LMFDB - Embedded newform 5120.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5120,2,Mod(1,5120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5120, 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("5120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5120 = 2^{10} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5120.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:	$40.8834058349$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} - 8x^{5} + 21x^{4} + 12x^{3} - 10x^{2} - 4x + 1 $ x^8 - 12x^6 - 8x^5 + 21x^4 + 12x^3 - 10x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 80)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.06941$ of defining polynomial
Character	$\chi$	$=$	5120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.28980 q^{3} +1.00000 q^{5} +0.982011 q^{7} +7.82281 q^{9} +O(q^{10})$	$q-3.28980 q^{3} +1.00000 q^{5} +0.982011 q^{7} +7.82281 q^{9} -2.30015 q^{11} -0.976603 q^{13} -3.28980 q^{15} +2.19577 q^{17} -2.72461 q^{19} -3.23062 q^{21} +2.01442 q^{23} +1.00000 q^{25} -15.8661 q^{27} +7.45547 q^{29} -0.435286 q^{31} +7.56703 q^{33} +0.982011 q^{35} -8.19969 q^{37} +3.21283 q^{39} -3.93139 q^{41} -0.717844 q^{43} +7.82281 q^{45} -9.21960 q^{47} -6.03565 q^{49} -7.22365 q^{51} -8.90045 q^{53} -2.30015 q^{55} +8.96345 q^{57} +8.02959 q^{59} +5.09542 q^{61} +7.68209 q^{63} -0.976603 q^{65} +6.41435 q^{67} -6.62705 q^{69} -10.3984 q^{71} +9.24439 q^{73} -3.28980 q^{75} -2.25877 q^{77} +15.4493 q^{79} +28.7280 q^{81} +0.966253 q^{83} +2.19577 q^{85} -24.5271 q^{87} +5.44401 q^{89} -0.959035 q^{91} +1.43201 q^{93} -2.72461 q^{95} +5.54540 q^{97} -17.9936 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9	$ 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59} - 20 q^{63} - 36 q^{67} - 16 q^{69} + 8 q^{71} - 4 q^{75} + 8 q^{79} + 8 q^{81} - 20 q^{83} - 40 q^{91} - 16 q^{95} - 40 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^15 - 16 * q^19 - 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 + 8 * q^45 - 20 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^55 - 16 * q^59 - 20 * q^63 - 36 * q^67 - 16 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 - 16 * q^95 - 40 * 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
$3$	−3.28980	−1.89937	−0.949685	−	0.313207i	$-0.898597\pi$
$3$	−3.28980	−1.89937	−0.949685	+	0.313207i	$0.898597\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.982011	0.371165	0.185583	−	0.982629i	$-0.440583\pi$
$7$	0.982011	0.371165	0.185583	+	0.982629i	$0.440583\pi$
$8$	0	0
$9$	7.82281	2.60760
$10$	0	0
$11$	−2.30015	−0.693520	−0.346760	−	0.937954i	$-0.612718\pi$
$11$	−2.30015	−0.693520	−0.346760	+	0.937954i	$0.612718\pi$
$12$	0	0
$13$	−0.976603	−0.270861	−0.135430	−	0.990787i	$-0.543242\pi$
$13$	−0.976603	−0.270861	−0.135430	+	0.990787i	$0.543242\pi$
$14$	0	0
$15$	−3.28980	−0.849424
$16$	0	0
$17$	2.19577	0.532552	0.266276	−	0.963897i	$-0.414207\pi$
$17$	2.19577	0.532552	0.266276	+	0.963897i	$0.414207\pi$
$18$	0	0
$19$	−2.72461	−0.625069	−0.312535	−	0.949906i	$-0.601178\pi$
$19$	−2.72461	−0.625069	−0.312535	+	0.949906i	$0.601178\pi$
$20$	0	0
$21$	−3.23062	−0.704980
$22$	0	0
$23$	2.01442	0.420035	0.210018	−	0.977698i	$-0.432648\pi$
$23$	2.01442	0.420035	0.210018	+	0.977698i	$0.432648\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−15.8661	−3.05344
$28$	0	0
$29$	7.45547	1.38445	0.692223	−	0.721683i	$-0.256631\pi$
$29$	7.45547	1.38445	0.692223	+	0.721683i	$0.256631\pi$
$30$	0	0
$31$	−0.435286	−0.0781797	−0.0390898	−	0.999236i	$-0.512446\pi$
$31$	−0.435286	−0.0781797	−0.0390898	+	0.999236i	$0.512446\pi$
$32$	0	0
$33$	7.56703	1.31725
$34$	0	0
$35$	0.982011	0.165990
$36$	0	0
$37$	−8.19969	−1.34802	−0.674010	−	0.738722i	$-0.735430\pi$
$37$	−8.19969	−1.34802	−0.674010	+	0.738722i	$0.735430\pi$
$38$	0	0
$39$	3.21283	0.514465
$40$	0	0
$41$	−3.93139	−0.613980	−0.306990	−	0.951713i	$-0.599322\pi$
$41$	−3.93139	−0.613980	−0.306990	+	0.951713i	$0.599322\pi$
$42$	0	0
$43$	−0.717844	−0.109470	−0.0547351	−	0.998501i	$-0.517431\pi$
$43$	−0.717844	−0.109470	−0.0547351	+	0.998501i	$0.517431\pi$
$44$	0	0
$45$	7.82281	1.16616
$46$	0	0
$47$	−9.21960	−1.34482	−0.672409	−	0.740180i	$-0.734740\pi$
$47$	−9.21960	−1.34482	−0.672409	+	0.740180i	$0.734740\pi$
$48$	0	0
$49$	−6.03565	−0.862236
$50$	0	0
$51$	−7.22365	−1.01151
$52$	0	0
$53$	−8.90045	−1.22257	−0.611285	−	0.791410i	$-0.709347\pi$
$53$	−8.90045	−1.22257	−0.611285	+	0.791410i	$0.709347\pi$
$54$	0	0
$55$	−2.30015	−0.310152
$56$	0	0
$57$	8.96345	1.18724
$58$	0	0
$59$	8.02959	1.04536	0.522682	−	0.852528i	$-0.324932\pi$
$59$	8.02959	1.04536	0.522682	+	0.852528i	$0.324932\pi$
$60$	0	0
$61$	5.09542	0.652402	0.326201	−	0.945300i	$-0.394231\pi$
$61$	5.09542	0.652402	0.326201	+	0.945300i	$0.394231\pi$
$62$	0	0
$63$	7.68209	0.967852
$64$	0	0
$65$	−0.976603	−0.121133
$66$	0	0
$67$	6.41435	0.783638	0.391819	−	0.920042i	$-0.371846\pi$
