LMFDB - Embedded newform 1339.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50812 q^{2} -2.94446 q^{3} +4.29069 q^{4} -0.0167965 q^{5} +7.38506 q^{6} -1.24099 q^{7} -5.74532 q^{8} +5.66982 q^{9} +O(q^{10})$	\(q-2.50812 q^{2} -2.94446 q^{3} +4.29069 q^{4} -0.0167965 q^{5} +7.38506 q^{6} -1.24099 q^{7} -5.74532 q^{8} +5.66982 q^{9} +0.0421278 q^{10} +2.84375 q^{11} -12.6337 q^{12} +1.00000 q^{13} +3.11255 q^{14} +0.0494566 q^{15} +5.82861 q^{16} +4.10466 q^{17} -14.2206 q^{18} +3.44861 q^{19} -0.0720686 q^{20} +3.65403 q^{21} -7.13247 q^{22} +2.88935 q^{23} +16.9168 q^{24} -4.99972 q^{25} -2.50812 q^{26} -7.86115 q^{27} -5.32469 q^{28} +4.49623 q^{29} -0.124043 q^{30} +1.61917 q^{31} -3.12823 q^{32} -8.37329 q^{33} -10.2950 q^{34} +0.0208443 q^{35} +24.3274 q^{36} +2.41480 q^{37} -8.64955 q^{38} -2.94446 q^{39} +0.0965014 q^{40} +3.78254 q^{41} -9.16477 q^{42} -7.32703 q^{43} +12.2016 q^{44} -0.0952332 q^{45} -7.24684 q^{46} -6.06522 q^{47} -17.1621 q^{48} -5.45995 q^{49} +12.5399 q^{50} -12.0860 q^{51} +4.29069 q^{52} -2.66148 q^{53} +19.7167 q^{54} -0.0477651 q^{55} +7.12987 q^{56} -10.1543 q^{57} -11.2771 q^{58} -1.64595 q^{59} +0.212203 q^{60} -0.0716635 q^{61} -4.06109 q^{62} -7.03617 q^{63} -3.81123 q^{64} -0.0167965 q^{65} +21.0012 q^{66} -10.0766 q^{67} +17.6118 q^{68} -8.50755 q^{69} -0.0522800 q^{70} +4.13636 q^{71} -32.5749 q^{72} +8.42853 q^{73} -6.05661 q^{74} +14.7214 q^{75} +14.7969 q^{76} -3.52905 q^{77} +7.38506 q^{78} -7.18106 q^{79} -0.0979004 q^{80} +6.13737 q^{81} -9.48709 q^{82} -5.99056 q^{83} +15.6783 q^{84} -0.0689441 q^{85} +18.3771 q^{86} -13.2390 q^{87} -16.3382 q^{88} +8.41012 q^{89} +0.238857 q^{90} -1.24099 q^{91} +12.3973 q^{92} -4.76758 q^{93} +15.2123 q^{94} -0.0579247 q^{95} +9.21093 q^{96} +5.15632 q^{97} +13.6942 q^{98} +16.1235 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) $ 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * 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 $

$2$	−2.50812	−1.77351	−0.886756	−	0.462238i	$-0.847046\pi$
$2$	−2.50812	−1.77351	−0.886756	+	0.462238i	$0.847046\pi$
$3$	−2.94446	−1.69998	−0.849991	−	0.526797i	$-0.823393\pi$
$3$	−2.94446	−1.69998	−0.849991	+	0.526797i	$0.823393\pi$
$4$	4.29069	2.14534
$5$	−0.0167965	−0.00751163	−0.00375582	−	0.999993i	$-0.501196\pi$
$5$	−0.0167965	−0.00751163	−0.00375582	+	0.999993i	$0.501196\pi$
$6$	7.38506	3.01494
$7$	−1.24099	−0.469049	−0.234525	−	0.972110i	$-0.575353\pi$
$7$	−1.24099	−0.469049	−0.234525	+	0.972110i	$0.575353\pi$
$8$	−5.74532	−2.03128
$9$	5.66982	1.88994
$10$	0.0421278	0.0133220
$11$	2.84375	0.857422	0.428711	−	0.903442i	$-0.358968\pi$
$11$	2.84375	0.857422	0.428711	+	0.903442i	$0.358968\pi$
$12$	−12.6337	−3.64704
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	3.11255	0.831864
$15$	0.0494566	0.0127696
$16$	5.82861	1.45715
$17$	4.10466	0.995527	0.497764	−	0.867313i	$-0.334155\pi$
$17$	4.10466	0.995527	0.497764	+	0.867313i	$0.334155\pi$
$18$	−14.2206	−3.35183
$19$	3.44861	0.791166	0.395583	−	0.918430i	$-0.370543\pi$
$19$	3.44861	0.791166	0.395583	+	0.918430i	$0.370543\pi$
$20$	−0.0720686	−0.0161150
$21$	3.65403	0.797375
$22$	−7.13247	−1.52065
$23$	2.88935	0.602470	0.301235	−	0.953550i	$-0.402601\pi$
$23$	2.88935	0.602470	0.301235	+	0.953550i	$0.402601\pi$
$24$	16.9168	3.45314
$25$	−4.99972	−0.999944
$26$	−2.50812	−0.491884
$27$	−7.86115	−1.51288
$28$	−5.32469	−1.00627
$29$	4.49623	0.834929	0.417465	−	0.908693i	$-0.362919\pi$
$29$	4.49623	0.834929	0.417465	+	0.908693i	$0.362919\pi$
$30$	−0.124043	−0.0226471
$31$	1.61917	0.290812	0.145406	−	0.989372i	$-0.453551\pi$
$31$	1.61917	0.290812	0.145406	+	0.989372i	$0.453551\pi$
$32$	−3.12823	−0.552998
$33$	−8.37329	−1.45760
$34$	−10.2950	−1.76558
$35$	0.0208443	0.00352333
$36$	24.3274	4.05457
$37$	2.41480	0.396990	0.198495	−	0.980102i	$-0.436395\pi$
$37$	2.41480	0.396990	0.198495	+	0.980102i	$0.436395\pi$
$38$	−8.64955	−1.40314
$39$	−2.94446	−0.471490
$40$	0.0965014	0.0152582
$41$	3.78254	0.590734	0.295367	−	0.955384i	$-0.404558\pi$
$41$	3.78254	0.590734	0.295367	+	0.955384i	$0.404558\pi$
$42$	−9.16477	−1.41415
$43$	−7.32703	−1.11736	−0.558680	−	0.829383i	$-0.688692\pi$
$43$	−7.32703	−1.11736	−0.558680	+	0.829383i	$0.688692\pi$
$44$	12.2016	1.83946
$45$	−0.0952332	−0.0141965
$46$	−7.24684	−1.06849
$47$	−6.06522	−0.884703	−0.442351	−	0.896842i	$-0.645856\pi$
$47$	−6.06522	−0.884703	−0.442351	+	0.896842i	$0.645856\pi$
$48$	−17.1621	−2.47713
$49$	−5.45995	−0.779993
$50$	12.5399	1.77341
$51$	−12.0860	−1.69238
$52$	4.29069	0.595011
$53$	−2.66148	−0.365583	−0.182791	−	0.983152i	$-0.558513\pi$
$53$	−2.66148	−0.365583	−0.182791	+	0.983152i	$0.558513\pi$
$54$	19.7167	2.68311
$55$	−0.0477651	−0.00644064
$56$	7.12987	0.952769
$57$	−10.1543	−1.34497
$58$	−11.2771	−1.48076
$59$	−1.64595	−0.214285	−0.107142	−	0.994244i	$-0.534170\pi$
$59$	−1.64595	−0.214285	−0.107142	+	0.994244i	$0.534170\pi$
$60$	0.212203	0.0273953
$61$	−0.0716635	−0.00917557	−0.00458779	−	0.999989i	$-0.501460\pi$
$61$	−0.0716635	−0.00917557	−0.00458779	+	0.999989i	$0.501460\pi$
$62$	−4.06109	−0.515758
$63$	−7.03617	−0.886474
$64$	−3.81123	−0.476404
$65$	−0.0167965	−0.00208335
$66$	21.0012	2.58507
$67$	−10.0766	−1.23105	−0.615526	−	0.788117i	$-0.711056\pi$
$67$	−10.0766	−1.23105	−0.615526	+	0.788117i	$0.711056\pi$
$68$	17.6118	2.13575
$69$	−8.50755	−1.02419
$70$	−0.0522800	−0.00624866
$71$	4.13636	0.490896	0.245448	−	0.969410i	$-0.421065\pi$
$71$	4.13636	0.490896	0.245448	+	0.969410i	$0.421065\pi$
$72$	−32.5749	−3.83899
$73$	8.42853	0.986485	0.493243	−	0.869892i	$-0.335812\pi$
