LMFDB - Embedded newform 8470.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.73205 q^{12} -5.73205 q^{13} +1.00000 q^{14} -1.73205 q^{15} +1.00000 q^{16} -0.732051 q^{17} -0.267949 q^{19} -1.00000 q^{20} +1.73205 q^{21} -5.00000 q^{23} +1.73205 q^{24} +1.00000 q^{25} -5.73205 q^{26} -5.19615 q^{27} +1.00000 q^{28} +8.00000 q^{29} -1.73205 q^{30} -2.73205 q^{31} +1.00000 q^{32} -0.732051 q^{34} -1.00000 q^{35} +6.00000 q^{37} -0.267949 q^{38} -9.92820 q^{39} -1.00000 q^{40} -6.19615 q^{41} +1.73205 q^{42} -5.26795 q^{43} -5.00000 q^{46} -4.73205 q^{47} +1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.26795 q^{51} -5.73205 q^{52} -6.19615 q^{53} -5.19615 q^{54} +1.00000 q^{56} -0.464102 q^{57} +8.00000 q^{58} -5.53590 q^{59} -1.73205 q^{60} -2.92820 q^{61} -2.73205 q^{62} +1.00000 q^{64} +5.73205 q^{65} -9.66025 q^{67} -0.732051 q^{68} -8.66025 q^{69} -1.00000 q^{70} +6.92820 q^{71} +9.12436 q^{73} +6.00000 q^{74} +1.73205 q^{75} -0.267949 q^{76} -9.92820 q^{78} -7.00000 q^{79} -1.00000 q^{80} -9.00000 q^{81} -6.19615 q^{82} +5.00000 q^{83} +1.73205 q^{84} +0.732051 q^{85} -5.26795 q^{86} +13.8564 q^{87} -13.6603 q^{89} -5.73205 q^{91} -5.00000 q^{92} -4.73205 q^{93} -4.73205 q^{94} +0.267949 q^{95} +1.73205 q^{96} +10.9282 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} - 10 q^{23} + 2 q^{25} - 8 q^{26} + 2 q^{28} + 16 q^{29} - 2 q^{31} + 2 q^{32} + 2 q^{34} - 2 q^{35} + 12 q^{37} - 4 q^{38} - 6 q^{39} - 2 q^{40} - 2 q^{41} - 14 q^{43} - 10 q^{46} - 6 q^{47} + 2 q^{49} + 2 q^{50} - 6 q^{51} - 8 q^{52} - 2 q^{53} + 2 q^{56} + 6 q^{57} + 16 q^{58} - 18 q^{59} + 8 q^{61} - 2 q^{62} + 2 q^{64} + 8 q^{65} - 2 q^{67} + 2 q^{68} - 2 q^{70} - 6 q^{73} + 12 q^{74} - 4 q^{76} - 6 q^{78} - 14 q^{79} - 2 q^{80} - 18 q^{81} - 2 q^{82} + 10 q^{83} - 2 q^{85} - 14 q^{86} - 10 q^{89} - 8 q^{91} - 10 q^{92} - 6 q^{93} - 6 q^{94} + 4 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 - 8 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 - 2 * q^20 - 10 * q^23 + 2 * q^25 - 8 * q^26 + 2 * q^28 + 16 * q^29 - 2 * q^31 + 2 * q^32 + 2 * q^34 - 2 * q^35 + 12 * q^37 - 4 * q^38 - 6 * q^39 - 2 * q^40 - 2 * q^41 - 14 * q^43 - 10 * q^46 - 6 * q^47 + 2 * q^49 + 2 * q^50 - 6 * q^51 - 8 * q^52 - 2 * q^53 + 2 * q^56 + 6 * q^57 + 16 * q^58 - 18 * q^59 + 8 * q^61 - 2 * q^62 + 2 * q^64 + 8 * q^65 - 2 * q^67 + 2 * q^68 - 2 * q^70 - 6 * q^73 + 12 * q^74 - 4 * q^76 - 6 * q^78 - 14 * q^79 - 2 * q^80 - 18 * q^81 - 2 * q^82 + 10 * q^83 - 2 * q^85 - 14 * q^86 - 10 * q^89 - 8 * q^91 - 10 * q^92 - 6 * q^93 - 6 * q^94 + 4 * q^95 + 8 * q^97 + 2 * q^98

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
$3$	1.73205	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$3$	1.73205	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.73205	0.707107
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.73205	0.500000
$13$	−5.73205	−1.58978	−0.794892	−	0.606750i	$-0.792473\pi$
$13$	−5.73205	−1.58978	−0.794892	+	0.606750i	$0.792473\pi$
$14$	1.00000	0.267261
$15$	−1.73205	−0.447214
$16$	1.00000	0.250000
$17$	−0.732051	−0.177548	−0.0887742	−	0.996052i	$-0.528295\pi$
$17$	−0.732051	−0.177548	−0.0887742	+	0.996052i	$0.528295\pi$
$18$	0	0
$19$	−0.267949	−0.0614718	−0.0307359	−	0.999528i	$-0.509785\pi$
$19$	−0.267949	−0.0614718	−0.0307359	+	0.999528i	$0.509785\pi$
$20$	−1.00000	−0.223607
$21$	1.73205	0.377964
$22$	0	0
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	1.73205	0.353553
$25$	1.00000	0.200000
$26$	−5.73205	−1.12415
$27$	−5.19615	−1.00000
$28$	1.00000	0.188982
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	−1.73205	−0.316228
$31$	−2.73205	−0.490691	−0.245345	−	0.969436i	$-0.578901\pi$
$31$	−2.73205	−0.490691	−0.245345	+	0.969436i	$0.578901\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.732051	−0.125546
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−0.267949	−0.0434671
$39$	−9.92820	−1.58978
$40$	−1.00000	−0.158114
$41$	−6.19615	−0.967676	−0.483838	−	0.875157i	$-0.660758\pi$
$41$	−6.19615	−0.967676	−0.483838	+	0.875157i	$0.660758\pi$
$42$	1.73205	0.267261
$43$	−5.26795	−0.803355	−0.401677	−	0.915781i	$-0.631573\pi$
$43$	−5.26795	−0.803355	−0.401677	+	0.915781i	$0.631573\pi$
$44$	0	0
$45$	0	0
$46$	−5.00000	−0.737210
$47$	−4.73205	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$47$	−4.73205	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$48$	1.73205	0.250000
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−1.26795	−0.177548
$52$	−5.73205	−0.794892
$53$	−6.19615	−0.851107	−0.425553	−	0.904933i	$-0.639921\pi$
$53$	−6.19615	−0.851107	−0.425553	+	0.904933i	$0.639921\pi$
$54$	−5.19615	−0.707107
$55$	0	0
$56$	1.00000	0.133631
$57$	−0.464102	−0.0614718
$58$	8.00000	1.05045
$59$	−5.53590	−0.720713	−0.360356	−	0.932815i	$-0.617345\pi$
$59$	−5.53590	−0.720713	−0.360356	+	0.932815i	$0.617345\pi$
$60$	−1.73205	−0.223607
$61$	−2.92820	−0.374918	−0.187459	−	0.982272i	$-0.560025\pi$
$61$	−2.92820	−0.374918	−0.187459	+	0.982272i	$0.560025\pi$
$62$	−2.73205	−0.346971
$63$	0	0
$64$	1.00000	0.125000
$65$	5.73205	0.710973
$66$	0	0
$67$	−9.66025	−1.18019	−0.590094	−	0.807335i	$-0.700909\pi$
$67$	−9.66025	−1.18019	−0.590094	+	0.807335i	$0.700909\pi$
$68$	−0.732051	−0.0887742
$69$	−8.66025	−1.04257
$70$	−1.00000	−0.119523
