LMFDB - Embedded newform 3040.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.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:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
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			1.3
Root			$-0.787711$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.787711 q^{3} +1.00000 q^{5} +1.37951 q^{7} -2.37951 q^{9} -6.04159 q^{11} +3.25388 q^{13} +0.787711 q^{15} -4.23750 q^{17} -1.00000 q^{19} +1.08666 q^{21} -0.0553572 q^{23} +1.00000 q^{25} -4.23750 q^{27} +2.66208 q^{29} -0.816397 q^{31} -4.75902 q^{33} +1.37951 q^{35} -6.30777 q^{37} +2.56311 q^{39} +1.57542 q^{41} -0.717435 q^{43} -2.37951 q^{45} -7.22519 q^{47} -5.09695 q^{49} -3.33792 q^{51} -10.7719 q^{53} -6.04159 q^{55} -0.787711 q^{57} +3.84568 q^{59} +8.66006 q^{61} -3.28257 q^{63} +3.25388 q^{65} -5.40618 q^{67} -0.0436054 q^{69} -12.5078 q^{71} +4.48876 q^{73} +0.787711 q^{75} -8.33445 q^{77} -7.43487 q^{79} +3.80061 q^{81} -3.28257 q^{83} -4.23750 q^{85} +2.09695 q^{87} -1.74873 q^{89} +4.48876 q^{91} -0.643085 q^{93} -1.00000 q^{95} +4.59180 q^{97} +14.3760 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

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
$3$	0.787711	0.454785	0.227392	−	0.973803i	$-0.426980\pi$
$3$	0.787711	0.454785	0.227392	+	0.973803i	$0.426980\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.37951	0.521407	0.260703	−	0.965419i	$-0.416046\pi$
$7$	1.37951	0.521407	0.260703	+	0.965419i	$0.416046\pi$
$8$	0	0
$9$	−2.37951	−0.793171
$10$	0	0
$11$	−6.04159	−1.82161	−0.910804	−	0.412839i	$-0.864537\pi$
$11$	−6.04159	−1.82161	−0.910804	+	0.412839i	$0.864537\pi$
$12$	0	0
$13$	3.25388	0.902464	0.451232	−	0.892407i	$-0.350985\pi$
$13$	3.25388	0.902464	0.451232	+	0.892407i	$0.350985\pi$
$14$	0	0
$15$	0.787711	0.203386
$16$	0	0
$17$	−4.23750	−1.02774	−0.513872	−	0.857867i	$-0.671789\pi$
$17$	−4.23750	−1.02774	−0.513872	+	0.857867i	$0.671789\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.08666	0.237128
$22$	0	0
$23$	−0.0553572	−0.0115428	−0.00577138	−	0.999983i	$-0.501837\pi$
$23$	−0.0553572	−0.0115428	−0.00577138	+	0.999983i	$0.501837\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.23750	−0.815507
$28$	0	0
$29$	2.66208	0.494335	0.247168	−	0.968973i	$-0.420500\pi$
$29$	2.66208	0.494335	0.247168	+	0.968973i	$0.420500\pi$
$30$	0	0
$31$	−0.816397	−0.146629	−0.0733146	−	0.997309i	$-0.523358\pi$
$31$	−0.816397	−0.146629	−0.0733146	+	0.997309i	$0.523358\pi$
$32$	0	0
$33$	−4.75902	−0.828440
$34$	0	0
$35$	1.37951	0.233180
$36$	0	0
$37$	−6.30777	−1.03699	−0.518496	−	0.855080i	$-0.673508\pi$
$37$	−6.30777	−1.03699	−0.518496	+	0.855080i	$0.673508\pi$
$38$	0	0
$39$	2.56311	0.410427
$40$	0	0
$41$	1.57542	0.246039	0.123020	−	0.992404i	$-0.460742\pi$
$41$	1.57542	0.246039	0.123020	+	0.992404i	$0.460742\pi$
$42$	0	0
$43$	−0.717435	−0.109408	−0.0547039	−	0.998503i	$-0.517421\pi$
$43$	−0.717435	−0.109408	−0.0547039	+	0.998503i	$0.517421\pi$
$44$	0	0
$45$	−2.37951	−0.354717
$46$	0	0
$47$	−7.22519	−1.05390	−0.526951	−	0.849895i	$-0.676665\pi$
$47$	−7.22519	−1.05390	−0.526951	+	0.849895i	$0.676665\pi$
$48$	0	0
$49$	−5.09695	−0.728135
$50$	0	0
$51$	−3.33792	−0.467403
$52$	0	0
$53$	−10.7719	−1.47964	−0.739819	−	0.672806i	$-0.765089\pi$
$53$	−10.7719	−1.47964	−0.739819	+	0.672806i	$0.765089\pi$
$54$	0	0
$55$	−6.04159	−0.814648
$56$	0	0
$57$	−0.787711	−0.104335
$58$	0	0
$59$	3.84568	0.500665	0.250332	−	0.968160i	$-0.419460\pi$
$59$	3.84568	0.500665	0.250332	+	0.968160i	$0.419460\pi$
$60$	0	0
$61$	8.66006	1.10881	0.554404	−	0.832248i	$-0.312946\pi$
$61$	8.66006	1.10881	0.554404	+	0.832248i	$0.312946\pi$
$62$	0	0
$63$	−3.28257	−0.413564
$64$	0	0
$65$	3.25388	0.403594
$66$	0	0
$67$	−5.40618	−0.660470	−0.330235	−	0.943899i	$-0.607128\pi$
$67$	−5.40618	−0.660470	−0.330235	+	0.943899i	$0.607128\pi$
$68$	0	0
$69$	−0.0436054	−0.00524948
$70$	0	0
$71$	−12.5078	−1.48440	−0.742199	−	0.670180i	$-0.766217\pi$
$71$	−12.5078	−1.48440	−0.742199	+	0.670180i	$0.766217\pi$
$72$	0	0
$73$	4.48876	0.525370	0.262685	−	0.964882i	$-0.415392\pi$
$73$	4.48876	0.525370	0.262685	+	0.964882i	$0.415392\pi$
$74$	0	0
$75$	0.787711	0.0909570
$76$	0	0
$77$	−8.33445	−0.949798
$78$	0	0
$79$	−7.43487	−0.836488	−0.418244	−	0.908335i	$-0.637354\pi$
$79$	−7.43487	−0.836488	−0.418244	+	0.908335i	$0.637354\pi$
$80$	0	0
$81$	3.80061	0.422290
$82$	0	0
$83$	−3.28257	−0.360308	−0.180154	−	0.983638i	$-0.557660\pi$
$83$	−3.28257	−0.360308	−0.180154	+	0.983638i	$0.557660\pi$
$84$	0	0
$85$	−4.23750	−0.459621
$86$	0	0
