Embedded newform 49.6.e.a.8.8

• Label: 49.6.e.a.8.8
• Level: $49$
• Weight: $6$
• Character: 49.8
• Analytic conductor: $7.859$
• Analytic rank: $0$
• Dimension: $138$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85880717084\)
Analytic rank:	\(0\)
Dimension:	\(138\)
Relative dimension:	\(23\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			8.8
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.16704 - 3.97134i) q^{2} +(-14.8323 + 7.14284i) q^{3} +(1.37924 - 6.04286i) q^{4} +(68.5349 - 33.0046i) q^{5} +(75.3410 + 36.2823i) q^{6} +(129.221 - 10.4405i) q^{7} +(-174.815 + 84.1863i) q^{8} +(17.4677 - 21.9038i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.16704	−	3.97134i	−0.559859	−	0.702041i	0.418673	−	0.908137i	\(-0.362495\pi\)
							−0.978532	+	0.206096i	\(0.933924\pi\)
\(3\)	−14.8323	+	7.14284i	−0.951490	+	0.458213i	−0.844208	−	0.536015i	\(-0.819929\pi\)
							−0.107282	+	0.994229i	\(0.534215\pi\)
\(5\)	68.5349	−	33.0046i	1.22599	−	0.590405i	0.295015	−	0.955492i	\(-0.404675\pi\)
							0.930973	+	0.365087i	\(0.118961\pi\)
\(7\)	129.221	−	10.4405i	0.996752	−	0.0805334i
							\(11\)	−262.918	−	329.688i	−0.655146	−	0.821527i	0.337659	−	0.941269i	\(-0.390365\pi\)
−0.992805	+	0.119741i	\(0.961794\pi\)
\(13\)	−595.913	−	747.251i	−0.977968	−	1.22633i	−0.974048	−	0.226340i	\(-0.927324\pi\)
							−0.00391925	−	0.999992i	\(-0.501248\pi\)
\(17\)	25.7189	+	112.682i	0.0215839	+	0.0945654i	0.984573	−	0.174976i	\(-0.0559849\pi\)
							−0.962989	+	0.269542i	\(0.913128\pi\)
\(19\)	−727.751			−0.462486			−0.231243	−	0.972896i	\(-0.574279\pi\)
							−0.231243	+	0.972896i	\(0.574279\pi\)
\(23\)	−259.896	+	1138.68i	−0.102442	+	0.448830i	0.897527	+	0.440960i	\(0.145362\pi\)
							−0.999969	+	0.00786947i	\(0.997495\pi\)
\(29\)	−877.635	−	3845.17i	−0.193784	−	0.849025i	−0.974544	−	0.224194i	\(-0.928025\pi\)
							0.780760	−	0.624831i	\(-0.214832\pi\)
\(31\)	−5441.18			−1.01692			−0.508462	−	0.861084i	\(-0.669786\pi\)
							−0.508462	+	0.861084i	\(0.669786\pi\)
\(37\)	−2670.60	−	11700.7i	−0.320705	−	1.40510i	−0.836303	−	0.548268i	\(-0.815287\pi\)
							0.515598	−	0.856831i	\(-0.327570\pi\)
\(41\)	−9481.70	+	4566.15i	−0.880900	+	0.424219i	−0.818953	−	0.573861i	\(-0.805445\pi\)
							−0.0619467	+	0.998079i	\(0.519731\pi\)
\(43\)	2286.84	+	1101.29i	0.188610	+	0.0908299i	0.525806	−	0.850605i	\(-0.323764\pi\)
							−0.337195	+	0.941435i	\(0.609478\pi\)
\(47\)	6492.08	+	8140.81i	0.428686	+	0.537555i	0.948522	−	0.316711i	\(-0.102578\pi\)
							−0.519836	+	0.854266i	\(0.674007\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.8	✓	138
49.43	even	7	inner	49.6.e.a.43.8	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.8	✓	138	1.1	even	1	trivial
49.6.e.a.43.8	yes	138	49.43	even	7	inner