LMFDB - Newform orbit 529.2.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [529,2,Mod(1,529)] 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("529.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(529, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 529 = 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	529.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,-2,4,-4] 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:	$4.22408626693$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of $\beta = \sqrt{2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (2 \beta - 1) q^{5} + (\beta + 2) q^{6} + ( - \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9} + (\beta + 3) q^{10} + \beta q^{11} + (\beta + 4) q^{12}+ \cdots - \beta q^{99} +O(q^{100}) $ q + (b + 1) * q^2 + b * q^3 + (2b + 1) q^4 + (2b - 1) q^5 + (b + 2) * q^6 + (-b - 2) * q^7 + (b + 3) * q^8 - q^9 + (b + 3) * q^10 + b * q^11 + (b + 4) * q^12 - 3 * q^13 + (-3b - 4) q^14 + (-b + 4) * q^15 + 3 * q^16 - 2b q^17 + (-b - 1) * q^18 + (3b - 2) q^19 + 7 * q^20 + (-2b - 2) q^21 + (b + 2) * q^22 + (3b + 2) q^24 + (-4b + 4) q^25 + (-3b - 3) q^26 - 4b q^27 + (-5b - 6) q^28 + 3 * q^29 + (3b + 2) q^30 + 6 * q^31 + (b - 3) * q^32 + 2 * q^33 + (-2b - 4) q^34 + (-3b - 2) q^35 + (-2b - 1) q^36 + (4b + 4) q^37 + (b + 4) * q^38 - 3b q^39 + (5b + 1) q^40 + (2b + 7) q^41 + (-4b - 6) q^42 - 6 * q^43 + (b + 4) * q^44 + (-2b + 1) q^45 - 5b q^47 + 3b q^48 + (4b - 1) q^49 - 4 * q^50 - 4 * q^51 + (-6b - 3) q^52 + (-2b - 7) q^53 + (-4b - 8) q^54 + (-b + 4) * q^55 + (-5b - 8) q^56 + (-2b + 6) q^57 + (3b + 3) q^58 + (b + 2) * q^59 + 7b q^60 + (4b - 1) q^61 + (6b + 6) q^62 + (b + 2) * q^63 + (-2b - 7) q^64 + (-6b + 3) q^65 + (2b + 2) q^66 + (8b + 4) q^67 + (-2b - 8) q^68 + (-5b - 8) q^70 + (5b - 2) q^71 + (-b - 3) * q^72 + (2b - 1) q^73 + (8b + 12) q^74 + (4b - 8) q^75 + (-b + 10) * q^76 + (-2b - 2) q^77 + (-3b - 6) q^78 + (4b + 4) q^79 + (6b - 3) q^80 - 5 * q^81 + (9b + 11) q^82 + (-4b + 10) q^83 + (-6b - 10) q^84 + (2b - 8) q^85 + (-6b - 6) q^86 + 3b q^87 + (3b + 2) q^88 + (-4b - 3) q^89 + (-b - 3) * q^90 + (3b + 6) q^91 + 6b q^93 + (-5b - 10) q^94 + (-7b + 14) q^95 + (-3b + 2) q^96 + (-6b + 5) q^97 + (3b + 7) q^98 - b * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} + 6 q^{10} + 8 q^{12} - 6 q^{13} - 8 q^{14} + 8 q^{15} + 6 q^{16} - 2 q^{18} - 4 q^{19} + 14 q^{20} - 4 q^{21} + 4 q^{22} + 4 q^{24}+ \cdots + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9 + 6 * q^10 + 8 * q^12 - 6 * q^13 - 8 * q^14 + 8 * q^15 + 6 * q^16 - 2 * q^18 - 4 * q^19 + 14 * q^20 - 4 * q^21 + 4 * q^22 + 4 * q^24 + 8 * q^25 - 6 * q^26 - 12 * q^28 + 6 * q^29 + 4 * q^30 + 12 * q^31 - 6 * q^32 + 4 * q^33 - 8 * q^34 - 4 * q^35 - 2 * q^36 + 8 * q^37 + 8 * q^38 + 2 * q^40 + 14 * q^41 - 12 * q^42 - 12 * q^43 + 8 * q^44 + 2 * q^45 - 2 * q^49 - 8 * q^50 - 8 * q^51 - 6 * q^52 - 14 * q^53 - 16 * q^54 + 8 * q^55 - 16 * q^56 + 12 * q^57 + 6 * q^58 + 4 * q^59 - 2 * q^61 + 12 * q^62 + 4 * q^63 - 14 * q^64 + 6 * q^65 + 4 * q^66 + 8 * q^67 - 16 * q^68 - 16 * q^70 - 4 * q^71 - 6 * q^72 - 2 * q^73 + 24 * q^74 - 16 * q^75 + 20 * q^76 - 4 * q^77 - 12 * q^78 + 8 * q^79 - 6 * q^80 - 10 * q^81 + 22 * q^82 + 20 * q^83 - 20 * q^84 - 16 * q^85 - 12 * q^86 + 4 * q^88 - 6 * q^89 - 6 * q^90 + 12 * q^91 - 20 * q^94 + 28 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^98

