Newform orbit 3969.2.a.ba

• Label: 3969.2.a.ba
• Level: $3969$
• Weight: $2$
• Character orbit: 3969.a
• Self dual: yes
• Analytic conductor: $31.693$
• Analytic rank: $0$
• Dimension: $5$
• 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:	\(5\)
Coefficient field:	5.5.574857.1
Defining polynomial:	\( x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 5x^{3} + 9x^{2} + 3x - 3 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( \nu^{2} - 3 \)
\(\beta_{3}\)	\(=\)	\( \nu^{3} - 4\nu - 1 \)
\(\beta_{4}\)	\(=\)	\( \nu^{4} - \nu^{3} - 5\nu^{2} + 4\nu + 2 \)

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

2.38687
1.84124
0.495868
−0.670333
−2.05365

−2.38687

0

3.69714

2.92087

0

0

−4.05086

0

−6.97172

1.2

−1.84124

0

1.39017

−1.33475

0

0

1.12285

0

2.45760

1.3

−0.495868

0

−1.75411

3.69258

0

0

1.86155

0

−1.83103

1.4

0.670333

0

−1.55065

−1.42494

0

0

−2.38012

0

−0.955182

1.5

2.05365

0

2.21746

0.146246

0

0

0.446582

0

0.300337

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.ba		5
3.b	odd	2	1		3969.2.a.bb		5
7.b	odd	2	1		3969.2.a.z		5
7.d	odd	6	2		567.2.e.f		10
9.c	even	3	2		441.2.f.f		10
9.d	odd	6	2		1323.2.f.f		10
21.c	even	2	1		3969.2.a.bc		5
21.g	even	6	2		567.2.e.e		10
63.g	even	3	2		441.2.g.f		10
63.h	even	3	2		441.2.h.f		10
63.i	even	6	2		189.2.h.b		10
63.j	odd	6	2		1323.2.h.f		10
63.k	odd	6	2		63.2.g.b	✓	10
63.l	odd	6	2		441.2.f.e		10
63.n	odd	6	2		1323.2.g.f		10
63.o	even	6	2		1323.2.f.e		10
63.s	even	6	2		189.2.g.b		10
63.t	odd	6	2		63.2.h.b	yes	10
252.n	even	6	2		1008.2.t.i		10
252.r	odd	6	2		3024.2.q.i		10
252.bj	even	6	2		1008.2.q.i		10
252.bn	odd	6	2		3024.2.t.i		10

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.2.g.b	✓	10	63.k	odd	6	2
63.2.h.b	yes	10	63.t	odd	6	2
189.2.g.b		10	63.s	even	6	2
189.2.h.b		10	63.i	even	6	2
441.2.f.e		10	63.l	odd	6	2
441.2.f.f		10	9.c	even	3	2
441.2.g.f		10	63.g	even	3	2
441.2.h.f		10	63.h	even	3	2
567.2.e.e		10	21.g	even	6	2
567.2.e.f		10	7.d	odd	6	2
1008.2.q.i		10	252.bj	even	6	2
1008.2.t.i		10	252.n	even	6	2
1323.2.f.e		10	63.o	even	6	2
1323.2.f.f		10	9.d	odd	6	2
1323.2.g.f		10	63.n	odd	6	2
1323.2.h.f		10	63.j	odd	6	2
3024.2.q.i		10	252.r	odd	6	2
3024.2.t.i		10	252.bn	odd	6	2
3969.2.a.z		5	7.b	odd	2	1
3969.2.a.ba		5	1.a	even	1	1	trivial
3969.2.a.bb		5	3.b	odd	2	1
3969.2.a.bc		5	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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3969))\):

\( T_{2}^{5} + 2T_{2}^{4} - 5T_{2}^{3} - 9T_{2}^{2} + 3T_{2} + 3 \)

\( T_{5}^{5} - 4T_{5}^{4} - 5T_{5}^{3} + 18T_{5}^{2} + 18T_{5} - 3 \)

\( T_{11}^{5} + 4T_{11}^{4} - 8T_{11}^{3} - 15T_{11}^{2} + 12T_{11} + 15 \)

\( T_{13}^{5} + 8T_{13}^{4} + 13T_{13}^{3} - 13T_{13}^{2} - 23T_{13} + 5 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^5 + 2*T^4 - 5*T^3 - 9*T^2 + 3*T + 3
$3$	\( T^{5} \)
$5$	T^5 - 4*T^4 - 5*T^3 + 18*T^2 + 18*T - 3
$7$	\( T^{5} \)
$11$	T^5 + 4*T^4 - 8*T^3 - 15*T^2 + 12*T + 15