LMFDB - Embedded newform 8007.2.a.f.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.17906 q^{2} -1.00000 q^{3} +2.74829 q^{4} -2.11197 q^{5} +2.17906 q^{6} +2.69680 q^{7} -1.63058 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.17906 q^{2} -1.00000 q^{3} +2.74829 q^{4} -2.11197 q^{5} +2.17906 q^{6} +2.69680 q^{7} -1.63058 q^{8} +1.00000 q^{9} +4.60211 q^{10} +3.86808 q^{11} -2.74829 q^{12} -2.95745 q^{13} -5.87648 q^{14} +2.11197 q^{15} -1.94347 q^{16} -1.00000 q^{17} -2.17906 q^{18} +2.38202 q^{19} -5.80432 q^{20} -2.69680 q^{21} -8.42877 q^{22} +6.54103 q^{23} +1.63058 q^{24} -0.539577 q^{25} +6.44446 q^{26} -1.00000 q^{27} +7.41160 q^{28} -0.373077 q^{29} -4.60211 q^{30} +0.0561826 q^{31} +7.49608 q^{32} -3.86808 q^{33} +2.17906 q^{34} -5.69556 q^{35} +2.74829 q^{36} +2.78351 q^{37} -5.19056 q^{38} +2.95745 q^{39} +3.44373 q^{40} -4.44920 q^{41} +5.87648 q^{42} -8.21989 q^{43} +10.6306 q^{44} -2.11197 q^{45} -14.2533 q^{46} +9.32591 q^{47} +1.94347 q^{48} +0.272727 q^{49} +1.17577 q^{50} +1.00000 q^{51} -8.12794 q^{52} +4.89515 q^{53} +2.17906 q^{54} -8.16927 q^{55} -4.39734 q^{56} -2.38202 q^{57} +0.812957 q^{58} +8.24937 q^{59} +5.80432 q^{60} -11.0605 q^{61} -0.122425 q^{62} +2.69680 q^{63} -12.4475 q^{64} +6.24605 q^{65} +8.42877 q^{66} -3.51722 q^{67} -2.74829 q^{68} -6.54103 q^{69} +12.4110 q^{70} -12.0755 q^{71} -1.63058 q^{72} -3.35786 q^{73} -6.06544 q^{74} +0.539577 q^{75} +6.54650 q^{76} +10.4314 q^{77} -6.44446 q^{78} -9.02614 q^{79} +4.10454 q^{80} +1.00000 q^{81} +9.69506 q^{82} -13.8275 q^{83} -7.41160 q^{84} +2.11197 q^{85} +17.9116 q^{86} +0.373077 q^{87} -6.30721 q^{88} +4.41269 q^{89} +4.60211 q^{90} -7.97565 q^{91} +17.9767 q^{92} -0.0561826 q^{93} -20.3217 q^{94} -5.03076 q^{95} -7.49608 q^{96} +17.3963 q^{97} -0.594287 q^{98} +3.86808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.17906	−1.54083	−0.770413	−	0.637545i	$-0.779950\pi$
$2$	−2.17906	−1.54083	−0.770413	+	0.637545i	$0.779950\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.74829	1.37415
$5$	−2.11197	−0.944502	−0.472251	−	0.881464i	$-0.656558\pi$
$5$	−2.11197	−0.944502	−0.472251	+	0.881464i	$0.656558\pi$
$6$	2.17906	0.889597
$7$	2.69680	1.01929	0.509647	−	0.860383i	$-0.329776\pi$
$7$	2.69680	1.01929	0.509647	+	0.860383i	$0.329776\pi$
$8$	−1.63058	−0.576496
$9$	1.00000	0.333333
$10$	4.60211	1.45531
$11$	3.86808	1.16627	0.583135	−	0.812375i	$-0.301826\pi$
$11$	3.86808	1.16627	0.583135	+	0.812375i	$0.301826\pi$
$12$	−2.74829	−0.793364
$13$	−2.95745	−0.820249	−0.410125	−	0.912030i	$-0.634515\pi$
$13$	−2.95745	−0.820249	−0.410125	+	0.912030i	$0.634515\pi$
$14$	−5.87648	−1.57056
$15$	2.11197	0.545309
$16$	−1.94347	−0.485866
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.17906	−0.513609
$19$	2.38202	0.546473	0.273237	−	0.961947i	$-0.411906\pi$
$19$	2.38202	0.546473	0.273237	+	0.961947i	$0.411906\pi$
$20$	−5.80432	−1.29789
$21$	−2.69680	−0.588490
$22$	−8.42877	−1.79702
$23$	6.54103	1.36390	0.681949	−	0.731399i	$-0.261132\pi$
$23$	6.54103	1.36390	0.681949	+	0.731399i	$0.261132\pi$
$24$	1.63058	0.332840
$25$	−0.539577	−0.107915
$26$	6.44446	1.26386
$27$	−1.00000	−0.192450
$28$	7.41160	1.40066
$29$	−0.373077	−0.0692787	−0.0346393	−	0.999400i	$-0.511028\pi$
$29$	−0.373077	−0.0692787	−0.0346393	+	0.999400i	$0.511028\pi$
$30$	−4.60211	−0.840226
$31$	0.0561826	0.0100907	0.00504535	−	0.999987i	$-0.498394\pi$
$31$	0.0561826	0.0100907	0.00504535	+	0.999987i	$0.498394\pi$
$32$	7.49608	1.32513
$33$	−3.86808	−0.673346
$34$	2.17906	0.373705
$35$	−5.69556	−0.962726
$36$	2.74829	0.458049
$37$	2.78351	0.457607	0.228803	−	0.973473i	$-0.426519\pi$
$37$	2.78351	0.457607	0.228803	+	0.973473i	$0.426519\pi$
$38$	−5.19056	−0.842020
$39$	2.95745	0.473571
$40$	3.44373	0.544502
$41$	−4.44920	−0.694848	−0.347424	−	0.937708i	$-0.612943\pi$
$41$	−4.44920	−0.694848	−0.347424	+	0.937708i	$0.612943\pi$
$42$	5.87648	0.906761
$43$	−8.21989	−1.25352	−0.626761	−	0.779212i	$-0.715620\pi$
$43$	−8.21989	−1.25352	−0.626761	+	0.779212i	$0.715620\pi$
$44$	10.6306	1.60263
$45$	−2.11197	−0.314834
$46$	−14.2533	−2.10153
$47$	9.32591	1.36032	0.680162	−	0.733062i	$-0.261909\pi$
$47$	9.32591	1.36032	0.680162	+	0.733062i	$0.261909\pi$
$48$	1.94347	0.280515
$49$	0.272727	0.0389609
$50$	1.17577	0.166279
$51$	1.00000	0.140028
$52$	−8.12794	−1.12714
$53$	4.89515	0.672400	0.336200	−	0.941791i	$-0.390858\pi$
$53$	4.89515	0.672400	0.336200	+	0.941791i	$0.390858\pi$
$54$	2.17906	0.296532
$55$	−8.16927	−1.10154
$56$	−4.39734	−0.587620
$57$	−2.38202	−0.315506
$58$	0.812957	0.106746
$59$	8.24937	1.07398	0.536989	−	0.843590i	$-0.319562\pi$
$59$	8.24937	1.07398	0.536989	+	0.843590i	$0.319562\pi$
$60$	5.80432	0.749335
$61$	−11.0605	−1.41615	−0.708077	−	0.706135i	$-0.750437\pi$
$61$	−11.0605	−1.41615	−0.708077	+	0.706135i	$0.750437\pi$
$62$	−0.122425	−0.0155480
$63$	2.69680	0.339765
$64$	−12.4475	−1.55593
$65$	6.24605	0.774727
$66$	8.42877	1.03751
$67$	−3.51722	−0.429697	−0.214848	−	0.976647i	$-0.568926\pi$
$67$	−3.51722	−0.429697	−0.214848	+	0.976647i	$0.568926\pi$
$68$	−2.74829	−0.333280
$69$	−6.54103	−0.787447
$70$	12.4110	1.48339
$71$	−12.0755	−1.43310	−0.716550	−	0.697536i	$-0.754280\pi$
$71$	−12.0755	−1.43310	−0.716550	+	0.697536i	$0.754280\pi$
$72$	−1.63058	−0.192165
$73$	−3.35786	−0.393008	−0.196504	−	0.980503i	$-0.562959\pi$
$73$	−3.35786	−0.393008	−0.196504	+	0.980503i	$0.562959\pi$
$74$	−6.06544	−0.705093
$75$	0.539577	0.0623050
$76$	6.54650	0.750935
$77$	10.4314	1.18877
$78$	−6.44446	−0.729691
