LMFDB - Embedded newform 8043.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8043,2,Mod(1,8043)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8043, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8043.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8043 = 3 \cdot 7 \cdot 383 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8043.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:	$64.2236783457$
Analytic rank:	$0$
Dimension:	$52$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8043.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56328 q^{2} -1.00000 q^{3} +4.57042 q^{4} -4.02293 q^{5} +2.56328 q^{6} +1.00000 q^{7} -6.58873 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.56328 q^{2} -1.00000 q^{3} +4.57042 q^{4} -4.02293 q^{5} +2.56328 q^{6} +1.00000 q^{7} -6.58873 q^{8} +1.00000 q^{9} +10.3119 q^{10} -2.04432 q^{11} -4.57042 q^{12} -3.98824 q^{13} -2.56328 q^{14} +4.02293 q^{15} +7.74793 q^{16} -2.64291 q^{17} -2.56328 q^{18} +0.955824 q^{19} -18.3865 q^{20} -1.00000 q^{21} +5.24017 q^{22} +5.68481 q^{23} +6.58873 q^{24} +11.1839 q^{25} +10.2230 q^{26} -1.00000 q^{27} +4.57042 q^{28} +2.11501 q^{29} -10.3119 q^{30} -3.54335 q^{31} -6.68269 q^{32} +2.04432 q^{33} +6.77453 q^{34} -4.02293 q^{35} +4.57042 q^{36} +7.26134 q^{37} -2.45005 q^{38} +3.98824 q^{39} +26.5060 q^{40} -2.24510 q^{41} +2.56328 q^{42} +9.17675 q^{43} -9.34340 q^{44} -4.02293 q^{45} -14.5718 q^{46} -10.5562 q^{47} -7.74793 q^{48} +1.00000 q^{49} -28.6676 q^{50} +2.64291 q^{51} -18.2280 q^{52} +0.903669 q^{53} +2.56328 q^{54} +8.22414 q^{55} -6.58873 q^{56} -0.955824 q^{57} -5.42137 q^{58} -5.90876 q^{59} +18.3865 q^{60} -4.64238 q^{61} +9.08261 q^{62} +1.00000 q^{63} +1.63378 q^{64} +16.0444 q^{65} -5.24017 q^{66} -5.76668 q^{67} -12.0792 q^{68} -5.68481 q^{69} +10.3119 q^{70} -5.68274 q^{71} -6.58873 q^{72} -12.0298 q^{73} -18.6129 q^{74} -11.1839 q^{75} +4.36852 q^{76} -2.04432 q^{77} -10.2230 q^{78} -2.09234 q^{79} -31.1694 q^{80} +1.00000 q^{81} +5.75484 q^{82} -0.102376 q^{83} -4.57042 q^{84} +10.6322 q^{85} -23.5226 q^{86} -2.11501 q^{87} +13.4695 q^{88} -9.25679 q^{89} +10.3119 q^{90} -3.98824 q^{91} +25.9820 q^{92} +3.54335 q^{93} +27.0585 q^{94} -3.84521 q^{95} +6.68269 q^{96} +17.4113 q^{97} -2.56328 q^{98} -2.04432 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) $ 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * 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 $

$2$	−2.56328	−1.81252	−0.906258	−	0.422726i	$-0.861073\pi$
$2$	−2.56328	−1.81252	−0.906258	+	0.422726i	$0.861073\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.57042	2.28521
$5$	−4.02293	−1.79911	−0.899554	−	0.436810i	$-0.856108\pi$
$5$	−4.02293	−1.79911	−0.899554	+	0.436810i	$0.856108\pi$
$6$	2.56328	1.04646
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.58873	−2.32947
$9$	1.00000	0.333333
$10$	10.3119	3.26091
$11$	−2.04432	−0.616385	−0.308192	−	0.951324i	$-0.599724\pi$
$11$	−2.04432	−0.616385	−0.308192	+	0.951324i	$0.599724\pi$
$12$	−4.57042	−1.31937
$13$	−3.98824	−1.10614	−0.553070	−	0.833135i	$-0.686544\pi$
$13$	−3.98824	−1.10614	−0.553070	+	0.833135i	$0.686544\pi$
$14$	−2.56328	−0.685066
$15$	4.02293	1.03872
$16$	7.74793	1.93698
$17$	−2.64291	−0.641000	−0.320500	−	0.947248i	$-0.603851\pi$
$17$	−2.64291	−0.641000	−0.320500	+	0.947248i	$0.603851\pi$
$18$	−2.56328	−0.604172
$19$	0.955824	0.219281	0.109640	−	0.993971i	$-0.465030\pi$
$19$	0.955824	0.219281	0.109640	+	0.993971i	$0.465030\pi$
$20$	−18.3865	−4.11134
$21$	−1.00000	−0.218218
$22$	5.24017	1.11721
$23$	5.68481	1.18536	0.592682	−	0.805437i	$-0.298069\pi$
$23$	5.68481	1.18536	0.592682	+	0.805437i	$0.298069\pi$
$24$	6.58873	1.34492
$25$	11.1839	2.23679
$26$	10.2230	2.00489
$27$	−1.00000	−0.192450
$28$	4.57042	0.863729
$29$	2.11501	0.392747	0.196374	−	0.980529i	$-0.437083\pi$
$29$	2.11501	0.392747	0.196374	+	0.980529i	$0.437083\pi$
$30$	−10.3119	−1.88269
$31$	−3.54335	−0.636404	−0.318202	−	0.948023i	$-0.603079\pi$
$31$	−3.54335	−0.636404	−0.318202	+	0.948023i	$0.603079\pi$
$32$	−6.68269	−1.18134
$33$	2.04432	0.355870
$34$	6.77453	1.16182
$35$	−4.02293	−0.679999
$36$	4.57042	0.761737
$37$	7.26134	1.19376	0.596878	−	0.802332i	$-0.296407\pi$
$37$	7.26134	1.19376	0.596878	+	0.802332i	$0.296407\pi$
$38$	−2.45005	−0.397450
$39$	3.98824	0.638630
$40$	26.5060	4.19096
$41$	−2.24510	−0.350626	−0.175313	−	0.984513i	$-0.556094\pi$
$41$	−2.24510	−0.350626	−0.175313	+	0.984513i	$0.556094\pi$
$42$	2.56328	0.395523
$43$	9.17675	1.39944	0.699721	−	0.714416i	$-0.253308\pi$
$43$	9.17675	1.39944	0.699721	+	0.714416i	$0.253308\pi$
$44$	−9.34340	−1.40857
$45$	−4.02293	−0.599703
$46$	−14.5718	−2.14849
$47$	−10.5562	−1.53978	−0.769890	−	0.638177i	$-0.779689\pi$
$47$	−10.5562	−1.53978	−0.769890	+	0.638177i	$0.779689\pi$
$48$	−7.74793	−1.11832
$49$	1.00000	0.142857
$50$	−28.6676	−4.05421
$51$	2.64291	0.370082
$52$	−18.2280	−2.52776
$53$	0.903669	0.124128	0.0620642	−	0.998072i	$-0.480232\pi$
$53$	0.903669	0.124128	0.0620642	+	0.998072i	$0.480232\pi$
$54$	2.56328	0.348819
$55$	8.22414	1.10894
$56$	−6.58873	−0.880456
$57$	−0.955824	−0.126602
$58$	−5.42137	−0.711861
$59$	−5.90876	−0.769255	−0.384627	−	0.923072i	$-0.625670\pi$
$59$	−5.90876	−0.769255	−0.384627	+	0.923072i	$0.625670\pi$
$60$	18.3865	2.37369
$61$	−4.64238	−0.594395	−0.297198	−	0.954816i	$-0.596052\pi$
$61$	−4.64238	−0.594395	−0.297198	+	0.954816i	$0.596052\pi$
$62$	9.08261	1.15349
$63$	1.00000	0.125988
$64$	1.63378	0.204222
$65$	16.0444	1.99006
$66$	−5.24017	−0.645020
$67$	−5.76668	−0.704512	−0.352256	−	0.935904i	$-0.614585\pi$
$67$	−5.76668	−0.704512	−0.352256	+	0.935904i	$0.614585\pi$
$68$	−12.0792	−1.46482
$69$	−5.68481	−0.684370
$70$	10.3119	1.23251
$71$	−5.68274	−0.674417	−0.337208	−	0.941430i	$-0.609483\pi$
$71$	−5.68274	−0.674417	−0.337208	+	0.941430i	$0.609483\pi$
$72$	−6.58873	−0.776489
