LMFDB - Embedded newform 8007.2.a.f.1.27

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			1.27
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)

$f(q)$	$=$	$q+0.390980 q^{2} -1.00000 q^{3} -1.84713 q^{4} +3.91390 q^{5} -0.390980 q^{6} +2.71351 q^{7} -1.50415 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.390980 q^{2} -1.00000 q^{3} -1.84713 q^{4} +3.91390 q^{5} -0.390980 q^{6} +2.71351 q^{7} -1.50415 q^{8} +1.00000 q^{9} +1.53026 q^{10} -5.03642 q^{11} +1.84713 q^{12} -2.69008 q^{13} +1.06093 q^{14} -3.91390 q^{15} +3.10618 q^{16} -1.00000 q^{17} +0.390980 q^{18} -4.89634 q^{19} -7.22950 q^{20} -2.71351 q^{21} -1.96914 q^{22} +3.69816 q^{23} +1.50415 q^{24} +10.3186 q^{25} -1.05177 q^{26} -1.00000 q^{27} -5.01222 q^{28} +0.627384 q^{29} -1.53026 q^{30} +2.22657 q^{31} +4.22276 q^{32} +5.03642 q^{33} -0.390980 q^{34} +10.6204 q^{35} -1.84713 q^{36} -4.11141 q^{37} -1.91437 q^{38} +2.69008 q^{39} -5.88710 q^{40} -3.46252 q^{41} -1.06093 q^{42} +10.0757 q^{43} +9.30295 q^{44} +3.91390 q^{45} +1.44590 q^{46} -7.23632 q^{47} -3.10618 q^{48} +0.363126 q^{49} +4.03438 q^{50} +1.00000 q^{51} +4.96893 q^{52} -0.926864 q^{53} -0.390980 q^{54} -19.7121 q^{55} -4.08153 q^{56} +4.89634 q^{57} +0.245295 q^{58} +0.268502 q^{59} +7.22950 q^{60} -2.95056 q^{61} +0.870546 q^{62} +2.71351 q^{63} -4.56134 q^{64} -10.5287 q^{65} +1.96914 q^{66} +11.4515 q^{67} +1.84713 q^{68} -3.69816 q^{69} +4.15236 q^{70} -4.07057 q^{71} -1.50415 q^{72} -12.2324 q^{73} -1.60748 q^{74} -10.3186 q^{75} +9.04419 q^{76} -13.6664 q^{77} +1.05177 q^{78} -13.1141 q^{79} +12.1573 q^{80} +1.00000 q^{81} -1.35377 q^{82} +12.0126 q^{83} +5.01222 q^{84} -3.91390 q^{85} +3.93939 q^{86} -0.627384 q^{87} +7.57555 q^{88} +2.79877 q^{89} +1.53026 q^{90} -7.29954 q^{91} -6.83100 q^{92} -2.22657 q^{93} -2.82926 q^{94} -19.1638 q^{95} -4.22276 q^{96} -11.8418 q^{97} +0.141975 q^{98} -5.03642 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.390980	0.276464	0.138232	−	0.990400i	$-0.455858\pi$
$2$	0.390980	0.276464	0.138232	+	0.990400i	$0.455858\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.84713	−0.923567
$5$	3.91390	1.75035	0.875175	−	0.483806i	$-0.160746\pi$
$5$	3.91390	1.75035	0.875175	+	0.483806i	$0.160746\pi$
$6$	−0.390980	−0.159617
$7$	2.71351	1.02561	0.512805	−	0.858505i	$-0.328606\pi$
$7$	2.71351	1.02561	0.512805	+	0.858505i	$0.328606\pi$
$8$	−1.50415	−0.531798
$9$	1.00000	0.333333
$10$	1.53026	0.483910
$11$	−5.03642	−1.51854	−0.759269	−	0.650777i	$-0.774443\pi$
$11$	−5.03642	−1.51854	−0.759269	+	0.650777i	$0.774443\pi$
$12$	1.84713	0.533222
$13$	−2.69008	−0.746093	−0.373046	−	0.927813i	$-0.621687\pi$
$13$	−2.69008	−0.746093	−0.373046	+	0.927813i	$0.621687\pi$
$14$	1.06093	0.283545
$15$	−3.91390	−1.01057
$16$	3.10618	0.776544
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.390980	0.0921548
$19$	−4.89634	−1.12330	−0.561648	−	0.827376i	$-0.689833\pi$
$19$	−4.89634	−1.12330	−0.561648	+	0.827376i	$0.689833\pi$
$20$	−7.22950	−1.61657
$21$	−2.71351	−0.592136
$22$	−1.96914	−0.419822
$23$	3.69816	0.771119	0.385560	−	0.922683i	$-0.374008\pi$
$23$	3.69816	0.771119	0.385560	+	0.922683i	$0.374008\pi$
$24$	1.50415	0.307034
$25$	10.3186	2.06373
$26$	−1.05177	−0.206268
$27$	−1.00000	−0.192450
$28$	−5.01222	−0.947220
$29$	0.627384	0.116502	0.0582512	−	0.998302i	$-0.481448\pi$
$29$	0.627384	0.116502	0.0582512	+	0.998302i	$0.481448\pi$
$30$	−1.53026	−0.279385
$31$	2.22657	0.399905	0.199952	−	0.979806i	$-0.435921\pi$
$31$	2.22657	0.399905	0.199952	+	0.979806i	$0.435921\pi$
$32$	4.22276	0.746485
$33$	5.03642	0.876729
$34$	−0.390980	−0.0670525
$35$	10.6204	1.79518
$36$	−1.84713	−0.307856
$37$	−4.11141	−0.675911	−0.337956	−	0.941162i	$-0.609735\pi$
$37$	−4.11141	−0.675911	−0.337956	+	0.941162i	$0.609735\pi$
$38$	−1.91437	−0.310552
$39$	2.69008	0.430757
$40$	−5.88710	−0.930833
$41$	−3.46252	−0.540754	−0.270377	−	0.962754i	$-0.587148\pi$
$41$	−3.46252	−0.540754	−0.270377	+	0.962754i	$0.587148\pi$
$42$	−1.06093	−0.163705
$43$	10.0757	1.53653	0.768265	−	0.640132i	$-0.221120\pi$
$43$	10.0757	1.53653	0.768265	+	0.640132i	$0.221120\pi$
$44$	9.30295	1.40247
$45$	3.91390	0.583450
$46$	1.44590	0.213187
$47$	−7.23632	−1.05553	−0.527763	−	0.849392i	$-0.676969\pi$
$47$	−7.23632	−1.05553	−0.527763	+	0.849392i	$0.676969\pi$
$48$	−3.10618	−0.448338
$49$	0.363126	0.0518751
$50$	4.03438	0.570547
$51$	1.00000	0.140028
$52$	4.96893	0.689067
$53$	−0.926864	−0.127315	−0.0636573	−	0.997972i	$-0.520276\pi$
$53$	−0.926864	−0.127315	−0.0636573	+	0.997972i	$0.520276\pi$
$54$	−0.390980	−0.0532056
$55$	−19.7121	−2.65797
$56$	−4.08153	−0.545417
$57$	4.89634	0.648536
$58$	0.245295	0.0322088
$59$	0.268502	0.0349560	0.0174780	−	0.999847i	$-0.494436\pi$
$59$	0.268502	0.0349560	0.0174780	+	0.999847i	$0.494436\pi$
$60$	7.22950	0.933325
$61$	−2.95056	−0.377780	−0.188890	−	0.981998i	$-0.560489\pi$
$61$	−2.95056	−0.377780	−0.188890	+	0.981998i	$0.560489\pi$
$62$	0.870546	0.110559
$63$	2.71351	0.341870
$64$	−4.56134	−0.570168
$65$	−10.5287	−1.30592
$66$	1.96914	0.242384
$67$	11.4515	1.39902	0.699509	−	0.714624i	$-0.253402\pi$
$67$	11.4515	1.39902	0.699509	+	0.714624i	$0.253402\pi$
$68$	1.84713	0.223998
$69$	−3.69816	−0.445206
$70$	4.15236	0.496302
$71$	−4.07057	−0.483088	−0.241544	−	0.970390i	$-0.577654\pi$
$71$	−4.07057	−0.483088	−0.241544	+	0.970390i	$0.577654\pi$
$72$	−1.50415	−0.177266
$73$	−12.2324	−1.43169	−0.715845	−	0.698260i	$-0.753958\pi$
$73$	−12.2324	−1.43169	−0.715845	+	0.698260i	$0.753958\pi$
$74$	−1.60748	−0.186865
