LMFDB - Embedded newform 8007.2.a.f.1.32

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			1.32
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)

$f(q)$	$=$	$q+0.941352 q^{2} -1.00000 q^{3} -1.11386 q^{4} -2.91330 q^{5} -0.941352 q^{6} -1.76114 q^{7} -2.93124 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.941352 q^{2} -1.00000 q^{3} -1.11386 q^{4} -2.91330 q^{5} -0.941352 q^{6} -1.76114 q^{7} -2.93124 q^{8} +1.00000 q^{9} -2.74244 q^{10} -4.88957 q^{11} +1.11386 q^{12} -2.39917 q^{13} -1.65785 q^{14} +2.91330 q^{15} -0.531613 q^{16} -1.00000 q^{17} +0.941352 q^{18} +6.33142 q^{19} +3.24500 q^{20} +1.76114 q^{21} -4.60281 q^{22} +3.09254 q^{23} +2.93124 q^{24} +3.48733 q^{25} -2.25846 q^{26} -1.00000 q^{27} +1.96165 q^{28} +5.08146 q^{29} +2.74244 q^{30} +4.97869 q^{31} +5.36204 q^{32} +4.88957 q^{33} -0.941352 q^{34} +5.13072 q^{35} -1.11386 q^{36} +5.58869 q^{37} +5.96010 q^{38} +2.39917 q^{39} +8.53958 q^{40} +3.89049 q^{41} +1.65785 q^{42} +1.41729 q^{43} +5.44628 q^{44} -2.91330 q^{45} +2.91117 q^{46} -3.41885 q^{47} +0.531613 q^{48} -3.89840 q^{49} +3.28281 q^{50} +1.00000 q^{51} +2.67233 q^{52} -5.53042 q^{53} -0.941352 q^{54} +14.2448 q^{55} +5.16230 q^{56} -6.33142 q^{57} +4.78344 q^{58} +8.82698 q^{59} -3.24500 q^{60} -10.1434 q^{61} +4.68670 q^{62} -1.76114 q^{63} +6.11079 q^{64} +6.98951 q^{65} +4.60281 q^{66} -7.53148 q^{67} +1.11386 q^{68} -3.09254 q^{69} +4.82982 q^{70} +9.14600 q^{71} -2.93124 q^{72} -13.4556 q^{73} +5.26092 q^{74} -3.48733 q^{75} -7.05229 q^{76} +8.61120 q^{77} +2.25846 q^{78} +1.67678 q^{79} +1.54875 q^{80} +1.00000 q^{81} +3.66232 q^{82} +3.13668 q^{83} -1.96165 q^{84} +2.91330 q^{85} +1.33417 q^{86} -5.08146 q^{87} +14.3325 q^{88} +3.90643 q^{89} -2.74244 q^{90} +4.22526 q^{91} -3.44464 q^{92} -4.97869 q^{93} -3.21834 q^{94} -18.4454 q^{95} -5.36204 q^{96} +12.4025 q^{97} -3.66977 q^{98} -4.88957 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.941352	0.665637	0.332818	−	0.942991i	$-0.392000\pi$
$2$	0.941352	0.665637	0.332818	+	0.942991i	$0.392000\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.11386	−0.556928
$5$	−2.91330	−1.30287	−0.651434	−	0.758705i	$-0.725832\pi$
$5$	−2.91330	−1.30287	−0.651434	+	0.758705i	$0.725832\pi$
$6$	−0.941352	−0.384305
$7$	−1.76114	−0.665647	−0.332823	−	0.942989i	$-0.608001\pi$
$7$	−1.76114	−0.665647	−0.332823	+	0.942989i	$0.608001\pi$
$8$	−2.93124	−1.03635
$9$	1.00000	0.333333
$10$	−2.74244	−0.867237
$11$	−4.88957	−1.47426	−0.737131	−	0.675750i	$-0.763820\pi$
$11$	−4.88957	−1.47426	−0.737131	+	0.675750i	$0.763820\pi$
$12$	1.11386	0.321543
$13$	−2.39917	−0.665410	−0.332705	−	0.943031i	$-0.607961\pi$
$13$	−2.39917	−0.665410	−0.332705	+	0.943031i	$0.607961\pi$
$14$	−1.65785	−0.443079
$15$	2.91330	0.752212
$16$	−0.531613	−0.132903
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.941352	0.221879
$19$	6.33142	1.45253	0.726264	−	0.687416i	$-0.241255\pi$
$19$	6.33142	1.45253	0.726264	+	0.687416i	$0.241255\pi$
$20$	3.24500	0.725604
$21$	1.76114	0.384311
$22$	−4.60281	−0.981323
$23$	3.09254	0.644839	0.322420	−	0.946597i	$-0.395504\pi$
$23$	3.09254	0.644839	0.322420	+	0.946597i	$0.395504\pi$
$24$	2.93124	0.598336
$25$	3.48733	0.697467
$26$	−2.25846	−0.442921
$27$	−1.00000	−0.192450
$28$	1.96165	0.370717
$29$	5.08146	0.943603	0.471801	−	0.881705i	$-0.343604\pi$
$29$	5.08146	0.943603	0.471801	+	0.881705i	$0.343604\pi$
$30$	2.74244	0.500700
$31$	4.97869	0.894198	0.447099	−	0.894484i	$-0.352457\pi$
$31$	4.97869	0.894198	0.447099	+	0.894484i	$0.352457\pi$
$32$	5.36204	0.947883
$33$	4.88957	0.851166
$34$	−0.941352	−0.161441
$35$	5.13072	0.867250
$36$	−1.11386	−0.185643
$37$	5.58869	0.918774	0.459387	−	0.888236i	$-0.348069\pi$
$37$	5.58869	0.918774	0.459387	+	0.888236i	$0.348069\pi$
$38$	5.96010	0.966856
$39$	2.39917	0.384175
$40$	8.53958	1.35023
$41$	3.89049	0.607592	0.303796	−	0.952737i	$-0.401746\pi$
$41$	3.89049	0.607592	0.303796	+	0.952737i	$0.401746\pi$
$42$	1.65785	0.255812
$43$	1.41729	0.216135	0.108068	−	0.994144i	$-0.465534\pi$
$43$	1.41729	0.216135	0.108068	+	0.994144i	$0.465534\pi$
$44$	5.44628	0.821058
$45$	−2.91330	−0.434290
$46$	2.91117	0.429228
$47$	−3.41885	−0.498691	−0.249345	−	0.968415i	$-0.580215\pi$
$47$	−3.41885	−0.498691	−0.249345	+	0.968415i	$0.580215\pi$
$48$	0.531613	0.0767317
$49$	−3.89840	−0.556914
$50$	3.28281	0.464259
$51$	1.00000	0.140028
$52$	2.67233	0.370586
$53$	−5.53042	−0.759662	−0.379831	−	0.925056i	$-0.624018\pi$
$53$	−5.53042	−0.759662	−0.379831	+	0.925056i	$0.624018\pi$
$54$	−0.941352	−0.128102
$55$	14.2448	1.92077
$56$	5.16230	0.689842
$57$	−6.33142	−0.838617
$58$	4.78344	0.628097
$59$	8.82698	1.14917	0.574587	−	0.818443i	$-0.305163\pi$
$59$	8.82698	1.14917	0.574587	+	0.818443i	$0.305163\pi$
$60$	−3.24500	−0.418928
$61$	−10.1434	−1.29873	−0.649366	−	0.760476i	$-0.724966\pi$
$61$	−10.1434	−1.29873	−0.649366	+	0.760476i	$0.724966\pi$
$62$	4.68670	0.595211
$63$	−1.76114	−0.221882
$64$	6.11079	0.763849
$65$	6.98951	0.866942
$66$	4.60281	0.566567
$67$	−7.53148	−0.920117	−0.460058	−	0.887889i	$-0.652171\pi$
$67$	−7.53148	−0.920117	−0.460058	+	0.887889i	$0.652171\pi$
$68$	1.11386	0.135075
$69$	−3.09254	−0.372298
$70$	4.82982	0.577273
$71$	9.14600	1.08543	0.542715	−	0.839917i	$-0.317396\pi$
$71$	9.14600	1.08543	0.542715	+	0.839917i	$0.317396\pi$
$72$	−2.93124	−0.345449
$73$	−13.4556	−1.57485	−0.787427	−	0.616408i	$-0.788587\pi$
$73$	−13.4556	−1.57485	−0.787427	+	0.616408i	$0.788587\pi$
$74$	5.26092	0.611570
$75$	−3.48733	−0.402683
$76$	−7.05229	−0.808953