$79$	−9.02614	−1.01552	−0.507760	−	0.861499i	$-0.669526\pi$
$79$	−9.02614	−1.01552	−0.507760	+	0.861499i	$0.669526\pi$
$80$	4.10454	0.458902
$81$	1.00000	0.111111
$82$	9.69506	1.07064
$83$	−13.8275	−1.51776	−0.758882	−	0.651229i	$-0.774254\pi$
$83$	−13.8275	−1.51776	−0.758882	+	0.651229i	$0.774254\pi$
$84$	−7.41160	−0.808672
$85$	2.11197	0.229075
$86$	17.9116	1.93146
$87$	0.373077	0.0399981
$88$	−6.30721	−0.672350
$89$	4.41269	0.467744	0.233872	−	0.972267i	$-0.424860\pi$
$89$	4.41269	0.467744	0.233872	+	0.972267i	$0.424860\pi$
$90$	4.60211	0.485105
$91$	−7.97565	−0.836075
$92$	17.9767	1.87420
$93$	−0.0561826	−0.00582587
$94$	−20.3217	−2.09602
$95$	−5.03076	−0.516145
$96$	−7.49608	−0.765065
$97$	17.3963	1.76633	0.883164	−	0.469064i	$-0.155409\pi$
$97$	17.3963	1.76633	0.883164	+	0.469064i	$0.155409\pi$
$98$	−0.594287	−0.0600321
$99$	3.86808	0.388757
$100$	−1.48292	−0.148292
$101$	−8.36277	−0.832126	−0.416063	−	0.909336i	$-0.636590\pi$
$101$	−8.36277	−0.832126	−0.416063	+	0.909336i	$0.636590\pi$
$102$	−2.17906	−0.215759
$103$	−14.3905	−1.41794	−0.708969	−	0.705239i	$-0.750839\pi$
$103$	−14.3905	−1.41794	−0.708969	+	0.705239i	$0.750839\pi$
$104$	4.82235	0.472871
$105$	5.69556	0.555830
$106$	−10.6668	−1.03605
$107$	3.26151	0.315302	0.157651	−	0.987495i	$-0.449608\pi$
$107$	3.26151	0.315302	0.157651	+	0.987495i	$0.449608\pi$
$108$	−2.74829	−0.264455
$109$	−3.48691	−0.333985	−0.166992	−	0.985958i	$-0.553406\pi$
$109$	−3.48691	−0.333985	−0.166992	+	0.985958i	$0.553406\pi$
$110$	17.8013	1.69729
$111$	−2.78351	−0.264200
$112$	−5.24114	−0.495241
$113$	−16.9344	−1.59306	−0.796529	−	0.604600i	$-0.793333\pi$
$113$	−16.9344	−1.59306	−0.796529	+	0.604600i	$0.793333\pi$
$114$	5.19056	0.486141
$115$	−13.8145	−1.28821
$116$	−1.02533	−0.0951991
$117$	−2.95745	−0.273416
$118$	−17.9759	−1.65481
$119$	−2.69680	−0.247215
$120$	−3.44373	−0.314368
$121$	3.96204	0.360185
$122$	24.1015	2.18205
$123$	4.44920	0.401170
$124$	0.154406	0.0138661
$125$	11.6994	1.04643
$126$	−5.87648	−0.523519
$127$	−14.9097	−1.32302	−0.661510	−	0.749936i	$-0.730084\pi$
$127$	−14.9097	−1.32302	−0.661510	+	0.749936i	$0.730084\pi$
$128$	12.1316	1.07229
$129$	8.21989	0.723721
$130$	−13.6105	−1.19372
$131$	9.53654	0.833211	0.416606	−	0.909087i	$-0.363220\pi$
$131$	9.53654	0.833211	0.416606	+	0.909087i	$0.363220\pi$
$132$	−10.6306	−0.925277
$133$	6.42383	0.557017
$134$	7.66423	0.662088
$135$	2.11197	0.181770
$136$	1.63058	0.139821
$137$	−18.3801	−1.57032	−0.785159	−	0.619295i	$-0.787419\pi$
$137$	−18.3801	−1.57032	−0.785159	+	0.619295i	$0.787419\pi$
$138$	14.2533	1.21332
$139$	16.8849	1.43216	0.716078	−	0.698020i	$-0.245935\pi$
$139$	16.8849	1.43216	0.716078	+	0.698020i	$0.245935\pi$
$140$	−15.6531	−1.32293
$141$	−9.32591	−0.785384
$142$	26.3132	2.20816
$143$	−11.4397	−0.956632
$144$	−1.94347	−0.161955
$145$	0.787928	0.0654339
$146$	7.31698	0.605557
$147$	−0.272727	−0.0224941
$148$	7.64992	0.628819
$149$	−4.52276	−0.370519	−0.185259	−	0.982690i	$-0.559312\pi$
$149$	−4.52276	−0.370519	−0.185259	+	0.982690i	$0.559312\pi$
$150$	−1.17577	−0.0960012
$151$	−0.926814	−0.0754230	−0.0377115	−	0.999289i	$-0.512007\pi$
$151$	−0.926814	−0.0754230	−0.0377115	+	0.999289i	$0.512007\pi$
$152$	−3.88407	−0.315040
$153$	−1.00000	−0.0808452
$154$	−22.7307	−1.83169
$155$	−0.118656	−0.00953068
$156$	8.12794	0.650756
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	19.6685	1.56474
$159$	−4.89515	−0.388210
$160$	−15.8315	−1.25159
$161$	17.6398	1.39021
$162$	−2.17906	−0.171203
$163$	−21.5271	−1.68613	−0.843066	−	0.537811i	$-0.819252\pi$
$163$	−21.5271	−1.68613	−0.843066	+	0.537811i	$0.819252\pi$
$164$	−12.2277	−0.954823
$165$	8.16927	0.635977
$166$	30.1309	2.33861
$167$	6.11273	0.473017	0.236509	−	0.971629i	$-0.423997\pi$
$167$	6.11273	0.473017	0.236509	+	0.971629i	$0.423997\pi$
$168$	4.39734	0.339262
$169$	−4.25349	−0.327192
$170$	−4.60211	−0.352966
$171$	2.38202	0.182158
$172$	−22.5907	−1.72252
$173$	23.4820	1.78530	0.892650	−	0.450750i	$-0.148843\pi$
$173$	23.4820	1.78530	0.892650	+	0.450750i	$0.148843\pi$
$174$	−0.812957	−0.0616301
$175$	−1.45513	−0.109998
$176$	−7.51748	−0.566651
$177$	−8.24937	−0.620061
$178$	−9.61550	−0.720712
$179$	9.24636	0.691106	0.345553	−	0.938399i	$-0.387691\pi$
$179$	9.24636	0.691106	0.345553	+	0.938399i	$0.387691\pi$
$180$	−5.80432	−0.432628
$181$	25.9507	1.92890	0.964451	−	0.264261i	$-0.0851281\pi$
$181$	25.9507	1.92890	0.964451	+	0.264261i	$0.0851281\pi$
$182$	17.3794	1.28825
$183$	11.0605	0.817617
$184$	−10.6657	−0.786283
$185$	−5.87870	−0.432211
$186$	0.122425	0.00897665
$187$	−3.86808	−0.282862
$188$	25.6304	1.86929
$189$	−2.69680	−0.196163
$190$	10.9623	0.795290
$191$	7.58483	0.548819	0.274410	−	0.961613i	$-0.411518\pi$
$191$	7.58483	0.548819	0.274410	+	0.961613i	$0.411518\pi$
$192$	12.4475	0.898318
$193$	−9.64509	−0.694269	−0.347134	−	0.937815i	$-0.612845\pi$
$193$	−9.64509	−0.694269	−0.347134	+	0.937815i	$0.612845\pi$
$194$	−37.9076	−2.72161
$195$	−6.24605	−0.447289
$196$	0.749533	0.0535381
$197$	−18.5587	−1.32225	−0.661126	−	0.750275i	$-0.729921\pi$
$197$	−18.5587	−1.32225	−0.661126	+	0.750275i	$0.729921\pi$
$198$	−8.42877	−0.599007
$199$	−10.6954	−0.758180	−0.379090	−	0.925360i	$-0.623763\pi$
$199$	−10.6954	−0.758180	−0.379090	+	0.925360i	$0.623763\pi$
$200$	0.879822	0.0622128
$201$	3.51722	0.248086
$202$	18.2230	1.28216
$203$	−1.00611	−0.0706153
$204$	2.74829	0.192419
$205$	9.39657	0.656285
$206$	31.3577	2.18480
$207$	6.54103	0.454633
$208$	5.74770	0.398531