$87$	2.09695	0.224816
$88$	0	0
$89$	−1.74873	−0.185365	−0.0926827	−	0.995696i	$-0.529544\pi$
$89$	−1.74873	−0.185365	−0.0926827	+	0.995696i	$0.529544\pi$
$90$	0	0
$91$	4.48876	0.470550
$92$	0	0
$93$	−0.643085	−0.0666848
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	4.59180	0.466227	0.233113	−	0.972450i	$-0.425109\pi$
$97$	4.59180	0.466227	0.233113	+	0.972450i	$0.425109\pi$
$98$	0	0
$99$	14.3760	1.44485
$100$	0	0
$101$	−2.54412	−0.253150	−0.126575	−	0.991957i	$-0.540398\pi$
$101$	−2.54412	−0.253150	−0.126575	+	0.991957i	$0.540398\pi$
$102$	0	0
$103$	−7.44979	−0.734049	−0.367025	−	0.930211i	$-0.619624\pi$
$103$	−7.44979	−0.734049	−0.367025	+	0.930211i	$0.619624\pi$
$104$	0	0
$105$	1.08666	0.106047
$106$	0	0
$107$	−4.11187	−0.397509	−0.198755	−	0.980049i	$-0.563690\pi$
$107$	−4.11187	−0.397509	−0.198755	+	0.980049i	$0.563690\pi$
$108$	0	0
$109$	12.1474	1.16351	0.581753	−	0.813365i	$-0.302367\pi$
$109$	12.1474	1.16351	0.581753	+	0.813365i	$0.302367\pi$
$110$	0	0
$111$	−4.96870	−0.471608
$112$	0	0
$113$	−10.4237	−0.980581	−0.490290	−	0.871559i	$-0.663109\pi$
$113$	−10.4237	−0.980581	−0.490290	+	0.871559i	$0.663109\pi$
$114$	0	0
$115$	−0.0553572	−0.00516208
$116$	0	0
$117$	−7.74264	−0.715808
$118$	0	0
$119$	−5.84568	−0.535873
$120$	0	0
$121$	25.5008	2.31825
$122$	0	0
$123$	1.24098	0.111895
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.266185	0.0236201	0.0118101	−	0.999930i	$-0.496241\pi$
$127$	0.266185	0.0236201	0.0118101	+	0.999930i	$0.496241\pi$
$128$	0	0
$129$	−0.565131	−0.0497570
$130$	0	0
$131$	−13.7693	−1.20303	−0.601515	−	0.798861i	$-0.705436\pi$
$131$	−13.7693	−1.20303	−0.601515	+	0.798861i	$0.705436\pi$
$132$	0	0
$133$	−1.37951	−0.119619
$134$	0	0
$135$	−4.23750	−0.364706
$136$	0	0
$137$	8.12678	0.694318	0.347159	−	0.937806i	$-0.387146\pi$
$137$	8.12678	0.694318	0.347159	+	0.937806i	$0.387146\pi$
$138$	0	0
$139$	19.5596	1.65903	0.829513	−	0.558487i	$-0.188618\pi$
$139$	19.5596	1.65903	0.829513	+	0.558487i	$0.188618\pi$
$140$	0	0
$141$	−5.69136	−0.479299
$142$	0	0
$143$	−19.6586	−1.64393
$144$	0	0
$145$	2.66208	0.221073
$146$	0	0
$147$	−4.01492	−0.331145
$148$	0	0
$149$	6.57396	0.538560	0.269280	−	0.963062i	$-0.413214\pi$
$149$	6.57396	0.538560	0.269280	+	0.963062i	$0.413214\pi$
$150$	0	0
$151$	−5.49224	−0.446952	−0.223476	−	0.974709i	$-0.571740\pi$
$151$	−5.49224	−0.446952	−0.223476	+	0.974709i	$0.571740\pi$
$152$	0	0
$153$	10.0832	0.815177
$154$	0	0
$155$	−0.816397	−0.0655746
$156$	0	0
$157$	−3.43487	−0.274132	−0.137066	−	0.990562i	$-0.543767\pi$
$157$	−3.43487	−0.274132	−0.137066	+	0.990562i	$0.543767\pi$
$158$	0	0
$159$	−8.48516	−0.672917
$160$	0	0
$161$	−0.0763659	−0.00601847
$162$	0	0
$163$	−5.93088	−0.464542	−0.232271	−	0.972651i	$-0.574616\pi$
$163$	−5.93088	−0.464542	−0.232271	+	0.972651i	$0.574616\pi$
$164$	0	0
$165$	−4.75902	−0.370490
$166$	0	0
$167$	19.0510	1.47421	0.737106	−	0.675777i	$-0.236192\pi$
$167$	19.0510	1.47421	0.737106	+	0.675777i	$0.236192\pi$
$168$	0	0
$169$	−2.41227	−0.185559
$170$	0	0
$171$	2.37951	0.181966
$172$	0	0
$173$	17.9365	1.36369	0.681845	−	0.731497i	$-0.261178\pi$
$173$	17.9365	1.36369	0.681845	+	0.731497i	$0.261178\pi$
$174$	0	0
$175$	1.37951	0.104281
$176$	0	0
$177$	3.02928	0.227695
$178$	0	0
$179$	−21.7090	−1.62261	−0.811304	−	0.584624i	$-0.801242\pi$
$179$	−21.7090	−1.62261	−0.811304	+	0.584624i	$0.801242\pi$
$180$	0	0
$181$	−0.367206	−0.0272942	−0.0136471	−	0.999907i	$-0.504344\pi$
$181$	−0.367206	−0.0272942	−0.0136471	+	0.999907i	$0.504344\pi$
$182$	0	0
$183$	6.82162	0.504269
$184$	0	0
$185$	−6.30777	−0.463757
$186$	0	0
$187$	25.6012	1.87215
$188$	0	0
$189$	−5.84568	−0.425211
$190$	0	0
$191$	−16.8026	−1.21580	−0.607898	−	0.794015i	$-0.707987\pi$
$191$	−16.8026	−1.21580	−0.607898	+	0.794015i	$0.707987\pi$
$192$	0	0
$193$	19.9090	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$193$	19.9090	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$194$	0	0
$195$	2.56311	0.183548
$196$	0	0
$197$	10.9857	0.782697	0.391349	−	0.920243i	$-0.372009\pi$
$197$	10.9857	0.782697	0.391349	+	0.920243i	$0.372009\pi$
$198$	0	0
$199$	−14.2375	−1.00927	−0.504635	−	0.863333i	$-0.668373\pi$
$199$	−14.2375	−1.00927	−0.504635	+	0.863333i	$0.668373\pi$
$200$	0	0
$201$	−4.25851	−0.300372
$202$	0	0
$203$	3.67237	0.257750
$204$	0	0
$205$	1.57542	0.110032
$206$	0	0
$207$	0.131723	0.00915538
$208$	0	0
$209$	6.04159	0.417905
$210$	0	0
$211$	3.42110	0.235518	0.117759	−	0.993042i	$-0.462429\pi$
$211$	3.42110	0.235518	0.117759	+	0.993042i	$0.462429\pi$
$212$	0	0
$213$	−9.85249	−0.675082
