LMFDB - Embedded newform 8038.2.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.27949 q^{3} +1.00000 q^{4} -1.91910 q^{5} -3.27949 q^{6} +3.63095 q^{7} +1.00000 q^{8} +7.75505 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.27949 q^{3} +1.00000 q^{4} -1.91910 q^{5} -3.27949 q^{6} +3.63095 q^{7} +1.00000 q^{8} +7.75505 q^{9} -1.91910 q^{10} +5.12513 q^{11} -3.27949 q^{12} +3.75020 q^{13} +3.63095 q^{14} +6.29366 q^{15} +1.00000 q^{16} +1.11680 q^{17} +7.75505 q^{18} -4.33396 q^{19} -1.91910 q^{20} -11.9077 q^{21} +5.12513 q^{22} -8.09062 q^{23} -3.27949 q^{24} -1.31707 q^{25} +3.75020 q^{26} -15.5941 q^{27} +3.63095 q^{28} -2.41941 q^{29} +6.29366 q^{30} -8.08705 q^{31} +1.00000 q^{32} -16.8078 q^{33} +1.11680 q^{34} -6.96815 q^{35} +7.75505 q^{36} +3.51955 q^{37} -4.33396 q^{38} -12.2987 q^{39} -1.91910 q^{40} -5.59752 q^{41} -11.9077 q^{42} +10.7732 q^{43} +5.12513 q^{44} -14.8827 q^{45} -8.09062 q^{46} -0.283525 q^{47} -3.27949 q^{48} +6.18382 q^{49} -1.31707 q^{50} -3.66252 q^{51} +3.75020 q^{52} -11.3354 q^{53} -15.5941 q^{54} -9.83562 q^{55} +3.63095 q^{56} +14.2132 q^{57} -2.41941 q^{58} -8.10932 q^{59} +6.29366 q^{60} -13.6307 q^{61} -8.08705 q^{62} +28.1582 q^{63} +1.00000 q^{64} -7.19700 q^{65} -16.8078 q^{66} +3.28752 q^{67} +1.11680 q^{68} +26.5331 q^{69} -6.96815 q^{70} +5.06592 q^{71} +7.75505 q^{72} +3.57492 q^{73} +3.51955 q^{74} +4.31931 q^{75} -4.33396 q^{76} +18.6091 q^{77} -12.2987 q^{78} -13.5129 q^{79} -1.91910 q^{80} +27.8756 q^{81} -5.59752 q^{82} -12.7730 q^{83} -11.9077 q^{84} -2.14324 q^{85} +10.7732 q^{86} +7.93444 q^{87} +5.12513 q^{88} -16.2700 q^{89} -14.8827 q^{90} +13.6168 q^{91} -8.09062 q^{92} +26.5214 q^{93} -0.283525 q^{94} +8.31729 q^{95} -3.27949 q^{96} +11.8767 q^{97} +6.18382 q^{98} +39.7456 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) $ 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.27949	−1.89341	−0.946707	−	0.322096i	$-0.895613\pi$
$3$	−3.27949	−1.89341	−0.946707	+	0.322096i	$0.895613\pi$
$4$	1.00000	0.500000
$5$	−1.91910	−0.858246	−0.429123	−	0.903246i	$-0.641177\pi$
$5$	−1.91910	−0.858246	−0.429123	+	0.903246i	$0.641177\pi$
$6$	−3.27949	−1.33885
$7$	3.63095	1.37237	0.686186	−	0.727426i	$-0.259284\pi$
$7$	3.63095	1.37237	0.686186	+	0.727426i	$0.259284\pi$
$8$	1.00000	0.353553
$9$	7.75505	2.58502
$10$	−1.91910	−0.606872
$11$	5.12513	1.54528	0.772642	−	0.634842i	$-0.218935\pi$
$11$	5.12513	1.54528	0.772642	+	0.634842i	$0.218935\pi$
$12$	−3.27949	−0.946707
$13$	3.75020	1.04012	0.520059	−	0.854130i	$-0.325910\pi$
$13$	3.75020	1.04012	0.520059	+	0.854130i	$0.325910\pi$
$14$	3.63095	0.970413
$15$	6.29366	1.62502
$16$	1.00000	0.250000
$17$	1.11680	0.270863	0.135431	−	0.990787i	$-0.456758\pi$
$17$	1.11680	0.270863	0.135431	+	0.990787i	$0.456758\pi$
$18$	7.75505	1.82788
$19$	−4.33396	−0.994279	−0.497139	−	0.867671i	$-0.665616\pi$
$19$	−4.33396	−0.994279	−0.497139	+	0.867671i	$0.665616\pi$
$20$	−1.91910	−0.429123
$21$	−11.9077	−2.59847
$22$	5.12513	1.09268
$23$	−8.09062	−1.68701	−0.843505	−	0.537121i	$-0.819512\pi$
$23$	−8.09062	−1.68701	−0.843505	+	0.537121i	$0.819512\pi$
$24$	−3.27949	−0.669423
$25$	−1.31707	−0.263414
$26$	3.75020	0.735475
$27$	−15.5941	−3.00109
$28$	3.63095	0.686186
$29$	−2.41941	−0.449274	−0.224637	−	0.974443i	$-0.572120\pi$
$29$	−2.41941	−0.449274	−0.224637	+	0.974443i	$0.572120\pi$
$30$	6.29366	1.14906
$31$	−8.08705	−1.45248	−0.726239	−	0.687443i	$-0.758733\pi$
$31$	−8.08705	−1.45248	−0.726239	+	0.687443i	$0.758733\pi$
$32$	1.00000	0.176777
$33$	−16.8078	−2.92586
$34$	1.11680	0.191529
$35$	−6.96815	−1.17783
$36$	7.75505	1.29251
$37$	3.51955	0.578610	0.289305	−	0.957237i	$-0.406576\pi$
$37$	3.51955	0.578610	0.289305	+	0.957237i	$0.406576\pi$
$38$	−4.33396	−0.703061
$39$	−12.2987	−1.96937
$40$	−1.91910	−0.303436
$41$	−5.59752	−0.874186	−0.437093	−	0.899416i	$-0.643992\pi$
$41$	−5.59752	−0.874186	−0.437093	+	0.899416i	$0.643992\pi$
$42$	−11.9077	−1.83739
$43$	10.7732	1.64290	0.821448	−	0.570283i	$-0.193167\pi$
$43$	10.7732	1.64290	0.821448	+	0.570283i	$0.193167\pi$
$44$	5.12513	0.772642
$45$	−14.8827	−2.21858
$46$	−8.09062	−1.19290
$47$	−0.283525	−0.0413564	−0.0206782	−	0.999786i	$-0.506583\pi$
$47$	−0.283525	−0.0413564	−0.0206782	+	0.999786i	$0.506583\pi$
$48$	−3.27949	−0.473353
$49$	6.18382	0.883403
$50$	−1.31707	−0.186262
$51$	−3.66252	−0.512855
$52$	3.75020	0.520059
$53$	−11.3354	−1.55704	−0.778521	−	0.627619i	$-0.784030\pi$
$53$	−11.3354	−1.55704	−0.778521	+	0.627619i	$0.784030\pi$
$54$	−15.5941	−2.12209
$55$	−9.83562	−1.32623
$56$	3.63095	0.485207
$57$	14.2132	1.88258
$58$	−2.41941	−0.317684
$59$	−8.10932	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$59$	−8.10932	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$60$	6.29366	0.812508
$61$	−13.6307	−1.74523	−0.872616	−	0.488408i	$-0.837578\pi$
$61$	−13.6307	−1.74523	−0.872616	+	0.488408i	$0.837578\pi$
$62$	−8.08705	−1.02706
$63$	28.1582	3.54760
$64$	1.00000	0.125000
$65$	−7.19700	−0.892678
$66$	−16.8078	−2.06890
$67$	3.28752	0.401635	0.200817	−	0.979629i	$-0.435640\pi$
$67$	3.28752	0.401635	0.200817	+	0.979629i	$0.435640\pi$
$68$	1.11680	0.135431
$69$	26.5331	3.19421
$70$	−6.96815	−0.832853
$71$	5.06592	0.601214	0.300607	−	0.953748i	$-0.402811\pi$
$71$	5.06592	0.601214	0.300607	+	0.953748i	$0.402811\pi$
$72$	7.75505	0.913941
$73$	3.57492	0.418412	0.209206	−	0.977872i	$-0.432912\pi$
$73$	3.57492	0.418412	0.209206	+	0.977872i	$0.432912\pi$
$74$	3.51955	0.409139
$75$	4.31931	0.498751
$76$	−4.33396	−0.497139
$77$	18.6091	2.12070
$78$	−12.2987	−1.39256
$79$	−13.5129	−1.52032	−0.760161	−	0.649735i	$-0.774880\pi$