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	9.12436	1.06793	0.533963	−	0.845508i	$-0.320702\pi$
$73$	9.12436	1.06793	0.533963	+	0.845508i	$0.320702\pi$
$74$	6.00000	0.697486
$75$	1.73205	0.200000
$76$	−0.267949	−0.0307359
$77$	0	0
$78$	−9.92820	−1.12415
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	−1.00000	−0.111803
$81$	−9.00000	−1.00000
$82$	−6.19615	−0.684251
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	1.73205	0.188982
$85$	0.732051	0.0794021
$86$	−5.26795	−0.568058
$87$	13.8564	1.48556
$88$	0	0
$89$	−13.6603	−1.44798	−0.723992	−	0.689808i	$-0.757695\pi$
$89$	−13.6603	−1.44798	−0.723992	+	0.689808i	$0.757695\pi$
$90$	0	0
$91$	−5.73205	−0.600882
$92$	−5.00000	−0.521286
$93$	−4.73205	−0.490691
$94$	−4.73205	−0.488074
$95$	0.267949	0.0274910
$96$	1.73205	0.176777
$97$	10.9282	1.10959	0.554795	−	0.831987i	$-0.312797\pi$
$97$	10.9282	1.10959	0.554795	+	0.831987i	$0.312797\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−11.5359	−1.14786	−0.573932	−	0.818903i	$-0.694583\pi$
$101$	−11.5359	−1.14786	−0.573932	+	0.818903i	$0.694583\pi$
$102$	−1.26795	−0.125546
$103$	−10.7321	−1.05746	−0.528730	−	0.848790i	$-0.677332\pi$
$103$	−10.7321	−1.05746	−0.528730	+	0.848790i	$0.677332\pi$
$104$	−5.73205	−0.562074
$105$	−1.73205	−0.169031
$106$	−6.19615	−0.601824
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	−5.19615	−0.500000
$109$	20.0526	1.92069	0.960343	−	0.278820i	$-0.0899433\pi$
$109$	20.0526	1.92069	0.960343	+	0.278820i	$0.0899433\pi$
$110$	0	0
$111$	10.3923	0.986394
$112$	1.00000	0.0944911
$113$	−14.1244	−1.32871	−0.664354	−	0.747418i	$-0.731293\pi$
$113$	−14.1244	−1.32871	−0.664354	+	0.747418i	$0.731293\pi$
$114$	−0.464102	−0.0434671
$115$	5.00000	0.466252
$116$	8.00000	0.742781
$117$	0	0
$118$	−5.53590	−0.509621
$119$	−0.732051	−0.0671070
$120$	−1.73205	−0.158114
$121$	0	0
$122$	−2.92820	−0.265107
$123$	−10.7321	−0.967676
$124$	−2.73205	−0.245345
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.12436	−0.543449	−0.271724	−	0.962375i	$-0.587594\pi$
$127$	−6.12436	−0.543449	−0.271724	+	0.962375i	$0.587594\pi$
$128$	1.00000	0.0883883
$129$	−9.12436	−0.803355
$130$	5.73205	0.502734
$131$	−13.1962	−1.15295	−0.576477	−	0.817114i	$-0.695573\pi$
$131$	−13.1962	−1.15295	−0.576477	+	0.817114i	$0.695573\pi$
$132$	0	0
$133$	−0.267949	−0.0232341
$134$	−9.66025	−0.834519
$135$	5.19615	0.447214
$136$	−0.732051	−0.0627728
$137$	14.6603	1.25251	0.626255	−	0.779618i	$-0.284587\pi$
$137$	14.6603	1.25251	0.626255	+	0.779618i	$0.284587\pi$
$138$	−8.66025	−0.737210
$139$	−17.1962	−1.45856	−0.729279	−	0.684216i	$-0.760144\pi$
$139$	−17.1962	−1.45856	−0.729279	+	0.684216i	$0.760144\pi$
$140$	−1.00000	−0.0845154
$141$	−8.19615	−0.690241
$142$	6.92820	0.581402
$143$	0	0
$144$	0	0
$145$	−8.00000	−0.664364
$146$	9.12436	0.755137
$147$	1.73205	0.142857
$148$	6.00000	0.493197
$149$	19.4641	1.59456	0.797281	−	0.603609i	$-0.206271\pi$
$149$	19.4641	1.59456	0.797281	+	0.603609i	$0.206271\pi$
$150$	1.73205	0.141421
$151$	−18.8564	−1.53451	−0.767256	−	0.641341i	$-0.778379\pi$
$151$	−18.8564	−1.53451	−0.767256	+	0.641341i	$0.778379\pi$
$152$	−0.267949	−0.0217335
$153$	0	0
$154$	0	0
$155$	2.73205	0.219444
$156$	−9.92820	−0.794892
$157$	−5.39230	−0.430353	−0.215176	−	0.976575i	$-0.569033\pi$
$157$	−5.39230	−0.430353	−0.215176	+	0.976575i	$0.569033\pi$
$158$	−7.00000	−0.556890
$159$	−10.7321	−0.851107
$160$	−1.00000	−0.0790569
$161$	−5.00000	−0.394055
$162$	−9.00000	−0.707107
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	−6.19615	−0.483838
$165$	0	0
$166$	5.00000	0.388075
$167$	−0.339746	−0.0262903	−0.0131452	−	0.999914i	$-0.504184\pi$
$167$	−0.339746	−0.0262903	−0.0131452	+	0.999914i	$0.504184\pi$
$168$	1.73205	0.133631
$169$	19.8564	1.52742
$170$	0.732051	0.0561457
$171$	0	0
$172$	−5.26795	−0.401677
$173$	7.85641	0.597312	0.298656	−	0.954361i	$-0.403462\pi$
$173$	7.85641	0.597312	0.298656	+	0.954361i	$0.403462\pi$
$174$	13.8564	1.05045
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.58846	−0.720713
$178$	−13.6603	−1.02388
$179$	−16.1962	−1.21056	−0.605279	−	0.796014i	$-0.706938\pi$
$179$	−16.1962	−1.21056	−0.605279	+	0.796014i	$0.706938\pi$
$180$	0	0
$181$	−22.6603	−1.68432	−0.842162	−	0.539225i	$-0.818717\pi$
$181$	−22.6603	−1.68432	−0.842162	+	0.539225i	$0.818717\pi$
$182$	−5.73205	−0.424888
$183$	−5.07180	−0.374918
$184$	−5.00000	−0.368605
$185$	−6.00000	−0.441129
$186$	−4.73205	−0.346971
$187$	0	0
$188$	−4.73205	−0.345120
$189$	−5.19615	−0.377964
$190$	0.267949	0.0194391
$191$	−8.66025	−0.626634	−0.313317	−	0.949649i	$-0.601440\pi$
$191$	−8.66025	−0.626634	−0.313317	+	0.949649i	$0.601440\pi$
$192$	1.73205	0.125000
$193$	11.3923	0.820036	0.410018	−	0.912077i	$-0.365522\pi$
$193$	11.3923	0.820036	0.410018	+	0.912077i	$0.365522\pi$
$194$	10.9282	0.784599
$195$	9.92820	0.710973
$196$	1.00000	0.0714286
$197$	−9.46410	−0.674289	−0.337145	−	0.941453i	$-0.609461\pi$
$197$	−9.46410	−0.674289	−0.337145	+	0.941453i	$0.609461\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	1.00000	0.0707107
$201$	−16.7321	−1.18019
$202$	−11.5359	−0.811663
$203$	8.00000	0.561490
$204$	−1.26795	−0.0887742
$205$	6.19615	0.432758
$206$	−10.7321	−0.747737
$207$	0	0
$208$	−5.73205	−0.397446
$209$	0	0
$210$	−1.73205	−0.119523
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−6.19615	−0.425553