$214$	0	0
$215$	−0.717435	−0.0489286
$216$	0	0
$217$	−1.12623	−0.0764535
$218$	0	0
$219$	3.53585	0.238930
$220$	0	0
$221$	−13.7883	−0.927502
$222$	0	0
$223$	−22.4355	−1.50239	−0.751195	−	0.660080i	$-0.770522\pi$
$223$	−22.4355	−1.50239	−0.751195	+	0.660080i	$0.770522\pi$
$224$	0	0
$225$	−2.37951	−0.158634
$226$	0	0
$227$	6.07911	0.403484	0.201742	−	0.979439i	$-0.435340\pi$
$227$	6.07911	0.403484	0.201742	+	0.979439i	$0.435340\pi$
$228$	0	0
$229$	7.01912	0.463836	0.231918	−	0.972735i	$-0.425500\pi$
$229$	7.01912	0.463836	0.231918	+	0.972735i	$0.425500\pi$
$230$	0	0
$231$	−6.56513	−0.431954
$232$	0	0
$233$	−26.4831	−1.73497	−0.867484	−	0.497465i	$-0.834264\pi$
$233$	−26.4831	−1.73497	−0.867484	+	0.497465i	$0.834264\pi$
$234$	0	0
$235$	−7.22519	−0.471320
$236$	0	0
$237$	−5.85653	−0.380422
$238$	0	0
$239$	10.2633	0.663878	0.331939	−	0.943301i	$-0.392297\pi$
$239$	10.2633	0.663878	0.331939	+	0.943301i	$0.392297\pi$
$240$	0	0
$241$	−19.4853	−1.25516	−0.627579	−	0.778553i	$-0.715954\pi$
$241$	−19.4853	−1.25516	−0.627579	+	0.778553i	$0.715954\pi$
$242$	0	0
$243$	15.7063	1.00756
$244$	0	0
$245$	−5.09695	−0.325632
$246$	0	0
$247$	−3.25388	−0.207039
$248$	0	0
$249$	−2.58571	−0.163863
$250$	0	0
$251$	1.71597	0.108311	0.0541556	−	0.998533i	$-0.482753\pi$
$251$	1.71597	0.108311	0.0541556	+	0.998533i	$0.482753\pi$
$252$	0	0
$253$	0.334445	0.0210264
$254$	0	0
$255$	−3.33792	−0.209029
$256$	0	0
$257$	−11.8258	−0.737675	−0.368837	−	0.929494i	$-0.620244\pi$
$257$	−11.8258	−0.737675	−0.368837	+	0.929494i	$0.620244\pi$
$258$	0	0
$259$	−8.70165	−0.540694
$260$	0	0
$261$	−6.33445	−0.392092
$262$	0	0
$263$	9.44370	0.582323	0.291162	−	0.956674i	$-0.405958\pi$
$263$	9.44370	0.582323	0.291162	+	0.956674i	$0.405958\pi$
$264$	0	0
$265$	−10.7719	−0.661714
$266$	0	0
$267$	−1.37750	−0.0843014
$268$	0	0
$269$	18.7767	1.14483	0.572417	−	0.819962i	$-0.306006\pi$
$269$	18.7767	1.14483	0.572417	+	0.819962i	$0.306006\pi$
$270$	0	0
$271$	5.40532	0.328350	0.164175	−	0.986431i	$-0.447504\pi$
$271$	5.40532	0.328350	0.164175	+	0.986431i	$0.447504\pi$
$272$	0	0
$273$	3.53585	0.213999
$274$	0	0
$275$	−6.04159	−0.364322
$276$	0	0
$277$	5.88929	0.353853	0.176926	−	0.984224i	$-0.443385\pi$
$277$	5.88929	0.353853	0.176926	+	0.984224i	$0.443385\pi$
$278$	0	0
$279$	1.94263	0.116302
$280$	0	0
$281$	−17.4607	−1.04162	−0.520808	−	0.853674i	$-0.674369\pi$
$281$	−17.4607	−1.04162	−0.520808	+	0.853674i	$0.674369\pi$
$282$	0	0
$283$	11.0015	0.653969	0.326984	−	0.945030i	$-0.393968\pi$
$283$	11.0015	0.653969	0.326984	+	0.945030i	$0.393968\pi$
$284$	0	0
$285$	−0.787711	−0.0466599
$286$	0	0
$287$	2.17331	0.128287
$288$	0	0
$289$	0.956395	0.0562585
$290$	0	0
$291$	3.61701	0.212033
$292$	0	0
$293$	1.50515	0.0879315	0.0439658	−	0.999033i	$-0.486001\pi$
$293$	1.50515	0.0879315	0.0439658	+	0.999033i	$0.486001\pi$
$294$	0	0
$295$	3.84568	0.223904
$296$	0	0
$297$	25.6012	1.48553
$298$	0	0
$299$	−0.180125	−0.0104169
$300$	0	0
$301$	−0.989710	−0.0570459
$302$	0	0
$303$	−2.00403	−0.115129
$304$	0	0
$305$	8.66006	0.495874
$306$	0	0
$307$	2.38505	0.136122	0.0680609	−	0.997681i	$-0.478319\pi$
$307$	2.38505	0.136122	0.0680609	+	0.997681i	$0.478319\pi$
$308$	0	0
$309$	−5.86828	−0.333835
$310$	0	0
$311$	−20.3446	−1.15364	−0.576818	−	0.816872i	$-0.695706\pi$
$311$	−20.3446	−1.15364	−0.576818	+	0.816872i	$0.695706\pi$
$312$	0	0
$313$	8.65512	0.489216	0.244608	−	0.969622i	$-0.421341\pi$
$313$	8.65512	0.489216	0.244608	+	0.969622i	$0.421341\pi$
$314$	0	0
$315$	−3.28257	−0.184952
$316$	0	0
$317$	6.01290	0.337718	0.168859	−	0.985640i	$-0.445992\pi$
$317$	6.01290	0.337718	0.168859	+	0.985640i	$0.445992\pi$
$318$	0	0
$319$	−16.0832	−0.900485
$320$	0	0
$321$	−3.23896	−0.180781
$322$	0	0
$323$	4.23750	0.235781
$324$	0	0
$325$	3.25388	0.180493
$326$	0	0
$327$	9.56861	0.529145
$328$	0	0
$329$	−9.96724	−0.549512
$330$	0	0
$331$	−35.3240	−1.94158	−0.970792	−	0.239924i	$-0.922877\pi$
$331$	−35.3240	−1.94158	−0.970792	+	0.239924i	$0.922877\pi$
$332$	0	0
$333$	15.0094	0.822511
$334$	0	0
$335$	−5.40618	−0.295371
$336$	0	0
$337$	−24.2681	−1.32197	−0.660983	−	0.750401i	$-0.729860\pi$
$337$	−24.2681	−1.32197	−0.660983	+	0.750401i	$0.729860\pi$
$338$	0	0
$339$	−8.21087	−0.445953
$340$	0	0
$341$	4.93234	0.267101
$342$	0	0
$343$	−16.6879	−0.901061
$344$	0	0
$345$	−0.0436054	−0.00234764
$346$	0	0
$347$	13.3330	0.715752	0.357876	−	0.933769i	$-0.383501\pi$
$347$	13.3330	0.715752	0.357876	+	0.933769i	$0.383501\pi$
$348$	0	0
$349$	19.2844	1.03227	0.516136	−	0.856507i	$-0.327370\pi$
