Embedded newform 448.2.bn.a.205.47

• Label: 448.2.bn.a.205.47
• 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			205.47
Character	\(\chi\)	\(=\)	448.205
Dual form			448.2.bn.a.389.47

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.972612 - 1.02666i) q^{2} +(2.23224 + 1.10082i) q^{3} +(-0.108053 - 1.99708i) q^{4} +(0.305510 - 0.900004i) q^{5} +(3.30127 - 1.22108i) q^{6} +(-2.05568 - 1.66558i) q^{7} +(-2.15541 - 1.83145i) q^{8} +(1.94481 + 2.53452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 992 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 32 q^{6} - 16 q^{7} - 32 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.972612	−	1.02666i	0.687740	−	0.725957i
							\(3\)	2.23224	+	1.10082i	1.28878	+	0.635558i	0.952021	−	0.306032i	\(-0.0990013\pi\)
0.336763	+	0.941590i	\(0.390668\pi\)
\(5\)	0.305510	−	0.900004i	0.136628	−	0.402494i	−0.856984	−	0.515343i	\(-0.827664\pi\)
							0.993612	+	0.112849i	\(0.0359978\pi\)
\(7\)	−2.05568	−	1.66558i	−0.776975	−	0.629531i
							\(11\)	3.48209	+	0.228228i	1.04989	+	0.0688133i	0.580566	−	0.814213i	\(-0.302831\pi\)
0.469323	+	0.883027i	\(0.344498\pi\)
\(13\)	0.0548905	+	0.0109184i	0.0152239	+	0.00302822i	0.202696	−	0.979242i	\(-0.435030\pi\)
							−0.187473	+	0.982270i	\(0.560030\pi\)
\(17\)	−0.0737522	+	0.275247i	−0.0178875	+	0.0667572i	−0.974293	−	0.225286i	\(-0.927668\pi\)
							0.956405	+	0.292043i	\(0.0943351\pi\)
\(19\)	1.71564	+	1.50457i	0.393594	+	0.345173i	0.833155	−	0.553040i	\(-0.186532\pi\)
							−0.439560	+	0.898213i	\(0.644866\pi\)
\(23\)	−5.42410	+	4.16206i	−1.13100	+	0.867849i	−0.992467	−	0.122508i	\(-0.960906\pi\)
							−0.138535	+	0.990357i	\(0.544239\pi\)
\(29\)	4.02940	+	6.03042i	0.748241	+	1.11982i	0.988809	+	0.149189i	\(0.0476663\pi\)
							−0.240568	+	0.970632i	\(0.577334\pi\)
\(31\)	−2.23289	+	1.28916i	−0.401039	+	0.231540i	−0.686932	−	0.726722i	\(-0.741043\pi\)
							0.285893	+	0.958261i	\(0.407710\pi\)
\(37\)	−2.66284	−	0.903911i	−0.437768	−	0.148602i	0.0938504	−	0.995586i	\(-0.470082\pi\)
							−0.531618	+	0.846984i	\(0.678416\pi\)
\(41\)	−4.03550	−	1.67156i	−0.630239	−	0.261054i	0.0446155	−	0.999004i	\(-0.485794\pi\)
							−0.674855	+	0.737951i	\(0.735794\pi\)
\(43\)	5.56797	+	3.72040i	0.849107	+	0.567355i	0.902241	−	0.431232i	\(-0.141921\pi\)
							−0.0531336	+	0.998587i	\(0.516921\pi\)
\(47\)	−4.41673	+	1.18346i	−0.644246	+	0.172625i	−0.566126	−	0.824319i	\(-0.691558\pi\)
							−0.0781199	+	0.996944i	\(0.524892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bn.a.205.47	✓	992
7.4	even	3	inner	448.2.bn.a.333.36	yes	992
64.5	even	16	inner	448.2.bn.a.261.36	yes	992
448.389	even	48	inner	448.2.bn.a.389.47	yes	992

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bn.a.205.47	✓	992	1.1	even	1	trivial
448.2.bn.a.261.36	yes	992	64.5	even	16	inner
448.2.bn.a.333.36	yes	992	7.4	even	3	inner
448.2.bn.a.389.47	yes	992	448.389	even	48	inner