LMFDB - Newform orbit 6069.2.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6069,2,Mod(1,6069)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6069, 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("6069.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,1,-2,-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:	$48.4612089867$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 357)
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} - 2 q^{4} - q^{5} - q^{7} + q^{9} + 5 q^{11} - 2 q^{12} - 5 q^{13} - q^{15} + 4 q^{16} - 5 q^{19} + 2 q^{20} - q^{21} + q^{23} - 4 q^{25} + q^{27} + 2 q^{28} + 6 q^{29} + 6 q^{31} + 5 q^{33}+ \cdots + 5 q^{99}+O(q^{100}) $

q + q^3 - 2 * q^4 - q^5 - q^7 + q^9 + 5 * q^11 - 2 * q^12 - 5 * q^13 - q^15 + 4 * q^16 - 5 * q^19 + 2 * q^20 - q^21 + q^23 - 4 * q^25 + q^27 + 2 * q^28 + 6 * q^29 + 6 * q^31 + 5 * q^33 + q^35 - 2 * q^36 - 4 * q^37 - 5 * q^39 - 7 * q^41 - 7 * q^43 - 10 * q^44 - q^45 + 6 * q^47 + 4 * q^48 + q^49 + 10 * q^52 + 6 * q^53 - 5 * q^55 - 5 * q^57 + 14 * q^59 + 2 * q^60 - q^63 - 8 * q^64 + 5 * q^65 - 12 * q^67 + q^69 - 4 * q^71 - 6 * q^73 - 4 * q^75 + 10 * q^76 - 5 * q^77 + 6 * q^79 - 4 * q^80 + q^81 - 6 * q^83 + 2 * q^84 + 6 * q^87 + 12 * q^89 + 5 * q^91 - 2 * q^92 + 6 * q^93 + 5 * q^95 - 8 * q^97 + 5 * 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

−2.00000

−1.00000

1.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$7$	$ +1 $
$17$	$ +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	6069.2.a.c		1
17.b	even	2	1		357.2.a.c	✓	1
51.c	odd	2	1		1071.2.a.c		1
68.d	odd	2	1		5712.2.a.u		1
85.c	even	2	1		8925.2.a.o		1
119.d	odd	2	1		2499.2.a.g		1
357.c	even	2	1		7497.2.a.i		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
357.2.a.c	✓	1	17.b	even	2	1
1071.2.a.c		1	51.c	odd	2	1
2499.2.a.g		1	119.d	odd	2	1
5712.2.a.u		1	68.d	odd	2	1
6069.2.a.c		1	1.a	even	1	1	trivial
7497.2.a.i		1	357.c	even	2	1
8925.2.a.o		1	85.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6069))$:

$ T_{2} $

T2

$ T_{5} + 1 $

T5 + 1

$ T_{11} - 5 $

T11 - 5

$ T_{23} - 1 $

T23 - 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 - 5 $ T - 5
$13$	$ T + 5 $ T + 5
$17$	$ T $ T
$19$	$ T + 5 $ T + 5
$23$	$ T - 1 $ T - 1
$29$	$ T - 6 $ T - 6
$31$	$ T - 6 $ T - 6
$37$	$ T + 4 $ T + 4
$41$	$ T + 7 $ T + 7
$43$	$ T + 7 $ T + 7
$47$	$ T - 6 $ T - 6
$53$	$ T - 6 $ T - 6
$59$	$ T - 14 $ T - 14
$61$	$ T $ T
$67$	$ T + 12 $ T + 12
$71$	$ T + 4 $ T + 4
$73$	$ T + 6 $ T + 6
$79$	$ T - 6 $ T - 6
$83$	$ T + 6 $ T + 6
$89$	$ T - 12 $ T - 12
$97$	$ T + 8 $ T + 8
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Level:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

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