Newform orbit 112.5.c.a

• Label: 112.5.c.a
• Level: $112$
• Weight: $5$
• Character orbit: 112.c
• Self dual: yes
• Analytic conductor: $11.577$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -7
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	112.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(11.5774358654\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 49 q^{7} + 81 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(17\)	\(85\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

97.1

0

0

0

0

0

0

−49.0000

0

81.0000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	112.5.c.a		1
3.b	odd	2	1		1008.5.f.a		1
4.b	odd	2	1		7.5.b.a	✓	1
7.b	odd	2	1	CM	112.5.c.a		1
8.b	even	2	1		448.5.c.a		1
8.d	odd	2	1		448.5.c.b		1
12.b	even	2	1		63.5.d.a		1
20.d	odd	2	1		175.5.d.a		1
20.e	even	4	2		175.5.c.a		2
21.c	even	2	1		1008.5.f.a		1
28.d	even	2	1		7.5.b.a	✓	1
28.f	even	6	2		49.5.d.a		2
28.g	odd	6	2		49.5.d.a		2
56.e	even	2	1		448.5.c.b		1
56.h	odd	2	1		448.5.c.a		1
84.h	odd	2	1		63.5.d.a		1
140.c	even	2	1		175.5.d.a		1
140.j	odd	4	2		175.5.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
7.5.b.a	✓	1	4.b	odd	2	1
7.5.b.a	✓	1	28.d	even	2	1
49.5.d.a		2	28.f	even	6	2
49.5.d.a		2	28.g	odd	6	2
63.5.d.a		1	12.b	even	2	1
63.5.d.a		1	84.h	odd	2	1
112.5.c.a		1	1.a	even	1	1	trivial
112.5.c.a		1	7.b	odd	2	1	CM
175.5.c.a		2	20.e	even	4	2
175.5.c.a		2	140.j	odd	4	2
175.5.d.a		1	20.d	odd	2	1
175.5.d.a		1	140.c	even	2	1
448.5.c.a		1	8.b	even	2	1
448.5.c.a		1	56.h	odd	2	1
448.5.c.b		1	8.d	odd	2	1
448.5.c.b		1	56.e	even	2	1
1008.5.f.a		1	3.b	odd	2	1
1008.5.f.a		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{5}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T + 49 \)
$11$	\( T - 206 \)