Newform orbit 1521.2.a.g

• Label: 1521.2.a.g
• Level: $1521$
• Weight: $2$
• Character orbit: 1521.a
• Self dual: yes
• Analytic conductor: $12.145$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(12.1452461474\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + (\beta + 2) q^{4} + ( - \beta + 2) q^{5} + ( - \beta - 1) q^{7} + ( - \beta - 4) q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} - 9 q^{8}+O(q^{10}) \)

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

2.56155
−1.56155

−2.56155

0

4.56155

−0.561553

0

−3.56155

−6.56155

0

1.43845

1.2

1.56155

0

0.438447

3.56155

0

0.561553

−2.43845

0

5.56155

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(13\)	\(1\)

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	1521.2.a.g		2
3.b	odd	2	1		507.2.a.g		2
12.b	even	2	1		8112.2.a.bk		2
13.b	even	2	1		1521.2.a.m		2
13.c	even	3	2		117.2.g.c		4
13.d	odd	4	2		1521.2.b.h		4
39.d	odd	2	1		507.2.a.d		2
39.f	even	4	2		507.2.b.d		4
39.h	odd	6	2		507.2.e.g		4
39.i	odd	6	2		39.2.e.b	✓	4
39.k	even	12	4		507.2.j.g		8
52.j	odd	6	2		1872.2.t.r		4
156.h	even	2	1		8112.2.a.bo		2
156.p	even	6	2		624.2.q.h		4
195.x	odd	6	2		975.2.i.k		4
195.bl	even	12	4		975.2.bb.i		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.e.b	✓	4	39.i	odd	6	2
117.2.g.c		4	13.c	even	3	2
507.2.a.d		2	39.d	odd	2	1
507.2.a.g		2	3.b	odd	2	1
507.2.b.d		4	39.f	even	4	2
507.2.e.g		4	39.h	odd	6	2
507.2.j.g		8	39.k	even	12	4
624.2.q.h		4	156.p	even	6	2
975.2.i.k		4	195.x	odd	6	2
975.2.bb.i		8	195.bl	even	12	4
1521.2.a.g		2	1.a	even	1	1	trivial
1521.2.a.m		2	13.b	even	2	1
1521.2.b.h		4	13.d	odd	4	2
1872.2.t.r		4	52.j	odd	6	2
8112.2.a.bk		2	12.b	even	2	1
8112.2.a.bo		2	156.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1521))\):

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

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

\( T_{7}^{2} + 3T_{7} - 2 \)

Hecke characteristic polynomials

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