LMFDB - Embedded newform 8007.2.a.e.1.20

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:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q-0.576474 q^{2} +1.00000 q^{3} -1.66768 q^{4} +0.828352 q^{5} -0.576474 q^{6} -0.479856 q^{7} +2.11432 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.576474 q^{2} +1.00000 q^{3} -1.66768 q^{4} +0.828352 q^{5} -0.576474 q^{6} -0.479856 q^{7} +2.11432 q^{8} +1.00000 q^{9} -0.477524 q^{10} +5.91719 q^{11} -1.66768 q^{12} -2.47116 q^{13} +0.276625 q^{14} +0.828352 q^{15} +2.11650 q^{16} -1.00000 q^{17} -0.576474 q^{18} +2.75787 q^{19} -1.38142 q^{20} -0.479856 q^{21} -3.41111 q^{22} +4.50266 q^{23} +2.11432 q^{24} -4.31383 q^{25} +1.42456 q^{26} +1.00000 q^{27} +0.800245 q^{28} -6.59210 q^{29} -0.477524 q^{30} -2.35371 q^{31} -5.44875 q^{32} +5.91719 q^{33} +0.576474 q^{34} -0.397490 q^{35} -1.66768 q^{36} -9.35003 q^{37} -1.58984 q^{38} -2.47116 q^{39} +1.75140 q^{40} -7.84656 q^{41} +0.276625 q^{42} -10.6454 q^{43} -9.86797 q^{44} +0.828352 q^{45} -2.59567 q^{46} -10.0395 q^{47} +2.11650 q^{48} -6.76974 q^{49} +2.48681 q^{50} -1.00000 q^{51} +4.12110 q^{52} +10.7649 q^{53} -0.576474 q^{54} +4.90152 q^{55} -1.01457 q^{56} +2.75787 q^{57} +3.80018 q^{58} +0.266454 q^{59} -1.38142 q^{60} -7.69049 q^{61} +1.35685 q^{62} -0.479856 q^{63} -1.09194 q^{64} -2.04699 q^{65} -3.41111 q^{66} -12.4138 q^{67} +1.66768 q^{68} +4.50266 q^{69} +0.229143 q^{70} +7.86135 q^{71} +2.11432 q^{72} +2.29870 q^{73} +5.39005 q^{74} -4.31383 q^{75} -4.59924 q^{76} -2.83940 q^{77} +1.42456 q^{78} -6.28776 q^{79} +1.75321 q^{80} +1.00000 q^{81} +4.52334 q^{82} +11.7711 q^{83} +0.800245 q^{84} -0.828352 q^{85} +6.13679 q^{86} -6.59210 q^{87} +12.5108 q^{88} +10.1686 q^{89} -0.477524 q^{90} +1.18580 q^{91} -7.50899 q^{92} -2.35371 q^{93} +5.78749 q^{94} +2.28449 q^{95} -5.44875 q^{96} +0.346481 q^{97} +3.90258 q^{98} +5.91719 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * 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 $

$2$	−0.576474	−0.407629	−0.203814	−	0.979010i	$-0.565334\pi$
$2$	−0.576474	−0.407629	−0.203814	+	0.979010i	$0.565334\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.66768	−0.833839
$5$	0.828352	0.370450	0.185225	−	0.982696i	$-0.440699\pi$
$5$	0.828352	0.370450	0.185225	+	0.982696i	$0.440699\pi$
$6$	−0.576474	−0.235345
$7$	−0.479856	−0.181369	−0.0906843	−	0.995880i	$-0.528905\pi$
$7$	−0.479856	−0.181369	−0.0906843	+	0.995880i	$0.528905\pi$
$8$	2.11432	0.747525
$9$	1.00000	0.333333
$10$	−0.477524	−0.151006
$11$	5.91719	1.78410	0.892050	−	0.451937i	$-0.149267\pi$
$11$	5.91719	1.78410	0.892050	+	0.451937i	$0.149267\pi$
$12$	−1.66768	−0.481417
$13$	−2.47116	−0.685377	−0.342688	−	0.939449i	$-0.611337\pi$
$13$	−2.47116	−0.685377	−0.342688	+	0.939449i	$0.611337\pi$
$14$	0.276625	0.0739310
$15$	0.828352	0.213880
$16$	2.11650	0.529126
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.576474	−0.135876
$19$	2.75787	0.632699	0.316349	−	0.948643i	$-0.397543\pi$
$19$	2.75787	0.632699	0.316349	+	0.948643i	$0.397543\pi$
$20$	−1.38142	−0.308896
$21$	−0.479856	−0.104713
$22$	−3.41111	−0.727250
$23$	4.50266	0.938870	0.469435	−	0.882967i	$-0.344458\pi$
$23$	4.50266	0.938870	0.469435	+	0.882967i	$0.344458\pi$
$24$	2.11432	0.431584
$25$	−4.31383	−0.862767
$26$	1.42456	0.279379
$27$	1.00000	0.192450
$28$	0.800245	0.151232
$29$	−6.59210	−1.22412	−0.612061	−	0.790810i	$-0.709659\pi$
$29$	−6.59210	−1.22412	−0.612061	+	0.790810i	$0.709659\pi$
$30$	−0.477524	−0.0871835
$31$	−2.35371	−0.422739	−0.211370	−	0.977406i	$-0.567792\pi$
$31$	−2.35371	−0.422739	−0.211370	+	0.977406i	$0.567792\pi$
$32$	−5.44875	−0.963212
$33$	5.91719	1.03005
$34$	0.576474	0.0988645
$35$	−0.397490	−0.0671880
$36$	−1.66768	−0.277946
$37$	−9.35003	−1.53713	−0.768567	−	0.639769i	$-0.779030\pi$
$37$	−9.35003	−1.53713	−0.768567	+	0.639769i	$0.779030\pi$
$38$	−1.58984	−0.257906
$39$	−2.47116	−0.395702
$40$	1.75140	0.276921
$41$	−7.84656	−1.22543	−0.612713	−	0.790305i	$-0.709922\pi$
$41$	−7.84656	−1.22543	−0.612713	+	0.790305i	$0.709922\pi$
$42$	0.276625	0.0426841
$43$	−10.6454	−1.62341	−0.811703	−	0.584071i	$-0.801459\pi$
$43$	−10.6454	−1.62341	−0.811703	+	0.584071i	$0.801459\pi$
$44$	−9.86797	−1.48765
$45$	0.828352	0.123483
$46$	−2.59567	−0.382711
$47$	−10.0395	−1.46441	−0.732203	−	0.681087i	$-0.761508\pi$
$47$	−10.0395	−1.46441	−0.732203	+	0.681087i	$0.761508\pi$
$48$	2.11650	0.305491
$49$	−6.76974	−0.967105
$50$	2.48681	0.351688
$51$	−1.00000	−0.140028
$52$	4.12110	0.571494
$53$	10.7649	1.47867	0.739335	−	0.673338i	$-0.235140\pi$
$53$	10.7649	1.47867	0.739335	+	0.673338i	$0.235140\pi$
$54$	−0.576474	−0.0784482
$55$	4.90152	0.660920
$56$	−1.01457	−0.135578
$57$	2.75787	0.365289
$58$	3.80018	0.498988
$59$	0.266454	0.0346893	0.0173447	−	0.999850i	$-0.494479\pi$
$59$	0.266454	0.0346893	0.0173447	+	0.999850i	$0.494479\pi$
$60$	−1.38142	−0.178341
$61$	−7.69049	−0.984666	−0.492333	−	0.870407i	$-0.663856\pi$
$61$	−7.69049	−0.984666	−0.492333	+	0.870407i	$0.663856\pi$
$62$	1.35685	0.172321
$63$	−0.479856	−0.0604562
$64$	−1.09194	−0.136493
$65$	−2.04699	−0.253898
$66$	−3.41111	−0.419878
$67$	−12.4138	−1.51659	−0.758293	−	0.651914i	$-0.773966\pi$
$67$	−12.4138	−1.51659	−0.758293	+	0.651914i	$0.773966\pi$
$68$	1.66768	0.202236
$69$	4.50266	0.542057
$70$	0.229143	0.0273878
$71$	7.86135	0.932970	0.466485	−	0.884529i	$-0.345520\pi$
$71$	7.86135	0.932970	0.466485	+	0.884529i	$0.345520\pi$
$72$	2.11432	0.249175
$73$	2.29870	0.269043	0.134521	−	0.990911i	$-0.457050\pi$
$73$	2.29870	0.269043	0.134521	+	0.990911i	$0.457050\pi$
