Embedded newform 441.2.y.a.151.54

• Label: 441.2.y.a.151.54
• Level: $441$
• Weight: $2$
• Character: 441.151
• 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			151.54
Character	\(\chi\)	\(=\)	441.151
Dual form			441.2.y.a.184.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.52625 - 1.21658i) q^{2} +(-1.72157 - 0.190260i) q^{3} +(3.65489 - 4.58308i) q^{4} +(-1.41356 - 1.31159i) q^{5} +(-4.58058 + 1.61378i) q^{6} +(-2.36573 + 1.18462i) q^{7} +(2.40961 - 10.5572i) q^{8} +(2.92760 + 0.655091i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 13 q^{3} - 109 q^{4} - 12 q^{5} + 2 q^{6} - 7 q^{7} - 16 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{5}{21}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.52625	−	1.21658i	1.78633	−	0.860250i	0.836174	−	0.548464i	\(-0.184787\pi\)
							0.950152	−	0.311786i	\(-0.100927\pi\)
\(3\)	−1.72157	−	0.190260i	−0.993949	−	0.109847i
							\(5\)	−1.41356	−	1.31159i	−0.632164	−	0.586563i	0.297613	−	0.954687i	\(-0.403810\pi\)
−0.929777	+	0.368124i	\(0.880000\pi\)
\(7\)	−2.36573	+	1.18462i	−0.894162	+	0.447744i
							\(11\)	−1.01262	−	0.690394i	−0.305317	−	0.208162i	0.400963	−	0.916094i	\(-0.368676\pi\)
−0.706280	+	0.707933i	\(0.749628\pi\)
\(13\)	−0.415037	−	0.282968i	−0.115111	−	0.0784811i	0.504396	−	0.863472i	\(-0.331715\pi\)
							−0.619507	+	0.784991i	\(0.712667\pi\)
\(17\)	5.37911	−	0.810771i	1.30463	−	0.196641i	0.540301	−	0.841472i	\(-0.318310\pi\)
							0.764325	+	0.644831i	\(0.223072\pi\)
\(19\)	−1.07046	−	1.85408i	−0.245579	−	0.425356i	0.716715	−	0.697366i	\(-0.245645\pi\)
							−0.962294	+	0.272010i	\(0.912312\pi\)
\(23\)	3.17735	+	8.09576i	0.662523	+	1.68808i	0.723844	+	0.689964i	\(0.242374\pi\)
							−0.0613202	+	0.998118i	\(0.519531\pi\)
\(29\)	0.0826482	−	0.0124572i	0.0153474	−	0.00231325i	−0.141364	−	0.989958i	\(-0.545149\pi\)
							0.156711	+	0.987644i	\(0.449911\pi\)
\(31\)	8.79333			1.57933			0.789664	−	0.613540i	\(-0.210255\pi\)
							0.789664	+	0.613540i	\(0.210255\pi\)
\(37\)	−0.726269	+	1.85050i	−0.119398	+	0.304221i	−0.978061	−	0.208320i	\(-0.933200\pi\)
							0.858663	+	0.512541i	\(0.171296\pi\)
\(41\)	6.47639	−	1.99770i	1.01144	−	0.311989i	0.255628	−	0.966775i	\(-0.417718\pi\)
							0.755814	+	0.654787i	\(0.227242\pi\)
\(43\)	−0.972326	−	0.299923i	−0.148278	−	0.0457378i	0.219727	−	0.975561i	\(-0.429483\pi\)
							−0.368006	+	0.929823i	\(0.619959\pi\)
\(47\)	−3.57030	+	1.71937i	−0.520783	+	0.250796i	−0.675759	−	0.737122i	\(-0.736184\pi\)
							0.154977	+	0.987918i	\(0.450470\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.y.a.151.54	✓	648
9.4	even	3		441.2.z.a.4.1	yes	648
49.37	even	21		441.2.z.a.331.1	yes	648
441.184	even	21	inner	441.2.y.a.184.54	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.y.a.151.54	✓	648	1.1	even	1	trivial
441.2.y.a.184.54	yes	648	441.184	even	21	inner
441.2.z.a.4.1	yes	648	9.4	even	3
441.2.z.a.331.1	yes	648	49.37	even	21