LMFDB - Embedded newform 476.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [476,2,Mod(1,476)] 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("476.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(476, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.80087913621$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{3} -2.30278 q^{5} +1.00000 q^{7} -1.30278 q^{9} +6.60555 q^{13} +3.00000 q^{15} +1.00000 q^{17} +6.60555 q^{19} -1.30278 q^{21} +0.302776 q^{25} +5.60555 q^{27} +4.30278 q^{31} -2.30278 q^{35} -2.60555 q^{37} -8.60555 q^{39} +3.90833 q^{41} +7.30278 q^{43} +3.00000 q^{45} +4.60555 q^{47} +1.00000 q^{49} -1.30278 q^{51} -3.69722 q^{53} -8.60555 q^{57} +9.21110 q^{59} -7.90833 q^{61} -1.30278 q^{63} -15.2111 q^{65} -1.69722 q^{67} -7.81665 q^{71} -7.90833 q^{73} -0.394449 q^{75} +12.6056 q^{79} -3.39445 q^{81} +6.00000 q^{83} -2.30278 q^{85} -16.6056 q^{89} +6.60555 q^{91} -5.60555 q^{93} -15.2111 q^{95} -6.30278 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9} + 6 q^{13} + 6 q^{15} + 2 q^{17} + 6 q^{19} + q^{21} - 3 q^{25} + 4 q^{27} + 5 q^{31} - q^{35} + 2 q^{37} - 10 q^{39} - 3 q^{41} + 11 q^{43} + 6 q^{45} + 2 q^{47}+ \cdots - 9 q^{97}+O(q^{100}) $ 2 * q + q^3 - q^5 + 2 * q^7 + q^9 + 6 * q^13 + 6 * q^15 + 2 * q^17 + 6 * q^19 + q^21 - 3 * q^25 + 4 * q^27 + 5 * q^31 - q^35 + 2 * q^37 - 10 * q^39 - 3 * q^41 + 11 * q^43 + 6 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 11 * q^53 - 10 * q^57 + 4 * q^59 - 5 * q^61 + q^63 - 16 * q^65 - 7 * q^67 + 6 * q^71 - 5 * q^73 - 8 * q^75 + 18 * q^79 - 14 * q^81 + 12 * q^83 - q^85 - 26 * q^89 + 6 * q^91 - 4 * q^93 - 16 * q^95 - 9 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	0	0
$5$	−2.30278	−1.02983	−0.514916	−	0.857240i	$-0.672177\pi$
$5$	−2.30278	−1.02983	−0.514916	+	0.857240i	$0.672177\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	6.60555	1.83205	0.916025	−	0.401121i	$-0.131379\pi$
$13$	6.60555	1.83205	0.916025	+	0.401121i	$0.131379\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	6.60555	1.51542	0.757709	−	0.652593i	$-0.226319\pi$
$19$	6.60555	1.51542	0.757709	+	0.652593i	$0.226319\pi$
$20$	0	0
$21$	−1.30278	−0.284289
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	5.60555	1.07879
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	4.30278	0.772801	0.386401	−	0.922331i	$-0.373718\pi$
$31$	4.30278	0.772801	0.386401	+	0.922331i	$0.373718\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.30278	−0.389240
$36$	0	0
$37$	−2.60555	−0.428350	−0.214175	−	0.976795i	$-0.568706\pi$
$37$	−2.60555	−0.428350	−0.214175	+	0.976795i	$0.568706\pi$
$38$	0	0
$39$	−8.60555	−1.37799
$40$	0	0
$41$	3.90833	0.610378	0.305189	−	0.952292i	$-0.401280\pi$
$41$	3.90833	0.610378	0.305189	+	0.952292i	$0.401280\pi$
$42$	0	0
$43$	7.30278	1.11366	0.556831	−	0.830626i	$-0.312017\pi$
$43$	7.30278	1.11366	0.556831	+	0.830626i	$0.312017\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	4.60555	0.671789	0.335894	−	0.941900i	$-0.390961\pi$
$47$	4.60555	0.671789	0.335894	+	0.941900i	$0.390961\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.30278	−0.182425
$52$	0	0
$53$	−3.69722	−0.507853	−0.253926	−	0.967224i	$-0.581722\pi$
$53$	−3.69722	−0.507853	−0.253926	+	0.967224i	$0.581722\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.60555	−1.13983
$58$	0	0
