LMFDB - Embedded newform 8047.2.a.d.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$0$
Dimension:	$156$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-2.35417 q^{2} +1.10999 q^{3} +3.54214 q^{4} -1.30122 q^{5} -2.61310 q^{6} -1.47122 q^{7} -3.63046 q^{8} -1.76793 q^{9} +O(q^{10})$	\(q-2.35417 q^{2} +1.10999 q^{3} +3.54214 q^{4} -1.30122 q^{5} -2.61310 q^{6} -1.47122 q^{7} -3.63046 q^{8} -1.76793 q^{9} +3.06330 q^{10} -1.68819 q^{11} +3.93172 q^{12} -1.00000 q^{13} +3.46351 q^{14} -1.44433 q^{15} +1.46246 q^{16} -6.23597 q^{17} +4.16202 q^{18} +1.84147 q^{19} -4.60910 q^{20} -1.63303 q^{21} +3.97429 q^{22} +3.49602 q^{23} -4.02976 q^{24} -3.30683 q^{25} +2.35417 q^{26} -5.29233 q^{27} -5.21127 q^{28} +8.66870 q^{29} +3.40021 q^{30} -5.69790 q^{31} +3.81803 q^{32} -1.87386 q^{33} +14.6806 q^{34} +1.91438 q^{35} -6.26226 q^{36} +6.95414 q^{37} -4.33515 q^{38} -1.10999 q^{39} +4.72403 q^{40} +6.33093 q^{41} +3.84445 q^{42} -5.81334 q^{43} -5.97980 q^{44} +2.30047 q^{45} -8.23024 q^{46} -3.11362 q^{47} +1.62331 q^{48} -4.83551 q^{49} +7.78485 q^{50} -6.92184 q^{51} -3.54214 q^{52} -9.02298 q^{53} +12.4591 q^{54} +2.19670 q^{55} +5.34121 q^{56} +2.04401 q^{57} -20.4076 q^{58} -11.5665 q^{59} -5.11603 q^{60} +4.32867 q^{61} +13.4139 q^{62} +2.60102 q^{63} -11.9132 q^{64} +1.30122 q^{65} +4.41140 q^{66} -8.15753 q^{67} -22.0887 q^{68} +3.88053 q^{69} -4.50679 q^{70} -7.85371 q^{71} +6.41841 q^{72} +10.5191 q^{73} -16.3712 q^{74} -3.67053 q^{75} +6.52275 q^{76} +2.48370 q^{77} +2.61310 q^{78} -6.63005 q^{79} -1.90299 q^{80} -0.570615 q^{81} -14.9041 q^{82} +8.15800 q^{83} -5.78443 q^{84} +8.11437 q^{85} +13.6856 q^{86} +9.62213 q^{87} +6.12890 q^{88} -5.43175 q^{89} -5.41570 q^{90} +1.47122 q^{91} +12.3834 q^{92} -6.32459 q^{93} +7.33000 q^{94} -2.39616 q^{95} +4.23795 q^{96} -10.6591 q^{97} +11.3836 q^{98} +2.98460 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) $ 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.35417	−1.66465	−0.832326	−	0.554286i	$-0.812991\pi$
$2$	−2.35417	−1.66465	−0.832326	+	0.554286i	$0.812991\pi$
$3$	1.10999	0.640850	0.320425	−	0.947274i	$-0.396174\pi$
$3$	1.10999	0.640850	0.320425	+	0.947274i	$0.396174\pi$
$4$	3.54214	1.77107
$5$	−1.30122	−0.581923	−0.290961	−	0.956735i	$-0.593975\pi$
$5$	−1.30122	−0.581923	−0.290961	+	0.956735i	$0.593975\pi$
$6$	−2.61310	−1.06679
$7$	−1.47122	−0.556069	−0.278035	−	0.960571i	$-0.589683\pi$
$7$	−1.47122	−0.556069	−0.278035	+	0.960571i	$0.589683\pi$
$8$	−3.63046	−1.28356
$9$	−1.76793	−0.589311
$10$	3.06330	0.968700
$11$	−1.68819	−0.509008	−0.254504	−	0.967072i	$-0.581912\pi$
$11$	−1.68819	−0.509008	−0.254504	+	0.967072i	$0.581912\pi$
$12$	3.93172	1.13499
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	3.46351	0.925662
$15$	−1.44433	−0.372925
$16$	1.46246	0.365616
$17$	−6.23597	−1.51245	−0.756223	−	0.654314i	$-0.772958\pi$
$17$	−6.23597	−1.51245	−0.756223	+	0.654314i	$0.772958\pi$
$18$	4.16202	0.980998
$19$	1.84147	0.422463	0.211232	−	0.977436i	$-0.432253\pi$
$19$	1.84147	0.422463	0.211232	+	0.977436i	$0.432253\pi$
$20$	−4.60910	−1.03063
$21$	−1.63303	−0.356357
$22$	3.97429	0.847322
$23$	3.49602	0.728970	0.364485	−	0.931209i	$-0.381245\pi$
$23$	3.49602	0.728970	0.364485	+	0.931209i	$0.381245\pi$
$24$	−4.02976	−0.822571
$25$	−3.30683	−0.661366
$26$	2.35417	0.461692
$27$	−5.29233	−1.01851
$28$	−5.21127	−0.984837
$29$	8.66870	1.60974	0.804868	−	0.593453i	$-0.202236\pi$
$29$	8.66870	1.60974	0.804868	+	0.593453i	$0.202236\pi$
$30$	3.40021	0.620791
$31$	−5.69790	−1.02337	−0.511687	−	0.859172i	$-0.670979\pi$
$31$	−5.69790	−1.02337	−0.511687	+	0.859172i	$0.670979\pi$
$32$	3.81803	0.674938
$33$	−1.87386	−0.326198
$34$	14.6806	2.51770
$35$	1.91438	0.323590
$36$	−6.26226	−1.04371
$37$	6.95414	1.14325	0.571626	−	0.820514i	$-0.306313\pi$
$37$	6.95414	1.14325	0.571626	+	0.820514i	$0.306313\pi$
$38$	−4.33515	−0.703254
$39$	−1.10999	−0.177740
$40$	4.72403	0.746934
$41$	6.33093	0.988725	0.494363	−	0.869256i	$-0.335402\pi$
$41$	6.33093	0.988725	0.494363	+	0.869256i	$0.335402\pi$
$42$	3.84445	0.593211
$43$	−5.81334	−0.886526	−0.443263	−	0.896392i	$-0.646179\pi$
$43$	−5.81334	−0.886526	−0.443263	+	0.896392i	$0.646179\pi$
$44$	−5.97980	−0.901488
$45$	2.30047	0.342934
$46$	−8.23024	−1.21348
$47$	−3.11362	−0.454168	−0.227084	−	0.973875i	$-0.572919\pi$
$47$	−3.11362	−0.454168	−0.227084	+	0.973875i	$0.572919\pi$
$48$	1.62331	0.234305
$49$	−4.83551	−0.690787
$50$	7.78485	1.10094
$51$	−6.92184	−0.969251
$52$	−3.54214	−0.491206
$53$	−9.02298	−1.23940	−0.619701	−	0.784838i	$-0.712746\pi$
$53$	−9.02298	−1.23940	−0.619701	+	0.784838i	$0.712746\pi$
$54$	12.4591	1.69547
$55$	2.19670	0.296203
$56$	5.34121	0.713749
$57$	2.04401	0.270736
$58$	−20.4076	−2.67965
$59$	−11.5665	−1.50584	−0.752918	−	0.658115i	$-0.771354\pi$
$59$	−11.5665	−1.50584	−0.752918	+	0.658115i	$0.771354\pi$
$60$	−5.11603	−0.660477
$61$	4.32867	0.554230	0.277115	−	0.960837i	$-0.410622\pi$
$61$	4.32867	0.554230	0.277115	+	0.960837i	$0.410622\pi$
$62$	13.4139	1.70356
$63$	2.60102	0.327698
$64$	−11.9132	−1.48915
$65$	1.30122	0.161396
$66$	4.41140	0.543006
$67$	−8.15753	−0.996601	−0.498301	−	0.867004i	$-0.666042\pi$
$67$	−8.15753	−0.996601	−0.498301	+	0.867004i	$0.666042\pi$
$68$	−22.0887	−2.67865
$69$	3.88053	0.467161
$70$	−4.50679	−0.538664
$71$	−7.85371	−0.932064	−0.466032	−	0.884768i	$-0.654317\pi$
$71$	−7.85371	−0.932064	−0.466032	+	0.884768i	$0.654317\pi$
$72$	6.41841	0.756417
$73$	10.5191	1.23117	0.615586	−	0.788070i	$-0.288919\pi$
$73$	10.5191	1.23117	0.615586	+	0.788070i	$0.288919\pi$
$74$	−16.3712	−1.90312
$75$	−3.67053	−0.423836
$76$	6.52275	0.748211
$77$	2.48370	0.283044
$78$	2.61310	0.295875
$79$	−6.63005	−0.745939	−0.372969	−	0.927844i	$-0.621660\pi$
$79$	−6.63005	−0.745939	−0.372969	+	0.927844i	$0.621660\pi$
