Embedded newform 49.6.e.a.8.14

• Label: 49.6.e.a.8.14
• 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			8.14
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.63882 + 2.05502i) q^{2} +(-16.7655 + 8.07386i) q^{3} +(5.58330 - 24.4621i) q^{4} +(39.0863 - 18.8230i) q^{5} +(-44.0677 - 21.2219i) q^{6} +(49.2203 + 119.935i) q^{7} +(135.202 - 65.1096i) q^{8} +(64.3879 - 80.7399i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.63882	+	2.05502i	0.289706	+	0.363280i	0.905292	−	0.424790i	\(-0.139652\pi\)
							−0.615586	+	0.788070i	\(0.711081\pi\)
\(3\)	−16.7655	+	8.07386i	−1.07551	+	0.517938i	−0.885878	−	0.463917i	\(-0.846443\pi\)
							−0.189631	+	0.981855i	\(0.560729\pi\)
\(5\)	39.0863	−	18.8230i	0.699197	−	0.336715i	−0.0502901	−	0.998735i	\(-0.516015\pi\)
							0.749487	+	0.662019i	\(0.230300\pi\)
\(7\)	49.2203	+	119.935i	0.379664	+	0.925125i
							\(11\)	160.232	+	200.925i	0.399272	+	0.500671i	0.940306	−	0.340329i	\(-0.110539\pi\)
−0.541034	+	0.841000i	\(0.681967\pi\)
\(13\)	421.730	+	528.832i	0.692111	+	0.867880i	0.996407	−	0.0846980i	\(-0.0269926\pi\)
							−0.304296	+	0.952578i	\(0.598421\pi\)
\(17\)	205.342	+	899.663i	0.172328	+	0.755019i	0.985036	+	0.172348i	\(0.0551352\pi\)
							−0.812708	+	0.582671i	\(0.802008\pi\)
\(19\)	2282.51			1.45054			0.725268	−	0.688466i	\(-0.241716\pi\)
							0.725268	+	0.688466i	\(0.241716\pi\)
\(23\)	994.380	−	4356.67i	0.391952	−	1.71725i	−0.265807	−	0.964026i	\(-0.585638\pi\)
							0.657759	−	0.753228i	\(-0.271504\pi\)
\(29\)	1727.95	+	7570.66i	0.381538	+	1.67163i	0.692667	+	0.721258i	\(0.256436\pi\)
							−0.311129	+	0.950368i	\(0.600707\pi\)
\(31\)	2660.84			0.497296			0.248648	−	0.968594i	\(-0.420014\pi\)
							0.248648	+	0.968594i	\(0.420014\pi\)
\(37\)	−2121.53	−	9295.03i	−0.254768	−	1.11621i	−0.926760	−	0.375653i	\(-0.877418\pi\)
							0.671992	−	0.740558i	\(-0.265439\pi\)
\(41\)	−11047.1	+	5319.99i	−1.02633	+	0.494255i	−0.869793	−	0.493416i	\(-0.835748\pi\)
							−0.156539	+	0.987672i	\(0.550034\pi\)
\(43\)	−13204.6	−	6359.01i	−1.08907	−	0.524467i	−0.198863	−	0.980027i	\(-0.563725\pi\)
							−0.890205	+	0.455560i	\(0.849439\pi\)
\(47\)	5447.99	+	6831.57i	0.359743	+	0.451103i	0.928461	−	0.371429i	\(-0.121132\pi\)
							−0.568719	+	0.822532i	\(0.692561\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.14	✓	138
49.43	even	7	inner	49.6.e.a.43.14	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.14	✓	138	1.1	even	1	trivial
49.6.e.a.43.14	yes	138	49.43	even	7	inner