$77$	8.61120	0.981338
$78$	2.25846	0.255721
$79$	1.67678	0.188653	0.0943264	−	0.995541i	$-0.469930\pi$
$79$	1.67678	0.188653	0.0943264	+	0.995541i	$0.469930\pi$
$80$	1.54875	0.173155
$81$	1.00000	0.111111
$82$	3.66232	0.404436
$83$	3.13668	0.344295	0.172148	−	0.985071i	$-0.444929\pi$
$83$	3.13668	0.344295	0.172148	+	0.985071i	$0.444929\pi$
$84$	−1.96165	−0.214034
$85$	2.91330	0.315992
$86$	1.33417	0.143867
$87$	−5.08146	−0.544789
$88$	14.3325	1.52785
$89$	3.90643	0.414081	0.207040	−	0.978332i	$-0.433617\pi$
$89$	3.90643	0.414081	0.207040	+	0.978332i	$0.433617\pi$
$90$	−2.74244	−0.289079
$91$	4.22526	0.442928
$92$	−3.44464	−0.359129
$93$	−4.97869	−0.516266
$94$	−3.21834	−0.331947
$95$	−18.4454	−1.89245
$96$	−5.36204	−0.547260
$97$	12.4025	1.25929	0.629644	−	0.776884i	$-0.283201\pi$
$97$	12.4025	1.25929	0.629644	+	0.776884i	$0.283201\pi$
$98$	−3.66977	−0.370703
$99$	−4.88957	−0.491421
$100$	−3.88439	−0.388439
$101$	19.6558	1.95583	0.977914	−	0.209007i	$-0.0670230\pi$
$101$	19.6558	1.95583	0.977914	+	0.209007i	$0.0670230\pi$
$102$	0.941352	0.0932078
$103$	−8.10501	−0.798611	−0.399305	−	0.916818i	$-0.630749\pi$
$103$	−8.10501	−0.798611	−0.399305	+	0.916818i	$0.630749\pi$
$104$	7.03253	0.689597
$105$	−5.13072	−0.500707
$106$	−5.20607	−0.505659
$107$	−18.1965	−1.75912	−0.879560	−	0.475788i	$-0.842163\pi$
$107$	−18.1965	−1.75912	−0.879560	+	0.475788i	$0.842163\pi$
$108$	1.11386	0.107181
$109$	−6.36457	−0.609615	−0.304808	−	0.952414i	$-0.598592\pi$
$109$	−6.36457	−0.609615	−0.304808	+	0.952414i	$0.598592\pi$
$110$	13.4094	1.27853
$111$	−5.58869	−0.530455
$112$	0.936242	0.0884666
$113$	4.11886	0.387470	0.193735	−	0.981054i	$-0.437940\pi$
$113$	4.11886	0.387470	0.193735	+	0.981054i	$0.437940\pi$
$114$	−5.96010	−0.558214
$115$	−9.00951	−0.840141
$116$	−5.66001	−0.525519
$117$	−2.39917	−0.221803
$118$	8.30929	0.764933
$119$	1.76114	0.161443
$120$	−8.53958	−0.779553
$121$	12.9079	1.17345
$122$	−9.54854	−0.864484
$123$	−3.89049	−0.350794
$124$	−5.54554	−0.498004
$125$	4.40685	0.394161
$126$	−1.65785	−0.147693
$127$	−18.1323	−1.60899	−0.804493	−	0.593962i	$-0.797563\pi$
$127$	−18.1323	−1.60899	−0.804493	+	0.593962i	$0.797563\pi$
$128$	−4.97167	−0.439437
$129$	−1.41729	−0.124786
$130$	6.57959	0.577068
$131$	−7.50153	−0.655412	−0.327706	−	0.944780i	$-0.606276\pi$
$131$	−7.50153	−0.655412	−0.327706	+	0.944780i	$0.606276\pi$
$132$	−5.44628	−0.474038
$133$	−11.1505	−0.966870
$134$	−7.08977	−0.612463
$135$	2.91330	0.250737
$136$	2.93124	0.251351
$137$	9.16605	0.783109	0.391554	−	0.920155i	$-0.371937\pi$
$137$	9.16605	0.783109	0.391554	+	0.920155i	$0.371937\pi$
$138$	−2.91117	−0.247815
$139$	−14.3118	−1.21391	−0.606956	−	0.794735i	$-0.707610\pi$
$139$	−14.3118	−1.21391	−0.606956	+	0.794735i	$0.707610\pi$
$140$	−5.71489	−0.482996
$141$	3.41885	0.287919
$142$	8.60961	0.722502
$143$	11.7309	0.980989
$144$	−0.531613	−0.0443011
$145$	−14.8038	−1.22939
$146$	−12.6664	−1.04828
$147$	3.89840	0.321535
$148$	−6.22499	−0.511691
$149$	3.10954	0.254743	0.127372	−	0.991855i	$-0.459346\pi$
$149$	3.10954	0.254743	0.127372	+	0.991855i	$0.459346\pi$
$150$	−3.28281	−0.268040
$151$	−5.37928	−0.437760	−0.218880	−	0.975752i	$-0.570240\pi$
$151$	−5.37928	−0.437760	−0.218880	+	0.975752i	$0.570240\pi$
$152$	−18.5589	−1.50532
$153$	−1.00000	−0.0808452
$154$	8.10618	0.653214
$155$	−14.5044	−1.16502
$156$	−2.67233	−0.213958
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	1.57844	0.125574
$159$	5.53042	0.438591
$160$	−15.6212	−1.23497
$161$	−5.44638	−0.429235
$162$	0.941352	0.0739596
$163$	9.43076	0.738674	0.369337	−	0.929296i	$-0.379585\pi$
$163$	9.43076	0.738674	0.369337	+	0.929296i	$0.379585\pi$
$164$	−4.33344	−0.338385
$165$	−14.2448	−1.10896
$166$	2.95272	0.229175
$167$	−21.2647	−1.64551	−0.822757	−	0.568393i	$-0.807565\pi$
$167$	−21.2647	−1.64551	−0.822757	+	0.568393i	$0.807565\pi$
$168$	−5.16230	−0.398280
$169$	−7.24398	−0.557229
$170$	2.74244	0.210336
$171$	6.33142	0.484176
$172$	−1.57866	−0.120372
$173$	15.0481	1.14409	0.572044	−	0.820223i	$-0.306151\pi$
$173$	15.0481	1.14409	0.572044	+	0.820223i	$0.306151\pi$
$174$	−4.78344	−0.362632
$175$	−6.14167	−0.464267
$176$	2.59936	0.195934
$177$	−8.82698	−0.663476
$178$	3.67733	0.275627
$179$	23.4409	1.75205	0.876027	−	0.482263i	$-0.160185\pi$
$179$	23.4409	1.75205	0.876027	+	0.482263i	$0.160185\pi$
$180$	3.24500	0.241868
$181$	17.6018	1.30833	0.654165	−	0.756352i	$-0.273020\pi$
$181$	17.6018	1.30833	0.654165	+	0.756352i	$0.273020\pi$
$182$	3.97746	0.294829
$183$	10.1434	0.749824
$184$	−9.06496	−0.668278
$185$	−16.2815	−1.19704
$186$	−4.68670	−0.343645
$187$	4.88957	0.357561
$188$	3.80811	0.277735
$189$	1.76114	0.128104
$190$	−17.3636	−1.25969
$191$	12.0554	0.872295	0.436148	−	0.899875i	$-0.356343\pi$
$191$	12.0554	0.872295	0.436148	+	0.899875i	$0.356343\pi$
$192$	−6.11079	−0.441008
$193$	−6.84319	−0.492584	−0.246292	−	0.969196i	$-0.579212\pi$
$193$	−6.84319	−0.492584	−0.246292	+	0.969196i	$0.579212\pi$
$194$	11.6752	0.838228
$195$	−6.98951	−0.500529
$196$	4.34226	0.310161
$197$	−10.4592	−0.745184	−0.372592	−	0.927995i	$-0.621531\pi$
$197$	−10.4592	−0.745184	−0.372592	+	0.927995i	$0.621531\pi$
$198$	−4.60281	−0.327108
$199$	−1.97722	−0.140161	−0.0700807	−	0.997541i	$-0.522326\pi$
$199$	−1.97722	−0.140161	−0.0700807	+	0.997541i	$0.522326\pi$
$200$	−10.2222	−0.722819
$201$	7.53148	0.531230
$202$	18.5031	1.30187
$203$	−8.94914	−0.628106
$204$	−1.11386	−0.0779855
$205$	−11.3342	−0.791613
$206$	−7.62967	−0.531585
$207$	3.09254	0.214946
$208$	1.27543	0.0884351
