LMFDB - Embedded newform 8470.2.a.bz.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.73205 q^{12} -2.26795 q^{13} +1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +2.73205 q^{17} -3.73205 q^{19} -1.00000 q^{20} -1.73205 q^{21} -5.00000 q^{23} -1.73205 q^{24} +1.00000 q^{25} -2.26795 q^{26} +5.19615 q^{27} +1.00000 q^{28} +8.00000 q^{29} +1.73205 q^{30} +0.732051 q^{31} +1.00000 q^{32} +2.73205 q^{34} -1.00000 q^{35} +6.00000 q^{37} -3.73205 q^{38} +3.92820 q^{39} -1.00000 q^{40} +4.19615 q^{41} -1.73205 q^{42} -8.73205 q^{43} -5.00000 q^{46} -1.26795 q^{47} -1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.73205 q^{51} -2.26795 q^{52} +4.19615 q^{53} +5.19615 q^{54} +1.00000 q^{56} +6.46410 q^{57} +8.00000 q^{58} -12.4641 q^{59} +1.73205 q^{60} +10.9282 q^{61} +0.732051 q^{62} +1.00000 q^{64} +2.26795 q^{65} +7.66025 q^{67} +2.73205 q^{68} +8.66025 q^{69} -1.00000 q^{70} -6.92820 q^{71} -15.1244 q^{73} +6.00000 q^{74} -1.73205 q^{75} -3.73205 q^{76} +3.92820 q^{78} -7.00000 q^{79} -1.00000 q^{80} -9.00000 q^{81} +4.19615 q^{82} +5.00000 q^{83} -1.73205 q^{84} -2.73205 q^{85} -8.73205 q^{86} -13.8564 q^{87} +3.66025 q^{89} -2.26795 q^{91} -5.00000 q^{92} -1.26795 q^{93} -1.26795 q^{94} +3.73205 q^{95} -1.73205 q^{96} -2.92820 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.666667\pi$
$3$	−1.73205	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$4$	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
$13$	−2.26795	−0.629016	−0.314508	−	0.949255i	$-0.601840\pi$
$13$	−2.26795	−0.629016	−0.314508	+	0.949255i	$0.601840\pi$
$14$	1.00000	0.267261
$15$	1.73205	0.447214
$16$	1.00000	0.250000
$17$	2.73205	0.662620	0.331310	−	0.943522i	$-0.392509\pi$
$17$	2.73205	0.662620	0.331310	+	0.943522i	$0.392509\pi$
$18$	0	0
$19$	−3.73205	−0.856191	−0.428096	−	0.903733i	$-0.640815\pi$
$19$	−3.73205	−0.856191	−0.428096	+	0.903733i	$0.640815\pi$
$20$	−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
$26$	−2.26795	−0.444781
$27$	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
$31$	0.732051	0.131480	0.0657401	−	0.997837i	$-0.479059\pi$
$31$	0.732051	0.131480	0.0657401	+	0.997837i	$0.479059\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.73205	0.468543
$35$	−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	$0.335828\pi$
$38$	−3.73205	−0.605419
$39$	3.92820	0.629016
$40$	−1.00000	−0.158114
$41$	4.19615	0.655329	0.327664	−	0.944794i	$-0.393738\pi$
$41$	4.19615	0.655329	0.327664	+	0.944794i	$0.393738\pi$
$42$	−1.73205	−0.267261
$43$	−8.73205	−1.33163	−0.665813	−	0.746119i	$-0.731915\pi$
$43$	−8.73205	−1.33163	−0.665813	+	0.746119i	$0.731915\pi$
$44$	0	0
$45$	0	0
$46$	−5.00000	−0.737210
$47$	−1.26795	−0.184949	−0.0924747	−	0.995715i	$-0.529478\pi$
$47$	−1.26795	−0.184949	−0.0924747	+	0.995715i	$0.529478\pi$
$48$	−1.73205	−0.250000
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−4.73205	−0.662620
$52$	−2.26795	−0.314508
$53$	4.19615	0.576386	0.288193	−	0.957572i	$-0.406946\pi$
$53$	4.19615	0.576386	0.288193	+	0.957572i	$0.406946\pi$
$54$	5.19615	0.707107
$55$	0	0
$56$	1.00000	0.133631
$57$	6.46410	0.856191
$58$	8.00000	1.05045
$59$	−12.4641	−1.62269	−0.811344	−	0.584569i	$-0.801264\pi$
$59$	−12.4641	−1.62269	−0.811344	+	0.584569i	$0.801264\pi$
$60$	1.73205	0.223607
$61$	10.9282	1.39921	0.699607	−	0.714528i	$-0.253359\pi$
$61$	10.9282	1.39921	0.699607	+	0.714528i	$0.253359\pi$
$62$	0.732051	0.0929705
$63$	0	0
$64$	1.00000	0.125000
$65$	2.26795	0.281304
$66$	0	0
$67$	7.66025	0.935849	0.467924	−	0.883768i	$-0.345002\pi$
$67$	7.66025	0.935849	0.467924	+	0.883768i	$0.345002\pi$
$68$	2.73205	0.331310
$69$	8.66025	1.04257
$70$	−1.00000	−0.119523
$71$	−6.92820	−0.822226	−0.411113	−	0.911584i	$-0.634860\pi$
$71$	−6.92820	−0.822226	−0.411113	+	0.911584i	$0.634860\pi$
$72$	0	0
$73$	−15.1244	−1.77017	−0.885086	−	0.465428i	$-0.845901\pi$
$73$	−15.1244	−1.77017	−0.885086	+	0.465428i	$0.845901\pi$
$74$	6.00000	0.697486
$75$	−1.73205	−0.200000
$76$	−3.73205	−0.428096
$77$	0	0
$78$	3.92820	0.444781
$79$	−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
$82$	4.19615	0.463388
$83$	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
$85$	−2.73205	−0.296333
$86$	−8.73205	−0.941601
$87$	−13.8564	−1.48556
$88$	0	0
$89$	3.66025	0.387986	0.193993	−	0.981003i	$-0.437856\pi$
$89$	3.66025	0.387986	0.193993	+	0.981003i	$0.437856\pi$
$90$	0	0
$91$	−2.26795	−0.237746
$92$	−5.00000	−0.521286
$93$	−1.26795	−0.131480
$94$	−1.26795	−0.130779
$95$	3.73205	0.382900
$96$	−1.73205	−0.176777
$97$	−2.92820	−0.297314	−0.148657	−	0.988889i	$-0.547495\pi$
$97$	−2.92820	−0.297314	−0.148657	+	0.988889i	$0.547495\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−18.4641	−1.83725	−0.918623	−	0.395134i	$-0.870698\pi$
$101$	−18.4641	−1.83725	−0.918623	+	0.395134i	$0.870698\pi$
$102$	−4.73205	−0.468543
$103$	−7.26795	−0.716132	−0.358066	−	0.933696i	$-0.616564\pi$
