Newform orbit 112.4.i.a

• Label: 112.4.i.a
• Level: $112$
• Weight: $4$
• Character orbit: 112.i
• Analytic conductor: $6.608$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.60821392064\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
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.

\(f(q)\)	\(=\)	\( q + (5 \zeta_{6} - 5) q^{3} + 9 \zeta_{6} q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{3} + 9 q^{5} + 28 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(17\)	\(85\)
\(\chi(n)\)	\(1\)	\(-\zeta_{6}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

65.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

−2.50000

−

4.33013i

0

4.50000

−

7.79423i

0

14.0000

+

12.1244i

0

1.00000

−

1.73205i

0

81.1

0

−2.50000

+

4.33013i

0

4.50000

+

7.79423i

0

14.0000

−

12.1244i

0

1.00000

+

1.73205i

0

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	112.4.i.a		2
4.b	odd	2	1		14.4.c.a	✓	2
7.c	even	3	1	inner	112.4.i.a		2
7.c	even	3	1		784.4.a.p		1
7.d	odd	6	1		784.4.a.c		1
8.b	even	2	1		448.4.i.e		2
8.d	odd	2	1		448.4.i.b		2
12.b	even	2	1		126.4.g.d		2
20.d	odd	2	1		350.4.e.e		2
20.e	even	4	2		350.4.j.b		4
28.d	even	2	1		98.4.c.a		2
28.f	even	6	1		98.4.a.f		1
28.f	even	6	1		98.4.c.a		2
28.g	odd	6	1		14.4.c.a	✓	2
28.g	odd	6	1		98.4.a.d		1
56.k	odd	6	1		448.4.i.b		2
56.p	even	6	1		448.4.i.e		2
84.h	odd	2	1		882.4.g.u		2
84.j	odd	6	1		882.4.a.c		1
84.j	odd	6	1		882.4.g.u		2
84.n	even	6	1		126.4.g.d		2
84.n	even	6	1		882.4.a.f		1
140.p	odd	6	1		350.4.e.e		2
140.p	odd	6	1		2450.4.a.q		1
140.s	even	6	1		2450.4.a.d		1
140.w	even	12	2		350.4.j.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.c.a	✓	2	4.b	odd	2	1
14.4.c.a	✓	2	28.g	odd	6	1
98.4.a.d		1	28.g	odd	6	1
98.4.a.f		1	28.f	even	6	1
98.4.c.a		2	28.d	even	2	1
98.4.c.a		2	28.f	even	6	1
112.4.i.a		2	1.a	even	1	1	trivial
112.4.i.a		2	7.c	even	3	1	inner
126.4.g.d		2	12.b	even	2	1
126.4.g.d		2	84.n	even	6	1
350.4.e.e		2	20.d	odd	2	1
350.4.e.e		2	140.p	odd	6	1
350.4.j.b		4	20.e	even	4	2
350.4.j.b		4	140.w	even	12	2
448.4.i.b		2	8.d	odd	2	1
448.4.i.b		2	56.k	odd	6	1
448.4.i.e		2	8.b	even	2	1
448.4.i.e		2	56.p	even	6	1
784.4.a.c		1	7.d	odd	6	1
784.4.a.p		1	7.c	even	3	1
882.4.a.c		1	84.j	odd	6	1
882.4.a.f		1	84.n	even	6	1
882.4.g.u		2	84.h	odd	2	1
882.4.g.u		2	84.j	odd	6	1
2450.4.a.d		1	140.s	even	6	1
2450.4.a.q		1	140.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 5T + 25 \)
$5$	\( T^{2} - 9T + 81 \)
$7$	\( T^{2} - 28T + 343 \)
$11$	\( T^{2} + 57T + 3249 \)