$213$	12.0000	0.822226
$214$	2.00000	0.136717
$215$	5.26795	0.359271
$216$	−5.19615	−0.353553
$217$	−2.73205	−0.185464
$218$	20.0526	1.35813
$219$	15.8038	1.06793
$220$	0	0
$221$	4.19615	0.282264
$222$	10.3923	0.697486
$223$	−11.2679	−0.754558	−0.377279	−	0.926100i	$-0.623140\pi$
$223$	−11.2679	−0.754558	−0.377279	+	0.926100i	$0.623140\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.1244	−0.939538
$227$	17.3205	1.14960	0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205	1.14960	0.574801	+	0.818293i	$0.305079\pi$
$228$	−0.464102	−0.0307359
$229$	5.85641	0.387002	0.193501	−	0.981100i	$-0.438016\pi$
$229$	5.85641	0.387002	0.193501	+	0.981100i	$0.438016\pi$
$230$	5.00000	0.329690
$231$	0	0
$232$	8.00000	0.525226
$233$	5.53590	0.362669	0.181334	−	0.983421i	$-0.441958\pi$
$233$	5.53590	0.362669	0.181334	+	0.983421i	$0.441958\pi$
$234$	0	0
$235$	4.73205	0.308685
$236$	−5.53590	−0.360356
$237$	−12.1244	−0.787562
$238$	−0.732051	−0.0474518
$239$	−4.46410	−0.288759	−0.144379	−	0.989522i	$-0.546119\pi$
$239$	−4.46410	−0.288759	−0.144379	+	0.989522i	$0.546119\pi$
$240$	−1.73205	−0.111803
$241$	17.4641	1.12496	0.562481	−	0.826810i	$-0.309847\pi$
$241$	17.4641	1.12496	0.562481	+	0.826810i	$0.309847\pi$
$242$	0	0
$243$	0	0
$244$	−2.92820	−0.187459
$245$	−1.00000	−0.0638877
$246$	−10.7321	−0.684251
$247$	1.53590	0.0977269
$248$	−2.73205	−0.173485
$249$	8.66025	0.548821
$250$	−1.00000	−0.0632456
$251$	−6.39230	−0.403479	−0.201739	−	0.979439i	$-0.564659\pi$
$251$	−6.39230	−0.403479	−0.201739	+	0.979439i	$0.564659\pi$
$252$	0	0
$253$	0	0
$254$	−6.12436	−0.384276
$255$	1.26795	0.0794021
$256$	1.00000	0.0625000
$257$	8.19615	0.511262	0.255631	−	0.966774i	$-0.417717\pi$
$257$	8.19615	0.511262	0.255631	+	0.966774i	$0.417717\pi$
$258$	−9.12436	−0.568058
$259$	6.00000	0.372822
$260$	5.73205	0.355487
$261$	0	0
$262$	−13.1962	−0.815261
$263$	10.1244	0.624295	0.312147	−	0.950034i	$-0.398952\pi$
$263$	10.1244	0.624295	0.312147	+	0.950034i	$0.398952\pi$
$264$	0	0
$265$	6.19615	0.380627
$266$	−0.267949	−0.0164290
$267$	−23.6603	−1.44798
$268$	−9.66025	−0.590094
$269$	−21.7321	−1.32503	−0.662513	−	0.749050i	$-0.730510\pi$
$269$	−21.7321	−1.32503	−0.662513	+	0.749050i	$0.730510\pi$
$270$	5.19615	0.316228
$271$	21.4641	1.30385	0.651926	−	0.758283i	$-0.273961\pi$
$271$	21.4641	1.30385	0.651926	+	0.758283i	$0.273961\pi$
$272$	−0.732051	−0.0443871
$273$	−9.92820	−0.600882
$274$	14.6603	0.885658
$275$	0	0
$276$	−8.66025	−0.521286
$277$	1.80385	0.108383	0.0541913	−	0.998531i	$-0.482742\pi$
$277$	1.80385	0.108383	0.0541913	+	0.998531i	$0.482742\pi$
$278$	−17.1962	−1.03136
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	7.73205	0.461255	0.230628	−	0.973042i	$-0.425922\pi$
$281$	7.73205	0.461255	0.230628	+	0.973042i	$0.425922\pi$
$282$	−8.19615	−0.488074
$283$	25.7846	1.53274	0.766368	−	0.642402i	$-0.222062\pi$
$283$	25.7846	1.53274	0.766368	+	0.642402i	$0.222062\pi$
$284$	6.92820	0.411113
$285$	0.464102	0.0274910
$286$	0	0
$287$	−6.19615	−0.365747
$288$	0	0
$289$	−16.4641	−0.968477
$290$	−8.00000	−0.469776
$291$	18.9282	1.10959
$292$	9.12436	0.533963
$293$	11.7321	0.685394	0.342697	−	0.939446i	$-0.388660\pi$
$293$	11.7321	0.685394	0.342697	+	0.939446i	$0.388660\pi$
$294$	1.73205	0.101015
$295$	5.53590	0.322312
$296$	6.00000	0.348743
$297$	0	0
$298$	19.4641	1.12753
$299$	28.6603	1.65747
$300$	1.73205	0.100000
$301$	−5.26795	−0.303640
$302$	−18.8564	−1.08506
$303$	−19.9808	−1.14786
$304$	−0.267949	−0.0153679
$305$	2.92820	0.167668
$306$	0	0
$307$	−7.07180	−0.403609	−0.201804	−	0.979426i	$-0.564681\pi$
$307$	−7.07180	−0.403609	−0.201804	+	0.979426i	$0.564681\pi$
$308$	0	0
$309$	−18.5885	−1.05746
$310$	2.73205	0.155170
$311$	12.7321	0.721968	0.360984	−	0.932572i	$-0.382441\pi$
$311$	12.7321	0.721968	0.360984	+	0.932572i	$0.382441\pi$
$312$	−9.92820	−0.562074
$313$	−5.60770	−0.316966	−0.158483	−	0.987362i	$-0.550660\pi$
$313$	−5.60770	−0.316966	−0.158483	+	0.987362i	$0.550660\pi$
$314$	−5.39230	−0.304305
$315$	0	0
$316$	−7.00000	−0.393781
$317$	5.12436	0.287812	0.143906	−	0.989591i	$-0.454034\pi$
$317$	5.12436	0.287812	0.143906	+	0.989591i	$0.454034\pi$
$318$	−10.7321	−0.601824
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	3.46410	0.193347
$322$	−5.00000	−0.278639
$323$	0.196152	0.0109142
$324$	−9.00000	−0.500000
$325$	−5.73205	−0.317957
$326$	10.0000	0.553849
$327$	34.7321	1.92069
$328$	−6.19615	−0.342125
$329$	−4.73205	−0.260886
$330$	0	0
$331$	34.0526	1.87170	0.935849	−	0.352401i	$-0.114635\pi$
$331$	34.0526	1.87170	0.935849	+	0.352401i	$0.114635\pi$
$332$	5.00000	0.274411
$333$	0	0
$334$	−0.339746	−0.0185901
$335$	9.66025	0.527796
$336$	1.73205	0.0944911
$337$	−9.39230	−0.511631	−0.255816	−	0.966726i	$-0.582344\pi$
$337$	−9.39230	−0.511631	−0.255816	+	0.966726i	$0.582344\pi$
$338$	19.8564	1.08005
$339$	−24.4641	−1.32871
$340$	0.732051	0.0397010
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.26795	−0.284029
$345$	8.66025	0.466252
$346$	7.85641	0.422363
$347$	−15.4641	−0.830156	−0.415078	−	0.909786i	$-0.636246\pi$
$347$	−15.4641	−0.830156	−0.415078	+	0.909786i	$0.636246\pi$
$348$	13.8564	0.742781
$349$	0.464102	0.0248428	0.0124214	−	0.999923i	$-0.496046\pi$
$349$	0.464102	0.0248428	0.0124214	+	0.999923i	$0.496046\pi$
$350$	1.00000	0.0534522
$351$	29.7846	1.58978
$352$	0	0
