Newform orbit 112.2.p.c

• Label: 112.2.p.c
• Level: $112$
• Weight: $2$
• Character orbit: 112.p
• Analytic conductor: $0.894$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.894324502638\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial:	\( x^{4} + 7x^{2} + 49 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
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.

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + ( - \beta_{2} - 2) q^{5} - \beta_{3} q^{7} + 4 \beta_{2} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 7 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 7 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)	\(85\)
\(\chi(n)\)	\(-1\)	\(-\beta_{2}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

31.1

−1.32288	+	2.29129i
1.32288	−	2.29129i
−1.32288	−	2.29129i
1.32288	+	2.29129i

0

−1.32288

+

2.29129i

0

−1.50000

+

0.866025i

0

−2.64575

0

−2.00000

−

3.46410i

0

31.2

0

1.32288

−

2.29129i

0

−1.50000

+

0.866025i

0

2.64575

0

−2.00000

−

3.46410i

0

47.1

0

−1.32288

−

2.29129i

0

−1.50000

−

0.866025i

0

−2.64575

0

−2.00000

+

3.46410i

0

47.2

0

1.32288

+

2.29129i

0

−1.50000

−

0.866025i

0

2.64575

0

−2.00000

+

3.46410i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.d	odd	6	1	inner
28.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	112.2.p.c	✓	4
3.b	odd	2	1		1008.2.cs.q		4
4.b	odd	2	1	inner	112.2.p.c	✓	4
7.b	odd	2	1		784.2.p.g		4
7.c	even	3	1		784.2.f.d		4
7.c	even	3	1		784.2.p.g		4
7.d	odd	6	1	inner	112.2.p.c	✓	4
7.d	odd	6	1		784.2.f.d		4
8.b	even	2	1		448.2.p.c		4
8.d	odd	2	1		448.2.p.c		4
12.b	even	2	1		1008.2.cs.q		4
21.g	even	6	1		1008.2.cs.q		4
21.g	even	6	1		7056.2.b.s		4
21.h	odd	6	1		7056.2.b.s		4
28.d	even	2	1		784.2.p.g		4
28.f	even	6	1	inner	112.2.p.c	✓	4
28.f	even	6	1		784.2.f.d		4
28.g	odd	6	1		784.2.f.d		4
28.g	odd	6	1		784.2.p.g		4
56.j	odd	6	1		448.2.p.c		4
56.j	odd	6	1		3136.2.f.f		4
56.k	odd	6	1		3136.2.f.f		4
56.m	even	6	1		448.2.p.c		4
56.m	even	6	1		3136.2.f.f		4
56.p	even	6	1		3136.2.f.f		4
84.j	odd	6	1		1008.2.cs.q		4
84.j	odd	6	1		7056.2.b.s		4
84.n	even	6	1		7056.2.b.s		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.p.c	✓	4	1.a	even	1	1	trivial
112.2.p.c	✓	4	4.b	odd	2	1	inner
112.2.p.c	✓	4	7.d	odd	6	1	inner
112.2.p.c	✓	4	28.f	even	6	1	inner
448.2.p.c		4	8.b	even	2	1
448.2.p.c		4	8.d	odd	2	1
448.2.p.c		4	56.j	odd	6	1
448.2.p.c		4	56.m	even	6	1
784.2.f.d		4	7.c	even	3	1
784.2.f.d		4	7.d	odd	6	1
784.2.f.d		4	28.f	even	6	1
784.2.f.d		4	28.g	odd	6	1
784.2.p.g		4	7.b	odd	2	1
784.2.p.g		4	7.c	even	3	1
784.2.p.g		4	28.d	even	2	1
784.2.p.g		4	28.g	odd	6	1
1008.2.cs.q		4	3.b	odd	2	1
1008.2.cs.q		4	12.b	even	2	1
1008.2.cs.q		4	21.g	even	6	1
1008.2.cs.q		4	84.j	odd	6	1
3136.2.f.f		4	56.j	odd	6	1
3136.2.f.f		4	56.k	odd	6	1
3136.2.f.f		4	56.m	even	6	1
3136.2.f.f		4	56.p	even	6	1
7056.2.b.s		4	21.g	even	6	1
7056.2.b.s		4	21.h	odd	6	1
7056.2.b.s		4	84.j	odd	6	1
7056.2.b.s		4	84.n	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 7T_{3}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( T^{4} + 7T^{2} + 49 \)
$5$	\( (T^{2} + 3 T + 3)^{2} \)
$7$	\( (T^{2} - 7)^{2} \)
$11$	\( T^{4} - 21T^{2} + 441 \)