Embedded newform 448.2.bm.a.227.61

• Label: 448.2.bm.a.227.61
• Level: $448$
• Weight: $2$
• Character: 448.227
• 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.bm (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			227.61
Character	\(\chi\)	\(=\)	448.227
Dual form			448.2.bm.a.75.61

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41224 - 0.0746945i) q^{2} +(-0.296012 + 0.337537i) q^{3} +(1.98884 - 0.210973i) q^{4} +(-0.244088 + 3.72406i) q^{5} +(-0.392827 + 0.498793i) q^{6} +(-0.768622 - 2.53164i) q^{7} +(2.79296 - 0.446500i) q^{8} +(0.365270 + 2.77450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 24 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41224	−	0.0746945i	0.998604	−	0.0528170i
							\(3\)	−0.296012	+	0.337537i	−0.170902	+	0.194877i	−0.830950	−	0.556347i	\(-0.812203\pi\)
0.660048	+	0.751224i	\(0.270536\pi\)
\(5\)	−0.244088	+	3.72406i	−0.109159	+	1.66545i	0.495232	+	0.868761i	\(0.335083\pi\)
							−0.604391	+	0.796688i	\(0.706584\pi\)
\(7\)	−0.768622	−	2.53164i	−0.290512	−	0.956871i
							\(11\)	0.474280	−	1.39718i	0.143001	−	0.421267i	−0.851706	−	0.524020i	\(-0.824432\pi\)
0.994707	+	0.102753i	\(0.0327651\pi\)
\(13\)	0.929228	−	0.620890i	0.257721	−	0.172204i	−0.419997	−	0.907525i	\(-0.637969\pi\)
							0.677718	+	0.735322i	\(0.262969\pi\)
\(17\)	1.64469	+	6.13806i	0.398895	+	1.48870i	0.815043	+	0.579401i	\(0.196713\pi\)
							−0.416147	+	0.909297i	\(0.636620\pi\)
\(19\)	−2.81947	−	1.39041i	−0.646832	−	0.318982i	0.0891151	−	0.996021i	\(-0.471596\pi\)
							−0.735947	+	0.677039i	\(0.763263\pi\)
\(23\)	0.639051	+	4.85407i	0.133251	+	1.01214i	0.919183	+	0.393831i	\(0.128850\pi\)
							−0.785932	+	0.618313i	\(0.787816\pi\)
\(29\)	1.84400	−	9.27043i	0.342423	−	1.72148i	−0.298970	−	0.954263i	\(-0.596643\pi\)
							0.641392	−	0.767213i	\(-0.278357\pi\)
\(31\)	2.70615	+	1.56240i	0.486039	+	0.280615i	0.722930	−	0.690921i	\(-0.242795\pi\)
							−0.236891	+	0.971536i	\(0.576128\pi\)
\(37\)	0.447850	−	6.83287i	0.0736261	−	1.12332i	−0.787967	−	0.615718i	\(-0.788866\pi\)
							0.861593	−	0.507600i	\(-0.169467\pi\)
\(41\)	−4.77174	−	1.97652i	−0.745220	−	0.308680i	−0.0224307	−	0.999748i	\(-0.507141\pi\)
							−0.722790	+	0.691068i	\(0.757141\pi\)
\(43\)	−1.47234	−	7.40194i	−0.224529	−	1.12878i	−0.914387	−	0.404840i	\(-0.867327\pi\)
							0.689858	−	0.723944i	\(-0.257673\pi\)
\(47\)	2.18905	−	8.16964i	0.319306	−	1.19166i	−0.600608	−	0.799544i	\(-0.705075\pi\)
							0.919914	−	0.392121i	\(-0.128259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bm.a.227.61	yes	992
7.5	odd	6	inner	448.2.bm.a.355.21	yes	992
64.11	odd	16	inner	448.2.bm.a.395.21	yes	992
448.75	even	48	inner	448.2.bm.a.75.61	✓	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bm.a.75.61	✓	992	448.75	even	48	inner
448.2.bm.a.227.61	yes	992	1.1	even	1	trivial
448.2.bm.a.355.21	yes	992	7.5	odd	6	inner
448.2.bm.a.395.21	yes	992	64.11	odd	16	inner