$75$	−10.3186	−1.19149
$76$	9.04419	1.03744
$77$	−13.6664	−1.55743
$78$	1.05177	0.119089
$79$	−13.1141	−1.47545	−0.737723	−	0.675103i	$-0.764099\pi$
$79$	−13.1141	−1.47545	−0.737723	+	0.675103i	$0.764099\pi$
$80$	12.1573	1.35922
$81$	1.00000	0.111111
$82$	−1.35377	−0.149499
$83$	12.0126	1.31856	0.659280	−	0.751898i	$-0.270861\pi$
$83$	12.0126	1.31856	0.659280	+	0.751898i	$0.270861\pi$
$84$	5.01222	0.546878
$85$	−3.91390	−0.424522
$86$	3.93939	0.424796
$87$	−0.627384	−0.0672627
$88$	7.57555	0.807556
$89$	2.79877	0.296669	0.148335	−	0.988937i	$-0.452609\pi$
$89$	2.79877	0.296669	0.148335	+	0.988937i	$0.452609\pi$
$90$	1.53026	0.161303
$91$	−7.29954	−0.765200
$92$	−6.83100	−0.712180
$93$	−2.22657	−0.230885
$94$	−2.82926	−0.291815
$95$	−19.1638	−1.96616
$96$	−4.22276	−0.430983
$97$	−11.8418	−1.20235	−0.601175	−	0.799118i	$-0.705300\pi$
$97$	−11.8418	−1.20235	−0.601175	+	0.799118i	$0.705300\pi$
$98$	0.141975	0.0143416
$99$	−5.03642	−0.506179
$100$	−19.0599	−1.90599
$101$	−18.3679	−1.82768	−0.913839	−	0.406077i	$-0.866897\pi$
$101$	−18.3679	−1.82768	−0.913839	+	0.406077i	$0.866897\pi$
$102$	0.390980	0.0387128
$103$	−6.88240	−0.678143	−0.339071	−	0.940761i	$-0.610113\pi$
$103$	−6.88240	−0.678143	−0.339071	+	0.940761i	$0.610113\pi$
$104$	4.04628	0.396771
$105$	−10.6204	−1.03645
$106$	−0.362385	−0.0351980
$107$	8.90217	0.860606	0.430303	−	0.902685i	$-0.358407\pi$
$107$	8.90217	0.860606	0.430303	+	0.902685i	$0.358407\pi$
$108$	1.84713	0.177741
$109$	−7.57002	−0.725076	−0.362538	−	0.931969i	$-0.618090\pi$
$109$	−7.57002	−0.725076	−0.362538	+	0.931969i	$0.618090\pi$
$110$	−7.70702	−0.734835
$111$	4.11141	0.390237
$112$	8.42863	0.796431
$113$	6.73057	0.633159	0.316579	−	0.948566i	$-0.397466\pi$
$113$	6.73057	0.633159	0.316579	+	0.948566i	$0.397466\pi$
$114$	1.91437	0.179297
$115$	14.4742	1.34973
$116$	−1.15886	−0.107598
$117$	−2.69008	−0.248698
$118$	0.104979	0.00966409
$119$	−2.71351	−0.248747
$120$	5.88710	0.537417
$121$	14.3656	1.30596
$122$	−1.15361	−0.104443
$123$	3.46252	0.312205
$124$	−4.11278	−0.369339
$125$	20.8166	1.86189
$126$	1.06093	0.0945149
$127$	7.85630	0.697134	0.348567	−	0.937284i	$-0.386668\pi$
$127$	7.85630	0.697134	0.348567	+	0.937284i	$0.386668\pi$
$128$	−10.2289	−0.904116
$129$	−10.0757	−0.887116
$130$	−4.11651	−0.361041
$131$	−0.368603	−0.0322050	−0.0161025	−	0.999870i	$-0.505126\pi$
$131$	−0.368603	−0.0322050	−0.0161025	+	0.999870i	$0.505126\pi$
$132$	−9.30295	−0.809718
$133$	−13.2862	−1.15206
$134$	4.47729	0.386779
$135$	−3.91390	−0.336855
$136$	1.50415	0.128980
$137$	−14.3166	−1.22315	−0.611574	−	0.791187i	$-0.709463\pi$
$137$	−14.3166	−1.22315	−0.611574	+	0.791187i	$0.709463\pi$
$138$	−1.44590	−0.123084
$139$	−21.0100	−1.78204	−0.891021	−	0.453962i	$-0.850010\pi$
$139$	−21.0100	−1.78204	−0.891021	+	0.453962i	$0.850010\pi$
$140$	−19.6173	−1.65797
$141$	7.23632	0.609408
$142$	−1.59151	−0.133557
$143$	13.5484	1.13297
$144$	3.10618	0.258848
$145$	2.45552	0.203920
$146$	−4.78261	−0.395811
$147$	−0.363126	−0.0299501
$148$	7.59432	0.624249
$149$	10.1025	0.827626	0.413813	−	0.910362i	$-0.364197\pi$
$149$	10.1025	0.827626	0.413813	+	0.910362i	$0.364197\pi$
$150$	−4.03438	−0.329405
$151$	0.503267	0.0409553	0.0204776	−	0.999790i	$-0.493481\pi$
$151$	0.503267	0.0409553	0.0204776	+	0.999790i	$0.493481\pi$
$152$	7.36483	0.597367
$153$	−1.00000	−0.0808452
$154$	−5.34328	−0.430573
$155$	8.71460	0.699973
$156$	−4.96893	−0.397833
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−5.12733	−0.407908
$159$	0.926864	0.0735051
$160$	16.5275	1.30661
$161$	10.0350	0.790867
$162$	0.390980	0.0307183
$163$	−14.5475	−1.13945	−0.569724	−	0.821836i	$-0.692950\pi$
$163$	−14.5475	−1.13945	−0.569724	+	0.821836i	$0.692950\pi$
$164$	6.39574	0.499423
$165$	19.7121	1.53458
$166$	4.69670	0.364535
$167$	−16.3017	−1.26146	−0.630731	−	0.776002i	$-0.717245\pi$
$167$	−16.3017	−1.26146	−0.630731	+	0.776002i	$0.717245\pi$
$168$	4.08153	0.314897
$169$	−5.76349	−0.443346
$170$	−1.53026	−0.117365
$171$	−4.89634	−0.374432
$172$	−18.6112	−1.41909
$173$	18.7711	1.42714	0.713569	−	0.700585i	$-0.247077\pi$
$173$	18.7711	1.42714	0.713569	+	0.700585i	$0.247077\pi$
$174$	−0.245295	−0.0185957
$175$	27.9997	2.11658
$176$	−15.6440	−1.17921
$177$	−0.268502	−0.0201818
$178$	1.09426	0.0820186
$179$	−17.3903	−1.29981	−0.649907	−	0.760014i	$-0.725192\pi$
$179$	−17.3903	−1.29981	−0.649907	+	0.760014i	$0.725192\pi$
$180$	−7.22950	−0.538855
$181$	0.951024	0.0706891	0.0353445	−	0.999375i	$-0.488747\pi$
$181$	0.951024	0.0706891	0.0353445	+	0.999375i	$0.488747\pi$
$182$	−2.85397	−0.211551
$183$	2.95056	0.218112
$184$	−5.56259	−0.410080
$185$	−16.0916	−1.18308
$186$	−0.870546	−0.0638315
$187$	5.03642	0.368300
$188$	13.3665	0.974849
$189$	−2.71351	−0.197379
$190$	−7.49265	−0.543574
$191$	−25.5087	−1.84575	−0.922874	−	0.385102i	$-0.874166\pi$
$191$	−25.5087	−1.84575	−0.922874	+	0.385102i	$0.874166\pi$
$192$	4.56134	0.329186
$193$	12.5600	0.904087	0.452043	−	0.891996i	$-0.350695\pi$
$193$	12.5600	0.904087	0.452043	+	0.891996i	$0.350695\pi$
$194$	−4.62989	−0.332407
$195$	10.5287	0.753975
$196$	−0.670743	−0.0479102
$197$	2.66907	0.190163	0.0950817	−	0.995469i	$-0.469689\pi$
$197$	2.66907	0.190163	0.0950817	+	0.995469i	$0.469689\pi$
$198$	−1.96914	−0.139941
$199$	8.32187	0.589922	0.294961	−	0.955509i	$-0.404693\pi$
$199$	8.32187	0.589922	0.294961	+	0.955509i	$0.404693\pi$
$200$	−15.5208	−1.09749
$201$	−11.4515	−0.807724
$202$	−7.18149	−0.505288
$203$	1.70241	0.119486
$204$	−1.84713	−0.129325
$205$	−13.5520	−0.946510
$206$	−2.69088	−0.187482