$79$	−13.5129	−1.52032	−0.760161	+	0.649735i	$0.774880\pi$
$80$	−1.91910	−0.214562
$81$	27.8756	3.09729
$82$	−5.59752	−0.618143
$83$	−12.7730	−1.40202	−0.701010	−	0.713151i	$-0.747267\pi$
$83$	−12.7730	−1.40202	−0.701010	+	0.713151i	$0.747267\pi$
$84$	−11.9077	−1.29923
$85$	−2.14324	−0.232467
$86$	10.7732	1.16170
$87$	7.93444	0.850661
$88$	5.12513	0.546341
$89$	−16.2700	−1.72461	−0.862306	−	0.506387i	$-0.830981\pi$
$89$	−16.2700	−1.72461	−0.862306	+	0.506387i	$0.830981\pi$
$90$	−14.8827	−1.56877
$91$	13.6168	1.42743
$92$	−8.09062	−0.843505
$93$	26.5214	2.75014
$94$	−0.283525	−0.0292434
$95$	8.31729	0.853336
$96$	−3.27949	−0.334711
$97$	11.8767	1.20589	0.602947	−	0.797781i	$-0.293993\pi$
$97$	11.8767	1.20589	0.602947	+	0.797781i	$0.293993\pi$
$98$	6.18382	0.624661
$99$	39.7456	3.99459
$100$	−1.31707	−0.131707
$101$	4.72412	0.470067	0.235034	−	0.971987i	$-0.424480\pi$
$101$	4.72412	0.470067	0.235034	+	0.971987i	$0.424480\pi$
$102$	−3.66252	−0.362643
$103$	−15.8486	−1.56160	−0.780802	−	0.624778i	$-0.785189\pi$
$103$	−15.8486	−1.56160	−0.780802	+	0.624778i	$0.785189\pi$
$104$	3.75020	0.367737
$105$	22.8520	2.23012
$106$	−11.3354	−1.10099
$107$	2.30768	0.223092	0.111546	−	0.993759i	$-0.464420\pi$
$107$	2.30768	0.223092	0.111546	+	0.993759i	$0.464420\pi$
$108$	−15.5941	−1.50055
$109$	8.49749	0.813912	0.406956	−	0.913448i	$-0.366590\pi$
$109$	8.49749	0.813912	0.406956	+	0.913448i	$0.366590\pi$
$110$	−9.83562	−0.937789
$111$	−11.5423	−1.09555
$112$	3.63095	0.343093
$113$	2.38345	0.224216	0.112108	−	0.993696i	$-0.464240\pi$
$113$	2.38345	0.224216	0.112108	+	0.993696i	$0.464240\pi$
$114$	14.2132	1.33119
$115$	15.5267	1.44787
$116$	−2.41941	−0.224637
$117$	29.0830	2.68872
$118$	−8.10932	−0.746523
$119$	4.05503	0.371724
$120$	6.29366	0.574530
$121$	15.2669	1.38790
$122$	−13.6307	−1.23406
$123$	18.3570	1.65520
$124$	−8.08705	−0.726239
$125$	12.1231	1.08432
$126$	28.1582	2.50853
$127$	−3.35545	−0.297748	−0.148874	−	0.988856i	$-0.547565\pi$
$127$	−3.35545	−0.297748	−0.148874	+	0.988856i	$0.547565\pi$
$128$	1.00000	0.0883883
$129$	−35.3306	−3.11068
$130$	−7.19700	−0.631218
$131$	−10.2506	−0.895600	−0.447800	−	0.894134i	$-0.647792\pi$
$131$	−10.2506	−0.895600	−0.447800	+	0.894134i	$0.647792\pi$
$132$	−16.8078	−1.46293
$133$	−15.7364	−1.36452
$134$	3.28752	0.283999
$135$	29.9266	2.57568
$136$	1.11680	0.0957644
$137$	−12.2209	−1.04410	−0.522052	−	0.852913i	$-0.674833\pi$
$137$	−12.2209	−1.04410	−0.522052	+	0.852913i	$0.674833\pi$
$138$	26.5331	2.25865
$139$	−12.1671	−1.03200	−0.515999	−	0.856589i	$-0.672579\pi$
$139$	−12.1671	−1.03200	−0.515999	+	0.856589i	$0.672579\pi$
$140$	−6.96815	−0.588916
$141$	0.929818	0.0783048
$142$	5.06592	0.425123
$143$	19.2203	1.60728
$144$	7.75505	0.646254
$145$	4.64309	0.385587
$146$	3.57492	0.295862
$147$	−20.2798	−1.67265
$148$	3.51955	0.289305
$149$	−18.4250	−1.50944	−0.754719	−	0.656048i	$-0.772227\pi$
$149$	−18.4250	−1.50944	−0.754719	+	0.656048i	$0.772227\pi$
$150$	4.31931	0.352670
$151$	−9.34409	−0.760412	−0.380206	−	0.924902i	$-0.624147\pi$
$151$	−9.34409	−0.760412	−0.380206	+	0.924902i	$0.624147\pi$
$152$	−4.33396	−0.351531
$153$	8.66080	0.700184
$154$	18.6091	1.49956
$155$	15.5198	1.24658
$156$	−12.2987	−0.984687
$157$	9.25962	0.738998	0.369499	−	0.929231i	$-0.379529\pi$
$157$	9.25962	0.738998	0.369499	+	0.929231i	$0.379529\pi$
$158$	−13.5129	−1.07503
$159$	37.1744	2.94812
$160$	−1.91910	−0.151718
$161$	−29.3767	−2.31521
$162$	27.8756	2.19012
$163$	−6.79637	−0.532333	−0.266166	−	0.963927i	$-0.585757\pi$
$163$	−6.79637	−0.532333	−0.266166	+	0.963927i	$0.585757\pi$
$164$	−5.59752	−0.437093
$165$	32.2558	2.51111
$166$	−12.7730	−0.991378
$167$	−14.1859	−1.09773	−0.548867	−	0.835910i	$-0.684941\pi$
$167$	−14.1859	−1.09773	−0.548867	+	0.835910i	$0.684941\pi$
$168$	−11.9077	−0.918697
$169$	1.06400	0.0818462
$170$	−2.14324	−0.164379
$171$	−33.6101	−2.57023
$172$	10.7732	0.821448
$173$	−3.35840	−0.255335	−0.127667	−	0.991817i	$-0.540749\pi$
$173$	−3.35840	−0.255335	−0.127667	+	0.991817i	$0.540749\pi$
$174$	7.93444	0.601508
$175$	−4.78221	−0.361501
$176$	5.12513	0.386321
$177$	26.5944	1.99896
$178$	−16.2700	−1.21949
$179$	25.3384	1.89388	0.946941	−	0.321408i	$-0.104156\pi$
$179$	25.3384	1.89388	0.946941	+	0.321408i	$0.104156\pi$
$180$	−14.8827	−1.10929
$181$	18.3920	1.36706	0.683532	−	0.729920i	$-0.260443\pi$
$181$	18.3920	1.36706	0.683532	+	0.729920i	$0.260443\pi$
$182$	13.6168	1.00934
$183$	44.7017	3.30445
$184$	−8.09062	−0.596448
$185$	−6.75435	−0.496590
$186$	26.5214	1.94464
$187$	5.72372	0.418560
$188$	−0.283525	−0.0206782
$189$	−56.6216	−4.11861
$190$	8.31729	0.603400
$191$	2.76917	0.200370	0.100185	−	0.994969i	$-0.468057\pi$
$191$	2.76917	0.200370	0.100185	+	0.994969i	$0.468057\pi$
$192$	−3.27949	−0.236677
$193$	14.1703	1.02000	0.510002	−	0.860173i	$-0.329645\pi$
$193$	14.1703	1.02000	0.510002	+	0.860173i	$0.329645\pi$
$194$	11.8767	0.852696
$195$	23.6025	1.69021
$196$	6.18382	0.441702
$197$	−21.9401	−1.56317	−0.781583	−	0.623802i	$-0.785587\pi$
$197$	−21.9401	−1.56317	−0.781583	+	0.623802i	$0.785587\pi$
$198$	39.7456	2.82460
$199$	19.8899	1.40996	0.704980	−	0.709228i	$-0.250956\pi$
$199$	19.8899	1.40996	0.704980	+	0.709228i	$0.250956\pi$
$200$	−1.31707	−0.0931308
$201$	−10.7814	−0.760461
$202$	4.72412	0.332388
$203$	−8.78477	−0.616570
$204$	−3.66252	−0.256427
$205$	10.7422	0.750267
$206$	−15.8486	−1.10422
$207$	−62.7432	−4.36095
$208$	3.75020	0.260030
$209$	−22.2121	−1.53644
$210$	22.8520	1.57694
$211$	−6.34249	−0.436635	−0.218318	−	0.975878i	$-0.570057\pi$
$211$	−6.34249	−0.436635	−0.218318	+	0.975878i	$0.570057\pi$
$212$	−11.3354	−0.778521
$213$	−16.6136	−1.13835
