LMFDB - Embedded newform 6400.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6400,2,Mod(1,6400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6400, 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("6400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6400 = 2^{8} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6400.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:	$51.1042572936$
Analytic rank:	$0$
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:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -0.732051 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -0.732051 q^{7} +4.46410 q^{9} +2.00000 q^{11} -3.46410 q^{13} -3.46410 q^{17} +0.535898 q^{19} +2.00000 q^{21} +6.19615 q^{23} -4.00000 q^{27} -6.92820 q^{29} +5.46410 q^{31} -5.46410 q^{33} +2.00000 q^{37} +9.46410 q^{39} -1.46410 q^{41} -5.26795 q^{43} +3.26795 q^{47} -6.46410 q^{49} +9.46410 q^{51} -11.4641 q^{53} -1.46410 q^{57} +7.46410 q^{59} +8.92820 q^{61} -3.26795 q^{63} -10.7321 q^{67} -16.9282 q^{69} +5.46410 q^{71} +7.46410 q^{73} -1.46410 q^{77} +1.07180 q^{79} -2.46410 q^{81} -1.26795 q^{83} +18.9282 q^{87} -8.92820 q^{89} +2.53590 q^{91} -14.9282 q^{93} +14.3923 q^{97} +8.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{19} + 4 q^{21} + 2 q^{23} - 8 q^{27} + 4 q^{31} - 4 q^{33} + 4 q^{37} + 12 q^{39} + 4 q^{41} - 14 q^{43} + 10 q^{47} - 6 q^{49} + 12 q^{51} - 16 q^{53} + 4 q^{57} + 8 q^{59} + 4 q^{61} - 10 q^{63} - 18 q^{67} - 20 q^{69} + 4 q^{71} + 8 q^{73} + 4 q^{77} + 16 q^{79} + 2 q^{81} - 6 q^{83} + 24 q^{87} - 4 q^{89} + 12 q^{91} - 16 q^{93} + 8 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^19 + 4 * q^21 + 2 * q^23 - 8 * q^27 + 4 * q^31 - 4 * q^33 + 4 * q^37 + 12 * q^39 + 4 * q^41 - 14 * q^43 + 10 * q^47 - 6 * q^49 + 12 * q^51 - 16 * q^53 + 4 * q^57 + 8 * q^59 + 4 * q^61 - 10 * q^63 - 18 * q^67 - 20 * q^69 + 4 * q^71 + 8 * q^73 + 4 * q^77 + 16 * q^79 + 2 * q^81 - 6 * q^83 + 24 * q^87 - 4 * q^89 + 12 * q^91 - 16 * q^93 + 8 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−3.46410	−0.960769	−0.480384	−	0.877058i	$-0.659503\pi$
$13$	−3.46410	−0.960769	−0.480384	+	0.877058i	$0.659503\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	0.535898	0.122944	0.0614718	−	0.998109i	$-0.480421\pi$
$19$	0.535898	0.122944	0.0614718	+	0.998109i	$0.480421\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	6.19615	1.29199	0.645994	−	0.763343i	$-0.276443\pi$
$23$	6.19615	1.29199	0.645994	+	0.763343i	$0.276443\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−6.92820	−1.28654	−0.643268	−	0.765641i	$-0.722422\pi$
$29$	−6.92820	−1.28654	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0	0
$31$	5.46410	0.981382	0.490691	−	0.871334i	$-0.336744\pi$
$31$	5.46410	0.981382	0.490691	+	0.871334i	$0.336744\pi$
$32$	0	0
$33$	−5.46410	−0.951178
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	9.46410	1.51547
$40$	0	0
$41$	−1.46410	−0.228654	−0.114327	−	0.993443i	$-0.536471\pi$
$41$	−1.46410	−0.228654	−0.114327	+	0.993443i	$0.536471\pi$
$42$	0	0
$43$	−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
$46$	0	0
$47$	3.26795	0.476679	0.238340	−	0.971182i	$-0.423397\pi$
$47$	3.26795	0.476679	0.238340	+	0.971182i	$0.423397\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	9.46410	1.32524
$52$	0	0
$53$	−11.4641	−1.57472	−0.787358	−	0.616496i	$-0.788551\pi$
$53$	−11.4641	−1.57472	−0.787358	+	0.616496i	$0.788551\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.46410	−0.193925
$58$	0	0
$59$	7.46410	0.971743	0.485872	−	0.874030i	$-0.338502\pi$
$59$	7.46410	0.971743	0.485872	+	0.874030i	$0.338502\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	0	0
$63$	−3.26795	−0.411723
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.7321	−1.31113	−0.655564	−	0.755139i	$-0.727569\pi$
$67$	−10.7321	−1.31113	−0.655564	+	0.755139i	$0.727569\pi$
$68$	0	0
$69$	−16.9282	−2.03792
$70$	0	0
$71$	5.46410	0.648470	0.324235	−	0.945977i	$-0.394893\pi$
$71$	5.46410	0.648470	0.324235	+	0.945977i	$0.394893\pi$
$72$	0	0
$73$	7.46410	0.873607	0.436804	−	0.899557i	$-0.356111\pi$
$73$	7.46410	0.873607	0.436804	+	0.899557i	$0.356111\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.46410	−0.166850
$78$	0	0
$79$	1.07180	0.120587	0.0602933	−	0.998181i	$-0.480796\pi$
$79$	1.07180	0.120587	0.0602933	+	0.998181i	$0.480796\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−1.26795	−0.139176	−0.0695878	−	0.997576i	$-0.522168\pi$
$83$	−1.26795	−0.139176	−0.0695878	+	0.997576i	$0.522168\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.9282	2.02932
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	2.53590	0.265834
