Embedded newform 441.2.z.a.268.41

• Label: 441.2.z.a.268.41
• Level: $441$
• Weight: $2$
• Character: 441.268
• 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			268.41
Character	\(\chi\)	\(=\)	441.268
Dual form			441.2.z.a.130.41

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.14	✓	648	9.4	even	3
441.2.y.a.277.14	yes	648	49.32	even	21
441.2.z.a.130.41	yes	648	441.130	even	21	inner
441.2.z.a.268.41	yes	648	1.1	even	1	trivial