$353$	12.1962	0.649136	0.324568	−	0.945862i	$-0.394781\pi$
$353$	12.1962	0.649136	0.324568	+	0.945862i	$0.394781\pi$
$354$	−9.58846	−0.509621
$355$	−6.92820	−0.367711
$356$	−13.6603	−0.723992
$357$	−1.26795	−0.0671070
$358$	−16.1962	−0.855993
$359$	21.7128	1.14596	0.572979	−	0.819570i	$-0.305788\pi$
$359$	21.7128	1.14596	0.572979	+	0.819570i	$0.305788\pi$
$360$	0	0
$361$	−18.9282	−0.996221
$362$	−22.6603	−1.19100
$363$	0	0
$364$	−5.73205	−0.300441
$365$	−9.12436	−0.477591
$366$	−5.07180	−0.265107
$367$	−17.2679	−0.901380	−0.450690	−	0.892681i	$-0.648822\pi$
$367$	−17.2679	−0.901380	−0.450690	+	0.892681i	$0.648822\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	−6.00000	−0.311925
$371$	−6.19615	−0.321688
$372$	−4.73205	−0.245345
$373$	21.3205	1.10393	0.551967	−	0.833866i	$-0.313877\pi$
$373$	21.3205	1.10393	0.551967	+	0.833866i	$0.313877\pi$
$374$	0	0
$375$	−1.73205	−0.0894427
$376$	−4.73205	−0.244037
$377$	−45.8564	−2.36173
$378$	−5.19615	−0.267261
$379$	−6.78461	−0.348502	−0.174251	−	0.984701i	$-0.555750\pi$
$379$	−6.78461	−0.348502	−0.174251	+	0.984701i	$0.555750\pi$
$380$	0.267949	0.0137455
$381$	−10.6077	−0.543449
$382$	−8.66025	−0.443097
$383$	25.6603	1.31118	0.655589	−	0.755118i	$-0.272420\pi$
$383$	25.6603	1.31118	0.655589	+	0.755118i	$0.272420\pi$
$384$	1.73205	0.0883883
$385$	0	0
$386$	11.3923	0.579853
$387$	0	0
$388$	10.9282	0.554795
$389$	−23.1244	−1.17245	−0.586226	−	0.810148i	$-0.699387\pi$
$389$	−23.1244	−1.17245	−0.586226	+	0.810148i	$0.699387\pi$
$390$	9.92820	0.502734
$391$	3.66025	0.185107
$392$	1.00000	0.0505076
$393$	−22.8564	−1.15295
$394$	−9.46410	−0.476795
$395$	7.00000	0.352208
$396$	0	0
$397$	12.7846	0.641641	0.320821	−	0.947140i	$-0.396041\pi$
$397$	12.7846	0.641641	0.320821	+	0.947140i	$0.396041\pi$
$398$	−4.00000	−0.200502
$399$	−0.464102	−0.0232341
$400$	1.00000	0.0500000
$401$	−18.5359	−0.925639	−0.462819	−	0.886453i	$-0.653162\pi$
$401$	−18.5359	−0.925639	−0.462819	+	0.886453i	$0.653162\pi$
$402$	−16.7321	−0.834519
$403$	15.6603	0.780093
$404$	−11.5359	−0.573932
$405$	9.00000	0.447214
$406$	8.00000	0.397033
$407$	0	0
$408$	−1.26795	−0.0627728
$409$	35.7128	1.76588	0.882942	−	0.469481i	$-0.155559\pi$
$409$	35.7128	1.76588	0.882942	+	0.469481i	$0.155559\pi$
$410$	6.19615	0.306006
$411$	25.3923	1.25251
$412$	−10.7321	−0.528730
$413$	−5.53590	−0.272404
$414$	0	0
$415$	−5.00000	−0.245440
$416$	−5.73205	−0.281037
$417$	−29.7846	−1.45856
$418$	0	0
$419$	15.9282	0.778144	0.389072	−	0.921207i	$-0.372796\pi$
$419$	15.9282	0.778144	0.389072	+	0.921207i	$0.372796\pi$
$420$	−1.73205	−0.0845154
$421$	−7.60770	−0.370776	−0.185388	−	0.982665i	$-0.559354\pi$
$421$	−7.60770	−0.370776	−0.185388	+	0.982665i	$0.559354\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	−6.19615	−0.300912
$425$	−0.732051	−0.0355097
$426$	12.0000	0.581402
$427$	−2.92820	−0.141706
$428$	2.00000	0.0966736
$429$	0	0
$430$	5.26795	0.254043
$431$	−34.4641	−1.66008	−0.830039	−	0.557706i	$-0.811682\pi$
$431$	−34.4641	−1.66008	−0.830039	+	0.557706i	$0.811682\pi$
$432$	−5.19615	−0.250000
$433$	−20.0526	−0.963664	−0.481832	−	0.876263i	$-0.660029\pi$
$433$	−20.0526	−0.963664	−0.481832	+	0.876263i	$0.660029\pi$
$434$	−2.73205	−0.131143
$435$	−13.8564	−0.664364
$436$	20.0526	0.960343
$437$	1.33975	0.0640887
$438$	15.8038	0.755137
$439$	−14.7321	−0.703122	−0.351561	−	0.936165i	$-0.614349\pi$
$439$	−14.7321	−0.703122	−0.351561	+	0.936165i	$0.614349\pi$
$440$	0	0
$441$	0	0
$442$	4.19615	0.199591
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	10.3923	0.493197
$445$	13.6603	0.647558
$446$	−11.2679	−0.533553
$447$	33.7128	1.59456
$448$	1.00000	0.0472456
$449$	31.3923	1.48149	0.740747	−	0.671784i	$-0.234472\pi$
$449$	31.3923	1.48149	0.740747	+	0.671784i	$0.234472\pi$
$450$	0	0
$451$	0	0
$452$	−14.1244	−0.664354
$453$	−32.6603	−1.53451
$454$	17.3205	0.812892
$455$	5.73205	0.268723
$456$	−0.464102	−0.0217335
$457$	−3.39230	−0.158685	−0.0793427	−	0.996847i	$-0.525282\pi$
$457$	−3.39230	−0.158685	−0.0793427	+	0.996847i	$0.525282\pi$
$458$	5.85641	0.273652
$459$	3.80385	0.177548
$460$	5.00000	0.233126
$461$	10.5359	0.490706	0.245353	−	0.969434i	$-0.421096\pi$
$461$	10.5359	0.490706	0.245353	+	0.969434i	$0.421096\pi$
$462$	0	0
$463$	26.1769	1.21654	0.608272	−	0.793729i	$-0.291863\pi$
$463$	26.1769	1.21654	0.608272	+	0.793729i	$0.291863\pi$
$464$	8.00000	0.371391
$465$	4.73205	0.219444
$466$	5.53590	0.256446
$467$	13.5885	0.628799	0.314399	−	0.949291i	$-0.398197\pi$
$467$	13.5885	0.628799	0.314399	+	0.949291i	$0.398197\pi$
$468$	0	0
$469$	−9.66025	−0.446069
$470$	4.73205	0.218273
$471$	−9.33975	−0.430353
$472$	−5.53590	−0.254810
$473$	0	0
$474$	−12.1244	−0.556890
$475$	−0.267949	−0.0122944
$476$	−0.732051	−0.0335535
$477$	0	0
$478$	−4.46410	−0.204183
$479$	28.9808	1.32416	0.662082	−	0.749431i	$-0.269673\pi$
$479$	28.9808	1.32416	0.662082	+	0.749431i	$0.269673\pi$
$480$	−1.73205	−0.0790569
$481$	−34.3923	−1.56815
$482$	17.4641	0.795468
$483$	−8.66025	−0.394055
$484$	0	0
$485$	−10.9282	−0.496224
$486$	0	0
$487$	17.3923	0.788121	0.394060	−	0.919085i	$-0.371070\pi$
$487$	17.3923	0.788121	0.394060	+	0.919085i	$0.371070\pi$
$488$	−2.92820	−0.132554
$489$	17.3205	0.783260
$490$	−1.00000	−0.0451754
$491$	9.66025	0.435961	0.217981	−	0.975953i	$-0.430053\pi$
