LMFDB - Embedded newform 8007.2.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28381 q^{2} -1.00000 q^{3} +3.21580 q^{4} +1.64317 q^{5} +2.28381 q^{6} +3.93496 q^{7} -2.77667 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.28381 q^{2} -1.00000 q^{3} +3.21580 q^{4} +1.64317 q^{5} +2.28381 q^{6} +3.93496 q^{7} -2.77667 q^{8} +1.00000 q^{9} -3.75270 q^{10} -0.343655 q^{11} -3.21580 q^{12} +0.744706 q^{13} -8.98670 q^{14} -1.64317 q^{15} -0.0902101 q^{16} -1.00000 q^{17} -2.28381 q^{18} -7.57478 q^{19} +5.28412 q^{20} -3.93496 q^{21} +0.784844 q^{22} -3.77572 q^{23} +2.77667 q^{24} -2.29998 q^{25} -1.70077 q^{26} -1.00000 q^{27} +12.6540 q^{28} +5.07234 q^{29} +3.75270 q^{30} -4.98225 q^{31} +5.75937 q^{32} +0.343655 q^{33} +2.28381 q^{34} +6.46581 q^{35} +3.21580 q^{36} +4.72599 q^{37} +17.2994 q^{38} -0.744706 q^{39} -4.56255 q^{40} -2.83772 q^{41} +8.98670 q^{42} -0.185855 q^{43} -1.10513 q^{44} +1.64317 q^{45} +8.62304 q^{46} +8.35281 q^{47} +0.0902101 q^{48} +8.48387 q^{49} +5.25274 q^{50} +1.00000 q^{51} +2.39483 q^{52} +8.10439 q^{53} +2.28381 q^{54} -0.564685 q^{55} -10.9261 q^{56} +7.57478 q^{57} -11.5843 q^{58} -7.50731 q^{59} -5.28412 q^{60} -14.1241 q^{61} +11.3785 q^{62} +3.93496 q^{63} -12.9729 q^{64} +1.22368 q^{65} -0.784844 q^{66} +2.48169 q^{67} -3.21580 q^{68} +3.77572 q^{69} -14.7667 q^{70} -1.23832 q^{71} -2.77667 q^{72} -9.29357 q^{73} -10.7933 q^{74} +2.29998 q^{75} -24.3590 q^{76} -1.35227 q^{77} +1.70077 q^{78} +9.60307 q^{79} -0.148231 q^{80} +1.00000 q^{81} +6.48081 q^{82} +14.1340 q^{83} -12.6540 q^{84} -1.64317 q^{85} +0.424458 q^{86} -5.07234 q^{87} +0.954217 q^{88} -3.05836 q^{89} -3.75270 q^{90} +2.93038 q^{91} -12.1420 q^{92} +4.98225 q^{93} -19.0763 q^{94} -12.4467 q^{95} -5.75937 q^{96} -6.82046 q^{97} -19.3756 q^{98} -0.343655 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	−2.28381	−1.61490	−0.807450	−	0.589936i	$-0.799153\pi$
$2$	−2.28381	−1.61490	−0.807450	+	0.589936i	$0.799153\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.21580	1.60790
$5$	1.64317	0.734849	0.367425	−	0.930053i	$-0.380240\pi$
$5$	1.64317	0.734849	0.367425	+	0.930053i	$0.380240\pi$
$6$	2.28381	0.932363
$7$	3.93496	1.48727	0.743637	−	0.668584i	$-0.233099\pi$
$7$	3.93496	1.48727	0.743637	+	0.668584i	$0.233099\pi$
$8$	−2.77667	−0.981701
$9$	1.00000	0.333333
$10$	−3.75270	−1.18671
$11$	−0.343655	−0.103616	−0.0518079	−	0.998657i	$-0.516498\pi$
$11$	−0.343655	−0.103616	−0.0518079	+	0.998657i	$0.516498\pi$
$12$	−3.21580	−0.928323
$13$	0.744706	0.206544	0.103272	−	0.994653i	$-0.467069\pi$
$13$	0.744706	0.206544	0.103272	+	0.994653i	$0.467069\pi$
$14$	−8.98670	−2.40180
$15$	−1.64317	−0.424265
$16$	−0.0902101	−0.0225525
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.28381	−0.538300
$19$	−7.57478	−1.73777	−0.868887	−	0.495010i	$-0.835164\pi$
$19$	−7.57478	−1.73777	−0.868887	+	0.495010i	$0.835164\pi$
$20$	5.28412	1.18157
$21$	−3.93496	−0.858678
$22$	0.784844	0.167329
$23$	−3.77572	−0.787292	−0.393646	−	0.919262i	$-0.628786\pi$
$23$	−3.77572	−0.787292	−0.393646	+	0.919262i	$0.628786\pi$
$24$	2.77667	0.566786
$25$	−2.29998	−0.459997
$26$	−1.70077	−0.333548
$27$	−1.00000	−0.192450
$28$	12.6540	2.39139
$29$	5.07234	0.941910	0.470955	−	0.882157i	$-0.343909\pi$
$29$	5.07234	0.941910	0.470955	+	0.882157i	$0.343909\pi$
$30$	3.75270	0.685146
$31$	−4.98225	−0.894838	−0.447419	−	0.894324i	$-0.647657\pi$
$31$	−4.98225	−0.894838	−0.447419	+	0.894324i	$0.647657\pi$
$32$	5.75937	1.01812
$33$	0.343655	0.0598227
$34$	2.28381	0.391671
$35$	6.46581	1.09292
$36$	3.21580	0.535967
$37$	4.72599	0.776949	0.388474	−	0.921460i	$-0.373002\pi$
$37$	4.72599	0.776949	0.388474	+	0.921460i	$0.373002\pi$
$38$	17.2994	2.80633
$39$	−0.744706	−0.119248
$40$	−4.56255	−0.721402
$41$	−2.83772	−0.443177	−0.221588	−	0.975140i	$-0.571124\pi$
$41$	−2.83772	−0.443177	−0.221588	+	0.975140i	$0.571124\pi$
$42$	8.98670	1.38668
$43$	−0.185855	−0.0283426	−0.0141713	−	0.999900i	$-0.504511\pi$
$43$	−0.185855	−0.0283426	−0.0141713	+	0.999900i	$0.504511\pi$
$44$	−1.10513	−0.166604
$45$	1.64317	0.244950
$46$	8.62304	1.27140
$47$	8.35281	1.21838	0.609191	−	0.793023i	$-0.291494\pi$
$47$	8.35281	1.21838	0.609191	+	0.793023i	$0.291494\pi$
$48$	0.0902101	0.0130207
$49$	8.48387	1.21198
$50$	5.25274	0.742849
$51$	1.00000	0.140028
$52$	2.39483	0.332103
$53$	8.10439	1.11322	0.556611	−	0.830773i	$-0.312101\pi$
$53$	8.10439	1.11322	0.556611	+	0.830773i	$0.312101\pi$
$54$	2.28381	0.310788
$55$	−0.564685	−0.0761420
$56$	−10.9261	−1.46006
$57$	7.57478	1.00330
$58$	−11.5843	−1.52109
$59$	−7.50731	−0.977369	−0.488684	−	0.872461i	$-0.662523\pi$
$59$	−7.50731	−0.977369	−0.488684	+	0.872461i	$0.662523\pi$
$60$	−5.28412	−0.682177
$61$	−14.1241	−1.80840	−0.904201	−	0.427107i	$-0.859533\pi$
$61$	−14.1241	−1.80840	−0.904201	+	0.427107i	$0.859533\pi$
$62$	11.3785	1.44507
$63$	3.93496	0.495758
$64$	−12.9729	−1.62161
$65$	1.22368	0.151779
$66$	−0.784844	−0.0966076
$67$	2.48169	0.303187	0.151594	−	0.988443i	$-0.451559\pi$
$67$	2.48169	0.303187	0.151594	+	0.988443i	$0.451559\pi$
$68$	−3.21580	−0.389974
$69$	3.77572	0.454543
$70$	−14.7667	−1.76496
$71$	−1.23832	−0.146962	−0.0734809	−	0.997297i	$-0.523411\pi$
$71$	−1.23832	−0.146962	−0.0734809	+	0.997297i	$0.523411\pi$
$72$	−2.77667	−0.327234
$73$	−9.29357	−1.08773	−0.543865	−	0.839173i	$-0.683040\pi$
$73$	−9.29357	−1.08773	−0.543865	+	0.839173i	$0.683040\pi$
$74$	−10.7933	−1.25469
$75$	2.29998	0.265579
$76$	−24.3590	−2.79417
$77$	−1.35227	−0.154105
$78$	1.70077	0.192574
