Embedded newform 448.2.bd.b.195.24

• Label: 448.2.bd.b.195.24
• 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			195.24
Character	\(\chi\)	\(=\)	448.195
Dual form			448.2.bd.b.363.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.771491 - 1.18524i) q^{2} +(0.440187 - 2.21297i) q^{3} +(-0.809603 + 1.82881i) q^{4} +(0.656009 - 0.981787i) q^{5} +(-2.96251 + 1.18556i) q^{6} +(2.05706 + 1.66389i) q^{7} +(2.79219 - 0.451334i) q^{8} +(-1.93183 - 0.800189i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 16 q^{2} - 16 q^{4} - 8 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.771491	−	1.18524i	−0.545527	−	0.838094i
							\(3\)	0.440187	−	2.21297i	0.254142	−	1.27766i	−0.617131	−	0.786860i	\(-0.711705\pi\)
0.871273	−	0.490798i	\(-0.163295\pi\)
\(5\)	0.656009	−	0.981787i	0.293376	−	0.439068i	−0.655276	−	0.755389i	\(-0.727448\pi\)
							0.948652	+	0.316321i	\(0.102448\pi\)
\(7\)	2.05706	+	1.66389i	0.777494	+	0.628890i
							\(11\)	−0.644094	−	3.23808i	−0.194202	−	0.976317i	−0.947772	−	0.318950i	\(-0.896670\pi\)
0.753570	−	0.657368i	\(-0.228330\pi\)
\(13\)	−3.09507	−	4.63210i	−0.858418	−	1.28471i	−0.957148	−	0.289599i	\(-0.906478\pi\)
							0.0987301	−	0.995114i	\(-0.468522\pi\)
\(17\)	2.13907	−	2.13907i	0.518800	−	0.518800i	−0.398408	−	0.917208i	\(-0.630437\pi\)
							0.917208	+	0.398408i	\(0.130437\pi\)
\(19\)	1.46472	+	2.19211i	0.336031	+	0.502905i	0.960551	−	0.278103i	\(-0.0897056\pi\)
							−0.624521	+	0.781008i	\(0.714706\pi\)
\(23\)	−8.43558	−	3.49413i	−1.75894	−	0.728577i	−0.996690	−	0.0812930i	\(-0.974095\pi\)
							−0.762249	−	0.647284i	\(-0.775905\pi\)
\(29\)	1.04683	+	0.208227i	0.194391	+	0.0386669i	0.291326	−	0.956624i	\(-0.405904\pi\)
							−0.0969343	+	0.995291i	\(0.530904\pi\)
\(31\)			4.18208i			0.751123i	0.926797	+	0.375562i	\(0.122550\pi\)
							−0.926797	+	0.375562i	\(0.877450\pi\)
\(37\)	−0.883143	+	1.32172i	−0.145188	+	0.217289i	−0.896935	−	0.442163i	\(-0.854211\pi\)
							0.751747	+	0.659452i	\(0.229211\pi\)
\(41\)	11.2138	+	4.64490i	1.75130	+	0.725412i	0.997678	+	0.0681022i	\(0.0216944\pi\)
							0.753621	+	0.657310i	\(0.228306\pi\)
\(43\)	−1.78841	+	0.355737i	−0.272730	+	0.0542494i	−0.329561	−	0.944134i	\(-0.606901\pi\)
							0.0568303	+	0.998384i	\(0.481901\pi\)
\(47\)	−2.46672	−	2.46672i	−0.359808	−	0.359808i	0.503934	−	0.863742i	\(-0.331885\pi\)
							−0.863742	+	0.503934i	\(0.831885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bd.b.195.24	yes	480
7.6	odd	2	inner	448.2.bd.b.195.23	✓	480
64.43	odd	16	inner	448.2.bd.b.363.23	yes	480
448.363	even	16	inner	448.2.bd.b.363.24	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bd.b.195.23	✓	480	7.6	odd	2	inner
448.2.bd.b.195.24	yes	480	1.1	even	1	trivial
448.2.bd.b.363.23	yes	480	64.43	odd	16	inner
448.2.bd.b.363.24	yes	480	448.363	even	16	inner