Newform orbit 147.3.b.b

• Label: 147.3.b.b
• Level: $147$
• Weight: $3$
• Character orbit: 147.b
• Self dual: yes
• Analytic conductor: $4.005$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	147.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(4.00545988610\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q + 3 q^{3} + 4 q^{4} + 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

50.1

0

0

3.00000

4.00000

0

0

0

0

9.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.3.b.b		1
3.b	odd	2	1	CM	147.3.b.b		1
7.b	odd	2	1		147.3.b.a		1
7.c	even	3	2		147.3.h.a		2
7.d	odd	6	2		21.3.h.a	✓	2
21.c	even	2	1		147.3.b.a		1
21.g	even	6	2		21.3.h.a	✓	2
21.h	odd	6	2		147.3.h.a		2
28.f	even	6	2		336.3.bn.b		2
84.j	odd	6	2		336.3.bn.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.3.h.a	✓	2	7.d	odd	6	2
21.3.h.a	✓	2	21.g	even	6	2
147.3.b.a		1	7.b	odd	2	1
147.3.b.a		1	21.c	even	2	1
147.3.b.b		1	1.a	even	1	1	trivial
147.3.b.b		1	3.b	odd	2	1	CM
147.3.h.a		2	7.c	even	3	2
147.3.h.a		2	21.h	odd	6	2
336.3.bn.b		2	28.f	even	6	2
336.3.bn.b		2	84.j	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\):

\( T_{2} \)

\( T_{5} \)

\( T_{13} + 23 \)

Hecke characteristic polynomials

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