Newform orbit 169.4.b.b

• Label: 169.4.b.b
• Level: $169$
• Weight: $4$
• Character orbit: 169.b
• Analytic conductor: $9.971$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.97132279097\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

168.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−

3.46410i

−7.00000

−4.00000

−

13.8564i

24.2487i

−

22.5167i

−

13.8564i

22.0000

−48.0000

168.2

3.46410i

−7.00000

−4.00000

13.8564i

−

24.2487i

22.5167i

13.8564i

22.0000

−48.0000

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

    

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

Hecke kernels

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

Hecke characteristic polynomials

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