$80$	−1.90299	−0.212760
$81$	−0.570615	−0.0634017
$82$	−14.9041	−1.64588
$83$	8.15800	0.895457	0.447728	−	0.894170i	$-0.352233\pi$
$83$	8.15800	0.895457	0.447728	+	0.894170i	$0.352233\pi$
$84$	−5.78443	−0.631133
$85$	8.11437	0.880127
$86$	13.6856	1.47576
$87$	9.62213	1.03160
$88$	6.12890	0.653343
$89$	−5.43175	−0.575764	−0.287882	−	0.957666i	$-0.592951\pi$
$89$	−5.43175	−0.575764	−0.287882	+	0.957666i	$0.592951\pi$
$90$	−5.41570	−0.570865
$91$	1.47122	0.154226
$92$	12.3834	1.29106
$93$	−6.32459	−0.655829
$94$	7.33000	0.756031
$95$	−2.39616	−0.245841
$96$	4.23795	0.432534
$97$	−10.6591	−1.08227	−0.541135	−	0.840936i	$-0.682005\pi$
$97$	−10.6591	−1.08227	−0.541135	+	0.840936i	$0.682005\pi$
$98$	11.3836	1.14992
$99$	2.98460	0.299964
$100$	−11.7132	−1.17132
$101$	0.226198	0.0225075	0.0112538	−	0.999937i	$-0.496418\pi$
$101$	0.226198	0.0225075	0.0112538	+	0.999937i	$0.496418\pi$
$102$	16.2952	1.61347
$103$	−4.29506	−0.423205	−0.211602	−	0.977356i	$-0.567868\pi$
$103$	−4.29506	−0.423205	−0.211602	+	0.977356i	$0.567868\pi$
$104$	3.63046	0.355996
$105$	2.12494	0.207372
$106$	21.2417	2.06317
$107$	−8.94365	−0.864615	−0.432308	−	0.901726i	$-0.642301\pi$
$107$	−8.94365	−0.864615	−0.432308	+	0.901726i	$0.642301\pi$
$108$	−18.7462	−1.80385
$109$	−4.49608	−0.430646	−0.215323	−	0.976543i	$-0.569080\pi$
$109$	−4.49608	−0.430646	−0.215323	+	0.976543i	$0.569080\pi$
$110$	−5.17142	−0.493076
$111$	7.71899	0.732654
$112$	−2.15161	−0.203308
$113$	5.48133	0.515640	0.257820	−	0.966193i	$-0.416996\pi$
$113$	5.48133	0.515640	0.257820	+	0.966193i	$0.416996\pi$
$114$	−4.81195	−0.450681
$115$	−4.54909	−0.424205
$116$	30.7057	2.85095
$117$	1.76793	0.163445
$118$	27.2297	2.50669
$119$	9.17450	0.841025
$120$	5.24360	0.478673
$121$	−8.15002	−0.740911
$122$	−10.1905	−0.922600
$123$	7.02724	0.633625
$124$	−20.1828	−1.81246
$125$	10.8090	0.966787
$126$	−6.12326	−0.545503
$127$	−12.0161	−1.06626	−0.533128	−	0.846034i	$-0.678984\pi$
$127$	−12.0161	−1.06626	−0.533128	+	0.846034i	$0.678984\pi$
$128$	20.4098	1.80399
$129$	−6.45272	−0.568131
$130$	−3.06330	−0.268669
$131$	−11.9880	−1.04740	−0.523699	−	0.851903i	$-0.675448\pi$
$131$	−11.9880	−1.04740	−0.523699	+	0.851903i	$0.675448\pi$
$132$	−6.63749	−0.577719
$133$	−2.70922	−0.234919
$134$	19.2043	1.65899
$135$	6.88649	0.592695
$136$	22.6395	1.94132
$137$	−2.27123	−0.194044	−0.0970222	−	0.995282i	$-0.530932\pi$
$137$	−2.27123	−0.194044	−0.0970222	+	0.995282i	$0.530932\pi$
$138$	−9.13544	−0.777661
$139$	−8.71651	−0.739325	−0.369662	−	0.929166i	$-0.620527\pi$
$139$	−8.71651	−0.739325	−0.369662	+	0.929166i	$0.620527\pi$
$140$	6.78100	0.573099
$141$	−3.45607	−0.291053
$142$	18.4890	1.55156
$143$	1.68819	0.141173
$144$	−2.58554	−0.215462
$145$	−11.2799	−0.936743
$146$	−24.7639	−2.04947
$147$	−5.36734	−0.442691
$148$	24.6325	2.02478
$149$	7.67177	0.628496	0.314248	−	0.949341i	$-0.398248\pi$
$149$	7.67177	0.628496	0.314248	+	0.949341i	$0.398248\pi$
$150$	8.64107	0.705540
$151$	−16.6271	−1.35310	−0.676548	−	0.736399i	$-0.736525\pi$
$151$	−16.6271	−1.35310	−0.676548	+	0.736399i	$0.736525\pi$
$152$	−6.68540	−0.542258
$153$	11.0248	0.891301
$154$	−5.84706	−0.471170
$155$	7.41422	0.595524
$156$	−3.93172	−0.314790
$157$	6.10732	0.487417	0.243709	−	0.969848i	$-0.421636\pi$
$157$	6.10732	0.487417	0.243709	+	0.969848i	$0.421636\pi$
$158$	15.6083	1.24173
$159$	−10.0154	−0.794271
$160$	−4.96809	−0.392762
$161$	−5.14342	−0.405358
$162$	1.34333	0.105542
$163$	1.44083	0.112855	0.0564274	−	0.998407i	$-0.482029\pi$
$163$	1.44083	0.112855	0.0564274	+	0.998407i	$0.482029\pi$
$164$	22.4250	1.75110
$165$	2.43831	0.189822
$166$	−19.2054	−1.49062
$167$	5.16977	0.400049	0.200024	−	0.979791i	$-0.435898\pi$
$167$	5.16977	0.400049	0.200024	+	0.979791i	$0.435898\pi$
$168$	5.92867	0.457407
$169$	1.00000	0.0769231
$170$	−19.1026	−1.46511
$171$	−3.25560	−0.248962
$172$	−20.5917	−1.57010
$173$	13.1758	1.00174	0.500868	−	0.865524i	$-0.333014\pi$
$173$	13.1758	1.00174	0.500868	+	0.865524i	$0.333014\pi$
$174$	−22.6522	−1.71726
$175$	4.86508	0.367765
$176$	−2.46892	−0.186102
$177$	−12.8387	−0.965015
$178$	12.7873	0.958447
$179$	5.52614	0.413043	0.206522	−	0.978442i	$-0.433786\pi$
$179$	5.52614	0.413043	0.206522	+	0.978442i	$0.433786\pi$
$180$	8.14858	0.607359
$181$	−12.6621	−0.941166	−0.470583	−	0.882356i	$-0.655956\pi$
$181$	−12.6621	−0.941166	−0.470583	+	0.882356i	$0.655956\pi$
$182$	−3.46351	−0.256733
$183$	4.80476	0.355178
$184$	−12.6922	−0.935679
$185$	−9.04886	−0.665285
$186$	14.8892	1.09173
$187$	10.5275	0.769847
$188$	−11.0289	−0.804362
$189$	7.78620	0.566362
$190$	5.64098	0.409240
$191$	−0.823491	−0.0595857	−0.0297928	−	0.999556i	$-0.509485\pi$
$191$	−0.823491	−0.0595857	−0.0297928	+	0.999556i	$0.509485\pi$
$192$	−13.2235	−0.954325
$193$	17.8562	1.28532	0.642658	−	0.766153i	$-0.277832\pi$
$193$	17.8562	1.28532	0.642658	+	0.766153i	$0.277832\pi$
$194$	25.0934	1.80160
$195$	1.44433	0.103431
$196$	−17.1280	−1.22343
$197$	24.0040	1.71022	0.855108	−	0.518450i	$-0.173491\pi$
$197$	24.0040	1.71022	0.855108	+	0.518450i	$0.173491\pi$
$198$	−7.02628	−0.499336
$199$	−4.21446	−0.298755	−0.149378	−	0.988780i	$-0.547727\pi$
$199$	−4.21446	−0.298755	−0.149378	+	0.988780i	$0.547727\pi$
$200$	12.0053	0.848904
$201$	−9.05474	−0.638672
$202$	−0.532509	−0.0374672
$203$	−12.7536	−0.895125
$204$	−24.5181	−1.71661
$205$	−8.23793	−0.575362
$206$	10.1113	0.704489
$207$	−6.18073	−0.429590
$208$	−1.46246	−0.101404
$209$	−3.10875	−0.215037
$210$	−5.00247	−0.345203
$211$	−9.23199	−0.635557	−0.317778	−	0.948165i	$-0.602937\pi$
$211$	−9.23199	−0.635557	−0.317778	+	0.948165i	$0.602937\pi$
$212$	−31.9606	−2.19507
$213$	−8.71751	−0.597314
$214$	21.0549	1.43928
$215$	7.56443	0.515890
$216$	19.2136	1.30732
$217$	8.38287	0.569067
$218$	10.5845	0.716876
$219$	11.6761	0.788997
$220$	7.78103	0.524597
$221$	6.23597	0.419477
