5.36854
$481$	7.99296	0.364448
$482$	68.6138	3.12527
$483$	−20.6132	−0.937932
$484$	5.26456	0.239298
$485$	18.0474	0.819490
$486$	−20.3981	−0.925278
$487$	−5.38172	−0.243869	−0.121935	−	0.992538i	$-0.538910\pi$
$487$	−5.38172	−0.243869	−0.121935	+	0.992538i	$0.538910\pi$
$488$	7.64265	0.345967
$489$	9.03011	0.408356
$490$	−21.6026	−0.975904
$491$	21.4758	0.969189	0.484595	−	0.874739i	$-0.338967\pi$
$491$	21.4758	0.969189	0.484595	+	0.874739i	$0.338967\pi$
$492$	79.8749	3.60104
$493$	−1.12944	−0.0508675
$494$	−6.39577	−0.287759
$495$	9.19338	0.413212
$496$	−10.6703	−0.479112
$497$	22.5929	1.01343
$498$	−21.7286	−0.973683
$499$	24.0262	1.07556	0.537781	−	0.843085i	$-0.319263\pi$
$499$	24.0262	1.07556	0.537781	+	0.843085i	$0.319263\pi$
$500$	−59.0210	−2.63950
$501$	−1.56357	−0.0698550
$502$	−14.7727	−0.659339
$503$	36.0506	1.60742	0.803709	−	0.595023i	$-0.202857\pi$
$503$	36.0506	1.60742	0.803709	+	0.595023i	$0.202857\pi$
$504$	15.2477	0.679185
$505$	−20.5104	−0.912699
$506$	59.0382	2.62457
$507$	1.50429	0.0668077
$508$	69.8671	3.09985
$509$	−36.5216	−1.61879	−0.809395	−	0.587264i	$-0.800205\pi$
$509$	−36.5216	−1.61879	−0.809395	+	0.587264i	$0.800205\pi$
$510$	2.62605	0.116283
$511$	−31.3982	−1.38897
$512$	−41.7935	−1.84703
$513$	−13.1679	−0.581376
$514$	77.4106	3.41443
$515$	−67.7951	−2.98741
$516$	33.9061	1.49263
$517$	−9.03615	−0.397410
$518$	−47.8447	−2.10217
$519$	−3.48987	−0.153188
$520$	34.0219	1.49196
$521$	−32.1039	−1.40650	−0.703250	−	0.710943i	$-0.748268\pi$
$521$	−32.1039	−1.40650	−0.703250	+	0.710943i	$0.748268\pi$
$522$	−12.8203	−0.561131
$523$	11.1476	0.487451	0.243726	−	0.969844i	$-0.421630\pi$
$523$	11.1476	0.487451	0.243726	+	0.969844i	$0.421630\pi$
$524$	−97.5434	−4.26120
$525$	26.3829	1.15145
$526$	−40.5320	−1.76728
$527$	−0.127381	−0.00554881
$528$	77.2875	3.36350
$529$	16.0721	0.698785
$530$	−89.3392	−3.88065
$531$	−2.97530	−0.129117
$532$	28.0143	1.21458
$533$	9.73260	0.421566
$534$	−36.8444	−1.59441
$535$	1.86412	0.0805930
$536$	122.782	5.30337
$537$	18.9795	0.819027
$538$	−30.6538	−1.32158
$539$	−7.58994	−0.326922
$540$	110.585	4.75883
$541$	1.14779	0.0493475	0.0246738	−	0.999696i	$-0.492145\pi$
$541$	1.14779	0.0493475	0.0246738	+	0.999696i	$0.492145\pi$
$542$	11.7755	0.505800
$543$	37.5909	1.61318
$544$	−3.84520	−0.164861
$545$	53.4874	2.29115
$546$	−9.00443	−0.385354
$547$	−41.5609	−1.77702	−0.888508	−	0.458862i	$-0.848257\pi$
$547$	−41.5609	−1.77702	−0.888508	+	0.458862i	$0.848257\pi$
$548$	−34.9591	−1.49338
$549$	0.597041	0.0254811
$550$	−75.5634	−3.22203
$551$	−14.9198	−0.635603
$552$	88.7247	3.77637
$553$	−14.1972	−0.603727
$554$	34.6787	1.47336
$555$	−43.3527	−1.84022
$556$	−89.4989	−3.79560