$103$	−7.26795	−0.716132	−0.358066	+	0.933696i	$0.616564\pi$
$104$	−2.26795	−0.222391
$105$	1.73205	0.169031
$106$	4.19615	0.407566
$107$	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
$109$	−18.0526	−1.72912	−0.864561	−	0.502528i	$-0.832403\pi$
$109$	−18.0526	−1.72912	−0.864561	+	0.502528i	$0.832403\pi$
$110$	0	0
$111$	−10.3923	−0.986394
$112$	1.00000	0.0944911
$113$	10.1244	0.952419	0.476210	−	0.879332i	$-0.342010\pi$
$113$	10.1244	0.952419	0.476210	+	0.879332i	$0.342010\pi$
$114$	6.46410	0.605419
$115$	5.00000	0.466252
$116$	8.00000	0.742781
$117$	0	0
$118$	−12.4641	−1.14741
$119$	2.73205	0.250447
$120$	1.73205	0.158114
$121$	0	0
$122$	10.9282	0.989393
$123$	−7.26795	−0.655329
$124$	0.732051	0.0657401
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.1244	1.60828	0.804138	−	0.594442i	$-0.202627\pi$
$127$	18.1244	1.60828	0.804138	+	0.594442i	$0.202627\pi$
$128$	1.00000	0.0883883
$129$	15.1244	1.33163
$130$	2.26795	0.198912
$131$	−2.80385	−0.244973	−0.122487	−	0.992470i	$-0.539087\pi$
$131$	−2.80385	−0.244973	−0.122487	+	0.992470i	$0.539087\pi$
$132$	0	0
$133$	−3.73205	−0.323610
$134$	7.66025	0.661745
$135$	−5.19615	−0.447214
$136$	2.73205	0.234271
$137$	−2.66025	−0.227281	−0.113640	−	0.993522i	$-0.536251\pi$
$137$	−2.66025	−0.227281	−0.113640	+	0.993522i	$0.536251\pi$
$138$	8.66025	0.737210
$139$	−6.80385	−0.577095	−0.288547	−	0.957466i	$-0.593172\pi$
$139$	−6.80385	−0.577095	−0.288547	+	0.957466i	$0.593172\pi$
$140$	−1.00000	−0.0845154
$141$	2.19615	0.184949
$142$	−6.92820	−0.581402
$143$	0	0
$144$	0	0
$145$	−8.00000	−0.664364
$146$	−15.1244	−1.25170
$147$	−1.73205	−0.142857
$148$	6.00000	0.493197
$149$	12.5359	1.02698	0.513490	−	0.858095i	$-0.328352\pi$
$149$	12.5359	1.02698	0.513490	+	0.858095i	$0.328352\pi$
$150$	−1.73205	−0.141421
$151$	8.85641	0.720724	0.360362	−	0.932813i	$-0.382653\pi$
$151$	8.85641	0.720724	0.360362	+	0.932813i	$0.382653\pi$
$152$	−3.73205	−0.302709
$153$	0	0
$154$	0	0
$155$	−0.732051	−0.0587997
$156$	3.92820	0.314508
$157$	15.3923	1.22844	0.614220	−	0.789135i	$-0.289471\pi$
$157$	15.3923	1.22844	0.614220	+	0.789135i	$0.289471\pi$
$158$	−7.00000	−0.556890
$159$	−7.26795	−0.576386
$160$	−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	$0.371911\pi$
$164$	4.19615	0.327664
$165$	0	0
$166$	5.00000	0.388075
$167$	−17.6603	−1.36659	−0.683296	−	0.730142i	$-0.739454\pi$
$167$	−17.6603	−1.36659	−0.683296	+	0.730142i	$0.739454\pi$
$168$	−1.73205	−0.133631
$169$	−7.85641	−0.604339
$170$	−2.73205	−0.209539
$171$	0	0
$172$	−8.73205	−0.665813
$173$	−19.8564	−1.50965	−0.754827	−	0.655924i	$-0.772279\pi$
$173$	−19.8564	−1.50965	−0.754827	+	0.655924i	$0.772279\pi$
$174$	−13.8564	−1.05045
$175$	1.00000	0.0755929
$176$	0	0
$177$	21.5885	1.62269
$178$	3.66025	0.274348
$179$	−5.80385	−0.433800	−0.216900	−	0.976194i	$-0.569595\pi$
$179$	−5.80385	−0.433800	−0.216900	+	0.976194i	$0.569595\pi$
$180$	0	0
$181$	−5.33975	−0.396900	−0.198450	−	0.980111i	$-0.563591\pi$
$181$	−5.33975	−0.396900	−0.198450	+	0.980111i	$0.563591\pi$
$182$	−2.26795	−0.168112
$183$	−18.9282	−1.39921
$184$	−5.00000	−0.368605
$185$	−6.00000	−0.441129
$186$	−1.26795	−0.0929705
$187$	0	0
$188$	−1.26795	−0.0924747
$189$	5.19615	0.377964
$190$	3.73205	0.270751
$191$	8.66025	0.626634	0.313317	−	0.949649i	$-0.398560\pi$
$191$	8.66025	0.626634	0.313317	+	0.949649i	$0.398560\pi$
$192$	−1.73205	−0.125000
$193$	−9.39230	−0.676073	−0.338036	−	0.941133i	$-0.609763\pi$
$193$	−9.39230	−0.676073	−0.338036	+	0.941133i	$0.609763\pi$
$194$	−2.92820	−0.210233
$195$	−3.92820	−0.281304
$196$	1.00000	0.0714286
$197$	−2.53590	−0.180675	−0.0903376	−	0.995911i	$-0.528795\pi$
$197$	−2.53590	−0.180675	−0.0903376	+	0.995911i	$0.528795\pi$
$198$	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
$201$	−13.2679	−0.935849
$202$	−18.4641	−1.29913
$203$	8.00000	0.561490
$204$	−4.73205	−0.331310
$205$	−4.19615	−0.293072
$206$	−7.26795	−0.506382
$207$	0	0
$208$	−2.26795	−0.157254
$209$	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	$0.588801\pi$
$212$	4.19615	0.288193
$213$	12.0000	0.822226
$214$	2.00000	0.136717
$215$	8.73205	0.595521
$216$	5.19615	0.353553
$217$	0.732051	0.0496948
$218$	−18.0526	−1.22267
$219$	26.1962	1.77017
$220$	0	0
$221$	−6.19615	−0.416798
$222$	−10.3923	−0.697486
$223$	−14.7321	−0.986531	−0.493266	−	0.869879i	$-0.664197\pi$
$223$	−14.7321	−0.986531	−0.493266	+	0.869879i	$0.664197\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	10.1244	0.673462
$227$	−17.3205	−1.14960	−0.574801	−	0.818293i	$-0.694921\pi$
$227$	−17.3205	−1.14960	−0.574801	+	0.818293i	$0.694921\pi$
$228$	6.46410	0.428096
$229$	−21.8564	−1.44431	−0.722156	−	0.691730i	$-0.756849\pi$
$229$	−21.8564	−1.44431	−0.722156	+	0.691730i	$0.756849\pi$
$230$	5.00000	0.329690
$231$	0	0
$232$	8.00000	0.525226
$233$	12.4641	0.816550	0.408275	−	0.912859i	$-0.366130\pi$
$233$	12.4641	0.816550	0.408275	+	0.912859i	$0.366130\pi$
$234$	0	0
$235$	1.26795	0.0827119