$92$	0	0
$93$	−14.9282	−1.54798
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.3923	1.46132	0.730659	−	0.682743i	$-0.239213\pi$
$97$	14.3923	1.46132	0.730659	+	0.682743i	$0.239213\pi$
$98$	0	0
$99$	8.92820	0.897318
$100$	0	0
$101$	2.92820	0.291367	0.145684	−	0.989331i	$-0.453462\pi$
$101$	2.92820	0.291367	0.145684	+	0.989331i	$0.453462\pi$
$102$	0	0
$103$	−15.6603	−1.54305	−0.771525	−	0.636199i	$-0.780506\pi$
$103$	−15.6603	−1.54305	−0.771525	+	0.636199i	$0.780506\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.73205	0.264117	0.132059	−	0.991242i	$-0.457841\pi$
$107$	2.73205	0.264117	0.132059	+	0.991242i	$0.457841\pi$
$108$	0	0
$109$	−16.9282	−1.62143	−0.810714	−	0.585443i	$-0.800921\pi$
$109$	−16.9282	−1.62143	−0.810714	+	0.585443i	$0.800921\pi$
$110$	0	0
$111$	−5.46410	−0.518630
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15.4641	−1.42966
$118$	0	0
$119$	2.53590	0.232465
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	4.00000	0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.7321	1.48473	0.742365	−	0.669996i	$-0.233704\pi$
$127$	16.7321	1.48473	0.742365	+	0.669996i	$0.233704\pi$
$128$	0	0
$129$	14.3923	1.26717
$130$	0	0
$131$	−19.8564	−1.73486	−0.867431	−	0.497557i	$-0.834230\pi$
$131$	−19.8564	−1.73486	−0.867431	+	0.497557i	$0.834230\pi$
$132$	0	0
$133$	−0.392305	−0.0340171
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.92820	0.421045	0.210522	−	0.977589i	$-0.432484\pi$
$137$	4.92820	0.421045	0.210522	+	0.977589i	$0.432484\pi$
$138$	0	0
$139$	0.535898	0.0454543	0.0227272	−	0.999742i	$-0.492765\pi$
$139$	0.535898	0.0454543	0.0227272	+	0.999742i	$0.492765\pi$
$140$	0	0
$141$	−8.92820	−0.751890
$142$	0	0
$143$	−6.92820	−0.579365
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.6603	1.45659
$148$	0	0
$149$	−7.85641	−0.643622	−0.321811	−	0.946804i	$-0.604292\pi$
$149$	−7.85641	−0.643622	−0.321811	+	0.946804i	$0.604292\pi$
$150$	0	0
$151$	−12.3923	−1.00847	−0.504236	−	0.863566i	$-0.668226\pi$
$151$	−12.3923	−1.00847	−0.504236	+	0.863566i	$0.668226\pi$
$152$	0	0
$153$	−15.4641	−1.25020
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.07180	−0.245156	−0.122578	−	0.992459i	$-0.539116\pi$
$157$	−3.07180	−0.245156	−0.122578	+	0.992459i	$0.539116\pi$
$158$	0	0
$159$	31.3205	2.48388
$160$	0	0
$161$	−4.53590	−0.357479
$162$	0	0
$163$	0.196152	0.0153638	0.00768192	−	0.999970i	$-0.497555\pi$
$163$	0.196152	0.0153638	0.00768192	+	0.999970i	$0.497555\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.80385	0.758645	0.379322	−	0.925265i	$-0.376157\pi$
$167$	9.80385	0.758645	0.379322	+	0.925265i	$0.376157\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	2.39230	0.182944
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.3923	−1.53278
$178$	0	0
$179$	−8.53590	−0.638003	−0.319002	−	0.947754i	$-0.603348\pi$
$179$	−8.53590	−0.638003	−0.319002	+	0.947754i	$0.603348\pi$
$180$	0	0
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	−24.3923	−1.80313
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.92820	−0.506640
$188$	0	0
$189$	2.92820	0.212995
$190$	0	0
$191$	−15.3205	−1.10855	−0.554277	−	0.832333i	$-0.687005\pi$
$191$	−15.3205	−1.10855	−0.554277	+	0.832333i	$0.687005\pi$
$192$	0	0
$193$	−0.535898	−0.0385748	−0.0192874	−	0.999814i	$-0.506140\pi$
$193$	−0.535898	−0.0385748	−0.0192874	+	0.999814i	$0.506140\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.4641	1.38676	0.693380	−	0.720572i	$-0.256121\pi$
$197$	19.4641	1.38676	0.693380	+	0.720572i	$0.256121\pi$
$198$	0	0
$199$	−1.85641	−0.131597	−0.0657986	−	0.997833i	$-0.520959\pi$
$199$	−1.85641	−0.131597	−0.0657986	+	0.997833i	$0.520959\pi$
$200$	0	0
$201$	29.3205	2.06811
$202$	0	0
$203$	5.07180	0.355970
$204$	0	0
$205$	0	0
$206$	0	0
$207$	27.6603	1.92252
$208$	0	0
$209$	1.07180	0.0741377
$210$	0	0
$211$	26.7846	1.84393	0.921964	−	0.387275i	$-0.126584\pi$
$211$	26.7846	1.84393	0.921964	+	0.387275i	$0.126584\pi$
$212$	0	0
$213$	−14.9282	−1.02286
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	−20.3923	−1.37798
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−5.80385	−0.388654	−0.194327	−	0.980937i	$-0.562252\pi$
$223$	−5.80385	−0.388654	−0.194327	+	0.980937i	$0.562252\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.0526	−0.667212	−0.333606	−	0.942713i	$-0.608265\pi$
$227$	−10.0526	−0.667212	−0.333606	+	0.942713i	$0.608265\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	4.00000	0.263181
$232$	0	0
$233$	−5.32051	−0.348558	−0.174279	−	0.984696i	$-0.555759\pi$
$233$	−5.32051	−0.348558	−0.174279	+	0.984696i	$0.555759\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.92820	−0.190207