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−1.41421
1.41421

−0.414214

−1.41421

−1.82843

−3.82843

0.585786

−0.585786

1.58579

−1.00000

1.58579

1.2

2.41421

1.41421

3.82843

1.82843

3.41421

−3.41421

4.41421

−1.00000

4.41421

Atkin-Lehner signs

$ p $	Sign
$23$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	529.2.a.d	✓	2
3.b	odd	2	1		4761.2.a.j		2
4.b	odd	2	1		8464.2.a.z		2
23.b	odd	2	1		529.2.a.e	yes	2
23.c	even	11	10		529.2.c.k		20
23.d	odd	22	10		529.2.c.j		20
69.c	even	2	1		4761.2.a.i		2
92.b	even	2	1		8464.2.a.ba		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
529.2.a.d	✓	2	1.a	even	1	1	trivial
529.2.a.e	yes	2	23.b	odd	2	1
529.2.c.j		20	23.d	odd	22	10
529.2.c.k		20	23.c	even	11	10
4761.2.a.i		2	69.c	even	2	1
4761.2.a.j		2	3.b	odd	2	1
8464.2.a.z		2	4.b	odd	2	1
8464.2.a.ba		2	92.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(529))$:

$ T_{2}^{2} - 2T_{2} - 1 $

T2^2 - 2*T2 - 1

$ T_{5}^{2} + 2T_{5} - 7 $

T5^2 + 2*T5 - 7

$ T_{7}^{2} + 4T_{7} + 2 $

T7^2 + 4*T7 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 2T - 1 $ T^2 - 2*T - 1
$3$	$ T^{2} - 2 $ T^2 - 2
$5$	$ T^{2} + 2T - 7 $ T^2 + 2*T - 7
$7$	$ T^{2} + 4T + 2 $ T^2 + 4*T + 2
$11$	$ T^{2} - 2 $ T^2 - 2
$13$	$ (T + 3)^{2} $ (T + 3)^2
$17$	$ T^{2} - 8 $ T^2 - 8
$19$	$ T^{2} + 4T - 14 $ T^2 + 4*T - 14
$23$	$ T^{2} $ T^2
$29$	$ (T - 3)^{2} $ (T - 3)^2
$31$	$ (T - 6)^{2} $ (T - 6)^2
$37$	$ T^{2} - 8T - 16 $ T^2 - 8*T - 16
$41$	$ T^{2} - 14T + 41 $ T^2 - 14*T + 41
$43$	$ (T + 6)^{2} $ (T + 6)^2
$47$	$ T^{2} - 50 $ T^2 - 50
$53$	$ T^{2} + 14T + 41 $ T^2 + 14*T + 41
$59$	$ T^{2} - 4T + 2 $ T^2 - 4*T + 2
$61$	$ T^{2} + 2T - 31 $ T^2 + 2*T - 31
$67$	$ T^{2} - 8T - 112 $ T^2 - 8*T - 112
$71$	$ T^{2} + 4T - 46 $ T^2 + 4*T - 46
$73$	$ T^{2} + 2T - 7 $ T^2 + 2*T - 7
$79$	$ T^{2} - 8T - 16 $ T^2 - 8*T - 16
$83$	$ T^{2} - 20T + 68 $ T^2 - 20*T + 68
$89$	$ T^{2} + 6T - 23 $ T^2 + 6*T - 23
$97$	$ T^{2} - 10T - 47 $ T^2 - 10*T - 47
show more
show less

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:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	529.a (trivial)

