Embedded newform 441.2.y.a.247.16

• Label: 441.2.y.a.247.16
• Level: $441$
• Weight: $2$
• Character: 441.247
• 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			247.16
Character	\(\chi\)	\(=\)	441.247
Dual form			441.2.y.a.25.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.302994 - 1.32750i) q^{2} +(-1.73138 + 0.0482649i) q^{3} +(0.131481 - 0.0633181i) q^{4} +(1.28284 - 3.26863i) q^{5} +(0.588668 + 2.28378i) q^{6} +(2.63656 - 0.220319i) q^{7} +(-1.82183 - 2.28450i) q^{8} +(2.99534 - 0.167130i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.302994	−	1.32750i	−0.214249	−	0.938686i	−0.961643	−	0.274305i	\(-0.911552\pi\)
							0.747394	−	0.664381i	\(-0.231305\pi\)
\(3\)	−1.73138	+	0.0482649i	−0.999612	+	0.0278658i
							\(5\)	1.28284	−	3.26863i	0.573705	−	1.46178i	−0.289626	−	0.957140i	\(-0.593531\pi\)
0.863331	−	0.504638i	\(-0.168374\pi\)
\(7\)	2.63656	−	0.220319i	0.996527	−	0.0832727i
							\(11\)	−0.233466	−	0.0720147i	−0.0703926	−	0.0217132i	0.259359	−	0.965781i	\(-0.416489\pi\)
−0.329751	+	0.944068i	\(0.606965\pi\)
\(13\)	5.88468	+	1.81518i	1.63212	+	0.503441i	0.969412	−	0.245439i	\(-0.0789322\pi\)
							0.662705	+	0.748881i	\(0.269408\pi\)
\(17\)	0.132058	+	1.76219i	0.0320287	+	0.427393i	0.990052	+	0.140703i	\(0.0449362\pi\)
							−0.958023	+	0.286691i	\(0.907445\pi\)
\(19\)	1.79733	−	3.11307i	0.412337	−	0.714188i	−0.582808	−	0.812610i	\(-0.698046\pi\)
							0.995145	+	0.0984216i	\(0.0313794\pi\)
\(23\)	−3.69823	−	2.52141i	−0.771133	−	0.525750i	0.112656	−	0.993634i	\(-0.464064\pi\)
							−0.883790	+	0.467884i	\(0.845016\pi\)
\(29\)	0.455042	+	6.07211i	0.0844992	+	1.12756i	0.864786	+	0.502141i	\(0.167454\pi\)
							−0.780287	+	0.625422i	\(0.784927\pi\)
\(31\)	−0.258105			−0.0463571			−0.0231785	−	0.999731i	\(-0.507379\pi\)
							−0.0231785	+	0.999731i	\(0.507379\pi\)
\(37\)	−6.45066	+	4.39799i	−1.06048	+	0.723025i	−0.962258	−	0.272140i	\(-0.912269\pi\)
							−0.0982246	+	0.995164i	\(0.531316\pi\)
\(41\)	−1.90167	+	0.286631i	−0.296991	+	0.0447642i	−0.295848	−	0.955235i	\(-0.595602\pi\)
							−0.00114356	+	0.999999i	\(0.500364\pi\)
\(43\)	−8.64273	−	1.30268i	−1.31800	−	0.198657i	−0.547883	−	0.836555i	\(-0.684566\pi\)
							−0.770121	+	0.637898i	\(0.779804\pi\)
\(47\)	−0.106911	−	0.468406i	−0.0155945	−	0.0683241i	0.966532	−	0.256546i	\(-0.0825846\pi\)
							−0.982126	+	0.188222i	\(0.939727\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.247.16	yes	648
9.7	even	3		441.2.z.a.394.16	yes	648
49.25	even	21		441.2.z.a.319.16	yes	648
441.25	even	21	inner	441.2.y.a.25.16	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.25.16	✓	648	441.25	even	21	inner
441.2.y.a.247.16	yes	648	1.1	even	1	trivial
441.2.z.a.319.16	yes	648	49.25	even	21
441.2.z.a.394.16	yes	648	9.7	even	3