$79$	9.60307	1.08043	0.540215	−	0.841527i	$-0.318343\pi$
$79$	9.60307	1.08043	0.540215	+	0.841527i	$0.318343\pi$
$80$	−0.148231	−0.0165727
$81$	1.00000	0.111111
$82$	6.48081	0.715686
$83$	14.1340	1.55140	0.775702	−	0.631099i	$-0.217396\pi$
$83$	14.1340	1.55140	0.775702	+	0.631099i	$0.217396\pi$
$84$	−12.6540	−1.38067
$85$	−1.64317	−0.178227
$86$	0.424458	0.0457705
$87$	−5.07234	−0.543812
$88$	0.954217	0.101720
$89$	−3.05836	−0.324186	−0.162093	−	0.986776i	$-0.551824\pi$
$89$	−3.05836	−0.324186	−0.162093	+	0.986776i	$0.551824\pi$
$90$	−3.75270	−0.395569
$91$	2.93038	0.307188
$92$	−12.1420	−1.26589
$93$	4.98225	0.516635
$94$	−19.0763	−1.96757
$95$	−12.4467	−1.27700
$96$	−5.75937	−0.587813
$97$	−6.82046	−0.692513	−0.346257	−	0.938140i	$-0.612547\pi$
$97$	−6.82046	−0.692513	−0.346257	+	0.938140i	$0.612547\pi$
$98$	−19.3756	−1.95723
$99$	−0.343655	−0.0345386
$100$	−7.39630	−0.739630
$101$	14.1430	1.40728	0.703639	−	0.710558i	$-0.251557\pi$
$101$	14.1430	1.40728	0.703639	+	0.710558i	$0.251557\pi$
$102$	−2.28381	−0.226131
$103$	4.71884	0.464961	0.232480	−	0.972601i	$-0.425316\pi$
$103$	4.71884	0.464961	0.232480	+	0.972601i	$0.425316\pi$
$104$	−2.06780	−0.202765
$105$	−6.46581	−0.630998
$106$	−18.5089	−1.79774
$107$	−16.2355	−1.56955	−0.784774	−	0.619783i	$-0.787221\pi$
$107$	−16.2355	−1.56955	−0.784774	+	0.619783i	$0.787221\pi$
$108$	−3.21580	−0.309441
$109$	−18.7232	−1.79336	−0.896680	−	0.442680i	$-0.854028\pi$
$109$	−18.7232	−1.79336	−0.896680	+	0.442680i	$0.854028\pi$
$110$	1.28963	0.122962
$111$	−4.72599	−0.448572
$112$	−0.354973	−0.0335418
$113$	−15.7168	−1.47851	−0.739255	−	0.673426i	$-0.764822\pi$
$113$	−15.7168	−1.47851	−0.739255	+	0.673426i	$0.764822\pi$
$114$	−17.2994	−1.62024
$115$	−6.20416	−0.578541
$116$	16.3116	1.51450
$117$	0.744706	0.0688481
$118$	17.1453	1.57835
$119$	−3.93496	−0.360717
$120$	4.56255	0.416502
$121$	−10.8819	−0.989264
$122$	32.2568	2.92039
$123$	2.83772	0.255868
$124$	−16.0219	−1.43881
$125$	−11.9951	−1.07288
$126$	−8.98670	−0.800599
$127$	−1.73892	−0.154305	−0.0771523	−	0.997019i	$-0.524583\pi$
$127$	−1.73892	−0.154305	−0.0771523	+	0.997019i	$0.524583\pi$
$128$	18.1089	1.60062
$129$	0.185855	0.0163636
$130$	−2.79466	−0.245108
$131$	−3.19032	−0.278739	−0.139370	−	0.990240i	$-0.544508\pi$
$131$	−3.19032	−0.278739	−0.139370	+	0.990240i	$0.544508\pi$
$132$	1.10513	0.0961890
$133$	−29.8064	−2.58455
$134$	−5.66773	−0.489617
$135$	−1.64317	−0.141422
$136$	2.77667	0.238098
$137$	13.0663	1.11633	0.558164	−	0.829731i	$-0.311506\pi$
$137$	13.0663	1.11633	0.558164	+	0.829731i	$0.311506\pi$
$138$	−8.62304	−0.734042
$139$	5.19269	0.440439	0.220219	−	0.975450i	$-0.429323\pi$
$139$	5.19269	0.440439	0.220219	+	0.975450i	$0.429323\pi$
$140$	20.7928	1.75731
$141$	−8.35281	−0.703433
$142$	2.82810	0.237329
$143$	−0.255922	−0.0214013
$144$	−0.0902101	−0.00751751
$145$	8.33473	0.692161
$146$	21.2248	1.75658
$147$	−8.48387	−0.699738
$148$	15.1979	1.24926
$149$	10.4192	0.853575	0.426788	−	0.904352i	$-0.359645\pi$
$149$	10.4192	0.853575	0.426788	+	0.904352i	$0.359645\pi$
$150$	−5.25274	−0.428884
$151$	11.8289	0.962624	0.481312	−	0.876549i	$-0.340160\pi$
$151$	11.8289	0.962624	0.481312	+	0.876549i	$0.340160\pi$
$152$	21.0327	1.70598
$153$	−1.00000	−0.0808452
$154$	3.08833	0.248864
$155$	−8.18669	−0.657571
$156$	−2.39483	−0.191740
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−21.9316	−1.74479
$159$	−8.10439	−0.642720
$160$	9.46363	0.748166
$161$	−14.8573	−1.17092
$162$	−2.28381	−0.179433
$163$	−2.45136	−0.192005	−0.0960027	−	0.995381i	$-0.530606\pi$
$163$	−2.45136	−0.192005	−0.0960027	+	0.995381i	$0.530606\pi$
$164$	−9.12554	−0.712585
$165$	0.564685	0.0439606
$166$	−32.2793	−2.50536
$167$	11.2918	0.873786	0.436893	−	0.899514i	$-0.356079\pi$
$167$	11.2918	0.873786	0.436893	+	0.899514i	$0.356079\pi$
$168$	10.9261	0.842965
$169$	−12.4454	−0.957340
$170$	3.75270	0.287819
$171$	−7.57478	−0.579258
$172$	−0.597673	−0.0455721
$173$	1.69657	0.128988	0.0644939	−	0.997918i	$-0.479457\pi$
$173$	1.69657	0.128988	0.0644939	+	0.997918i	$0.479457\pi$
$174$	11.5843	0.878202
$175$	−9.05033	−0.684141
$176$	0.0310012	0.00233680
$177$	7.50731	0.564284
$178$	6.98473	0.523528
$179$	5.23063	0.390955	0.195478	−	0.980708i	$-0.437374\pi$
$179$	5.23063	0.390955	0.195478	+	0.980708i	$0.437374\pi$
$180$	5.28412	0.393855
$181$	−11.8843	−0.883352	−0.441676	−	0.897175i	$-0.645616\pi$
$181$	−11.8843	−0.883352	−0.441676	+	0.897175i	$0.645616\pi$
$182$	−6.69245	−0.496077
$183$	14.1241	1.04408
$184$	10.4839	0.772886
$185$	7.76562	0.570940
$186$	−11.3785	−0.834314
$187$	0.343655	0.0251305
$188$	26.8610	1.95904
$189$	−3.93496	−0.286226
$190$	28.4259	2.06223
$191$	−8.14955	−0.589681	−0.294840	−	0.955547i	$-0.595266\pi$
$191$	−8.14955	−0.589681	−0.294840	+	0.955547i	$0.595266\pi$
$192$	12.9729	0.936238
$193$	−23.0373	−1.65826	−0.829131	−	0.559055i	$-0.811164\pi$
$193$	−23.0373	−1.65826	−0.829131	+	0.559055i	$0.811164\pi$
$194$	15.5767	1.11834
$195$	−1.22368	−0.0876295
$196$	27.2825	1.94875
$197$	−5.11546	−0.364462	−0.182231	−	0.983256i	$-0.558332\pi$
$197$	−5.11546	−0.364462	−0.182231	+	0.983256i	$0.558332\pi$
$198$	0.784844	0.0557764
$199$	15.1202	1.07184	0.535922	−	0.844267i	$-0.319964\pi$
$199$	15.1202	1.07184	0.535922	+	0.844267i	$0.319964\pi$
$200$	6.38630	0.451580
$201$	−2.48169	−0.175045
$202$	−32.2999	−2.27261
$203$	19.9594	1.40088
$204$	3.21580	0.225151
$205$	−4.66286	−0.325668
$206$	−10.7769	−0.750865
$207$	−3.77572	−0.262431
$208$	−0.0671800	−0.00465809
