Embedded newform 448.2.bn.a.205.55

• Label: 448.2.bn.a.205.55
• Level: $448$
• Weight: $2$
• Character: 448.205
• Analytic conductor: $3.577$
• Analytic rank: $0$
• Dimension: $992$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.bn (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.57729801055\)
Analytic rank:	\(0\)
Dimension:	\(992\)
Relative dimension:	\(62\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			205.55
Character	\(\chi\)	\(=\)	448.205
Dual form			448.2.bn.a.389.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.26406 - 0.634147i) q^{2} +(-0.519235 - 0.256058i) q^{3} +(1.19572 - 1.60320i) q^{4} +(0.777812 - 2.29136i) q^{5} +(-0.818725 + 0.00559701i) q^{6} +(-2.41192 + 1.08749i) q^{7} +(0.494795 - 2.78481i) q^{8} +(-1.62225 - 2.11415i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{15}{16}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.26406	−	0.634147i	0.893828	−	0.448409i
							\(3\)	−0.519235	−	0.256058i	−0.299781	−	0.147835i	0.286178	−	0.958176i	\(-0.407615\pi\)
−0.585958	+	0.810341i	\(0.699282\pi\)
\(5\)	0.777812	−	2.29136i	0.347848	−	1.02473i	−0.622418	−	0.782685i	\(-0.713849\pi\)
							0.970266	−	0.242042i	\(-0.0778172\pi\)
\(7\)	−2.41192	+	1.08749i	−0.911620	+	0.411034i
							\(11\)	−1.67256	−	0.109625i	−0.504296	−	0.0330533i	−0.188867	−	0.982003i	\(-0.560482\pi\)
−0.315428	+	0.948949i	\(0.602148\pi\)
\(13\)	2.52674	+	0.502600i	0.700792	+	0.139396i	0.532611	−	0.846360i	\(-0.321211\pi\)
							0.168180	+	0.985756i	\(0.446211\pi\)
\(17\)	0.486715	−	1.81644i	0.118046	−	0.440553i	−0.881451	−	0.472275i	\(-0.843433\pi\)
							0.999497	+	0.0317229i	\(0.0100994\pi\)
\(19\)	3.08896	+	2.70895i	0.708657	+	0.621475i	0.935860	−	0.352372i	\(-0.114625\pi\)
							−0.227203	+	0.973847i	\(0.572958\pi\)
\(23\)	−1.15854	+	0.888978i	−0.241572	+	0.185365i	−0.722451	−	0.691422i	\(-0.756984\pi\)
							0.480879	+	0.876787i	\(0.340318\pi\)
\(29\)	−1.24110	−	1.85744i	−0.230467	−	0.344919i	0.698154	−	0.715948i	\(-0.254005\pi\)
							−0.928621	+	0.371029i	\(0.879005\pi\)
\(31\)	7.35594	−	4.24695i	1.32117	−	0.762775i	0.337251	−	0.941415i	\(-0.390503\pi\)
							0.983915	+	0.178640i	\(0.0571697\pi\)
\(37\)	5.51623	+	1.87251i	0.906863	+	0.307839i	0.735623	−	0.677391i	\(-0.236889\pi\)
							0.171240	+	0.985229i	\(0.445223\pi\)
\(41\)	4.59103	+	1.90167i	0.716998	+	0.296990i	0.711197	−	0.702993i	\(-0.248153\pi\)
							0.00580114	+	0.999983i	\(0.498153\pi\)
\(43\)	−2.95289	−	1.97306i	−0.450312	−	0.300889i	0.309657	−	0.950848i	\(-0.399786\pi\)
							−0.759969	+	0.649960i	\(0.774786\pi\)
\(47\)	0.904315	−	0.242310i	0.131908	−	0.0353446i	−0.192261	−	0.981344i	\(-0.561582\pi\)
							0.324169	+	0.945999i	\(0.394915\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.205.55	✓	992
7.4	even	3	inner	448.2.bn.a.333.29	yes	992
64.5	even	16	inner	448.2.bn.a.261.29	yes	992
448.389	even	48	inner	448.2.bn.a.389.55	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.205.55	✓	992	1.1	even	1	trivial
448.2.bn.a.261.29	yes	992	64.5	even	16	inner
448.2.bn.a.333.29	yes	992	7.4	even	3	inner
448.2.bn.a.389.55	yes	992	448.389	even	48	inner