$491$	9.66025	0.435961	0.217981	+	0.975953i	$0.430053\pi$
$492$	−10.7321	−0.483838
$493$	−5.85641	−0.263759
$494$	1.53590	0.0691033
$495$	0	0
$496$	−2.73205	−0.122673
$497$	6.92820	0.310772
$498$	8.66025	0.388075
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−1.00000	−0.0447214
$501$	−0.588457	−0.0262903
$502$	−6.39230	−0.285303
$503$	18.1962	0.811326	0.405663	−	0.914023i	$-0.367041\pi$
$503$	18.1962	0.811326	0.405663	+	0.914023i	$0.367041\pi$
$504$	0	0
$505$	11.5359	0.513341
$506$	0	0
$507$	34.3923	1.52742
$508$	−6.12436	−0.271724
$509$	−32.6603	−1.44764	−0.723820	−	0.689989i	$-0.757615\pi$
$509$	−32.6603	−1.44764	−0.723820	+	0.689989i	$0.757615\pi$
$510$	1.26795	0.0561457
$511$	9.12436	0.403638
$512$	1.00000	0.0441942
$513$	1.39230	0.0614718
$514$	8.19615	0.361517
$515$	10.7321	0.472911
$516$	−9.12436	−0.401677
$517$	0	0
$518$	6.00000	0.263625
$519$	13.6077	0.597312
$520$	5.73205	0.251367
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	0	0
$523$	−23.3923	−1.02287	−0.511437	−	0.859321i	$-0.670887\pi$
$523$	−23.3923	−1.02287	−0.511437	+	0.859321i	$0.670887\pi$
$524$	−13.1962	−0.576477
$525$	1.73205	0.0755929
$526$	10.1244	0.441443
$527$	2.00000	0.0871214
$528$	0	0
$529$	2.00000	0.0869565
$530$	6.19615	0.269144
$531$	0	0
$532$	−0.267949	−0.0116171
$533$	35.5167	1.53840
$534$	−23.6603	−1.02388
$535$	−2.00000	−0.0864675
$536$	−9.66025	−0.417259
$537$	−28.0526	−1.21056
$538$	−21.7321	−0.936935
$539$	0	0
$540$	5.19615	0.223607
$541$	30.3923	1.30667	0.653334	−	0.757070i	$-0.273370\pi$
$541$	30.3923	1.30667	0.653334	+	0.757070i	$0.273370\pi$
$542$	21.4641	0.921962
$543$	−39.2487	−1.68432
$544$	−0.732051	−0.0313864
$545$	−20.0526	−0.858957
$546$	−9.92820	−0.424888
$547$	−7.51666	−0.321389	−0.160695	−	0.987004i	$-0.551373\pi$
$547$	−7.51666	−0.321389	−0.160695	+	0.987004i	$0.551373\pi$
$548$	14.6603	0.626255
$549$	0	0
$550$	0	0
$551$	−2.14359	−0.0913202
$552$	−8.66025	−0.368605
$553$	−7.00000	−0.297670
$554$	1.80385	0.0766381
$555$	−10.3923	−0.441129
$556$	−17.1962	−0.729279
$557$	−45.3731	−1.92252	−0.961259	−	0.275646i	$-0.911108\pi$
$557$	−45.3731	−1.92252	−0.961259	+	0.275646i	$0.911108\pi$
$558$	0	0
$559$	30.1962	1.27716
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	7.73205	0.326157
$563$	−8.46410	−0.356719	−0.178360	−	0.983965i	$-0.557079\pi$
$563$	−8.46410	−0.356719	−0.178360	+	0.983965i	$0.557079\pi$
$564$	−8.19615	−0.345120
$565$	14.1244	0.594216
$566$	25.7846	1.08381
$567$	−9.00000	−0.377964
$568$	6.92820	0.290701
$569$	13.8756	0.581697	0.290849	−	0.956769i	$-0.406062\pi$
$569$	13.8756	0.581697	0.290849	+	0.956769i	$0.406062\pi$
$570$	0.464102	0.0194391
$571$	6.53590	0.273519	0.136759	−	0.990604i	$-0.456331\pi$
$571$	6.53590	0.273519	0.136759	+	0.990604i	$0.456331\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	−6.19615	−0.258622
$575$	−5.00000	−0.208514
$576$	0	0
$577$	−40.4449	−1.68374	−0.841871	−	0.539679i	$-0.818546\pi$
$577$	−40.4449	−1.68374	−0.841871	+	0.539679i	$0.818546\pi$
$578$	−16.4641	−0.684816
$579$	19.7321	0.820036
$580$	−8.00000	−0.332182
$581$	5.00000	0.207435
$582$	18.9282	0.784599
$583$	0	0
$584$	9.12436	0.377569
$585$	0	0
$586$	11.7321	0.484647
$587$	−30.5167	−1.25956	−0.629779	−	0.776775i	$-0.716854\pi$
$587$	−30.5167	−1.25956	−0.629779	+	0.776775i	$0.716854\pi$
$588$	1.73205	0.0714286
$589$	0.732051	0.0301636
$590$	5.53590	0.227909
$591$	−16.3923	−0.674289
$592$	6.00000	0.246598
$593$	−2.14359	−0.0880268	−0.0440134	−	0.999031i	$-0.514014\pi$
$593$	−2.14359	−0.0880268	−0.0440134	+	0.999031i	$0.514014\pi$
$594$	0	0
$595$	0.732051	0.0300112
$596$	19.4641	0.797281
$597$	−6.92820	−0.283552
$598$	28.6603	1.17200
$599$	19.9808	0.816392	0.408196	−	0.912894i	$-0.366158\pi$
$599$	19.9808	0.816392	0.408196	+	0.912894i	$0.366158\pi$
$600$	1.73205	0.0707107
$601$	−13.6603	−0.557213	−0.278607	−	0.960405i	$-0.589873\pi$
$601$	−13.6603	−0.557213	−0.278607	+	0.960405i	$0.589873\pi$
$602$	−5.26795	−0.214706
$603$	0	0
$604$	−18.8564	−0.767256
$605$	0	0
$606$	−19.9808	−0.811663
$607$	48.2487	1.95835	0.979177	−	0.203006i	$-0.0650713\pi$
$607$	48.2487	1.95835	0.979177	+	0.203006i	$0.0650713\pi$
$608$	−0.267949	−0.0108668
$609$	13.8564	0.561490
$610$	2.92820	0.118559
$611$	27.1244	1.09733
$612$	0	0
$613$	40.7321	1.64515	0.822576	−	0.568655i	$-0.192536\pi$
$613$	40.7321	1.64515	0.822576	+	0.568655i	$0.192536\pi$
$614$	−7.07180	−0.285394
$615$	10.7321	0.432758
$616$	0	0
$617$	34.3923	1.38458	0.692291	−	0.721618i	$-0.256601\pi$
$617$	34.3923	1.38458	0.692291	+	0.721618i	$0.256601\pi$
$618$	−18.5885	−0.747737
$619$	−22.7128	−0.912905	−0.456453	−	0.889748i	$-0.650880\pi$
$619$	−22.7128	−0.912905	−0.456453	+	0.889748i	$0.650880\pi$
$620$	2.73205	0.109722
$621$	25.9808	1.04257
$622$	12.7321	0.510509
$623$	−13.6603	−0.547287
$624$	−9.92820	−0.397446
$625$	1.00000	0.0400000
$626$	−5.60770	−0.224129
$627$	0	0
$628$	−5.39230	−0.215176
$629$	−4.39230	−0.175133
$630$	0	0
$631$	−4.67949	−0.186288	−0.0931438	−	0.995653i	$-0.529692\pi$
$631$	−4.67949	−0.186288	−0.0931438	+	0.995653i	$0.529692\pi$
$632$	−7.00000	−0.278445
$633$	−13.8564	−0.550743
$634$	5.12436	0.203514
$635$	6.12436	0.243038
$636$	−10.7321	−0.425553
$637$	−5.73205	−0.227112
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−46.4641	−1.83522	−0.917611	−	0.397480i	$-0.869885\pi$