$207$	3.69816	0.257040
$208$	−8.35585	−0.579374
$209$	24.6600	1.70577
$210$	−4.15236	−0.286540
$211$	−9.85334	−0.678332	−0.339166	−	0.940727i	$-0.610145\pi$
$211$	−9.85334	−0.678332	−0.339166	+	0.940727i	$0.610145\pi$
$212$	1.71204	0.117584
$213$	4.07057	0.278911
$214$	3.48057	0.237927
$215$	39.4353	2.68946
$216$	1.50415	0.102345
$217$	6.04183	0.410146
$218$	−2.95972	−0.200458
$219$	12.2324	0.826586
$220$	36.4108	2.45482
$221$	2.69008	0.180954
$222$	1.60748	0.107887
$223$	24.2147	1.62153	0.810767	−	0.585368i	$-0.199050\pi$
$223$	24.2147	1.62153	0.810767	+	0.585368i	$0.199050\pi$
$224$	11.4585	0.765602
$225$	10.3186	0.687909
$226$	2.63152	0.175046
$227$	8.04202	0.533767	0.266884	−	0.963729i	$-0.414006\pi$
$227$	8.04202	0.533767	0.266884	+	0.963729i	$0.414006\pi$
$228$	−9.04419	−0.598966
$229$	16.1158	1.06496	0.532482	−	0.846441i	$-0.321259\pi$
$229$	16.1158	1.06496	0.532482	+	0.846441i	$0.321259\pi$
$230$	5.65913	0.373152
$231$	13.6664	0.899181
$232$	−0.943681	−0.0619557
$233$	−16.3553	−1.07147	−0.535736	−	0.844385i	$-0.679966\pi$
$233$	−16.3553	−1.07147	−0.535736	+	0.844385i	$0.679966\pi$
$234$	−1.05177	−0.0687560
$235$	−28.3223	−1.84754
$236$	−0.495959	−0.0322842
$237$	13.1141	0.851849
$238$	−1.06093	−0.0687697
$239$	25.1365	1.62595	0.812973	−	0.582302i	$-0.197848\pi$
$239$	25.1365	1.62595	0.812973	+	0.582302i	$0.197848\pi$
$240$	−12.1573	−0.784748
$241$	−15.1972	−0.978938	−0.489469	−	0.872021i	$-0.662809\pi$
$241$	−15.1972	−0.978938	−0.489469	+	0.872021i	$0.662809\pi$
$242$	5.61664	0.361051
$243$	−1.00000	−0.0641500
$244$	5.45008	0.348906
$245$	1.42124	0.0907997
$246$	1.35377	0.0863135
$247$	13.1715	0.838083
$248$	−3.34911	−0.212669
$249$	−12.0126	−0.761271
$250$	8.13887	0.514747
$251$	6.79441	0.428859	0.214430	−	0.976739i	$-0.431211\pi$
$251$	6.79441	0.428859	0.214430	+	0.976739i	$0.431211\pi$
$252$	−5.01222	−0.315740
$253$	−18.6255	−1.17097
$254$	3.07166	0.192733
$255$	3.91390	0.245098
$256$	5.12339	0.320212
$257$	2.97235	0.185410	0.0927051	−	0.995694i	$-0.470449\pi$
$257$	2.97235	0.185410	0.0927051	+	0.995694i	$0.470449\pi$
$258$	−3.93939	−0.245256
$259$	−11.1563	−0.693221
$260$	19.4479	1.20611
$261$	0.627384	0.0388341
$262$	−0.144117	−0.00890355
$263$	−11.9656	−0.737830	−0.368915	−	0.929463i	$-0.620271\pi$
$263$	−11.9656	−0.737830	−0.368915	+	0.929463i	$0.620271\pi$
$264$	−7.57555	−0.466243
$265$	−3.62766	−0.222845
$266$	−5.19466	−0.318505
$267$	−2.79877	−0.171282
$268$	−21.1524	−1.29209
$269$	−25.5460	−1.55757	−0.778783	−	0.627294i	$-0.784163\pi$
$269$	−25.5460	−1.55757	−0.778783	+	0.627294i	$0.784163\pi$
$270$	−1.53026	−0.0931285
$271$	23.9568	1.45527	0.727635	−	0.685965i	$-0.240620\pi$
$271$	23.9568	1.45527	0.727635	+	0.685965i	$0.240620\pi$
$272$	−3.10618	−0.188340
$273$	7.29954	0.441788
$274$	−5.59749	−0.338157
$275$	−51.9690	−3.13385
$276$	6.83100	0.411178
$277$	21.8448	1.31252	0.656262	−	0.754533i	$-0.272136\pi$
$277$	21.8448	1.31252	0.656262	+	0.754533i	$0.272136\pi$
$278$	−8.21447	−0.492671
$279$	2.22657	0.133302
$280$	−15.9747	−0.954671
$281$	−13.1790	−0.786192	−0.393096	−	0.919498i	$-0.628596\pi$
$281$	−13.1790	−0.786192	−0.393096	+	0.919498i	$0.628596\pi$
$282$	2.82926	0.168480
$283$	−17.8527	−1.06123	−0.530617	−	0.847612i	$-0.678040\pi$
$283$	−17.8527	−1.06123	−0.530617	+	0.847612i	$0.678040\pi$
$284$	7.51889	0.446164
$285$	19.1638	1.13516
$286$	5.29713	0.313226
$287$	−9.39557	−0.554603
$288$	4.22276	0.248828
$289$	1.00000	0.0588235
$290$	0.960059	0.0563766
$291$	11.8418	0.694177
$292$	22.5948	1.32226
$293$	−14.5044	−0.847353	−0.423677	−	0.905813i	$-0.639261\pi$
$293$	−14.5044	−0.847353	−0.423677	+	0.905813i	$0.639261\pi$
$294$	−0.141975	−0.00828015
$295$	1.05089	0.0611852
$296$	6.18418	0.359448
$297$	5.03642	0.292243
$298$	3.94986	0.228809
$299$	−9.94832	−0.575326
$300$	19.0599	1.10042
$301$	27.3405	1.57588
$302$	0.196767	0.0113227
$303$	18.3679	1.05521
$304$	−15.2089	−0.872289
$305$	−11.5482	−0.661248
$306$	−0.390980	−0.0223508
$307$	31.5035	1.79800	0.899000	−	0.437949i	$-0.144295\pi$
$307$	31.5035	1.79800	0.899000	+	0.437949i	$0.144295\pi$
$308$	25.2436	1.43839
$309$	6.88240	0.391526
$310$	3.40723	0.193518
$311$	−2.25888	−0.128090	−0.0640448	−	0.997947i	$-0.520400\pi$
$311$	−2.25888	−0.128090	−0.0640448	+	0.997947i	$0.520400\pi$
$312$	−4.04628	−0.229076
$313$	−25.5127	−1.44206	−0.721032	−	0.692902i	$-0.756332\pi$
$313$	−25.5127	−1.44206	−0.721032	+	0.692902i	$0.756332\pi$
$314$	−0.390980	−0.0220643
$315$	10.6204	0.598392
$316$	24.2234	1.36267
$317$	−20.5528	−1.15436	−0.577180	−	0.816617i	$-0.695847\pi$
$317$	−20.5528	−1.15436	−0.577180	+	0.816617i	$0.695847\pi$
$318$	0.362385	0.0203215
$319$	−3.15977	−0.176913
$320$	−17.8526	−0.997993
$321$	−8.90217	−0.496871
$322$	3.92347	0.218647
$323$	4.89634	0.272439
$324$	−1.84713	−0.102619
$325$	−27.7579	−1.53973
$326$	−5.68778	−0.315017
$327$	7.57002	0.418623
$328$	5.20815	0.287572
$329$	−19.6358	−1.08256
$330$	7.70702	0.424257
$331$	−5.22816	−0.287366	−0.143683	−	0.989624i	$-0.545895\pi$
$331$	−5.22816	−0.287366	−0.143683	+	0.989624i	$0.545895\pi$
$332$	−22.1890	−1.21778
$333$	−4.11141	−0.225304
$334$	−6.37362	−0.348749
$335$	44.8199	2.44877
$336$	−8.42863	−0.459820
$337$	19.6234	1.06896	0.534478	−	0.845182i	$-0.320508\pi$
$337$	19.6234	1.06896	0.534478	+	0.845182i	$0.320508\pi$
$338$	−2.25341	−0.122569
$339$	−6.73057	−0.365554
$340$	7.22950	0.392075
$341$	−11.2140	−0.607271
$342$	−1.91437	−0.103517
$343$	−18.0092	−0.972406
$344$	−15.1554	−0.817123
$345$	−14.4742	−0.779266
$346$	7.33911	0.394553
