Embedded newform 448.2.bd.b.195.21

• Label: 448.2.bd.b.195.21
• Level: $448$
• Weight: $2$
• Character: 448.195
• Analytic conductor: $3.577$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.bd (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.57729801055\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(60\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			195.21
Character	\(\chi\)	\(=\)	448.195
Dual form			448.2.bd.b.363.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.786344 - 1.17544i) q^{2} +(-0.214098 + 1.07635i) q^{3} +(-0.763327 + 1.84860i) q^{4} +(0.565204 - 0.845888i) q^{5} +(1.43354 - 0.594717i) q^{6} +(-2.61659 - 0.391738i) q^{7} +(2.77316 - 0.556391i) q^{8} +(1.65896 + 0.687163i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 16 q^{2} - 16 q^{4} - 8 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.786344	−	1.17544i	−0.556029	−	0.831163i
							\(3\)	−0.214098	+	1.07635i	−0.123610	+	0.621428i	0.868462	+	0.495756i	\(0.165109\pi\)
−0.992072	+	0.125672i	\(0.959891\pi\)
\(5\)	0.565204	−	0.845888i	0.252767	−	0.378293i	−0.683286	−	0.730150i	\(-0.739450\pi\)
							0.936053	+	0.351858i	\(0.114450\pi\)
\(7\)	−2.61659	−	0.391738i	−0.988978	−	0.148063i
							\(11\)	0.888804	+	4.46832i	0.267985	+	1.34725i	0.846852	+	0.531828i	\(0.178495\pi\)
−0.578868	+	0.815421i	\(0.696505\pi\)
\(13\)	−1.97229	−	2.95174i	−0.547015	−	0.818666i	0.450225	−	0.892915i	\(-0.351344\pi\)
							−0.997240	+	0.0742495i	\(0.976344\pi\)
\(17\)	−2.62951	+	2.62951i	−0.637751	+	0.637751i	−0.950000	−	0.312249i	\(-0.898918\pi\)
							0.312249	+	0.950000i	\(0.398918\pi\)
\(19\)	2.34910	+	3.51568i	0.538921	+	0.806553i	0.996585	−	0.0825728i	\(-0.0263137\pi\)
							−0.457664	+	0.889125i	\(0.651314\pi\)
\(23\)	0.433796	+	0.179684i	0.0904528	+	0.0374668i	0.427451	−	0.904039i	\(-0.359412\pi\)
							−0.336998	+	0.941505i	\(0.609412\pi\)
\(29\)	2.49523	+	0.496332i	0.463352	+	0.0921665i	0.421246	−	0.906947i	\(-0.361593\pi\)
							0.0421068	+	0.999113i	\(0.486593\pi\)
\(31\)			9.75448i			1.75196i	0.482351	+	0.875978i	\(0.339783\pi\)
							−0.482351	+	0.875978i	\(0.660217\pi\)
\(37\)	−0.0742910	+	0.111184i	−0.0122134	+	0.0182786i	−0.837526	−	0.546397i	\(-0.815999\pi\)
							0.825313	+	0.564676i	\(0.190999\pi\)
\(41\)	1.09700	+	0.454392i	0.171322	+	0.0709641i	0.466696	−	0.884418i	\(-0.345444\pi\)
							−0.295374	+	0.955382i	\(0.595444\pi\)
\(43\)	3.45999	−	0.688235i	0.527643	−	0.104955i	0.0759218	−	0.997114i	\(-0.475810\pi\)
							0.451722	+	0.892159i	\(0.350810\pi\)
\(47\)	0.494402	+	0.494402i	0.0721159	+	0.0721159i	0.742245	−	0.670129i	\(-0.233761\pi\)
							−0.670129	+	0.742245i	\(0.733761\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bd.b.195.21	✓	480
7.6	odd	2	inner	448.2.bd.b.195.22	yes	480
64.43	odd	16	inner	448.2.bd.b.363.22	yes	480
448.363	even	16	inner	448.2.bd.b.363.21	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bd.b.195.21	✓	480	1.1	even	1	trivial
448.2.bd.b.195.22	yes	480	7.6	odd	2	inner
448.2.bd.b.363.21	yes	480	448.363	even	16	inner
448.2.bd.b.363.22	yes	480	64.43	odd	16	inner