$641$	−46.4641	−1.83522	−0.917611	+	0.397480i	$0.869885\pi$
$642$	3.46410	0.136717
$643$	−39.1769	−1.54499	−0.772493	−	0.635023i	$-0.780991\pi$
$643$	−39.1769	−1.54499	−0.772493	+	0.635023i	$0.780991\pi$
$644$	−5.00000	−0.197028
$645$	9.12436	0.359271
$646$	0.196152	0.00771751
$647$	−32.1962	−1.26576	−0.632881	−	0.774249i	$-0.718128\pi$
$647$	−32.1962	−1.26576	−0.632881	+	0.774249i	$0.718128\pi$
$648$	−9.00000	−0.353553
$649$	0	0
$650$	−5.73205	−0.224830
$651$	−4.73205	−0.185464
$652$	10.0000	0.391630
$653$	8.19615	0.320740	0.160370	−	0.987057i	$-0.448731\pi$
$653$	8.19615	0.320740	0.160370	+	0.987057i	$0.448731\pi$
$654$	34.7321	1.35813
$655$	13.1962	0.515616
$656$	−6.19615	−0.241919
$657$	0	0
$658$	−4.73205	−0.184475
$659$	26.4449	1.03015	0.515073	−	0.857146i	$-0.327765\pi$
$659$	26.4449	1.03015	0.515073	+	0.857146i	$0.327765\pi$
$660$	0	0
$661$	20.6603	0.803591	0.401795	−	0.915729i	$-0.368386\pi$
$661$	20.6603	0.803591	0.401795	+	0.915729i	$0.368386\pi$
$662$	34.0526	1.32349
$663$	7.26795	0.282264
$664$	5.00000	0.194038
$665$	0.267949	0.0103906
$666$	0	0
$667$	−40.0000	−1.54881
$668$	−0.339746	−0.0131452
$669$	−19.5167	−0.754558
$670$	9.66025	0.373208
$671$	0	0
$672$	1.73205	0.0668153
$673$	−29.1051	−1.12192	−0.560960	−	0.827843i	$-0.689568\pi$
$673$	−29.1051	−1.12192	−0.560960	+	0.827843i	$0.689568\pi$
$674$	−9.39230	−0.361778
$675$	−5.19615	−0.200000
$676$	19.8564	0.763708
$677$	−5.98076	−0.229859	−0.114930	−	0.993374i	$-0.536664\pi$
$677$	−5.98076	−0.229859	−0.114930	+	0.993374i	$0.536664\pi$
$678$	−24.4641	−0.939538
$679$	10.9282	0.419386
$680$	0.732051	0.0280729
$681$	30.0000	1.14960
$682$	0	0
$683$	−25.0718	−0.959346	−0.479673	−	0.877447i	$-0.659245\pi$
$683$	−25.0718	−0.959346	−0.479673	+	0.877447i	$0.659245\pi$
$684$	0	0
$685$	−14.6603	−0.560140
$686$	1.00000	0.0381802
$687$	10.1436	0.387002
$688$	−5.26795	−0.200839
$689$	35.5167	1.35308
$690$	8.66025	0.329690
$691$	−25.8564	−0.983624	−0.491812	−	0.870701i	$-0.663665\pi$
$691$	−25.8564	−0.983624	−0.491812	+	0.870701i	$0.663665\pi$
$692$	7.85641	0.298656
$693$	0	0
$694$	−15.4641	−0.587009
$695$	17.1962	0.652287
$696$	13.8564	0.525226
$697$	4.53590	0.171809
$698$	0.464102	0.0175665
$699$	9.58846	0.362669
$700$	1.00000	0.0377964
$701$	−17.9090	−0.676412	−0.338206	−	0.941072i	$-0.609820\pi$
$701$	−17.9090	−0.676412	−0.338206	+	0.941072i	$0.609820\pi$
$702$	29.7846	1.12415
$703$	−1.60770	−0.0606354
$704$	0	0
$705$	8.19615	0.308685
$706$	12.1962	0.459008
$707$	−11.5359	−0.433852
$708$	−9.58846	−0.360356
$709$	−33.3205	−1.25138	−0.625689	−	0.780073i	$-0.715182\pi$
$709$	−33.3205	−1.25138	−0.625689	+	0.780073i	$0.715182\pi$
$710$	−6.92820	−0.260011
$711$	0	0
$712$	−13.6603	−0.511940
$713$	13.6603	0.511581
$714$	−1.26795	−0.0474518
$715$	0	0
$716$	−16.1962	−0.605279
$717$	−7.73205	−0.288759
$718$	21.7128	0.810315
$719$	−21.0718	−0.785845	−0.392923	−	0.919572i	$-0.628536\pi$
$719$	−21.0718	−0.785845	−0.392923	+	0.919572i	$0.628536\pi$
$720$	0	0
$721$	−10.7321	−0.399682
$722$	−18.9282	−0.704435
$723$	30.2487	1.12496
$724$	−22.6603	−0.842162
$725$	8.00000	0.297113
$726$	0	0
$727$	19.3205	0.716558	0.358279	−	0.933615i	$-0.383364\pi$
$727$	19.3205	0.716558	0.358279	+	0.933615i	$0.383364\pi$
$728$	−5.73205	−0.212444
$729$	27.0000	1.00000
$730$	−9.12436	−0.337708
$731$	3.85641	0.142634
$732$	−5.07180	−0.187459
$733$	7.58846	0.280286	0.140143	−	0.990131i	$-0.455244\pi$
$733$	7.58846	0.280286	0.140143	+	0.990131i	$0.455244\pi$
$734$	−17.2679	−0.637372
$735$	−1.73205	−0.0638877
$736$	−5.00000	−0.184302
$737$	0	0
$738$	0	0
$739$	−24.9808	−0.918932	−0.459466	−	0.888195i	$-0.651959\pi$
$739$	−24.9808	−0.918932	−0.459466	+	0.888195i	$0.651959\pi$
$740$	−6.00000	−0.220564
$741$	2.66025	0.0977269
$742$	−6.19615	−0.227468
$743$	19.4641	0.714069	0.357034	−	0.934091i	$-0.383788\pi$
$743$	19.4641	0.714069	0.357034	+	0.934091i	$0.383788\pi$
$744$	−4.73205	−0.173485
$745$	−19.4641	−0.713110
$746$	21.3205	0.780599
$747$	0	0
$748$	0	0
$749$	2.00000	0.0730784
$750$	−1.73205	−0.0632456
$751$	30.3731	1.10833	0.554164	−	0.832407i	$-0.313038\pi$
$751$	30.3731	1.10833	0.554164	+	0.832407i	$0.313038\pi$
$752$	−4.73205	−0.172560
$753$	−11.0718	−0.403479
$754$	−45.8564	−1.66999
$755$	18.8564	0.686255
$756$	−5.19615	−0.188982
$757$	39.9090	1.45052	0.725258	−	0.688477i	$-0.241720\pi$
$757$	39.9090	1.45052	0.725258	+	0.688477i	$0.241720\pi$
$758$	−6.78461	−0.246428
$759$	0	0
$760$	0.267949	0.00971954
$761$	18.4449	0.668626	0.334313	−	0.942462i	$-0.391496\pi$
$761$	18.4449	0.668626	0.334313	+	0.942462i	$0.391496\pi$
$762$	−10.6077	−0.384276
$763$	20.0526	0.725951
$764$	−8.66025	−0.313317
$765$	0	0
$766$	25.6603	0.927143
$767$	31.7321	1.14578
$768$	1.73205	0.0625000
$769$	−6.78461	−0.244659	−0.122330	−	0.992490i	$-0.539037\pi$
$769$	−6.78461	−0.244659	−0.122330	+	0.992490i	$0.539037\pi$
$770$	0	0
$771$	14.1962	0.511262
$772$	11.3923	0.410018
$773$	34.4641	1.23959	0.619794	−	0.784765i	$-0.287216\pi$
$773$	34.4641	1.23959	0.619794	+	0.784765i	$0.287216\pi$
$774$	0	0
$775$	−2.73205	−0.0981382
$776$	10.9282	0.392300
$777$	10.3923	0.372822
$778$	−23.1244	−0.829048
$779$	1.66025	0.0594848
$780$	9.92820	0.355487
$781$	0	0
$782$	3.66025	0.130890
$783$	−41.5692	−1.48556
$784$	1.00000	0.0357143
$785$	5.39230	0.192460
$786$	−22.8564	−0.815261