$209$	2.60311	0.180061
$210$	14.7667	1.01900
$211$	3.68584	0.253744	0.126872	−	0.991919i	$-0.459506\pi$
$211$	3.68584	0.253744	0.126872	+	0.991919i	$0.459506\pi$
$212$	26.0621	1.78995
$213$	1.23832	0.0848484
$214$	37.0789	2.53466
$215$	−0.305392	−0.0208275
$216$	2.77667	0.188929
$217$	−19.6049	−1.33087
$218$	42.7603	2.89610
$219$	9.29357	0.628002
$220$	−1.81592	−0.122429
$221$	−0.744706	−0.0500943
$222$	10.7933	0.724398
$223$	16.2493	1.08814	0.544068	−	0.839041i	$-0.316883\pi$
$223$	16.2493	1.08814	0.544068	+	0.839041i	$0.316883\pi$
$224$	22.6628	1.51422
$225$	−2.29998	−0.153332
$226$	35.8942	2.38765
$227$	−12.9106	−0.856903	−0.428452	−	0.903565i	$-0.640941\pi$
$227$	−12.9106	−0.856903	−0.428452	+	0.903565i	$0.640941\pi$
$228$	24.3590	1.61322
$229$	−22.3888	−1.47949	−0.739747	−	0.672885i	$-0.765055\pi$
$229$	−22.3888	−1.47949	−0.739747	+	0.672885i	$0.765055\pi$
$230$	14.1691	0.934286
$231$	1.35227	0.0889726
$232$	−14.0842	−0.924674
$233$	−27.3862	−1.79413	−0.897065	−	0.441898i	$-0.854305\pi$
$233$	−27.3862	−1.79413	−0.897065	+	0.441898i	$0.854305\pi$
$234$	−1.70077	−0.111183
$235$	13.7251	0.895327
$236$	−24.1420	−1.57151
$237$	−9.60307	−0.623787
$238$	8.98670	0.582521
$239$	−26.4691	−1.71214	−0.856071	−	0.516858i	$-0.827102\pi$
$239$	−26.4691	−1.71214	−0.856071	+	0.516858i	$0.827102\pi$
$240$	0.148231	0.00956825
$241$	5.03232	0.324160	0.162080	−	0.986778i	$-0.448180\pi$
$241$	5.03232	0.324160	0.162080	+	0.986778i	$0.448180\pi$
$242$	24.8522	1.59756
$243$	−1.00000	−0.0641500
$244$	−45.4203	−2.90773
$245$	13.9405	0.890624
$246$	−6.48081	−0.413202
$247$	−5.64098	−0.358927
$248$	13.8341	0.878464
$249$	−14.1340	−0.895703
$250$	27.3946	1.73259
$251$	14.7615	0.931736	0.465868	−	0.884854i	$-0.345742\pi$
$251$	14.7615	0.931736	0.465868	+	0.884854i	$0.345742\pi$
$252$	12.6540	0.797130
$253$	1.29755	0.0815760
$254$	3.97138	0.249186
$255$	1.64317	0.102899
$256$	−15.4117	−0.963229
$257$	−3.34200	−0.208468	−0.104234	−	0.994553i	$-0.533239\pi$
$257$	−3.34200	−0.208468	−0.104234	+	0.994553i	$0.533239\pi$
$258$	−0.424458	−0.0264256
$259$	18.5966	1.15554
$260$	3.93511	0.244045
$261$	5.07234	0.313970
$262$	7.28609	0.450136
$263$	−15.8904	−0.979843	−0.489921	−	0.871767i	$-0.662975\pi$
$263$	−15.8904	−0.979843	−0.489921	+	0.871767i	$0.662975\pi$
$264$	−0.954217	−0.0587280
$265$	13.3169	0.818051
$266$	68.0723	4.17378
$267$	3.05836	0.187169
$268$	7.98064	0.487495
$269$	−16.1573	−0.985127	−0.492564	−	0.870276i	$-0.663940\pi$
$269$	−16.1573	−0.985127	−0.492564	+	0.870276i	$0.663940\pi$
$270$	3.75270	0.228382
$271$	0.735116	0.0446551	0.0223276	−	0.999751i	$-0.492892\pi$
$271$	0.735116	0.0446551	0.0223276	+	0.999751i	$0.492892\pi$
$272$	0.0902101	0.00546979
$273$	−2.93038	−0.177355
$274$	−29.8410	−1.80276
$275$	0.790401	0.0476630
$276$	12.1420	0.730861
$277$	21.7686	1.30795	0.653974	−	0.756517i	$-0.273101\pi$
$277$	21.7686	1.30795	0.653974	+	0.756517i	$0.273101\pi$
$278$	−11.8591	−0.711264
$279$	−4.98225	−0.298279
$280$	−17.9534	−1.07292
$281$	28.2341	1.68431	0.842153	−	0.539239i	$-0.181288\pi$
$281$	28.2341	1.68431	0.842153	+	0.539239i	$0.181288\pi$
$282$	19.0763	1.13597
$283$	−22.8906	−1.36071	−0.680353	−	0.732885i	$-0.738174\pi$
$283$	−22.8906	−1.36071	−0.680353	+	0.732885i	$0.738174\pi$
$284$	−3.98220	−0.236300
$285$	12.4467	0.737277
$286$	0.584478	0.0345609
$287$	−11.1663	−0.659125
$288$	5.75937	0.339374
$289$	1.00000	0.0588235
$290$	−19.0350	−1.11777
$291$	6.82046	0.399823
$292$	−29.8863	−1.74896
$293$	−22.7044	−1.32641	−0.663203	−	0.748439i	$-0.730804\pi$
$293$	−22.7044	−1.32641	−0.663203	+	0.748439i	$0.730804\pi$
$294$	19.3756	1.13001
$295$	−12.3358	−0.718219
$296$	−13.1225	−0.762732
$297$	0.343655	0.0199409
$298$	−23.7955	−1.37844
$299$	−2.81180	−0.162611
$300$	7.39630	0.427026
$301$	−0.731330	−0.0421532
$302$	−27.0150	−1.55454
$303$	−14.1430	−0.812492
$304$	0.683322	0.0391912
$305$	−23.2083	−1.32890
$306$	2.28381	0.130557
$307$	−21.9407	−1.25222	−0.626112	−	0.779733i	$-0.715355\pi$
$307$	−21.9407	−1.25222	−0.626112	+	0.779733i	$0.715355\pi$
$308$	−4.34863	−0.247786
$309$	−4.71884	−0.268445
$310$	18.6969	1.06191
$311$	−6.35520	−0.360370	−0.180185	−	0.983633i	$-0.557670\pi$
$311$	−6.35520	−0.360370	−0.180185	+	0.983633i	$0.557670\pi$
$312$	2.06780	0.117066
$313$	−14.8247	−0.837939	−0.418970	−	0.908000i	$-0.637609\pi$
$313$	−14.8247	−0.837939	−0.418970	+	0.908000i	$0.637609\pi$
$314$	2.28381	0.128883
$315$	6.46581	0.364307
$316$	30.8816	1.73723
$317$	−11.3622	−0.638167	−0.319083	−	0.947727i	$-0.603375\pi$
$317$	−11.3622	−0.638167	−0.319083	+	0.947727i	$0.603375\pi$
$318$	18.5089	1.03793
$319$	−1.74313	−0.0975968
$320$	−21.3167	−1.19164
$321$	16.2355	0.906178
$322$	33.9313	1.89092
$323$	7.57478	0.421472
$324$	3.21580	0.178656
$325$	−1.71281	−0.0950097
$326$	5.59845	0.310070
$327$	18.7232	1.03540
$328$	7.87940	0.435067
$329$	32.8679	1.81207
$330$	−1.28963	−0.0709920
$331$	−21.8740	−1.20230	−0.601151	−	0.799135i	$-0.705291\pi$
$331$	−21.8740	−1.20230	−0.601151	+	0.799135i	$0.705291\pi$
$332$	45.4521	2.49451
$333$	4.72599	0.258983
$334$	−25.7884	−1.41108
$335$	4.07785	0.222797
$336$	0.354973	0.0193653
$337$	31.6522	1.72421	0.862103	−	0.506734i	$-0.169147\pi$
$337$	31.6522	1.72421	0.862103	+	0.506734i	$0.169147\pi$
$338$	28.4230	1.54601
$339$	15.7168	0.853618
$340$	−5.28412	−0.286572
$341$	1.71217	0.0927194
$342$	17.2994	0.935444
$343$	5.83897	0.315275
$344$	0.516058	0.0278240
$345$	6.20416	0.334021
$346$	−3.87465	−0.208303
$347$	15.0487	0.807856	0.403928	−	0.914791i	$-0.367645\pi$