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	16.3923	1.05592	0.527961	−	0.849269i	$-0.322957\pi$
$241$	16.3923	1.05592	0.527961	+	0.849269i	$0.322957\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.85641	−0.118120
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	−24.9282	−1.57345	−0.786727	−	0.617301i	$-0.788226\pi$
$251$	−24.9282	−1.57345	−0.786727	+	0.617301i	$0.788226\pi$
$252$	0	0
$253$	12.3923	0.779098
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	0	0
$259$	−1.46410	−0.0909748
$260$	0	0
$261$	−30.9282	−1.91441
$262$	0	0
$263$	11.6603	0.719002	0.359501	−	0.933145i	$-0.382947\pi$
$263$	11.6603	0.719002	0.359501	+	0.933145i	$0.382947\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	24.3923	1.49278
$268$	0	0
$269$	8.92820	0.544362	0.272181	−	0.962246i	$-0.412255\pi$
$269$	8.92820	0.544362	0.272181	+	0.962246i	$0.412255\pi$
$270$	0	0
$271$	19.3205	1.17364	0.586819	−	0.809718i	$-0.300380\pi$
$271$	19.3205	1.17364	0.586819	+	0.809718i	$0.300380\pi$
$272$	0	0
$273$	−6.92820	−0.419314
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	0	0
$279$	24.3923	1.46033
$280$	0	0
$281$	−10.5359	−0.628519	−0.314260	−	0.949337i	$-0.601756\pi$
$281$	−10.5359	−0.628519	−0.314260	+	0.949337i	$0.601756\pi$
$282$	0	0
$283$	−9.66025	−0.574242	−0.287121	−	0.957894i	$-0.592698\pi$
$283$	−9.66025	−0.574242	−0.287121	+	0.957894i	$0.592698\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.07180	0.0632662
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	−39.3205	−2.30501
$292$	0	0
$293$	15.8564	0.926341	0.463171	−	0.886269i	$-0.346712\pi$
$293$	15.8564	0.926341	0.463171	+	0.886269i	$0.346712\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−8.00000	−0.464207
$298$	0	0
$299$	−21.4641	−1.24130
$300$	0	0
$301$	3.85641	0.222280
$302$	0	0
$303$	−8.00000	−0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.9808	1.42573	0.712864	−	0.701303i	$-0.247398\pi$
$307$	24.9808	1.42573	0.712864	+	0.701303i	$0.247398\pi$
$308$	0	0
$309$	42.7846	2.43393
$310$	0	0
$311$	31.3205	1.77602	0.888012	−	0.459821i	$-0.152086\pi$
$311$	31.3205	1.77602	0.888012	+	0.459821i	$0.152086\pi$
$312$	0	0
$313$	−4.14359	−0.234210	−0.117105	−	0.993120i	$-0.537361\pi$
$313$	−4.14359	−0.234210	−0.117105	+	0.993120i	$0.537361\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.53590	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$317$	8.53590	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$318$	0	0
$319$	−13.8564	−0.775810
$320$	0	0
$321$	−7.46410	−0.416606
$322$	0	0
$323$	−1.85641	−0.103293
$324$	0	0
$325$	0	0
$326$	0	0
$327$	46.2487	2.55756
$328$	0	0
$329$	−2.39230	−0.131892
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	8.92820	0.489263
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.8564	1.08165	0.540824	−	0.841136i	$-0.318113\pi$
$337$	19.8564	1.08165	0.540824	+	0.841136i	$0.318113\pi$
$338$	0	0
$339$	−35.3205	−1.91835
$340$	0	0
$341$	10.9282	0.591795
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.66025	−0.0891271	−0.0445636	−	0.999007i	$-0.514190\pi$
$347$	−1.66025	−0.0891271	−0.0445636	+	0.999007i	$0.514190\pi$
$348$	0	0
$349$	28.0000	1.49881	0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000	1.49881	0.749403	+	0.662114i	$0.230341\pi$
$350$	0	0
$351$	13.8564	0.739600
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.92820	−0.366679
$358$	0	0
$359$	18.9282	0.998992	0.499496	−	0.866316i	$-0.333518\pi$
$359$	18.9282	0.998992	0.499496	+	0.866316i	$0.333518\pi$
$360$	0	0
$361$	−18.7128	−0.984885
$362$	0	0
$363$	19.1244	1.00377
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.87564	0.150107	0.0750537	−	0.997179i	$-0.476087\pi$
$367$	2.87564	0.150107	0.0750537	+	0.997179i	$0.476087\pi$
$368$	0	0
$369$	−6.53590	−0.340245
$370$	0	0
$371$	8.39230	0.435707
$372$	0	0
$373$	25.7128	1.33136	0.665679	−	0.746238i	$-0.268142\pi$
$373$	25.7128	1.33136	0.665679	+	0.746238i	$0.268142\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	36.2487	1.86197	0.930986	−	0.365056i	$-0.118950\pi$
$379$	36.2487	1.86197	0.930986	+	0.365056i	$0.118950\pi$
$380$	0	0
$381$	−45.7128	−2.34194
$382$	0	0
$383$	21.1244	1.07940	0.539702	−	0.841856i	$-0.318537\pi$
$383$	21.1244	1.07940	0.539702	+	0.841856i	$0.318537\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−23.5167	−1.19542
$388$	0	0
$389$	6.78461	0.343993	0.171997	−	0.985098i	$-0.444978\pi$
$389$	6.78461	0.343993	0.171997	+	0.985098i	$0.444978\pi$
$390$	0	0
$391$	−21.4641	−1.08549
$392$	0	0
$393$	54.2487	2.73649
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.2487	1.61852	0.809258	−	0.587453i	$-0.199869\pi$
$397$	32.2487	1.61852	0.809258	+	0.587453i	$0.199869\pi$
