LMFDB - Embedded newform 5733.2.a.bm.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5733,2,Mod(1,5733)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5733.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5733, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-4,0,8,2,0,0,-9,0,5,-11,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.746052.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 8x + 2 $ x^5 - x^4 - 7x^3 + 8x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.72525$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.72525 q^{2} +5.42699 q^{4} +2.18716 q^{5} -9.33940 q^{8} -5.96057 q^{10} +1.04815 q^{11} -1.00000 q^{13} +14.5982 q^{16} -5.29125 q^{17} -0.756906 q^{19} +11.8697 q^{20} -2.85648 q^{22} -0.653584 q^{23} -0.216314 q^{25} +2.72525 q^{26} +3.10408 q^{29} -1.02791 q^{31} -21.1050 q^{32} +14.4200 q^{34} -10.8932 q^{37} +2.06276 q^{38} -20.4268 q^{40} +7.32040 q^{41} +0.887771 q^{43} +5.68833 q^{44} +1.78118 q^{46} +2.33751 q^{47} +0.589510 q^{50} -5.42699 q^{52} -4.88814 q^{53} +2.29249 q^{55} -8.45941 q^{58} -1.04815 q^{59} +12.4998 q^{61} +2.80132 q^{62} +28.3200 q^{64} -2.18716 q^{65} +4.47889 q^{67} -28.7155 q^{68} +6.60274 q^{71} +8.28347 q^{73} +29.6868 q^{74} -4.10772 q^{76} +2.14014 q^{79} +31.9287 q^{80} -19.9499 q^{82} -6.66558 q^{83} -11.5728 q^{85} -2.41940 q^{86} -9.78914 q^{88} -5.76777 q^{89} -3.54699 q^{92} -6.37030 q^{94} -1.65548 q^{95} +2.88777 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} + 8 q^{4} + 2 q^{5} - 9 q^{8} + 5 q^{10} - 11 q^{11} - 5 q^{13} + 10 q^{16} - 5 q^{17} - 9 q^{19} + q^{20} + 8 q^{22} - 10 q^{23} + 9 q^{25} + 4 q^{26} + 3 q^{29} + 6 q^{31} - 22 q^{32} + 22 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) $ 5 * q - 4 * q^2 + 8 * q^4 + 2 * q^5 - 9 * q^8 + 5 * q^10 - 11 * q^11 - 5 * q^13 + 10 * q^16 - 5 * q^17 - 9 * q^19 + q^20 + 8 * q^22 - 10 * q^23 + 9 * q^25 + 4 * q^26 + 3 * q^29 + 6 * q^31 - 22 * q^32 + 22 * q^34 + 4 * q^37 - 10 * q^38 - 28 * q^40 + 14 * q^41 + 2 * q^43 + 3 * q^46 + q^47 - 9 * q^50 - 8 * q^52 - 17 * q^53 - 27 * q^58 + 11 * q^59 + 11 * q^61 - 23 * q^62 + 9 * q^64 - 2 * q^65 + 13 * q^67 - 32 * q^68 - 15 * q^71 + 33 * q^74 - 8 * q^76 + 2 * q^79 + 55 * q^80 - 34 * q^82 + 6 * q^83 - 22 * q^85 - 28 * q^86 - 3 * q^88 - 4 * q^89 - 21 * q^92 - 20 * q^94 + 12 * q^95 + 12 * q^97

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	−2.72525	−1.92704	−0.963521	−	0.267631i	$-0.913759\pi$
$2$	−2.72525	−1.92704	−0.963521	+	0.267631i	$0.913759\pi$
$3$	0	0
$3$	0	0
$4$	5.42699	2.71349
$5$	2.18716	0.978129	0.489065	−	0.872247i	$-0.337338\pi$
$5$	2.18716	0.978129	0.489065	+	0.872247i	$0.337338\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−9.33940	−3.30198
$9$	0	0
$10$	−5.96057	−1.88490
$11$	1.04815	0.316031	0.158015	−	0.987437i	$-0.449490\pi$
$11$	1.04815	0.316031	0.158015	+	0.987437i	$0.449490\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	14.5982	3.64956
$17$	−5.29125	−1.28332	−0.641658	−	0.766991i	$-0.721753\pi$
$17$	−5.29125	−1.28332	−0.641658	+	0.766991i	$0.721753\pi$
$18$	0	0