$347$	15.0487	0.807856	0.403928	+	0.914791i	$0.367645\pi$
$348$	−16.3116	−0.874396
$349$	−10.8723	−0.581982	−0.290991	−	0.956726i	$-0.593985\pi$
$349$	−10.8723	−0.581982	−0.290991	+	0.956726i	$0.593985\pi$
$350$	20.6693	1.10482
$351$	−0.744706	−0.0397494
$352$	−1.97923	−0.105494
$353$	−8.45806	−0.450177	−0.225088	−	0.974338i	$-0.572267\pi$
$353$	−8.45806	−0.450177	−0.225088	+	0.974338i	$0.572267\pi$
$354$	−17.1453	−0.911263
$355$	−2.03478	−0.107995
$356$	−9.83510	−0.521259
$357$	3.93496	0.208260
$358$	−11.9458	−0.631354
$359$	16.6960	0.881183	0.440591	−	0.897708i	$-0.354769\pi$
$359$	16.6960	0.881183	0.440591	+	0.897708i	$0.354769\pi$
$360$	−4.56255	−0.240467
$361$	38.3773	2.01986
$362$	27.1415	1.42653
$363$	10.8819	0.571152
$364$	9.42354	0.493928
$365$	−15.2709	−0.799318
$366$	−32.2568	−1.68609
$367$	−25.1638	−1.31354	−0.656770	−	0.754091i	$-0.728078\pi$
$367$	−25.1638	−1.31354	−0.656770	+	0.754091i	$0.728078\pi$
$368$	0.340608	0.0177554
$369$	−2.83772	−0.147726
$370$	−17.7352	−0.922011
$371$	31.8904	1.65567
$372$	16.0219	0.830698
$373$	4.42595	0.229167	0.114583	−	0.993414i	$-0.463447\pi$
$373$	4.42595	0.229167	0.114583	+	0.993414i	$0.463447\pi$
$374$	−0.784844	−0.0405833
$375$	11.9951	0.619426
$376$	−23.1930	−1.19609
$377$	3.77740	0.194546
$378$	8.98670	0.462226
$379$	21.9939	1.12975	0.564877	−	0.825175i	$-0.308924\pi$
$379$	21.9939	1.12975	0.564877	+	0.825175i	$0.308924\pi$
$380$	−40.0261	−2.05329
$381$	1.73892	0.0890878
$382$	18.6121	0.952275
$383$	8.03045	0.410337	0.205168	−	0.978727i	$-0.434226\pi$
$383$	8.03045	0.410337	0.205168	+	0.978727i	$0.434226\pi$
$384$	−18.1089	−0.924118
$385$	−2.22201	−0.113244
$386$	52.6129	2.67793
$387$	−0.185855	−0.00944753
$388$	−21.9333	−1.11349
$389$	13.8051	0.699947	0.349973	−	0.936760i	$-0.386191\pi$
$389$	13.8051	0.699947	0.349973	+	0.936760i	$0.386191\pi$
$390$	2.79466	0.141513
$391$	3.77572	0.190946
$392$	−23.5569	−1.18980
$393$	3.19032	0.160930
$394$	11.6828	0.588569
$395$	15.7795	0.793953
$396$	−1.10513	−0.0555347
$397$	20.1434	1.01097	0.505483	−	0.862836i	$-0.331314\pi$
$397$	20.1434	1.01097	0.505483	+	0.862836i	$0.331314\pi$
$398$	−34.5318	−1.73092
$399$	29.8064	1.49219
$400$	0.207482	0.0103741
$401$	9.08698	0.453782	0.226891	−	0.973920i	$-0.427144\pi$
$401$	9.08698	0.453782	0.226891	+	0.973920i	$0.427144\pi$
$402$	5.66773	0.282681
$403$	−3.71031	−0.184824
$404$	45.4810	2.26277
$405$	1.64317	0.0816499
$406$	−45.5836	−2.26228
$407$	−1.62411	−0.0805042
$408$	−2.77667	−0.137466
$409$	21.5349	1.06483	0.532416	−	0.846483i	$-0.321284\pi$
$409$	21.5349	1.06483	0.532416	+	0.846483i	$0.321284\pi$
$410$	10.6491	0.525921
$411$	−13.0663	−0.644512
$412$	15.1749	0.747612
$413$	−29.5409	−1.45361
$414$	8.62304	0.423799
$415$	23.2245	1.14005
$416$	4.28903	0.210287
$417$	−5.19269	−0.254287
$418$	−5.94502	−0.290781
$419$	12.9785	0.634042	0.317021	−	0.948419i	$-0.397317\pi$
$419$	12.9785	0.634042	0.317021	+	0.948419i	$0.397317\pi$
$420$	−20.7928	−1.01458
$421$	−30.2124	−1.47246	−0.736232	−	0.676729i	$-0.763397\pi$
$421$	−30.2124	−1.47246	−0.736232	+	0.676729i	$0.763397\pi$
$422$	−8.41777	−0.409771
$423$	8.35281	0.406127
$424$	−22.5032	−1.09285
$425$	2.29998	0.111566
$426$	−2.82810	−0.137022
$427$	−55.5776	−2.68959
$428$	−52.2103	−2.52368
$429$	0.255922	0.0123560
$430$	0.697457	0.0336344
$431$	0.789997	0.0380528	0.0190264	−	0.999819i	$-0.493943\pi$
$431$	0.789997	0.0380528	0.0190264	+	0.999819i	$0.493943\pi$
$432$	0.0902101	0.00434024
$433$	−11.6288	−0.558845	−0.279423	−	0.960168i	$-0.590143\pi$
$433$	−11.6288	−0.558845	−0.279423	+	0.960168i	$0.590143\pi$
$434$	44.7740	2.14922
$435$	−8.33473	−0.399620
$436$	−60.2102	−2.88355
$437$	28.6003	1.36814
$438$	−21.2248	−1.01416
$439$	3.63175	0.173334	0.0866670	−	0.996237i	$-0.472378\pi$
$439$	3.63175	0.173334	0.0866670	+	0.996237i	$0.472378\pi$
$440$	1.56794	0.0747488
$441$	8.48387	0.403994
$442$	1.70077	0.0808973
$443$	−23.4637	−1.11479	−0.557396	−	0.830247i	$-0.688199\pi$
$443$	−23.4637	−1.11479	−0.557396	+	0.830247i	$0.688199\pi$
$444$	−15.1979	−0.721259
$445$	−5.02542	−0.238228
$446$	−37.1104	−1.75723
$447$	−10.4192	−0.492812
$448$	−51.0478	−2.41178
$449$	−5.79842	−0.273644	−0.136822	−	0.990596i	$-0.543689\pi$
$449$	−5.79842	−0.273644	−0.136822	+	0.990596i	$0.543689\pi$
$450$	5.25274	0.247616
$451$	0.975195	0.0459202
$452$	−50.5421	−2.37730
$453$	−11.8289	−0.555771
$454$	29.4853	1.38381
$455$	4.81512	0.225737
$456$	−21.0327	−0.984945
$457$	−26.5566	−1.24226	−0.621132	−	0.783706i	$-0.713327\pi$
$457$	−26.5566	−1.24226	−0.621132	+	0.783706i	$0.713327\pi$
$458$	51.1319	2.38924
$459$	1.00000	0.0466760
$460$	−19.9514	−0.930237
$461$	−10.1533	−0.472888	−0.236444	−	0.971645i	$-0.575982\pi$
$461$	−10.1533	−0.472888	−0.236444	+	0.971645i	$0.575982\pi$
$462$	−3.08833	−0.143682
$463$	31.2143	1.45065	0.725326	−	0.688405i	$-0.241689\pi$
$463$	31.2143	1.45065	0.725326	+	0.688405i	$0.241689\pi$
$464$	−0.457576	−0.0212424
$465$	8.18669	0.379649
$466$	62.5450	2.89734
$467$	−10.6906	−0.494701	−0.247350	−	0.968926i	$-0.579560\pi$
$467$	−10.6906	−0.494701	−0.247350	+	0.968926i	$0.579560\pi$
$468$	2.39483	0.110701
$469$	9.76536	0.450922
$470$	−31.3456	−1.44586
$471$	1.00000	0.0460776
$472$	20.8453	0.959484
$473$	0.0638700	0.00293674
$474$	21.9316	1.00735
$475$	17.4219	0.799371
$476$	−12.6540	−0.579997
$477$	8.10439	0.371074
$478$	60.4504	2.76494
$479$	5.50305	0.251441	0.125720	−	0.992066i	$-0.459876\pi$
$479$	5.50305	0.251441	0.125720	+	0.992066i	$0.459876\pi$
$480$	−9.46363	−0.431954
$481$	3.51947	0.160474
