Embedded newform 49.6.g.a.11.12

• Label: 49.6.g.a.11.12
• Level: $49$
• Weight: $6$
• Character: 49.11
• 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			11.12
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.457373 - 0.424380i) q^{2} +(-22.4874 + 3.38943i) q^{3} +(-2.36227 + 31.5223i) q^{4} +(-1.17229 + 2.98696i) q^{5} +(-8.84671 + 11.0934i) q^{6} +(-27.2101 - 126.754i) q^{7} +(24.7454 + 31.0298i) q^{8} +(261.990 - 80.8133i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.457373	−	0.424380i	0.0808528	−	0.0750205i	−0.638711	−	0.769447i	\(-0.720532\pi\)
							0.719564	+	0.694426i	\(0.244342\pi\)
\(3\)	−22.4874	+	3.38943i	−1.44257	+	0.217432i	−0.823237	−	0.567698i	\(-0.807834\pi\)
							−0.619330	+	0.785130i	\(0.712596\pi\)
\(5\)	−1.17229	+	2.98696i	−0.0209706	+	0.0534323i	−0.940976	−	0.338474i	\(-0.890089\pi\)
							0.920005	+	0.391907i	\(0.128184\pi\)
\(7\)	−27.2101	−	126.754i	−0.209887	−	0.977726i
							\(11\)	379.432	+	117.039i	0.945480	+	0.291642i	0.728898	−	0.684623i	\(-0.240033\pi\)
0.216582	+	0.976264i	\(0.430509\pi\)
\(13\)	−249.005	−	1090.96i	−0.408648	−	1.79040i	−0.590455	−	0.807071i	\(-0.701052\pi\)
							0.181807	−	0.983334i	\(-0.441806\pi\)
\(17\)	833.692	−	568.401i	0.699654	−	0.477016i	−0.160510	−	0.987034i	\(-0.551314\pi\)
							0.860164	+	0.510018i	\(0.170361\pi\)
\(19\)	303.050	+	524.898i	0.192588	+	0.333573i	0.946107	−	0.323853i	\(-0.104978\pi\)
							−0.753519	+	0.657426i	\(0.771645\pi\)
\(23\)	−3143.97	−	2143.52i	−1.23925	−	0.844906i	−0.246982	−	0.969020i	\(-0.579439\pi\)
							−0.992268	+	0.124114i	\(0.960391\pi\)
\(29\)	1428.72	+	688.033i	0.315465	+	0.151920i	0.584915	−	0.811094i	\(-0.301128\pi\)
							−0.269451	+	0.963014i	\(0.586842\pi\)
\(31\)	516.505	−	894.613i	0.0965318	−	0.167198i	−0.813715	−	0.581264i	\(-0.802558\pi\)
							0.910247	+	0.414066i	\(0.135892\pi\)
\(37\)	−720.654	−	9616.46i	−0.0865411	−	1.15481i	−0.856301	−	0.516477i	\(-0.827243\pi\)
							0.769760	−	0.638334i	\(-0.220376\pi\)
\(41\)	−9484.42	−	11893.1i	−0.881153	−	1.10493i	−0.993787	−	0.111294i	\(-0.964500\pi\)
							0.112635	−	0.993636i	\(-0.464071\pi\)
\(43\)	−5076.05	+	6365.17i	−0.418654	+	0.524975i	−0.945778	−	0.324813i	\(-0.894699\pi\)
							0.527124	+	0.849788i	\(0.323270\pi\)
\(47\)	−3471.33	+	3220.92i	−0.229219	+	0.212684i	−0.786418	−	0.617695i	\(-0.788067\pi\)
							0.557199	+	0.830379i	\(0.311876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.12	yes	264
49.9	even	21	inner	49.6.g.a.9.12	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.12	✓	264	49.9	even	21	inner
49.6.g.a.11.12	yes	264	1.1	even	1	trivial