$222$	−18.1718	−1.21961
$223$	26.7999	1.79466	0.897328	−	0.441364i	$-0.145505\pi$
$223$	26.7999	1.79466	0.897328	+	0.441364i	$0.145505\pi$
$224$	−5.61716	−0.375312
$225$	5.84625	0.389750
$226$	−12.9040	−0.858361
$227$	−13.9505	−0.925924	−0.462962	−	0.886378i	$-0.653213\pi$
$227$	−13.9505	−0.925924	−0.462962	+	0.886378i	$0.653213\pi$
$228$	7.24016	0.479491
$229$	13.9470	0.921643	0.460822	−	0.887493i	$-0.347555\pi$
$229$	13.9470	0.921643	0.460822	+	0.887493i	$0.347555\pi$
$230$	10.7093	0.706153
$231$	2.75687	0.181389
$232$	−31.4714	−2.06620
$233$	20.2422	1.32611	0.663056	−	0.748570i	$-0.269259\pi$
$233$	20.2422	1.32611	0.663056	+	0.748570i	$0.269259\pi$
$234$	−4.16202	−0.272080
$235$	4.05150	0.264291
$236$	−40.9703	−2.66694
$237$	−7.35926	−0.478035
$238$	−21.5984	−1.40001
$239$	22.1386	1.43203	0.716014	−	0.698086i	$-0.245965\pi$
$239$	22.1386	1.43203	0.716014	+	0.698086i	$0.245965\pi$
$240$	−2.11229	−0.136348
$241$	4.74920	0.305923	0.152961	−	0.988232i	$-0.451119\pi$
$241$	4.74920	0.305923	0.152961	+	0.988232i	$0.451119\pi$
$242$	19.1866	1.23336
$243$	15.2436	0.977879
$244$	15.3328	0.981579
$245$	6.29206	0.401985
$246$	−16.5433	−1.05477
$247$	−1.84147	−0.117170
$248$	20.6860	1.31356
$249$	9.05526	0.573854
$250$	−25.4463	−1.60936
$251$	1.69740	0.107139	0.0535695	−	0.998564i	$-0.482940\pi$
$251$	1.69740	0.107139	0.0535695	+	0.998564i	$0.482940\pi$
$252$	9.21317	0.580375
$253$	−5.90194	−0.371052
$254$	28.2880	1.77495
$255$	9.00683	0.564030
$256$	−24.2217	−1.51386
$257$	−14.9239	−0.930926	−0.465463	−	0.885067i	$-0.654112\pi$
$257$	−14.9239	−0.930926	−0.465463	+	0.885067i	$0.654112\pi$
$258$	15.1908	0.945740
$259$	−10.2311	−0.635728
$260$	4.60910	0.285844
$261$	−15.3257	−0.948635
$262$	28.2219	1.74356
$263$	14.7590	0.910078	0.455039	−	0.890472i	$-0.349625\pi$
$263$	14.7590	0.910078	0.455039	+	0.890472i	$0.349625\pi$
$264$	6.80299	0.418695
$265$	11.7409	0.721236
$266$	6.37797	0.391058
$267$	−6.02916	−0.368979
$268$	−28.8951	−1.76505
$269$	15.5825	0.950081	0.475041	−	0.879964i	$-0.342433\pi$
$269$	15.5825	0.950081	0.475041	+	0.879964i	$0.342433\pi$
$270$	−16.2120	−0.986631
$271$	18.9996	1.15414	0.577072	−	0.816693i	$-0.304195\pi$
$271$	18.9996	1.15414	0.577072	+	0.816693i	$0.304195\pi$
$272$	−9.11989	−0.552975
$273$	1.63303	0.0988357
$274$	5.34687	0.323017
$275$	5.58255	0.336640
$276$	13.7454	0.827374
$277$	−18.5837	−1.11659	−0.558293	−	0.829644i	$-0.688543\pi$
$277$	−18.5837	−1.11659	−0.558293	+	0.829644i	$0.688543\pi$
$278$	20.5202	1.23072
$279$	10.0735	0.603085
$280$	−6.95009	−0.415347
$281$	−13.7189	−0.818401	−0.409200	−	0.912445i	$-0.634192\pi$
$281$	−13.7189	−0.818401	−0.409200	+	0.912445i	$0.634192\pi$
$282$	8.13619	0.484503
$283$	15.8061	0.939574	0.469787	−	0.882780i	$-0.344331\pi$
$283$	15.8061	0.939574	0.469787	+	0.882780i	$0.344331\pi$
$284$	−27.8189	−1.65075
$285$	−2.65970	−0.157547
$286$	−3.97429	−0.235005
$287$	−9.31420	−0.549800
$288$	−6.75001	−0.397748
$289$	21.8874	1.28749
$290$	26.5548	1.55935
$291$	−11.8315	−0.693573
$292$	37.2602	2.18049
$293$	15.1824	0.886964	0.443482	−	0.896283i	$-0.353743\pi$
$293$	15.1824	0.886964	0.443482	+	0.896283i	$0.353743\pi$
$294$	12.6357	0.736927
$295$	15.0506	0.876280
$296$	−25.2467	−1.46744
$297$	8.93446	0.518430
$298$	−18.0607	−1.04623
$299$	−3.49602	−0.202180
$300$	−13.0015	−0.750643
$301$	8.55271	0.492970
$302$	39.1431	2.25243
$303$	0.251076	0.0144240
$304$	2.69309	0.154459
$305$	−5.63255	−0.322519
$306$	−25.9543	−1.48371
$307$	−14.6965	−0.838774	−0.419387	−	0.907807i	$-0.637755\pi$
$307$	−14.6965	−0.838774	−0.419387	+	0.907807i	$0.637755\pi$
$308$	8.79760	0.501290
$309$	−4.76745	−0.271211
$310$	−17.4544	−0.991341
$311$	27.7653	1.57443	0.787213	−	0.616682i	$-0.211523\pi$
$311$	27.7653	1.57443	0.787213	+	0.616682i	$0.211523\pi$
$312$	4.02976	0.228140
$313$	7.12472	0.402713	0.201357	−	0.979518i	$-0.435465\pi$
$313$	7.12472	0.402713	0.201357	+	0.979518i	$0.435465\pi$
$314$	−14.3777	−0.811381
$315$	−3.38450	−0.190695
$316$	−23.4845	−1.32111
$317$	−2.02999	−0.114015	−0.0570077	−	0.998374i	$-0.518156\pi$
$317$	−2.02999	−0.114015	−0.0570077	+	0.998374i	$0.518156\pi$
$318$	23.5779	1.32219
$319$	−14.6344	−0.819369
$320$	15.5017	0.866573
$321$	−9.92732	−0.554089
$322$	12.1085	0.674780
$323$	−11.4834	−0.638953
$324$	−2.02120	−0.112289
$325$	3.30683	0.183430
$326$	−3.39197	−0.187864
$327$	−4.99058	−0.275980
$328$	−22.9842	−1.26909
$329$	4.58082	0.252549
$330$	−5.74020	−0.315988
$331$	30.7522	1.69029	0.845146	−	0.534536i	$-0.179514\pi$
$331$	30.7522	1.69029	0.845146	+	0.534536i	$0.179514\pi$
$332$	28.8968	1.58592
$333$	−12.2944	−0.673731
$334$	−12.1705	−0.665943
$335$	10.6147	0.579945
$336$	−2.38825	−0.130290
$337$	−19.8815	−1.08302	−0.541508	−	0.840695i	$-0.682147\pi$
$337$	−19.8815	−1.08302	−0.541508	+	0.840695i	$0.682147\pi$
$338$	−2.35417	−0.128050
$339$	6.08419	0.330448
$340$	28.7422	1.55877
$341$	9.61913	0.520905
$342$	7.66426	0.414435
$343$	17.4127	0.940195
$344$	21.1051	1.13791
$345$	−5.04942	−0.271852
$346$	−31.0181	−1.66754
$347$	25.2613	1.35610	0.678049	−	0.735016i	$-0.262826\pi$
$347$	25.2613	1.35610	0.678049	+	0.735016i	$0.262826\pi$
$348$	34.0829	1.82703
$349$	6.74472	0.361037	0.180518	−	0.983572i	$-0.442223\pi$
$349$	6.74472	0.361037	0.180518	+	0.983572i	$0.442223\pi$
$350$	−11.4532	−0.612201
$351$	5.29233	0.282484
$352$	−6.44555	−0.343549
$353$	2.15397	0.114644	0.0573222	−	0.998356i	$-0.481744\pi$
$353$	2.15397	0.114644	0.0573222	+	0.998356i	$0.481744\pi$
$354$	30.2245	1.60642
$355$	10.2194	0.542390
$356$	−19.2400	−1.01972
$357$	10.1836	0.538971
$358$	−13.0095	−0.687574
$359$	−10.9119	−0.575906	−0.287953	−	0.957645i	$-0.592975\pi$
$359$	−10.9119	−0.575906	−0.287953	+	0.957645i	$0.592975\pi$
$360$	−8.35176	−0.440176
$361$	−15.6090	−0.821525
$362$	29.8088	1.56671
$363$	−9.04640	−0.474813
