Embedded newform 448.2.bm.a.171.2

• Label: 448.2.bm.a.171.2
• Level: $448$
• Weight: $2$
• Character: 448.171
• 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			171.2
Character	\(\chi\)	\(=\)	448.171
Dual form			448.2.bm.a.131.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41304 - 0.0575359i) q^{2} +(2.52363 + 0.856658i) q^{3} +(1.99338 + 0.162601i) q^{4} +(0.753756 - 1.52847i) q^{5} +(-3.51671 - 1.35569i) q^{6} +(-2.49233 - 0.887850i) q^{7} +(-2.80737 - 0.344453i) q^{8} +(3.25479 + 2.49749i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 24 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{13}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41304	−	0.0575359i	−0.999172	−	0.0406840i
							\(3\)	2.52363	+	0.856658i	1.45702	+	0.494591i	0.933859	−	0.357642i	\(-0.116419\pi\)
0.523161	+	0.852234i	\(0.324753\pi\)
\(5\)	0.753756	−	1.52847i	0.337090	−	0.683551i	−0.660654	−	0.750691i	\(-0.729721\pi\)
							0.997744	+	0.0671401i	\(0.0213875\pi\)
\(7\)	−2.49233	−	0.887850i	−0.942013	−	0.335576i
							\(11\)	1.91310	+	1.67774i	0.576822	+	0.505859i	0.897314	−	0.441392i	\(-0.145515\pi\)
−0.320493	+	0.947251i	\(0.603848\pi\)
\(13\)	2.02502	−	3.03065i	0.561638	−	0.840551i	−0.436614	−	0.899649i	\(-0.643822\pi\)
							0.998252	+	0.0590980i	\(0.0188224\pi\)
\(17\)	4.36301	−	1.16906i	1.05819	−	0.283540i	0.312555	−	0.949900i	\(-0.398815\pi\)
							0.745630	+	0.666360i	\(0.232149\pi\)
\(19\)	0.337527	+	0.0221227i	0.0774339	+	0.00507529i	0.104071	−	0.994570i	\(-0.466813\pi\)
							−0.0266369	+	0.999645i	\(0.508480\pi\)
\(23\)	3.85107	+	2.95503i	0.803003	+	0.616166i	0.926549	−	0.376173i	\(-0.122760\pi\)
							−0.123547	+	0.992339i	\(0.539427\pi\)
\(29\)	1.73100	−	0.344318i	0.321439	−	0.0639382i	−0.0317336	−	0.999496i	\(-0.510103\pi\)
							0.353173	+	0.935558i	\(0.385103\pi\)
\(31\)	−6.04110	−	3.48783i	−1.08501	−	0.626433i	−0.152769	−	0.988262i	\(-0.548819\pi\)
							−0.932244	+	0.361829i	\(0.882152\pi\)
\(37\)	−3.47516	+	7.04694i	−0.571314	+	1.15851i	0.399066	+	0.916922i	\(0.369335\pi\)
							−0.970379	+	0.241587i	\(0.922332\pi\)
\(41\)	3.24329	−	1.34341i	0.506517	−	0.209806i	−0.114766	−	0.993393i	\(-0.536612\pi\)
							0.621283	+	0.783586i	\(0.286612\pi\)
\(43\)	−10.4785	−	2.08430i	−1.59796	−	0.317853i	−0.685829	−	0.727762i	\(-0.740560\pi\)
							−0.912126	+	0.409909i	\(0.865560\pi\)
\(47\)	−2.27167	−	0.608693i	−0.331358	−	0.0887870i	0.0893046	−	0.996004i	\(-0.471536\pi\)
							−0.420662	+	0.907217i	\(0.638202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bm.a.171.2	yes	992
7.5	odd	6	inner	448.2.bm.a.299.42	yes	992
64.3	odd	16	inner	448.2.bm.a.3.42	✓	992
448.131	even	48	inner	448.2.bm.a.131.2	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bm.a.3.42	✓	992	64.3	odd	16	inner
448.2.bm.a.131.2	yes	992	448.131	even	48	inner
448.2.bm.a.171.2	yes	992	1.1	even	1	trivial
448.2.bm.a.299.42	yes	992	7.5	odd	6	inner