$398$	0	0
$399$	1.07180	0.0536570
$400$	0	0
$401$	−7.85641	−0.392330	−0.196165	−	0.980571i	$-0.562849\pi$
$401$	−7.85641	−0.392330	−0.196165	+	0.980571i	$0.562849\pi$
$402$	0	0
$403$	−18.9282	−0.942881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	11.3205	0.559763	0.279882	−	0.960035i	$-0.409705\pi$
$409$	11.3205	0.559763	0.279882	+	0.960035i	$0.409705\pi$
$410$	0	0
$411$	−13.4641	−0.664135
$412$	0	0
$413$	−5.46410	−0.268871
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.46410	−0.0716974
$418$	0	0
$419$	18.3923	0.898523	0.449261	−	0.893400i	$-0.351687\pi$
$419$	18.3923	0.898523	0.449261	+	0.893400i	$0.351687\pi$
$420$	0	0
$421$	0.143594	0.00699832	0.00349916	−	0.999994i	$-0.498886\pi$
$421$	0.143594	0.00699832	0.00349916	+	0.999994i	$0.498886\pi$
$422$	0	0
$423$	14.5885	0.709315
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.53590	−0.316294
$428$	0	0
$429$	18.9282	0.913862
$430$	0	0
$431$	21.4641	1.03389	0.516945	−	0.856019i	$-0.327069\pi$
$431$	21.4641	1.03389	0.516945	+	0.856019i	$0.327069\pi$
$432$	0	0
$433$	19.4641	0.935385	0.467693	−	0.883891i	$-0.345085\pi$
$433$	19.4641	0.935385	0.467693	+	0.883891i	$0.345085\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.32051	0.158841
$438$	0	0
$439$	40.7846	1.94654	0.973272	−	0.229657i	$-0.0737605\pi$
$439$	40.7846	1.94654	0.973272	+	0.229657i	$0.0737605\pi$
$440$	0	0
$441$	−28.8564	−1.37411
$442$	0	0
$443$	20.9808	0.996826	0.498413	−	0.866940i	$-0.333916\pi$
$443$	20.9808	0.996826	0.498413	+	0.866940i	$0.333916\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.4641	1.01522
$448$	0	0
$449$	−23.3205	−1.10056	−0.550281	−	0.834979i	$-0.685480\pi$
$449$	−23.3205	−1.10056	−0.550281	+	0.834979i	$0.685480\pi$
$450$	0	0
$451$	−2.92820	−0.137884
$452$	0	0
$453$	33.8564	1.59071
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.7846	−1.25293	−0.626466	−	0.779449i	$-0.715499\pi$
$457$	−26.7846	−1.25293	−0.626466	+	0.779449i	$0.715499\pi$
$458$	0	0
$459$	13.8564	0.646762
$460$	0	0
$461$	10.9282	0.508977	0.254489	−	0.967076i	$-0.418093\pi$
$461$	10.9282	0.508977	0.254489	+	0.967076i	$0.418093\pi$
$462$	0	0
$463$	−11.2679	−0.523666	−0.261833	−	0.965113i	$-0.584327\pi$
$463$	−11.2679	−0.523666	−0.261833	+	0.965113i	$0.584327\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.6603	1.18741	0.593707	−	0.804681i	$-0.297664\pi$
$467$	25.6603	1.18741	0.593707	+	0.804681i	$0.297664\pi$
$468$	0	0
$469$	7.85641	0.362775
$470$	0	0
$471$	8.39230	0.386697
$472$	0	0
$473$	−10.5359	−0.484441
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−51.1769	−2.34323
$478$	0	0
$479$	−5.85641	−0.267586	−0.133793	−	0.991009i	$-0.542716\pi$
$479$	−5.85641	−0.267586	−0.133793	+	0.991009i	$0.542716\pi$
$480$	0	0
$481$	−6.92820	−0.315899
$482$	0	0
$483$	12.3923	0.563869
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.58846	−0.298551	−0.149276	−	0.988796i	$-0.547694\pi$
$487$	−6.58846	−0.298551	−0.149276	+	0.988796i	$0.547694\pi$
$488$	0	0
$489$	−0.535898	−0.0242342
$490$	0	0
$491$	16.9282	0.763959	0.381980	−	0.924171i	$-0.375242\pi$
$491$	16.9282	0.763959	0.381980	+	0.924171i	$0.375242\pi$
$492$	0	0
$493$	24.0000	1.08091
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.00000	−0.179425
$498$	0	0
$499$	31.4641	1.40853	0.704263	−	0.709939i	$-0.251277\pi$
$499$	31.4641	1.40853	0.704263	+	0.709939i	$0.251277\pi$
$500$	0	0
$501$	−26.7846	−1.19665
$502$	0	0
$503$	0.339746	0.0151485	0.00757426	−	0.999971i	$-0.497589\pi$
$503$	0.339746	0.0151485	0.00757426	+	0.999971i	$0.497589\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.73205	0.121335
$508$	0	0
$509$	1.85641	0.0822838	0.0411419	−	0.999153i	$-0.486900\pi$
$509$	1.85641	0.0822838	0.0411419	+	0.999153i	$0.486900\pi$
$510$	0	0
$511$	−5.46410	−0.241718
$512$	0	0
$513$	−2.14359	−0.0946420
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.53590	0.287448
$518$	0	0
$519$	−5.46410	−0.239847
$520$	0	0
$521$	43.8564	1.92138	0.960692	−	0.277616i	$-0.0895444\pi$
$521$	43.8564	1.92138	0.960692	+	0.277616i	$0.0895444\pi$
$522$	0	0
$523$	−11.8038	−0.516146	−0.258073	−	0.966125i	$-0.583088\pi$
$523$	−11.8038	−0.516146	−0.258073	+	0.966125i	$0.583088\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.9282	−0.824525
$528$	0	0
$529$	15.3923	0.669231
$530$	0	0
$531$	33.3205	1.44599
$532$	0	0
$533$	5.07180	0.219684
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3205	1.00635
$538$	0	0
$539$	−12.9282	−0.556857
$540$	0	0
$541$	26.9282	1.15773	0.578867	−	0.815422i	$-0.303495\pi$
$541$	26.9282	1.15773	0.578867	+	0.815422i	$0.303495\pi$
$542$	0	0
$543$	−43.7128	−1.87590