$214$	2.30768	0.157750
$215$	−20.6748	−1.41001
$216$	−15.5941	−1.06105
$217$	−29.3637	−1.99334
$218$	8.49749	0.575523
$219$	−11.7239	−0.792228
$220$	−9.83562	−0.663117
$221$	4.18820	0.281729
$222$	−11.5423	−0.774670
$223$	12.8708	0.861896	0.430948	−	0.902377i	$-0.358179\pi$
$223$	12.8708	0.861896	0.430948	+	0.902377i	$0.358179\pi$
$224$	3.63095	0.242603
$225$	−10.2139	−0.680929
$226$	2.38345	0.158545
$227$	1.90572	0.126487	0.0632435	−	0.997998i	$-0.479856\pi$
$227$	1.90572	0.126487	0.0632435	+	0.997998i	$0.479856\pi$
$228$	14.2132	0.941291
$229$	−21.6974	−1.43380	−0.716902	−	0.697174i	$-0.754440\pi$
$229$	−21.6974	−1.43380	−0.716902	+	0.697174i	$0.754440\pi$
$230$	15.5267	1.02380
$231$	−61.0284	−4.01537
$232$	−2.41941	−0.158842
$233$	−20.3313	−1.33195	−0.665974	−	0.745975i	$-0.731984\pi$
$233$	−20.3313	−1.33195	−0.665974	+	0.745975i	$0.731984\pi$
$234$	29.0830	1.90121
$235$	0.544112	0.0354940
$236$	−8.10932	−0.527872
$237$	44.3155	2.87860
$238$	4.05503	0.262849
$239$	0.107089	0.00692703	0.00346351	−	0.999994i	$-0.498898\pi$
$239$	0.107089	0.00692703	0.00346351	+	0.999994i	$0.498898\pi$
$240$	6.29366	0.406254
$241$	1.29818	0.0836228	0.0418114	−	0.999126i	$-0.486687\pi$
$241$	1.29818	0.0836228	0.0418114	+	0.999126i	$0.486687\pi$
$242$	15.2669	0.981397
$243$	−44.6355	−2.86337
$244$	−13.6307	−0.872616
$245$	−11.8674	−0.758178
$246$	18.3570	1.17040
$247$	−16.2532	−1.03417
$248$	−8.08705	−0.513528
$249$	41.8890	2.65460
$250$	12.1231	0.766730
$251$	26.2036	1.65396	0.826978	−	0.562234i	$-0.190058\pi$
$251$	26.2036	1.65396	0.826978	+	0.562234i	$0.190058\pi$
$252$	28.1582	1.77380
$253$	−41.4655	−2.60691
$254$	−3.35545	−0.210540
$255$	7.02872	0.440156
$256$	1.00000	0.0625000
$257$	13.5624	0.845998	0.422999	−	0.906130i	$-0.360977\pi$
$257$	13.5624	0.845998	0.422999	+	0.906130i	$0.360977\pi$
$258$	−35.3306	−2.19958
$259$	12.7793	0.794068
$260$	−7.19700	−0.446339
$261$	−18.7627	−1.16138
$262$	−10.2506	−0.633285
$263$	5.56401	0.343091	0.171546	−	0.985176i	$-0.445124\pi$
$263$	5.56401	0.343091	0.171546	+	0.985176i	$0.445124\pi$
$264$	−16.8078	−1.03445
$265$	21.7538	1.33632
$266$	−15.7364	−0.964861
$267$	53.3572	3.26541
$268$	3.28752	0.200817
$269$	13.2157	0.805773	0.402886	−	0.915250i	$-0.368007\pi$
$269$	13.2157	0.805773	0.402886	+	0.915250i	$0.368007\pi$
$270$	29.9266	1.82128
$271$	16.4995	1.00227	0.501136	−	0.865369i	$-0.332916\pi$
$271$	16.4995	1.00227	0.501136	+	0.865369i	$0.332916\pi$
$272$	1.11680	0.0677156
$273$	−44.6562	−2.70271
$274$	−12.2209	−0.738294
$275$	−6.75014	−0.407049
$276$	26.5331	1.59710
$277$	0.758381	0.0455667	0.0227834	−	0.999740i	$-0.492747\pi$
$277$	0.758381	0.0455667	0.0227834	+	0.999740i	$0.492747\pi$
$278$	−12.1671	−0.729733
$279$	−62.7155	−3.75468
$280$	−6.96815	−0.416427
$281$	12.9484	0.772437	0.386218	−	0.922407i	$-0.373781\pi$
$281$	12.9484	0.772437	0.386218	+	0.922407i	$0.373781\pi$
$282$	0.929818	0.0553699
$283$	12.2506	0.728222	0.364111	−	0.931355i	$-0.381373\pi$
$283$	12.2506	0.728222	0.364111	+	0.931355i	$0.381373\pi$
$284$	5.06592	0.300607
$285$	−27.2765	−1.61572
$286$	19.2203	1.13652
$287$	−20.3243	−1.19971
$288$	7.75505	0.456971
$289$	−15.7528	−0.926633
$290$	4.64309	0.272651
$291$	−38.9495	−2.28326
$292$	3.57492	0.209206
$293$	18.1413	1.05982	0.529912	−	0.848052i	$-0.322225\pi$
$293$	18.1413	1.05982	0.529912	+	0.848052i	$0.322225\pi$
$294$	−20.2798	−1.18274
$295$	15.5626	0.906088
$296$	3.51955	0.204570
$297$	−79.9219	−4.63754
$298$	−18.4250	−1.06733
$299$	−30.3414	−1.75469
$300$	4.31931	0.249376
$301$	39.1170	2.25466
$302$	−9.34409	−0.537692
$303$	−15.4927	−0.890032
$304$	−4.33396	−0.248570
$305$	26.1586	1.49784
$306$	8.66080	0.495105
$307$	−29.8552	−1.70393	−0.851963	−	0.523602i	$-0.824588\pi$
$307$	−29.8552	−1.70393	−0.851963	+	0.523602i	$0.824588\pi$
$308$	18.6091	1.06035
$309$	51.9752	2.95676
$310$	15.5198	0.881467
$311$	−2.10541	−0.119387	−0.0596935	−	0.998217i	$-0.519012\pi$
$311$	−2.10541	−0.119387	−0.0596935	+	0.998217i	$0.519012\pi$
$312$	−12.2987	−0.696279
$313$	−19.5998	−1.10785	−0.553923	−	0.832568i	$-0.686870\pi$
$313$	−19.5998	−1.10785	−0.553923	+	0.832568i	$0.686870\pi$
$314$	9.25962	0.522550
$315$	−54.0384	−3.04472
$316$	−13.5129	−0.760161
$317$	3.98974	0.224086	0.112043	−	0.993703i	$-0.464261\pi$
$317$	3.98974	0.224086	0.112043	+	0.993703i	$0.464261\pi$
$318$	37.1744	2.08464
$319$	−12.3998	−0.694256
$320$	−1.91910	−0.107281
$321$	−7.56801	−0.422405
$322$	−29.3767	−1.63710
$323$	−4.84015	−0.269313
$324$	27.8756	1.54865
$325$	−4.93927	−0.273981
$326$	−6.79637	−0.376416
$327$	−27.8674	−1.54107
$328$	−5.59752	−0.309071
$329$	−1.02947	−0.0567563
$330$	32.2558	1.77562
$331$	−22.9248	−1.26006	−0.630031	−	0.776570i	$-0.716958\pi$
$331$	−22.9248	−1.26006	−0.630031	+	0.776570i	$0.716958\pi$
$332$	−12.7730	−0.701010
$333$	27.2943	1.49572
$334$	−14.1859	−0.776215
$335$	−6.30908	−0.344702
$336$	−11.9077	−0.649617
$337$	−23.6421	−1.28787	−0.643933	−	0.765082i	$-0.722698\pi$
$337$	−23.6421	−1.28787	−0.643933	+	0.765082i	$0.722698\pi$
$338$	1.06400	0.0578740
$339$	−7.81649	−0.424533
$340$	−2.14324	−0.116233
$341$	−41.4472	−2.24449
$342$	−33.6101	−1.81743
$343$	−2.96350	−0.160014
$344$	10.7732	0.580851
$345$	−50.9196	−2.74142
$346$	−3.35840	−0.180549
$347$	−0.840567	−0.0451240	−0.0225620	−	0.999745i	$-0.507182\pi$
$347$	−0.840567	−0.0451240	−0.0225620	+	0.999745i	$0.507182\pi$
$348$	7.93444	0.425330
$349$	13.2972	0.711780	0.355890	−	0.934528i	$-0.384178\pi$
$349$	13.2972	0.711780	0.355890	+	0.934528i	$0.384178\pi$
$350$	−4.78221	−0.255620
$351$	−58.4811	−3.12149
$352$	5.12513	0.273170
$353$	−26.1560	−1.39214	−0.696072	−	0.717972i	$-0.745070\pi$
