Newform orbit 112.2.i.a

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

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.894324502638\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \)

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

−0.500000

−

0.866025i

0

0.500000

−

0.866025i

0

2.00000

−

1.73205i

0

1.00000

−

1.73205i

0

81.1

0

−0.500000

+

0.866025i

0

0.500000

+

0.866025i

0

2.00000

+

1.73205i

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	112.2.i.a		2
3.b	odd	2	1		1008.2.s.g		2
4.b	odd	2	1		56.2.i.b	✓	2
7.b	odd	2	1		784.2.i.h		2
7.c	even	3	1	inner	112.2.i.a		2
7.c	even	3	1		784.2.a.h		1
7.d	odd	6	1		784.2.a.c		1
7.d	odd	6	1		784.2.i.h		2
8.b	even	2	1		448.2.i.d		2
8.d	odd	2	1		448.2.i.b		2
12.b	even	2	1		504.2.s.c		2
20.d	odd	2	1		1400.2.q.d		2
20.e	even	4	2		1400.2.bh.a		4
21.g	even	6	1		7056.2.a.u		1
21.h	odd	6	1		1008.2.s.g		2
21.h	odd	6	1		7056.2.a.bj		1
28.d	even	2	1		392.2.i.b		2
28.f	even	6	1		392.2.a.e		1
28.f	even	6	1		392.2.i.b		2
28.g	odd	6	1		56.2.i.b	✓	2
28.g	odd	6	1		392.2.a.c		1
56.j	odd	6	1		3136.2.a.t		1
56.k	odd	6	1		448.2.i.b		2
56.k	odd	6	1		3136.2.a.u		1
56.m	even	6	1		3136.2.a.i		1
56.p	even	6	1		448.2.i.d		2
56.p	even	6	1		3136.2.a.j		1
84.h	odd	2	1		3528.2.s.q		2
84.j	odd	6	1		3528.2.a.j		1
84.j	odd	6	1		3528.2.s.q		2
84.n	even	6	1		504.2.s.c		2
84.n	even	6	1		3528.2.a.p		1
140.p	odd	6	1		1400.2.q.d		2
140.p	odd	6	1		9800.2.a.be		1
140.s	even	6	1		9800.2.a.s		1
140.w	even	12	2		1400.2.bh.a		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.i.b	✓	2	4.b	odd	2	1
56.2.i.b	✓	2	28.g	odd	6	1
112.2.i.a		2	1.a	even	1	1	trivial
112.2.i.a		2	7.c	even	3	1	inner
392.2.a.c		1	28.g	odd	6	1
392.2.a.e		1	28.f	even	6	1
392.2.i.b		2	28.d	even	2	1
392.2.i.b		2	28.f	even	6	1
448.2.i.b		2	8.d	odd	2	1
448.2.i.b		2	56.k	odd	6	1
448.2.i.d		2	8.b	even	2	1
448.2.i.d		2	56.p	even	6	1
504.2.s.c		2	12.b	even	2	1
504.2.s.c		2	84.n	even	6	1
784.2.a.c		1	7.d	odd	6	1
784.2.a.h		1	7.c	even	3	1
784.2.i.h		2	7.b	odd	2	1
784.2.i.h		2	7.d	odd	6	1
1008.2.s.g		2	3.b	odd	2	1
1008.2.s.g		2	21.h	odd	6	1
1400.2.q.d		2	20.d	odd	2	1
1400.2.q.d		2	140.p	odd	6	1
1400.2.bh.a		4	20.e	even	4	2
1400.2.bh.a		4	140.w	even	12	2
3136.2.a.i		1	56.m	even	6	1
3136.2.a.j		1	56.p	even	6	1
3136.2.a.t		1	56.j	odd	6	1
3136.2.a.u		1	56.k	odd	6	1
3528.2.a.j		1	84.j	odd	6	1
3528.2.a.p		1	84.n	even	6	1
3528.2.s.q		2	84.h	odd	2	1
3528.2.s.q		2	84.j	odd	6	1
7056.2.a.u		1	21.g	even	6	1
7056.2.a.bj		1	21.h	odd	6	1
9800.2.a.s		1	140.s	even	6	1
9800.2.a.be		1	140.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} - T + 1 \)
$7$	\( T^{2} - 4T + 7 \)
$11$	\( T^{2} - 3T + 9 \)