$209$	9.21385	0.637335
$210$	−12.4110	−0.856438
$211$	22.3890	1.54132	0.770662	−	0.637245i	$-0.219926\pi$
$211$	22.3890	1.54132	0.770662	+	0.637245i	$0.219926\pi$
$212$	13.4533	0.923977
$213$	12.0755	0.827401
$214$	−7.10702	−0.485826
$215$	17.3602	1.18395
$216$	1.63058	0.110947
$217$	0.151513	0.0102854
$218$	7.59817	0.514613
$219$	3.35786	0.226903
$220$	−22.4516	−1.51368
$221$	2.95745	0.198940
$222$	6.06544	0.407086
$223$	−14.6092	−0.978302	−0.489151	−	0.872199i	$-0.662693\pi$
$223$	−14.6092	−0.978302	−0.489151	+	0.872199i	$0.662693\pi$
$224$	20.2154	1.35070
$225$	−0.539577	−0.0359718
$226$	36.9011	2.45463
$227$	−3.22125	−0.213802	−0.106901	−	0.994270i	$-0.534093\pi$
$227$	−3.22125	−0.213802	−0.106901	+	0.994270i	$0.534093\pi$
$228$	−6.54650	−0.433552
$229$	3.84496	0.254082	0.127041	−	0.991897i	$-0.459452\pi$
$229$	3.84496	0.254082	0.127041	+	0.991897i	$0.459452\pi$
$230$	30.1025	1.98490
$231$	−10.4314	−0.686338
$232$	0.608331	0.0399389
$233$	9.64403	0.631801	0.315901	−	0.948792i	$-0.397693\pi$
$233$	9.64403	0.631801	0.315901	+	0.948792i	$0.397693\pi$
$234$	6.44446	0.421287
$235$	−19.6961	−1.28483
$236$	22.6717	1.47580
$237$	9.02614	0.586311
$238$	5.87648	0.380916
$239$	21.5983	1.39708	0.698540	−	0.715571i	$-0.253833\pi$
$239$	21.5983	1.39708	0.698540	+	0.715571i	$0.253833\pi$
$240$	−4.10454	−0.264947
$241$	−0.331039	−0.0213241	−0.0106620	−	0.999943i	$-0.503394\pi$
$241$	−0.331039	−0.0213241	−0.0106620	+	0.999943i	$0.503394\pi$
$242$	−8.63351	−0.554983
$243$	−1.00000	−0.0641500
$244$	−30.3976	−1.94600
$245$	−0.575991	−0.0367987
$246$	−9.69506	−0.618134
$247$	−7.04471	−0.448244
$248$	−0.0916101	−0.00581725
$249$	13.8275	0.876281
$250$	−25.4937	−1.61237
$251$	−16.3893	−1.03448	−0.517242	−	0.855839i	$-0.673041\pi$
$251$	−16.3893	−1.03448	−0.517242	+	0.855839i	$0.673041\pi$
$252$	7.41160	0.466887
$253$	25.3012	1.59067
$254$	32.4891	2.03855
$255$	−2.11197	−0.132257
$256$	−1.54051	−0.0962821
$257$	7.34696	0.458291	0.229145	−	0.973392i	$-0.426407\pi$
$257$	7.34696	0.458291	0.229145	+	0.973392i	$0.426407\pi$
$258$	−17.9116	−1.11513
$259$	7.50658	0.466436
$260$	17.1660	1.06459
$261$	−0.373077	−0.0230929
$262$	−20.7807	−1.28383
$263$	10.9172	0.673185	0.336593	−	0.941650i	$-0.390726\pi$
$263$	10.9172	0.673185	0.336593	+	0.941650i	$0.390726\pi$
$264$	6.30721	0.388182
$265$	−10.3384	−0.635083
$266$	−13.9979	−0.858267
$267$	−4.41269	−0.270052
$268$	−9.66636	−0.590467
$269$	−26.3169	−1.60457	−0.802285	−	0.596942i	$-0.796382\pi$
$269$	−26.3169	−1.60457	−0.802285	+	0.596942i	$0.796382\pi$
$270$	−4.60211	−0.280075
$271$	15.7062	0.954081	0.477041	−	0.878881i	$-0.341709\pi$
$271$	15.7062	0.954081	0.477041	+	0.878881i	$0.341709\pi$
$272$	1.94347	0.117840
$273$	7.97565	0.482708
$274$	40.0513	2.41959
$275$	−2.08713	−0.125858
$276$	−17.9767	−1.08207
$277$	21.1409	1.27023	0.635116	−	0.772416i	$-0.280952\pi$
$277$	21.1409	1.27023	0.635116	+	0.772416i	$0.280952\pi$
$278$	−36.7931	−2.20671
$279$	0.0561826	0.00336356
$280$	9.28706	0.555008
$281$	25.1756	1.50185	0.750926	−	0.660387i	$-0.229608\pi$
$281$	25.1756	1.50185	0.750926	+	0.660387i	$0.229608\pi$
$282$	20.3217	1.21014
$283$	10.0335	0.596427	0.298214	−	0.954499i	$-0.403609\pi$
$283$	10.0335	0.596427	0.298214	+	0.954499i	$0.403609\pi$
$284$	−33.1871	−1.96929
$285$	5.03076	0.297997
$286$	24.9277	1.47400
$287$	−11.9986	−0.708254
$288$	7.49608	0.441711
$289$	1.00000	0.0588235
$290$	−1.71694	−0.100822
$291$	−17.3963	−1.01979
$292$	−9.22839	−0.540051
$293$	−17.1601	−1.00251	−0.501253	−	0.865301i	$-0.667127\pi$
$293$	−17.1601	−1.00251	−0.501253	+	0.865301i	$0.667127\pi$
$294$	0.594287	0.0346595
$295$	−17.4224	−1.01437
$296$	−4.53874	−0.263809
$297$	−3.86808	−0.224449
$298$	9.85535	0.570905
$299$	−19.3448	−1.11874
$300$	1.48292	0.0856162
$301$	−22.1674	−1.27771
$302$	2.01958	0.116214
$303$	8.36277	0.480428
$304$	−4.62937	−0.265513
$305$	23.3595	1.33756
$306$	2.17906	0.124568
$307$	−12.3845	−0.706820	−0.353410	−	0.935468i	$-0.614978\pi$
$307$	−12.3845	−0.706820	−0.353410	+	0.935468i	$0.614978\pi$
$308$	28.6687	1.63355
$309$	14.3905	0.818647
$310$	0.258558	0.0146851
$311$	0.513125	0.0290966	0.0145483	−	0.999894i	$-0.495369\pi$
$311$	0.513125	0.0290966	0.0145483	+	0.999894i	$0.495369\pi$
$312$	−4.82235	−0.273012
$313$	12.5013	0.706617	0.353309	−	0.935507i	$-0.385057\pi$
$313$	12.5013	0.706617	0.353309	+	0.935507i	$0.385057\pi$
$314$	2.17906	0.122971
$315$	−5.69556	−0.320909
$316$	−24.8065	−1.39547
$317$	8.32697	0.467689	0.233845	−	0.972274i	$-0.424869\pi$
$317$	8.32697	0.467689	0.233845	+	0.972274i	$0.424869\pi$
$318$	10.6668	0.598165
$319$	−1.44309	−0.0807976
$320$	26.2887	1.46958
$321$	−3.26151	−0.182040
$322$	−38.4382	−2.14208
$323$	−2.38202	−0.132539
$324$	2.74829	0.152683
$325$	1.59577	0.0885175
$326$	46.9088	2.59804
$327$	3.48691	0.192826
$328$	7.25476	0.400577
$329$	25.1501	1.38657
$330$	−17.8013	−0.979930
$331$	−11.5544	−0.635089	−0.317545	−	0.948243i	$-0.602858\pi$
$331$	−11.5544	−0.635089	−0.317545	+	0.948243i	$0.602858\pi$
$332$	−38.0020	−2.08563
$333$	2.78351	0.152536
$334$	−13.3200	−0.728837
$335$	7.42827	0.405850
$336$	5.24114	0.285927
$337$	19.0497	1.03771	0.518853	−	0.854864i	$-0.326359\pi$
$337$	19.0497	1.03771	0.518853	+	0.854864i	$0.326359\pi$
$338$	9.26860	0.504145
$339$	16.9344	0.919753
$340$	5.80432	0.314783
$341$	0.217319	0.0117685
$342$	−5.19056	−0.280673
$343$	−18.1421	−0.979582
$344$	13.4032	0.722651
$345$	13.8145	0.743746
$346$	−51.1686	−2.75084
$347$	−18.0836	−0.970778	−0.485389	−	0.874298i	$-0.661322\pi$