$557$	19.7512	0.836883	0.418442	−	0.908244i	$-0.362576\pi$
$557$	19.7512	0.836883	0.418442	+	0.908244i	$0.362576\pi$
$558$	−1.44591	−0.0612101
$559$	4.13139	0.174739
$560$	−117.404	−4.96122
$561$	0.922648	0.0389542
$562$	62.2772	2.62701
$563$	−34.4356	−1.45129	−0.725644	−	0.688070i	$-0.758458\pi$
$563$	−34.4356	−1.45129	−0.725644	+	0.688070i	$0.758458\pi$
$564$	−21.4392	−0.902754
$565$	49.6749	2.08984
$566$	32.0740	1.34817
$567$	−13.6909	−0.574965
$568$	−97.2460	−4.08035
$569$	−33.5770	−1.40762	−0.703811	−	0.710387i	$-0.748520\pi$
$569$	−33.5770	−1.40762	−0.703811	+	0.710387i	$0.748520\pi$
$570$	34.6897	1.45299
$571$	−19.1520	−0.801486	−0.400743	−	0.916191i	$-0.631248\pi$
$571$	−19.1520	−0.801486	−0.400743	+	0.916191i	$0.631248\pi$
$572$	18.8715	0.789057
$573$	−5.04188	−0.210627
$574$	−58.2579	−2.43164
$575$	−50.0086	−2.08550
$576$	−21.7495	−0.906228
$577$	−27.1311	−1.12948	−0.564741	−	0.825268i	$-0.691024\pi$
$577$	−27.1311	−1.12948	−0.564741	+	0.825268i	$0.691024\pi$
$578$	46.3329	1.92719
$579$	16.3764	0.680578
$580$	125.298	5.20270
$581$	11.5968	0.481117
$582$	20.5594	0.852216
$583$	−31.3888	−1.29999
$584$	135.146	5.59239
$585$	2.65778	0.109886
$586$	−11.5670	−0.477828
$587$	−35.7984	−1.47756	−0.738780	−	0.673947i	$-0.764598\pi$
$587$	−35.7984	−1.47756	−0.738780	+	0.673947i	$0.764598\pi$
$588$	−18.0079	−0.742634
$589$	−1.68269	−0.0693338
$590$	39.7386	1.63601
$591$	−16.1637	−0.664885
$592$	118.722	4.87944
$593$	25.4766	1.04620	0.523100	−	0.852271i	$-0.324775\pi$
$593$	25.4766	1.04620	0.523100	+	0.852271i	$0.324775\pi$
$594$	53.0968	2.17859
$595$	−1.40155	−0.0574581
$596$	40.0465	1.64037
$597$	−25.7605	−1.05431
$598$	17.0678	0.697954
$599$	40.4181	1.65144	0.825720	−	0.564080i	$-0.190769\pi$
$599$	40.4181	1.65144	0.825720	+	0.564080i	$0.190769\pi$
$600$	−113.559	−4.63604
$601$	40.4371	1.64946	0.824732	−	0.565524i	$-0.191326\pi$
$601$	40.4371	1.64946	0.824732	+	0.565524i	$0.191326\pi$
$602$	−24.7299	−1.00792
$603$	9.59168	0.390603
$604$	0.713358	0.0290261
$605$	−3.47928	−0.141453
$606$	−23.3652	−0.949147
$607$	4.20442	0.170652	0.0853262	−	0.996353i	$-0.472807\pi$
$607$	4.20442	0.170652	0.0853262	+	0.996353i	$0.472807\pi$
$608$	−50.7945	−2.05999
$609$	−21.0051	−0.851171
$610$	−7.97418	−0.322865
$611$	−2.61233	−0.105684
$612$	−0.713086	−0.0288248
$613$	1.09015	0.0440309	0.0220155	−	0.999758i	$-0.492992\pi$
$613$	1.09015	0.0440309	0.0220155	+	0.999758i	$0.492992\pi$
$614$	37.6742	1.52041
$615$	−52.7883	−2.12863
$616$	−71.5514	−2.88289
$617$	4.16647	0.167736	0.0838679	−	0.996477i	$-0.473273\pi$
$617$	4.16647	0.167736	0.0838679	+	0.996477i	$0.473273\pi$
$618$	−77.2315	−3.10671
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	14.1314	0.567529
$621$	35.1399	1.41012
