Newform orbit 169.4.a.h

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

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:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\)	\(=\)	\( q + 2 \beta q^{2} - 7 q^{3} + 4 q^{4} + 8 \beta q^{5} - 14 \beta q^{6} - 13 \beta q^{7} - 8 \beta q^{8} + 22 q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} + 8 q^{4} + 44 q^{9}+O(q^{10}) \)

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

−3.46410

−7.00000

4.00000

−13.8564

24.2487

22.5167

13.8564

22.0000

48.0000

1.2

3.46410

−7.00000

4.00000

13.8564

−24.2487

−22.5167

−13.8564

22.0000

48.0000

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	169.4.a.h		2
3.b	odd	2	1		1521.4.a.q		2
13.b	even	2	1	inner	169.4.a.h		2
13.c	even	3	2		169.4.c.i		4
13.d	odd	4	2		169.4.b.b		2
13.e	even	6	2		169.4.c.i		4
13.f	odd	12	2		13.4.e.a	✓	2
13.f	odd	12	2		169.4.e.b		2
39.d	odd	2	1		1521.4.a.q		2
39.k	even	12	2		117.4.q.c		2
52.l	even	12	2		208.4.w.a		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 12 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 12 \)
$3$	\( (T + 7)^{2} \)
$5$	\( T^{2} - 192 \)
$7$	\( T^{2} - 507 \)
$11$	\( T^{2} - 507 \)