$73$	8.42853	0.986485	0.493243	+	0.869892i	$0.335812\pi$
$74$	−6.05661	−0.704067
$75$	14.7214	1.69989
$76$	14.7969	1.69732
$77$	−3.52905	−0.402173
$78$	7.38506	0.836193
$79$	−7.18106	−0.807932	−0.403966	−	0.914774i	$-0.632369\pi$
$79$	−7.18106	−0.807932	−0.403966	+	0.914774i	$0.632369\pi$
$80$	−0.0979004	−0.0109456
$81$	6.13737	0.681930
$82$	−9.48709	−1.04767
$83$	−5.99056	−0.657549	−0.328774	−	0.944408i	$-0.606636\pi$
$83$	−5.99056	−0.657549	−0.328774	+	0.944408i	$0.606636\pi$
$84$	15.6783	1.71064
$85$	−0.0689441	−0.00747804
$86$	18.3771	1.98165
$87$	−13.2390	−1.41936
$88$	−16.3382	−1.74166
$89$	8.41012	0.891471	0.445735	−	0.895165i	$-0.352942\pi$
$89$	8.41012	0.891471	0.445735	+	0.895165i	$0.352942\pi$
$90$	0.238857	0.0251777
$91$	−1.24099	−0.130091
$92$	12.3973	1.29250
$93$	−4.76758	−0.494375
$94$	15.2123	1.56903
$95$	−0.0579247	−0.00594295
$96$	9.21093	0.940087
$97$	5.15632	0.523545	0.261772	−	0.965130i	$-0.415693\pi$
$97$	5.15632	0.523545	0.261772	+	0.965130i	$0.415693\pi$
$98$	13.6942	1.38333
$99$	16.1235	1.62047
$100$	−21.4522	−2.14522
$101$	7.58575	0.754810	0.377405	−	0.926048i	$-0.376816\pi$
$101$	7.58575	0.754810	0.377405	+	0.926048i	$0.376816\pi$
$102$	30.3132	3.00145
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−5.74532	−0.563375
$105$	−0.0613750	−0.00598959
$106$	6.67533	0.648365
$107$	19.7477	1.90908	0.954539	−	0.298086i	$-0.0963482\pi$
$107$	19.7477	1.90908	0.954539	+	0.298086i	$0.0963482\pi$
$108$	−33.7297	−3.24565
$109$	16.9500	1.62352	0.811758	−	0.583994i	$-0.198511\pi$
$109$	16.9500	1.62352	0.811758	+	0.583994i	$0.198511\pi$
$110$	0.119801	0.0114225
$111$	−7.11026	−0.674876
$112$	−7.23323	−0.683476
$113$	14.2637	1.34182	0.670910	−	0.741539i	$-0.265904\pi$
$113$	14.2637	1.34182	0.670910	+	0.741539i	$0.265904\pi$
$114$	25.4682	2.38532
$115$	−0.0485310	−0.00452554
$116$	19.2919	1.79121
$117$	5.66982	0.524175
$118$	4.12825	0.380036
$119$	−5.09384	−0.466951
$120$	−0.284144	−0.0259387
$121$	−2.91310	−0.264828
$122$	0.179741	0.0162730
$123$	−11.1375	−1.00424
$124$	6.94736	0.623891
$125$	0.167961	0.0150228
$126$	17.6476	1.57217
$127$	−0.604900	−0.0536762	−0.0268381	−	0.999640i	$-0.508544\pi$
$127$	−0.604900	−0.0536762	−0.0268381	+	0.999640i	$0.508544\pi$
$128$	15.8155	1.39791
$129$	21.5741	1.89949
$130$	0.0421278	0.00369485
$131$	−1.69185	−0.147817	−0.0739087	−	0.997265i	$-0.523547\pi$
$131$	−1.69185	−0.147817	−0.0739087	+	0.997265i	$0.523547\pi$
$132$	−35.9271	−3.12706
$133$	−4.27969	−0.371096
$134$	25.2733	2.18328
$135$	0.132040	0.0113642
$136$	−23.5826	−2.02219
$137$	1.91917	0.163965	0.0819827	−	0.996634i	$-0.473875\pi$
$137$	1.91917	0.163965	0.0819827	+	0.996634i	$0.473875\pi$
$138$	21.3380	1.81641
$139$	−13.3256	−1.13027	−0.565133	−	0.825000i	$-0.691175\pi$
$139$	−13.3256	−1.13027	−0.565133	+	0.825000i	$0.691175\pi$
$140$	0.0894362	0.00755874
$141$	17.8588	1.50398
$142$	−10.3745	−0.870609
$143$	2.84375	0.237806
$144$	33.0471	2.75393
$145$	−0.0755211	−0.00627168
$146$	−21.1398	−1.74954
$147$	16.0766	1.32597
$148$	10.3611	0.851680
$149$	−3.91624	−0.320831	−0.160416	−	0.987050i	$-0.551283\pi$
$149$	−3.91624	−0.320831	−0.160416	+	0.987050i	$0.551283\pi$
$150$	−36.9232	−3.01477
$151$	20.3344	1.65479	0.827396	−	0.561619i	$-0.189821\pi$
$151$	20.3344	1.65479	0.827396	+	0.561619i	$0.189821\pi$
$152$	−19.8134	−1.60708
$153$	23.2727	1.88149
$154$	8.85131	0.713259
$155$	−0.0271965	−0.00218447
$156$	−12.6337	−1.01151
$157$	−1.84781	−0.147471	−0.0737355	−	0.997278i	$-0.523492\pi$
$157$	−1.84781	−0.147471	−0.0737355	+	0.997278i	$0.523492\pi$
$158$	18.0110	1.43288
$159$	7.83662	0.621484
$160$	0.0525434	0.00415392
$161$	−3.58564	−0.282588
$162$	−15.3933	−1.20941
$163$	1.81048	0.141808	0.0709038	−	0.997483i	$-0.477412\pi$
$163$	1.81048	0.141808	0.0709038	+	0.997483i	$0.477412\pi$
$164$	16.2297	1.26733
$165$	0.140642	0.0109490
$166$	15.0251	1.16617
$167$	−16.7558	−1.29660	−0.648302	−	0.761383i	$-0.724521\pi$
$167$	−16.7558	−1.29660	−0.648302	+	0.761383i	$0.724521\pi$
$168$	−20.9936	−1.61969
$169$	1.00000	0.0769231
$170$	0.172920	0.0132624
$171$	19.5530	1.49526
$172$	−31.4380	−2.39712
$173$	4.26215	0.324045	0.162022	−	0.986787i	$-0.448198\pi$
$173$	4.26215	0.324045	0.162022	+	0.986787i	$0.448198\pi$
$174$	33.2049	2.51726
$175$	6.20459	0.469023
$176$	16.5751	1.24939
$177$	4.84643	0.364280
$178$	−21.0936	−1.58103
$179$	21.2651	1.58943	0.794713	−	0.606986i	$-0.207622\pi$
$179$	21.2651	1.58943	0.794713	+	0.606986i	$0.207622\pi$
$180$	−0.408616	−0.0304564
$181$	4.23605	0.314863	0.157432	−	0.987530i	$-0.449679\pi$
$181$	4.23605	0.314863	0.157432	+	0.987530i	$0.449679\pi$
$182$	3.11255	0.230718
$183$	0.211010	0.0155983
$184$	−16.6002	−1.22378
$185$	−0.0405602	−0.00298205
$186$	11.9577	0.876780
$187$	11.6726	0.853587
$188$	−26.0239	−1.89799
$189$	9.75559	0.709615
$190$	0.145282	0.0105399
$191$	−26.4156	−1.91137	−0.955683	−	0.294397i	$-0.904881\pi$
$191$	−26.4156	−1.91137	−0.955683	+	0.294397i	$0.904881\pi$
$192$	11.2220	0.809878
$193$	3.10538	0.223530	0.111765	−	0.993735i	$-0.464350\pi$
$193$	3.10538	0.223530	0.111765	+	0.993735i	$0.464350\pi$
$194$	−12.9327	−0.928512
$195$	0.0494566	0.00354166
$196$	−23.4269	−1.67335
$197$	21.4746	1.53000	0.765001	−	0.644030i	$-0.222739\pi$
$197$	21.4746	1.53000	0.765001	+	0.644030i	$0.222739\pi$
$198$	−40.4398	−2.87393
$199$	−0.844191	−0.0598431	−0.0299216	−	0.999552i	$-0.509526\pi$
$199$	−0.844191	−0.0598431	−0.0299216	+	0.999552i	$0.509526\pi$
$200$	28.7250	2.03116
$201$	29.6701	2.09277
$202$	−19.0260	−1.33866
$203$	−5.57977	−0.391623
$204$	−51.8572	−3.63073
