LMFDB - Embedded newform 1472.2.a.u.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,2,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$11.7539791775$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} -2.00000 q^{5} +3.56155 q^{9} -5.12311 q^{11} -4.56155 q^{13} -5.12311 q^{15} -3.12311 q^{17} -5.12311 q^{19} -1.00000 q^{23} -1.00000 q^{25} +1.43845 q^{27} +0.561553 q^{29} -6.56155 q^{31} -13.1231 q^{33} +8.24621 q^{37} -11.6847 q^{39} +10.8078 q^{41} +8.00000 q^{43} -7.12311 q^{45} +11.6847 q^{47} -7.00000 q^{49} -8.00000 q^{51} -2.00000 q^{53} +10.2462 q^{55} -13.1231 q^{57} +6.24621 q^{59} -12.2462 q^{61} +9.12311 q^{65} +5.12311 q^{67} -2.56155 q^{69} +9.43845 q^{71} -2.31534 q^{73} -2.56155 q^{75} -5.12311 q^{79} -7.00000 q^{81} +2.24621 q^{83} +6.24621 q^{85} +1.43845 q^{87} -13.3693 q^{89} -16.8078 q^{93} +10.2462 q^{95} -13.3693 q^{97} -18.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 4 q^{5} + 3 q^{9} - 2 q^{11} - 5 q^{13} - 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{23} - 2 q^{25} + 7 q^{27} - 3 q^{29} - 9 q^{31} - 18 q^{33} - 11 q^{39} + q^{41} + 16 q^{43} - 6 q^{45} + 11 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) $ 2 * q + q^3 - 4 * q^5 + 3 * q^9 - 2 * q^11 - 5 * q^13 - 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^23 - 2 * q^25 + 7 * q^27 - 3 * q^29 - 9 * q^31 - 18 * q^33 - 11 * q^39 + q^41 + 16 * q^43 - 6 * q^45 + 11 * q^47 - 14 * q^49 - 16 * q^51 - 4 * q^53 + 4 * q^55 - 18 * q^57 - 4 * q^59 - 8 * q^61 + 10 * q^65 + 2 * q^67 - q^69 + 23 * q^71 - 17 * q^73 - q^75 - 2 * q^79 - 14 * q^81 - 12 * q^83 - 4 * q^85 + 7 * q^87 - 2 * q^89 - 13 * q^93 + 4 * q^95 - 2 * q^97 - 20 * 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 $\theta_p = \textrm{Arg}(\alpha_p)$. 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 $

$2$	0	0
$2$	0	0
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	−4.56155	−1.26515	−0.632574	−	0.774500i	$-0.718001\pi$
$13$	−4.56155	−1.26515	−0.632574	+	0.774500i	$0.718001\pi$
$14$	0	0
$15$	−5.12311	−1.32278
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	0.561553	0.104278	0.0521389	−	0.998640i	$-0.483396\pi$
$29$	0.561553	0.104278	0.0521389	+	0.998640i	$0.483396\pi$
$30$	0	0
$31$	−6.56155	−1.17849	−0.589245	−	0.807955i	$-0.700575\pi$
$31$	−6.56155	−1.17849	−0.589245	+	0.807955i	$0.700575\pi$
$32$	0	0
$33$	−13.1231	−2.28444
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	−11.6847	−1.87104
$40$	0	0
$41$	10.8078	1.68789	0.843945	−	0.536430i	$-0.180228\pi$
$41$	10.8078	1.68789	0.843945	+	0.536430i	$0.180228\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−7.12311	−1.06185
$46$	0	0
$47$	11.6847	1.70438	0.852191	−	0.523230i	$-0.175273\pi$
$47$	11.6847	1.70438	0.852191	+	0.523230i	$0.175273\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−8.00000	−1.12022
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	10.2462	1.38160
$56$	0	0
$57$	−13.1231	−1.73820
$58$	0	0
$59$	6.24621	0.813187	0.406594	−	0.913609i	$-0.366716\pi$
$59$	6.24621	0.813187	0.406594	+	0.913609i	$0.366716\pi$
$60$	0	0
$61$	−12.2462	−1.56797	−0.783983	−	0.620782i	$-0.786815\pi$
$61$	−12.2462	−1.56797	−0.783983	+	0.620782i	$0.786815\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.12311	1.13158
$66$	0	0
$67$	5.12311	0.625887	0.312943	−	0.949772i	$-0.398685\pi$
$67$	5.12311	0.625887	0.312943	+	0.949772i	$0.398685\pi$
$68$	0	0
$69$	−2.56155	−0.308375
$70$	0	0
$71$	9.43845	1.12014	0.560069	−	0.828446i	$-0.310775\pi$
$71$	9.43845	1.12014	0.560069	+	0.828446i	$0.310775\pi$
$72$	0	0
$73$	−2.31534	−0.270990	−0.135495	−	0.990778i	$-0.543263\pi$
$73$	−2.31534	−0.270990	−0.135495	+	0.990778i	$0.543263\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	2.24621	0.246554	0.123277	−	0.992372i	$-0.460660\pi$
$83$	2.24621	0.246554	0.123277	+	0.992372i	$0.460660\pi$
$84$	0	0
$85$	6.24621	0.677497
$86$	0	0
$87$	1.43845	0.154218
$88$	0	0
$89$	−13.3693	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$89$	−13.3693	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−16.8078	−1.74288
$94$	0	0
$95$	10.2462	1.05124
$96$	0	0
$97$	−13.3693	−1.35745	−0.678724	−	0.734393i	$-0.737467\pi$
$97$	−13.3693	−1.35745	−0.678724	+	0.734393i	$0.737467\pi$
$98$	0	0
$99$	−18.2462	−1.83381

