Embedded newform 441.2.ba.a.337.40

• Label: 441.2.ba.a.337.40
• Level: $441$
• Weight: $2$
• Character: 441.337
• Analytic conductor: $3.521$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.ba (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			337.40
Character	\(\chi\)	\(=\)	441.337
Dual form			441.2.ba.a.106.40

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46084 + 0.220186i) q^{2} +(1.61318 - 0.630586i) q^{3} +(0.174420 + 0.0538014i) q^{4} +(-0.764060 + 0.520927i) q^{5} +(2.49545 - 0.565984i) q^{6} +(1.69903 - 2.02813i) q^{7} +(-2.41912 - 1.16499i) q^{8} +(2.20472 - 2.03450i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46084	+	0.220186i	1.03297	+	0.155695i	0.643580	−	0.765379i	\(-0.277448\pi\)
							0.389388	+	0.921074i	\(0.372686\pi\)
\(3\)	1.61318	−	0.630586i	0.931372	−	0.364069i
							\(5\)	−0.764060	+	0.520927i	−0.341698	+	0.232966i	−0.721994	−	0.691899i	\(-0.756774\pi\)
0.380296	+	0.924865i	\(0.375822\pi\)
\(7\)	1.69903	−	2.02813i	0.642173	−	0.766560i
							\(11\)	2.99886	+	0.452005i	0.904189	+	0.136285i	0.584647	−	0.811288i	\(-0.301233\pi\)
0.319542	+	0.947572i	\(0.396471\pi\)
\(13\)	1.91339	+	4.87524i	0.530679	+	1.35215i	0.905222	+	0.424939i	\(0.139704\pi\)
							−0.374543	+	0.927210i	\(0.622200\pi\)
\(17\)	0.918942	−	4.02615i	0.222876	−	0.976485i	−0.732424	−	0.680848i	\(-0.761611\pi\)
							0.955301	−	0.295636i	\(-0.0955316\pi\)
\(19\)	0.0896998			0.0205785			0.0102893	−	0.999947i	\(-0.496725\pi\)
							0.0102893	+	0.999947i	\(0.496725\pi\)
\(23\)	−3.67388	−	1.13324i	−0.766058	−	0.236297i	−0.112989	−	0.993596i	\(-0.536042\pi\)
							−0.653069	+	0.757299i	\(0.726519\pi\)
\(29\)	−1.76461	+	0.544310i	−0.327680	+	0.101076i	−0.454230	−	0.890884i	\(-0.650086\pi\)
							0.126551	+	0.991960i	\(0.459609\pi\)
\(31\)	−3.48931	+	6.04367i	−0.626699	+	1.08547i	0.361511	+	0.932368i	\(0.382261\pi\)
							−0.988210	+	0.153107i	\(0.951072\pi\)
\(37\)	−1.83024	+	8.01881i	−0.300890	+	1.31828i	0.567897	+	0.823099i	\(0.307757\pi\)
							−0.868787	+	0.495185i	\(0.835100\pi\)
\(41\)	−7.22928	+	4.92884i	−1.12902	+	0.769755i	−0.975850	−	0.218444i	\(-0.929902\pi\)
							−0.153174	+	0.988199i	\(0.548950\pi\)
\(43\)	−9.44672	−	6.44066i	−1.44061	−	0.982192i	−0.996266	−	0.0863362i	\(-0.972484\pi\)
							−0.444345	−	0.895856i	\(-0.646564\pi\)
\(47\)	3.21684	+	0.484861i	0.469225	+	0.0707242i	0.379399	−	0.925233i	\(-0.376131\pi\)
							0.0898258	+	0.995957i	\(0.471369\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.337.40	yes	648
9.7	even	3	inner	441.2.ba.a.43.15	✓	648
49.8	even	7	inner	441.2.ba.a.400.15	yes	648
441.106	even	21	inner	441.2.ba.a.106.40	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.43.15	✓	648	9.7	even	3	inner
441.2.ba.a.106.40	yes	648	441.106	even	21	inner
441.2.ba.a.337.40	yes	648	1.1	even	1	trivial
441.2.ba.a.400.15	yes	648	49.8	even	7	inner