$205$	−0.0635336	−0.00443738
$206$	2.50812	0.174749
$207$	16.3821	1.13863
$208$	5.82861	0.404141
$209$	9.80699	0.678363
$210$	0.153936	0.0106226
$211$	−13.9323	−0.959139	−0.479569	−	0.877504i	$-0.659207\pi$
$211$	−13.9323	−0.959139	−0.479569	+	0.877504i	$0.659207\pi$
$212$	−11.4196	−0.784300
$213$	−12.1793	−0.834514
$214$	−49.5296	−3.38577
$215$	0.123069	0.00839321
$216$	45.1649	3.07308
$217$	−2.00937	−0.136405
$218$	−42.5127	−2.87932
$219$	−24.8174	−1.67701
$220$	−0.204945	−0.0138174
$221$	4.10466	0.276110
$222$	17.8334	1.19690
$223$	5.34613	0.358003	0.179002	−	0.983849i	$-0.442713\pi$
$223$	5.34613	0.358003	0.179002	+	0.983849i	$0.442713\pi$
$224$	3.88209	0.259383
$225$	−28.3475	−1.88983
$226$	−35.7752	−2.37973
$227$	2.85224	0.189310	0.0946550	−	0.995510i	$-0.469825\pi$
$227$	2.85224	0.189310	0.0946550	+	0.995510i	$0.469825\pi$
$228$	−43.5689	−2.88542
$229$	−0.138474	−0.00915065	−0.00457533	−	0.999990i	$-0.501456\pi$
$229$	−0.138474	−0.00915065	−0.00457533	+	0.999990i	$0.501456\pi$
$230$	0.121722	0.00802609
$231$	10.3911	0.683687
$232$	−25.8323	−1.69597
$233$	−20.3146	−1.33086	−0.665428	−	0.746462i	$-0.731751\pi$
$233$	−20.3146	−1.33086	−0.665428	+	0.746462i	$0.731751\pi$
$234$	−14.2206	−0.929630
$235$	0.101875	0.00664556
$236$	−7.06226	−0.459714
$237$	21.1443	1.37347
$238$	12.7760	0.828144
$239$	−13.7342	−0.888390	−0.444195	−	0.895930i	$-0.646510\pi$
$239$	−13.7342	−0.888390	−0.444195	+	0.895930i	$0.646510\pi$
$240$	0.288263	0.0186073
$241$	24.9266	1.60566	0.802832	−	0.596206i	$-0.203326\pi$
$241$	24.9266	1.60566	0.802832	+	0.596206i	$0.203326\pi$
$242$	7.30642	0.469675
$243$	5.51226	0.353612
$244$	−0.307486	−0.0196847
$245$	0.0917082	0.00585902
$246$	27.9343	1.78103
$247$	3.44861	0.219430
$248$	−9.30267	−0.590720
$249$	17.6389	1.11782
$250$	−0.421266	−0.0266432
$251$	22.6061	1.42689	0.713444	−	0.700713i	$-0.247134\pi$
$251$	22.6061	1.42689	0.713444	+	0.700713i	$0.247134\pi$
$252$	−30.1900	−1.90179
$253$	8.21657	0.516571
$254$	1.51716	0.0951953
$255$	0.203003	0.0127125
$256$	−32.0448	−2.00280
$257$	14.0340	0.875416	0.437708	−	0.899117i	$-0.355790\pi$
$257$	14.0340	0.875416	0.437708	+	0.899117i	$0.355790\pi$
$258$	−54.1105	−3.36877
$259$	−2.99673	−0.186208
$260$	−0.0720686	−0.00446950
$261$	25.4928	1.57796
$262$	4.24336	0.262156
$263$	−4.57484	−0.282097	−0.141049	−	0.990003i	$-0.545047\pi$
$263$	−4.57484	−0.282097	−0.141049	+	0.990003i	$0.545047\pi$
$264$	48.1072	2.96080
$265$	0.0447037	0.00274612
$266$	10.7340	0.658143
$267$	−24.7632	−1.51548
$268$	−43.2355	−2.64103
$269$	−13.2862	−0.810073	−0.405036	−	0.914301i	$-0.632741\pi$
$269$	−13.2862	−0.810073	−0.405036	+	0.914301i	$0.632741\pi$
$270$	−0.331173	−0.0201545
$271$	−27.7302	−1.68449	−0.842246	−	0.539094i	$-0.818767\pi$
$271$	−27.7302	−1.68449	−0.842246	+	0.539094i	$0.818767\pi$
$272$	23.9245	1.45064
$273$	3.65403	0.221152
$274$	−4.81350	−0.290794
$275$	−14.2179	−0.857374
$276$	−36.5032	−2.19723
$277$	12.2158	0.733976	0.366988	−	0.930226i	$-0.380389\pi$
$277$	12.2158	0.733976	0.366988	+	0.930226i	$0.380389\pi$
$278$	33.4223	2.00454
$279$	9.18041	0.549617
$280$	−0.119757	−0.00715686
$281$	16.2835	0.971394	0.485697	−	0.874127i	$-0.338566\pi$
$281$	16.2835	0.971394	0.485697	+	0.874127i	$0.338566\pi$
$282$	−44.7920	−2.66732
$283$	0.527323	0.0313461	0.0156731	−	0.999877i	$-0.495011\pi$
$283$	0.527323	0.0313461	0.0156731	+	0.999877i	$0.495011\pi$
$284$	17.7478	1.05314
$285$	0.170557	0.0101029
$286$	−7.13247	−0.421752
$287$	−4.69409	−0.277083
$288$	−17.7365	−1.04513
$289$	−0.151726	−0.00892505
$290$	0.189416	0.0111229
$291$	−15.1825	−0.890017
$292$	36.1642	2.11635
$293$	−6.09565	−0.356111	−0.178056	−	0.984020i	$-0.556981\pi$
$293$	−6.09565	−0.356111	−0.178056	+	0.984020i	$0.556981\pi$
$294$	−40.3220	−2.35163
$295$	0.0276463	0.00160963
$296$	−13.8738	−0.806398
$297$	−22.3551	−1.29718
$298$	9.82242	0.568998
$299$	2.88935	0.167095
$300$	63.1651	3.64684
$301$	9.09275	0.524097
$302$	−51.0013	−2.93479
$303$	−22.3359	−1.28316
$304$	20.1006	1.15285
$305$	0.00120370	6.89235e−5 0
$306$	−58.3708	−3.33684
$307$	0.00333324	0.000190238 0	9.51190e−5	−	1.00000i	$-0.499970\pi$
$307$	0.00333324	0.000190238 0	9.51190e−5		1.00000i	$0.499970\pi$
$308$	−15.1421	−0.862799
$309$	2.94446	0.167504
$310$	0.0682121	0.00387419
$311$	−10.1902	−0.577832	−0.288916	−	0.957354i	$-0.593295\pi$
$311$	−10.1902	−0.577832	−0.288916	+	0.957354i	$0.593295\pi$
$312$	16.9168	0.957728
$313$	−7.58052	−0.428477	−0.214238	−	0.976781i	$-0.568727\pi$
$313$	−7.58052	−0.428477	−0.214238	+	0.976781i	$0.568727\pi$
$314$	4.63453	0.261541
$315$	0.118183	0.00665887
$316$	−30.8117	−1.73329
$317$	−10.1667	−0.571018	−0.285509	−	0.958376i	$-0.592163\pi$
$317$	−10.1667	−0.571018	−0.285509	+	0.958376i	$0.592163\pi$
$318$	−19.6552	−1.10221
$319$	12.7861	0.715887
$320$	0.0640154	0.00357857
$321$	−58.1461	−3.24540
$322$	8.99323	0.501173
$323$	14.1554	0.787628
$324$	26.3335	1.46297
$325$	−4.99972	−0.277334
$326$	−4.54091	−0.251498
$327$	−49.9085	−2.75995
$328$	−21.7319	−1.19995
$329$	7.52686	0.414969
$330$	−0.352748	−0.0194181
$331$	19.3356	1.06278	0.531392	−	0.847126i	$-0.321669\pi$
$331$	19.3356	1.06278	0.531392	+	0.847126i	$0.321669\pi$
$332$	−25.7036	−1.41067
$333$	13.6915	0.750287
$334$	42.0257	2.29954
$335$	0.169252	0.00924721
$336$	21.2979	1.16190
$337$	−24.7169	−1.34642	−0.673208	−	0.739453i	$-0.735084\pi$
$337$	−24.7169	−1.34642	−0.673208	+	0.739453i	$0.735084\pi$
$338$	−2.50812	−0.136424
$339$	−41.9989	−2.28107
$340$	−0.295817	−0.0160430
$341$	4.60452	0.249349
$342$	−49.0414	−2.65185
$343$	15.4626	0.834904