$364$	5.21127	0.273145
$365$	−13.6877	−0.716447
$366$	−11.3113	−0.591249
$367$	−20.9353	−1.09282	−0.546408	−	0.837519i	$-0.684005\pi$
$367$	−20.9353	−1.09282	−0.546408	+	0.837519i	$0.684005\pi$
$368$	5.11280	0.266523
$369$	−11.1927	−0.582667
$370$	21.3026	1.10747
$371$	13.2748	0.689193
$372$	−22.4026	−1.16152
$373$	−0.253784	−0.0131404	−0.00657022	−	0.999978i	$-0.502091\pi$
$373$	−0.253784	−0.0131404	−0.00657022	+	0.999978i	$0.502091\pi$
$374$	−24.7836	−1.28153
$375$	11.9978	0.619566
$376$	11.3039	0.582952
$377$	−8.66870	−0.446461
$378$	−18.3301	−0.942797
$379$	1.98731	0.102081	0.0510406	−	0.998697i	$-0.483746\pi$
$379$	1.98731	0.102081	0.0510406	+	0.998697i	$0.483746\pi$
$380$	−8.48753	−0.435401
$381$	−13.3377	−0.683311
$382$	1.93864	0.0991895
$383$	−7.52696	−0.384610	−0.192305	−	0.981335i	$-0.561596\pi$
$383$	−7.52696	−0.384610	−0.192305	+	0.981335i	$0.561596\pi$
$384$	22.6545	1.15608
$385$	−3.23184	−0.164710
$386$	−42.0366	−2.13960
$387$	10.2776	0.522440
$388$	−37.7561	−1.91678
$389$	25.9486	1.31565	0.657823	−	0.753173i	$-0.271478\pi$
$389$	25.9486	1.31565	0.657823	+	0.753173i	$0.271478\pi$
$390$	−3.40021	−0.172177
$391$	−21.8011	−1.10253
$392$	17.5551	0.886668
$393$	−13.3065	−0.671226
$394$	−56.5096	−2.84691
$395$	8.62715	0.434079
$396$	10.5719	0.531257
$397$	17.4409	0.875335	0.437668	−	0.899137i	$-0.355805\pi$
$397$	17.4409	0.875335	0.437668	+	0.899137i	$0.355805\pi$
$398$	9.92158	0.497324
$399$	−3.00719	−0.150548
$400$	−4.83612	−0.241806
$401$	1.07479	0.0536727	0.0268363	−	0.999640i	$-0.491457\pi$
$401$	1.07479	0.0536727	0.0268363	+	0.999640i	$0.491457\pi$
$402$	21.3164	1.06317
$403$	5.69790	0.283833
$404$	0.801224	0.0398624
$405$	0.742496	0.0368949
$406$	30.0241	1.49007
$407$	−11.7399	−0.581925
$408$	25.1295	1.24409
$409$	13.6295	0.673935	0.336968	−	0.941516i	$-0.390599\pi$
$409$	13.6295	0.673935	0.336968	+	0.941516i	$0.390599\pi$
$410$	19.3935	0.957778
$411$	−2.52103	−0.124353
$412$	−15.2137	−0.749524
$413$	17.0169	0.837349
$414$	14.5505	0.715119
$415$	−10.6153	−0.521087
$416$	−3.81803	−0.187194
$417$	−9.67520	−0.473796
$418$	7.31855	0.357962
$419$	−0.243395	−0.0118906	−0.00594532	−	0.999982i	$-0.501892\pi$
$419$	−0.243395	−0.0118906	−0.00594532	+	0.999982i	$0.501892\pi$
$420$	7.52681	0.367271
$421$	22.2323	1.08354	0.541768	−	0.840528i	$-0.317755\pi$
$421$	22.2323	1.08354	0.541768	+	0.840528i	$0.317755\pi$
$422$	21.7337	1.05798
$423$	5.50466	0.267646
$424$	32.7576	1.59085
$425$	20.6213	1.00028
$426$	20.5225	0.994320
$427$	−6.36844	−0.308190
$428$	−31.6796	−1.53129
$429$	1.87386	0.0904710
$430$	−17.8080	−0.858778
$431$	26.2671	1.26524	0.632622	−	0.774461i	$-0.281979\pi$
$431$	26.2671	1.26524	0.632622	+	0.774461i	$0.281979\pi$
$432$	−7.73985	−0.372384
$433$	−38.7106	−1.86031	−0.930156	−	0.367164i	$-0.880329\pi$
$433$	−38.7106	−1.86031	−0.930156	+	0.367164i	$0.880329\pi$
$434$	−19.7347	−0.947298
$435$	−12.5205	−0.600312
$436$	−15.9257	−0.762704
$437$	6.43783	0.307963
$438$	−27.4876	−1.31341
$439$	−12.2099	−0.582746	−0.291373	−	0.956610i	$-0.594112\pi$
$439$	−12.2099	−0.582746	−0.291373	+	0.956610i	$0.594112\pi$
$440$	−7.97505	−0.380195
$441$	8.54885	0.407088
$442$	−14.6806	−0.698284
$443$	1.05271	0.0500156	0.0250078	−	0.999687i	$-0.492039\pi$
$443$	1.05271	0.0500156	0.0250078	+	0.999687i	$0.492039\pi$
$444$	27.3417	1.29758
$445$	7.06789	0.335050
$446$	−63.0917	−2.98748
$447$	8.51555	0.402772
$448$	17.5270	0.828073
$449$	15.7735	0.744396	0.372198	−	0.928153i	$-0.378604\pi$
$449$	15.7735	0.744396	0.372198	+	0.928153i	$0.378604\pi$
$450$	−13.7631	−0.648798
$451$	−10.6878	−0.503269
$452$	19.4156	0.913233
$453$	−18.4558	−0.867131
$454$	32.8418	1.54134
$455$	−1.91438	−0.0897476
$456$	−7.42070	−0.347506
$457$	−27.4514	−1.28412	−0.642062	−	0.766653i	$-0.721921\pi$
$457$	−27.4514	−1.28412	−0.642062	+	0.766653i	$0.721921\pi$
$458$	−32.8337	−1.53422
$459$	33.0029	1.54044
$460$	−16.1135	−0.751296
$461$	−32.5611	−1.51652	−0.758262	−	0.651950i	$-0.773951\pi$
$461$	−32.5611	−1.51652	−0.758262	+	0.651950i	$0.773951\pi$
$462$	−6.49015	−0.301949
$463$	−16.0768	−0.747154	−0.373577	−	0.927599i	$-0.621869\pi$
$463$	−16.0768	−0.747154	−0.373577	+	0.927599i	$0.621869\pi$
$464$	12.6777	0.588546
$465$	8.22967	0.381642
$466$	−47.6537	−2.20751
$467$	14.4057	0.666616	0.333308	−	0.942818i	$-0.391835\pi$
$467$	14.4057	0.666616	0.333308	+	0.942818i	$0.391835\pi$
$468$	6.26226	0.289473
$469$	12.0015	0.554179
$470$	−9.53793	−0.439952
$471$	6.77904	0.312362
$472$	41.9919	1.93283
$473$	9.81402	0.451249
$474$	17.3250	0.795762
$475$	−6.08944	−0.279403
$476$	32.4973	1.48951
$477$	15.9520	0.730393
$478$	−52.1181	−2.38383
$479$	−29.6970	−1.35689	−0.678446	−	0.734650i	$-0.737346\pi$
$479$	−29.6970	−1.35689	−0.678446	+	0.734650i	$0.737346\pi$
$480$	−5.51451	−0.251702
$481$	−6.95414	−0.317081
$482$	−11.1804	−0.509255
$483$	−5.70912	−0.259774
$484$	−28.8685	−1.31220
$485$	13.8699	0.629798
$486$	−35.8862	−1.62783
$487$	28.8189	1.30591	0.652954	−	0.757398i	$-0.273530\pi$
$487$	28.8189	1.30591	0.652954	+	0.757398i	$0.273530\pi$
$488$	−15.7151	−0.711388
$489$	1.59930	0.0723230
$490$	−14.8126	−0.669165
$491$	33.4482	1.50950	0.754748	−	0.656014i	$-0.227759\pi$
$491$	33.4482	1.50950	0.754748	+	0.656014i	$0.227759\pi$
$492$	24.8914	1.12219
$493$	−54.0578	−2.43464
$494$	4.33515	0.195048
$495$	−3.88362	−0.174556
$496$	−8.33298	−0.374162
$497$	11.5545	0.518292
$498$	−21.3177	−0.955267
$499$	−17.7672	−0.795371	−0.397685	−	0.917522i	$-0.630186\pi$
$499$	−17.7672	−0.795371	−0.397685	+	0.917522i	$0.630186\pi$
$500$	38.2870	1.71225
$501$	5.73837	0.256371
$502$	−3.99598	−0.178349
$503$	−17.0059	−0.758257	−0.379128	−	0.925344i	$-0.623776\pi$
$503$	−17.0059	−0.758257	−0.379128	+	0.925344i	$0.623776\pi$
$504$	−9.44290	−0.420620
$505$	−0.294333	−0.0130976
$506$	13.8942	0.617672
$507$	1.10999	0.0492962
