Embedded newform 49.6.g.a.2.20

• Label: 49.6.g.a.2.20
• Level: $49$
• Weight: $6$
• Character: 49.2
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(264\)
Relative dimension:	\(22\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			2.20
Character	\(\chi\)	\(=\)	49.2
Dual form			49.6.g.a.25.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.25626 + 2.85518i) q^{2} +(4.05332 - 10.3277i) q^{3} +(51.0867 + 34.8304i) q^{4} +(20.5483 - 3.09715i) q^{5} +(67.0061 - 84.0230i) q^{6} +(13.7032 - 128.916i) q^{7} +(180.161 + 225.915i) q^{8} +(87.8996 + 81.5589i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{13}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.25626	+	2.85518i	1.63629	+	0.504729i	0.970438	−	0.241353i	\(-0.0775910\pi\)
							0.665854	+	0.746082i	\(0.268067\pi\)
\(3\)	4.05332	−	10.3277i	0.260021	−	0.662522i	−0.739927	−	0.672687i	\(-0.765140\pi\)
							0.999948	+	0.0101643i	\(0.00323546\pi\)
\(5\)	20.5483	−	3.09715i	0.367579	−	0.0554036i	0.0373441	−	0.999302i	\(-0.488110\pi\)
							0.330235	+	0.943899i	\(0.392872\pi\)
\(7\)	13.7032	−	128.916i	0.105700	−	0.994398i
							\(11\)	−385.204	+	357.417i	−0.959862	+	0.890622i	−0.994064	−	0.108793i	\(-0.965301\pi\)
0.0342025	+	0.999415i	\(0.489111\pi\)
\(13\)	67.0386	+	293.715i	0.110019	+	0.482023i	0.999677	+	0.0253974i	\(0.00808510\pi\)
							−0.889659	+	0.456626i	\(0.849058\pi\)
\(17\)	−109.067	−	1455.40i	−0.0915317	−	1.22141i	−0.834092	−	0.551626i	\(-0.814008\pi\)
							0.742560	−	0.669779i	\(-0.233611\pi\)
\(19\)	−633.873	+	1097.90i	−0.402826	+	0.697716i	−0.994066	−	0.108780i	\(-0.965306\pi\)
							0.591239	+	0.806496i	\(0.298639\pi\)
\(23\)	−158.579	+	2116.09i	−0.0625067	+	0.834095i	0.874554	+	0.484928i	\(0.161154\pi\)
							−0.937061	+	0.349166i	\(0.886465\pi\)
\(29\)	−5332.39	−	2567.94i	−1.17741	−	0.567010i	−0.260253	−	0.965541i	\(-0.583806\pi\)
							−0.917155	+	0.398531i	\(0.869520\pi\)
\(31\)	510.456	+	884.136i	0.0954013	+	0.165240i	0.909776	−	0.415099i	\(-0.136253\pi\)
							−0.814375	+	0.580339i	\(0.802920\pi\)
\(37\)	−1392.86	+	949.635i	−0.167264	+	0.114039i	−0.644059	−	0.764976i	\(-0.722751\pi\)
							0.476795	+	0.879015i	\(0.341798\pi\)
\(41\)	−2687.30	−	3369.77i	−0.249665	−	0.313069i	0.641169	−	0.767400i	\(-0.278450\pi\)
							−0.890833	+	0.454331i	\(0.849878\pi\)
\(43\)	4842.94	−	6072.86i	0.399428	−	0.500867i	−0.540923	−	0.841072i	\(-0.681925\pi\)
							0.940351	+	0.340205i	\(0.110497\pi\)
\(47\)	1000.10	+	308.490i	0.0660387	+	0.0203702i	0.327599	−	0.944817i	\(-0.393761\pi\)
							−0.261560	+	0.965187i	\(0.584237\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.2.20	✓	264
49.25	even	21	inner	49.6.g.a.25.20	yes	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.2.20	✓	264	1.1	even	1	trivial
49.6.g.a.25.20	yes	264	49.25	even	21	inner