$349$	19.2844	1.03227	0.516136	+	0.856507i	$0.327370\pi$
$350$	0	0
$351$	−13.7883	−0.735965
$352$	0	0
$353$	10.6047	0.564431	0.282216	−	0.959351i	$-0.408931\pi$
$353$	10.6047	0.564431	0.282216	+	0.959351i	$0.408931\pi$
$354$	0	0
$355$	−12.5078	−0.663843
$356$	0	0
$357$	−4.60470	−0.243707
$358$	0	0
$359$	−30.0742	−1.58726	−0.793628	−	0.608403i	$-0.791810\pi$
$359$	−30.0742	−1.58726	−0.793628	+	0.608403i	$0.791810\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	20.0873	1.05431
$364$	0	0
$365$	4.48876	0.234953
$366$	0	0
$367$	−0.541199	−0.0282504	−0.0141252	−	0.999900i	$-0.504496\pi$
$367$	−0.541199	−0.0282504	−0.0141252	+	0.999900i	$0.504496\pi$
$368$	0	0
$369$	−3.74873	−0.195151
$370$	0	0
$371$	−14.8600	−0.771493
$372$	0	0
$373$	−2.49485	−0.129179	−0.0645893	−	0.997912i	$-0.520574\pi$
$373$	−2.49485	−0.129179	−0.0645893	+	0.997912i	$0.520574\pi$
$374$	0	0
$375$	0.787711	0.0406772
$376$	0	0
$377$	8.66208	0.446120
$378$	0	0
$379$	35.7555	1.83664	0.918320	−	0.395840i	$-0.129546\pi$
$379$	35.7555	1.83664	0.918320	+	0.395840i	$0.129546\pi$
$380$	0	0
$381$	0.209677	0.0107421
$382$	0	0
$383$	−21.5002	−1.09861	−0.549305	−	0.835622i	$-0.685107\pi$
$383$	−21.5002	−1.09861	−0.549305	+	0.835622i	$0.685107\pi$
$384$	0	0
$385$	−8.33445	−0.424763
$386$	0	0
$387$	1.70714	0.0867790
$388$	0	0
$389$	25.4578	1.29076	0.645380	−	0.763862i	$-0.276699\pi$
$389$	25.4578	1.29076	0.645380	+	0.763862i	$0.276699\pi$
$390$	0	0
$391$	0.234576	0.0118630
$392$	0	0
$393$	−10.8462	−0.547120
$394$	0	0
$395$	−7.43487	−0.374089
$396$	0	0
$397$	0.507758	0.0254836	0.0127418	−	0.999919i	$-0.495944\pi$
$397$	0.507758	0.0254836	0.0127418	+	0.999919i	$0.495944\pi$
$398$	0	0
$399$	−1.08666	−0.0544009
$400$	0	0
$401$	−11.6012	−0.579338	−0.289669	−	0.957127i	$-0.593545\pi$
$401$	−11.6012	−0.579338	−0.289669	+	0.957127i	$0.593545\pi$
$402$	0	0
$403$	−2.65646	−0.132328
$404$	0	0
$405$	3.80061	0.188854
$406$	0	0
$407$	38.1090	1.88899
$408$	0	0
$409$	3.52500	0.174300	0.0871501	−	0.996195i	$-0.472224\pi$
$409$	3.52500	0.174300	0.0871501	+	0.996195i	$0.472224\pi$
$410$	0	0
$411$	6.40155	0.315765
$412$	0	0
$413$	5.30516	0.261050
$414$	0	0
$415$	−3.28257	−0.161135
$416$	0	0
$417$	15.4073	0.754500
$418$	0	0
$419$	24.8750	1.21522	0.607611	−	0.794235i	$-0.292128\pi$
$419$	24.8750	1.21522	0.607611	+	0.794235i	$0.292128\pi$
$420$	0	0
$421$	−13.5289	−0.659358	−0.329679	−	0.944093i	$-0.606941\pi$
$421$	−13.5289	−0.659358	−0.329679	+	0.944093i	$0.606941\pi$
$422$	0	0
$423$	17.1924	0.835925
$424$	0	0
$425$	−4.23750	−0.205549
$426$	0	0
$427$	11.9467	0.578139
$428$	0	0
$429$	−15.4853	−0.747637
$430$	0	0
$431$	−2.23069	−0.107448	−0.0537242	−	0.998556i	$-0.517109\pi$
$431$	−2.23069	−0.107448	−0.0537242	+	0.998556i	$0.517109\pi$
$432$	0	0
$433$	30.2556	1.45399	0.726996	−	0.686641i	$-0.240916\pi$
$433$	30.2556	1.45399	0.726996	+	0.686641i	$0.240916\pi$
$434$	0	0
$435$	2.09695	0.100541
$436$	0	0
$437$	0.0553572	0.00264809
$438$	0	0
$439$	−12.5828	−0.600544	−0.300272	−	0.953854i	$-0.597077\pi$
$439$	−12.5828	−0.600544	−0.300272	+	0.953854i	$0.597077\pi$
$440$	0	0
$441$	12.1282	0.577536
$442$	0	0
$443$	33.1907	1.57694	0.788469	−	0.615075i	$-0.210874\pi$
$443$	33.1907	1.57694	0.788469	+	0.615075i	$0.210874\pi$
$444$	0	0
$445$	−1.74873	−0.0828979
$446$	0	0
$447$	5.17838	0.244929
$448$	0	0
$449$	35.2720	1.66459	0.832294	−	0.554334i	$-0.187027\pi$
$449$	35.2720	1.66459	0.832294	+	0.554334i	$0.187027\pi$
$450$	0	0
$451$	−9.51805	−0.448187
$452$	0	0
$453$	−4.32630	−0.203267
$454$	0	0
$455$	4.48876	0.210437
$456$	0	0
$457$	−16.2100	−0.758270	−0.379135	−	0.925341i	$-0.623778\pi$
$457$	−16.2100	−0.758270	−0.379135	+	0.925341i	$0.623778\pi$
$458$	0	0
$459$	17.9564	0.838133
$460$	0	0
$461$	5.03610	0.234554	0.117277	−	0.993099i	$-0.462583\pi$
$461$	5.03610	0.234554	0.117277	+	0.993099i	$0.462583\pi$
$462$	0	0
$463$	29.6932	1.37996	0.689981	−	0.723828i	$-0.257619\pi$
$463$	29.6932	1.37996	0.689981	+	0.723828i	$0.257619\pi$
$464$	0	0
$465$	−0.643085	−0.0298223
$466$	0	0
$467$	2.37603	0.109950	0.0549749	−	0.998488i	$-0.482492\pi$
$467$	2.37603	0.109950	0.0549749	+	0.998488i	$0.482492\pi$
$468$	0	0
$469$	−7.45789	−0.344374
$470$	0	0
$471$	−2.70568	−0.124671
$472$	0	0
$473$	4.33445	0.199298
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	25.6319	1.17361
$478$	0	0
$479$	25.7054	1.17451	0.587255	−	0.809402i	$-0.300209\pi$
$479$	25.7054	1.17451	0.587255	+	0.809402i	$0.300209\pi$
$480$	0	0
$481$	−20.5247	−0.935847
$482$	0	0
$483$	−0.0601542	−0.00273711
$484$	0	0
$485$	4.59180	0.208503
$486$	0	0
