LMFDB - Newform orbit 9075.2.a.bt

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9075,2,Mod(1,9075)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9075.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9075 = 3 \cdot 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,-2,-1,0,-1,6,0,2,0,0,1,-2,8,0,-3,-4,1,-10,0,-6,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.4642398343$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 825)
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 $\beta = \frac{1}{2}(1 + \sqrt{5})$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + (2 \beta + 2) q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + ( - \beta + 1) q^{12} - 2 \beta q^{13} + (4 \beta + 2) q^{14} - 3 \beta q^{16} - 2 q^{17} + \cdots + (13 \beta + 12) q^{98} +O(q^{100}) $ q + b * q^2 - q^3 + (b - 1) * q^4 - b * q^6 + (2b + 2) q^7 + (-2b + 1) q^8 + q^9 + (-b + 1) * q^12 - 2b q^13 + (4b + 2) q^14 - 3b q^16 - 2 * q^17 + b * q^18 - 5 * q^19 + (-2b - 2) q^21 + (b + 3) * q^23 + (2b - 1) q^24 + (-2b - 2) q^26 - q^27 + 2b q^28 + (-3b + 4) q^29 + 7 * q^31 + (b - 5) * q^32 - 2b q^34 + (b - 1) * q^36 + (4b - 5) q^37 - 5b q^38 + 2b q^39 + (-4b - 5) q^41 + (-4b - 2) q^42 + (-5b - 1) q^43 + (4b + 1) q^46 + (-5b + 2) q^47 + 3b q^48 + (12b + 1) q^49 + 2 * q^51 - 2 * q^52 + (b - 7) * q^53 - b * q^54 + (-6b - 2) q^56 + 5 * q^57 + (b - 3) * q^58 + (6b - 3) q^59 - 7 * q^61 + 7b q^62 + (2b + 2) q^63 + (2b + 1) q^64 + (-b - 10) * q^67 + (-2b + 2) q^68 + (-b - 3) * q^69 - 8 * q^71 + (-2b + 1) q^72 + (-4b + 11) q^73 + (-b + 4) * q^74 + (-5b + 5) q^76 + (2b + 2) q^78 + (-5b + 5) q^79 + q^81 + (-9b - 4) q^82 + (-6b + 2) q^83 - 2b q^84 + (-6b - 5) q^86 + (3b - 4) q^87 + (-3b + 9) q^89 + (-8b - 4) q^91 + (3b - 2) q^92 - 7 * q^93 + (-3b - 5) q^94 + (-b + 5) * q^96 + (-4b + 9) q^97 + (13b + 12) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} - 2 q^{13} + 8 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 10 q^{19} - 6 q^{21} + 7 q^{23} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 5 q^{29}+ \cdots + 37 q^{98}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 - 2 * q^13 + 8 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 10 * q^19 - 6 * q^21 + 7 * q^23 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 5 * q^29 + 14 * q^31 - 9 * q^32 - 2 * q^34 - q^36 - 6 * q^37 - 5 * q^38 + 2 * q^39 - 14 * q^41 - 8 * q^42 - 7 * q^43 + 6 * q^46 - q^47 + 3 * q^48 + 14 * q^49 + 4 * q^51 - 4 * q^52 - 13 * q^53 - q^54 - 10 * q^56 + 10 * q^57 - 5 * q^58 - 14 * q^61 + 7 * q^62 + 6 * q^63 + 4 * q^64 - 21 * q^67 + 2 * q^68 - 7 * q^69 - 16 * q^71 + 18 * q^73 + 7 * q^74 + 5 * q^76 + 6 * q^78 + 5 * q^79 + 2 * q^81 - 17 * q^82 - 2 * q^83 - 2 * q^84 - 16 * q^86 - 5 * q^87 + 15 * q^89 - 16 * q^91 - q^92 - 14 * q^93 - 13 * q^94 + 9 * q^96 + 14 * q^97 + 37 * 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

−0.618034
1.61803

−0.618034

−1.00000

−1.61803

0.618034

0.763932

2.23607

1.00000

1.2

1.61803

−1.00000

0.618034

−1.61803

5.23607

−2.23607

1.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$5$	$ -1 $
$11$	$ -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	9075.2.a.bt		2
5.b	even	2	1		9075.2.a.bc		2
11.b	odd	2	1		9075.2.a.z		2
11.d	odd	10	2		825.2.n.b	✓	4
55.d	odd	2	1		9075.2.a.by		2
55.h	odd	10	2		825.2.n.d	yes	4
55.l	even	20	4		825.2.bx.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.n.b	✓	4	11.d	odd	10	2
825.2.n.d	yes	4	55.h	odd	10	2
825.2.bx.c		8	55.l	even	20	4
9075.2.a.z		2	11.b	odd	2	1
9075.2.a.bc		2	5.b	even	2	1
9075.2.a.bt		2	1.a	even	1	1	trivial
9075.2.a.by		2	55.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9075))$:

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

T2^2 - T2 - 1

$ T_{7}^{2} - 6T_{7} + 4 $

T7^2 - 6*T7 + 4

$ T_{13}^{2} + 2T_{13} - 4 $

T13^2 + 2*T13 - 4

$ T_{17} + 2 $

T17 + 2

$ T_{19} + 5 $

T19 + 5

$ T_{23}^{2} - 7T_{23} + 11 $

T23^2 - 7*T23 + 11

$ T_{37}^{2} + 6T_{37} - 11 $

T37^2 + 6*T37 - 11

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - T - 1 $ T^2 - T - 1
$3$	$ (T + 1)^{2} $ (T + 1)^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} - 6T + 4 $ T^2 - 6*T + 4
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 2T - 4 $ T^2 + 2*T - 4
$17$	$ (T + 2)^{2} $ (T + 2)^2
$19$	$ (T + 5)^{2} $ (T + 5)^2
$23$	$ T^{2} - 7T + 11 $ T^2 - 7*T + 11
$29$	$ T^{2} - 5T - 5 $ T^2 - 5*T - 5
$31$	$ (T - 7)^{2} $ (T - 7)^2
$37$	$ T^{2} + 6T - 11 $ T^2 + 6*T - 11
$41$	$ T^{2} + 14T + 29 $ T^2 + 14*T + 29
$43$	$ T^{2} + 7T - 19 $ T^2 + 7*T - 19
$47$	$ T^{2} + T - 31 $ T^2 + T - 31
$53$	$ T^{2} + 13T + 41 $ T^2 + 13*T + 41
$59$	$ T^{2} - 45 $ T^2 - 45
$61$	$ (T + 7)^{2} $ (T + 7)^2
$67$	$ T^{2} + 21T + 109 $ T^2 + 21*T + 109
$71$	$ (T + 8)^{2} $ (T + 8)^2
$73$	$ T^{2} - 18T + 61 $ T^2 - 18*T + 61
$79$	$ T^{2} - 5T - 25 $ T^2 - 5*T - 25
$83$	$ T^{2} + 2T - 44 $ T^2 + 2*T - 44
$89$	$ T^{2} - 15T + 45 $ T^2 - 15*T + 45
$97$	$ T^{2} - 14T + 29 $ T^2 - 14*T + 29
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:	\( N \)	\(=\)	\( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9075.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta q^{2} - q^{3} + (\beta - 1) q^{4} - \beta q^{6} + (2 \beta + 2) q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + ( - \beta + 1) q^{12} - 2 \beta q^{13} + (4 \beta + 2) q^{14} - 3 \beta q^{16} - 2 q^{17} + \cdots + (13 \beta + 12) q^{98} +O(q^{100}) \) q + b * q^2 - q^3 + (b - 1) * q^4 - b * q^6 + (2b + 2) q^7 + (-2b + 1) q^8 + q^9 + (-b + 1) * q^12 - 2b q^13 + (4b + 2) q^14 - 3b q^16 - 2 * q^17 + b * q^18 - 5 * q^19 + (-2b - 2) q^21 + (b + 3) * q^23 + (2b - 1) q^24 + (-2b - 2) q^26 - q^27 + 2b q^28 + (-3b + 4) q^29 + 7 * q^31 + (b - 5) * q^32 - 2b q^34 + (b - 1) * q^36 + (4b - 5) q^37 - 5b q^38 + 2b q^39 + (-4b - 5) q^41 + (-4b - 2) q^42 + (-5b - 1) q^43 + (4b + 1) q^46 + (-5b + 2) q^47 + 3b q^48 + (12b + 1) q^49 + 2 * q^51 - 2 * q^52 + (b - 7) * q^53 - b * q^54 + (-6b - 2) q^56 + 5 * q^57 + (b - 3) * q^58 + (6b - 3) q^59 - 7 * q^61 + 7b q^62 + (2b + 2) q^63 + (2b + 1) q^64 + (-b - 10) * q^67 + (-2b + 2) q^68 + (-b - 3) * q^69 - 8 * q^71 + (-2b + 1) q^72 + (-4b + 11) q^73 + (-b + 4) * q^74 + (-5b + 5) q^76 + (2b + 2) q^78 + (-5b + 5) q^79 + q^81 + (-9b - 4) q^82 + (-6b + 2) q^83 - 2b q^84 + (-6b - 5) q^86 + (3b - 4) q^87 + (-3b + 9) q^89 + (-8b - 4) q^91 + (3b - 2) q^92 - 7 * q^93 + (-3b - 5) q^94 + (-b + 5) * q^96 + (-4b + 9) q^97 + (13b + 12) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + 6 q^{7} + 2 q^{9} + q^{12} - 2 q^{13} + 8 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 10 q^{19} - 6 q^{21} + 7 q^{23} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 5 q^{29}+ \cdots + 37 q^{98}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + 6 * q^7 + 2 * q^9 + q^12 - 2 * q^13 + 8 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 10 * q^19 - 6 * q^21 + 7 * q^23 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 5 * q^29 + 14 * q^31 - 9 * q^32 - 2 * q^34 - q^36 - 6 * q^37 - 5 * q^38 + 2 * q^39 - 14 * q^41 - 8 * q^42 - 7 * q^43 + 6 * q^46 - q^47 + 3 * q^48 + 14 * q^49 + 4 * q^51 - 4 * q^52 - 13 * q^53 - q^54 - 10 * q^56 + 10 * q^57 - 5 * q^58 - 14 * q^61 + 7 * q^62 + 6 * q^63 + 4 * q^64 - 21 * q^67 + 2 * q^68 - 7 * q^69 - 16 * q^71 + 18 * q^73 + 7 * q^74 + 5 * q^76 + 6 * q^78 + 5 * q^79 + 2 * q^81 - 17 * q^82 - 2 * q^83 - 2 * q^84 - 16 * q^86 - 5 * q^87 + 15 * q^89 - 16 * q^91 - q^92 - 14 * q^93 - 13 * q^94 + 9 * q^96 + 14 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more