$209$	−30.9580	−2.14141
$210$	−4.82982	−0.333289
$211$	22.2420	1.53120	0.765600	−	0.643317i	$-0.222442\pi$
$211$	22.2420	1.53120	0.765600	+	0.643317i	$0.222442\pi$
$212$	6.16009	0.423077
$213$	−9.14600	−0.626674
$214$	−17.1293	−1.17093
$215$	−4.12900	−0.281596
$216$	2.93124	0.199445
$217$	−8.76814	−0.595220
$218$	−5.99130	−0.405782
$219$	13.4556	0.909242
$220$	−15.8667	−1.06973
$221$	2.39917	0.161386
$222$	−5.26092	−0.353090
$223$	−14.8466	−0.994201	−0.497101	−	0.867693i	$-0.665602\pi$
$223$	−14.8466	−0.994201	−0.497101	+	0.867693i	$0.665602\pi$
$224$	−9.44327	−0.630955
$225$	3.48733	0.232489
$226$	3.87730	0.257914
$227$	1.35159	0.0897083	0.0448541	−	0.998994i	$-0.485718\pi$
$227$	1.35159	0.0897083	0.0448541	+	0.998994i	$0.485718\pi$
$228$	7.05229	0.467049
$229$	9.70817	0.641534	0.320767	−	0.947158i	$-0.396059\pi$
$229$	9.70817	0.641534	0.320767	+	0.947158i	$0.396059\pi$
$230$	−8.48112	−0.559228
$231$	−8.61120	−0.566576
$232$	−14.8949	−0.977901
$233$	14.2832	0.935722	0.467861	−	0.883802i	$-0.345025\pi$
$233$	14.2832	0.935722	0.467861	+	0.883802i	$0.345025\pi$
$234$	−2.25846	−0.147640
$235$	9.96015	0.649728
$236$	−9.83198	−0.640007
$237$	−1.67678	−0.108919
$238$	1.65785	0.107462
$239$	−3.50372	−0.226637	−0.113318	−	0.993559i	$-0.536148\pi$
$239$	−3.50372	−0.226637	−0.113318	+	0.993559i	$0.536148\pi$
$240$	−1.54875	−0.0999713
$241$	−21.0468	−1.35574	−0.677871	−	0.735180i	$-0.737097\pi$
$241$	−21.0468	−1.35574	−0.677871	+	0.735180i	$0.737097\pi$
$242$	12.1509	0.781090
$243$	−1.00000	−0.0641500
$244$	11.2983	0.723301
$245$	11.3572	0.725586
$246$	−3.66232	−0.233501
$247$	−15.1902	−0.966527
$248$	−14.5937	−0.926701
$249$	−3.13668	−0.198779
$250$	4.14840	0.262368
$251$	18.0563	1.13970	0.569851	−	0.821748i	$-0.307001\pi$
$251$	18.0563	1.13970	0.569851	+	0.821748i	$0.307001\pi$
$252$	1.96165	0.123572
$253$	−15.1212	−0.950662
$254$	−17.0689	−1.07100
$255$	−2.91330	−0.182438
$256$	−16.9017	−1.05635
$257$	29.4736	1.83851	0.919255	−	0.393663i	$-0.128792\pi$
$257$	29.4736	1.83851	0.919255	+	0.393663i	$0.128792\pi$
$258$	−1.33417	−0.0830619
$259$	−9.84243	−0.611579
$260$	−7.78531	−0.482824
$261$	5.08146	0.314534
$262$	−7.06158	−0.436266
$263$	30.4681	1.87874	0.939372	−	0.342901i	$-0.111409\pi$
$263$	30.4681	1.87874	0.939372	+	0.342901i	$0.111409\pi$
$264$	−14.3325	−0.882104
$265$	16.1118	0.989740
$266$	−10.4965	−0.643584
$267$	−3.90643	−0.239070
$268$	8.38898	0.512439
$269$	−15.1101	−0.921279	−0.460639	−	0.887587i	$-0.652380\pi$
$269$	−15.1101	−0.921279	−0.460639	+	0.887587i	$0.652380\pi$
$270$	2.74244	0.166900
$271$	23.2237	1.41074	0.705371	−	0.708838i	$-0.250780\pi$
$271$	23.2237	1.41074	0.705371	+	0.708838i	$0.250780\pi$
$272$	0.531613	0.0322338
$273$	−4.22526	−0.255725
$274$	8.62848	0.521266
$275$	−17.0516	−1.02825
$276$	3.44464	0.207343
$277$	−2.13483	−0.128269	−0.0641347	−	0.997941i	$-0.520429\pi$
$277$	−2.13483	−0.128269	−0.0641347	+	0.997941i	$0.520429\pi$
$278$	−13.4725	−0.808025
$279$	4.97869	0.298066
$280$	−15.0394	−0.898773
$281$	17.4436	1.04060	0.520298	−	0.853985i	$-0.325821\pi$
$281$	17.4436	1.04060	0.520298	+	0.853985i	$0.325821\pi$
$282$	3.21834	0.191649
$283$	−9.09374	−0.540567	−0.270283	−	0.962781i	$-0.587117\pi$
$283$	−9.09374	−0.540567	−0.270283	+	0.962781i	$0.587117\pi$
$284$	−10.1873	−0.604507
$285$	18.4454	1.09261
$286$	11.0429	0.652982
$287$	−6.85168	−0.404442
$288$	5.36204	0.315961
$289$	1.00000	0.0588235
$290$	−13.9356	−0.818327
$291$	−12.4025	−0.727050
$292$	14.9876	0.877080
$293$	−0.0560033	−0.00327175	−0.00163587	−	0.999999i	$-0.500521\pi$
$293$	−0.0560033	−0.00327175	−0.00163587	+	0.999999i	$0.500521\pi$
$294$	3.66977	0.214025
$295$	−25.7157	−1.49722
$296$	−16.3818	−0.952170
$297$	4.88957	0.283722
$298$	2.92717	0.169566
$299$	−7.41953	−0.429082
$300$	3.88439	0.224265
$301$	−2.49604	−0.143870
$302$	−5.06380	−0.291389
$303$	−19.6558	−1.12920
$304$	−3.36586	−0.193046
$305$	29.5509	1.69208
$306$	−0.941352	−0.0538135
$307$	−11.5596	−0.659741	−0.329871	−	0.944026i	$-0.607005\pi$
$307$	−11.5596	−0.659741	−0.329871	+	0.944026i	$0.607005\pi$
$308$	−9.59164	−0.546534
$309$	8.10501	0.461078
$310$	−13.6538	−0.775482
$311$	−34.0306	−1.92970	−0.964849	−	0.262804i	$-0.915353\pi$
$311$	−34.0306	−1.92970	−0.964849	+	0.262804i	$0.915353\pi$
$312$	−7.03253	−0.398139
$313$	−28.8680	−1.63172	−0.815858	−	0.578253i	$-0.803735\pi$
$313$	−28.8680	−1.63172	−0.815858	+	0.578253i	$0.803735\pi$
$314$	−0.941352	−0.0531236
$315$	5.13072	0.289083
$316$	−1.86770	−0.105066
$317$	14.2126	0.798261	0.399130	−	0.916894i	$-0.369312\pi$
$317$	14.2126	0.798261	0.399130	+	0.916894i	$0.369312\pi$
$318$	5.20607	0.291942
$319$	−24.8462	−1.39112
$320$	−17.8026	−0.995195
$321$	18.1965	1.01563
$322$	−5.12696	−0.285715
$323$	−6.33142	−0.352290
$324$	−1.11386	−0.0618809
$325$	−8.36671	−0.464101
$326$	8.87766	0.491688
$327$	6.36457	0.351962
$328$	−11.4039	−0.629677
$329$	6.02106	0.331952
$330$	−13.4094	−0.738162
$331$	7.38597	0.405970	0.202985	−	0.979182i	$-0.434936\pi$
$331$	7.38597	0.405970	0.202985	+	0.979182i	$0.434936\pi$
$332$	−3.49381	−0.191748
$333$	5.58869	0.306258
$334$	−20.0176	−1.09531
$335$	21.9415	1.19879
$336$	−0.936242	−0.0510762
$337$	−9.42884	−0.513622	−0.256811	−	0.966462i	$-0.582672\pi$
$337$	−9.42884	−0.513622	−0.256811	+	0.966462i	$0.582672\pi$
$338$	−6.81914	−0.370912
$339$	−4.11886	−0.223706
$340$	−3.24500	−0.175985
$341$	−24.3437	−1.31828
$342$	5.96010	0.322285
$343$	19.1936	1.03635
$344$	−4.15442	−0.223991
$345$	9.00951	0.485055
$346$	14.1656	0.761547