$508$	−42.5627	−1.88841
$509$	15.8891	0.704273	0.352137	−	0.935949i	$-0.385455\pi$
$509$	15.8891	0.704273	0.352137	+	0.935949i	$0.385455\pi$
$510$	−21.2037	−0.938914
$511$	−15.4760	−0.684617
$512$	16.2026	0.716059
$513$	−9.74570	−0.430283
$514$	35.1334	1.54967
$515$	5.58881	0.246272
$516$	−22.8564	−1.00620
$517$	5.25637	0.231175
$518$	24.0857	1.05827
$519$	14.6249	0.641963
$520$	−4.72403	−0.207162
$521$	−35.5244	−1.55635	−0.778176	−	0.628047i	$-0.783855\pi$
$521$	−35.5244	−1.55635	−0.778176	+	0.628047i	$0.783855\pi$
$522$	36.0793	1.57915
$523$	20.6106	0.901237	0.450618	−	0.892717i	$-0.351204\pi$
$523$	20.6106	0.901237	0.450618	+	0.892717i	$0.351204\pi$
$524$	−42.4632	−1.85502
$525$	5.40016	0.235682
$526$	−34.7452	−1.51496
$527$	35.5320	1.54780
$528$	−2.74046	−0.119263
$529$	−10.7778	−0.468602
$530$	−27.6401	−1.20061
$531$	20.4489	0.887405
$532$	−9.59641	−0.416057
$533$	−6.33093	−0.274223
$534$	14.1937	0.614221
$535$	11.6377	0.503139
$536$	29.6156	1.27920
$537$	6.13394	0.264699
$538$	−36.6839	−1.58156
$539$	8.16325	0.351616
$540$	24.3929	1.04970
$541$	−5.48971	−0.236021	−0.118011	−	0.993012i	$-0.537652\pi$
$541$	−5.48971	−0.236021	−0.118011	+	0.993012i	$0.537652\pi$
$542$	−44.7284	−1.92125
$543$	−14.0547	−0.603146
$544$	−23.8091	−1.02081
$545$	5.85038	0.250603
$546$	−3.84445	−0.164527
$547$	12.3333	0.527336	0.263668	−	0.964614i	$-0.415068\pi$
$547$	12.3333	0.527336	0.263668	+	0.964614i	$0.415068\pi$
$548$	−8.04501	−0.343666
$549$	−7.65280	−0.326614
$550$	−13.1423	−0.560389
$551$	15.9632	0.680054
$552$	−14.0881	−0.599630
$553$	9.75427	0.414794
$554$	43.7492	1.85873
$555$	−10.0441	−0.426348
$556$	−30.8751	−1.30939
$557$	−3.94998	−0.167366	−0.0836829	−	0.996492i	$-0.526668\pi$
$557$	−3.94998	−0.167366	−0.0836829	+	0.996492i	$0.526668\pi$
$558$	−23.7148	−1.00393
$559$	5.81334	0.245878
$560$	2.79971	0.118310
$561$	11.6854	0.493357
$562$	32.2967	1.36235
$563$	−21.2797	−0.896832	−0.448416	−	0.893825i	$-0.648012\pi$
$563$	−21.2797	−0.896832	−0.448416	+	0.893825i	$0.648012\pi$
$564$	−12.2419	−0.515476
$565$	−7.13241	−0.300063
$566$	−37.2103	−1.56406
$567$	0.839501	0.0352557
$568$	28.5126	1.19636
$569$	−21.7493	−0.911777	−0.455888	−	0.890037i	$-0.650678\pi$
$569$	−21.7493	−0.911777	−0.455888	+	0.890037i	$0.650678\pi$
$570$	6.26141	0.262261
$571$	−4.83912	−0.202511	−0.101255	−	0.994860i	$-0.532286\pi$
$571$	−4.83912	−0.202511	−0.101255	+	0.994860i	$0.532286\pi$
$572$	5.97980	0.250028
$573$	−0.914063	−0.0381855
$574$	21.9272	0.915226
$575$	−11.5607	−0.482116
$576$	21.0618	0.877575
$577$	−26.0659	−1.08514	−0.542569	−	0.840011i	$-0.682548\pi$
$577$	−26.0659	−1.08514	−0.542569	+	0.840011i	$0.682548\pi$
$578$	−51.5267	−2.14323
$579$	19.8201	0.823695
$580$	−39.9549	−1.65904
$581$	−12.0022	−0.497936
$582$	27.8533	1.15456
$583$	15.2325	0.630865
$584$	−38.1893	−1.58029
$585$	−2.30047	−0.0951127
$586$	−35.7420	−1.47649
$587$	8.18377	0.337780	0.168890	−	0.985635i	$-0.445982\pi$
$587$	8.18377	0.337780	0.168890	+	0.985635i	$0.445982\pi$
$588$	−19.0119	−0.784036
$589$	−10.4925	−0.432337
$590$	−35.4318	−1.45870
$591$	26.6441	1.09599
$592$	10.1702	0.417992
$593$	39.5742	1.62512	0.812558	−	0.582881i	$-0.198074\pi$
$593$	39.5742	1.62512	0.812558	+	0.582881i	$0.198074\pi$
$594$	−21.0333	−0.863006
$595$	−11.9380	−0.489412
$596$	27.1745	1.11311
$597$	−4.67799	−0.191457
$598$	8.23024	0.336560
$599$	−1.36047	−0.0555872	−0.0277936	−	0.999614i	$-0.508848\pi$
$599$	−1.36047	−0.0555872	−0.0277936	+	0.999614i	$0.508848\pi$
$600$	13.3257	0.544020
$601$	−27.4133	−1.11821	−0.559106	−	0.829096i	$-0.688856\pi$
$601$	−27.4133	−1.11821	−0.559106	+	0.829096i	$0.688856\pi$
$602$	−20.1346	−0.820624
$603$	14.4220	0.587308
$604$	−58.8955	−2.39642
$605$	10.6050	0.431153
$606$	−0.591077	−0.0240109
$607$	−5.31268	−0.215635	−0.107817	−	0.994171i	$-0.534386\pi$
$607$	−5.31268	−0.215635	−0.107817	+	0.994171i	$0.534386\pi$
$608$	7.03080	0.285136
$609$	−14.1563	−0.573641
$610$	13.2600	0.536882
$611$	3.11362	0.125963
$612$	39.0513	1.57856
$613$	16.0502	0.648261	0.324131	−	0.946012i	$-0.394928\pi$
$613$	16.0502	0.648261	0.324131	+	0.946012i	$0.394928\pi$
$614$	34.5982	1.39627
$615$	−9.14398	−0.368721
$616$	−9.01697	−0.363304
$617$	44.2165	1.78009	0.890045	−	0.455874i	$-0.150673\pi$
$617$	44.2165	1.78009	0.890045	+	0.455874i	$0.150673\pi$
$618$	11.2234	0.451472
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	26.2622	1.05471
$621$	−18.5021	−0.742464
$622$	−65.3643	−2.62087
$623$	7.99130	0.320165
$624$	−1.62331	−0.0649846
$625$	2.46926	0.0987702
$626$	−16.7728	−0.670377
$627$	−3.45067	−0.137807
$628$	21.6330	0.863250
$629$	−43.3658	−1.72911
$630$	7.96770	0.317441
$631$	27.4790	1.09392	0.546961	−	0.837158i	$-0.315785\pi$
$631$	27.4790	1.09392	0.546961	+	0.837158i	$0.315785\pi$
$632$	24.0701	0.957458
$633$	−10.2474	−0.407297
$634$	4.77894	0.189796
$635$	15.6356	0.620479
$636$	−35.4758	−1.40671
$637$	4.83551	0.191590
$638$	34.4519	1.36396
$639$	13.8848	0.549276
$640$	−26.5576	−1.04978
$641$	25.6546	1.01330	0.506649	−	0.862153i	$-0.330884\pi$
$641$	25.6546	1.01330	0.506649	+	0.862153i	$0.330884\pi$
$642$	23.3706	0.922366
$643$	−25.7785	−1.01661	−0.508303	−	0.861178i	$-0.669727\pi$
$643$	−25.7785	−1.01661	−0.508303	+	0.861178i	$0.669727\pi$
$644$	−18.2187	−0.717917
$645$	8.39641	0.330608
$646$	27.0339	1.06363
$647$	−17.3144	−0.680698	−0.340349	−	0.940299i	$-0.610545\pi$
$647$	−17.3144	−0.680698	−0.340349	+	0.940299i	$0.610545\pi$
$648$	2.07160	0.0813800
$649$	19.5265	0.766482
$650$	−7.78485	−0.305347
$651$	9.30487	0.364686
$652$	5.10363	0.199874
$653$	43.4743	1.70128	0.850641	−	0.525746i	$-0.176214\pi$
$653$	43.4743	1.70128	0.850641	+	0.525746i	$0.176214\pi$
$654$	11.7487	0.459410
$655$	15.5991	0.609505
$656$	9.25876	0.361494
$657$	−18.5971	−0.725543
$658$	−10.7840	−0.420406