$622$	−6.83201	−0.273939
$623$	19.6643	0.787833
$624$	22.3436	0.894460
$625$	−0.995698	−0.0398279
$626$	−13.6482	−0.545490
$627$	12.1880	0.486744
$628$	0.889303	0.0354870
$629$	1.41729	0.0565109
$630$	−15.9091	−0.633833
$631$	−28.2352	−1.12403	−0.562014	−	0.827128i	$-0.689973\pi$
$631$	−28.2352	−1.12403	−0.562014	+	0.827128i	$0.689973\pi$
$632$	61.1087	2.43077
$633$	11.6613	0.463494
$634$	48.0849	1.90969
$635$	−46.1743	−1.83237
$636$	−74.4733	−2.95306
$637$	−2.19423	−0.0869386
$638$	60.1608	2.38179
$639$	−7.59682	−0.300526
$640$	134.111	5.30120
$641$	−36.2825	−1.43307	−0.716537	−	0.697549i	$-0.754274\pi$
$641$	−36.2825	−1.43307	−0.716537	+	0.697549i	$0.754274\pi$
$642$	2.12359	0.0838114
$643$	−11.6968	−0.461278	−0.230639	−	0.973039i	$-0.574082\pi$
$643$	−11.6968	−0.461278	−0.230639	+	0.973039i	$0.574082\pi$
$644$	−74.7593	−2.94593
$645$	−22.4081	−0.882318
$646$	−1.13408	−0.0446197
$647$	−33.9983	−1.33661	−0.668305	−	0.743887i	$-0.732980\pi$
$647$	−33.9983	−1.33661	−0.668305	+	0.743887i	$0.732980\pi$
$648$	58.9294	2.31497
$649$	13.9619	0.548054
$650$	−21.8452	−0.856838
$651$	−2.36901	−0.0928487
$652$	32.7502	1.28260
$653$	25.4578	0.996240	0.498120	−	0.867108i	$-0.334024\pi$
$653$	25.4578	0.996240	0.498120	+	0.867108i	$0.334024\pi$
$654$	60.9324	2.38264
$655$	64.4652	2.51886
$656$	144.561	5.64417
$657$	10.5576	0.411890
$658$	15.6370	0.609594
$659$	−49.2889	−1.92002	−0.960012	−	0.279959i	$-0.909679\pi$
$659$	−49.2889	−1.92002	−0.960012	+	0.279959i	$0.909679\pi$
$660$	−102.356	−3.98422
$661$	−11.1707	−0.434491	−0.217246	−	0.976117i	$-0.569707\pi$
$661$	−11.1707	−0.434491	−0.217246	+	0.976117i	$0.569707\pi$
$662$	91.2716	3.54737
$663$	0.266735	0.0103591
$664$	−49.9159	−1.93711
$665$	−18.5143	−0.717955
$666$	16.0877	0.623385
$667$	39.8150	1.54164
$668$	−5.67070	−0.219406
$669$	10.3283	0.399317
$670$	−128.108	−4.94924
$671$	−2.80168	−0.108158
$672$	−71.5122	−2.75864
$673$	1.64252	0.0633145	0.0316573	−	0.999499i	$-0.489921\pi$
$673$	1.64252	0.0633145	0.0316573	+	0.999499i	$0.489921\pi$
$674$	74.6973	2.87723
$675$	−44.9758	−1.73112
$676$	5.45570	0.209835
$677$	4.72021	0.181413	0.0907063	−	0.995878i	$-0.471088\pi$
$677$	4.72021	0.181413	0.0907063	+	0.995878i	$0.471088\pi$
$678$	56.5892	2.17330
$679$	−10.9728	−0.421098
$680$	6.03267	0.231342
$681$	−27.6293	−1.05876
$682$	6.78508	0.259814
$683$	−23.0772	−0.883026	−0.441513	−	0.897255i	$-0.645558\pi$
$683$	−23.0772	−0.883026	−0.441513	+	0.897255i	$0.645558\pi$
$684$	−9.41977	−0.360174
$685$	23.1040	0.882759
$686$	55.0353	2.10126
$687$	33.2419	1.26826
$688$	61.3648	2.33951
$689$	−9.07443	−0.345708
$690$	−92.5733	−3.52421
$691$	6.34271	0.241288	0.120644	−	0.992696i	$-0.461504\pi$