$67$	6.41435	0.783638	0.391819	+	0.920042i	$0.371846\pi$
$68$	0	0
$69$	−6.62705	−0.797803
$70$	0	0
$71$	−10.3984	−1.23407	−0.617033	−	0.786937i	$-0.711665\pi$
$71$	−10.3984	−1.23407	−0.617033	+	0.786937i	$0.711665\pi$
$72$	0	0
$73$	9.24439	1.08197	0.540987	−	0.841031i	$-0.318051\pi$
$73$	9.24439	1.08197	0.540987	+	0.841031i	$0.318051\pi$
$74$	0	0
$75$	−3.28980	−0.379874
$76$	0	0
$77$	−2.25877	−0.257411
$78$	0	0
$79$	15.4493	1.73818	0.869091	−	0.494653i	$-0.164705\pi$
$79$	15.4493	1.73818	0.869091	+	0.494653i	$0.164705\pi$
$80$	0	0
$81$	28.7280	3.19200
$82$	0	0
$83$	0.966253	0.106060	0.0530300	−	0.998593i	$-0.483112\pi$
$83$	0.966253	0.106060	0.0530300	+	0.998593i	$0.483112\pi$
$84$	0	0
$85$	2.19577	0.238165
$86$	0	0
$87$	−24.5271	−2.62958
$88$	0	0
$89$	5.44401	0.577064	0.288532	−	0.957470i	$-0.406833\pi$
$89$	5.44401	0.577064	0.288532	+	0.957470i	$0.406833\pi$
$90$	0	0
$91$	−0.959035	−0.100534
$92$	0	0
$93$	1.43201	0.148492
$94$	0	0
$95$	−2.72461	−0.279540
$96$	0	0
$97$	5.54540	0.563050	0.281525	−	0.959554i	$-0.409160\pi$
$97$	5.54540	0.563050	0.281525	+	0.959554i	$0.409160\pi$
$98$	0	0
$99$	−17.9936	−1.80843
$100$	0	0
$101$	−0.411952	−0.0409907	−0.0204954	−	0.999790i	$-0.506524\pi$
$101$	−0.411952	−0.0409907	−0.0204954	+	0.999790i	$0.506524\pi$
$102$	0	0
$103$	−4.50219	−0.443614	−0.221807	−	0.975091i	$-0.571195\pi$
$103$	−4.50219	−0.443614	−0.221807	+	0.975091i	$0.571195\pi$
$104$	0	0
$105$	−3.23062	−0.315277
$106$	0	0
$107$	−9.19083	−0.888511	−0.444255	−	0.895900i	$-0.646532\pi$
$107$	−9.19083	−0.888511	−0.444255	+	0.895900i	$0.646532\pi$
$108$	0	0
$109$	−3.56311	−0.341284	−0.170642	−	0.985333i	$-0.554584\pi$
$109$	−3.56311	−0.341284	−0.170642	+	0.985333i	$0.554584\pi$
$110$	0	0
$111$	26.9754	2.56039
$112$	0	0
$113$	−5.38101	−0.506203	−0.253102	−	0.967440i	$-0.581451\pi$
$113$	−5.38101	−0.506203	−0.253102	+	0.967440i	$0.581451\pi$
$114$	0	0
$115$	2.01442	0.187846
$116$	0	0
$117$	−7.63978	−0.706298
$118$	0	0
$119$	2.15627	0.197665
$120$	0	0
$121$	−5.70933	−0.519030
$122$	0	0
$123$	12.9335	1.16617
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.86578	−0.431768	−0.215884	−	0.976419i	$-0.569263\pi$
$127$	−4.86578	−0.431768	−0.215884	+	0.976419i	$0.569263\pi$
$128$	0	0
$129$	2.36157	0.207924
$130$	0	0
$131$	11.3219	0.989201	0.494601	−	0.869120i	$-0.335314\pi$
$131$	11.3219	0.989201	0.494601	+	0.869120i	$0.335314\pi$
$132$	0	0
$133$	−2.67560	−0.232004
$134$	0	0
$135$	−15.8661	−1.36554
$136$	0	0
$137$	13.5567	1.15822	0.579112	−	0.815248i	$-0.303399\pi$
$137$	13.5567	1.15822	0.579112	+	0.815248i	$0.303399\pi$
$138$	0	0
$139$	11.6340	0.986779	0.493390	−	0.869808i	$-0.335758\pi$
$139$	11.6340	0.986779	0.493390	+	0.869808i	$0.335758\pi$
$140$	0	0
$141$	30.3307	2.55431
$142$	0	0
$143$	2.24633	0.187847
$144$	0	0
$145$	7.45547	0.619143
$146$	0	0
$147$	19.8561	1.63771
$148$	0	0
$149$	−17.8705	−1.46401	−0.732003	−	0.681301i	$-0.761414\pi$
$149$	−17.8705	−1.46401	−0.732003	+	0.681301i	$0.761414\pi$
$150$	0	0
$151$	15.1562	1.23339	0.616696	−	0.787201i	$-0.288471\pi$
$151$	15.1562	1.23339	0.616696	+	0.787201i	$0.288471\pi$
$152$	0	0
$153$	17.1771	1.38869
$154$	0	0
$155$	−0.435286	−0.0349630
$156$	0	0
$157$	2.48641	0.198437	0.0992186	−	0.995066i	$-0.468366\pi$
$157$	2.48641	0.198437	0.0992186	+	0.995066i	$0.468366\pi$
$158$	0	0
$159$	29.2807	2.32211
$160$	0	0
$161$	1.97818	0.155903
$162$	0	0
$163$	−19.6720	−1.54083	−0.770416	−	0.637542i	$-0.779951\pi$
$163$	−19.6720	−1.54083	−0.770416	+	0.637542i	$0.779951\pi$
$164$	0	0
$165$	7.56703	0.589092
$166$	0	0
$167$	−18.8620	−1.45958	−0.729792	−	0.683669i	$-0.760383\pi$
$167$	−18.8620	−1.45958	−0.729792	+	0.683669i	$0.760383\pi$
$168$	0	0
$169$	−12.0462	−0.926634
$170$	0	0
$171$	−21.3142	−1.62993
$172$	0	0
$173$	22.7298	1.72812	0.864059	−	0.503391i	$-0.167914\pi$
$173$	22.7298	1.72812	0.864059	+	0.503391i	$0.167914\pi$
$174$	0	0
$175$	0.982011	0.0742331
$176$	0	0
$177$	−26.4158	−1.98553
$178$	0	0
$179$	−23.2413	−1.73714	−0.868568	−	0.495570i	$-0.834959\pi$
$179$	−23.2413	−1.73714	−0.868568	+	0.495570i	$0.834959\pi$
$180$	0	0
$181$	21.8552	1.62448	0.812241	−	0.583322i	$-0.198247\pi$
$181$	21.8552	1.62448	0.812241	+	0.583322i	$0.198247\pi$
$182$	0	0
$183$	−16.7629	−1.23915
$184$	0	0
$185$	−8.19969	−0.602853
$186$	0	0
$187$	−5.05059	−0.369335
$188$	0	0
$189$	−15.5807	−1.13333
$190$	0	0
$191$	14.7872	1.06997	0.534983	−	0.844863i	$-0.320318\pi$
$191$	14.7872	1.06997	0.534983	+	0.844863i	$0.320318\pi$
$192$	0	0
$193$	−11.2912	−0.812758	−0.406379	−	0.913705i	$-0.633209\pi$
$193$	−11.2912	−0.812758	−0.406379	+	0.913705i	$0.633209\pi$
$194$	0	0
$195$	3.21283	0.230076
$196$	0	0
$197$	−15.0122	−1.06957	−0.534786	−	0.844987i	$-0.679608\pi$
$197$	−15.0122	−1.06957	−0.534786	+	0.844987i	$0.679608\pi$
$198$	0	0
$199$	4.68789	0.332316	0.166158	−	0.986099i	$-0.446864\pi$
$199$	4.68789	0.332316	0.166158	+	0.986099i	$0.446864\pi$
$200$	0	0
$201$	−21.1020	−1.48842