$347$	−18.0836	−0.970778	−0.485389	+	0.874298i	$0.661322\pi$
$348$	1.02533	0.0549632
$349$	36.4607	1.95170	0.975848	−	0.218452i	$-0.0701007\pi$
$349$	36.4607	1.95170	0.975848	+	0.218452i	$0.0701007\pi$
$350$	3.17081	0.169487
$351$	2.95745	0.157857
$352$	28.9954	1.54546
$353$	−28.3352	−1.50813	−0.754066	−	0.656798i	$-0.771910\pi$
$353$	−28.3352	−1.50813	−0.754066	+	0.656798i	$0.771910\pi$
$354$	17.9759	0.955407
$355$	25.5031	1.35357
$356$	12.1274	0.642749
$357$	2.69680	0.142730
$358$	−20.1484	−1.06487
$359$	−34.3831	−1.81467	−0.907336	−	0.420407i	$-0.861887\pi$
$359$	−34.3831	−1.81467	−0.907336	+	0.420407i	$0.861887\pi$
$360$	3.44373	0.181501
$361$	−13.3260	−0.701367
$362$	−56.5481	−2.97210
$363$	−3.96204	−0.207953
$364$	−21.9194	−1.14889
$365$	7.09171	0.371197
$366$	−24.1015	−1.25981
$367$	−5.82738	−0.304187	−0.152093	−	0.988366i	$-0.548601\pi$
$367$	−5.82738	−0.304187	−0.152093	+	0.988366i	$0.548601\pi$
$368$	−12.7123	−0.662672
$369$	−4.44920	−0.231616
$370$	12.8100	0.665962
$371$	13.2012	0.685373
$372$	−0.154406	−0.00800560
$373$	4.44885	0.230353	0.115176	−	0.993345i	$-0.463257\pi$
$373$	4.44885	0.230353	0.115176	+	0.993345i	$0.463257\pi$
$374$	8.42877	0.435841
$375$	−11.6994	−0.604156
$376$	−15.2066	−0.784222
$377$	1.10336	0.0568258
$378$	5.87648	0.302254
$379$	−19.1773	−0.985073	−0.492537	−	0.870292i	$-0.663930\pi$
$379$	−19.1773	−0.985073	−0.492537	+	0.870292i	$0.663930\pi$
$380$	−13.8260	−0.709259
$381$	14.9097	0.763846
$382$	−16.5278	−0.845636
$383$	−31.8856	−1.62928	−0.814638	−	0.579970i	$-0.803064\pi$
$383$	−31.8856	−1.62928	−0.814638	+	0.579970i	$0.803064\pi$
$384$	−12.1316	−0.619088
$385$	−22.0309	−1.12280
$386$	21.0172	1.06975
$387$	−8.21989	−0.417841
$388$	47.8102	2.42719
$389$	31.0338	1.57348	0.786738	−	0.617287i	$-0.211768\pi$
$389$	31.0338	1.57348	0.786738	+	0.617287i	$0.211768\pi$
$390$	13.6105	0.689195
$391$	−6.54103	−0.330794
$392$	−0.444702	−0.0224608
$393$	−9.53654	−0.481055
$394$	40.4405	2.03736
$395$	19.0629	0.959161
$396$	10.6306	0.534209
$397$	−22.2317	−1.11578	−0.557888	−	0.829916i	$-0.688388\pi$
$397$	−22.2317	−1.11578	−0.557888	+	0.829916i	$0.688388\pi$
$398$	23.3060	1.16822
$399$	−6.42383	−0.321594
$400$	1.04865	0.0524324
$401$	34.7042	1.73304	0.866521	−	0.499140i	$-0.166351\pi$
$401$	34.7042	1.73304	0.866521	+	0.499140i	$0.166351\pi$
$402$	−7.66423	−0.382257
$403$	−0.166157	−0.00827688
$404$	−22.9833	−1.14346
$405$	−2.11197	−0.104945
$406$	2.19238	0.108806
$407$	10.7669	0.533693
$408$	−1.63058	−0.0807256
$409$	−14.3081	−0.707492	−0.353746	−	0.935342i	$-0.615092\pi$
$409$	−14.3081	−0.707492	−0.353746	+	0.935342i	$0.615092\pi$
$410$	−20.4757	−1.01122
$411$	18.3801	0.906623
$412$	−39.5493	−1.94846
$413$	22.2469	1.09470
$414$	−14.2533	−0.700511
$415$	29.2032	1.43353
$416$	−22.1693	−1.08694
$417$	−16.8849	−0.826856
$418$	−20.0775	−0.982023
$419$	−17.4127	−0.850668	−0.425334	−	0.905036i	$-0.639843\pi$
$419$	−17.4127	−0.850668	−0.425334	+	0.905036i	$0.639843\pi$
$420$	15.6531	0.763792
$421$	−19.8586	−0.967848	−0.483924	−	0.875110i	$-0.660789\pi$
$421$	−19.8586	−0.967848	−0.483924	+	0.875110i	$0.660789\pi$
$422$	−48.7870	−2.37491
$423$	9.32591	0.453441
$424$	−7.98192	−0.387636
$425$	0.539577	0.0261733
$426$	−26.3132	−1.27488
$427$	−29.8280	−1.44348
$428$	8.96360	0.433272
$429$	11.4397	0.552312
$430$	−37.8288	−1.82427
$431$	−25.2344	−1.21550	−0.607750	−	0.794128i	$-0.707928\pi$
$431$	−25.2344	−1.21550	−0.607750	+	0.794128i	$0.707928\pi$
$432$	1.94347	0.0935050
$433$	−19.8315	−0.953043	−0.476521	−	0.879163i	$-0.658103\pi$
$433$	−19.8315	−0.953043	−0.476521	+	0.879163i	$0.658103\pi$
$434$	−0.330156	−0.0158480
$435$	−0.787928	−0.0377783
$436$	−9.58304	−0.458945
$437$	15.5809	0.745334
$438$	−7.31698	−0.349619
$439$	−10.0525	−0.479779	−0.239889	−	0.970800i	$-0.577111\pi$
$439$	−10.0525	−0.479779	−0.239889	+	0.970800i	$0.577111\pi$
$440$	13.3206	0.635036
$441$	0.272727	0.0129870
$442$	−6.44446	−0.306532
$443$	31.3857	1.49118	0.745591	−	0.666404i	$-0.232167\pi$
$443$	31.3857	1.49118	0.745591	+	0.666404i	$0.232167\pi$
$444$	−7.64992	−0.363049
$445$	−9.31946	−0.441785
$446$	31.8342	1.50739
$447$	4.52276	0.213919
$448$	−33.5683	−1.58595
$449$	33.3758	1.57510	0.787551	−	0.616249i	$-0.211349\pi$
$449$	33.3758	1.57510	0.787551	+	0.616249i	$0.211349\pi$
$450$	1.17577	0.0554263
$451$	−17.2098	−0.810380
$452$	−46.5408	−2.18910
$453$	0.926814	0.0435455
$454$	7.01929	0.329432
$455$	16.8443	0.789675
$456$	3.88407	0.181888
$457$	4.89199	0.228838	0.114419	−	0.993433i	$-0.463499\pi$
$457$	4.89199	0.228838	0.114419	+	0.993433i	$0.463499\pi$
$458$	−8.37839	−0.391496
$459$	1.00000	0.0466760
$460$	−37.9662	−1.77018
$461$	18.3178	0.853145	0.426572	−	0.904453i	$-0.359721\pi$
$461$	18.3178	0.853145	0.426572	+	0.904453i	$0.359721\pi$
$462$	22.7307	1.05753
$463$	−13.7118	−0.637240	−0.318620	−	0.947882i	$-0.603219\pi$
$463$	−13.7118	−0.637240	−0.318620	+	0.947882i	$0.603219\pi$
$464$	0.725062	0.0336602
$465$	0.118656	0.00550254
$466$	−21.0149	−0.973497
$467$	−3.37767	−0.156300	−0.0781501	−	0.996942i	$-0.524901\pi$
$467$	−3.37767	−0.156300	−0.0781501	+	0.996942i	$0.524901\pi$
$468$	−8.12794	−0.375714
$469$	−9.48524	−0.437988
$470$	42.9189	1.97970
$471$	1.00000	0.0460776
$472$	−13.4512	−0.619144
$473$	−31.7952	−1.46194
$474$	−19.6685	−0.903403
$475$	−1.28528	−0.0589728
$476$	−7.41160	−0.339710
$477$	4.89515	0.224133
$478$	−47.0640	−2.15266
$479$	20.2762	0.926445	0.463222	−	0.886242i	$-0.346693\pi$
$479$	20.2762	0.926445	0.463222	+	0.886242i	$0.346693\pi$