$73$	−12.0298	−1.40798	−0.703989	−	0.710211i	$-0.748599\pi$
$73$	−12.0298	−1.40798	−0.703989	+	0.710211i	$0.748599\pi$
$74$	−18.6129	−2.16370
$75$	−11.1839	−1.29141
$76$	4.36852	0.501104
$77$	−2.04432	−0.232972
$78$	−10.2230	−1.15753
$79$	−2.09234	−0.235406	−0.117703	−	0.993049i	$-0.537553\pi$
$79$	−2.09234	−0.235406	−0.117703	+	0.993049i	$0.537553\pi$
$80$	−31.1694	−3.48484
$81$	1.00000	0.111111
$82$	5.75484	0.635516
$83$	−0.102376	−0.0112372	−0.00561860	−	0.999984i	$-0.501788\pi$
$83$	−0.102376	−0.0112372	−0.00561860	+	0.999984i	$0.501788\pi$
$84$	−4.57042	−0.498674
$85$	10.6322	1.15323
$86$	−23.5226	−2.53651
$87$	−2.11501	−0.226753
$88$	13.4695	1.43585
$89$	−9.25679	−0.981218	−0.490609	−	0.871380i	$-0.663226\pi$
$89$	−9.25679	−0.981218	−0.490609	+	0.871380i	$0.663226\pi$
$90$	10.3119	1.08697
$91$	−3.98824	−0.418081
$92$	25.9820	2.70881
$93$	3.54335	0.367428
$94$	27.0585	2.79087
$95$	−3.84521	−0.394510
$96$	6.68269	0.682049
$97$	17.4113	1.76785	0.883923	−	0.467632i	$-0.154893\pi$
$97$	17.4113	1.76785	0.883923	+	0.467632i	$0.154893\pi$
$98$	−2.56328	−0.258931
$99$	−2.04432	−0.205462
$100$	51.1154	5.11154
$101$	4.05370	0.403358	0.201679	−	0.979452i	$-0.435360\pi$
$101$	4.05370	0.403358	0.201679	+	0.979452i	$0.435360\pi$
$102$	−6.77453	−0.670779
$103$	−6.68168	−0.658365	−0.329183	−	0.944266i	$-0.606773\pi$
$103$	−6.68168	−0.658365	−0.329183	+	0.944266i	$0.606773\pi$
$104$	26.2774	2.57671
$105$	4.02293	0.392597
$106$	−2.31636	−0.224985
$107$	15.9187	1.53892	0.769460	−	0.638695i	$-0.220525\pi$
$107$	15.9187	1.53892	0.769460	+	0.638695i	$0.220525\pi$
$108$	−4.57042	−0.439789
$109$	0.670528	0.0642249	0.0321124	−	0.999484i	$-0.489777\pi$
$109$	0.670528	0.0642249	0.0321124	+	0.999484i	$0.489777\pi$
$110$	−21.0808	−2.00998
$111$	−7.26134	−0.689216
$112$	7.74793	0.732111
$113$	5.49468	0.516896	0.258448	−	0.966025i	$-0.416789\pi$
$113$	5.49468	0.516896	0.258448	+	0.966025i	$0.416789\pi$
$114$	2.45005	0.229468
$115$	−22.8696	−2.13260
$116$	9.66649	0.897511
$117$	−3.98824	−0.368713
$118$	15.1458	1.39429
$119$	−2.64291	−0.242275
$120$	−26.5060	−2.41965
$121$	−6.82077	−0.620070
$122$	11.8997	1.07735
$123$	2.24510	0.202434
$124$	−16.1946	−1.45432
$125$	−24.8776	−2.22512
$126$	−2.56328	−0.228355
$127$	−7.83776	−0.695489	−0.347744	−	0.937589i	$-0.613052\pi$
$127$	−7.83776	−0.695489	−0.347744	+	0.937589i	$0.613052\pi$
$128$	9.17755	0.811189
$129$	−9.17675	−0.807968
$130$	−41.1264	−3.60702
$131$	13.3048	1.16244	0.581221	−	0.813746i	$-0.302575\pi$
$131$	13.3048	1.16244	0.581221	+	0.813746i	$0.302575\pi$
$132$	9.34340	0.813238
$133$	0.955824	0.0828804
$134$	14.7816	1.27694
$135$	4.02293	0.346238
$136$	17.4134	1.49319
$137$	−12.5012	−1.06805	−0.534024	−	0.845469i	$-0.679321\pi$
$137$	−12.5012	−1.06805	−0.534024	+	0.845469i	$0.679321\pi$
$138$	14.5718	1.24043
$139$	3.94608	0.334702	0.167351	−	0.985897i	$-0.446479\pi$
$139$	3.94608	0.334702	0.167351	+	0.985897i	$0.446479\pi$
$140$	−18.3865	−1.55394
$141$	10.5562	0.888992
$142$	14.5665	1.22239
$143$	8.15323	0.681807
$144$	7.74793	0.645661
$145$	−8.50853	−0.706595
$146$	30.8357	2.55198
$147$	−1.00000	−0.0824786
$148$	33.1874	2.72799
$149$	−14.7408	−1.20761	−0.603807	−	0.797130i	$-0.706350\pi$
$149$	−14.7408	−1.20761	−0.603807	+	0.797130i	$0.706350\pi$
$150$	28.6676	2.34070
$151$	−0.942868	−0.0767295	−0.0383647	−	0.999264i	$-0.512215\pi$
$151$	−0.942868	−0.0767295	−0.0383647	+	0.999264i	$0.512215\pi$
$152$	−6.29766	−0.510808
$153$	−2.64291	−0.213667
$154$	5.24017	0.422265
$155$	14.2546	1.14496
$156$	18.2280	1.45940
$157$	−1.74511	−0.139275	−0.0696376	−	0.997572i	$-0.522184\pi$
$157$	−1.74511	−0.139275	−0.0696376	+	0.997572i	$0.522184\pi$
$158$	5.36325	0.426678
$159$	−0.903669	−0.0716656
$160$	26.8840	2.12537
$161$	5.68481	0.448025
$162$	−2.56328	−0.201391
$163$	5.76830	0.451808	0.225904	−	0.974150i	$-0.427466\pi$
$163$	5.76830	0.451808	0.225904	+	0.974150i	$0.427466\pi$
$164$	−10.2611	−0.801256
$165$	−8.22414	−0.640248
$166$	0.262418	0.0203676
$167$	−9.38778	−0.726448	−0.363224	−	0.931702i	$-0.618324\pi$
$167$	−9.38778	−0.726448	−0.363224	+	0.931702i	$0.618324\pi$
$168$	6.58873	0.508331
$169$	2.90607	0.223544
$170$	−27.2534	−2.09024
$171$	0.955824	0.0730937
$172$	41.9417	3.19802
$173$	−1.87702	−0.142707	−0.0713535	−	0.997451i	$-0.522732\pi$
$173$	−1.87702	−0.142707	−0.0713535	+	0.997451i	$0.522732\pi$
$174$	5.42137	0.410993
$175$	11.1839	0.845427
$176$	−15.8392	−1.19393
$177$	5.90876	0.444130
$178$	23.7278	1.77847
$179$	−20.4454	−1.52816	−0.764079	−	0.645123i	$-0.776806\pi$
$179$	−20.4454	−1.52816	−0.764079	+	0.645123i	$0.776806\pi$
$180$	−18.3865	−1.37045
$181$	8.66454	0.644030	0.322015	−	0.946735i	$-0.395640\pi$
$181$	8.66454	0.644030	0.322015	+	0.946735i	$0.395640\pi$
$182$	10.2230	0.757779
$183$	4.64238	0.343174
$184$	−37.4556	−2.76127
$185$	−29.2118	−2.14770
$186$	−9.08261	−0.665969
$187$	5.40295	0.395103
$188$	−48.2463	−3.51872
$189$	−1.00000	−0.0727393
$190$	9.85636	0.715056
$191$	−21.0559	−1.52355	−0.761775	−	0.647842i	$-0.775672\pi$
$191$	−21.0559	−1.52355	−0.761775	+	0.647842i	$0.775672\pi$
$192$	−1.63378	−0.117908
$193$	−24.3282	−1.75118	−0.875590	−	0.483054i	$-0.839527\pi$
$193$	−24.3282	−1.75118	−0.875590	+	0.483054i	$0.839527\pi$
$194$	−44.6300	−3.20425
$195$	−16.0444	−1.14896
$196$	4.57042	0.326459
$197$	9.16186	0.652755	0.326378	−	0.945239i	$-0.394172\pi$
$197$	9.16186	0.652755	0.326378	+	0.945239i	$0.394172\pi$
$198$	5.24017	0.372402
$199$	−15.8809	−1.12576	−0.562882	−	0.826537i	$-0.690308\pi$
$199$	−15.8809	−1.12576	−0.562882	+	0.826537i	$0.690308\pi$
$200$	−73.6880	−5.21053
$201$	5.76668	0.406750
$202$	−10.3908	−0.731093
$203$	2.11501	0.148445
$204$	12.0792	0.845715
$205$	9.03189	0.630815
$206$	17.1270	1.19330
$207$	5.68481	0.395121
$208$	−30.9006	−2.14257
$209$	−1.95401	−0.135161
$210$	−10.3119	−0.711589
$211$	11.3669	0.782528	0.391264	−	0.920278i	$-0.372038\pi$
