Embedded newform 441.2.bn.a.5.12

• Label: 441.2.bn.a.5.12
• Level: $441$
• Weight: $2$
• Character: 441.5
• 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.bn (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			5.12
Character	\(\chi\)	\(=\)	441.5
Dual form			441.2.bn.a.353.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.50264 + 1.19832i) q^{2} +(-1.36206 - 1.06995i) q^{3} +(0.376930 - 1.65144i) q^{4} +(2.30957 + 1.57464i) q^{5} +(3.32883 - 0.0244174i) q^{6} +(1.77434 + 1.96258i) q^{7} +(-0.255253 - 0.530039i) q^{8} +(0.710396 + 2.91468i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 21 q^{2} - 11 q^{3} + 99 q^{4} - 18 q^{5} - 34 q^{6} - 5 q^{7} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(199\)	\(344\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.50264	+	1.19832i	−1.06253	+	0.847339i	−0.988696	−	0.149931i	\(-0.952095\pi\)
							−0.0738333	+	0.997271i	\(0.523523\pi\)
\(3\)	−1.36206	−	1.06995i	−0.786384	−	0.617738i
							\(5\)	2.30957	+	1.57464i	1.03287	+	0.704199i	0.956157	−	0.292854i	\(-0.0946051\pi\)
0.0767130	+	0.997053i	\(0.475557\pi\)
\(7\)	1.77434	+	1.96258i	0.670638	+	0.741785i
							\(11\)	−0.840064	−	5.57346i	−0.253289	−	1.68046i	−0.647988	−	0.761651i	\(-0.724389\pi\)
0.394699	−	0.918811i	\(-0.370849\pi\)
\(13\)	0.450458	+	2.98860i	0.124935	+	0.828888i	0.960255	+	0.279124i	\(0.0900441\pi\)
							−0.835320	+	0.549764i	\(0.814718\pi\)
\(17\)	4.82026	−	4.47255i	1.16908	−	1.08475i	0.174099	−	0.984728i	\(-0.444299\pi\)
							0.994985	−	0.100024i	\(-0.0318919\pi\)
\(19\)	−2.47887	−	1.43118i	−0.568693	−	0.328335i	0.187934	−	0.982182i	\(-0.439821\pi\)
							−0.756627	+	0.653847i	\(0.773154\pi\)
\(23\)	1.47045	+	4.76708i	0.306610	+	0.994006i	0.969459	+	0.245253i	\(0.0788709\pi\)
							−0.662849	+	0.748753i	\(0.730653\pi\)
\(29\)	0.592843	+	0.638933i	0.110088	+	0.118647i	0.785692	−	0.618618i	\(-0.212307\pi\)
							−0.675603	+	0.737265i	\(0.736117\pi\)
\(31\)			8.06962i			1.44935i	0.689093	+	0.724673i	\(0.258009\pi\)
							−0.689093	+	0.724673i	\(0.741991\pi\)
\(37\)	10.1293	+	3.12447i	1.66524	+	0.513659i	0.977256	−	0.212065i	\(-0.0680188\pi\)
							0.687986	+	0.725724i	\(0.258495\pi\)
\(41\)	−0.367879	+	4.90900i	−0.0574530	+	0.766657i	0.891857	+	0.452317i	\(0.149402\pi\)
							−0.949310	+	0.314340i	\(0.898217\pi\)
\(43\)	−0.0740286	−	0.987843i	−0.0112893	−	0.150645i	−1.00000		0.000805731i	\(-0.999744\pi\)
							0.988710	−	0.149839i	\(-0.0478755\pi\)
\(47\)	−1.62834	−	2.04188i	−0.237518	−	0.297839i	0.648759	−	0.760994i	\(-0.275288\pi\)
							−0.886277	+	0.463156i	\(0.846717\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bn.a.5.12	yes	648
9.2	odd	6		441.2.bd.a.299.12	yes	648
49.10	odd	42		441.2.bd.a.59.12	✓	648
441.353	even	42	inner	441.2.bn.a.353.12	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.59.12	✓	648	49.10	odd	42
441.2.bd.a.299.12	yes	648	9.2	odd	6
441.2.bn.a.5.12	yes	648	1.1	even	1	trivial
441.2.bn.a.353.12	yes	648	441.353	even	42	inner