LMFDB - Newform orbit 3042.2.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3042,2,Mod(1,3042)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3042, 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("3042.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,-3,0,1,-1,0,3,6,0,0,-1,0,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
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^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8} + 3 q^{10} + 6 q^{11} - q^{14} + q^{16} + 3 q^{17} - 2 q^{19} - 3 q^{20} - 6 q^{22} + 4 q^{25} + q^{28} - 6 q^{29} + 4 q^{31} - q^{32} - 3 q^{34} - 3 q^{35}+ \cdots + 6 q^{98}+O(q^{100}) $

q - q^2 + q^4 - 3 * q^5 + q^7 - q^8 + 3 * q^10 + 6 * q^11 - q^14 + q^16 + 3 * q^17 - 2 * q^19 - 3 * q^20 - 6 * q^22 + 4 * q^25 + q^28 - 6 * q^29 + 4 * q^31 - q^32 - 3 * q^34 - 3 * q^35 + 7 * q^37 + 2 * q^38 + 3 * q^40 - q^43 + 6 * q^44 + 3 * q^47 - 6 * q^49 - 4 * q^50 - 18 * q^55 - q^56 + 6 * q^58 - 6 * q^59 + 8 * q^61 - 4 * q^62 + q^64 - 14 * q^67 + 3 * q^68 + 3 * q^70 - 3 * q^71 - 2 * q^73 - 7 * q^74 - 2 * q^76 + 6 * q^77 + 8 * q^79 - 3 * q^80 + 12 * q^83 - 9 * q^85 + q^86 - 6 * q^88 - 6 * q^89 - 3 * q^94 + 6 * q^95 + 10 * q^97 + 6 * 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

−1.00000

1.00000

−3.00000

1.00000

−1.00000

3.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -1 $
$13$	$ +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	3042.2.a.a		1
3.b	odd	2	1		338.2.a.f		1
12.b	even	2	1		2704.2.a.f		1
13.b	even	2	1		234.2.a.e		1
13.d	odd	4	2		3042.2.b.a		2
15.d	odd	2	1		8450.2.a.c		1
39.d	odd	2	1		26.2.a.a	✓	1
39.f	even	4	2		338.2.b.c		2
39.h	odd	6	2		338.2.c.d		2
39.i	odd	6	2		338.2.c.a		2
39.k	even	12	4		338.2.e.a		4
52.b	odd	2	1		1872.2.a.q		1
65.d	even	2	1		5850.2.a.p		1
65.h	odd	4	2		5850.2.e.a		2
104.e	even	2	1		7488.2.a.g		1
104.h	odd	2	1		7488.2.a.h		1
117.n	odd	6	2		2106.2.e.ba		2
117.t	even	6	2		2106.2.e.b		2
156.h	even	2	1		208.2.a.a		1
156.l	odd	4	2		2704.2.f.d		2
195.e	odd	2	1		650.2.a.j		1
195.s	even	4	2		650.2.b.d		2
273.g	even	2	1		1274.2.a.d		1
273.w	odd	6	2		1274.2.f.p		2
273.ba	even	6	2		1274.2.f.r		2
312.b	odd	2	1		832.2.a.d		1
312.h	even	2	1		832.2.a.i		1
429.e	even	2	1		3146.2.a.n		1
624.v	even	4	2		3328.2.b.j		2
624.bi	odd	4	2		3328.2.b.m		2
663.g	odd	2	1		7514.2.a.c		1
741.d	even	2	1		9386.2.a.j		1
780.d	even	2	1		5200.2.a.x		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
26.2.a.a	✓	1	39.d	odd	2	1
208.2.a.a		1	156.h	even	2	1
234.2.a.e		1	13.b	even	2	1
338.2.a.f		1	3.b	odd	2	1
338.2.b.c		2	39.f	even	4	2
338.2.c.a		2	39.i	odd	6	2
338.2.c.d		2	39.h	odd	6	2
338.2.e.a		4	39.k	even	12	4
650.2.a.j		1	195.e	odd	2	1
650.2.b.d		2	195.s	even	4	2
832.2.a.d		1	312.b	odd	2	1
832.2.a.i		1	312.h	even	2	1
1274.2.a.d		1	273.g	even	2	1
1274.2.f.p		2	273.w	odd	6	2
1274.2.f.r		2	273.ba	even	6	2
1872.2.a.q		1	52.b	odd	2	1
2106.2.e.b		2	117.t	even	6	2
2106.2.e.ba		2	117.n	odd	6	2
2704.2.a.f		1	12.b	even	2	1
2704.2.f.d		2	156.l	odd	4	2
3042.2.a.a		1	1.a	even	1	1	trivial
3042.2.b.a		2	13.d	odd	4	2
3146.2.a.n		1	429.e	even	2	1
3328.2.b.j		2	624.v	even	4	2
3328.2.b.m		2	624.bi	odd	4	2
5200.2.a.x		1	780.d	even	2	1
5850.2.a.p		1	65.d	even	2	1
5850.2.e.a		2	65.h	odd	4	2
7488.2.a.g		1	104.e	even	2	1
7488.2.a.h		1	104.h	odd	2	1
7514.2.a.c		1	663.g	odd	2	1
8450.2.a.c		1	15.d	odd	2	1
9386.2.a.j		1	741.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3042))$:

$ T_{5} + 3 $

T5 + 3

$ T_{7} - 1 $

T7 - 1

$ T_{11} - 6 $

T11 - 6

$ T_{17} - 3 $

T17 - 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T $ T
$5$	$ T + 3 $ T + 3
$7$	$ T - 1 $ T - 1
$11$	$ T - 6 $ T - 6
$13$	$ T $ T
$17$	$ T - 3 $ T - 3
$19$	$ T + 2 $ T + 2
$23$	$ T $ T
$29$	$ T + 6 $ T + 6
$31$	$ T - 4 $ T - 4
$37$	$ T - 7 $ T - 7
$41$	$ T $ T
$43$	$ T + 1 $ T + 1
$47$	$ T - 3 $ T - 3
$53$	$ T $ T
$59$	$ T + 6 $ T + 6
$61$	$ T - 8 $ T - 8
$67$	$ T + 14 $ T + 14
$71$	$ T + 3 $ T + 3
$73$	$ T + 2 $ T + 2
$79$	$ T - 8 $ T - 8
$83$	$ T - 12 $ T - 12
$89$	$ T + 6 $ T + 6
$97$	$ T - 10 $ T - 10
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

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