Embedded newform 441.2.y.a.25.2

• Label: 441.2.y.a.25.2
• Level: $441$
• Weight: $2$
• Character: 441.25
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.y (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			25.2
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.603411 + 2.64372i) q^{2} +(0.233708 + 1.71621i) q^{3} +(-4.82320 - 2.32273i) q^{4} +(-0.317680 - 0.809436i) q^{5} +(-4.67820 - 0.417724i) q^{6} +(-2.63556 - 0.231947i) q^{7} +(5.66957 - 7.10941i) q^{8} +(-2.89076 + 0.802183i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{8}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.603411	+	2.64372i	−0.426676	+	1.86939i	0.0638433	+	0.997960i	\(0.479664\pi\)
							−0.490519	+	0.871430i	\(0.663193\pi\)
\(3\)	0.233708	+	1.71621i	0.134931	+	0.990855i
							\(5\)	−0.317680	−	0.809436i	−0.142071	−	0.361991i	0.842073	−	0.539364i	\(-0.181335\pi\)
−0.984144	+	0.177373i	\(0.943240\pi\)
\(7\)	−2.63556	−	0.231947i	−0.996150	−	0.0876678i
							\(11\)	−0.758751	+	0.234044i	−0.228772	+	0.0705668i	−0.407022	−	0.913418i	\(-0.633433\pi\)
0.178250	+	0.983985i	\(0.442956\pi\)
\(13\)	−1.10661	+	0.341344i	−0.306918	+	0.0946718i	−0.444389	−	0.895834i	\(-0.646579\pi\)
							0.137470	+	0.990506i	\(0.456103\pi\)
\(17\)	0.355570	−	4.74475i	0.0862383	−	1.15077i	−0.771342	−	0.636421i	\(-0.780414\pi\)
							0.857580	−	0.514350i	\(-0.171967\pi\)
\(19\)	2.79052	+	4.83333i	0.640190	+	1.10884i	0.985390	+	0.170312i	\(0.0544776\pi\)
							−0.345200	+	0.938529i	\(0.612189\pi\)
\(23\)	−1.95767	+	1.33471i	−0.408202	+	0.278307i	−0.749963	−	0.661479i	\(-0.769929\pi\)
							0.341762	+	0.939787i	\(0.388976\pi\)
\(29\)	0.434819	−	5.80226i	0.0807439	−	1.07745i	−0.798788	−	0.601612i	\(-0.794525\pi\)
							0.879532	−	0.475840i	\(-0.157856\pi\)
\(31\)	−9.04671			−1.62484			−0.812419	−	0.583074i	\(-0.801850\pi\)
							−0.812419	+	0.583074i	\(0.801850\pi\)
\(37\)	−0.366013	−	0.249544i	−0.0601722	−	0.0410247i	0.532860	−	0.846203i	\(-0.321117\pi\)
							−0.593033	+	0.805178i	\(0.702069\pi\)
\(41\)	−2.24549	−	0.338453i	−0.350686	−	0.0528574i	−0.0286628	−	0.999589i	\(-0.509125\pi\)
							−0.322023	+	0.946732i	\(0.604363\pi\)
\(43\)	−10.9947	+	1.65718i	−1.67668	+	0.252718i	−0.917376	−	0.398022i	\(-0.869697\pi\)
							−0.759301	+	0.650740i	\(0.774459\pi\)
\(47\)	0.635372	−	2.78375i	0.0926785	−	0.406051i	−0.907215	−	0.420668i	\(-0.861796\pi\)
							0.999893	+	0.0146170i	\(0.00465290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.2	✓	648
9.4	even	3		441.2.z.a.319.2	yes	648
49.2	even	21		441.2.z.a.394.2	yes	648
441.247	even	21	inner	441.2.y.a.247.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.2	✓	648	1.1	even	1	trivial
441.2.y.a.247.2	yes	648	441.247	even	21	inner
441.2.z.a.319.2	yes	648	9.4	even	3
441.2.z.a.394.2	yes	648	49.2	even	21