Newform orbit 169.4.a.g

• Label: 169.4.a.g
• 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.

\(f(q)\)	\(=\)	\( q - \beta q^{2} + ( - 3 \beta + 4) q^{3} + (\beta - 4) q^{4} + ( - \beta + 2) q^{5} + ( - \beta + 12) q^{6} + ( - 11 \beta + 10) q^{7} + (11 \beta - 4) q^{8} + ( - 15 \beta + 25) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 5 q^{3} - 7 q^{4} + 3 q^{5} + 23 q^{6} + 9 q^{7} + 3 q^{8} + 35 q^{9}+O(q^{10}) \)

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

−2.56155

−3.68466

−1.43845

−0.561553

9.43845

−18.1771

24.1771

−13.4233

1.43845

1.2

1.56155

8.68466

−5.56155

3.56155

13.5616

27.1771

−21.1771

48.4233

5.56155

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.g		2
3.b	odd	2	1		1521.4.a.r		2
13.b	even	2	1		13.4.a.b	✓	2
13.c	even	3	2		169.4.c.j		4
13.d	odd	4	2		169.4.b.f		4
13.e	even	6	2		169.4.c.g		4
13.f	odd	12	4		169.4.e.f		8
39.d	odd	2	1		117.4.a.d		2
52.b	odd	2	1		208.4.a.h		2
65.d	even	2	1		325.4.a.f		2
65.h	odd	4	2		325.4.b.e		4
91.b	odd	2	1		637.4.a.b		2
104.e	even	2	1		832.4.a.s		2
104.h	odd	2	1		832.4.a.z		2
143.d	odd	2	1		1573.4.a.b		2
156.h	even	2	1		1872.4.a.bb		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.a.b	✓	2	13.b	even	2	1
117.4.a.d		2	39.d	odd	2	1
169.4.a.g		2	1.a	even	1	1	trivial
169.4.b.f		4	13.d	odd	4	2
169.4.c.g		4	13.e	even	6	2
169.4.c.j		4	13.c	even	3	2
169.4.e.f		8	13.f	odd	12	4
208.4.a.h		2	52.b	odd	2	1
325.4.a.f		2	65.d	even	2	1
325.4.b.e		4	65.h	odd	4	2
637.4.a.b		2	91.b	odd	2	1
832.4.a.s		2	104.e	even	2	1
832.4.a.z		2	104.h	odd	2	1
1521.4.a.r		2	3.b	odd	2	1
1573.4.a.b		2	143.d	odd	2	1
1872.4.a.bb		2	156.h	even	2	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T - 4 \)
$3$	\( T^{2} - 5T - 32 \)
$5$	\( T^{2} - 3T - 2 \)
$7$	\( T^{2} - 9T - 494 \)
$11$	\( T^{2} + 80T + 988 \)