$480$	15.8315	0.722606
$481$	−8.23210	−0.375352
$482$	0.721353	0.0328567
$483$	−17.6398	−0.802641
$484$	10.8888	0.494948
$485$	−36.7405	−1.66830
$486$	2.17906	0.0988441
$487$	−19.4435	−0.881069	−0.440534	−	0.897736i	$-0.645211\pi$
$487$	−19.4435	−0.881069	−0.440534	+	0.897736i	$0.645211\pi$
$488$	18.0350	0.816408
$489$	21.5271	0.973488
$490$	1.25512	0.0567004
$491$	−31.0238	−1.40008	−0.700042	−	0.714102i	$-0.746835\pi$
$491$	−31.0238	−1.40008	−0.700042	+	0.714102i	$0.746835\pi$
$492$	12.2277	0.551267
$493$	0.373077	0.0168025
$494$	15.3508	0.690666
$495$	−8.16927	−0.367182
$496$	−0.109189	−0.00490273
$497$	−32.5652	−1.46075
$498$	−30.1309	−1.35020
$499$	20.7560	0.929165	0.464582	−	0.885530i	$-0.346204\pi$
$499$	20.7560	0.929165	0.464582	+	0.885530i	$0.346204\pi$
$500$	32.1535	1.43795
$501$	−6.11273	−0.273097
$502$	35.7133	1.59396
$503$	−12.9342	−0.576708	−0.288354	−	0.957524i	$-0.593108\pi$
$503$	−12.9342	−0.576708	−0.288354	+	0.957524i	$0.593108\pi$
$504$	−4.39734	−0.195873
$505$	17.6619	0.785945
$506$	−55.1328	−2.45095
$507$	4.25349	0.188904
$508$	−40.9762	−1.81802
$509$	−10.3352	−0.458102	−0.229051	−	0.973414i	$-0.573562\pi$
$509$	−10.3352	−0.458102	−0.229051	+	0.973414i	$0.573562\pi$
$510$	4.60211	0.203785
$511$	−9.05548	−0.400591
$512$	−20.9063	−0.923937
$513$	−2.38202	−0.105169
$514$	−16.0095	−0.706147
$515$	30.3923	1.33925
$516$	22.5907	0.994499
$517$	36.0734	1.58651
$518$	−16.3573	−0.718697
$519$	−23.4820	−1.03074
$520$	−10.1847	−0.446627
$521$	14.8498	0.650584	0.325292	−	0.945614i	$-0.394537\pi$
$521$	14.8498	0.650584	0.325292	+	0.945614i	$0.394537\pi$
$522$	0.812957	0.0355821
$523$	0.00955388	0.000417762 0	0.000208881	−	1.00000i	$-0.499934\pi$
$523$	0.00955388	0.000417762 0	0.000208881		1.00000i	$0.499934\pi$
$524$	26.2092	1.14496
$525$	1.45513	0.0635071
$526$	−23.7893	−1.03726
$527$	−0.0561826	−0.00244735
$528$	7.51748	0.327156
$529$	19.7851	0.860220
$530$	22.5280	0.978553
$531$	8.24937	0.357992
$532$	17.6546	0.765423
$533$	13.1583	0.569948
$534$	9.61550	0.416103
$535$	−6.88822	−0.297804
$536$	5.73510	0.247719
$537$	−9.24636	−0.399010
$538$	57.3460	2.47236
$539$	1.05493	0.0454390
$540$	5.80432	0.249778
$541$	−38.4825	−1.65449	−0.827247	−	0.561839i	$-0.810094\pi$
$541$	−38.4825	−1.65449	−0.827247	+	0.561839i	$0.810094\pi$
$542$	−34.2246	−1.47007
$543$	−25.9507	−1.11365
$544$	−7.49608	−0.321392
$545$	7.36424	0.315450
$546$	−17.3794	−0.743770
$547$	3.96629	0.169586	0.0847931	−	0.996399i	$-0.472977\pi$
$547$	3.96629	0.169586	0.0847931	+	0.996399i	$0.472977\pi$
$548$	−50.5139	−2.15785
$549$	−11.0605	−0.472051
$550$	4.54797	0.193926
$551$	−0.888677	−0.0378589
$552$	10.6657	0.453961
$553$	−24.3417	−1.03511
$554$	−46.0672	−1.95721
$555$	5.87870	0.249537
$556$	46.4046	1.96799
$557$	40.3010	1.70761	0.853805	−	0.520593i	$-0.174289\pi$
$557$	40.3010	1.70761	0.853805	+	0.520593i	$0.174289\pi$
$558$	−0.122425	−0.00518267
$559$	24.3099	1.02820
$560$	11.0691	0.467756
$561$	3.86808	0.163310
$562$	−54.8591	−2.31409
$563$	3.76673	0.158749	0.0793744	−	0.996845i	$-0.474708\pi$
$563$	3.76673	0.158749	0.0793744	+	0.996845i	$0.474708\pi$
$564$	−25.6304	−1.07923
$565$	35.7651	1.50465
$566$	−21.8635	−0.918991
$567$	2.69680	0.113255
$568$	19.6901	0.826177
$569$	−4.76646	−0.199820	−0.0999102	−	0.994996i	$-0.531856\pi$
$569$	−4.76646	−0.199820	−0.0999102	+	0.994996i	$0.531856\pi$
$570$	−10.9623	−0.459161
$571$	−31.0140	−1.29790	−0.648949	−	0.760832i	$-0.724791\pi$
$571$	−31.0140	−1.29790	−0.648949	+	0.760832i	$0.724791\pi$
$572$	−31.4395	−1.31455
$573$	−7.58483	−0.316861
$574$	26.1456	1.09130
$575$	−3.52939	−0.147186
$576$	−12.4475	−0.518644
$577$	−25.9928	−1.08210	−0.541048	−	0.840992i	$-0.681972\pi$
$577$	−25.9928	−1.08210	−0.541048	+	0.840992i	$0.681972\pi$
$578$	−2.17906	−0.0906369
$579$	9.64509	0.400836
$580$	2.16546	0.0899158
$581$	−37.2899	−1.54705
$582$	37.9076	1.57132
$583$	18.9348	0.784200
$584$	5.47526	0.226568
$585$	6.24605	0.258242
$586$	37.3929	1.54469
$587$	9.46799	0.390786	0.195393	−	0.980725i	$-0.437402\pi$
$587$	9.46799	0.390786	0.195393	+	0.980725i	$0.437402\pi$
$588$	−0.749533	−0.0309102
$589$	0.133828	0.00551429
$590$	37.9645	1.56297
$591$	18.5587	0.763403
$592$	−5.40966	−0.222336
$593$	−0.566854	−0.0232779	−0.0116389	−	0.999932i	$-0.503705\pi$
$593$	−0.566854	−0.0232779	−0.0116389	+	0.999932i	$0.503705\pi$
$594$	8.42877	0.345837
$595$	5.69556	0.233495
$596$	−12.4299	−0.509147
$597$	10.6954	0.437735
$598$	42.1534	1.72378
$599$	7.16888	0.292912	0.146456	−	0.989217i	$-0.453213\pi$
$599$	7.16888	0.292912	0.146456	+	0.989217i	$0.453213\pi$
$600$	−0.879822	−0.0359186
$601$	5.03026	0.205189	0.102594	−	0.994723i	$-0.467286\pi$
$601$	5.03026	0.205189	0.102594	+	0.994723i	$0.467286\pi$
$602$	48.3040	1.96873
$603$	−3.51722	−0.143232
$604$	−2.54716	−0.103642
$605$	−8.36771	−0.340196
$606$	−18.2230	−0.740257
$607$	−0.718841	−0.0291768	−0.0145884	−	0.999894i	$-0.504644\pi$
$607$	−0.718841	−0.0291768	−0.0145884	+	0.999894i	$0.504644\pi$
$608$	17.8558	0.724149
$609$	1.00611	0.0407698
$610$	−50.9017	−2.06095
$611$	−27.5809	−1.11580
$612$	−2.74829	−0.111093
$613$	−9.89319	−0.399582	−0.199791	−	0.979839i	$-0.564026\pi$
$613$	−9.89319	−0.399582	−0.199791	+	0.979839i	$0.564026\pi$
$614$	26.9865	1.08909
$615$	−9.39657	−0.378906
$616$	−17.0093	−0.685323
$617$	−26.8405	−1.08056	−0.540280	−	0.841486i	$-0.681682\pi$
$617$	−26.8405	−1.08056	−0.540280	+	0.841486i	$0.681682\pi$
$618$	−31.3577	−1.26139
$619$	38.1813	1.53464	0.767318	−	0.641266i	$-0.221591\pi$
