Newform orbit 169.4.a.j

• Label: 169.4.a.j
• Level: $169$
• Weight: $4$
• Character orbit: 169.a
• Self dual: yes
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(9.97132279097\)
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 13)
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.

\(f(q)\)	\(=\)	\( q + ( - \beta + 3) q^{2} + ( - 3 \beta + 4) q^{3} + ( - 5 \beta + 5) q^{4} + (5 \beta + 5) q^{5} + ( - 10 \beta + 24) q^{6} + (\beta - 8) q^{7} + ( - 7 \beta + 11) q^{8} + ( - 15 \beta + 25) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 5 q^{2} + 5 q^{3} + 5 q^{4} + 15 q^{5} + 38 q^{6} - 15 q^{7} + 15 q^{8} + 35 q^{9}+O(q^{10}) \)

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

2.56155
−1.56155

0.438447

−3.68466

−7.80776

17.8078

−1.61553

−5.43845

−6.93087

−13.4233

7.80776

1.2

4.56155

8.68466

12.8078

−2.80776

39.6155

−9.56155

21.9309

48.4233

−12.8078

Atkin-Lehner signs

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

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	169.4.a.j		2
3.b	odd	2	1		1521.4.a.l		2
13.b	even	2	1		169.4.a.f		2
13.c	even	3	2		169.4.c.f		4
13.d	odd	4	2		169.4.b.e		4
13.e	even	6	2		13.4.c.b	✓	4
13.f	odd	12	4		169.4.e.g		8
39.d	odd	2	1		1521.4.a.t		2
39.h	odd	6	2		117.4.g.d		4
52.i	odd	6	2		208.4.i.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.c.b	✓	4	13.e	even	6	2
117.4.g.d		4	39.h	odd	6	2
169.4.a.f		2	13.b	even	2	1
169.4.a.j		2	1.a	even	1	1	trivial
169.4.b.e		4	13.d	odd	4	2
169.4.c.f		4	13.c	even	3	2
169.4.e.g		8	13.f	odd	12	4
208.4.i.e		4	52.i	odd	6	2
1521.4.a.l		2	3.b	odd	2	1
1521.4.a.t		2	39.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5T_{2} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 5T + 2 \)
$3$	\( T^{2} - 5T - 32 \)
$5$	\( T^{2} - 15T - 50 \)
$7$	\( T^{2} + 15T + 52 \)
$11$	\( T^{2} + 17T - 884 \)