Embedded newform 441.2.y.a.25.9

• Label: 441.2.y.a.25.9
• 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			25.9
Character	\(\chi\)	\(=\)	441.25
Dual form			441.2.y.a.247.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.474737 + 2.07996i) q^{2} +(-1.49636 + 0.872292i) q^{3} +(-2.29891 - 1.10710i) q^{4} +(-0.320939 - 0.817739i) q^{5} +(-1.10395 - 3.52648i) q^{6} +(-0.354961 - 2.62183i) q^{7} +(0.733726 - 0.920064i) q^{8} +(1.47821 - 2.61053i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.474737	+	2.07996i	−0.335690	+	1.47075i	0.472238	+	0.881471i	\(0.343446\pi\)
							−0.807928	+	0.589281i	\(0.799411\pi\)
\(3\)	−1.49636	+	0.872292i	−0.863926	+	0.503618i
							\(5\)	−0.320939	−	0.817739i	−0.143528	−	0.365704i	0.840974	−	0.541075i	\(-0.181983\pi\)
−0.984502	+	0.175371i	\(0.943887\pi\)
\(7\)	−0.354961	−	2.62183i	−0.134163	−	0.990959i
							\(11\)	5.64014	−	1.73975i	1.70057	−	0.524555i	0.715781	−	0.698325i	\(-0.246071\pi\)
0.984786	+	0.173770i	\(0.0555949\pi\)
\(13\)	−1.14108	+	0.351977i	−0.316479	+	0.0976210i	−0.448925	−	0.893569i	\(-0.648193\pi\)
							0.132446	+	0.991190i	\(0.457717\pi\)
\(17\)	−0.127912	+	1.70687i	−0.0310232	+	0.413976i	0.959974	+	0.280090i	\(0.0903644\pi\)
							−0.990997	+	0.133885i	\(0.957255\pi\)
\(19\)	−0.539504	−	0.934449i	−0.123771	−	0.214377i	0.797481	−	0.603344i	\(-0.206165\pi\)
							−0.921252	+	0.388967i	\(0.872832\pi\)
\(23\)	0.143637	−	0.0979298i	0.0299503	−	0.0204198i	−0.548252	−	0.836313i	\(-0.684707\pi\)
							0.578203	+	0.815893i	\(0.303754\pi\)
\(29\)	0.438482	−	5.85113i	0.0814240	−	1.08653i	−0.795517	−	0.605931i	\(-0.792801\pi\)
							0.876941	−	0.480597i	\(-0.159580\pi\)
\(31\)	−0.284492			−0.0510962			−0.0255481	−	0.999674i	\(-0.508133\pi\)
							−0.0255481	+	0.999674i	\(0.508133\pi\)
\(37\)	−4.72151	−	3.21907i	−0.776211	−	0.529212i	0.109201	−	0.994020i	\(-0.465171\pi\)
							−0.885411	+	0.464808i	\(0.846123\pi\)
\(41\)	10.2461	+	1.54435i	1.60017	+	0.241186i	0.887667	−	0.460486i	\(-0.152325\pi\)
							0.712500	+	0.701672i	\(0.247563\pi\)
\(43\)	−2.90796	+	0.438305i	−0.443460	+	0.0668408i	−0.366977	−	0.930230i	\(-0.619607\pi\)
							−0.0764830	+	0.997071i	\(0.524369\pi\)
\(47\)	−0.554740	+	2.43047i	−0.0809171	+	0.354521i	−0.999136	−	0.0415514i	\(-0.986770\pi\)
							0.918219	+	0.396072i	\(0.129627\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.25.9	✓	648
9.4	even	3		441.2.z.a.319.9	yes	648
49.2	even	21		441.2.z.a.394.9	yes	648
441.247	even	21	inner	441.2.y.a.247.9	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.9	✓	648	1.1	even	1	trivial
441.2.y.a.247.9	yes	648	441.247	even	21	inner
441.2.z.a.319.9	yes	648	9.4	even	3
441.2.z.a.394.9	yes	648	49.2	even	21