$19$	−0.756906	−0.173646	−0.0868231	−	0.996224i	$-0.527671\pi$
$19$	−0.756906	−0.173646	−0.0868231	+	0.996224i	$0.527671\pi$
$20$	11.8697	2.65415
$21$	0	0
$22$	−2.85648	−0.609005
$23$	−0.653584	−0.136282	−0.0681408	−	0.997676i	$-0.521707\pi$
$23$	−0.653584	−0.136282	−0.0681408	+	0.997676i	$0.521707\pi$
$24$	0	0
$25$	−0.216314	−0.0432628
$26$	2.72525	0.534466
$27$	0	0
$28$	0	0
$29$	3.10408	0.576414	0.288207	−	0.957568i	$-0.406941\pi$
$29$	3.10408	0.576414	0.288207	+	0.957568i	$0.406941\pi$
$30$	0	0
$31$	−1.02791	−0.184618	−0.0923092	−	0.995730i	$-0.529425\pi$
$31$	−1.02791	−0.184618	−0.0923092	+	0.995730i	$0.529425\pi$
$32$	−21.1050	−3.73088
$33$	0	0
$34$	14.4200	2.47301
$35$	0	0
$36$	0	0
$37$	−10.8932	−1.79084	−0.895418	−	0.445227i	$-0.853123\pi$
$37$	−10.8932	−1.79084	−0.895418	+	0.445227i	$0.853123\pi$
$38$	2.06276	0.334624
$39$	0	0
$40$	−20.4268	−3.22976
$41$	7.32040	1.14325	0.571627	−	0.820514i	$-0.306312\pi$
$41$	7.32040	1.14325	0.571627	+	0.820514i	$0.306312\pi$
$42$	0	0
$43$	0.887771	0.135384	0.0676919	−	0.997706i	$-0.478437\pi$
$43$	0.887771	0.135384	0.0676919	+	0.997706i	$0.478437\pi$
$44$	5.68833	0.857547
$45$	0	0
$46$	1.78118	0.262621
$47$	2.33751	0.340961	0.170480	−	0.985361i	$-0.445468\pi$
$47$	2.33751	0.340961	0.170480	+	0.985361i	$0.445468\pi$
$48$	0	0
$49$	0	0
$50$	0.589510	0.0833692
$51$	0	0
$52$	−5.42699	−0.752588
$53$	−4.88814	−0.671438	−0.335719	−	0.941962i	$-0.608979\pi$
$53$	−4.88814	−0.671438	−0.335719	+	0.941962i	$0.608979\pi$
$54$	0	0
$55$	2.29249	0.309119
$56$	0	0
$57$	0	0
$58$	−8.45941	−1.11077
$59$	−1.04815	−0.136458	−0.0682291	−	0.997670i	$-0.521735\pi$
$59$	−1.04815	−0.136458	−0.0682291	+	0.997670i	$0.521735\pi$
$60$	0	0
$61$	12.4998	1.60043	0.800217	−	0.599711i	$-0.204718\pi$
$61$	12.4998	1.60043	0.800217	+	0.599711i	$0.204718\pi$
$62$	2.80132	0.355768
$63$	0	0
$64$	28.3200	3.54000
$65$	−2.18716	−0.271284
$66$	0	0
$67$	4.47889	0.547183	0.273592	−	0.961846i	$-0.411788\pi$
$67$	4.47889	0.547183	0.273592	+	0.961846i	$0.411788\pi$
$68$	−28.7155	−3.48227
$69$	0	0
$70$	0	0
$71$	6.60274	0.783601	0.391801	−	0.920050i	$-0.371852\pi$
$71$	6.60274	0.783601	0.391801	+	0.920050i	$0.371852\pi$
$72$	0	0
$73$	8.28347	0.969507	0.484754	−	0.874651i	$-0.338909\pi$
$73$	8.28347	0.969507	0.484754	+	0.874651i	$0.338909\pi$
$74$	29.6868	3.45102
$75$	0	0
$76$	−4.10772	−0.471188
$77$	0	0
$78$	0	0
$79$	2.14014	0.240785	0.120392	−	0.992726i	$-0.461585\pi$
$79$	2.14014	0.240785	0.120392	+	0.992726i	$0.461585\pi$
$80$	31.9287	3.56974
$81$	0	0
$82$	−19.9499	−2.20310
$83$	−6.66558	−0.731642	−0.365821	−	0.930685i	$-0.619212\pi$
$83$	−6.66558	−0.731642	−0.365821	+	0.930685i	$0.619212\pi$
$84$	0	0
$85$	−11.5728	−1.25525
$86$	−2.41940	−0.260890
$87$	0	0
$88$	−9.78914	−1.04353
$89$	−5.76777	−0.611382	−0.305691	−	0.952131i	$-0.598887\pi$
$89$	−5.76777	−0.611382	−0.305691	+	0.952131i	$0.598887\pi$
$90$	0	0
$91$	0	0
$92$	−3.54699	−0.369800
$93$	0	0
$94$	−6.37030	−0.657046
$95$	−1.65548	−0.169848
$96$	0	0
$97$	2.88777	0.293209	0.146604	−	0.989195i	$-0.453166\pi$
$97$	2.88777	0.293209	0.146604	+	0.989195i	$0.453166\pi$
$98$	0	0