$487$	1.62483	0.0736281	0.0368140	−	0.999322i	$-0.488279\pi$
$487$	1.62483	0.0736281	0.0368140	+	0.999322i	$0.488279\pi$
$488$	0	0
$489$	−4.67181	−0.211267
$490$	0	0
$491$	22.9254	1.03461	0.517304	−	0.855802i	$-0.326936\pi$
$491$	22.9254	1.03461	0.517304	+	0.855802i	$0.326936\pi$
$492$	0	0
$493$	−11.2805	−0.508050
$494$	0	0
$495$	14.3760	0.646155
$496$	0	0
$497$	−17.2546	−0.773975
$498$	0	0
$499$	6.15230	0.275415	0.137707	−	0.990473i	$-0.456027\pi$
$499$	6.15230	0.275415	0.137707	+	0.990473i	$0.456027\pi$
$500$	0	0
$501$	15.0067	0.670449
$502$	0	0
$503$	−7.48730	−0.333842	−0.166921	−	0.985970i	$-0.553383\pi$
$503$	−7.48730	−0.333842	−0.166921	+	0.985970i	$0.553383\pi$
$504$	0	0
$505$	−2.54412	−0.113212
$506$	0	0
$507$	−1.90017	−0.0843896
$508$	0	0
$509$	−36.3655	−1.61187	−0.805936	−	0.592003i	$-0.798337\pi$
$509$	−36.3655	−1.61187	−0.805936	+	0.592003i	$0.798337\pi$
$510$	0	0
$511$	6.19231	0.273931
$512$	0	0
$513$	4.23750	0.187090
$514$	0	0
$515$	−7.44979	−0.328277
$516$	0	0
$517$	43.6516	1.91980
$518$	0	0
$519$	14.1288	0.620185
$520$	0	0
$521$	14.6237	0.640676	0.320338	−	0.947303i	$-0.396204\pi$
$521$	14.6237	0.640676	0.320338	+	0.947303i	$0.396204\pi$
$522$	0	0
$523$	−1.73771	−0.0759845	−0.0379923	−	0.999278i	$-0.512096\pi$
$523$	−1.73771	−0.0759845	−0.0379923	+	0.999278i	$0.512096\pi$
$524$	0	0
$525$	1.08666	0.0474256
$526$	0	0
$527$	3.45948	0.150697
$528$	0	0
$529$	−22.9969	−0.999867
$530$	0	0
$531$	−9.15084	−0.397113
$532$	0	0
$533$	5.12623	0.222042
$534$	0	0
$535$	−4.11187	−0.177771
$536$	0	0
$537$	−17.1004	−0.737938
$538$	0	0
$539$	30.7937	1.32638
$540$	0	0
$541$	−39.5281	−1.69944	−0.849722	−	0.527231i	$-0.823230\pi$
$541$	−39.5281	−1.69944	−0.849722	+	0.527231i	$0.823230\pi$
$542$	0	0
$543$	−0.289252	−0.0124130
$544$	0	0
$545$	12.1474	0.520336
$546$	0	0
$547$	33.7147	1.44154	0.720768	−	0.693177i	$-0.243789\pi$
$547$	33.7147	1.44154	0.720768	+	0.693177i	$0.243789\pi$
$548$	0	0
$549$	−20.6067	−0.879473
$550$	0	0
$551$	−2.66208	−0.113408
$552$	0	0
$553$	−10.2565	−0.436150
$554$	0	0
$555$	−4.96870	−0.210910
$556$	0	0
$557$	−36.1417	−1.53137	−0.765687	−	0.643213i	$-0.777601\pi$
$557$	−36.1417	−1.53137	−0.765687	+	0.643213i	$0.777601\pi$
$558$	0	0
$559$	−2.33445	−0.0987365
$560$	0	0
$561$	20.1664	0.851424
$562$	0	0
$563$	14.9403	0.629659	0.314829	−	0.949148i	$-0.398053\pi$
$563$	14.9403	0.629659	0.314829	+	0.949148i	$0.398053\pi$
$564$	0	0
$565$	−10.4237	−0.438529
$566$	0	0
$567$	5.24299	0.220185
$568$	0	0
$569$	25.7992	1.08156	0.540778	−	0.841165i	$-0.318130\pi$
$569$	25.7992	1.08156	0.540778	+	0.841165i	$0.318130\pi$
$570$	0	0
$571$	−12.8183	−0.536428	−0.268214	−	0.963359i	$-0.586433\pi$
$571$	−12.8183	−0.536428	−0.268214	+	0.963359i	$0.586433\pi$
$572$	0	0
$573$	−13.2356	−0.552926
$574$	0	0
$575$	−0.0553572	−0.00230855
$576$	0	0
$577$	−35.6912	−1.48584	−0.742922	−	0.669378i	$-0.766561\pi$
$577$	−35.6912	−1.48584	−0.742922	+	0.669378i	$0.766561\pi$
$578$	0	0
$579$	15.6825	0.651744
$580$	0	0
$581$	−4.52834	−0.187867
$582$	0	0
$583$	65.0796	2.69532
$584$	0	0
$585$	−7.74264	−0.320119
$586$	0	0
$587$	−30.3363	−1.25211	−0.626057	−	0.779777i	$-0.715332\pi$
$587$	−30.3363	−1.25211	−0.626057	+	0.779777i	$0.715332\pi$
$588$	0	0
$589$	0.816397	0.0336391
$590$	0	0
$591$	8.65353	0.355959
$592$	0	0
$593$	−12.5828	−0.516713	−0.258357	−	0.966050i	$-0.583181\pi$
$593$	−12.5828	−0.516713	−0.258357	+	0.966050i	$0.583181\pi$
$594$	0	0
$595$	−5.84568	−0.239650
$596$	0	0
$597$	−11.2150	−0.459001
$598$	0	0
$599$	21.4319	0.875686	0.437843	−	0.899052i	$-0.355743\pi$
$599$	21.4319	0.875686	0.437843	+	0.899052i	$0.355743\pi$
$600$	0	0
$601$	14.2267	0.580317	0.290159	−	0.956979i	$-0.406292\pi$
$601$	14.2267	0.580317	0.290159	+	0.956979i	$0.406292\pi$
$602$	0	0
$603$	12.8641	0.523866
$604$	0	0
$605$	25.5008	1.03676
$606$	0	0
$607$	24.1761	0.981276	0.490638	−	0.871364i	$-0.336764\pi$
$607$	24.1761	0.981276	0.490638	+	0.871364i	$0.336764\pi$
$608$	0	0
$609$	2.89276	0.117221
$610$	0	0
$611$	−23.5099	−0.951109
$612$	0	0
$613$	−39.5204	−1.59621	−0.798106	−	0.602517i	$-0.794165\pi$
$613$	−39.5204	−1.59621	−0.798106	+	0.602517i	$0.794165\pi$
$614$	0	0
$615$	1.24098	0.0500410
$616$	0	0
$617$	−31.5479	−1.27007	−0.635035	−	0.772483i	$-0.719014\pi$
$617$	−31.5479	−1.27007	−0.635035	+	0.772483i	$0.719014\pi$
$618$	0	0
$619$	45.6399	1.83442	0.917211	−	0.398402i	$-0.130435\pi$
$619$	45.6399	1.83442	0.917211	+	0.398402i	$0.130435\pi$
$620$	0	0
$621$	0.234576	0.00941321
$622$	0	0
$623$	−2.41240	−0.0966507
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.75902	0.190057
