Embedded newform 448.2.bd.b.195.12

• Label: 448.2.bd.b.195.12
• 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			195.12
Character	\(\chi\)	\(=\)	448.195
Dual form			448.2.bd.b.363.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19968 - 0.748839i) q^{2} +(0.166288 - 0.835986i) q^{3} +(0.878480 + 1.79674i) q^{4} +(1.20282 - 1.80015i) q^{5} +(-0.825512 + 0.878396i) q^{6} +(-2.37920 + 1.15732i) q^{7} +(0.291570 - 2.81336i) q^{8} +(2.10042 + 0.870021i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 480 q - 16 q^{2} - 16 q^{4} - 8 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19968	−	0.748839i	−0.848304	−	0.529509i
							\(3\)	0.166288	−	0.835986i	0.0960064	−	0.482657i	−0.902628	−	0.430421i	\(-0.858365\pi\)
0.998635	−	0.0522361i	\(-0.0166348\pi\)
\(5\)	1.20282	−	1.80015i	0.537918	−	0.805051i	−0.458582	−	0.888652i	\(-0.651642\pi\)
							0.996499	+	0.0836018i	\(0.0266424\pi\)
\(7\)	−2.37920	+	1.15732i	−0.899255	+	0.437426i
							\(11\)	−0.819782	−	4.12132i	−0.247174	−	1.24263i	−0.882474	−	0.470361i	\(-0.844124\pi\)
0.635301	−	0.772265i	\(-0.280876\pi\)
\(13\)	0.179266	+	0.268291i	0.0497195	+	0.0744105i	0.855497	−	0.517808i	\(-0.173252\pi\)
							−0.805777	+	0.592219i	\(0.798252\pi\)
\(17\)	2.03374	−	2.03374i	0.493253	−	0.493253i	−0.416076	−	0.909330i	\(-0.636595\pi\)
							0.909330	+	0.416076i	\(0.136595\pi\)
\(19\)	−4.57571	−	6.84804i	−1.04974	−	1.57105i	−0.797440	−	0.603398i	\(-0.793813\pi\)
							−0.252300	−	0.967649i	\(-0.581187\pi\)
\(23\)	6.17219	+	2.55660i	1.28699	+	0.533089i	0.918087	−	0.396379i	\(-0.129733\pi\)
							0.368903	+	0.929468i	\(0.379733\pi\)
\(29\)	−5.29479	−	1.05320i	−0.983218	−	0.195574i	−0.322796	−	0.946469i	\(-0.604623\pi\)
							−0.660422	+	0.750894i	\(0.729623\pi\)
\(31\)		−	3.69210i		−	0.663121i	−0.943434	−	0.331561i	\(-0.892425\pi\)
							0.943434	−	0.331561i	\(-0.107575\pi\)
\(37\)	2.50665	−	3.75146i	0.412090	−	0.616736i	−0.566128	−	0.824317i	\(-0.691559\pi\)
							0.978218	+	0.207581i	\(0.0665591\pi\)
\(41\)	−3.81140	−	1.57873i	−0.595240	−	0.246556i	0.0646631	−	0.997907i	\(-0.479403\pi\)
							−0.659903	+	0.751351i	\(0.729403\pi\)
\(43\)	−9.39351	+	1.86849i	−1.43250	+	0.284941i	−0.849524	−	0.527550i	\(-0.823111\pi\)
							−0.582973	+	0.812491i	\(0.698111\pi\)
\(47\)	1.73831	+	1.73831i	0.253559	+	0.253559i	0.822428	−	0.568869i	\(-0.192619\pi\)
							−0.568869	+	0.822428i	\(0.692619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.bd.b.195.12	yes	480
7.6	odd	2	inner	448.2.bd.b.195.11	✓	480
64.43	odd	16	inner	448.2.bd.b.363.11	yes	480
448.363	even	16	inner	448.2.bd.b.363.12	yes	480

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.bd.b.195.11	✓	480	7.6	odd	2	inner
448.2.bd.b.195.12	yes	480	1.1	even	1	trivial
448.2.bd.b.363.11	yes	480	64.43	odd	16	inner
448.2.bd.b.363.12	yes	480	448.363	even	16	inner