$74$	5.39005	0.626580
$75$	−4.31383	−0.498118
$76$	−4.59924	−0.527569
$77$	−2.83940	−0.323580
$78$	1.42456	0.161300
$79$	−6.28776	−0.707428	−0.353714	−	0.935354i	$-0.615081\pi$
$79$	−6.28776	−0.707428	−0.353714	+	0.935354i	$0.615081\pi$
$80$	1.75321	0.196015
$81$	1.00000	0.111111
$82$	4.52334	0.499519
$83$	11.7711	1.29204	0.646022	−	0.763319i	$-0.276432\pi$
$83$	11.7711	1.29204	0.646022	+	0.763319i	$0.276432\pi$
$84$	0.800245	0.0873139
$85$	−0.828352	−0.0898474
$86$	6.13679	0.661747
$87$	−6.59210	−0.706748
$88$	12.5108	1.33366
$89$	10.1686	1.07787	0.538934	−	0.842348i	$-0.318827\pi$
$89$	10.1686	1.07787	0.538934	+	0.842348i	$0.318827\pi$
$90$	−0.477524	−0.0503354
$91$	1.18580	0.124306
$92$	−7.50899	−0.782867
$93$	−2.35371	−0.244069
$94$	5.78749	0.596934
$95$	2.28449	0.234384
$96$	−5.44875	−0.556111
$97$	0.346481	0.0351798	0.0175899	−	0.999845i	$-0.494401\pi$
$97$	0.346481	0.0351798	0.0175899	+	0.999845i	$0.494401\pi$
$98$	3.90258	0.394220
$99$	5.91719	0.594700
$100$	7.19408	0.719408
$101$	11.5452	1.14879	0.574397	−	0.818577i	$-0.305237\pi$
$101$	11.5452	1.14879	0.574397	+	0.818577i	$0.305237\pi$
$102$	0.576474	0.0570794
$103$	7.18519	0.707978	0.353989	−	0.935250i	$-0.384825\pi$
$103$	7.18519	0.707978	0.353989	+	0.935250i	$0.384825\pi$
$104$	−5.22483	−0.512336
$105$	−0.397490	−0.0387910
$106$	−6.20568	−0.602749
$107$	−2.35202	−0.227379	−0.113689	−	0.993516i	$-0.536267\pi$
$107$	−2.35202	−0.227379	−0.113689	+	0.993516i	$0.536267\pi$
$108$	−1.66768	−0.160472
$109$	3.67918	0.352402	0.176201	−	0.984354i	$-0.443619\pi$
$109$	3.67918	0.352402	0.176201	+	0.984354i	$0.443619\pi$
$110$	−2.82560	−0.269410
$111$	−9.35003	−0.887465
$112$	−1.01562	−0.0959668
$113$	10.6804	1.00473	0.502366	−	0.864655i	$-0.332463\pi$
$113$	10.6804	1.00473	0.502366	+	0.864655i	$0.332463\pi$
$114$	−1.58984	−0.148902
$115$	3.72979	0.347805
$116$	10.9935	1.02072
$117$	−2.47116	−0.228459
$118$	−0.153604	−0.0141404
$119$	0.479856	0.0439883
$120$	1.75140	0.159880
$121$	24.0131	2.18301
$122$	4.43337	0.401378
$123$	−7.84656	−0.707500
$124$	3.92523	0.352496
$125$	−7.71513	−0.690063
$126$	0.276625	0.0246437
$127$	−7.23512	−0.642013	−0.321007	−	0.947077i	$-0.604021\pi$
$127$	−7.23512	−0.642013	−0.321007	+	0.947077i	$0.604021\pi$
$128$	11.5270	1.01885
$129$	−10.6454	−0.937274
$130$	1.18004	0.103496
$131$	−16.6001	−1.45036	−0.725180	−	0.688560i	$-0.758243\pi$
$131$	−16.6001	−1.45036	−0.725180	+	0.688560i	$0.758243\pi$
$132$	−9.86797	−0.858896
$133$	−1.32338	−0.114752
$134$	7.15623	0.618204
$135$	0.828352	0.0712932
$136$	−2.11432	−0.181302
$137$	−8.81614	−0.753214	−0.376607	−	0.926373i	$-0.622909\pi$
$137$	−8.81614	−0.753214	−0.376607	+	0.926373i	$0.622909\pi$
$138$	−2.59567	−0.220958
$139$	−1.60980	−0.136542	−0.0682709	−	0.997667i	$-0.521748\pi$
$139$	−1.60980	−0.136542	−0.0682709	+	0.997667i	$0.521748\pi$
$140$	0.662885	0.0560240
$141$	−10.0395	−0.845475
$142$	−4.53186	−0.380306
$143$	−14.6223	−1.22278
$144$	2.11650	0.176375
$145$	−5.46058	−0.453477
$146$	−1.32514	−0.109670
$147$	−6.76974	−0.558359
$148$	15.5928	1.28172
$149$	1.20969	0.0991016	0.0495508	−	0.998772i	$-0.484221\pi$
$149$	1.20969	0.0991016	0.0495508	+	0.998772i	$0.484221\pi$
$150$	2.48681	0.203047
$151$	−19.6838	−1.60184	−0.800922	−	0.598769i	$-0.795657\pi$
$151$	−19.6838	−1.60184	−0.800922	+	0.598769i	$0.795657\pi$
$152$	5.83103	0.472959
$153$	−1.00000	−0.0808452
$154$	1.63684	0.131900
$155$	−1.94970	−0.156604
$156$	4.12110	0.329952
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	3.62473	0.288368
$159$	10.7649	0.853711
$160$	−4.51349	−0.356822
$161$	−2.16063	−0.170282
$162$	−0.576474	−0.0452921
$163$	−13.6542	−1.06948	−0.534742	−	0.845016i	$-0.679591\pi$
$163$	−13.6542	−1.06948	−0.534742	+	0.845016i	$0.679591\pi$
$164$	13.0855	1.02181
$165$	4.90152	0.381583
$166$	−6.78572	−0.526674
$167$	18.4779	1.42986	0.714930	−	0.699196i	$-0.246459\pi$
$167$	18.4779	1.42986	0.714930	+	0.699196i	$0.246459\pi$
$168$	−1.01457	−0.0782758
$169$	−6.89337	−0.530259
$170$	0.477524	0.0366244
$171$	2.75787	0.210900
$172$	17.7531	1.35366
$173$	12.7582	0.969989	0.484995	−	0.874517i	$-0.338822\pi$
$173$	12.7582	0.969989	0.484995	+	0.874517i	$0.338822\pi$
$174$	3.80018	0.288091
$175$	2.07002	0.156479
$176$	12.5238	0.944013
$177$	0.266454	0.0200279
$178$	−5.86193	−0.439370
$179$	9.56385	0.714836	0.357418	−	0.933945i	$-0.383657\pi$
$179$	9.56385	0.714836	0.357418	+	0.933945i	$0.383657\pi$
$180$	−1.38142	−0.102965
$181$	−18.9044	−1.40516	−0.702578	−	0.711607i	$-0.747968\pi$
$181$	−18.9044	−1.40516	−0.702578	+	0.711607i	$0.747968\pi$
$182$	−0.683584	−0.0506706
$183$	−7.69049	−0.568497
$184$	9.52008	0.701830
$185$	−7.74511	−0.569432
$186$	1.35685	0.0994894
$187$	−5.91719	−0.432708
$188$	16.7426	1.22108
$189$	−0.479856	−0.0349044
$190$	−1.31695	−0.0955415
$191$	−19.8281	−1.43471	−0.717355	−	0.696707i	$-0.754648\pi$
$191$	−19.8281	−1.43471	−0.717355	+	0.696707i	$0.754648\pi$
$192$	−1.09194	−0.0788041
$193$	−23.4458	−1.68766	−0.843831	−	0.536608i	$-0.819705\pi$
$193$	−23.4458	−1.68766	−0.843831	+	0.536608i	$0.819705\pi$
$194$	−0.199737	−0.0143403
$195$	−2.04699	−0.146588
$196$	11.2897	0.806410
$197$	9.98421	0.711346	0.355673	−	0.934611i	$-0.384252\pi$
$197$	9.98421	0.711346	0.355673	+	0.934611i	$0.384252\pi$
$198$	−3.41111	−0.242417
$199$	4.12796	0.292623	0.146311	−	0.989239i	$-0.453260\pi$
$199$	4.12796	0.292623	0.146311	+	0.989239i	$0.453260\pi$
$200$	−9.12083	−0.644940
$201$	−12.4138	−0.875601
$202$	−6.65553	−0.468282
$203$	3.16326	0.222017
$204$	1.66768	0.116761
$205$	−6.49971	−0.453960