$347$	−9.67098	−0.519166	−0.259583	−	0.965721i	$-0.583585\pi$
$347$	−9.67098	−0.519166	−0.259583	+	0.965721i	$0.583585\pi$
$348$	1.15886	0.0621216
$349$	−8.50332	−0.455172	−0.227586	−	0.973758i	$-0.573083\pi$
$349$	−8.50332	−0.455172	−0.227586	+	0.973758i	$0.573083\pi$
$350$	10.9473	0.585158
$351$	2.69008	0.143586
$352$	−21.2676	−1.13357
$353$	−13.4112	−0.713806	−0.356903	−	0.934141i	$-0.616167\pi$
$353$	−13.4112	−0.713806	−0.356903	+	0.934141i	$0.616167\pi$
$354$	−0.104979	−0.00557956
$355$	−15.9318	−0.845573
$356$	−5.16971	−0.273994
$357$	2.71351	0.143614
$358$	−6.79927	−0.359352
$359$	−16.1583	−0.852802	−0.426401	−	0.904534i	$-0.640219\pi$
$359$	−16.1583	−0.852802	−0.426401	+	0.904534i	$0.640219\pi$
$360$	−5.88710	−0.310278
$361$	4.97411	0.261795
$362$	0.371831	0.0195430
$363$	−14.3656	−0.753996
$364$	13.4832	0.706714
$365$	−47.8763	−2.50596
$366$	1.15361	0.0603001
$367$	−2.09072	−0.109135	−0.0545674	−	0.998510i	$-0.517378\pi$
$367$	−2.09072	−0.109135	−0.0545674	+	0.998510i	$0.517378\pi$
$368$	11.4871	0.598808
$369$	−3.46252	−0.180251
$370$	−6.29151	−0.327080
$371$	−2.51505	−0.130575
$372$	4.11278	0.213238
$373$	−14.5443	−0.753075	−0.376538	−	0.926401i	$-0.622885\pi$
$373$	−14.5443	−0.753075	−0.376538	+	0.926401i	$0.622885\pi$
$374$	1.96914	0.101822
$375$	−20.8166	−1.07496
$376$	10.8845	0.561327
$377$	−1.68771	−0.0869215
$378$	−1.06093	−0.0545682
$379$	12.5409	0.644181	0.322091	−	0.946709i	$-0.395614\pi$
$379$	12.5409	0.644181	0.322091	+	0.946709i	$0.395614\pi$
$380$	35.3981	1.81588
$381$	−7.85630	−0.402491
$382$	−9.97340	−0.510284
$383$	33.8549	1.72990	0.864951	−	0.501856i	$-0.167349\pi$
$383$	33.8549	1.72990	0.864951	+	0.501856i	$0.167349\pi$
$384$	10.2289	0.521992
$385$	−53.4888	−2.72604
$386$	4.91070	0.249948
$387$	10.0757	0.512176
$388$	21.8733	1.11045
$389$	−34.4262	−1.74548	−0.872740	−	0.488186i	$-0.837659\pi$
$389$	−34.4262	−1.74548	−0.872740	+	0.488186i	$0.837659\pi$
$390$	4.11651	0.208447
$391$	−3.69816	−0.187024
$392$	−0.546197	−0.0275871
$393$	0.368603	0.0185936
$394$	1.04355	0.0525734
$395$	−51.3271	−2.58255
$396$	9.30295	0.467491
$397$	−12.9328	−0.649078	−0.324539	−	0.945872i	$-0.605209\pi$
$397$	−12.9328	−0.649078	−0.324539	+	0.945872i	$0.605209\pi$
$398$	3.25368	0.163092
$399$	13.2862	0.665144
$400$	32.0515	1.60257
$401$	13.0276	0.650566	0.325283	−	0.945617i	$-0.394540\pi$
$401$	13.0276	0.650566	0.325283	+	0.945617i	$0.394540\pi$
$402$	−4.47729	−0.223307
$403$	−5.98965	−0.298366
$404$	33.9281	1.68798
$405$	3.91390	0.194483
$406$	0.665609	0.0330336
$407$	20.7068	1.02640
$408$	−1.50415	−0.0744666
$409$	−28.1191	−1.39040	−0.695200	−	0.718817i	$-0.744684\pi$
$409$	−28.1191	−1.39040	−0.695200	+	0.718817i	$0.744684\pi$
$410$	−5.29854	−0.261676
$411$	14.3166	0.706185
$412$	12.7127	0.626311
$413$	0.728582	0.0358512
$414$	1.44590	0.0710623
$415$	47.0163	2.30794
$416$	−11.3595	−0.556947
$417$	21.0100	1.02886
$418$	9.64157	0.471585
$419$	14.6967	0.717980	0.358990	−	0.933341i	$-0.383121\pi$
$419$	14.6967	0.717980	0.358990	+	0.933341i	$0.383121\pi$
$420$	19.6173	0.957227
$421$	−30.7408	−1.49821	−0.749107	−	0.662449i	$-0.769517\pi$
$421$	−30.7408	−1.49821	−0.749107	+	0.662449i	$0.769517\pi$
$422$	−3.85246	−0.187535
$423$	−7.23632	−0.351842
$424$	1.39414	0.0677056
$425$	−10.3186	−0.500527
$426$	1.59151	0.0771089
$427$	−8.00637	−0.387455
$428$	−16.4435	−0.794827
$429$	−13.5484	−0.654121
$430$	15.4184	0.743541
$431$	23.6442	1.13890	0.569451	−	0.822025i	$-0.307156\pi$
$431$	23.6442	1.13890	0.569451	+	0.822025i	$0.307156\pi$
$432$	−3.10618	−0.149446
$433$	9.03598	0.434242	0.217121	−	0.976145i	$-0.430333\pi$
$433$	9.03598	0.434242	0.217121	+	0.976145i	$0.430333\pi$
$434$	2.36223	0.113391
$435$	−2.45552	−0.117733
$436$	13.9828	0.669657
$437$	−18.1074	−0.866196
$438$	4.78261	0.228522
$439$	−3.12261	−0.149034	−0.0745170	−	0.997220i	$-0.523741\pi$
$439$	−3.12261	−0.149034	−0.0745170	+	0.997220i	$0.523741\pi$
$440$	29.6499	1.41351
$441$	0.363126	0.0172917
$442$	1.05177	0.0500274
$443$	−18.1452	−0.862106	−0.431053	−	0.902327i	$-0.641858\pi$
$443$	−18.1452	−0.862106	−0.431053	+	0.902327i	$0.641858\pi$
$444$	−7.59432	−0.360411
$445$	10.9541	0.519275
$446$	9.46745	0.448297
$447$	−10.1025	−0.477830
$448$	−12.3772	−0.584769
$449$	−16.5972	−0.783269	−0.391634	−	0.920121i	$-0.628090\pi$
$449$	−16.5972	−0.783269	−0.391634	+	0.920121i	$0.628090\pi$
$450$	4.03438	0.190182
$451$	17.4387	0.821156
$452$	−12.4323	−0.584765
$453$	−0.503267	−0.0236455
$454$	3.14427	0.147568
$455$	−28.5697	−1.33937
$456$	−7.36483	−0.344890
$457$	−3.14747	−0.147233	−0.0736163	−	0.997287i	$-0.523454\pi$
$457$	−3.14747	−0.147233	−0.0736163	+	0.997287i	$0.523454\pi$
$458$	6.30097	0.294425
$459$	1.00000	0.0466760
$460$	−26.7358	−1.24657
$461$	19.8597	0.924957	0.462478	−	0.886631i	$-0.346960\pi$
$461$	19.8597	0.924957	0.462478	+	0.886631i	$0.346960\pi$
$462$	5.34328	0.248592
$463$	18.2132	0.846438	0.423219	−	0.906027i	$-0.360900\pi$
$463$	18.2132	0.846438	0.423219	+	0.906027i	$0.360900\pi$
$464$	1.94877	0.0904692
$465$	−8.71460	−0.404130
$466$	−6.39460	−0.296224
$467$	8.44237	0.390666	0.195333	−	0.980737i	$-0.437421\pi$
$467$	8.44237	0.390666	0.195333	+	0.980737i	$0.437421\pi$
$468$	4.96893	0.229689
$469$	31.0736	1.43485
$470$	−11.0734	−0.510779
$471$	1.00000	0.0460776
$472$	−0.403868	−0.0185895
$473$	−50.7455	−2.33328
$474$	5.12733	0.235506
$475$	−50.5235	−2.31818
$476$	5.01222	0.229735
$477$	−0.926864	−0.0424382
$478$	9.82787	0.449516
$479$	−27.0597	−1.23639	−0.618195	−	0.786025i	$-0.712136\pi$