$619$	38.1813	1.53464	0.767318	+	0.641266i	$0.221591\pi$
$620$	−0.326102	−0.0130966
$621$	−6.54103	−0.262482
$622$	−1.11813	−0.0448329
$623$	11.9001	0.476769
$624$	−5.74770	−0.230092
$625$	−22.0110	−0.880439
$626$	−27.2411	−1.08877
$627$	−9.21385	−0.367966
$628$	−2.74829	−0.109669
$629$	−2.78351	−0.110986
$630$	12.4110	0.494465
$631$	26.6293	1.06009	0.530047	−	0.847968i	$-0.322174\pi$
$631$	26.6293	1.06009	0.530047	+	0.847968i	$0.322174\pi$
$632$	14.7178	0.585443
$633$	−22.3890	−0.889883
$634$	−18.1450	−0.720628
$635$	31.4888	1.24960
$636$	−13.4533	−0.533458
$637$	−0.806575	−0.0319577
$638$	3.14458	0.124495
$639$	−12.0755	−0.477700
$640$	−25.6216	−1.01278
$641$	−6.95415	−0.274673	−0.137336	−	0.990524i	$-0.543854\pi$
$641$	−6.95415	−0.274673	−0.137336	+	0.990524i	$0.543854\pi$
$642$	7.10702	0.280492
$643$	−0.313263	−0.0123539	−0.00617695	−	0.999981i	$-0.501966\pi$
$643$	−0.313263	−0.0123539	−0.00617695	+	0.999981i	$0.501966\pi$
$644$	48.4795	1.91036
$645$	−17.3602	−0.683556
$646$	5.19056	0.204220
$647$	−19.8133	−0.778943	−0.389471	−	0.921039i	$-0.627342\pi$
$647$	−19.8133	−0.778943	−0.389471	+	0.921039i	$0.627342\pi$
$648$	−1.63058	−0.0640552
$649$	31.9092	1.25255
$650$	−3.47728	−0.136390
$651$	−0.151513	−0.00593827
$652$	−59.1628	−2.31699
$653$	−39.9324	−1.56268	−0.781338	−	0.624108i	$-0.785463\pi$
$653$	−39.9324	−1.56268	−0.781338	+	0.624108i	$0.785463\pi$
$654$	−7.59817	−0.297112
$655$	−20.1409	−0.786970
$656$	8.64686	0.337603
$657$	−3.35786	−0.131003
$658$	−54.8036	−2.13647
$659$	26.8018	1.04405	0.522025	−	0.852930i	$-0.325177\pi$
$659$	26.8018	1.04405	0.522025	+	0.852930i	$0.325177\pi$
$660$	22.4516	0.873926
$661$	15.5641	0.605373	0.302686	−	0.953090i	$-0.402117\pi$
$661$	15.5641	0.605373	0.302686	+	0.953090i	$0.402117\pi$
$662$	25.1778	0.978562
$663$	−2.95745	−0.114858
$664$	22.5468	0.874985
$665$	−13.5670	−0.526104
$666$	−6.06544	−0.235031
$667$	−2.44031	−0.0944891
$668$	16.7996	0.649995
$669$	14.6092	0.564823
$670$	−16.1866	−0.625344
$671$	−42.7830	−1.65162
$672$	−20.2154	−0.779827
$673$	7.90781	0.304824	0.152412	−	0.988317i	$-0.451296\pi$
$673$	7.90781	0.304824	0.152412	+	0.988317i	$0.451296\pi$
$674$	−41.5105	−1.59893
$675$	0.539577	0.0207683
$676$	−11.6898	−0.449609
$677$	−21.3072	−0.818902	−0.409451	−	0.912332i	$-0.634280\pi$
$677$	−21.3072	−0.818902	−0.409451	+	0.912332i	$0.634280\pi$
$678$	−36.9011	−1.41718
$679$	46.9144	1.80041
$680$	−3.44373	−0.132061
$681$	3.22125	0.123439
$682$	−0.473550	−0.0181332
$683$	1.16602	0.0446163	0.0223082	−	0.999751i	$-0.492899\pi$
$683$	1.16602	0.0446163	0.0223082	+	0.999751i	$0.492899\pi$
$684$	6.54650	0.250312
$685$	38.8182	1.48317
$686$	39.5327	1.50937
$687$	−3.84496	−0.146694
$688$	15.9751	0.609044
$689$	−14.4771	−0.551535
$690$	−30.1025	−1.14598
$691$	0.680408	0.0258839	0.0129420	−	0.999916i	$-0.495880\pi$
$691$	0.680408	0.0258839	0.0129420	+	0.999916i	$0.495880\pi$
$692$	64.5354	2.45327
$693$	10.4314	0.396257
$694$	39.4052	1.49580
$695$	−35.6604	−1.35268
$696$	−0.608331	−0.0230587
$697$	4.44920	0.168525
$698$	−79.4499	−3.00722
$699$	−9.64403	−0.364771
$700$	−3.99913	−0.151153
$701$	26.9917	1.01946	0.509732	−	0.860333i	$-0.329745\pi$
$701$	26.9917	1.01946	0.509732	+	0.860333i	$0.329745\pi$
$702$	−6.44446	−0.243230
$703$	6.63039	0.250070
$704$	−48.1478	−1.81464
$705$	19.6961	0.741797
$706$	61.7441	2.32377
$707$	−22.5527	−0.848182
$708$	−22.6717	−0.852055
$709$	−2.54356	−0.0955253	−0.0477626	−	0.998859i	$-0.515209\pi$
$709$	−2.54356	−0.0955253	−0.0477626	+	0.998859i	$0.515209\pi$
$710$	−55.5728	−2.08561
$711$	−9.02614	−0.338507
$712$	−7.19523	−0.269653
$713$	0.367492	0.0137627
$714$	−5.87648	−0.219922
$715$	24.1602	0.903541
$716$	25.4117	0.949681
$717$	−21.5983	−0.806604
$718$	74.9228	2.79609
$719$	−0.366136	−0.0136546	−0.00682728	−	0.999977i	$-0.502173\pi$
$719$	−0.366136	−0.0136546	−0.00682728	+	0.999977i	$0.502173\pi$
$720$	4.10454	0.152967
$721$	−38.8083	−1.44530
$722$	29.0381	1.08069
$723$	0.331039	0.0123115
$724$	71.3203	2.65060
$725$	0.201304	0.00747623
$726$	8.63351	0.320420
$727$	−30.4594	−1.12968	−0.564839	−	0.825201i	$-0.691062\pi$
$727$	−30.4594	−1.12968	−0.564839	+	0.825201i	$0.691062\pi$
$728$	13.0049	0.481994
$729$	1.00000	0.0370370
$730$	−15.4532	−0.571950
$731$	8.21989	0.304024
$732$	30.3976	1.12353
$733$	34.1123	1.25997	0.629984	−	0.776608i	$-0.283061\pi$
$733$	34.1123	1.25997	0.629984	+	0.776608i	$0.283061\pi$
$734$	12.6982	0.468699
$735$	0.575991	0.0212457
$736$	49.0321	1.80735
$737$	−13.6049	−0.501142
$738$	9.69506	0.356880
$739$	34.7478	1.27822	0.639108	−	0.769117i	$-0.279303\pi$
$739$	34.7478	1.27822	0.639108	+	0.769117i	$0.279303\pi$
$740$	−16.1564	−0.593921
$741$	7.04471	0.258794
$742$	−28.7662	−1.05604
$743$	13.4799	0.494530	0.247265	−	0.968948i	$-0.420468\pi$
$743$	13.4799	0.494530	0.247265	+	0.968948i	$0.420468\pi$
$744$	0.0916101	0.00335859
$745$	9.55193	0.349956
$746$	−9.69429	−0.354933
$747$	−13.8275	−0.505921
$748$	−10.6306	−0.388694
$749$	8.79564	0.321386
$750$	25.4937	0.930900
$751$	40.7030	1.48527	0.742636	−	0.669695i	$-0.233575\pi$
$751$	40.7030	1.48527	0.742636	+	0.669695i	$0.233575\pi$
$752$	−18.1246	−0.660936
$753$	16.3893	0.597260
$754$	−2.40428	−0.0875587
$755$	1.95740	0.0712372
$756$	−7.41160	−0.269557
$757$	−14.6769	−0.533440	−0.266720	−	0.963774i	$-0.585940\pi$
$757$	−14.6769	−0.533440	−0.266720	+	0.963774i	$0.585940\pi$
$758$	41.7885	1.51783
$759$	−25.3012	−0.918376
$760$	8.20305	0.297556
$761$	−20.1385	−0.730019	−0.365009	−	0.931004i	$-0.618934\pi$