$211$	11.3669	0.782528	0.391264	+	0.920278i	$0.372038\pi$
$212$	4.13015	0.283660
$213$	5.68274	0.389375
$214$	−40.8042	−2.78932
$215$	−36.9174	−2.51775
$216$	6.58873	0.448306
$217$	−3.54335	−0.240538
$218$	−1.71875	−0.116409
$219$	12.0298	0.812896
$220$	37.5878	2.53417
$221$	10.5406	0.709035
$222$	18.6129	1.24921
$223$	16.2233	1.08639	0.543196	−	0.839606i	$-0.317214\pi$
$223$	16.2233	1.08639	0.543196	+	0.839606i	$0.317214\pi$
$224$	−6.68269	−0.446506
$225$	11.1839	0.745596
$226$	−14.0844	−0.936882
$227$	−6.93735	−0.460448	−0.230224	−	0.973138i	$-0.573946\pi$
$227$	−6.93735	−0.460448	−0.230224	+	0.973138i	$0.573946\pi$
$228$	−4.36852	−0.289312
$229$	1.57459	0.104052	0.0520259	−	0.998646i	$-0.483432\pi$
$229$	1.57459	0.104052	0.0520259	+	0.998646i	$0.483432\pi$
$230$	58.6212	3.86537
$231$	2.04432	0.134506
$232$	−13.9352	−0.914892
$233$	−6.63984	−0.434990	−0.217495	−	0.976061i	$-0.569789\pi$
$233$	−6.63984	−0.434990	−0.217495	+	0.976061i	$0.569789\pi$
$234$	10.2230	0.668298
$235$	42.4668	2.77023
$236$	−27.0055	−1.75791
$237$	2.09234	0.135912
$238$	6.77453	0.439128
$239$	7.77321	0.502807	0.251403	−	0.967882i	$-0.419108\pi$
$239$	7.77321	0.502807	0.251403	+	0.967882i	$0.419108\pi$
$240$	31.1694	2.01197
$241$	−17.0364	−1.09741	−0.548707	−	0.836015i	$-0.684880\pi$
$241$	−17.0364	−1.09741	−0.548707	+	0.836015i	$0.684880\pi$
$242$	17.4836	1.12389
$243$	−1.00000	−0.0641500
$244$	−21.2176	−1.35832
$245$	−4.02293	−0.257015
$246$	−5.75484	−0.366915
$247$	−3.81205	−0.242555
$248$	23.3462	1.48248
$249$	0.102376	0.00648780
$250$	63.7682	4.03306
$251$	−29.1693	−1.84115	−0.920576	−	0.390563i	$-0.872280\pi$
$251$	−29.1693	−1.84115	−0.920576	+	0.390563i	$0.872280\pi$
$252$	4.57042	0.287910
$253$	−11.6215	−0.730640
$254$	20.0904	1.26058
$255$	−10.6322	−0.665817
$256$	−26.7922	−1.67451
$257$	15.1827	0.947070	0.473535	−	0.880775i	$-0.342978\pi$
$257$	15.1827	0.947070	0.473535	+	0.880775i	$0.342978\pi$
$258$	23.5226	1.46445
$259$	7.26134	0.451198
$260$	73.3297	4.54772
$261$	2.11501	0.130916
$262$	−34.1039	−2.10694
$263$	27.1481	1.67403	0.837013	−	0.547183i	$-0.184300\pi$
$263$	27.1481	1.67403	0.837013	+	0.547183i	$0.184300\pi$
$264$	−13.4695	−0.828987
$265$	−3.63539	−0.223320
$266$	−2.45005	−0.150222
$267$	9.25679	0.566507
$268$	−26.3562	−1.60996
$269$	2.49848	0.152335	0.0761675	−	0.997095i	$-0.475732\pi$
$269$	2.49848	0.152335	0.0761675	+	0.997095i	$0.475732\pi$
$270$	−10.3119	−0.627563
$271$	−5.52726	−0.335757	−0.167878	−	0.985808i	$-0.553692\pi$
$271$	−5.52726	−0.335757	−0.167878	+	0.985808i	$0.553692\pi$
$272$	−20.4771	−1.24161
$273$	3.98824	0.241379
$274$	32.0441	1.93585
$275$	−22.8635	−1.37872
$276$	−25.9820	−1.56393
$277$	4.50271	0.270541	0.135271	−	0.990809i	$-0.456810\pi$
$277$	4.50271	0.270541	0.135271	+	0.990809i	$0.456810\pi$
$278$	−10.1149	−0.606653
$279$	−3.54335	−0.212135
$280$	26.5060	1.58403
$281$	1.30509	0.0778550	0.0389275	−	0.999242i	$-0.487606\pi$
$281$	1.30509	0.0778550	0.0389275	+	0.999242i	$0.487606\pi$
$282$	−27.0585	−1.61131
$283$	−4.46319	−0.265309	−0.132655	−	0.991162i	$-0.542350\pi$
$283$	−4.46319	−0.265309	−0.132655	+	0.991162i	$0.542350\pi$
$284$	−25.9725	−1.54119
$285$	3.84521	0.227771
$286$	−20.8990	−1.23579
$287$	−2.24510	−0.132524
$288$	−6.68269	−0.393781
$289$	−10.0150	−0.589119
$290$	21.8098	1.28071
$291$	−17.4113	−1.02067
$292$	−54.9811	−3.21753
$293$	0.00623872	0.000364470 0	0.000182235	−	1.00000i	$-0.499942\pi$
$293$	0.00623872	0.000364470 0	0.000182235		1.00000i	$0.499942\pi$
$294$	2.56328	0.149494
$295$	23.7705	1.38397
$296$	−47.8430	−2.78082
$297$	2.04432	0.118623
$298$	37.7849	2.18882
$299$	−22.6724	−1.31118
$300$	−51.1154	−2.95115
$301$	9.17675	0.528939
$302$	2.41684	0.139073
$303$	−4.05370	−0.232879
$304$	7.40566	0.424743
$305$	18.6759	1.06938
$306$	6.77453	0.387274
$307$	−25.8509	−1.47539	−0.737694	−	0.675135i	$-0.764085\pi$
$307$	−25.8509	−1.47539	−0.737694	+	0.675135i	$0.764085\pi$
$308$	−9.34340	−0.532389
$309$	6.68168	0.380107
$310$	−36.5387	−2.07526
$311$	8.78160	0.497959	0.248979	−	0.968509i	$-0.419905\pi$
$311$	8.78160	0.497959	0.248979	+	0.968509i	$0.419905\pi$
$312$	−26.2774	−1.48767
$313$	−18.0846	−1.02220	−0.511101	−	0.859521i	$-0.670762\pi$
$313$	−18.0846	−1.02220	−0.511101	+	0.859521i	$0.670762\pi$
$314$	4.47322	0.252439
$315$	−4.02293	−0.226666
$316$	−9.56287	−0.537953
$317$	4.96903	0.279088	0.139544	−	0.990216i	$-0.455436\pi$
$317$	4.96903	0.279088	0.139544	+	0.990216i	$0.455436\pi$
$318$	2.31636	0.129895
$319$	−4.32375	−0.242083
$320$	−6.57256	−0.367417
$321$	−15.9187	−0.888496
$322$	−14.5718	−0.812053
$323$	−2.52616	−0.140559
$324$	4.57042	0.253912
$325$	−44.6043	−2.47420
$326$	−14.7858	−0.818909
$327$	−0.670528	−0.0370803
$328$	14.7924	0.816773
$329$	−10.5562	−0.581982
$330$	21.0808	1.16046
$331$	−9.09624	−0.499974	−0.249987	−	0.968249i	$-0.580426\pi$
$331$	−9.09624	−0.499974	−0.249987	+	0.968249i	$0.580426\pi$
$332$	−0.467901	−0.0256794
$333$	7.26134	0.397919
$334$	24.0635	1.31670
$335$	23.1989	1.26749
$336$	−7.74793	−0.422684
$337$	−19.0399	−1.03717	−0.518585	−	0.855026i	$-0.673541\pi$
$337$	−19.0399	−1.03717	−0.518585	+	0.855026i	$0.673541\pi$
$338$	−7.44907	−0.405176
$339$	−5.49468	−0.298430
$340$	48.5938	2.63537
$341$	7.24373	0.392270
$342$	−2.45005	−0.132483
$343$	1.00000	0.0539949
$344$	−60.4631	−3.25995
$345$	22.8696	1.23126
$346$	4.81133	0.258659
$347$	16.9033	0.907414	0.453707	−	0.891151i	$-0.350101\pi$
$347$	16.9033	0.907414	0.453707	+	0.891151i	$0.350101\pi$
$348$	−9.66649	−0.518178
$349$	15.3604	0.822221	0.411110	−	0.911586i	$-0.365141\pi$
$349$	15.3604	0.822221	0.411110	+	0.911586i	$0.365141\pi$
$350$	−28.6676	−1.53235
$351$	3.98824	0.212877
$352$	13.6615	0.728163
$353$	−2.26559	−0.120585	−0.0602925	−	0.998181i	$-0.519203\pi$
$353$	−2.26559	−0.120585	−0.0602925	+	0.998181i	$0.519203\pi$
$354$	−15.1458	−0.804992
$355$	22.8612	1.21335
$356$	−42.3075	−2.24229