$206$	−4.14208	−0.288592
$207$	4.50266	0.312957
$208$	−5.23022	−0.362650
$209$	16.3188	1.12880
$210$	0.229143	0.0158123
$211$	−8.50287	−0.585362	−0.292681	−	0.956210i	$-0.594547\pi$
$211$	−8.50287	−0.585362	−0.292681	+	0.956210i	$0.594547\pi$
$212$	−17.9524	−1.23297
$213$	7.86135	0.538651
$214$	1.35588	0.0926861
$215$	−8.81813	−0.601391
$216$	2.11432	0.143861
$217$	1.12944	0.0766716
$218$	−2.12095	−0.143649
$219$	2.29870	0.155332
$220$	−8.17415	−0.551101
$221$	2.47116	0.166228
$222$	5.39005	0.361756
$223$	−15.1528	−1.01471	−0.507354	−	0.861738i	$-0.669376\pi$
$223$	−15.1528	−1.01471	−0.507354	+	0.861738i	$0.669376\pi$
$224$	2.61462	0.174696
$225$	−4.31383	−0.287589
$226$	−6.15700	−0.409557
$227$	−22.2243	−1.47508	−0.737539	−	0.675305i	$-0.764012\pi$
$227$	−22.2243	−1.47508	−0.737539	+	0.675305i	$0.764012\pi$
$228$	−4.59924	−0.304592
$229$	11.3305	0.748742	0.374371	−	0.927279i	$-0.377859\pi$
$229$	11.3305	0.748742	0.374371	+	0.927279i	$0.377859\pi$
$230$	−2.15013	−0.141775
$231$	−2.83940	−0.186819
$232$	−13.9378	−0.915063
$233$	4.18406	0.274107	0.137053	−	0.990564i	$-0.456237\pi$
$233$	4.18406	0.274107	0.137053	+	0.990564i	$0.456237\pi$
$234$	1.42456	0.0931264
$235$	−8.31621	−0.542490
$236$	−0.444359	−0.0289253
$237$	−6.28776	−0.408434
$238$	−0.276625	−0.0179309
$239$	9.78127	0.632698	0.316349	−	0.948643i	$-0.397543\pi$
$239$	9.78127	0.632698	0.316349	+	0.948643i	$0.397543\pi$
$240$	1.75321	0.113169
$241$	−21.7377	−1.40025	−0.700125	−	0.714020i	$-0.746873\pi$
$241$	−21.7377	−1.40025	−0.700125	+	0.714020i	$0.746873\pi$
$242$	−13.8430	−0.889859
$243$	1.00000	0.0641500
$244$	12.8253	0.821053
$245$	−5.60773	−0.358265
$246$	4.52334	0.288397
$247$	−6.81514	−0.433637
$248$	−4.97651	−0.316008
$249$	11.7711	0.745962
$250$	4.44758	0.281289
$251$	−23.3319	−1.47270	−0.736349	−	0.676602i	$-0.763452\pi$
$251$	−23.3319	−1.47270	−0.736349	+	0.676602i	$0.763452\pi$
$252$	0.800245	0.0504107
$253$	26.6431	1.67504
$254$	4.17086	0.261703
$255$	−0.828352	−0.0518734
$256$	−4.46112	−0.278820
$257$	13.3221	0.831011	0.415505	−	0.909591i	$-0.363605\pi$
$257$	13.3221	0.831011	0.415505	+	0.909591i	$0.363605\pi$
$258$	6.13679	0.382060
$259$	4.48667	0.278788
$260$	3.41372	0.211710
$261$	−6.59210	−0.408041
$262$	9.56954	0.591208
$263$	−13.5052	−0.832765	−0.416382	−	0.909190i	$-0.636702\pi$
$263$	−13.5052	−0.832765	−0.416382	+	0.909190i	$0.636702\pi$
$264$	12.5108	0.769989
$265$	8.91711	0.547774
$266$	0.762895	0.0467761
$267$	10.1686	0.622308
$268$	20.7022	1.26459
$269$	−11.2883	−0.688259	−0.344129	−	0.938922i	$-0.611826\pi$
$269$	−11.2883	−0.688259	−0.344129	+	0.938922i	$0.611826\pi$
$270$	−0.477524	−0.0290612
$271$	8.30026	0.504205	0.252103	−	0.967701i	$-0.418878\pi$
$271$	8.30026	0.504205	0.252103	+	0.967701i	$0.418878\pi$
$272$	−2.11650	−0.128332
$273$	1.18580	0.0717680
$274$	5.08228	0.307032
$275$	−25.5258	−1.53926
$276$	−7.50899	−0.451988
$277$	23.1419	1.39046	0.695232	−	0.718786i	$-0.255302\pi$
$277$	23.1419	1.39046	0.695232	+	0.718786i	$0.255302\pi$
$278$	0.928010	0.0556583
$279$	−2.35371	−0.140913
$280$	−0.840421	−0.0502248
$281$	−10.0789	−0.601255	−0.300627	−	0.953742i	$-0.597196\pi$
$281$	−10.0789	−0.601255	−0.300627	+	0.953742i	$0.597196\pi$
$282$	5.78749	0.344640
$283$	−13.2197	−0.785828	−0.392914	−	0.919575i	$-0.628533\pi$
$283$	−13.2197	−0.785828	−0.392914	+	0.919575i	$0.628533\pi$
$284$	−13.1102	−0.777947
$285$	2.28449	0.135321
$286$	8.42939	0.498440
$287$	3.76522	0.222254
$288$	−5.44875	−0.321071
$289$	1.00000	0.0588235
$290$	3.14788	0.184850
$291$	0.346481	0.0203111
$292$	−3.83349	−0.224338
$293$	18.9435	1.10669	0.553346	−	0.832952i	$-0.313351\pi$
$293$	18.9435	1.10669	0.553346	+	0.832952i	$0.313351\pi$
$294$	3.90258	0.227603
$295$	0.220718	0.0128507
$296$	−19.7690	−1.14905
$297$	5.91719	0.343350
$298$	−0.697354	−0.0403966
$299$	−11.1268	−0.643480
$300$	7.19408	0.415351
$301$	5.10825	0.294435
$302$	11.3472	0.652958
$303$	11.5452	0.663257
$304$	5.83704	0.334777
$305$	−6.37043	−0.364770
$306$	0.576474	0.0329548
$307$	22.9698	1.31096	0.655479	−	0.755213i	$-0.272467\pi$
$307$	22.9698	1.31096	0.655479	+	0.755213i	$0.272467\pi$
$308$	4.73520	0.269813
$309$	7.18519	0.408751
$310$	1.12395	0.0638363
$311$	21.1653	1.20017	0.600086	−	0.799935i	$-0.295133\pi$
$311$	21.1653	1.20017	0.600086	+	0.799935i	$0.295133\pi$
$312$	−5.22483	−0.295798
$313$	13.2039	0.746329	0.373164	−	0.927765i	$-0.378273\pi$
$313$	13.2039	0.746329	0.373164	+	0.927765i	$0.378273\pi$
$314$	−0.576474	−0.0325323
$315$	−0.397490	−0.0223960
$316$	10.4860	0.589881
$317$	−10.0021	−0.561775	−0.280887	−	0.959741i	$-0.590629\pi$
$317$	−10.0021	−0.561775	−0.280887	+	0.959741i	$0.590629\pi$
$318$	−6.20568	−0.347997
$319$	−39.0067	−2.18396
$320$	−0.904513	−0.0505638
$321$	−2.35202	−0.131277
$322$	1.24555	0.0694117
$323$	−2.75787	−0.153452
$324$	−1.66768	−0.0926488
$325$	10.6602	0.591320
$326$	7.87132	0.435952
$327$	3.67918	0.203459
$328$	−16.5901	−0.916037
$329$	4.81750	0.265597
$330$	−2.82560	−0.155544
$331$	12.4144	0.682360	0.341180	−	0.939998i	$-0.389173\pi$
$331$	12.4144	0.682360	0.341180	+	0.939998i	$0.389173\pi$
$332$	−19.6304	−1.07736
$333$	−9.35003	−0.512378
$334$	−10.6520	−0.582852
$335$	−10.2830	−0.561820
$336$	−1.01562	−0.0554065
$337$	−5.55921	−0.302829	−0.151415	−	0.988470i	$-0.548383\pi$
$337$	−5.55921	−0.302829	−0.151415	+	0.988470i	$0.548383\pi$
$338$	3.97385	0.216149
$339$	10.6804	0.580082
$340$	1.38142	0.0749183
$341$	−13.9274	−0.754209
$342$	−1.58984	−0.0859688
$343$	6.60749	0.356771
$344$	−22.5078	−1.21354
$345$	3.72979	0.200805
