Embedded newform 441.2.y.a.25.14

• Label: 441.2.y.a.25.14
• 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			25.14
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.322970 + 1.41503i) q^{2} +(-1.49665 - 0.871802i) q^{3} +(-0.0960492 - 0.0462548i) q^{4} +(-1.42246 - 3.62437i) q^{5} +(1.71700 - 1.83623i) q^{6} +(1.71510 - 2.01455i) q^{7} +(-1.71341 + 2.14855i) q^{8} +(1.47992 + 2.60956i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.322970	+	1.41503i	−0.228374	+	1.00057i	0.722591	+	0.691276i	\(0.242951\pi\)
							−0.950965	+	0.309298i	\(0.899906\pi\)
\(3\)	−1.49665	−	0.871802i	−0.864091	−	0.503335i
							\(5\)	−1.42246	−	3.62437i	−0.636144	−	1.62087i	−0.776027	−	0.630700i	\(-0.782768\pi\)
0.139883	−	0.990168i	\(-0.455327\pi\)
\(7\)	1.71510	−	2.01455i	0.648247	−	0.761430i
							\(11\)	−4.66480	+	1.43890i	−1.40649	+	0.433844i	−0.902807	−	0.430045i	\(-0.858498\pi\)
−0.503682	+	0.863889i	\(0.668022\pi\)
\(13\)	−3.94889	+	1.21807i	−1.09523	+	0.337832i	−0.789166	−	0.614180i	\(-0.789487\pi\)
							−0.306060	+	0.952012i	\(0.599011\pi\)
\(17\)	−0.361640	+	4.82575i	−0.0877106	+	1.17042i	0.763577	+	0.645717i	\(0.223441\pi\)
							−0.851288	+	0.524699i	\(0.824178\pi\)
\(19\)	1.88223	+	3.26011i	0.431812	+	0.747921i	0.997029	−	0.0770211i	\(-0.0245409\pi\)
							−0.565217	+	0.824942i	\(0.691208\pi\)
\(23\)	−2.23791	+	1.52578i	−0.466637	+	0.318148i	−0.773712	−	0.633538i	\(-0.781602\pi\)
							0.307075	+	0.951685i	\(0.400650\pi\)
\(29\)	0.300063	−	4.00407i	0.0557204	−	0.743537i	−0.897453	−	0.441111i	\(-0.854585\pi\)
							0.953173	−	0.302426i	\(-0.0977964\pi\)
\(31\)	−2.61784			−0.470179			−0.235089	−	0.971974i	\(-0.575538\pi\)
							−0.235089	+	0.971974i	\(0.575538\pi\)
\(37\)	0.460558	+	0.314003i	0.0757153	+	0.0516218i	0.600586	−	0.799560i	\(-0.294934\pi\)
							−0.524871	+	0.851182i	\(0.675886\pi\)
\(41\)	5.09253	+	0.767576i	0.795320	+	0.119875i	0.534119	−	0.845409i	\(-0.320643\pi\)
							0.261201	+	0.965284i	\(0.415881\pi\)
\(43\)	−0.856278	+	0.129063i	−0.130581	+	0.0196820i	−0.214008	−	0.976832i	\(-0.568652\pi\)
							0.0834264	+	0.996514i	\(0.473414\pi\)
\(47\)	1.52696	−	6.69007i	0.222731	−	0.975847i	−0.732682	−	0.680572i	\(-0.761731\pi\)
							0.955412	−	0.295275i	\(-0.0954114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.14	✓	648
9.4	even	3		441.2.z.a.319.14	yes	648
49.2	even	21		441.2.z.a.394.14	yes	648
441.247	even	21	inner	441.2.y.a.247.14	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.14	✓	648	1.1	even	1	trivial
441.2.y.a.247.14	yes	648	441.247	even	21	inner
441.2.z.a.319.14	yes	648	9.4	even	3
441.2.z.a.394.14	yes	648	49.2	even	21