$353$	−26.1560	−1.39214	−0.696072	+	0.717972i	$0.745070\pi$
$354$	26.5944	1.41348
$355$	−9.72199	−0.515990
$356$	−16.2700	−0.862306
$357$	−13.2984	−0.703828
$358$	25.3384	1.33918
$359$	33.9757	1.79317	0.896585	−	0.442872i	$-0.146040\pi$
$359$	33.9757	1.79317	0.896585	+	0.442872i	$0.146040\pi$
$360$	−14.8827	−0.784387
$361$	−0.216778	−0.0114094
$362$	18.3920	0.966661
$363$	−50.0678	−2.62788
$364$	13.6168	0.713714
$365$	−6.86061	−0.359101
$366$	44.7017	2.33660
$367$	−20.9897	−1.09565	−0.547827	−	0.836592i	$-0.684545\pi$
$367$	−20.9897	−1.09565	−0.547827	+	0.836592i	$0.684545\pi$
$368$	−8.09062	−0.421753
$369$	−43.4091	−2.25979
$370$	−6.75435	−0.351142
$371$	−41.1584	−2.13684
$372$	26.5214	1.37507
$373$	20.2623	1.04914	0.524571	−	0.851367i	$-0.324226\pi$
$373$	20.2623	1.04914	0.524571	+	0.851367i	$0.324226\pi$
$374$	5.72372	0.295966
$375$	−39.7575	−2.05307
$376$	−0.283525	−0.0146217
$377$	−9.07328	−0.467298
$378$	−56.6216	−2.91230
$379$	−9.73342	−0.499972	−0.249986	−	0.968249i	$-0.580426\pi$
$379$	−9.73342	−0.499972	−0.249986	+	0.968249i	$0.580426\pi$
$380$	8.31729	0.426668
$381$	11.0042	0.563760
$382$	2.76917	0.141683
$383$	−35.2214	−1.79973	−0.899865	−	0.436168i	$-0.856335\pi$
$383$	−35.2214	−1.79973	−0.899865	+	0.436168i	$0.856335\pi$
$384$	−3.27949	−0.167356
$385$	−35.7127	−1.82009
$386$	14.1703	0.721251
$387$	83.5466	4.24691
$388$	11.8767	0.602947
$389$	15.3146	0.776482	0.388241	−	0.921558i	$-0.373083\pi$
$389$	15.3146	0.776482	0.388241	+	0.921558i	$0.373083\pi$
$390$	23.6025	1.19516
$391$	−9.03556	−0.456948
$392$	6.18382	0.312330
$393$	33.6168	1.69574
$394$	−21.9401	−1.10532
$395$	25.9326	1.30481
$396$	39.7456	1.99729
$397$	21.5473	1.08143	0.540713	−	0.841207i	$-0.318155\pi$
$397$	21.5473	1.08143	0.540713	+	0.841207i	$0.318155\pi$
$398$	19.8899	0.996992
$399$	51.6074	2.58360
$400$	−1.31707	−0.0658534
$401$	−14.4040	−0.719303	−0.359651	−	0.933087i	$-0.617104\pi$
$401$	−14.4040	−0.719303	−0.359651	+	0.933087i	$0.617104\pi$
$402$	−10.7814	−0.537727
$403$	−30.3281	−1.51075
$404$	4.72412	0.235034
$405$	−53.4961	−2.65824
$406$	−8.78477	−0.435981
$407$	18.0381	0.894117
$408$	−3.66252	−0.181322
$409$	34.8044	1.72097	0.860484	−	0.509478i	$-0.170161\pi$
$409$	34.8044	1.72097	0.860484	+	0.509478i	$0.170161\pi$
$410$	10.7422	0.530519
$411$	40.0784	1.97692
$412$	−15.8486	−0.780802
$413$	−29.4446	−1.44887
$414$	−62.7432	−3.08366
$415$	24.5126	1.20328
$416$	3.75020	0.183869
$417$	39.9018	1.95400
$418$	−22.2121	−1.08643
$419$	14.1266	0.690128	0.345064	−	0.938579i	$-0.387857\pi$
$419$	14.1266	0.690128	0.345064	+	0.938579i	$0.387857\pi$
$420$	22.8520	1.11506
$421$	−0.742022	−0.0361639	−0.0180820	−	0.999837i	$-0.505756\pi$
$421$	−0.742022	−0.0361639	−0.0180820	+	0.999837i	$0.505756\pi$
$422$	−6.34249	−0.308748
$423$	−2.19875	−0.106907
$424$	−11.3354	−0.550497
$425$	−1.47090	−0.0713489
$426$	−16.6136	−0.804933
$427$	−49.4924	−2.39511
$428$	2.30768	0.111546
$429$	−63.0326	−3.04324
$430$	−20.6748	−0.997027
$431$	−24.6578	−1.18773	−0.593863	−	0.804567i	$-0.702398\pi$
$431$	−24.6578	−1.18773	−0.593863	+	0.804567i	$0.702398\pi$
$432$	−15.5941	−0.750273
$433$	−35.3633	−1.69945	−0.849727	−	0.527224i	$-0.823233\pi$
$433$	−35.3633	−1.69945	−0.849727	+	0.527224i	$0.823233\pi$
$434$	−29.3637	−1.40950
$435$	−15.2270	−0.730076
$436$	8.49749	0.406956
$437$	35.0644	1.67736
$438$	−11.7239	−0.560190
$439$	−7.34899	−0.350748	−0.175374	−	0.984502i	$-0.556113\pi$
$439$	−7.34899	−0.350748	−0.175374	+	0.984502i	$0.556113\pi$
$440$	−9.83562	−0.468895
$441$	47.9559	2.28361
$442$	4.18820	0.199213
$443$	−15.1735	−0.720914	−0.360457	−	0.932776i	$-0.617379\pi$
$443$	−15.1735	−0.720914	−0.360457	+	0.932776i	$0.617379\pi$
$444$	−11.5423	−0.547774
$445$	31.2236	1.48014
$446$	12.8708	0.609452
$447$	60.4247	2.85799
$448$	3.63095	0.171546
$449$	−12.7065	−0.599659	−0.299829	−	0.953993i	$-0.596930\pi$
$449$	−12.7065	−0.599659	−0.299829	+	0.953993i	$0.596930\pi$
$450$	−10.2139	−0.481489
$451$	−28.6880	−1.35087
$452$	2.38345	0.112108
$453$	30.6439	1.43977
$454$	1.90572	0.0894397
$455$	−26.1320	−1.22509
$456$	14.2132	0.665593
$457$	19.4287	0.908834	0.454417	−	0.890789i	$-0.349848\pi$
$457$	19.4287	0.908834	0.454417	+	0.890789i	$0.349848\pi$
$458$	−21.6974	−1.01385
$459$	−17.4155	−0.812884
$460$	15.5267	0.723935
$461$	29.0394	1.35250	0.676251	−	0.736672i	$-0.263604\pi$
$461$	29.0394	1.35250	0.676251	+	0.736672i	$0.263604\pi$
$462$	−61.0284	−2.83930
$463$	−17.9372	−0.833611	−0.416806	−	0.908996i	$-0.636850\pi$
$463$	−17.9372	−0.833611	−0.416806	+	0.908996i	$0.636850\pi$
$464$	−2.41941	−0.112318
$465$	−50.8971	−2.36030
$466$	−20.3313	−0.941829
$467$	9.87664	0.457036	0.228518	−	0.973540i	$-0.426612\pi$
$467$	9.87664	0.457036	0.228518	+	0.973540i	$0.426612\pi$
$468$	29.0830	1.34436
$469$	11.9368	0.551192
$470$	0.544112	0.0250980
$471$	−30.3668	−1.39923
$472$	−8.10932	−0.373262
$473$	55.2140	2.53874
$474$	44.3155	2.03548
$475$	5.70812	0.261907
$476$	4.05503	0.185862
$477$	−87.9069	−4.02498
$478$	0.107089	0.00489815
$479$	14.2009	0.648856	0.324428	−	0.945910i	$-0.394828\pi$
$479$	14.2009	0.648856	0.324428	+	0.945910i	$0.394828\pi$
$480$	6.29366	0.287265
$481$	13.1990	0.601823
$482$	1.29818	0.0591303
$483$	96.3405	4.38364
$484$	15.2669	0.693952
$485$	−22.7925	−1.03495
$486$	−44.6355	−2.02471
$487$	27.4733	1.24494	0.622468	−	0.782645i	$-0.286130\pi$
$487$	27.4733	1.24494	0.622468	+	0.782645i	$0.286130\pi$
$488$	−13.6307	−0.617032
$489$	22.2886	1.00793
$490$	−11.8674	−0.536112
$491$	5.96478	0.269187	0.134593	−	0.990901i	$-0.457027\pi$
$491$	5.96478	0.269187	0.134593	+	0.990901i	$0.457027\pi$
$492$	18.3570	0.827598
$493$	−2.70199	−0.121691