$357$	2.64291	0.139878
$358$	52.4072	2.76981
$359$	−7.90961	−0.417453	−0.208727	−	0.977974i	$-0.566932\pi$
$359$	−7.90961	−0.417453	−0.208727	+	0.977974i	$0.566932\pi$
$360$	26.5060	1.39699
$361$	−18.0864	−0.951916
$362$	−22.2097	−1.16731
$363$	6.82077	0.357997
$364$	−18.2280	−0.955404
$365$	48.3949	2.53310
$366$	−11.8997	−0.622009
$367$	23.0178	1.20152	0.600758	−	0.799431i	$-0.294865\pi$
$367$	23.0178	1.20152	0.600758	+	0.799431i	$0.294865\pi$
$368$	44.0455	2.29603
$369$	−2.24510	−0.116875
$370$	74.8782	3.89273
$371$	0.903669	0.0469161
$372$	16.1946	0.839652
$373$	0.910604	0.0471493	0.0235746	−	0.999722i	$-0.492495\pi$
$373$	0.910604	0.0471493	0.0235746	+	0.999722i	$0.492495\pi$
$374$	−13.8493	−0.716130
$375$	24.8776	1.28467
$376$	69.5519	3.58687
$377$	−8.43517	−0.434433
$378$	2.56328	0.131841
$379$	12.6111	0.647788	0.323894	−	0.946093i	$-0.395008\pi$
$379$	12.6111	0.647788	0.323894	+	0.946093i	$0.395008\pi$
$380$	−17.5742	−0.901539
$381$	7.83776	0.401541
$382$	53.9722	2.76146
$383$	−1.00000	−0.0510976
$383$	−1.00000	−0.0510976
$384$	−9.17755	−0.468340
$385$	8.22414	0.419141
$386$	62.3600	3.17404
$387$	9.17675	0.466481
$388$	79.5769	4.03991
$389$	−6.87183	−0.348416	−0.174208	−	0.984709i	$-0.555736\pi$
$389$	−6.87183	−0.348416	−0.174208	+	0.984709i	$0.555736\pi$
$390$	41.1264	2.08251
$391$	−15.0244	−0.759818
$392$	−6.58873	−0.332781
$393$	−13.3048	−0.671136
$394$	−23.4844	−1.18313
$395$	8.41732	0.423521
$396$	−9.34340	−0.469523
$397$	−18.1901	−0.912935	−0.456467	−	0.889740i	$-0.650886\pi$
$397$	−18.1901	−0.912935	−0.456467	+	0.889740i	$0.650886\pi$
$398$	40.7072	2.04047
$399$	−0.955824	−0.0478510
$400$	86.6524	4.33262
$401$	−13.1933	−0.658840	−0.329420	−	0.944183i	$-0.606853\pi$
$401$	−13.1933	−0.658840	−0.329420	+	0.944183i	$0.606853\pi$
$402$	−14.7816	−0.737241
$403$	14.1317	0.703952
$404$	18.5271	0.921759
$405$	−4.02293	−0.199901
$406$	−5.42137	−0.269058
$407$	−14.8445	−0.735814
$408$	−17.4134	−0.862093
$409$	−30.4427	−1.50530	−0.752648	−	0.658423i	$-0.771224\pi$
$409$	−30.4427	−1.50530	−0.752648	+	0.658423i	$0.771224\pi$
$410$	−23.1513	−1.14336
$411$	12.5012	0.616638
$412$	−30.5381	−1.50450
$413$	−5.90876	−0.290751
$414$	−14.5718	−0.716163
$415$	0.411850	0.0202169
$416$	26.6522	1.30673
$417$	−3.94608	−0.193240
$418$	5.00867	0.244982
$419$	−3.12119	−0.152480	−0.0762400	−	0.997089i	$-0.524292\pi$
$419$	−3.12119	−0.152480	−0.0762400	+	0.997089i	$0.524292\pi$
$420$	18.3865	0.897169
$421$	−27.5179	−1.34114	−0.670570	−	0.741846i	$-0.733950\pi$
$421$	−27.5179	−1.34114	−0.670570	+	0.741846i	$0.733950\pi$
$422$	−29.1365	−1.41834
$423$	−10.5562	−0.513260
$424$	−5.95403	−0.289153
$425$	−29.5582	−1.43378
$426$	−14.5665	−0.705748
$427$	−4.64238	−0.224660
$428$	72.7553	3.51676
$429$	−8.15323	−0.393642
$430$	94.6298	4.56345
$431$	28.6289	1.37900	0.689502	−	0.724284i	$-0.257829\pi$
$431$	28.6289	1.37900	0.689502	+	0.724284i	$0.257829\pi$
$432$	−7.74793	−0.372773
$433$	2.89495	0.139123	0.0695613	−	0.997578i	$-0.477840\pi$
$433$	2.89495	0.139123	0.0695613	+	0.997578i	$0.477840\pi$
$434$	9.08261	0.435979
$435$	8.50853	0.407953
$436$	3.06460	0.146768
$437$	5.43367	0.259928
$438$	−30.8357	−1.47339
$439$	37.3897	1.78451	0.892256	−	0.451530i	$-0.149122\pi$
$439$	37.3897	1.78451	0.892256	+	0.451530i	$0.149122\pi$
$440$	−54.1866	−2.58325
$441$	1.00000	0.0476190
$442$	−27.0185	−1.28514
$443$	−2.85854	−0.135813	−0.0679066	−	0.997692i	$-0.521632\pi$
$443$	−2.85854	−0.135813	−0.0679066	+	0.997692i	$0.521632\pi$
$444$	−33.1874	−1.57500
$445$	37.2394	1.76532
$446$	−41.5849	−1.96910
$447$	14.7408	0.697217
$448$	1.63378	0.0771886
$449$	−6.77794	−0.319871	−0.159935	−	0.987127i	$-0.551129\pi$
$449$	−6.77794	−0.319871	−0.159935	+	0.987127i	$0.551129\pi$
$450$	−28.6676	−1.35140
$451$	4.58970	0.216121
$452$	25.1130	1.18122
$453$	0.942868	0.0442998
$454$	17.7824	0.834570
$455$	16.0444	0.752173
$456$	6.29766	0.294915
$457$	3.47991	0.162783	0.0813916	−	0.996682i	$-0.474064\pi$
$457$	3.47991	0.162783	0.0813916	+	0.996682i	$0.474064\pi$
$458$	−4.03612	−0.188595
$459$	2.64291	0.123361
$460$	−104.524	−4.87344
$461$	17.4648	0.813415	0.406708	−	0.913558i	$-0.366677\pi$
$461$	17.4648	0.813415	0.406708	+	0.913558i	$0.366677\pi$
$462$	−5.24017	−0.243795
$463$	6.87390	0.319457	0.159729	−	0.987161i	$-0.448938\pi$
$463$	6.87390	0.319457	0.159729	+	0.987161i	$0.448938\pi$
$464$	16.3869	0.760745
$465$	−14.2546	−0.661043
$466$	17.0198	0.788427
$467$	17.8400	0.825535	0.412768	−	0.910836i	$-0.364562\pi$
$467$	17.8400	0.825535	0.412768	+	0.910836i	$0.364562\pi$
$468$	−18.2280	−0.842588
$469$	−5.76668	−0.266280
$470$	−108.854	−5.02108
$471$	1.74511	0.0804106
$472$	38.9312	1.79195
$473$	−18.7602	−0.862595
$474$	−5.36325	−0.246342
$475$	10.6899	0.490485
$476$	−12.0792	−0.553650
$477$	0.903669	0.0413761
$478$	−19.9249	−0.911345
$479$	−7.67759	−0.350798	−0.175399	−	0.984497i	$-0.556122\pi$
$479$	−7.67759	−0.350798	−0.175399	+	0.984497i	$0.556122\pi$
$480$	−26.8840	−1.22708
$481$	−28.9600	−1.32046
$482$	43.6693	1.98908
$483$	−5.68481	−0.258668
$484$	−31.1738	−1.41699
$485$	−70.0443	−3.18055
$486$	2.56328	0.116273
$487$	5.06733	0.229622	0.114811	−	0.993387i	$-0.463374\pi$
$487$	5.06733	0.229622	0.114811	+	0.993387i	$0.463374\pi$
$488$	30.5874	1.38462
$489$	−5.76830	−0.260851
$490$	10.3119	0.465844
$491$	28.4086	1.28206	0.641031	−	0.767515i	$-0.278507\pi$
$491$	28.4086	1.28206	0.641031	+	0.767515i	$0.278507\pi$
$492$	10.2611	0.462605
$493$	−5.58978	−0.251751
$494$	9.77138	0.439635
$495$	8.22414	0.369648
$496$	−27.4536	−1.23270
$497$	−5.68274	−0.254906
$498$	−0.262418	−0.0117592
$499$	−20.3883	−0.912706	−0.456353	−	0.889799i	$-0.650844\pi$
$499$	−20.3883	−0.912706	−0.456353	+	0.889799i	$0.650844\pi$
$500$	−113.701	−5.08486
$501$	9.38778	0.419415
$502$	74.7693	3.33712
$503$	26.0575	1.16185	0.580923	−	0.813958i	$-0.302692\pi$
$503$	26.0575	1.16185	0.580923	+	0.813958i	$0.302692\pi$