$787$	−50.4974	−1.80004	−0.900019	−	0.435850i	$-0.856448\pi$
$787$	−50.4974	−1.80004	−0.900019	+	0.435850i	$0.856448\pi$
$788$	−9.46410	−0.337145
$789$	17.5359	0.624295
$790$	7.00000	0.249049
$791$	−14.1244	−0.502204
$792$	0	0
$793$	16.7846	0.596039
$794$	12.7846	0.453709
$795$	10.7321	0.380627
$796$	−4.00000	−0.141776
$797$	−28.1769	−0.998078	−0.499039	−	0.866580i	$-0.666313\pi$
$797$	−28.1769	−0.998078	−0.499039	+	0.866580i	$0.666313\pi$
$798$	−0.464102	−0.0164290
$799$	3.46410	0.122551
$800$	1.00000	0.0353553
$801$	0	0
$802$	−18.5359	−0.654525
$803$	0	0
$804$	−16.7321	−0.590094
$805$	5.00000	0.176227
$806$	15.6603	0.551609
$807$	−37.6410	−1.32503
$808$	−11.5359	−0.405831
$809$	−50.7846	−1.78549	−0.892746	−	0.450561i	$-0.851224\pi$
$809$	−50.7846	−1.78549	−0.892746	+	0.450561i	$0.851224\pi$
$810$	9.00000	0.316228
$811$	−37.4641	−1.31554	−0.657771	−	0.753218i	$-0.728501\pi$
$811$	−37.4641	−1.31554	−0.657771	+	0.753218i	$0.728501\pi$
$812$	8.00000	0.280745
$813$	37.1769	1.30385
$814$	0	0
$815$	−10.0000	−0.350285
$816$	−1.26795	−0.0443871
$817$	1.41154	0.0493836
$818$	35.7128	1.24867
$819$	0	0
$820$	6.19615	0.216379
$821$	16.1436	0.563415	0.281708	−	0.959500i	$-0.409099\pi$
$821$	16.1436	0.563415	0.281708	+	0.959500i	$0.409099\pi$
$822$	25.3923	0.885658
$823$	1.21539	0.0423658	0.0211829	−	0.999776i	$-0.493257\pi$
$823$	1.21539	0.0423658	0.0211829	+	0.999776i	$0.493257\pi$
$824$	−10.7321	−0.373869
$825$	0	0
$826$	−5.53590	−0.192619
$827$	−45.9090	−1.59641	−0.798206	−	0.602385i	$-0.794217\pi$
$827$	−45.9090	−1.59641	−0.798206	+	0.602385i	$0.794217\pi$
$828$	0	0
$829$	24.1244	0.837874	0.418937	−	0.908015i	$-0.362403\pi$
$829$	24.1244	0.837874	0.418937	+	0.908015i	$0.362403\pi$
$830$	−5.00000	−0.173553
$831$	3.12436	0.108383
$832$	−5.73205	−0.198723
$833$	−0.732051	−0.0253641
$834$	−29.7846	−1.03136
$835$	0.339746	0.0117574
$836$	0	0
$837$	14.1962	0.490691
$838$	15.9282	0.550231
$839$	1.80385	0.0622757	0.0311379	−	0.999515i	$-0.490087\pi$
$839$	1.80385	0.0622757	0.0311379	+	0.999515i	$0.490087\pi$
$840$	−1.73205	−0.0597614
$841$	35.0000	1.20690
$842$	−7.60770	−0.262178
$843$	13.3923	0.461255
$844$	−8.00000	−0.275371
$845$	−19.8564	−0.683081
$846$	0	0
$847$	0	0
$848$	−6.19615	−0.212777
$849$	44.6603	1.53274
$850$	−0.732051	−0.0251091
$851$	−30.0000	−1.02839
$852$	12.0000	0.411113
$853$	−35.8372	−1.22704	−0.613521	−	0.789679i	$-0.710247\pi$
$853$	−35.8372	−1.22704	−0.613521	+	0.789679i	$0.710247\pi$
$854$	−2.92820	−0.100201
$855$	0	0
$856$	2.00000	0.0683586
$857$	−36.3923	−1.24314	−0.621569	−	0.783360i	$-0.713504\pi$
$857$	−36.3923	−1.24314	−0.621569	+	0.783360i	$0.713504\pi$
$858$	0	0
$859$	31.8564	1.08693	0.543463	−	0.839433i	$-0.317113\pi$
$859$	31.8564	1.08693	0.543463	+	0.839433i	$0.317113\pi$
$860$	5.26795	0.179636
$861$	−10.7321	−0.365747
$862$	−34.4641	−1.17385
$863$	−37.8564	−1.28865	−0.644324	−	0.764753i	$-0.722861\pi$
$863$	−37.8564	−1.28865	−0.644324	+	0.764753i	$0.722861\pi$
$864$	−5.19615	−0.176777
$865$	−7.85641	−0.267126
$866$	−20.0526	−0.681414
$867$	−28.5167	−0.968477
$868$	−2.73205	−0.0927318
$869$	0	0
$870$	−13.8564	−0.469776
$871$	55.3731	1.87624
$872$	20.0526	0.679065
$873$	0	0
$874$	1.33975	0.0453176
$875$	−1.00000	−0.0338062
$876$	15.8038	0.533963
$877$	48.0526	1.62262	0.811310	−	0.584616i	$-0.198755\pi$
$877$	48.0526	1.62262	0.811310	+	0.584616i	$0.198755\pi$
$878$	−14.7321	−0.497183
$879$	20.3205	0.685394
$880$	0	0
$881$	57.4256	1.93472	0.967359	−	0.253409i	$-0.0815518\pi$
$881$	57.4256	1.93472	0.967359	+	0.253409i	$0.0815518\pi$
$882$	0	0
$883$	36.6410	1.23307	0.616534	−	0.787328i	$-0.288536\pi$
$883$	36.6410	1.23307	0.616534	+	0.787328i	$0.288536\pi$
$884$	4.19615	0.141132
$885$	9.58846	0.322312
$886$	4.00000	0.134383
$887$	19.6603	0.660127	0.330063	−	0.943959i	$-0.392930\pi$
$887$	19.6603	0.660127	0.330063	+	0.943959i	$0.392930\pi$
$888$	10.3923	0.348743
$889$	−6.12436	−0.205404
$890$	13.6603	0.457893
$891$	0	0
$892$	−11.2679	−0.377279
$893$	1.26795	0.0424303
$894$	33.7128	1.12753
$895$	16.1962	0.541378
$896$	1.00000	0.0334077
$897$	49.6410	1.65747
$898$	31.3923	1.04757
$899$	−21.8564	−0.728952
$900$	0	0
$901$	4.53590	0.151113
$902$	0	0
$903$	−9.12436	−0.303640
$904$	−14.1244	−0.469769
$905$	22.6603	0.753252
$906$	−32.6603	−1.08506
$907$	36.0000	1.19536	0.597680	−	0.801735i	$-0.296089\pi$
$907$	36.0000	1.19536	0.597680	+	0.801735i	$0.296089\pi$
$908$	17.3205	0.574801
$909$	0	0
$910$	5.73205	0.190016
$911$	−21.9808	−0.728255	−0.364128	−	0.931349i	$-0.618633\pi$
$911$	−21.9808	−0.728255	−0.364128	+	0.931349i	$0.618633\pi$
$912$	−0.464102	−0.0153679
$913$	0	0
$914$	−3.39230	−0.112207
$915$	5.07180	0.167668
$916$	5.85641	0.193501
$917$	−13.1962	−0.435775
$918$	3.80385	0.125546
$919$	39.8564	1.31474	0.657371	−	0.753567i	$-0.271669\pi$
$919$	39.8564	1.31474	0.657371	+	0.753567i	$0.271669\pi$
$920$	5.00000	0.164845
$921$	−12.2487	−0.403609
$922$	10.5359	0.346981
$923$	−39.7128	−1.30716
$924$	0	0
$925$	6.00000	0.197279
$926$	26.1769	0.860227
$927$	0	0
$928$	8.00000	0.262613
$929$	−12.6795	−0.416001	−0.208000	−	0.978129i	$-0.566696\pi$
$929$	−12.6795	−0.416001	−0.208000	+	0.978129i	$0.566696\pi$
$930$	4.73205	0.155170
$931$	−0.267949	−0.00878168
$932$	5.53590	0.181334
$933$	22.0526	0.721968
$934$	13.5885	0.444628
$935$	0	0
$936$	0	0
$937$	17.4641	0.570527	0.285264	−	0.958449i	$-0.407919\pi$