$482$	−11.4929	−0.523486
$483$	14.8573	0.676030
$484$	−34.9941	−1.59064
$485$	−11.2072	−0.508893
$486$	2.28381	0.103596
$487$	9.02380	0.408907	0.204454	−	0.978876i	$-0.434458\pi$
$487$	9.02380	0.408907	0.204454	+	0.978876i	$0.434458\pi$
$488$	39.2179	1.77531
$489$	2.45136	0.110854
$490$	−31.8374	−1.43827
$491$	−7.47171	−0.337194	−0.168597	−	0.985685i	$-0.553924\pi$
$491$	−7.47171	−0.337194	−0.168597	+	0.985685i	$0.553924\pi$
$492$	9.12554	0.411411
$493$	−5.07234	−0.228447
$494$	12.8830	0.579631
$495$	−0.564685	−0.0253807
$496$	0.449449	0.0201809
$497$	−4.87274	−0.218572
$498$	32.2793	1.44647
$499$	−36.9614	−1.65462	−0.827311	−	0.561745i	$-0.810130\pi$
$499$	−36.9614	−1.65462	−0.827311	+	0.561745i	$0.810130\pi$
$500$	−38.5740	−1.72508
$501$	−11.2918	−0.504480
$502$	−33.7125	−1.50466
$503$	27.2962	1.21708	0.608538	−	0.793525i	$-0.291756\pi$
$503$	27.2962	1.21708	0.608538	+	0.793525i	$0.291756\pi$
$504$	−10.9261	−0.486686
$505$	23.2393	1.03414
$506$	−2.96335	−0.131737
$507$	12.4454	0.552720
$508$	−5.59204	−0.248107
$509$	18.2765	0.810093	0.405047	−	0.914296i	$-0.367255\pi$
$509$	18.2765	0.810093	0.405047	+	0.914296i	$0.367255\pi$
$510$	−3.75270	−0.166172
$511$	−36.5698	−1.61775
$512$	−1.02052	−0.0451011
$513$	7.57478	0.334435
$514$	7.63251	0.336656
$515$	7.75386	0.341676
$516$	0.597673	0.0263111
$517$	−2.87048	−0.126244
$518$	−42.4711	−1.86607
$519$	−1.69657	−0.0744712
$520$	−3.39776	−0.149001
$521$	−32.9305	−1.44271	−0.721355	−	0.692566i	$-0.756480\pi$
$521$	−32.9305	−1.44271	−0.721355	+	0.692566i	$0.756480\pi$
$522$	−11.5843	−0.507030
$523$	17.1929	0.751793	0.375896	−	0.926662i	$-0.377335\pi$
$523$	17.1929	0.751793	0.375896	+	0.926662i	$0.377335\pi$
$524$	−10.2594	−0.448186
$525$	9.05033	0.394989
$526$	36.2907	1.58235
$527$	4.98225	0.217030
$528$	−0.0310012	−0.00134915
$529$	−8.74393	−0.380171
$530$	−30.4133	−1.32107
$531$	−7.50731	−0.325790
$532$	−95.8516	−4.15570
$533$	−2.11326	−0.0915356
$534$	−6.98473	−0.302259
$535$	−26.6778	−1.15338
$536$	−6.89085	−0.297639
$537$	−5.23063	−0.225718
$538$	36.9002	1.59088
$539$	−2.91553	−0.125581
$540$	−5.28412	−0.227392
$541$	−26.7196	−1.14876	−0.574382	−	0.818588i	$-0.694758\pi$
$541$	−26.7196	−1.14876	−0.574382	+	0.818588i	$0.694758\pi$
$542$	−1.67887	−0.0721136
$543$	11.8843	0.510004
$544$	−5.75937	−0.246931
$545$	−30.7655	−1.31785
$546$	6.69245	0.286410
$547$	−9.68769	−0.414216	−0.207108	−	0.978318i	$-0.566405\pi$
$547$	−9.68769	−0.414216	−0.207108	+	0.978318i	$0.566405\pi$
$548$	42.0186	1.79495
$549$	−14.1241	−0.602801
$550$	−1.80513	−0.0769710
$551$	−38.4219	−1.63683
$552$	−10.4839	−0.446226
$553$	37.7877	1.60689
$554$	−49.7154	−2.11221
$555$	−7.76562	−0.329632
$556$	16.6987	0.708182
$557$	15.7084	0.665586	0.332793	−	0.943000i	$-0.392009\pi$
$557$	15.7084	0.665586	0.332793	+	0.943000i	$0.392009\pi$
$558$	11.3785	0.481691
$559$	−0.138407	−0.00585400
$560$	−0.583281	−0.0246481
$561$	−0.343655	−0.0145091
$562$	−64.4814	−2.71998
$563$	−0.931120	−0.0392420	−0.0196210	−	0.999807i	$-0.506246\pi$
$563$	−0.931120	−0.0392420	−0.0196210	+	0.999807i	$0.506246\pi$
$564$	−26.8610	−1.13105
$565$	−25.8254	−1.08648
$566$	52.2779	2.19740
$567$	3.93496	0.165253
$568$	3.43841	0.144273
$569$	−8.07417	−0.338487	−0.169243	−	0.985574i	$-0.554132\pi$
$569$	−8.07417	−0.338487	−0.169243	+	0.985574i	$0.554132\pi$
$570$	−28.4259	−1.19063
$571$	−10.2170	−0.427569	−0.213785	−	0.976881i	$-0.568579\pi$
$571$	−10.2170	−0.427569	−0.213785	+	0.976881i	$0.568579\pi$
$572$	−0.822995	−0.0344111
$573$	8.14955	0.340452
$574$	25.5017	1.06442
$575$	8.68410	0.362152
$576$	−12.9729	−0.540537
$577$	3.95276	0.164556	0.0822778	−	0.996609i	$-0.473781\pi$
$577$	3.95276	0.164556	0.0822778	+	0.996609i	$0.473781\pi$
$578$	−2.28381	−0.0949941
$579$	23.0373	0.957398
$580$	26.8029	1.11293
$581$	55.6165	2.30736
$582$	−15.5767	−0.645674
$583$	−2.78511	−0.115348
$584$	25.8052	1.06783
$585$	1.22368	0.0505929
$586$	51.8527	2.14201
$587$	−4.52030	−0.186573	−0.0932863	−	0.995639i	$-0.529737\pi$
$587$	−4.52030	−0.186573	−0.0932863	+	0.995639i	$0.529737\pi$
$588$	−27.2825	−1.12511
$589$	37.7394	1.55503
$590$	28.1727	1.15985
$591$	5.11546	0.210422
$592$	−0.426332	−0.0175222
$593$	−5.17192	−0.212385	−0.106193	−	0.994346i	$-0.533866\pi$
$593$	−5.17192	−0.212385	−0.106193	+	0.994346i	$0.533866\pi$
$594$	−0.784844	−0.0322025
$595$	−6.46581	−0.265072
$596$	33.5062	1.37247
$597$	−15.1202	−0.618830
$598$	6.42163	0.262600
$599$	−31.7199	−1.29604	−0.648021	−	0.761622i	$-0.724403\pi$
$599$	−31.7199	−1.29604	−0.648021	+	0.761622i	$0.724403\pi$
$600$	−6.38630	−0.260720
$601$	−16.6431	−0.678888	−0.339444	−	0.940626i	$-0.610239\pi$
$601$	−16.6431	−0.678888	−0.339444	+	0.940626i	$0.610239\pi$
$602$	1.67022	0.0680732
$603$	2.48169	0.101062
$604$	38.0395	1.54781
$605$	−17.8808	−0.726960
$606$	32.2999	1.31209
$607$	−42.2975	−1.71680	−0.858401	−	0.512979i	$-0.828542\pi$
$607$	−42.2975	−1.71680	−0.858401	+	0.512979i	$0.828542\pi$
$608$	−43.6259	−1.76927
$609$	−19.9594	−0.808797
$610$	53.0034	2.14605
$611$	6.22038	0.251650
$612$	−3.21580	−0.129991
$613$	−11.7368	−0.474046	−0.237023	−	0.971504i	$-0.576172\pi$
$613$	−11.7368	−0.474046	−0.237023	+	0.971504i	$0.576172\pi$
$614$	50.1085	2.02222
$615$	4.66286	0.188025
$616$	3.75480	0.151285
$617$	−5.37427	−0.216360	−0.108180	−	0.994131i	$-0.534502\pi$
$617$	−5.37427	−0.216360	−0.108180	+	0.994131i	$0.534502\pi$
$618$	10.7769	0.433512
$619$	23.5041	0.944710	0.472355	−	0.881408i	$-0.343404\pi$
$619$	23.5041	0.944710	0.472355	+	0.881408i	$0.343404\pi$
