Embedded newform 441.2.bd.a.47.9

• Label: 441.2.bd.a.47.9
• Level: $441$
• Weight: $2$
• Character: 441.47
• 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			47.9
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34426 - 1.44876i) q^{2} +(-1.69107 + 0.374554i) q^{3} +(-0.142431 + 1.90061i) q^{4} +(0.213240 + 0.267394i) q^{5} +(2.81587 + 1.94646i) q^{6} +(-2.30163 - 1.30480i) q^{7} +(-0.145343 + 0.115907i) q^{8} +(2.71942 - 1.26679i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34426	−	1.44876i	−0.950532	−	1.02443i	−0.999671	−	0.0256382i	\(-0.991838\pi\)
							0.0491390	−	0.998792i	\(-0.484352\pi\)
\(3\)	−1.69107	+	0.374554i	−0.976338	+	0.216249i
							\(5\)	0.213240	+	0.267394i	0.0953638	+	0.119582i	0.827224	−	0.561873i	\(-0.189919\pi\)
−0.731860	+	0.681455i	\(0.761347\pi\)
\(7\)	−2.30163	−	1.30480i	−0.869933	−	0.493170i
							\(11\)	−3.98812	+	0.910263i	−1.20246	+	0.274455i	−0.776401	−	0.630239i	\(-0.782957\pi\)
−0.426063	+	0.904694i	\(0.640100\pi\)
\(13\)	1.48090	+	1.59603i	0.410727	+	0.442659i	0.904204	−	0.427100i	\(-0.140465\pi\)
							−0.493477	+	0.869759i	\(0.664274\pi\)
\(17\)	0.0517311	+	0.690304i	0.0125466	+	0.167423i	0.999957	+	0.00924535i	\(0.00294293\pi\)
							−0.987411	+	0.158178i	\(0.949438\pi\)
\(19\)	2.01961	+	1.16603i	0.463331	+	0.267504i	0.713444	−	0.700712i	\(-0.247134\pi\)
							−0.250113	+	0.968217i	\(0.580468\pi\)
\(23\)	0.884031	+	1.83571i	0.184333	+	0.382772i	0.972574	−	0.232596i	\(-0.0747219\pi\)
							−0.788240	+	0.615368i	\(0.789008\pi\)
\(29\)	1.35923	+	1.99362i	0.252402	+	0.370206i	0.931439	−	0.363898i	\(-0.118554\pi\)
							−0.679036	+	0.734105i	\(0.737602\pi\)
\(31\)	8.36372	+	4.82880i	1.50217	+	0.867277i	0.999997	+	0.00250934i	\(0.000798747\pi\)
							0.502172	+	0.864768i	\(0.332535\pi\)
\(37\)	5.08083	−	3.46405i	0.835284	−	0.569487i	−0.0684145	−	0.997657i	\(-0.521794\pi\)
							0.903698	+	0.428170i	\(0.140842\pi\)
\(41\)	3.12207	−	7.95490i	0.487585	−	1.24235i	−0.449511	−	0.893275i	\(-0.648402\pi\)
							0.937096	−	0.349072i	\(-0.113503\pi\)
\(43\)	−0.747643	−	1.90496i	−0.114015	−	0.290504i	0.862464	−	0.506118i	\(-0.168920\pi\)
							−0.976479	+	0.215614i	\(0.930825\pi\)
\(47\)	1.28570	−	1.19296i	0.187539	−	0.174010i	−0.580816	−	0.814035i	\(-0.697266\pi\)
							0.768354	+	0.640025i	\(0.221076\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.9	✓	648
9.5	odd	6		441.2.bn.a.194.9	yes	648
49.24	odd	42		441.2.bn.a.416.9	yes	648
441.122	even	42	inner	441.2.bd.a.122.9	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.9	✓	648	1.1	even	1	trivial
441.2.bd.a.122.9	yes	648	441.122	even	42	inner
441.2.bn.a.194.9	yes	648	9.5	odd	6
441.2.bn.a.416.9	yes	648	49.24	odd	42