Embedded newform 49.6.e.a.8.11

• Label: 49.6.e.a.8.11
• 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			8.11
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16235 - 1.45754i) q^{2} +(17.4856 - 8.42061i) q^{3} +(6.34730 - 27.8093i) q^{4} +(97.8327 - 47.1137i) q^{5} +(-32.5978 - 15.6983i) q^{6} +(-91.6882 + 91.6530i) q^{7} +(-101.660 + 48.9568i) q^{8} +(83.3305 - 104.493i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16235	−	1.45754i	−0.205477	−	0.257660i	0.668406	−	0.743797i	\(-0.266977\pi\)
							−0.873883	+	0.486137i	\(0.838406\pi\)
\(3\)	17.4856	−	8.42061i	1.12170	−	0.540182i	0.221283	−	0.975210i	\(-0.428976\pi\)
							0.900417	+	0.435028i	\(0.143261\pi\)
\(5\)	97.8327	−	47.1137i	1.75008	−	0.842796i	0.771704	−	0.635982i	\(-0.219405\pi\)
							0.978380	−	0.206815i	\(-0.0663097\pi\)
\(7\)	−91.6882	+	91.6530i	−0.707243	+	0.706971i
							\(11\)	283.100	+	354.996i	0.705437	+	0.884590i	0.997417	−	0.0718344i	\(-0.0228853\pi\)
−0.291979	+	0.956425i	\(0.594314\pi\)
\(13\)	−292.275	−	366.501i	−0.479660	−	0.601474i	0.481847	−	0.876255i	\(-0.339966\pi\)
							−0.961507	+	0.274781i	\(0.911395\pi\)
\(17\)	309.299	+	1355.13i	0.259571	+	1.13726i	0.921712	+	0.387876i	\(0.126791\pi\)
							−0.662141	+	0.749380i	\(0.730352\pi\)
\(19\)	−528.127			−0.335625			−0.167813	−	0.985819i	\(-0.553670\pi\)
							−0.167813	+	0.985819i	\(0.553670\pi\)
\(23\)	290.550	−	1272.98i	0.114525	−	0.501768i	−0.884832	−	0.465911i	\(-0.845727\pi\)
							0.999357	−	0.0358571i	\(-0.0114161\pi\)
\(29\)	252.818	+	1107.67i	0.0558231	+	0.244577i	0.995140	−	0.0984750i	\(-0.0313965\pi\)
							−0.939316	+	0.343052i	\(0.888539\pi\)
\(31\)	−8080.28			−1.51016			−0.755079	−	0.655634i	\(-0.772402\pi\)
							−0.755079	+	0.655634i	\(0.772402\pi\)
\(37\)	2952.41	+	12935.4i	0.354546	+	1.55337i	0.766550	+	0.642185i	\(0.221972\pi\)
							−0.412004	+	0.911182i	\(0.635171\pi\)
\(41\)	13015.0	−	6267.67i	1.20916	−	0.582300i	0.282885	−	0.959154i	\(-0.408708\pi\)
							0.926273	+	0.376854i	\(0.122994\pi\)
\(43\)	54.5174	+	26.2542i	0.00449639	+	0.00216535i	0.436131	−	0.899883i	\(-0.356349\pi\)
							−0.431634	+	0.902049i	\(0.642063\pi\)
\(47\)	−11094.1	−	13911.6i	−0.732568	−	0.918612i	0.266407	−	0.963861i	\(-0.414163\pi\)
							−0.998976	+	0.0452488i	\(0.985592\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.11	✓	138
49.43	even	7	inner	49.6.e.a.43.11	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.11	✓	138	1.1	even	1	trivial
49.6.e.a.43.11	yes	138	49.43	even	7	inner