$494$	−16.2532	−0.731267
$495$	−76.2757	−3.42834
$496$	−8.08705	−0.363119
$497$	18.3941	0.825089
$498$	41.8890	1.87709
$499$	24.1507	1.08113	0.540566	−	0.841301i	$-0.318210\pi$
$499$	24.1507	1.08113	0.540566	+	0.841301i	$0.318210\pi$
$500$	12.1231	0.542160
$501$	46.5224	2.07847
$502$	26.2036	1.16952
$503$	−7.76587	−0.346263	−0.173132	−	0.984899i	$-0.555389\pi$
$503$	−7.76587	−0.346263	−0.173132	+	0.984899i	$0.555389\pi$
$504$	28.1582	1.25427
$505$	−9.06604	−0.403434
$506$	−41.4655	−1.84337
$507$	−3.48938	−0.154969
$508$	−3.35545	−0.148874
$509$	34.9533	1.54928	0.774638	−	0.632405i	$-0.217932\pi$
$509$	34.9533	1.54928	0.774638	+	0.632405i	$0.217932\pi$
$510$	7.02872	0.311237
$511$	12.9804	0.574217
$512$	1.00000	0.0441942
$513$	67.5844	2.98392
$514$	13.5624	0.598211
$515$	30.4149	1.34024
$516$	−35.3306	−1.55534
$517$	−1.45310	−0.0639074
$518$	12.7793	0.561491
$519$	11.0138	0.483454
$520$	−7.19700	−0.315609
$521$	−1.72877	−0.0757387	−0.0378693	−	0.999283i	$-0.512057\pi$
$521$	−1.72877	−0.0757387	−0.0378693	+	0.999283i	$0.512057\pi$
$522$	−18.7627	−0.821219
$523$	9.86100	0.431192	0.215596	−	0.976483i	$-0.430831\pi$
$523$	9.86100	0.431192	0.215596	+	0.976483i	$0.430831\pi$
$524$	−10.2506	−0.447800
$525$	15.6832	0.684472
$526$	5.56401	0.242602
$527$	−9.03158	−0.393422
$528$	−16.8078	−0.731466
$529$	42.4581	1.84601
$530$	21.7538	0.944924
$531$	−62.8882	−2.72911
$532$	−15.7364	−0.682260
$533$	−20.9918	−0.909257
$534$	53.3572	2.30899
$535$	−4.42866	−0.191468
$536$	3.28752	0.141999
$537$	−83.0971	−3.58590
$538$	13.2157	0.569767
$539$	31.6929	1.36511
$540$	29.9266	1.28784
$541$	9.15727	0.393702	0.196851	−	0.980433i	$-0.436929\pi$
$541$	9.15727	0.393702	0.196851	+	0.980433i	$0.436929\pi$
$542$	16.4995	0.708713
$543$	−60.3163	−2.58842
$544$	1.11680	0.0478822
$545$	−16.3075	−0.698537
$546$	−44.6562	−1.91111
$547$	9.34268	0.399464	0.199732	−	0.979851i	$-0.435993\pi$
$547$	9.34268	0.399464	0.199732	+	0.979851i	$0.435993\pi$
$548$	−12.2209	−0.522052
$549$	−105.707	−4.51145
$550$	−6.75014	−0.287827
$551$	10.4856	0.446703
$552$	26.5331	1.12932
$553$	−49.0648	−2.08645
$554$	0.758381	0.0322205
$555$	22.1508	0.940250
$556$	−12.1671	−0.515999
$557$	27.8827	1.18143	0.590715	−	0.806881i	$-0.298846\pi$
$557$	27.8827	1.18143	0.590715	+	0.806881i	$0.298846\pi$
$558$	−62.7155	−2.65496
$559$	40.4016	1.70881
$560$	−6.96815	−0.294458
$561$	−18.7709	−0.792507
$562$	12.9484	0.546195
$563$	−30.7791	−1.29719	−0.648593	−	0.761135i	$-0.724642\pi$
$563$	−30.7791	−1.29719	−0.648593	+	0.761135i	$0.724642\pi$
$564$	0.929818	0.0391524
$565$	−4.57406	−0.192432
$566$	12.2506	0.514931
$567$	101.215	4.25064
$568$	5.06592	0.212561
$569$	3.23095	0.135449	0.0677243	−	0.997704i	$-0.478426\pi$
$569$	3.23095	0.135449	0.0677243	+	0.997704i	$0.478426\pi$
$570$	−27.2765	−1.14249
$571$	18.9887	0.794652	0.397326	−	0.917677i	$-0.369938\pi$
$571$	18.9887	0.794652	0.397326	+	0.917677i	$0.369938\pi$
$572$	19.2203	0.803639
$573$	−9.08147	−0.379384
$574$	−20.3243	−0.848322
$575$	10.6559	0.444382
$576$	7.75505	0.323127
$577$	15.8599	0.660256	0.330128	−	0.943936i	$-0.392908\pi$
$577$	15.8599	0.660256	0.330128	+	0.943936i	$0.392908\pi$
$578$	−15.7528	−0.655229
$579$	−46.4715	−1.93129
$580$	4.64309	0.192794
$581$	−46.3782	−1.92409
$582$	−38.9495	−1.61451
$583$	−58.0956	−2.40607
$584$	3.57492	0.147931
$585$	−55.8131	−2.30759
$586$	18.1413	0.749409
$587$	1.37295	0.0566677	0.0283338	−	0.999599i	$-0.490980\pi$
$587$	1.37295	0.0566677	0.0283338	+	0.999599i	$0.490980\pi$
$588$	−20.2798	−0.836324
$589$	35.0490	1.44417
$590$	15.5626	0.640701
$591$	71.9522	2.95972
$592$	3.51955	0.144652
$593$	35.4701	1.45658	0.728290	−	0.685269i	$-0.240315\pi$
$593$	35.4701	1.45658	0.728290	+	0.685269i	$0.240315\pi$
$594$	−79.9219	−3.27924
$595$	−7.78200	−0.319031
$596$	−18.4250	−0.754719
$597$	−65.2288	−2.66964
$598$	−30.3414	−1.24075
$599$	14.6372	0.598061	0.299031	−	0.954243i	$-0.403337\pi$
$599$	14.6372	0.598061	0.299031	+	0.954243i	$0.403337\pi$
$600$	4.31931	0.176335
$601$	−16.3224	−0.665806	−0.332903	−	0.942961i	$-0.608028\pi$
$601$	−16.3224	−0.665806	−0.332903	+	0.942961i	$0.608028\pi$
$602$	39.1170	1.59429
$603$	25.4949	1.03823
$604$	−9.34409	−0.380206
$605$	−29.2988	−1.19116
$606$	−15.4927	−0.629348
$607$	−4.76409	−0.193368	−0.0966841	−	0.995315i	$-0.530824\pi$
$607$	−4.76409	−0.193368	−0.0966841	+	0.995315i	$0.530824\pi$
$608$	−4.33396	−0.175765
$609$	28.8096	1.16742
$610$	26.1586	1.05913
$611$	−1.06328	−0.0430156
$612$	8.66080	0.350092
$613$	34.6852	1.40092	0.700462	−	0.713690i	$-0.252977\pi$
$613$	34.6852	1.40092	0.700462	+	0.713690i	$0.252977\pi$
$614$	−29.8552	−1.20486
$615$	−35.2289	−1.42057
$616$	18.6091	0.749782
$617$	7.76079	0.312438	0.156219	−	0.987722i	$-0.450069\pi$
$617$	7.76079	0.312438	0.156219	+	0.987722i	$0.450069\pi$
$618$	51.9752	2.09075
$619$	−4.25918	−0.171191	−0.0855955	−	0.996330i	$-0.527279\pi$
$619$	−4.25918	−0.171191	−0.0855955	+	0.996330i	$0.527279\pi$
$620$	15.5198	0.623292
$621$	126.166	5.06288
$622$	−2.10541	−0.0844193
$623$	−59.0755	−2.36681
$624$	−12.2987	−0.492344
$625$	−16.6800	−0.667200
$626$	−19.5998	−0.783365
$627$	72.8444	2.90912
$628$	9.25962	0.369499
$629$	3.93061	0.156724
$630$	−54.0384	−2.15294
$631$	−24.7167	−0.983958	−0.491979	−	0.870607i	$-0.663726\pi$
$631$	−24.7167	−0.983958	−0.491979	+	0.870607i	$0.663726\pi$
$632$	−13.5129	−0.537515
$633$	20.8001	0.826731
$634$	3.98974	0.158453
$635$	6.43943	0.255541
$636$	37.1744	1.47406
$637$	23.1906	0.918844
$638$	−12.3998	−0.490913
$639$	39.2865	1.55415
$640$	−1.91910	−0.0758590
$641$	−24.5506	−0.969689	−0.484845	−	0.874600i	$-0.661124\pi$
$641$	−24.5506	−0.969689	−0.484845	+	0.874600i	$0.661124\pi$