$691$	6.34271	0.241288	0.120644	+	0.992696i	$0.461504\pi$
$692$	−12.6570	−0.481146
$693$	−5.58957	−0.212330
$694$	8.83733	0.335461
$695$	59.1487	2.24364
$696$	90.4117	3.42705
$697$	1.72575	0.0653676
$698$	16.0709	0.608292
$699$	0.645743	0.0244242
$700$	95.6849	3.61655
$701$	−6.86129	−0.259147	−0.129574	−	0.991570i	$-0.541361\pi$
$701$	−6.86129	−0.259147	−0.129574	+	0.991570i	$0.541361\pi$
$702$	15.3501	0.579354
$703$	18.7222	0.706119
$704$	102.062	3.84660
$705$	14.1689	0.533632
$706$	16.0538	0.604193
$707$	12.4703	0.468993
$708$	33.1261	1.24496
$709$	−34.8672	−1.30946	−0.654732	−	0.755861i	$-0.727218\pi$
$709$	−34.8672	−1.30946	−0.654732	+	0.755861i	$0.727218\pi$
$710$	101.464	3.80789
$711$	4.77379	0.179031
$712$	−84.6404	−3.17203
$713$	4.49043	0.168168
$714$	−1.59664	−0.0597526
$715$	−12.4719	−0.466424
$716$	68.8345	2.57247
$717$	40.9915	1.53086
$718$	−55.3518	−2.06571
$719$	−18.9910	−0.708246	−0.354123	−	0.935199i	$-0.615221\pi$
$719$	−18.9910	−0.708246	−0.354123	+	0.935199i	$0.615221\pi$
$720$	39.4768	1.47122
$721$	41.2194	1.53509
$722$	36.8988	1.37323
$723$	−37.8005	−1.40581
$724$	136.334	5.06680
$725$	−50.9595	−1.89259
$726$	−3.96356	−0.147102
$727$	−17.4529	−0.647294	−0.323647	−	0.946178i	$-0.604909\pi$
$727$	−17.4529	−0.647294	−0.323647	+	0.946178i	$0.604909\pi$
$728$	−20.6853	−0.766649
$729$	29.9735	1.11013
$730$	−141.009	−5.21896
$731$	0.732566	0.0270949
$732$	−6.64729	−0.245691
$733$	14.3534	0.530154	0.265077	−	0.964227i	$-0.414603\pi$
$733$	14.3534	0.530154	0.265077	+	0.964227i	$0.414603\pi$
$734$	19.3373	0.713754
$735$	11.9012	0.438983
$736$	135.551	4.99646
$737$	−45.0100	−1.65797
$738$	19.5891	0.721085
$739$	−43.0377	−1.58317	−0.791585	−	0.611060i	$-0.790744\pi$
$739$	−43.0377	−1.58317	−0.791585	+	0.611060i	$0.790744\pi$
$740$	−157.230	−5.77991
$741$	3.52353	0.129440
$742$	54.3182	1.99408
$743$	−34.4552	−1.26404	−0.632019	−	0.774953i	$-0.717774\pi$
$743$	−34.4552	−1.26404	−0.632019	+	0.774953i	$0.717774\pi$
$744$	10.1968	0.373835
$745$	−26.4662	−0.969647
$746$	−9.03551	−0.330814
$747$	−3.89941	−0.142672
$748$	3.34624	0.122351
$749$	−1.13338	−0.0414130
$750$	44.4356	1.62256
$751$	20.4139	0.744913	0.372456	−	0.928050i	$-0.378516\pi$
$751$	20.4139	0.744913	0.372456	+	0.928050i	$0.378516\pi$
$752$	−38.8017	−1.41495
$753$	8.13855	0.296585
$754$	17.3923	0.633392
$755$	−0.471449	−0.0171578
$756$	−67.2357	−2.44534
$757$	11.2119	0.407503	0.203752	−	0.979023i	$-0.434687\pi$
$757$	11.2119	0.407503	0.203752	+	0.979023i	$0.434687\pi$
$758$	−86.2350	−3.13220
$759$	−32.5251	−1.18059
$760$	79.6906	2.89068
$761$	51.0964	1.85224	0.926122	−	0.377223i	$-0.123121\pi$
$761$	51.0964	1.85224	0.926122	+	0.377223i	$0.123121\pi$
$762$	−52.6013	−1.90555
$763$	−32.5203	−1.17731