$346$	−7.35478	−0.395396
$347$	−12.7321	−0.683494	−0.341747	−	0.939792i	$-0.611019\pi$
$347$	−12.7321	−0.683494	−0.341747	+	0.939792i	$0.611019\pi$
$348$	10.9935	0.589313
$349$	19.5735	1.04775	0.523874	−	0.851796i	$-0.324486\pi$
$349$	19.5735	1.04775	0.523874	+	0.851796i	$0.324486\pi$
$350$	−1.19331	−0.0637852
$351$	−2.47116	−0.131901
$352$	−32.2413	−1.71847
$353$	−16.4742	−0.876834	−0.438417	−	0.898772i	$-0.644461\pi$
$353$	−16.4742	−0.876834	−0.438417	+	0.898772i	$0.644461\pi$
$354$	−0.153604	−0.00816395
$355$	6.51197	0.345619
$356$	−16.9579	−0.898769
$357$	0.479856	0.0253967
$358$	−5.51331	−0.291388
$359$	−10.4095	−0.549391	−0.274695	−	0.961531i	$-0.588577\pi$
$359$	−10.4095	−0.549391	−0.274695	+	0.961531i	$0.588577\pi$
$360$	1.75140	0.0923070
$361$	−11.3941	−0.599692
$362$	10.8979	0.572782
$363$	24.0131	1.26036
$364$	−1.97753	−0.103651
$365$	1.90413	0.0996669
$366$	4.43337	0.231736
$367$	−22.6560	−1.18263	−0.591316	−	0.806440i	$-0.701391\pi$
$367$	−22.6560	−1.18263	−0.591316	+	0.806440i	$0.701391\pi$
$368$	9.52991	0.496781
$369$	−7.84656	−0.408475
$370$	4.46486	0.232117
$371$	−5.16559	−0.268184
$372$	3.92523	0.203514
$373$	23.1860	1.20052	0.600262	−	0.799803i	$-0.295063\pi$
$373$	23.1860	1.20052	0.600262	+	0.799803i	$0.295063\pi$
$374$	3.41111	0.176384
$375$	−7.71513	−0.398408
$376$	−21.2266	−1.09468
$377$	16.2901	0.838985
$378$	0.276625	0.0142280
$379$	18.9940	0.975654	0.487827	−	0.872940i	$-0.337790\pi$
$379$	18.9940	0.975654	0.487827	+	0.872940i	$0.337790\pi$
$380$	−3.80979	−0.195438
$381$	−7.23512	−0.370667
$382$	11.4304	0.584829
$383$	1.12690	0.0575818	0.0287909	−	0.999585i	$-0.490834\pi$
$383$	1.12690	0.0575818	0.0287909	+	0.999585i	$0.490834\pi$
$384$	11.5270	0.588234
$385$	−2.35202	−0.119870
$386$	13.5159	0.687940
$387$	−10.6454	−0.541135
$388$	−0.577818	−0.0293343
$389$	−28.1550	−1.42751	−0.713757	−	0.700393i	$-0.753008\pi$
$389$	−28.1550	−1.42751	−0.713757	+	0.700393i	$0.753008\pi$
$390$	1.18004	0.0597535
$391$	−4.50266	−0.227710
$392$	−14.3134	−0.722936
$393$	−16.6001	−0.837365
$394$	−5.75564	−0.289965
$395$	−5.20848	−0.262067
$396$	−9.86797	−0.495884
$397$	−28.7826	−1.44456	−0.722278	−	0.691603i	$-0.756905\pi$
$397$	−28.7826	−1.44456	−0.722278	+	0.691603i	$0.756905\pi$
$398$	−2.37966	−0.119282
$399$	−1.32338	−0.0662519
$400$	−9.13024	−0.456512
$401$	19.0968	0.953649	0.476825	−	0.878998i	$-0.341788\pi$
$401$	19.0968	0.953649	0.476825	+	0.878998i	$0.341788\pi$
$402$	7.15623	0.356920
$403$	5.81640	0.289736
$404$	−19.2537	−0.957909
$405$	0.828352	0.0411612
$406$	−1.82354	−0.0905007
$407$	−55.3259	−2.74240
$408$	−2.11432	−0.104675
$409$	30.9140	1.52860	0.764300	−	0.644861i	$-0.223085\pi$
$409$	30.9140	1.52860	0.764300	+	0.644861i	$0.223085\pi$
$410$	3.74692	0.185047
$411$	−8.81614	−0.434868
$412$	−11.9826	−0.590339
$413$	−0.127860	−0.00629156
$414$	−2.59567	−0.127570
$415$	9.75060	0.478638
$416$	13.4647	0.660163
$417$	−1.60980	−0.0788324
$418$	−9.40739	−0.460131
$419$	14.4095	0.703952	0.351976	−	0.936009i	$-0.385510\pi$
$419$	14.4095	0.703952	0.351976	+	0.936009i	$0.385510\pi$
$420$	0.662885	0.0323455
$421$	6.23018	0.303640	0.151820	−	0.988408i	$-0.451487\pi$
$421$	6.23018	0.303640	0.151820	+	0.988408i	$0.451487\pi$
$422$	4.90168	0.238610
$423$	−10.0395	−0.488135
$424$	22.7604	1.10534
$425$	4.31383	0.209252
$426$	−4.53186	−0.219570
$427$	3.69033	0.178587
$428$	3.92242	0.189597
$429$	−14.6223	−0.705973
$430$	5.08342	0.245144
$431$	8.93404	0.430337	0.215169	−	0.976577i	$-0.430970\pi$
$431$	8.93404	0.430337	0.215169	+	0.976577i	$0.430970\pi$
$432$	2.11650	0.101830
$433$	−38.2328	−1.83735	−0.918677	−	0.395010i	$-0.870741\pi$
$433$	−38.2328	−1.83735	−0.918677	+	0.395010i	$0.870741\pi$
$434$	−0.651095	−0.0312536
$435$	−5.46058	−0.261815
$436$	−6.13569	−0.293846
$437$	12.4178	0.594022
$438$	−1.32514	−0.0633177
$439$	−7.98160	−0.380941	−0.190471	−	0.981693i	$-0.561001\pi$
$439$	−7.98160	−0.380941	−0.190471	+	0.981693i	$0.561001\pi$
$440$	10.3634	0.494055
$441$	−6.76974	−0.322368
$442$	−1.42456	−0.0677594
$443$	−6.80632	−0.323378	−0.161689	−	0.986842i	$-0.551694\pi$
$443$	−6.80632	−0.323378	−0.161689	+	0.986842i	$0.551694\pi$
$444$	15.5928	0.740003
$445$	8.42318	0.399297
$446$	8.73521	0.413624
$447$	1.20969	0.0572163
$448$	0.523975	0.0247555
$449$	−2.60492	−0.122934	−0.0614669	−	0.998109i	$-0.519578\pi$
$449$	−2.60492	−0.122934	−0.0614669	+	0.998109i	$0.519578\pi$
$450$	2.48681	0.117229
$451$	−46.4296	−2.18628
$452$	−17.8115	−0.837784
$453$	−19.6838	−0.924825
$454$	12.8117	0.601284
$455$	0.982261	0.0460491
$456$	5.83103	0.273063
$457$	36.9769	1.72971	0.864853	−	0.502026i	$-0.167412\pi$
$457$	36.9769	1.72971	0.864853	+	0.502026i	$0.167412\pi$
$458$	−6.53176	−0.305209
$459$	−1.00000	−0.0466760
$460$	−6.22009	−0.290013
$461$	−11.9000	−0.554239	−0.277119	−	0.960835i	$-0.589380\pi$
$461$	−11.9000	−0.554239	−0.277119	+	0.960835i	$0.589380\pi$
$462$	1.63684	0.0761527
$463$	−18.0312	−0.837983	−0.418991	−	0.907990i	$-0.637616\pi$
$463$	−18.0312	−0.837983	−0.418991	+	0.907990i	$0.637616\pi$
$464$	−13.9522	−0.647715
$465$	−1.94970	−0.0904153
$466$	−2.41200	−0.111734
$467$	−36.7066	−1.69858	−0.849289	−	0.527929i	$-0.822969\pi$
$467$	−36.7066	−1.69858	−0.849289	+	0.527929i	$0.822969\pi$
$468$	4.12110	0.190498
$469$	5.95683	0.275061
$470$	4.79408	0.221134
$471$	1.00000	0.0460776
$472$	0.563369	0.0259312
$473$	−62.9908	−2.89632
$474$	3.62473	0.166489
$475$	−11.8970	−0.545872
$476$	−0.800245	−0.0366792
$477$	10.7649	0.492890
$478$	−5.63865	−0.257906
$479$	−39.4505	−1.80254	−0.901269	−	0.433260i	$-0.857363\pi$
