Newform orbit 147.6.e.k

• Label: 147.6.e.k
• Level: $147$
• Weight: $6$
• Character orbit: 147.e
• Analytic conductor: $23.576$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5764215125\)
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 3)
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.

\(f(q)\)	\(=\)	\( q + 6 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} + 54 q^{6} + 168 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{2} + 9 q^{3} - 4 q^{4} + 6 q^{5} + 108 q^{6} + 336 q^{8} - 81 q^{9}+O(q^{10}) \)

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

3.00000

+

5.19615i

4.50000

−

7.79423i

−2.00000

+

3.46410i

3.00000

+

5.19615i

54.0000

0

168.000

−40.5000

−

70.1481i

−18.0000

+

31.1769i

79.1

3.00000

−

5.19615i

4.50000

+

7.79423i

−2.00000

−

3.46410i

3.00000

−

5.19615i

54.0000

0

168.000

−40.5000

+

70.1481i

−18.0000

−

31.1769i

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	147.6.e.k		2
7.b	odd	2	1		147.6.e.h		2
7.c	even	3	1		147.6.a.a		1
7.c	even	3	1	inner	147.6.e.k		2
7.d	odd	6	1		3.6.a.a	✓	1
7.d	odd	6	1		147.6.e.h		2
21.g	even	6	1		9.6.a.a		1
21.h	odd	6	1		441.6.a.i		1
28.f	even	6	1		48.6.a.a		1
35.i	odd	6	1		75.6.a.e		1
35.k	even	12	2		75.6.b.b		2
56.j	odd	6	1		192.6.a.d		1
56.m	even	6	1		192.6.a.l		1
63.i	even	6	1		81.6.c.a		2
63.k	odd	6	1		81.6.c.c		2
63.s	even	6	1		81.6.c.a		2
63.t	odd	6	1		81.6.c.c		2
77.i	even	6	1		363.6.a.d		1
84.j	odd	6	1		144.6.a.f		1
91.s	odd	6	1		507.6.a.b		1
105.p	even	6	1		225.6.a.a		1
105.w	odd	12	2		225.6.b.b		2
112.v	even	12	2		768.6.d.h		2
112.x	odd	12	2		768.6.d.k		2
119.h	odd	6	1		867.6.a.a		1
133.o	even	6	1		1083.6.a.c		1
168.ba	even	6	1		576.6.a.s		1
168.be	odd	6	1		576.6.a.t		1
231.k	odd	6	1		1089.6.a.b		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.6.a.a	✓	1	7.d	odd	6	1
9.6.a.a		1	21.g	even	6	1
48.6.a.a		1	28.f	even	6	1
75.6.a.e		1	35.i	odd	6	1
75.6.b.b		2	35.k	even	12	2
81.6.c.a		2	63.i	even	6	1
81.6.c.a		2	63.s	even	6	1
81.6.c.c		2	63.k	odd	6	1
81.6.c.c		2	63.t	odd	6	1
144.6.a.f		1	84.j	odd	6	1
147.6.a.a		1	7.c	even	3	1
147.6.e.h		2	7.b	odd	2	1
147.6.e.h		2	7.d	odd	6	1
147.6.e.k		2	1.a	even	1	1	trivial
147.6.e.k		2	7.c	even	3	1	inner
192.6.a.d		1	56.j	odd	6	1
192.6.a.l		1	56.m	even	6	1
225.6.a.a		1	105.p	even	6	1
225.6.b.b		2	105.w	odd	12	2
363.6.a.d		1	77.i	even	6	1
441.6.a.i		1	21.h	odd	6	1
507.6.a.b		1	91.s	odd	6	1
576.6.a.s		1	168.ba	even	6	1
576.6.a.t		1	168.be	odd	6	1
768.6.d.h		2	112.v	even	12	2
768.6.d.k		2	112.x	odd	12	2
867.6.a.a		1	119.h	odd	6	1
1083.6.a.c		1	133.o	even	6	1
1089.6.a.b		1	231.k	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(147, [\chi])\):

\( T_{2}^{2} - 6T_{2} + 36 \)

\( T_{5}^{2} - 6T_{5} + 36 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 6T + 36 \)
$3$	\( T^{2} - 9T + 81 \)
$5$	\( T^{2} - 6T + 36 \)
$7$	\( T^{2} \)
$11$	\( T^{2} - 564T + 318096 \)