Newform orbit 448.4.q.a

• Label: 448.4.q.a
• Level: $448$
• Weight: $4$
• Character orbit: 448.q
• Analytic conductor: $26.433$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(26.4328556826\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 24 q^{5} + 176 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1		0		−8.78932	−	5.07452i		0		−8.37041	−	14.4980i		0		0.901615	−	18.4983i		0		38.0015	+	65.8205i		0
31.2		0		−6.84052	−	3.94938i		0		−9.21856	−	15.9670i		0		−14.3277	+	11.7353i		0		17.6951	+	30.6489i		0
31.3		0		−6.23656	−	3.60068i		0		3.36366	+	5.82602i		0		4.34160	−	18.0042i		0		12.4298	+	21.5290i		0
31.4		0		−5.65076	−	3.26247i		0		8.63222	+	14.9514i		0		−17.0056	−	7.33551i		0		7.78736	+	13.4881i		0
31.5		0		−4.17280	−	2.40917i		0		3.69991	+	6.40843i		0		13.0239	+	13.1673i		0		−1.89183	−	3.27674i		0
31.6		0		−3.34210	−	1.92956i		0		3.71621	+	6.43666i		0		6.86804	+	17.1997i		0		−6.05359	−	10.4851i		0
31.7		0		−1.64284	−	0.948492i		0		−0.518458	−	0.897996i		0		18.1631	−	3.61980i		0		−11.7007	−	20.2663i		0
31.8		0		−1.35962	−	0.784974i		0		−7.30456	−	12.6519i		0		−14.7369	+	11.2171i		0		−12.2676	−	21.2482i		0
31.9		0		1.35962	+	0.784974i		0		−7.30456	−	12.6519i		0		14.7369	−	11.2171i		0		−12.2676	−	21.2482i		0
31.10		0		1.64284	+	0.948492i		0		−0.518458	−	0.897996i		0		−18.1631	+	3.61980i		0		−11.7007	−	20.2663i		0
31.11		0		3.34210	+	1.92956i		0		3.71621	+	6.43666i		0		−6.86804	−	17.1997i		0		−6.05359	−	10.4851i		0
31.12		0		4.17280	+	2.40917i		0		3.69991	+	6.40843i		0		−13.0239	−	13.1673i		0		−1.89183	−	3.27674i		0
31.13		0		5.65076	+	3.26247i		0		8.63222	+	14.9514i		0		17.0056	+	7.33551i		0		7.78736	+	13.4881i		0
31.14		0		6.23656	+	3.60068i		0		3.36366	+	5.82602i		0		−4.34160	+	18.0042i		0		12.4298	+	21.5290i		0
31.15		0		6.84052	+	3.94938i		0		−9.21856	−	15.9670i		0		14.3277	−	11.7353i		0		17.6951	+	30.6489i		0
31.16		0		8.78932	+	5.07452i		0		−8.37041	−	14.4980i		0		−0.901615	+	18.4983i		0		38.0015	+	65.8205i		0
159.1		0		−8.78932	+	5.07452i		0		−8.37041	+	14.4980i		0		0.901615	+	18.4983i		0		38.0015	−	65.8205i		0
159.2		0		−6.84052	+	3.94938i		0		−9.21856	+	15.9670i		0		−14.3277	−	11.7353i		0		17.6951	−	30.6489i		0
159.3		0		−6.23656	+	3.60068i		0		3.36366	−	5.82602i		0		4.34160	+	18.0042i		0		12.4298	−	21.5290i		0
159.4		0		−5.65076	+	3.26247i		0		8.63222	−	14.9514i		0		−17.0056	+	7.33551i		0		7.78736	−	13.4881i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
56.j	odd	6	1	inner
56.m	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.4.q.a	✓	32
4.b	odd	2	1	inner	448.4.q.a	✓	32
7.d	odd	6	1		448.4.q.c	yes	32
8.b	even	2	1		448.4.q.c	yes	32
8.d	odd	2	1		448.4.q.c	yes	32
28.f	even	6	1		448.4.q.c	yes	32
56.j	odd	6	1	inner	448.4.q.a	✓	32
56.m	even	6	1	inner	448.4.q.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
448.4.q.a	✓	32	1.a	even	1	1	trivial
448.4.q.a	✓	32	4.b	odd	2	1	inner
448.4.q.a	✓	32	56.j	odd	6	1	inner
448.4.q.a	✓	32	56.m	even	6	1	inner
448.4.q.c	yes	32	7.d	odd	6	1
448.4.q.c	yes	32	8.b	even	2	1
448.4.q.c	yes	32	8.d	odd	2	1
448.4.q.c	yes	32	28.f	even	6	1