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−33.2679	−1.42243	−0.711217	−	0.702972i	$-0.751856\pi$
$547$	−33.2679	−1.42243	−0.711217	+	0.702972i	$0.751856\pi$
$548$	0	0
$549$	39.8564	1.70103
$550$	0	0
$551$	−3.71281	−0.158171
$552$	0	0
$553$	−0.784610	−0.0333650
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7846	0.626444	0.313222	−	0.949680i	$-0.398592\pi$
$557$	14.7846	0.626444	0.313222	+	0.949680i	$0.398592\pi$
$558$	0	0
$559$	18.2487	0.771838
$560$	0	0
$561$	18.9282	0.799149
$562$	0	0
$563$	22.0526	0.929405	0.464702	−	0.885467i	$-0.346161\pi$
$563$	22.0526	0.929405	0.464702	+	0.885467i	$0.346161\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.80385	0.0757545
$568$	0	0
$569$	13.4641	0.564445	0.282222	−	0.959349i	$-0.408928\pi$
$569$	13.4641	0.564445	0.282222	+	0.959349i	$0.408928\pi$
$570$	0	0
$571$	−6.78461	−0.283927	−0.141964	−	0.989872i	$-0.545342\pi$
$571$	−6.78461	−0.283927	−0.141964	+	0.989872i	$0.545342\pi$
$572$	0	0
$573$	41.8564	1.74858
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.5692	−1.64729	−0.823644	−	0.567107i	$-0.808063\pi$
$577$	−39.5692	−1.64729	−0.823644	+	0.567107i	$0.808063\pi$
$578$	0	0
$579$	1.46410	0.0608460
$580$	0	0
$581$	0.928203	0.0385084
$582$	0	0
$583$	−22.9282	−0.949589
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.80385	−0.157002	−0.0785008	−	0.996914i	$-0.525013\pi$
$587$	−3.80385	−0.157002	−0.0785008	+	0.996914i	$0.525013\pi$
$588$	0	0
$589$	2.92820	0.120655
$590$	0	0
$591$	−53.1769	−2.18741
$592$	0	0
$593$	−32.6410	−1.34041	−0.670203	−	0.742178i	$-0.733793\pi$
$593$	−32.6410	−1.34041	−0.670203	+	0.742178i	$0.733793\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.07180	0.207575
$598$	0	0
$599$	−34.6410	−1.41539	−0.707697	−	0.706516i	$-0.750266\pi$
$599$	−34.6410	−1.41539	−0.707697	+	0.706516i	$0.750266\pi$
$600$	0	0
$601$	−18.5359	−0.756095	−0.378048	−	0.925786i	$-0.623404\pi$
$601$	−18.5359	−0.756095	−0.378048	+	0.925786i	$0.623404\pi$
$602$	0	0
$603$	−47.9090	−1.95100
$604$	0	0
$605$	0	0
$606$	0	0
$607$	30.9808	1.25747	0.628735	−	0.777619i	$-0.283573\pi$
$607$	30.9808	1.25747	0.628735	+	0.777619i	$0.283573\pi$
$608$	0	0
$609$	−13.8564	−0.561490
$610$	0	0
$611$	−11.3205	−0.457979
$612$	0	0
$613$	−26.3923	−1.06598	−0.532988	−	0.846123i	$-0.678931\pi$
$613$	−26.3923	−1.06598	−0.532988	+	0.846123i	$0.678931\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.5359	−0.826744	−0.413372	−	0.910562i	$-0.635649\pi$
$617$	−20.5359	−0.826744	−0.413372	+	0.910562i	$0.635649\pi$
$618$	0	0
$619$	−1.32051	−0.0530757	−0.0265379	−	0.999648i	$-0.508448\pi$
$619$	−1.32051	−0.0530757	−0.0265379	+	0.999648i	$0.508448\pi$
$620$	0	0
$621$	−24.7846	−0.994572
$622$	0	0
$623$	6.53590	0.261855
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.92820	−0.116941
$628$	0	0
$629$	−6.92820	−0.276246
$630$	0	0
$631$	−23.3205	−0.928375	−0.464187	−	0.885737i	$-0.653654\pi$
$631$	−23.3205	−0.928375	−0.464187	+	0.885737i	$0.653654\pi$
$632$	0	0
$633$	−73.1769	−2.90852
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.3923	0.887215
$638$	0	0
$639$	24.3923	0.964945
$640$	0	0
$641$	0.392305	0.0154951	0.00774755	−	0.999970i	$-0.497534\pi$
$641$	0.392305	0.0154951	0.00774755	+	0.999970i	$0.497534\pi$
$642$	0	0
$643$	−39.1244	−1.54291	−0.771457	−	0.636281i	$-0.780472\pi$
$643$	−39.1244	−1.54291	−0.771457	+	0.636281i	$0.780472\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.7321	−0.657805	−0.328902	−	0.944364i	$-0.606679\pi$
$647$	−16.7321	−0.657805	−0.328902	+	0.944364i	$0.606679\pi$
$648$	0	0
$649$	14.9282	0.585983
$650$	0	0
$651$	10.9282	0.428310
$652$	0	0
$653$	12.2487	0.479329	0.239665	−	0.970856i	$-0.422963\pi$
$653$	12.2487	0.479329	0.239665	+	0.970856i	$0.422963\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	33.3205	1.29996
$658$	0	0
$659$	−17.3205	−0.674711	−0.337356	−	0.941377i	$-0.609532\pi$
$659$	−17.3205	−0.674711	−0.337356	+	0.941377i	$0.609532\pi$
$660$	0	0
$661$	−8.14359	−0.316749	−0.158375	−	0.987379i	$-0.550625\pi$
$661$	−8.14359	−0.316749	−0.158375	+	0.987379i	$0.550625\pi$
$662$	0	0
$663$	−32.7846	−1.27325
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−42.9282	−1.66219
$668$	0	0
$669$	15.8564	0.613044
$670$	0	0
$671$	17.8564	0.689339
$672$	0	0
$673$	−12.5359	−0.483223	−0.241612	−	0.970373i	$-0.577676\pi$
$673$	−12.5359	−0.483223	−0.241612	+	0.970373i	$0.577676\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−17.6077	−0.676719	−0.338359	−	0.941017i	$-0.609872\pi$
$677$	−17.6077	−0.676719	−0.338359	+	0.941017i	$0.609872\pi$
$678$	0	0
$679$	−10.5359	−0.404331
$680$	0	0
$681$	27.4641	1.05243
$682$	0	0
$683$	−16.9808	−0.649751	−0.324875	−	0.945757i	$-0.605322\pi$