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bm.1.1		5
3.2	odd	2		637.2.a.k.1.5		5
7.3	odd	6		819.2.j.h.352.5		10
7.5	odd	6		819.2.j.h.235.5		10
7.6	odd	2		5733.2.a.bl.1.1		5
21.2	odd	6		637.2.e.m.508.1		10
21.5	even	6		91.2.e.c.53.1	✓	10
21.11	odd	6		637.2.e.m.79.1		10
21.17	even	6		91.2.e.c.79.1	yes	10
21.20	even	2		637.2.a.l.1.5		5
39.38	odd	2		8281.2.a.bx.1.1		5
84.47	odd	6		1456.2.r.p.417.2		10
84.59	odd	6		1456.2.r.p.625.2		10
273.38	even	6		1183.2.e.f.170.5		10
273.194	even	6		1183.2.e.f.508.5		10
273.272	even	2		8281.2.a.bw.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.c.53.1	✓	10	21.5	even	6
91.2.e.c.79.1	yes	10	21.17	even	6
637.2.a.k.1.5		5	3.2	odd	2
637.2.a.l.1.5		5	21.20	even	2
637.2.e.m.79.1		10	21.11	odd	6
637.2.e.m.508.1		10	21.2	odd	6
819.2.j.h.235.5		10	7.5	odd	6
819.2.j.h.352.5		10	7.3	odd	6
1183.2.e.f.170.5		10	273.38	even	6
1183.2.e.f.508.5		10	273.194	even	6
1456.2.r.p.417.2		10	84.47	odd	6
1456.2.r.p.625.2		10	84.59	odd	6
5733.2.a.bl.1.1		5	7.6	odd	2
5733.2.a.bm.1.1		5	1.1	even	1	trivial
8281.2.a.bw.1.1		5	273.272	even	2
8281.2.a.bx.1.1		5	39.38	odd	2

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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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q-2.72525 q^{2} +5.42699 q^{4} +2.18716 q^{5} -9.33940 q^{8} -5.96057 q^{10} +1.04815 q^{11} -1.00000 q^{13} +14.5982 q^{16} -5.29125 q^{17} -0.756906 q^{19} +11.8697 q^{20} -2.85648 q^{22} -0.653584 q^{23} -0.216314 q^{25} +2.72525 q^{26} +3.10408 q^{29} -1.02791 q^{31} -21.1050 q^{32} +14.4200 q^{34} -10.8932 q^{37} +2.06276 q^{38} -20.4268 q^{40} +7.32040 q^{41} +0.887771 q^{43} +5.68833 q^{44} +1.78118 q^{46} +2.33751 q^{47} +0.589510 q^{50} -5.42699 q^{52} -4.88814 q^{53} +2.29249 q^{55} -8.45941 q^{58} -1.04815 q^{59} +12.4998 q^{61} +2.80132 q^{62} +28.3200 q^{64} -2.18716 q^{65} +4.47889 q^{67} -28.7155 q^{68} +6.60274 q^{71} +8.28347 q^{73} +29.6868 q^{74} -4.10772 q^{76} +2.14014 q^{79} +31.9287 q^{80} -19.9499 q^{82} -6.66558 q^{83} -11.5728 q^{85} -2.41940 q^{86} -9.78914 q^{88} -5.76777 q^{89} -3.54699 q^{92} -6.37030 q^{94} -1.65548 q^{95} +2.88777 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} + 8 q^{4} + 2 q^{5} - 9 q^{8} + 5 q^{10} - 11 q^{11} - 5 q^{13} + 10 q^{16} - 5 q^{17} - 9 q^{19} + q^{20} + 8 q^{22} - 10 q^{23} + 9 q^{25} + 4 q^{26} + 3 q^{29} + 6 q^{31} - 22 q^{32} + 22 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) \) 5 * q - 4 * q^2 + 8 * q^4 + 2 * q^5 - 9 * q^8 + 5 * q^10 - 11 * q^11 - 5 * q^13 + 10 * q^16 - 5 * q^17 - 9 * q^19 + q^20 + 8 * q^22 - 10 * q^23 + 9 * q^25 + 4 * q^26 + 3 * q^29 + 6 * q^31 - 22 * q^32 + 22 * q^34 + 4 * q^37 - 10 * q^38 - 28 * q^40 + 14 * q^41 + 2 * q^43 + 3 * q^46 + q^47 - 9 * q^50 - 8 * q^52 - 17 * q^53 - 27 * q^58 + 11 * q^59 + 11 * q^61 - 23 * q^62 + 9 * q^64 - 2 * q^65 + 13 * q^67 - 32 * q^68 - 15 * q^71 + 33 * q^74 - 8 * q^76 + 2 * q^79 + 55 * q^80 - 34 * q^82 + 6 * q^83 - 22 * q^85 - 28 * q^86 - 3 * q^88 - 4 * q^89 - 21 * q^92 - 20 * q^94 + 12 * q^95 + 12 * q^97

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