$764$	−18.2858	−0.661555
$765$	0.471270	0.0170388
$766$	−32.8528	−1.18702
$767$	4.03636	0.145744
$768$	64.0076	2.30967
$769$	40.1807	1.44895	0.724476	−	0.689300i	$-0.242082\pi$
$769$	40.1807	1.44895	0.724476	+	0.689300i	$0.242082\pi$
$770$	74.6551	2.69038
$771$	−42.6468	−1.53589
$772$	59.3933	2.13761
$773$	−21.6166	−0.777497	−0.388748	−	0.921344i	$-0.627092\pi$
$773$	−21.6166	−0.777497	−0.388748	+	0.921344i	$0.627092\pi$
$774$	8.31538	0.298890
$775$	−5.74733	−0.206450
$776$	47.2299	1.69546
$777$	26.3584	0.945603
$778$	−68.8077	−2.46688
$779$	22.7970	0.816786
$780$	−29.5910	−1.05953
$781$	35.6489	1.27562
$782$	3.02641	0.108224
$783$	35.8081	1.27968
$784$	−32.5916	−1.16399
$785$	−0.587729	−0.0209769
$786$	73.4381	2.61945
$787$	5.33104	0.190031	0.0950156	−	0.995476i	$-0.469710\pi$
$787$	5.33104	0.190031	0.0950156	+	0.995476i	$0.469710\pi$
$788$	−58.6220	−2.08832
$789$	22.3297	0.794960
$790$	−63.7594	−2.26846
$791$	−30.2023	−1.07387
$792$	24.0590	0.854900
$793$	−0.809959	−0.0287625
$794$	−58.0888	−2.06149
$795$	49.2185	1.74560
$796$	−93.4275	−3.31145
$797$	−5.94980	−0.210753	−0.105376	−	0.994432i	$-0.533605\pi$
$797$	−5.94980	−0.210753	−0.105376	+	0.994432i	$0.533605\pi$
$798$	−21.0913	−0.746625
$799$	−0.463210	−0.0163872
$800$	−173.492	−6.13387
$801$	−6.61208	−0.233626
$802$	−49.4335	−1.74556
$803$	−49.5426	−1.74832
$804$	−106.791	−3.76623
$805$	49.4075	1.74138
$806$	1.96155	0.0690926
$807$	16.8877	0.594474
$808$	−53.6755	−1.88830
$809$	−19.2018	−0.675100	−0.337550	−	0.941308i	$-0.609598\pi$
$809$	−19.2018	−0.675100	−0.337550	+	0.941308i	$0.609598\pi$
$810$	−61.4857	−2.16039
$811$	−12.5498	−0.440683	−0.220341	−	0.975423i	$-0.570717\pi$
$811$	−12.5498	−0.440683	−0.220341	+	0.975423i	$0.570717\pi$
$812$	−76.1809	−2.67342
$813$	−6.48730	−0.227520
$814$	−75.4932	−2.64603
$815$	−21.6442	−0.758162
$816$	3.96190	0.138694
$817$	9.67709	0.338558
$818$	30.5697	1.06884
$819$	−1.61593	−0.0564652
$820$	−191.451	−6.68577
$821$	49.8121	1.73846	0.869228	−	0.494412i	$-0.164616\pi$
$821$	49.8121	1.73846	0.869228	+	0.494412i	$0.164616\pi$
$822$	26.3199	0.918011
$823$	−32.6883	−1.13944	−0.569722	−	0.821838i	$-0.692949\pi$
$823$	−32.6883	−1.13944	−0.569722	+	0.821838i	$0.692949\pi$
$824$	−177.419	−6.18070
$825$	41.6291	1.44934
$826$	−24.1610	−0.840670
$827$	8.17642	0.284322	0.142161	−	0.989844i	$-0.454595\pi$
$827$	8.17642	0.284322	0.142161	+	0.989844i	$0.454595\pi$
$828$	25.1377	0.873595
$829$	37.1214	1.28928	0.644640	−	0.764487i	$-0.277007\pi$
$829$	37.1214	1.28928	0.644640	+	0.764487i	$0.277007\pi$
$830$	52.0811	1.80776
$831$	−19.1051	−0.662747
$832$	29.5058	1.02293
$833$	−0.389074	−0.0134806
$834$	67.3816	2.33323
$835$	3.74770	0.129694
$836$	44.2033	1.52880