$479$	−39.4505	−1.80254	−0.901269	+	0.433260i	$0.857363\pi$
$480$	−4.51349	−0.206011
$481$	23.1054	1.05352
$482$	12.5312	0.570782
$483$	−2.16063	−0.0983121
$484$	−40.0462	−1.82028
$485$	0.287008	0.0130324
$486$	−0.576474	−0.0261494
$487$	37.0843	1.68045	0.840225	−	0.542239i	$-0.182423\pi$
$487$	37.0843	1.68045	0.840225	+	0.542239i	$0.182423\pi$
$488$	−16.2602	−0.736063
$489$	−13.6542	−0.617466
$490$	3.23271	0.146039
$491$	35.3651	1.59601	0.798003	−	0.602653i	$-0.205890\pi$
$491$	35.3651	1.59601	0.798003	+	0.602653i	$0.205890\pi$
$492$	13.0855	0.589941
$493$	6.59210	0.296893
$494$	3.92875	0.176763
$495$	4.90152	0.220307
$496$	−4.98164	−0.223682
$497$	−3.77232	−0.169211
$498$	−6.78572	−0.304075
$499$	−23.6503	−1.05873	−0.529366	−	0.848394i	$-0.677570\pi$
$499$	−23.6503	−1.05873	−0.529366	+	0.848394i	$0.677570\pi$
$500$	12.8664	0.575401
$501$	18.4779	0.825530
$502$	13.4503	0.600314
$503$	5.15704	0.229941	0.114971	−	0.993369i	$-0.463323\pi$
$503$	5.15704	0.229941	0.114971	+	0.993369i	$0.463323\pi$
$504$	−1.01457	−0.0451925
$505$	9.56353	0.425571
$506$	−15.3591	−0.682794
$507$	−6.89337	−0.306145
$508$	12.0659	0.535336
$509$	14.8015	0.656066	0.328033	−	0.944666i	$-0.393614\pi$
$509$	14.8015	0.656066	0.328033	+	0.944666i	$0.393614\pi$
$510$	0.477524	0.0211451
$511$	−1.10305	−0.0487959
$512$	−20.4822	−0.905196
$513$	2.75787	0.121763
$514$	−7.67986	−0.338744
$515$	5.95187	0.262271
$516$	17.7531	0.781535
$517$	−59.4054	−2.61265
$518$	−2.58645	−0.113642
$519$	12.7582	0.560024
$520$	−4.32800	−0.189795
$521$	−6.86900	−0.300936	−0.150468	−	0.988615i	$-0.548078\pi$
$521$	−6.86900	−0.300936	−0.150468	+	0.988615i	$0.548078\pi$
$522$	3.80018	0.166329
$523$	−43.1892	−1.88853	−0.944267	−	0.329181i	$-0.893227\pi$
$523$	−43.1892	−1.88853	−0.944267	+	0.329181i	$0.893227\pi$
$524$	27.6837	1.20937
$525$	2.07002	0.0903430
$526$	7.78538	0.339459
$527$	2.35371	0.102529
$528$	12.5238	0.545026
$529$	−2.72601	−0.118522
$530$	−5.14049	−0.223288
$531$	0.266454	0.0115631
$532$	2.20697	0.0956844
$533$	19.3901	0.839878
$534$	−5.86193	−0.253671
$535$	−1.94830	−0.0842325
$536$	−26.2467	−1.13369
$537$	9.56385	0.412711
$538$	6.50740	0.280554
$539$	−40.0578	−1.72541
$540$	−1.38142	−0.0594470
$541$	44.4463	1.91089	0.955447	−	0.295162i	$-0.0953737\pi$
$541$	44.4463	1.91089	0.955447	+	0.295162i	$0.0953737\pi$
$542$	−4.78489	−0.205528
$543$	−18.9044	−0.811267
$544$	5.44875	0.233613
$545$	3.04766	0.130547
$546$	−0.683584	−0.0292547
$547$	12.2060	0.521890	0.260945	−	0.965354i	$-0.415966\pi$
$547$	12.2060	0.521890	0.260945	+	0.965354i	$0.415966\pi$
$548$	14.7025	0.628059
$549$	−7.69049	−0.328222
$550$	14.7149	0.627447
$551$	−18.1802	−0.774501
$552$	9.52008	0.405202
$553$	3.01722	0.128305
$554$	−13.3407	−0.566793
$555$	−7.74511	−0.328762
$556$	2.68463	0.113854
$557$	42.0547	1.78191	0.890956	−	0.454089i	$-0.150035\pi$
$557$	42.0547	1.78191	0.890956	+	0.454089i	$0.150035\pi$
$558$	1.35685	0.0574402
$559$	26.3064	1.11264
$560$	−0.841289	−0.0355509
$561$	−5.91719	−0.249824
$562$	5.81021	0.245089
$563$	−9.38469	−0.395518	−0.197759	−	0.980251i	$-0.563366\pi$
$563$	−9.38469	−0.395518	−0.197759	+	0.980251i	$0.563366\pi$
$564$	16.7426	0.704990
$565$	8.84717	0.372203
$566$	7.62080	0.320326
$567$	−0.479856	−0.0201521
$568$	16.6214	0.697419
$569$	26.4031	1.10688	0.553438	−	0.832890i	$-0.313316\pi$
$569$	26.4031	1.10688	0.553438	+	0.832890i	$0.313316\pi$
$570$	−1.31695	−0.0551609
$571$	45.7159	1.91315	0.956575	−	0.291487i	$-0.0941500\pi$
$571$	45.7159	1.91315	0.956575	+	0.291487i	$0.0941500\pi$
$572$	24.3853	1.01960
$573$	−19.8281	−0.828331
$574$	−2.17055	−0.0905970
$575$	−19.4237	−0.810026
$576$	−1.09194	−0.0454976
$577$	−39.1130	−1.62830	−0.814148	−	0.580658i	$-0.802795\pi$
$577$	−39.1130	−1.62830	−0.814148	+	0.580658i	$0.802795\pi$
$578$	−0.576474	−0.0239782
$579$	−23.4458	−0.974372
$580$	9.10649	0.378126
$581$	−5.64842	−0.234336
$582$	−0.199737	−0.00827937
$583$	63.6979	2.63810
$584$	4.86019	0.201116
$585$	−2.04699	−0.0846327
$586$	−10.9204	−0.451119
$587$	−15.3040	−0.631662	−0.315831	−	0.948815i	$-0.602283\pi$
$587$	−15.3040	−0.631662	−0.315831	+	0.948815i	$0.602283\pi$
$588$	11.2897	0.465581
$589$	−6.49124	−0.267467
$590$	−0.127238	−0.00523831
$591$	9.98421	0.410696
$592$	−19.7894	−0.813338
$593$	17.9368	0.736574	0.368287	−	0.929712i	$-0.379944\pi$
$593$	17.9368	0.736574	0.368287	+	0.929712i	$0.379944\pi$
$594$	−3.41111	−0.139959
$595$	0.397490	0.0162955
$596$	−2.01737	−0.0826347
$597$	4.12796	0.168946
$598$	6.41432	0.262301
$599$	34.6784	1.41692	0.708460	−	0.705751i	$-0.249390\pi$
$599$	34.6784	1.41692	0.708460	+	0.705751i	$0.249390\pi$
$600$	−9.12083	−0.372356
$601$	−8.64543	−0.352655	−0.176327	−	0.984332i	$-0.556422\pi$
$601$	−8.64543	−0.352655	−0.176327	+	0.984332i	$0.556422\pi$
$602$	−2.94477	−0.120020
$603$	−12.4138	−0.505529
$604$	32.8262	1.33568
$605$	19.8913	0.808698
$606$	−6.65553	−0.270363
$607$	9.90149	0.401889	0.200945	−	0.979603i	$-0.435599\pi$
$607$	9.90149	0.401889	0.200945	+	0.979603i	$0.435599\pi$
$608$	−15.0270	−0.609424
$609$	3.16326	0.128182
$610$	3.67239	0.148691
$611$	24.8091	1.00367
$612$	1.66768	0.0674119
$613$	−24.5725	−0.992473	−0.496236	−	0.868188i	$-0.665285\pi$
$613$	−24.5725	−0.992473	−0.496236	+	0.868188i	$0.665285\pi$
$614$	−13.2415	−0.534384
$615$	−6.49971	−0.262094
$616$	−6.00340	−0.241884
$617$	−44.2221	−1.78031	−0.890157	−	0.455654i	$-0.849406\pi$
$617$	−44.2221	−1.78031	−0.890157	+	0.455654i	$0.849406\pi$
$618$	−4.14208	−0.166619
$619$	−0.932339	−0.0374739	−0.0187369	−	0.999824i	$-0.505965\pi$