\(f(q)\)	\(=\)	\( q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (2 \beta - 1) q^{5} + (\beta + 2) q^{6} + ( - \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9} + (\beta + 3) q^{10} + \beta q^{11} + (\beta + 4) q^{12}+ \cdots - \beta q^{99} +O(q^{100}) \) q + (b + 1) * q^2 + b * q^3 + (2b + 1) q^4 + (2b - 1) q^5 + (b + 2) * q^6 + (-b - 2) * q^7 + (b + 3) * q^8 - q^9 + (b + 3) * q^10 + b * q^11 + (b + 4) * q^12 - 3 * q^13 + (-3b - 4) q^14 + (-b + 4) * q^15 + 3 * q^16 - 2b q^17 + (-b - 1) * q^18 + (3b - 2) q^19 + 7 * q^20 + (-2b - 2) q^21 + (b + 2) * q^22 + (3b + 2) q^24 + (-4b + 4) q^25 + (-3b - 3) q^26 - 4b q^27 + (-5b - 6) q^28 + 3 * q^29 + (3b + 2) q^30 + 6 * q^31 + (b - 3) * q^32 + 2 * q^33 + (-2b - 4) q^34 + (-3b - 2) q^35 + (-2b - 1) q^36 + (4b + 4) q^37 + (b + 4) * q^38 - 3b q^39 + (5b + 1) q^40 + (2b + 7) q^41 + (-4b - 6) q^42 - 6 * q^43 + (b + 4) * q^44 + (-2b + 1) q^45 - 5b q^47 + 3b q^48 + (4b - 1) q^49 - 4 * q^50 - 4 * q^51 + (-6b - 3) q^52 + (-2b - 7) q^53 + (-4b - 8) q^54 + (-b + 4) * q^55 + (-5b - 8) q^56 + (-2b + 6) q^57 + (3b + 3) q^58 + (b + 2) * q^59 + 7b q^60 + (4b - 1) q^61 + (6b + 6) q^62 + (b + 2) * q^63 + (-2b - 7) q^64 + (-6b + 3) q^65 + (2b + 2) q^66 + (8b + 4) q^67 + (-2b - 8) q^68 + (-5b - 8) q^70 + (5b - 2) q^71 + (-b - 3) * q^72 + (2b - 1) q^73 + (8b + 12) q^74 + (4b - 8) q^75 + (-b + 10) * q^76 + (-2b - 2) q^77 + (-3b - 6) q^78 + (4b + 4) q^79 + (6b - 3) q^80 - 5 * q^81 + (9b + 11) q^82 + (-4b + 10) q^83 + (-6b - 10) q^84 + (2b - 8) q^85 + (-6b - 6) q^86 + 3b q^87 + (3b + 2) q^88 + (-4b - 3) q^89 + (-b - 3) * q^90 + (3b + 6) q^91 + 6b q^93 + (-5b - 10) q^94 + (-7b + 14) q^95 + (-3b + 2) q^96 + (-6b + 5) q^97 + (3b + 7) q^98 - b * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} + 6 q^{10} + 8 q^{12} - 6 q^{13} - 8 q^{14} + 8 q^{15} + 6 q^{16} - 2 q^{18} - 4 q^{19} + 14 q^{20} - 4 q^{21} + 4 q^{22} + 4 q^{24}+ \cdots + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9 + 6 * q^10 + 8 * q^12 - 6 * q^13 - 8 * q^14 + 8 * q^15 + 6 * q^16 - 2 * q^18 - 4 * q^19 + 14 * q^20 - 4 * q^21 + 4 * q^22 + 4 * q^24 + 8 * q^25 - 6 * q^26 - 12 * q^28 + 6 * q^29 + 4 * q^30 + 12 * q^31 - 6 * q^32 + 4 * q^33 - 8 * q^34 - 4 * q^35 - 2 * q^36 + 8 * q^37 + 8 * q^38 + 2 * q^40 + 14 * q^41 - 12 * q^42 - 12 * q^43 + 8 * q^44 + 2 * q^45 - 2 * q^49 - 8 * q^50 - 8 * q^51 - 6 * q^52 - 14 * q^53 - 16 * q^54 + 8 * q^55 - 16 * q^56 + 12 * q^57 + 6 * q^58 + 4 * q^59 - 2 * q^61 + 12 * q^62 + 4 * q^63 - 14 * q^64 + 6 * q^65 + 4 * q^66 + 8 * q^67 - 16 * q^68 - 16 * q^70 - 4 * q^71 - 6 * q^72 - 2 * q^73 + 24 * q^74 - 16 * q^75 + 20 * q^76 - 4 * q^77 - 12 * q^78 + 8 * q^79 - 6 * q^80 - 10 * q^81 + 22 * q^82 + 20 * q^83 - 20 * q^84 - 16 * q^85 - 12 * q^86 + 4 * q^88 - 6 * q^89 - 6 * q^90 + 12 * q^91 - 20 * q^94 + 28 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more