LMFDB - Newform orbit 2299.2.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,2,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2299.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2299.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$18.3576074247$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
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)

$f(q)$

$=$

$ q - 2 q^{3} - 2 q^{4} + 3 q^{5} + q^{7} + q^{9} + 4 q^{12} + 4 q^{13} - 6 q^{15} + 4 q^{16} + 3 q^{17} - q^{19} - 6 q^{20} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 2 q^{28} - 6 q^{29} - 4 q^{31} + 3 q^{35}+ \cdots + 8 q^{97}+O(q^{100}) $

q - 2 * q^3 - 2 * q^4 + 3 * q^5 + q^7 + q^9 + 4 * q^12 + 4 * q^13 - 6 * q^15 + 4 * q^16 + 3 * q^17 - q^19 - 6 * q^20 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 2 * q^28 - 6 * q^29 - 4 * q^31 + 3 * q^35 - 2 * q^36 + 2 * q^37 - 8 * q^39 + 6 * q^41 + q^43 + 3 * q^45 - 3 * q^47 - 8 * q^48 - 6 * q^49 - 6 * q^51 - 8 * q^52 + 12 * q^53 + 2 * q^57 - 6 * q^59 + 12 * q^60 + q^61 + q^63 - 8 * q^64 + 12 * q^65 - 4 * q^67 - 6 * q^68 + 6 * q^71 + 7 * q^73 - 8 * q^75 + 2 * q^76 - 8 * q^79 + 12 * q^80 - 11 * q^81 - 12 * q^83 + 4 * q^84 + 9 * q^85 + 12 * q^87 + 12 * q^89 + 4 * q^91 + 8 * q^93 - 3 * q^95 + 8 * q^97

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

−2.00000

3.00000

1.00000

Atkin-Lehner signs

$ p $	Sign
$11$	$ -1 $
$19$	$ +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	2299.2.a.b		1
11.b	odd	2	1		19.2.a.a	✓	1
33.d	even	2	1		171.2.a.b		1
44.c	even	2	1		304.2.a.f		1
55.d	odd	2	1		475.2.a.b		1
55.e	even	4	2		475.2.b.a		2
77.b	even	2	1		931.2.a.a		1
77.h	odd	6	2		931.2.f.c		2
77.i	even	6	2		931.2.f.b		2
88.b	odd	2	1		1216.2.a.o		1
88.g	even	2	1		1216.2.a.b		1
132.d	odd	2	1		2736.2.a.c		1
143.d	odd	2	1		3211.2.a.a		1
165.d	even	2	1		4275.2.a.i		1
187.b	odd	2	1		5491.2.a.b		1
209.d	even	2	1		361.2.a.b		1
209.g	even	6	2		361.2.c.a		2
209.h	odd	6	2		361.2.c.c		2
209.p	even	18	6		361.2.e.e		6
209.q	odd	18	6		361.2.e.d		6
220.g	even	2	1		7600.2.a.c		1
231.h	odd	2	1		8379.2.a.j		1
627.b	odd	2	1		3249.2.a.d		1
836.h	odd	2	1		5776.2.a.c		1
1045.e	even	2	1		9025.2.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.a.a	✓	1	11.b	odd	2	1
171.2.a.b		1	33.d	even	2	1
304.2.a.f		1	44.c	even	2	1
361.2.a.b		1	209.d	even	2	1
361.2.c.a		2	209.g	even	6	2
361.2.c.c		2	209.h	odd	6	2
361.2.e.d		6	209.q	odd	18	6
361.2.e.e		6	209.p	even	18	6
475.2.a.b		1	55.d	odd	2	1
475.2.b.a		2	55.e	even	4	2
931.2.a.a		1	77.b	even	2	1
931.2.f.b		2	77.i	even	6	2
931.2.f.c		2	77.h	odd	6	2
1216.2.a.b		1	88.g	even	2	1
1216.2.a.o		1	88.b	odd	2	1
2299.2.a.b		1	1.a	even	1	1	trivial
2736.2.a.c		1	132.d	odd	2	1
3211.2.a.a		1	143.d	odd	2	1
3249.2.a.d		1	627.b	odd	2	1
4275.2.a.i		1	165.d	even	2	1
5491.2.a.b		1	187.b	odd	2	1
5776.2.a.c		1	836.h	odd	2	1
7600.2.a.c		1	220.g	even	2	1
8379.2.a.j		1	231.h	odd	2	1
9025.2.a.d		1	1045.e	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2299))$:

$ T_{2} $

T2

$ T_{3} + 2 $

T3 + 2

$ T_{7} - 1 $

T7 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 2 $ T + 2
$5$	$ T - 3 $ T - 3
$7$	$ T - 1 $ T - 1
$11$	$ T $ T
$13$	$ T - 4 $ T - 4
$17$	$ T - 3 $ T - 3
$19$	$ T + 1 $ T + 1
$23$	$ T $ T
$29$	$ T + 6 $ T + 6
$31$	$ T + 4 $ T + 4
$37$	$ T - 2 $ T - 2
$41$	$ T - 6 $ T - 6
$43$	$ T - 1 $ T - 1
$47$	$ T + 3 $ T + 3
$53$	$ T - 12 $ T - 12
$59$	$ T + 6 $ T + 6
$61$	$ T - 1 $ T - 1
$67$	$ T + 4 $ T + 4
$71$	$ T - 6 $ T - 6
$73$	$ T - 7 $ T - 7
$79$	$ T + 8 $ T + 8
$83$	$ T + 12 $ T + 12
$89$	$ T - 12 $ T - 12
$97$	$ T - 8 $ T - 8
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Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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