$504$	−6.58873	−0.293485
$505$	−16.3077	−0.725684
$506$	29.7893	1.32430
$507$	−2.90607	−0.129063
$508$	−35.8219	−1.58934
$509$	34.9571	1.54945	0.774724	−	0.632300i	$-0.217889\pi$
$509$	34.9571	1.54945	0.774724	+	0.632300i	$0.217889\pi$
$510$	27.2534	1.20680
$511$	−12.0298	−0.532165
$512$	50.3210	2.22389
$513$	−0.955824	−0.0422006
$514$	−38.9175	−1.71658
$515$	26.8799	1.18447
$516$	−41.9417	−1.84638
$517$	21.5802	0.949097
$518$	−18.6129	−0.817803
$519$	1.87702	0.0823919
$520$	−105.712	−4.63579
$521$	5.04523	0.221036	0.110518	−	0.993874i	$-0.464749\pi$
$521$	5.04523	0.221036	0.110518	+	0.993874i	$0.464749\pi$
$522$	−5.42137	−0.237287
$523$	−39.3182	−1.71927	−0.859633	−	0.510912i	$-0.829308\pi$
$523$	−39.3182	−1.71927	−0.859633	+	0.510912i	$0.829308\pi$
$524$	60.8084	2.65643
$525$	−11.1839	−0.488107
$526$	−69.5884	−3.03420
$527$	9.36476	0.407935
$528$	15.8392	0.689314
$529$	9.31701	0.405087
$530$	9.31854	0.404772
$531$	−5.90876	−0.256418
$532$	4.36852	0.189399
$533$	8.95402	0.387841
$534$	−23.7278	−1.02680
$535$	−64.0398	−2.76868
$536$	37.9951	1.64114
$537$	20.4454	0.882282
$538$	−6.40431	−0.276109
$539$	−2.04432	−0.0880550
$540$	18.3865	0.791228
$541$	−15.1529	−0.651474	−0.325737	−	0.945460i	$-0.605612\pi$
$541$	−15.1529	−0.651474	−0.325737	+	0.945460i	$0.605612\pi$
$542$	14.1679	0.608564
$543$	−8.66454	−0.371831
$544$	17.6618	0.757242
$545$	−2.69748	−0.115547
$546$	−10.2230	−0.437504
$547$	1.85970	0.0795149	0.0397575	−	0.999209i	$-0.487341\pi$
$547$	1.85970	0.0795149	0.0397575	+	0.999209i	$0.487341\pi$
$548$	−57.1357	−2.44072
$549$	−4.64238	−0.198132
$550$	58.6057	2.49896
$551$	2.02158	0.0861220
$552$	37.4556	1.59422
$553$	−2.09234	−0.0889752
$554$	−11.5417	−0.490361
$555$	29.2118	1.23997
$556$	18.0353	0.764866
$557$	38.2706	1.62158	0.810790	−	0.585338i	$-0.199038\pi$
$557$	38.2706	1.62158	0.810790	+	0.585338i	$0.199038\pi$
$558$	9.08261	0.384498
$559$	−36.5991	−1.54798
$560$	−31.1694	−1.31715
$561$	−5.40295	−0.228113
$562$	−3.34531	−0.141113
$563$	−21.6875	−0.914021	−0.457011	−	0.889461i	$-0.651080\pi$
$563$	−21.6875	−0.914021	−0.457011	+	0.889461i	$0.651080\pi$
$564$	48.2463	2.03154
$565$	−22.1047	−0.929951
$566$	11.4404	0.480877
$567$	1.00000	0.0419961
$568$	37.4420	1.57103
$569$	−18.5523	−0.777755	−0.388877	−	0.921289i	$-0.627137\pi$
$569$	−18.5523	−0.777755	−0.388877	+	0.921289i	$0.627137\pi$
$570$	−9.85636	−0.412838
$571$	−0.274191	−0.0114745	−0.00573726	−	0.999984i	$-0.501826\pi$
$571$	−0.274191	−0.0114745	−0.00573726	+	0.999984i	$0.501826\pi$
$572$	37.2637	1.55807
$573$	21.0559	0.879622
$574$	5.75484	0.240202
$575$	63.5785	2.65141
$576$	1.63378	0.0680740
$577$	41.8550	1.74245	0.871224	−	0.490886i	$-0.163327\pi$
$577$	41.8550	1.74245	0.871224	+	0.490886i	$0.163327\pi$
$578$	25.6713	1.06779
$579$	24.3282	1.01104
$580$	−38.8876	−1.61472
$581$	−0.102376	−0.00424726
$582$	44.6300	1.84997
$583$	−1.84739	−0.0765109
$584$	79.2608	3.27984
$585$	16.0444	0.663354
$586$	−0.0159916	−0.000660607 0
$587$	22.8612	0.943584	0.471792	−	0.881710i	$-0.343607\pi$
$587$	22.8612	0.943584	0.471792	+	0.881710i	$0.343607\pi$
$588$	−4.57042	−0.188481
$589$	−3.38682	−0.139551
$590$	−60.9306	−2.50847
$591$	−9.16186	−0.376868
$592$	56.2604	2.31229
$593$	−12.5229	−0.514253	−0.257127	−	0.966378i	$-0.582776\pi$
$593$	−12.5229	−0.514253	−0.257127	+	0.966378i	$0.582776\pi$
$594$	−5.24017	−0.215007
$595$	10.6322	0.435879
$596$	−67.3718	−2.75966
$597$	15.8809	0.649961
$598$	58.1157	2.37653
$599$	−13.0209	−0.532019	−0.266009	−	0.963970i	$-0.585705\pi$
$599$	−13.0209	−0.532019	−0.266009	+	0.963970i	$0.585705\pi$
$600$	73.6880	3.00830
$601$	11.7006	0.477279	0.238640	−	0.971108i	$-0.423299\pi$
$601$	11.7006	0.477279	0.238640	+	0.971108i	$0.423299\pi$
$602$	−23.5226	−0.958711
$603$	−5.76668	−0.234837
$604$	−4.30931	−0.175343
$605$	27.4394	1.11557
$606$	10.3908	0.422096
$607$	−32.1572	−1.30522	−0.652610	−	0.757694i	$-0.726326\pi$
$607$	−32.1572	−1.30522	−0.652610	+	0.757694i	$0.726326\pi$
$608$	−6.38748	−0.259046
$609$	−2.11501	−0.0857045
$610$	−47.8717	−1.93827
$611$	42.1007	1.70321
$612$	−12.0792	−0.488274
$613$	−13.8065	−0.557639	−0.278819	−	0.960344i	$-0.589943\pi$
$613$	−13.8065	−0.557639	−0.278819	+	0.960344i	$0.589943\pi$
$614$	66.2632	2.67416
$615$	−9.03189	−0.364201
$616$	13.4695	0.542700
$617$	30.2406	1.21744	0.608720	−	0.793385i	$-0.291683\pi$
$617$	30.2406	1.21744	0.608720	+	0.793385i	$0.291683\pi$
$618$	−17.1270	−0.688951
$619$	38.6627	1.55398	0.776992	−	0.629511i	$-0.216745\pi$
$619$	38.6627	1.55398	0.776992	+	0.629511i	$0.216745\pi$
$620$	65.1498	2.61648
$621$	−5.68481	−0.228123
$622$	−22.5097	−0.902558
$623$	−9.25679	−0.370866
$624$	30.9006	1.23701
$625$	44.1609	1.76643
$626$	46.3560	1.85276
$627$	1.95401	0.0780355
$628$	−7.97591	−0.318273
$629$	−19.1911	−0.765198
$630$	10.3119	0.410836
$631$	−27.1322	−1.08011	−0.540057	−	0.841628i	$-0.681597\pi$
$631$	−27.1322	−1.08011	−0.540057	+	0.841628i	$0.681597\pi$
$632$	13.7858	0.548371
$633$	−11.3669	−0.451793
$634$	−12.7370	−0.505852
$635$	31.5307	1.25126
$636$	−4.13015	−0.163771
$637$	−3.98824	−0.158020
$638$	11.0830	0.438780
$639$	−5.68274	−0.224806
$640$	−36.9206	−1.45942
$641$	−44.4324	−1.75497	−0.877487	−	0.479601i	$-0.840781\pi$
$641$	−44.4324	−1.75497	−0.877487	+	0.479601i	$0.840781\pi$
$642$	40.8042	1.61041
$643$	30.7327	1.21198	0.605989	−	0.795473i	$-0.292778\pi$
$643$	30.7327	1.21198	0.605989	+	0.795473i	$0.292778\pi$
$644$	25.9820	1.02383
$645$	36.9174	1.45362
$646$	6.47526	0.254766
$647$	−45.6093	−1.79309	−0.896543	−	0.442956i	$-0.853930\pi$
$647$	−45.6093	−1.79309	−0.896543	+	0.442956i	$0.853930\pi$
$648$	−6.58873	−0.258830
$649$	12.0794	0.474157
$650$	114.333	4.48452
$651$	3.54335	0.138875
$652$	26.3636	1.03248
$653$	−8.67504	−0.339480	−0.169740	−	0.985489i	$-0.554293\pi$
$653$	−8.67504	−0.339480	−0.169740	+	0.985489i	$0.554293\pi$