$628$	0	0
$629$	26.7292	1.06576
$630$	0	0
$631$	−32.3537	−1.28798	−0.643991	−	0.765033i	$-0.722722\pi$
$631$	−32.3537	−1.28798	−0.643991	+	0.765033i	$0.722722\pi$
$632$	0	0
$633$	2.69484	0.107110
$634$	0	0
$635$	0.266185	0.0105632
$636$	0	0
$637$	−16.5848	−0.657116
$638$	0	0
$639$	29.7624	1.17738
$640$	0	0
$641$	22.4802	0.887915	0.443958	−	0.896048i	$-0.353574\pi$
$641$	22.4802	0.887915	0.443958	+	0.896048i	$0.353574\pi$
$642$	0	0
$643$	40.7884	1.60854	0.804269	−	0.594265i	$-0.202557\pi$
$643$	40.7884	1.60854	0.804269	+	0.594265i	$0.202557\pi$
$644$	0	0
$645$	−0.565131	−0.0222520
$646$	0	0
$647$	13.6963	0.538457	0.269229	−	0.963076i	$-0.413231\pi$
$647$	13.6963	0.538457	0.269229	+	0.963076i	$0.413231\pi$
$648$	0	0
$649$	−23.2340	−0.912015
$650$	0	0
$651$	−0.887143	−0.0347699
$652$	0	0
$653$	−27.2392	−1.06595	−0.532977	−	0.846130i	$-0.678927\pi$
$653$	−27.2392	−1.06595	−0.532977	+	0.846130i	$0.678927\pi$
$654$	0	0
$655$	−13.7693	−0.538012
$656$	0	0
$657$	−10.6811	−0.416708
$658$	0	0
$659$	−42.1578	−1.64224	−0.821118	−	0.570759i	$-0.806649\pi$
$659$	−42.1578	−1.64224	−0.821118	+	0.570759i	$0.806649\pi$
$660$	0	0
$661$	29.9822	1.16617	0.583086	−	0.812410i	$-0.301845\pi$
$661$	29.9822	1.16617	0.583086	+	0.812410i	$0.301845\pi$
$662$	0	0
$663$	−10.8612	−0.421814
$664$	0	0
$665$	−1.37951	−0.0534952
$666$	0	0
$667$	−0.147365	−0.00570600
$668$	0	0
$669$	−17.6727	−0.683264
$670$	0	0
$671$	−52.3205	−2.01981
$672$	0	0
$673$	1.32502	0.0510758	0.0255379	−	0.999674i	$-0.491870\pi$
$673$	1.32502	0.0510758	0.0255379	+	0.999674i	$0.491870\pi$
$674$	0	0
$675$	−4.23750	−0.163101
$676$	0	0
$677$	22.7362	0.873825	0.436912	−	0.899504i	$-0.356072\pi$
$677$	22.7362	0.873825	0.436912	+	0.899504i	$0.356072\pi$
$678$	0	0
$679$	6.33445	0.243094
$680$	0	0
$681$	4.78858	0.183499
$682$	0	0
$683$	−42.5743	−1.62906	−0.814530	−	0.580122i	$-0.803005\pi$
$683$	−42.5743	−1.62906	−0.814530	+	0.580122i	$0.803005\pi$
$684$	0	0
$685$	8.12678	0.310508
$686$	0	0
$687$	5.52903	0.210946
$688$	0	0
$689$	−35.0505	−1.33532
$690$	0	0
$691$	34.7661	1.32257	0.661283	−	0.750137i	$-0.270012\pi$
$691$	34.7661	1.32257	0.661283	+	0.750137i	$0.270012\pi$
$692$	0	0
$693$	19.8319	0.753352
$694$	0	0
$695$	19.5596	0.741939
$696$	0	0
$697$	−6.67585	−0.252866
$698$	0	0
$699$	−20.8611	−0.789038
$700$	0	0
$701$	−24.0920	−0.909943	−0.454971	−	0.890506i	$-0.650350\pi$
$701$	−24.0920	−0.909943	−0.454971	+	0.890506i	$0.650350\pi$
$702$	0	0
$703$	6.30777	0.237902
$704$	0	0
$705$	−5.69136	−0.214349
$706$	0	0
$707$	−3.50965	−0.131994
$708$	0	0
$709$	−28.0328	−1.05279	−0.526396	−	0.850239i	$-0.676457\pi$
$709$	−28.0328	−1.05279	−0.526396	+	0.850239i	$0.676457\pi$
$710$	0	0
$711$	17.6914	0.663478
$712$	0	0
$713$	0.0451934	0.00169251
$714$	0	0
$715$	−19.6586	−0.735190
$716$	0	0
$717$	8.08451	0.301922
$718$	0	0
$719$	−45.5102	−1.69724	−0.848622	−	0.529000i	$-0.822567\pi$
$719$	−45.5102	−1.69724	−0.848622	+	0.529000i	$0.822567\pi$
$720$	0	0
$721$	−10.2771	−0.382738
$722$	0	0
$723$	−15.3488	−0.570827
$724$	0	0
$725$	2.66208	0.0988671
$726$	0	0
$727$	33.1694	1.23019	0.615093	−	0.788455i	$-0.289119\pi$
$727$	33.1694	1.23019	0.615093	+	0.788455i	$0.289119\pi$
$728$	0	0
$729$	0.970162	0.0359319
$730$	0	0
$731$	3.04013	0.112443
$732$	0	0
$733$	39.4783	1.45817	0.729083	−	0.684426i	$-0.239947\pi$
$733$	39.4783	1.45817	0.729083	+	0.684426i	$0.239947\pi$
$734$	0	0
$735$	−4.01492	−0.148093
$736$	0	0
$737$	32.6619	1.20312
$738$	0	0
$739$	28.8216	1.06022	0.530110	−	0.847929i	$-0.322151\pi$
$739$	28.8216	1.06022	0.530110	+	0.847929i	$0.322151\pi$
$740$	0	0
$741$	−2.56311	−0.0941584
$742$	0	0
$743$	−11.7691	−0.431768	−0.215884	−	0.976419i	$-0.569263\pi$
$743$	−11.7691	−0.431768	−0.215884	+	0.976419i	$0.569263\pi$
$744$	0	0
$745$	6.57396	0.240851
$746$	0	0
$747$	7.81090	0.285786
$748$	0	0
$749$	−5.67237	−0.207264
$750$	0	0
$751$	−7.72704	−0.281964	−0.140982	−	0.990012i	$-0.545026\pi$
$751$	−7.72704	−0.281964	−0.140982	+	0.990012i	$0.545026\pi$
$752$	0	0
$753$	1.35169	0.0492583
$754$	0	0
$755$	−5.49224	−0.199883
$756$	0	0
$757$	−22.8709	−0.831258	−0.415629	−	0.909534i	$-0.636439\pi$
$757$	−22.8709	−0.831258	−0.415629	+	0.909534i	$0.636439\pi$
$758$	0	0
$759$	0.263446	0.00956249
$760$	0	0
$761$	18.1890	0.659349	0.329675	−	0.944095i	$-0.393061\pi$
$761$	18.1890	0.659349	0.329675	+	0.944095i	$0.393061\pi$
$762$	0	0
$763$	16.7574	0.606660
$764$	0	0
$765$	10.0832	0.364558
$766$	0	0
$767$	12.5134	0.451832
$768$	0	0
$769$	−7.91884	−0.285561	−0.142780	−	0.989754i	$-0.545604\pi$
