Newform orbit 3249.2.a.z

• Label: 3249.2.a.z
• Level: $3249$
• Weight: $2$
• Character orbit: 3249.a
• Self dual: yes
• Analytic conductor: $25.943$
• Analytic rank: $0$
• Dimension: $3$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(25.9433956167\)
Analytic rank:	\(0\)
Dimension:	\(3\)
Coefficient field:	\(\Q(\zeta_{18})^+\)
Defining polynomial:	\( x^{3} - 3x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - 2 \beta_1 + 1) q^{4} + (\beta_{2} + 1) q^{5} - \beta_1 q^{7} + (3 \beta_{2} - 2 \beta_1 + 2) q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 6 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 2 \)

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.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

1.87939
−0.347296
−1.53209

−0.879385

0

−1.22668

2.53209

0

−1.87939

2.83750

0

−2.22668

1.2

1.34730

0

−0.184793

−0.879385

0

0.347296

−2.94356

0

−1.18479

1.3

2.53209

0

4.41147

1.34730

0

1.53209

6.10607

0

3.41147

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(19\)	\(1\)

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	3249.2.a.z		3
3.b	odd	2	1		361.2.a.g		3
12.b	even	2	1		5776.2.a.br		3
15.d	odd	2	1		9025.2.a.bd		3
19.b	odd	2	1		3249.2.a.s		3
19.e	even	9	2		171.2.u.c		6
57.d	even	2	1		361.2.a.h		3
57.f	even	6	2		361.2.c.h		6
57.h	odd	6	2		361.2.c.i		6
57.j	even	18	2		361.2.e.a		6
57.j	even	18	2		361.2.e.b		6
57.j	even	18	2		361.2.e.h		6
57.l	odd	18	2		19.2.e.a	✓	6
57.l	odd	18	2		361.2.e.f		6
57.l	odd	18	2		361.2.e.g		6
228.b	odd	2	1		5776.2.a.bi		3
228.v	even	18	2		304.2.u.b		6
285.b	even	2	1		9025.2.a.x		3
285.bd	odd	18	2		475.2.l.a		6
285.bi	even	36	4		475.2.u.a		12
399.ca	odd	18	2		931.2.x.a		6
399.cb	even	18	2		931.2.x.b		6
399.ch	odd	18	2		931.2.v.b		6
399.ci	even	18	2		931.2.v.a		6
399.cj	even	18	2		931.2.w.a		6

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
19.2.e.a	✓	6	57.l	odd	18	2
171.2.u.c		6	19.e	even	9	2
304.2.u.b		6	228.v	even	18	2
361.2.a.g		3	3.b	odd	2	1
361.2.a.h		3	57.d	even	2	1
361.2.c.h		6	57.f	even	6	2
361.2.c.i		6	57.h	odd	6	2
361.2.e.a		6	57.j	even	18	2
361.2.e.b		6	57.j	even	18	2
361.2.e.f		6	57.l	odd	18	2
361.2.e.g		6	57.l	odd	18	2
361.2.e.h		6	57.j	even	18	2
475.2.l.a		6	285.bd	odd	18	2
475.2.u.a		12	285.bi	even	36	4
931.2.v.a		6	399.ci	even	18	2
931.2.v.b		6	399.ch	odd	18	2
931.2.w.a		6	399.cj	even	18	2
931.2.x.a		6	399.ca	odd	18	2
931.2.x.b		6	399.cb	even	18	2
3249.2.a.s		3	19.b	odd	2	1
3249.2.a.z		3	1.a	even	1	1	trivial
5776.2.a.bi		3	228.b	odd	2	1
5776.2.a.br		3	12.b	even	2	1
9025.2.a.x		3	285.b	even	2	1
9025.2.a.bd		3	15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):

\( T_{2}^{3} - 3T_{2}^{2} + 3 \)

\( T_{5}^{3} - 3T_{5}^{2} + 3 \)

\( T_{13}^{3} - 21T_{13} + 37 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{3} - 3T^{2} + 3 \)
$3$	\( T^{3} \)
$5$	\( T^{3} - 3T^{2} + 3 \)
$7$	\( T^{3} - 3T + 1 \)
$11$	\( T^{3} - 9T + 9 \)