$620$	−26.3268	−1.05731
$621$	3.77572	0.151514
$622$	14.5141	0.581962
$623$	−12.0345	−0.482153
$624$	0.0671800	0.00268935
$625$	−8.21015	−0.328406
$626$	33.8568	1.35319
$627$	−2.60311	−0.103958
$628$	−3.21580	−0.128325
$629$	−4.72599	−0.188438
$630$	−14.7667	−0.588320
$631$	−31.6032	−1.25810	−0.629051	−	0.777364i	$-0.716556\pi$
$631$	−31.6032	−1.25810	−0.629051	+	0.777364i	$0.716556\pi$
$632$	−26.6646	−1.06066
$633$	−3.68584	−0.146499
$634$	25.9492	1.03058
$635$	−2.85735	−0.113391
$636$	−26.0621	−1.03343
$637$	6.31799	0.250328
$638$	3.98099	0.157609
$639$	−1.23832	−0.0489873
$640$	29.7561	1.17621
$641$	−12.1560	−0.480132	−0.240066	−	0.970757i	$-0.577169\pi$
$641$	−12.1560	−0.480132	−0.240066	+	0.970757i	$0.577169\pi$
$642$	−37.0789	−1.46339
$643$	25.8277	1.01854	0.509272	−	0.860606i	$-0.329915\pi$
$643$	25.8277	1.01854	0.509272	+	0.860606i	$0.329915\pi$
$644$	−47.7782	−1.88272
$645$	0.305392	0.0120248
$646$	−17.2994	−0.680635
$647$	8.69894	0.341991	0.170995	−	0.985272i	$-0.445302\pi$
$647$	8.69894	0.341991	0.170995	+	0.985272i	$0.445302\pi$
$648$	−2.77667	−0.109078
$649$	2.57993	0.101271
$650$	3.91174	0.153431
$651$	19.6049	0.768377
$652$	−7.88310	−0.308726
$653$	−17.8507	−0.698553	−0.349276	−	0.937020i	$-0.613573\pi$
$653$	−17.8507	−0.698553	−0.349276	+	0.937020i	$0.613573\pi$
$654$	−42.7603	−1.67206
$655$	−5.24224	−0.204831
$656$	0.255991	0.00999476
$657$	−9.29357	−0.362577
$658$	−75.0642	−2.92631
$659$	29.0028	1.12979	0.564894	−	0.825163i	$-0.308917\pi$
$659$	29.0028	1.12979	0.564894	+	0.825163i	$0.308917\pi$
$660$	1.81592	0.0706844
$661$	−21.9879	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$661$	−21.9879	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$662$	49.9561	1.94160
$663$	0.744706	0.0289220
$664$	−39.2454	−1.52302
$665$	−48.9771	−1.89925
$666$	−10.7933	−0.418232
$667$	−19.1517	−0.741558
$668$	36.3122	1.40496
$669$	−16.2493	−0.628235
$670$	−9.31305	−0.359795
$671$	4.85381	0.187379
$672$	−22.6628	−0.874238
$673$	−26.6462	−1.02713	−0.513567	−	0.858049i	$-0.671676\pi$
$673$	−26.6462	−1.02713	−0.513567	+	0.858049i	$0.671676\pi$
$674$	−72.2877	−2.78442
$675$	2.29998	0.0885264
$676$	−40.0220	−1.53931
$677$	5.43009	0.208695	0.104348	−	0.994541i	$-0.466725\pi$
$677$	5.43009	0.208695	0.104348	+	0.994541i	$0.466725\pi$
$678$	−35.8942	−1.37851
$679$	−26.8382	−1.02996
$680$	4.56255	0.174966
$681$	12.9106	0.494733
$682$	−3.91029	−0.149733
$683$	20.0529	0.767303	0.383652	−	0.923478i	$-0.374666\pi$
$683$	20.0529	0.767303	0.383652	+	0.923478i	$0.374666\pi$
$684$	−24.3590	−0.931390
$685$	21.4702	0.820333
$686$	−13.3351	−0.509137
$687$	22.3888	0.854186
$688$	0.0167660	0.000639197 0
$689$	6.03538	0.229930
$690$	−14.1691	−0.539410
$691$	6.41006	0.243850	0.121925	−	0.992539i	$-0.461093\pi$
$691$	6.41006	0.243850	0.121925	+	0.992539i	$0.461093\pi$
$692$	5.45584	0.207400
$693$	−1.35227	−0.0513684
$694$	−34.3684	−1.30461
$695$	8.53249	0.323656
$696$	14.0842	0.533861
$697$	2.83772	0.107486
$698$	24.8304	0.939843
$699$	27.3862	1.03584
$700$	−29.1041	−1.10003
$701$	−17.2133	−0.650137	−0.325068	−	0.945691i	$-0.605387\pi$
$701$	−17.2133	−0.650137	−0.325068	+	0.945691i	$0.605387\pi$
$702$	1.70077	0.0641914
$703$	−35.7984	−1.35016
$704$	4.45820	0.168025
$705$	−13.7251	−0.516917
$706$	19.3166	0.726991
$707$	55.6519	2.09301
$708$	24.1420	0.907314
$709$	28.4357	1.06793	0.533963	−	0.845508i	$-0.320702\pi$
$709$	28.4357	1.06793	0.533963	+	0.845508i	$0.320702\pi$
$710$	4.64705	0.174401
$711$	9.60307	0.360143
$712$	8.49207	0.318254
$713$	18.8116	0.704499
$714$	−8.98670	−0.336319
$715$	−0.420524	−0.0157267
$716$	16.8207	0.628618
$717$	26.4691	0.988506
$718$	−38.1306	−1.42302
$719$	14.7261	0.549191	0.274595	−	0.961560i	$-0.411456\pi$
$719$	14.7261	0.549191	0.274595	+	0.961560i	$0.411456\pi$
$720$	−0.148231	−0.00552423
$721$	18.5684	0.691524
$722$	−87.6467	−3.26187
$723$	−5.03232	−0.187154
$724$	−38.2175	−1.42034
$725$	−11.6663	−0.433275
$726$	−24.8522	−0.922353
$727$	26.8968	0.997546	0.498773	−	0.866733i	$-0.333784\pi$
$727$	26.8968	0.997546	0.498773	+	0.866733i	$0.333784\pi$
$728$	−8.13671	−0.301567
$729$	1.00000	0.0370370
$730$	34.8760	1.29082
$731$	0.185855	0.00687409
$732$	45.4203	1.67878
$733$	−17.9577	−0.663282	−0.331641	−	0.943406i	$-0.607602\pi$
$733$	−17.9577	−0.663282	−0.331641	+	0.943406i	$0.607602\pi$
$734$	57.4695	2.12124
$735$	−13.9405	−0.514202
$736$	−21.7458	−0.801559
$737$	−0.852847	−0.0314150
$738$	6.48081	0.238562
$739$	−15.0489	−0.553582	−0.276791	−	0.960930i	$-0.589271\pi$
$739$	−15.0489	−0.553582	−0.276791	+	0.960930i	$0.589271\pi$
$740$	24.9727	0.918016
$741$	5.64098	0.207227
$742$	−72.8317	−2.67374
$743$	−7.99002	−0.293125	−0.146563	−	0.989201i	$-0.546821\pi$
$743$	−7.99002	−0.293125	−0.146563	+	0.989201i	$0.546821\pi$
$744$	−13.8341	−0.507181
$745$	17.1206	0.627249
$746$	−10.1080	−0.370081
$747$	14.1340	0.517135
$748$	1.10513	0.0404075
$749$	−63.8860	−2.33435
$750$	−27.3946	−1.00031
$751$	2.65760	0.0969774	0.0484887	−	0.998824i	$-0.484560\pi$
$751$	2.65760	0.0969774	0.0484887	+	0.998824i	$0.484560\pi$
$752$	−0.753508	−0.0274776
$753$	−14.7615	−0.537938
$754$	−8.62687	−0.314172
$755$	19.4370	0.707383
$756$	−12.6540	−0.460223
$757$	−15.5147	−0.563891	−0.281945	−	0.959430i	$-0.590980\pi$
$757$	−15.5147	−0.563891	−0.281945	+	0.959430i	$0.590980\pi$
$758$	−50.2301	−1.82444
$759$	−1.29755	−0.0470979
$760$	34.5603	1.25363
$761$	−9.83730	−0.356602	−0.178301	−	0.983976i	$-0.557060\pi$
$761$	−9.83730	−0.356602	−0.178301	+	0.983976i	$0.557060\pi$