$837$	4.03852	0.139592
$838$	58.2282	2.01146
$839$	−51.3616	−1.77320	−0.886599	−	0.462538i	$-0.846939\pi$
$839$	−51.3616	−1.77320	−0.886599	+	0.462538i	$0.846939\pi$
$840$	112.194	3.87107
$841$	11.5721	0.399037
$842$	61.5301	2.12047
$843$	−34.3096	−1.18168
$844$	42.2928	1.45578
$845$	−3.60561	−0.124037
$846$	−5.25791	−0.180771
$847$	2.11540	0.0726860
$848$	−134.785	−4.62855
$849$	−17.6701	−0.606437
$850$	−3.87352	−0.132861
$851$	−49.9621	−1.71268
$852$	84.5808	2.89769
$853$	−49.8969	−1.70844	−0.854219	−	0.519914i	$-0.825964\pi$
$853$	−49.8969	−1.70844	−0.854219	+	0.519914i	$0.825964\pi$
$854$	4.84829	0.165905
$855$	6.22540	0.212904
$856$	4.87839	0.166740
$857$	37.5103	1.28133	0.640664	−	0.767821i	$-0.278659\pi$
$857$	37.5103	1.28133	0.640664	+	0.767821i	$0.278659\pi$
$858$	−14.2079	−0.485050
$859$	34.7413	1.18536	0.592679	−	0.805439i	$-0.298070\pi$
$859$	34.7413	1.18536	0.592679	+	0.805439i	$0.298070\pi$
$860$	−81.2691	−2.77125
$861$	32.0952	1.09380
$862$	49.1623	1.67448
$863$	−0.427901	−0.0145659	−0.00728296	−	0.999973i	$-0.502318\pi$
$863$	−0.427901	−0.0145659	−0.00728296	+	0.999973i	$0.502318\pi$
$864$	121.909	4.14743
$865$	8.36484	0.284413
$866$	−52.8101	−1.79456
$867$	−25.5256	−0.866893
$868$	−8.59186	−0.291627
$869$	−22.4015	−0.759919
$870$	−94.3336	−3.19821
$871$	−13.0123	−0.440904
$872$	139.976	4.74020
$873$	3.68958	0.124874
$874$	39.9784	1.35229
$875$	−23.7158	−0.801740
$876$	−117.545	−3.97148
$877$	−47.1146	−1.59095	−0.795473	−	0.605989i	$-0.792778\pi$
$877$	−47.1146	−1.59095	−0.795473	+	0.605989i	$0.792778\pi$
$878$	92.6061	3.12531
$879$	6.37245	0.214937
$880$	−185.249	−6.24476
$881$	−33.0147	−1.11229	−0.556147	−	0.831084i	$-0.687721\pi$
$881$	−33.0147	−1.11229	−0.556147	+	0.831084i	$0.687721\pi$
$882$	−4.41640	−0.148708
$883$	−27.4187	−0.922714	−0.461357	−	0.887215i	$-0.652637\pi$
$883$	−27.4187	−0.922714	−0.461357	+	0.887215i	$0.652637\pi$
$884$	0.967388	0.0325368
$885$	−21.8926	−0.735913
$886$	−10.7586	−0.361441
$887$	−28.6524	−0.962055	−0.481027	−	0.876706i	$-0.659736\pi$
$887$	−28.6524	−0.962055	−0.481027	+	0.876706i	$0.659736\pi$
$888$	−113.454	−3.80726
$889$	28.0739	0.941570
$890$	88.3119	2.96022
$891$	−21.6027	−0.723716
$892$	37.4586	1.25421
$893$	−6.11894	−0.204762
$894$	−30.1500	−1.00837
$895$	−45.4918	−1.52062
$896$	−81.5394	−2.72404
$897$	−9.40293	−0.313955
$898$	−15.6782	−0.523189
$899$	4.57581	0.152612
$900$	−32.1739	−1.07246
$901$	−1.60905	−0.0536052
$902$	−91.9240	−3.06074
$903$	13.6241	0.453382
$904$	129.999	4.32370
$905$	−90.1011	−2.99506
$906$	−0.537070	−0.0178430
$907$	−50.3220	−1.67091	−0.835457	−	0.549555i	$-0.814797\pi$
$907$	−50.3220	−1.67091	−0.835457	+	0.549555i	$0.814797\pi$
$908$	−100.205	−3.32543