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	1472.2.a.u.1.2		2
4.3	odd	2		1472.2.a.p.1.1		2
8.3	odd	2		368.2.a.i.1.2		2
8.5	even	2		184.2.a.e.1.1	✓	2
24.5	odd	2		1656.2.a.j.1.1		2
24.11	even	2		3312.2.a.t.1.2		2
40.13	odd	4		4600.2.e.o.4049.1		4
40.19	odd	2		9200.2.a.br.1.1		2
40.29	even	2		4600.2.a.s.1.2		2
40.37	odd	4		4600.2.e.o.4049.4		4
56.13	odd	2		9016.2.a.w.1.2		2
184.45	odd	2		4232.2.a.o.1.1		2
184.91	even	2		8464.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.a.e.1.1	✓	2	8.5	even	2
368.2.a.i.1.2		2	8.3	odd	2
1472.2.a.p.1.1		2	4.3	odd	2
1472.2.a.u.1.2		2	1.1	even	1	trivial
1656.2.a.j.1.1		2	24.5	odd	2
3312.2.a.t.1.2		2	24.11	even	2
4232.2.a.o.1.1		2	184.45	odd	2
4600.2.a.s.1.2		2	40.29	even	2
4600.2.e.o.4049.1		4	40.13	odd	4
4600.2.e.o.4049.4		4	40.37	odd	4
8464.2.a.bd.1.2		2	184.91	even	2
9016.2.a.w.1.2		2	56.13	odd	2
9200.2.a.br.1.1		2	40.19	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} -2.00000 q^{5} +3.56155 q^{9} -5.12311 q^{11} -4.56155 q^{13} -5.12311 q^{15} -3.12311 q^{17} -5.12311 q^{19} -1.00000 q^{23} -1.00000 q^{25} +1.43845 q^{27} +0.561553 q^{29} -6.56155 q^{31} -13.1231 q^{33} +8.24621 q^{37} -11.6847 q^{39} +10.8078 q^{41} +8.00000 q^{43} -7.12311 q^{45} +11.6847 q^{47} -7.00000 q^{49} -8.00000 q^{51} -2.00000 q^{53} +10.2462 q^{55} -13.1231 q^{57} +6.24621 q^{59} -12.2462 q^{61} +9.12311 q^{65} +5.12311 q^{67} -2.56155 q^{69} +9.43845 q^{71} -2.31534 q^{73} -2.56155 q^{75} -5.12311 q^{79} -7.00000 q^{81} +2.24621 q^{83} +6.24621 q^{85} +1.43845 q^{87} -13.3693 q^{89} -16.8078 q^{93} +10.2462 q^{95} -13.3693 q^{97} -18.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 4 q^{5} + 3 q^{9} - 2 q^{11} - 5 q^{13} - 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{23} - 2 q^{25} + 7 q^{27} - 3 q^{29} - 9 q^{31} - 18 q^{33} - 11 q^{39} + q^{41} + 16 q^{43} - 6 q^{45} + 11 q^{47}+ \cdots - 20 q^{99}+O(q^{100}) \) 2 * q + q^3 - 4 * q^5 + 3 * q^9 - 2 * q^11 - 5 * q^13 - 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^23 - 2 * q^25 + 7 * q^27 - 3 * q^29 - 9 * q^31 - 18 * q^33 - 11 * q^39 + q^41 + 16 * q^43 - 6 * q^45 + 11 * q^47 - 14 * q^49 - 16 * q^51 - 4 * q^53 + 4 * q^55 - 18 * q^57 - 4 * q^59 - 8 * q^61 + 10 * q^65 + 2 * q^67 - q^69 + 23 * q^71 - 17 * q^73 - q^75 - 2 * q^79 - 14 * q^81 - 12 * q^83 - 4 * q^85 + 7 * q^87 - 2 * q^89 - 13 * q^93 + 4 * q^95 - 2 * q^97 - 20 * q^99

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