$659$	18.2543	0.711087	0.355544	−	0.934660i	$-0.384296\pi$
$659$	18.2543	0.711087	0.355544	+	0.934660i	$0.384296\pi$
$660$	8.63682	0.336188
$661$	5.51618	0.214555	0.107277	−	0.994229i	$-0.465787\pi$
$661$	5.51618	0.214555	0.107277	+	0.994229i	$0.465787\pi$
$662$	−72.3959	−2.81375
$663$	6.92184	0.268822
$664$	−29.6173	−1.14937
$665$	3.52528	0.136705
$666$	28.9433	1.12153
$667$	30.3059	1.17345
$668$	18.3120	0.708514
$669$	29.7475	1.15011
$670$	−24.9889	−0.965407
$671$	−7.30762	−0.282107
$672$	−6.23497	−0.240519
$673$	−21.2768	−0.820159	−0.410079	−	0.912050i	$-0.634499\pi$
$673$	−21.2768	−0.820159	−0.410079	+	0.912050i	$0.634499\pi$
$674$	46.8046	1.80285
$675$	17.5008	0.673608
$676$	3.54214	0.136236
$677$	21.1664	0.813491	0.406745	−	0.913542i	$-0.366664\pi$
$677$	21.1664	0.813491	0.406745	+	0.913542i	$0.366664\pi$
$678$	−14.3232	−0.550081
$679$	15.6819	0.601817
$680$	−29.4589	−1.12970
$681$	−15.4848	−0.593379
$682$	−22.6451	−0.867126
$683$	−3.02228	−0.115644	−0.0578222	−	0.998327i	$-0.518416\pi$
$683$	−3.02228	−0.115644	−0.0578222	+	0.998327i	$0.518416\pi$
$684$	−11.5318	−0.440929
$685$	2.95537	0.112919
$686$	−40.9924	−1.56510
$687$	15.4810	0.590635
$688$	−8.50181	−0.324128
$689$	9.02298	0.343748
$690$	11.8872	0.452539
$691$	33.2256	1.26396	0.631981	−	0.774984i	$-0.282242\pi$
$691$	33.2256	1.26396	0.631981	+	0.774984i	$0.282242\pi$
$692$	46.6704	1.77414
$693$	−4.39101	−0.166801
$694$	−59.4696	−2.25743
$695$	11.3421	0.430230
$696$	−34.9328	−1.32412
$697$	−39.4795	−1.49539
$698$	−15.8782	−0.601001
$699$	22.4686	0.849839
$700$	17.2328	0.651337
$701$	−26.4767	−1.00001	−0.500007	−	0.866022i	$-0.666669\pi$
$701$	−26.4767	−1.00001	−0.500007	+	0.866022i	$0.666669\pi$
$702$	−12.4591	−0.470238
$703$	12.8059	0.482982
$704$	20.1118	0.757991
$705$	4.49710	0.169371
$706$	−5.07083	−0.190843
$707$	−0.332787	−0.0125157
$708$	−45.4764	−1.70911
$709$	−3.68946	−0.138561	−0.0692803	−	0.997597i	$-0.522070\pi$
$709$	−3.68946	−0.138561	−0.0692803	+	0.997597i	$0.522070\pi$
$710$	−24.0583	−0.902890
$711$	11.7215	0.439590
$712$	19.7197	0.739029
$713$	−19.9200	−0.746009
$714$	−23.9739	−0.897200
$715$	−2.19670	−0.0821520
$716$	19.5744	0.731528
$717$	24.5735	0.917715
$718$	25.6884	0.958684
$719$	15.4041	0.574475	0.287238	−	0.957859i	$-0.407263\pi$
$719$	15.4041	0.574475	0.287238	+	0.957859i	$0.407263\pi$
$720$	3.36435	0.125382
$721$	6.31898	0.235331
$722$	36.7462	1.36755
$723$	5.27154	0.196051
$724$	−44.8509	−1.66687
$725$	−28.6659	−1.06462
$726$	21.2968	0.790399
$727$	47.3115	1.75469	0.877343	−	0.479864i	$-0.159314\pi$
$727$	47.3115	1.75469	0.877343	+	0.479864i	$0.159314\pi$
$728$	−5.34121	−0.197958
$729$	18.6321	0.690076
$730$	32.2233	1.19264
$731$	36.2519	1.34082
$732$	17.0191	0.629045
$733$	31.4180	1.16045	0.580224	−	0.814457i	$-0.302965\pi$
$733$	31.4180	1.16045	0.580224	+	0.814457i	$0.302965\pi$
$734$	49.2854	1.81916
$735$	6.98409	0.257612
$736$	13.3479	0.492010
$737$	13.7715	0.507278
$738$	26.3495	0.969937
$739$	−26.1414	−0.961626	−0.480813	−	0.876823i	$-0.659658\pi$
$739$	−26.1414	−0.961626	−0.480813	+	0.876823i	$0.659658\pi$
$740$	−32.0523	−1.17827
$741$	−2.04401	−0.0750885
$742$	−31.2512	−1.14727
$743$	−29.6061	−1.08614	−0.543071	−	0.839687i	$-0.682738\pi$
$743$	−29.6061	−1.08614	−0.543071	+	0.839687i	$0.682738\pi$
$744$	22.9612	0.841797
$745$	−9.98266	−0.365736
$746$	0.597452	0.0218743
$747$	−14.4228	−0.527703
$748$	37.2899	1.36345
$749$	13.1581	0.480786
$750$	−28.2450	−1.03136
$751$	−45.7204	−1.66836	−0.834180	−	0.551492i	$-0.814059\pi$
$751$	−45.7204	−1.66836	−0.834180	+	0.551492i	$0.814059\pi$
$752$	−4.55355	−0.166051
$753$	1.88409	0.0686600
$754$	20.4076	0.743202
$755$	21.6355	0.787397
$756$	27.5798	1.00307
$757$	27.0166	0.981935	0.490968	−	0.871178i	$-0.336643\pi$
$757$	27.0166	0.981935	0.490968	+	0.871178i	$0.336643\pi$
$758$	−4.67847	−0.169930
$759$	−6.55107	−0.237789
$760$	8.69917	0.315552
$761$	−17.0462	−0.617923	−0.308961	−	0.951075i	$-0.599981\pi$
$761$	−17.0462	−0.617923	−0.308961	+	0.951075i	$0.599981\pi$
$762$	31.3993	1.13748
$763$	6.61472	0.239469
$764$	−2.91692	−0.105530
$765$	−14.3457	−0.518669
$766$	17.7198	0.640242
$767$	11.5665	0.417644
$768$	−26.8857	−0.970155
$769$	17.4697	0.629973	0.314987	−	0.949096i	$-0.398000\pi$
$769$	17.4697	0.629973	0.314987	+	0.949096i	$0.398000\pi$
$770$	7.60831	0.274184
$771$	−16.5653	−0.596584
$772$	63.2490	2.27638
$773$	−25.6788	−0.923602	−0.461801	−	0.886984i	$-0.652797\pi$
$773$	−25.6788	−0.923602	−0.461801	+	0.886984i	$0.652797\pi$
$774$	−24.1953	−0.869681
$775$	18.8420	0.676824
$776$	38.6975	1.38916
$777$	−11.3563	−0.407406
$778$	−61.0875	−2.19009
$779$	11.6582	0.417700
$780$	5.11603	0.183183
$781$	13.2585	0.474428
$782$	51.3236	1.83533
$783$	−45.8776	−1.63953
$784$	−7.07176	−0.252563
$785$	−7.94697	−0.283639
$786$	31.3259	1.11736
$787$	39.4507	1.40626	0.703132	−	0.711059i	$-0.251784\pi$
$787$	39.4507	1.40626	0.703132	+	0.711059i	$0.251784\pi$
$788$	85.0255	3.02891
$789$	16.3823	0.583224
$790$	−20.3098	−0.722591
$791$	−8.06424	−0.286731
$792$	−10.8355	−0.385022
$793$	−4.32867	−0.153716
$794$	−41.0590	−1.45713
$795$	13.0322	0.462205
$796$	−14.9282	−0.529116
$797$	−26.2592	−0.930149	−0.465075	−	0.885271i	$-0.653973\pi$
$797$	−26.2592	−0.930149	−0.465075	+	0.885271i	$0.653973\pi$
$798$	7.07945	0.250610
$799$	19.4164	0.686904
$800$	−12.6256	−0.446381
$801$	9.60296	0.339304
$802$	−2.53025	−0.0893463
$803$	−17.7583	−0.626677
$804$	−32.0731	−1.13113
$805$	6.69271	0.235887
$806$	−13.4139	−0.472483
$807$	17.2963	0.608860
$808$	−0.821203	−0.0288898
$809$	−9.76089	−0.343175	−0.171587	−	0.985169i	$-0.554890\pi$
$809$	−9.76089	−0.343175	−0.171587	+	0.985169i	$0.554890\pi$
$810$	−1.74796	−0.0614172
$811$	−53.0645	−1.86334	−0.931672	−	0.363299i	$-0.881650\pi$
$811$	−53.0645	−1.86334	−0.931672	+	0.363299i	$0.881650\pi$
$812$	−45.1749	−1.58533
