Newform orbit 169.2.a.a

• Label: 169.2.a.a
• Level: $169$
• Weight: $2$
• Character orbit: 169.a
• Self dual: yes
• Analytic conductor: $1.349$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.34947179416\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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

−1.73205
1.73205

−1.73205

2.00000

1.00000

1.73205

−3.46410

0

1.73205

1.00000

−3.00000

1.2

1.73205

2.00000

1.00000

−1.73205

3.46410

0

−1.73205

1.00000

−3.00000

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	169.2.a.a		2
3.b	odd	2	1		1521.2.a.k		2
4.b	odd	2	1		2704.2.a.o		2
5.b	even	2	1		4225.2.a.v		2
7.b	odd	2	1		8281.2.a.q		2
13.b	even	2	1	inner	169.2.a.a		2
13.c	even	3	2		169.2.c.a		4
13.d	odd	4	2		169.2.b.a		2
13.e	even	6	2		169.2.c.a		4
13.f	odd	12	2		13.2.e.a	✓	2
13.f	odd	12	2		169.2.e.a		2
39.d	odd	2	1		1521.2.a.k		2
39.f	even	4	2		1521.2.b.a		2
39.k	even	12	2		117.2.q.c		2
52.b	odd	2	1		2704.2.a.o		2
52.f	even	4	2		2704.2.f.b		2
52.l	even	12	2		208.2.w.b		2
65.d	even	2	1		4225.2.a.v		2
65.o	even	12	2		325.2.m.a		4
65.s	odd	12	2		325.2.n.a		2
65.t	even	12	2		325.2.m.a		4
91.b	odd	2	1		8281.2.a.q		2
91.w	even	12	2		637.2.u.b		2
91.x	odd	12	2		637.2.k.a		2
91.ba	even	12	2		637.2.k.c		2
91.bc	even	12	2		637.2.q.a		2
91.bd	odd	12	2		637.2.u.c		2
104.u	even	12	2		832.2.w.a		2
104.x	odd	12	2		832.2.w.d		2
156.v	odd	12	2		1872.2.by.d		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.2.e.a	✓	2	13.f	odd	12	2
117.2.q.c		2	39.k	even	12	2
169.2.a.a		2	1.a	even	1	1	trivial
169.2.a.a		2	13.b	even	2	1	inner
169.2.b.a		2	13.d	odd	4	2
169.2.c.a		4	13.c	even	3	2
169.2.c.a		4	13.e	even	6	2
169.2.e.a		2	13.f	odd	12	2
208.2.w.b		2	52.l	even	12	2
325.2.m.a		4	65.o	even	12	2
325.2.m.a		4	65.t	even	12	2
325.2.n.a		2	65.s	odd	12	2
637.2.k.a		2	91.x	odd	12	2
637.2.k.c		2	91.ba	even	12	2
637.2.q.a		2	91.bc	even	12	2
637.2.u.b		2	91.w	even	12	2
637.2.u.c		2	91.bd	odd	12	2
832.2.w.a		2	104.u	even	12	2
832.2.w.d		2	104.x	odd	12	2
1521.2.a.k		2	3.b	odd	2	1
1521.2.a.k		2	39.d	odd	2	1
1521.2.b.a		2	39.f	even	4	2
1872.2.by.d		2	156.v	odd	12	2
2704.2.a.o		2	4.b	odd	2	1
2704.2.a.o		2	52.b	odd	2	1
2704.2.f.b		2	52.f	even	4	2
4225.2.a.v		2	5.b	even	2	1
4225.2.a.v		2	65.d	even	2	1
8281.2.a.q		2	7.b	odd	2	1
8281.2.a.q		2	91.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

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