$654$	1.71875	0.0672085
$655$	−53.5241	−2.09136
$656$	−17.3949	−0.679157
$657$	−12.0298	−0.469326
$658$	27.0585	1.05485
$659$	−12.9070	−0.502784	−0.251392	−	0.967885i	$-0.580888\pi$
$659$	−12.9070	−0.502784	−0.251392	+	0.967885i	$0.580888\pi$
$660$	−37.5878	−1.46310
$661$	22.2481	0.865350	0.432675	−	0.901550i	$-0.357570\pi$
$661$	22.2481	0.865350	0.432675	+	0.901550i	$0.357570\pi$
$662$	23.3162	0.906211
$663$	−10.5406	−0.409362
$664$	0.674526	0.0261767
$665$	−3.84521	−0.149111
$666$	−18.6129	−0.721234
$667$	12.0234	0.465548
$668$	−42.9061	−1.66009
$669$	−16.2233	−0.627228
$670$	−59.4654	−2.29735
$671$	9.49049	0.366376
$672$	6.68269	0.257790
$673$	−3.96523	−0.152849	−0.0764243	−	0.997075i	$-0.524350\pi$
$673$	−3.96523	−0.152849	−0.0764243	+	0.997075i	$0.524350\pi$
$674$	48.8047	1.87989
$675$	−11.1839	−0.430470
$676$	13.2820	0.510844
$677$	8.87621	0.341140	0.170570	−	0.985346i	$-0.445439\pi$
$677$	8.87621	0.341140	0.170570	+	0.985346i	$0.445439\pi$
$678$	14.0844	0.540909
$679$	17.4113	0.668183
$680$	−70.0529	−2.68641
$681$	6.93735	0.265840
$682$	−18.5677	−0.710996
$683$	−0.742743	−0.0284203	−0.0142101	−	0.999899i	$-0.504523\pi$
$683$	−0.742743	−0.0284203	−0.0142101	+	0.999899i	$0.504523\pi$
$684$	4.36852	0.167035
$685$	50.2913	1.92153
$686$	−2.56328	−0.0978666
$687$	−1.57459	−0.0600743
$688$	71.1009	2.71069
$689$	−3.60405	−0.137303
$690$	−58.6212	−2.23167
$691$	41.0374	1.56114	0.780568	−	0.625071i	$-0.214930\pi$
$691$	41.0374	1.56114	0.780568	+	0.625071i	$0.214930\pi$
$692$	−8.57877	−0.326116
$693$	−2.04432	−0.0776572
$694$	−43.3278	−1.64470
$695$	−15.8748	−0.602166
$696$	13.9352	0.528213
$697$	5.93361	0.224752
$698$	−39.3729	−1.49029
$699$	6.63984	0.251142
$700$	51.1154	1.93198
$701$	−12.6077	−0.476186	−0.238093	−	0.971242i	$-0.576522\pi$
$701$	−12.6077	−0.476186	−0.238093	+	0.971242i	$0.576522\pi$
$702$	−10.2230	−0.385842
$703$	6.94056	0.261768
$704$	−3.33996	−0.125879
$705$	−42.4668	−1.59939
$706$	5.80734	0.218562
$707$	4.05370	0.152455
$708$	27.0055	1.01493
$709$	−25.6422	−0.963014	−0.481507	−	0.876442i	$-0.659910\pi$
$709$	−25.6422	−0.963014	−0.481507	+	0.876442i	$0.659910\pi$
$710$	−58.5998	−2.19921
$711$	−2.09234	−0.0784688
$712$	60.9905	2.28572
$713$	−20.1433	−0.754371
$714$	−6.77453	−0.253530
$715$	−32.7998	−1.22664
$716$	−93.4439	−3.49216
$717$	−7.77321	−0.290296
$718$	20.2746	0.756640
$719$	44.4331	1.65708	0.828538	−	0.559933i	$-0.189173\pi$
$719$	44.4331	1.65708	0.828538	+	0.559933i	$0.189173\pi$
$720$	−31.1694	−1.16161
$721$	−6.68168	−0.248839
$722$	46.3606	1.72536
$723$	17.0364	0.633592
$724$	39.6006	1.47175
$725$	23.6541	0.878493
$726$	−17.4836	−0.648876
$727$	−13.7089	−0.508434	−0.254217	−	0.967147i	$-0.581818\pi$
$727$	−13.7089	−0.508434	−0.254217	+	0.967147i	$0.581818\pi$
$728$	26.2774	0.973907
$729$	1.00000	0.0370370
$730$	−124.050	−4.59129
$731$	−24.2533	−0.897042
$732$	21.2176	0.784226
$733$	−5.94860	−0.219716	−0.109858	−	0.993947i	$-0.535040\pi$
$733$	−5.94860	−0.219716	−0.109858	+	0.993947i	$0.535040\pi$
$734$	−59.0010	−2.17777
$735$	4.02293	0.148388
$736$	−37.9898	−1.40032
$737$	11.7889	0.434250
$738$	5.75484	0.211839
$739$	−30.8869	−1.13619	−0.568096	−	0.822962i	$-0.692320\pi$
$739$	−30.8869	−1.13619	−0.568096	+	0.822962i	$0.692320\pi$
$740$	−133.511	−4.90794
$741$	3.81205	0.140039
$742$	−2.31636	−0.0850362
$743$	−12.5600	−0.460781	−0.230391	−	0.973098i	$-0.574000\pi$
$743$	−12.5600	−0.460781	−0.230391	+	0.973098i	$0.574000\pi$
$744$	−23.3462	−0.855912
$745$	59.3012	2.17263
$746$	−2.33414	−0.0854588
$747$	−0.102376	−0.00374573
$748$	24.6938	0.902894
$749$	15.9187	0.581657
$750$	−63.7682	−2.32849
$751$	27.2977	0.996108	0.498054	−	0.867146i	$-0.334048\pi$
$751$	27.2977	0.996108	0.498054	+	0.867146i	$0.334048\pi$
$752$	−81.7887	−2.98253
$753$	29.1693	1.06299
$754$	21.6217	0.787417
$755$	3.79309	0.138045
$756$	−4.57042	−0.166225
$757$	−5.20830	−0.189299	−0.0946495	−	0.995511i	$-0.530173\pi$
$757$	−5.20830	−0.189299	−0.0946495	+	0.995511i	$0.530173\pi$
$758$	−32.3258	−1.17413
$759$	11.6215	0.421835
$760$	25.3350	0.918998
$761$	35.6890	1.29373	0.646863	−	0.762607i	$-0.276081\pi$
$761$	35.6890	1.29373	0.646863	+	0.762607i	$0.276081\pi$
$762$	−20.0904	−0.727799
$763$	0.670528	0.0242747
$764$	−96.2343	−3.48163
$765$	10.6322	0.384409
$766$	2.56328	0.0926152
$767$	23.5656	0.850903
$768$	26.7922	0.966781
$769$	14.3693	0.518168	0.259084	−	0.965855i	$-0.416579\pi$
$769$	14.3693	0.518168	0.259084	+	0.965855i	$0.416579\pi$
$770$	−21.0808	−0.759699
$771$	−15.1827	−0.546791
$772$	−111.190	−4.00182
$773$	41.4087	1.48937	0.744684	−	0.667417i	$-0.232600\pi$
$773$	41.4087	1.48937	0.744684	+	0.667417i	$0.232600\pi$
$774$	−23.5226	−0.845503
$775$	−39.6286	−1.42350
$776$	−114.718	−4.11814
$777$	−7.26134	−0.260499
$778$	17.6145	0.631509
$779$	−2.14592	−0.0768857
$780$	−73.3297	−2.62563
$781$	11.6173	0.415700
$782$	38.5119	1.37718
$783$	−2.11501	−0.0755843
$784$	7.74793	0.276712
$785$	7.02047	0.250571
$786$	34.1039	1.21644
$787$	−8.15910	−0.290840	−0.145420	−	0.989370i	$-0.546453\pi$
$787$	−8.15910	−0.290840	−0.145420	+	0.989370i	$0.546453\pi$
$788$	41.8736	1.49168
$789$	−27.1481	−0.966499
$790$	−21.5760	−0.767639
$791$	5.49468	0.195368
$792$	13.4695	0.478616
$793$	18.5149	0.657484
$794$	46.6264	1.65471
$795$	3.63539	0.128934
$796$	−72.5823	−2.57261
$797$	45.9056	1.62606	0.813029	−	0.582223i	$-0.197817\pi$
$797$	45.9056	1.62606	0.813029	+	0.582223i	$0.197817\pi$
$798$	2.45005	0.0867307
$799$	27.8991	0.986999
$800$	−74.7389	−2.64242
$801$	−9.25679	−0.327073
$802$	33.8181	1.19416
$803$	24.5927	0.867856
$804$	26.3562	0.929510
$805$	−22.8696	−0.806046
$806$	−36.2236	−1.27592
$807$	−2.49848	−0.0879506
$808$	−26.7087	−0.939609
$809$	−42.2943	−1.48699	−0.743493	−	0.668743i	$-0.766833\pi$
$809$	−42.2943	−1.48699	−0.743493	+	0.668743i	$0.766833\pi$
$810$	10.3119	0.362323
$811$	−12.2425	−0.429891	−0.214945	−	0.976626i	$-0.568957\pi$