$202$	0	0
$203$	7.32136	0.513859
$204$	0	0
$205$	−3.93139	−0.274580
$206$	0	0
$207$	15.7584	1.09529
$208$	0	0
$209$	6.26701	0.433498
$210$	0	0
$211$	3.72242	0.256262	0.128131	−	0.991757i	$-0.459102\pi$
$211$	3.72242	0.256262	0.128131	+	0.991757i	$0.459102\pi$
$212$	0	0
$213$	34.2088	2.34395
$214$	0	0
$215$	−0.717844	−0.0489565
$216$	0	0
$217$	−0.427456	−0.0290176
$218$	0	0
$219$	−30.4123	−2.05507
$220$	0	0
$221$	−2.14439	−0.144248
$222$	0	0
$223$	−3.45644	−0.231461	−0.115730	−	0.993281i	$-0.536921\pi$
$223$	−3.45644	−0.231461	−0.115730	+	0.993281i	$0.536921\pi$
$224$	0	0
$225$	7.82281	0.521521
$226$	0	0
$227$	−6.71115	−0.445434	−0.222717	−	0.974883i	$-0.571493\pi$
$227$	−6.71115	−0.445434	−0.222717	+	0.974883i	$0.571493\pi$
$228$	0	0
$229$	−18.8905	−1.24832	−0.624161	−	0.781296i	$-0.714559\pi$
$229$	−18.8905	−1.24832	−0.624161	+	0.781296i	$0.714559\pi$
$230$	0	0
$231$	7.43091	0.488918
$232$	0	0
$233$	4.82691	0.316222	0.158111	−	0.987421i	$-0.449460\pi$
$233$	4.82691	0.316222	0.158111	+	0.987421i	$0.449460\pi$
$234$	0	0
$235$	−9.21960	−0.601421
$236$	0	0
$237$	−50.8252	−3.30145
$238$	0	0
$239$	−8.82497	−0.570840	−0.285420	−	0.958403i	$-0.592133\pi$
$239$	−8.82497	−0.570840	−0.285420	+	0.958403i	$0.592133\pi$
$240$	0	0
$241$	3.74147	0.241009	0.120504	−	0.992713i	$-0.461549\pi$
$241$	3.74147	0.241009	0.120504	+	0.992713i	$0.461549\pi$
$242$	0	0
$243$	−46.9111	−3.00935
$244$	0	0
$245$	−6.03565	−0.385604
$246$	0	0
$247$	2.66087	0.169307
$248$	0	0
$249$	−3.17878	−0.201447
$250$	0	0
$251$	8.47569	0.534981	0.267490	−	0.963561i	$-0.413806\pi$
$251$	8.47569	0.534981	0.267490	+	0.963561i	$0.413806\pi$
$252$	0	0
$253$	−4.63346	−0.291303
$254$	0	0
$255$	−7.22365	−0.452362
$256$	0	0
$257$	−14.7662	−0.921091	−0.460545	−	0.887636i	$-0.652346\pi$
$257$	−14.7662	−0.921091	−0.460545	+	0.887636i	$0.652346\pi$
$258$	0	0
$259$	−8.05218	−0.500338
$260$	0	0
$261$	58.3228	3.61009
$262$	0	0
$263$	6.79486	0.418989	0.209494	−	0.977810i	$-0.432818\pi$
$263$	6.79486	0.418989	0.209494	+	0.977810i	$0.432818\pi$
$264$	0	0
$265$	−8.90045	−0.546750
$266$	0	0
$267$	−17.9097	−1.09606
$268$	0	0
$269$	−8.54170	−0.520797	−0.260398	−	0.965501i	$-0.583854\pi$
$269$	−8.54170	−0.520797	−0.260398	+	0.965501i	$0.583854\pi$
$270$	0	0
$271$	−24.6221	−1.49568	−0.747842	−	0.663877i	$-0.768910\pi$
$271$	−24.6221	−1.49568	−0.747842	+	0.663877i	$0.768910\pi$
$272$	0	0
$273$	3.15504	0.190952
$274$	0	0
$275$	−2.30015	−0.138704
$276$	0	0
$277$	−14.1141	−0.848034	−0.424017	−	0.905654i	$-0.639380\pi$
$277$	−14.1141	−0.848034	−0.424017	+	0.905654i	$0.639380\pi$
$278$	0	0
$279$	−3.40516	−0.203862
$280$	0	0
$281$	−14.4611	−0.862675	−0.431337	−	0.902191i	$-0.641958\pi$
$281$	−14.4611	−0.862675	−0.431337	+	0.902191i	$0.641958\pi$
$282$	0	0
$283$	−28.3950	−1.68791	−0.843953	−	0.536416i	$-0.819778\pi$
$283$	−28.3950	−1.68791	−0.843953	+	0.536416i	$0.819778\pi$
$284$	0	0
$285$	8.96345	0.530949
$286$	0	0
$287$	−3.86067	−0.227888
$288$	0	0
$289$	−12.1786	−0.716388
$290$	0	0
$291$	−18.2433	−1.06944
$292$	0	0
$293$	21.7842	1.27265	0.636325	−	0.771421i	$-0.280454\pi$
$293$	21.7842	1.27265	0.636325	+	0.771421i	$0.280454\pi$
$294$	0	0
$295$	8.02959	0.467501
$296$	0	0
$297$	36.4944	2.11762
$298$	0	0
$299$	−1.96729	−0.113771
$300$	0	0
$301$	−0.704930	−0.0406315
$302$	0	0
$303$	1.35524	0.0778566
$304$	0	0
$305$	5.09542	0.291763
$306$	0	0
$307$	−12.9033	−0.736428	−0.368214	−	0.929741i	$-0.620031\pi$
$307$	−12.9033	−0.736428	−0.368214	+	0.929741i	$0.620031\pi$
$308$	0	0
$309$	14.8113	0.842587
$310$	0	0
$311$	−0.642911	−0.0364561	−0.0182281	−	0.999834i	$-0.505802\pi$
$311$	−0.642911	−0.0364561	−0.0182281	+	0.999834i	$0.505802\pi$
$312$	0	0
$313$	−21.3775	−1.20833	−0.604164	−	0.796860i	$-0.706493\pi$
$313$	−21.3775	−1.20833	−0.604164	+	0.796860i	$0.706493\pi$
$314$	0	0
$315$	7.68209	0.432837
$316$	0	0
$317$	−12.2499	−0.688024	−0.344012	−	0.938965i	$-0.611786\pi$
$317$	−12.2499	−0.688024	−0.344012	+	0.938965i	$0.611786\pi$
$318$	0	0
$319$	−17.1487	−0.960141
$320$	0	0
$321$	30.2360	1.68761
$322$	0	0
$323$	−5.98262	−0.332882
$324$	0	0
$325$	−0.976603	−0.0541722
$326$	0	0
$327$	11.7219	0.648224
$328$	0	0
$329$	−9.05375	−0.499150
$330$	0	0
$331$	−11.9351	−0.656014	−0.328007	−	0.944675i	$-0.606377\pi$
$331$	−11.9351	−0.656014	−0.328007	+	0.944675i	$0.606377\pi$
$332$	0	0
$333$	−64.1446	−3.51510
$334$	0	0
$335$	6.41435	0.350454
$336$	0	0
$337$	−30.7047	−1.67259	−0.836295	−	0.548280i	$-0.815283\pi$
$337$	−30.7047	−1.67259	−0.836295	+	0.548280i	$0.815283\pi$
$338$	0	0
$339$	17.7025	0.961467
$340$	0	0
$341$	1.00122	0.0542191
$342$	0	0
$343$	−12.8012	−0.691197
$344$	0	0
$345$	−6.62705	−0.356788
$346$	0	0
$347$	−19.2924	−1.03567	−0.517834	−	0.855481i	$-0.673262\pi$
$347$	−19.2924	−1.03567	−0.517834	+	0.855481i	$0.673262\pi$
$348$	0	0
$349$	−14.1042	−0.754983	−0.377492	−	0.926013i	$-0.623213\pi$
$349$	−14.1042	−0.754983	−0.377492	+	0.926013i	$0.623213\pi$