$59$	9.21110	1.19918	0.599592	−	0.800306i	$-0.295330\pi$
$59$	9.21110	1.19918	0.599592	+	0.800306i	$0.295330\pi$
$60$	0	0
$61$	−7.90833	−1.01256	−0.506279	−	0.862370i	$-0.668979\pi$
$61$	−7.90833	−1.01256	−0.506279	+	0.862370i	$0.668979\pi$
$62$	0	0
$63$	−1.30278	−0.164134
$64$	0	0
$65$	−15.2111	−1.88671
$66$	0	0
$67$	−1.69722	−0.207349	−0.103674	−	0.994611i	$-0.533060\pi$
$67$	−1.69722	−0.207349	−0.103674	+	0.994611i	$0.533060\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.81665	−0.927666	−0.463833	−	0.885923i	$-0.653526\pi$
$71$	−7.81665	−0.927666	−0.463833	+	0.885923i	$0.653526\pi$
$72$	0	0
$73$	−7.90833	−0.925600	−0.462800	−	0.886463i	$-0.653155\pi$
$73$	−7.90833	−0.925600	−0.462800	+	0.886463i	$0.653155\pi$
$74$	0	0
$75$	−0.394449	−0.0455470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.6056	1.41824	0.709118	−	0.705090i	$-0.249093\pi$
$79$	12.6056	1.41824	0.709118	+	0.705090i	$0.249093\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−2.30278	−0.249771
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−16.6056	−1.76018	−0.880092	−	0.474802i	$-0.842520\pi$
$89$	−16.6056	−1.76018	−0.880092	+	0.474802i	$0.842520\pi$
$90$	0	0
$91$	6.60555	0.692450
$92$	0	0
$93$	−5.60555	−0.581269
$94$	0	0
$95$	−15.2111	−1.56063
$96$	0	0
$97$	−6.30278	−0.639950	−0.319975	−	0.947426i	$-0.603675\pi$
$97$	−6.30278	−0.639950	−0.319975	+	0.947426i	$0.603675\pi$
$98$	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	476.2.a.d.1.1	✓	2
3.2	odd	2		4284.2.a.n.1.2		2
4.3	odd	2		1904.2.a.h.1.2		2
7.6	odd	2		3332.2.a.j.1.2		2
8.3	odd	2		7616.2.a.v.1.1		2
8.5	even	2		7616.2.a.q.1.2		2
17.16	even	2		8092.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.d.1.1	✓	2	1.1	even	1	trivial
1904.2.a.h.1.2		2	4.3	odd	2
3332.2.a.j.1.2		2	7.6	odd	2
4284.2.a.n.1.2		2	3.2	odd	2
7616.2.a.q.1.2		2	8.5	even	2
7616.2.a.v.1.1		2	8.3	odd	2
8092.2.a.k.1.2		2	17.16	even	2

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Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{3} -2.30278 q^{5} +1.00000 q^{7} -1.30278 q^{9} +6.60555 q^{13} +3.00000 q^{15} +1.00000 q^{17} +6.60555 q^{19} -1.30278 q^{21} +0.302776 q^{25} +5.60555 q^{27} +4.30278 q^{31} -2.30278 q^{35} -2.60555 q^{37} -8.60555 q^{39} +3.90833 q^{41} +7.30278 q^{43} +3.00000 q^{45} +4.60555 q^{47} +1.00000 q^{49} -1.30278 q^{51} -3.69722 q^{53} -8.60555 q^{57} +9.21110 q^{59} -7.90833 q^{61} -1.30278 q^{63} -15.2111 q^{65} -1.69722 q^{67} -7.81665 q^{71} -7.90833 q^{73} -0.394449 q^{75} +12.6056 q^{79} -3.39445 q^{81} +6.00000 q^{83} -2.30278 q^{85} -16.6056 q^{89} +6.60555 q^{91} -5.60555 q^{93} -15.2111 q^{95} -6.30278 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9} + 6 q^{13} + 6 q^{15} + 2 q^{17} + 6 q^{19} + q^{21} - 3 q^{25} + 4 q^{27} + 5 q^{31} - q^{35} + 2 q^{37} - 10 q^{39} - 3 q^{41} + 11 q^{43} + 6 q^{45} + 2 q^{47}+ \cdots - 9 q^{97}+O(q^{100}) \) 2 * q + q^3 - q^5 + 2 * q^7 + q^9 + 6 * q^13 + 6 * q^15 + 2 * q^17 + 6 * q^19 + q^21 - 3 * q^25 + 4 * q^27 + 5 * q^31 - q^35 + 2 * q^37 - 10 * q^39 - 3 * q^41 + 11 * q^43 + 6 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 11 * q^53 - 10 * q^57 + 4 * q^59 - 5 * q^61 + q^63 - 16 * q^65 - 7 * q^67 + 6 * q^71 - 5 * q^73 - 8 * q^75 + 18 * q^79 - 14 * q^81 + 12 * q^83 - q^85 - 26 * q^89 + 6 * q^91 - 4 * q^93 - 16 * q^95 - 9 * q^97

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