$762$	−3.97138	−0.143868
$763$	−73.6750	−2.66721
$764$	−26.2074	−0.948149
$765$	−1.64317	−0.0594090
$766$	−18.3401	−0.662653
$767$	−5.59074	−0.201870
$768$	15.4117	0.556121
$769$	−4.02433	−0.145121	−0.0725606	−	0.997364i	$-0.523117\pi$
$769$	−4.02433	−0.145121	−0.0725606	+	0.997364i	$0.523117\pi$
$770$	5.07465	0.182878
$771$	3.34200	0.120359
$772$	−74.0835	−2.66632
$773$	52.4555	1.88669	0.943347	−	0.331809i	$-0.107659\pi$
$773$	52.4555	1.88669	0.943347	+	0.331809i	$0.107659\pi$
$774$	0.424458	0.0152568
$775$	11.4591	0.411623
$776$	18.9382	0.679841
$777$	−18.5966	−0.667148
$778$	−31.5283	−1.13034
$779$	21.4951	0.770141
$780$	−3.93511	−0.140900
$781$	0.425556	0.0152276
$782$	−8.62304	−0.308359
$783$	−5.07234	−0.181271
$784$	−0.765331	−0.0273332
$785$	−1.64317	−0.0586473
$786$	−7.28609	−0.259886
$787$	27.9913	0.997784	0.498892	−	0.866664i	$-0.333740\pi$
$787$	27.9913	0.997784	0.498892	+	0.866664i	$0.333740\pi$
$788$	−16.4503	−0.586019
$789$	15.8904	0.565712
$790$	−36.0374	−1.28215
$791$	−61.8448	−2.19895
$792$	0.954217	0.0339066
$793$	−10.5183	−0.373515
$794$	−46.0037	−1.63261
$795$	−13.3169	−0.472302
$796$	48.6237	1.72342
$797$	21.5375	0.762897	0.381448	−	0.924390i	$-0.375425\pi$
$797$	21.5375	0.762897	0.381448	+	0.924390i	$0.375425\pi$
$798$	−68.0723	−2.40973
$799$	−8.35281	−0.295501
$800$	−13.2464	−0.468333
$801$	−3.05836	−0.108062
$802$	−20.7530	−0.732813
$803$	3.19378	0.112706
$804$	−7.98064	−0.281456
$805$	−24.4131	−0.860449
$806$	8.47365	0.298472
$807$	16.1573	0.568764
$808$	−39.2704	−1.38153
$809$	16.7563	0.589122	0.294561	−	0.955633i	$-0.404827\pi$
$809$	16.7563	0.589122	0.294561	+	0.955633i	$0.404827\pi$
$810$	−3.75270	−0.131856
$811$	5.50520	0.193314	0.0966569	−	0.995318i	$-0.469185\pi$
$811$	5.50520	0.193314	0.0966569	+	0.995318i	$0.469185\pi$
$812$	64.1856	2.25247
$813$	−0.735116	−0.0257816
$814$	3.70917	0.130006
$815$	−4.02801	−0.141095
$816$	−0.0902101	−0.00315799
$817$	1.40781	0.0492530
$818$	−49.1817	−1.71960
$819$	2.93038	0.102396
$820$	−14.9948	−0.523642
$821$	0.976847	0.0340922	0.0170461	−	0.999855i	$-0.494574\pi$
$821$	0.976847	0.0340922	0.0170461	+	0.999855i	$0.494574\pi$
$822$	29.8410	1.04082
$823$	−27.6090	−0.962389	−0.481195	−	0.876614i	$-0.659797\pi$
$823$	−27.6090	−0.962389	−0.481195	+	0.876614i	$0.659797\pi$
$824$	−13.1027	−0.456453
$825$	−0.790401	−0.0275182
$826$	67.4660	2.34744
$827$	−16.3078	−0.567077	−0.283539	−	0.958961i	$-0.591508\pi$
$827$	−16.3078	−0.567077	−0.283539	+	0.958961i	$0.591508\pi$
$828$	−12.1420	−0.421963
$829$	46.7561	1.62391	0.811954	−	0.583722i	$-0.198404\pi$
$829$	46.7561	1.62391	0.811954	+	0.583722i	$0.198404\pi$
$830$	−53.0405	−1.84106
$831$	−21.7686	−0.755144
$832$	−9.66099	−0.334934
$833$	−8.48387	−0.293949
$834$	11.8591	0.410649
$835$	18.5544	0.642101
$836$	8.37110	0.289521
$837$	4.98225	0.172212
$838$	−29.6405	−1.02391
$839$	35.8232	1.23676	0.618378	−	0.785881i	$-0.287790\pi$
$839$	35.8232	1.23676	0.618378	+	0.785881i	$0.287790\pi$
$840$	17.9534	0.619452
$841$	−3.27139	−0.112806
$842$	68.9996	2.37788
$843$	−28.2341	−0.972434
$844$	11.8529	0.407995
$845$	−20.4500	−0.703500
$846$	−19.0763	−0.655855
$847$	−42.8198	−1.47131
$848$	−0.731097	−0.0251060
$849$	22.8906	0.785604
$850$	−5.25274	−0.180167
$851$	−17.8440	−0.611686
$852$	3.98220	0.136428
$853$	22.0479	0.754906	0.377453	−	0.926029i	$-0.376800\pi$
$853$	22.0479	0.754906	0.377453	+	0.926029i	$0.376800\pi$
$854$	126.929	4.34342
$855$	−12.4467	−0.425667
$856$	45.0807	1.54083
$857$	36.8458	1.25863	0.629314	−	0.777151i	$-0.283336\pi$
$857$	36.8458	1.25863	0.629314	+	0.777151i	$0.283336\pi$
$858$	−0.584478	−0.0199537
$859$	−46.3277	−1.58068	−0.790341	−	0.612667i	$-0.790097\pi$
$859$	−46.3277	−1.58068	−0.790341	+	0.612667i	$0.790097\pi$
$860$	−0.982080	−0.0334886
$861$	11.1663	0.380546
$862$	−1.80421	−0.0614515
$863$	−34.1582	−1.16276	−0.581379	−	0.813633i	$-0.697487\pi$
$863$	−34.1582	−1.16276	−0.581379	+	0.813633i	$0.697487\pi$
$864$	−5.75937	−0.195938
$865$	2.78776	0.0947866
$866$	26.5581	0.902480
$867$	−1.00000	−0.0339618
$868$	−63.0456	−2.13991
$869$	−3.30014	−0.111950
$870$	19.0350	0.645346
$871$	1.84813	0.0626216
$872$	51.9882	1.76054
$873$	−6.82046	−0.230838
$874$	−65.3177	−2.20940
$875$	−47.2003	−1.59566
$876$	29.8863	1.00977
$877$	14.2785	0.482151	0.241076	−	0.970506i	$-0.422500\pi$
$877$	14.2785	0.482151	0.241076	+	0.970506i	$0.422500\pi$
$878$	−8.29424	−0.279917
$879$	22.7044	0.765801
$880$	0.0509402	0.00171720
$881$	−2.72212	−0.0917107	−0.0458554	−	0.998948i	$-0.514601\pi$
$881$	−2.72212	−0.0917107	−0.0458554	+	0.998948i	$0.514601\pi$
$882$	−19.3756	−0.652410
$883$	8.36424	0.281479	0.140740	−	0.990047i	$-0.455052\pi$
$883$	8.36424	0.281479	0.140740	+	0.990047i	$0.455052\pi$
$884$	−2.39483	−0.0805468
$885$	12.3358	0.414664
$886$	53.5866	1.80028
$887$	−22.1991	−0.745374	−0.372687	−	0.927957i	$-0.621564\pi$
$887$	−22.1991	−0.745374	−0.372687	+	0.927957i	$0.621564\pi$
$888$	13.1225	0.440363
$889$	−6.84259	−0.229493
$890$	11.4771	0.384714
$891$	−0.343655	−0.0115129
$892$	52.2547	1.74962
$893$	−63.2707	−2.11727
$894$	23.7955	0.795842
$895$	8.59482	0.287293
$896$	71.2579	2.38056
$897$	2.81180	0.0938833
$898$	13.2425	0.441908
$899$	−25.2716	−0.842856
$900$	−7.39630	−0.246543
$901$	−8.10439	−0.269996
$902$	−2.22716	−0.0741565
$903$	0.731330	0.0243372
$904$	43.6403	1.45146
$905$	−19.5279	−0.649130
$906$	27.0150	0.897515
$907$	11.0560	0.367108	0.183554	−	0.983010i	$-0.441240\pi$
$907$	11.0560	0.367108	0.183554	+	0.983010i	$0.441240\pi$