$769$	−7.91884	−0.285561	−0.142780	+	0.989754i	$0.545604\pi$
$770$	0	0
$771$	−9.31533	−0.335483
$772$	0	0
$773$	2.92174	0.105088	0.0525438	−	0.998619i	$-0.483267\pi$
$773$	2.92174	0.105088	0.0525438	+	0.998619i	$0.483267\pi$
$774$	0	0
$775$	−0.816397	−0.0293259
$776$	0	0
$777$	−6.85438	−0.245900
$778$	0	0
$779$	−1.57542	−0.0564453
$780$	0	0
$781$	75.5667	2.70399
$782$	0	0
$783$	−11.2805	−0.403134
$784$	0	0
$785$	−3.43487	−0.122596
$786$	0	0
$787$	−26.1829	−0.933318	−0.466659	−	0.884437i	$-0.654543\pi$
$787$	−26.1829	−0.933318	−0.466659	+	0.884437i	$0.654543\pi$
$788$	0	0
$789$	7.43890	0.264832
$790$	0	0
$791$	−14.3796	−0.511281
$792$	0	0
$793$	28.1788	1.00066
$794$	0	0
$795$	−8.48516	−0.300938
$796$	0	0
$797$	45.2658	1.60340	0.801698	−	0.597729i	$-0.203930\pi$
$797$	45.2658	1.60340	0.801698	+	0.597729i	$0.203930\pi$
$798$	0	0
$799$	30.6167	1.08314
$800$	0	0
$801$	4.16113	0.147026
$802$	0	0
$803$	−27.1193	−0.957018
$804$	0	0
$805$	−0.0763659	−0.00269154
$806$	0	0
$807$	14.7906	0.520653
$808$	0	0
$809$	25.7814	0.906424	0.453212	−	0.891403i	$-0.350278\pi$
$809$	25.7814	0.906424	0.453212	+	0.891403i	$0.350278\pi$
$810$	0	0
$811$	51.6677	1.81430	0.907149	−	0.420809i	$-0.138254\pi$
$811$	51.6677	1.81430	0.907149	+	0.420809i	$0.138254\pi$
$812$	0	0
$813$	4.25783	0.149328
$814$	0	0
$815$	−5.93088	−0.207749
$816$	0	0
$817$	0.717435	0.0250999
$818$	0	0
$819$	−10.6811	−0.373227
$820$	0	0
$821$	1.47166	0.0513613	0.0256807	−	0.999670i	$-0.491825\pi$
$821$	1.47166	0.0513613	0.0256807	+	0.999670i	$0.491825\pi$
$822$	0	0
$823$	−45.5021	−1.58610	−0.793052	−	0.609154i	$-0.791509\pi$
$823$	−45.5021	−1.58610	−0.793052	+	0.609154i	$0.791509\pi$
$824$	0	0
$825$	−4.75902	−0.165688
$826$	0	0
$827$	22.4423	0.780394	0.390197	−	0.920731i	$-0.372407\pi$
$827$	22.4423	0.780394	0.390197	+	0.920731i	$0.372407\pi$
$828$	0	0
$829$	−35.6585	−1.23847	−0.619235	−	0.785206i	$-0.712557\pi$
$829$	−35.6585	−1.23847	−0.619235	+	0.785206i	$0.712557\pi$
$830$	0	0
$831$	4.63905	0.160927
$832$	0	0
$833$	21.5983	0.748337
$834$	0	0
$835$	19.0510	0.659288
$836$	0	0
$837$	3.45948	0.119577
$838$	0	0
$839$	39.8783	1.37675	0.688376	−	0.725354i	$-0.258324\pi$
$839$	39.8783	1.37675	0.688376	+	0.725354i	$0.258324\pi$
$840$	0	0
$841$	−21.9133	−0.755633
$842$	0	0
$843$	−13.7540	−0.473711
$844$	0	0
$845$	−2.41227	−0.0829847
$846$	0	0
$847$	35.1787	1.20875
$848$	0	0
$849$	8.66597	0.297415
$850$	0	0
$851$	0.349180	0.0119698
$852$	0	0
$853$	−29.6873	−1.01647	−0.508237	−	0.861217i	$-0.669703\pi$
$853$	−29.6873	−1.01647	−0.508237	+	0.861217i	$0.669703\pi$
$854$	0	0
$855$	2.37951	0.0813776
$856$	0	0
$857$	24.3542	0.831922	0.415961	−	0.909382i	$-0.363445\pi$
$857$	24.3542	0.831922	0.415961	+	0.909382i	$0.363445\pi$
$858$	0	0
$859$	45.6722	1.55832	0.779158	−	0.626827i	$-0.215647\pi$
$859$	45.6722	1.55832	0.779158	+	0.626827i	$0.215647\pi$
$860$	0	0
$861$	1.71194	0.0583428
$862$	0	0
$863$	6.40793	0.218128	0.109064	−	0.994035i	$-0.465215\pi$
$863$	6.40793	0.218128	0.109064	+	0.994035i	$0.465215\pi$
$864$	0	0
$865$	17.9365	0.609861
$866$	0	0
$867$	0.753362	0.0255855
$868$	0	0
$869$	44.9184	1.52375
$870$	0	0
$871$	−17.5911	−0.596050
$872$	0	0
$873$	−10.9262	−0.369797
$874$	0	0
$875$	1.37951	0.0466360
$876$	0	0
$877$	−30.4398	−1.02788	−0.513939	−	0.857827i	$-0.671814\pi$
$877$	−30.4398	−1.02788	−0.513939	+	0.857827i	$0.671814\pi$
$878$	0	0
$879$	1.18562	0.0399899
$880$	0	0
$881$	18.7914	0.633097	0.316548	−	0.948576i	$-0.397476\pi$
$881$	18.7914	0.633097	0.316548	+	0.948576i	$0.397476\pi$
$882$	0	0
$883$	19.8805	0.669031	0.334515	−	0.942390i	$-0.391427\pi$
$883$	19.8805	0.669031	0.334515	+	0.942390i	$0.391427\pi$
$884$	0	0
$885$	3.02928	0.101828
$886$	0	0
$887$	48.7641	1.63734	0.818669	−	0.574266i	$-0.194713\pi$
$887$	48.7641	1.63734	0.818669	+	0.574266i	$0.194713\pi$
$888$	0	0
$889$	0.367206	0.0123157
$890$	0	0
$891$	−22.9617	−0.769247
$892$	0	0
$893$	7.22519	0.241782
$894$	0	0
$895$	−21.7090	−0.725652
$896$	0	0
$897$	−0.141887	−0.00473746
$898$	0	0
$899$	−2.17331	−0.0724840
$900$	0	0
$901$	45.6460	1.52069
$902$	0	0
$903$	−0.779605	−0.0259436
$904$	0	0
$905$	−0.367206	−0.0122063
$906$	0	0
$907$	42.0855	1.39743	0.698713	−	0.715402i	$-0.253756\pi$
$907$	42.0855	1.39743	0.698713	+	0.715402i	$0.253756\pi$
$908$	0	0
$909$	6.05377	0.200791
$910$	0	0
$911$	−9.44705	−0.312995	−0.156497	−	0.987678i	$-0.550020\pi$
$911$	−9.44705	−0.312995	−0.156497	+	0.987678i	$0.550020\pi$
$912$	0	0
$913$	19.8319	0.656341
$914$	0	0
$915$	6.82162	0.225516
$916$	0	0
$917$	−18.9949	−0.627268
$918$	0	0
$919$	0.485842	0.0160265	0.00801323	−	0.999968i	$-0.497449\pi$
