Embedded newform 441.2.bd.a.122.9

• Label: 441.2.bd.a.122.9
• Level: $441$
• Weight: $2$
• Character: 441.122
• 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			122.9
Character	\(\chi\)	\(=\)	441.122
Dual form			441.2.bd.a.47.9

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{37}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.9	✓	648	441.47	even	42	inner
441.2.bd.a.122.9	yes	648	1.1	even	1	trivial
441.2.bn.a.194.9	yes	648	49.47	odd	42
441.2.bn.a.416.9	yes	648	9.2	odd	6