$813$	21.0893	0.739634
$814$	27.6378	0.968703
$815$	−1.87484	−0.0656728
$816$	−10.1229	−0.354374
$817$	−10.7051	−0.374525
$818$	−32.0862	−1.12187
$819$	−2.60102	−0.0908870
$820$	−29.1799	−1.01901
$821$	−6.00863	−0.209703	−0.104851	−	0.994488i	$-0.533437\pi$
$821$	−6.00863	−0.209703	−0.104851	+	0.994488i	$0.533437\pi$
$822$	5.93495	0.207005
$823$	−7.05007	−0.245750	−0.122875	−	0.992422i	$-0.539211\pi$
$823$	−7.05007	−0.245750	−0.122875	+	0.992422i	$0.539211\pi$
$824$	15.5930	0.543209
$825$	6.19655	0.215736
$826$	−40.0609	−1.39390
$827$	−13.3926	−0.465706	−0.232853	−	0.972512i	$-0.574806\pi$
$827$	−13.3926	−0.465706	−0.232853	+	0.972512i	$0.574806\pi$
$828$	−21.8930	−0.760834
$829$	−27.0152	−0.938276	−0.469138	−	0.883125i	$-0.655435\pi$
$829$	−27.0152	−0.938276	−0.469138	+	0.883125i	$0.655435\pi$
$830$	24.9904	0.867429
$831$	−20.6276	−0.715564
$832$	11.9132	0.413017
$833$	30.1541	1.04478
$834$	22.7771	0.788707
$835$	−6.72701	−0.232798
$836$	−11.0116	−0.380845
$837$	30.1552	1.04232
$838$	0.572995	0.0197938
$839$	7.10009	0.245122	0.122561	−	0.992461i	$-0.460889\pi$
$839$	7.10009	0.245122	0.122561	+	0.992461i	$0.460889\pi$
$840$	−7.71449	−0.266175
$841$	46.1463	1.59125
$842$	−52.3387	−1.80371
$843$	−15.2278	−0.524472
$844$	−32.7010	−1.12561
$845$	−1.30122	−0.0447633
$846$	−12.9589	−0.445538
$847$	11.9905	0.411998
$848$	−13.1958	−0.453145
$849$	17.5445	0.602126
$850$	−48.5461	−1.66512
$851$	24.3118	0.833398
$852$	−30.8786	−1.05788
$853$	−12.7776	−0.437498	−0.218749	−	0.975781i	$-0.570198\pi$
$853$	−12.7776	−0.437498	−0.218749	+	0.975781i	$0.570198\pi$
$854$	14.9924	0.513030
$855$	4.23625	0.144877
$856$	32.4696	1.10979
$857$	−12.3781	−0.422829	−0.211414	−	0.977397i	$-0.567807\pi$
$857$	−12.3781	−0.422829	−0.211414	+	0.977397i	$0.567807\pi$
$858$	−4.41140	−0.150603
$859$	4.80511	0.163948	0.0819741	−	0.996634i	$-0.473878\pi$
$859$	4.80511	0.163948	0.0819741	+	0.996634i	$0.473878\pi$
$860$	26.7943	0.913677
$861$	−10.3386	−0.352339
$862$	−61.8375	−2.10619
$863$	23.8400	0.811523	0.405762	−	0.913979i	$-0.367006\pi$
$863$	23.8400	0.811523	0.405762	+	0.913979i	$0.367006\pi$
$864$	−20.2063	−0.687431
$865$	−17.1446	−0.582933
$866$	91.1315	3.09677
$867$	24.2947	0.825090
$868$	29.6933	1.00786
$869$	11.1928	0.379689
$870$	29.4754	0.999311
$871$	8.15753	0.276407
$872$	16.3228	0.552761
$873$	18.8446	0.637794
$874$	−15.1558	−0.512652
$875$	−15.9024	−0.537600
$876$	41.3583	1.39737
$877$	−12.7158	−0.429381	−0.214691	−	0.976682i	$-0.568874\pi$
$877$	−12.7158	−0.429381	−0.214691	+	0.976682i	$0.568874\pi$
$878$	28.7442	0.970069
$879$	16.8522	0.568411
$880$	3.21260	0.108297
$881$	−18.0621	−0.608528	−0.304264	−	0.952588i	$-0.598411\pi$
$881$	−18.0621	−0.608528	−0.304264	+	0.952588i	$0.598411\pi$
$882$	−20.1255	−0.677661
$883$	−11.7661	−0.395961	−0.197981	−	0.980206i	$-0.563438\pi$
$883$	−11.7661	−0.395961	−0.197981	+	0.980206i	$0.563438\pi$
$884$	22.0887	0.742923
$885$	16.7060	0.561564
$886$	−2.47825	−0.0832586
$887$	−16.5534	−0.555807	−0.277904	−	0.960609i	$-0.589640\pi$
$887$	−16.5534	−0.555807	−0.277904	+	0.960609i	$0.589640\pi$
$888$	−28.0235	−0.940407
$889$	17.6783	0.592913
$890$	−16.6391	−0.557742
$891$	0.963306	0.0322720
$892$	94.9291	3.17846
$893$	−5.73364	−0.191869
$894$	−20.0471	−0.670475
$895$	−7.19072	−0.240359
$896$	−30.0273	−1.00314
$897$	−3.88053	−0.129567
$898$	−37.1335	−1.23916
$899$	−49.3934	−1.64736
$900$	20.7082	0.690274
$901$	56.2671	1.87453
$902$	25.1610	0.837768
$903$	9.49338	0.315920
$904$	−19.8997	−0.661856
$905$	16.4762	0.547686
$906$	43.4483	1.44347
$907$	8.20441	0.272423	0.136212	−	0.990680i	$-0.456507\pi$
$907$	8.20441	0.272423	0.136212	+	0.990680i	$0.456507\pi$
$908$	−49.4144	−1.63988
$909$	−0.399903	−0.0132639
$910$	4.50679	0.149399
$911$	−7.96092	−0.263757	−0.131878	−	0.991266i	$-0.542101\pi$
$911$	−7.96092	−0.263757	−0.131878	+	0.991266i	$0.542101\pi$
$912$	2.98929	0.0989853
$913$	−13.7722	−0.455795
$914$	64.6255	2.13762
$915$	−6.25205	−0.206686
$916$	49.4022	1.63229
$917$	17.6370	0.582426
$918$	−77.6945	−2.56430
$919$	16.1156	0.531604	0.265802	−	0.964028i	$-0.414363\pi$
$919$	16.1156	0.531604	0.265802	+	0.964028i	$0.414363\pi$
$920$	16.5153	0.544493
$921$	−16.3129	−0.537529
$922$	76.6546	2.52449
$923$	7.85371	0.258508
$924$	9.76521	0.321252
$925$	−22.9961	−0.756108
$926$	37.8477	1.24375
$927$	7.59337	0.249399
$928$	33.0973	1.08647
$929$	26.3212	0.863571	0.431786	−	0.901976i	$-0.357884\pi$
$929$	26.3212	0.863571	0.431786	+	0.901976i	$0.357884\pi$
$930$	−19.3741	−0.635301
$931$	−8.90446	−0.291832
$932$	71.7007	2.34863
$933$	30.8191	1.00897
$934$	−33.9135	−1.10968
$935$	−13.6986	−0.447992
$936$	−6.41841	−0.209792
$937$	28.4810	0.930435	0.465218	−	0.885196i	$-0.345976\pi$
$937$	28.4810	0.930435	0.465218	+	0.885196i	$0.345976\pi$
$938$	−28.2537	−0.922516
$939$	7.90834	0.258079
$940$	14.3510	0.468077
$941$	−2.81430	−0.0917436	−0.0458718	−	0.998947i	$-0.514607\pi$
$941$	−2.81430	−0.0917436	−0.0458718	+	0.998947i	$0.514607\pi$
$942$	−15.9590	−0.519974
$943$	22.1331	0.720751
$944$	−16.9157	−0.550558
$945$	−10.1315	−0.329579
$946$	−23.1039	−0.751173
$947$	−32.6557	−1.06117	−0.530583	−	0.847633i	$-0.678027\pi$
$947$	−32.6557	−1.06117	−0.530583	+	0.847633i	$0.678027\pi$
$948$	−26.0675	−0.846633
$949$	−10.5191	−0.341466
$950$	14.3356	0.465108
$951$	−2.25325	−0.0730668
$952$	−33.3077	−1.07951
$953$	−35.4717	−1.14904	−0.574520	−	0.818490i	$-0.694811\pi$
$953$	−35.4717	−1.14904	−0.574520	+	0.818490i	$0.694811\pi$
$954$	−37.5538	−1.21585
$955$	1.07154	0.0346743
$956$	78.4180	2.53622
$957$	−16.2440	−0.525093
$958$	69.9120	2.25875
$959$	3.34148	0.107902
$960$	17.2067	0.555343
$961$	1.46609	0.0472931
$962$	16.3712	0.527830
$963$	15.8118	0.509527
$964$	16.8223	0.541810
$965$	−23.2348	−0.747955
$966$	13.4403	0.432433
$967$	7.02824	0.226013	0.113006	−	0.993594i	$-0.463952\pi$