$619$	−0.932339	−0.0374739	−0.0187369	+	0.999824i	$0.505965\pi$
$620$	3.25148	0.130582
$621$	4.50266	0.180686
$622$	−12.2012	−0.489225
$623$	−4.87946	−0.195491
$624$	−5.23022	−0.209376
$625$	15.1783	0.607133
$626$	−7.61171	−0.304225
$627$	16.3188	0.651712
$628$	−1.66768	−0.0665476
$629$	9.35003	0.372810
$630$	0.229143	0.00912926
$631$	41.6049	1.65627	0.828133	−	0.560532i	$-0.189403\pi$
$631$	41.6049	1.65627	0.828133	+	0.560532i	$0.189403\pi$
$632$	−13.2943	−0.528821
$633$	−8.50287	−0.337959
$634$	5.76596	0.228995
$635$	−5.99323	−0.237834
$636$	−17.9524	−0.711857
$637$	16.7291	0.662831
$638$	22.4864	0.890244
$639$	7.86135	0.310990
$640$	9.54840	0.377434
$641$	34.7701	1.37334	0.686669	−	0.726970i	$-0.259072\pi$
$641$	34.7701	1.37334	0.686669	+	0.726970i	$0.259072\pi$
$642$	1.35588	0.0535124
$643$	−0.624812	−0.0246402	−0.0123201	−	0.999924i	$-0.503922\pi$
$643$	−0.624812	−0.0246402	−0.0123201	+	0.999924i	$0.503922\pi$
$644$	3.60324	0.141987
$645$	−8.81813	−0.347213
$646$	1.58984	0.0625515
$647$	−13.1851	−0.518360	−0.259180	−	0.965829i	$-0.583452\pi$
$647$	−13.1851	−0.518360	−0.259180	+	0.965829i	$0.583452\pi$
$648$	2.11432	0.0830584
$649$	1.57666	0.0618893
$650$	−6.14531	−0.241039
$651$	1.12944	0.0442664
$652$	22.7709	0.891777
$653$	−12.6116	−0.493529	−0.246765	−	0.969075i	$-0.579367\pi$
$653$	−12.6116	−0.493529	−0.246765	+	0.969075i	$0.579367\pi$
$654$	−2.12095	−0.0829358
$655$	−13.7507	−0.537286
$656$	−16.6073	−0.648405
$657$	2.29870	0.0896809
$658$	−2.77716	−0.108265
$659$	29.2162	1.13810	0.569051	−	0.822302i	$-0.307311\pi$
$659$	29.2162	1.13810	0.569051	+	0.822302i	$0.307311\pi$
$660$	−8.17415	−0.318178
$661$	−18.3845	−0.715073	−0.357536	−	0.933899i	$-0.616383\pi$
$661$	−18.3845	−0.715073	−0.357536	+	0.933899i	$0.616383\pi$
$662$	−7.15661	−0.278149
$663$	2.47116	0.0959719
$664$	24.8878	0.965835
$665$	−1.09623	−0.0425098
$666$	5.39005	0.208860
$667$	−29.6820	−1.14929
$668$	−30.8151	−1.19227
$669$	−15.1528	−0.585842
$670$	5.92788	0.229014
$671$	−45.5061	−1.75674
$672$	2.61462	0.100861
$673$	−21.1181	−0.814043	−0.407021	−	0.913419i	$-0.633433\pi$
$673$	−21.1181	−0.814043	−0.407021	+	0.913419i	$0.633433\pi$
$674$	3.20474	0.123442
$675$	−4.31383	−0.166039
$676$	11.4959	0.442150
$677$	3.95528	0.152014	0.0760068	−	0.997107i	$-0.475783\pi$
$677$	3.95528	0.152014	0.0760068	+	0.997107i	$0.475783\pi$
$678$	−6.15700	−0.236458
$679$	−0.166261	−0.00638051
$680$	−1.75140	−0.0671632
$681$	−22.2243	−0.851636
$682$	8.02877	0.307437
$683$	−10.9882	−0.420453	−0.210227	−	0.977653i	$-0.567420\pi$
$683$	−10.9882	−0.420453	−0.210227	+	0.977653i	$0.567420\pi$
$684$	−4.59924	−0.175856
$685$	−7.30287	−0.279028
$686$	−3.80905	−0.145430
$687$	11.3305	0.432287
$688$	−22.5310	−0.858986
$689$	−26.6018	−1.01345
$690$	−2.15013	−0.0818540
$691$	44.1142	1.67818	0.839092	−	0.543989i	$-0.183087\pi$
$691$	44.1142	1.67818	0.839092	+	0.543989i	$0.183087\pi$
$692$	−21.2766	−0.808815
$693$	−2.83940	−0.107860
$694$	7.33971	0.278612
$695$	−1.33348	−0.0505819
$696$	−13.9378	−0.528312
$697$	7.84656	0.297210
$698$	−11.2836	−0.427092
$699$	4.18406	0.158256
$700$	−3.45212	−0.130478
$701$	−26.1733	−0.988553	−0.494277	−	0.869305i	$-0.664567\pi$
$701$	−26.1733	−0.988553	−0.494277	+	0.869305i	$0.664567\pi$
$702$	1.42456	0.0537666
$703$	−25.7862	−0.972544
$704$	−6.46123	−0.243517
$705$	−8.31621	−0.313207
$706$	9.49696	0.357423
$707$	−5.54005	−0.208355
$708$	−0.444359	−0.0167000
$709$	−1.69026	−0.0634792	−0.0317396	−	0.999496i	$-0.510105\pi$
$709$	−1.69026	−0.0634792	−0.0317396	+	0.999496i	$0.510105\pi$
$710$	−3.75398	−0.140884
$711$	−6.28776	−0.235809
$712$	21.4997	0.805734
$713$	−10.5980	−0.396897
$714$	−0.276625	−0.0103524
$715$	−12.1124	−0.452979
$716$	−15.9494	−0.596058
$717$	9.78127	0.365288
$718$	6.00079	0.223948
$719$	3.93316	0.146682	0.0733411	−	0.997307i	$-0.476634\pi$
$719$	3.93316	0.146682	0.0733411	+	0.997307i	$0.476634\pi$
$720$	1.75321	0.0653383
$721$	−3.44786	−0.128405
$722$	6.56843	0.244452
$723$	−21.7377	−0.808435
$724$	31.5265	1.17167
$725$	28.4372	1.05613
$726$	−13.8430	−0.513760
$727$	−29.0999	−1.07926	−0.539628	−	0.841903i	$-0.681435\pi$
$727$	−29.0999	−1.07926	−0.539628	+	0.841903i	$0.681435\pi$
$728$	2.50716	0.0929217
$729$	1.00000	0.0370370
$730$	−1.09768	−0.0406271
$731$	10.6454	0.393734
$732$	12.8253	0.474035
$733$	−52.2535	−1.93003	−0.965013	−	0.262203i	$-0.915551\pi$
$733$	−52.2535	−1.93003	−0.965013	+	0.262203i	$0.915551\pi$
$734$	13.0606	0.482075
$735$	−5.60773	−0.206844
$736$	−24.5339	−0.904332
$737$	−73.4548	−2.70574
$738$	4.52334	0.166506
$739$	−4.32924	−0.159254	−0.0796268	−	0.996825i	$-0.525373\pi$
$739$	−4.32924	−0.159254	−0.0796268	+	0.996825i	$0.525373\pi$
$740$	12.9164	0.474815
$741$	−6.81514	−0.250360
$742$	2.97783	0.109320
$743$	36.1019	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$743$	36.1019	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$744$	−4.97651	−0.182448
$745$	1.00205	0.0367122
$746$	−13.3661	−0.489369
$747$	11.7711	0.430681
$748$	9.86797	0.360809
$749$	1.12863	0.0412394
$750$	4.44758	0.162402
$751$	−2.95922	−0.107983	−0.0539917	−	0.998541i	$-0.517194\pi$
$751$	−2.95922	−0.107983	−0.0539917	+	0.998541i	$0.517194\pi$
$752$	−21.2486	−0.774855
$753$	−23.3319	−0.850263
$754$	−9.39085	−0.341994
$755$	−16.3051	−0.593404
$756$	0.800245	0.0291046
$757$	49.0765	1.78372	0.891859	−	0.452314i	$-0.149401\pi$
$757$	49.0765	1.78372	0.891859	+	0.452314i	$0.149401\pi$
$758$	−10.9495	−0.397705
$759$	26.6431	0.967084
$760$	4.83014	0.175208
$761$	51.0121	1.84919	0.924593	−	0.380955i	$-0.124405\pi$