$909$	−4.19311	−0.139077
$910$	21.5826	0.715457
$911$	−30.9739	−1.02621	−0.513106	−	0.858326i	$-0.671505\pi$
$911$	−30.9739	−1.02621	−0.513106	+	0.858326i	$0.671505\pi$
$912$	52.3361	1.73302
$913$	18.2984	0.605589
$914$	−30.0014	−0.992358
$915$	4.39311	0.145232
$916$	120.561	3.98344
$917$	−39.1948	−1.29433
$918$	2.72184	0.0898340
$919$	−16.4893	−0.543933	−0.271966	−	0.962307i	$-0.587674\pi$
$919$	−16.4893	−0.543933	−0.271966	+	0.962307i	$0.587674\pi$
$920$	−212.663	−7.01130
$921$	−20.7553	−0.683911
$922$	−23.3598	−0.769315
$923$	10.3060	0.339227
$924$	62.2326	2.04730
$925$	63.9468	2.10256
$926$	−9.35025	−0.307268
$927$	−13.8599	−0.455220
$928$	138.128	4.53428
$929$	−18.9491	−0.621701	−0.310851	−	0.950459i	$-0.600614\pi$
$929$	−18.9491	−0.621701	−0.310851	+	0.950459i	$0.600614\pi$
$930$	−10.6392	−0.348872
$931$	−5.13962	−0.168444
$932$	2.34196	0.0767135
$933$	3.76387	0.123223
$934$	58.2683	1.90660
$935$	−2.21148	−0.0723233
$936$	6.95540	0.227344
$937$	22.5115	0.735420	0.367710	−	0.929941i	$-0.380142\pi$
$937$	22.5115	0.735420	0.367710	+	0.929941i	$0.380142\pi$
$938$	77.8895	2.54318
$939$	7.51900	0.245373
$940$	51.3874	1.67607
$941$	39.3552	1.28294	0.641471	−	0.767147i	$-0.278324\pi$
$941$	39.3552	1.28294	0.641471	+	0.767147i	$0.278324\pi$
$942$	−0.669535	−0.0218146
$943$	−60.8362	−1.98110
$944$	59.9532	1.95131
$945$	44.4352	1.44548
$946$	−39.0208	−1.26868
$947$	−35.1050	−1.14076	−0.570380	−	0.821381i	$-0.693204\pi$
$947$	−35.1050	−1.14076	−0.570380	+	0.821381i	$0.693204\pi$
$948$	−53.1500	−1.72623
$949$	−14.3226	−0.464933
$950$	−51.1686	−1.66013
$951$	−26.4907	−0.859021
$952$	−3.66786	−0.118876
$953$	−32.8639	−1.06457	−0.532283	−	0.846567i	$-0.678666\pi$
$953$	−32.8639	−1.06457	−0.532283	+	0.846567i	$0.678666\pi$
$954$	−18.2644	−0.591331
$955$	12.0848	0.391056
$956$	148.667	4.80824
$957$	−33.1436	−1.07138
$958$	−18.6566	−0.602768
$959$	−14.0472	−0.453609
$960$	−160.035	−5.16512
$961$	−30.4839	−0.983353
$962$	−21.8249	−0.703663
$963$	0.381098	0.0122807
$964$	−137.094	−4.41549
$965$	−39.2523	−1.26358
$966$	56.2845	1.81093
$967$	1.94298	0.0624822	0.0312411	−	0.999512i	$-0.490054\pi$
$967$	1.94298	0.0624822	0.0312411	+	0.999512i	$0.490054\pi$
$968$	−9.10526	−0.292654
$969$	0.624782	0.0200709
$970$	−49.2787	−1.58224
$971$	40.7934	1.30912	0.654561	−	0.756009i	$-0.272853\pi$
$971$	40.7934	1.30912	0.654561	+	0.756009i	$0.272853\pi$
$972$	40.7565	1.30727
$973$	−35.9624	−1.15290
$974$	14.6949	0.470854
$975$	12.0349	0.385424
$976$	−12.0306	−0.385089
$977$	46.6703	1.49312	0.746558	−	0.665321i	$-0.231705\pi$
$977$	46.6703	1.49312	0.746558	+	0.665321i	$0.231705\pi$
$978$	−24.6568	−0.788439
$979$	31.0279	0.991656
$980$	43.1630	1.37879
$981$	10.9349	0.349124