$350$	0	0
$351$	15.4949	0.827056
$352$	0	0
$353$	−26.7843	−1.42559	−0.712793	−	0.701374i	$-0.752570\pi$
$353$	−26.7843	−1.42559	−0.712793	+	0.701374i	$0.752570\pi$
$354$	0	0
$355$	−10.3984	−0.551891
$356$	0	0
$357$	−7.09370	−0.375439
$358$	0	0
$359$	19.1190	1.00906	0.504532	−	0.863393i	$-0.331665\pi$
$359$	19.1190	1.00906	0.504532	+	0.863393i	$0.331665\pi$
$360$	0	0
$361$	−11.5765	−0.609288
$362$	0	0
$363$	18.7826	0.985830
$364$	0	0
$365$	9.24439	0.483874
$366$	0	0
$367$	4.24385	0.221527	0.110764	−	0.993847i	$-0.464670\pi$
$367$	4.24385	0.221527	0.110764	+	0.993847i	$0.464670\pi$
$368$	0	0
$369$	−30.7545	−1.60102
$370$	0	0
$371$	−8.74034	−0.453776
$372$	0	0
$373$	−33.8724	−1.75385	−0.876923	−	0.480631i	$-0.840408\pi$
$373$	−33.8724	−1.75385	−0.876923	+	0.480631i	$0.840408\pi$
$374$	0	0
$375$	−3.28980	−0.169885
$376$	0	0
$377$	−7.28104	−0.374992
$378$	0	0
$379$	10.5456	0.541690	0.270845	−	0.962623i	$-0.412697\pi$
$379$	10.5456	0.541690	0.270845	+	0.962623i	$0.412697\pi$
$380$	0	0
$381$	16.0075	0.820088
$382$	0	0
$383$	−5.19667	−0.265538	−0.132769	−	0.991147i	$-0.542387\pi$
$383$	−5.19667	−0.265538	−0.132769	+	0.991147i	$0.542387\pi$
$384$	0	0
$385$	−2.25877	−0.115117
$386$	0	0
$387$	−5.61556	−0.285455
$388$	0	0
$389$	14.6861	0.744615	0.372307	−	0.928109i	$-0.378567\pi$
$389$	14.6861	0.744615	0.372307	+	0.928109i	$0.378567\pi$
$390$	0	0
$391$	4.42320	0.223691
$392$	0	0
$393$	−37.2469	−1.87886
$394$	0	0
$395$	15.4493	0.777338
$396$	0	0
$397$	14.0446	0.704879	0.352439	−	0.935835i	$-0.385352\pi$
$397$	14.0446	0.704879	0.352439	+	0.935835i	$0.385352\pi$
$398$	0	0
$399$	8.80221	0.440662
$400$	0	0
$401$	−9.51392	−0.475102	−0.237551	−	0.971375i	$-0.576345\pi$
$401$	−9.51392	−0.475102	−0.237551	+	0.971375i	$0.576345\pi$
$402$	0	0
$403$	0.425101	0.0211758
$404$	0	0
$405$	28.7280	1.42750
$406$	0	0
$407$	18.8605	0.934879
$408$	0	0
$409$	4.81799	0.238234	0.119117	−	0.992880i	$-0.461994\pi$
$409$	4.81799	0.238234	0.119117	+	0.992880i	$0.461994\pi$
$410$	0	0
$411$	−44.5988	−2.19990
$412$	0	0
$413$	7.88514	0.388003
$414$	0	0
$415$	0.966253	0.0474315
$416$	0	0
$417$	−38.2734	−1.87426
$418$	0	0
$419$	30.3179	1.48113	0.740564	−	0.671986i	$-0.234559\pi$
$419$	30.3179	1.48113	0.740564	+	0.671986i	$0.234559\pi$
$420$	0	0
$421$	−6.79027	−0.330938	−0.165469	−	0.986215i	$-0.552914\pi$
$421$	−6.79027	−0.330938	−0.165469	+	0.986215i	$0.552914\pi$
$422$	0	0
$423$	−72.1232	−3.50675
$424$	0	0
$425$	2.19577	0.106510
$426$	0	0
$427$	5.00376	0.242149
$428$	0	0
$429$	−7.38998	−0.356792
$430$	0	0
$431$	13.2369	0.637597	0.318799	−	0.947822i	$-0.396721\pi$
$431$	13.2369	0.637597	0.318799	+	0.947822i	$0.396721\pi$
$432$	0	0
$433$	1.50709	0.0724259	0.0362129	−	0.999344i	$-0.488471\pi$
$433$	1.50709	0.0724259	0.0362129	+	0.999344i	$0.488471\pi$
$434$	0	0
$435$	−24.5271	−1.17598
$436$	0	0
$437$	−5.48852	−0.262551
$438$	0	0
$439$	−10.3092	−0.492033	−0.246016	−	0.969266i	$-0.579122\pi$
$439$	−10.3092	−0.492033	−0.246016	+	0.969266i	$0.579122\pi$
$440$	0	0
$441$	−47.2158	−2.24837
$442$	0	0
$443$	20.1739	0.958491	0.479245	−	0.877681i	$-0.340910\pi$
$443$	20.1739	0.958491	0.479245	+	0.877681i	$0.340910\pi$
$444$	0	0
$445$	5.44401	0.258071
$446$	0	0
$447$	58.7904	2.78069
$448$	0	0
$449$	−19.5711	−0.923618	−0.461809	−	0.886979i	$-0.652799\pi$
$449$	−19.5711	−0.923618	−0.461809	+	0.886979i	$0.652799\pi$
$450$	0	0
$451$	9.04276	0.425807
$452$	0	0
$453$	−49.8608	−2.34267
$454$	0	0
$455$	−0.959035	−0.0449602
$456$	0	0
$457$	−39.0185	−1.82521	−0.912604	−	0.408845i	$-0.865932\pi$
$457$	−39.0185	−1.82521	−0.912604	+	0.408845i	$0.865932\pi$
$458$	0	0
$459$	−34.8383	−1.62611
$460$	0	0
$461$	27.8516	1.29718	0.648590	−	0.761138i	$-0.275359\pi$
$461$	27.8516	1.29718	0.648590	+	0.761138i	$0.275359\pi$
$462$	0	0
$463$	−14.9979	−0.697009	−0.348505	−	0.937307i	$-0.613310\pi$
$463$	−14.9979	−0.697009	−0.348505	+	0.937307i	$0.613310\pi$
$464$	0	0
$465$	1.43201	0.0664077
$466$	0	0
$467$	−6.91367	−0.319926	−0.159963	−	0.987123i	$-0.551138\pi$
$467$	−6.91367	−0.319926	−0.159963	+	0.987123i	$0.551138\pi$
$468$	0	0
$469$	6.29897	0.290859
$470$	0	0
$471$	−8.17980	−0.376905
$472$	0	0
$473$	1.65114	0.0759197
$474$	0	0
$475$	−2.72461	−0.125014
$476$	0	0
$477$	−69.6266	−3.18798
$478$	0	0
$479$	27.3381	1.24911	0.624555	−	0.780981i	$-0.285280\pi$
$479$	27.3381	1.24911	0.624555	+	0.780981i	$0.285280\pi$
$480$	0	0
$481$	8.00784	0.365126
$482$	0	0
$483$	−6.50783	−0.296117
$484$	0	0
$485$	5.54540	0.251803
$486$	0	0
$487$	35.4769	1.60761	0.803806	−	0.594892i	$-0.202805\pi$
$487$	35.4769	1.60761	0.803806	+	0.594892i	$0.202805\pi$
$488$	0	0
$489$	64.7171	2.92661
$490$	0	0
$491$	5.02914	0.226962	0.113481	−	0.993540i	$-0.463800\pi$
$491$	5.02914	0.226962	0.113481	+	0.993540i	$0.463800\pi$
$492$	0	0
$493$	16.3705	0.737290
$494$	0	0
$495$	−17.9936	−0.808753
$496$	0	0
$497$	−10.2114	−0.458042
$498$	0	0
$499$	24.9639	1.11754	0.558768	−	0.829324i	$-0.311274\pi$