$479$	−27.0597	−1.23639	−0.618195	+	0.786025i	$0.712136\pi$
$480$	−16.5275	−0.754372
$481$	11.0600	0.504292
$482$	−5.94180	−0.270642
$483$	−10.0350	−0.456607
$484$	−26.5351	−1.20614
$485$	−46.3475	−2.10453
$486$	−0.390980	−0.0177352
$487$	−30.8115	−1.39620	−0.698101	−	0.715999i	$-0.745971\pi$
$487$	−30.8115	−1.39620	−0.698101	+	0.715999i	$0.745971\pi$
$488$	4.43809	0.200903
$489$	14.5475	0.657861
$490$	0.555676	0.0251029
$491$	−37.9651	−1.71334	−0.856672	−	0.515862i	$-0.827472\pi$
$491$	−37.9651	−1.71334	−0.856672	+	0.515862i	$0.827472\pi$
$492$	−6.39574	−0.288342
$493$	−0.627384	−0.0282560
$494$	5.14980	0.231700
$495$	−19.7121	−0.885991
$496$	6.91613	0.310544
$497$	−11.0455	−0.495459
$498$	−4.69670	−0.210464
$499$	−1.11532	−0.0499286	−0.0249643	−	0.999688i	$-0.507947\pi$
$499$	−1.11532	−0.0499286	−0.0249643	+	0.999688i	$0.507947\pi$
$500$	−38.4511	−1.71958
$501$	16.3017	0.728305
$502$	2.65648	0.118564
$503$	28.7185	1.28049	0.640247	−	0.768169i	$-0.278832\pi$
$503$	28.7185	1.28049	0.640247	+	0.768169i	$0.278832\pi$
$504$	−4.08153	−0.181806
$505$	−71.8903	−3.19908
$506$	−7.28219	−0.323733
$507$	5.76349	0.255966
$508$	−14.5116	−0.643850
$509$	−1.10663	−0.0490505	−0.0245252	−	0.999699i	$-0.507807\pi$
$509$	−1.10663	−0.0490505	−0.0245252	+	0.999699i	$0.507807\pi$
$510$	1.53026	0.0677609
$511$	−33.1926	−1.46835
$512$	22.4610	0.992643
$513$	4.89634	0.216179
$514$	1.16213	0.0512594
$515$	−26.9370	−1.18699
$516$	18.6112	0.819311
$517$	36.4452	1.60286
$518$	−4.36190	−0.191651
$519$	−18.7711	−0.823959
$520$	15.8368	0.694488
$521$	−8.89276	−0.389599	−0.194800	−	0.980843i	$-0.562406\pi$
$521$	−8.89276	−0.389599	−0.194800	+	0.980843i	$0.562406\pi$
$522$	0.245295	0.0107363
$523$	−29.5908	−1.29392	−0.646959	−	0.762525i	$-0.723959\pi$
$523$	−29.5908	−1.29392	−0.646959	+	0.762525i	$0.723959\pi$
$524$	0.680860	0.0297435
$525$	−27.9997	−1.22201
$526$	−4.67831	−0.203984
$527$	−2.22657	−0.0969911
$528$	15.6440	0.680818
$529$	−9.32363	−0.405375
$530$	−1.41834	−0.0616088
$531$	0.268502	0.0116520
$532$	24.5415	1.06401
$533$	9.31443	0.403453
$534$	−1.09426	−0.0473534
$535$	34.8422	1.50636
$536$	−17.2247	−0.743995
$537$	17.3903	0.750448
$538$	−9.98796	−0.430612
$539$	−1.82886	−0.0787744
$540$	7.22950	0.311108
$541$	27.9717	1.20260	0.601298	−	0.799025i	$-0.294650\pi$
$541$	27.9717	1.20260	0.601298	+	0.799025i	$0.294650\pi$
$542$	9.36661	0.402330
$543$	−0.951024	−0.0408124
$544$	−4.22276	−0.181049
$545$	−29.6283	−1.26914
$546$	2.85397	0.122139
$547$	−6.33546	−0.270885	−0.135442	−	0.990785i	$-0.543246\pi$
$547$	−6.33546	−0.270885	−0.135442	+	0.990785i	$0.543246\pi$
$548$	26.4446	1.12966
$549$	−2.95056	−0.125927
$550$	−20.3188	−0.866398
$551$	−3.07188	−0.130867
$552$	5.56259	0.236760
$553$	−35.5851	−1.51323
$554$	8.54086	0.362866
$555$	16.0916	0.683052
$556$	38.8082	1.64584
$557$	9.51901	0.403333	0.201667	−	0.979454i	$-0.435364\pi$
$557$	9.51901	0.403333	0.201667	+	0.979454i	$0.435364\pi$
$558$	0.870546	0.0368531
$559$	−27.1044	−1.14639
$560$	32.9889	1.39403
$561$	−5.03642	−0.212638
$562$	−5.15271	−0.217354
$563$	24.6388	1.03840	0.519201	−	0.854652i	$-0.326230\pi$
$563$	24.6388	1.03840	0.519201	+	0.854652i	$0.326230\pi$
$564$	−13.3665	−0.562830
$565$	26.3428	1.10825
$566$	−6.98005	−0.293393
$567$	2.71351	0.113957
$568$	6.12275	0.256905
$569$	−34.8417	−1.46064	−0.730320	−	0.683106i	$-0.760629\pi$
$569$	−34.8417	−1.46064	−0.730320	+	0.683106i	$0.760629\pi$
$570$	7.49265	0.313833
$571$	−31.1425	−1.30327	−0.651637	−	0.758531i	$-0.725917\pi$
$571$	−31.1425	−1.30327	−0.651637	+	0.758531i	$0.725917\pi$
$572$	−25.0256	−1.04637
$573$	25.5087	1.06564
$574$	−3.67348	−0.153328
$575$	38.1599	1.59138
$576$	−4.56134	−0.190056
$577$	24.5020	1.02003	0.510016	−	0.860165i	$-0.329639\pi$
$577$	24.5020	1.02003	0.510016	+	0.860165i	$0.329639\pi$
$578$	0.390980	0.0162626
$579$	−12.5600	−0.521975
$580$	−4.53568	−0.188334
$581$	32.5964	1.35233
$582$	4.62989	0.191915
$583$	4.66808	0.193332
$584$	18.3993	0.761370
$585$	−10.5287	−0.435308
$586$	−5.67091	−0.234263
$587$	10.4206	0.430106	0.215053	−	0.976602i	$-0.431008\pi$
$587$	10.4206	0.430106	0.215053	+	0.976602i	$0.431008\pi$
$588$	0.670743	0.0276610
$589$	−10.9021	−0.449212
$590$	0.410877	0.0169155
$591$	−2.66907	−0.109791
$592$	−12.7708	−0.524875
$593$	0.863996	0.0354801	0.0177400	−	0.999843i	$-0.494353\pi$
$593$	0.863996	0.0354801	0.0177400	+	0.999843i	$0.494353\pi$
$594$	1.96914	0.0807948
$595$	−10.6204	−0.435394
$596$	−18.6606	−0.764368
$597$	−8.32187	−0.340592
$598$	−3.88959	−0.159057
$599$	15.4502	0.631277	0.315639	−	0.948879i	$-0.397781\pi$
$599$	15.4502	0.631277	0.315639	+	0.948879i	$0.397781\pi$
$600$	15.5208	0.633634
$601$	2.71877	0.110901	0.0554505	−	0.998461i	$-0.482340\pi$
$601$	2.71877	0.110901	0.0554505	+	0.998461i	$0.482340\pi$
$602$	10.6896	0.435675
$603$	11.4515	0.466339
$604$	−0.929602	−0.0378250
$605$	56.2254	2.28589
$606$	7.18149	0.291728
$607$	−4.92228	−0.199789	−0.0998947	−	0.994998i	$-0.531851\pi$
$607$	−4.92228	−0.199789	−0.0998947	+	0.994998i	$0.531851\pi$
$608$	−20.6760	−0.838524
$609$	−1.70241	−0.0689852
$610$	−4.51511	−0.182812
$611$	19.4663	0.787520
$612$	1.84713	0.0746660
$613$	−45.0334	−1.81888	−0.909440	−	0.415835i	$-0.863489\pi$
$613$	−45.0334	−1.81888	−0.909440	+	0.415835i	$0.863489\pi$
$614$	12.3172	0.497083
$615$	13.5520	0.546468
$616$	20.5563	0.828237
$617$	7.28965	0.293470	0.146735	−	0.989176i	$-0.453124\pi$
$617$	7.28965	0.293470	0.146735	+	0.989176i	$0.453124\pi$
$618$	2.69088	0.108243
$619$	35.8709	1.44177	0.720886	−	0.693054i	$-0.243735\pi$