$642$	−7.56801	−0.298686
$643$	−33.8777	−1.33600	−0.668002	−	0.744159i	$-0.732850\pi$
$643$	−33.8777	−1.33600	−0.668002	+	0.744159i	$0.732850\pi$
$644$	−29.3767	−1.15760
$645$	67.8028	2.66973
$646$	−4.84015	−0.190433
$647$	−40.1594	−1.57883	−0.789414	−	0.613861i	$-0.789615\pi$
$647$	−40.1594	−1.57883	−0.789414	+	0.613861i	$0.789615\pi$
$648$	27.8756	1.09506
$649$	−41.5613	−1.63142
$650$	−4.93927	−0.193734
$651$	96.2980	3.77422
$652$	−6.79637	−0.266166
$653$	23.1120	0.904444	0.452222	−	0.891905i	$-0.350631\pi$
$653$	23.1120	0.904444	0.452222	+	0.891905i	$0.350631\pi$
$654$	−27.8674	−1.08970
$655$	19.6719	0.768645
$656$	−5.59752	−0.218546
$657$	27.7237	1.08160
$658$	−1.02947	−0.0401328
$659$	−18.4589	−0.719057	−0.359529	−	0.933134i	$-0.617063\pi$
$659$	−18.4589	−0.719057	−0.359529	+	0.933134i	$0.617063\pi$
$660$	32.2558	1.25556
$661$	8.26129	0.321327	0.160664	−	0.987009i	$-0.448637\pi$
$661$	8.26129	0.321327	0.160664	+	0.987009i	$0.448637\pi$
$662$	−22.9248	−0.890998
$663$	−13.7352	−0.533430
$664$	−12.7730	−0.495689
$665$	30.1997	1.17109
$666$	27.2943	1.05763
$667$	19.5745	0.757930
$668$	−14.1859	−0.548867
$669$	−42.2098	−1.63193
$670$	−6.30908	−0.243741
$671$	−69.8591	−2.69688
$672$	−11.9077	−0.459348
$673$	−29.7823	−1.14802	−0.574012	−	0.818847i	$-0.694614\pi$
$673$	−29.7823	−1.14802	−0.574012	+	0.818847i	$0.694614\pi$
$674$	−23.6421	−0.910659
$675$	20.5385	0.790529
$676$	1.06400	0.0409231
$677$	6.82505	0.262308	0.131154	−	0.991362i	$-0.458132\pi$
$677$	6.82505	0.262308	0.131154	+	0.991362i	$0.458132\pi$
$678$	−7.81649	−0.300190
$679$	43.1237	1.65494
$680$	−2.14324	−0.0821894
$681$	−6.24978	−0.239492
$682$	−41.4472	−1.58709
$683$	−35.6014	−1.36225	−0.681125	−	0.732167i	$-0.738509\pi$
$683$	−35.6014	−1.36225	−0.681125	+	0.732167i	$0.738509\pi$
$684$	−33.6101	−1.28511
$685$	23.4532	0.896099
$686$	−2.96350	−0.113147
$687$	71.1564	2.71478
$688$	10.7732	0.410724
$689$	−42.5101	−1.61951
$690$	−50.9196	−1.93848
$691$	30.7556	1.17000	0.584999	−	0.811034i	$-0.301095\pi$
$691$	30.7556	1.17000	0.584999	+	0.811034i	$0.301095\pi$
$692$	−3.35840	−0.127667
$693$	144.315	5.48206
$694$	−0.840567	−0.0319075
$695$	23.3498	0.885709
$696$	7.93444	0.300754
$697$	−6.25128	−0.236784
$698$	13.2972	0.503305
$699$	66.6763	2.52193
$700$	−4.78221	−0.180751
$701$	5.80041	0.219078	0.109539	−	0.993982i	$-0.465062\pi$
$701$	5.80041	0.219078	0.109539	+	0.993982i	$0.465062\pi$
$702$	−58.4811	−2.20723
$703$	−15.2536	−0.575300
$704$	5.12513	0.193161
$705$	−1.78441	−0.0672048
$706$	−26.1560	−0.984394
$707$	17.1531	0.645107
$708$	26.5944	0.999480
$709$	35.8350	1.34581	0.672905	−	0.739729i	$-0.265046\pi$
$709$	35.8350	1.34581	0.672905	+	0.739729i	$0.265046\pi$
$710$	−9.72199	−0.364860
$711$	−104.793	−3.93006
$712$	−16.2700	−0.609743
$713$	65.4293	2.45035
$714$	−13.2984	−0.497681
$715$	−36.8855	−1.37944
$716$	25.3384	0.946941
$717$	−0.351198	−0.0131157
$718$	33.9757	1.26796
$719$	−19.8308	−0.739562	−0.369781	−	0.929119i	$-0.620567\pi$
$719$	−19.8308	−0.739562	−0.369781	+	0.929119i	$0.620567\pi$
$720$	−14.8827	−0.554645
$721$	−57.5454	−2.14310
$722$	−0.216778	−0.00806763
$723$	−4.25735	−0.158333
$724$	18.3920	0.683532
$725$	3.18653	0.118345
$726$	−50.0678	−1.85819
$727$	−0.772382	−0.0286461	−0.0143230	−	0.999897i	$-0.504559\pi$
$727$	−0.772382	−0.0286461	−0.0143230	+	0.999897i	$0.504559\pi$
$728$	13.6168	0.504672
$729$	62.7546	2.32424
$730$	−6.86061	−0.253923
$731$	12.0314	0.444999
$732$	44.7017	1.65222
$733$	−48.1430	−1.77820	−0.889102	−	0.457710i	$-0.848670\pi$
$733$	−48.1430	−1.77820	−0.889102	+	0.457710i	$0.848670\pi$
$734$	−20.9897	−0.774744
$735$	38.9189	1.43554
$736$	−8.09062	−0.298224
$737$	16.8490	0.620640
$738$	−43.4091	−1.59791
$739$	20.5092	0.754441	0.377221	−	0.926123i	$-0.376880\pi$
$739$	20.5092	0.754441	0.377221	+	0.926123i	$0.376880\pi$
$740$	−6.75435	−0.248295
$741$	53.3023	1.95811
$742$	−41.1584	−1.51097
$743$	1.98076	0.0726671	0.0363336	−	0.999340i	$-0.488432\pi$
$743$	1.98076	0.0726671	0.0363336	+	0.999340i	$0.488432\pi$
$744$	26.5214	0.972322
$745$	35.3594	1.29547
$746$	20.2623	0.741855
$747$	−99.0553	−3.62425
$748$	5.72372	0.209280
$749$	8.37908	0.306165
$750$	−39.7575	−1.45174
$751$	−5.43265	−0.198240	−0.0991201	−	0.995075i	$-0.531603\pi$
$751$	−5.43265	−0.198240	−0.0991201	+	0.995075i	$0.531603\pi$
$752$	−0.283525	−0.0103391
$753$	−85.9344	−3.13162
$754$	−9.07328	−0.330429
$755$	17.9322	0.652620
$756$	−56.6216	−2.05931
$757$	40.8393	1.48433	0.742164	−	0.670218i	$-0.233799\pi$
$757$	40.8393	1.48433	0.742164	+	0.670218i	$0.233799\pi$
$758$	−9.73342	−0.353534
$759$	135.986	4.93596
$760$	8.31729	0.301700
$761$	27.1288	0.983420	0.491710	−	0.870759i	$-0.336372\pi$
$761$	27.1288	0.983420	0.491710	+	0.870759i	$0.336372\pi$
$762$	11.0042	0.398639
$763$	30.8540	1.11699
$764$	2.76917	0.100185
$765$	−16.6209	−0.600930
$766$	−35.2214	−1.27260
$767$	−30.4116	−1.09810
$768$	−3.27949	−0.118338
$769$	1.60153	0.0577525	0.0288762	−	0.999583i	$-0.490807\pi$
$769$	1.60153	0.0577525	0.0288762	+	0.999583i	$0.490807\pi$
$770$	−35.7127	−1.28700
$771$	−44.4777	−1.60182
$772$	14.1703	0.510002
$773$	−40.7695	−1.46638	−0.733189	−	0.680025i	$-0.761969\pi$
$773$	−40.7695	−1.46638	−0.733189	+	0.680025i	$0.761969\pi$
$774$	83.5466	3.00302
$775$	10.6512	0.382602
$776$	11.8767	0.426348
$777$	−41.9096	−1.50350
$778$	15.3146	0.549056
$779$	24.2594	0.869185
$780$	23.6025	0.845104
$781$	25.9635	0.929047
$782$	−9.03556	−0.323111
$783$	37.7286	1.34831
$784$	6.18382	0.220851
$785$	−17.7701	−0.634242
$786$	33.6168	1.19907
$787$	−41.9945	−1.49694	−0.748471	−	0.663167i	$-0.769212\pi$