$344$	42.0961	2.26967
$345$	0.142897	0.00769333
$346$	−10.6900	−0.574697
$347$	5.89294	0.316350	0.158175	−	0.987411i	$-0.449439\pi$
$347$	5.89294	0.316350	0.158175	+	0.987411i	$0.449439\pi$
$348$	−56.8042	−3.04502
$349$	12.8488	0.687783	0.343891	−	0.939009i	$-0.388255\pi$
$349$	12.8488	0.687783	0.343891	+	0.939009i	$0.388255\pi$
$350$	−15.5619	−0.831817
$351$	−7.86115	−0.419597
$352$	−8.89589	−0.474153
$353$	32.0455	1.70561	0.852804	−	0.522231i	$-0.174900\pi$
$353$	32.0455	1.70561	0.852804	+	0.522231i	$0.174900\pi$
$354$	−12.1555	−0.646055
$355$	−0.0694765	−0.00368743
$356$	36.0852	1.91251
$357$	14.9986	0.793809
$358$	−53.3354	−2.81886
$359$	23.3003	1.22974	0.614871	−	0.788628i	$-0.289208\pi$
$359$	23.3003	1.22974	0.614871	+	0.788628i	$0.289208\pi$
$360$	0.547145	0.0288371
$361$	−7.10706	−0.374056
$362$	−10.6245	−0.558414
$363$	8.57750	0.450202
$364$	−5.32469	−0.279089
$365$	−0.141570	−0.00741012
$366$	−0.529239	−0.0276638
$367$	36.8139	1.92167	0.960835	−	0.277122i	$-0.0893807\pi$
$367$	36.8139	1.92167	0.960835	+	0.277122i	$0.0893807\pi$
$368$	16.8409	0.877891
$369$	21.4463	1.11645
$370$	0.101730	0.00528869
$371$	3.30287	0.171476
$372$	−20.4562	−1.06060
$373$	22.2974	1.15452	0.577258	−	0.816562i	$-0.304123\pi$
$373$	22.2974	1.15452	0.577258	+	0.816562i	$0.304123\pi$
$374$	−29.2764	−1.51385
$375$	−0.494552	−0.0255386
$376$	34.8466	1.79708
$377$	4.49623	0.231568
$378$	−24.4682	−1.25851
$379$	−4.05720	−0.208404	−0.104202	−	0.994556i	$-0.533229\pi$
$379$	−4.05720	−0.208404	−0.104202	+	0.994556i	$0.533229\pi$
$380$	−0.248537	−0.0127497
$381$	1.78110	0.0912486
$382$	66.2536	3.38983
$383$	29.4735	1.50603	0.753013	−	0.658005i	$-0.228600\pi$
$383$	29.4735	1.50603	0.753013	+	0.658005i	$0.228600\pi$
$384$	−46.5680	−2.37641
$385$	0.0592759	0.00302098
$386$	−7.78867	−0.396433
$387$	−41.5429	−2.11174
$388$	22.1241	1.12318
$389$	−3.99226	−0.202415	−0.101208	−	0.994865i	$-0.532271\pi$
$389$	−3.99226	−0.202415	−0.101208	+	0.994865i	$0.532271\pi$
$390$	−0.124043	−0.00628118
$391$	11.8598	0.599776
$392$	31.3692	1.58438
$393$	4.98157	0.251287
$394$	−53.8609	−2.71347
$395$	0.120617	0.00606889
$396$	69.1810	3.47647
$397$	34.8944	1.75130	0.875651	−	0.482945i	$-0.160433\pi$
$397$	34.8944	1.75130	0.875651	+	0.482945i	$0.160433\pi$
$398$	2.11734	0.106132
$399$	12.6013	0.630856
$400$	−29.1414	−1.45707
$401$	−31.9794	−1.59697	−0.798487	−	0.602013i	$-0.794366\pi$
$401$	−31.9794	−1.59697	−0.798487	+	0.602013i	$0.794366\pi$
$402$	−74.4162	−3.71154
$403$	1.61917	0.0806568
$404$	32.5480	1.61933
$405$	−0.103086	−0.00512241
$406$	13.9947	0.694548
$407$	6.86707	0.340388
$408$	69.4380	3.43769
$409$	−18.4811	−0.913833	−0.456917	−	0.889510i	$-0.651046\pi$
$409$	−18.4811	−0.913833	−0.456917	+	0.889510i	$0.651046\pi$
$410$	0.159350	0.00786974
$411$	−5.65090	−0.278738
$412$	−4.29069	−0.211387
$413$	2.04261	0.100510
$414$	−41.0882	−2.01938
$415$	0.100621	0.00493927
$416$	−3.12823	−0.153374
$417$	39.2367	1.92143
$418$	−24.5971	−1.20309
$419$	34.3829	1.67971	0.839856	−	0.542809i	$-0.182639\pi$
$419$	34.3829	1.67971	0.839856	+	0.542809i	$0.182639\pi$
$420$	−0.263341	−0.0128497
$421$	14.0810	0.686265	0.343133	−	0.939287i	$-0.388512\pi$
$421$	14.0810	0.686265	0.343133	+	0.939287i	$0.388512\pi$
$422$	34.9439	1.70104
$423$	−34.3887	−1.67203
$424$	15.2911	0.742600
$425$	−20.5222	−0.995471
$426$	30.5473	1.48002
$427$	0.0889335	0.00430380
$428$	84.7310	4.09563
$429$	−8.37329	−0.404266
$430$	−0.308671	−0.0148854
$431$	18.2789	0.880461	0.440231	−	0.897885i	$-0.354897\pi$
$431$	18.2789	0.880461	0.440231	+	0.897885i	$0.354897\pi$
$432$	−45.8196	−2.20450
$433$	−4.60533	−0.221318	−0.110659	−	0.993858i	$-0.535296\pi$
$433$	−4.60533	−0.221318	−0.110659	+	0.993858i	$0.535296\pi$
$434$	5.03976	0.241916
$435$	0.222368	0.0106617
$436$	72.7271	3.48300
$437$	9.96424	0.476654
$438$	62.2452	2.97419
$439$	14.8383	0.708196	0.354098	−	0.935208i	$-0.384788\pi$
$439$	14.8383	0.708196	0.354098	+	0.935208i	$0.384788\pi$
$440$	0.274426	0.0130827
$441$	−30.9569	−1.47414
$442$	−10.2950	−0.489684
$443$	−21.0474	−0.999990	−0.499995	−	0.866028i	$-0.666665\pi$
$443$	−21.0474	−0.999990	−0.499995	+	0.866028i	$0.666665\pi$
$444$	−30.5079	−1.44784
$445$	−0.141261	−0.00669640
$446$	−13.4087	−0.634923
$447$	11.5312	0.545407
$448$	4.72969	0.223457
$449$	−5.88381	−0.277674	−0.138837	−	0.990315i	$-0.544336\pi$
$449$	−5.88381	−0.277674	−0.138837	+	0.990315i	$0.544336\pi$
$450$	71.0990	3.35164
$451$	10.7566	0.506508
$452$	61.2012	2.87866
$453$	−59.8738	−2.81312
$454$	−7.15378	−0.335743
$455$	0.0208443	0.000977195 0
$456$	58.3397	2.73201
$457$	33.8154	1.58182	0.790909	−	0.611934i	$-0.209608\pi$
$457$	33.8154	1.58182	0.790909	+	0.611934i	$0.209608\pi$
$458$	0.347311	0.0162288
$459$	−32.2674	−1.50611
$460$	−0.208231	−0.00970882
$461$	26.4033	1.22973	0.614863	−	0.788634i	$-0.289211\pi$
$461$	26.4033	1.22973	0.614863	+	0.788634i	$0.289211\pi$
$462$	−26.0623	−1.21253
$463$	−39.3030	−1.82656	−0.913282	−	0.407328i	$-0.866461\pi$
$463$	−39.3030	−1.82656	−0.913282	+	0.407328i	$0.866461\pi$
$464$	26.2068	1.21662
$465$	0.0800788	0.00371357
$466$	50.9516	2.36029
$467$	29.5928	1.36939	0.684695	−	0.728830i	$-0.259935\pi$
$467$	29.5928	1.36939	0.684695	+	0.728830i	$0.259935\pi$
$468$	24.3274	1.12453
$469$	12.5049	0.577424
$470$	−0.255514	−0.0117860
$471$	5.44078	0.250698
$472$	9.45652	0.435272
$473$	−20.8362	−0.958050
$474$	−53.0325	−2.43587
$475$	−17.2421	−0.791122
$476$	−21.8561	−1.00177
$477$	−15.0901	−0.690929
$478$	34.4470	1.57557
$479$	−5.88215	−0.268763	−0.134381	−	0.990930i	$-0.542905\pi$