$619$	35.8709	1.44177	0.720886	+	0.693054i	$0.243735\pi$
$620$	−16.0970	−0.646472
$621$	−3.69816	−0.148402
$622$	−0.883178	−0.0354122
$623$	7.59450	0.304267
$624$	8.35585	0.334502
$625$	29.8810	1.19524
$626$	−9.97495	−0.398679
$627$	−24.6600	−0.984826
$628$	1.84713	0.0737087
$629$	4.11141	0.163933
$630$	4.15236	0.165434
$631$	−5.46917	−0.217724	−0.108862	−	0.994057i	$-0.534721\pi$
$631$	−5.46917	−0.217724	−0.108862	+	0.994057i	$0.534721\pi$
$632$	19.7255	0.784639
$633$	9.85334	0.391635
$634$	−8.03573	−0.319140
$635$	30.7488	1.22023
$636$	−1.71204	−0.0678869
$637$	−0.976837	−0.0387037
$638$	−1.23541	−0.0489102
$639$	−4.07057	−0.161029
$640$	−40.0349	−1.58252
$641$	−9.80582	−0.387307	−0.193653	−	0.981070i	$-0.562034\pi$
$641$	−9.80582	−0.387307	−0.193653	+	0.981070i	$0.562034\pi$
$642$	−3.48057	−0.137367
$643$	−0.935603	−0.0368966	−0.0184483	−	0.999830i	$-0.505873\pi$
$643$	−0.935603	−0.0368966	−0.0184483	+	0.999830i	$0.505873\pi$
$644$	−18.5360	−0.730419
$645$	−39.4353	−1.55276
$646$	1.91437	0.0753198
$647$	34.9678	1.37473	0.687363	−	0.726315i	$-0.258768\pi$
$647$	34.9678	1.37473	0.687363	+	0.726315i	$0.258768\pi$
$648$	−1.50415	−0.0590887
$649$	−1.35229	−0.0530820
$650$	−10.8528	−0.425681
$651$	−6.04183	−0.236798
$652$	26.8712	1.05236
$653$	9.78767	0.383021	0.191510	−	0.981491i	$-0.438661\pi$
$653$	9.78767	0.383021	0.191510	+	0.981491i	$0.438661\pi$
$654$	2.95972	0.115734
$655$	−1.44268	−0.0563701
$656$	−10.7552	−0.419920
$657$	−12.2324	−0.477230
$658$	−7.67721	−0.299289
$659$	27.2614	1.06196	0.530978	−	0.847386i	$-0.321825\pi$
$659$	27.2614	1.06196	0.530978	+	0.847386i	$0.321825\pi$
$660$	−36.4108	−1.41729
$661$	−29.7500	−1.15714	−0.578571	−	0.815632i	$-0.696389\pi$
$661$	−29.7500	−1.15714	−0.578571	+	0.815632i	$0.696389\pi$
$662$	−2.04410	−0.0794464
$663$	−2.69008	−0.104474
$664$	−18.0689	−0.701207
$665$	−52.0011	−2.01652
$666$	−1.60748	−0.0622885
$667$	2.32017	0.0898372
$668$	30.1114	1.16504
$669$	−24.2147	−0.936194
$670$	17.5237	0.676998
$671$	14.8603	0.573674
$672$	−11.4585	−0.442021
$673$	8.44710	0.325612	0.162806	−	0.986658i	$-0.447946\pi$
$673$	8.44710	0.325612	0.162806	+	0.986658i	$0.447946\pi$
$674$	7.67236	0.295528
$675$	−10.3186	−0.397164
$676$	10.6459	0.409460
$677$	44.8786	1.72482	0.862412	−	0.506207i	$-0.168953\pi$
$677$	44.8786	1.72482	0.862412	+	0.506207i	$0.168953\pi$
$678$	−2.63152	−0.101063
$679$	−32.1327	−1.23314
$680$	5.88710	0.225760
$681$	−8.04202	−0.308171
$682$	−4.38444	−0.167889
$683$	−20.4485	−0.782442	−0.391221	−	0.920297i	$-0.627947\pi$
$683$	−20.4485	−0.782442	−0.391221	+	0.920297i	$0.627947\pi$
$684$	9.04419	0.345813
$685$	−56.0337	−2.14094
$686$	−7.04124	−0.268836
$687$	−16.1158	−0.614858
$688$	31.2969	1.19318
$689$	2.49333	0.0949885
$690$	−5.65913	−0.215439
$691$	−12.8532	−0.488960	−0.244480	−	0.969654i	$-0.578617\pi$
$691$	−12.8532	−0.488960	−0.244480	+	0.969654i	$0.578617\pi$
$692$	−34.6727	−1.31806
$693$	−13.6664	−0.519143
$694$	−3.78116	−0.143531
$695$	−82.2309	−3.11920
$696$	0.943681	0.0357701
$697$	3.46252	0.131152
$698$	−3.32463	−0.125839
$699$	16.3553	0.618615
$700$	−51.7192	−1.95480
$701$	19.0313	0.718804	0.359402	−	0.933183i	$-0.382981\pi$
$701$	19.0313	0.718804	0.359402	+	0.933183i	$0.382981\pi$
$702$	1.05177	0.0396963
$703$	20.1308	0.759249
$704$	22.9728	0.865821
$705$	28.3223	1.06668
$706$	−5.24351	−0.197342
$707$	−49.8415	−1.87448
$708$	0.495959	0.0186393
$709$	−28.1634	−1.05770	−0.528850	−	0.848715i	$-0.677377\pi$
$709$	−28.1634	−1.05770	−0.528850	+	0.848715i	$0.677377\pi$
$710$	−6.22901	−0.233771
$711$	−13.1141	−0.491815
$712$	−4.20978	−0.157768
$713$	8.23422	0.308374
$714$	1.06093	0.0397042
$715$	53.0269	1.98310
$716$	32.1223	1.20047
$717$	−25.1365	−0.938740
$718$	−6.31757	−0.235770
$719$	−29.6726	−1.10660	−0.553300	−	0.832982i	$-0.686632\pi$
$719$	−29.6726	−1.10660	−0.553300	+	0.832982i	$0.686632\pi$
$720$	12.1573	0.453075
$721$	−18.6754	−0.695510
$722$	1.94478	0.0723771
$723$	15.1972	0.565190
$724$	−1.75667	−0.0652861
$725$	6.47375	0.240429
$726$	−5.61664	−0.208453
$727$	47.0603	1.74537	0.872685	−	0.488283i	$-0.162377\pi$
$727$	47.0603	1.74537	0.872685	+	0.488283i	$0.162377\pi$
$728$	10.9796	0.406932
$729$	1.00000	0.0370370
$730$	−18.7187	−0.692808
$731$	−10.0757	−0.372663
$732$	−5.45008	−0.201441
$733$	−12.7425	−0.470656	−0.235328	−	0.971916i	$-0.575616\pi$
$733$	−12.7425	−0.470656	−0.235328	+	0.971916i	$0.575616\pi$
$734$	−0.817430	−0.0301719
$735$	−1.42124	−0.0524232
$736$	15.6164	0.575629
$737$	−57.6744	−2.12446
$738$	−1.35377	−0.0498331
$739$	−4.08206	−0.150161	−0.0750804	−	0.997177i	$-0.523921\pi$
$739$	−4.08206	−0.150161	−0.0750804	+	0.997177i	$0.523921\pi$
$740$	29.7234	1.09266
$741$	−13.1715	−0.483868
$742$	−0.983335	−0.0360994
$743$	−45.7473	−1.67831	−0.839154	−	0.543894i	$-0.816949\pi$
$743$	−45.7473	−1.67831	−0.839154	+	0.543894i	$0.816949\pi$
$744$	3.34911	0.122784
$745$	39.5400	1.44864
$746$	−5.68653	−0.208199
$747$	12.0126	0.439520
$748$	−9.30295	−0.340150
$749$	24.1561	0.882645
$750$	−8.13887	−0.297190
$751$	10.1158	0.369130	0.184565	−	0.982820i	$-0.440912\pi$
$751$	10.1158	0.369130	0.184565	+	0.982820i	$0.440912\pi$
$752$	−22.4773	−0.819663
$753$	−6.79441	−0.247602
$754$	−0.659861	−0.0240307
$755$	1.96974	0.0716861
$756$	5.01222	0.182293
$757$	−44.3243	−1.61100	−0.805498	−	0.592599i	$-0.798102\pi$
$757$	−44.3243	−1.61100	−0.805498	+	0.592599i	$0.798102\pi$
$758$	4.90323	0.178093
$759$	18.6255	0.676062
$760$	28.8252	1.04560
$761$	−4.31678	−0.156483	−0.0782417	−	0.996934i	$-0.524931\pi$
