Embedded newform 441.2.y.a.121.11

• Label: 441.2.y.a.121.11
• Level: $441$
• Weight: $2$
• Character: 441.121
• 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			121.11
Character	\(\chi\)	\(=\)	441.121
Dual form			441.2.y.a.277.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58912 + 0.765282i) q^{2} +(-0.452198 - 1.67198i) q^{3} +(0.692680 - 0.868593i) q^{4} +(-4.13369 + 1.27507i) q^{5} +(1.99813 + 2.31093i) q^{6} +(-1.26212 + 2.32531i) q^{7} +(0.348927 - 1.52875i) q^{8} +(-2.59103 + 1.51213i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{19}{21}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58912	+	0.765282i	−1.12368	+	0.541136i	−0.901028	−	0.433762i	\(-0.857186\pi\)
							−0.222653	+	0.974898i	\(0.571472\pi\)
\(3\)	−0.452198	−	1.67198i	−0.261076	−	0.965318i
							\(5\)	−4.13369	+	1.27507i	−1.84864	+	0.570230i	−0.849554	+	0.527501i	\(0.823129\pi\)
−0.999087	+	0.0427290i	\(0.986395\pi\)
\(7\)	−1.26212	+	2.32531i	−0.477037	+	0.878883i
							\(11\)	0.172299	−	2.29917i	0.0519500	−	0.693224i	−0.909052	−	0.416683i	\(-0.863193\pi\)
0.961002	−	0.276542i	\(-0.0891884\pi\)
\(13\)	0.199209	−	2.65826i	0.0552507	−	0.737270i	−0.898940	−	0.438071i	\(-0.855662\pi\)
							0.954191	−	0.299198i	\(-0.0967192\pi\)
\(17\)	−2.36226	+	6.01895i	−0.572933	+	1.45981i	0.291257	+	0.956645i	\(0.405927\pi\)
							−0.864190	+	0.503165i	\(0.832169\pi\)
\(19\)	0.136351	−	0.236166i	0.0312810	−	0.0541803i	−0.849961	−	0.526845i	\(-0.823375\pi\)
							0.881242	+	0.472665i	\(0.156708\pi\)
\(23\)	−1.89144	−	0.285088i	−0.394392	−	0.0594450i	−0.0511475	−	0.998691i	\(-0.516288\pi\)
							−0.343245	+	0.939246i	\(0.611526\pi\)
\(29\)	3.51908	−	8.96647i	0.653476	−	1.66503i	−0.0897279	−	0.995966i	\(-0.528600\pi\)
							0.743204	−	0.669064i	\(-0.233305\pi\)
\(31\)	5.53237			0.993642			0.496821	−	0.867853i	\(-0.334501\pi\)
							0.496821	+	0.867853i	\(0.334501\pi\)
\(37\)	5.97807	−	0.901049i	0.982789	−	0.148132i	0.362058	−	0.932155i	\(-0.382074\pi\)
							0.620731	+	0.784024i	\(0.286836\pi\)
\(41\)	1.00789	+	0.935188i	0.157406	+	0.146052i	0.754944	−	0.655789i	\(-0.227664\pi\)
							−0.597538	+	0.801841i	\(0.703854\pi\)
\(43\)	−3.15575	+	2.92810i	−0.481247	+	0.446532i	−0.883109	−	0.469169i	\(-0.844554\pi\)
							0.401862	+	0.915700i	\(0.368363\pi\)
\(47\)	3.44421	−	1.65864i	0.502390	−	0.241938i	−0.165484	−	0.986212i	\(-0.552919\pi\)
							0.667874	+	0.744274i	\(0.267204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.121.11	✓	648
9.7	even	3		441.2.z.a.268.44	yes	648
49.32	even	21		441.2.z.a.130.44	yes	648
441.277	even	21	inner	441.2.y.a.277.11	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.121.11	✓	648	1.1	even	1	trivial
441.2.y.a.277.11	yes	648	441.277	even	21	inner
441.2.z.a.130.44	yes	648	49.32	even	21
441.2.z.a.268.44	yes	648	9.7	even	3