$236$	−12.4641	−0.811344
$237$	12.1244	0.787562
$238$	2.73205	0.177093
$239$	2.46410	0.159389	0.0796947	−	0.996819i	$-0.474605\pi$
$239$	2.46410	0.159389	0.0796947	+	0.996819i	$0.474605\pi$
$240$	1.73205	0.111803
$241$	10.5359	0.678677	0.339338	−	0.940664i	$-0.389797\pi$
$241$	10.5359	0.678677	0.339338	+	0.940664i	$0.389797\pi$
$242$	0	0
$243$	0	0
$244$	10.9282	0.699607
$245$	−1.00000	−0.0638877
$246$	−7.26795	−0.463388
$247$	8.46410	0.538558
$248$	0.732051	0.0464853
$249$	−8.66025	−0.548821
$250$	−1.00000	−0.0632456
$251$	14.3923	0.908434	0.454217	−	0.890891i	$-0.349919\pi$
$251$	14.3923	0.908434	0.454217	+	0.890891i	$0.349919\pi$
$252$	0	0
$253$	0	0
$254$	18.1244	1.13722
$255$	4.73205	0.296333
$256$	1.00000	0.0625000
$257$	−2.19615	−0.136992	−0.0684961	−	0.997651i	$-0.521820\pi$
$257$	−2.19615	−0.136992	−0.0684961	+	0.997651i	$0.521820\pi$
$258$	15.1244	0.941601
$259$	6.00000	0.372822
$260$	2.26795	0.140652
$261$	0	0
$262$	−2.80385	−0.173222
$263$	−14.1244	−0.870945	−0.435473	−	0.900202i	$-0.643419\pi$
$263$	−14.1244	−0.870945	−0.435473	+	0.900202i	$0.643419\pi$
$264$	0	0
$265$	−4.19615	−0.257768
$266$	−3.73205	−0.228827
$267$	−6.33975	−0.387986
$268$	7.66025	0.467924
$269$	−18.2679	−1.11382	−0.556908	−	0.830574i	$-0.688013\pi$
$269$	−18.2679	−1.11382	−0.556908	+	0.830574i	$0.688013\pi$
$270$	−5.19615	−0.316228
$271$	14.5359	0.882993	0.441496	−	0.897263i	$-0.354448\pi$
$271$	14.5359	0.882993	0.441496	+	0.897263i	$0.354448\pi$
$272$	2.73205	0.165655
$273$	3.92820	0.237746
$274$	−2.66025	−0.160712
$275$	0	0
$276$	8.66025	0.521286
$277$	12.1962	0.732796	0.366398	−	0.930458i	$-0.380591\pi$
$277$	12.1962	0.732796	0.366398	+	0.930458i	$0.380591\pi$
$278$	−6.80385	−0.408068
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	4.26795	0.254605	0.127302	−	0.991864i	$-0.459368\pi$
$281$	4.26795	0.254605	0.127302	+	0.991864i	$0.459368\pi$
$282$	2.19615	0.130779
$283$	−15.7846	−0.938298	−0.469149	−	0.883119i	$-0.655439\pi$
$283$	−15.7846	−0.938298	−0.469149	+	0.883119i	$0.655439\pi$
$284$	−6.92820	−0.411113
$285$	−6.46410	−0.382900
$286$	0	0
$287$	4.19615	0.247691
$288$	0	0
$289$	−9.53590	−0.560935
$290$	−8.00000	−0.469776
$291$	5.07180	0.297314
$292$	−15.1244	−0.885086
$293$	8.26795	0.483019	0.241509	−	0.970398i	$-0.422358\pi$
$293$	8.26795	0.483019	0.241509	+	0.970398i	$0.422358\pi$
$294$	−1.73205	−0.101015
$295$	12.4641	0.725688
$296$	6.00000	0.348743
$297$	0	0
$298$	12.5359	0.726185
$299$	11.3397	0.655794
$300$	−1.73205	−0.100000
$301$	−8.73205	−0.503307
$302$	8.85641	0.509629
$303$	31.9808	1.83725
$304$	−3.73205	−0.214048
$305$	−10.9282	−0.625747
$306$	0	0
$307$	−20.9282	−1.19444	−0.597218	−	0.802079i	$-0.703727\pi$
$307$	−20.9282	−1.19444	−0.597218	+	0.802079i	$0.703727\pi$
$308$	0	0
$309$	12.5885	0.716132
$310$	−0.732051	−0.0415777
$311$	9.26795	0.525537	0.262769	−	0.964859i	$-0.415364\pi$
$311$	9.26795	0.525537	0.262769	+	0.964859i	$0.415364\pi$
$312$	3.92820	0.222391
$313$	−26.3923	−1.49178	−0.745891	−	0.666068i	$-0.767976\pi$
$313$	−26.3923	−1.49178	−0.745891	+	0.666068i	$0.767976\pi$
$314$	15.3923	0.868638
$315$	0	0
$316$	−7.00000	−0.393781
$317$	−19.1244	−1.07413	−0.537065	−	0.843541i	$-0.680467\pi$
$317$	−19.1244	−1.07413	−0.537065	+	0.843541i	$0.680467\pi$
$318$	−7.26795	−0.407566
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−3.46410	−0.193347
$322$	−5.00000	−0.278639
$323$	−10.1962	−0.567329
$324$	−9.00000	−0.500000
$325$	−2.26795	−0.125803
$326$	10.0000	0.553849
$327$	31.2679	1.72912
$328$	4.19615	0.231694
$329$	−1.26795	−0.0699043
$330$	0	0
$331$	−4.05256	−0.222749	−0.111374	−	0.993779i	$-0.535525\pi$
$331$	−4.05256	−0.222749	−0.111374	+	0.993779i	$0.535525\pi$
$332$	5.00000	0.274411
$333$	0	0
$334$	−17.6603	−0.966326
$335$	−7.66025	−0.418524
$336$	−1.73205	−0.0944911
$337$	11.3923	0.620578	0.310289	−	0.950642i	$-0.399574\pi$
$337$	11.3923	0.620578	0.310289	+	0.950642i	$0.399574\pi$
$338$	−7.85641	−0.427332
$339$	−17.5359	−0.952419
$340$	−2.73205	−0.148166
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.73205	−0.470801
$345$	−8.66025	−0.466252
$346$	−19.8564	−1.06749
$347$	−8.53590	−0.458231	−0.229116	−	0.973399i	$-0.573583\pi$
$347$	−8.53590	−0.458231	−0.229116	+	0.973399i	$0.573583\pi$
$348$	−13.8564	−0.742781
$349$	−6.46410	−0.346015	−0.173008	−	0.984920i	$-0.555349\pi$
$349$	−6.46410	−0.346015	−0.173008	+	0.984920i	$0.555349\pi$
$350$	1.00000	0.0534522
$351$	−11.7846	−0.629016
$352$	0	0
$353$	1.80385	0.0960091	0.0480046	−	0.998847i	$-0.484714\pi$
$353$	1.80385	0.0960091	0.0480046	+	0.998847i	$0.484714\pi$
$354$	21.5885	1.14741
$355$	6.92820	0.367711
$356$	3.66025	0.193993
$357$	−4.73205	−0.250447
$358$	−5.80385	−0.306743
$359$	−33.7128	−1.77929	−0.889647	−	0.456649i	$-0.849050\pi$
$359$	−33.7128	−1.77929	−0.889647	+	0.456649i	$0.849050\pi$
$360$	0	0
$361$	−5.07180	−0.266937
$362$	−5.33975	−0.280651
$363$	0	0
$364$	−2.26795	−0.118873
$365$	15.1244	0.791645