$499$	24.9639	1.11754	0.558768	+	0.829324i	$0.311274\pi$
$500$	0	0
$501$	62.0523	2.77229
$502$	0	0
$503$	−31.8567	−1.42042	−0.710210	−	0.703990i	$-0.751400\pi$
$503$	−31.8567	−1.42042	−0.710210	+	0.703990i	$0.751400\pi$
$504$	0	0
$505$	−0.411952	−0.0183316
$506$	0	0
$507$	39.6298	1.76002
$508$	0	0
$509$	7.93451	0.351691	0.175845	−	0.984418i	$-0.443734\pi$
$509$	7.93451	0.351691	0.175845	+	0.984418i	$0.443734\pi$
$510$	0	0
$511$	9.07810	0.401591
$512$	0	0
$513$	43.2291	1.90861
$514$	0	0
$515$	−4.50219	−0.198390
$516$	0	0
$517$	21.2064	0.932658
$518$	0	0
$519$	−74.7767	−3.28233
$520$	0	0
$521$	−33.1977	−1.45442	−0.727208	−	0.686417i	$-0.759182\pi$
$521$	−33.1977	−1.45442	−0.727208	+	0.686417i	$0.759182\pi$
$522$	0	0
$523$	−3.68695	−0.161219	−0.0806095	−	0.996746i	$-0.525687\pi$
$523$	−3.68695	−0.161219	−0.0806095	+	0.996746i	$0.525687\pi$
$524$	0	0
$525$	−3.23062	−0.140996
$526$	0	0
$527$	−0.955787	−0.0416347
$528$	0	0
$529$	−18.9421	−0.823570
$530$	0	0
$531$	62.8140	2.72589
$532$	0	0
$533$	3.83940	0.166303
$534$	0	0
$535$	−9.19083	−0.397354
$536$	0	0
$537$	76.4593	3.29946
$538$	0	0
$539$	13.8829	0.597978
$540$	0	0
$541$	32.0799	1.37922	0.689611	−	0.724180i	$-0.257782\pi$
$541$	32.0799	1.37922	0.689611	+	0.724180i	$0.257782\pi$
$542$	0	0
$543$	−71.8992	−3.08549
$544$	0	0
$545$	−3.56311	−0.152627
$546$	0	0
$547$	−4.27495	−0.182784	−0.0913918	−	0.995815i	$-0.529132\pi$
$547$	−4.27495	−0.182784	−0.0913918	+	0.995815i	$0.529132\pi$
$548$	0	0
$549$	39.8605	1.70121
$550$	0	0
$551$	−20.3133	−0.865375
$552$	0	0
$553$	15.1714	0.645153
$554$	0	0
$555$	26.9754	1.14504
$556$	0	0
$557$	13.1168	0.555775	0.277888	−	0.960614i	$-0.410366\pi$
$557$	13.1168	0.555775	0.277888	+	0.960614i	$0.410366\pi$
$558$	0	0
$559$	0.701048	0.0296512
$560$	0	0
$561$	16.6154	0.701504
$562$	0	0
$563$	28.7121	1.21007	0.605035	−	0.796199i	$-0.293159\pi$
$563$	28.7121	1.21007	0.605035	+	0.796199i	$0.293159\pi$
$564$	0	0
$565$	−5.38101	−0.226381
$566$	0	0
$567$	28.2112	1.18476
$568$	0	0
$569$	−14.3362	−0.601005	−0.300503	−	0.953781i	$-0.597154\pi$
$569$	−14.3362	−0.601005	−0.300503	+	0.953781i	$0.597154\pi$
$570$	0	0
$571$	−12.0826	−0.505641	−0.252820	−	0.967513i	$-0.581358\pi$
$571$	−12.0826	−0.505641	−0.252820	+	0.967513i	$0.581358\pi$
$572$	0	0
$573$	−48.6471	−2.03226
$574$	0	0
$575$	2.01442	0.0840071
$576$	0	0
$577$	−8.68179	−0.361428	−0.180714	−	0.983536i	$-0.557841\pi$
$577$	−8.68179	−0.361428	−0.180714	+	0.983536i	$0.557841\pi$
$578$	0	0
$579$	37.1458	1.54373
$580$	0	0
$581$	0.948871	0.0393658
$582$	0	0
$583$	20.4723	0.847877
$584$	0	0
$585$	−7.63978	−0.315866
$586$	0	0
$587$	30.9772	1.27856	0.639282	−	0.768972i	$-0.279231\pi$
$587$	30.9772	1.27856	0.639282	+	0.768972i	$0.279231\pi$
$588$	0	0
$589$	1.18599	0.0488677
$590$	0	0
$591$	49.3871	2.03151
$592$	0	0
$593$	17.5142	0.719222	0.359611	−	0.933102i	$-0.382909\pi$
$593$	17.5142	0.719222	0.359611	+	0.933102i	$0.382909\pi$
$594$	0	0
$595$	2.15627	0.0883984
$596$	0	0
$597$	−15.4222	−0.631190
$598$	0	0
$599$	−19.0276	−0.777447	−0.388724	−	0.921354i	$-0.627084\pi$
$599$	−19.0276	−0.777447	−0.388724	+	0.921354i	$0.627084\pi$
$600$	0	0
$601$	5.52545	0.225388	0.112694	−	0.993630i	$-0.464052\pi$
$601$	5.52545	0.225388	0.112694	+	0.993630i	$0.464052\pi$
$602$	0	0
$603$	50.1783	2.04342
$604$	0	0
$605$	−5.70933	−0.232117
$606$	0	0
$607$	12.1064	0.491384	0.245692	−	0.969348i	$-0.420985\pi$
$607$	12.1064	0.491384	0.245692	+	0.969348i	$0.420985\pi$
$608$	0	0
$609$	−24.0858	−0.976007
$610$	0	0
$611$	9.00389	0.364258
$612$	0	0
$613$	25.1833	1.01714	0.508572	−	0.861019i	$-0.330173\pi$
$613$	25.1833	1.01714	0.508572	+	0.861019i	$0.330173\pi$
$614$	0	0
$615$	12.9335	0.521529
$616$	0	0
$617$	1.10944	0.0446642	0.0223321	−	0.999751i	$-0.492891\pi$
$617$	1.10944	0.0446642	0.0223321	+	0.999751i	$0.492891\pi$
$618$	0	0
$619$	−45.0713	−1.81157	−0.905785	−	0.423738i	$-0.860718\pi$
$619$	−45.0713	−1.81157	−0.905785	+	0.423738i	$0.860718\pi$
$620$	0	0
$621$	−31.9610	−1.28255
$622$	0	0
$623$	5.34608	0.214186
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−20.6172	−0.823373
$628$	0	0
$629$	−18.0046	−0.717891
$630$	0	0
$631$	6.80064	0.270729	0.135365	−	0.990796i	$-0.456779\pi$
$631$	6.80064	0.270729	0.135365	+	0.990796i	$0.456779\pi$
$632$	0	0
$633$	−12.2460	−0.486736
$634$	0	0
$635$	−4.86578	−0.193093
$636$	0	0
$637$	5.89444	0.233546
$638$	0	0
$639$	−81.3449	−3.21796
$640$	0	0
$641$	14.2566	0.563100	0.281550	−	0.959547i	$-0.409151\pi$
$641$	14.2566	0.563100	0.281550	+	0.959547i	$0.409151\pi$
$642$	0	0
$643$	20.3841	0.803870	0.401935	−	0.915668i	$-0.368338\pi$
$643$	20.3841	0.803870	0.401935	+	0.915668i	$0.368338\pi$
$644$	0	0
$645$	2.36157	0.0929866
$646$	0	0
$647$	−20.5723	−0.808782	−0.404391	−	0.914586i	$-0.632517\pi$
$647$	−20.5723	−0.808782	−0.404391	+	0.914586i	$0.632517\pi$
$648$	0	0
$649$	−18.4692	−0.724980
$650$	0	0
$651$	1.40625	0.0551151
$652$	0	0