$761$	51.0121	1.84919	0.924593	+	0.380955i	$0.124405\pi$
$762$	4.17086	0.151094
$763$	−1.76548	−0.0639146
$764$	33.0669	1.19632
$765$	−0.828352	−0.0299491
$766$	−0.649628	−0.0234720
$767$	−0.658450	−0.0237753
$768$	−4.46112	−0.160977
$769$	−7.49780	−0.270378	−0.135189	−	0.990820i	$-0.543164\pi$
$769$	−7.49780	−0.270378	−0.135189	+	0.990820i	$0.543164\pi$
$770$	1.35588	0.0488625
$771$	13.3221	0.479784
$772$	39.1000	1.40724
$773$	53.9391	1.94005	0.970027	−	0.242996i	$-0.0781303\pi$
$773$	53.9391	1.94005	0.970027	+	0.242996i	$0.0781303\pi$
$774$	6.13679	0.220582
$775$	10.1535	0.364725
$776$	0.732572	0.0262978
$777$	4.48667	0.160958
$778$	16.2306	0.581896
$779$	−21.6398	−0.775326
$780$	3.41372	0.122231
$781$	46.5171	1.66451
$782$	2.59567	0.0928210
$783$	−6.59210	−0.235583
$784$	−14.3282	−0.511721
$785$	0.828352	0.0295652
$786$	9.56954	0.341334
$787$	29.7518	1.06054	0.530268	−	0.847830i	$-0.322091\pi$
$787$	29.7518	1.06054	0.530268	+	0.847830i	$0.322091\pi$
$788$	−16.6504	−0.593148
$789$	−13.5052	−0.480797
$790$	3.00255	0.106826
$791$	−5.12507	−0.182227
$792$	12.5108	0.444553
$793$	19.0044	0.674867
$794$	16.5924	0.588842
$795$	8.91711	0.316257
$796$	−6.88410	−0.244000
$797$	−41.0298	−1.45335	−0.726675	−	0.686981i	$-0.758936\pi$
$797$	−41.0298	−1.45335	−0.726675	+	0.686981i	$0.758936\pi$
$798$	0.762895	0.0270062
$799$	10.0395	0.355171
$800$	23.5050	0.831027
$801$	10.1686	0.359290
$802$	−11.0088	−0.388735
$803$	13.6019	0.479999
$804$	20.7022	0.730110
$805$	−1.78976	−0.0630809
$806$	−3.35301	−0.118105
$807$	−11.2883	−0.397366
$808$	24.4103	0.858753
$809$	14.8953	0.523692	0.261846	−	0.965110i	$-0.415669\pi$
$809$	14.8953	0.523692	0.261846	+	0.965110i	$0.415669\pi$
$810$	−0.477524	−0.0167785
$811$	−20.0677	−0.704673	−0.352336	−	0.935873i	$-0.614613\pi$
$811$	−20.0677	−0.704673	−0.352336	+	0.935873i	$0.614613\pi$
$812$	−5.27530	−0.185127
$813$	8.30026	0.291103
$814$	31.8939	1.11788
$815$	−11.3105	−0.396190
$816$	−2.11650	−0.0740924
$817$	−29.3586	−1.02713
$818$	−17.8211	−0.623101
$819$	1.18580	0.0414352
$820$	10.8394	0.378529
$821$	−37.3729	−1.30432	−0.652161	−	0.758080i	$-0.726137\pi$
$821$	−37.3729	−1.30432	−0.652161	+	0.758080i	$0.726137\pi$
$822$	5.08228	0.177265
$823$	−25.4583	−0.887419	−0.443709	−	0.896171i	$-0.646338\pi$
$823$	−25.4583	−0.887419	−0.443709	+	0.896171i	$0.646338\pi$
$824$	15.1918	0.529231
$825$	−25.5258	−0.888693
$826$	0.0737077	0.00256462
$827$	−12.9303	−0.449631	−0.224815	−	0.974401i	$-0.572178\pi$
$827$	−12.9303	−0.449631	−0.224815	+	0.974401i	$0.572178\pi$
$828$	−7.50899	−0.260956
$829$	−14.7331	−0.511700	−0.255850	−	0.966716i	$-0.582355\pi$
$829$	−14.7331	−0.511700	−0.255850	+	0.966716i	$0.582355\pi$
$830$	−5.62097	−0.195107
$831$	23.1419	0.802785
$832$	2.69836	0.0935490
$833$	6.76974	0.234558
$834$	0.928010	0.0321344
$835$	15.3062	0.529692
$836$	−27.2146	−0.941236
$837$	−2.35371	−0.0813562
$838$	−8.30672	−0.286951
$839$	17.6593	0.609667	0.304834	−	0.952406i	$-0.401399\pi$
$839$	17.6593	0.609667	0.304834	+	0.952406i	$0.401399\pi$
$840$	−0.840421	−0.0289973
$841$	14.4558	0.498476
$842$	−3.59154	−0.123773
$843$	−10.0789	−0.347135
$844$	14.1800	0.488097
$845$	−5.71014	−0.196435
$846$	5.78749	0.198978
$847$	−11.5228	−0.395930
$848$	22.7839	0.782403
$849$	−13.2197	−0.453698
$850$	−2.48681	−0.0852970
$851$	−42.1000	−1.44317
$852$	−13.1102	−0.449148
$853$	−30.9734	−1.06051	−0.530255	−	0.847838i	$-0.677904\pi$
$853$	−30.9734	−1.06051	−0.530255	+	0.847838i	$0.677904\pi$
$854$	−2.12738	−0.0727974
$855$	2.28449	0.0781279
$856$	−4.97293	−0.169971
$857$	−39.7544	−1.35799	−0.678993	−	0.734145i	$-0.737583\pi$
$857$	−39.7544	−1.35799	−0.678993	+	0.734145i	$0.737583\pi$
$858$	8.42939	0.287775
$859$	24.0479	0.820504	0.410252	−	0.911972i	$-0.365441\pi$
$859$	24.0479	0.820504	0.410252	+	0.911972i	$0.365441\pi$
$860$	14.7058	0.501463
$861$	3.76522	0.128318
$862$	−5.15024	−0.175418
$863$	20.2167	0.688184	0.344092	−	0.938936i	$-0.388187\pi$
$863$	20.2167	0.688184	0.344092	+	0.938936i	$0.388187\pi$
$864$	−5.44875	−0.185370
$865$	10.5683	0.359333
$866$	22.0402	0.748958
$867$	1.00000	0.0339618
$868$	−1.88355	−0.0639318
$869$	−37.2059	−1.26212
$870$	3.14788	0.106723
$871$	30.6765	1.03943
$872$	7.77897	0.263429
$873$	0.346481	0.0117266
$874$	−7.15852	−0.242141
$875$	3.70215	0.125156
$876$	−3.83349	−0.129522
$877$	20.0885	0.678341	0.339171	−	0.940725i	$-0.389854\pi$
$877$	20.0885	0.678341	0.339171	+	0.940725i	$0.389854\pi$
$878$	4.60119	0.155283
$879$	18.9435	0.638949
$880$	10.3741	0.349710
$881$	−9.64903	−0.325084	−0.162542	−	0.986702i	$-0.551969\pi$
$881$	−9.64903	−0.325084	−0.162542	+	0.986702i	$0.551969\pi$
$882$	3.90258	0.131407
$883$	−6.79260	−0.228589	−0.114295	−	0.993447i	$-0.536461\pi$
$883$	−6.79260	−0.228589	−0.114295	+	0.993447i	$0.536461\pi$
$884$	−4.12110	−0.138608
$885$	0.220718	0.00741934
$886$	3.92367	0.131818
$887$	54.1860	1.81939	0.909693	−	0.415280i	$-0.136317\pi$
$887$	54.1860	1.81939	0.909693	+	0.415280i	$0.136317\pi$
$888$	−19.7690	−0.663403
$889$	3.47182	0.116441
$890$	−4.85574	−0.162765
$891$	5.91719	0.198233
$892$	25.2700	0.846103
$893$	−27.6875	−0.926528
$894$	−0.697354	−0.0233230
$895$	7.92224	0.264811
$896$	−5.53129	−0.184787
$897$	−11.1268	−0.371513
$898$	1.50167	0.0501114
$899$	15.5159	0.517485
$900$	7.19408	0.239803
$901$	−10.7649	−0.358630
$902$	26.7654	0.891192
$903$	5.10825	0.169992
$904$	22.5819	0.751062
$905$	−15.6595	−0.520540
$906$	11.3472	0.376985
$907$	−20.9908	−0.696987	−0.348493	−	0.937311i	$-0.613307\pi$
$907$	−20.9908	−0.696987	−0.348493	+	0.937311i	$0.613307\pi$
$908$	37.0629	1.22998