$366$	−18.9282	−0.989393
$367$	−20.7321	−1.08220	−0.541102	−	0.840957i	$-0.681993\pi$
$367$	−20.7321	−1.08220	−0.541102	+	0.840957i	$0.681993\pi$
$368$	−5.00000	−0.260643
$369$	0	0
$370$	−6.00000	−0.311925
$371$	4.19615	0.217853
$372$	−1.26795	−0.0657401
$373$	−13.3205	−0.689710	−0.344855	−	0.938656i	$-0.612072\pi$
$373$	−13.3205	−0.689710	−0.344855	+	0.938656i	$0.612072\pi$
$374$	0	0
$375$	1.73205	0.0894427
$376$	−1.26795	−0.0653895
$377$	−18.1436	−0.934443
$378$	5.19615	0.267261
$379$	34.7846	1.78677	0.893383	−	0.449297i	$-0.148325\pi$
$379$	34.7846	1.78677	0.893383	+	0.449297i	$0.148325\pi$
$380$	3.73205	0.191450
$381$	−31.3923	−1.60828
$382$	8.66025	0.443097
$383$	8.33975	0.426141	0.213071	−	0.977037i	$-0.431654\pi$
$383$	8.33975	0.426141	0.213071	+	0.977037i	$0.431654\pi$
$384$	−1.73205	−0.0883883
$385$	0	0
$386$	−9.39230	−0.478056
$387$	0	0
$388$	−2.92820	−0.148657
$389$	1.12436	0.0570071	0.0285035	−	0.999594i	$-0.490926\pi$
$389$	1.12436	0.0570071	0.0285035	+	0.999594i	$0.490926\pi$
$390$	−3.92820	−0.198912
$391$	−13.6603	−0.690829
$392$	1.00000	0.0505076
$393$	4.85641	0.244973
$394$	−2.53590	−0.127757
$395$	7.00000	0.352208
$396$	0	0
$397$	−28.7846	−1.44466	−0.722329	−	0.691549i	$-0.756928\pi$
$397$	−28.7846	−1.44466	−0.722329	+	0.691549i	$0.756928\pi$
$398$	−4.00000	−0.200502
$399$	6.46410	0.323610
$400$	1.00000	0.0500000
$401$	−25.4641	−1.27162	−0.635808	−	0.771847i	$-0.719333\pi$
$401$	−25.4641	−1.27162	−0.635808	+	0.771847i	$0.719333\pi$
$402$	−13.2679	−0.661745
$403$	−1.66025	−0.0827031
$404$	−18.4641	−0.918623
$405$	9.00000	0.447214
$406$	8.00000	0.397033
$407$	0	0
$408$	−4.73205	−0.234271
$409$	−19.7128	−0.974736	−0.487368	−	0.873197i	$-0.662043\pi$
$409$	−19.7128	−0.974736	−0.487368	+	0.873197i	$0.662043\pi$
$410$	−4.19615	−0.207233
$411$	4.60770	0.227281
$412$	−7.26795	−0.358066
$413$	−12.4641	−0.613318
$414$	0	0
$415$	−5.00000	−0.245440
$416$	−2.26795	−0.111195
$417$	11.7846	0.577095
$418$	0	0
$419$	2.07180	0.101214	0.0506069	−	0.998719i	$-0.483884\pi$
$419$	2.07180	0.101214	0.0506069	+	0.998719i	$0.483884\pi$
$420$	1.73205	0.0845154
$421$	−28.3923	−1.38376	−0.691878	−	0.722014i	$-0.743216\pi$
$421$	−28.3923	−1.38376	−0.691878	+	0.722014i	$0.743216\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	4.19615	0.203783
$425$	2.73205	0.132524
$426$	12.0000	0.581402
$427$	10.9282	0.528853
$428$	2.00000	0.0966736
$429$	0	0
$430$	8.73205	0.421097
$431$	−27.5359	−1.32636	−0.663179	−	0.748461i	$-0.730793\pi$
$431$	−27.5359	−1.32636	−0.663179	+	0.748461i	$0.730793\pi$
$432$	5.19615	0.250000
$433$	18.0526	0.867551	0.433775	−	0.901021i	$-0.357181\pi$
$433$	18.0526	0.867551	0.433775	+	0.901021i	$0.357181\pi$
$434$	0.732051	0.0351396
$435$	13.8564	0.664364
$436$	−18.0526	−0.864561
$437$	18.6603	0.892641
$438$	26.1962	1.25170
$439$	−11.2679	−0.537790	−0.268895	−	0.963170i	$-0.586658\pi$
$439$	−11.2679	−0.537790	−0.268895	+	0.963170i	$0.586658\pi$
$440$	0	0
$441$	0	0
$442$	−6.19615	−0.294721
$443$	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
$445$	−3.66025	−0.173513
$446$	−14.7321	−0.697583
$447$	−21.7128	−1.02698
$448$	1.00000	0.0472456
$449$	10.6077	0.500608	0.250304	−	0.968167i	$-0.419469\pi$
$449$	10.6077	0.500608	0.250304	+	0.968167i	$0.419469\pi$
$450$	0	0
$451$	0	0
$452$	10.1244	0.476210
$453$	−15.3397	−0.720724
$454$	−17.3205	−0.812892
$455$	2.26795	0.106323
$456$	6.46410	0.302709
$457$	17.3923	0.813578	0.406789	−	0.913522i	$-0.366648\pi$
$457$	17.3923	0.813578	0.406789	+	0.913522i	$0.366648\pi$
$458$	−21.8564	−1.02128
$459$	14.1962	0.662620
$460$	5.00000	0.233126
$461$	17.4641	0.813384	0.406692	−	0.913565i	$-0.366682\pi$
$461$	17.4641	0.813384	0.406692	+	0.913565i	$0.366682\pi$
$462$	0	0
$463$	−36.1769	−1.68128	−0.840642	−	0.541591i	$-0.817822\pi$
$463$	−36.1769	−1.68128	−0.840642	+	0.541591i	$0.817822\pi$
$464$	8.00000	0.371391
$465$	1.26795	0.0587997
$466$	12.4641	0.577388
$467$	−17.5885	−0.813897	−0.406948	−	0.913451i	$-0.633407\pi$
$467$	−17.5885	−0.813897	−0.406948	+	0.913451i	$0.633407\pi$
$468$	0	0
$469$	7.66025	0.353718
$470$	1.26795	0.0584861
$471$	−26.6603	−1.22844
$472$	−12.4641	−0.573707
$473$	0	0
$474$	12.1244	0.556890
$475$	−3.73205	−0.171238
$476$	2.73205	0.125223
$477$	0	0
$478$	2.46410	0.112705
$479$	−22.9808	−1.05002	−0.525009	−	0.851097i	$-0.675938\pi$
$479$	−22.9808	−1.05002	−0.525009	+	0.851097i	$0.675938\pi$
$480$	1.73205	0.0790569
$481$	−13.6077	−0.620457
$482$	10.5359	0.479897
$483$	8.66025	0.394055
$484$	0	0
$485$	2.92820	0.132963
$486$	0	0
$487$	−3.39230	−0.153720	−0.0768600	−	0.997042i	$-0.524489\pi$
$487$	−3.39230	−0.153720	−0.0768600	+	0.997042i	$0.524489\pi$
$488$	10.9282	0.494697
$489$	−17.3205	−0.783260
$490$	−1.00000	−0.0451754
$491$	−7.66025	−0.345702	−0.172851	−	0.984948i	$-0.555298\pi$
$491$	−7.66025	−0.345702	−0.172851	+	0.984948i	$0.555298\pi$
$492$	−7.26795	−0.327664
$493$	21.8564	0.984363
$494$	8.46410	0.380818