$761$	−20.1385	−0.730019	−0.365009	+	0.931004i	$0.618934\pi$
$762$	−32.4891	−1.17695
$763$	−9.40348	−0.340429
$764$	20.8454	0.754159
$765$	2.11197	0.0763585
$766$	69.4805	2.51043
$767$	−24.3971	−0.880929
$768$	1.54051	0.0555885
$769$	17.8489	0.643649	0.321825	−	0.946799i	$-0.395704\pi$
$769$	17.8489	0.643649	0.321825	+	0.946799i	$0.395704\pi$
$770$	48.0066	1.73004
$771$	−7.34696	−0.264594
$772$	−26.5075	−0.954027
$773$	−15.7529	−0.566593	−0.283296	−	0.959032i	$-0.591428\pi$
$773$	−15.7529	−0.566593	−0.283296	+	0.959032i	$0.591428\pi$
$774$	17.9116	0.643820
$775$	−0.0303148	−0.00108894
$776$	−28.3660	−1.01828
$777$	−7.50658	−0.269297
$778$	−67.6245	−2.42445
$779$	−10.5981	−0.379716
$780$	−17.1660	−0.614641
$781$	−46.7090	−1.67138
$782$	14.2533	0.509696
$783$	0.373077	0.0133327
$784$	−0.530035	−0.0189298
$785$	2.11197	0.0753795
$786$	20.7807	0.741222
$787$	3.85152	0.137292	0.0686459	−	0.997641i	$-0.478132\pi$
$787$	3.85152	0.137292	0.0686459	+	0.997641i	$0.478132\pi$
$788$	−51.0048	−1.81697
$789$	−10.9172	−0.388664
$790$	−41.5393	−1.47790
$791$	−45.6688	−1.62380
$792$	−6.30721	−0.224117
$793$	32.7109	1.16160
$794$	48.4441	1.71922
$795$	10.3384	0.366666
$796$	−29.3942	−1.04185
$797$	−29.9733	−1.06171	−0.530854	−	0.847463i	$-0.678129\pi$
$797$	−29.9733	−1.06171	−0.530854	+	0.847463i	$0.678129\pi$
$798$	13.9979	0.495520
$799$	−9.32591	−0.329927
$800$	−4.04471	−0.143002
$801$	4.41269	0.155915
$802$	−75.6224	−2.67032
$803$	−12.9885	−0.458353
$804$	9.66636	0.340906
$805$	−37.2548	−1.31306
$806$	0.362066	0.0127532
$807$	26.3169	0.926399
$808$	13.6361	0.479718
$809$	0.559803	0.0196816	0.00984080	−	0.999952i	$-0.496868\pi$
$809$	0.559803	0.0196816	0.00984080	+	0.999952i	$0.496868\pi$
$810$	4.60211	0.161702
$811$	−31.0479	−1.09024	−0.545120	−	0.838358i	$-0.683516\pi$
$811$	−31.0479	−1.09024	−0.545120	+	0.838358i	$0.683516\pi$
$812$	−2.76510	−0.0970359
$813$	−15.7062	−0.550839
$814$	−23.4616	−0.822329
$815$	45.4646	1.59256
$816$	−1.94347	−0.0680349
$817$	−19.5800	−0.685016
$818$	31.1783	1.09012
$819$	−7.97565	−0.278692
$820$	25.8246	0.901833
$821$	−34.5665	−1.20638	−0.603189	−	0.797598i	$-0.706104\pi$
$821$	−34.5665	−1.20638	−0.603189	+	0.797598i	$0.706104\pi$
$822$	−40.0513	−1.39695
$823$	27.4537	0.956977	0.478489	−	0.878094i	$-0.341185\pi$
$823$	27.4537	0.956977	0.478489	+	0.878094i	$0.341185\pi$
$824$	23.4648	0.817436
$825$	2.08713	0.0726644
$826$	−48.4773	−1.68674
$827$	−36.4227	−1.26654	−0.633271	−	0.773930i	$-0.718288\pi$
$827$	−36.4227	−1.26654	−0.633271	+	0.773930i	$0.718288\pi$
$828$	17.9767	0.624733
$829$	−6.58206	−0.228604	−0.114302	−	0.993446i	$-0.536463\pi$
$829$	−6.58206	−0.228604	−0.114302	+	0.993446i	$0.536463\pi$
$830$	−63.6356	−2.20882
$831$	−21.1409	−0.733369
$832$	36.8128	1.27625
$833$	−0.272727	−0.00944942
$834$	36.7931	1.27404
$835$	−12.9099	−0.446766
$836$	25.3224	0.875792
$837$	−0.0561826	−0.00194196
$838$	37.9434	1.31073
$839$	−49.4206	−1.70619	−0.853094	−	0.521758i	$-0.825276\pi$
$839$	−49.4206	−1.70619	−0.853094	+	0.521758i	$0.825276\pi$
$840$	−9.28706	−0.320434
$841$	−28.8608	−0.995200
$842$	43.2730	1.49129
$843$	−25.1756	−0.867094
$844$	61.5316	2.11801
$845$	8.98325	0.309033
$846$	−20.3217	−0.698675
$847$	10.6848	0.367135
$848$	−9.51354	−0.326696
$849$	−10.0335	−0.344347
$850$	−1.17577	−0.0403286
$851$	18.2070	0.624130
$852$	33.1871	1.13697
$853$	0.889324	0.0304499	0.0152249	−	0.999884i	$-0.495154\pi$
$853$	0.889324	0.0304499	0.0152249	+	0.999884i	$0.495154\pi$
$854$	64.9969	2.22415
$855$	−5.03076	−0.172048
$856$	−5.31815	−0.181771
$857$	27.6391	0.944135	0.472067	−	0.881562i	$-0.343508\pi$
$857$	27.6391	0.944135	0.472067	+	0.881562i	$0.343508\pi$
$858$	−24.9277	−0.851016
$859$	44.6674	1.52403	0.762016	−	0.647558i	$-0.224210\pi$
$859$	44.6674	1.52403	0.762016	+	0.647558i	$0.224210\pi$
$860$	47.7109	1.62693
$861$	11.9986	0.408911
$862$	54.9873	1.87287
$863$	−48.8577	−1.66313	−0.831567	−	0.555424i	$-0.812556\pi$
$863$	−48.8577	−1.66313	−0.831567	+	0.555424i	$0.812556\pi$
$864$	−7.49608	−0.255022
$865$	−49.5932	−1.68622
$866$	43.2141	1.46847
$867$	−1.00000	−0.0339618
$868$	0.416403	0.0141336
$869$	−34.9138	−1.18437
$870$	1.71694	0.0582098
$871$	10.4020	0.352458
$872$	5.68567	0.192541
$873$	17.3963	0.588776
$874$	−33.9516	−1.14843
$875$	31.5510	1.06662
$876$	9.22839	0.311799
$877$	−31.1375	−1.05144	−0.525719	−	0.850659i	$-0.676204\pi$
$877$	−31.1375	−1.05144	−0.525719	+	0.850659i	$0.676204\pi$
$878$	21.9049	0.739256
$879$	17.1601	0.578797
$880$	15.8767	0.535203
$881$	−20.5395	−0.691993	−0.345996	−	0.938236i	$-0.612459\pi$
$881$	−20.5395	−0.691993	−0.345996	+	0.938236i	$0.612459\pi$
$882$	−0.594287	−0.0200107
$883$	−22.9662	−0.772874	−0.386437	−	0.922316i	$-0.626294\pi$
$883$	−22.9662	−0.772874	−0.386437	+	0.922316i	$0.626294\pi$
$884$	8.12794	0.273372
$885$	17.4224	0.585649
$886$	−68.3914	−2.29765
$887$	−15.9416	−0.535267	−0.267634	−	0.963521i	$-0.586242\pi$
$887$	−15.9416	−0.535267	−0.267634	+	0.963521i	$0.586242\pi$
$888$	4.53874	0.152310
$889$	−40.2084	−1.34855
$890$	20.3077	0.680714
$891$	3.86808	0.129586
$892$	−40.1503	−1.34433
$893$	22.2145	0.743381
$894$	−9.85535	−0.329612
$895$	−19.5281	−0.652751
$896$	32.7165	1.09298
$897$	19.3448	0.645903
$898$	−72.7279	−2.42696
$899$	−0.0209604	−0.000699070 0
$900$	−1.48292	−0.0494305
$901$	−4.89515	−0.163081
$902$	37.5012	1.24865
$903$	22.1674	0.737685
$904$	27.6129	0.918392
$905$	−54.8072	−1.82185
$906$	−2.01958	−0.0670961
$907$	−46.6453	−1.54883	−0.774416	−	0.632676i	$-0.781956\pi$