$683$	−16.9808	−0.649751	−0.324875	+	0.945757i	$0.605322\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.9282	0.416937
$688$	0	0
$689$	39.7128	1.51294
$690$	0	0
$691$	18.0000	0.684752	0.342376	−	0.939563i	$-0.388768\pi$
$691$	18.0000	0.684752	0.342376	+	0.939563i	$0.388768\pi$
$692$	0	0
$693$	−6.53590	−0.248278
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.07180	0.192108
$698$	0	0
$699$	14.5359	0.549798
$700$	0	0
$701$	−19.0718	−0.720332	−0.360166	−	0.932888i	$-0.617280\pi$
$701$	−19.0718	−0.720332	−0.360166	+	0.932888i	$0.617280\pi$
$702$	0	0
$703$	1.07180	0.0404236
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2.14359	−0.0806181
$708$	0	0
$709$	−12.7846	−0.480136	−0.240068	−	0.970756i	$-0.577170\pi$
$709$	−12.7846	−0.480136	−0.240068	+	0.970756i	$0.577170\pi$
$710$	0	0
$711$	4.78461	0.179437
$712$	0	0
$713$	33.8564	1.26793
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−54.6410	−2.04061
$718$	0	0
$719$	1.85641	0.0692323	0.0346161	−	0.999401i	$-0.488979\pi$
$719$	1.85641	0.0692323	0.0346161	+	0.999401i	$0.488979\pi$
$720$	0	0
$721$	11.4641	0.426945
$722$	0	0
$723$	−44.7846	−1.66556
$724$	0	0
$725$	0	0
$726$	0	0
$727$	24.0526	0.892060	0.446030	−	0.895018i	$-0.352837\pi$
$727$	24.0526	0.892060	0.446030	+	0.895018i	$0.352837\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	18.2487	0.674953
$732$	0	0
$733$	−35.0718	−1.29541	−0.647703	−	0.761893i	$-0.724270\pi$
$733$	−35.0718	−1.29541	−0.647703	+	0.761893i	$0.724270\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.4641	−0.790640
$738$	0	0
$739$	29.3205	1.07857	0.539286	−	0.842123i	$-0.318694\pi$
$739$	29.3205	1.07857	0.539286	+	0.842123i	$0.318694\pi$
$740$	0	0
$741$	5.07180	0.186317
$742$	0	0
$743$	−10.9808	−0.402845	−0.201423	−	0.979504i	$-0.564556\pi$
$743$	−10.9808	−0.402845	−0.201423	+	0.979504i	$0.564556\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−5.66025	−0.207098
$748$	0	0
$749$	−2.00000	−0.0730784
$750$	0	0
$751$	−26.2487	−0.957829	−0.478915	−	0.877862i	$-0.658970\pi$
$751$	−26.2487	−0.957829	−0.478915	+	0.877862i	$0.658970\pi$
$752$	0	0
$753$	68.1051	2.48189
$754$	0	0
$755$	0	0
$756$	0	0
$757$	19.0718	0.693176	0.346588	−	0.938017i	$-0.387340\pi$
$757$	19.0718	0.693176	0.346588	+	0.938017i	$0.387340\pi$
$758$	0	0
$759$	−33.8564	−1.22891
$760$	0	0
$761$	5.71281	0.207089	0.103545	−	0.994625i	$-0.466982\pi$
$761$	5.71281	0.207089	0.103545	+	0.994625i	$0.466982\pi$
$762$	0	0
$763$	12.3923	0.448632
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.8564	−0.933621
$768$	0	0
$769$	12.9282	0.466203	0.233101	−	0.972452i	$-0.425113\pi$
$769$	12.9282	0.466203	0.233101	+	0.972452i	$0.425113\pi$
$770$	0	0
$771$	−5.46410	−0.196785
$772$	0	0
$773$	22.3923	0.805395	0.402698	−	0.915333i	$-0.368073\pi$
$773$	22.3923	0.805395	0.402698	+	0.915333i	$0.368073\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	−0.784610	−0.0281116
$780$	0	0
$781$	10.9282	0.391042
$782$	0	0
$783$	27.7128	0.990375
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16.5885	−0.591315	−0.295657	−	0.955294i	$-0.595539\pi$
$787$	−16.5885	−0.591315	−0.295657	+	0.955294i	$0.595539\pi$
$788$	0	0
$789$	−31.8564	−1.13412
$790$	0	0
$791$	−9.46410	−0.336505
$792$	0	0
$793$	−30.9282	−1.09829
$794$	0	0
$795$	0	0
$796$	0	0
$797$	50.1051	1.77481	0.887407	−	0.460986i	$-0.152504\pi$
$797$	50.1051	1.77481	0.887407	+	0.460986i	$0.152504\pi$
$798$	0	0
$799$	−11.3205	−0.400491
$800$	0	0
$801$	−39.8564	−1.40826
$802$	0	0
$803$	14.9282	0.526805
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.3923	−0.858650
$808$	0	0
$809$	−23.8564	−0.838747	−0.419373	−	0.907814i	$-0.637750\pi$
$809$	−23.8564	−0.838747	−0.419373	+	0.907814i	$0.637750\pi$
$810$	0	0
$811$	28.9282	1.01581	0.507903	−	0.861414i	$-0.330421\pi$
$811$	28.9282	1.01581	0.507903	+	0.861414i	$0.330421\pi$
$812$	0	0
$813$	−52.7846	−1.85124
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.82309	−0.0987673
$818$	0	0
$819$	11.3205	0.395571
$820$	0	0
$821$	34.7846	1.21399	0.606996	−	0.794705i	$-0.292375\pi$
$821$	34.7846	1.21399	0.606996	+	0.794705i	$0.292375\pi$
$822$	0	0
$823$	9.12436	0.318055	0.159028	−	0.987274i	$-0.449164\pi$
$823$	9.12436	0.318055	0.159028	+	0.987274i	$0.449164\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	23.1244	0.804113	0.402056	−	0.915615i	$-0.368296\pi$
$827$	23.1244	0.804113	0.402056	+	0.915615i	$0.368296\pi$
$828$	0	0
$829$	−28.9282	−1.00472	−0.502359	−	0.864659i	$-0.667534\pi$
$829$	−28.9282	−1.00472	−0.502359	+	0.864659i	$0.667534\pi$
$830$	0	0
$831$	5.46410	0.189548
$832$	0	0
$833$	22.3923	0.775847
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−21.8564	−0.755468
$838$	0	0
