Newform orbit 448.2.z.a

• Label: 448.2.z.a
• Level: $448$
• Weight: $2$
• Character orbit: 448.z
• Analytic conductor: $3.577$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.z (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.57729801055\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(14\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{3} - 6 q^{5} + 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1		0		−0.743580	+	2.77508i		0		−3.38680	+	0.907491i		0		−0.0791552	−	2.64457i		0		−4.55008	−	2.62699i		0
47.2		0		−0.615336	+	2.29647i		0		−0.835825	+	0.223959i		0		1.25761	+	2.32775i		0		−2.29704	−	1.32620i		0
47.3		0		−0.524583	+	1.95777i		0		2.48890	−	0.666898i		0		2.38027	+	1.15512i		0		−0.959602	−	0.554026i		0
47.4		0		−0.449868	+	1.67893i		0		0.731029	−	0.195879i		0		−2.52163	−	0.800849i		0		−0.0183525	−	0.0105958i		0
47.5		0		−0.350301	+	1.30734i		0		1.09872	−	0.294400i		0		1.56831	−	2.13082i		0		1.01165	+	0.584076i		0
47.6		0		−0.0530773	+	0.198087i		0		−1.82029	+	0.487744i		0		−1.84933	+	1.89208i		0		2.56165	+	1.47897i		0
47.7		0		0.0661591	−	0.246909i		0		−0.499339	+	0.133797i		0		−2.45011	−	0.998472i		0		2.54149	+	1.46733i		0
47.8		0		0.204601	−	0.763582i		0		−3.81370	+	1.02188i		0		2.64575	+	0.00379639i		0		2.05688	+	1.18754i		0
47.9		0		0.237312	−	0.885661i		0		3.05579	−	0.818796i		0		0.640837	+	2.56697i		0		1.87000	+	1.07964i		0
47.10		0		0.282507	−	1.05433i		0		−1.39922	+	0.374919i		0		0.298614	+	2.62885i		0		1.56627	+	0.904287i		0
47.11		0		0.388227	−	1.44888i		0		2.81809	−	0.755106i		0		1.47726	−	2.19493i		0		0.649537	+	0.375010i		0
47.12		0		0.665801	−	2.48480i		0		−3.12401	+	0.837076i		0		−1.56215	−	2.13534i		0		−3.13287	−	1.80877i		0
47.13		0		0.700587	−	2.61463i		0		0.450647	−	0.120751i		0		2.37326	−	1.16946i		0		−3.74737	−	2.16355i		0
47.14		0		0.825526	−	3.08091i		0		1.86998	−	0.501060i		0		−2.17953	+	1.49988i		0		−6.21241	−	3.58674i		0
143.1		0		−0.743580	−	2.77508i		0		−3.38680	−	0.907491i		0		−0.0791552	+	2.64457i		0		−4.55008	+	2.62699i		0
143.2		0		−0.615336	−	2.29647i		0		−0.835825	−	0.223959i		0		1.25761	−	2.32775i		0		−2.29704	+	1.32620i		0
143.3		0		−0.524583	−	1.95777i		0		2.48890	+	0.666898i		0		2.38027	−	1.15512i		0		−0.959602	+	0.554026i		0
143.4		0		−0.449868	−	1.67893i		0		0.731029	+	0.195879i		0		−2.52163	+	0.800849i		0		−0.0183525	+	0.0105958i		0
143.5		0		−0.350301	−	1.30734i		0		1.09872	+	0.294400i		0		1.56831	+	2.13082i		0		1.01165	−	0.584076i		0
143.6		0		−0.0530773	−	0.198087i		0		−1.82029	−	0.487744i		0		−1.84933	−	1.89208i		0		2.56165	−	1.47897i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
16.f	odd	4	1	inner
112.v	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.2.z.a		56
4.b	odd	2	1		112.2.v.a	✓	56
7.d	odd	6	1	inner	448.2.z.a		56
8.b	even	2	1		896.2.z.a		56
8.d	odd	2	1		896.2.z.b		56
16.e	even	4	1		112.2.v.a	✓	56
16.e	even	4	1		896.2.z.b		56
16.f	odd	4	1	inner	448.2.z.a		56
16.f	odd	4	1		896.2.z.a		56
28.d	even	2	1		784.2.w.f		56
28.f	even	6	1		112.2.v.a	✓	56
28.f	even	6	1		784.2.j.a		56
28.g	odd	6	1		784.2.j.a		56
28.g	odd	6	1		784.2.w.f		56
56.j	odd	6	1		896.2.z.a		56
56.m	even	6	1		896.2.z.b		56
112.l	odd	4	1		784.2.w.f		56
112.v	even	12	1	inner	448.2.z.a		56
112.v	even	12	1		896.2.z.a		56
112.w	even	12	1		784.2.j.a		56
112.w	even	12	1		784.2.w.f		56
112.x	odd	12	1		112.2.v.a	✓	56
112.x	odd	12	1		784.2.j.a		56
112.x	odd	12	1		896.2.z.b		56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.2.v.a	✓	56	4.b	odd	2	1
112.2.v.a	✓	56	16.e	even	4	1
112.2.v.a	✓	56	28.f	even	6	1
112.2.v.a	✓	56	112.x	odd	12	1
448.2.z.a		56	1.a	even	1	1	trivial
448.2.z.a		56	7.d	odd	6	1	inner
448.2.z.a		56	16.f	odd	4	1	inner
448.2.z.a		56	112.v	even	12	1	inner
784.2.j.a		56	28.f	even	6	1
784.2.j.a		56	28.g	odd	6	1
784.2.j.a		56	112.w	even	12	1
784.2.j.a		56	112.x	odd	12	1
784.2.w.f		56	28.d	even	2	1
784.2.w.f		56	28.g	odd	6	1
784.2.w.f		56	112.l	odd	4	1
784.2.w.f		56	112.w	even	12	1
896.2.z.a		56	8.b	even	2	1
896.2.z.a		56	16.f	odd	4	1
896.2.z.a		56	56.j	odd	6	1
896.2.z.a		56	112.v	even	12	1
896.2.z.b		56	8.d	odd	2	1
896.2.z.b		56	16.e	even	4	1
896.2.z.b		56	56.m	even	6	1
896.2.z.b		56	112.x	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(448, [\chi])\).