Embedded newform 441.2.z.a.130.41

• Label: 441.2.z.a.130.41
• Level: $441$
• Weight: $2$
• Character: 441.130
• 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.z (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			130.41
Character	\(\chi\)	\(=\)	441.130
Dual form			441.2.z.a.268.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.22614 - 0.835967i) q^{2} +(0.435367 + 1.67644i) q^{3} +(0.0738918 - 0.188273i) q^{4} +(-0.839497 - 3.67808i) q^{5} +(1.93527 + 1.69160i) q^{6} +(2.50624 + 0.847796i) q^{7} +(0.593653 + 2.60096i) q^{8} +(-2.62091 + 1.45974i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 8 q^{2} - 13 q^{3} + 44 q^{4} + 3 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 39 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{2}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.22614	−	0.835967i	0.867010	−	0.591118i	−0.0460513	−	0.998939i	\(-0.514664\pi\)
							0.913062	+	0.407821i	\(0.133711\pi\)
\(3\)	0.435367	+	1.67644i	0.251359	+	0.967894i
							\(5\)	−0.839497	−	3.67808i	−0.375434	−	1.64489i	−0.711237	−	0.702953i	\(-0.751865\pi\)
0.335802	−	0.941933i	\(-0.390993\pi\)
\(7\)	2.50624	+	0.847796i	0.947270	+	0.320437i
							\(11\)	3.06016	+	1.47370i	0.922673	+	0.444336i	0.834025	−	0.551727i	\(-0.186031\pi\)
0.0886483	+	0.996063i	\(0.471745\pi\)
\(13\)	3.79355	−	2.58640i	1.05214	−	0.717337i	0.0916901	−	0.995788i	\(-0.470773\pi\)
							0.960451	+	0.278450i	\(0.0898207\pi\)
\(17\)	−1.28155	−	3.26534i	−0.310822	−	0.791961i	−0.997863	−	0.0653426i	\(-0.979186\pi\)
							0.687041	−	0.726619i	\(-0.258909\pi\)
\(19\)	2.34967	+	4.06975i	0.539052	+	0.933665i	0.998955	+	0.0456960i	\(0.0145506\pi\)
							−0.459904	+	0.887969i	\(0.652116\pi\)
\(23\)	−3.10268	−	3.89064i	−0.646954	−	0.811254i	0.344900	−	0.938640i	\(-0.387913\pi\)
							−0.991853	+	0.127385i	\(0.959341\pi\)
\(29\)	−2.21725	−	0.334196i	−0.411732	−	0.0620587i	−0.0600897	−	0.998193i	\(-0.519139\pi\)
							−0.351643	+	0.936134i	\(0.614377\pi\)
\(31\)	2.86225	+	4.95757i	0.514076	+	0.890406i	0.999867	+	0.0163307i	\(0.00519844\pi\)
							−0.485791	+	0.874075i	\(0.661468\pi\)
\(37\)	−8.60873	−	1.29756i	−1.41527	−	0.213317i	−0.603530	−	0.797340i	\(-0.706240\pi\)
							−0.811737	+	0.584023i	\(0.801478\pi\)
\(41\)	−3.21425	−	0.991463i	−0.501981	−	0.154841i	0.0334155	−	0.999442i	\(-0.489362\pi\)
							−0.535396	+	0.844601i	\(0.679838\pi\)
\(43\)	−2.48336	+	0.766017i	−0.378710	+	0.116816i	−0.478265	−	0.878216i	\(-0.658734\pi\)
							0.0995554	+	0.995032i	\(0.468258\pi\)
\(47\)	5.44048	−	3.70925i	0.793575	−	0.541050i	−0.0973234	−	0.995253i	\(-0.531028\pi\)
							0.890899	+	0.454202i	\(0.150076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.z.a.130.41	yes	648
9.7	even	3		441.2.y.a.277.14	yes	648
49.23	even	21		441.2.y.a.121.14	✓	648
441.268	even	21	inner	441.2.z.a.268.41	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.14	✓	648	49.23	even	21
441.2.y.a.277.14	yes	648	9.7	even	3
441.2.z.a.130.41	yes	648	1.1	even	1	trivial
441.2.z.a.268.41	yes	648	441.268	even	21	inner