Embedded newform 441.2.bd.a.437.49

• Label: 441.2.bd.a.437.49
• Level: $441$
• Weight: $2$
• Character: 441.437
• 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			437.49
Character	\(\chi\)	\(=\)	441.437
Dual form			441.2.bd.a.110.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.28939 + 0.171566i) q^{2} +(0.623167 + 1.61606i) q^{3} +(3.23419 + 0.487476i) q^{4} +(0.442795 - 1.94001i) q^{5} +(1.14941 + 3.80671i) q^{6} +(2.56732 + 0.639424i) q^{7} +(2.84419 + 0.649167i) q^{8} +(-2.22333 + 2.01416i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{31}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.28939	+	0.171566i	1.61884	+	0.121315i	0.853069	−	0.521798i	\(-0.174739\pi\)
							0.765771	+	0.643113i	\(0.222358\pi\)
\(3\)	0.623167	+	1.61606i	0.359786	+	0.933035i
							\(5\)	0.442795	−	1.94001i	0.198024	−	0.867599i	−0.774087	−	0.633079i	\(-0.781791\pi\)
0.972111	−	0.234520i	\(-0.0753520\pi\)
\(7\)	2.56732	+	0.639424i	0.970356	+	0.241679i
							\(11\)	−1.84568	−	3.83259i	−0.556493	−	1.15557i	−0.969556	−	0.244868i	\(-0.921255\pi\)
0.413063	−	0.910702i	\(-0.364459\pi\)
\(13\)	−4.17934	−	0.313198i	−1.15914	−	0.0868656i	−0.518758	−	0.854921i	\(-0.673605\pi\)
							−0.640383	+	0.768056i	\(0.721224\pi\)
\(17\)	1.78250	−	0.268669i	0.432320	−	0.0651617i	0.0707222	−	0.997496i	\(-0.477470\pi\)
							0.361597	+	0.932334i	\(0.382232\pi\)
\(19\)	−5.84088	+	3.37223i	−1.33999	+	0.773643i	−0.986805	−	0.161911i	\(-0.948234\pi\)
							−0.353183	+	0.935554i	\(0.614901\pi\)
\(23\)	3.85793	+	3.07660i	0.804434	+	0.641515i	0.936870	−	0.349677i	\(-0.113709\pi\)
							−0.132436	+	0.991192i	\(0.542280\pi\)
\(29\)	0.822724	+	0.322895i	0.152776	+	0.0599601i	0.440497	−	0.897754i	\(-0.354802\pi\)
							−0.287721	+	0.957714i	\(0.592898\pi\)
\(31\)	−8.22765	+	4.75023i	−1.47773	+	0.853167i	−0.999683	−	0.0251681i	\(-0.991988\pi\)
							−0.478045	+	0.878335i	\(0.658655\pi\)
\(37\)	2.25356	−	5.74197i	0.370482	−	0.943974i	−0.617102	−	0.786883i	\(-0.711693\pi\)
							0.987584	−	0.157091i	\(-0.0502114\pi\)
\(41\)	7.66437	+	7.11150i	1.19697	+	1.11063i	0.991247	+	0.132020i	\(0.0421464\pi\)
							0.205727	+	0.978609i	\(0.434044\pi\)
\(43\)	5.11074	−	4.74207i	0.779381	−	0.723159i	−0.186413	−	0.982471i	\(-0.559686\pi\)
							0.965794	+	0.259312i	\(0.0834958\pi\)
\(47\)	0.524137	−	6.99413i	0.0764533	−	1.02020i	−0.818636	−	0.574312i	\(-0.805270\pi\)
							0.895090	−	0.445886i	\(-0.147111\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.437.49	yes	648
9.2	odd	6		441.2.bn.a.290.49	yes	648
49.12	odd	42		441.2.bn.a.257.49	yes	648
441.110	even	42	inner	441.2.bd.a.110.49	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.110.49	✓	648	441.110	even	42	inner
441.2.bd.a.437.49	yes	648	1.1	even	1	trivial
441.2.bn.a.257.49	yes	648	49.12	odd	42
441.2.bn.a.290.49	yes	648	9.2	odd	6