$495$	0	0
$496$	0.732051	0.0328701
$497$	−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
$501$	30.5885	1.36659
$502$	14.3923	0.642360
$503$	7.80385	0.347956	0.173978	−	0.984750i	$-0.444338\pi$
$503$	7.80385	0.347956	0.173978	+	0.984750i	$0.444338\pi$
$504$	0	0
$505$	18.4641	0.821642
$506$	0	0
$507$	13.6077	0.604339
$508$	18.1244	0.804138
$509$	−15.3397	−0.679922	−0.339961	−	0.940439i	$-0.610414\pi$
$509$	−15.3397	−0.679922	−0.339961	+	0.940439i	$0.610414\pi$
$510$	4.73205	0.209539
$511$	−15.1244	−0.669062
$512$	1.00000	0.0441942
$513$	−19.3923	−0.856191
$514$	−2.19615	−0.0968681
$515$	7.26795	0.320264
$516$	15.1244	0.665813
$517$	0	0
$518$	6.00000	0.263625
$519$	34.3923	1.50965
$520$	2.26795	0.0994562
$521$	−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
$523$	−2.60770	−0.114027	−0.0570133	−	0.998373i	$-0.518158\pi$
$523$	−2.60770	−0.114027	−0.0570133	+	0.998373i	$0.518158\pi$
$524$	−2.80385	−0.122487
$525$	−1.73205	−0.0755929
$526$	−14.1244	−0.615851
$527$	2.00000	0.0871214
$528$	0	0
$529$	2.00000	0.0869565
$530$	−4.19615	−0.182269
$531$	0	0
$532$	−3.73205	−0.161805
$533$	−9.51666	−0.412212
$534$	−6.33975	−0.274348
$535$	−2.00000	−0.0864675
$536$	7.66025	0.330873
$537$	10.0526	0.433800
$538$	−18.2679	−0.787587
$539$	0	0
$540$	−5.19615	−0.223607
$541$	9.60770	0.413067	0.206534	−	0.978440i	$-0.433782\pi$
$541$	9.60770	0.413067	0.206534	+	0.978440i	$0.433782\pi$
$542$	14.5359	0.624370
$543$	9.24871	0.396900
$544$	2.73205	0.117136
$545$	18.0526	0.773287
$546$	3.92820	0.168112
$547$	37.5167	1.60410	0.802048	−	0.597259i	$-0.203744\pi$
$547$	37.5167	1.60410	0.802048	+	0.597259i	$0.203744\pi$
$548$	−2.66025	−0.113640
$549$	0	0
$550$	0	0
$551$	−29.8564	−1.27193
$552$	8.66025	0.368605
$553$	−7.00000	−0.297670
$554$	12.1962	0.518165
$555$	10.3923	0.441129
$556$	−6.80385	−0.288547
$557$	27.3731	1.15983	0.579917	−	0.814676i	$-0.303085\pi$
$557$	27.3731	1.15983	0.579917	+	0.814676i	$0.303085\pi$
$558$	0	0
$559$	19.8038	0.837614
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	4.26795	0.180033
$563$	−1.53590	−0.0647304	−0.0323652	−	0.999476i	$-0.510304\pi$
$563$	−1.53590	−0.0647304	−0.0323652	+	0.999476i	$0.510304\pi$
$564$	2.19615	0.0924747
$565$	−10.1244	−0.425935
$566$	−15.7846	−0.663477
$567$	−9.00000	−0.377964
$568$	−6.92820	−0.290701
$569$	38.1244	1.59826	0.799128	−	0.601161i	$-0.205295\pi$
$569$	38.1244	1.59826	0.799128	+	0.601161i	$0.205295\pi$
$570$	−6.46410	−0.270751
$571$	13.4641	0.563455	0.281728	−	0.959494i	$-0.409093\pi$
$571$	13.4641	0.563455	0.281728	+	0.959494i	$0.409093\pi$
$572$	0	0
$573$	−15.0000	−0.626634
$574$	4.19615	0.175144
$575$	−5.00000	−0.208514
$576$	0	0
$577$	18.4449	0.767870	0.383935	−	0.923360i	$-0.374569\pi$
$577$	18.4449	0.767870	0.383935	+	0.923360i	$0.374569\pi$
$578$	−9.53590	−0.396641
$579$	16.2679	0.676073
$580$	−8.00000	−0.332182
$581$	5.00000	0.207435
$582$	5.07180	0.210233
$583$	0	0
$584$	−15.1244	−0.625850
$585$	0	0
$586$	8.26795	0.341546
$587$	14.5167	0.599167	0.299583	−	0.954070i	$-0.403152\pi$
$587$	14.5167	0.599167	0.299583	+	0.954070i	$0.403152\pi$
$588$	−1.73205	−0.0714286
$589$	−2.73205	−0.112572
$590$	12.4641	0.513139
$591$	4.39230	0.180675
$592$	6.00000	0.246598
$593$	−29.8564	−1.22606	−0.613028	−	0.790061i	$-0.710049\pi$
$593$	−29.8564	−1.22606	−0.613028	+	0.790061i	$0.710049\pi$
$594$	0	0
$595$	−2.73205	−0.112003
$596$	12.5359	0.513490
$597$	6.92820	0.283552
$598$	11.3397	0.463717
$599$	−31.9808	−1.30670	−0.653349	−	0.757057i	$-0.726637\pi$
$599$	−31.9808	−1.30670	−0.653349	+	0.757057i	$0.726637\pi$
$600$	−1.73205	−0.0707107
$601$	3.66025	0.149305	0.0746524	−	0.997210i	$-0.476215\pi$
$601$	3.66025	0.149305	0.0746524	+	0.997210i	$0.476215\pi$
$602$	−8.73205	−0.355892
$603$	0	0
$604$	8.85641	0.360362
$605$	0	0
$606$	31.9808	1.29913
$607$	−0.248711	−0.0100949	−0.00504744	−	0.999987i	$-0.501607\pi$
$607$	−0.248711	−0.0100949	−0.00504744	+	0.999987i	$0.501607\pi$
$608$	−3.73205	−0.151355
$609$	−13.8564	−0.561490
$610$	−10.9282	−0.442470
$611$	2.87564	0.116336
$612$	0	0
$613$	37.2679	1.50524	0.752619	−	0.658456i	$-0.228790\pi$
$613$	37.2679	1.50524	0.752619	+	0.658456i	$0.228790\pi$
$614$	−20.9282	−0.844594
$615$	7.26795	0.293072
$616$	0	0
$617$	13.6077	0.547825	0.273913	−	0.961755i	$-0.411682\pi$
$617$	13.6077	0.547825	0.273913	+	0.961755i	$0.411682\pi$
$618$	12.5885	0.506382
$619$	32.7128	1.31484	0.657419	−	0.753525i	$-0.271648\pi$
$619$	32.7128	1.31484	0.657419	+	0.753525i	$0.271648\pi$
$620$	−0.732051	−0.0293999
$621$	−25.9808	−1.04257
$622$	9.26795	0.371611
$623$	3.66025	0.146645
$624$	3.92820	0.157254
$625$	1.00000	0.0400000
$626$	−26.3923	−1.05485
$627$	0	0
$628$	15.3923	0.614220
$629$	16.3923	0.653604
$630$	0	0
$631$	−39.3205	−1.56532	−0.782662	−	0.622446i	$-0.786139\pi$
$631$	−39.3205	−1.56532	−0.782662	+	0.622446i	$0.786139\pi$
$632$	−7.00000	−0.278445
$633$	13.8564	0.550743
$634$	−19.1244	−0.759525