$967$	7.02824	0.226013	0.113006	+	0.993594i	$0.463952\pi$
$968$	29.5883	0.951005
$969$	−12.7464	−0.409473
$970$	−32.6521	−1.04839
$971$	23.8971	0.766893	0.383447	−	0.923563i	$-0.374737\pi$
$971$	23.8971	0.766893	0.383447	+	0.923563i	$0.374737\pi$
$972$	53.9950	1.73189
$973$	12.8239	0.411116
$974$	−67.8446	−2.17388
$975$	3.67053	0.117551
$976$	6.33053	0.202635
$977$	8.22417	0.263115	0.131557	−	0.991309i	$-0.458002\pi$
$977$	8.22417	0.263115	0.131557	+	0.991309i	$0.458002\pi$
$978$	−3.76504	−0.120393
$979$	9.16981	0.293068
$980$	22.2873	0.711943
$981$	7.94876	0.253784
$982$	−78.7429	−2.51279
$983$	−13.9610	−0.445288	−0.222644	−	0.974900i	$-0.571469\pi$
$983$	−13.9610	−0.445288	−0.222644	+	0.974900i	$0.571469\pi$
$984$	−25.5121	−0.813297
$985$	−31.2345	−0.995214
$986$	127.261	4.05283
$987$	5.08464	0.161846
$988$	−6.52275	−0.207516
$989$	−20.3236	−0.646251
$990$	9.14273	0.290575
$991$	−6.87872	−0.218510	−0.109255	−	0.994014i	$-0.534846\pi$
$991$	−6.87872	−0.218510	−0.109255	+	0.994014i	$0.534846\pi$
$992$	−21.7547	−0.690714
$993$	34.1344	1.08322
$994$	−27.2014	−0.862777
$995$	5.48394	0.173853
$996$	32.0750	1.01633
$997$	32.3940	1.02593	0.512965	−	0.858410i	$-0.328547\pi$
$997$	32.3940	1.02593	0.512965	+	0.858410i	$0.328547\pi$
$998$	41.8272	1.32402
$999$	−36.8036	−1.16441

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.d.1.16	✓	156

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.d.1.16	✓	156	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.35417 q^{2} +1.10999 q^{3} +3.54214 q^{4} -1.30122 q^{5} -2.61310 q^{6} -1.47122 q^{7} -3.63046 q^{8} -1.76793 q^{9} +O(q^{10})\)	\(q-2.35417 q^{2} +1.10999 q^{3} +3.54214 q^{4} -1.30122 q^{5} -2.61310 q^{6} -1.47122 q^{7} -3.63046 q^{8} -1.76793 q^{9} +3.06330 q^{10} -1.68819 q^{11} +3.93172 q^{12} -1.00000 q^{13} +3.46351 q^{14} -1.44433 q^{15} +1.46246 q^{16} -6.23597 q^{17} +4.16202 q^{18} +1.84147 q^{19} -4.60910 q^{20} -1.63303 q^{21} +3.97429 q^{22} +3.49602 q^{23} -4.02976 q^{24} -3.30683 q^{25} +2.35417 q^{26} -5.29233 q^{27} -5.21127 q^{28} +8.66870 q^{29} +3.40021 q^{30} -5.69790 q^{31} +3.81803 q^{32} -1.87386 q^{33} +14.6806 q^{34} +1.91438 q^{35} -6.26226 q^{36} +6.95414 q^{37} -4.33515 q^{38} -1.10999 q^{39} +4.72403 q^{40} +6.33093 q^{41} +3.84445 q^{42} -5.81334 q^{43} -5.97980 q^{44} +2.30047 q^{45} -8.23024 q^{46} -3.11362 q^{47} +1.62331 q^{48} -4.83551 q^{49} +7.78485 q^{50} -6.92184 q^{51} -3.54214 q^{52} -9.02298 q^{53} +12.4591 q^{54} +2.19670 q^{55} +5.34121 q^{56} +2.04401 q^{57} -20.4076 q^{58} -11.5665 q^{59} -5.11603 q^{60} +4.32867 q^{61} +13.4139 q^{62} +2.60102 q^{63} -11.9132 q^{64} +1.30122 q^{65} +4.41140 q^{66} -8.15753 q^{67} -22.0887 q^{68} +3.88053 q^{69} -4.50679 q^{70} -7.85371 q^{71} +6.41841 q^{72} +10.5191 q^{73} -16.3712 q^{74} -3.67053 q^{75} +6.52275 q^{76} +2.48370 q^{77} +2.61310 q^{78} -6.63005 q^{79} -1.90299 q^{80} -0.570615 q^{81} -14.9041 q^{82} +8.15800 q^{83} -5.78443 q^{84} +8.11437 q^{85} +13.6856 q^{86} +9.62213 q^{87} +6.12890 q^{88} -5.43175 q^{89} -5.41570 q^{90} +1.47122 q^{91} +12.3834 q^{92} -6.32459 q^{93} +7.33000 q^{94} -2.39616 q^{95} +4.23795 q^{96} -10.6591 q^{97} +11.3836 q^{98} +2.98460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9}+O(q^{10}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9	\( 156 q + 13 q^{2} + 23 q^{3} + 161 q^{4} + 39 q^{5} + 25 q^{6} + 19 q^{7} + 42 q^{8} + 169 q^{9} + 11 q^{10} + 23 q^{11} + 57 q^{12} - 156 q^{13} + 18 q^{14} + 32 q^{15} + 159 q^{16} + 119 q^{17} + 36 q^{18} + 35 q^{19} + 109 q^{20} + 33 q^{21} + 11 q^{22} + 55 q^{23} + 63 q^{24} + 189 q^{25} - 13 q^{26} + 89 q^{27} + 54 q^{28} - 55 q^{29} + 47 q^{31} + 112 q^{32} + 109 q^{33} + 51 q^{34} + 25 q^{35} + 162 q^{36} + 53 q^{37} + 37 q^{38} - 23 q^{39} + 25 q^{40} + 113 q^{41} + 26 q^{42} + 31 q^{43} + 86 q^{44} + 144 q^{45} + 37 q^{46} + 115 q^{47} + 129 q^{48} + 189 q^{49} + 72 q^{50} - 4 q^{51} - 161 q^{52} + 51 q^{53} + 108 q^{54} + 22 q^{55} + 39 q^{56} + 102 q^{57} + 31 q^{58} + 75 q^{59} + 97 q^{60} + 7 q^{61} + 77 q^{62} + 94 q^{63} + 158 q^{64} - 39 q^{65} + 48 q^{66} + 37 q^{67} + 235 q^{68} + 27 q^{69} + 38 q^{70} + 70 q^{71} + 152 q^{72} + 155 q^{73} - 18 q^{74} + 80 q^{75} + 21 q^{76} + 101 q^{77} - 25 q^{78} + 10 q^{79} + 211 q^{80} + 220 q^{81} + 45 q^{82} + 132 q^{83} + 86 q^{84} + 74 q^{85} + 35 q^{86} + 53 q^{87} + 51 q^{88} + 190 q^{89} - 27 q^{90} - 19 q^{91} + 125 q^{92} + 96 q^{93} - 19 q^{94} + 72 q^{95} + 146 q^{96} + 155 q^{97} + 135 q^{98} + 89 q^{99}+O(q^{100}) \) 156 * q + 13 * q^2 + 23 * q^3 + 161 * q^4 + 39 * q^5 + 25 * q^6 + 19 * q^7 + 42 * q^8 + 169 * q^9 + 11 * q^10 + 23 * q^11 + 57 * q^12 - 156 * q^13 + 18 * q^14 + 32 * q^15 + 159 * q^16 + 119 * q^17 + 36 * q^18 + 35 * q^19 + 109 * q^20 + 33 * q^21 + 11 * q^22 + 55 * q^23 + 63 * q^24 + 189 * q^25 - 13 * q^26 + 89 * q^27 + 54 * q^28 - 55 * q^29 + 47 * q^31 + 112 * q^32 + 109 * q^33 + 51 * q^34 + 25 * q^35 + 162 * q^36 + 53 * q^37 + 37 * q^38 - 23 * q^39 + 25 * q^40 + 113 * q^41 + 26 * q^42 + 31 * q^43 + 86 * q^44 + 144 * q^45 + 37 * q^46 + 115 * q^47 + 129 * q^48 + 189 * q^49 + 72 * q^50 - 4 * q^51 - 161 * q^52 + 51 * q^53 + 108 * q^54 + 22 * q^55 + 39 * q^56 + 102 * q^57 + 31 * q^58 + 75 * q^59 + 97 * q^60 + 7 * q^61 + 77 * q^62 + 94 * q^63 + 158 * q^64 - 39 * q^65 + 48 * q^66 + 37 * q^67 + 235 * q^68 + 27 * q^69 + 38 * q^70 + 70 * q^71 + 152 * q^72 + 155 * q^73 - 18 * q^74 + 80 * q^75 + 21 * q^76 + 101 * q^77 - 25 * q^78 + 10 * q^79 + 211 * q^80 + 220 * q^81 + 45 * q^82 + 132 * q^83 + 86 * q^84 + 74 * q^85 + 35 * q^86 + 53 * q^87 + 51 * q^88 + 190 * q^89 - 27 * q^90 - 19 * q^91 + 125 * q^92 + 96 * q^93 - 19 * q^94 + 72 * q^95 + 146 * q^96 + 155 * q^97 + 135 * q^98 + 89 * q^99

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