$347$	−19.4280	−1.04295	−0.521474	−	0.853267i	$-0.674618\pi$
$347$	−19.4280	−1.04295	−0.521474	+	0.853267i	$0.674618\pi$
$348$	5.66001	0.303408
$349$	20.0175	1.07151	0.535757	−	0.844372i	$-0.320026\pi$
$349$	20.0175	1.07151	0.535757	+	0.844372i	$0.320026\pi$
$350$	−5.78147	−0.309033
$351$	2.39917	0.128058
$352$	−26.2181	−1.39743
$353$	2.03912	0.108532	0.0542658	−	0.998527i	$-0.482718\pi$
$353$	2.03912	0.108532	0.0542658	+	0.998527i	$0.482718\pi$
$354$	−8.30929	−0.441634
$355$	−26.6451	−1.41417
$356$	−4.35120	−0.230613
$357$	−1.76114	−0.0932092
$358$	22.0661	1.16623
$359$	−11.6934	−0.617156	−0.308578	−	0.951199i	$-0.599853\pi$
$359$	−11.6934	−0.617156	−0.308578	+	0.951199i	$0.599853\pi$
$360$	8.53958	0.450075
$361$	21.0869	1.10984
$362$	16.5695	0.870873
$363$	−12.9079	−0.677491
$364$	−4.70634	−0.246679
$365$	39.2001	2.05183
$366$	9.54854	0.499110
$367$	−5.17491	−0.270128	−0.135064	−	0.990837i	$-0.543124\pi$
$367$	−5.17491	−0.270128	−0.135064	+	0.990837i	$0.543124\pi$
$368$	−1.64403	−0.0857012
$369$	3.89049	0.202531
$370$	−15.3267	−0.796795
$371$	9.73982	0.505666
$372$	5.54554	0.287523
$373$	15.9628	0.826524	0.413262	−	0.910612i	$-0.364389\pi$
$373$	15.9628	0.826524	0.413262	+	0.910612i	$0.364389\pi$
$374$	4.60281	0.238006
$375$	−4.40685	−0.227569
$376$	10.0215	0.516817
$377$	−12.1913	−0.627883
$378$	1.65785	0.0852705
$379$	19.5606	1.00476	0.502381	−	0.864647i	$-0.332458\pi$
$379$	19.5606	1.00476	0.502381	+	0.864647i	$0.332458\pi$
$380$	20.5455	1.05396
$381$	18.1323	0.928948
$382$	11.3483	0.580632
$383$	−16.4324	−0.839657	−0.419829	−	0.907603i	$-0.637910\pi$
$383$	−16.4324	−0.839657	−0.419829	+	0.907603i	$0.637910\pi$
$384$	4.97167	0.253709
$385$	−25.0870	−1.27855
$386$	−6.44186	−0.327882
$387$	1.41729	0.0720450
$388$	−13.8147	−0.701333
$389$	−5.65157	−0.286546	−0.143273	−	0.989683i	$-0.545763\pi$
$389$	−5.65157	−0.286546	−0.143273	+	0.989683i	$0.545763\pi$
$390$	−6.57959	−0.333171
$391$	−3.09254	−0.156396
$392$	11.4271	0.577157
$393$	7.50153	0.378402
$394$	−9.84575	−0.496021
$395$	−4.88498	−0.245790
$396$	5.44628	0.273686
$397$	−17.9417	−0.900470	−0.450235	−	0.892910i	$-0.648660\pi$
$397$	−17.9417	−0.900470	−0.450235	+	0.892910i	$0.648660\pi$
$398$	−1.86126	−0.0932965
$399$	11.1505	0.558223
$400$	−1.85391	−0.0926956
$401$	−7.57649	−0.378352	−0.189176	−	0.981943i	$-0.560582\pi$
$401$	−7.57649	−0.378352	−0.189176	+	0.981943i	$0.560582\pi$
$402$	7.08977	0.353606
$403$	−11.9447	−0.595009
$404$	−21.8938	−1.08926
$405$	−2.91330	−0.144763
$406$	−8.42429	−0.418090
$407$	−27.3263	−1.35451
$408$	−2.93124	−0.145118
$409$	−7.71869	−0.381665	−0.190832	−	0.981623i	$-0.561119\pi$
$409$	−7.71869	−0.381665	−0.190832	+	0.981623i	$0.561119\pi$
$410$	−10.6694	−0.526926
$411$	−9.16605	−0.452128
$412$	9.02782	0.444769
$413$	−15.5455	−0.764944
$414$	2.91117	0.143076
$415$	−9.13809	−0.448571
$416$	−12.8644	−0.630731
$417$	14.3118	0.700853
$418$	−29.1423	−1.42540
$419$	29.0791	1.42061	0.710303	−	0.703896i	$-0.248558\pi$
$419$	29.0791	1.42061	0.710303	+	0.703896i	$0.248558\pi$
$420$	5.71489	0.278858
$421$	−11.1478	−0.543308	−0.271654	−	0.962395i	$-0.587571\pi$
$421$	−11.1478	−0.543308	−0.271654	+	0.962395i	$0.587571\pi$
$422$	20.9375	1.01922
$423$	−3.41885	−0.166230
$424$	16.2110	0.787274
$425$	−3.48733	−0.169161
$426$	−8.60961	−0.417137
$427$	17.8639	0.864497
$428$	20.2683	0.979703
$429$	−11.7309	−0.566374
$430$	−3.88685	−0.187440
$431$	32.1924	1.55065	0.775326	−	0.631561i	$-0.217585\pi$
$431$	32.1924	1.55065	0.775326	+	0.631561i	$0.217585\pi$
$432$	0.531613	0.0255772
$433$	22.1845	1.06612	0.533059	−	0.846078i	$-0.321042\pi$
$433$	22.1845	1.06612	0.533059	+	0.846078i	$0.321042\pi$
$434$	−8.25391	−0.396200
$435$	14.8038	0.709789
$436$	7.08922	0.339512
$437$	19.5802	0.936647
$438$	12.6664	0.605225
$439$	32.0274	1.52859	0.764293	−	0.644869i	$-0.223088\pi$
$439$	32.0274	1.52859	0.764293	+	0.644869i	$0.223088\pi$
$440$	−41.7549	−1.99059
$441$	−3.89840	−0.185638
$442$	2.25846	0.107424
$443$	28.2636	1.34284	0.671422	−	0.741075i	$-0.265684\pi$
$443$	28.2636	1.34284	0.671422	+	0.741075i	$0.265684\pi$
$444$	6.22499	0.295425
$445$	−11.3806	−0.539493
$446$	−13.9759	−0.661777
$447$	−3.10954	−0.147076
$448$	−10.7619	−0.508453
$449$	−37.2522	−1.75804	−0.879019	−	0.476786i	$-0.841802\pi$
$449$	−37.2522	−1.75804	−0.879019	+	0.476786i	$0.841802\pi$
$450$	3.28281	0.154753
$451$	−19.0228	−0.895750
$452$	−4.58782	−0.215793
$453$	5.37928	0.252741
$454$	1.27232	0.0597131
$455$	−12.3095	−0.577077
$456$	18.5589	0.869100
$457$	20.3854	0.953589	0.476794	−	0.879015i	$-0.341799\pi$
$457$	20.3854	0.953589	0.476794	+	0.879015i	$0.341799\pi$
$458$	9.13881	0.427028
$459$	1.00000	0.0466760
$460$	10.0353	0.467898
$461$	−25.0140	−1.16502	−0.582509	−	0.812824i	$-0.697929\pi$
$461$	−25.0140	−1.16502	−0.582509	+	0.812824i	$0.697929\pi$
$462$	−8.10618	−0.377133
$463$	−1.30736	−0.0607582	−0.0303791	−	0.999538i	$-0.509671\pi$
$463$	−1.30736	−0.0607582	−0.0303791	+	0.999538i	$0.509671\pi$
$464$	−2.70137	−0.125408
$465$	14.5044	0.672626
$466$	13.4455	0.622850
$467$	10.9088	0.504797	0.252399	−	0.967623i	$-0.418781\pi$
$467$	10.9088	0.504797	0.252399	+	0.967623i	$0.418781\pi$
$468$	2.67233	0.123529
$469$	13.2640	0.612473
$470$	9.37601	0.432483
$471$	1.00000	0.0460776
$472$	−25.8739	−1.19094
$473$	−6.92996	−0.318640
$474$	−1.57844	−0.0725003
$475$	22.0798	1.01309
$476$	−1.96165	−0.0899121
$477$	−5.53042	−0.253221
$478$	−3.29823	−0.150858
$479$	−14.0791	−0.643291	−0.321645	−	0.946860i	$-0.604236\pi$
$479$	−14.0791	−0.643291	−0.321645	+	0.946860i	$0.604236\pi$
