Newform orbit 224.3.g.a

• Label: 224.3.g.a
• Level: $224$
• Weight: $3$
• Character orbit: 224.g
• Analytic conductor: $6.104$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.10355792167\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Coefficient field:	\(\Q(\sqrt{2}, \sqrt{-7})\)
Defining polynomial:	\( x^{4} + 6x^{2} + 16 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_1 - 2) q^{3} + (2 \beta_{3} - \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - 4 \beta_1 - 3) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{3} - 12 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 16 \) :

\(\beta_{1}\)	\(=\)	\( ( \nu^{3} + 2\nu ) / 4 \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{3} + 10\nu ) / 4 \)
\(\beta_{3}\)	\(=\)	\( \nu^{2} + 3 \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

15.1

0.707107	−	1.87083i
0.707107	+	1.87083i
−0.707107	+	1.87083i
−0.707107	−	1.87083i

0

−3.41421

0

−

1.54985i

0

2.64575i

0

2.65685

0

15.2

0

−3.41421

0

1.54985i

0

−

2.64575i

0

2.65685

0

15.3

0

−0.585786

0

−

9.03316i

0

2.64575i

0

−8.65685

0

15.4

0

−0.585786

0

9.03316i

0

−

2.64575i

0

−8.65685

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	224.3.g.a		4
3.b	odd	2	1		2016.3.g.a		4
4.b	odd	2	1		56.3.g.a	✓	4
7.b	odd	2	1		1568.3.g.h		4
8.b	even	2	1		56.3.g.a	✓	4
8.d	odd	2	1	inner	224.3.g.a		4
12.b	even	2	1		504.3.g.a		4
16.e	even	4	2		1792.3.d.g		8
16.f	odd	4	2		1792.3.d.g		8
24.f	even	2	1		2016.3.g.a		4
24.h	odd	2	1		504.3.g.a		4
28.d	even	2	1		392.3.g.h		4
28.f	even	6	2		392.3.k.j		8
28.g	odd	6	2		392.3.k.i		8
56.e	even	2	1		1568.3.g.h		4
56.h	odd	2	1		392.3.g.h		4
56.j	odd	6	2		392.3.k.j		8
56.p	even	6	2		392.3.k.i		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.3.g.a	✓	4	4.b	odd	2	1
56.3.g.a	✓	4	8.b	even	2	1
224.3.g.a		4	1.a	even	1	1	trivial
224.3.g.a		4	8.d	odd	2	1	inner
392.3.g.h		4	28.d	even	2	1
392.3.g.h		4	56.h	odd	2	1
392.3.k.i		8	28.g	odd	6	2
392.3.k.i		8	56.p	even	6	2
392.3.k.j		8	28.f	even	6	2
392.3.k.j		8	56.j	odd	6	2
504.3.g.a		4	12.b	even	2	1
504.3.g.a		4	24.h	odd	2	1
1568.3.g.h		4	7.b	odd	2	1
1568.3.g.h		4	56.e	even	2	1
1792.3.d.g		8	16.e	even	4	2
1792.3.d.g		8	16.f	odd	4	2
2016.3.g.a		4	3.b	odd	2	1
2016.3.g.a		4	24.f	even	2	1

Hecke kernels

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

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} \)
$3$	\( (T^{2} + 4 T + 2)^{2} \)
$5$	\( T^{4} + 84T^{2} + 196 \)
$7$	\( (T^{2} + 7)^{2} \)
$11$	\( (T^{2} + 8 T - 56)^{2} \)