$982$	−58.6399	−1.87128
$983$	24.3825	0.777681	0.388841	−	0.921305i	$-0.372876\pi$
$983$	24.3825	0.777681	0.388841	+	0.921305i	$0.372876\pi$
$984$	−138.147	−4.40395
$985$	38.7425	1.23444
$986$	3.08396	0.0982131
$987$	−8.61469	−0.274209
$988$	12.7791	0.406556
$989$	−25.8244	−0.821167
$990$	−25.1026	−0.797814
$991$	13.5031	0.428941	0.214471	−	0.976730i	$-0.431197\pi$
$991$	13.5031	0.428941	0.214471	+	0.976730i	$0.431197\pi$
$992$	15.5784	0.494615
$993$	−50.2831	−1.59568
$994$	−61.6903	−1.95670
$995$	61.7450	1.95745
$996$	43.4149	1.37565
$997$	23.3919	0.740828	0.370414	−	0.928867i	$-0.379216\pi$
$997$	23.3919	0.740828	0.370414	+	0.928867i	$0.379216\pi$
$998$	−65.6039	−2.07666
$999$	−44.9341	−1.42165

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.4	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.4	✓	142	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73051 q^{2} +1.50429 q^{3} +5.45570 q^{4} -3.60561 q^{5} -4.10747 q^{6} +2.19221 q^{7} -9.43585 q^{8} -0.737125 q^{9} +O(q^{10})\)	\(q-2.73051 q^{2} +1.50429 q^{3} +5.45570 q^{4} -3.60561 q^{5} -4.10747 q^{6} +2.19221 q^{7} -9.43585 q^{8} -0.737125 q^{9} +9.84515 q^{10} +3.45904 q^{11} +8.20694 q^{12} +1.00000 q^{13} -5.98585 q^{14} -5.42386 q^{15} +14.8533 q^{16} +0.177317 q^{17} +2.01273 q^{18} +2.34233 q^{19} -19.6711 q^{20} +3.29770 q^{21} -9.44496 q^{22} -6.25076 q^{23} -14.1942 q^{24} +8.00039 q^{25} -2.73051 q^{26} -5.62170 q^{27} +11.9600 q^{28} -6.36962 q^{29} +14.8099 q^{30} -0.718381 q^{31} -21.6854 q^{32} +5.20339 q^{33} -0.484166 q^{34} -7.90423 q^{35} -4.02154 q^{36} +7.99296 q^{37} -6.39577 q^{38} +1.50429 q^{39} +34.0219 q^{40} +9.73260 q^{41} -9.00443 q^{42} +4.13139 q^{43} +18.8715 q^{44} +2.65778 q^{45} +17.0678 q^{46} -2.61233 q^{47} +22.3436 q^{48} -2.19423 q^{49} -21.8452 q^{50} +0.266735 q^{51} +5.45570 q^{52} -9.07443 q^{53} +15.3501 q^{54} -12.4719 q^{55} -20.6853 q^{56} +3.52353 q^{57} +17.3923 q^{58} +4.03636 q^{59} -29.5910 q^{60} -0.809959 q^{61} +1.96155 q^{62} -1.61593 q^{63} +29.5058 q^{64} -3.60561 q^{65} -14.2079 q^{66} -13.0123 q^{67} +0.967388 q^{68} -9.40293 q^{69} +21.5826 q^{70} +10.3060 q^{71} +6.95540 q^{72} -14.3226 q^{73} -21.8249 q^{74} +12.0349 q^{75} +12.7791 q^{76} +7.58293 q^{77} -4.10747 q^{78} -6.47622 q^{79} -53.5552 q^{80} -6.24527 q^{81} -26.5750 q^{82} +5.29002 q^{83} +17.9913 q^{84} -0.639335 q^{85} -11.2808 q^{86} -9.58173 q^{87} -32.6390 q^{88} +8.97009 q^{89} -7.25711 q^{90} +2.19221 q^{91} -34.1023 q^{92} -1.08065 q^{93} +7.13300 q^{94} -8.44552 q^{95} -32.6211 q^{96} -5.00537 q^{97} +5.99138 q^{98} -2.54975 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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