Newform orbit 224.2.a.d

• Label: 224.2.a.d
• Level: $224$
• Weight: $2$
• Character orbit: 224.a
• Self dual: yes
• Analytic conductor: $1.789$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.78864900528\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−0.618034
1.61803

0

−1.23607

0

3.23607

0

−1.00000

0

−1.47214

0

1.2

0

3.23607

0

−1.23607

0

−1.00000

0

7.47214

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(7\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.2.a.d	yes	2
3.b	odd	2	1		2016.2.a.o		2
4.b	odd	2	1		224.2.a.c	✓	2
5.b	even	2	1		5600.2.a.z		2
7.b	odd	2	1		1568.2.a.k		2
7.c	even	3	2		1568.2.i.m		4
7.d	odd	6	2		1568.2.i.w		4
8.b	even	2	1		448.2.a.i		2
8.d	odd	2	1		448.2.a.j		2
12.b	even	2	1		2016.2.a.r		2
16.e	even	4	2		1792.2.b.m		4
16.f	odd	4	2		1792.2.b.k		4
20.d	odd	2	1		5600.2.a.bk		2
24.f	even	2	1		4032.2.a.bw		2
24.h	odd	2	1		4032.2.a.bv		2
28.d	even	2	1		1568.2.a.v		2
28.f	even	6	2		1568.2.i.n		4
28.g	odd	6	2		1568.2.i.v		4
56.e	even	2	1		3136.2.a.bf		2
56.h	odd	2	1		3136.2.a.by		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
224.2.a.c	✓	2	4.b	odd	2	1
224.2.a.d	yes	2	1.a	even	1	1	trivial
448.2.a.i		2	8.b	even	2	1
448.2.a.j		2	8.d	odd	2	1
1568.2.a.k		2	7.b	odd	2	1
1568.2.a.v		2	28.d	even	2	1
1568.2.i.m		4	7.c	even	3	2
1568.2.i.n		4	28.f	even	6	2
1568.2.i.v		4	28.g	odd	6	2
1568.2.i.w		4	7.d	odd	6	2
1792.2.b.k		4	16.f	odd	4	2
1792.2.b.m		4	16.e	even	4	2
2016.2.a.o		2	3.b	odd	2	1
2016.2.a.r		2	12.b	even	2	1
3136.2.a.bf		2	56.e	even	2	1
3136.2.a.by		2	56.h	odd	2	1
4032.2.a.bv		2	24.h	odd	2	1
4032.2.a.bw		2	24.f	even	2	1
5600.2.a.z		2	5.b	even	2	1
5600.2.a.bk		2	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(224))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 2T - 4 \)
$5$	\( T^{2} - 2T - 4 \)
$7$	\( (T + 1)^{2} \)
$11$	\( T^{2} - 4T - 16 \)