$635$	−18.1244	−0.719243
$636$	−7.26795	−0.288193
$637$	−2.26795	−0.0898594
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−39.5359	−1.56157	−0.780787	−	0.624797i	$-0.785182\pi$
$641$	−39.5359	−1.56157	−0.780787	+	0.624797i	$0.785182\pi$
$642$	−3.46410	−0.136717
$643$	23.1769	0.914008	0.457004	−	0.889465i	$-0.348922\pi$
$643$	23.1769	0.914008	0.457004	+	0.889465i	$0.348922\pi$
$644$	−5.00000	−0.197028
$645$	−15.1244	−0.595521
$646$	−10.1962	−0.401162
$647$	−21.8038	−0.857198	−0.428599	−	0.903495i	$-0.640993\pi$
$647$	−21.8038	−0.857198	−0.428599	+	0.903495i	$0.640993\pi$
$648$	−9.00000	−0.353553
$649$	0	0
$650$	−2.26795	−0.0889563
$651$	−1.26795	−0.0496948
$652$	10.0000	0.391630
$653$	−2.19615	−0.0859421	−0.0429710	−	0.999076i	$-0.513682\pi$
$653$	−2.19615	−0.0859421	−0.0429710	+	0.999076i	$0.513682\pi$
$654$	31.2679	1.22267
$655$	2.80385	0.109555
$656$	4.19615	0.163832
$657$	0	0
$658$	−1.26795	−0.0494298
$659$	−32.4449	−1.26387	−0.631936	−	0.775020i	$-0.717740\pi$
$659$	−32.4449	−1.26387	−0.631936	+	0.775020i	$0.717740\pi$
$660$	0	0
$661$	3.33975	0.129901	0.0649505	−	0.997888i	$-0.479311\pi$
$661$	3.33975	0.129901	0.0649505	+	0.997888i	$0.479311\pi$
$662$	−4.05256	−0.157507
$663$	10.7321	0.416798
$664$	5.00000	0.194038
$665$	3.73205	0.144723
$666$	0	0
$667$	−40.0000	−1.54881
$668$	−17.6603	−0.683296
$669$	25.5167	0.986531
$670$	−7.66025	−0.295941
$671$	0	0
$672$	−1.73205	−0.0668153
$673$	47.1051	1.81577	0.907884	−	0.419221i	$-0.137697\pi$
$673$	47.1051	1.81577	0.907884	+	0.419221i	$0.137697\pi$
$674$	11.3923	0.438815
$675$	5.19615	0.200000
$676$	−7.85641	−0.302169
$677$	45.9808	1.76718	0.883592	−	0.468257i	$-0.155118\pi$
$677$	45.9808	1.76718	0.883592	+	0.468257i	$0.155118\pi$
$678$	−17.5359	−0.673462
$679$	−2.92820	−0.112374
$680$	−2.73205	−0.104769
$681$	30.0000	1.14960
$682$	0	0
$683$	−38.9282	−1.48955	−0.744773	−	0.667318i	$-0.767442\pi$
$683$	−38.9282	−1.48955	−0.744773	+	0.667318i	$0.767442\pi$
$684$	0	0
$685$	2.66025	0.101643
$686$	1.00000	0.0381802
$687$	37.8564	1.44431
$688$	−8.73205	−0.332906
$689$	−9.51666	−0.362556
$690$	−8.66025	−0.329690
$691$	1.85641	0.0706210	0.0353105	−	0.999376i	$-0.488758\pi$
$691$	1.85641	0.0706210	0.0353105	+	0.999376i	$0.488758\pi$
$692$	−19.8564	−0.754827
$693$	0	0
$694$	−8.53590	−0.324018
$695$	6.80385	0.258085
$696$	−13.8564	−0.525226
$697$	11.4641	0.434234
$698$	−6.46410	−0.244670
$699$	−21.5885	−0.816550
$700$	1.00000	0.0377964
$701$	47.9090	1.80950	0.904748	−	0.425947i	$-0.140059\pi$
$701$	47.9090	1.80950	0.904748	+	0.425947i	$0.140059\pi$
$702$	−11.7846	−0.444781
$703$	−22.3923	−0.844542
$704$	0	0
$705$	−2.19615	−0.0827119
$706$	1.80385	0.0678887
$707$	−18.4641	−0.694414
$708$	21.5885	0.811344
$709$	1.32051	0.0495927	0.0247964	−	0.999693i	$-0.492106\pi$
$709$	1.32051	0.0495927	0.0247964	+	0.999693i	$0.492106\pi$
$710$	6.92820	0.260011
$711$	0	0
$712$	3.66025	0.137174
$713$	−3.66025	−0.137078
$714$	−4.73205	−0.177093
$715$	0	0
$716$	−5.80385	−0.216900
$717$	−4.26795	−0.159389
$718$	−33.7128	−1.25815
$719$	−34.9282	−1.30260	−0.651301	−	0.758819i	$-0.725777\pi$
$719$	−34.9282	−1.30260	−0.651301	+	0.758819i	$0.725777\pi$
$720$	0	0
$721$	−7.26795	−0.270673
$722$	−5.07180	−0.188753
$723$	−18.2487	−0.678677
$724$	−5.33975	−0.198450
$725$	8.00000	0.297113
$726$	0	0
$727$	−15.3205	−0.568206	−0.284103	−	0.958794i	$-0.591696\pi$
$727$	−15.3205	−0.568206	−0.284103	+	0.958794i	$0.591696\pi$
$728$	−2.26795	−0.0840558
$729$	27.0000	1.00000
$730$	15.1244	0.559778
$731$	−23.8564	−0.882361
$732$	−18.9282	−0.699607
$733$	−23.5885	−0.871260	−0.435630	−	0.900126i	$-0.643474\pi$
$733$	−23.5885	−0.871260	−0.435630	+	0.900126i	$0.643474\pi$
$734$	−20.7321	−0.765234
$735$	1.73205	0.0638877
$736$	−5.00000	−0.184302
$737$	0	0
$738$	0	0
$739$	26.9808	0.992503	0.496252	−	0.868179i	$-0.334709\pi$
$739$	26.9808	0.992503	0.496252	+	0.868179i	$0.334709\pi$
$740$	−6.00000	−0.220564
$741$	−14.6603	−0.538558
$742$	4.19615	0.154046
$743$	12.5359	0.459898	0.229949	−	0.973203i	$-0.426144\pi$
$743$	12.5359	0.459898	0.229949	+	0.973203i	$0.426144\pi$
$744$	−1.26795	−0.0464853
$745$	−12.5359	−0.459280
$746$	−13.3205	−0.487698
$747$	0	0
$748$	0	0
$749$	2.00000	0.0730784
$750$	1.73205	0.0632456
$751$	−42.3731	−1.54622	−0.773108	−	0.634275i	$-0.781299\pi$
$751$	−42.3731	−1.54622	−0.773108	+	0.634275i	$0.781299\pi$
$752$	−1.26795	−0.0462373
$753$	−24.9282	−0.908434
$754$	−18.1436	−0.660751
$755$	−8.85641	−0.322318
$756$	5.19615	0.188982
$757$	−25.9090	−0.941677	−0.470839	−	0.882219i	$-0.656049\pi$
$757$	−25.9090	−0.941677	−0.470839	+	0.882219i	$0.656049\pi$
$758$	34.7846	1.26343
$759$	0	0
$760$	3.73205	0.135376
$761$	−40.4449	−1.46613	−0.733063	−	0.680161i	$-0.761910\pi$
$761$	−40.4449	−1.46613	−0.733063	+	0.680161i	$0.761910\pi$
$762$	−31.3923	−1.13722
$763$	−18.0526	−0.653547
$764$	8.66025	0.313317
$765$	0	0
$766$	8.33975	0.301327
$767$	28.2679	1.02070
$768$	−1.73205	−0.0625000
