Embedded newform 448.2.bn.a.205.17

• Label: 448.2.bn.a.205.17
• 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			205.17
Character	\(\chi\)	\(=\)	448.205
Dual form			448.2.bn.a.389.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00781 + 0.992129i) q^{2} +(1.26266 + 0.622677i) q^{3} +(0.0313618 - 1.99975i) q^{4} +(0.932836 - 2.74805i) q^{5} +(-1.89030 + 0.625185i) q^{6} +(-2.32622 - 1.26043i) q^{7} +(1.95241 + 2.04649i) q^{8} +(-0.619692 - 0.807598i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00781	+	0.992129i	−0.712629	+	0.701541i
							\(3\)	1.26266	+	0.622677i	0.728999	+	0.359502i	0.768614	−	0.639713i	\(-0.220947\pi\)
−0.0396153	+	0.999215i	\(0.512613\pi\)
\(5\)	0.932836	−	2.74805i	0.417177	−	1.22896i	−0.511890	−	0.859051i	\(-0.671054\pi\)
							0.929067	−	0.369912i	\(-0.120612\pi\)
\(7\)	−2.32622	−	1.26043i	−0.879229	−	0.476399i
							\(11\)	−3.25247	−	0.213178i	−0.980655	−	0.0642755i	−0.433365	−	0.901218i	\(-0.642674\pi\)
−0.547290	+	0.836943i	\(0.684341\pi\)
\(13\)	1.92976	+	0.383853i	0.535219	+	0.106462i	0.455297	−	0.890340i	\(-0.349533\pi\)
							0.0799215	+	0.996801i	\(0.474533\pi\)
\(17\)	1.01233	−	3.77806i	0.245525	−	0.916313i	−0.727593	−	0.686009i	\(-0.759361\pi\)
							0.973119	−	0.230304i	\(-0.0739722\pi\)
\(19\)	−4.66993	−	4.09542i	−1.07136	−	0.939553i	−0.0730259	−	0.997330i	\(-0.523266\pi\)
							−0.998330	+	0.0577766i	\(0.981599\pi\)
\(23\)	−2.63926	+	2.02517i	−0.550323	+	0.422278i	−0.846135	−	0.532968i	\(-0.821077\pi\)
							0.295812	+	0.955246i	\(0.404410\pi\)
\(29\)	1.79099	+	2.68040i	0.332578	+	0.497738i	0.959640	−	0.281231i	\(-0.0907428\pi\)
							−0.627062	+	0.778969i	\(0.715743\pi\)
\(31\)	2.95022	−	1.70331i	0.529875	−	0.305923i	−0.211091	−	0.977466i	\(-0.567702\pi\)
							0.740965	+	0.671543i	\(0.234368\pi\)
\(37\)	4.72553	+	1.60410i	0.776872	+	0.263713i	0.681587	−	0.731737i	\(-0.261290\pi\)
							0.0952857	+	0.995450i	\(0.469624\pi\)
\(41\)	1.87936	+	0.778454i	0.293506	+	0.121574i	0.524578	−	0.851363i	\(-0.324223\pi\)
							−0.231072	+	0.972937i	\(0.574223\pi\)
\(43\)	−5.22158	−	3.48895i	−0.796284	−	0.532060i	0.0895943	−	0.995978i	\(-0.471443\pi\)
							−0.885878	+	0.463919i	\(0.846443\pi\)
\(47\)	9.22133	−	2.47085i	1.34507	−	0.360410i	0.486757	−	0.873538i	\(-0.338180\pi\)
							0.858312	+	0.513128i	\(0.171513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.205.17	✓	992
7.4	even	3	inner	448.2.bn.a.333.27	yes	992
64.5	even	16	inner	448.2.bn.a.261.27	yes	992
448.389	even	48	inner	448.2.bn.a.389.17	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.205.17	✓	992	1.1	even	1	trivial
448.2.bn.a.261.27	yes	992	64.5	even	16	inner
448.2.bn.a.333.27	yes	992	7.4	even	3	inner
448.2.bn.a.389.17	yes	992	448.389	even	48	inner