$653$	−13.8585	−0.542326	−0.271163	−	0.962533i	$-0.587408\pi$
$653$	−13.8585	−0.542326	−0.271163	+	0.962533i	$0.587408\pi$
$654$	0	0
$655$	11.3219	0.442384
$656$	0	0
$657$	72.3172	2.82136
$658$	0	0
$659$	12.3131	0.479651	0.239826	−	0.970816i	$-0.422910\pi$
$659$	12.3131	0.479651	0.239826	+	0.970816i	$0.422910\pi$
$660$	0	0
$661$	−27.9871	−1.08857	−0.544286	−	0.838900i	$-0.683199\pi$
$661$	−27.9871	−1.08857	−0.544286	+	0.838900i	$0.683199\pi$
$662$	0	0
$663$	7.05464	0.273979
$664$	0	0
$665$	−2.67560	−0.103755
$666$	0	0
$667$	15.0184	0.581517
$668$	0	0
$669$	11.3710	0.439629
$670$	0	0
$671$	−11.7202	−0.452454
$672$	0	0
$673$	−14.0829	−0.542857	−0.271429	−	0.962459i	$-0.587496\pi$
$673$	−14.0829	−0.542857	−0.271429	+	0.962459i	$0.587496\pi$
$674$	0	0
$675$	−15.8661	−0.610687
$676$	0	0
$677$	−42.1670	−1.62061	−0.810305	−	0.586009i	$-0.800698\pi$
$677$	−42.1670	−1.62061	−0.810305	+	0.586009i	$0.800698\pi$
$678$	0	0
$679$	5.44564	0.208984
$680$	0	0
$681$	22.0784	0.846045
$682$	0	0
$683$	−17.0620	−0.652858	−0.326429	−	0.945222i	$-0.605845\pi$
$683$	−17.0620	−0.652858	−0.326429	+	0.945222i	$0.605845\pi$
$684$	0	0
$685$	13.5567	0.517974
$686$	0	0
$687$	62.1462	2.37102
$688$	0	0
$689$	8.69221	0.331147
$690$	0	0
$691$	−3.66093	−0.139268	−0.0696342	−	0.997573i	$-0.522183\pi$
$691$	−3.66093	−0.139268	−0.0696342	+	0.997573i	$0.522183\pi$
$692$	0	0
$693$	−17.6699	−0.671225
$694$	0	0
$695$	11.6340	0.441301
$696$	0	0
$697$	−8.63242	−0.326976
$698$	0	0
$699$	−15.8796	−0.600622
$700$	0	0
$701$	38.1758	1.44188	0.720939	−	0.692998i	$-0.243711\pi$
$701$	38.1758	1.44188	0.720939	+	0.692998i	$0.243711\pi$
$702$	0	0
$703$	22.3410	0.842606
$704$	0	0
$705$	30.3307	1.14232
$706$	0	0
$707$	−0.404541	−0.0152143
$708$	0	0
$709$	−49.5878	−1.86231	−0.931155	−	0.364624i	$-0.881198\pi$
$709$	−49.5878	−1.86231	−0.931155	+	0.364624i	$0.881198\pi$
$710$	0	0
$711$	120.857	4.53249
$712$	0	0
$713$	−0.876848	−0.0328382
$714$	0	0
$715$	2.24633	0.0840079
$716$	0	0
$717$	29.0324	1.08424
$718$	0	0
$719$	−0.436840	−0.0162914	−0.00814568	−	0.999967i	$-0.502593\pi$
$719$	−0.436840	−0.0162914	−0.00814568	+	0.999967i	$0.502593\pi$
$720$	0	0
$721$	−4.42120	−0.164654
$722$	0	0
$723$	−12.3087	−0.457765
$724$	0	0
$725$	7.45547	0.276889
$726$	0	0
$727$	38.8072	1.43928	0.719640	−	0.694348i	$-0.244307\pi$
$727$	38.8072	1.43928	0.719640	+	0.694348i	$0.244307\pi$
$728$	0	0
$729$	68.1444	2.52387
$730$	0	0
$731$	−1.57622	−0.0582985
$732$	0	0
$733$	34.3737	1.26962	0.634811	−	0.772668i	$-0.281078\pi$
$733$	34.3737	1.26962	0.634811	+	0.772668i	$0.281078\pi$
$734$	0	0
$735$	19.8561	0.732404
$736$	0	0
$737$	−14.7539	−0.543469
$738$	0	0
$739$	−38.2208	−1.40597	−0.702987	−	0.711203i	$-0.748151\pi$
$739$	−38.2208	−1.40597	−0.702987	+	0.711203i	$0.748151\pi$
$740$	0	0
$741$	−8.75373	−0.321576
$742$	0	0
$743$	−12.1663	−0.446337	−0.223169	−	0.974780i	$-0.571640\pi$
$743$	−12.1663	−0.446337	−0.223169	+	0.974780i	$0.571640\pi$
$744$	0	0
$745$	−17.8705	−0.654724
$746$	0	0
$747$	7.55881	0.276563
$748$	0	0
$749$	−9.02549	−0.329784
$750$	0	0
$751$	−40.8606	−1.49102	−0.745512	−	0.666492i	$-0.767795\pi$
$751$	−40.8606	−1.49102	−0.745512	+	0.666492i	$0.767795\pi$
$752$	0	0
$753$	−27.8834	−1.01613
$754$	0	0
$755$	15.1562	0.551590
$756$	0	0
$757$	0.00564513	0.000205176 0	0.000102588	−	1.00000i	$-0.499967\pi$
$757$	0.00564513	0.000205176 0	0.000102588		1.00000i	$0.499967\pi$
$758$	0	0
$759$	15.2432	0.553292
$760$	0	0
$761$	0.751325	0.0272355	0.0136178	−	0.999907i	$-0.495665\pi$
$761$	0.751325	0.0272355	0.0136178	+	0.999907i	$0.495665\pi$
$762$	0	0
$763$	−3.49901	−0.126673
$764$	0	0
$765$	17.1771	0.621039
$766$	0	0
$767$	−7.84172	−0.283148
$768$	0	0
$769$	35.4522	1.27844	0.639219	−	0.769025i	$-0.279258\pi$
$769$	35.4522	1.27844	0.639219	+	0.769025i	$0.279258\pi$
$770$	0	0
$771$	48.5779	1.74949
$772$	0	0
$773$	−7.78080	−0.279856	−0.139928	−	0.990162i	$-0.544687\pi$
$773$	−7.78080	−0.279856	−0.139928	+	0.990162i	$0.544687\pi$
$774$	0	0
$775$	−0.435286	−0.0156359
$776$	0	0
$777$	26.4901	0.950327
$778$	0	0
$779$	10.7115	0.383780
$780$	0	0
$781$	23.9179	0.855849
$782$	0	0
$783$	−118.289	−4.22732
$784$	0	0
$785$	2.48641	0.0887438
$786$	0	0
$787$	−40.5800	−1.44652	−0.723260	−	0.690576i	$-0.757357\pi$
$787$	−40.5800	−1.44652	−0.723260	+	0.690576i	$0.757357\pi$
$788$	0	0
$789$	−22.3538	−0.795815
$790$	0	0
$791$	−5.28422	−0.187885
$792$	0	0
$793$	−4.97620	−0.176710
$794$	0	0
$795$	29.2807	1.03848
$796$	0	0
$797$	8.89932	0.315230	0.157615	−	0.987501i	$-0.449619\pi$
$797$	8.89932	0.315230	0.157615	+	0.987501i	$0.449619\pi$
$798$	0	0
$799$	−20.2441	−0.716185
$800$	0	0
$801$	42.5875	1.50476
$802$	0	0
$803$	−21.2635	−0.750371
$804$	0	0
$805$	1.97818	0.0697218
$806$	0	0
$807$	28.1005	0.989186
$808$	0	0
$809$	−27.0850	−0.952257	−0.476128	−	0.879376i	$-0.657960\pi$
$809$	−27.0850	−0.952257	−0.476128	+	0.879376i	$0.657960\pi$
$810$	0	0
$811$	−20.7451	−0.728458	−0.364229	−	0.931309i	$-0.618668\pi$
