Newform orbit 224.2.b.a

• Label: 224.2.b.a
• Level: $224$
• Weight: $2$
• Character orbit: 224.b
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

113.1

	−	1.41421i
		1.41421i

0

−

1.41421i

0

−

1.41421i

0

−1.00000

0

1.00000

0

113.2

0

1.41421i

0

1.41421i

0

−1.00000

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.2.b.a		2
3.b	odd	2	1		2016.2.c.a		2
4.b	odd	2	1		56.2.b.a	✓	2
7.b	odd	2	1		1568.2.b.a		2
7.c	even	3	2		1568.2.t.c		4
7.d	odd	6	2		1568.2.t.b		4
8.b	even	2	1	inner	224.2.b.a		2
8.d	odd	2	1		56.2.b.a	✓	2
12.b	even	2	1		504.2.c.a		2
16.e	even	4	2		1792.2.a.p		2
16.f	odd	4	2		1792.2.a.n		2
24.f	even	2	1		504.2.c.a		2
24.h	odd	2	1		2016.2.c.a		2
28.d	even	2	1		392.2.b.b		2
28.f	even	6	2		392.2.p.b		4
28.g	odd	6	2		392.2.p.a		4
56.e	even	2	1		392.2.b.b		2
56.h	odd	2	1		1568.2.b.a		2
56.j	odd	6	2		1568.2.t.b		4
56.k	odd	6	2		392.2.p.a		4
56.m	even	6	2		392.2.p.b		4
56.p	even	6	2		1568.2.t.c		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.b.a	✓	2	4.b	odd	2	1
56.2.b.a	✓	2	8.d	odd	2	1
224.2.b.a		2	1.a	even	1	1	trivial
224.2.b.a		2	8.b	even	2	1	inner
392.2.b.b		2	28.d	even	2	1
392.2.b.b		2	56.e	even	2	1
392.2.p.a		4	28.g	odd	6	2
392.2.p.a		4	56.k	odd	6	2
392.2.p.b		4	28.f	even	6	2
392.2.p.b		4	56.m	even	6	2
504.2.c.a		2	12.b	even	2	1
504.2.c.a		2	24.f	even	2	1
1568.2.b.a		2	7.b	odd	2	1
1568.2.b.a		2	56.h	odd	2	1
1568.2.t.b		4	7.d	odd	6	2
1568.2.t.b		4	56.j	odd	6	2
1568.2.t.c		4	7.c	even	3	2
1568.2.t.c		4	56.p	even	6	2
1792.2.a.n		2	16.f	odd	4	2
1792.2.a.p		2	16.e	even	4	2
2016.2.c.a		2	3.b	odd	2	1
2016.2.c.a		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2 \)
$5$	\( T^{2} + 2 \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} + 8 \)