$479$	−5.88215	−0.268763	−0.134381	+	0.990930i	$0.542905\pi$
$480$	−0.154712	−0.00706159
$481$	2.41480	0.110105
$482$	−62.5190	−2.84766
$483$	10.5578	0.480395
$484$	−12.4992	−0.568146
$485$	−0.0866082	−0.00393268
$486$	−13.8254	−0.627134
$487$	9.26909	0.420022	0.210011	−	0.977699i	$-0.432650\pi$
$487$	9.26909	0.420022	0.210011	+	0.977699i	$0.432650\pi$
$488$	0.411730	0.0186381
$489$	−5.33087	−0.241070
$490$	−0.230015	−0.0103910
$491$	−39.3574	−1.77618	−0.888088	−	0.459673i	$-0.847967\pi$
$491$	−39.3574	−1.77618	−0.888088	+	0.459673i	$0.847967\pi$
$492$	−47.7876	−2.15443
$493$	18.4555	0.831195
$494$	−8.64955	−0.389162
$495$	−0.270819	−0.0121724
$496$	9.43753	0.423757
$497$	−5.13317	−0.230254
$498$	−44.2406	−1.98247
$499$	−13.6956	−0.613100	−0.306550	−	0.951855i	$-0.599175\pi$
$499$	−13.6956	−0.613100	−0.306550	+	0.951855i	$0.599175\pi$
$500$	0.720666	0.0322291
$501$	49.3368	2.20420
$502$	−56.6990	−2.53060
$503$	−9.95079	−0.443684	−0.221842	−	0.975083i	$-0.571207\pi$
$503$	−9.95079	−0.443684	−0.221842	+	0.975083i	$0.571207\pi$
$504$	40.4251	1.80068
$505$	−0.127414	−0.00566986
$506$	−20.6082	−0.916145
$507$	−2.94446	−0.130768
$508$	−2.59544	−0.115154
$509$	−20.4747	−0.907524	−0.453762	−	0.891123i	$-0.649918\pi$
$509$	−20.4747	−0.907524	−0.453762	+	0.891123i	$0.649918\pi$
$510$	−0.509156	−0.0225458
$511$	−10.4597	−0.462710
$512$	48.7413	2.15408
$513$	−27.1101	−1.19694
$514$	−35.1990	−1.55256
$515$	0.0167965	0.000740143 0
$516$	92.5677	4.07506
$517$	−17.2479	−0.758563
$518$	7.51618	0.330242
$519$	−12.5497	−0.550871
$520$	0.0965014	0.00423187
$521$	−8.27749	−0.362643	−0.181322	−	0.983424i	$-0.558038\pi$
$521$	−8.27749	−0.362643	−0.181322	+	0.983424i	$0.558038\pi$
$522$	−63.9391	−2.79854
$523$	−16.6176	−0.726636	−0.363318	−	0.931665i	$-0.618356\pi$
$523$	−16.6176	−0.726636	−0.363318	+	0.931665i	$0.618356\pi$
$524$	−7.25919	−0.317119
$525$	−18.2691	−0.797330
$526$	11.4743	0.500302
$527$	6.64616	0.289511
$528$	−48.8046	−2.12395
$529$	−14.6517	−0.637030
$530$	−0.112122	−0.00487028
$531$	−9.33224	−0.404985
$532$	−18.3628	−0.796128
$533$	3.78254	0.163840
$534$	62.1092	2.68773
$535$	−0.331692	−0.0143403
$536$	57.8933	2.50061
$537$	−62.6140	−2.70199
$538$	33.3234	1.43667
$539$	−15.5267	−0.668783
$540$	0.566542	0.0243801
$541$	−15.7886	−0.678805	−0.339403	−	0.940641i	$-0.610225\pi$
$541$	−15.7886	−0.678805	−0.339403	+	0.940641i	$0.610225\pi$
$542$	69.5509	2.98746
$543$	−12.4729	−0.535262
$544$	−12.8403	−0.550525
$545$	−0.284701	−0.0121953
$546$	−9.16477	−0.392216
$547$	−30.8455	−1.31886	−0.659429	−	0.751766i	$-0.729202\pi$
$547$	−30.8455	−1.31886	−0.659429	+	0.751766i	$0.729202\pi$
$548$	8.23454	0.351762
$549$	−0.406319	−0.0173413
$550$	35.6603	1.52056
$551$	15.5058	0.660568
$552$	48.8786	2.08041
$553$	8.91160	0.378960
$554$	−30.6387	−1.30172
$555$	0.119428	0.00506942
$556$	−57.1761	−2.42481
$557$	30.0446	1.27303	0.636516	−	0.771264i	$-0.280375\pi$
$557$	30.0446	1.27303	0.636516	+	0.771264i	$0.280375\pi$
$558$	−23.0256	−0.974752
$559$	−7.32703	−0.309900
$560$	0.121493	0.00513402
$561$	−34.3695	−1.45108
$562$	−40.8411	−1.72278
$563$	19.4139	0.818197	0.409099	−	0.912490i	$-0.365843\pi$
$563$	19.4139	0.818197	0.409099	+	0.912490i	$0.365843\pi$
$564$	76.6263	3.22655
$565$	−0.239581	−0.0100793
$566$	−1.32259	−0.0555927
$567$	−7.61639	−0.319859
$568$	−23.7647	−0.997145
$569$	−2.38212	−0.0998638	−0.0499319	−	0.998753i	$-0.515900\pi$
$569$	−2.38212	−0.0998638	−0.0499319	+	0.998753i	$0.515900\pi$
$570$	−0.427777	−0.0179176
$571$	31.9203	1.33582	0.667912	−	0.744241i	$-0.267188\pi$
$571$	31.9203	1.33582	0.667912	+	0.744241i	$0.267188\pi$
$572$	12.2016	0.510176
$573$	77.7796	3.24929
$574$	11.7734	0.491411
$575$	−14.4459	−0.602436
$576$	−21.6090	−0.900374
$577$	−7.09160	−0.295227	−0.147614	−	0.989045i	$-0.547159\pi$
$577$	−7.09160	−0.295227	−0.147614	+	0.989045i	$0.547159\pi$
$578$	0.380547	0.0158287
$579$	−9.14364	−0.379997
$580$	−0.324037	−0.0134549
$581$	7.43421	0.308423
$582$	38.0797	1.57845
$583$	−7.56858	−0.313459
$584$	−48.4246	−2.00383
$585$	−0.0952332	−0.00393741
$586$	15.2886	0.631568
$587$	−13.5034	−0.557346	−0.278673	−	0.960386i	$-0.589895\pi$
$587$	−13.5034	−0.557346	−0.278673	+	0.960386i	$0.589895\pi$
$588$	68.9795	2.84467
$589$	5.58390	0.230081
$590$	−0.0693403	−0.00285469
$591$	−63.2310	−2.60097
$592$	14.0749	0.578475
$593$	−20.2985	−0.833559	−0.416779	−	0.909008i	$-0.636841\pi$
$593$	−20.2985	−0.833559	−0.416779	+	0.909008i	$0.636841\pi$
$594$	56.0694	2.30056
$595$	0.0855588	0.00350757
$596$	−16.8034	−0.688293
$597$	2.48568	0.101732
$598$	−7.24684	−0.296345
$599$	3.97514	0.162420	0.0812099	−	0.996697i	$-0.474122\pi$
$599$	3.97514	0.162420	0.0812099	+	0.996697i	$0.474122\pi$
$600$	−84.5794	−3.45294
$601$	−45.2847	−1.84720	−0.923601	−	0.383354i	$-0.874769\pi$
$601$	−45.2847	−1.84720	−0.923601	+	0.383354i	$0.874769\pi$
$602$	−22.8057	−0.929492
$603$	−57.1324	−2.32661
$604$	87.2486	3.55010
$605$	0.0489300	0.00198929
$606$	56.0212	2.27570
$607$	8.87394	0.360182	0.180091	−	0.983650i	$-0.442361\pi$
$607$	8.87394	0.360182	0.180091	+	0.983650i	$0.442361\pi$
$608$	−10.7881	−0.437514
$609$	16.4294	0.665752
$610$	−0.00301902	−0.000122237 0
$611$	−6.06522	−0.245372
$612$	99.8558	4.03643
$613$	2.69096	0.108687	0.0543435	−	0.998522i	$-0.482693\pi$
$613$	2.69096	0.108687	0.0543435	+	0.998522i	$0.482693\pi$
$614$	−0.00836018	−0.000337389 0
$615$	0.187072	0.00754346
$616$	20.2756	0.816925
$617$	−32.0722	−1.29118	−0.645589	−	0.763685i	$-0.723388\pi$
$617$	−32.0722	−1.29118	−0.645589	+	0.763685i	$0.723388\pi$
$618$	−7.38506	−0.297071