$480$	15.6212	0.713009
$481$	−13.4082	−0.611362
$482$	−19.8124	−0.902432
$483$	5.44638	0.247819
$484$	−14.3776	−0.653527
$485$	−36.1324	−1.64069
$486$	−0.941352	−0.0427006
$487$	−5.50370	−0.249397	−0.124698	−	0.992195i	$-0.539796\pi$
$487$	−5.50370	−0.249397	−0.124698	+	0.992195i	$0.539796\pi$
$488$	29.7328	1.34594
$489$	−9.43076	−0.426474
$490$	10.6911	0.482977
$491$	−0.659256	−0.0297518	−0.0148759	−	0.999889i	$-0.504735\pi$
$491$	−0.659256	−0.0297518	−0.0148759	+	0.999889i	$0.504735\pi$
$492$	4.33344	0.195367
$493$	−5.08146	−0.228857
$494$	−14.2993	−0.643355
$495$	14.2448	0.640257
$496$	−2.64673	−0.118842
$497$	−16.1073	−0.722513
$498$	−2.95272	−0.132314
$499$	−19.3147	−0.864645	−0.432323	−	0.901719i	$-0.642306\pi$
$499$	−19.3147	−0.864645	−0.432323	+	0.901719i	$0.642306\pi$
$500$	−4.90860	−0.219519
$501$	21.2647	0.950038
$502$	16.9973	0.758628
$503$	−23.6373	−1.05393	−0.526967	−	0.849886i	$-0.676671\pi$
$503$	−23.6373	−1.05393	−0.526967	+	0.849886i	$0.676671\pi$
$504$	5.16230	0.229947
$505$	−57.2634	−2.54819
$506$	−14.2344	−0.632795
$507$	7.24398	0.321717
$508$	20.1968	0.896089
$509$	26.2537	1.16368	0.581838	−	0.813304i	$-0.302334\pi$
$509$	26.2537	1.16368	0.581838	+	0.813304i	$0.302334\pi$
$510$	−2.74244	−0.121437
$511$	23.6971	1.04830
$512$	−5.96709	−0.263711
$513$	−6.33142	−0.279539
$514$	27.7450	1.22378
$515$	23.6124	1.04049
$516$	1.57866	0.0694966
$517$	16.7167	0.735201
$518$	−9.26520	−0.407089
$519$	−15.0481	−0.660540
$520$	−20.4879	−0.898454
$521$	−9.34762	−0.409527	−0.204763	−	0.978811i	$-0.565642\pi$
$521$	−9.34762	−0.409527	−0.204763	+	0.978811i	$0.565642\pi$
$522$	4.78344	0.209366
$523$	38.8854	1.70034	0.850170	−	0.526509i	$-0.176499\pi$
$523$	38.8854	1.70034	0.850170	+	0.526509i	$0.176499\pi$
$524$	8.35562	0.365017
$525$	6.14167	0.268044
$526$	28.6812	1.25056
$527$	−4.97869	−0.216875
$528$	−2.59936	−0.113123
$529$	−13.4362	−0.584183
$530$	15.1669	0.658807
$531$	8.82698	0.383058
$532$	12.4200	0.538477
$533$	−9.33394	−0.404298
$534$	−3.67733	−0.159134
$535$	53.0118	2.29190
$536$	22.0765	0.953561
$537$	−23.4409	−1.01155
$538$	−14.2239	−0.613237
$539$	19.0615	0.821038
$540$	−3.24500	−0.139643
$541$	−34.1261	−1.46720	−0.733598	−	0.679584i	$-0.762160\pi$
$541$	−34.1261	−1.46720	−0.733598	+	0.679584i	$0.762160\pi$
$542$	21.8617	0.939042
$543$	−17.6018	−0.755365
$544$	−5.36204	−0.229895
$545$	18.5419	0.794249
$546$	−3.97746	−0.170220
$547$	7.94965	0.339902	0.169951	−	0.985452i	$-0.445639\pi$
$547$	7.94965	0.339902	0.169951	+	0.985452i	$0.445639\pi$
$548$	−10.2097	−0.436135
$549$	−10.1434	−0.432911
$550$	−16.0515	−0.684440
$551$	32.1729	1.37061
$552$	9.06496	0.385830
$553$	−2.95304	−0.125576
$554$	−2.00962	−0.0853808
$555$	16.2815	0.691113
$556$	15.9413	0.676062
$557$	−41.5792	−1.76177	−0.880885	−	0.473331i	$-0.843051\pi$
$557$	−41.5792	−1.76177	−0.880885	+	0.473331i	$0.843051\pi$
$558$	4.68670	0.198404
$559$	−3.40033	−0.143818
$560$	−2.72756	−0.115260
$561$	−4.88957	−0.206438
$562$	16.4205	0.692659
$563$	18.3520	0.773446	0.386723	−	0.922196i	$-0.373607\pi$
$563$	18.3520	0.773446	0.386723	+	0.922196i	$0.373607\pi$
$564$	−3.80811	−0.160350
$565$	−11.9995	−0.504822
$566$	−8.56041	−0.359821
$567$	−1.76114	−0.0739607
$568$	−26.8091	−1.12488
$569$	−29.8361	−1.25079	−0.625397	−	0.780307i	$-0.715063\pi$
$569$	−29.8361	−1.25079	−0.625397	+	0.780307i	$0.715063\pi$
$570$	17.3636	0.727280
$571$	−12.2002	−0.510561	−0.255280	−	0.966867i	$-0.582168\pi$
$571$	−12.2002	−0.510561	−0.255280	+	0.966867i	$0.582168\pi$
$572$	−13.0666	−0.546340
$573$	−12.0554	−0.503620
$574$	−6.44984	−0.269211
$575$	10.7847	0.449754
$576$	6.11079	0.254616
$577$	−22.7635	−0.947658	−0.473829	−	0.880617i	$-0.657128\pi$
$577$	−22.7635	−0.947658	−0.473829	+	0.880617i	$0.657128\pi$
$578$	0.941352	0.0391551
$579$	6.84319	0.284393
$580$	16.4893	0.684682
$581$	−5.52411	−0.229179
$582$	−11.6752	−0.483951
$583$	27.0414	1.11994
$584$	39.4414	1.63210
$585$	6.98951	0.288981
$586$	−0.0527188	−0.00217779
$587$	−20.8736	−0.861545	−0.430773	−	0.902461i	$-0.641759\pi$
$587$	−20.8736	−0.861545	−0.430773	+	0.902461i	$0.641759\pi$
$588$	−4.34226	−0.179072
$589$	31.5222	1.29885
$590$	−24.2075	−0.996607
$591$	10.4592	0.430232
$592$	−2.97102	−0.122108
$593$	−12.8052	−0.525847	−0.262923	−	0.964817i	$-0.584687\pi$
$593$	−12.8052	−0.525847	−0.262923	+	0.964817i	$0.584687\pi$
$594$	4.60281	0.188856
$595$	−5.13072	−0.210339
$596$	−3.46358	−0.141874
$597$	1.97722	0.0809222
$598$	−6.98439	−0.285613
$599$	−23.2934	−0.951742	−0.475871	−	0.879515i	$-0.657867\pi$
$599$	−23.2934	−0.951742	−0.475871	+	0.879515i	$0.657867\pi$
$600$	10.2222	0.417319
$601$	−21.3993	−0.872895	−0.436448	−	0.899730i	$-0.643764\pi$
$601$	−21.3993	−0.872895	−0.436448	+	0.899730i	$0.643764\pi$
$602$	−2.34966	−0.0957648
$603$	−7.53148	−0.306706
$604$	5.99175	0.243801
$605$	−37.6047	−1.52885
$606$	−18.5031	−0.751635
$607$	−36.6012	−1.48560	−0.742798	−	0.669515i	$-0.766502\pi$
$607$	−36.6012	−1.48560	−0.742798	+	0.669515i	$0.766502\pi$
$608$	33.9493	1.37683
$609$	8.94914	0.362637
$610$	27.8178	1.12631
$611$	8.20240	0.331834
$612$	1.11386	0.0450250
$613$	−10.3878	−0.419559	−0.209780	−	0.977749i	$-0.567275\pi$
$613$	−10.3878	−0.419559	−0.209780	+	0.977749i	$0.567275\pi$
$614$	−10.8817	−0.439148
$615$	11.3342	0.457038
$616$	−25.2415	−1.01701
$617$	−43.1873	−1.73865	−0.869327	−	0.494237i	$-0.835448\pi$
$617$	−43.1873	−1.73865	−0.869327	+	0.494237i	$0.835448\pi$
$618$	7.62967	0.306910
$619$	−2.47567	−0.0995055	−0.0497528	−	0.998762i	$-0.515843\pi$
