Newform orbit 162.7.d.c

• Label: 162.7.d.c
• Level: $162$
• Weight: $7$
• Character orbit: 162.d
• Analytic conductor: $37.269$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(37.2687615464\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial:	\( x^{4} - 2x^{2} + 4 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{3} - \beta_1) q^{2} + ( - 32 \beta_{2} + 32) q^{4} + 6 \beta_1 q^{5} + 205 \beta_{2} q^{7} + 32 \beta_{3} q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 64 q^{4} + 410 q^{7}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) :

\(\beta_{1}\)	\(=\)	\( 4\nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \)
\(\beta_{3}\)	\(=\)	\( 2\nu^{3} \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\)	\(83\)
\(\chi(n)\)	\(1 - \beta_{2}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

53.1

1.22474	+	0.707107i
−1.22474	−	0.707107i
1.22474	−	0.707107i
−1.22474	+	0.707107i

−4.89898

+

2.82843i

0

16.0000

−

27.7128i

29.3939

+

16.9706i

0

102.500

+

177.535i

181.019i

0

−192.000

53.2

4.89898

−

2.82843i

0

16.0000

−

27.7128i

−29.3939

−

16.9706i

0

102.500

+

177.535i

−

181.019i

0

−192.000

107.1

−4.89898

−

2.82843i

0

16.0000

+

27.7128i

29.3939

−

16.9706i

0

102.500

−

177.535i

−

181.019i

0

−192.000

107.2

4.89898

+

2.82843i

0

16.0000

+

27.7128i

−29.3939

+

16.9706i

0

102.500

−

177.535i

181.019i

0

−192.000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	162.7.d.c		4
3.b	odd	2	1	inner	162.7.d.c		4
9.c	even	3	1		54.7.b.a	✓	2
9.c	even	3	1	inner	162.7.d.c		4
9.d	odd	6	1		54.7.b.a	✓	2
9.d	odd	6	1	inner	162.7.d.c		4
36.f	odd	6	1		432.7.e.f		2
36.h	even	6	1		432.7.e.f		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.7.b.a	✓	2	9.c	even	3	1
54.7.b.a	✓	2	9.d	odd	6	1
162.7.d.c		4	1.a	even	1	1	trivial
162.7.d.c		4	3.b	odd	2	1	inner
162.7.d.c		4	9.c	even	3	1	inner
162.7.d.c		4	9.d	odd	6	1	inner
432.7.e.f		2	36.f	odd	6	1
432.7.e.f		2	36.h	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 1152T_{5}^{2} + 1327104 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} - 32T^{2} + 1024 \)
$3$	\( T^{4} \)
$5$	T^4 - 1152*T^2 + 1327104
$7$	\( (T^{2} - 205 T + 42025)^{2} \)
$11$	T^4 - 1107072*T^2 + 1225608413184