$619$	31.5351	1.26750	0.633752	−	0.773537i	$-0.281514\pi$
$619$	31.5351	1.26750	0.633752	+	0.773537i	$0.281514\pi$
$620$	−0.116692	−0.00468644
$621$	−22.7136	−0.911465
$622$	25.5582	1.02479
$623$	−10.4369	−0.418144
$624$	−17.1621	−0.687033
$625$	24.9958	0.999831
$626$	19.0129	0.759908
$627$	−28.8762	−1.15321
$628$	−7.92835	−0.316376
$629$	9.91193	0.395215
$630$	−0.296418	−0.0118096
$631$	21.0616	0.838450	0.419225	−	0.907882i	$-0.362302\pi$
$631$	21.0616	0.838450	0.419225	+	0.907882i	$0.362302\pi$
$632$	41.2575	1.64114
$633$	41.0230	1.63052
$634$	25.4993	1.01271
$635$	0.0101602	0.000403196 0
$636$	33.6245	1.33330
$637$	−5.45995	−0.216331
$638$	−32.0692	−1.26963
$639$	23.4524	0.927763
$640$	−0.265645	−0.0105006
$641$	−15.7692	−0.622848	−0.311424	−	0.950271i	$-0.600806\pi$
$641$	−15.7692	−0.622848	−0.311424	+	0.950271i	$0.600806\pi$
$642$	145.838	5.75575
$643$	38.4844	1.51768	0.758839	−	0.651278i	$-0.225767\pi$
$643$	38.4844	1.51768	0.758839	+	0.651278i	$0.225767\pi$
$644$	−15.3849	−0.606248
$645$	−0.362370	−0.0142683
$646$	−35.5035	−1.39687
$647$	18.5860	0.730690	0.365345	−	0.930872i	$-0.380951\pi$
$647$	18.5860	0.730690	0.365345	+	0.930872i	$0.380951\pi$
$648$	−35.2611	−1.38519
$649$	−4.68067	−0.183732
$650$	12.5399	0.491856
$651$	5.91651	0.231886
$652$	7.76820	0.304226
$653$	18.7651	0.734337	0.367168	−	0.930154i	$-0.380327\pi$
$653$	18.7651	0.734337	0.367168	+	0.930154i	$0.380327\pi$
$654$	125.177	4.89480
$655$	0.0284172	0.00111035
$656$	22.0470	0.860790
$657$	47.7882	1.86440
$658$	−18.8783	−0.735952
$659$	−29.8839	−1.16411	−0.582056	−	0.813149i	$-0.697751\pi$
$659$	−29.8839	−1.16411	−0.582056	+	0.813149i	$0.697751\pi$
$660$	0.603451	0.0234893
$661$	22.5016	0.875212	0.437606	−	0.899167i	$-0.355826\pi$
$661$	22.5016	0.875212	0.437606	+	0.899167i	$0.355826\pi$
$662$	−48.4962	−1.88486
$663$	−12.0860	−0.469381
$664$	34.4177	1.33566
$665$	0.0718839	0.00278754
$666$	−34.3399	−1.33064
$667$	12.9912	0.503020
$668$	−71.8940	−2.78166
$669$	−15.7414	−0.608599
$670$	−0.424504	−0.0164000
$671$	−0.203793	−0.00786734
$672$	−11.4307	−0.440947
$673$	−21.7873	−0.839839	−0.419919	−	0.907561i	$-0.637942\pi$
$673$	−21.7873	−0.839839	−0.419919	+	0.907561i	$0.637942\pi$
$674$	61.9931	2.38789
$675$	39.3036	1.51279
$676$	4.29069	0.165026
$677$	1.44295	0.0554570	0.0277285	−	0.999615i	$-0.491173\pi$
$677$	1.44295	0.0554570	0.0277285	+	0.999615i	$0.491173\pi$
$678$	105.338	4.04550
$679$	−6.39893	−0.245568
$680$	0.396106	0.0151900
$681$	−8.39830	−0.321823
$682$	−11.5487	−0.442223
$683$	26.2957	1.00618	0.503089	−	0.864235i	$-0.332197\pi$
$683$	26.2957	1.00618	0.503089	+	0.864235i	$0.332197\pi$
$684$	83.8958	3.20784
$685$	−0.0322353	−0.00123165
$686$	−38.7822	−1.48071
$687$	0.407732	0.0155559
$688$	−42.7064	−1.62817
$689$	−2.66148	−0.101394
$690$	−0.358404	−0.0136442
$691$	49.0488	1.86591	0.932953	−	0.359998i	$-0.117223\pi$
$691$	49.0488	1.86591	0.932953	+	0.359998i	$0.117223\pi$
$692$	18.2875	0.695187
$693$	−20.0091	−0.760082
$694$	−14.7802	−0.561050
$695$	0.223824	0.00849014
$696$	76.0620	2.88312
$697$	15.5261	0.588092
$698$	−32.2265	−1.21979
$699$	59.8155	2.26243
$700$	26.6219	1.00621
$701$	−5.76167	−0.217615	−0.108808	−	0.994063i	$-0.534703\pi$
$701$	−5.76167	−0.217615	−0.108808	+	0.994063i	$0.534703\pi$
$702$	19.7167	0.744161
$703$	8.32770	0.314085
$704$	−10.8382	−0.408479
$705$	−0.299965	−0.0112973
$706$	−80.3740	−3.02492
$707$	−9.41382	−0.354043
$708$	20.7945	0.781506
$709$	15.6283	0.586932	0.293466	−	0.955969i	$-0.405191\pi$
$709$	15.6283	0.586932	0.293466	+	0.955969i	$0.405191\pi$
$710$	0.174256	0.00653970
$711$	−40.7153	−1.52694
$712$	−48.3188	−1.81083
$713$	4.67835	0.175206
$714$	−37.6183	−1.40783
$715$	−0.0477651	−0.00178631
$716$	91.2417	3.40986
$717$	40.4396	1.51025
$718$	−58.4400	−2.18096
$719$	50.9469	1.90000	0.949999	−	0.312251i	$-0.101083\pi$
$719$	50.9469	1.90000	0.949999	+	0.312251i	$0.101083\pi$
$720$	−0.555077	−0.0206865
$721$	1.24099	0.0462168
$722$	17.8254	0.663392
$723$	−73.3952	−2.72960
$724$	18.1756	0.675490
$725$	−22.4799	−0.834882
$726$	−21.5134	−0.798438
$727$	4.18666	0.155275	0.0776373	−	0.996982i	$-0.475262\pi$
$727$	4.18666	0.155275	0.0776373	+	0.996982i	$0.475262\pi$
$728$	7.12987	0.264251
$729$	−34.6427	−1.28306
$730$	0.355075	0.0131419
$731$	−30.0750	−1.11236
$732$	0.905378	0.0334637
$733$	−3.17767	−0.117370	−0.0586849	−	0.998277i	$-0.518691\pi$
$733$	−3.17767	−0.117370	−0.0586849	+	0.998277i	$0.518691\pi$
$734$	−92.3338	−3.40810
$735$	−0.270031	−0.00996023
$736$	−9.03854	−0.333165
$737$	−28.6553	−1.05553
$738$	−53.7900	−1.98004
$739$	39.0067	1.43488	0.717442	−	0.696618i	$-0.245313\pi$
$739$	39.0067	1.43488	0.717442	+	0.696618i	$0.245313\pi$
$740$	−0.174031	−0.00639751
$741$	−10.1543	−0.373027
$742$	−8.28400	−0.304115
$743$	−1.27884	−0.0469159	−0.0234580	−	0.999725i	$-0.507468\pi$
$743$	−1.27884	−0.0469159	−0.0234580	+	0.999725i	$0.507468\pi$
$744$	27.3913	1.00421
$745$	0.0657793	0.00240997
$746$	−55.9247	−2.04755
$747$	−33.9653	−1.24273
$748$	50.0836	1.83124
$749$	−24.5066	−0.895452
$750$	1.24040	0.0452929
$751$	−48.2098	−1.75920	−0.879601	−	0.475712i	$-0.842190\pi$
$751$	−48.2098	−1.75920	−0.879601	+	0.475712i	$0.842190\pi$
$752$	−35.3518	−1.28915
$753$	−66.5628	−2.42568
$754$	−11.2771	−0.410688
$755$	−0.341548	−0.0124302
$756$	41.8582	1.52237
$757$	30.3589	1.10341	0.551706	−	0.834038i	$-0.313977\pi$
$757$	30.3589	1.10341	0.551706	+	0.834038i	$0.313977\pi$
$758$	10.1760	0.369608
$759$	−24.1933	−0.878162
$760$	0.332796	0.0120718
$761$	45.8426	1.66179	0.830896	−	0.556428i	$-0.187828\pi$
$761$	45.8426	1.66179	0.830896	+	0.556428i	$0.187828\pi$