$619$	−2.47567	−0.0995055	−0.0497528	+	0.998762i	$0.515843\pi$
$620$	16.1558	0.648834
$621$	−3.09254	−0.124099
$622$	−32.0348	−1.28448
$623$	−6.87975	−0.275632
$624$	−1.27543	−0.0510580
$625$	−30.2752	−1.21101
$626$	−27.1749	−1.08613
$627$	30.9580	1.23634
$628$	1.11386	0.0444477
$629$	−5.58869	−0.222836
$630$	4.82982	0.192424
$631$	−12.6402	−0.503199	−0.251600	−	0.967831i	$-0.580957\pi$
$631$	−12.6402	−0.503199	−0.251600	+	0.967831i	$0.580957\pi$
$632$	−4.91505	−0.195510
$633$	−22.2420	−0.884039
$634$	13.3791	0.531351
$635$	52.8250	2.09630
$636$	−6.16009	−0.244264
$637$	9.35293	0.370577
$638$	−23.3890	−0.925979
$639$	9.14600	0.361810
$640$	14.4840	0.572529
$641$	−35.5654	−1.40475	−0.702375	−	0.711807i	$-0.747877\pi$
$641$	−35.5654	−1.40475	−0.702375	+	0.711807i	$0.747877\pi$
$642$	17.1293	0.676039
$643$	3.17037	0.125027	0.0625137	−	0.998044i	$-0.480088\pi$
$643$	3.17037	0.125027	0.0625137	+	0.998044i	$0.480088\pi$
$644$	6.06649	0.239053
$645$	4.12900	0.162579
$646$	−5.96010	−0.234497
$647$	22.7973	0.896253	0.448127	−	0.893970i	$-0.352091\pi$
$647$	22.7973	0.896253	0.448127	+	0.893970i	$0.352091\pi$
$648$	−2.93124	−0.115150
$649$	−43.1602	−1.69418
$650$	−7.87602	−0.308923
$651$	8.76814	0.343651
$652$	−10.5045	−0.411388
$653$	−27.3049	−1.06852	−0.534261	−	0.845319i	$-0.679410\pi$
$653$	−27.3049	−1.06852	−0.534261	+	0.845319i	$0.679410\pi$
$654$	5.99130	0.234279
$655$	21.8542	0.853915
$656$	−2.06823	−0.0807509
$657$	−13.4556	−0.524951
$658$	5.66794	0.220959
$659$	−28.2346	−1.09987	−0.549933	−	0.835209i	$-0.685347\pi$
$659$	−28.2346	−1.09987	−0.549933	+	0.835209i	$0.685347\pi$
$660$	15.8667	0.617609
$661$	6.81184	0.264950	0.132475	−	0.991186i	$-0.457708\pi$
$661$	6.81184	0.264950	0.132475	+	0.991186i	$0.457708\pi$
$662$	6.95280	0.270228
$663$	−2.39917	−0.0931760
$664$	−9.19434	−0.356810
$665$	32.4848	1.25971
$666$	5.26092	0.203857
$667$	15.7146	0.608472
$668$	23.6858	0.916433
$669$	14.8466	0.574002
$670$	20.6547	0.797959
$671$	49.5970	1.91467
$672$	9.44327	0.364282
$673$	−0.0174381	−0.000672190 0	−0.000336095	−	1.00000i	$-0.500107\pi$
$673$	−0.0174381	−0.000672190 0	−0.000336095		1.00000i	$0.500107\pi$
$674$	−8.87586	−0.341885
$675$	−3.48733	−0.134228
$676$	8.06875	0.310337
$677$	6.08369	0.233815	0.116908	−	0.993143i	$-0.462702\pi$
$677$	6.08369	0.233815	0.116908	+	0.993143i	$0.462702\pi$
$678$	−3.87730	−0.148907
$679$	−21.8426	−0.838241
$680$	−8.53958	−0.327478
$681$	−1.35159	−0.0517931
$682$	−22.9160	−0.877497
$683$	−39.1783	−1.49911	−0.749557	−	0.661940i	$-0.769734\pi$
$683$	−39.1783	−1.49911	−0.749557	+	0.661940i	$0.769734\pi$
$684$	−7.05229	−0.269651
$685$	−26.7035	−1.02029
$686$	18.0679	0.689836
$687$	−9.70817	−0.370390
$688$	−0.753451	−0.0287250
$689$	13.2684	0.505487
$690$	8.48112	0.322871
$691$	−45.4727	−1.72986	−0.864931	−	0.501891i	$-0.832638\pi$
$691$	−45.4727	−1.72986	−0.864931	+	0.501891i	$0.832638\pi$
$692$	−16.7615	−0.637175
$693$	8.61120	0.327113
$694$	−18.2886	−0.694224
$695$	41.6947	1.58157
$696$	14.8949	0.564592
$697$	−3.89049	−0.147363
$698$	18.8435	0.713238
$699$	−14.2832	−0.540239
$700$	6.84093	0.258563
$701$	43.2405	1.63317	0.816585	−	0.577225i	$-0.195864\pi$
$701$	43.2405	1.63317	0.816585	+	0.577225i	$0.195864\pi$
$702$	2.25846	0.0852402
$703$	35.3843	1.33455
$704$	−29.8792	−1.12611
$705$	−9.96015	−0.375121
$706$	1.91953	0.0722426
$707$	−34.6166	−1.30189
$708$	9.83198	0.369508
$709$	−25.7634	−0.967566	−0.483783	−	0.875188i	$-0.660738\pi$
$709$	−25.7634	−0.967566	−0.483783	+	0.875188i	$0.660738\pi$
$710$	−25.0824	−0.941325
$711$	1.67678	0.0628843
$712$	−11.4507	−0.429132
$713$	15.3968	0.576614
$714$	−1.65785	−0.0620434
$715$	−34.1757	−1.27810
$716$	−26.1098	−0.975768
$717$	3.50372	0.130849
$718$	−11.0076	−0.410802
$719$	−37.4482	−1.39658	−0.698291	−	0.715814i	$-0.746056\pi$
$719$	−37.4482	−1.39658	−0.698291	+	0.715814i	$0.746056\pi$
$720$	1.54875	0.0577185
$721$	14.2740	0.531593
$722$	19.8502	0.738748
$723$	21.0468	0.782739
$724$	−19.6059	−0.728646
$725$	17.7207	0.658132
$726$	−12.1509	−0.450963
$727$	−25.1190	−0.931613	−0.465806	−	0.884887i	$-0.654236\pi$
$727$	−25.1190	−0.931613	−0.465806	+	0.884887i	$0.654236\pi$
$728$	−12.3852	−0.459028
$729$	1.00000	0.0370370
$730$	36.9011	1.36577
$731$	−1.41729	−0.0524204
$732$	−11.2983	−0.417598
$733$	−50.4468	−1.86330	−0.931648	−	0.363362i	$-0.881629\pi$
$733$	−50.4468	−1.86330	−0.931648	+	0.363362i	$0.881629\pi$
$734$	−4.87142	−0.179807
$735$	−11.3572	−0.418918
$736$	16.5823	0.611232
$737$	36.8257	1.35649
$738$	3.66232	0.134812
$739$	37.1419	1.36629	0.683143	−	0.730284i	$-0.260612\pi$
$739$	37.1419	1.36629	0.683143	+	0.730284i	$0.260612\pi$
$740$	18.1353	0.666666
$741$	15.1902	0.558024
$742$	9.16860	0.336590
$743$	−50.4784	−1.85187	−0.925937	−	0.377678i	$-0.876723\pi$
$743$	−50.4784	−1.85187	−0.925937	+	0.377678i	$0.876723\pi$
$744$	14.5937	0.535031
$745$	−9.05902	−0.331897
$746$	15.0266	0.550165
$747$	3.13668	0.114765
$748$	−5.44628	−0.199136
$749$	32.0465	1.17095
$750$	−4.14840	−0.151478
$751$	−12.7440	−0.465034	−0.232517	−	0.972592i	$-0.574696\pi$
$751$	−12.7440	−0.465034	−0.232517	+	0.972592i	$0.574696\pi$
$752$	1.81750	0.0662776
$753$	−18.0563	−0.658007
$754$	−11.4763	−0.417942
$755$	15.6715	0.570344
$756$	−1.96165	−0.0713446
$757$	12.9725	0.471492	0.235746	−	0.971815i	$-0.424247\pi$
$757$	12.9725	0.471492	0.235746	+	0.971815i	$0.424247\pi$
$758$	18.4134	0.668806
$759$	15.1212	0.548865
$760$	54.0677	1.96124
$761$	40.4763	1.46726	0.733632	−	0.679547i	$-0.237824\pi$
$761$	40.4763	1.46726	0.733632	+	0.679547i	$0.237824\pi$
