Embedded newform 49.6.e.a.8.9

• Label: 49.6.e.a.8.9
• 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			8.9
Character	\(\chi\)	\(=\)	49.8
Dual form			49.6.e.a.43.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.27658 - 2.85474i) q^{2} +(-14.5247 + 6.99475i) q^{3} +(4.15395 - 18.1996i) q^{4} +(0.00127143 - 0.000612289i) q^{5} +(53.0349 + 25.5402i) q^{6} +(-61.2225 - 114.275i) q^{7} +(-166.684 + 80.2708i) q^{8} +(10.5335 - 13.2086i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 138 q - 5 q^{2} + 13 q^{3} - 389 q^{4} + 67 q^{5} + 331 q^{6} - 56 q^{7} + 305 q^{8} - 918 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.27658	−	2.85474i	−0.402446	−	0.504651i	0.538772	−	0.842452i	\(-0.318889\pi\)
							−0.941218	+	0.337801i	\(0.890317\pi\)
\(3\)	−14.5247	+	6.99475i	−0.931762	+	0.448713i	−0.837256	−	0.546811i	\(-0.815842\pi\)
							−0.0945064	+	0.995524i	\(0.530127\pi\)
\(5\)	0.00127143		0.000612289i	2.27441e−5		1.09530e-5i	−0.433872	−	0.900974i	\(-0.642853\pi\)
							0.433895	+	0.900963i	\(0.357139\pi\)
\(7\)	−61.2225	−	114.275i	−0.472244	−	0.881468i
							\(11\)	455.909	+	571.691i	1.13605	+	1.42456i	0.890389	+	0.455200i	\(0.150432\pi\)
0.245657	+	0.969357i	\(0.420996\pi\)
\(13\)	336.361	+	421.784i	0.552011	+	0.692200i	0.977059	−	0.212971i	\(-0.0683140\pi\)
							−0.425047	+	0.905171i	\(0.639743\pi\)
\(17\)	47.6683	+	208.849i	0.0400044	+	0.175271i	0.990983	−	0.133985i	\(-0.0427773\pi\)
							−0.950979	+	0.309255i	\(0.899920\pi\)
\(19\)	586.900			0.372975			0.186488	−	0.982457i	\(-0.440290\pi\)
							0.186488	+	0.982457i	\(0.440290\pi\)
\(23\)	−741.014	+	3246.59i	−0.292083	+	1.27970i	0.589537	+	0.807741i	\(0.299310\pi\)
							−0.881620	+	0.471959i	\(0.843547\pi\)
\(29\)	−339.779	−	1488.67i	−0.0750242	−	0.328703i	0.923463	−	0.383688i	\(-0.125346\pi\)
							−0.998487	+	0.0549850i	\(0.982489\pi\)
\(31\)	−3012.97			−0.563106			−0.281553	−	0.959546i	\(-0.590849\pi\)
							−0.281553	+	0.959546i	\(0.590849\pi\)
\(37\)	744.035	+	3259.83i	0.0893489	+	0.391463i	0.999752	−	0.0222587i	\(-0.00708576\pi\)
							−0.910403	+	0.413722i	\(0.864229\pi\)
\(41\)	10828.8	−	5214.87i	1.00605	−	0.484489i	0.143063	−	0.989714i	\(-0.454305\pi\)
							0.862988	+	0.505225i	\(0.168590\pi\)
\(43\)	38.1515	+	18.3728i	0.00314660	+	0.00151532i	0.435456	−	0.900210i	\(-0.356587\pi\)
							−0.432310	+	0.901725i	\(0.642301\pi\)
\(47\)	−15278.8	−	19159.0i	−1.00889	−	1.26511i	−0.963939	−	0.266123i	\(-0.914257\pi\)
							−0.0449539	−	0.998989i	\(-0.514314\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.e.a.8.9	✓	138
49.43	even	7	inner	49.6.e.a.43.9	yes	138

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.6.e.a.8.9	✓	138	1.1	even	1	trivial
49.6.e.a.43.9	yes	138	49.43	even	7	inner