$762$	−4.46722	−0.161830
$763$	−21.0347	−0.761509
$764$	−113.341	−4.10054
$765$	−0.390900	−0.0141330
$766$	−73.9232	−2.67096
$767$	−1.64595	−0.0594319
$768$	94.3544	3.40472
$769$	40.3728	1.45588	0.727941	−	0.685640i	$-0.240478\pi$
$769$	40.3728	1.45588	0.727941	+	0.685640i	$0.240478\pi$
$770$	−0.148671	−0.00535774
$771$	−41.3225	−1.48819
$772$	13.3242	0.479548
$773$	44.2697	1.59227	0.796135	−	0.605120i	$-0.206875\pi$
$773$	44.2697	1.59227	0.796135	+	0.605120i	$0.206875\pi$
$774$	104.195	3.74520
$775$	−8.09541	−0.290796
$776$	−29.6247	−1.06346
$777$	8.82375	0.316550
$778$	10.0131	0.358986
$779$	13.0445	0.467369
$780$	0.212203	0.00759808
$781$	11.7628	0.420905
$782$	−29.7458	−1.06371
$783$	−35.3456	−1.26315
$784$	−31.8239	−1.13657
$785$	0.0310367	0.00110775
$786$	−12.4944	−0.445660
$787$	50.6781	1.80648	0.903240	−	0.429135i	$-0.141182\pi$
$787$	50.6781	1.80648	0.903240	+	0.429135i	$0.141182\pi$
$788$	92.1407	3.28238
$789$	13.4704	0.479560
$790$	−0.302522	−0.0107632
$791$	−17.7011	−0.629379
$792$	−92.6348	−3.29164
$793$	−0.0716635	−0.00254485
$794$	−87.5195	−3.10595
$795$	−0.131628	−0.00466836
$796$	−3.62216	−0.128384
$797$	3.19675	0.113235	0.0566174	−	0.998396i	$-0.481968\pi$
$797$	3.19675	0.113235	0.0566174	+	0.998396i	$0.481968\pi$
$798$	−31.6057	−1.11883
$799$	−24.8957	−0.880746
$800$	15.6403	0.552967
$801$	47.6838	1.68483
$802$	80.2082	2.83225
$803$	23.9686	0.845834
$804$	127.305	4.48970
$805$	0.0602263	0.00212270
$806$	−4.06109	−0.143046
$807$	39.1206	1.37711
$808$	−43.5826	−1.53323
$809$	−13.3676	−0.469978	−0.234989	−	0.971998i	$-0.575505\pi$
$809$	−13.3676	−0.469978	−0.234989	+	0.971998i	$0.575505\pi$
$810$	0.258554	0.00908464
$811$	−41.2406	−1.44815	−0.724077	−	0.689719i	$-0.757734\pi$
$811$	−41.2406	−1.44815	−0.724077	+	0.689719i	$0.757734\pi$
$812$	−23.9410	−0.840165
$813$	81.6504	2.86361
$814$	−17.2235	−0.603682
$815$	−0.0304098	−0.00106521
$816$	−70.4446	−2.46605
$817$	−25.2681	−0.884018
$818$	46.3530	1.62069
$819$	−7.03617	−0.245864
$820$	−0.272603	−0.00951970
$821$	−10.1858	−0.355487	−0.177743	−	0.984077i	$-0.556880\pi$
$821$	−10.1858	−0.355487	−0.177743	+	0.984077i	$0.556880\pi$
$822$	14.1731	0.494345
$823$	3.08848	0.107658	0.0538289	−	0.998550i	$-0.482857\pi$
$823$	3.08848	0.107658	0.0538289	+	0.998550i	$0.482857\pi$
$824$	5.74532	0.200148
$825$	41.8641	1.45752
$826$	−5.12311	−0.178256
$827$	−14.4419	−0.502195	−0.251098	−	0.967962i	$-0.580791\pi$
$827$	−14.4419	−0.502195	−0.251098	+	0.967962i	$0.580791\pi$
$828$	70.2902	2.44275
$829$	−4.53173	−0.157394	−0.0786968	−	0.996899i	$-0.525076\pi$
$829$	−4.53173	−0.157394	−0.0786968	+	0.996899i	$0.525076\pi$
$830$	−0.252369	−0.00875985
$831$	−35.9689	−1.24775
$832$	−3.81123	−0.132131
$833$	−22.4113	−0.776504
$834$	−98.4106	−3.40768
$835$	0.281440	0.00973962
$836$	42.0787	1.45532
$837$	−12.7286	−0.439964
$838$	−86.2365	−2.97899
$839$	−36.2393	−1.25112	−0.625560	−	0.780176i	$-0.715129\pi$
$839$	−36.2393	−1.25112	−0.625560	+	0.780176i	$0.715129\pi$
$840$	0.352619	0.0121665
$841$	−8.78391	−0.302893
$842$	−35.3169	−1.21710
$843$	−47.9461	−1.65135
$844$	−59.7791	−2.05768
$845$	−0.0167965	−0.000577818 0
$846$	86.2510	2.96537
$847$	3.61512	0.124217
$848$	−15.5127	−0.532710
$849$	−1.55268	−0.0532878
$850$	51.4721	1.76548
$851$	6.97718	0.239175
$852$	−52.2577	−1.79032
$853$	−44.1298	−1.51098	−0.755488	−	0.655162i	$-0.772600\pi$
$853$	−44.1298	−1.51098	−0.755488	+	0.655162i	$0.772600\pi$
$854$	−0.223056	−0.00763283
$855$	−0.328423	−0.0112318
$856$	−113.457	−3.87787
$857$	−15.7064	−0.536522	−0.268261	−	0.963346i	$-0.586449\pi$
$857$	−15.7064	−0.536522	−0.268261	+	0.963346i	$0.586449\pi$
$858$	21.0012	0.716970
$859$	26.5177	0.904773	0.452387	−	0.891822i	$-0.350573\pi$
$859$	26.5177	0.904773	0.452387	+	0.891822i	$0.350573\pi$
$860$	0.528048	0.0180063
$861$	13.8215	0.471037
$862$	−45.8456	−1.56151
$863$	−2.49446	−0.0849126	−0.0424563	−	0.999098i	$-0.513518\pi$
$863$	−2.49446	−0.0849126	−0.0424563	+	0.999098i	$0.513518\pi$
$864$	24.5915	0.836620
$865$	−0.0715892	−0.00243411
$866$	11.5507	0.392510
$867$	0.446750	0.0151724
$868$	−8.62159	−0.292636
$869$	−20.4211	−0.692739
$870$	−0.557727	−0.0189087
$871$	−10.0766	−0.341432
$872$	−97.3832	−3.29781
$873$	29.2354	0.989467
$874$	−24.9915	−0.845351
$875$	−0.208437	−0.00704645
$876$	−106.484	−3.59775
$877$	−31.5610	−1.06574	−0.532870	−	0.846197i	$-0.678886\pi$
$877$	−31.5610	−1.06574	−0.532870	+	0.846197i	$0.678886\pi$
$878$	−37.2164	−1.25599
$879$	17.9484	0.605383
$880$	−0.278404	−0.00938499
$881$	36.5106	1.23007	0.615037	−	0.788498i	$-0.289141\pi$
$881$	36.5106	1.23007	0.615037	+	0.788498i	$0.289141\pi$
$882$	77.6438	2.61440
$883$	−1.96597	−0.0661600	−0.0330800	−	0.999453i	$-0.510532\pi$
$883$	−1.96597	−0.0661600	−0.0330800	+	0.999453i	$0.510532\pi$
$884$	17.6118	0.592350
$885$	−0.0814032	−0.00273634
$886$	52.7894	1.77349
$887$	23.3988	0.785655	0.392828	−	0.919612i	$-0.371497\pi$
$887$	23.3988	0.785655	0.392828	+	0.919612i	$0.371497\pi$
$888$	40.8507	1.37086
$889$	0.750673	0.0251768
$890$	0.354299	0.0118761
$891$	17.4531	0.584701
$892$	22.9385	0.768039
$893$	−20.9166	−0.699947
$894$	−28.9217	−0.967286
$895$	−0.357179	−0.0119392
$896$	−19.6268	−0.655687
$897$	−8.50755	−0.284059
$898$	14.7573	0.492458
$899$	7.28018	0.242807
$900$	−121.630	−4.05434
$901$	−10.9245	−0.363948
$902$	−26.9789	−0.898298
$903$	−26.7732	−0.890956
$904$	−81.9497	−2.72561
$905$	−0.0711509	−0.00236514
$906$	150.171	4.98909
$907$	50.0676	1.66247	0.831233	−	0.555924i	$-0.187635\pi$
$907$	50.0676	1.66247	0.831233	+	0.555924i	$0.187635\pi$