$907$	−46.6453	−1.54883	−0.774416	+	0.632676i	$0.781956\pi$
$908$	−8.85295	−0.293795
$909$	−8.36277	−0.277375
$910$	−36.7048	−1.21675
$911$	31.6895	1.04992	0.524960	−	0.851127i	$-0.324080\pi$
$911$	31.6895	1.04992	0.524960	+	0.851127i	$0.324080\pi$
$912$	4.62937	0.153294
$913$	−53.4858	−1.77012
$914$	−10.6599	−0.352600
$915$	−23.3595	−0.772241
$916$	10.5671	0.349146
$917$	25.7181	0.849287
$918$	−2.17906	−0.0719196
$919$	−48.2394	−1.59127	−0.795636	−	0.605775i	$-0.792863\pi$
$919$	−48.2394	−1.59127	−0.795636	+	0.605775i	$0.792863\pi$
$920$	22.5256	0.742646
$921$	12.3845	0.408083
$922$	−39.9155	−1.31455
$923$	35.7127	1.17550
$924$	−28.6687	−0.943130
$925$	−1.50192	−0.0493828
$926$	29.8788	0.981877
$927$	−14.3905	−0.472646
$928$	−2.79662	−0.0918034
$929$	−42.5217	−1.39509	−0.697546	−	0.716540i	$-0.745725\pi$
$929$	−42.5217	−1.39509	−0.697546	+	0.716540i	$0.745725\pi$
$930$	−0.258558	−0.00847847
$931$	0.649641	0.0212911
$932$	26.5046	0.868188
$933$	−0.513125	−0.0167990
$934$	7.36015	0.240831
$935$	8.16927	0.267164
$936$	4.82235	0.157624
$937$	−25.2527	−0.824971	−0.412485	−	0.910964i	$-0.635339\pi$
$937$	−25.2527	−0.824971	−0.412485	+	0.910964i	$0.635339\pi$
$938$	20.6689	0.674863
$939$	−12.5013	−0.407966
$940$	−54.1306	−1.76555
$941$	15.0814	0.491640	0.245820	−	0.969315i	$-0.420943\pi$
$941$	15.0814	0.491640	0.245820	+	0.969315i	$0.420943\pi$
$942$	−2.17906	−0.0709976
$943$	−29.1023	−0.947702
$944$	−16.0324	−0.521809
$945$	5.69556	0.185277
$946$	69.2836	2.25260
$947$	45.3069	1.47228	0.736139	−	0.676831i	$-0.236647\pi$
$947$	45.3069	1.47228	0.736139	+	0.676831i	$0.236647\pi$
$948$	24.8065	0.805677
$949$	9.93071	0.322364
$950$	2.80071	0.0908669
$951$	−8.32697	−0.270021
$952$	4.39734	0.142519
$953$	−41.7836	−1.35350	−0.676751	−	0.736212i	$-0.736613\pi$
$953$	−41.7836	−1.35350	−0.676751	+	0.736212i	$0.736613\pi$
$954$	−10.6668	−0.345351
$955$	−16.0190	−0.518361
$956$	59.3586	1.91979
$957$	1.44309	0.0466485
$958$	−44.1831	−1.42749
$959$	−49.5674	−1.60062
$960$	−26.2887	−0.848464
$961$	−30.9968	−0.999898
$962$	17.9382	0.578352
$963$	3.26151	0.105101
$964$	−0.909793	−0.0293025
$965$	20.3701	0.655738
$966$	38.4382	1.23673
$967$	−43.7585	−1.40718	−0.703589	−	0.710607i	$-0.748420\pi$
$967$	−43.7585	−1.40718	−0.703589	+	0.710607i	$0.748420\pi$
$968$	−6.46041	−0.207645
$969$	2.38202	0.0765215
$970$	80.0597	2.57056
$971$	49.8280	1.59906	0.799528	−	0.600629i	$-0.205083\pi$
$971$	49.8280	1.59906	0.799528	+	0.600629i	$0.205083\pi$
$972$	−2.74829	−0.0881516
$973$	45.5351	1.45979
$974$	42.3685	1.35757
$975$	−1.59577	−0.0511056
$976$	21.4957	0.688062
$977$	−54.2989	−1.73718	−0.868588	−	0.495536i	$-0.834972\pi$
$977$	−54.2989	−1.73718	−0.868588	+	0.495536i	$0.834972\pi$
$978$	−46.9088	−1.49998
$979$	17.0686	0.545515
$980$	−1.58299	−0.0505668
$981$	−3.48691	−0.111328
$982$	67.6026	2.15729
$983$	−38.3446	−1.22300	−0.611502	−	0.791243i	$-0.709434\pi$
$983$	−38.3446	−1.22300	−0.611502	+	0.791243i	$0.709434\pi$
$984$	−7.25476	−0.231273
$985$	39.1954	1.24887
$986$	−0.812957	−0.0258898
$987$	−25.1501	−0.800537
$988$	−19.3609	−0.615953
$989$	−53.7665	−1.70968
$990$	17.8013	0.565763
$991$	−48.3036	−1.53442	−0.767208	−	0.641399i	$-0.778354\pi$
$991$	−48.3036	−1.53442	−0.767208	+	0.641399i	$0.778354\pi$
$992$	0.421149	0.0133715
$993$	11.5544	0.366669
$994$	70.9616	2.25076
$995$	22.5885	0.716102
$996$	38.0020	1.20414
$997$	−9.66132	−0.305977	−0.152989	−	0.988228i	$-0.548890\pi$
$997$	−9.66132	−0.305977	−0.152989	+	0.988228i	$0.548890\pi$
$998$	−45.2285	−1.43168
$999$	−2.78351	−0.0880665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.8	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-2.17906 q^{2} -1.00000 q^{3} +2.74829 q^{4} -2.11197 q^{5} +2.17906 q^{6} +2.69680 q^{7} -1.63058 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.17906 q^{2} -1.00000 q^{3} +2.74829 q^{4} -2.11197 q^{5} +2.17906 q^{6} +2.69680 q^{7} -1.63058 q^{8} +1.00000 q^{9} +4.60211 q^{10} +3.86808 q^{11} -2.74829 q^{12} -2.95745 q^{13} -5.87648 q^{14} +2.11197 q^{15} -1.94347 q^{16} -1.00000 q^{17} -2.17906 q^{18} +2.38202 q^{19} -5.80432 q^{20} -2.69680 q^{21} -8.42877 q^{22} +6.54103 q^{23} +1.63058 q^{24} -0.539577 q^{25} +6.44446 q^{26} -1.00000 q^{27} +7.41160 q^{28} -0.373077 q^{29} -4.60211 q^{30} +0.0561826 q^{31} +7.49608 q^{32} -3.86808 q^{33} +2.17906 q^{34} -5.69556 q^{35} +2.74829 q^{36} +2.78351 q^{37} -5.19056 q^{38} +2.95745 q^{39} +3.44373 q^{40} -4.44920 q^{41} +5.87648 q^{42} -8.21989 q^{43} +10.6306 q^{44} -2.11197 q^{45} -14.2533 q^{46} +9.32591 q^{47} +1.94347 q^{48} +0.272727 q^{49} +1.17577 q^{50} +1.00000 q^{51} -8.12794 q^{52} +4.89515 q^{53} +2.17906 q^{54} -8.16927 q^{55} -4.39734 q^{56} -2.38202 q^{57} +0.812957 q^{58} +8.24937 q^{59} +5.80432 q^{60} -11.0605 q^{61} -0.122425 q^{62} +2.69680 q^{63} -12.4475 q^{64} +6.24605 q^{65} +8.42877 q^{66} -3.51722 q^{67} -2.74829 q^{68} -6.54103 q^{69} +12.4110 q^{70} -12.0755 q^{71} -1.63058 q^{72} -3.35786 q^{73} -6.06544 q^{74} +0.539577 q^{75} +6.54650 q^{76} +10.4314 q^{77} -6.44446 q^{78} -9.02614 q^{79} +4.10454 q^{80} +1.00000 q^{81} +9.69506 q^{82} -13.8275 q^{83} -7.41160 q^{84} +2.11197 q^{85} +17.9116 q^{86} +0.373077 q^{87} -6.30721 q^{88} +4.41269 q^{89} +4.60211 q^{90} -7.97565 q^{91} +17.9767 q^{92} -0.0561826 q^{93} -20.3217 q^{94} -5.03076 q^{95} -7.49608 q^{96} +17.3963 q^{97} -0.594287 q^{98} +3.86808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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