$787$	−41.9945	−1.49694	−0.748471	+	0.663167i	$0.769212\pi$
$788$	−21.9401	−0.781583
$789$	−18.2471	−0.649614
$790$	25.9326	0.922640
$791$	8.65418	0.307707
$792$	39.7456	1.41230
$793$	−51.1178	−1.81525
$794$	21.5473	0.764683
$795$	−71.3413	−2.53022
$796$	19.8899	0.704980
$797$	−5.45728	−0.193307	−0.0966534	−	0.995318i	$-0.530814\pi$
$797$	−5.45728	−0.193307	−0.0966534	+	0.995318i	$0.530814\pi$
$798$	51.6074	1.82688
$799$	−0.316640	−0.0112019
$800$	−1.31707	−0.0465654
$801$	−126.174	−4.45815
$802$	−14.4040	−0.508624
$803$	18.3219	0.646566
$804$	−10.7814	−0.380231
$805$	56.3767	1.98702
$806$	−30.3281	−1.06826
$807$	−43.3406	−1.52566
$808$	4.72412	0.166194
$809$	−28.9653	−1.01837	−0.509183	−	0.860658i	$-0.670052\pi$
$809$	−28.9653	−1.01837	−0.509183	+	0.860658i	$0.670052\pi$
$810$	−53.4961	−1.87966
$811$	12.1382	0.426229	0.213114	−	0.977027i	$-0.431639\pi$
$811$	12.1382	0.426229	0.213114	+	0.977027i	$0.431639\pi$
$812$	−8.78477	−0.308285
$813$	−54.1098	−1.89772
$814$	18.0381	0.632236
$815$	13.0429	0.456872
$816$	−3.66252	−0.128214
$817$	−46.6906	−1.63350
$818$	34.8044	1.21691
$819$	105.599	3.68993
$820$	10.7422	0.375133
$821$	14.7130	0.513488	0.256744	−	0.966479i	$-0.417350\pi$
$821$	14.7130	0.513488	0.256744	+	0.966479i	$0.417350\pi$
$822$	40.0784	1.39790
$823$	−37.3953	−1.30352	−0.651759	−	0.758426i	$-0.725969\pi$
$823$	−37.3953	−1.30352	−0.651759	+	0.758426i	$0.725969\pi$
$824$	−15.8486	−0.552110
$825$	22.1370	0.770712
$826$	−29.4446	−1.02451
$827$	9.19080	0.319595	0.159798	−	0.987150i	$-0.448916\pi$
$827$	9.19080	0.319595	0.159798	+	0.987150i	$0.448916\pi$
$828$	−62.7432	−2.18048
$829$	53.2455	1.84929	0.924646	−	0.380827i	$-0.124361\pi$
$829$	53.2455	1.84929	0.924646	+	0.380827i	$0.124361\pi$
$830$	24.5126	0.850846
$831$	−2.48710	−0.0862766
$832$	3.75020	0.130015
$833$	6.90606	0.239281
$834$	39.9018	1.38169
$835$	27.2240	0.942126
$836$	−22.2121	−0.768222
$837$	126.111	4.35902
$838$	14.1266	0.487994
$839$	34.0083	1.17410	0.587048	−	0.809552i	$-0.300290\pi$
$839$	34.0083	1.17410	0.587048	+	0.809552i	$0.300290\pi$
$840$	22.8520	0.788468
$841$	−23.1464	−0.798153
$842$	−0.742022	−0.0255718
$843$	−42.4641	−1.46254
$844$	−6.34249	−0.218318
$845$	−2.04192	−0.0702442
$846$	−2.19875	−0.0755947
$847$	55.4336	1.90472
$848$	−11.3354	−0.389260
$849$	−40.1757	−1.37883
$850$	−1.47090	−0.0504513
$851$	−28.4753	−0.976121
$852$	−16.6136	−0.569174
$853$	6.47079	0.221556	0.110778	−	0.993845i	$-0.464666\pi$
$853$	6.47079	0.221556	0.110778	+	0.993845i	$0.464666\pi$
$854$	−49.4924	−1.69360
$855$	64.5010	2.20589
$856$	2.30768	0.0788749
$857$	−5.52987	−0.188897	−0.0944484	−	0.995530i	$-0.530109\pi$
$857$	−5.52987	−0.188897	−0.0944484	+	0.995530i	$0.530109\pi$
$858$	−63.0326	−2.15190
$859$	−18.5142	−0.631697	−0.315848	−	0.948810i	$-0.602289\pi$
$859$	−18.5142	−0.631697	−0.315848	+	0.948810i	$0.602289\pi$
$860$	−20.6748	−0.705005
$861$	66.6535	2.27154
$862$	−24.6578	−0.839848
$863$	−4.29467	−0.146192	−0.0730961	−	0.997325i	$-0.523288\pi$
$863$	−4.29467	−0.146192	−0.0730961	+	0.997325i	$0.523288\pi$
$864$	−15.5941	−0.530523
$865$	6.44510	0.219140
$866$	−35.3633	−1.20169
$867$	51.6610	1.75450
$868$	−29.3637	−0.996669
$869$	−69.2554	−2.34933
$870$	−15.2270	−0.516242
$871$	12.3289	0.417748
$872$	8.49749	0.287761
$873$	92.1043	3.11726
$874$	35.0644	1.18607
$875$	44.0183	1.48809
$876$	−11.7239	−0.396114
$877$	1.90145	0.0642075	0.0321037	−	0.999485i	$-0.489779\pi$
$877$	1.90145	0.0642075	0.0321037	+	0.999485i	$0.489779\pi$
$878$	−7.34899	−0.248016
$879$	−59.4941	−2.00669
$880$	−9.83562	−0.331559
$881$	7.01655	0.236394	0.118197	−	0.992990i	$-0.462289\pi$
$881$	7.01655	0.236394	0.118197	+	0.992990i	$0.462289\pi$
$882$	47.9559	1.61476
$883$	−0.261756	−0.00880880	−0.00440440	−	0.999990i	$-0.501402\pi$
$883$	−0.261756	−0.00880880	−0.00440440	+	0.999990i	$0.501402\pi$
$884$	4.18820	0.140865
$885$	−51.0373	−1.71560
$886$	−15.1735	−0.509763
$887$	−24.3888	−0.818894	−0.409447	−	0.912334i	$-0.634278\pi$
$887$	−24.3888	−0.818894	−0.409447	+	0.912334i	$0.634278\pi$
$888$	−11.5423	−0.387335
$889$	−12.1835	−0.408621
$890$	31.2236	1.04662
$891$	142.866	4.78620
$892$	12.8708	0.430948
$893$	1.22879	0.0411198
$894$	60.4247	2.02091
$895$	−48.6269	−1.62542
$896$	3.63095	0.121302
$897$	99.5044	3.32236
$898$	−12.7065	−0.424023
$899$	19.5659	0.652560
$900$	−10.2139	−0.340464
$901$	−12.6594	−0.421744
$902$	−28.6880	−0.955207
$903$	−128.284	−4.26901
$904$	2.38345	0.0792723
$905$	−35.2960	−1.17328
$906$	30.6439	1.01807
$907$	44.3574	1.47286	0.736432	−	0.676512i	$-0.236509\pi$
$907$	44.3574	1.47286	0.736432	+	0.676512i	$0.236509\pi$
$908$	1.90572	0.0632435
$909$	36.6358	1.21513
$910$	−26.1320	−0.866266
$911$	−46.4297	−1.53829	−0.769143	−	0.639077i	$-0.779316\pi$
$911$	−46.4297	−1.53829	−0.769143	+	0.639077i	$0.779316\pi$
$912$	14.2132	0.470645
$913$	−65.4633	−2.16652
$914$	19.4287	0.642643
$915$	−85.7869	−2.83603
$916$	−21.6974	−0.716902
$917$	−37.2195	−1.22910
$918$	−17.4155	−0.574796
$919$	36.8480	1.21550	0.607751	−	0.794127i	$-0.292072\pi$
$919$	36.8480	1.21550	0.607751	+	0.794127i	$0.292072\pi$
$920$	15.5267	0.511900
$921$	97.9098	3.22624
$922$	29.0394	0.956363
$923$	18.9982	0.625334
$924$	−61.0284	−2.00769
$925$	−4.63548	−0.152414
$926$	−17.9372	−0.589452
$927$	−122.906	−4.03677
$928$	−2.41941	−0.0794211
$929$	22.7079	0.745021	0.372511	−	0.928028i	$-0.378497\pi$
$929$	22.7079	0.745021	0.372511	+	0.928028i	$0.378497\pi$
$930$	−50.8971	−1.66898
$931$	−26.8005	−0.878349
$932$	−20.3313	−0.665974
$933$	6.90468	0.226049
$934$	9.87664	0.323174
$935$	−10.9844	−0.359227
$936$	29.0830	0.950607
$937$	9.31861	0.304426	0.152213	−	0.988348i	$-0.451360\pi$
