Embedded newform 49.6.g.a.11.2

• Label: 49.6.g.a.11.2
• Level: $49$
• Weight: $6$
• Character: 49.11
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $264$
• 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			11.2
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.89591 + 6.39847i) q^{2} +(-17.4072 + 2.62371i) q^{3} +(4.22180 - 56.3360i) q^{4} +(6.46740 - 16.4787i) q^{5} +(103.251 - 129.472i) q^{6} +(128.609 - 16.3294i) q^{7} +(143.663 + 180.148i) q^{8} +(63.9213 - 19.7171i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.89591	+	6.39847i	−1.21904	+	1.13110i	−0.231645	+	0.972800i	\(0.574411\pi\)
							−0.987391	+	0.158300i	\(0.949399\pi\)
\(3\)	−17.4072	+	2.62371i	−1.11667	+	0.168311i	−0.681345	−	0.731962i	\(-0.738605\pi\)
							−0.435325	+	0.900273i	\(0.643367\pi\)
\(5\)	6.46740	−	16.4787i	0.115692	−	0.294779i	−0.861285	−	0.508122i	\(-0.830340\pi\)
							0.976977	+	0.213343i	\(0.0684351\pi\)
\(7\)	128.609	−	16.3294i	0.992036	−	0.125958i
							\(11\)	−200.348	−	61.7993i	−0.499234	−	0.153993i	0.0349038	−	0.999391i	\(-0.488888\pi\)
−0.534138	+	0.845397i	\(0.679364\pi\)
\(13\)	−120.978	−	530.037i	−0.198539	−	0.869857i	−0.971807	−	0.235778i	\(-0.924236\pi\)
							0.773268	−	0.634080i	\(-0.218621\pi\)
\(17\)	−578.232	+	394.232i	−0.485266	+	0.330849i	−0.781116	−	0.624386i	\(-0.785349\pi\)
							0.295850	+	0.955234i	\(0.404397\pi\)
\(19\)	1410.66	+	2443.33i	0.896475	+	1.55274i	0.831969	+	0.554823i	\(0.187214\pi\)
							0.0645061	+	0.997917i	\(0.479453\pi\)
\(23\)	−800.359	−	545.675i	−0.315475	−	0.215087i	0.395222	−	0.918586i	\(-0.370668\pi\)
							−0.710697	+	0.703499i	\(0.751620\pi\)
\(29\)	3751.00	+	1806.38i	0.828231	+	0.398855i	0.799451	−	0.600731i	\(-0.205124\pi\)
							0.0287797	+	0.999586i	\(0.490838\pi\)
\(31\)	682.072	−	1181.38i	0.127475	−	0.220794i	−0.795222	−	0.606318i	\(-0.792646\pi\)
							0.922698	+	0.385524i	\(0.125979\pi\)
\(37\)	21.5323	+	287.329i	0.00258575	+	0.0345044i	0.998347	−	0.0574667i	\(-0.0183023\pi\)
							−0.995762	+	0.0919711i	\(0.970683\pi\)
\(41\)	4593.80	+	5760.44i	0.426788	+	0.535175i	0.948008	−	0.318247i	\(-0.103094\pi\)
							−0.521220	+	0.853422i	\(0.674523\pi\)
\(43\)	6352.29	−	7965.52i	0.523913	−	0.656966i	−0.447522	−	0.894273i	\(-0.647693\pi\)
							0.971435	+	0.237307i	\(0.0762647\pi\)
\(47\)	−6046.65	+	5610.47i	−0.399273	+	0.370471i	−0.854100	−	0.520108i	\(-0.825892\pi\)
							0.454827	+	0.890580i	\(0.349701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.2	yes	264
49.9	even	21	inner	49.6.g.a.9.2	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.2	✓	264	49.9	even	21	inner
49.6.g.a.11.2	yes	264	1.1	even	1	trivial