$761$	−4.31678	−0.156483	−0.0782417	+	0.996934i	$0.524931\pi$
$762$	−3.07166	−0.111274
$763$	−20.5413	−0.743645
$764$	47.1181	1.70467
$765$	−3.91390	−0.141507
$766$	13.2366	0.478257
$767$	−0.722291	−0.0260804
$768$	−5.12339	−0.184874
$769$	−40.1856	−1.44913	−0.724565	−	0.689206i	$-0.757959\pi$
$769$	−40.1856	−1.44913	−0.724565	+	0.689206i	$0.757959\pi$
$770$	−20.9131	−0.753654
$771$	−2.97235	−0.107047
$772$	−23.2000	−0.834985
$773$	−41.6002	−1.49625	−0.748127	−	0.663555i	$-0.769047\pi$
$773$	−41.6002	−1.49625	−0.748127	+	0.663555i	$0.769047\pi$
$774$	3.93939	0.141599
$775$	22.9752	0.825294
$776$	17.8118	0.639407
$777$	11.1563	0.400231
$778$	−13.4600	−0.482563
$779$	16.9537	0.607428
$780$	−19.4479	−0.696347
$781$	20.5011	0.733587
$782$	−1.44590	−0.0517055
$783$	−0.627384	−0.0224209
$784$	1.12793	0.0402833
$785$	−3.91390	−0.139693
$786$	0.144117	0.00514047
$787$	−4.67893	−0.166786	−0.0833930	−	0.996517i	$-0.526576\pi$
$787$	−4.67893	−0.166786	−0.0833930	+	0.996517i	$0.526576\pi$
$788$	−4.93013	−0.175629
$789$	11.9656	0.425987
$790$	−20.0679	−0.713983
$791$	18.2635	0.649374
$792$	7.57555	0.269185
$793$	7.93723	0.281859
$794$	−5.05646	−0.179447
$795$	3.62766	0.128660
$796$	−15.3716	−0.544833
$797$	27.8694	0.987185	0.493592	−	0.869693i	$-0.335683\pi$
$797$	27.8694	0.987185	0.493592	+	0.869693i	$0.335683\pi$
$798$	5.19466	0.183889
$799$	7.23632	0.256003
$800$	43.5731	1.54054
$801$	2.79877	0.0988898
$802$	5.09352	0.179858
$803$	61.6073	2.17408
$804$	21.1524	0.745987
$805$	39.2759	1.38429
$806$	−2.34183	−0.0824876
$807$	25.5460	0.899261
$808$	27.6282	0.971956
$809$	−22.4890	−0.790670	−0.395335	−	0.918537i	$-0.629371\pi$
$809$	−22.4890	−0.790670	−0.395335	+	0.918537i	$0.629371\pi$
$810$	1.53026	0.0537677
$811$	28.3476	0.995420	0.497710	−	0.867343i	$-0.334174\pi$
$811$	28.3476	0.995420	0.497710	+	0.867343i	$0.334174\pi$
$812$	−3.14459	−0.110353
$813$	−23.9568	−0.840200
$814$	8.09593	0.283762
$815$	−56.9375	−1.99443
$816$	3.10618	0.108738
$817$	−49.3340	−1.72598
$818$	−10.9940	−0.384396
$819$	−7.29954	−0.255067
$820$	25.0323	0.874165
$821$	21.4621	0.749031	0.374515	−	0.927221i	$-0.377809\pi$
$821$	21.4621	0.749031	0.374515	+	0.927221i	$0.377809\pi$
$822$	5.59749	0.195235
$823$	16.2167	0.565279	0.282639	−	0.959226i	$-0.408790\pi$
$823$	16.2167	0.565279	0.282639	+	0.959226i	$0.408790\pi$
$824$	10.3522	0.360635
$825$	51.9690	1.80933
$826$	0.284861	0.00991158
$827$	44.8546	1.55975	0.779874	−	0.625936i	$-0.215283\pi$
$827$	44.8546	1.55975	0.779874	+	0.625936i	$0.215283\pi$
$828$	−6.83100	−0.237393
$829$	12.0470	0.418409	0.209204	−	0.977872i	$-0.432913\pi$
$829$	12.0470	0.418409	0.209204	+	0.977872i	$0.432913\pi$
$830$	18.3824	0.638064
$831$	−21.8448	−0.757786
$832$	12.2704	0.425398
$833$	−0.363126	−0.0125816
$834$	8.21447	0.284444
$835$	−63.8031	−2.20800
$836$	−45.5504	−1.57539
$837$	−2.22657	−0.0769617
$838$	5.74611	0.198496
$839$	−6.52612	−0.225307	−0.112653	−	0.993634i	$-0.535935\pi$
$839$	−6.52612	−0.225307	−0.112653	+	0.993634i	$0.535935\pi$
$840$	15.9747	0.551180
$841$	−28.6064	−0.986427
$842$	−12.0190	−0.414203
$843$	13.1790	0.453908
$844$	18.2005	0.626486
$845$	−22.5577	−0.776010
$846$	−2.82926	−0.0972718
$847$	38.9810	1.33940
$848$	−2.87900	−0.0988654
$849$	17.8527	0.612703
$850$	−4.03438	−0.138378
$851$	−15.2046	−0.521208
$852$	−7.51889	−0.257593
$853$	22.5464	0.771975	0.385988	−	0.922504i	$-0.373861\pi$
$853$	22.5464	0.771975	0.385988	+	0.922504i	$0.373861\pi$
$854$	−3.13033	−0.107118
$855$	−19.1638	−0.655388
$856$	−13.3902	−0.457668
$857$	33.3447	1.13903	0.569516	−	0.821980i	$-0.307131\pi$
$857$	33.3447	1.13903	0.569516	+	0.821980i	$0.307131\pi$
$858$	−5.29713	−0.180841
$859$	2.66269	0.0908498	0.0454249	−	0.998968i	$-0.485536\pi$
$859$	2.66269	0.0908498	0.0454249	+	0.998968i	$0.485536\pi$
$860$	−72.8423	−2.48390
$861$	9.39557	0.320200
$862$	9.24441	0.314866
$863$	−35.0239	−1.19223	−0.596114	−	0.802900i	$-0.703290\pi$
$863$	−35.0239	−1.19223	−0.596114	+	0.802900i	$0.703290\pi$
$864$	−4.22276	−0.143661
$865$	73.4682	2.49799
$866$	3.53289	0.120052
$867$	−1.00000	−0.0339618
$868$	−11.1601	−0.378798
$869$	66.0479	2.24052
$870$	−0.960059	−0.0325491
$871$	−30.8053	−1.04380
$872$	11.3865	0.385594
$873$	−11.8418	−0.400783
$874$	−7.07964	−0.239472
$875$	56.4860	1.90958
$876$	−22.5948	−0.763408
$877$	4.94513	0.166985	0.0834927	−	0.996508i	$-0.473392\pi$
$877$	4.94513	0.166985	0.0834927	+	0.996508i	$0.473392\pi$
$878$	−1.22088	−0.0412026
$879$	14.5044	0.489220
$880$	−61.2292	−2.06403
$881$	−19.2954	−0.650080	−0.325040	−	0.945700i	$-0.605378\pi$
$881$	−19.2954	−0.650080	−0.325040	+	0.945700i	$0.605378\pi$
$882$	0.141975	0.00478055
$883$	53.6574	1.80572	0.902858	−	0.429940i	$-0.141465\pi$
$883$	53.6574	1.80572	0.902858	+	0.429940i	$0.141465\pi$
$884$	−4.96893	−0.167123
$885$	−1.05089	−0.0353253
$886$	−7.09442	−0.238342
$887$	34.5970	1.16165	0.580827	−	0.814027i	$-0.302729\pi$
$887$	34.5970	1.16165	0.580827	+	0.814027i	$0.302729\pi$
$888$	−6.18418	−0.207527
$889$	21.3181	0.714987
$890$	4.28284	0.143561
$891$	−5.03642	−0.168726
$892$	−44.7278	−1.49760
$893$	35.4315	1.18567
$894$	−3.94986	−0.132103
$895$	−68.0641	−2.27513
$896$	−27.7562	−0.927270
$897$	9.94832	0.332165
$898$	−6.48916	−0.216546
$899$	1.39692	0.0465898
$900$	−19.0599	−0.635330
$901$	0.926864	0.0308783
$902$	6.81818	0.227021
$903$	−27.3405	−0.909834
$904$	−10.1238	−0.336713
$905$	3.72222	0.123731
$906$	−0.196767	−0.00653715
$907$	3.96175	0.131548	0.0657739	−	0.997835i	$-0.479048\pi$
$907$	3.96175	0.131548	0.0657739	+	0.997835i	$0.479048\pi$