$937$	17.4641	0.570527	0.285264	+	0.958449i	$0.407919\pi$
$938$	−9.66025	−0.315418
$939$	−9.71281	−0.316966
$940$	4.73205	0.154342
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−9.33975	−0.304305
$943$	30.9808	1.00887
$944$	−5.53590	−0.180178
$945$	5.19615	0.169031
$946$	0	0
$947$	−21.9474	−0.713196	−0.356598	−	0.934258i	$-0.616063\pi$
$947$	−21.9474	−0.713196	−0.356598	+	0.934258i	$0.616063\pi$
$948$	−12.1244	−0.393781
$949$	−52.3013	−1.69777
$950$	−0.267949	−0.00869342
$951$	8.87564	0.287812
$952$	−0.732051	−0.0237259
$953$	−23.3923	−0.757751	−0.378876	−	0.925448i	$-0.623689\pi$
$953$	−23.3923	−0.757751	−0.378876	+	0.925448i	$0.623689\pi$
$954$	0	0
$955$	8.66025	0.280239
$956$	−4.46410	−0.144379
$957$	0	0
$958$	28.9808	0.936326
$959$	14.6603	0.473404
$960$	−1.73205	−0.0559017
$961$	−23.5359	−0.759223
$962$	−34.3923	−1.10885
$963$	0	0
$964$	17.4641	0.562481
$965$	−11.3923	−0.366731
$966$	−8.66025	−0.278639
$967$	−28.7846	−0.925651	−0.462825	−	0.886450i	$-0.653164\pi$
$967$	−28.7846	−0.925651	−0.462825	+	0.886450i	$0.653164\pi$
$968$	0	0
$969$	0.339746	0.0109142
$970$	−10.9282	−0.350883
$971$	50.7128	1.62745	0.813726	−	0.581249i	$-0.197436\pi$
$971$	50.7128	1.62745	0.813726	+	0.581249i	$0.197436\pi$
$972$	0	0
$973$	−17.1962	−0.551283
$974$	17.3923	0.557285
$975$	−9.92820	−0.317957
$976$	−2.92820	−0.0937295
$977$	−14.2679	−0.456472	−0.228236	−	0.973606i	$-0.573296\pi$
$977$	−14.2679	−0.456472	−0.228236	+	0.973606i	$0.573296\pi$
$978$	17.3205	0.553849
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	9.66025	0.308271
$983$	37.6077	1.19950	0.599750	−	0.800188i	$-0.295267\pi$
$983$	37.6077	1.19950	0.599750	+	0.800188i	$0.295267\pi$
$984$	−10.7321	−0.342125
$985$	9.46410	0.301551
$986$	−5.85641	−0.186506
$987$	−8.19615	−0.260886
$988$	1.53590	0.0488634
$989$	26.3397	0.837555
$990$	0	0
$991$	−1.87564	−0.0595818	−0.0297909	−	0.999556i	$-0.509484\pi$
$991$	−1.87564	−0.0595818	−0.0297909	+	0.999556i	$0.509484\pi$
$992$	−2.73205	−0.0867427
$993$	58.9808	1.87170
$994$	6.92820	0.219749
$995$	4.00000	0.126809
$996$	8.66025	0.274411
$997$	38.6603	1.22438	0.612191	−	0.790710i	$-0.290288\pi$
$997$	38.6603	1.22438	0.612191	+	0.790710i	$0.290288\pi$
$998$	−22.0000	−0.696398
$999$	−31.1769	−0.986394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.bz.1.2	yes	2
11.10	odd	2		8470.2.a.bm.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bm.1.2	✓	2	11.10	odd	2
8470.2.a.bz.1.2	yes	2	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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.73205 q^{12} -5.73205 q^{13} +1.00000 q^{14} -1.73205 q^{15} +1.00000 q^{16} -0.732051 q^{17} -0.267949 q^{19} -1.00000 q^{20} +1.73205 q^{21} -5.00000 q^{23} +1.73205 q^{24} +1.00000 q^{25} -5.73205 q^{26} -5.19615 q^{27} +1.00000 q^{28} +8.00000 q^{29} -1.73205 q^{30} -2.73205 q^{31} +1.00000 q^{32} -0.732051 q^{34} -1.00000 q^{35} +6.00000 q^{37} -0.267949 q^{38} -9.92820 q^{39} -1.00000 q^{40} -6.19615 q^{41} +1.73205 q^{42} -5.26795 q^{43} -5.00000 q^{46} -4.73205 q^{47} +1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.26795 q^{51} -5.73205 q^{52} -6.19615 q^{53} -5.19615 q^{54} +1.00000 q^{56} -0.464102 q^{57} +8.00000 q^{58} -5.53590 q^{59} -1.73205 q^{60} -2.92820 q^{61} -2.73205 q^{62} +1.00000 q^{64} +5.73205 q^{65} -9.66025 q^{67} -0.732051 q^{68} -8.66025 q^{69} -1.00000 q^{70} +6.92820 q^{71} +9.12436 q^{73} +6.00000 q^{74} +1.73205 q^{75} -0.267949 q^{76} -9.92820 q^{78} -7.00000 q^{79} -1.00000 q^{80} -9.00000 q^{81} -6.19615 q^{82} +5.00000 q^{83} +1.73205 q^{84} +0.732051 q^{85} -5.26795 q^{86} +13.8564 q^{87} -13.6603 q^{89} -5.73205 q^{91} -5.00000 q^{92} -4.73205 q^{93} -4.73205 q^{94} +0.267949 q^{95} +1.73205 q^{96} +10.9282 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8} - 2 q^{10} - 8 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{20} - 10 q^{23} + 2 q^{25} - 8 q^{26} + 2 q^{28} + 16 q^{29} - 2 q^{31} + 2 q^{32} + 2 q^{34} - 2 q^{35} + 12 q^{37} - 4 q^{38} - 6 q^{39} - 2 q^{40} - 2 q^{41} - 14 q^{43} - 10 q^{46} - 6 q^{47} + 2 q^{49} + 2 q^{50} - 6 q^{51} - 8 q^{52} - 2 q^{53} + 2 q^{56} + 6 q^{57} + 16 q^{58} - 18 q^{59} + 8 q^{61} - 2 q^{62} + 2 q^{64} + 8 q^{65} - 2 q^{67} + 2 q^{68} - 2 q^{70} - 6 q^{73} + 12 q^{74} - 4 q^{76} - 6 q^{78} - 14 q^{79} - 2 q^{80} - 18 q^{81} - 2 q^{82} + 10 q^{83} - 2 q^{85} - 14 q^{86} - 10 q^{89} - 8 q^{91} - 10 q^{92} - 6 q^{93} - 6 q^{94} + 4 q^{95} + 8 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 2 * q^7 + 2 * q^8 - 2 * q^10 - 8 * q^13 + 2 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 - 2 * q^20 - 10 * q^23 + 2 * q^25 - 8 * q^26 + 2 * q^28 + 16 * q^29 - 2 * q^31 + 2 * q^32 + 2 * q^34 - 2 * q^35 + 12 * q^37 - 4 * q^38 - 6 * q^39 - 2 * q^40 - 2 * q^41 - 14 * q^43 - 10 * q^46 - 6 * q^47 + 2 * q^49 + 2 * q^50 - 6 * q^51 - 8 * q^52 - 2 * q^53 + 2 * q^56 + 6 * q^57 + 16 * q^58 - 18 * q^59 + 8 * q^61 - 2 * q^62 + 2 * q^64 + 8 * q^65 - 2 * q^67 + 2 * q^68 - 2 * q^70 - 6 * q^73 + 12 * q^74 - 4 * q^76 - 6 * q^78 - 14 * q^79 - 2 * q^80 - 18 * q^81 - 2 * q^82 + 10 * q^83 - 2 * q^85 - 14 * q^86 - 10 * q^89 - 8 * q^91 - 10 * q^92 - 6 * q^93 - 6 * q^94 + 4 * q^95 + 8 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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