$919$	0.485842	0.0160265	0.00801323	+	0.999968i	$0.497449\pi$
$920$	0	0
$921$	1.87873	0.0619062
$922$	0	0
$923$	−40.6987	−1.33961
$924$	0	0
$925$	−6.30777	−0.207398
$926$	0	0
$927$	17.7269	0.582226
$928$	0	0
$929$	17.9494	0.588902	0.294451	−	0.955667i	$-0.404863\pi$
$929$	17.9494	0.588902	0.294451	+	0.955667i	$0.404863\pi$
$930$	0	0
$931$	5.09695	0.167046
$932$	0	0
$933$	−16.0257	−0.524657
$934$	0	0
$935$	25.6012	0.837250
$936$	0	0
$937$	45.4249	1.48397	0.741983	−	0.670419i	$-0.233886\pi$
$937$	45.4249	1.48397	0.741983	+	0.670419i	$0.233886\pi$
$938$	0	0
$939$	6.81773	0.222488
$940$	0	0
$941$	−28.9484	−0.943691	−0.471846	−	0.881681i	$-0.656412\pi$
$941$	−28.9484	−0.943691	−0.471846	+	0.881681i	$0.656412\pi$
$942$	0	0
$943$	−0.0872108	−0.00283998
$944$	0	0
$945$	−5.84568	−0.190160
$946$	0	0
$947$	−55.2664	−1.79592	−0.897958	−	0.440082i	$-0.854949\pi$
$947$	−55.2664	−1.79592	−0.897958	+	0.440082i	$0.854949\pi$
$948$	0	0
$949$	14.6059	0.474127
$950$	0	0
$951$	4.73643	0.153589
$952$	0	0
$953$	−52.1029	−1.68778	−0.843889	−	0.536518i	$-0.819740\pi$
$953$	−52.1029	−1.68778	−0.843889	+	0.536518i	$0.819740\pi$
$954$	0	0
$955$	−16.8026	−0.543720
$956$	0	0
$957$	−12.6689	−0.409527
$958$	0	0
$959$	11.2110	0.362022
$960$	0	0
$961$	−30.3335	−0.978500
$962$	0	0
$963$	9.78423	0.315293
$964$	0	0
$965$	19.9090	0.640893
$966$	0	0
$967$	45.5075	1.46342	0.731711	−	0.681615i	$-0.238722\pi$
$967$	45.5075	1.46342	0.731711	+	0.681615i	$0.238722\pi$
$968$	0	0
$969$	3.33792	0.107230
$970$	0	0
$971$	−34.9581	−1.12186	−0.560930	−	0.827863i	$-0.689556\pi$
$971$	−34.9581	−1.12186	−0.560930	+	0.827863i	$0.689556\pi$
$972$	0	0
$973$	26.9828	0.865027
$974$	0	0
$975$	2.56311	0.0820854
$976$	0	0
$977$	−8.89752	−0.284657	−0.142328	−	0.989819i	$-0.545459\pi$
$977$	−8.89752	−0.284657	−0.142328	+	0.989819i	$0.545459\pi$
$978$	0	0
$979$	10.5651	0.337663
$980$	0	0
$981$	−28.9048	−0.922859
$982$	0	0
$983$	58.6974	1.87216	0.936079	−	0.351791i	$-0.114427\pi$
$983$	58.6974	1.87216	0.936079	+	0.351791i	$0.114427\pi$
$984$	0	0
$985$	10.9857	0.350033
$986$	0	0
$987$	−7.85130	−0.249910
$988$	0	0
$989$	0.0397151	0.00126287
$990$	0	0
$991$	15.0103	0.476818	0.238409	−	0.971165i	$-0.423374\pi$
$991$	15.0103	0.476818	0.238409	+	0.971165i	$0.423374\pi$
$992$	0	0
$993$	−27.8251	−0.883003
$994$	0	0
$995$	−14.2375	−0.451359
$996$	0	0
$997$	29.4836	0.933754	0.466877	−	0.884322i	$-0.345379\pi$
$997$	29.4836	0.933754	0.466877	+	0.884322i	$0.345379\pi$
$998$	0	0
$999$	26.7292	0.845674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.r.1.3	✓	4
4.3	odd	2		3040.2.a.t.1.2	yes	4
8.3	odd	2		6080.2.a.cd.1.3		4
8.5	even	2		6080.2.a.cf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.r.1.3	✓	4	1.1	even	1	trivial
3040.2.a.t.1.2	yes	4	4.3	odd	2
6080.2.a.cd.1.3		4	8.3	odd	2
6080.2.a.cf.1.2		4	8.5	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.787711 q^{3} +1.00000 q^{5} +1.37951 q^{7} -2.37951 q^{9} -6.04159 q^{11} +3.25388 q^{13} +0.787711 q^{15} -4.23750 q^{17} -1.00000 q^{19} +1.08666 q^{21} -0.0553572 q^{23} +1.00000 q^{25} -4.23750 q^{27} +2.66208 q^{29} -0.816397 q^{31} -4.75902 q^{33} +1.37951 q^{35} -6.30777 q^{37} +2.56311 q^{39} +1.57542 q^{41} -0.717435 q^{43} -2.37951 q^{45} -7.22519 q^{47} -5.09695 q^{49} -3.33792 q^{51} -10.7719 q^{53} -6.04159 q^{55} -0.787711 q^{57} +3.84568 q^{59} +8.66006 q^{61} -3.28257 q^{63} +3.25388 q^{65} -5.40618 q^{67} -0.0436054 q^{69} -12.5078 q^{71} +4.48876 q^{73} +0.787711 q^{75} -8.33445 q^{77} -7.43487 q^{79} +3.80061 q^{81} -3.28257 q^{83} -4.23750 q^{85} +2.09695 q^{87} -1.74873 q^{89} +4.48876 q^{91} -0.643085 q^{93} -1.00000 q^{95} +4.59180 q^{97} +14.3760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} - 5 q^{7} + q^{9} - 6 q^{11} - q^{13} - q^{15} - q^{17} - 4 q^{19} + 5 q^{21} - 5 q^{23} + 4 q^{25} - q^{27} + 3 q^{29} - 16 q^{31} + 2 q^{33} - 5 q^{35} - 8 q^{37} - 13 q^{39}+ \cdots + 10 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 - 5 * q^7 + q^9 - 6 * q^11 - q^13 - q^15 - q^17 - 4 * q^19 + 5 * q^21 - 5 * q^23 + 4 * q^25 - q^27 + 3 * q^29 - 16 * q^31 + 2 * q^33 - 5 * q^35 - 8 * q^37 - 13 * q^39 - 2 * q^41 + q^45 + 2 * q^47 - 7 * q^49 - 21 * q^51 + 13 * q^53 - 6 * q^55 + q^57 - 5 * q^59 - 2 * q^61 - 16 * q^63 - q^65 + q^67 - 11 * q^69 - 22 * q^71 + 9 * q^73 - q^75 - 4 * q^77 - 24 * q^79 - 24 * q^81 - 16 * q^83 - q^85 - 5 * q^87 + 9 * q^91 - 14 * q^93 - 4 * q^95 + 12 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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