Embedded newform 448.2.bd.b.195.45

• Label: 448.2.bd.b.195.45
• 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			195.45
Character	\(\chi\)	\(=\)	448.195
Dual form			448.2.bd.b.363.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00633 - 0.993634i) q^{2} +(-0.376307 + 1.89183i) q^{3} +(0.0253828 - 1.99984i) q^{4} +(1.44394 - 2.16101i) q^{5} +(1.50109 + 2.27770i) q^{6} +(-2.16386 - 1.52240i) q^{7} +(-1.96156 - 2.03771i) q^{8} +(-0.665756 - 0.275765i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 16 q^{2} - 16 q^{4} - 8 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00633	−	0.993634i	0.711580	−	0.702605i
							\(3\)	−0.376307	+	1.89183i	−0.217261	+	1.09225i	0.706041	+	0.708171i	\(0.250479\pi\)
−0.923302	+	0.384075i	\(0.874521\pi\)
\(5\)	1.44394	−	2.16101i	0.645750	−	0.966433i	−0.353766	−	0.935334i	\(-0.615099\pi\)
							0.999516	−	0.0310995i	\(-0.00990088\pi\)
\(7\)	−2.16386	−	1.52240i	−0.817863	−	0.575413i
							\(11\)	−0.739199	−	3.71621i	−0.222877	−	1.12048i	−0.916468	−	0.400109i	\(-0.868972\pi\)
0.693591	−	0.720369i	\(-0.256028\pi\)
\(13\)	−0.636249	−	0.952214i	−0.176464	−	0.264097i	0.732685	−	0.680568i	\(-0.238267\pi\)
							−0.909149	+	0.416471i	\(0.863267\pi\)
\(17\)	2.60346	−	2.60346i	0.631432	−	0.631432i	−0.316995	−	0.948427i	\(-0.602674\pi\)
							0.948427	+	0.316995i	\(0.102674\pi\)
\(19\)	2.14951	+	3.21697i	0.493132	+	0.738024i	0.991664	−	0.128847i	\(-0.0411276\pi\)
							−0.498533	+	0.866871i	\(0.666128\pi\)
\(23\)	3.20877	+	1.32912i	0.669076	+	0.277140i	0.691252	−	0.722614i	\(-0.257059\pi\)
							−0.0221762	+	0.999754i	\(0.507059\pi\)
\(29\)	7.43807	+	1.47952i	1.38122	+	0.274741i	0.829134	−	0.559049i	\(-0.188834\pi\)
							0.552081	+	0.833790i	\(0.313834\pi\)
\(31\)			0.278443i			0.0500098i	0.999687	+	0.0250049i	\(0.00796014\pi\)
							−0.999687	+	0.0250049i	\(0.992040\pi\)
\(37\)	1.27557	−	1.90903i	0.209703	−	0.313843i	−0.711676	−	0.702508i	\(-0.752064\pi\)
							0.921379	+	0.388665i	\(0.127064\pi\)
\(41\)	−9.87484	−	4.09029i	−1.54219	−	0.638796i	−0.560307	−	0.828285i	\(-0.689317\pi\)
							−0.981883	+	0.189489i	\(0.939317\pi\)
\(43\)	−4.62111	+	0.919197i	−0.704713	+	0.140176i	−0.534422	−	0.845218i	\(-0.679471\pi\)
							−0.170291	+	0.985394i	\(0.554471\pi\)
\(47\)	−2.08202	−	2.08202i	−0.303694	−	0.303694i	0.538763	−	0.842457i	\(-0.318892\pi\)
							−0.842457	+	0.538763i	\(0.818892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bd.b.195.45	✓	480
7.6	odd	2	inner	448.2.bd.b.195.46	yes	480
64.43	odd	16	inner	448.2.bd.b.363.46	yes	480
448.363	even	16	inner	448.2.bd.b.363.45	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bd.b.195.45	✓	480	1.1	even	1	trivial
448.2.bd.b.195.46	yes	480	7.6	odd	2	inner
448.2.bd.b.363.45	yes	480	448.363	even	16	inner
448.2.bd.b.363.46	yes	480	64.43	odd	16	inner