Newform orbit 147.4.e.g

• Label: 147.4.e.g
• Level: $147$
• Weight: $4$
• Character orbit: 147.e
• Analytic conductor: $8.673$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.67328077084\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + 3 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (\zeta_{6} - 1) q^{4} - 18 \zeta_{6} q^{5} - 9 q^{6} + 21 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 3 q^{3} - q^{4} - 18 q^{5} - 18 q^{6} + 42 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(50\)	\(52\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

67.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.50000

+

2.59808i

−1.50000

+

2.59808i

−0.500000

+

0.866025i

−9.00000

−

15.5885i

−9.00000

0

21.0000

−4.50000

−

7.79423i

27.0000

−

46.7654i

79.1

1.50000

−

2.59808i

−1.50000

−

2.59808i

−0.500000

−

0.866025i

−9.00000

+

15.5885i

−9.00000

0

21.0000

−4.50000

+

7.79423i

27.0000

+

46.7654i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	147.4.e.g		2
3.b	odd	2	1		441.4.e.d		2
7.b	odd	2	1		147.4.e.i		2
7.c	even	3	1		147.4.a.c		1
7.c	even	3	1	inner	147.4.e.g		2
7.d	odd	6	1		21.4.a.a	✓	1
7.d	odd	6	1		147.4.e.i		2
21.c	even	2	1		441.4.e.b		2
21.g	even	6	1		63.4.a.c		1
21.g	even	6	1		441.4.e.b		2
21.h	odd	6	1		441.4.a.j		1
21.h	odd	6	1		441.4.e.d		2
28.f	even	6	1		336.4.a.f		1
28.g	odd	6	1		2352.4.a.r		1
35.i	odd	6	1		525.4.a.g		1
35.k	even	12	2		525.4.d.c		2
56.j	odd	6	1		1344.4.a.ba		1
56.m	even	6	1		1344.4.a.n		1
84.j	odd	6	1		1008.4.a.v		1
105.p	even	6	1		1575.4.a.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.a	✓	1	7.d	odd	6	1
63.4.a.c		1	21.g	even	6	1
147.4.a.c		1	7.c	even	3	1
147.4.e.g		2	1.a	even	1	1	trivial
147.4.e.g		2	7.c	even	3	1	inner
147.4.e.i		2	7.b	odd	2	1
147.4.e.i		2	7.d	odd	6	1
336.4.a.f		1	28.f	even	6	1
441.4.a.j		1	21.h	odd	6	1
441.4.e.b		2	21.c	even	2	1
441.4.e.b		2	21.g	even	6	1
441.4.e.d		2	3.b	odd	2	1
441.4.e.d		2	21.h	odd	6	1
525.4.a.g		1	35.i	odd	6	1
525.4.d.c		2	35.k	even	12	2
1008.4.a.v		1	84.j	odd	6	1
1344.4.a.n		1	56.m	even	6	1
1344.4.a.ba		1	56.j	odd	6	1
1575.4.a.b		1	105.p	even	6	1
2352.4.a.r		1	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{2} - 3T_{2} + 9 \)

\( T_{5}^{2} + 18T_{5} + 324 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3T + 9 \)
$3$	\( T^{2} + 3T + 9 \)
$5$	\( T^{2} + 18T + 324 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 36T + 1296 \)