$908$	−41.5178	−1.37782
$909$	14.1430	0.469093
$910$	−10.9968	−0.364542
$911$	−6.18810	−0.205021	−0.102510	−	0.994732i	$-0.532687\pi$
$911$	−6.18810	−0.205021	−0.102510	+	0.994732i	$0.532687\pi$
$912$	−0.683322	−0.0226271
$913$	−4.85721	−0.160750
$914$	60.6502	2.00613
$915$	23.2083	0.767242
$916$	−71.9980	−2.37888
$917$	−12.5538	−0.414562
$918$	−2.28381	−0.0753771
$919$	−32.5750	−1.07455	−0.537274	−	0.843407i	$-0.680546\pi$
$919$	−32.5750	−1.07455	−0.537274	+	0.843407i	$0.680546\pi$
$920$	17.2269	0.567955
$921$	21.9407	0.722972
$922$	23.1883	0.763667
$923$	−0.922185	−0.0303541
$924$	4.34863	0.143059
$925$	−10.8697	−0.357394
$926$	−71.2877	−2.34266
$927$	4.71884	0.154987
$928$	29.2134	0.958978
$929$	30.0554	0.986084	0.493042	−	0.870005i	$-0.335885\pi$
$929$	30.0554	0.986084	0.493042	+	0.870005i	$0.335885\pi$
$930$	−18.6969	−0.613095
$931$	−64.2635	−2.10615
$932$	−88.0687	−2.88479
$933$	6.35520	0.208060
$934$	24.4153	0.798892
$935$	0.564685	0.0184672
$936$	−2.06780	−0.0675882
$937$	35.5638	1.16182	0.580909	−	0.813969i	$-0.302697\pi$
$937$	35.5638	1.16182	0.580909	+	0.813969i	$0.302697\pi$
$938$	−22.3023	−0.728194
$939$	14.8247	0.483785
$940$	44.1373	1.43960
$941$	18.7705	0.611902	0.305951	−	0.952047i	$-0.401026\pi$
$941$	18.7705	0.611902	0.305951	+	0.952047i	$0.401026\pi$
$942$	−2.28381	−0.0744107
$943$	10.7144	0.348910
$944$	0.677235	0.0220421
$945$	−6.46581	−0.210333
$946$	−0.145867	−0.00474255
$947$	−52.5018	−1.70608	−0.853039	−	0.521847i	$-0.825243\pi$
$947$	−52.5018	−1.70608	−0.853039	+	0.521847i	$0.825243\pi$
$948$	−30.8816	−1.00299
$949$	−6.92098	−0.224664
$950$	−39.7883	−1.29090
$951$	11.3622	0.368446
$952$	10.9261	0.354116
$953$	−31.7200	−1.02751	−0.513756	−	0.857936i	$-0.671746\pi$
$953$	−31.7200	−1.02751	−0.513756	+	0.857936i	$0.671746\pi$
$954$	−18.5089	−0.599248
$955$	−13.3911	−0.433326
$956$	−85.1194	−2.75296
$957$	1.74313	0.0563475
$958$	−12.5679	−0.406052
$959$	51.4152	1.66029
$960$	21.3167	0.687994
$961$	−6.17722	−0.199265
$962$	−8.03782	−0.259150
$963$	−16.2355	−0.523182
$964$	16.1829	0.521218
$965$	−37.8543	−1.21857
$966$	−33.9313	−1.09172
$967$	0.957467	0.0307901	0.0153950	−	0.999881i	$-0.495099\pi$
$967$	0.957467	0.0307901	0.0153950	+	0.999881i	$0.495099\pi$
$968$	30.2155	0.971162
$969$	−7.57478	−0.243337
$970$	25.5952	0.821811
$971$	−40.2360	−1.29124	−0.645618	−	0.763660i	$-0.723400\pi$
$971$	−40.2360	−1.29124	−0.645618	+	0.763660i	$0.723400\pi$
$972$	−3.21580	−0.103147
$973$	20.4330	0.655052
$974$	−20.6087	−0.660345
$975$	1.71281	0.0548539
$976$	1.27413	0.0407840
$977$	−24.2585	−0.776099	−0.388050	−	0.921639i	$-0.626851\pi$
$977$	−24.2585	−0.776099	−0.388050	+	0.921639i	$0.626851\pi$
$978$	−5.59845	−0.179019
$979$	1.05102	0.0335908
$980$	44.8298	1.43204
$981$	−18.7232	−0.597786
$982$	17.0640	0.544534
$983$	−8.98399	−0.286545	−0.143272	−	0.989683i	$-0.545763\pi$
$983$	−8.98399	−0.286545	−0.143272	+	0.989683i	$0.545763\pi$
$984$	−7.87940	−0.251186
$985$	−8.40559	−0.267824
$986$	11.5843	0.368918
$987$	−32.8679	−1.04620
$988$	−18.1403	−0.577120
$989$	0.701736	0.0223139
$990$	1.28963	0.0409873
$991$	21.5905	0.685845	0.342922	−	0.939364i	$-0.388583\pi$
$991$	21.5905	0.685845	0.342922	+	0.939364i	$0.388583\pi$
$992$	−28.6946	−0.911054
$993$	21.8740	0.694150
$994$	11.1284	0.352972
$995$	24.8451	0.787644
$996$	−45.4521	−1.44020
$997$	−42.2012	−1.33653	−0.668263	−	0.743925i	$-0.732962\pi$
$997$	−42.2012	−1.33653	−0.668263	+	0.743925i	$0.732962\pi$
$998$	84.4130	2.67205
$999$	−4.72599	−0.149524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.6	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.6	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.28381 q^{2} -1.00000 q^{3} +3.21580 q^{4} +1.64317 q^{5} +2.28381 q^{6} +3.93496 q^{7} -2.77667 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.28381 q^{2} -1.00000 q^{3} +3.21580 q^{4} +1.64317 q^{5} +2.28381 q^{6} +3.93496 q^{7} -2.77667 q^{8} +1.00000 q^{9} -3.75270 q^{10} -0.343655 q^{11} -3.21580 q^{12} +0.744706 q^{13} -8.98670 q^{14} -1.64317 q^{15} -0.0902101 q^{16} -1.00000 q^{17} -2.28381 q^{18} -7.57478 q^{19} +5.28412 q^{20} -3.93496 q^{21} +0.784844 q^{22} -3.77572 q^{23} +2.77667 q^{24} -2.29998 q^{25} -1.70077 q^{26} -1.00000 q^{27} +12.6540 q^{28} +5.07234 q^{29} +3.75270 q^{30} -4.98225 q^{31} +5.75937 q^{32} +0.343655 q^{33} +2.28381 q^{34} +6.46581 q^{35} +3.21580 q^{36} +4.72599 q^{37} +17.2994 q^{38} -0.744706 q^{39} -4.56255 q^{40} -2.83772 q^{41} +8.98670 q^{42} -0.185855 q^{43} -1.10513 q^{44} +1.64317 q^{45} +8.62304 q^{46} +8.35281 q^{47} +0.0902101 q^{48} +8.48387 q^{49} +5.25274 q^{50} +1.00000 q^{51} +2.39483 q^{52} +8.10439 q^{53} +2.28381 q^{54} -0.564685 q^{55} -10.9261 q^{56} +7.57478 q^{57} -11.5843 q^{58} -7.50731 q^{59} -5.28412 q^{60} -14.1241 q^{61} +11.3785 q^{62} +3.93496 q^{63} -12.9729 q^{64} +1.22368 q^{65} -0.784844 q^{66} +2.48169 q^{67} -3.21580 q^{68} +3.77572 q^{69} -14.7667 q^{70} -1.23832 q^{71} -2.77667 q^{72} -9.29357 q^{73} -10.7933 q^{74} +2.29998 q^{75} -24.3590 q^{76} -1.35227 q^{77} +1.70077 q^{78} +9.60307 q^{79} -0.148231 q^{80} +1.00000 q^{81} +6.48081 q^{82} +14.1340 q^{83} -12.6540 q^{84} -1.64317 q^{85} +0.424458 q^{86} -5.07234 q^{87} +0.954217 q^{88} -3.05836 q^{89} -3.75270 q^{90} +2.93038 q^{91} -12.1420 q^{92} +4.98225 q^{93} -19.0763 q^{94} -12.4467 q^{95} -5.75937 q^{96} -6.82046 q^{97} -19.3756 q^{98} -0.343655 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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