$908$	12.2381	0.406135
$909$	43.0098	1.42654
$910$	−0.0522800	−0.00173307
$911$	−23.5259	−0.779448	−0.389724	−	0.920932i	$-0.627430\pi$
$911$	−23.5259	−0.779448	−0.389724	+	0.920932i	$0.627430\pi$
$912$	−59.1854	−1.95982
$913$	−17.0356	−0.563797
$914$	−84.8132	−2.80537
$915$	−0.00354423	−0.000117169 0
$916$	−0.594150	−0.0196313
$917$	2.09956	0.0693337
$918$	80.9306	2.67111
$919$	2.13222	0.0703353	0.0351676	−	0.999381i	$-0.488803\pi$
$919$	2.13222	0.0703353	0.0351676	+	0.999381i	$0.488803\pi$
$920$	0.278826	0.00919262
$921$	−0.00981458	−0.000323401 0
$922$	−66.2228	−2.18093
$923$	4.13636	0.136150
$924$	44.5851	1.46674
$925$	−12.0733	−0.396968
$926$	98.5767	3.23943
$927$	−5.66982	−0.186221
$928$	−14.0652	−0.461714
$929$	8.17795	0.268310	0.134155	−	0.990960i	$-0.457168\pi$
$929$	8.17795	0.268310	0.134155	+	0.990960i	$0.457168\pi$
$930$	−0.200848	−0.00658605
$931$	−18.8293	−0.617104
$932$	−87.1637	−2.85514
$933$	30.0045	0.982304
$934$	−74.2223	−2.42863
$935$	−0.196060	−0.00641183
$936$	−32.5749	−1.06474
$937$	−25.1913	−0.822964	−0.411482	−	0.911418i	$-0.634989\pi$
$937$	−25.1913	−0.822964	−0.411482	+	0.911418i	$0.634989\pi$
$938$	−31.3639	−1.02407
$939$	22.3205	0.728402
$940$	0.437112	0.0142570
$941$	−39.2456	−1.27937	−0.639685	−	0.768637i	$-0.720935\pi$
$941$	−39.2456	−1.27937	−0.639685	+	0.768637i	$0.720935\pi$
$942$	−13.6462	−0.444616
$943$	10.9291	0.355900
$944$	−9.59361	−0.312245
$945$	−0.163860	−0.00533037
$946$	52.2598	1.69911
$947$	−20.4814	−0.665555	−0.332778	−	0.943005i	$-0.607986\pi$
$947$	−20.4814	−0.665555	−0.332778	+	0.943005i	$0.607986\pi$
$948$	90.7236	2.94656
$949$	8.42853	0.273602
$950$	43.2453	1.40306
$951$	29.9354	0.970721
$952$	29.2657	0.948508
$953$	17.3624	0.562422	0.281211	−	0.959646i	$-0.409264\pi$
$953$	17.3624	0.562422	0.281211	+	0.959646i	$0.409264\pi$
$954$	37.8479	1.22537
$955$	0.443690	0.0143575
$956$	−58.9290	−1.90590
$957$	−37.6482	−1.21699
$958$	14.7532	0.476653
$959$	−2.38166	−0.0769078
$960$	−0.188491	−0.00608351
$961$	−28.3783	−0.915428
$962$	−6.05661	−0.195273
$963$	111.966	3.60804
$964$	106.952	3.44470
$965$	−0.0521595	−0.00167907
$966$	−26.4802	−0.851986
$967$	32.2534	1.03720	0.518600	−	0.855017i	$-0.326453\pi$
$967$	32.2534	1.03720	0.518600	+	0.855017i	$0.326453\pi$
$968$	16.7367	0.537938
$969$	−41.6800	−1.33895
$970$	0.217224	0.00697465
$971$	48.2751	1.54922	0.774611	−	0.632438i	$-0.217946\pi$
$971$	48.2751	1.54922	0.774611	+	0.632438i	$0.217946\pi$
$972$	23.6514	0.758618
$973$	16.5369	0.530150
$974$	−23.2480	−0.744914
$975$	14.7214	0.471464
$976$	−0.417699	−0.0133702
$977$	41.7528	1.33579	0.667895	−	0.744256i	$-0.267196\pi$
$977$	41.7528	1.33579	0.667895	+	0.744256i	$0.267196\pi$
$978$	13.3705	0.427541
$979$	23.9162	0.764367
$980$	0.393491	0.0125696
$981$	96.1034	3.06835
$982$	98.7133	3.15007
$983$	−2.45004	−0.0781440	−0.0390720	−	0.999236i	$-0.512440\pi$
$983$	−2.45004	−0.0781440	−0.0390720	+	0.999236i	$0.512440\pi$
$984$	63.9887	2.03989
$985$	−0.360698	−0.0114928
$986$	−46.2887	−1.47413
$987$	−22.1625	−0.705440
$988$	14.7969	0.470753
$989$	−21.1703	−0.673177
$990$	0.679248	0.0215879
$991$	7.30709	0.232117	0.116059	−	0.993242i	$-0.462974\pi$
$991$	7.30709	0.232117	0.116059	+	0.993242i	$0.462974\pi$
$992$	−5.06515	−0.160819
$993$	−56.9329	−1.80671
$994$	12.8746	0.408358
$995$	0.0141795	0.000449520 0
$996$	75.6831	2.39811
$997$	53.8677	1.70601	0.853004	−	0.521904i	$-0.174778\pi$
$997$	53.8677	1.70601	0.853004	+	0.521904i	$0.174778\pi$
$998$	34.3503	1.08734
$999$	−18.9831	−0.600598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.f.1.1	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.f.1.1	✓	28	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50812 q^{2} -2.94446 q^{3} +4.29069 q^{4} -0.0167965 q^{5} +7.38506 q^{6} -1.24099 q^{7} -5.74532 q^{8} +5.66982 q^{9} +O(q^{10})\)	\(q-2.50812 q^{2} -2.94446 q^{3} +4.29069 q^{4} -0.0167965 q^{5} +7.38506 q^{6} -1.24099 q^{7} -5.74532 q^{8} +5.66982 q^{9} +0.0421278 q^{10} +2.84375 q^{11} -12.6337 q^{12} +1.00000 q^{13} +3.11255 q^{14} +0.0494566 q^{15} +5.82861 q^{16} +4.10466 q^{17} -14.2206 q^{18} +3.44861 q^{19} -0.0720686 q^{20} +3.65403 q^{21} -7.13247 q^{22} +2.88935 q^{23} +16.9168 q^{24} -4.99972 q^{25} -2.50812 q^{26} -7.86115 q^{27} -5.32469 q^{28} +4.49623 q^{29} -0.124043 q^{30} +1.61917 q^{31} -3.12823 q^{32} -8.37329 q^{33} -10.2950 q^{34} +0.0208443 q^{35} +24.3274 q^{36} +2.41480 q^{37} -8.64955 q^{38} -2.94446 q^{39} +0.0965014 q^{40} +3.78254 q^{41} -9.16477 q^{42} -7.32703 q^{43} +12.2016 q^{44} -0.0952332 q^{45} -7.24684 q^{46} -6.06522 q^{47} -17.1621 q^{48} -5.45995 q^{49} +12.5399 q^{50} -12.0860 q^{51} +4.29069 q^{52} -2.66148 q^{53} +19.7167 q^{54} -0.0477651 q^{55} +7.12987 q^{56} -10.1543 q^{57} -11.2771 q^{58} -1.64595 q^{59} +0.212203 q^{60} -0.0716635 q^{61} -4.06109 q^{62} -7.03617 q^{63} -3.81123 q^{64} -0.0167965 q^{65} +21.0012 q^{66} -10.0766 q^{67} +17.6118 q^{68} -8.50755 q^{69} -0.0522800 q^{70} +4.13636 q^{71} -32.5749 q^{72} +8.42853 q^{73} -6.05661 q^{74} +14.7214 q^{75} +14.7969 q^{76} -3.52905 q^{77} +7.38506 q^{78} -7.18106 q^{79} -0.0979004 q^{80} +6.13737 q^{81} -9.48709 q^{82} -5.99056 q^{83} +15.6783 q^{84} -0.0689441 q^{85} +18.3771 q^{86} -13.2390 q^{87} -16.3382 q^{88} +8.41012 q^{89} +0.238857 q^{90} -1.24099 q^{91} +12.3973 q^{92} -4.76758 q^{93} +15.2123 q^{94} -0.0579247 q^{95} +9.21093 q^{96} +5.15632 q^{97} +13.6942 q^{98} +16.1235 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * q^99

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