$811$	−12.2425	−0.429891	−0.214945	+	0.976626i	$0.568957\pi$
$812$	9.66649	0.339227
$813$	5.52726	0.193849
$814$	38.0506	1.33367
$815$	−23.2054	−0.812851
$816$	20.4771	0.716842
$817$	8.77136	0.306871
$818$	78.0334	2.72837
$819$	−3.98824	−0.139360
$820$	41.2796	1.44155
$821$	−43.1325	−1.50533	−0.752667	−	0.658401i	$-0.771233\pi$
$821$	−43.1325	−1.50533	−0.752667	+	0.658401i	$0.771233\pi$
$822$	−32.0441	−1.11767
$823$	−22.2274	−0.774797	−0.387398	−	0.921912i	$-0.626626\pi$
$823$	−22.2274	−0.774797	−0.387398	+	0.921912i	$0.626626\pi$
$824$	44.0238	1.53364
$825$	22.8635	0.796006
$826$	15.1458	0.526991
$827$	18.5947	0.646603	0.323301	−	0.946296i	$-0.395207\pi$
$827$	18.5947	0.646603	0.323301	+	0.946296i	$0.395207\pi$
$828$	25.9820	0.902936
$829$	42.8336	1.48767	0.743837	−	0.668362i	$-0.233004\pi$
$829$	42.8336	1.48767	0.743837	+	0.668362i	$0.233004\pi$
$830$	−1.05569	−0.0366435
$831$	−4.50271	−0.156197
$832$	−6.51589	−0.225898
$833$	−2.64291	−0.0915714
$834$	10.1149	0.350251
$835$	37.7663	1.30696
$836$	−8.93064	−0.308873
$837$	3.54335	0.122476
$838$	8.00049	0.276372
$839$	16.0338	0.553548	0.276774	−	0.960935i	$-0.410735\pi$
$839$	16.0338	0.553548	0.276774	+	0.960935i	$0.410735\pi$
$840$	−26.5060	−0.914543
$841$	−24.5267	−0.845750
$842$	70.5362	2.43084
$843$	−1.30509	−0.0449496
$844$	51.9514	1.78824
$845$	−11.6909	−0.402179
$846$	27.0585	0.930291
$847$	−6.82077	−0.234364
$848$	7.00156	0.240435
$849$	4.46319	0.153176
$850$	75.7660	2.59875
$851$	41.2793	1.41504
$852$	25.9725	0.889804
$853$	10.2883	0.352265	0.176132	−	0.984367i	$-0.443641\pi$
$853$	10.2883	0.352265	0.176132	+	0.984367i	$0.443641\pi$
$854$	11.8997	0.407200
$855$	−3.84521	−0.131503
$856$	−104.884	−3.58486
$857$	31.2063	1.06599	0.532994	−	0.846119i	$-0.321067\pi$
$857$	31.2063	1.06599	0.532994	+	0.846119i	$0.321067\pi$
$858$	20.8990	0.713482
$859$	45.8315	1.56375	0.781876	−	0.623434i	$-0.214263\pi$
$859$	45.8315	1.56375	0.781876	+	0.623434i	$0.214263\pi$
$860$	−168.728	−5.75359
$861$	2.24510	0.0765129
$862$	−73.3839	−2.49947
$863$	55.0210	1.87294	0.936469	−	0.350750i	$-0.114073\pi$
$863$	55.0210	1.87294	0.936469	+	0.350750i	$0.114073\pi$
$864$	6.68269	0.227350
$865$	7.55110	0.256745
$866$	−7.42059	−0.252162
$867$	10.0150	0.340128
$868$	−16.1946	−0.549681
$869$	4.27740	0.145101
$870$	−21.8098	−0.739420
$871$	22.9989	0.779288
$872$	−4.41792	−0.149610
$873$	17.4113	0.589282
$874$	−13.9280	−0.471123
$875$	−24.8776	−0.841015
$876$	54.9811	1.85764
$877$	−31.9097	−1.07751	−0.538756	−	0.842462i	$-0.681106\pi$
$877$	−31.9097	−1.07751	−0.538756	+	0.842462i	$0.681106\pi$
$878$	−95.8404	−3.23446
$879$	−0.00623872	−0.000210427 0
$880$	63.7201	2.14800
$881$	41.2680	1.39035	0.695177	−	0.718839i	$-0.255326\pi$
$881$	41.2680	1.39035	0.695177	+	0.718839i	$0.255326\pi$
$882$	−2.56328	−0.0863103
$883$	−8.13370	−0.273721	−0.136860	−	0.990590i	$-0.543701\pi$
$883$	−8.13370	−0.273721	−0.136860	+	0.990590i	$0.543701\pi$
$884$	48.1749	1.62030
$885$	−23.7705	−0.799037
$886$	7.32725	0.246164
$887$	−5.50769	−0.184930	−0.0924651	−	0.995716i	$-0.529475\pi$
$887$	−5.50769	−0.184930	−0.0924651	+	0.995716i	$0.529475\pi$
$888$	47.8430	1.60551
$889$	−7.83776	−0.262870
$890$	−95.4552	−3.19966
$891$	−2.04432	−0.0684872
$892$	74.1473	2.48263
$893$	−10.0899	−0.337644
$894$	−37.7849	−1.26372
$895$	82.2502	2.74932
$896$	9.17755	0.306601
$897$	22.6724	0.757008
$898$	17.3738	0.579771
$899$	−7.49422	−0.249946
$900$	51.1154	1.70385
$901$	−2.38832	−0.0795663
$902$	−11.7647	−0.391722
$903$	−9.17675	−0.305383
$904$	−36.2029	−1.20409
$905$	−34.8568	−1.15868
$906$	−2.41684	−0.0802940
$907$	34.5792	1.14818	0.574091	−	0.818791i	$-0.305355\pi$
$907$	34.5792	1.14818	0.574091	+	0.818791i	$0.305355\pi$
$908$	−31.7067	−1.05222
$909$	4.05370	0.134453
$910$	−41.1264	−1.36333
$911$	29.1614	0.966159	0.483080	−	0.875576i	$-0.339518\pi$
$911$	29.1614	0.966159	0.483080	+	0.875576i	$0.339518\pi$
$912$	−7.40566	−0.245226
$913$	0.209289	0.00692644
$914$	−8.91999	−0.295047
$915$	−18.6759	−0.617408
$916$	7.19654	0.237780
$917$	13.3048	0.439362
$918$	−6.77453	−0.223593
$919$	−14.1196	−0.465762	−0.232881	−	0.972505i	$-0.574815\pi$
$919$	−14.1196	−0.465762	−0.232881	+	0.972505i	$0.574815\pi$
$920$	150.681	4.96781
$921$	25.8509	0.851816
$922$	−44.7672	−1.47433
$923$	22.6641	0.745999
$924$	9.34340	0.307375
$925$	81.2104	2.67018
$926$	−17.6198	−0.579021
$927$	−6.68168	−0.219455
$928$	−14.1340	−0.463970
$929$	−8.43203	−0.276646	−0.138323	−	0.990387i	$-0.544171\pi$
$929$	−8.43203	−0.276646	−0.138323	+	0.990387i	$0.544171\pi$
$930$	36.5387	1.19815
$931$	0.955824	0.0313259
$932$	−30.3469	−0.994045
$933$	−8.78160	−0.287497
$934$	−45.7289	−1.49630
$935$	−21.7357	−0.710832
$936$	26.2774	0.858905
$937$	24.2574	0.792455	0.396227	−	0.918152i	$-0.370319\pi$
$937$	24.2574	0.792455	0.396227	+	0.918152i	$0.370319\pi$
$938$	14.7816	0.482637
$939$	18.0846	0.590169
$940$	194.091	6.33056
$941$	−37.8015	−1.23229	−0.616146	−	0.787632i	$-0.711307\pi$
$941$	−37.8015	−1.23229	−0.616146	+	0.787632i	$0.711307\pi$
$942$	−4.47322	−0.145745
$943$	−12.7630	−0.415620
$944$	−45.7807	−1.49003
$945$	4.02293	0.130866
$946$	48.0877	1.56347
$947$	−0.353923	−0.0115010	−0.00575048	−	0.999983i	$-0.501830\pi$
$947$	−0.353923	−0.0115010	−0.00575048	+	0.999983i	$0.501830\pi$
$948$	9.56287	0.310588
$949$	47.9776	1.55742
$950$	−27.4012	−0.889012
$951$	−4.96903	−0.161132
$952$	17.4134	0.564372
$953$	4.13854	0.134060	0.0670302	−	0.997751i	$-0.478648\pi$
$953$	4.13854	0.134060	0.0670302	+	0.997751i	$0.478648\pi$
$954$	−2.31636	−0.0749949
$955$	84.7063	2.74103
$956$	35.5268	1.14902
$957$	4.32375	0.139767
$958$	19.6799	0.635827
$959$	−12.5012	−0.403684
$960$	6.57256	0.212128
$961$	−18.4447	−0.594989
$962$	74.2326	2.39336
$963$	15.9187	0.512973
$964$	−77.8638	−2.50782
$965$	97.8705	3.15056
$966$	14.5718	0.468839
$967$	−47.0116	−1.51179	−0.755895	−	0.654693i	$-0.772798\pi$