$909$	11.5452	0.382931
$910$	−0.566248	−0.0187709
$911$	33.3813	1.10597	0.552985	−	0.833191i	$-0.313489\pi$
$911$	33.3813	1.10597	0.552985	+	0.833191i	$0.313489\pi$
$912$	5.83704	0.193284
$913$	69.6517	2.30513
$914$	−21.3162	−0.705078
$915$	−6.37043	−0.210600
$916$	−18.8957	−0.624330
$917$	7.96567	0.263050
$918$	0.576474	0.0190265
$919$	37.4810	1.23639	0.618193	−	0.786027i	$-0.287865\pi$
$919$	37.4810	1.23639	0.618193	+	0.786027i	$0.287865\pi$
$920$	7.88598	0.259993
$921$	22.9698	0.756882
$922$	6.86005	0.225924
$923$	−19.4267	−0.639436
$924$	4.73520	0.155777
$925$	40.3344	1.32619
$926$	10.3945	0.341586
$927$	7.18519	0.235993
$928$	35.9187	1.17909
$929$	21.0780	0.691548	0.345774	−	0.938318i	$-0.387616\pi$
$929$	21.0780	0.691548	0.345774	+	0.938318i	$0.387616\pi$
$930$	1.12395	0.0368559
$931$	−18.6701	−0.611887
$932$	−6.97767	−0.228561
$933$	21.1653	0.692920
$934$	21.1604	0.692389
$935$	−4.90152	−0.160297
$936$	−5.22483	−0.170779
$937$	−11.3980	−0.372356	−0.186178	−	0.982516i	$-0.559610\pi$
$937$	−11.3980	−0.372356	−0.186178	+	0.982516i	$0.559610\pi$
$938$	−3.43396	−0.112123
$939$	13.2039	0.430893
$940$	13.8688	0.452349
$941$	−12.5444	−0.408935	−0.204468	−	0.978873i	$-0.565546\pi$
$941$	−12.5444	−0.408935	−0.204468	+	0.978873i	$0.565546\pi$
$942$	−0.576474	−0.0187825
$943$	−35.3304	−1.15052
$944$	0.563951	0.0183550
$945$	−0.397490	−0.0129303
$946$	36.3125	1.18062
$947$	−0.0831248	−0.00270119	−0.00135060	−	0.999999i	$-0.500430\pi$
$947$	−0.0831248	−0.00270119	−0.00135060	+	0.999999i	$0.500430\pi$
$948$	10.4860	0.340568
$949$	−5.68046	−0.184395
$950$	6.85831	0.222513
$951$	−10.0021	−0.324341
$952$	1.01457	0.0328824
$953$	−56.8291	−1.84088	−0.920438	−	0.390888i	$-0.872168\pi$
$953$	−56.8291	−1.84088	−0.920438	+	0.390888i	$0.872168\pi$
$954$	−6.20568	−0.200916
$955$	−16.4246	−0.531489
$956$	−16.3120	−0.527568
$957$	−39.0067	−1.26091
$958$	22.7422	0.734766
$959$	4.23048	0.136609
$960$	−0.904513	−0.0291930
$961$	−25.4600	−0.821291
$962$	−13.3197	−0.429444
$963$	−2.35202	−0.0757929
$964$	36.2515	1.16758
$965$	−19.4213	−0.625195
$966$	1.24555	0.0400748
$967$	−48.6775	−1.56536	−0.782682	−	0.622422i	$-0.786149\pi$
$967$	−48.6775	−1.56536	−0.782682	+	0.622422i	$0.786149\pi$
$968$	50.7715	1.63186
$969$	−2.75787	−0.0885956
$970$	−0.165453	−0.00531237
$971$	−32.6531	−1.04789	−0.523943	−	0.851753i	$-0.675540\pi$
$971$	−32.6531	−1.04789	−0.523943	+	0.851753i	$0.675540\pi$
$972$	−1.66768	−0.0534908
$973$	0.772474	0.0247644
$974$	−21.3781	−0.684999
$975$	10.6602	0.341399
$976$	−16.2769	−0.521012
$977$	1.95573	0.0625694	0.0312847	−	0.999511i	$-0.490040\pi$
$977$	1.95573	0.0625694	0.0312847	+	0.999511i	$0.490040\pi$
$978$	7.87132	0.251697
$979$	60.1695	1.92303
$980$	9.35188	0.298735
$981$	3.67918	0.117467
$982$	−20.3871	−0.650578
$983$	−15.5246	−0.495157	−0.247578	−	0.968868i	$-0.579635\pi$
$983$	−15.5246	−0.495157	−0.247578	+	0.968868i	$0.579635\pi$
$984$	−16.5901	−0.528874
$985$	8.27044	0.263518
$986$	−3.80018	−0.121022
$987$	4.81750	0.153343
$988$	11.3655	0.361583
$989$	−47.9326	−1.52417
$990$	−2.82560	−0.0898034
$991$	−27.8062	−0.883293	−0.441646	−	0.897189i	$-0.645605\pi$
$991$	−27.8062	−0.883293	−0.441646	+	0.897189i	$0.645605\pi$
$992$	12.8248	0.407188
$993$	12.4144	0.393961
$994$	2.17464	0.0689755
$995$	3.41940	0.108402
$996$	−19.6304	−0.622012
$997$	58.8387	1.86344	0.931721	−	0.363175i	$-0.118307\pi$
$997$	58.8387	1.86344	0.931721	+	0.363175i	$0.118307\pi$
$998$	13.6338	0.431569
$999$	−9.35003	−0.295822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.e.1.20	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.e.1.20	✓	46	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)

\(f(q)\)	\(=\)	\(q-0.576474 q^{2} +1.00000 q^{3} -1.66768 q^{4} +0.828352 q^{5} -0.576474 q^{6} -0.479856 q^{7} +2.11432 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.576474 q^{2} +1.00000 q^{3} -1.66768 q^{4} +0.828352 q^{5} -0.576474 q^{6} -0.479856 q^{7} +2.11432 q^{8} +1.00000 q^{9} -0.477524 q^{10} +5.91719 q^{11} -1.66768 q^{12} -2.47116 q^{13} +0.276625 q^{14} +0.828352 q^{15} +2.11650 q^{16} -1.00000 q^{17} -0.576474 q^{18} +2.75787 q^{19} -1.38142 q^{20} -0.479856 q^{21} -3.41111 q^{22} +4.50266 q^{23} +2.11432 q^{24} -4.31383 q^{25} +1.42456 q^{26} +1.00000 q^{27} +0.800245 q^{28} -6.59210 q^{29} -0.477524 q^{30} -2.35371 q^{31} -5.44875 q^{32} +5.91719 q^{33} +0.576474 q^{34} -0.397490 q^{35} -1.66768 q^{36} -9.35003 q^{37} -1.58984 q^{38} -2.47116 q^{39} +1.75140 q^{40} -7.84656 q^{41} +0.276625 q^{42} -10.6454 q^{43} -9.86797 q^{44} +0.828352 q^{45} -2.59567 q^{46} -10.0395 q^{47} +2.11650 q^{48} -6.76974 q^{49} +2.48681 q^{50} -1.00000 q^{51} +4.12110 q^{52} +10.7649 q^{53} -0.576474 q^{54} +4.90152 q^{55} -1.01457 q^{56} +2.75787 q^{57} +3.80018 q^{58} +0.266454 q^{59} -1.38142 q^{60} -7.69049 q^{61} +1.35685 q^{62} -0.479856 q^{63} -1.09194 q^{64} -2.04699 q^{65} -3.41111 q^{66} -12.4138 q^{67} +1.66768 q^{68} +4.50266 q^{69} +0.229143 q^{70} +7.86135 q^{71} +2.11432 q^{72} +2.29870 q^{73} +5.39005 q^{74} -4.31383 q^{75} -4.59924 q^{76} -2.83940 q^{77} +1.42456 q^{78} -6.28776 q^{79} +1.75321 q^{80} +1.00000 q^{81} +4.52334 q^{82} +11.7711 q^{83} +0.800245 q^{84} -0.828352 q^{85} +6.13679 q^{86} -6.59210 q^{87} +12.5108 q^{88} +10.1686 q^{89} -0.477524 q^{90} +1.18580 q^{91} -7.50899 q^{92} -2.35371 q^{93} +5.78749 q^{94} +2.28449 q^{95} -5.44875 q^{96} +0.346481 q^{97} +3.90258 q^{98} +5.91719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

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