$839$	24.7846	0.855660	0.427830	−	0.903859i	$-0.359278\pi$
$839$	24.7846	0.855660	0.427830	+	0.903859i	$0.359278\pi$
$840$	0	0
$841$	19.0000	0.655172
$842$	0	0
$843$	28.7846	0.991395
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.12436	0.176075
$848$	0	0
$849$	26.3923	0.905782
$850$	0	0
$851$	12.3923	0.424803
$852$	0	0
$853$	21.6077	0.739833	0.369917	−	0.929065i	$-0.379386\pi$
$853$	21.6077	0.739833	0.369917	+	0.929065i	$0.379386\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.8564	−0.678282	−0.339141	−	0.940736i	$-0.610136\pi$
$857$	−19.8564	−0.678282	−0.339141	+	0.940736i	$0.610136\pi$
$858$	0	0
$859$	−28.2487	−0.963834	−0.481917	−	0.876217i	$-0.660059\pi$
$859$	−28.2487	−0.963834	−0.481917	+	0.876217i	$0.660059\pi$
$860$	0	0
$861$	−2.92820	−0.0997929
$862$	0	0
$863$	47.6603	1.62237	0.811187	−	0.584787i	$-0.198822\pi$
$863$	47.6603	1.62237	0.811187	+	0.584787i	$0.198822\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.6603	0.463927
$868$	0	0
$869$	2.14359	0.0727164
$870$	0	0
$871$	37.1769	1.25969
$872$	0	0
$873$	64.2487	2.17449
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.71281	−0.0578376	−0.0289188	−	0.999582i	$-0.509206\pi$
$877$	−1.71281	−0.0578376	−0.0289188	+	0.999582i	$0.509206\pi$
$878$	0	0
$879$	−43.3205	−1.46116
$880$	0	0
$881$	9.46410	0.318854	0.159427	−	0.987210i	$-0.449035\pi$
$881$	9.46410	0.318854	0.159427	+	0.987210i	$0.449035\pi$
$882$	0	0
$883$	27.9090	0.939211	0.469606	−	0.882876i	$-0.344396\pi$
$883$	27.9090	0.939211	0.469606	+	0.882876i	$0.344396\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.9090	0.467017	0.233509	−	0.972355i	$-0.424979\pi$
$887$	13.9090	0.467017	0.233509	+	0.972355i	$0.424979\pi$
$888$	0	0
$889$	−12.2487	−0.410809
$890$	0	0
$891$	−4.92820	−0.165101
$892$	0	0
$893$	1.75129	0.0586046
$894$	0	0
$895$	0	0
$896$	0	0
$897$	58.6410	1.95797
$898$	0	0
$899$	−37.8564	−1.26258
$900$	0	0
$901$	39.7128	1.32303
$902$	0	0
$903$	−10.5359	−0.350613
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.87564	0.161893	0.0809466	−	0.996718i	$-0.474206\pi$
$907$	4.87564	0.161893	0.0809466	+	0.996718i	$0.474206\pi$
$908$	0	0
$909$	13.0718	0.433564
$910$	0	0
$911$	49.1769	1.62930	0.814652	−	0.579950i	$-0.196928\pi$
$911$	49.1769	1.62930	0.814652	+	0.579950i	$0.196928\pi$
$912$	0	0
$913$	−2.53590	−0.0839260
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.5359	0.480018
$918$	0	0
$919$	−38.9282	−1.28412	−0.642061	−	0.766653i	$-0.721921\pi$
$919$	−38.9282	−1.28412	−0.642061	+	0.766653i	$0.721921\pi$
$920$	0	0
$921$	−68.2487	−2.24887
$922$	0	0
$923$	−18.9282	−0.623029
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−69.9090	−2.29611
$928$	0	0
$929$	−17.4641	−0.572979	−0.286489	−	0.958083i	$-0.592488\pi$
$929$	−17.4641	−0.572979	−0.286489	+	0.958083i	$0.592488\pi$
$930$	0	0
$931$	−3.46410	−0.113531
$932$	0	0
$933$	−85.5692	−2.80141
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.24871	0.138799	0.0693997	−	0.997589i	$-0.477892\pi$
$937$	4.24871	0.138799	0.0693997	+	0.997589i	$0.477892\pi$
$938$	0	0
$939$	11.3205	0.369431
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	0	0
$943$	−9.07180	−0.295418
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.12436	0.101528	0.0507640	−	0.998711i	$-0.483834\pi$
$947$	3.12436	0.101528	0.0507640	+	0.998711i	$0.483834\pi$
$948$	0	0
$949$	−25.8564	−0.839334
$950$	0	0
$951$	−23.3205	−0.756219
$952$	0	0
$953$	−17.2154	−0.557661	−0.278831	−	0.960340i	$-0.589947\pi$
$953$	−17.2154	−0.557661	−0.278831	+	0.960340i	$0.589947\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	37.8564	1.22372
$958$	0	0
$959$	−3.60770	−0.116499
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	12.1962	0.393016
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−16.3397	−0.525451	−0.262725	−	0.964871i	$-0.584621\pi$
$967$	−16.3397	−0.525451	−0.262725	+	0.964871i	$0.584621\pi$
$968$	0	0
$969$	5.07180	0.162930
$970$	0	0
$971$	−36.9282	−1.18508	−0.592541	−	0.805540i	$-0.701875\pi$
$971$	−36.9282	−1.18508	−0.592541	+	0.805540i	$0.701875\pi$
$972$	0	0
$973$	−0.392305	−0.0125767
$974$	0	0
$975$	0	0
$976$	0	0
$977$	24.5359	0.784973	0.392486	−	0.919758i	$-0.371615\pi$
$977$	24.5359	0.784973	0.392486	+	0.919758i	$0.371615\pi$
$978$	0	0
$979$	−17.8564	−0.570693
$980$	0	0
$981$	−75.5692	−2.41274
$982$	0	0
$983$	48.7321	1.55431	0.777156	−	0.629309i	$-0.216662\pi$
$983$	48.7321	1.55431	0.777156	+	0.629309i	$0.216662\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.53590	0.208040
$988$	0	0
$989$	−32.6410	−1.03792
$990$	0	0
$991$	−41.4641	−1.31715	−0.658575	−	0.752515i	$-0.728841\pi$
$991$	−41.4641	−1.31715	−0.658575	+	0.752515i	$0.728841\pi$