$811$	−20.7451	−0.728458	−0.364229	+	0.931309i	$0.618668\pi$
$812$	0	0
$813$	81.0018	2.84086
$814$	0	0
$815$	−19.6720	−0.689081
$816$	0	0
$817$	1.95585	0.0684264
$818$	0	0
$819$	−7.50235	−0.262153
$820$	0	0
$821$	21.8803	0.763626	0.381813	−	0.924240i	$-0.375300\pi$
$821$	21.8803	0.763626	0.381813	+	0.924240i	$0.375300\pi$
$822$	0	0
$823$	7.64319	0.266425	0.133212	−	0.991088i	$-0.457471\pi$
$823$	7.64319	0.266425	0.133212	+	0.991088i	$0.457471\pi$
$824$	0	0
$825$	7.56703	0.263450
$826$	0	0
$827$	1.10476	0.0384163	0.0192082	−	0.999816i	$-0.493885\pi$
$827$	1.10476	0.0384163	0.0192082	+	0.999816i	$0.493885\pi$
$828$	0	0
$829$	−40.8881	−1.42010	−0.710050	−	0.704151i	$-0.751328\pi$
$829$	−40.8881	−1.42010	−0.710050	+	0.704151i	$0.751328\pi$
$830$	0	0
$831$	46.4326	1.61073
$832$	0	0
$833$	−13.2529	−0.459186
$834$	0	0
$835$	−18.8620	−0.652746
$836$	0	0
$837$	6.90630	0.238717
$838$	0	0
$839$	−35.9665	−1.24170	−0.620851	−	0.783928i	$-0.713213\pi$
$839$	−35.9665	−1.24170	−0.620851	+	0.783928i	$0.713213\pi$
$840$	0	0
$841$	26.5841	0.916692
$842$	0	0
$843$	47.5741	1.63854
$844$	0	0
$845$	−12.0462	−0.414403
$846$	0	0
$847$	−5.60663	−0.192646
$848$	0	0
$849$	93.4140	3.20596
$850$	0	0
$851$	−16.5176	−0.566216
$852$	0	0
$853$	−12.0656	−0.413118	−0.206559	−	0.978434i	$-0.566227\pi$
$853$	−12.0656	−0.413118	−0.206559	+	0.978434i	$0.566227\pi$
$854$	0	0
$855$	−21.3142	−0.728929
$856$	0	0
$857$	−20.6681	−0.706010	−0.353005	−	0.935621i	$-0.614840\pi$
$857$	−20.6681	−0.706010	−0.353005	+	0.935621i	$0.614840\pi$
$858$	0	0
$859$	−37.6185	−1.28353	−0.641763	−	0.766903i	$-0.721797\pi$
$859$	−37.6185	−1.28353	−0.641763	+	0.766903i	$0.721797\pi$
$860$	0	0
$861$	12.7008	0.432843
$862$	0	0
$863$	−24.2911	−0.826880	−0.413440	−	0.910531i	$-0.635673\pi$
$863$	−24.2911	−0.826880	−0.413440	+	0.910531i	$0.635673\pi$
$864$	0	0
$865$	22.7298	0.772838
$866$	0	0
$867$	40.0652	1.36069
$868$	0	0
$869$	−35.5356	−1.20546
$870$	0	0
$871$	−6.26428	−0.212257
$872$	0	0
$873$	43.3806	1.46821
$874$	0	0
$875$	0.982011	0.0331980
$876$	0	0
$877$	−24.7919	−0.837162	−0.418581	−	0.908179i	$-0.637472\pi$
$877$	−24.7919	−0.837162	−0.418581	+	0.908179i	$0.637472\pi$
$878$	0	0
$879$	−71.6659	−2.41723
$880$	0	0
$881$	−35.1334	−1.18367	−0.591837	−	0.806058i	$-0.701597\pi$
$881$	−35.1334	−1.18367	−0.591837	+	0.806058i	$0.701597\pi$
$882$	0	0
$883$	−25.5924	−0.861251	−0.430626	−	0.902531i	$-0.641707\pi$
$883$	−25.5924	−0.861251	−0.430626	+	0.902531i	$0.641707\pi$
$884$	0	0
$885$	−26.4158	−0.887957
$886$	0	0
$887$	14.6666	0.492455	0.246228	−	0.969212i	$-0.420809\pi$
$887$	14.6666	0.492455	0.246228	+	0.969212i	$0.420809\pi$
$888$	0	0
$889$	−4.77825	−0.160257
$890$	0	0
$891$	−66.0785	−2.21371
$892$	0	0
$893$	25.1199	0.840604
$894$	0	0
$895$	−23.2413	−0.776871
$896$	0	0
$897$	6.47199	0.216094
$898$	0	0
$899$	−3.24526	−0.108236
$900$	0	0
$901$	−19.5433	−0.651083
$902$	0	0
$903$	2.31908	0.0771743
$904$	0	0
$905$	21.8552	0.726490
$906$	0	0
$907$	−35.9817	−1.19475	−0.597376	−	0.801961i	$-0.703790\pi$
$907$	−35.9817	−1.19475	−0.597376	+	0.801961i	$0.703790\pi$
$908$	0	0
$909$	−3.22262	−0.106888
$910$	0	0
$911$	14.6852	0.486542	0.243271	−	0.969958i	$-0.421780\pi$
$911$	14.6852	0.486542	0.243271	+	0.969958i	$0.421780\pi$
$912$	0	0
$913$	−2.22252	−0.0735547
$914$	0	0
$915$	−16.7629	−0.554166
$916$	0	0
$917$	11.1183	0.367157
$918$	0	0
$919$	46.2157	1.52451	0.762257	−	0.647274i	$-0.224091\pi$
$919$	46.2157	1.52451	0.762257	+	0.647274i	$0.224091\pi$
$920$	0	0
$921$	42.4492	1.39875
$922$	0	0
$923$	10.1551	0.334260
$924$	0	0
$925$	−8.19969	−0.269604
$926$	0	0
$927$	−35.2198	−1.15677
$928$	0	0
$929$	52.0543	1.70785	0.853923	−	0.520400i	$-0.174217\pi$
$929$	52.0543	1.70785	0.853923	+	0.520400i	$0.174217\pi$
$930$	0	0
$931$	16.4448	0.538958
$932$	0	0
$933$	2.11505	0.0692436
$934$	0	0
$935$	−5.05059	−0.165172
$936$	0	0
$937$	33.7454	1.10241	0.551207	−	0.834368i	$-0.314167\pi$
$937$	33.7454	1.10241	0.551207	+	0.834368i	$0.314167\pi$
$938$	0	0
$939$	70.3278	2.29506
$940$	0	0
$941$	−20.6213	−0.672234	−0.336117	−	0.941820i	$-0.609114\pi$
$941$	−20.6213	−0.672234	−0.336117	+	0.941820i	$0.609114\pi$
$942$	0	0
$943$	−7.91946	−0.257893
$944$	0	0
$945$	−15.5807	−0.506840
$946$	0	0
$947$	53.3982	1.73521	0.867604	−	0.497256i	$-0.165659\pi$
$947$	53.3982	1.73521	0.867604	+	0.497256i	$0.165659\pi$
$948$	0	0
$949$	−9.02810	−0.293065
$950$	0	0
$951$	40.2998	1.30681
$952$	0	0
$953$	48.6441	1.57574	0.787868	−	0.615844i	$-0.211185\pi$
$953$	48.6441	1.57574	0.787868	+	0.615844i	$0.211185\pi$
$954$	0	0
$955$	14.7872	0.478504
$956$	0	0
$957$	56.4158	1.82366
$958$	0	0
$959$	13.3128	0.429893
$960$	0	0
$961$	−30.8105	−0.993888
$962$	0	0
$963$	−71.8981	−2.31688
$964$	0	0
$965$	−11.2912	−0.363476
$966$	0	0
$967$	12.4521	0.400433	0.200216	−	0.979752i	$-0.435835\pi$
$967$	12.4521	0.400433	0.200216	+	0.979752i	$0.435835\pi$
$968$	0	0
$969$	19.6817	0.632266
$970$	0	0
