LMFDB - Newform orbit 1340.2.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1340,2,Mod(1,1340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1340, 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("1340.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1340 = 2^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1340.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,0,-1,0,1] 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:	$10.6999538709$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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)

$f(q)$

$=$

$ q + q^{3} - q^{5} + q^{7} - 2 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 4 q^{17} - 8 q^{19} + q^{21} + 4 q^{23} + q^{25} - 5 q^{27} - 7 q^{29} - 4 q^{31} - 6 q^{33} - q^{35} - 2 q^{37} + 2 q^{39} - 4 q^{41}+ \cdots + 12 q^{99}+O(q^{100}) $

q + q^3 - q^5 + q^7 - 2 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 4 * q^17 - 8 * q^19 + q^21 + 4 * q^23 + q^25 - 5 * q^27 - 7 * q^29 - 4 * q^31 - 6 * q^33 - q^35 - 2 * q^37 + 2 * q^39 - 4 * q^41 - 11 * q^43 + 2 * q^45 + 8 * q^47 - 6 * q^49 + 4 * q^51 + 3 * q^53 + 6 * q^55 - 8 * q^57 + 9 * q^59 - 12 * q^61 - 2 * q^63 - 2 * q^65 + q^67 + 4 * q^69 + 12 * q^71 + 14 * q^73 + q^75 - 6 * q^77 - 6 * q^79 + q^81 - 16 * q^83 - 4 * q^85 - 7 * q^87 + q^89 + 2 * q^91 - 4 * q^93 + 8 * q^95 - 11 * q^97 + 12 * q^99

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

1.00000

−1.00000

1.00000

−2.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ +1 $
$67$	$ -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	1340.2.a.b	✓	1
4.b	odd	2	1		5360.2.a.c		1
5.b	even	2	1		6700.2.a.e		1
5.c	odd	4	2		6700.2.d.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1340.2.a.b	✓	1	1.a	even	1	1	trivial
5360.2.a.c		1	4.b	odd	2	1
6700.2.a.e		1	5.b	even	2	1
6700.2.d.e		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1340))$:

$ T_{3} - 1 $

T3 - 1

$ T_{7} - 1 $

T7 - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 1 $ T - 1
$5$	$ T + 1 $ T + 1
$7$	$ T - 1 $ T - 1
$11$	$ T + 6 $ T + 6
$13$	$ T - 2 $ T - 2
$17$	$ T - 4 $ T - 4
$19$	$ T + 8 $ T + 8
$23$	$ T - 4 $ T - 4
$29$	$ T + 7 $ T + 7
$31$	$ T + 4 $ T + 4
$37$	$ T + 2 $ T + 2
$41$	$ T + 4 $ T + 4
$43$	$ T + 11 $ T + 11
$47$	$ T - 8 $ T - 8
$53$	$ T - 3 $ T - 3
$59$	$ T - 9 $ T - 9
$61$	$ T + 12 $ T + 12
$67$	$ T - 1 $ T - 1
$71$	$ T - 12 $ T - 12
$73$	$ T - 14 $ T - 14
$79$	$ T + 6 $ T + 6
$83$	$ T + 16 $ T + 16
$89$	$ T - 1 $ T - 1
$97$	$ T + 11 $ T + 11
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Level:	\( N \)	\(=\)	\( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1340.a (trivial)

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