$937$	9.31861	0.304426	0.152213	+	0.988348i	$0.451360\pi$
$938$	11.9368	0.389752
$939$	64.2773	2.09761
$940$	0.544112	0.0177470
$941$	8.95845	0.292037	0.146018	−	0.989282i	$-0.453354\pi$
$941$	8.95845	0.292037	0.146018	+	0.989282i	$0.453354\pi$
$942$	−30.3668	−0.989404
$943$	45.2874	1.47476
$944$	−8.10932	−0.263936
$945$	108.662	3.53478
$946$	55.2140	1.79516
$947$	−45.6930	−1.48482	−0.742412	−	0.669944i	$-0.766318\pi$
$947$	−45.6930	−1.48482	−0.742412	+	0.669944i	$0.766318\pi$
$948$	44.3155	1.43930
$949$	13.4067	0.435198
$950$	5.70812	0.185196
$951$	−13.0843	−0.424288
$952$	4.05503	0.131424
$953$	4.35441	0.141053	0.0705266	−	0.997510i	$-0.477532\pi$
$953$	4.35441	0.141053	0.0705266	+	0.997510i	$0.477532\pi$
$954$	−87.9069	−2.84609
$955$	−5.31431	−0.171967
$956$	0.107089	0.00346351
$957$	40.6650	1.31451
$958$	14.2009	0.458811
$959$	−44.3736	−1.43290
$960$	6.29366	0.203127
$961$	34.4004	1.10969
$962$	13.1990	0.425553
$963$	17.8962	0.576696
$964$	1.29818	0.0418114
$965$	−27.1943	−0.875414
$966$	96.3405	3.09970
$967$	−43.4850	−1.39838	−0.699192	−	0.714934i	$-0.746457\pi$
$967$	−43.4850	−1.39838	−0.699192	+	0.714934i	$0.746457\pi$
$968$	15.2669	0.490698
$969$	15.8732	0.509921
$970$	−22.7925	−0.731823
$971$	38.0445	1.22090	0.610452	−	0.792053i	$-0.290988\pi$
$971$	38.0445	1.22090	0.610452	+	0.792053i	$0.290988\pi$
$972$	−44.6355	−1.43168
$973$	−44.1781	−1.41629
$974$	27.4733	0.880302
$975$	16.1983	0.518760
$976$	−13.6307	−0.436308
$977$	11.4289	0.365642	0.182821	−	0.983146i	$-0.441477\pi$
$977$	11.4289	0.365642	0.182821	+	0.983146i	$0.441477\pi$
$978$	22.2886	0.712711
$979$	−83.3857	−2.66502
$980$	−11.8674	−0.379089
$981$	65.8985	2.10398
$982$	5.96478	0.190344
$983$	−31.0880	−0.991552	−0.495776	−	0.868451i	$-0.665116\pi$
$983$	−31.0880	−0.991552	−0.495776	+	0.868451i	$0.665116\pi$
$984$	18.3570	0.585200
$985$	42.1051	1.34158
$986$	−2.70199	−0.0860488
$987$	3.37613	0.107463
$988$	−16.2532	−0.517084
$989$	−87.1618	−2.77158
$990$	−76.2757	−2.42420
$991$	−3.27125	−0.103915	−0.0519574	−	0.998649i	$-0.516546\pi$
$991$	−3.27125	−0.103915	−0.0519574	+	0.998649i	$0.516546\pi$
$992$	−8.08705	−0.256764
$993$	75.1816	2.38582
$994$	18.3941	0.583426
$995$	−38.1707	−1.21009
$996$	41.8890	1.32730
$997$	6.33250	0.200552	0.100276	−	0.994960i	$-0.468027\pi$
$997$	6.33250	0.200552	0.100276	+	0.994960i	$0.468027\pi$
$998$	24.1507	0.764476
$999$	−54.8843	−1.73646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.a.1.4	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.a.1.4	✓	75	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.27949 q^{3} +1.00000 q^{4} -1.91910 q^{5} -3.27949 q^{6} +3.63095 q^{7} +1.00000 q^{8} +7.75505 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.27949 q^{3} +1.00000 q^{4} -1.91910 q^{5} -3.27949 q^{6} +3.63095 q^{7} +1.00000 q^{8} +7.75505 q^{9} -1.91910 q^{10} +5.12513 q^{11} -3.27949 q^{12} +3.75020 q^{13} +3.63095 q^{14} +6.29366 q^{15} +1.00000 q^{16} +1.11680 q^{17} +7.75505 q^{18} -4.33396 q^{19} -1.91910 q^{20} -11.9077 q^{21} +5.12513 q^{22} -8.09062 q^{23} -3.27949 q^{24} -1.31707 q^{25} +3.75020 q^{26} -15.5941 q^{27} +3.63095 q^{28} -2.41941 q^{29} +6.29366 q^{30} -8.08705 q^{31} +1.00000 q^{32} -16.8078 q^{33} +1.11680 q^{34} -6.96815 q^{35} +7.75505 q^{36} +3.51955 q^{37} -4.33396 q^{38} -12.2987 q^{39} -1.91910 q^{40} -5.59752 q^{41} -11.9077 q^{42} +10.7732 q^{43} +5.12513 q^{44} -14.8827 q^{45} -8.09062 q^{46} -0.283525 q^{47} -3.27949 q^{48} +6.18382 q^{49} -1.31707 q^{50} -3.66252 q^{51} +3.75020 q^{52} -11.3354 q^{53} -15.5941 q^{54} -9.83562 q^{55} +3.63095 q^{56} +14.2132 q^{57} -2.41941 q^{58} -8.10932 q^{59} +6.29366 q^{60} -13.6307 q^{61} -8.08705 q^{62} +28.1582 q^{63} +1.00000 q^{64} -7.19700 q^{65} -16.8078 q^{66} +3.28752 q^{67} +1.11680 q^{68} +26.5331 q^{69} -6.96815 q^{70} +5.06592 q^{71} +7.75505 q^{72} +3.57492 q^{73} +3.51955 q^{74} +4.31931 q^{75} -4.33396 q^{76} +18.6091 q^{77} -12.2987 q^{78} -13.5129 q^{79} -1.91910 q^{80} +27.8756 q^{81} -5.59752 q^{82} -12.7730 q^{83} -11.9077 q^{84} -2.14324 q^{85} +10.7732 q^{86} +7.93444 q^{87} +5.12513 q^{88} -16.2700 q^{89} -14.8827 q^{90} +13.6168 q^{91} -8.09062 q^{92} +26.5214 q^{93} -0.283525 q^{94} +8.31729 q^{95} -3.27949 q^{96} +11.8767 q^{97} +6.18382 q^{98} +39.7456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9	\( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 75 * q + 75 * q^2 - 30 * q^3 + 75 * q^4 - 29 * q^5 - 30 * q^6 - 31 * q^7 + 75 * q^8 + 55 * q^9 - 29 * q^10 - 30 * q^11 - 30 * q^12 - 23 * q^13 - 31 * q^14 - 22 * q^15 + 75 * q^16 - 48 * q^17 + 55 * q^18 - 58 * q^19 - 29 * q^20 - 5 * q^21 - 30 * q^22 - 79 * q^23 - 30 * q^24 + 42 * q^25 - 23 * q^26 - 108 * q^27 - 31 * q^28 - 39 * q^29 - 22 * q^30 - 95 * q^31 + 75 * q^32 - 44 * q^33 - 48 * q^34 - 60 * q^35 + 55 * q^36 - 16 * q^37 - 58 * q^38 - 57 * q^39 - 29 * q^40 - 85 * q^41 - 5 * q^42 - 55 * q^43 - 30 * q^44 - 48 * q^45 - 79 * q^46 - 88 * q^47 - 30 * q^48 + 26 * q^49 + 42 * q^50 - 39 * q^51 - 23 * q^52 - 79 * q^53 - 108 * q^54 - 78 * q^55 - 31 * q^56 - 40 * q^57 - 39 * q^58 - 74 * q^59 - 22 * q^60 - 21 * q^61 - 95 * q^62 - 97 * q^63 + 75 * q^64 - 63 * q^65 - 44 * q^66 - 59 * q^67 - 48 * q^68 + 3 * q^69 - 60 * q^70 - 72 * q^71 + 55 * q^72 - 91 * q^73 - 16 * q^74 - 95 * q^75 - 58 * q^76 - 60 * q^77 - 57 * q^78 - 64 * q^79 - 29 * q^80 + 47 * q^81 - 85 * q^82 - 105 * q^83 - 5 * q^84 + 14 * q^85 - 55 * q^86 - 75 * q^87 - 30 * q^88 - 78 * q^89 - 48 * q^90 - 89 * q^91 - 79 * q^92 + 27 * q^93 - 88 * q^94 - 53 * q^95 - 30 * q^96 - 79 * q^97 + 26 * q^98 - 57 * q^99

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