Embedded newform 49.6.g.a.11.10

• Label: 49.6.g.a.11.10
• 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			11.10
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27390 + 1.18201i) q^{2} +(18.3349 - 2.76355i) q^{3} +(-2.16568 + 28.8990i) q^{4} +(-17.7966 + 45.3450i) q^{5} +(-20.0903 + 25.1925i) q^{6} +(-121.833 + 44.3138i) q^{7} +(-66.0720 - 82.8516i) q^{8} +(96.3284 - 29.7134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27390	+	1.18201i	−0.225196	+	0.208951i	−0.784699	−	0.619877i	\(-0.787182\pi\)
							0.559503	+	0.828828i	\(0.310992\pi\)
\(3\)	18.3349	−	2.76355i	1.17619	−	0.177282i	0.468275	−	0.883583i	\(-0.344876\pi\)
							0.707911	+	0.706301i	\(0.249638\pi\)
\(5\)	−17.7966	+	45.3450i	−0.318355	+	0.811155i	0.678696	+	0.734419i	\(0.262545\pi\)
							−0.997051	+	0.0767363i	\(0.975550\pi\)
\(7\)	−121.833	+	44.3138i	−0.939766	+	0.341817i
							\(11\)	172.750	+	53.2864i	0.430464	+	0.132781i	0.502413	−	0.864628i	\(-0.332446\pi\)
−0.0719489	+	0.997408i	\(0.522922\pi\)
\(13\)	−0.626877	−	2.74653i	−0.00102878	−	0.00450739i	0.974411	−	0.224774i	\(-0.0721644\pi\)
							−0.975440	+	0.220267i	\(0.929307\pi\)
\(17\)	−314.709	+	214.565i	−0.264111	+	0.180068i	−0.688134	−	0.725583i	\(-0.741570\pi\)
							0.424023	+	0.905651i	\(0.360618\pi\)
\(19\)	999.330	+	1730.89i	0.635075	+	1.09998i	0.986499	+	0.163765i	\(0.0523640\pi\)
							−0.351425	+	0.936216i	\(0.614303\pi\)
\(23\)	1114.75	+	760.021i	0.439397	+	0.299575i	0.762739	−	0.646707i	\(-0.223854\pi\)
							−0.323342	+	0.946282i	\(0.604807\pi\)
\(29\)	6882.12	+	3314.25i	1.51959	+	0.731797i	0.992977	−	0.118306i	\(-0.0377464\pi\)
							0.526616	+	0.850103i	\(0.323461\pi\)
\(31\)	5200.12	−	9006.88i	0.971873	−	1.68333i	0.281982	−	0.959420i	\(-0.409008\pi\)
							0.689891	−	0.723913i	\(-0.257659\pi\)
\(37\)	9.82899	+	131.159i	0.00118033	+	0.0157504i	0.997763	−	0.0668525i	\(-0.0212957\pi\)
							−0.996583	+	0.0826030i	\(0.973677\pi\)
\(41\)	−3626.80	−	4547.87i	−0.336949	−	0.422521i	0.584273	−	0.811557i	\(-0.301380\pi\)
							−0.921223	+	0.389036i	\(0.872808\pi\)
\(43\)	−2979.60	+	3736.30i	−0.245746	+	0.308156i	−0.889372	−	0.457185i	\(-0.848858\pi\)
							0.643626	+	0.765340i	\(0.277429\pi\)
\(47\)	−5997.65	+	5565.01i	−0.396038	+	0.367469i	−0.852906	−	0.522065i	\(-0.825162\pi\)
							0.456868	+	0.889535i	\(0.348971\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.10	yes	264
49.9	even	21	inner	49.6.g.a.9.10	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.10	✓	264	49.9	even	21	inner
49.6.g.a.11.10	yes	264	1.1	even	1	trivial