$992$	0	0
$993$	−38.2487	−1.21379
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.1769	0.353976	0.176988	−	0.984213i	$-0.443365\pi$
$997$	11.1769	0.353976	0.176988	+	0.984213i	$0.443365\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6400.2.a.be.1.1		2
4.3	odd	2		6400.2.a.ce.1.2		2
5.4	even	2		1280.2.a.n.1.2		2
8.3	odd	2		6400.2.a.z.1.1		2
8.5	even	2		6400.2.a.cj.1.2		2
16.3	odd	4		200.2.d.f.101.3		4
16.5	even	4		800.2.d.e.401.4		4
16.11	odd	4		200.2.d.f.101.4		4
16.13	even	4		800.2.d.e.401.1		4
20.19	odd	2		1280.2.a.a.1.1		2
40.19	odd	2		1280.2.a.o.1.2		2
40.29	even	2		1280.2.a.d.1.1		2
48.5	odd	4		7200.2.k.j.3601.3		4
48.11	even	4		1800.2.k.j.901.1		4
48.29	odd	4		7200.2.k.j.3601.4		4
48.35	even	4		1800.2.k.j.901.2		4
80.3	even	4		200.2.f.e.149.1		4
80.13	odd	4		800.2.f.e.49.3		4
80.19	odd	4		40.2.d.a.21.2	yes	4
80.27	even	4		200.2.f.e.149.2		4
80.29	even	4		160.2.d.a.81.4		4
80.37	odd	4		800.2.f.e.49.4		4
80.43	even	4		200.2.f.c.149.3		4
80.53	odd	4		800.2.f.c.49.1		4
80.59	odd	4		40.2.d.a.21.1	✓	4
80.67	even	4		200.2.f.c.149.4		4
80.69	even	4		160.2.d.a.81.1		4
80.77	odd	4		800.2.f.c.49.2		4
240.29	odd	4		1440.2.k.e.721.1		4
240.53	even	4		7200.2.d.o.2449.2		4
240.59	even	4		360.2.k.e.181.4		4
240.77	even	4		7200.2.d.o.2449.3		4
240.83	odd	4		1800.2.d.l.1549.4		4
240.107	odd	4		1800.2.d.l.1549.3		4
240.149	odd	4		1440.2.k.e.721.3		4
240.173	even	4		7200.2.d.n.2449.2		4
240.179	even	4		360.2.k.e.181.3		4
240.197	even	4		7200.2.d.n.2449.3		4
240.203	odd	4		1800.2.d.p.1549.2		4
240.227	odd	4		1800.2.d.p.1549.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.d.a.21.1	✓	4	80.59	odd	4
40.2.d.a.21.2	yes	4	80.19	odd	4
160.2.d.a.81.1		4	80.69	even	4
160.2.d.a.81.4		4	80.29	even	4
200.2.d.f.101.3		4	16.3	odd	4
200.2.d.f.101.4		4	16.11	odd	4
200.2.f.c.149.3		4	80.43	even	4
200.2.f.c.149.4		4	80.67	even	4
200.2.f.e.149.1		4	80.3	even	4
200.2.f.e.149.2		4	80.27	even	4
360.2.k.e.181.3		4	240.179	even	4
360.2.k.e.181.4		4	240.59	even	4
800.2.d.e.401.1		4	16.13	even	4
800.2.d.e.401.4		4	16.5	even	4
800.2.f.c.49.1		4	80.53	odd	4
800.2.f.c.49.2		4	80.77	odd	4
800.2.f.e.49.3		4	80.13	odd	4
800.2.f.e.49.4		4	80.37	odd	4
1280.2.a.a.1.1		2	20.19	odd	2
1280.2.a.d.1.1		2	40.29	even	2
1280.2.a.n.1.2		2	5.4	even	2
1280.2.a.o.1.2		2	40.19	odd	2
1440.2.k.e.721.1		4	240.29	odd	4
1440.2.k.e.721.3		4	240.149	odd	4
1800.2.d.l.1549.3		4	240.107	odd	4
1800.2.d.l.1549.4		4	240.83	odd	4
1800.2.d.p.1549.1		4	240.227	odd	4
1800.2.d.p.1549.2		4	240.203	odd	4
1800.2.k.j.901.1		4	48.11	even	4
1800.2.k.j.901.2		4	48.35	even	4
6400.2.a.z.1.1		2	8.3	odd	2
6400.2.a.be.1.1		2	1.1	even	1	trivial
6400.2.a.ce.1.2		2	4.3	odd	2
6400.2.a.cj.1.2		2	8.5	even	2
7200.2.d.n.2449.2		4	240.173	even	4
7200.2.d.n.2449.3		4	240.197	even	4
7200.2.d.o.2449.2		4	240.53	even	4
7200.2.d.o.2449.3		4	240.77	even	4
7200.2.k.j.3601.3		4	48.5	odd	4
7200.2.k.j.3601.4		4	48.29	odd	4

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:	\( N \)	\(=\)	\( 6400 = 2^{8} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6400.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -0.732051 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -0.732051 q^{7} +4.46410 q^{9} +2.00000 q^{11} -3.46410 q^{13} -3.46410 q^{17} +0.535898 q^{19} +2.00000 q^{21} +6.19615 q^{23} -4.00000 q^{27} -6.92820 q^{29} +5.46410 q^{31} -5.46410 q^{33} +2.00000 q^{37} +9.46410 q^{39} -1.46410 q^{41} -5.26795 q^{43} +3.26795 q^{47} -6.46410 q^{49} +9.46410 q^{51} -11.4641 q^{53} -1.46410 q^{57} +7.46410 q^{59} +8.92820 q^{61} -3.26795 q^{63} -10.7321 q^{67} -16.9282 q^{69} +5.46410 q^{71} +7.46410 q^{73} -1.46410 q^{77} +1.07180 q^{79} -2.46410 q^{81} -1.26795 q^{83} +18.9282 q^{87} -8.92820 q^{89} +2.53590 q^{91} -14.9282 q^{93} +14.3923 q^{97} +8.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 8 q^{19} + 4 q^{21} + 2 q^{23} - 8 q^{27} + 4 q^{31} - 4 q^{33} + 4 q^{37} + 12 q^{39} + 4 q^{41} - 14 q^{43} + 10 q^{47} - 6 q^{49} + 12 q^{51} - 16 q^{53} + 4 q^{57} + 8 q^{59} + 4 q^{61} - 10 q^{63} - 18 q^{67} - 20 q^{69} + 4 q^{71} + 8 q^{73} + 4 q^{77} + 16 q^{79} + 2 q^{81} - 6 q^{83} + 24 q^{87} - 4 q^{89} + 12 q^{91} - 16 q^{93} + 8 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 4 * q^11 + 8 * q^19 + 4 * q^21 + 2 * q^23 - 8 * q^27 + 4 * q^31 - 4 * q^33 + 4 * q^37 + 12 * q^39 + 4 * q^41 - 14 * q^43 + 10 * q^47 - 6 * q^49 + 12 * q^51 - 16 * q^53 + 4 * q^57 + 8 * q^59 + 4 * q^61 - 10 * q^63 - 18 * q^67 - 20 * q^69 + 4 * q^71 + 8 * q^73 + 4 * q^77 + 16 * q^79 + 2 * q^81 - 6 * q^83 + 24 * q^87 - 4 * q^89 + 12 * q^91 - 16 * q^93 + 8 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more