$769$	34.7846	1.25437	0.627183	−	0.778872i	$-0.284208\pi$
$769$	34.7846	1.25437	0.627183	+	0.778872i	$0.284208\pi$
$770$	0	0
$771$	3.80385	0.136992
$772$	−9.39230	−0.338036
$773$	27.5359	0.990397	0.495199	−	0.868780i	$-0.335095\pi$
$773$	27.5359	0.990397	0.495199	+	0.868780i	$0.335095\pi$
$774$	0	0
$775$	0.732051	0.0262960
$776$	−2.92820	−0.105116
$777$	−10.3923	−0.372822
$778$	1.12436	0.0403101
$779$	−15.6603	−0.561087
$780$	−3.92820	−0.140652
$781$	0	0
$782$	−13.6603	−0.488490
$783$	41.5692	1.48556
$784$	1.00000	0.0357143
$785$	−15.3923	−0.549375
$786$	4.85641	0.173222
$787$	46.4974	1.65745	0.828727	−	0.559653i	$-0.189066\pi$
$787$	46.4974	1.65745	0.828727	+	0.559653i	$0.189066\pi$
$788$	−2.53590	−0.0903376
$789$	24.4641	0.870945
$790$	7.00000	0.249049
$791$	10.1244	0.359981
$792$	0	0
$793$	−24.7846	−0.880127
$794$	−28.7846	−1.02153
$795$	7.26795	0.257768
$796$	−4.00000	−0.141776
$797$	34.1769	1.21061	0.605304	−	0.795994i	$-0.293051\pi$
$797$	34.1769	1.21061	0.605304	+	0.795994i	$0.293051\pi$
$798$	6.46410	0.228827
$799$	−3.46410	−0.122551
$800$	1.00000	0.0353553
$801$	0	0
$802$	−25.4641	−0.899169
$803$	0	0
$804$	−13.2679	−0.467924
$805$	5.00000	0.176227
$806$	−1.66025	−0.0584800
$807$	31.6410	1.11382
$808$	−18.4641	−0.649565
$809$	−9.21539	−0.323996	−0.161998	−	0.986791i	$-0.551794\pi$
$809$	−9.21539	−0.323996	−0.161998	+	0.986791i	$0.551794\pi$
$810$	9.00000	0.316228
$811$	−30.5359	−1.07226	−0.536130	−	0.844135i	$-0.680114\pi$
$811$	−30.5359	−1.07226	−0.536130	+	0.844135i	$0.680114\pi$
$812$	8.00000	0.280745
$813$	−25.1769	−0.882993
$814$	0	0
$815$	−10.0000	−0.350285
$816$	−4.73205	−0.165655
$817$	32.5885	1.14013
$818$	−19.7128	−0.689242
$819$	0	0
$820$	−4.19615	−0.146536
$821$	43.8564	1.53060	0.765300	−	0.643674i	$-0.222591\pi$
$821$	43.8564	1.53060	0.765300	+	0.643674i	$0.222591\pi$
$822$	4.60770	0.160712
$823$	42.7846	1.49138	0.745689	−	0.666294i	$-0.232121\pi$
$823$	42.7846	1.49138	0.745689	+	0.666294i	$0.232121\pi$
$824$	−7.26795	−0.253191
$825$	0	0
$826$	−12.4641	−0.433682
$827$	19.9090	0.692303	0.346151	−	0.938179i	$-0.387488\pi$
$827$	19.9090	0.692303	0.346151	+	0.938179i	$0.387488\pi$
$828$	0	0
$829$	−0.124356	−0.00431905	−0.00215953	−	0.999998i	$-0.500687\pi$
$829$	−0.124356	−0.00431905	−0.00215953	+	0.999998i	$0.500687\pi$
$830$	−5.00000	−0.173553
$831$	−21.1244	−0.732796
$832$	−2.26795	−0.0786270
$833$	2.73205	0.0946600
$834$	11.7846	0.408068
$835$	17.6603	0.611158
$836$	0	0
$837$	3.80385	0.131480
$838$	2.07180	0.0715690
$839$	12.1962	0.421058	0.210529	−	0.977588i	$-0.432481\pi$
$839$	12.1962	0.421058	0.210529	+	0.977588i	$0.432481\pi$
$840$	1.73205	0.0597614
$841$	35.0000	1.20690
$842$	−28.3923	−0.978463
$843$	−7.39230	−0.254605
$844$	−8.00000	−0.275371
$845$	7.85641	0.270269
$846$	0	0
$847$	0	0
$848$	4.19615	0.144096
$849$	27.3397	0.938298
$850$	2.73205	0.0937086
$851$	−30.0000	−1.02839
$852$	12.0000	0.411113
$853$	43.8372	1.50096	0.750478	−	0.660895i	$-0.229823\pi$
$853$	43.8372	1.50096	0.750478	+	0.660895i	$0.229823\pi$
$854$	10.9282	0.373955
$855$	0	0
$856$	2.00000	0.0683586
$857$	−15.6077	−0.533149	−0.266574	−	0.963814i	$-0.585892\pi$
$857$	−15.6077	−0.533149	−0.266574	+	0.963814i	$0.585892\pi$
$858$	0	0
$859$	4.14359	0.141378	0.0706888	−	0.997498i	$-0.477480\pi$
$859$	4.14359	0.141378	0.0706888	+	0.997498i	$0.477480\pi$
$860$	8.73205	0.297760
$861$	−7.26795	−0.247691
$862$	−27.5359	−0.937876
$863$	−10.1436	−0.345292	−0.172646	−	0.984984i	$-0.555232\pi$
$863$	−10.1436	−0.345292	−0.172646	+	0.984984i	$0.555232\pi$
$864$	5.19615	0.176777
$865$	19.8564	0.675138
$866$	18.0526	0.613451
$867$	16.5167	0.560935
$868$	0.732051	0.0248474
$869$	0	0
$870$	13.8564	0.469776
$871$	−17.3731	−0.588664
$872$	−18.0526	−0.611337
$873$	0	0
$874$	18.6603	0.631193
$875$	−1.00000	−0.0338062
$876$	26.1962	0.885086
$877$	9.94744	0.335901	0.167951	−	0.985795i	$-0.446285\pi$
$877$	9.94744	0.335901	0.167951	+	0.985795i	$0.446285\pi$
$878$	−11.2679	−0.380275
$879$	−14.3205	−0.483019
$880$	0	0
$881$	−53.4256	−1.79996	−0.899978	−	0.435936i	$-0.856417\pi$
$881$	−53.4256	−1.79996	−0.899978	+	0.435936i	$0.856417\pi$
$882$	0	0
$883$	−32.6410	−1.09846	−0.549229	−	0.835672i	$-0.685078\pi$
$883$	−32.6410	−1.09846	−0.549229	+	0.835672i	$0.685078\pi$
$884$	−6.19615	−0.208399
$885$	−21.5885	−0.725688
$886$	4.00000	0.134383
$887$	2.33975	0.0785610	0.0392805	−	0.999228i	$-0.487493\pi$
$887$	2.33975	0.0785610	0.0392805	+	0.999228i	$0.487493\pi$
$888$	−10.3923	−0.348743
$889$	18.1244	0.607871
$890$	−3.66025	−0.122692
$891$	0	0
$892$	−14.7321	−0.493266
$893$	4.73205	0.158352
$894$	−21.7128	−0.726185
$895$	5.80385	0.194001
$896$	1.00000	0.0334077
$897$	−19.6410	−0.655794
$898$	10.6077	0.353983
$899$	5.85641	0.195322
$900$	0	0
$901$	11.4641	0.381925
$902$	0	0
$903$	15.1244	0.503307
$904$	10.1244	0.336731
$905$	5.33975	0.177499
$906$	−15.3397	−0.509629
$907$	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