$762$	17.0689	0.618342
$763$	11.2089	0.405788
$764$	−13.4279	−0.485806
$765$	2.91330	0.105331
$766$	−15.4687	−0.558906
$767$	−21.1774	−0.764672
$768$	16.9017	0.609886
$769$	33.9064	1.22269	0.611347	−	0.791362i	$-0.290628\pi$
$769$	33.9064	1.22269	0.611347	+	0.791362i	$0.290628\pi$
$770$	−23.6157	−0.851052
$771$	−29.4736	−1.06146
$772$	7.62233	0.274334
$773$	26.7256	0.961252	0.480626	−	0.876926i	$-0.340409\pi$
$773$	26.7256	0.961252	0.480626	+	0.876926i	$0.340409\pi$
$774$	1.33417	0.0479558
$775$	17.3623	0.623674
$776$	−36.3548	−1.30506
$777$	9.84243	0.353095
$778$	−5.32012	−0.190736
$779$	24.6323	0.882545
$780$	7.78531	0.278759
$781$	−44.7200	−1.60021
$782$	−2.91117	−0.104103
$783$	−5.08146	−0.181596
$784$	2.07244	0.0740157
$785$	2.91330	0.103980
$786$	7.06158	0.251878
$787$	−30.8969	−1.10136	−0.550678	−	0.834718i	$-0.685631\pi$
$787$	−30.8969	−1.10136	−0.550678	+	0.834718i	$0.685631\pi$
$788$	11.6500	0.415014
$789$	−30.4681	−1.08469
$790$	−4.59848	−0.163607
$791$	−7.25387	−0.257918
$792$	14.3325	0.509283
$793$	24.3358	0.864190
$794$	−16.8895	−0.599385
$795$	−16.1118	−0.571426
$796$	2.20234	0.0780598
$797$	−12.6648	−0.448611	−0.224305	−	0.974519i	$-0.572011\pi$
$797$	−12.6648	−0.448611	−0.224305	+	0.974519i	$0.572011\pi$
$798$	10.4965	0.371574
$799$	3.41885	0.120950
$800$	18.6992	0.661117
$801$	3.90643	0.138027
$802$	−7.13214	−0.251845
$803$	65.7919	2.32175
$804$	−8.38898	−0.295857
$805$	15.8670	0.559237
$806$	−11.2442	−0.396059
$807$	15.1101	0.531901
$808$	−57.6159	−2.02692
$809$	−23.4934	−0.825985	−0.412992	−	0.910734i	$-0.635516\pi$
$809$	−23.4934	−0.825985	−0.412992	+	0.910734i	$0.635516\pi$
$810$	−2.74244	−0.0963597
$811$	9.71434	0.341117	0.170558	−	0.985348i	$-0.445443\pi$
$811$	9.71434	0.341117	0.170558	+	0.985348i	$0.445443\pi$
$812$	9.96805	0.349810
$813$	−23.2237	−0.814492
$814$	−25.7237	−0.901614
$815$	−27.4747	−0.962395
$816$	−0.531613	−0.0186102
$817$	8.97348	0.313942
$818$	−7.26601	−0.254050
$819$	4.22526	0.147643
$820$	12.6246	0.440871
$821$	−3.14106	−0.109624	−0.0548118	−	0.998497i	$-0.517456\pi$
$821$	−3.14106	−0.109624	−0.0548118	+	0.998497i	$0.517456\pi$
$822$	−8.62848	−0.300953
$823$	−16.6914	−0.581824	−0.290912	−	0.956750i	$-0.593959\pi$
$823$	−16.6914	−0.581824	−0.290912	+	0.956750i	$0.593959\pi$
$824$	23.7577	0.827639
$825$	17.0516	0.593660
$826$	−14.6338	−0.509175
$827$	25.9127	0.901072	0.450536	−	0.892758i	$-0.351233\pi$
$827$	25.9127	0.901072	0.450536	+	0.892758i	$0.351233\pi$
$828$	−3.44464	−0.119710
$829$	−8.25035	−0.286547	−0.143273	−	0.989683i	$-0.545763\pi$
$829$	−8.25035	−0.286547	−0.143273	+	0.989683i	$0.545763\pi$
$830$	−8.60216	−0.298585
$831$	2.13483	0.0740563
$832$	−14.6608	−0.508273
$833$	3.89840	0.135072
$834$	13.4725	0.466513
$835$	61.9506	2.14389
$836$	34.4827	1.19261
$837$	−4.97869	−0.172089
$838$	27.3737	0.945607
$839$	31.8573	1.09984	0.549918	−	0.835218i	$-0.314659\pi$
$839$	31.8573	1.09984	0.549918	+	0.835218i	$0.314659\pi$
$840$	15.0394	0.518907
$841$	−3.17879	−0.109613
$842$	−10.4940	−0.361646
$843$	−17.4436	−0.600788
$844$	−24.7743	−0.852768
$845$	21.1039	0.725997
$846$	−3.21834	−0.110649
$847$	−22.7326	−0.781102
$848$	2.94004	0.100961
$849$	9.09374	0.312096
$850$	−3.28281	−0.112599
$851$	17.2832	0.592462
$852$	10.1873	0.349012
$853$	50.7077	1.73620	0.868100	−	0.496389i	$-0.165341\pi$
$853$	50.7077	1.73620	0.868100	+	0.496389i	$0.165341\pi$
$854$	16.8163	0.575441
$855$	−18.4454	−0.630818
$856$	53.3381	1.82306
$857$	33.2486	1.13575	0.567875	−	0.823115i	$-0.307766\pi$
$857$	33.2486	1.13575	0.567875	+	0.823115i	$0.307766\pi$
$858$	−11.0429	−0.376999
$859$	−0.777383	−0.0265240	−0.0132620	−	0.999912i	$-0.504222\pi$
$859$	−0.777383	−0.0265240	−0.0132620	+	0.999912i	$0.504222\pi$
$860$	4.59911	0.156828
$861$	6.85168	0.233505
$862$	30.3044	1.03217
$863$	47.0297	1.60091	0.800456	−	0.599392i	$-0.204591\pi$
$863$	47.0297	1.60091	0.800456	+	0.599392i	$0.204591\pi$
$864$	−5.36204	−0.182420
$865$	−43.8398	−1.49060
$866$	20.8834	0.709647
$867$	−1.00000	−0.0339618
$868$	9.76645	0.331495
$869$	−8.19876	−0.278124
$870$	13.9356	0.472462
$871$	18.0693	0.612255
$872$	18.6561	0.631774
$873$	12.4025	0.419763
$874$	18.4318	0.623466
$875$	−7.76107	−0.262372
$876$	−14.9876	−0.506382
$877$	25.5912	0.864153	0.432077	−	0.901837i	$-0.357781\pi$
$877$	25.5912	0.864153	0.432077	+	0.901837i	$0.357781\pi$
$878$	30.1491	1.01748
$879$	0.0560033	0.00188894
$880$	−7.57272	−0.255276
$881$	−21.2156	−0.714771	−0.357386	−	0.933957i	$-0.616332\pi$
$881$	−21.2156	−0.714771	−0.357386	+	0.933957i	$0.616332\pi$
$882$	−3.66977	−0.123568
$883$	−22.6184	−0.761170	−0.380585	−	0.924746i	$-0.624277\pi$
$883$	−22.6184	−0.761170	−0.380585	+	0.924746i	$0.624277\pi$
$884$	−2.67233	−0.0898802
$885$	25.7157	0.864422
$886$	26.6060	0.893846
$887$	37.4665	1.25800	0.629002	−	0.777404i	$-0.283464\pi$
$887$	37.4665	1.25800	0.629002	+	0.777404i	$0.283464\pi$
$888$	16.3818	0.549736
$889$	31.9335	1.07102
$890$	−10.7132	−0.359106
$891$	−4.88957	−0.163807
$892$	16.5370	0.553698
$893$	−21.6462	−0.724362
$894$	−2.92717	−0.0978992
$895$	−68.2904	−2.28270
$896$	8.75578	0.292510
$897$	7.41953	0.247731
$898$	−35.0674	−1.17021
$899$	25.2990	0.843768
$900$	−3.88439	−0.129480
$901$	5.53042	0.184245
$902$	−17.9072	−0.596244
$903$	2.49604	0.0830631
$904$	−12.0734	−0.401554
$905$	−51.2793	−1.70458
$906$	5.06380	0.168233
$907$	21.8549	0.725678	0.362839	−	0.931852i	$-0.381807\pi$
$907$	21.8549	0.725678	0.362839	+	0.931852i	$0.381807\pi$
$908$	−1.50548	−0.0499610
$909$	19.6558	0.651943
