Embedded newform 441.2.ba.a.22.20

• Label: 441.2.ba.a.22.20
• Level: $441$
• Weight: $2$
• Character: 441.22
• 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			22.20
Character	\(\chi\)	\(=\)	441.22
Dual form			441.2.ba.a.421.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.819051 - 0.558419i) q^{2} +(-0.438134 - 1.67572i) q^{3} +(-0.371670 - 0.947000i) q^{4} +(-1.06857 - 0.991491i) q^{5} +(-0.576900 + 1.61716i) q^{6} +(0.639861 + 2.56721i) q^{7} +(-0.665577 + 2.91608i) q^{8} +(-2.61608 + 1.46838i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 5 q^{2} - 10 q^{3} + 47 q^{4} - 9 q^{5} - 34 q^{6} - 7 q^{7} - 28 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{4}{7}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.819051	−	0.558419i	−0.579156	−	0.394862i	0.238011	−	0.971262i	\(-0.423505\pi\)
							−0.817167	+	0.576400i	\(0.804457\pi\)
\(3\)	−0.438134	−	1.67572i	−0.252957	−	0.967478i
							\(5\)	−1.06857	−	0.991491i	−0.477881	−	0.443408i	0.404086	−	0.914721i	\(-0.367590\pi\)
−0.881966	+	0.471313i	\(0.843780\pi\)
\(7\)	0.639861	+	2.56721i	0.241845	+	0.970315i
							\(11\)	−2.98118	−	2.03253i	−0.898859	−	0.612832i	0.0232636	−	0.999729i	\(-0.492594\pi\)
−0.922123	+	0.386898i	\(0.873547\pi\)
\(13\)	−0.234013	+	3.12268i	−0.0649034	+	0.866076i	0.865821	+	0.500354i	\(0.166797\pi\)
							−0.930724	+	0.365722i	\(0.880822\pi\)
\(17\)	−0.540102	−	0.677266i	−0.130994	−	0.164261i	0.712009	−	0.702171i	\(-0.247786\pi\)
							−0.843003	+	0.537909i	\(0.819214\pi\)
\(19\)	−1.01771			−0.233478			−0.116739	−	0.993163i	\(-0.537244\pi\)
							−0.116739	+	0.993163i	\(0.537244\pi\)
\(23\)	0.574425	+	1.46361i	0.119776	+	0.305184i	0.978170	−	0.207807i	\(-0.0666326\pi\)
							−0.858394	+	0.512991i	\(0.828537\pi\)
\(29\)	−0.375015	+	0.955523i	−0.0696386	+	0.177436i	−0.961456	−	0.274960i	\(-0.911335\pi\)
							0.891817	+	0.452396i	\(0.149431\pi\)
\(31\)	−2.36673	+	4.09930i	−0.425078	+	0.736256i	−0.996428	−	0.0844502i	\(-0.973087\pi\)
							0.571350	+	0.820707i	\(0.306420\pi\)
\(37\)	4.88595	+	6.12679i	0.803245	+	1.00724i	0.999643	+	0.0267132i	\(0.00850407\pi\)
							−0.196398	+	0.980524i	\(0.562924\pi\)
\(41\)	−4.57930	−	4.24897i	−0.715167	−	0.663578i	0.236176	−	0.971710i	\(-0.424106\pi\)
							−0.951344	+	0.308132i	\(0.900296\pi\)
\(43\)	−0.576742	+	0.535138i	−0.0879523	+	0.0816078i	−0.722924	−	0.690928i	\(-0.757202\pi\)
							0.634972	+	0.772536i	\(0.281012\pi\)
\(47\)	−8.60475	−	5.86662i	−1.25513	−	0.855734i	−0.261152	−	0.965298i	\(-0.584102\pi\)
							−0.993980	+	0.109563i	\(0.965055\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.ba.a.22.20	✓	648
9.7	even	3	inner	441.2.ba.a.169.20	yes	648
49.29	even	7	inner	441.2.ba.a.274.20	yes	648
441.421	even	21	inner	441.2.ba.a.421.20	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.ba.a.22.20	✓	648	1.1	even	1	trivial
441.2.ba.a.169.20	yes	648	9.7	even	3	inner
441.2.ba.a.274.20	yes	648	49.29	even	7	inner
441.2.ba.a.421.20	yes	648	441.421	even	21	inner