$910$	2.26795	0.0751818
$911$	29.9808	0.993307	0.496653	−	0.867949i	$-0.334562\pi$
$911$	29.9808	0.993307	0.496653	+	0.867949i	$0.334562\pi$
$912$	6.46410	0.214048
$913$	0	0
$914$	17.3923	0.575286
$915$	18.9282	0.625747
$916$	−21.8564	−0.722156
$917$	−2.80385	−0.0925912
$918$	14.1962	0.468543
$919$	12.1436	0.400580	0.200290	−	0.979737i	$-0.435812\pi$
$919$	12.1436	0.400580	0.200290	+	0.979737i	$0.435812\pi$
$920$	5.00000	0.164845
$921$	36.2487	1.19444
$922$	17.4641	0.575150
$923$	15.7128	0.517194
$924$	0	0
$925$	6.00000	0.197279
$926$	−36.1769	−1.18885
$927$	0	0
$928$	8.00000	0.262613
$929$	−47.3205	−1.55254	−0.776268	−	0.630403i	$-0.782890\pi$
$929$	−47.3205	−1.55254	−0.776268	+	0.630403i	$0.782890\pi$
$930$	1.26795	0.0415777
$931$	−3.73205	−0.122313
$932$	12.4641	0.408275
$933$	−16.0526	−0.525537
$934$	−17.5885	−0.575512
$935$	0	0
$936$	0	0
$937$	10.5359	0.344193	0.172096	−	0.985080i	$-0.444946\pi$
$937$	10.5359	0.344193	0.172096	+	0.985080i	$0.444946\pi$
$938$	7.66025	0.250116
$939$	45.7128	1.49178
$940$	1.26795	0.0413559
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	−26.6603	−0.868638
$943$	−20.9808	−0.683228
$944$	−12.4641	−0.405672
$945$	−5.19615	−0.169031
$946$	0	0
$947$	−60.0526	−1.95145	−0.975723	−	0.219008i	$-0.929718\pi$
$947$	−60.0526	−1.95145	−0.975723	+	0.219008i	$0.929718\pi$
$948$	12.1244	0.393781
$949$	34.3013	1.11347
$950$	−3.73205	−0.121084
$951$	33.1244	1.07413
$952$	2.73205	0.0885463
$953$	−2.60770	−0.0844715	−0.0422358	−	0.999108i	$-0.513448\pi$
$953$	−2.60770	−0.0844715	−0.0422358	+	0.999108i	$0.513448\pi$
$954$	0	0
$955$	−8.66025	−0.280239
$956$	2.46410	0.0796947
$957$	0	0
$958$	−22.9808	−0.742475
$959$	−2.66025	−0.0859041
$960$	1.73205	0.0559017
$961$	−30.4641	−0.982713
$962$	−13.6077	−0.438730
$963$	0	0
$964$	10.5359	0.339338
$965$	9.39230	0.302349
$966$	8.66025	0.278639
$967$	12.7846	0.411125	0.205563	−	0.978644i	$-0.434098\pi$
$967$	12.7846	0.411125	0.205563	+	0.978644i	$0.434098\pi$
$968$	0	0
$969$	17.6603	0.567329
$970$	2.92820	0.0940189
$971$	−4.71281	−0.151241	−0.0756207	−	0.997137i	$-0.524094\pi$
$971$	−4.71281	−0.151241	−0.0756207	+	0.997137i	$0.524094\pi$
$972$	0	0
$973$	−6.80385	−0.218121
$974$	−3.39230	−0.108696
$975$	3.92820	0.125803
$976$	10.9282	0.349803
$977$	−17.7321	−0.567299	−0.283649	−	0.958928i	$-0.591545\pi$
$977$	−17.7321	−0.567299	−0.283649	+	0.958928i	$0.591545\pi$
$978$	−17.3205	−0.553849
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−7.66025	−0.244449
$983$	58.3923	1.86243	0.931213	−	0.364476i	$-0.118752\pi$
$983$	58.3923	1.86243	0.931213	+	0.364476i	$0.118752\pi$
$984$	−7.26795	−0.231694
$985$	2.53590	0.0808004
$986$	21.8564	0.696050
$987$	2.19615	0.0699043
$988$	8.46410	0.269279
$989$	43.6603	1.38832
$990$	0	0
$991$	−26.1244	−0.829868	−0.414934	−	0.909852i	$-0.636195\pi$
$991$	−26.1244	−0.829868	−0.414934	+	0.909852i	$0.636195\pi$
$992$	0.732051	0.0232426
$993$	7.01924	0.222749
$994$	−6.92820	−0.219749
$995$	4.00000	0.126809
$996$	−8.66025	−0.274411
$997$	21.3397	0.675837	0.337918	−	0.941175i	$-0.390277\pi$
$997$	21.3397	0.675837	0.337918	+	0.941175i	$0.390277\pi$
$998$	−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.1.1	yes	2
11.10	odd	2		8470.2.a.bm.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.bm.1.1	✓	2	11.10	odd	2
8470.2.a.bz.1.1	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.73205 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.73205 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.73205 q^{12} -2.26795 q^{13} +1.00000 q^{14} +1.73205 q^{15} +1.00000 q^{16} +2.73205 q^{17} -3.73205 q^{19} -1.00000 q^{20} -1.73205 q^{21} -5.00000 q^{23} -1.73205 q^{24} +1.00000 q^{25} -2.26795 q^{26} +5.19615 q^{27} +1.00000 q^{28} +8.00000 q^{29} +1.73205 q^{30} +0.732051 q^{31} +1.00000 q^{32} +2.73205 q^{34} -1.00000 q^{35} +6.00000 q^{37} -3.73205 q^{38} +3.92820 q^{39} -1.00000 q^{40} +4.19615 q^{41} -1.73205 q^{42} -8.73205 q^{43} -5.00000 q^{46} -1.26795 q^{47} -1.73205 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.73205 q^{51} -2.26795 q^{52} +4.19615 q^{53} +5.19615 q^{54} +1.00000 q^{56} +6.46410 q^{57} +8.00000 q^{58} -12.4641 q^{59} +1.73205 q^{60} +10.9282 q^{61} +0.732051 q^{62} +1.00000 q^{64} +2.26795 q^{65} +7.66025 q^{67} +2.73205 q^{68} +8.66025 q^{69} -1.00000 q^{70} -6.92820 q^{71} -15.1244 q^{73} +6.00000 q^{74} -1.73205 q^{75} -3.73205 q^{76} +3.92820 q^{78} -7.00000 q^{79} -1.00000 q^{80} -9.00000 q^{81} +4.19615 q^{82} +5.00000 q^{83} -1.73205 q^{84} -2.73205 q^{85} -8.73205 q^{86} -13.8564 q^{87} +3.66025 q^{89} -2.26795 q^{91} -5.00000 q^{92} -1.26795 q^{93} -1.26795 q^{94} +3.73205 q^{95} -1.73205 q^{96} -2.92820 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 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

Varieties

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Database

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