Newform orbit 1521.2.a.f

• Label: 1521.2.a.f
• 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:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−1.41421
1.41421

−2.41421

0

3.82843

2.82843

0

2.82843

−4.41421

0

−6.82843

1.2

0.414214

0

−1.82843

−2.82843

0

−2.82843

−1.58579

0

−1.17157

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.f		2
3.b	odd	2	1		507.2.a.h		2
12.b	even	2	1		8112.2.a.bm		2
13.b	even	2	1		117.2.a.c		2
13.d	odd	4	2		1521.2.b.j		4
39.d	odd	2	1		39.2.a.b	✓	2
39.f	even	4	2		507.2.b.e		4
39.h	odd	6	2		507.2.e.h		4
39.i	odd	6	2		507.2.e.d		4
39.k	even	12	4		507.2.j.f		8
52.b	odd	2	1		1872.2.a.w		2
65.d	even	2	1		2925.2.a.v		2
65.h	odd	4	2		2925.2.c.u		4
91.b	odd	2	1		5733.2.a.u		2
104.e	even	2	1		7488.2.a.cl		2
104.h	odd	2	1		7488.2.a.co		2
117.n	odd	6	2		1053.2.e.m		4
117.t	even	6	2		1053.2.e.e		4
156.h	even	2	1		624.2.a.k		2
195.e	odd	2	1		975.2.a.l		2
195.s	even	4	2		975.2.c.h		4
273.g	even	2	1		1911.2.a.h		2
312.b	odd	2	1		2496.2.a.bf		2
312.h	even	2	1		2496.2.a.bi		2
429.e	even	2	1		4719.2.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
39.2.a.b	✓	2	39.d	odd	2	1
117.2.a.c		2	13.b	even	2	1
507.2.a.h		2	3.b	odd	2	1
507.2.b.e		4	39.f	even	4	2
507.2.e.d		4	39.i	odd	6	2
507.2.e.h		4	39.h	odd	6	2
507.2.j.f		8	39.k	even	12	4
624.2.a.k		2	156.h	even	2	1
975.2.a.l		2	195.e	odd	2	1
975.2.c.h		4	195.s	even	4	2
1053.2.e.e		4	117.t	even	6	2
1053.2.e.m		4	117.n	odd	6	2
1521.2.a.f		2	1.a	even	1	1	trivial
1521.2.b.j		4	13.d	odd	4	2
1872.2.a.w		2	52.b	odd	2	1
1911.2.a.h		2	273.g	even	2	1
2496.2.a.bf		2	312.b	odd	2	1
2496.2.a.bi		2	312.h	even	2	1
2925.2.a.v		2	65.d	even	2	1
2925.2.c.u		4	65.h	odd	4	2
4719.2.a.p		2	429.e	even	2	1
5733.2.a.u		2	91.b	odd	2	1
7488.2.a.cl		2	104.e	even	2	1
7488.2.a.co		2	104.h	odd	2	1
8112.2.a.bm		2	12.b	even	2	1

Hecke kernels

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

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

\( T_{5}^{2} - 8 \)

\( T_{7}^{2} - 8 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T - 1 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 8 \)
$7$	\( T^{2} - 8 \)
$11$	\( (T + 2)^{2} \)