$971$	20.0721	0.644143	0.322072	−	0.946715i	$-0.395621\pi$
$971$	20.0721	0.644143	0.322072	+	0.946715i	$0.395621\pi$
$972$	0	0
$973$	11.4247	0.366258
$974$	0	0
$975$	3.21283	0.102893
$976$	0	0
$977$	−18.3144	−0.585929	−0.292965	−	0.956123i	$-0.594642\pi$
$977$	−18.3144	−0.585929	−0.292965	+	0.956123i	$0.594642\pi$
$978$	0	0
$979$	−12.5220	−0.400206
$980$	0	0
$981$	−27.8735	−0.889934
$982$	0	0
$983$	−27.0583	−0.863027	−0.431513	−	0.902107i	$-0.642020\pi$
$983$	−27.0583	−0.863027	−0.431513	+	0.902107i	$0.642020\pi$
$984$	0	0
$985$	−15.0122	−0.478328
$986$	0	0
$987$	29.7851	0.948069
$988$	0	0
$989$	−1.44604	−0.0459813
$990$	0	0
$991$	−25.7759	−0.818799	−0.409400	−	0.912355i	$-0.634262\pi$
$991$	−25.7759	−0.818799	−0.409400	+	0.912355i	$0.634262\pi$
$992$	0	0
$993$	39.2642	1.24601
$994$	0	0
$995$	4.68789	0.148616
$996$	0	0
$997$	15.7201	0.497861	0.248930	−	0.968521i	$-0.419921\pi$
$997$	15.7201	0.497861	0.248930	+	0.968521i	$0.419921\pi$
$998$	0	0
$999$	130.097	4.11609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5120.2.a.t.1.1		8
4.3	odd	2		5120.2.a.v.1.8		8
8.3	odd	2		5120.2.a.s.1.1		8
8.5	even	2		5120.2.a.u.1.8		8
32.3	odd	8		640.2.l.b.161.8		16
32.5	even	8		320.2.l.a.241.8		16
32.11	odd	8		640.2.l.b.481.8		16
32.13	even	8		320.2.l.a.81.8		16
32.19	odd	8		80.2.l.a.61.8	yes	16
32.21	even	8		640.2.l.a.481.1		16
32.27	odd	8		80.2.l.a.21.8	✓	16
32.29	even	8		640.2.l.a.161.1		16
96.5	odd	8		2880.2.t.c.2161.3		16
96.59	even	8		720.2.t.c.181.1		16
96.77	odd	8		2880.2.t.c.721.2		16
96.83	even	8		720.2.t.c.541.1		16
160.13	odd	8		1600.2.q.h.849.1		16
160.19	odd	8		400.2.l.h.301.1		16
160.27	even	8		400.2.q.g.149.6		16
160.37	odd	8		1600.2.q.h.49.1		16
160.59	odd	8		400.2.l.h.101.1		16
160.69	even	8		1600.2.l.i.1201.1		16
160.77	odd	8		1600.2.q.g.849.8		16
160.83	even	8		400.2.q.g.349.6		16
160.109	even	8		1600.2.l.i.401.1		16
160.123	even	8		400.2.q.h.149.3		16
160.133	odd	8		1600.2.q.g.49.8		16
160.147	even	8		400.2.q.h.349.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.l.a.21.8	✓	16	32.27	odd	8
80.2.l.a.61.8	yes	16	32.19	odd	8
320.2.l.a.81.8		16	32.13	even	8
320.2.l.a.241.8		16	32.5	even	8
400.2.l.h.101.1		16	160.59	odd	8
400.2.l.h.301.1		16	160.19	odd	8
400.2.q.g.149.6		16	160.27	even	8
400.2.q.g.349.6		16	160.83	even	8
400.2.q.h.149.3		16	160.123	even	8
400.2.q.h.349.3		16	160.147	even	8
640.2.l.a.161.1		16	32.29	even	8
640.2.l.a.481.1		16	32.21	even	8
640.2.l.b.161.8		16	32.3	odd	8
640.2.l.b.481.8		16	32.11	odd	8
720.2.t.c.181.1		16	96.59	even	8
720.2.t.c.541.1		16	96.83	even	8
1600.2.l.i.401.1		16	160.109	even	8
1600.2.l.i.1201.1		16	160.69	even	8
1600.2.q.g.49.8		16	160.133	odd	8
1600.2.q.g.849.8		16	160.77	odd	8
1600.2.q.h.49.1		16	160.37	odd	8
1600.2.q.h.849.1		16	160.13	odd	8
2880.2.t.c.721.2		16	96.77	odd	8
2880.2.t.c.2161.3		16	96.5	odd	8
5120.2.a.s.1.1		8	8.3	odd	2
5120.2.a.t.1.1		8	1.1	even	1	trivial
5120.2.a.u.1.8		8	8.5	even	2
5120.2.a.v.1.8		8	4.3	odd	2

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:	\( N \)	\(=\)	\( 5120 = 2^{10} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5120.a (trivial)

\(f(q)\)	\(=\)	\(q-3.28980 q^{3} +1.00000 q^{5} +0.982011 q^{7} +7.82281 q^{9} +O(q^{10})\)	\(q-3.28980 q^{3} +1.00000 q^{5} +0.982011 q^{7} +7.82281 q^{9} -2.30015 q^{11} -0.976603 q^{13} -3.28980 q^{15} +2.19577 q^{17} -2.72461 q^{19} -3.23062 q^{21} +2.01442 q^{23} +1.00000 q^{25} -15.8661 q^{27} +7.45547 q^{29} -0.435286 q^{31} +7.56703 q^{33} +0.982011 q^{35} -8.19969 q^{37} +3.21283 q^{39} -3.93139 q^{41} -0.717844 q^{43} +7.82281 q^{45} -9.21960 q^{47} -6.03565 q^{49} -7.22365 q^{51} -8.90045 q^{53} -2.30015 q^{55} +8.96345 q^{57} +8.02959 q^{59} +5.09542 q^{61} +7.68209 q^{63} -0.976603 q^{65} +6.41435 q^{67} -6.62705 q^{69} -10.3984 q^{71} +9.24439 q^{73} -3.28980 q^{75} -2.25877 q^{77} +15.4493 q^{79} +28.7280 q^{81} +0.966253 q^{83} +2.19577 q^{85} -24.5271 q^{87} +5.44401 q^{89} -0.959035 q^{91} +1.43201 q^{93} -2.72461 q^{95} +5.54540 q^{97} -17.9936 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9	\( 8 q - 4 q^{3} + 8 q^{5} - 4 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{15} - 16 q^{19} - 12 q^{23} + 8 q^{25} - 16 q^{27} - 4 q^{35} - 28 q^{43} + 8 q^{45} - 20 q^{47} + 8 q^{49} - 24 q^{51} - 8 q^{55} - 16 q^{59} - 20 q^{63} - 36 q^{67} - 16 q^{69} + 8 q^{71} - 4 q^{75} + 8 q^{79} + 8 q^{81} - 20 q^{83} - 40 q^{91} - 16 q^{95} - 40 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 + 8 * q^5 - 4 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^15 - 16 * q^19 - 12 * q^23 + 8 * q^25 - 16 * q^27 - 4 * q^35 - 28 * q^43 + 8 * q^45 - 20 * q^47 + 8 * q^49 - 24 * q^51 - 8 * q^55 - 16 * q^59 - 20 * q^63 - 36 * q^67 - 16 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^79 + 8 * q^81 - 20 * q^83 - 40 * q^91 - 16 * q^95 - 40 * q^99

Introduction

L-functions

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