Embedded newform 441.2.bd.a.248.2

• Label: 441.2.bd.a.248.2
• Level: $441$
• Weight: $2$
• Character: 441.248
• 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.bd (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.52140272914\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			248.2
Character	\(\chi\)	\(=\)	441.248
Dual form			441.2.bd.a.425.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.400606 - 2.65784i) q^{2} +(-0.811390 + 1.53024i) q^{3} +(-4.99251 + 1.53998i) q^{4} +(-2.29388 + 1.10467i) q^{5} +(4.39220 + 1.54352i) q^{6} +(2.64223 + 0.136501i) q^{7} +(3.76062 + 7.80901i) q^{8} +(-1.68329 - 2.48325i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.400606	−	2.65784i	−0.283271	−	1.87938i	−0.449482	−	0.893290i	\(-0.648391\pi\)
							0.166211	−	0.986090i	\(-0.446847\pi\)
\(3\)	−0.811390	+	1.53024i	−0.468456	+	0.883487i
							\(5\)	−2.29388	+	1.10467i	−1.02585	+	0.494025i	−0.869634	−	0.493697i	\(-0.835645\pi\)
−0.156220	+	0.987722i	\(0.549931\pi\)
\(7\)	2.64223	+	0.136501i	0.998668	+	0.0515926i
							\(11\)	−0.900817	+	0.718377i	−0.271606	+	0.216599i	−0.749815	−	0.661647i	\(-0.769858\pi\)
0.478209	+	0.878246i	\(0.341286\pi\)
\(13\)	−0.833557	−	5.53029i	−0.231187	−	1.53383i	−0.739797	−	0.672830i	\(-0.765079\pi\)
							0.508610	−	0.860997i	\(-0.330159\pi\)
\(17\)	2.91640	+	0.899589i	0.707330	+	0.218182i	0.627489	−	0.778625i	\(-0.284083\pi\)
							0.0798409	+	0.996808i	\(0.474559\pi\)
\(19\)	5.90833	−	3.41117i	1.35546	−	0.782577i	0.366454	−	0.930436i	\(-0.380572\pi\)
							0.989008	+	0.147859i	\(0.0472383\pi\)
\(23\)	−0.223996	−	0.0511256i	−0.0467064	−	0.0106604i	0.199104	−	0.979978i	\(-0.436197\pi\)
							−0.245810	+	0.969318i	\(0.579054\pi\)
\(29\)	−3.19874	−	3.44742i	−0.593991	−	0.640170i	0.362482	−	0.931991i	\(-0.381929\pi\)
							−0.956473	+	0.291821i	\(0.905739\pi\)
\(31\)	7.26169	−	4.19254i	1.30424	−	0.753002i	0.323110	−	0.946361i	\(-0.395272\pi\)
							0.981128	+	0.193359i	\(0.0619384\pi\)
\(37\)	0.586868	−	0.544534i	0.0964805	−	0.0895208i	−0.630458	−	0.776223i	\(-0.717133\pi\)
							0.726939	+	0.686702i	\(0.240942\pi\)
\(41\)	−0.447908	+	5.97691i	−0.0699514	+	0.933437i	0.846267	+	0.532759i	\(0.178845\pi\)
							−0.916219	+	0.400678i	\(0.868774\pi\)
\(43\)	0.517290	+	6.90275i	0.0788859	+	1.05266i	0.886433	+	0.462857i	\(0.153176\pi\)
							−0.807547	+	0.589803i	\(0.799205\pi\)
\(47\)	2.81658	−	0.424531i	0.410840	−	0.0619242i	0.0596295	−	0.998221i	\(-0.481008\pi\)
							0.351211	+	0.936296i	\(0.385770\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.248.2	✓	648
9.2	odd	6		441.2.bn.a.101.53	yes	648
49.33	odd	42		441.2.bn.a.131.53	yes	648
441.425	even	42	inner	441.2.bd.a.425.2	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.248.2	✓	648	1.1	even	1	trivial
441.2.bd.a.425.2	yes	648	441.425	even	42	inner
441.2.bn.a.101.53	yes	648	9.2	odd	6
441.2.bn.a.131.53	yes	648	49.33	odd	42