$908$	−14.8547	−0.492970
$909$	−18.3679	−0.609226
$910$	−11.1702	−0.370288
$911$	36.9384	1.22382	0.611912	−	0.790926i	$-0.290401\pi$
$911$	36.9384	1.22382	0.611912	+	0.790926i	$0.290401\pi$
$912$	15.2089	0.503617
$913$	−60.5008	−2.00228
$914$	−1.23060	−0.0407046
$915$	11.5482	0.381772
$916$	−29.7681	−0.983567
$917$	−1.00021	−0.0330298
$918$	0.390980	0.0129043
$919$	5.40711	0.178364	0.0891821	−	0.996015i	$-0.471575\pi$
$919$	5.40711	0.178364	0.0891821	+	0.996015i	$0.471575\pi$
$920$	−21.7714	−0.717783
$921$	−31.5035	−1.03808
$922$	7.76473	0.255718
$923$	10.9501	0.360428
$924$	−25.2436	−0.830455
$925$	−42.4241	−1.39490
$926$	7.12098	0.234010
$927$	−6.88240	−0.226048
$928$	2.64929	0.0869672
$929$	−1.13541	−0.0372515	−0.0186257	−	0.999827i	$-0.505929\pi$
$929$	−1.13541	−0.0372515	−0.0186257	+	0.999827i	$0.505929\pi$
$930$	−3.40723	−0.111728
$931$	−1.77799	−0.0582712
$932$	30.2105	0.989577
$933$	2.25888	0.0739526
$934$	3.30080	0.108005
$935$	19.7121	0.644653
$936$	4.04628	0.132257
$937$	13.5583	0.442929	0.221464	−	0.975168i	$-0.428916\pi$
$937$	13.5583	0.442929	0.221464	+	0.975168i	$0.428916\pi$
$938$	12.1492	0.396684
$939$	25.5127	0.832575
$940$	52.3150	1.70633
$941$	−5.52485	−0.180105	−0.0900525	−	0.995937i	$-0.528703\pi$
$941$	−5.52485	−0.180105	−0.0900525	+	0.995937i	$0.528703\pi$
$942$	0.390980	0.0127388
$943$	−12.8049	−0.416986
$944$	0.834015	0.0271449
$945$	−10.6204	−0.345482
$946$	−19.8405	−0.645069
$947$	−6.86093	−0.222950	−0.111475	−	0.993767i	$-0.535558\pi$
$947$	−6.86093	−0.222950	−0.111475	+	0.993767i	$0.535558\pi$
$948$	−24.2234	−0.786740
$949$	32.9060	1.06817
$950$	−19.7537	−0.640894
$951$	20.5528	0.666470
$952$	4.08153	0.132283
$953$	−12.1796	−0.394537	−0.197269	−	0.980349i	$-0.563207\pi$
$953$	−12.1796	−0.394537	−0.197269	+	0.980349i	$0.563207\pi$
$954$	−0.362385	−0.0117327
$955$	−99.8387	−3.23071
$956$	−46.4305	−1.50167
$957$	3.15977	0.102141
$958$	−10.5798	−0.341818
$959$	−38.8481	−1.25447
$960$	17.8526	0.576191
$961$	−26.0424	−0.840076
$962$	4.32423	0.139419
$963$	8.90217	0.286869
$964$	28.0713	0.904116
$965$	49.1585	1.58247
$966$	−3.92347	−0.126236
$967$	58.8927	1.89386	0.946930	−	0.321439i	$-0.104167\pi$
$967$	58.8927	1.89386	0.946930	+	0.321439i	$0.104167\pi$
$968$	−21.6080	−0.694507
$969$	−4.89634	−0.157293
$970$	−18.1209	−0.581828
$971$	30.9759	0.994063	0.497032	−	0.867732i	$-0.334423\pi$
$971$	30.9759	0.994063	0.497032	+	0.867732i	$0.334423\pi$
$972$	1.84713	0.0592469
$973$	−57.0107	−1.82768
$974$	−12.0467	−0.386000
$975$	27.7579	0.888964
$976$	−9.16496	−0.293363
$977$	4.33188	0.138589	0.0692945	−	0.997596i	$-0.477925\pi$
$977$	4.33188	0.138589	0.0692945	+	0.997596i	$0.477925\pi$
$978$	5.68778	0.181875
$979$	−14.0958	−0.450504
$980$	−2.62522	−0.0838596
$981$	−7.57002	−0.241692
$982$	−14.8436	−0.473679
$983$	−42.4941	−1.35535	−0.677675	−	0.735361i	$-0.737012\pi$
$983$	−42.4941	−1.35535	−0.677675	+	0.735361i	$0.737012\pi$
$984$	−5.20815	−0.166030
$985$	10.4465	0.332853
$986$	−0.245295	−0.00781177
$987$	19.6358	0.625015
$988$	−24.3296	−0.774027
$989$	37.2615	1.18485
$990$	−7.70702	−0.244945
$991$	−20.6176	−0.654941	−0.327471	−	0.944861i	$-0.606196\pi$
$991$	−20.6176	−0.654941	−0.327471	+	0.944861i	$0.606196\pi$
$992$	9.40228	0.298523
$993$	5.22816	0.165911
$994$	−4.31858	−0.136977
$995$	32.5710	1.03257
$996$	22.1890	0.703085
$997$	34.9737	1.10763	0.553814	−	0.832640i	$-0.313172\pi$
$997$	34.9737	1.10763	0.553814	+	0.832640i	$0.313172\pi$
$998$	−0.436068	−0.0138035
$999$	4.11141	0.130079

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.27	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.27	✓	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)

\(f(q)\)	\(=\)	\(q+0.390980 q^{2} -1.00000 q^{3} -1.84713 q^{4} +3.91390 q^{5} -0.390980 q^{6} +2.71351 q^{7} -1.50415 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.390980 q^{2} -1.00000 q^{3} -1.84713 q^{4} +3.91390 q^{5} -0.390980 q^{6} +2.71351 q^{7} -1.50415 q^{8} +1.00000 q^{9} +1.53026 q^{10} -5.03642 q^{11} +1.84713 q^{12} -2.69008 q^{13} +1.06093 q^{14} -3.91390 q^{15} +3.10618 q^{16} -1.00000 q^{17} +0.390980 q^{18} -4.89634 q^{19} -7.22950 q^{20} -2.71351 q^{21} -1.96914 q^{22} +3.69816 q^{23} +1.50415 q^{24} +10.3186 q^{25} -1.05177 q^{26} -1.00000 q^{27} -5.01222 q^{28} +0.627384 q^{29} -1.53026 q^{30} +2.22657 q^{31} +4.22276 q^{32} +5.03642 q^{33} -0.390980 q^{34} +10.6204 q^{35} -1.84713 q^{36} -4.11141 q^{37} -1.91437 q^{38} +2.69008 q^{39} -5.88710 q^{40} -3.46252 q^{41} -1.06093 q^{42} +10.0757 q^{43} +9.30295 q^{44} +3.91390 q^{45} +1.44590 q^{46} -7.23632 q^{47} -3.10618 q^{48} +0.363126 q^{49} +4.03438 q^{50} +1.00000 q^{51} +4.96893 q^{52} -0.926864 q^{53} -0.390980 q^{54} -19.7121 q^{55} -4.08153 q^{56} +4.89634 q^{57} +0.245295 q^{58} +0.268502 q^{59} +7.22950 q^{60} -2.95056 q^{61} +0.870546 q^{62} +2.71351 q^{63} -4.56134 q^{64} -10.5287 q^{65} +1.96914 q^{66} +11.4515 q^{67} +1.84713 q^{68} -3.69816 q^{69} +4.15236 q^{70} -4.07057 q^{71} -1.50415 q^{72} -12.2324 q^{73} -1.60748 q^{74} -10.3186 q^{75} +9.04419 q^{76} -13.6664 q^{77} +1.05177 q^{78} -13.1141 q^{79} +12.1573 q^{80} +1.00000 q^{81} -1.35377 q^{82} +12.0126 q^{83} +5.01222 q^{84} -3.91390 q^{85} +3.93939 q^{86} -0.627384 q^{87} +7.57555 q^{88} +2.79877 q^{89} +1.53026 q^{90} -7.29954 q^{91} -6.83100 q^{92} -2.22657 q^{93} -2.82926 q^{94} -19.1638 q^{95} -4.22276 q^{96} -11.8418 q^{97} +0.141975 q^{98} -5.03642 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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