Newform orbit 3969.2.a.c

• Label: 3969.2.a.c
• Level: $3969$
• Weight: $2$
• Character orbit: 3969.a
• Self dual: yes
• Analytic conductor: $31.693$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(31.6926245622\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{2} - q^{4} + q^{5} + 3 q^{8}+O(q^{10}) \)

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

0

−1.00000

0

−1.00000

1.00000

0

0

3.00000

0

−1.00000

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(7\)	\(1\)

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	3969.2.a.c		1
3.b	odd	2	1		3969.2.a.d		1
7.b	odd	2	1		3969.2.a.a		1
7.c	even	3	2		567.2.e.b		2
9.c	even	3	2		1323.2.f.a		2
9.d	odd	6	2		441.2.f.b		2
21.c	even	2	1		3969.2.a.f		1
21.h	odd	6	2		567.2.e.a		2
63.g	even	3	2		189.2.g.a		2
63.h	even	3	2		189.2.h.a		2
63.i	even	6	2		441.2.h.a		2
63.j	odd	6	2		63.2.h.a	yes	2
63.k	odd	6	2		1323.2.g.a		2
63.l	odd	6	2		1323.2.f.b		2
63.n	odd	6	2		63.2.g.a	✓	2
63.o	even	6	2		441.2.f.a		2
63.s	even	6	2		441.2.g.a		2
63.t	odd	6	2		1323.2.h.a		2
252.o	even	6	2		1008.2.t.d		2
252.u	odd	6	2		3024.2.q.b		2
252.bb	even	6	2		1008.2.q.c		2
252.bl	odd	6	2		3024.2.t.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.g.a	✓	2	63.n	odd	6	2
63.2.h.a	yes	2	63.j	odd	6	2
189.2.g.a		2	63.g	even	3	2
189.2.h.a		2	63.h	even	3	2
441.2.f.a		2	63.o	even	6	2
441.2.f.b		2	9.d	odd	6	2
441.2.g.a		2	63.s	even	6	2
441.2.h.a		2	63.i	even	6	2
567.2.e.a		2	21.h	odd	6	2
567.2.e.b		2	7.c	even	3	2
1008.2.q.c		2	252.bb	even	6	2
1008.2.t.d		2	252.o	even	6	2
1323.2.f.a		2	9.c	even	3	2
1323.2.f.b		2	63.l	odd	6	2
1323.2.g.a		2	63.k	odd	6	2
1323.2.h.a		2	63.t	odd	6	2
3024.2.q.b		2	252.u	odd	6	2
3024.2.t.d		2	252.bl	odd	6	2
3969.2.a.a		1	7.b	odd	2	1
3969.2.a.c		1	1.a	even	1	1	trivial
3969.2.a.d		1	3.b	odd	2	1
3969.2.a.f		1	21.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2} + 1 \)

\( T_{5} - 1 \)

\( T_{11} + 5 \)

\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 1 \)
$3$	\( T \)
$5$	\( T - 1 \)
$7$	\( T \)
$11$	\( T + 5 \)