$967$	−47.0116	−1.51179	−0.755895	+	0.654693i	$0.772798\pi$
$968$	44.9402	1.44443
$969$	2.52616	0.0811518
$970$	179.543	5.76479
$971$	34.4310	1.10494	0.552472	−	0.833531i	$-0.313685\pi$
$971$	34.4310	1.10494	0.552472	+	0.833531i	$0.313685\pi$
$972$	−4.57042	−0.146596
$973$	3.94608	0.126506
$974$	−12.9890	−0.416194
$975$	44.6043	1.42848
$976$	−35.9688	−1.15133
$977$	42.9239	1.37326	0.686628	−	0.727009i	$-0.259090\pi$
$977$	42.9239	1.37326	0.686628	+	0.727009i	$0.259090\pi$
$978$	14.7858	0.472797
$979$	18.9238	0.604808
$980$	−18.3865	−0.587335
$981$	0.670528	0.0214083
$982$	−72.8193	−2.32376
$983$	0.191557	0.00610972	0.00305486	−	0.999995i	$-0.499028\pi$
$983$	0.191557	0.00610972	0.00305486	+	0.999995i	$0.499028\pi$
$984$	−14.7924	−0.471564
$985$	−36.8575	−1.17438
$986$	14.3282	0.456303
$987$	10.5562	0.336007
$988$	−17.4227	−0.554290
$989$	52.1681	1.65885
$990$	−21.0808	−0.669992
$991$	54.9608	1.74589	0.872943	−	0.487822i	$-0.162208\pi$
$991$	54.9608	1.74589	0.872943	+	0.487822i	$0.162208\pi$
$992$	23.6791	0.751813
$993$	9.09624	0.288660
$994$	14.5665	0.462020
$995$	63.8876	2.02537
$996$	0.467901	0.0148260
$997$	55.9528	1.77204	0.886022	−	0.463644i	$-0.153458\pi$
$997$	55.9528	1.77204	0.886022	+	0.463644i	$0.153458\pi$
$998$	52.2610	1.65429
$999$	−7.26134	−0.229739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8043.2.a.t.1.3	✓	52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8043.2.a.t.1.3	✓	52	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:	\( N \)	\(=\)	\( 8043 = 3 \cdot 7 \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8043.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56328 q^{2} -1.00000 q^{3} +4.57042 q^{4} -4.02293 q^{5} +2.56328 q^{6} +1.00000 q^{7} -6.58873 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.56328 q^{2} -1.00000 q^{3} +4.57042 q^{4} -4.02293 q^{5} +2.56328 q^{6} +1.00000 q^{7} -6.58873 q^{8} +1.00000 q^{9} +10.3119 q^{10} -2.04432 q^{11} -4.57042 q^{12} -3.98824 q^{13} -2.56328 q^{14} +4.02293 q^{15} +7.74793 q^{16} -2.64291 q^{17} -2.56328 q^{18} +0.955824 q^{19} -18.3865 q^{20} -1.00000 q^{21} +5.24017 q^{22} +5.68481 q^{23} +6.58873 q^{24} +11.1839 q^{25} +10.2230 q^{26} -1.00000 q^{27} +4.57042 q^{28} +2.11501 q^{29} -10.3119 q^{30} -3.54335 q^{31} -6.68269 q^{32} +2.04432 q^{33} +6.77453 q^{34} -4.02293 q^{35} +4.57042 q^{36} +7.26134 q^{37} -2.45005 q^{38} +3.98824 q^{39} +26.5060 q^{40} -2.24510 q^{41} +2.56328 q^{42} +9.17675 q^{43} -9.34340 q^{44} -4.02293 q^{45} -14.5718 q^{46} -10.5562 q^{47} -7.74793 q^{48} +1.00000 q^{49} -28.6676 q^{50} +2.64291 q^{51} -18.2280 q^{52} +0.903669 q^{53} +2.56328 q^{54} +8.22414 q^{55} -6.58873 q^{56} -0.955824 q^{57} -5.42137 q^{58} -5.90876 q^{59} +18.3865 q^{60} -4.64238 q^{61} +9.08261 q^{62} +1.00000 q^{63} +1.63378 q^{64} +16.0444 q^{65} -5.24017 q^{66} -5.76668 q^{67} -12.0792 q^{68} -5.68481 q^{69} +10.3119 q^{70} -5.68274 q^{71} -6.58873 q^{72} -12.0298 q^{73} -18.6129 q^{74} -11.1839 q^{75} +4.36852 q^{76} -2.04432 q^{77} -10.2230 q^{78} -2.09234 q^{79} -31.1694 q^{80} +1.00000 q^{81} +5.75484 q^{82} -0.102376 q^{83} -4.57042 q^{84} +10.6322 q^{85} -23.5226 q^{86} -2.11501 q^{87} +13.4695 q^{88} -9.25679 q^{89} +10.3119 q^{90} -3.98824 q^{91} +25.9820 q^{92} +3.54335 q^{93} +27.0585 q^{94} -3.84521 q^{95} +6.68269 q^{96} +17.4113 q^{97} -2.56328 q^{98} -2.04432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9}+O(q^{10}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9	\( 52 q + 3 q^{2} - 52 q^{3} + 61 q^{4} - 7 q^{5} - 3 q^{6} + 52 q^{7} + 24 q^{8} + 52 q^{9} - 2 q^{10} + 9 q^{11} - 61 q^{12} + 44 q^{13} + 3 q^{14} + 7 q^{15} + 95 q^{16} - 6 q^{17} + 3 q^{18} + 7 q^{19} - 21 q^{20} - 52 q^{21} + 19 q^{22} - 4 q^{23} - 24 q^{24} + 83 q^{25} - 5 q^{26} - 52 q^{27} + 61 q^{28} + 31 q^{29} + 2 q^{30} + 11 q^{31} + 71 q^{32} - 9 q^{33} + 17 q^{34} - 7 q^{35} + 61 q^{36} + 71 q^{37} - 8 q^{38} - 44 q^{39} + 20 q^{40} - 25 q^{41} - 3 q^{42} + 75 q^{43} + 14 q^{44} - 7 q^{45} + 36 q^{46} - 20 q^{47} - 95 q^{48} + 52 q^{49} + 26 q^{50} + 6 q^{51} + 88 q^{52} + 70 q^{53} - 3 q^{54} + 12 q^{55} + 24 q^{56} - 7 q^{57} + 48 q^{58} - 27 q^{59} + 21 q^{60} + 59 q^{61} - 23 q^{62} + 52 q^{63} + 138 q^{64} + 44 q^{65} - 19 q^{66} + 65 q^{67} - 8 q^{68} + 4 q^{69} - 2 q^{70} - 11 q^{71} + 24 q^{72} + 34 q^{73} + 38 q^{74} - 83 q^{75} + 31 q^{76} + 9 q^{77} + 5 q^{78} + 74 q^{79} - 5 q^{80} + 52 q^{81} + 51 q^{82} - 30 q^{83} - 61 q^{84} + 70 q^{85} + 29 q^{86} - 31 q^{87} + 90 q^{88} - q^{89} - 2 q^{90} + 44 q^{91} + 34 q^{92} - 11 q^{93} + 27 q^{94} + 9 q^{95} - 71 q^{96} + 73 q^{97} + 3 q^{98} + 9 q^{99}+O(q^{100}) \) 52 * q + 3 * q^2 - 52 * q^3 + 61 * q^4 - 7 * q^5 - 3 * q^6 + 52 * q^7 + 24 * q^8 + 52 * q^9 - 2 * q^10 + 9 * q^11 - 61 * q^12 + 44 * q^13 + 3 * q^14 + 7 * q^15 + 95 * q^16 - 6 * q^17 + 3 * q^18 + 7 * q^19 - 21 * q^20 - 52 * q^21 + 19 * q^22 - 4 * q^23 - 24 * q^24 + 83 * q^25 - 5 * q^26 - 52 * q^27 + 61 * q^28 + 31 * q^29 + 2 * q^30 + 11 * q^31 + 71 * q^32 - 9 * q^33 + 17 * q^34 - 7 * q^35 + 61 * q^36 + 71 * q^37 - 8 * q^38 - 44 * q^39 + 20 * q^40 - 25 * q^41 - 3 * q^42 + 75 * q^43 + 14 * q^44 - 7 * q^45 + 36 * q^46 - 20 * q^47 - 95 * q^48 + 52 * q^49 + 26 * q^50 + 6 * q^51 + 88 * q^52 + 70 * q^53 - 3 * q^54 + 12 * q^55 + 24 * q^56 - 7 * q^57 + 48 * q^58 - 27 * q^59 + 21 * q^60 + 59 * q^61 - 23 * q^62 + 52 * q^63 + 138 * q^64 + 44 * q^65 - 19 * q^66 + 65 * q^67 - 8 * q^68 + 4 * q^69 - 2 * q^70 - 11 * q^71 + 24 * q^72 + 34 * q^73 + 38 * q^74 - 83 * q^75 + 31 * q^76 + 9 * q^77 + 5 * q^78 + 74 * q^79 - 5 * q^80 + 52 * q^81 + 51 * q^82 - 30 * q^83 - 61 * q^84 + 70 * q^85 + 29 * q^86 - 31 * q^87 + 90 * q^88 - q^89 - 2 * q^90 + 44 * q^91 + 34 * q^92 - 11 * q^93 + 27 * q^94 + 9 * q^95 - 71 * q^96 + 73 * q^97 + 3 * q^98 + 9 * q^99

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