Embedded newform 441.2.bd.a.47.6

• Label: 441.2.bd.a.47.6
• 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			47.6
Character	\(\chi\)	\(=\)	441.47
Dual form			441.2.bd.a.122.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59981 - 1.72418i) q^{2} +(1.65781 + 0.501676i) q^{3} +(-0.263960 + 3.52230i) q^{4} +(2.05109 + 2.57199i) q^{5} +(-1.78719 - 3.66094i) q^{6} +(-2.34210 + 1.23069i) q^{7} +(2.81755 - 2.24692i) q^{8} +(2.49664 + 1.66336i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 18 q^{2} - 11 q^{3} - 60 q^{4} - 15 q^{5} - 22 q^{6} - 5 q^{7} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59981	−	1.72418i	−1.13123	−	1.21918i	−0.972759	−	0.231817i	\(-0.925533\pi\)
							−0.158474	−	0.987363i	\(-0.550658\pi\)
\(3\)	1.65781	+	0.501676i	0.957135	+	0.289643i
							\(5\)	2.05109	+	2.57199i	0.917277	+	1.15023i	0.988265	+	0.152752i	\(0.0488136\pi\)
−0.0709874	+	0.997477i	\(0.522615\pi\)
\(7\)	−2.34210	+	1.23069i	−0.885229	+	0.465155i
							\(11\)	−1.51115	+	0.344909i	−0.455628	+	0.103994i	−0.444174	−	0.895940i	\(-0.646503\pi\)
−0.0114533	+	0.999934i	\(0.503646\pi\)
\(13\)	−3.29161	−	3.54752i	−0.912930	−	0.983904i	0.0869946	−	0.996209i	\(-0.472274\pi\)
							−0.999924	+	0.0123046i	\(0.996083\pi\)
\(17\)	0.234101	+	3.12386i	0.0567779	+	0.757648i	0.950836	+	0.309696i	\(0.100227\pi\)
							−0.894058	+	0.447952i	\(0.852154\pi\)
\(19\)	0.714075	+	0.412271i	0.163820	+	0.0945815i	0.579669	−	0.814852i	\(-0.303182\pi\)
							−0.415849	+	0.909434i	\(0.636515\pi\)
\(23\)	2.79678	+	5.80758i	0.583169	+	1.21096i	0.958770	+	0.284182i	\(0.0917220\pi\)
							−0.375601	+	0.926781i	\(0.622564\pi\)
\(29\)	3.54597	+	5.20098i	0.658470	+	0.965798i	0.999667	+	0.0258065i	\(0.00821538\pi\)
							−0.341197	+	0.939992i	\(0.610832\pi\)
\(31\)	−1.43893	−	0.830768i	−0.258440	−	0.149210i	0.365183	−	0.930936i	\(-0.381006\pi\)
							−0.623623	+	0.781725i	\(0.714340\pi\)
\(37\)	−0.855008	+	0.582935i	−0.140562	+	0.0958339i	−0.631560	−	0.775327i	\(-0.717585\pi\)
							0.490997	+	0.871161i	\(0.336633\pi\)
\(41\)	−3.89678	+	9.92883i	−0.608575	+	1.55062i	0.210900	+	0.977508i	\(0.432361\pi\)
							−0.819475	+	0.573116i	\(0.805735\pi\)
\(43\)	−0.251888	−	0.641800i	−0.0384126	−	0.0978736i	0.910380	−	0.413773i	\(-0.135789\pi\)
							−0.948793	+	0.315899i	\(0.897694\pi\)
\(47\)	6.65607	−	6.17593i	0.970887	−	0.900852i	−0.0242144	−	0.999707i	\(-0.507708\pi\)
							0.995102	+	0.0988549i	\(0.0315180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.2.bd.a.47.6	✓	648
9.5	odd	6		441.2.bn.a.194.6	yes	648
49.24	odd	42		441.2.bn.a.416.6	yes	648
441.122	even	42	inner	441.2.bd.a.122.6	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.bd.a.47.6	✓	648	1.1	even	1	trivial
441.2.bd.a.122.6	yes	648	441.122	even	42	inner
441.2.bn.a.194.6	yes	648	9.5	odd	6
441.2.bn.a.416.6	yes	648	49.24	odd	42