Embedded newform 49.6.g.a.11.21

• Label: 49.6.g.a.11.21
• 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			11.21
Character	\(\chi\)	\(=\)	49.11
Dual form			49.6.g.a.9.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.04911 - 6.54062i) q^{2} +(25.9061 - 3.90472i) q^{3} +(4.51890 - 60.3005i) q^{4} +(-32.3905 + 82.5297i) q^{5} +(157.076 - 196.967i) q^{6} +(-19.4619 - 128.173i) q^{7} +(-170.691 - 214.039i) q^{8} +(423.675 - 130.686i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 264 q - 13 q^{2} - 22 q^{3} + 307 q^{4} - 52 q^{5} - 130 q^{6} + 154 q^{7} - 320 q^{8} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.04911	−	6.54062i	1.24612	−	1.15623i	0.264703	−	0.964330i	\(-0.414726\pi\)
							0.981415	−	0.191898i	\(-0.0614644\pi\)
\(3\)	25.9061	−	3.90472i	1.66188	−	0.250488i	0.750145	−	0.661273i	\(-0.229984\pi\)
							0.911732	+	0.410786i	\(0.134746\pi\)
\(5\)	−32.3905	+	82.5297i	−0.579420	+	1.47634i	0.277417	+	0.960749i	\(0.410521\pi\)
							−0.856837	+	0.515587i	\(0.827574\pi\)
\(7\)	−19.4619	−	128.173i	−0.150121	−	0.988668i
							\(11\)	−348.614	−	107.533i	−0.868688	−	0.267955i	−0.171797	−	0.985132i	\(-0.554957\pi\)
−0.696890	+	0.717178i	\(0.745434\pi\)
\(13\)	147.414	+	645.863i	0.241925	+	1.05994i	0.939263	+	0.343200i	\(0.111511\pi\)
							−0.697338	+	0.716743i	\(0.745632\pi\)
\(17\)	−445.554	+	303.773i	−0.373919	+	0.254934i	−0.735669	−	0.677342i	\(-0.763132\pi\)
							0.361750	+	0.932275i	\(0.382180\pi\)
\(19\)	903.092	+	1564.20i	0.573915	+	0.994050i	0.996159	+	0.0875678i	\(0.0279095\pi\)
							−0.422243	+	0.906483i	\(0.638757\pi\)
\(23\)	−2988.28	−	2037.37i	−1.17788	−	0.803066i	−0.193845	−	0.981032i	\(-0.562096\pi\)
							−0.984037	+	0.177966i	\(0.943048\pi\)
\(29\)	3218.06	+	1549.74i	0.710557	+	0.342186i	0.754004	−	0.656870i	\(-0.228120\pi\)
							−0.0434473	+	0.999056i	\(0.513834\pi\)
\(31\)	−921.414	+	1595.94i	−0.172207	+	0.298271i	−0.939191	−	0.343395i	\(-0.888423\pi\)
							0.766984	+	0.641666i	\(0.221756\pi\)
\(37\)	351.679	+	4692.82i	0.0422320	+	0.563547i	0.977682	+	0.210089i	\(0.0673753\pi\)
							−0.935450	+	0.353458i	\(0.885006\pi\)
\(41\)	−7724.63	−	9686.38i	−0.717659	−	0.899916i	0.280544	−	0.959841i	\(-0.409485\pi\)
							−0.998203	+	0.0599254i	\(0.980914\pi\)
\(43\)	4444.10	−	5572.73i	0.366533	−	0.459618i	−0.564027	−	0.825756i	\(-0.690749\pi\)
							0.930561	+	0.366138i	\(0.119320\pi\)
\(47\)	12140.6	−	11264.8i	0.801670	−	0.743841i	−0.168647	−	0.985676i	\(-0.553940\pi\)
							0.970317	+	0.241836i	\(0.0777495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.g.a.11.21	yes	264
49.9	even	21	inner	49.6.g.a.9.21	✓	264

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.g.a.9.21	✓	264	49.9	even	21	inner
49.6.g.a.11.21	yes	264	1.1	even	1	trivial