Newform orbit 1519.2.a.a

• Label: 1519.2.a.a
• Level: $1519$
• Weight: $2$
• Character orbit: 1519.a
• Self dual: yes
• Analytic conductor: $12.129$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(12.1292760670\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} + 2 \beta q^{3} + (\beta - 1) q^{4} - q^{5} + (2 \beta + 2) q^{6} + ( - 2 \beta + 1) q^{8} + (4 \beta + 1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} - 2 q^{5} + 6 q^{6} + 6 q^{9}+O(q^{10}) \)

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

−0.618034
1.61803

−0.618034

−1.23607

−1.61803

−1.00000

0.763932

0

2.23607

−1.47214

0.618034

1.2

1.61803

3.23607

0.618034

−1.00000

5.23607

0

−2.23607

7.47214

−1.61803

Atkin-Lehner signs

\( p \)	Sign
\(7\)	\(-1\)
\(31\)	\(1\)

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	1519.2.a.a		2
7.b	odd	2	1		31.2.a.a	✓	2
21.c	even	2	1		279.2.a.a		2
28.d	even	2	1		496.2.a.i		2
35.c	odd	2	1		775.2.a.d		2
35.f	even	4	2		775.2.b.d		4
56.e	even	2	1		1984.2.a.n		2
56.h	odd	2	1		1984.2.a.r		2
77.b	even	2	1		3751.2.a.b		2
84.h	odd	2	1		4464.2.a.bf		2
91.b	odd	2	1		5239.2.a.f		2
105.g	even	2	1		6975.2.a.y		2
119.d	odd	2	1		8959.2.a.b		2
217.d	even	2	1		961.2.a.f		2
217.s	even	6	2		961.2.c.c		4
217.u	odd	6	2		961.2.c.e		4
217.v	even	10	2		961.2.d.a		4
217.v	even	10	2		961.2.d.g		4
217.w	odd	10	2		961.2.d.c		4
217.w	odd	10	2		961.2.d.d		4
217.bd	odd	30	4		961.2.g.a		8
217.bd	odd	30	4		961.2.g.h		8
217.be	even	30	4		961.2.g.d		8
217.be	even	30	4		961.2.g.e		8
651.e	odd	2	1		8649.2.a.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.2.a.a	✓	2	7.b	odd	2	1
279.2.a.a		2	21.c	even	2	1
496.2.a.i		2	28.d	even	2	1
775.2.a.d		2	35.c	odd	2	1
775.2.b.d		4	35.f	even	4	2
961.2.a.f		2	217.d	even	2	1
961.2.c.c		4	217.s	even	6	2
961.2.c.e		4	217.u	odd	6	2
961.2.d.a		4	217.v	even	10	2
961.2.d.c		4	217.w	odd	10	2
961.2.d.d		4	217.w	odd	10	2
961.2.d.g		4	217.v	even	10	2
961.2.g.a		8	217.bd	odd	30	4
961.2.g.d		8	217.be	even	30	4
961.2.g.e		8	217.be	even	30	4
961.2.g.h		8	217.bd	odd	30	4
1519.2.a.a		2	1.a	even	1	1	trivial
1984.2.a.n		2	56.e	even	2	1
1984.2.a.r		2	56.h	odd	2	1
3751.2.a.b		2	77.b	even	2	1
4464.2.a.bf		2	84.h	odd	2	1
5239.2.a.f		2	91.b	odd	2	1
6975.2.a.y		2	105.g	even	2	1
8649.2.a.c		2	651.e	odd	2	1
8959.2.a.b		2	119.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1519))\):

\( T_{2}^{2} - T_{2} - 1 \)

\( T_{3}^{2} - 2T_{3} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T - 1 \)
$3$	\( T^{2} - 2T - 4 \)
$5$	\( (T + 1)^{2} \)
$7$	\( T^{2} \)
$11$	\( (T - 2)^{2} \)