$910$	−11.5876	−0.384124
$911$	41.5959	1.37813	0.689066	−	0.724698i	$-0.258021\pi$
$911$	41.5959	1.37813	0.689066	+	0.724698i	$0.258021\pi$
$912$	3.36586	0.111455
$913$	−15.3370	−0.507581
$914$	19.1898	0.634743
$915$	−29.5509	−0.976922
$916$	−10.8135	−0.357288
$917$	13.2112	0.436273
$918$	0.941352	0.0310693
$919$	25.0396	0.825979	0.412989	−	0.910736i	$-0.364485\pi$
$919$	25.0396	0.825979	0.412989	+	0.910736i	$0.364485\pi$
$920$	26.4090	0.870678
$921$	11.5596	0.380902
$922$	−23.5470	−0.775479
$923$	−21.9428	−0.722256
$924$	9.59164	0.315542
$925$	19.4896	0.640815
$926$	−1.23069	−0.0404429
$927$	−8.10501	−0.266204
$928$	27.2470	0.894425
$929$	32.4361	1.06419	0.532097	−	0.846683i	$-0.321404\pi$
$929$	32.4361	1.06419	0.532097	+	0.846683i	$0.321404\pi$
$930$	13.6538	0.447725
$931$	−24.6824	−0.808934
$932$	−15.9094	−0.521130
$933$	34.0306	1.11411
$934$	10.2690	0.336011
$935$	−14.2448	−0.465855
$936$	7.03253	0.229866
$937$	30.5064	0.996600	0.498300	−	0.867005i	$-0.333958\pi$
$937$	30.5064	0.996600	0.498300	+	0.867005i	$0.333958\pi$
$938$	12.4861	0.407684
$939$	28.8680	0.942071
$940$	−11.0942	−0.361852
$941$	−34.1634	−1.11369	−0.556847	−	0.830615i	$-0.687989\pi$
$941$	−34.1634	−1.11369	−0.556847	+	0.830615i	$0.687989\pi$
$942$	0.941352	0.0306709
$943$	12.0315	0.391799
$944$	−4.69253	−0.152729
$945$	−5.13072	−0.166902
$946$	−6.52353	−0.212098
$947$	−57.8640	−1.88033	−0.940163	−	0.340725i	$-0.889328\pi$
$947$	−57.8640	−1.88033	−0.940163	+	0.340725i	$0.889328\pi$
$948$	1.86770	0.0606599
$949$	32.2822	1.04792
$950$	20.7849	0.674350
$951$	−14.2126	−0.460876
$952$	−5.16230	−0.167311
$953$	−40.0860	−1.29851	−0.649257	−	0.760569i	$-0.724920\pi$
$953$	−40.0860	−1.29851	−0.649257	+	0.760569i	$0.724920\pi$
$954$	−5.20607	−0.168553
$955$	−35.1209	−1.13649
$956$	3.90264	0.126220
$957$	24.8462	0.803162
$958$	−13.2534	−0.428198
$959$	−16.1427	−0.521274
$960$	17.8026	0.574576
$961$	−6.21268	−0.200409
$962$	−12.6218	−0.406945
$963$	−18.1965	−0.586373
$964$	23.4431	0.755051
$965$	19.9363	0.641772
$966$	5.12696	0.164957
$967$	15.0131	0.482790	0.241395	−	0.970427i	$-0.422395\pi$
$967$	15.0131	0.482790	0.241395	+	0.970427i	$0.422395\pi$
$968$	−37.8362	−1.21610
$969$	6.33142	0.203395
$970$	−34.0133	−1.09210
$971$	10.4450	0.335194	0.167597	−	0.985856i	$-0.446399\pi$
$971$	10.4450	0.335194	0.167597	+	0.985856i	$0.446399\pi$
$972$	1.11386	0.0357269
$973$	25.2051	0.808037
$974$	−5.18092	−0.166007
$975$	8.36671	0.267949
$976$	5.39237	0.172606
$977$	−2.94413	−0.0941912	−0.0470956	−	0.998890i	$-0.514997\pi$
$977$	−2.94413	−0.0941912	−0.0470956	+	0.998890i	$0.514997\pi$
$978$	−8.87766	−0.283876
$979$	−19.1008	−0.610464
$980$	−12.6503	−0.404099
$981$	−6.36457	−0.203205
$982$	−0.620592	−0.0198039
$983$	−56.1271	−1.79018	−0.895089	−	0.445888i	$-0.852888\pi$
$983$	−56.1271	−1.79018	−0.895089	+	0.445888i	$0.852888\pi$
$984$	11.4039	0.363544
$985$	30.4707	0.970877
$986$	−4.78344	−0.152336
$987$	−6.02106	−0.191652
$988$	16.9197	0.538286
$989$	4.38303	0.139372
$990$	13.4094	0.426178
$991$	2.77128	0.0880327	0.0440163	−	0.999031i	$-0.485985\pi$
$991$	2.77128	0.0880327	0.0440163	+	0.999031i	$0.485985\pi$
$992$	26.6959	0.847595
$993$	−7.38597	−0.234387
$994$	−15.1627	−0.480931
$995$	5.76024	0.182612
$996$	3.49381	0.110705
$997$	62.8680	1.99105	0.995524	−	0.0945050i	$-0.0301268\pi$
$997$	62.8680	1.99105	0.995524	+	0.0945050i	$0.0301268\pi$
$998$	−18.1819	−0.575540
$999$	−5.58869	−0.176818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.32	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.32	✓	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)

\(f(q)\)	\(=\)	\(q+0.941352 q^{2} -1.00000 q^{3} -1.11386 q^{4} -2.91330 q^{5} -0.941352 q^{6} -1.76114 q^{7} -2.93124 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.941352 q^{2} -1.00000 q^{3} -1.11386 q^{4} -2.91330 q^{5} -0.941352 q^{6} -1.76114 q^{7} -2.93124 q^{8} +1.00000 q^{9} -2.74244 q^{10} -4.88957 q^{11} +1.11386 q^{12} -2.39917 q^{13} -1.65785 q^{14} +2.91330 q^{15} -0.531613 q^{16} -1.00000 q^{17} +0.941352 q^{18} +6.33142 q^{19} +3.24500 q^{20} +1.76114 q^{21} -4.60281 q^{22} +3.09254 q^{23} +2.93124 q^{24} +3.48733 q^{25} -2.25846 q^{26} -1.00000 q^{27} +1.96165 q^{28} +5.08146 q^{29} +2.74244 q^{30} +4.97869 q^{31} +5.36204 q^{32} +4.88957 q^{33} -0.941352 q^{34} +5.13072 q^{35} -1.11386 q^{36} +5.58869 q^{37} +5.96010 q^{38} +2.39917 q^{39} +8.53958 q^{40} +3.89049 q^{41} +1.65785 q^{42} +1.41729 q^{43} +5.44628 q^{44} -2.91330 q^{45} +2.91117 q^{46} -3.41885 q^{47} +0.531613 q^{48} -3.89840 q^{49} +3.28281 q^{50} +1.00000 q^{51} +2.67233 q^{52} -5.53042 q^{53} -0.941352 q^{54} +14.2448 q^{55} +5.16230 q^{56} -6.33142 q^{57} +4.78344 q^{58} +8.82698 q^{59} -3.24500 q^{60} -10.1434 q^{61} +4.68670 q^{62} -1.76114 q^{63} +6.11079 q^{64} +6.98951 q^{65} +4.60281 q^{66} -7.53148 q^{67} +1.11386 q^{68} -3.09254 q^{69} +4.82982 q^{70} +9.14600 q^{71} -2.93124 q^{72} -13.4556 q^{73} +5.26092 q^{74} -3.48733 q^{75} -7.05229 q^{76} +8.61120 q^{77} +2.25846 q^{78} +1.67678 q^{79} +1.54875 q^{80} +1.00000 q^{81} +3.66232 q^{82} +3.13668 q^{83} -1.96165 q^{84} +2.91330 q^{85} +1.33417 q^{86} -5.08146 q^{87} +14.3325 q^{88} +3.90643 q^{89} -2.74244 q^{90} +4.22526 q^{91} -3.44464 q^{92} -4.97869 q^{93} -3.21834 q^{94} -18.4454 q^{95} -5.36204 q^{96} +12.4025 q^{97} -3.66977 q^{98} -4.88957 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 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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